1. Overlapping Generations with Labor Supply - Consider a variation of the overlapping generations
model as studied in class. In each period, the economy is still occupied by two cohorts of two gener-
ations of households - the young and the old - living for two periods. Instead of exogenous income,
however, assume that the output $Y_t^j$ of each cohort at time t is produced linearly using the labor effort
$L_t^j$ of each household, that is
$Y_t^j = A_t L_t^j$
where $A_t$ is the productivity parameter at time period t. Each cohort j household maximizes its
lifetime utility subject to the budget constraints
$\max_{\{C_t^j, C_{t+1}^j, L_t^j, L_{t+1}^j\}} u(C_t^j) + v(L_t^j) + \beta \left[ u(C_{t+1}^j) + v(L_{t+1}^j) \right]$
subject to
$C_t^j + S_{t+1}^j = Y_t^j$
$C_{t+1}^j = Y_{t+1}^j + (1 + r_{t+1}) S_{t+1}^j$
where $S_{t+1}^j$ is saving of cohort-j household decided when it is young. The interest rate at time t + 1 is
$r_{t+1}$. Let the utility functions be
$u(C) = \log(C)$
$v(L) = -0.5L^2$
There is no storage technology, no social security transfer and no fiat money.
