Question 1 (15 points) - Consumption-Savings Problem
Consider a household with the following lifetime utility function:
$U(C_t, C_{t+1}) = \ln C_t + \beta \ln C_{t+1}$
where $0 < \beta < 1$ denotes the household discount factor. The household's lifetime budget constraint is given by:
$C_t + \frac{C_{t+1}}{1+r_t} = Y_t + \frac{Y_{t+1}}{1+r_t}$
where $Y_t > 0$ denotes income in period $t$, $Y_{t+1} > 0$ denotes income in period $t+1$, and $r_t \geq 0$ denotes the real interest rate.
(a) Derive the household's period optimal consumption in period $t$ and in period $t+1$ (i.e., $C_t^*$ and $C_{t+1}^*$) as a function of $\{\beta, r_t, Y_t, Y_{t+1}\}$.
(b) Use your answer from part (a) to determine how optimal consumption in period $t+1$ responds to an increase in $\beta$. Provide some economic intuition.
(c) Use your answer from part (a) to determine how optimal consumption in period $t+1$ responds to an increase in $Y_t$. Provide some economic intuition.
Now assume that in addition to the budget constraint, the household is subject to a borrowing constraint of the following form:
$C_t \leq Y_t$
This constraint says that savings cannot be negative in the first period. Equivalently, this is saying consumers cannot borrow in the first period.
(d) Suppose $Y_t = 3$, $Y_{t+1} = 10$, $\beta = 0.95$ and $r_t = 0.1$. Is the borrowing constraint binding, i.e., is it impacting the household's optimal behavior?
(e) Suppose $Y_{t+1} = 10$, $\beta = 0.95$ and $r_t = 0.1$. Find the value for $Y_t$ such that the borrowing constraint is not binding, i.e., is not impacting the household's optimal behavior.
