everyone. I am studying Advanced Macroeconomics, by Derek Leslie, and I am having some troubles in understanding the result of a differentiation in the Chapter 1, section 5. This section approaches the concepts of seigniorage and inflation tax.
The author starts by a simple statement of a real money demand function:
$(1)$ $\frac{M_t}{P_t} = A + bY_t - cr$
Which represents the fact that real money demand responds positively from an increase in output ($Y_t$) and negatively from an increase in the nominal interest rate ($r$).
Then, $Z_t$ is introduced as the inflation tax:
$(2)$ $Z_t = \frac{[π /(1 + π)] M_t}{P_t}$
$Z_t$ is differentiated by $π$ in order to get the maximum possible value of seigniorage when $\frac{d Z_t}{d π} = 0$. There is an additional statement made by the author to support this calculation:
$(3)$ $\frac{M_t}{P_t} = A_0 - e π$
Where $A_0 = A + bY_t - c\rho$ and is considered as a constant for the purpose of the argument. $e$ is equal to $c(1 + \rho)$. $\rho$ is considered as the real interest rate.
After this new argument, the differentiation is calculated and it results in:
$(4)$ $\frac{d Z_t}{d π} = \frac{M_t/P_t}{1+π} - e π$
The author states that the fact that $\frac{d\frac{M_t}{P_t}}{d π} = - e$ in $(3)$ was necessary to calculate it. I see that, but in every way I try to use this info to help me in this differentiation, I don't get the right answer.
The closest I could get from it was when I thought $(2)$ as
$Z_t = \frac{π}{1 + π} * \frac{M_t}{P_t}$
And used the product rule ($u' * v + u * v'$), but I could only get to
$\frac{d Z_t}{d π} = \frac{M_t/P_t}{(1 + π)^2} - \frac{e π}{1 + π}$
I am probably miscalculating somewhere, but I do not know where exactly.
