Update OLG model #168
Update OLG model #168
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Would someone mind to look into this build error for me? |
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@Smit-create @shlff With apologies, I ended up cutting the quasi-linear exercise. Thanks for your efforts trying to make it work. Nonetheless, it was hard to clean up and it's a kind of unusual case. So I think what we have now is good. Perhaps @shlff could briefly review this and then @mmcky could merge? There's also a build error --- could someone please look into that (@mmcky or @Smit-create , if you have time?) |
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Thanks, @jstac for the fixes. I'm having a look at the failures. |
shlff
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Thanks @jstac and @Smit-create . This lecture looks very beautiful now.
Also great thanks for John's extremely helpful advice on academic writing.
I quickly reviewed the lecture. Please find my comments above and feel free to let me know if you like me to make any of those proposed changes @jstac .
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| Let's observe the dynamics of the equilibrium price $R^*_{t+1}$. | ||
| Let's redo our plot above but now inserting the equilibrium quantity and price. |
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Inserting is correct.
lectures/olg.md
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| cannot be obtained analytically. | ||
| To solve the equation we need to turn to Newton's method. | ||
| Instead, solve for $k^*$ using newton's method. |
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Thanks. Also, "Instead, we solve".
lectures/olg.md
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| :label: crra_newton_2 | ||
| g(k^*) = k^* \left [ 1 + \beta^{-1/\gamma} (\alpha (k^*)^{\alpha-1})^{(\gamma-1)/\gamma} \right ] - (1-\alpha)(k^*)^{\alpha} | ||
| ``` | ||
| First let define $f$. |
lectures/olg.md
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| ### Consumer's problem | ||
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| Suppose that utility for individuals born at time $t$ take the form |
lectures/olg.md
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| The first-order conditions for a maximum can be obtained by taking the derivative of the objective function with respect to capital and labor respectively and setting it to zero: | ||
| ### Demand | ||
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| Using our assumption $\ell_1 = 1$ allows us to write |
| ax.plot(k_grid, k_grid_next, lw=2, alpha=0.6, label='$g$') | ||
| ax.plot(k_grid, k_grid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$') | ||
| ax.plot(k_grid, k_grid_next, lw=2, alpha=0.6, label='$g$') |
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We need to define $g$ in the math expression above.
| ax.plot(np.arange(ts_length), np.full(ts_length, k_star), | ||
| alpha=0.6, color='red', label=r'$k^*$') | ||
| ax.legend(fontsize=10) | ||
| ax.plot(k_grid, k_grid_next, lw=2, alpha=0.6, label='$g$') |
lectures/olg.md
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| ```{code-cell} ipython3 | ||
| def find_Rstar(K_prev, model): | ||
| return optimize.newton(find_Rstar_newton, 0.5, args=(K_prev, model)) | ||
| We introduce a function $g$ such that |
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Should we consider using other notations than $g$ to distinguish it from the $g$ used in the plot code above?
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Good idea. I've changed it to
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Thanks @shlff , much appreciated. Merging. |
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