22. 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
22.1. 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:
\(V(c_y, l,c_o) = u(c_y,l) + \beta \times u(c_o)\)
22.2. Technology¶
Firms produce output \(Y\) using input capital \(K\) and labor \(L\):
\(Y(K,L) = A \times K^{0.3} \times L^{0.7}\)
22.3. 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 \(T_o\)
\(G + T_y + T_o = L \times \tau_L + K \times \tau_K + \tau_{Lump}\)
22.4. Household Problem¶
HHs maximize \(V(c_y,l,c_o)\) subject to their budget constraint in each period
\(c_y + s = w l + T_y\) \(c_o = R s + T_o\)
22.5. 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
22.6. 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
22.7. 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 \(l\)
\(\frac{\partial V}{\partial s}: \frac{1}{w l + T_y - s} = \frac{\beta R}{R s + T_o}\)
\(\frac{\partial V}{\partial l}: \frac{w}{w l + T_y - s} = \frac{1}{1-l}\)