\(1-\frac{\partial p}{\partial m} - L_{r+\pi^e} * \frac{\partial \pi^e}{\partial m} = \frac{\theta L_r - L_Y}{\theta} * \frac{\partial r}{\partial m}\)
\(\frac{\partial r}{\partial m}= \frac{\theta[1-\frac{\partial p}{\partial m} - L_{r+\pi^e} * \frac{\partial \pi^e}{\partial m}]}{\theta L_r - L_Y}\)
Since \(L_{r+\pi^e} < 0\) ; \(L_Y > 0\) ; \(0 < \frac{\partial p}{\partial m} < 1\) and \(\frac{\partial \pi^e}{\partial m} > 0\), an increase i the money supply will still lower the real interest rate. Compared to question (b), \(\frac{\partial r}{\partial m}\) is larger because of the effect of inflation expectation on \(\frac{\partial r}{\partial m}\) is positive. This means that to achieve a given change in r, a smaller change in m is required than in question (b).
d) Suppose there is complete and instantaneous price adjustment. \(\frac{\partial \pi^e}{\partial m} = 0\) and \(\frac{\partial p}{\partial m} = 1\). Find an expression for dr/dm. Does an increase in the money supply lower the real interest rate?
\(\frac{\partial r}{\partial m}= \frac{\theta[1-\frac{\partial p}{\partial m} - L_{r+\pi^e} * \frac{\partial \pi^e}{\partial m}]}{\theta L_r - L_Y}\)
\(\frac{\partial r}{\partial m}= \frac{\theta[1-1+0]}{\theta L_r - L_Y} = 0\)
Hence an increase in the money supply does not affect the real interest rate in the case of a complete and instantaneous price adjustment.