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<h1 id="Numerical-Simulations:-Modeling-assignment">Numerical Simulations: Modeling assignment<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Numerical-Simulations:-Modeling-assignment">¶</a></h1><p><em>Numerical Simulation of the financial development model based on Greenwood, Jovaovic (1990): Financial Development, Growth, and the Distribution of Income.</em></p>
<p>Members of the group are:</p>
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<li>Kshitiz Dahal (123934)</li>
<li>Tung Nguyen Huy (238052)</li>
<li>Justice A. Obilor (256456)</li>
</ul>
<p>April 14, 2016</p>
<p>The corresponding repository can be found <a href="https://github.com/numeraire92/third-assignment">here</a>.</p>
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<h3 id="ABSTRACT">ABSTRACT<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#ABSTRACT">¶</a></h3><p>This is a simulation of Greenwood-Jovanovic Model which made an important contribution to the field of Economics by laying out the path of financial development, economic growth, and distribution of income. It predicted that income gap widens in the initial phase of financial development, but this gap slowly subsides as economy advances along with advancement in financial intermediation, and eventually the economy converges to a higher level of growth with a stable level of income distribution. We follow an agent-based modeling approach implemented in Python 2.7 to simulate the financial development model described above. The aim of the simulation is to generate a time path of the average wealth, inequality, growth rate and development of the intermediary. We expect these time paths to reflect the predictions of the analytical model. We implement codes to define individuals as well as financial intermediary as objects. We also use the iteration process on the Bellman-equation by tailoring the codes implemented in Python by Sargent and Stachursky (2014). In order to analyse the time path of inequality in the economy, we rely on a function that produces the Gini-coefficient derived from Aaron Schumacher (2013). In order to obtain realistic results, we assumed a set of parameters for the model. Then we were able to simulate the dynamics of the economic environment where individuals decide whether to join the intermediary or not based on the parameters entered. Finally we calculate the growth rate path, inequality path, and the path traced by financial intermediary-thus efficiently simulating Greenwood-Jovanovic Model. We also provide plots for our results for easy analysis.</p>
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<h2 id="QUESTION">QUESTION<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#QUESTION">¶</a></h2><p>What does the time path of the average wealth, inequality, growth rate and development of the intermediary sector look like? Does it coincide with the predictions made by Greenwood-Jovanovic Model?</p>
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<h2 id="MOTIVATION">MOTIVATION<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#MOTIVATION">¶</a></h2><p>The motivation behind the simulation is to generate a time path of the average wealth, inequality, growth rate and development of the intermediary sector. Given a great deal of current interest in rising inequality as well as the importance attached to financial sectors, we believe this to be an important topic to simulate. Although the above mentioned questions could have been answered by analytically solving the model as well, the implementation of this simulation model has an added advantage in that it allows the analysis of further questions. For example, one of the proposed mechanism to lower the cost of access to financial intermediary is the provision of subsidies. But what would be the source of funding of such subsidy? Further expanding the simulation model would allow us to analyse the effect of a progressive tax or a tax on investment income produced by the financial intermediary. A tax on investment income would reduce the returns from the intermediary thus reducing the life-time expected benefit from the intermediary. Ironically, this could counteract the intended effect of subsidy, which is to enable earlier access to the intermediary.</p>
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<h1 id="1.-Introduction">1. Introduction<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#1.-Introduction">¶</a></h1><p>This model proposed by Jeremy Greenwood and Boyan Jovanovic presents the transition of a primitive slow-growing economy to a developed fast growing one through a development in the level and quality of financial intermediation. In short, it traces the path of financial development, economic growth, and distribution of income. The model is based on the assumption that institutions arise endogenously to facilitate trade. These intermediaries because of their advantage on colecting and analyzing information allow for optimal flow of investors' resources. The economy grows along with development of these financial intermediaries in a Kuznet curve resembling pattern. In the initial stages, an economy lacks any financial structure or is nonexistent and hence grows very slowly. As the economy approaches intermediate stage, financial structures start to develop. This stage is marked by increased growth and saving rates, but with a wider distribution of income across the rich and the poor. Finally, the economy matures and develops an extensive structure for financial intermediation. In this stage, the distribution of income becomes stable, the savings rate falls, and the economy's growth rate converges to a higher level.</p>
<h3 id="1.1-Theoretical-background-of-the-model">1.1 Theoretical background of the model<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#1.1-Theoretical-background-of-the-model">¶</a></h3><h4 id="I.-The-Economic-Environment">I. The Economic Environment<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#I.-The-Economic-Environment">¶</a></h4><p>The model considers an economy populated by a continuum of agents distributed over the interval [0,1]. Thus, each individual is represented by an index j, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>&#x2208;</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" role="math" style="width: 4.437em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.821em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1003.7em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-4" style="font-family: MathJax_Main; padding-left: 0.311em;">∈</span><span class="mo" id="MathJax-Span-5" style="font-family: MathJax_Main; padding-left: 0.311em;">[</span><span class="mn" id="MathJax-Span-6" style="font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-7" style="font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-8" style="font-family: MathJax_Main; padding-left: 0.188em;">1</span><span class="mo" id="MathJax-Span-9" style="font-family: MathJax_Main;">]</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-1"> j \in [0,1] </script>.
An agent's goal is maximizes his expected lifetime utility which is given by</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mrow><mo>[</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">&#x221E;</mi></mrow></munderover><msup><mi>&#x03B2;</mi><mi>t</mi></msup><mi>l</mi><mi>n</mi><msub><mi>c</mi><mi>t</mi></msub><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-10" role="math" style="width: 7.392em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.345em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(2.035em 1006.1em 5.36em -999.997em); top: -3.938em; left: 0.003em;"><span class="mrow" id="MathJax-Span-11"><span class="mi" id="MathJax-Span-12" style="font-family: MathJax_Math-italic;">E<span style="display: inline-block; 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font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-21" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.39em 1000.68em 4.19em -999.997em); top: -5.169em; left: 0.373em;"><span class="texatom" id="MathJax-Span-22"><span class="mrow" id="MathJax-Span-23"><span class="mi" id="MathJax-Span-24" style="font-size: 70.7%; font-family: MathJax_Main;">∞</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-25" style="padding-left: 0.188em;"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.56em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-26" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; 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top: -3.876em; left: 0.434em;"><span class="mi" id="MathJax-Span-32" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-33" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size4;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.944em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.496em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 3.646em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mrow><mo>[</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><msup><mi>β</mi><mi>t</mi></msup><mi>l</mi><mi>n</mi><msub><mi>c</mi><mi>t</mi></msub><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{equation}
E \left[\sum_{t=0}^{\infty}\beta^t ln c_t \right]
\end{equation}</script><p>with <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>&#x03B2;</mi><mo>&lt;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-34" role="math" style="width: 5.175em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.437em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1004.38em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-35"><span class="mn" id="MathJax-Span-36" style="font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-37" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="mi" id="MathJax-Span-38" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-39" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="mn" id="MathJax-Span-40" style="font-family: MathJax_Main; padding-left: 0.311em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-3">0<\beta<1</script></p>
<p>The <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-41" role="math" style="width: 0.927em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1000.8em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-42"><span class="msubsup" id="MathJax-Span-43"><span style="display: inline-block; position: relative; width: 0.742em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.43em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-44" style="font-family: MathJax_Math-italic;">c</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="mi" id="MathJax-Span-45" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.861em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mi>t</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-4"> c_t </script> is agent's period t consumption flow and <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B2;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-46" role="math" style="width: 0.742em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.619em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.62em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-47"><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-5">\beta</script> the discount factor
There are two types of one-period production projects. The first one is <strong>safe</strong>, but has a relatively low return on investment, and the second one offers a higher (unconditional) expected return but is more <strong>risky</strong>. The two projects are given by the following equations:-</p>
<p><strong>Safe Project</strong></p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>&#x03B4;</mi><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-49" role="math" style="width: 4.806em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.129em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1004.13em 2.528em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-50"><span class="msubsup" id="MathJax-Span-51"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-52" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-53"><span class="mrow" id="MathJax-Span-54"><span class="mi" id="MathJax-Span-55" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-56" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-57" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mi" id="MathJax-Span-58" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-59"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-60" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.557em;"><span class="texatom" id="MathJax-Span-61"><span class="mrow" id="MathJax-Span-62"><span class="mi" id="MathJax-Span-63" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-64" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>δ</mi><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-6"> y_{it} = \delta x_{it} </script><p>Here, an investment <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-65" role="math" style="width: 1.419em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.235em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1001.24em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-66"><span class="msubsup" id="MathJax-Span-67"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-68" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.557em;"><span class="texatom" id="MathJax-Span-69"><span class="mrow" id="MathJax-Span-70"><span class="mi" id="MathJax-Span-71" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-72" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.861em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-7">x_{it}</script> yields <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B4;</mi><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-73" role="math" style="width: 1.974em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.666em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.67em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-74"><span class="mi" id="MathJax-Span-75" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-76"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-77" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.557em;"><span class="texatom" id="MathJax-Span-78"><span class="mrow" id="MathJax-Span-79"><span class="mi" id="MathJax-Span-80" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-81" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-8">\delta x_{it}</script> units of output where <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B4;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-82" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1000.5em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-83"><span class="mi" id="MathJax-Span-84" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-9">\delta</script> is a technological constant.</p>
<p><strong>Risky Project</strong></p>
<p><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mi>t</mi></msub><mo>+</mo><msub><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-85" role="math" style="width: 8.747em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.515em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1007.52em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-86"><span class="msubsup" id="MathJax-Span-87"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-88" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-89"><span class="mrow" id="MathJax-Span-90"><span class="mi" id="MathJax-Span-91" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-92" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-93" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mo" id="MathJax-Span-94" style="font-family: MathJax_Main; padding-left: 0.311em;">(</span><span class="msubsup" id="MathJax-Span-95"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-96" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-97" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-98" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="msubsup" id="MathJax-Span-99" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-100" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-101"><span class="mrow" id="MathJax-Span-102"><span class="mi" id="MathJax-Span-103" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-104" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-105" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-106"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-107" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.557em;"><span class="texatom" id="MathJax-Span-108"><span class="mrow" id="MathJax-Span-109"><span class="mi" id="MathJax-Span-110" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-111" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>+</mo><msub><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-10"> y_{it} = (\theta_t + \epsilon_{it})x_{it}</script></p>
<p>where, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mi>t</mi></msub><mo>+</mo><msub><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-112" role="math" style="width: 4.498em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.882em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1003.76em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-113"><span class="mo" id="MathJax-Span-114" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-115"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-116" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-117" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-118" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="msubsup" id="MathJax-Span-119" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-120" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-121"><span class="mrow" id="MathJax-Span-122"><span class="mi" id="MathJax-Span-123" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-124" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-125" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>+</mo><msub><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-11">(\theta_t + \epsilon_{it})</script> represents a composite technology shock, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B8;</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-126" role="math" style="width: 1.05em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.87em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-127"><span class="msubsup" id="MathJax-Span-128"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-129" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-130" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mi>t</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-12">\theta_t</script> being the aggregate shock (common across both technologies) and <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-131" role="math" style="width: 1.235em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1001.05em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-132"><span class="msubsup" id="MathJax-Span-133"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-134" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-135"><span class="mrow" id="MathJax-Span-136"><span class="mi" id="MathJax-Span-137" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-138" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.861em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-13">\epsilon_{it}</script> being the idiosyncratic shock (project-specific shock) with <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-139" role="math" style="width: 4.991em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.313em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1004.19em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-140"><span class="mi" id="MathJax-Span-141" style="font-family: MathJax_Math-italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-142" style="font-family: MathJax_Main;">[</span><span class="msubsup" id="MathJax-Span-143"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-144" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-145"><span class="mrow" id="MathJax-Span-146"><span class="mi" id="MathJax-Span-147" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-148" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-149" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mn" id="MathJax-Span-150" style="font-family: MathJax_Main; padding-left: 0.311em;">0</span><span class="mo" id="MathJax-Span-151" style="font-family: MathJax_Main;">]</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-14">E[\epsilon_{it} = 0]</script>. It is to be noted that an agent can costlessly observe only the realized composite rate of return <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mi>t</mi></msub><mo>+</mo><msub><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-152" role="math" style="width: 4.498em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.882em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1003.76em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-153"><span class="mo" id="MathJax-Span-154" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-155"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-156" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-157" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-158" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="msubsup" id="MathJax-Span-159" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-160" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-161"><span class="mrow" id="MathJax-Span-162"><span class="mi" id="MathJax-Span-163" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-164" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-165" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>+</mo><msub><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-15">(\theta_t + \epsilon_{it})</script> on his own project.</p>
