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{\sc{\large
\noindent --- Problem Set 3 ---\\
\noindent MA41617: Monetary policy and business cycles\\
30 March 2017
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\noindent Antti Ripatti\\
\url{antti [at] ripatti.net}\\
\url{http://macro.ripatti.net/}
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\exerc{1 Eggertson-Krugman model}{
The appendix of the Eggertson and Krugman (2012) article contains a model whose details can be found from the appendix of the article and from the supplement listed in the QJE website of the article
(\url{http://qje.oxfordjournals.org/content/127/3/1469/suppl/DC1}. Code this model to dynare, but use the Calvo-pricing approach for price rigidities (no need to derive the details). (Hint! Since the appendix log-linearizes the model you can use this version directly.)
\begin{description}
\item[a)] Assume that $G=T^b=0$ (ie no fiscal policy), what does the deleverageing shocks does? (Calculate impulse responses!)
\item[b)] Assume balanced government budget in each period. Compare the two fiscal policies: (i) increase in $G$ (government purchases); (ii) decrease in $T^b$ (lump-sum social transfers to borrower). Which one is more effective? Provide intuition.
\end{description}
}
\exerc{2}{
Gali's book's exercise 5.4: As show in Steinsson (2003), in the presence of partial price indexation by firms, the second-order approximation to the household's welfare losses takes the form
\[
\frac{1}{2}\E_0 \sum_{t=0}^\infty \beta^t\left[\alpha_x x_t^2 + (\pi_t -\gamma\pi_{t-1})^2\right],
\]
where $\gamma$ denotes the degree of price indexation to past inflation (similar role as $\omega$ in the problem set 2). The equation describing the evolution of inflation is now given by
\[
\pi_t - \gamma\pi_{t-1} = \kappa x_t + \beta\E_t(\pi_{t+1}-\gamma\pi_t) + u_t,
\]
where $u_t$ represents an exogenous i.i.d. cost-push shock.
\begin{description}
\item[a)] Determine the optimal policy under discretion.
\item[b)] Determine the optimal policy under commitment.
\item[c)] Discuss how the degree of indexation $\gamma$ affects the optimal responses to a transitory cost-push shock under the previous two scenarios.
\end{description}
Note, that the problem will be much easier to solve if you make the following change of variable: $\pi^\star_t \equiv \pi_t -\gamma\pi_{t-1}$.
}
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