Add OLG lecture

in-work/OLG.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
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+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}
+```