The aggregate demand curve under flexible exchange rates
The derivation of the AD curve in the main text assumed that the expected 'normal' exchange rate is equal to the actual exchange rate observed during the last period. This exercise asks you to demonstrate that under certain assumptions, one can derive a similar AD curve under flexible exchange rates by simply assuming that the expected normal exchange rate is exogenous.
We start by assuming that the real interest rate affects aggregate demand mainly through its influence on business investment. Assuming that it only takes one period for firms to adjust their capital stock to its desired level $K^d$, and abstracting from depreciation on the existing capital stock, total gross investment in the current period, $I$, will equal the desired increase in the capital stock:
$$
I_l=K_1^{\mathrm{d}}-K_{1-1}^{\mathrm{d}} \text {. }
$$
Reflecting the underlying growth in aggregate demand, the desired capital stock varies positively with time, but at the same time it varies negatively with the real interest rate. Adopting a linear specification for convenience, we thus have:
$$
K_t^d=a t-b z_i \Rightarrow I_t=a-b\left(r_t-r_{t-1}\right) \text {, }
$$
where $a$ and $b$ are constants. In other words, the level of investment depends on the change in the real rate of interest. Dropping the time subscripts for the current period, we may therefore approximate the goods market equilibrium condition by:
$$
\begin{aligned}
& y-\hat{y}=\beta_1 e^f-\beta_2\left(r-x_{-1}\right)+\tilde{z}_{,} \\
& \bar{z}=\beta_3(g-\bar{g})+\beta_4\left(y^f-\bar{y}^f\right)+\beta_5(\ln \bar{\varepsilon}-\ln \bar{z}) .
\end{aligned}
$$
where we apply the usual notation (the constant $a$ has dropped out because (41) considers a deviation from trend, and the parameter $b$ is incorporated in $\beta_2$ ).
Now suppose that the central bank targets the foreign inflation rate by raising the real interest rate when domestic inflation is above foreign inflation, and vice versa:
$$
r-r_{-1}=h\left(\pi-\pi^f\right) \text {. }
$$
In addition we have the condition for uncovered interest rate parity:
$$
i=i^f+e_{+1}^e-e,
$$
where we assume that exchange rate expectations are regressive:
$$
e_{+1}^e-e=\theta(\tilde{e}-e), \quad \theta>0 .
$$
Here we treat the expected normal exchange rate, ef, as an exogenous variable, although it may change from time to time.
By definition, the ex ante domestic real interest rate is $r=i-\kappa_{+1}^n$. The central bank has credibility, so the expected domestic inflation rate equals the central bank's inflation target, that is, $\pi_{+1}^e=\pi_{+1}^f=\pi^f$. Hence we have:
$$
r=i-\pi f \text {. }
$$
Finally, we have the familiar definition of the foreign real interest rate,
$$
\Gamma^f=1^f-\pi^f \text {. }
$$
and the bookkeeping identity for the current real exchange rate:
$$
e^f=e_{-1}^c+e-e_{-1}+\pi^f-\pi .
$$
1. Use Eqs (45)-(51) to show that the aggregate demand curve takes the following form:
$$
\begin{aligned}
& y-\hat{y}=\beta_1 e_{-1}^e-\beta_1\left(\pi-\pi^f\right)+z, \quad \beta_1=\beta_2+h\left(\beta_2+\beta_1 \theta^{-1}\right), \\
& z=\beta_1 \Delta \theta+\beta_1 \theta^{-t} \Delta r^f+\beta_3(g-g)+\beta_4\left(y^f-y^f\right)+\beta_3(\ln s-\ln z) .
\end{aligned}
$$
where $\Delta \tilde{e}=\tilde{e}-\tilde{e}_{-1}$ and $\Delta r^f=r^f-x_{-1}^f$ (Hint: as an intermediate step, use (47)-(50) to derive an expression for $r-r_{-1}$ in terms of $e-e_{-1}, \Delta \bar{c}$, and $\Delta r^{\prime}$. Then insert (46) to obtain an expression for $\theta-\varepsilon_{-1}$ in terms of $\pi-\pi^f, \Delta \bar{\theta}_{\text {, and }} \Delta \Gamma^f$ ) Make a brief comparison between these results and the $A D$ curve (15) in the main text.
2. Suppose that expectations of a permanent weakening of the domestic currency arise. How will this affect the aggregate demand curve? Give some examples of economic events which might generate expectations of a permanent depreciation of the domestic currency.