<p>Furthermore, the model assumes that at the beginning of each period t, an agent will be endowed with a certain amount of wealth, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-166" role="math" style="width: 1.111em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.93em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-167"><span class="msubsup" id="MathJax-Span-168"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-169" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-170" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>t</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-16">k_t</script> that can be consumed or invested in either safe or risky production projects.</p>
<h4 id="II.-Competitive-Financial-Intermediation">II. Competitive Financial Intermediation<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#II.-Competitive-Financial-Intermediation">¶</a></h4><p>As the economy starts to grow from its primitive stage, a subset of agents establish networks to provide financial intermediation activities for some larger set of individuals. They charge competitive fees for their intermediation services.</p>
<p>To introduce that aspect of competitive financial intermediation, the model supposes that some individuals in period t have assumed the role of being an intermediary , which is represented by <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><msub><mi mathvariant="double-struck">A</mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">t</mi></mrow></msub></mrow><mo>&#x2208;</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-171" role="math" style="width: 5.237em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.498em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1004.38em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-172"><span class="texatom" id="MathJax-Span-173"><span class="mrow" id="MathJax-Span-174"><span class="msubsup" id="MathJax-Span-175"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.68em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-176" style="font-family: MathJax_AMS;">A</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.742em;"><span class="texatom" id="MathJax-Span-177"><span class="mrow" id="MathJax-Span-178"><span class="mi" id="MathJax-Span-179" style="font-size: 70.7%; font-family: MathJax_Main;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-180" style="font-family: MathJax_Main; padding-left: 0.311em;">∈</span><span class="mo" id="MathJax-Span-181" style="font-family: MathJax_Main; padding-left: 0.311em;">[</span><span class="mn" id="MathJax-Span-182" style="font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-183" style="font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-184" style="font-family: MathJax_Main; padding-left: 0.188em;">1</span><span class="mo" id="MathJax-Span-185" style="font-family: MathJax_Main;">]</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><msub><mi mathvariant="double-struck">A</mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">t</mi></mrow></msub></mrow><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-17">\mathbb{A_{t}}\in [0,1]</script>. It is also supposed that they bear a cost of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-186" role="math" style="width: 0.742em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.619em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1000.56em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-187"><span class="mi" id="MathJax-Span-188" style="font-family: MathJax_Math-italic;">α</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-18">\alpha</script> for joining the intermediary.</p>
<p>Now, the model assumes that in exchange for a once-and-for-all fee of q along with the rights to operate an individual's project, the intermediary promises a return of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-189" role="math" style="width: 3.39em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.897em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1002.77em 2.836em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-190"><span class="mi" id="MathJax-Span-191" style="font-family: MathJax_Math-italic;">r</span><span class="mo" id="MathJax-Span-192" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-193"><span style="display: inline-block; position: relative; width: 1.666em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-194" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-195"><span class="mrow" id="MathJax-Span-196"><span class="mi" id="MathJax-Span-197" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span class="mo" id="MathJax-Span-198" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-199" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-200" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-19">r(\theta_{t+j})</script> per unit of capital invested in any period t+j-1.</p>
<p>Since the goal of the intermediary is to maximize profits, it will adopt the most efficient scheme possible for intermediation.</p>
<h4 id="Financial-Intermediary's-Scheme">Financial Intermediary's Scheme<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Financial-Intermediary's-Scheme">¶</a></h4><p>To maximize the profits, financial intermediary draws a finite number of projects, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-201" role="math" style="width: 0.68em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1000.56em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-202"><span class="mi" id="MathJax-Span-203" style="font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math></span></span><script type="math/tex" id="MathJax-Element-20">\tau </script>, from the portfolio of individual projects to research aggregate productivity.</p>
<p>Thus, the aggregate amount of capital that the intermediary has to invest in time period t is given by,</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>=</mo><mfrac><mrow><msub><mo>&#x222B;</mo><mi>A</mi></msub><msub><mi>k</mi><mi>j</mi></msub><mi>d</mi><mi>&#x03BB;</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mo>&#x222B;</mo><mi>A</mi></msub><mi>d</mi><mi>&#x03BB;</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-204" role="math" style="width: 7.885em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.776em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(0.311em 1006.78em 3.513em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-205"><span class="texatom" id="MathJax-Span-206"><span class="mrow" id="MathJax-Span-207"><span class="munderover" id="MathJax-Span-208"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-209" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-210" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-211" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mfrac" id="MathJax-Span-212" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 4.56em; height: 0px; margin-right: 0.126em; margin-left: 0.126em;"><span style="position: absolute; clip: rect(3.02em 1004.38em 4.56em -999.997em); top: -4.861em; left: 50%; margin-left: -2.214em;"><span class="mrow" id="MathJax-Span-213"><span class="msubsup" id="MathJax-Span-214"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.02em 1000.62em 4.498em -999.997em); top: -3.999em; left: 0.003em;"><span class="mo" id="MathJax-Span-215" style="font-family: MathJax_Size1; vertical-align: 0.003em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.63em; left: 0.496em;"><span class="mi" id="MathJax-Span-216" style="font-size: 70.7%; font-family: MathJax_Math-italic;">A</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-217" style="padding-left: 0.188em;"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-218" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-219" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mi" id="MathJax-Span-220" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-221" style="font-family: MathJax_Math-italic;">λ</span><span class="mo" id="MathJax-Span-222" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-223" style="font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-224" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.02em 1003.45em 4.56em -999.997em); top: -3.199em; left: 50%; margin-left: -1.783em;"><span class="mrow" id="MathJax-Span-225"><span class="msubsup" id="MathJax-Span-226"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.02em 1000.62em 4.498em -999.997em); top: -3.999em; left: 0.003em;"><span class="mo" id="MathJax-Span-227" style="font-family: MathJax_Size1; vertical-align: 0.003em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.63em; left: 0.496em;"><span class="mi" id="MathJax-Span-228" style="font-size: 70.7%; font-family: MathJax_Math-italic;">A</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mi" id="MathJax-Span-229" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-230" style="font-family: MathJax_Math-italic;">λ</span><span class="mo" id="MathJax-Span-231" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-232" style="font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-233" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(0.804em 1004.56em 1.235em -999.997em); top: -1.29em; left: 0.003em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-top-width: 1.3px; border-top-style: solid; width: 4.56em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.05em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 3.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mfrac><mrow><msub><mo>∫</mo><mi>A</mi></msub><msub><mi>k</mi><mi>j</mi></msub><mi>d</mi><mi>λ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mo>∫</mo><mi>A</mi></msub><mi>d</mi><mi>λ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-21">\begin{equation}
\hat{k} = \frac {\int_A k_j d\lambda(j)} {\int_A d\lambda(j)}
\end{equation}</script><p>The intermediary then calculates the average net realized rate of return, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C4;</mi></mrow></msub><mo stretchy="false">&#x005E;</mo></mover></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-234" role="math" style="width: 1.173em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.988em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1000.99em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-235"><span class="texatom" id="MathJax-Span-236"><span class="mrow" id="MathJax-Span-237"><span class="munderover" id="MathJax-Span-238"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.93em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="msubsup" id="MathJax-Span-239"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-240" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-241"><span class="mrow" id="MathJax-Span-242"><span class="mi" id="MathJax-Span-243" style="font-size: 70.7%; font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.188em;"><span class="mo" id="MathJax-Span-244" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mi>τ</mi></mrow></msub><mo stretchy="false">^</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-22">\hat{\theta_{\tau}}</script>, on these projects, which is given by,</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>&#x03B8;</mi><mi>&#x03C4;</mi></msub><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>=</mo><mi>&#x03B8;</mi><mo>+</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mi>o</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C4;</mi></mrow></munderover><msub><mi>&#x03F5;</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>&#x03C4;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-245" role="math" style="width: 8.808em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.577em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(0.434em 1007.58em 3.698em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-246"><span class="texatom" id="MathJax-Span-247"><span class="mrow" id="MathJax-Span-248"><span class="munderover" id="MathJax-Span-249"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.93em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="msubsup" id="MathJax-Span-250"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-251" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-252" style="font-size: 70.7%; font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.188em;"><span class="mo" id="MathJax-Span-253" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-254" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mi" id="MathJax-Span-255" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">θ</span><span class="mo" id="MathJax-Span-256" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="munderover" id="MathJax-Span-257" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 1.419em; height: 0px;"><span style="position: absolute; clip: rect(2.897em 1001.42em 4.621em -999.997em); top: -3.999em; left: 0.003em;"><span class="mo" id="MathJax-Span-258" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∑</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1001.17em 4.437em -999.997em); top: -2.891em; left: 0.126em;"><span class="texatom" id="MathJax-Span-259"><span class="mrow" id="MathJax-Span-260"><span class="mi" id="MathJax-Span-261" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-262" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mi" id="MathJax-Span-263" style="font-size: 70.7%; font-family: MathJax_Math-italic;">o</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -5.169em; left: 0.557em;"><span class="texatom" id="MathJax-Span-264"><span class="mrow" id="MathJax-Span-265"><span class="mi" id="MathJax-Span-266" style="font-size: 70.7%; font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-267" style="padding-left: 0.188em;"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.37em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-268" style="font-family: MathJax_Math-italic;">ϵ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="mi" id="MathJax-Span-269" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="texatom" id="MathJax-Span-270"><span class="mrow" id="MathJax-Span-271"><span class="mo" id="MathJax-Span-272" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-273" style="font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.639em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 3.504em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>θ</mi><mi>τ</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mi>θ</mi><mo>+</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mi>o</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>τ</mi></mrow></munderover><msub><mi>ϵ</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>τ</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-23">\begin{equation}
\hat{\theta_\tau} = \theta + \sum_{j=o}^{\tau} \epsilon_j / \tau
\end{equation}</script><p>So, if the test statistic, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>&#x03B8;</mi><mi>&#x03C4;</mi></msub><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&gt;</mo><mi>&#x03B4;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-274" role="math" style="width: 3.328em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.836em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1002.84em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-275"><span class="texatom" id="MathJax-Span-276"><span class="mrow" id="MathJax-Span-277"><span class="munderover" id="MathJax-Span-278"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.93em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="msubsup" id="MathJax-Span-279"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-280" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-281" style="font-size: 70.7%; font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.188em;"><span class="mo" id="MathJax-Span-282" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-283" style="font-family: MathJax_Main; padding-left: 0.311em;">></span><span class="mi" id="MathJax-Span-284" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>θ</mi><mi>τ</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo>></mo><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">\hat{\theta_\tau} > \delta</script>, then all the remaining high-risk/return projects operated by intermediary are funded with <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x005E;</mo></mover></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-285" role="math" style="width: 0.68em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.173em 1000.56em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-286"><span class="texatom" id="MathJax-Span-287"><span class="mrow" id="MathJax-Span-288"><span class="munderover" id="MathJax-Span-289"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-290" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-291" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">^</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-25">\hat{k}</script>.</p>
<p>Otherwise, if <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>&#x03B8;</mi><mi>&#x03C4;</mi></msub><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&lt;</mo><mi>&#x03B4;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-292" role="math" style="width: 3.328em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.836em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1002.84em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-293"><span class="texatom" id="MathJax-Span-294"><span class="mrow" id="MathJax-Span-295"><span class="munderover" id="MathJax-Span-296"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.93em 4.375em -999.997em); top: -3.999em; left: 0.003em;"><span class="msubsup" id="MathJax-Span-297"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-298" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-299" style="font-size: 70.7%; font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.188em;"><span class="mo" id="MathJax-Span-300" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-301" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="mi" id="MathJax-Span-302" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>θ</mi><mi>τ</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo><</mo><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">\hat{\theta_\tau} < \delta</script>, the intermediary invests its resources in safe projects.</p>
<p>Then, the net return, <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mi>t</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>&#x03B8;</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-303" role="math" style="width: 3.636em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.143em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1003.02em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-304"><span class="mi" id="MathJax-Span-305" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-306" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-307"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-308" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-309" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-310" style="font-family: MathJax_Main;">,</span><span class="texatom" id="MathJax-Span-311" style="padding-left: 0.188em;"><span class="mrow" id="MathJax-Span-312"><span class="munderover" id="MathJax-Span-313"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-314" style="font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1000.37em 3.636em -999.997em); top: -4.307em; left: 0.065em;"><span class="mo" id="MathJax-Span-315" style="font-family: MathJax_Main;">^</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-316" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-27">z(\theta_t,\hat{\theta})</script> behaves in a way such that:
