23 An OLG Model with Labor-Leisure Choice
In this section we solve for an equilibrium using an overlapping generations (OLG) model. This model specifically models inter-cohort heterogeneity, i.e., the model allows for different behavior of young and old indivudals.
The model has the following components:
- Preferences
- Technology
- Government
- Equilibrium
23.1 Preferences
- Agents live for 2 periods: young and old
- They value consumption when young \(c_y\), leisure when young \(1-l\) and consumption when old \(c_o\)
- Their preferences are given via utility functions: \(u_t(c_t)\)
- Agents discount time with factor \(\beta\)
- Their life-time utility is:
\(V(c_y, l,c_o) = u(c_y,l) + \beta \times u(c_o)\)
23.2 Technology
- Firms produce output \(Y\) using input capital \(K\) and labor \(L\):
\(Y(K,L) = A \times K^{0.3} \times L^{0.7}\)
23.3 Government
- The government collects lump sum taxes from households \(\tau_{Lump}\), taxes on labor \(\tau_L\) and capital \(\tau_K\)
- The government pays for gov't consumption \(G\) and transfers to households T_y and \(T_o\)
\(G + T_y + T_o = L \times \tau_L + K \times \tau_K + \tau_{Lump}\)
23.4 Household Problem
- HHs maximize \(V(c_y,l,c_o)\) subject to their budget constraint in each period
\(c_y + s = w l + T_y\) \(c_o = R s + T_o\)
23.5 Equilibrium Definition
- * Given sequences of
-
- prices \(\left\{w_t, R_t \right\}\)
- government policies \(\left\{\tau_L, \tau_K, \tau_{Lump} \right\}\) and equilibrium is defined as an allocation of:
- * sequences of \(\left\{c_{y,t},l_t,c_{o,t},s_t\right\}\) so that
-
- the HH max problem is solved
- the firm maximization problem is solve
- the gov't budget constraint clears
23.6 Functional Forms and Solutions
- Preferences are given as: \(u(c_y,l) = log(c_y)+log(l)\) and \(u(c_o)=log(c_o)\)
- We can either set up a Lagrangian with two constraints or simply substitute consumption out of the utilities using the BC.
- We follow the second approach, since the form of the utility functions guarantees interior solutions
- Therefor we don't have to worry about corner solutions a la Kuhn-Tucker
23.7 Substitute in Preferences
\(max_s log(w l + T_y - s) + log(1-l) + \beta log(R s + T_o)\)
- This is now a function in 2 choice variables \(s\) and labor \(l\)
- Derive this function w.r.t. \(s\) and labor \(l\)
\(\frac{\partial V}{\partial s}: \frac{1}{w l + T_y - s} = \frac{\beta R}{R s + T_o}\)
\(\frac{\partial V}{\partial l}: \frac{w}{w l + T_y - s} = \frac{1}{1-l}\)