<p>I think that you are unnecessarily overthinking it. For any relationship of the form:</p>
<p><span class="math-container">$$\ln y = \beta \ln x  $$</span></p>
<p>the interpretation of beta coefficient is that <span class="math-container">$1\%$</span> increase in <span class="math-container">$x$</span> leads to <span class="math-container">$\beta$</span> <span class="math-container">$\%$</span> increase in <span class="math-container">$y$</span>. The mathematical reason why this relationship holds was already explored at cross-validated stack exchange and you can see it <a href="https://stats.stackexchange.com/questions/244199/why-is-it-that-natural-log-changes-are-percentage-changes-what-is-about-logs-th">here</a>, or on <a href="https://www.dummies.com/education/economics/econometrics/econometrics-and-the-log-log-model/" rel="nofollow noreferrer">this</a> website or in virtually any econometric textbook since the log-log form is important, so I wont be unnecessarily restating it.</p>
<p>Mankiw defines real money balances as <span class="math-container">$M/P$</span> or in logs <span class="math-container">$m-p$</span>, hence the left hand side of your equation is by definition the real money balance term expressed in logs <span class="math-container">$\ln (M_t/P_t)$</span>. The inflation is by definition change in price level again in your case expressed in logs <span class="math-container">$\ln(P_{t+1}/P_t)$</span>. So you can directly apply the interpretation from previous paragraph as Mankiw does.</p>
<p>Furthermore, you can actually derive it also with your calculations. To be more specific. Growth rate <span class="math-container">$g$</span> for variable <span class="math-container">$x$</span> is given as <span class="math-container">$g_x= \frac{x_{t+1}-x_t}{x_t}$</span>. Then we know that:</p>
<p><span class="math-container">$$\ln x_{t+1} = \ln ((1+g_x)x_t) \implies \ln x_{t+1} = \ln (1+g_x)+ \ln x_t  $$</span></p>
<p>Then since <span class="math-container">$\ln (1+g_x) \approx g_x$</span> we have:</p>
<p><span class="math-container">$$\ln x_{t+1} = g_x  + \ln x_t \implies \ln x_{t+1} -\ln x_t =g_x $$</span></p>
<p>as a consequence you can actually directly say that:</p>
<p><span class="math-container">$$\ln(M_t/P_t)=-\gamma \ln(P_{t+1}/P_t) \approx \ln(M_t/P_t)=-\gamma \left( \frac{P_{t+1}-P_t}{P_t} \right)$$</span></p>
<p>that 1 in your final expression should not be there. In this case the LHS gives you percentage in the real money balances and LHS percentage change in price level (inflation).</p>

CC BY-SA 4.0
2020-09-14T14:07:17.647Z
I thought inflation would be the difference in the price levels, especially since we're talking about a change in inflation measured in percentage points.
BagelEnthusiast
CC BY-SA 4.0
2020-09-14T14:41:06.623Z
Oh wait so in your final expression, you have it as a log-linear model, which - if my econometrics are correct - leads to Mankiw's interpretation?
BagelEnthusiast
CC BY-SA 4.0
2020-09-14T16:03:14.963Z
@BagelEnthusiast yes exactly.
1muflon1
CC BY-SA 4.0
2020-09-14T23:33:27.647Z
@1muflon1: You're answer was great but it took me a while  to notice the typo where you wrote $ln(x_{t+1}) = g_{x} \times ln(x_{t})$ instead of $g(x) + ln(x_{t})$. I would fix it but I doubt that's allowed. thanks.
mark leeds
CC BY-SA 4.0
2020-09-14T23:34:57.437Z
@markleeds thanks for pointing it out I corrected it. Also you can edit other people’s post if you see a obvious mistake. i don’t think most people would mind and they also get notification so if they would disagree they could revert it. This being said thanks again for pointing it out
1muflon1
CC BY-SA 4.0
2020-09-15T03:52:25.910Z
No problem. Like the OP,  I find the details of the economic models confusing so spelling it out the way you did was useful. I never understand why the papers ( or even sometimes textbooks ) don't want to be clear and spell it out like you did. It seems like the authors in economics are not all that interested in making the papers as clear as possible.
mark leeds
CC BY-SA 4.0
2020-09-14T13:18:58.333Z
2020-09-14T23:34:44.207Z
2020-09-14T23:34:44.207Z
1muflon1
<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 mt is the log of the quantity of money at time t, pt is the log of the price level at time t, and γ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, mt−pt is the log of real money balances, and pt+1−pt is the inflation rate between period t and period t+1. <strong>This equation states that if inflation goes up by 1 percentage point, real money balances fall by γ percent.</strong></p>
<ul>
<li>From Mankiw's Macroeconomics Textbook (Appendix, chapter 4).</li>
</ul>
<p>I'm struggling to understand how this expression leads to the interpretation in bold. Applying some of the results from this answer to the same question (<a href="https://economics.stackexchange.com/questions/27434/need-a-math-help-for-the-cagans-model-in-macroeconomics">Need a math help for the Cagan&#39;s model in macroeconomics</a>):</p>
<p><span class="math-container">$$\ln\left(\frac{m_t}{p_t}\right)=-\gamma \ln\left(\frac{p_{t+1}}{p_t}\right)$$</span> (re-writing the expression to include logs)</p>
<p><span class="math-container">$$RHS = -\gamma \ln\left(1+\frac{\Delta p_{t+1}}{p_t}\right) \approx -\gamma \frac{p_{t+1}-p_t}{p_t} $$</span>
<span class="math-container">$$\text{(using } \ln(1+x) \approx x)$$</span></p>
<p>Now sub this back to the first expression:</p>
<p><span class="math-container">$$\ln\left(\frac{m_t}{p_t}\right) \approx \gamma \frac{p_{t+1}-p_t}{p_t}  \approx \ln\left(1-\gamma \frac{p_{t+1}-p_t}{p_t}\right) $$</span>
<span class="math-container">$$ \text{using } \gamma x \approx \ln(1+ \gamma x) $$</span></p>
<p><span class="math-container">$$\frac{m_t}{p_t} \approx 1 - \gamma \frac{p_{t+1}-p_t}{p_t}$$</span></p>
<p>Am I correct and/or on the right lines? Not sure how to go about it from here.</p>