as <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">&#x2192;</mo><mi mathvariant="normal">&#x221E;</mi><mo>,</mo><mi>z</mi><mo stretchy="false">&#x2192;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-317" role="math" style="width: 11.456em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.855em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1009.73em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-318"><span class="mi" id="MathJax-Span-319" style="font-family: MathJax_Math-italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mo" id="MathJax-Span-320" style="font-family: MathJax_Main; padding-left: 0.311em;">→</span><span class="mi" id="MathJax-Span-321" style="font-family: MathJax_Main; padding-left: 0.311em;">∞</span><span class="mo" id="MathJax-Span-322" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-323" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-324" style="font-family: MathJax_Main; padding-left: 0.311em;">→</span><span class="mi" id="MathJax-Span-325" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">m</span><span class="mi" id="MathJax-Span-326" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-327" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-328" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-329" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-330" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-331" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-332" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mi>z</mi><mo stretchy="false">→</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-28">\tau\to\infty, z\to max(\delta, \theta)</script>.</p>
<p>Then, the intermediary repays <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-333" role="math" style="width: 3.02em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.589em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1002.59em 2.836em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-334"><span class="mi" id="MathJax-Span-335" style="font-family: MathJax_Math-italic;">r</span><span class="mo" id="MathJax-Span-336" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-337" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-338" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-339"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-340" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-341"><span class="mrow" id="MathJax-Span-342"><span class="mi" id="MathJax-Span-343" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-29">r(z)k_{j}</script> to individual j who joined intermediary; r(z) is determined endogenously as a function of z.</p>
<p>Thus, according to this model, the main benefits from financial intermediation are:-</p>
<ul>
<li>intermediaries produce information about the aggregate state of the economy</li>
<li>intermediaries enable pooling of cross-sectional risk</li>
<li>"intermediaries offer agents a rate of return on their investment that is (i)completely devoid of idiosyncratic production risk and (ii) safe-guarded from the potential losses that could occur when the aggregate return on the risky technology falls below the opportunity cost of the resources commited"</li>
<li>intermediaries enable optimal allocation of resources</li>
</ul>
<h3 id="III.-Market-Participation">III. Market Participation<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#III.-Market-Participation">¶</a></h3><p>Not all agents join the intermediation. Particularly, for some agents, it might be too costly for some agents to pay a lump-sum fee of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-344" role="math" style="width: 0.742em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.619em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1000.56em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-345"><span class="mi" id="MathJax-Span-346" style="font-family: MathJax_Math-italic;">α</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-30">\alpha</script>, and hence not worthwhile to gain access to the intermediation technology. Thus, outcomes of agents are determined by whether they join intermediation or not.</p>
<p>If s be the individual savings, then return to an agent outside of the intermediated sector is given by the outcome of following dynamic programming problem</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mi>d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-347" role="math" style="width: 40.397em; display: inline-block;"><span style="display: inline-block; position: relative; width: 34.794em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1034.55em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-348"><span class="mi" id="MathJax-Span-349" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-350" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-351" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-352" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-353" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-354" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-355" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-356"><span class="mrow" id="MathJax-Span-357"><span class="mn" id="MathJax-Span-358" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-359" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-360" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-361" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-362" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-363" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-364" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-365" style="font-family: MathJax_Math-italic;">l</span><span class="mi" id="MathJax-Span-366" style="font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-367" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-368" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-369" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-370" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-371" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-372" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-373" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-374" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-375" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">m</span><span class="mi" id="MathJax-Span-376" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-377" style="font-family: MathJax_Math-italic;">x</span><span class="mrow" id="MathJax-Span-378" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-379" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-380" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-381" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-382" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-383" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-384" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-385" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-386" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">ε</span><span class="mo" id="MathJax-Span-387" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-388" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-389" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-390" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-391" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-392" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-393" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-394" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-395" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-396" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">ε</span><span class="mo" id="MathJax-Span-397" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-398" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-399" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">α</span><span class="mo" id="MathJax-Span-400" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-401" style="font-family: MathJax_Main;">)</span></span><span class="mi" id="MathJax-Span-402" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-403" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-404" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-405" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-406" style="font-family: MathJax_Main;">)</span><span class="mi" id="MathJax-Span-407" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-408" style="font-family: MathJax_Math-italic;">G</span><span class="mo" id="MathJax-Span-409" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-410" style="font-family: MathJax_Math-italic;">ε</span><span class="mo" id="MathJax-Span-411" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-412" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>−</mo><mi>α</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-31">\begin{equation}
W(k) = \max_{0 \leq s \leq k} \left[ ln(k-s) + \beta \int max\left(W[s(\theta+\varepsilon)],V[s(\theta+\varepsilon)-\alpha]\right) dF(\theta) dG(\varepsilon) \right]
\end{equation}</script><p>Likewise, the return to an agent who joins the intermediary is given by the outcome of following dynamic programming problem:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-413" role="math" style="width: 39.227em; display: inline-block;"><span style="display: inline-block; position: relative; width: 33.808em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1033.56em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-414"><span class="mi" id="MathJax-Span-415" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-416" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-417" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-418" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-419" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-420" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-421" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-422"><span class="mrow" id="MathJax-Span-423"><span class="mn" id="MathJax-Span-424" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-425" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-426" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-427" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-428" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-429" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-430" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-431" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-432"></span><span class="mo" id="MathJax-Span-433" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-434" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-435" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-436" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-437" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-438" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-439" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-440" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-441" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">m</span><span class="mi" id="MathJax-Span-442" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-443" style="font-family: MathJax_Math-italic;">x</span><span class="mrow" id="MathJax-Span-444" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-445" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-446" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-447" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-448" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-449" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-450" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-451" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-452" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-453" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-454" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-455" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-456" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-457" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-458" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-459" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-460" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-461" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-462" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-463" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-464" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-465" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-466" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-467" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-468" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-469" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-470" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-471" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-472" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-473" style="font-family: MathJax_Main;">)</span></span><span class="mi" id="MathJax-Span-474" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-475" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-476" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-477" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-478" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-479" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-32">\begin{equation}
V(k) = \max_{0 \leq s \leq k} \left[ \ln (k-s) + \beta \int max\left(W[s\cdot max(\delta,\theta)],V[s \cdot max(\delta,\theta)]\right) dF(\theta) \right]
\end{equation}</script><p>.</p>
<h4 id="III.-Predictions/Results-of-the-model">III. Predictions/Results of the model<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#III.-Predictions/Results-of-the-model">¶</a></h4><p>(i). V(k) > W(k) for all k</p>
<p>The model predicts that agents get a higher expected return in the intermediated sector, all the while enjoying less risk. This implies that agents do not exit the financial intermediary sector, which in turn implies,</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-480" role="math" style="width: 26.85em; display: inline-block;"><span style="display: inline-block; position: relative; width: 23.156em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1022.91em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-481"><span class="mi" id="MathJax-Span-482" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-483" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-484" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-485" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-486" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-487" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-488" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-489"><span class="mrow" id="MathJax-Span-490"><span class="mn" id="MathJax-Span-491" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-492" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-493" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-494" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-495" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-496" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-497" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-498" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-499"></span><span class="mo" id="MathJax-Span-500" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-501" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-502" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-503" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-504" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-505" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-506" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-507" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-508" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-509" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-510" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-511" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-512" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-513" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-514" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-515" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-516" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-517" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-518" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-519" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-520" style="font-family: MathJax_Main;">]</span><span class="mi" id="MathJax-Span-521" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-522" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-523" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-524" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-525" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-526" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-33">\begin{equation}
V(k) = \max_{0 \leq s \leq k} \left[ \ln (k-s) + \beta \int V[s \cdot max(\delta,\theta)] dF(\theta) \right]
\end{equation}</script><p>This gives a solution for the individual saving as <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>&#x03B2;</mi><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-527" role="math" style="width: 3.513em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.02em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1003.02em 2.712em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-528"><span class="mi" id="MathJax-Span-529" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-530" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mi" id="MathJax-Span-531" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-532" style="font-family: MathJax_Math-italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>β</mi><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">s=\beta k</script>.</p>
<p>(ii). There exists <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x00AF;</mo></mover></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-533" role="math" style="width: 0.68em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1000.56em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-534"><span class="texatom" id="MathJax-Span-535"><span class="mrow" id="MathJax-Span-536"><span class="munderover" id="MathJax-Span-537"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-538" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.205em 1000.43em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-539" style="font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-35">\bar{k}</script> and <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>k</mi><mo accent="false">&#x00AF;</mo></mover></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-540" role="math" style="width: 0.742em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.619em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1000.62em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-541"><span class="munderover" id="MathJax-Span-542"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-543" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1000.56em 3.821em -999.997em); top: -4.492em; left: 0.003em;"><span class="mo" id="MathJax-Span-544" style=""><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; top: -3.999em; left: -0.058em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.249em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.065em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>k</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-36">\overline{k}</script> with <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x00AF;</mo></mover></mrow><mo>&lt;</mo><mover><mi>k</mi><mo accent="false">&#x00AF;</mo></mover></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-545" role="math" style="width: 5.175em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.437em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1004.44em 2.589em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-546"><span class="mn" id="MathJax-Span-547" style="font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-548" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="texatom" id="MathJax-Span-549" style="padding-left: 0.311em;"><span class="mrow" id="MathJax-Span-550"><span class="munderover" id="MathJax-Span-551"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-552" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.205em 1000.43em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-553" style="font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-554" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="munderover" id="MathJax-Span-555" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-556" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1000.56em 3.821em -999.997em); top: -4.492em; left: 0.003em;"><span class="mo" id="MathJax-Span-557" style=""><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; top: -3.999em; left: -0.058em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.249em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.065em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">¯</mo></mover></mrow><mo><</mo><mover><mi>k</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-37">0<\bar{k}< \overline{k}</script> such that:</p>
<ul>
<li><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-558" role="math" style="width: 8.87em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.639em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1007.52em 2.589em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-559"><span class="mi" id="MathJax-Span-560" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-561" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-562" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-563" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-564" style="font-family: MathJax_Main; padding-left: 0.311em;">></span><span class="mi" id="MathJax-Span-565" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-566" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-567" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-568" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-569" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">α</span><span class="mo" id="MathJax-Span-570" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>></mo><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-38"> W(k) > V(k-\alpha)</script> for $0<k<\bar{k}$< li="">
