c) Suppose increases in money also affect expected inflation. Specifically, assume that \(\frac{\partial \pi^e}{\partial m}\) is exogenous, with \(\frac{\partial \pi^e}{\partial m} > 0\). Continue to assume 0 < dp/dm < 1 . Find an expression for dr/dm. Does an increase in the money supply lower the interest rate? Does achieving a give change in r require a change in m smaller, larger, or the same as in part (b).
\(m-p=L(r+\pi^e , -\frac{r_t}{\theta})\)
Differentiating both sides with respect to m, we get:
\(1-\frac{\partial p}{\partial m} = \frac{\partial L}{\partial r+\pi^e} * \frac{\partial r}{\partial m} + \frac{\partial L}{\partial r+\pi^e} * \frac{\partial \pi^e}{\partial m} + \frac{\partial L}{\partial Y}*\frac{\partial (-\frac{r_t}{\theta})}{\partial m}\)
\(1-\frac{\partial p}{\partial m} = L_{r+\pi^e} * \frac{\partial r}{\partial m} + L_{r+\pi^e} * \frac{\partial \pi^e}{\partial m} + L_Y (-\frac{1}{\theta})*\frac{\partial r}{\partial m}\)
\(1-\frac{\partial p}{\partial m} = \frac{\theta L_r - L_Y}{\theta} * \frac{\partial r}{\partial m}+L_{r+\pi^e} * \frac{\partial \pi^e}{\partial m} \)