<mathematical-economics><monetary-policy>
Mankiw's version of the Cagan model - need help interpreting it
BagelEnthusiast
# Mankiw's version of the Cagan model - need help interpreting it

(Asked by: username_0 on 2020-09-14)

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 mt is the log of the quantity of money at time t, pt is the log of the price level at time t, and γ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, mt−pt is the log of real money balances, and pt+1−pt 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 γ percent.**

* From Mankiw's Macroeconomics Textbook (Appendix, chapter 4).

I'm struggling to understand how this expression leads to the interpretation in bold. Applying some of the results from this answer to the same question ([Need a math help for the Cagan's model in macroeconomics](https://economics.stackexchange.com/questions/27434/need-a-math-help-for-the-cagans-model-in-macroeconomics)):

$$\ln\left(\frac{m\_t}{p\_t}\right)=-\gamma \ln\left(\frac{p\_{t+1}}{p\_t}\right)$$ (re-writing the expression to include logs)

$$RHS = -\gamma \ln\left(1+\frac{\Delta p\_{t+1}}{p\_t}\right) \approx -\gamma \frac{p\_{t+1}-p\_t}{p\_t} $$
$$\text{(using } \ln(1+x) \approx x)$$

Now sub this back to the first expression:

$$\ln\left(\frac{m\_t}{p\_t}\right) \approx \gamma \frac{p\_{t+1}-p\_t}{p\_t} \approx \ln\left(1-\gamma \frac{p\_{t+1}-p\_t}{p\_t}\right) $$
$$ \text{using } \gamma x \approx \ln(1+ \gamma x) $$

$$\frac{m\_t}{p\_t} \approx 1 - \gamma \frac{p\_{t+1}-p\_t}{p\_t}$$

Am I correct and/or on the right lines? Not sure how to go about it from here.

**Tags:** mathematical-economics, monetary-policy


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## 1 answers

### Answer by username_1 on 2020-09-14. Score: 2 | Accepted answer

I think that you are unnecessarily overthinking it. For any relationship of the form:

$$\ln y = \beta \ln x $$

the interpretation of beta coefficient is that $1\%$ increase in $x$ leads to $\beta$ $\%$ increase in $y$. The mathematical reason why this relationship holds was already explored at cross-validated stack exchange and you can see it [here](https://stats.stackexchange.com/questions/244199/why-is-it-that-natural-log-changes-are-percentage-changes-what-is-about-logs-th), or on [this](https://www.dummies.com/education/economics/econometrics/econometrics-and-the-log-log-model/) website or in virtually any econometric textbook since the log-log form is important, so I wont be unnecessarily restating it.

Mankiw defines real money balances as $M/P$ or in logs $m-p$, hence the left hand side of your equation is by definition the real money balance term expressed in logs $\ln (M\_t/P\_t)$. The inflation is by definition change in price level again in your case expressed in logs $\ln(P\_{t+1}/P\_t)$. So you can directly apply the interpretation from previous paragraph as Mankiw does.

Furthermore, you can actually derive it also with your calculations. To be more specific. Growth rate $g$ for variable $x$ is given as $g\_x= \frac{x\_{t+1}-x\_t}{x\_t}$. Then we know that:

$$\ln x\_{t+1} = \ln ((1+g\_x)x\_t) \implies \ln x\_{t+1} = \ln (1+g\_x)+ \ln x\_t $$

Then since $\ln (1+g\_x) \approx g\_x$ we have:

$$\ln x\_{t+1} = g\_x + \ln x\_t \implies \ln x\_{t+1} -\ln x\_t =g\_x $$

as a consequence you can actually directly say that:

$$\ln(M\_t/P\_t)=-\gamma \ln(P\_{t+1}/P\_t) \approx \ln(M\_t/P\_t)=-\gamma \left( \frac{P\_{t+1}-P\_t}{P\_t} \right)$$

that 1 in your final expression should not be there. In this case the LHS gives you percentage in the real money balances and LHS percentage change in price level (inflation).

> **Comments:**
> **username_0**: I thought inflation would be the difference in the price levels, especially since we're talking about a change in inflation measured in percentage points.  
> **username_0**: Oh wait so in your final expression, you have it as a log-linear model, which - if my econometrics are correct - leads to Mankiw's interpretation?  
> **username_1**: @username_0 yes exactly.  
> **username_2**: @username_1: You're answer was great but it took me a while  to notice the typo where you wrote $ln(x_{t+1}) = g_{x} \times ln(x_{t})$ instead of $g(x) + ln(x_{t})$. I would fix it but I doubt that's allowed. thanks.  
> **username_1**: @username_2 thanks for pointing it out I corrected it. Also you can edit other people’s post if you see a obvious mistake. i don’t think most people would mind and they also get notification so if they would disagree they could revert it. This being said thanks again for pointing it out  
> **username_2**: No problem. Like the OP,  I find the details of the economic models confusing so spelling it out the way you did was useful. I never understand why the papers ( or even sometimes textbooks ) don't want to be clear and spell it out like you did. It seems like the authors in economics are not all that interested in making the papers as clear as possible.  

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