</k<\bar{k}$<></li><li><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-571" role="math" style="width: 8.87em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.639em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1007.52em 2.589em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-572"><span class="mi" id="MathJax-Span-573" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-574" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-575" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-576" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-577" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="mi" id="MathJax-Span-578" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-579" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-580" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-581" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-582" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">α</span><span class="mo" id="MathJax-Span-583" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo><</mo><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-39"> W(k) < V(k-\alpha)</script> for <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x00AF;</mo></mover></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-584" role="math" style="width: 2.897em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.466em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.05em 1002.47em 2.405em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-585"><span class="mi" id="MathJax-Span-586" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-587" style="font-family: MathJax_Main; padding-left: 0.311em;">></span><span class="texatom" id="MathJax-Span-588" style="padding-left: 0.311em;"><span class="mrow" id="MathJax-Span-589"><span class="munderover" id="MathJax-Span-590"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-591" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.205em 1000.43em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-592" style="font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-40">k>\bar{k}</script></li>
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<p>What this implies is that at low income level (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mo>&lt;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">&#x00AF;</mo></mover></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-593" role="math" style="width: 3.328em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.836em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1002.84em 2.836em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-594"><span class="msubsup" id="MathJax-Span-595"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-596" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-597"><span class="mrow" id="MathJax-Span-598"><span class="mi" id="MathJax-Span-599" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-600" style="font-family: MathJax_Main; padding-left: 0.311em;"><</span><span class="texatom" id="MathJax-Span-601" style="padding-left: 0.311em;"><span class="mrow" id="MathJax-Span-602"><span class="munderover" id="MathJax-Span-603"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-604" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.205em 1000.43em 3.636em -999.997em); top: -4.307em; left: 0.003em;"><span class="mo" id="MathJax-Span-605" style="font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mo><</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-41">k_{j}<\bar{k}</script>), the individual j is better off not participating in the intermediated sector given his inability to cover the fixed cost</p>
<p>However, at a higher level of income (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>j</mi></msub><mo>&gt;</mo><mover><mi>k</mi><mo accent="false">&#x00AF;</mo></mover></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-606" role="math" style="width: 3.39em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.897em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.111em 1002.9em 2.836em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-607"><span class="msubsup" id="MathJax-Span-608"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-609" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-610" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-611" style="font-family: MathJax_Main; padding-left: 0.311em;">></span><span class="munderover" id="MathJax-Span-612" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-613" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1000.56em 3.821em -999.997em); top: -4.492em; left: 0.003em;"><span class="mo" id="MathJax-Span-614" style=""><span style="display: inline-block; position: relative; width: 0.557em; height: 0px;"><span style="position: absolute; top: -3.999em; left: -0.058em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.249em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.999em; left: 0.065em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>j</mi></msub><mo>></mo><mover><mi>k</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-42">k_j > \overline{k}</script>) the fixed cost is insignificant and individual j is better off in the intermediated sector.</p>
<p>The model thus basically explains the initial widening income gap as some people can join the intermediated sector and some can not. It also shows that rich economies grow faster than poor countries.</p>
<h2 id="2.-Modeling-Greenwood-Jovanovic-Model-with-Python">2. Modeling Greenwood-Jovanovic Model with Python<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#2.-Modeling-Greenwood-Jovanovic-Model-with-Python">¶</a></h2><p>We show in detail how we model the Greenwood-Jovanovic model, and we present the findings of our simulations, and finally, the conclusion.</p>
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<h2 id="Estimation-of-the-value-functions---iteration-over-the-Bellman-equations">Estimation of the value functions - iteration over the Bellman equations<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Estimation-of-the-value-functions---iteration-over-the-Bellman-equations">¶</a></h2><p>This section focuses on how the value functions (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-615" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-616"><span class="mi" id="MathJax-Span-617" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-618" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-619" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-620" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-43">w(k)</script>,<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-621" role="math" style="width: 2.035em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.727em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.6em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-622"><span class="mi" id="MathJax-Span-623" style="font-family: MathJax_Math-italic;">v</span><span class="mo" id="MathJax-Span-624" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-625" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-626" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-44">v(k)</script>), which are essential in the decision on joining the financial intermediary, are determined. The following Bellman-equations represent the dynamic programmming problems. For an individual outside the financial intermediary, the life-time expected utility is:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mi>d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-627" role="math" style="width: 40.397em; display: inline-block;"><span style="display: inline-block; position: relative; width: 34.794em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1034.55em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-628"><span class="mi" id="MathJax-Span-629" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-630" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-631" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-632" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-633" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-634" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-635" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-636"><span class="mrow" id="MathJax-Span-637"><span class="mn" id="MathJax-Span-638" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-639" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-640" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-641" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-642" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-643" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-644" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-645" style="font-family: MathJax_Math-italic;">l</span><span class="mi" id="MathJax-Span-646" style="font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-647" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-648" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-649" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-650" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-651" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-652" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-653" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-654" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-655" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">m</span><span class="mi" id="MathJax-Span-656" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-657" style="font-family: MathJax_Math-italic;">x</span><span class="mrow" id="MathJax-Span-658" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-659" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-660" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-661" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-662" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-663" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-664" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-665" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-666" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">ε</span><span class="mo" id="MathJax-Span-667" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-668" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-669" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-670" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-671" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-672" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-673" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-674" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-675" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-676" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">ε</span><span class="mo" id="MathJax-Span-677" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-678" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-679" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">α</span><span class="mo" id="MathJax-Span-680" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-681" style="font-family: MathJax_Main;">)</span></span><span class="mi" id="MathJax-Span-682" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-683" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-684" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-685" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-686" style="font-family: MathJax_Main;">)</span><span class="mi" id="MathJax-Span-687" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-688" style="font-family: MathJax_Math-italic;">G</span><span class="mo" id="MathJax-Span-689" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-690" style="font-family: MathJax_Math-italic;">ε</span><span class="mo" id="MathJax-Span-691" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-692" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>−</mo><mi>α</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-45">\begin{equation}
W(k) = \max_{0 \leq s \leq k} \left[ ln(k-s) + \beta \int max\left(W[s(\theta+\varepsilon)],V[s(\theta+\varepsilon)-\alpha]\right) dF(\theta) dG(\varepsilon) \right]
\end{equation}</script><p>where <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03B5;</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">]</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-693" role="math" style="width: 7.946em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.838em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1006.72em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-694"><span class="mi" id="MathJax-Span-695" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-696" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-697" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-698" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-699" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-700" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-701" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">ε</span><span class="mo" id="MathJax-Span-702" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-703" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-704" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">α</span><span class="mo" id="MathJax-Span-705" style="font-family: MathJax_Main;">]</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>−</mo><mi>α</mi><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-46">V[s(\theta+\varepsilon)-\alpha]</script> is the life-time expected utility if the individual joins the intermediary in the next period. For the individual inside the financial intermediary:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-706" role="math" style="width: 39.227em; display: inline-block;"><span style="display: inline-block; position: relative; width: 33.808em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1033.56em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-707"><span class="mi" id="MathJax-Span-708" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-709" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-710" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-711" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-712" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-713" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-714" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-715"><span class="mrow" id="MathJax-Span-716"><span class="mn" id="MathJax-Span-717" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-718" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-719" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-720" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-721" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-722" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-723" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-724" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-725"></span><span class="mo" id="MathJax-Span-726" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-727" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-728" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-729" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-730" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-731" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-732" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-733" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-734" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">m</span><span class="mi" id="MathJax-Span-735" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-736" style="font-family: MathJax_Math-italic;">x</span><span class="mrow" id="MathJax-Span-737" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-738" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-739" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-740" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-741" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-742" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-743" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-744" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-745" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-746" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-747" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-748" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-749" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-750" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-751" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-752" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-753" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-754" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-755" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-756" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-757" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-758" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-759" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-760" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-761" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-762" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-763" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-764" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-765" style="font-family: MathJax_Main;">]</span><span class="mo" id="MathJax-Span-766" style="font-family: MathJax_Main;">)</span></span><span class="mi" id="MathJax-Span-767" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-768" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-769" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-770" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-771" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-772" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>m</mi><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mi>W</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>)</mo></mrow><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-47">\begin{equation}
V(k) = \max_{0 \leq s \leq k} \left[ \ln (k-s) + \beta \int max\left(W[s\cdot max(\delta,\theta)],V[s \cdot max(\delta,\theta)]\right) dF(\theta) \right]
\end{equation}</script><p>.</p>
