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\title{Econ 714: Final exam - Solution}
\author{Anton Babkin}
\begin{document}
{\Large Econ 714: Final exam - Solution\footnote{By Anton Babkin. \today.}}
\section[]{}
The correct budget constraint for this problem must be, in nominal terms:
\begin{equation*}
P_tc_t+M_t+\frac{\alpha_{Bt}}{I_t}+P_t\frac{\alpha_{bt}}{R_t}+P_tS_t\alpha_{St}=\tau_{t-1}+M_{t-1}+\alpha_{Bt-1}+P_t\alpha_{bt-1}+P_t(Y_t+S_t)\alpha_{St-1}
\end{equation*}
where $\tau_t = M_t^s-M_t$ is nominal transfer/tax from changing money supply.
At time $t$ decisions are made over $t$-indexed variables, and $t-1$
variables are states resulting from the previous period.
In real terms:
\begin{equation*}
c_t+m_t+\frac{\alpha_{Bt}}{P_tI_t}+\frac{\alpha_{bt}}{R_t}+S_t\alpha_{St}=\frac{\tau_{t-1}}{P_t}+m_{t-1}\frac{P_{t-1}}{P_t}+\frac{\alpha_{Bt-1}}{P_t}+\alpha_{bt-1}+(Y_t+S_t)\alpha_{St-1}
\end{equation*}
\begin{enumerate}[(a)]
\item With Lagrange multiplier on budget constraint $\lambda_t$, first
order conditions are:
\begin{align*}
[c_t]&:\beta^tu'(c_t)=\lambda_t\\
[\alpha_{bt}]&:\frac{\lambda_t}{R_t}=\E_t\lambda_{t+1}\\
[\alpha_{Bt}]&:\frac{\lambda_t}{P_tI_t}=\E_t\lambda_{t+1}\frac{1}{P_{t+1}}\\
[\alpha_{St}]&:\lambda_tS_t=\E_t\lambda_{t+1}(Y_{t+1}+S_{t+1})\\
[m_t]&:\beta^tv'(m_t)-\lambda_t+\E_t\lambda_{t+1}\frac{P_t}{P_{t+1}}=0
\end{align*}
Substitute out $\lambda_t$ and plug in goods market clearing
$c_t=Y_t$ to obtain equilibrium pricing conditions:
\begin{gather*}
\frac{1}{R_t}=\beta\E_t\frac{u'(Y_{t+1})}{u'(Y_t)}\\
\frac{1}{I_t}=\beta\E_t\frac{u'(Y_{t+1})}{u'(Y_t)}\frac{P_t}{P_{t+1}}\\
S_t=\beta\E_t\frac{u'(Y_{t+1})}{u'(Y_t)}(Y_{t+1}+S_{t+1})\\
1-\frac{v'(m_t)}{u'(Y_t)}=\beta\E_t\frac{u'(Y_{t+1})}{u'(Y_t)}\frac{P_t}{P_{t+1}}=\frac{1}{I_t}
\end{gather*}
\item Endowment process is
$\frac{Y_{t+1}}{Y_t}=\exp(\mu+\sigma W_{t+1})$. Using real bond
pricing equation:
\begin{align*}
\frac{1}{R_t}&=\beta\E_t\left(\frac{Y_{t+1}}{Y_t}\right)^{-\gamma}\\
&=\beta\E_t\exp(-\gamma\mu-\gamma\sigma W_{t+1})\\
&=\beta\exp(-\gamma\mu+\gamma^2\sigma^2/2)\\
-\log R_t&=\log\beta-\gamma\mu+\gamma^2\sigma^2/2\\
r_t&=\gamma\mu-\gamma^2\sigma^2/2-\log\beta
\end{align*}
Real bonds return positively depends on growth rate $\mu$ and
negatively on volatility $\sigma$ and patience $\beta$. Effect of
$\gamma$ is ambiguous.
\item
\begin{align*}
\pi_t&\equiv\log\E_t\frac{P_{t+1}}{P_{t}}\\
&=\log\E_t\left(\frac{Y_{t+1}}{Y_t}\right)^a\\
&=\log\E_t\exp(a\mu+a\sigma W_{t+1})\\
&=a\mu+a^2\sigma^2/2
\end{align*}
If $\sigma=0$, then simply $\pi_t=a\mu$. If $\sigma>0$, this is a
quadratic equation in $a$ which generally has two roots:
\begin{equation}\label{eq:a}
a=\frac{-\mu\pm\sqrt{\mu^2+2\sigma^2\pi_t}}{\sigma^2}
\end{equation}
\item Using nominal bond pricing equation and solution of class
$P_t=Y_t^a$:
\begin{align*}
\frac{1}{I_t}&=\beta\E_t\left(\frac{Y_{t+1}}{Y_t}\right)^{-\gamma-a}\\
&=\beta\exp(-(\gamma+a)\mu+(\gamma+a)^2\sigma^2/2)\\
-\log I_t&=\log\beta-(\gamma+a)\mu+(\gamma+a)^2\sigma^2/2\\
i_t&=(\gamma+a)\mu-(\gamma+a)^2\sigma^2/2-\log\beta\\
&=\gamma\mu-\gamma^2\sigma^2/2-\log\beta
+a\mu+a^2\sigma^2/2-\gamma a \sigma^2 - a^2\sigma^2\\
&=r_t+\pi_t-\gamma a \sigma^2 - a^2\sigma^2
\end{align*}
where $a$ is given by the equation~\eqref{eq:a}.
