Embedding the yield curve and risk premia in the aggregate demand curve
The aggregate demand curve in Eq. (33a) assumes that long-term and short-term market interest rates always move in parallel and that the central bank always strictly follows the Taylor rule. This exercise invites you to analyse how the AD curve may be modified to account for the facts that long-term interest rates do not always move one-to-one with short-term interest rates and that the central bank does not always strictly adhere to a fixed monetary policy rule. We start by restating the term structure equation (25), where we drop the common time subscript $t$ for convenience:
$$
i^l=\frac{1}{n}\left(i+i_{41}^n+i_{42}^*+\ldots+i_{+m-1}^n\right)
$$
Now subtract the expected inflation rate $\pi_{+1}^e$ from both sides of (42) to get:
$$
\begin{aligned}
& r=\frac{1}{\mathrm{n}}\left(I^s+I_{+1}^\sigma+I_{+2}^m+\ldots+I_{+5-2}^\sigma\right), \\
& r=i^1-\pi_{* 1}^*, \quad r^k=i-\pi_{+1}^*, \quad r_{+j}^{n j}=i_{+j}^*=\pi_{+1}^* \quad \text { for } 1 \leq j \leq n-1 \\
&
\end{aligned}
$$
The variable $r$ is the curvent long-term real interest rate, $r^s$ is the current short-term real interest rate, and $r_{+j}^{s p}$ is the expected short-term real interest rate for the future period $t+j$ on the simplifying assumption that the inflation rate is expected to stay constant. To keep things simple, suppose that
$$
\begin{array}{ll}
r_{* j}^m=r^s & \text { for } 1 \leq j \leq m-1, \\
r_{+j}^{* f}=r & \text { for } m \leq j \leq n-1 .
\end{array}
$$
Thus we assume that the short-term real interest rate is expected to remain at its current level until period $m$, and that it will return to its long-run equilibrium level $f$ from that time onwards. The parameter $m$ indicates how quickly the market expects the economy to return to long-run equilibrium.
1. Suppose $\vec{r}>r^t$. Draw a diagram to illustrate the qualitative properties of the yield curve implied by (43) and (44).
2. Use equations (43) and (44) to show that
$$
r=\omega\left(1-\pi_{+1}^s\right)+(1-\omega) \tilde{r}, \quad \omega=\frac{m}{n} .
$$
In the following, $i^p$ denotes the official interest rate set by the central bank, while $i$ denotes the current short-term interest rate formed in the interbank market. We assume that
$$
\begin{aligned}
& \hat{i}=\hat{\rho}^\rho+\rho, \\
& i^\eta=\tilde{f}-\rho+\pi_{+1}^e+h\left(\pi-\pi^*\right)+b(y-\hat{y}]+d, \quad \bar{\rho}=E[\rho], E[d]=0 .
\end{aligned}
$$
The variable $\rho$ is a risk premium and $\rho$ is its long-run mean value, so $\bar{f}-\bar{\rho}$ is the risk-free equilibrium real interest rate introduced in Section 16.4. The random variable d captures so-called discretionary monetary policy, reflecting that central bankers do not rigidly and mechanically follow a fixed policy rule. However, by restricting the mean value of $d$ to be zero, we assume that 'on average', the central bank does adhere to the Taylor rule.
3. Use (45), (46), (47) plus (12) from the chapter text to derive an AD curve of the same general form as (33) and (34). Specify how the definitions of $\alpha$ and $\mathrm{z}$ in (34) change when we incorporate the yield curve, risk premia and discretionary policy in the AD curve.
4. Write down a complete list of the types of shocks that may shift the $\mathrm{AD}$ curve derived in Question 3. Explain how the emergence of a risk premium in the interbank market will affect the AD curve. Explain also how a discretionary loosening of monetary policy (that is, a central bank interest rate cut that is not warranted by the Taylor rule) will affect the $A D$ curve.