<p>Since <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-773" role="math" style="width: 6.653em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.73em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1005.61em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-774"><span class="mi" id="MathJax-Span-775" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-776" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-777" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-778" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-779" style="font-family: MathJax_Main; padding-left: 0.311em;">></span><span class="mi" id="MathJax-Span-780" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-781" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-782" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-783" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>></mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-48">V(k)>W(k)</script> for <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x2200;</mi><mi>k</mi><mo>&#x2208;</mo><mi>K</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-784" role="math" style="width: 3.882em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.328em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1003.33em 2.589em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-785"><span class="mi" id="MathJax-Span-786" style="font-family: MathJax_Main;">∀</span><span class="mi" id="MathJax-Span-787" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-788" style="font-family: MathJax_Main; padding-left: 0.311em;">∈</span><span class="mi" id="MathJax-Span-789" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>k</mi><mo>∈</mo><mi>K</mi></math></span></span><script type="math/tex" id="MathJax-Element-49">\forall k \in K</script>, the second dynamic programming problem can be reduced to the following:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>&#x2264;</mo><mi>s</mi><mo>&#x2264;</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>k</mi><mo>&#x2212;</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>&#x03B2;</mi><mo>&#x222B;</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>&#x22C5;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-790" role="math" style="width: 26.85em; display: inline-block;"><span style="display: inline-block; position: relative; width: 23.156em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1022.91em 4.498em -999.997em); top: -3.384em; left: 0.003em;"><span class="mrow" id="MathJax-Span-791"><span class="mi" id="MathJax-Span-792" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-793" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-794" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-795" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-796" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="munderover" id="MathJax-Span-797" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.158em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1001.85em 4.19em -999.997em); top: -3.999em; left: 0.126em;"><span class="mo" id="MathJax-Span-798" style="font-family: MathJax_Main;">max</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.328em 1002.16em 4.375em -999.997em); top: -3.322em; left: 0.003em;"><span class="texatom" id="MathJax-Span-799"><span class="mrow" id="MathJax-Span-800"><span class="mn" id="MathJax-Span-801" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span class="mo" id="MathJax-Span-802" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-803" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-804" style="font-size: 70.7%; font-family: MathJax_Main;">≤</span><span class="mi" id="MathJax-Span-805" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mrow" id="MathJax-Span-806" style="padding-left: 0.188em;"><span class="mo" id="MathJax-Span-807" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mi" id="MathJax-Span-808" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-809"></span><span class="mo" id="MathJax-Span-810" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-811" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-812" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-813" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">s</span><span class="mo" id="MathJax-Span-814" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-815" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mi" id="MathJax-Span-816" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-817" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-818" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-819" style="font-family: MathJax_Main;">[</span><span class="mi" id="MathJax-Span-820" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-821" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-822" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">m</span><span class="mi" id="MathJax-Span-823" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-824" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-825" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-826" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-827" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-828" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-829" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-830" style="font-family: MathJax_Main;">]</span><span class="mi" id="MathJax-Span-831" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-832" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-833" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-834" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-835" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-836" style="vertical-align: 0.003em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.39em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.139em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="true" form="prefix">max</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></mrow></munder><mrow><mo>[</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>∫</mo><mi>V</mi><mo stretchy="false">[</mo><mi>s</mi><mo>⋅</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-50">\begin{equation}
V(k) = \max_{0 \leq s \leq k} \left[ \ln (k-s) + \beta \int V[s \cdot max(\delta,\theta)] dF(\theta) \right]
\end{equation}</script><p>Relying on this equation and the iteration process on the Bellman-equation (implemented in Python by <a href="http://quant-econ.net/py/optgrowth.html">Sargent and Stachursky (2014)</a>), <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-837" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-838"><span class="mi" id="MathJax-Span-839" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-840" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-841" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-842" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-51">V(k)</script> can be solved. By tailoring the code of <a href="http://quant-econ.net/py/optgrowth.html">Sargent and Stachursky (2014)</a>) to our case, we have managed to carry out the estimation by ourselves (see the code in the <a href="https://github.com/numeraire92">repository</a> in the file <code>[multiprocess_v_value_fun-FINAL.py]</code>). In order to increase the performance of the code, we relied on multi-processing that allows us to map the iterated value function at different values of <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-843" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.5em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-844"><span class="mi" id="MathJax-Span-845" style="font-family: MathJax_Math-italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-52">k</script> simultaneously.</p>
<p>However interestingly, the iteratively estimated function did not converged to the one suggested by the analytical results of the paper (see equation (2) and (3) in <a href="http://piketty.pse.ens.fr/files/GreenwoodJovanovicJPE1990.pdf">Greenwood, Jovanovic (1990)</a>). The analytical form of the value function <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-846" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-847"><span class="mi" id="MathJax-Span-848" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-849" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-850" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-851" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-53">V(k)</script> and the corresponding policy function are liear :</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B2;</mi></mrow></mfrac><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>&#x03B2;</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B2;</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>&#x03B2;</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B2;</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>&#x222B;</mo><mi>ln</mi><mo>&#x2061;</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03B4;</mi><mo>,</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B2;</mi></mrow></mfrac><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-852" role="math" style="width: 44.646em; display: inline-block;"><span style="display: inline-block; position: relative; width: 38.488em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.604em 1038.37em 4.375em -999.997em); top: -3.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-853"><span class="mi" id="MathJax-Span-854" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-855" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-856"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-857" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-858" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-859" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-860" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mfrac" id="MathJax-Span-861" style="padding-left: 0.311em;"><span style="display: inline-block; position: relative; width: 2.466em; height: 0px; margin-right: 0.126em; margin-left: 0.126em;"><span style="position: absolute; clip: rect(3.143em 1000.43em 4.19em -999.997em); top: -4.677em; left: 50%; margin-left: -0.243em;"><span class="mn" id="MathJax-Span-862" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1002.34em 4.375em -999.997em); top: -3.322em; left: 50%; margin-left: -1.167em;"><span class="mrow" id="MathJax-Span-863"><span class="mn" id="MathJax-Span-864" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-865" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-866" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(0.804em 1002.47em 1.235em -999.997em); top: -1.29em; left: 0.003em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-top-width: 1.3px; border-top-style: solid; width: 2.466em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.05em;"></span></span></span></span><span class="mi" id="MathJax-Span-867" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-868"></span><span class="mo" id="MathJax-Span-869" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-870" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-871" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-872" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-873" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-874" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mfrac" id="MathJax-Span-875" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 3.698em; height: 0px; margin-right: 0.126em; margin-left: 0.126em;"><span style="position: absolute; clip: rect(3.143em 1000.56em 4.375em -999.997em); top: -4.677em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-876" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.082em 1003.58em 4.437em -999.997em); top: -3.26em; left: 50%; margin-left: -1.783em;"><span class="mrow" id="MathJax-Span-877"><span class="mo" id="MathJax-Span-878" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-879" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-880" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-881" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-882"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.082em 1000.31em 4.437em -999.997em); top: -3.999em; left: 0.003em;"><span class="mo" id="MathJax-Span-883" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -4.307em; left: 0.373em;"><span class="mn" id="MathJax-Span-884" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(0.804em 1003.7em 1.235em -999.997em); top: -1.29em; left: 0.003em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-top-width: 1.3px; border-top-style: solid; width: 3.698em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.05em;"></span></span></span></span><span class="mi" id="MathJax-Span-885" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-886"></span><span class="mo" id="MathJax-Span-887" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-888" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-889" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-890" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mfrac" id="MathJax-Span-891" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 3.698em; height: 0px; margin-right: 0.126em; margin-left: 0.126em;"><span style="position: absolute; clip: rect(3.143em 1000.56em 4.375em -999.997em); top: -4.677em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-892" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.082em 1003.58em 4.437em -999.997em); top: -3.26em; left: 50%; margin-left: -1.783em;"><span class="mrow" id="MathJax-Span-893"><span class="mo" id="MathJax-Span-894" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-895" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-896" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-897" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-898"><span style="display: inline-block; position: relative; width: 0.804em; height: 0px;"><span style="position: absolute; clip: rect(3.082em 1000.31em 4.437em -999.997em); top: -3.999em; left: 0.003em;"><span class="mo" id="MathJax-Span-899" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -4.307em; left: 0.373em;"><span class="mn" id="MathJax-Span-900" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(0.804em 1003.7em 1.235em -999.997em); top: -1.29em; left: 0.003em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-top-width: 1.3px; border-top-style: solid; width: 3.698em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.05em;"></span></span></span></span><span class="mo" id="MathJax-Span-901" style="font-family: MathJax_Size2; vertical-align: 0.003em; padding-left: 0.188em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.373em;"></span></span><span class="mi" id="MathJax-Span-902" style="font-family: MathJax_Main; padding-left: 0.188em;">ln</span><span class="mo" id="MathJax-Span-903"></span><span class="mi" id="MathJax-Span-904" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">m</span><span class="mi" id="MathJax-Span-905" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-906" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-907" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-908" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-909" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-910" style="font-family: MathJax_Math-italic; padding-left: 0.188em;">θ</span><span class="mo" id="MathJax-Span-911" style="font-family: MathJax_Main;">)</span><span class="mi" id="MathJax-Span-912" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-913" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-914" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-915" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-916" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-917" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mfrac" id="MathJax-Span-918" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 2.466em; height: 0px; margin-right: 0.126em; margin-left: 0.126em;"><span style="position: absolute; clip: rect(3.143em 1000.43em 4.19em -999.997em); top: -4.677em; left: 50%; margin-left: -0.243em;"><span class="mn" id="MathJax-Span-919" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(3.143em 1002.34em 4.375em -999.997em); top: -3.322em; left: 50%; margin-left: -1.167em;"><span class="mrow" id="MathJax-Span-920"><span class="mn" id="MathJax-Span-921" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-922" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-923" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; clip: rect(0.804em 1002.47em 1.235em -999.997em); top: -1.29em; left: 0.003em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-top-width: 1.3px; border-top-style: solid; width: 2.466em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.05em;"></span></span></span></span><span class="mi" id="MathJax-Span-924" style="font-family: MathJax_Main;">ln</span><span class="mo" id="MathJax-Span-925"></span><span class="mo" id="MathJax-Span-926" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-927"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-928" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-929" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-930" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 3.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.211em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></mfrac><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>β</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>β</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>∫</mo><mi>ln</mi><mo></mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></mfrac><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-54">\begin{equation}
V(k_t) = \frac{1}{1-\beta} \ln(1-\beta) + \frac{\beta}{(1-\beta)^2} \ln (\beta)+ \frac{\beta}{(1-\beta)^2} \int \ln max(\delta,\theta)dF(\theta) + \frac{1}{1-\beta} \ln(k_t)
\end{equation}</script><span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>&#x03B2;</mi><msub><mi>k</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-931" role="math" style="width: 5.791em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.991em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1004.99em 2.589em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-932"><span class="mi" id="MathJax-Span-933" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-934" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-935"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-936" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-937" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-938" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-939" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mi" id="MathJax-Span-940" style="font-family: MathJax_Math-italic; padding-left: 0.311em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-941"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-942" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-943" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>β</mi><msub><mi>k</mi><mi>t</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-55">\begin{equation}
s(k_t) = \beta k_t
\end{equation}</script><p>By plugging in the parameters that we assumed for the model (see <code>[parameters.py]</code> module in the <a href="https://github.com/numeraire92">repository</a>), we get the following value function:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>76.9230769</mn><mo>&#x22C5;</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>w</mi><mi>e</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>h</mi><mi mathvariant="normal">&#x005F;</mi><mi>a</mi><mi>x</mi><mi>i</mi><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3569.764136</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-944" role="math" style="width: 26.358em; display: inline-block;"><span style="display: inline-block; position: relative; width: 22.725em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.235em 1022.66em 2.589em -999.997em); top: -2.152em; left: 0.003em;"><span class="mrow" id="MathJax-Span-945"><span class="mi" id="MathJax-Span-946" style="font-family: MathJax_Math-italic;">v</span><span class="mo" id="MathJax-Span-947" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-948"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px;"><span style="position: absolute; clip: rect(3.143em 1000.5em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-949" style="font-family: MathJax_Math-italic;">k</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="mi" id="MathJax-Span-950" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-951" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-952" style="font-family: MathJax_Main; padding-left: 0.311em;">=</span><span class="mn" id="MathJax-Span-953" style="font-family: MathJax_Main; padding-left: 0.311em;">76.9230769</span><span class="mo" id="MathJax-Span-954" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="mi" id="MathJax-Span-955" style="font-family: MathJax_Main; padding-left: 0.249em;">ln</span><span class="mo" id="MathJax-Span-956"></span><span class="mo" id="MathJax-Span-957" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-958" style="font-family: MathJax_Math-italic;">w</span><span class="mi" id="MathJax-Span-959" style="font-family: MathJax_Math-italic;">e</span><span class="mi" id="MathJax-Span-960" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-961" style="font-family: MathJax_Math-italic;">l</span><span class="mi" id="MathJax-Span-962" style="font-family: MathJax_Math-italic;">t</span><span class="mi" id="MathJax-Span-963" style="font-family: MathJax_Math-italic;">h</span><span class="mi" id="MathJax-Span-964" style="font-family: MathJax_Main;">_</span><span class="mi" id="MathJax-Span-965" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-966" style="font-family: MathJax_Math-italic;">x</span><span class="mi" id="MathJax-Span-967" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-968" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-969" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-970" style="font-family: MathJax_Main; padding-left: 0.249em;">+</span><span class="mn" id="MathJax-Span-971" style="font-family: MathJax_Main; padding-left: 0.249em;">3569.764136</span></span><span style="display: inline-block; width: 0px; height: 2.158em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>76.9230769</mn><mo>⋅</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>w</mi><mi>e</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>h</mi><mi mathvariant="normal">_</mi><mi>a</mi><mi>x</mi><mi>i</mi><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3569.764136</mn></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-56">\begin{equation}
v(k_t) = 76.9230769 \cdot \ln(wealth\_axis) + 3569.764136
\end{equation}</script><p>Since we could not explain why our estimates diverge away from this result, we assumed this value function for <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-972" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-973"><span class="mi" id="MathJax-Span-974" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-975" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-976" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-977" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-57">V(k)</script> and estimated in the other value function (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-978" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-979"><span class="mi" id="MathJax-Span-980" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-981" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-982" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-983" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-58">w(k)</script>) with this in a similar manner. Here we modified the code of <a href="http://quant-econ.net/py/optgrowth.html">Sargent and Stachursky (2014)</a>) again. The Python-notebook compatible version of the code of the following (this does not use multi-processing):</p>
