diff --git a/in-work/OLG.md b/in-work/OLG.md deleted file mode 100644 index bc45fbec2..000000000 --- a/in-work/OLG.md +++ /dev/null @@ -1,116 +0,0 @@ ---- -jupytext: - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.14.4 -kernelspec: - display_name: Python 3 (ipykernel) - language: python - name: python3 ---- - - -# The Overlapping Generations Model - -In this lecture we study the overlapping generations model. - -This model provide controllable alternative to infinite-horizon representative agent. -The dynamics of this model in some special cases are quite similar to Solow model. - -## The Model - -Let's assume that: - -- Time is discrete and runs to infinity. -- Each individual lives for two time periods. -- Individuals born at time $t$ live for dates $t$ and $t + 1$. - -Suppose that the utility functions take the familiar Constant Relative Risk Aversion (CRRA) form, given by: - -```{math} -:label: eq_crra - U(t) = \frac{c_1(t)^{1-\theta}-1}{1-\theta} + - \beta \left( \frac{c_2(1+t)^{1-\theta}-1}{1-\theta} \right ) -``` -where, - -- $\theta > 0$, and $\beta \in (0, 1)$ is the discount factor. -- $c_1(t)$: consumption of the individual born at $t$. -- $c_2(t)$: consumption of the individual at $t+1$. - -For each integer $t \geq 0$, output $Y(t)$ in period $t$ is given by -$$ -Y(t) = F(K(t), L(t)) -$$ -where $K(t)$ is capital, $L(t)$ is labor and $F$ is an aggregate -production function. - - -Without population growth, $L(t)$ equals some constant $L$. - -Setting $k(t) := K(t) / L$, $f(k)=F(K, 1)$ and using homogeneity of degree one now yields: - -$$ - 1 + r(t) = R(t) = f'(k(t)) -$$ -The gross rate of return to saving is equal to the rental rate of capital. - -And, the wage rate is given by, - -$$ - w(t) = f(k(t)) - k (t)f'(k(t)) -$$ - - -Savings by an individual of generation $t$, $s(t)$, is determined as a -solution to: - - -$$ - \begin{aligned} - \max_{c_1(t), c_2(t+1), s(t)} \ & u(c_1 (t)) + \beta u(c_2(t + 1)) \\ - \mbox{subject to } \ & c_1(t) + s(t) \le w(t) \\ - & c_2(t + 1) \le R (t + 1)s(t)\\ - \end{aligned} -$$ - -Second constraint incorporates notion that individuals only spend -money on their own end of life consumption. - -Solving for consumption and thus for savings, - -$$ - s(t) = s(w(t), R(t + 1)) -$$ - -Total savings in the economy will be equal to: - -$$ - S(t) = s(t) L(t) -$$ - -We assume a closed economy, so domestic investment equals aggregate domestic -saving. Therefore, we have - -$$ - K(t + 1) = L(t) s(w (t), R (t + 1)) -$$ - -Setting $k(t) := K(t) / L(t)$, where $L(t + 1) = (1 + n) L(t),$ and using homogeneity of degree one now yields: - -```{math} -:label: k_dyms - k(t) = \frac{s(w (t), R (t + 1))}{1 + n} -``` - - - -A steady state is given by a solution to this equation such that -$k(t + 1) = k (t) = k^*$, i.e, - -```{math} -:label: k_star - k^* = \frac{s(f(k^*)-k^*f'(k^*), f'(k^*))}{1+n} -``` diff --git a/lectures/_toc.yml b/lectures/_toc.yml index 0765a44e0..8feeadd5f 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -28,6 +28,7 @@ parts: - file: schelling - file: solow - file: cobweb + - file: olg - caption: Other numbered: true chapters: diff --git a/lectures/olg.md b/lectures/olg.md new file mode 100644 index 000000000..863dfadf7 --- /dev/null +++ b/lectures/olg.md @@ -0,0 +1,159 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst + format_version: 0.13 + jupytext_version: 1.14.4 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + + +# The Overlapping Generations Model + +In this lecture we study the