Exercise 9.7

Optimality of Constant Nominal Interest Rates, Continued

⬅ Return

Problem

Show that any constant interest-rate rule of the form \(R_t=\delta\), where \(\delta\) is a parameter, supports the full-employment equilibrium, provided \(\delta\ge \gamma/\beta\).

Answer

In full employment eqm: \(h_t = \bar{h}\) for all \(t\) and \(w_t = p_t F'(\bar{h})\) for all \(t\).

Start with the DNWR constraint: \(W_{t+1} \ge \gamma W_t\). This can be written as

\[ \begin{eqnarray*} \frac{ \mathcal{E}_{t+1} } { \mathcal{E}_t } \ge \gamma \frac{w_t}{w_{t+1}} = \gamma \frac{p_t}{p_{t+1}} = \gamma \left(\frac{c^T_t}{c^T_{t+1}}\right)^{1/\xi} \end{eqnarray*} \]

where the last equality uses the facts that \(U\) is CRRA and \(A\) is CES with \(\sigma = 1/\xi\) and that \(C^N_t = F(\bar{h})\).

Use the fact that the law of one price holds and that the foreign price of the traded good is unity, \(\mathcal{E}_t =P^T_t\), to eliminate \(\mathcal{E}_t\) from the above expression and rearrange to obtain:

\[ \begin{eqnarray*} \frac1\gamma \left(\frac{c^T_{t+1}}{c^T_{t}}\right)^{1/\xi} \ge \frac{P^T_t}{P^T_{t+1}} \end{eqnarray*} \]

The Euler equation is:

\[ (c^T_t)^{-1/\xi} = \beta R_t E_t (c^T_{t+1})^{-1/\xi} \frac{P^T_t} {P^T_{t+1} } \]

Rearrange terms to get

\[ \frac1R_t = \beta E_t (c^T_{t+1})^{-1/\xi} (c^T_t)^{1/\xi} \frac{P^T_t} {P^T_{t+1} } \le \frac{\beta}{\gamma } \]

Thus, we have shown that \(R_t \ge \gamma/\beta\) for all \(t\) is consistent with the full employment equilibrium.