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<div class=" highlight hl-ipython2"><pre><span class="c1"># Based on the approach taken by Sargent and Stachurski:</span>
<span class="c1"># http://quant-econ.net/py/optgrowth.html.</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">scipy</span> <span class="kn">as</span> <span class="nn">sp</span>
<span class="kn">import</span> <span class="nn">scipy.stats</span>
<span class="kn">import</span> <span class="nn">parameters</span> <span class="kn">as</span> <span class="nn">param</span>
<span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">dblquad</span>
<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">fminbound</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">log</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interp</span>
<span class="n">SAFE</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">SAFE_R</span>
<span class="n">BETA</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">BETA</span>
<span class="c1"># DISTRIBUTION PARAM (AGGREGATE)</span>
<span class="n">AG_MEAN</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">AG_MEAN</span>
<span class="n">AG_STDEV</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">AG_STDE</span>
<span class="n">MAX_VAL_AG</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">AG_MAXVAL</span>
<span class="n">MIN_VAL_AG</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">AG_MINVAL</span>
<span class="c1"># DISTRIBUTION PARAM (idios. shocks)</span>
<span class="n">ID_MEAN</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_MEAN</span>
<span class="n">ID_STDEV</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_STDE</span>
<span class="n">MAX_VAL_E</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_MAXVAL</span>
<span class="n">MIN_VAL_E</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_MINVAL</span>
<span class="c1"># Number of iterations</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">COST</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">COST_INT</span>
<span class="n">wealth_axis</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">1e-6</span><span class="p">,</span><span class="n">param</span><span class="o">.</span><span class="n">K_MAX</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">param</span><span class="o">.</span><span class="n">K_MAX</span><span class="p">)</span> <span class="c1"># np.ceil(param.K_MAX*0.75)</span>
<span class="c1"># We assume the following value function for v(k) based on the original paper</span>
<span class="n">v_func</span> <span class="o">=</span> <span class="mf">76.9230769</span><span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="n">wealth_axis</span><span class="p">)</span> <span class="o">+</span> <span class="mf">3569.764136</span>
<span class="c1"># or if the result from the estimated v(k) is used:</span>
<span class="c1"># v_func = numpy.genfromtxt(v_value_func.csv, delimiter=',')</span>
<span class="k">def</span> <span class="nf">aggPDF</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Returns the density of a given value from the distribution of the </span>
<span class="sd"> aggregate shock <theta>.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">scipy</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="n">AG_MEAN</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="n">AG_STDEV</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">idiPDF</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Returns the density of a given value from the distribution of the</span>
<span class="sd"> idiosynratic shock <eps = epsilon>.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">scipy</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="n">ID_MEAN</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="n">ID_STDEV</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">w_integral</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">W_ax</span><span class="p">,</span><span class="n">Y_ax</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> The integral part of the Bellman-equation.</span>
<span class="sd"> """</span>
<span class="n">function</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">theta</span><span class="p">,</span><span class="n">eps</span><span class="p">,</span><span class="n">s</span><span class="p">:</span> <span class="nb">max</span><span class="p">(</span><span class="n">interp</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="p">(</span><span class="n">theta</span><span class="o">+</span><span class="n">eps</span><span class="p">),</span><span class="n">W_ax</span><span class="p">,</span><span class="n">Y_ax</span><span class="p">),</span> \
<span class="n">interp</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="p">(</span><span class="n">theta</span><span class="o">+</span><span class="n">eps</span><span class="p">)</span><span class="o">-</span><span class="n">COST</span><span class="p">,</span><span class="n">W_ax</span><span class="p">,</span><span class="n">v_func</span><span class="p">))</span><span class="o">*</span><span class="n">aggPDF</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span><span class="n">idiPDF</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dblquad</span><span class="p">(</span><span class="n">function</span><span class="p">,</span><span class="n">MIN_VAL_E</span><span class="p">,</span><span class="n">MAX_VAL_E</span><span class="p">,</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">MIN_VAL_AG</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">MAX_VAL_AG</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">s</span><span class="p">,))</span>
<span class="k">def</span> <span class="nf">w_bellman_op</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the new value function form <Tw_e> at a given iteration stage.</span>
<span class="sd"> Do not run this function as this estimation procedures take a huge amount of time</span>
<span class="sd"> (days). Instead run the standalone 'multiprocess_w_func_FINAL.py' module as this</span>
<span class="sd"> incoporates the multiprocessing opportunities to increase operation rate.</span>
<span class="sd"> """</span>
<span class="n">wealth_obj</span> <span class="o">=</span> <span class="p">[[</span><span class="n">item</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">item</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">wealth_axis</span><span class="p">)]</span>
<span class="n">Tw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">wealth_axis</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">k</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">wealth_axis</span><span class="p">):</span>
<span class="c1"># Here we define the function to be optimized as a function of s and k.</span>
<span class="n">objective</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">s</span><span class="p">,</span><span class="n">k</span><span class="p">:</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="n">s</span><span class="p">)</span><span class="o">-</span><span class="n">BETA</span><span class="o">*</span><span class="n">w_integral</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">wealth_axis</span><span class="p">,</span><span class="n">w</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># The optimal saving for a given k and w_.</span>
<span class="n">s_star</span> <span class="o">=</span> <span class="n">fminbound</span><span class="p">(</span><span class="n">objective</span><span class="p">,</span> <span class="mf">0.3</span><span class="o">*</span><span class="n">k</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mf">1e-12</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">k</span><span class="p">,))</span>
<span class="c1"># The new value at k in the iterated value function.</span>
<span class="n">Tw</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">objective</span><span class="p">(</span><span class="n">s_star</span><span class="p">,</span><span class="n">k</span><span class="p">)</span>
<span class="k">return</span> <span class="n">Tw</span>
<span class="k">def</span> <span class="nf">policy</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Assuming that the value function takes the form stored in w,</span>
<span class="sd"> the function returns the w* greedy policy function, which gives</span>
<span class="sd"> the optimal saving decision for each level of wealth/bequest <k>.</span>
<span class="sd"> """</span>
<span class="n">policy_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">wealth_axis</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">k</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">wealth_axis</span><span class="p">):</span>
<span class="n">objective</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">s</span><span class="p">,</span><span class="n">k</span><span class="p">:</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="n">s</span><span class="p">)</span><span class="o">-</span><span class="n">BETA</span><span class="o">*</span><span class="n">w_integral</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">wealth_axis</span><span class="p">,</span><span class="n">w</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># Returning the optimal saving decision for a given k.</span>
<span class="n">policy_f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">fminbound</span><span class="p">(</span><span class="n">objective</span><span class="p">,</span><span class="mf">1e-12</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mf">1e-12</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">k</span><span class="p">,))</span>
<span class="k">return</span> <span class="n">policy_f</span>
<span class="c1"># We start with an initial naive assumption on the form of the value function w.</span>
<span class="c1"># We assume log-linear form given the form of v(k).</span>
<span class="n">w</span> <span class="o">=</span> <span class="mi">60</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">wealth_axis</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1500</span>
<span class="c1"># To follow how the estimated functional form changes, we log every functional form.</span>
<span class="n">time_path</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
<span class="c1"># Iteration on the Bellman equation.</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="k">print</span> <span class="s2">">>>> Iteration No. "</span><span class="p">,</span> <span class="n">i</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">w_bellman_op</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">time_path</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">w</span>
<span class="c1"># Obtain the w* greedy policy function.</span>
<span class="n">greedy_policy</span> <span class="o">=</span> <span class="n">policy</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="c1"># Export the estimated function and the log on the estimeted functions.</span>
<span class="c1"># We assume that the wealth grid is already exported in wealth_grid.csv, otherwise:</span>
<span class="c1">#np.savetxt('wealth_grid.csv',wealth_axis, delimiter=",")</span>
<span class="n">np</span><span class="o">.</span><span class="n">savetxt</span><span class="p">(</span><span class="s1">'w_value_func.csv'</span><span class="p">,</span><span class="n">w</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span>
<span class="k">print</span> <span class="s2">"Return value function w in 'w_value_func.csv'"</span>
<span class="n">np</span><span class="o">.</span><span class="n">savetxt</span><span class="p">(</span><span class="s2">"Return w greedy policy function in 'w_policy_func.csv'"</span><span class="p">)</span>
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<p><strong>We do not recommend to run this code directly as it takes a vast amount of time to run.</strong> (It take in net 24-25 hours to estimate the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-984" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-985"><span class="mi" id="MathJax-Span-986" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-987" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-988" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-989" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-59">w(k)</script> function with a processor capable to handle 4 threads simultaenously.) Instead we made a variant of this code that relies on multi-processing again, similarly to the estimation of (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-990" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-991"><span class="mi" id="MathJax-Span-992" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-993" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-994" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-995" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-60">V(k)</script>). This version can be found in the <a href="https://github.com/numeraire92">repository</a>: <code>[multiprocess_w_func_FINAL.py]</code>. In spite of our efforts, the estimation of the value function still took a vast amount of time, 1 iteration takes approximately 4 hours. For future use of this simulation, improvement of this code or implementation of this code to a more efficient software (e.g. MatLab for minimizing integrals) would be needed.</p>
<p>The program returns with a <code>.csv</code> file contaning the value of the value function <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-996" role="math" style="width: 2.712em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.343em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1002.22em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-997"><span class="mi" id="MathJax-Span-998" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-999" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1000" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1001" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-61">W(k)</script> for those <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1002" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.5em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1003"><span class="mi" id="MathJax-Span-1004" style="font-family: MathJax_Math-italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-62">k</script> values that we used to map the function. To produce two related <code>.csv</code> files that hold the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1005" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.5em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1006"><span class="mi" id="MathJax-Span-1007" style="font-family: MathJax_Math-italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-63">k</script>-values where mapping occured (<code>wealth_axis.csv</code>) and the value function <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1008" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1009"><span class="mi" id="MathJax-Span-1010" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-1011" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1012" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1013" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-64">V(k)</script> in a similar manner that for <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-65-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1014" role="math" style="width: 2.712em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.343em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1002.22em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1015"><span class="mi" id="MathJax-Span-1016" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-1017" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1018" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1019" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-65">W(k)</script> (<code>v_value_func.csv</code>), we run the code for estimating the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1020" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1021"><span class="mi" id="MathJax-Span-1022" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-1023" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1024" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1025" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-66">V(k)</script> function assuming the function suggested by the paper (see above) and with 0 iteration.</p>
<p>Finally to obtain the value of the value functions (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-67-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1026" role="math" style="width: 2.712em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.343em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1002.22em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1027"><span class="mi" id="MathJax-Span-1028" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span><span class="mo" id="MathJax-Span-1029" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1030" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1031" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-67">W(k)</script> and <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-68-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1032" role="math" style="width: 2.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.035em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.91em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1033"><span class="mi" id="MathJax-Span-1034" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.188em;"></span></span><span class="mo" id="MathJax-Span-1035" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1036" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1037" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-68">V(k)</script>) at <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-69-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1038" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.5em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1039"><span class="mi" id="MathJax-Span-1040" style="font-family: MathJax_Math-italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-69">k</script>-s where not mapping occured, we use linear interpolation. The employed value functions and related policy functions are graphed below.</p>
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<div class=" highlight hl-ipython2"><pre><span class="c1"># %load bellm_plot.py</span>
<span class="kn">import</span> <span class="nn">plotly.offline</span> <span class="kn">as</span> <span class="nn">ply</span>
<span class="kn">import</span> <span class="nn">plotly.graph_objs</span> <span class="kn">as</span> <span class="nn">go</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">genfromtxt</span>
<span class="n">ply</span><span class="o">.</span><span class="n">init_notebook_mode</span><span class="p">()</span>
<span class="n">pl_wealth_axis</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'wealth_grid.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The k values of the mapping</span>
<span class="n">pl_v_value_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'v_value_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The mapped V(k) value function</span>
<span class="n">pl_v_policy_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'v_policy_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The V* greedy policy function</span>
<span class="n">pl_w_value_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'w_value_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The mapped W(k) value function</span>
<span class="n">pl_w_policy_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'w_policy_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The W* greedy policy function</span>
<span class="c1">## Plotting the value functions</span>
<span class="n">pl_v_val_trace</span> <span class="o">=</span> <span class="n">go</span><span class="o">.</span><span class="n">Scatter</span><span class="p">(</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">pl_wealth_axis</span><span class="p">,</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pl_v_value_f</span><span class="p">,</span>
<span class="n">mode</span> <span class="o">=</span> <span class="s1">'lines+markers'</span><span class="p">,</span>
<span class="n">name</span> <span class="o">=</span> <span class="s1">'V value function (not est.)'</span><span class="p">,</span>
<span class="n">line</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
<span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'rgb(22, 96, 167)'</span><span class="p">),</span>
<span class="n">width</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="n">pl_w_val_trace</span> <span class="o">=</span> <span class="n">go</span><span class="o">.</span><span class="n">Scatter</span><span class="p">(</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">pl_wealth_axis</span><span class="p">,</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pl_w_value_f</span><span class="p">,</span>
