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\multicolumn{2}{c} {\bfseries Problem Set 14} \\
& \\
ECON 772001 - Math for Economists & Peter Ireland \\
Boston College, Department of Economics & Fall 2023 \\
& \\
\multicolumn{2}{c} {Practice for the Final -- Not Collected or Graded}
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{\bfseries 1. Stochastic Linear-Quadratic Dynamic Programming}
This problem asks you to use dynamic programming to characterize the solution to a stochastic version of the linear-quadratic problem that you studied previously, under conditions of perfect foresight, in problem set 12. The problem is to choose contingency plans for a flow variable $z_{t}$ for all $t=0,1,2,...$ and a stock variable $y_{t}$ for all $t=1,2,3,...$ to maximize the objective function
$$
E_{0} \sum_{t=0}^{\infty} \beta^{t}(Ry_{t}^{2}+Qz_{t}^{2}),
$$
subject to the constraints $y_{0}$ given and
\begin{equation}
Ay_{t}+Bz_{t} + \varepsilon_{t+1} \geq y_{t+1}, \tag{1}
\end{equation}
where $0<\beta<1$, $R<0$, $Q<0$, $A$, and $B$ are all constant, known parameters and $\varepsilon_{t+1}$ is an independently and identically distributed random shock that satisfies $E_{t}(\varepsilon_{t+1})=0$ and $E_{t}(\varepsilon_{t+1}^{2})=\sigma^{2}$, that is, which has zero mean and variance $\sigma^{2}$. The flow variable $z_{t}$ must be chosen at time $t$ before $\varepsilon_{t+1}$ is known, hence the constraint (1) must hold for all $t=0,1,2,...$ and for all possible realizations of $\varepsilon_{t+1}$.
\begin{enumerate}[label=\alph*.]
\item Guess that the value function for this problem depends only on $y_{t}$ and not $\varepsilon_{t}$ and takes the specific form
$$
v(y_{t},\varepsilon_{t}) = v(y_{t}) = Py_{t}^{2} + d,
$$
where $P$ and $d$ are unknown constants. Using this guess, write down the Bellman equation for the problem. In doing this, it might be helpful to note that the assumptions that $\varepsilon_{t+1}$ is iid and that $y_{t}$ and $z_{t}$ are known at time $t$ imply that
\begin{eqnarray*}
E_{t}[(Ay_{t}+Bz_{t} + \varepsilon_{t+1})^{2}] & = & E_{t}[(Ay_{t}+Bz_{t})^{2}] + 2E_{t}[(Ay_{t}+Bz_{t})\varepsilon_{t+1}] + E_{t}(\varepsilon_{t+1}^{2}) \\
& = & (Ay_{t}+Bz_{t})^{2} + 2(Ay_{t}+Bz_{t})E_{t}(\varepsilon_{t+1}) + E_{t}(\varepsilon_{t+1}^{2}) \\
& = & (Ay_{t}+Bz_{t})^{2} + \sigma^{2}.
\end{eqnarray*}
\item Next, write down the first-order condition for $z_{t}$ and the envelope condition for $y_{t}$.
\item Use your results from above to show that the unknown $P$ must satisfy the same Riccati equation
$$
P = R + \frac{\beta A^2 QP}{Q+\beta B^2P}
$$
that helps characterize the problem's solution in the nonstochastic case.
\item Finally, use your results from above to show how the new unknown parameter $d$ that enters the value function for the stochastic problem depends on the value of $P$ that solves the Riccati equation along with the model's other parameters.
\end{enumerate}
{\bfseries 2. Stochastic Growth}
This problem asks you to use dynamic programming to solve a stochastic version of the optimal growth model in which physical capital depreciates fully between periods; as in the nonstochastic case, that assumption of full depreciation allows an explicit solution for the value function to be found via the guess-and-verify method. In this version of the model, the representative consumer chooses contingency plans for consumption $c_{t}$ for all $t=0,1,2,...$ and physical capital $k_{t}$ for all $t=1,2,3,...$ to maximize the expected utility function
$$
E_{0} \sum_{t=0}^{\infty} \beta^{t} \ln(c_{t}),
$$
with $0<\beta<1$, subject to the constraints $k_{0}$ given and
\begin{equation}
z_{t}k_{t}^{\alpha} \geq c_{t} + k_{t+1}, \tag{2}
\end{equation}
where $0<\alpha<1$. In (2), $z_{t}$ represents a shock to the productivity of capital. The value of $z_{t}$ is known when $c_{t}$ and $k_{t+1}$ are chosen during period $t$, but the value of $z_{t+1}$ is random and satisfies $E_{t}[\ln(z_{t+1})]=0$ for all $t=0,1,2,...$. Then (2) must hold for all $t=0,1,2,...$ and all possible realizations of $z_{t}$.
\begin{enumerate}[label=\alph*.]
