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\multicolumn{2}{c} {\bfseries Problem Set 7} \\
& \\
ECON 772001 - Math for Economists & Peter Ireland \\
Boston College, Department of Economics & Fall 2023 \\
& \\
\multicolumn{2}{c} {For Extra Practice -- Not Collected or Graded}
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{\bfseries 1. The Permanent Income Hypothesis}
The permanent income hypothesis describes how a forward-looking consumer optimally saves or borrows to smooth out his or her consumption in the face of a fluctuating income stream. This problem formalizes the permanent income hypothesis using a two-period model. So consider a consumer who lives for two periods, earning income $w_{0}$ during period $t=0$ and $w_{1}$ during period $t=1$. Let $c_{0}$ and $c_{1}$ denote his or her consumption during periods $t=0$ and $t=1$ and let $s$ denote his or her amount saved (or borrowed, if negative) during period $t=0$. Suppose that savings earn interest between $t=0$ and $t=1$ at the constant rate $r$. Then the consumer faces the budget constraints
\begin{equation}
w_{0} \geq c_{0} + s \tag{1}
\end{equation}
at $t=0$ and
\begin{equation}
w_{1} + (1+r)s \geq c_{1} \tag{2}
\end{equation}
at $t=1$.
Finally, suppose that the consumer's preferences are described by the utility function
\begin{equation}
\ln(c_{0}) + \beta \ln(c_{1}), \tag{3}
\end{equation}
where the discount factor lies between zero and one: $0<\beta<1$.
\begin{enumerate}[label=\alph*.]
\item Find the values for $c_{0}^{*}$, $c_{1}^{*}$, and $s^{*}$ that solve the consumer's problem -- choose $c_{0}$, $c_{1}$, and $s$ to maximize the utility function in (3) subject to the constraints in (1) and (2) -- in terms of the parameters $w_{0}$, $w_{1}$, $r$, and $\beta$.
\item Suppose now that the market interest rate and the consumer's discount factor satisfy $\beta(1+r)=1$. Although it might seem that this condition will hold only through a rare coincidence, in fact as problem set 6 showed, it is often satisfied in more elaborate general equilibrium models that feature a large numbers of consumers who borrow and lend in a competitive market. With this extra condition imposed, what do your solutions from part (a) above imply for the optimal behavior of consumption: does it rise, fall, or stay the same moving from period $t=0$ to period $t=1$?
\item Continuing to assume that $\beta(1+r)=1$, what determines whether the consumer will borrow (choosing $s<0$) or save (choosing $s>0$) during period $t=0$?
\end{enumerate}
\pagebreak
{\bfseries 2. Habit Formation}
Consider the behavior of a consumer that faces the same budget constraints (1) and (2) shown above, but instead of (3) has the utility function
\begin{equation}
\ln(c_{0}) + \beta \ln(c_{1}-\gamma c_{0}), \tag{4}
\end{equation}
where $0 < \gamma < 1$. The preferences represented by this utility function display ``habit formation'' or capture an ``addictive'' element of consumption by implying that the more the individual consumers during period $t=0$ the more he or she will want to consume during period $t=1$.
\begin{enumerate}[label=\alph*.]
\item Assuming once more that $\beta$ and $r$ are separate parameters, find the values for $c_{0}^{*}$, $c_{1}^{*}$, and $s^{*}$ that solve the consumer's problem -- choose $c_{0}$, $c_{1}$, and $s$ to maximize the utility function in (4) subject to the constraints in (1) and (2) -- in terms of the parameters $w_{0}$, $w_{1}$, $r$, $\beta$, and $\gamma$.
\item Now assume that $\beta(1+r)=1$ holds again. What happens to the optimal behavior of consumption now: is $c_{1}^{*}$ larger than, smaller than, or the same as $c_{0}^{*}$?
\end{enumerate}
{\bfseries 3. Durable Consumption}
Now replace the utility function in (4) with
\begin{equation}
\ln(c_{0}) + \beta \ln(c_{1}+\theta c_{0}), \tag{5}
\end{equation}
where $0 < \theta < 1$. This specification might capture an aspect of ``durability'' in the consumption good, whereby a purchase made at $t=0$ continues to yield utility at $t=1$, in which case the new parameter $\theta$ can be interpreted as measuring the depreciation rate for the durable item.
\begin{enumerate}[label=\alph*.]
\item Assuming again that $\beta$ and $r$ are separate parameters, find the values for $c_{0}^{*}$, $c_{1}^{*}$, and $s^{*}$ that solve the consumer's problem -- choose $c_{0}$, $c_{1}$, and $s$ to maximize the utility function in (5) subject to the constraints in (1) and (2) -- in terms of the parameters $w_{0}$, $w_{1}$, $r$, $\beta$, and $\theta$.
\item Now assume for the last time that $\beta(1+r)=1$ holds. What happens to the optimal behavior of consumption in this case: is $c_{1}^{*}$ larger than, smaller than, or the same as $c_{0}^{*}$?
\end{enumerate}
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