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.


1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$
    – PGabriel96
    Jun 28, 2021 at 15:10

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