overlapping generations (OLG) model. + +The dynamics of this model are quite similar to Solow-Swan growth model. + +At the same time, the OLG model adds an important new feature: the choice of +how much to save is endogenous. + +To see why this is important, suppose, for example, that we are interested in +predicting the effect of a new tax on long-run growth. + +We could add a tax to the Solow-Swan model and look at the change in the +steady state. + +But this ignores something important: households will change their behavior +when they face the new tax rate. + +Some might decide to save less, and some might decide to save more. + +Such changes can substantially alter the predictions of the model. + +Hence, if we care about accurate predictions, we should model the decision +problems of the agents. + +In particular, households in the model should decide how much to save and how +much to consume, given the environment that they face (technology, taxes, +prices, etc.) + +The OLG model takes up this challenge. + +We will present a simple version of the OLG model that clarifies the decision +problem of households and studies the implications for long run growth. + + +## The Model + +We assume that + +- time is discrete, so that $t=0, 1, \ldots$ and +- individuals born at time $t$ live for two periods: dates $t$ and $t + 1$. + + +### Preferences + +Suppose that the utility functions take the familiar constant relative risk +aversion (CRRA) form, given by: + +```{math} +:label: eq_crra + U_t = \frac{c^1_t^{1-\gamma}-1}{1-\gamma} + + \beta \left( \frac{c^2_{t+1}^{1-\gamma}-1}{1-\gamma} \right ) +``` + +Here + +- $\gamma$ is a parameter and $\beta \in (0, 1)$ is the discount factor +- $c^1_t$ is time $t$ consumption of the individual born at time $t$ +- $c^2_t$ is time $t+1$ consumption of the same individual (born at time $t$) + +### Production + +For each integer $t \geq 0$, output $Y_t$ in period $t$ is given by + +$$ + Y_t = F(K_t, L_t) +$$ + +Here $K_t$ is capital, $L_t$ is labor and $F$ is an aggregate production function. + +Without population growth, $L_t$ equals some constant $L$. + +### Prices + +Setting $k_t := K_t / L$, $f(k)=F(K, 1)$ and using homogeneity of degree one now yields: + +$$ + 1 + r_t = R_t = f'(k_t) +$$ + +The gross rate of return to saving is equal to the rental rate of capital. + +The wage rate is given by + +$$ + w_t = f(k_t) - k_t f'(k_t) +$$ + + + +### Equilibrium + +Savings by an individual of generation $t$, $s_t$, is determined as a +solution to: + +$$ + \begin{aligned} + \max_{c^1_t, c^2_{t+1}, s_t} \ & u(c^1_t) + \beta u(c^2_{t+1}) \\ + \mbox{subject to } \ & c^1_t + s_t \le w_t \\ + & c^2_{t+1} \le R_{t+1}s_t\\ + \end{aligned} +$$ + +The second constraint incorporates notion that individuals only spend +money on their own end of life consumption. + +Solving for consumption and thus for savings, + +$$ + s_t = s(w_t, R_{t+1}) +$$ + +Total savings in the economy will be equal to: + +$$ + S_t = s_t L_t +$$ + + + +### Dynamics + +We assume a closed economy, so domestic investment equals aggregate domestic +saving. Therefore, we have + +$$ + K_{t+1} = L_t s(w_t, R_{t+1}) +$$ + +Setting $k_t := K_t / L_t$, where $L_{t+1} = (1 + n) L_t,$ and using homogeneity of degree one now yields: + +```{math} +:label: k_dyms + k_t = \frac{s(w_t, R_{t+1})}{1 + n} +``` + + + +A steady state is given by a solution to this equation such that +$k_{t+1} = k_t = k^*$, i.e, + +```{math} +:label: k_star + k^* = \frac{s(f(k^*)-k^*f'(k^*), f'(k^*))}{1+n} +```