<p>The answer to both your questions is that for small <span class="math-container">$x$</span> values
<span class="math-container">$$
\ln(1+x) \approx x,
$$</span>
the difference being less than <span class="math-container">$x^2/2$</span>. (Proof by Taylor-approximation.)</p>

<p>So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.</p>

<p>This should also answer your second question, as the approximation
<span class="math-container">$$
\gamma x \approx \ln(1+ \gamma x),
$$</span>
works as well.</p>

<p>It may also be worthwhile to look into <a href="https://en.wikipedia.org/wiki/Elasticity_(economics)#Mathematical_construct" rel="nofollow noreferrer">elasticity</a>.</p>

CC BY-SA 4.0
2019-03-24T10:19:06.357Z
2019-03-24T10:19:06.357Z
Giskard
<p>From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.</p>

<blockquote>
  <p>To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is</p>
  
  <p><span class="math-container">$m_t − p_t = −\gamma( p_{t+1} − p_t)$</span>,</p>
  
  <p>where <span class="math-container">$m_t$</span> is the log of the quantity of money at time t, <span class="math-container">$p_t$</span> is the log of the price level at time t, and <span class="math-container">$\gamma$</span> is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, <span class="math-container">$m_t − p_t$</span> is the log of real money balances, and <span class="math-container">$p_{t+1} − p_t$</span> is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by <span class="math-container">$\gamma$</span> percent.</p>
</blockquote>

<ol>
<li><p>Shouldn't <span class="math-container">$(p_{t+1} - p_t)$</span> be the <strong>log of</strong> inflation rate? Why it says just "the inflation rate"?</p></li>
<li><blockquote>
  <p>This equation states that if inflation goes up by 1 percentage point, real money balances fall by <span class="math-container">$\gamma$</span> percent.</p>
</blockquote>

<p>My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all. </p>

<p><span class="math-container">$\ln \frac{M}{P} = \ln (\frac{p_{t+1}}{p_t})^{-\gamma} \rightarrow \frac{M}{P} = (\frac{p_{t+1}}{p_t})^{-\gamma}$</span></p>

<p>So, if the <span class="math-container">$(p_{t+1} - p_t)$</span> is just the <strong>log</strong> of inflation rate, then <span class="math-container">$\frac{p_{t+1}}{p_t}$</span> is the inflation rate and,</p>

<blockquote>
  <p>inflation goes up by 1 percentage point</p>
</blockquote>

<p>would mean <span class="math-container">$\frac{p_{t+1}}{p_t}$</span> is going to get +1, right? But I couldn't possibly think it would result the fall of <span class="math-container">$\frac{M}{P}$</span> by the <span class="math-container">$\gamma$</span> point. What am I missing?</p>

<p>And secondly, if the <span class="math-container">$(p_{t+1} - p_t)$</span> is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "<span class="math-container">$−\gamma(1 + p_{t+1} − p_t)$</span>" at the right side, right? How could this be the case?</p></li>
</ol>

<mathematical-economics>
Need a math help for the Cagan's model in macroeconomics
dolco
# Need a math help for the Cagan's model in macroeconomics

(Asked by: username_0 on 2019-03-24)

From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.

> To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is
>
> $m\_t − p\_t = −\gamma( p\_{t+1} − p\_t)$,
>
> where $m\_t$ is the log of the quantity of money at time t, $p\_t$ is the log of the price level at time t, and $\gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m\_t − p\_t$ is the log of real money balances, and $p\_{t+1} − p\_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $\gamma$ percent.

1. Shouldn't $(p\_{t+1} - p\_t)$ be the **log of** inflation rate? Why it says just "the inflation rate"?
2. > This equation states that if inflation goes up by 1 percentage point, real money balances fall by $\gamma$ percent.

   My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.

   $\ln \frac{M}{P} = \ln (\frac{p\_{t+1}}{p\_t})^{-\gamma} \rightarrow \frac{M}{P} = (\frac{p\_{t+1}}{p\_t})^{-\gamma}$

   So, if the $(p\_{t+1} - p\_t)$ is just the **log** of inflation rate, then $\frac{p\_{t+1}}{p\_t}$ is the inflation rate and,

   > inflation goes up by 1 percentage point

   would mean $\frac{p\_{t+1}}{p\_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $\frac{M}{P}$ by the $\gamma$ point. What am I missing?

   And secondly, if the $(p\_{t+1} - p\_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−\gamma(1 + p\_{t+1} − p\_t)$" at the right side, right? How could this be the case?

**Tags:** mathematical-economics


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## 1 answers

### Answer by username_1 on 2019-03-24. Score: 3 | Accepted answer

The answer to both your questions is that for small $x$ values
$$
\ln(1+x) \approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)

So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.

This should also answer your second question, as the approximation
$$
\gamma x \approx \ln(1+ \gamma x),
$$
works as well.

It may also be worthwhile to look into [elasticity](https://en.wikipedia.org/wiki/Elasticity_(economics)#Mathematical_construct).


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