<p>Your calculation is correct. There's probably just a mistake in the book and (4) should really be <span class="math-container">$(1+π)\frac{d Z_t}{d π} = \frac{M_t/P_t}{1+π} - e π$</span>. Since presumably the RHS is then set to zero, this doesn't change the maximizer, so the mistake is innocent.</p>

CC BY-SA 4.0
2021-06-28T15:10:57.250Z
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.
PGabriel96
CC BY-SA 4.0
2021-06-28T09:14:00.290Z
2021-06-28T09:14:00.290Z
VARulle
<p>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.</p>
<p>The author starts by a simple statement of a real money demand function:</p>
<p><span class="math-container">$(1)$</span> <span class="math-container">$\frac{M_t}{P_t} = A + bY_t - cr$</span></p>
<p>Which represents the fact that real money demand responds positively from an increase in output (<span class="math-container">$Y_t$</span>) and negatively from an increase in the nominal interest rate (<span class="math-container">$r$</span>).</p>
<p>Then, <span class="math-container">$Z_t$</span> is introduced as the inflation tax:</p>
<p><span class="math-container">$(2)$</span> <span class="math-container">$Z_t = \frac{[π /(1 + π)] M_t}{P_t}$</span></p>
<p><span class="math-container">$Z_t$</span> is differentiated by <span class="math-container">$π$</span> in order to get the maximum possible value of seigniorage when <span class="math-container">$\frac{d Z_t}{d π} = 0$</span>. There is an additional statement made by the author to support this calculation:</p>
<p><span class="math-container">$(3)$</span> <span class="math-container">$\frac{M_t}{P_t} = A_0 - e π$</span></p>
<p>Where <span class="math-container">$A_0 = A + bY_t - c\rho$</span> and is considered as a constant for the purpose of the argument. <span class="math-container">$e$</span> is equal to <span class="math-container">$c(1 + \rho)$</span>. <span class="math-container">$\rho$</span> is considered as the real interest rate.</p>
<p>After this new argument, the differentiation is calculated and it results in:</p>
<p><span class="math-container">$(4)$</span> <span class="math-container">$\frac{d Z_t}{d π} = \frac{M_t/P_t}{1+π} - e π$</span></p>
<p>The author states that the fact that <span class="math-container">$\frac{d\frac{M_t}{P_t}}{d π} = - e$</span> in <span class="math-container">$(3)$</span> 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.</p>
<p>The closest I could get from it was when I thought <span class="math-container">$(2)$</span> as</p>
<p><span class="math-container">$Z_t = \frac{π}{1 + π} * \frac{M_t}{P_t}$</span></p>
<p>And used the product rule (<span class="math-container">$u' * v + u * v'$</span>), but I could only get to</p>
<p><span class="math-container">$\frac{d Z_t}{d π} = \frac{M_t/P_t}{(1 + π)^2} - \frac{e π}{1 + π}$</span></p>
<p>I am probably miscalculating somewhere, but I do not know where exactly.</p>

<macroeconomics><mathematical-economics><inflation>
Differentiation in a inflation tax calculation
PGabriel96
# Differentiation in a inflation tax calculation

(Asked by: username_0 on 2021-06-27)

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.

**Tags:** macroeconomics, mathematical-economics, inflation


---

## 1 answers

### Answer by username_1 on 2021-06-28. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: 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.  

---