\item Guess that the value function for this problem takes the form
$$
v(k_{t},z_{t}) = E + F\ln(k_{t}) + G\ln(z_{t}),
$$
where $E$, $F$, and $G$ are unknown constants. Using this guess, write down the Bellman equation for the problem. In doing this, it might be helpful to note that
$$
E_{t}[\ln(z_{t}k_{t}^{\alpha}-c_{t})] = \ln(z_{t}k_{t}^{\alpha}-c_{t})
$$
since all of the objects in this expression are known at time $t$ and to recall that
$$
E_{t}[\ln(z_{t+1})]=0
$$
by assumption.
\item Next, write down the first-order condition for $c_{t}$ and the envelope condition for $k_{t}$.
\item Use your results from above to derive an equation that shows how the optimal choice of $c_{t}$ depends on $k_{t}$, $z_{t}$, and the parameters $\alpha$ and $\beta$. Then substitute this expression into the binding constraint (2) to derive an equation that shows how the optimal choice of $k_{t+1}$ depends on $k_{t}$, $z_{t}$, and the parameters $\alpha$ and $\beta$.
\item For the sake of completeness, write down solutions that show how the unknown constants $E$, $F$, and $G$ depend on the parameters $\alpha$ and $\beta$.
\end{enumerate}
{\bfseries 3. Saving with a Random Return}
This problem asks you to use dynamic programming to solve a representative consumer's problem when savings earn a random rate of return. Let $A_{t}$ denote the consumer's assets at the beginning of each period $t=0,1,2,...$. During each period $t$, the consumer divides these assets up into an amount $c_{t}$ to be consumed and an amount $s_{t}$ to be saved. The consumer's savings earn interest at the gross rate $R_{t+1}$, where $R_{t+1}$ is random, possibly serially correlated, and does not become known until the beginning of period $t+1$. Thus, the consumer must choose $s_{t}$ before knowing the realized value of $R_{t+1}$. The consumer takes his or her initial assets $A_{0}$ as given and chooses contingency plans for $s_{t}$ for all $t=0,1,2,...$ and $A_{t}$ for $t=1,2,3,...$ to maximize the expected utility function
$$
E_{0} \sum_{t=0}^{\infty} \beta^{t} u(c_{t}) = E_{0} \sum_{t=0}^{\infty} \beta^{t} u(A_{t}-s_{t}),
$$
with $0<\beta<1$, subject to the constraints
$$
R_{t+1}s_{t} \geq A_{t+1},
$$
which must hold for all $t=0,1,2,...$ and all possible realizations of $R_{t+1}$.
\begin{enumerate}[label=\alph*.]
\item Write down the Bellman equation for this problem, using $A_{t}$ as the state variable, $s_{t}$ and the control variable, and allowing the value function for time $t$ to depend on $R_{t}$ as well as $A_{t}$.
\item Now suppose that the consumer's single-period utility function takes the form
$$
u(c_{t}) = u(A_{t}-s_{t}) = \frac{(A_{t}-s_{t})^{1-\sigma}}{1-\sigma},
$$
where $\sigma>0$ and $\sigma \neq 1$. Suppose also that the random interest rate $R_{t+1}$ is independently and identically distributed with
$$
E_{t}(R_{t+1}^{1-\sigma}) = 1
$$
for all $t=0,1,2,...$. Guess that under these conditions, the value function depends only on $A_{t}$ and takes the specific form
$$
v(A_{t}) = \frac{KA_{t}^{1-\sigma}}{1-\sigma},
$$
where $K$ is an unknown constant. Use this guess, together with the assumptions about the form of $u(c_{t})$ and the properties of $R_{t+1}$, to rewrite your Bellman equation from part (a) above. Then derive the first-order and envelope conditions for the consumer's problem.
\item Use your results from part (b) to solve for the unknown $K$ in terms of the parameters $\beta$ and $\sigma$.
\item Finally use your results from above to derive expressions that show how the optimal choices for $c_{t}$ and $s_{t}$ depend on the stock of assets $A_{t}$ and the parameters $\beta$ and $\sigma$.
\end{enumerate}
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