Hence the increase in government spending has only a temporary effect on output. The only case where an increase in G could have a permanent impact on output is if \(b>1\), that is if the propensity to consume is superior to 100%. Such a situation, however, is a quite irrealistic.
6.6. The central bank's ability to control the real interest rate. Suppose the economy is described by two equations. The first is the IS eqation, which for simplicity we assume takes the traditional form, \(Y_t=-\frac{r_t}{\theta}\). The second is the money market equilibrium condition, which we can write as \(m-p=L(r+\pi^e , Y)\); \(L_{r+\pi^e} < 0\) ; \(L_Y > 0\), where m and p denote ln(M) and ln(P).
a) Suppose \(P = \bar{P}\) and \(\pi^e = 0\). Find an expression for dr/dm. Does an increase in the money supply lower the real interest rate.
We substitute \(Y_t=-\frac{r_t}{\theta}\) into the the money market equilibrium equation:
\(m-p=L(r+\pi^e , -\frac{r_t}{\theta})\)
Since \(P = \bar{P}\) and \(\pi^e = 0\):
\(m-\bar{p}=L(r , -\frac{r_t}{\theta})\)
Differentiating both sides with respect to m, we get:
\(1 = \frac{\partial L}{\partial r} * \frac{\partial r}{\partial m} + \frac{\partial L}{\partial Y}*\frac{\partial (-\frac{r_t}{\theta})}{\partial m}\)
\(1 = L_r * \frac{\partial r}{\partial m} + L_Y (-\frac{1}{\theta})*\frac{\partial r}{\partial m}\)
\(1 = \frac{\theta L_r - L_Y}{\theta} * \frac{\partial r}{\partial m}\)
\(\frac{\partial r}{\partial m} = \frac{\theta}{\theta L_r - L_Y}\)
Since \(L_{r+\pi^e} < 0\) ; \(L_Y > 0\) , this means that an increase in the money supply lowers the real interest rate.
b) Suppose prices respond partially to increases in money. Specifically, assume dp/dm is exogenous, with 0 < dp/dm < 1. Continue to assume that \(\pi^e = 0\). Find an expression for dr/dm. Does an increase in the money supply lower the real interest rate? Does achieving a given change in r require a change in m smaller, larger or the same size as in part (a)?
We have:
\(m-p=L(r , -\frac{r_t}{\theta})\)
Differentiating both sides with respect to m, we get:
\(1-\frac{\partial p}{\partial m} = \frac{\partial L}{\partial r} * \frac{\partial r}{\partial m} + \frac{\partial L}{\partial Y}*\frac{\partial (-\frac{r_t}{\theta})}{\partial m}\)
\(1-\frac{\partial p}{\partial m} = L_r * \frac{\partial r}{\partial m} + L_Y (-\frac{1}{\theta})*\frac{\partial r}{\partial m}\)
\(1-\frac{\partial p}{\partial m} = \frac{\theta L_r - L_Y}{\theta} * \frac{\partial r}{\partial m}\)
\(\frac{\partial r}{\partial m} = \frac{\theta}{\theta L_r - L_Y}*[1-\frac{\partial p}{\partial m}]\)
Since \(L_{r+\pi^e} < 0\) ; \(L_Y > 0\) and \(0 < \frac{\partial p}{\partial m} < 1\) an increase in the money supply still lowers the real interest. However, it also implies that to achieve a given change in the real interest rate, a larger change in the money supply will be necessary.