<span class="n">mode</span> <span class="o">=</span> <span class="s1">'lines+markers'</span><span class="p">,</span>
<span class="n">name</span> <span class="o">=</span> <span class="s1">'W value function (est.)'</span><span class="p">,</span>
<span class="n">line</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
<span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'rgb(255, 165, 0)'</span><span class="p">),</span>
<span class="n">width</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="n">data_val_func</span> <span class="o">=</span> <span class="p">[</span><span class="n">pl_v_val_trace</span><span class="p">,</span> <span class="n">pl_w_val_trace</span><span class="p">]</span>
<span class="n">layout_value_f</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Value functions of the Bellman-equation'</span><span class="p">,</span>
<span class="n">xaxis</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Value function'</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span>
<span class="n">yaxis</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Wealth'</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span>
<span class="p">)</span>
<span class="n">fig_value_func</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">data_val_func</span><span class="p">,</span><span class="n">layout</span><span class="o">=</span><span class="n">layout_value_f</span><span class="p">)</span>
<span class="n">ply</span><span class="o">.</span><span class="n">iplot</span><span class="p">(</span><span class="n">fig_value_func</span><span class="p">)</span>
<span class="c1">## Plotting the policy functions</span>
<span class="n">pl_v_pol_trace</span> <span class="o">=</span> <span class="n">go</span><span class="o">.</span><span class="n">Scatter</span><span class="p">(</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">pl_wealth_axis</span><span class="p">,</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pl_v_policy_f</span><span class="p">,</span>
<span class="n">mode</span> <span class="o">=</span> <span class="s1">'lines+markers'</span><span class="p">,</span>
<span class="n">name</span> <span class="o">=</span> <span class="s1">'V policy function (est.)'</span><span class="p">,</span>
<span class="n">line</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
<span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'rgb(22, 96, 167)'</span><span class="p">),</span>
<span class="n">width</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="n">pl_w_pol_trace</span> <span class="o">=</span> <span class="n">go</span><span class="o">.</span><span class="n">Scatter</span><span class="p">(</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">pl_wealth_axis</span><span class="p">,</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pl_w_policy_f</span><span class="p">,</span>
<span class="n">mode</span> <span class="o">=</span> <span class="s1">'lines+markers'</span><span class="p">,</span>
<span class="n">name</span> <span class="o">=</span> <span class="s1">'W policy function (est.)'</span><span class="p">,</span>
<span class="n">line</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
<span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'rgb(255, 165, 0)'</span><span class="p">),</span>
<span class="n">width</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="n">data_policy_func</span> <span class="o">=</span> <span class="p">[</span><span class="n">pl_v_pol_trace</span><span class="p">,</span><span class="n">pl_w_pol_trace</span><span class="p">]</span>
<span class="n">layout_policy_f</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Policy function determining the optimal saving decision'</span><span class="p">,</span>
<span class="n">xaxis</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Saving'</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span>
<span class="n">yaxis</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">title</span> <span class="o">=</span> <span class="s1">'Wealth'</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span>
<span class="p">)</span>
<span class="n">fig_policy_func</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">data_policy_func</span><span class="p">,</span><span class="n">layout</span><span class="o">=</span><span class="n">layout_policy_f</span><span class="p">)</span>
<span class="n">ply</span><span class="o">.</span><span class="n">iplot</span><span class="p">(</span><span class="n">fig_policy_func</span><span class="p">)</span>
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<p>The graphs show unexpected results that seemingly reflects that we have not managed to estimate the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-70-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1041" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1042"><span class="mi" id="MathJax-Span-1043" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-1044" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1045" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1046" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-70">w(k)</script> value function adequately. According to the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-71-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1047" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1048"><span class="mi" id="MathJax-Span-1049" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-1050" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1051" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1052" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-71">w(k)</script> greedy policy function individuals outside the financial intermediary with lower amount of bequest than 2.586 should consume all their bequest and save none. This seems to be implausible as even outside the financial intermediary individual projects can provide higher returns that the safe project (+1%) on expected terms. Moreover, based on this, these individuals will always decide to join the financial intermediary as $v(k-JOINCOST)<w(k)$ for="" the="" relevant="" $k$="" itnerval.<="" p="">
</w(k)$></p><p>The problem is likely stem from the fact that the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-72-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1053" role="math" style="width: 2.035em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.727em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.6em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1054"><span class="mi" id="MathJax-Span-1055" style="font-family: MathJax_Math-italic;">v</span><span class="mo" id="MathJax-Span-1056" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1057" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1058" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-72">v(k)</script> value function is not mapped for negative values. This is relevant as <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-73-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#x2212;</mo><mi>J</mi><mi>O</mi><mi>I</mi><mi>N</mi><mi>C</mi><mi>O</mi><mi>S</mi><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1059" role="math" style="width: 8.685em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.454em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1007.45em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1060"><span class="mi" id="MathJax-Span-1061" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1062" style="font-family: MathJax_Main; padding-left: 0.249em;">−</span><span class="mi" id="MathJax-Span-1063" style="font-family: MathJax_Math-italic; padding-left: 0.249em;">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mi" id="MathJax-Span-1064" style="font-family: MathJax_Math-italic;">O</span><span class="mi" id="MathJax-Span-1065" style="font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mi" id="MathJax-Span-1066" style="font-family: MathJax_Math-italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mi" id="MathJax-Span-1067" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mi" id="MathJax-Span-1068" style="font-family: MathJax_Math-italic;">O</span><span class="mi" id="MathJax-Span-1069" style="font-family: MathJax_Math-italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.065em;"></span></span><span class="mi" id="MathJax-Span-1070" style="font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.126em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mi>J</mi><mi>O</mi><mi>I</mi><mi>N</mi><mi>C</mi><mi>O</mi><mi>S</mi><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-73">k-JOINCOST</script> can easily be negative at the individuals decision. However, we realized this issue too lately to be able to re-estimate the <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-74-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1071" role="math" style="width: 2.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.974em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.419em 1001.85em 2.774em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1072"><span class="mi" id="MathJax-Span-1073" style="font-family: MathJax_Math-italic;">w</span><span class="mo" id="MathJax-Span-1074" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1075" style="font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1076" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-74">w(k)</script> policy function in time. <strong>Because of this, we have decided to use these functions to demonstrate the capability of the simulation model, but we choose not to interpret the results as they are incorrect.</strong></p>
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<h2 id="The-simulation-model">The simulation model<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#The-simulation-model">¶</a></h2><p>We follow an agent-based modeling approach implemented in Python 2.7 to simulate the financial development model described above. The aim of the simulation is to generate a time path of the average wealth, inequality, growth rate and development of the intermediary. We expect these time paths to reflect the predictions of the analytical model. The parameters of the model can be found in the <code>[parameters.py]</code> module (the indicated parameters are the one we used). There are two types of agents in the model: the utility maximizing individuals of the economy and the financial intermediary that estimates the aggregate productivity of the economy and invest the involved capital in either safe or risky individual projects accordingly. This section starts with the implementation of these two types of agents. Then it presents ancillary function to produce the Gini coefficients for each period. Finally, the main element of the simulation, the dynamics will be discussed in detail.</p>
<p>The following header should be run before executing the codes in the subsequent sections. This header imports the <code>[NumPy]</code> module as well as the parameters of the model from the <code>[parameters.py]</code> module. It also assigns the relevant constants as well.</p>
<p><em>Before running the header, make sure that the most recent <code>[parameters.py]</code> is compiled to <code>.pyc</code>, otherwise the program will only read the old parameters. Recompile <code>[parameters.py]</code> by deleting <code>[parameters.pyc]</code> and running this header.</em></p>
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<div class=" highlight hl-ipython2"><pre><span class="c1">#%load_ext autoreload # This line is needed to read the newest changes fromt the parameters.py</span>
<span class="c1"># module</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">parameters</span> <span class="kn">as</span> <span class="nn">param</span>
<span class="c1">#%autoreload # This line is needed to read the newest changes fromt the parameters.py module</span>
<span class="c1"># Definition of the constants</span>
<span class="n">SAFE_PROJECT</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">SAFE_R</span> <span class="c1"># return on safe project 1%</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">T</span> <span class="c1"># number of periods</span>
<span class="n">IS_MEAN</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_MEAN</span> <span class="c1"># Idiosyncratic_shock parameters</span>
<span class="n">IS_STDE</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">IS_STDE</span> <span class="c1"># Std. deviation</span>
<span class="n">JOINCOST</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">COST_INT</span> <span class="c1"># Cost of joining the financial intermediary</span>
<span class="n">WE_MEAN</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">WE_MEAN</span> <span class="c1"># Mean of the wealth distribution</span>
<span class="n">WE_STDE</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">WE_STDE</span> <span class="c1"># Std. deviation</span>
<span class="n">aggr_shocks</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">param</span><span class="o">.</span><span class="n">AG_MEAN</span><span class="p">,</span><span class="n">param</span><span class="o">.</span><span class="n">AG_STDE</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="c1"># This list holds the time path of the</span>
<span class="c1"># aggregate productivity.</span>
<span class="c1"># Value functions:</span>
<span class="n">wealth_grid</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'wealth_grid.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The k values of the mapping</span>
<span class="n">v_value_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'v_value_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The mapped V(k) value function</span>
<span class="n">v_policy_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'v_policy_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The V* greedy policy function</span>
<span class="n">w_value_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'w_value_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The mapped W(k) value function</span>
<span class="n">w_policy_f</span> <span class="o">=</span> <span class="n">genfromtxt</span><span class="p">(</span><span class="s1">'w_policy_func.csv'</span><span class="p">,</span><span class="n">delimiter</span><span class="o">=</span><span class="s1">','</span><span class="p">)</span> <span class="c1"># The W* greedy policy function</span>
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<h3 id="Individuals-in-the-economy">Individuals in the economy<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Individuals-in-the-economy">¶</a></h3><p>The individuals of the economy are defined as <code>[objects]</code> in the code blocks below (the code is from the main module - <code>[main_m.py]</code>). Agents are homogenous in their impatience (discount_factor - <code>.pref</code>=0.987 attribute) but heterogenous in their initial starting bequest (wealth - first element of the <code>.wealth_path</code>, which is a list storing the available wealth of the given individual for each timeperiod <code>t</code>), which is drawn from a normal distribution with the mean and standard error of <code>WE_MEAN</code>=2 and <code>WE_STDE</code>=0.4, respectively . The time path of idiosyncratic productivity shocks is stored in the list under the <code>.idios_shocks</code> attribute and the values are drawn from a normal distribution with the mean and std. dev. of <code>IS_MEAN</code>=0 and <code>IS_STDE</code>=0.04. The <code>.member</code> and <code>.member_n</code> attributes indicates whether the individual is a member in the intermediary and if so, what is his/her ID in the list of member individuals in the intermediary (see <code>class Intermediary</code> and its <code>.agents</code> attribute later. The <code>.join_cost</code> attributes stores the expenses in each <code>t</code> period on joining the intermediary. The list contains <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-75-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1077" role="math" style="width: 0.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.496em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.481em 1000.43em 2.528em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1078"><span class="mn" id="MathJax-Span-1079" style="font-family: MathJax_Main;">0</span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-75">0</script>-ss except for the time period when the agent joins the intermediary.</p>
<p>The <code>.w(k)</code> and <code>v(k)</code> functions returns the value of the value function , while function <code>.w_sigma(k)</code> and <code>.v_sigma(k)</code> returns the value of the corresponding greedy function for a given wealth of <code>k</code>. The value <code>.w(k)</code> and <code>.v(k)</code> functions rely on the results produced in the previous section. Using the value functions, the <code>.join_(k)</code> function returns whether the individual will join the financial intermediary.</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">class</span> <span class="nc">Agent</span><span class="p">:</span>
<span class="sd">"""</span>
<span class="sd"> Class for an agent in the economy.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">iD</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> - randomly drawn initial wealth</span>
<span class="sd"> - randomly drawn initial idiosyncractic shock</span>
<span class="sd"> - setting time preference</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">ID</span> <span class="o">=</span> <span class="n">iD</span>
<span class="bp">self</span><span class="o">.</span><span class="n">wealth_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">wealth_path</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">WE_MEAN</span><span class="p">,</span><span class="n">WE_STDE</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">saving_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">idios_shocks</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">IS_MEAN</span><span class="p">,</span><span class="n">IS_STDE</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">pref</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">BETA</span>
<span class="bp">self</span><span class="o">.</span><span class="n">member</span> <span class="o">=</span> <span class="bp">False</span>
<span class="bp">self</span><span class="o">.</span><span class="n">member_n</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
<span class="bp">self</span><span class="o">.</span><span class="n">join_costs</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="k">def</span> <span class="nf">w</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Value function outside the intermediary.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">wealth_grid</span><span class="p">,</span><span class="n">w_value_f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">v</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Value function inside the intermediary.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">wealth_grid</span><span class="p">,</span><span class="n">v_value_f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">v_sigma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Mapping function for v(k). It determines the optimal amount</span>
<span class="sd"> of saving for a given amount of endowment (wealth). This function</span>
<span class="sd"> is the v*-greedy policy function.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">wealth_grid</span><span class="p">,</span><span class="n">v_policy_f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">w_sigma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Mapping function for w(k). It determines the optimal amount</span>
<span class="sd"> of saving for a given amount of endowment (wealth). This function</span>
<span class="sd"> is the w*-greedy policy function.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">wealth_grid</span><span class="p">,</span><span class="n">w_policy_f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">join_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function indicates whether the individual should join the</span>
<span class="sd"> financial intermediary or not. It compares the expected life time</span>
<span class="sd"> utilities for the two cases.</span>
<span class="sd"> """</span>
<span class="n">response</span> <span class="o">=</span> <span class="bp">False</span>
<span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">v</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="n">JOINCOST</span><span class="p">)):</span>
<span class="n">response</span> <span class="o">=</span> <span class="bp">True</span>
<span class="k">return</span> <span class="n">response</span>
