1. Determine the Euler equation in this case.
2. Determine the representative household’s optimal choices: $c_1^{\star}, c_2^{\star}$, and $b_1^{\star}$.
3. Determine the equilibrium interest rate $R^{\star}$.
4. Determine the effect on the equilibrium interest rate $R^{\star}$ of a permanent negative shock to the income of the representative household. (I.e., both $y_1$ and $y_2$ go down by an equal amount.) How does this relate to the case in which $u\left(c_t\right)=\ln \left(c_t\right)$ ?

1. Suppose the representative household becomes more impatient. Determine the direction of the change in the equilibrium interest rate. (Patience is measured by $\beta$. You should use calculus.)
2. Suppose the representative household gets a temporary negative shock to its period-1 income $y_1$. Determine the direction of the change in the equilibrium interest rate. (Again, use calculus.)

1. Write down Maxine’s maximization problem. (You should show her choice variables, her objective, and her constraints.)
2. Solve Maxine’s maximization problem. (This will give you her choices for given values of $e_1, e_2, \beta$, and $\delta$.)
3. How do Maxine’s choices change if she finds a way reduce the damage done by the rats? (You should use calculus to do comparative statics for changes in $\delta$.)

An agent lives for five periods and has an edible tree. The agent comes into the world at time $t=0$, at which time the tree is of size $x_0$. Let $c_t$ be the agent’s consumption at time $t$. If the agent eats the whole tree at time $t$, then $c_t=x_t$ and there will be nothing left to eat in subsequent periods. If the agent does not eat the whole tree, then the remainder grows at the simple growth rate $\alpha$ between periods. If at time $t$ the agent saves $100 s_t$ percent of the tree for the future, then $x_{t+1}=(1+\alpha) s_t x_t$. All the agent cares about is consumption during the five periods. Specifically, the agent’s preferences are: $U=\sum_{t=0}^4 \beta^t \ln \left(c_t\right)$. The tree is the only resource available to the agent.
Write out the agent’s optimization problem.