# Differentiation in a inflation tax calculation

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.

Your calculation is correct. There's probably just a mistake in the book and (4) should really be $$(1+π)\frac{d Z_t}{d π} = \frac{M_t/P_t}{1+π} - e π$$. Since presumably the RHS is then set to zero, this doesn't change the maximizer, so the mistake is innocent.
• Thank you, VARRulle. I was wondering if it could be a mistake in the book as you said, but even considering this I was stuck because I wasn't able to get the rationale behind that $(1 + π) \frac{dZ_t}{d π}$ would be "neutral" to the equation since the right hand side is set to zero. Now that you explained this, it is so clear to me. Jun 28 at 15:10