A flexible exchange rate regime with flexible inflation targeting
In the main text we assumed that the domestic central bank pursued so-called strict inflation targeting, reacting only to changes in the inflation gap. This exercise studies a regime of so-called flexible inflation targeting where interest rate policy follows a standard Taylor rule with a positive coefficient on the output gap. Our open economy with flexible exchange rates is thus described by the following equations (in our usual notation):
Goods market:
Inflation expectations:
$$
\pi_{+1}^e=\pi^e=\pi^{\prime},
$$
Monetary policy: $\quad i=x^f+\pi_{+1}^*+h\left(\pi-\pi^f\right)+b(y-\bar{y}), \quad b>0$,
Uncovered interest rate parity: $\quad l=i^f+e_{+1}^t-e$,
Exchange rate expectations: $\quad e_{41}^f-e=-\theta\left(e-e_{-1}\right)$,
Foreign real interest rate: $\quad T^f=i^f-\pi^{\prime}$.
SRAS: $\quad \pi=\pi^*+y(y-y)+s$,
Real exchange rate: $\quad e^t=e_{-1}^t+\varepsilon-\epsilon_{-1}+\pi^{\prime}-\pi$
1. Use Eqs (61)-(66) to show that the economy's aggregate demand curve is given by:
$$
\begin{aligned}
& \pi=\pi^f+\left(\frac{\beta_1}{\hat{\beta}_1}\right) e_{-1}^r-\left(\frac{1+b\left(\beta_2+\theta^{-1} \beta_2\right)}{\beta_2}\right)(y-\bar{y})+\frac{z}{\bar{\beta}_1} . \\
& \hat{\beta}_1=\beta_1+h\left(\beta_2+\beta_1 \theta^{-2}\right), \quad z=-\beta_2\left(r^f-\bar{l}^f\right)+\bar{z}_{.}
\end{aligned}
$$
2. Use a $(y, \pi)$ diagram to undertake a graphical analysis of the way the economy reacts to demand and supply shocks in the short run (i.e. in the first period), assuming that the economy is in long-run equilibrium in period 0 . Illustrate and explain what difference it makes for your results that $b$ is positive rather than zero. Can you think of a situation where policy makers would not want to choose a positive $b$ ?
3. Use (62)-(66) to show that
$$
e^r=\theta_{-1}^r-\left(1+\theta^{-1} h\right)\left(\pi-\pi^f\right)-\theta^{-1} b(y-\hat{y}) .
$$
Explain in economic terms why positive inflation and output gaps generate a real exchange rate appreciation (a fall in $e^r$ ).
Defining $\hat{y}_t=y_1-\hat{y}$ and $\hat{k}_t=\pi_2-\pi^{\prime}$ and using (62). (67), (69) and (70), we may summarize our model as follows:
SRAS: $\quad \hat{s}_{\mathrm{t}}=\gamma \hat{y}_{\mathrm{t}}+s_{\mathrm{t}}$.
Real exchange rate: $\quad e_t^f=e_{t-1}^f-\left(1+\theta^{-1} h\right) \hat{\pi}_t-\theta^{-1} b \hat{y}_t$.
4. Use (71)-(73) to show that the model may be condensed to the following difference equation in the output gap:
$$
\begin{aligned}
& \hat{y}_{t+1}=a \hat{y}_t+\beta\left[z_{t+1}-z_1\right)-\beta \hat{\beta}_1\left(s_{t+1}-s_1\right]-\beta \beta_1\left(1+\theta^{-1} h\right) s_t, \\
& \beta=\frac{1}{1+\gamma \hat{\beta}_1+b\left(\beta_2+\theta^{-1} \beta_1\right)}, \quad a=\beta\left[1+\beta_2(b+\gamma h)\right] .
\end{aligned}
$$
Is the economy's long-run equilibrium stable? Does an increase in the value of $b$ have an unambiguous effect on the economy's speed of adjustment? Try to give an economic explanation for your conclusion on the latter question, (Hint: it may be helpful for you to reconsider our explanation for the result found in equation (32) in the text.]