\item Without risk, $\sigma=0$, Fisher equation holds exactly and in
unique equilibrium $\pi_t=\bar{i}_t-r_t$, $a=\pi_t/\mu$.
If $\sigma>0$, there might be two possible inflation levels in
equilibrium that correspond to the two roots for $a$.
In these equilibria inflation is a function of the endowment growth,
so return on the real bond is correlated with
inflation. Decomposition of nominal interest rate into real interest
rate and inflation (Fisher equation) now includes an additional
covariance term - inflation risk - that can take two values for different $a$:
\begin{align*}
\frac{1}{I_t}&=\beta\E_t\frac{u'(Y_{t+1})}{u'(Y_t)}\frac{P_t}{P_{t+1}}\\
&=\E_t\beta\frac{u'(Y_{t+1})}{u'(Y_t)}\E_t\frac{P_t}{P_{t+1}}+Cov_t\left(\beta\frac{u'(Y_{t+1})}{u'(Y_t)},\E_t\frac{P_t}{P_{t+1}}\right)\\
&=\frac{1}{R_t}\E_t\frac{P_t}{P_{t+1}}+Cov_t\left(\beta\left(\frac{Y_{t+1}}{Y_t}\right)^{-\gamma},\left(\frac{Y_{t+1}}{Y_t}\right)^{-a}\right)
\end{align*}
\end{enumerate}
\section{}
\begin{enumerate}[(a)]
\item
\begin{gather*}
rW=w+\lambda(U(s)-W)\\
rU(s)=z-c(s)+q(s)(W-U(s))
\end{gather*}
Solve for $W-U = \frac{w-z+c(s)}{r+\lambda+q(s)}$. Substitute back
to get steady state values of $U(s)$ and $W$:
\begin{gather*}
rW = w-\lambda\frac{w-z+c(s)}{r+\lambda+q(s)}\\
rU(s)=z-c(s)+q(s)\frac{w-z+c(s)}{r+\lambda+q(s)}
\end{gather*}
\item First order condition of the unemployed with respect to $s$:
\begin{equation}\label{eq:s}
c'(s)=q'(s)(W-U)=q'(s)\frac{w-z+c(s)}{r+\lambda+q(s)}
\end{equation}
\item Rewrite \eqref{eq:s} as
\begin{equation*}
\frac{c'(s)(r+\lambda)}{q'(s)}+c'(s)s-c(s)=w-z
\end{equation*}
Derivative of the LHS with respect to $s$ is
\begin{align*}
(r+\lambda)\frac{c''(s)q'(s)-c'(s)q''(s)}{(q'(s))^2}+c''(s)s+c'(s)-c'(s)\\
=(r+\lambda)\frac{c''(s)q'(s)-c'(s)q''(s)}{(q'(s))^2}+c''(s)s
\end{align*}
and is positive since $c''(s)>0$ and $q''(s)<0$.
So the LHS is increasing in $s$ and the RHS is constant. So $s$
increases as $w$ increases.
\end{enumerate}
\section{}
\begin{enumerate}[(a)]
\item When price level fluctuates, and not all firms are able to
adjust, price dispersion results. This causes the relative prices of
the different goods to vary. If the price level rises, two things
happen:
\begin{itemize}
\item The relative price of firms who have not set their price for a
while falls, they experience an increase in demand and raise
output. Firms who have just reset their prices reduce output. This
production dispersion is inefficient.
\item Consumers increase consumption of the goods whose relative
price has fallend a reduce cunsumption of those goods whose
relative price has risen. This dispersion in consumption reduces
welfare.
\end{itemize}
\item Risk premium of a risky return over a risk-free one can be expressed as
\begin{equation*}
\frac{E(r_t)-r^f}{\sigma(r)}=\gamma\sigma(\Delta c_t)corr(\Delta c_t,r_t)
\end{equation*}
A puzzle is that empirical estimates of this equation imply that
$\gamma$ needs to be about 27. This is a puzzle, because such high
levels of risk-aversion imply implausibly high premiums
invidividuals will be willing to pay to avoid taking lotteries with
zero expected payoff. Existing micro-level studies suggest that
$\gamma$ should be in the order of 2 to 3.
But even if we allow risk-aversion to be very high, it won't resolve
the puzzle. With CRRA utility $\gamma$ is not only a coefficient of
relative risk-aversion, but also an inverse of the elastisity of
intertemporal substitution. For an observed levels of aggregate
consumption growth, this implies that risk-free rate must be much
higher than it historically was.
\item Time consistency problem arises when future plans that are
optimal at a particular point in time become not optimal when that
future actually comes.
In Ramsey problem, it is socially optimal to set zero tax on returns
to capital for all future periods except the initial one, because
then capital is already in place, and proportional tax effectively
becomes a lump sum tax and does not distort households'
incentives. However if the planner is allowed to reoptimize at some
time in the future, he would choose to deviate from the zero-tax
plan and tax capital in that period.
\end{enumerate}
\end{document}