Mankiw's version of the Cagan model - need help interpreting it

Question

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):

$$\ln\left(\frac{m_t}{p_t}\right)=-\gamma \ln\left(\frac{p_{t+1}}{p_t}\right)$$ (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.

Answer 1

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, or on this 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).