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<h3 id="Financial-intermediary">Financial intermediary<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Financial-intermediary">¶</a></h3><p>In a similar manner, the financial intermediary is defined as an object as well (the code is from the main module - <code>[main_m.py]</code>). The financial intermediary stores the attributes of member individuals in a list (<code>.agents</code>), which consists of <code>[objects]</code> representing the member individuals. The financial intermediary's object also stores the time path of the number of members (<code>.number_of_members</code>), of the estimates on the aggregate productivity produced by the sampling (<code>.estimates_path</code>), of the average return after the investment process (return on the sample and return on the reamining individual projects or the safe project) (<code>.avg_ret_path</code>) and some other indicators that help evaluating the performance of the intermediary.</p>
<p>These paths are useful for us to analyse the development and the performance of the financial intermediary over time.</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">class</span> <span class="nc">MemberIntermediary</span><span class="p">():</span>
<span class="sd">"""</span>
<span class="sd"> This object is used to represent members of the financial intermediary.</span>
<span class="sd"> </span>
<span class="sd"> - It stores the current idiosyncratic productivity shock and</span>
<span class="sd"> saving decision at a given period t. .sample idicates wether</span>
<span class="sd"> the member is selected into the sample in the period.</span>
<span class="sd"> </span>
<span class="sd"> - .r_project stores the return on the individuals idiosyncratic</span>
<span class="sd"> project if the average amount of capital (saving) of</span>
<span class="sd"> the fin. interm. is invested int he project.</span>
<span class="sd"> </span>
<span class="sd"> - .r_saving stores the return provided by the fin. interm.</span>
<span class="sd"> on the saving deposited by the member individual.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">agent</span><span class="p">,</span><span class="n">time</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">ID</span> <span class="o">=</span> <span class="n">agent</span><span class="o">.</span><span class="n">ID</span>
<span class="bp">self</span><span class="o">.</span><span class="n">idi_shock</span> <span class="o">=</span> <span class="n">agent</span><span class="o">.</span><span class="n">idios_shocks</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">saving</span> <span class="o">=</span> <span class="n">agent</span><span class="o">.</span><span class="n">saving_path</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">sample</span> <span class="o">=</span> <span class="mi">0</span>
<span class="bp">self</span><span class="o">.</span><span class="n">r_project</span> <span class="o">=</span> <span class="mi">0</span>
<span class="bp">self</span><span class="o">.</span><span class="n">r_saving</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">class</span> <span class="nc">Intermediary</span><span class="p">:</span>
<span class="sd">"""</span>
<span class="sd"> Class for the financial intermediary.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> - list for member agents</span>
<span class="sd"> - definin cost of joining the inst.</span>
<span class="sd"> - setting sampling parameter (% of the indiv.</span>
<span class="sd"> projects selected into the sample)</span>
<span class="sd"> - est_ret = the estimated return based on the</span>
<span class="sd"> sample</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># sampling parameter - this % of the individual</span>
<span class="bp">self</span><span class="o">.</span><span class="n">sampling_param</span> <span class="o">=</span> <span class="n">param</span><span class="o">.</span><span class="n">SAMPLING</span>
<span class="bp">self</span><span class="o">.</span><span class="n">number_of_members</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">estimates_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">avg_ret_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">investment_ret</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">tot_cap_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">tot_ret_path</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="p">)]</span>
<span class="c1"># self.est_ret = 1</span>
<span class="k">def</span> <span class="nf">add_member</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">the_Agent</span><span class="p">,</span><span class="n">time</span><span class="p">):</span>
<span class="n">entry</span> <span class="o">=</span> <span class="n">MemberIntermediary</span><span class="p">(</span><span class="n">the_Agent</span><span class="p">,</span><span class="n">time</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">entry</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span> <span class="c1"># Gives back the position of the agent in the member</span>
<span class="c1"># list.</span>
<span class="k">def</span> <span class="nf">depSaving</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mem_n</span><span class="p">,</span> <span class="n">saving_val</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function indicates that an agent deposit a certain</span>
<span class="sd"> sum. mem_n is the index of the agent within the institution</span>
<span class="sd"> and saving_val is the deposited sum.</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">mem_n</span><span class="p">]</span><span class="o">.</span><span class="n">saving</span> <span class="o">=</span> <span class="n">saving_val</span>
<span class="k">def</span> <span class="nf">sampling</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">all_Agents</span><span class="p">,</span><span class="n">time</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function does the sampling of the individual projects</span>
<span class="sd"> and return the estimate on the aggregate shock, the average</span>
<span class="sd"> capital per member in the intermediary, the total return</span>
<span class="sd"> on the sample investment, and the numble of the individual</span>
<span class="sd"> projects in the sample.</span>
<span class="sd"> """</span>
<span class="n">n_members</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span>
<span class="n">ag_shock</span> <span class="o">=</span> <span class="n">aggr_shocks</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="c1"># estimated aggregate shock</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># average capital</span>
<span class="n">avg_capital</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">agent</span><span class="o">.</span><span class="n">saving</span><span class="o">/</span><span class="n">n_members</span> <span class="k">for</span> <span class="n">agent</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">])</span>
<span class="n">totR_sampl</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># total return on sampling</span>
<span class="c1"># depending on the number of the members in the fin. int.</span>
<span class="c1"># there are 3 scenarios (0, 1, >1)</span>
<span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span><span class="o">></span><span class="mi">1</span><span class="p">):</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="n">ag_shock</span>
<span class="n">sample</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span><span class="o">*</span>
<span class="bp">self</span><span class="o">.</span><span class="n">sampling_param</span><span class="p">),</span><span class="n">replace</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
<span class="k">for</span> <span class="n">agent</span> <span class="ow">in</span> <span class="n">sample</span><span class="p">:</span>
<span class="n">id_intermediary</span> <span class="o">=</span> <span class="n">all_Agents</span><span class="p">[</span><span class="n">agent</span><span class="o">.</span><span class="n">ID</span><span class="p">]</span><span class="o">.</span><span class="n">member_n</span>
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">id_intermediary</span><span class="p">]</span><span class="o">.</span><span class="n">sample</span> <span class="o">=</span> <span class="mi">1</span>
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">id_intermediary</span><span class="p">]</span><span class="o">.</span><span class="n">r_project</span> <span class="o">=</span> <span class="p">(</span><span class="n">ag_shock</span> <span class="o">+</span> \
<span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">id_intermediary</span><span class="p">]</span><span class="o">.</span><span class="n">idi_shock</span><span class="p">)</span> <span class="o">*</span> <span class="n">avg_capital</span>
<span class="n">totR_sampl</span> <span class="o">=</span> <span class="n">totR_sampl</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">id_intermediary</span><span class="p">]</span><span class="o">.</span><span class="n">r_project</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="n">estimate</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="n">id_intermediary</span><span class="p">]</span><span class="o">.</span><span class="n">idi_shock</span> <span class="o">/</span> \
<span class="nb">len</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="o">==</span><span class="mi">1</span><span class="p">)):</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="n">Agents</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">.</span><span class="n">idios_shocks</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># the function gives back:</span>
<span class="c1"># - estiamted aggragate shock</span>
<span class="c1"># - average saving/capital per member in the intermediary</span>
<span class="c1"># - total return on the sample investment</span>
<span class="k">return</span> <span class="n">estimate</span><span class="p">,</span> <span class="n">avg_capital</span><span class="p">,</span> <span class="n">totR_sampl</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">invest</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">all_agents</span><span class="p">,</span> <span class="n">time</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> The intermediary's decision on which options to invest to.</span>
<span class="sd"> (sampling is done here)</span>
<span class="sd"> """</span>
<span class="n">ag_shock</span> <span class="o">=</span> <span class="n">aggr_shocks</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">tot_cap_path</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">member</span><span class="o">.</span><span class="n">saving</span> <span class="k">for</span> <span class="n">member</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">])</span>
<span class="n">est_agg_return</span><span class="p">,</span> <span class="n">avg_cap</span><span class="p">,</span> <span class="n">totR_sampl</span><span class="p">,</span> <span class="n">n_sampl</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sampling</span><span class="p">(</span><span class="n">all_agents</span><span class="p">,</span> <span class="n">time</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">estimates_path</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">est_agg_return</span>
<span class="n">total_return</span> <span class="o">=</span> <span class="n">totR_sampl</span> <span class="c1"># the total return on the whole portfolio</span>
<span class="k">if</span> <span class="n">est_agg_return</span> <span class="o">></span> <span class="n">SAFE_PROJECT</span><span class="p">:</span>
<span class="c1"># investing in the individual risky projects</span>
<span class="bp">self</span><span class="o">.</span><span class="n">investment_ret</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">est_agg_return</span>
<span class="k">for</span> <span class="n">agent</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">agent</span><span class="o">.</span><span class="n">sample</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
<span class="n">agent</span><span class="o">.</span><span class="n">r_project</span> <span class="o">=</span> <span class="n">avg_cap</span><span class="o">*</span><span class="p">(</span><span class="n">ag_shock</span><span class="o">+</span><span class="n">agent</span><span class="o">.</span><span class="n">idi_shock</span><span class="p">)</span>
<span class="n">total_return</span> <span class="o">=</span> <span class="n">total_return</span> <span class="o">+</span> <span class="n">agent</span><span class="o">.</span><span class="n">r_project</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># investing in the safe projects</span>
<span class="bp">self</span><span class="o">.</span><span class="n">investment_ret</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">SAFE_PROJECT</span>
<span class="n">total_return</span> <span class="o">=</span> <span class="n">total_return</span> <span class="o">+</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span><span class="o">-</span><span class="n">n_sampl</span><span class="p">)</span><span class="o">*</span>\
<span class="n">avg_cap</span><span class="o">*</span><span class="n">SAFE_PROJECT</span>
<span class="c1"># we have the total return on the whole portfolio: avg_return</span>
<span class="bp">self</span><span class="o">.</span><span class="n">tot_ret_path</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">total_return</span>
<span class="n">avg_return</span> <span class="o">=</span> <span class="p">(</span><span class="n">total_return</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">avg_cap</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">avg_ret_path</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">avg_return</span>
<span class="k">for</span> <span class="n">agent</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">:</span>
<span class="c1"># now we will have to establish the dividents for each member</span>
<span class="n">agent</span><span class="o">.</span><span class="n">r_saving</span> <span class="o">=</span> <span class="n">avg_return</span> <span class="o">*</span> <span class="n">agent</span><span class="o">.</span><span class="n">saving</span>
<span class="c1"># and pay out the divident</span>
<span class="k">if</span> <span class="p">(</span><span class="n">param</span><span class="o">.</span><span class="n">T</span> <span class="o">!=</span> <span class="n">time</span><span class="p">):</span>
<span class="n">tID</span> <span class="o">=</span> <span class="n">agent</span><span class="o">.</span><span class="n">ID</span>
<span class="n">all_agents</span><span class="p">[</span><span class="n">tID</span><span class="p">]</span><span class="o">.</span><span class="n">wealth_path</span><span class="p">[</span><span class="n">time</span><span class="p">]</span><span class="o">=</span><span class="n">agent</span><span class="o">.</span><span class="n">r_saving</span>
<span class="k">def</span> <span class="nf">newTick</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">all_Agents</span><span class="p">,</span> <span class="n">time</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> New tick - copy new idi_shocks and reset the rest.</span>
<span class="sd"> """</span>
<span class="k">for</span> <span class="n">agent</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">time</span><span class="o">!=</span><span class="n">T</span><span class="p">):</span>
<span class="n">agent</span><span class="o">.</span><span class="n">idi_shock</span> <span class="o">=</span> <span class="n">all_Agents</span><span class="p">[</span><span class="n">agent</span><span class="o">.</span><span class="n">ID</span><span class="p">]</span><span class="o">.</span><span class="n">idios_shocks</span><span class="p">[</span><span class="n">time</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">agent</span><span class="o">.</span><span class="n">idi_shock</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">agent</span><span class="o">.</span><span class="n">saving</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">agent</span><span class="o">.</span><span class="n">sample</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">agent</span><span class="o">.</span><span class="n">r_project</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">agent</span><span class="o">.</span><span class="n">r_saving</span> <span class="o">=</span> <span class="mi">0</span>
<span class="bp">self</span><span class="o">.</span><span class="n">number_of_members</span><span class="p">[</span><span class="n">time</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">agents</span><span class="p">)</span>
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<p>Finally, the fin. interm. object also have the functions to add new members returning the new member's ID in their list (<code>.add_member(...)</code>), to handle (store) the deposited saving of a member (<code>.depSaving(...)</code>), to draw a sample from the individual project and return the estimation on the aggregate productivity of the economy (<code>.sampling(...)</code>) and to execute the investment and pay the returns on savings (<code>.invest(...)</code>). The investment function first does the sampling and the estimation with the <code>.sampling</code> function, then decides whether to invest the remaining amount of capital in the risky or safe projects based on the relationship between the estimated aggregate productivity and the return on the safe project (<code>SAFE_PROJECT</code>=1.01), then finally determines the average return on the total portfolio and pays dividents (<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-76-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>&#x22C5;</mo><msub><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>v</mi><mi>g</mi></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1080" role="math" style="width: 4.067em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.513em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.727em 1003.51em 2.836em -999.997em); top: -2.337em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1081"><span class="msubsup" id="MathJax-Span-1082"><span style="display: inline-block; position: relative; width: 1.05em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.43em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-1083" style="font-family: MathJax_Math-italic;">s</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.496em;"><span class="texatom" id="MathJax-Span-1084"><span class="mrow" id="MathJax-Span-1085"><span class="mi" id="MathJax-Span-1086" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-1087" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span><span class="mo" id="MathJax-Span-1088" style="font-family: MathJax_Main; padding-left: 0.249em;">⋅</span><span class="msubsup" id="MathJax-Span-1089" style="padding-left: 0.249em;"><span style="display: inline-block; position: relative; width: 1.604em; height: 0px;"><span style="position: absolute; clip: rect(3.39em 1000.43em 4.19em -999.997em); top: -3.999em; left: 0.003em;"><span class="mi" id="MathJax-Span-1090" style="font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span><span style="position: absolute; top: -3.876em; left: 0.434em;"><span class="texatom" id="MathJax-Span-1091"><span class="mrow" id="MathJax-Span-1092"><span class="mi" id="MathJax-Span-1093" style="font-size: 70.7%; font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-1094" style="font-size: 70.7%; font-family: MathJax_Math-italic;">v</span><span class="mi" id="MathJax-Span-1095" style="font-size: 70.7%; font-family: MathJax_Math-italic;">g<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.006em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.343em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>t</mi></mrow></msub><mo>⋅</mo><msub><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>v</mi><mi>g</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-76">s_{it} \cdot r_{avg}</script>).</p>
<p>The final function (<code>.newTick(...)</code>) is responsible to set and reset the attributes for each member (see the ˙MemberIntermediary˙ object) to their initial values in the next period.</p>
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<h3 id="Calculation-of-the-Gini-coefficient">Calculation of the Gini-coefficient<a class="anchor-link" href="file:///C:/Users/Home/AppData/Local/Temp/Dahal_Nguyen_Huy_Obilor_-_Agent_based_modeling_approach_of_the_Greenwood-Jovanovic_model.html#Calculation-of-the-Gini-coefficient">¶</a></h3><p>In order to analyse the time path of inequality in the economy, we rely on a function that produces the Gini-coefficient for a given distribution of wealth (the code is part of the <code>[main_m.py]</code> module as well). It takes a list of values (representing the wealth of an individual) and returns a value between 0 and 1 (representing equality and inequality respectively). This part of the code is not ours, it was written by <a href="http://planspace.org/2013/06/21/how-to-calculate-gini-coefficient-from-raw-data-in-python/">Aaron Schumacher (2013)</a>.</p>
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