Leaning against the wind' under flexible exchange rates
This exercise invites you to explore the implications of a monetary policy regime where the central bank follows a policy of 'leaning against the wind' by setting the interest rate in accordance with the following policy rule, explained in the section on 'dirty floating';
$$
i-\pi_{+1}^e=r^t+h\left(\pi-\pi^*\right)+\lambda\left(e-e_{-2}\right), \quad h>0, \quad \lambda>0 .
$$
Because of perfect capital mobility and risk neutrality, the condition for uncovered interest parity must hold:
$$
i=i^f+e_{+1}^f-\theta, \quad e=\ln E, \quad e_{+1}^f=\ln E_{+1}^e .
$$
We also continue to assume regressive exchange rate expectations so that:
$$
e_{+1}^e-e=-\theta\left(e-e_{-1}\right),
$$
and we maintain the assumption that the central bank's inflation target is credible and equal to the foreign inflation rate:
$$
\pi_{+3}^t=\pi^t=\pi^{\star}=\pi^f .
$$
The goods market equilibrium condition corresponds to Eq. (12) in the main text, repeated here for convenience:
$$
\begin{aligned}
& \hat{z}=\beta_3(g-g)+\beta_4\left(y^f-y^f\right)+\beta_5(\ln \varepsilon-\ln \dot{e}) . \\
&
\end{aligned}
$$
Finally, we continue to work with an aggregate supply curve of the form:
$$
\pi=\pi^2+\gamma(y-\bar{y})+s .
$$
1. Discuss briefly why the authorities might want to adopt a policy of 'leaning against the wind'.
2. Use Eqs (54)-(59) to show that the economy's aggregate demand curve takes the form:
$$
\begin{aligned}
& \pi=\pi^f+\left(\frac{\beta_3}{\hat{\beta}_1}\right) e_{-1}^f-\left(\frac{1}{\tilde{\beta}_1}\right)(y-\hat{y}-z), \quad \tilde{\beta}_1=\beta_1+\frac{h\left(\beta_1+\theta \beta_2\right)}{\theta+\lambda}, \\
& z=-\beta_2\left(r^f-t^f\right)+\beta_2(g-g)+\beta_4\left(y_f-y_f\right)+\beta_s(\ln \varepsilon-\ln z) .
\end{aligned}
$$
(Hint: follow the procedure described in Section 25.2.) Explain the economic mechanisms underlying the negative slope of the AD curve. Explain how the policy of leaning against the wind affects the slope of the AD curve.
3. Derive the equations determing the change over time in the nominal and in the real exchange rate. Does the policy of leaning against the wind amplify or dampen the impact of the inflation gap $\pi-\pi^f$ on nominal and real exchange rates? Explain. Give a graphical illustration. of the economy's adjustment to long-run equilibrium and explain the adjustment mechanisms.
4. Does leaning against the wind slow down or speed up the economy's speed of adjustment to long-run equilibrium? (Hint: derive a difference equation of the same form as (28) in the main text and investigate by differentiation how the parameter $\lambda$ affects the coefficient $a$ on the lagged output gap.) Try to provide an economic explanation for your result.
5. Use the procedure described in Section 25.3 of the main text to investigate how the policy of leaning against the wind affects the economy's short-run reaction to supply and demand shocks. Does leaning against the wind amplify or dampen the short-run effects of the shocks? Give an economic explanation for your findings.
6. Demonstrate that if the policy of leaning against the wind is very aggressive (so the parameter $\lambda$ tends to infinity], the economy will work approximately as if the exchange rate were fully fixed. Give an intuitive explanation for your finding.