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):
$$ln(\frac{m_t}{p_t})=-\gamma ln(\frac{p_{t+1}}{p_t})$$$$\ln\left(\frac{m_t}{p_t}\right)=-\gamma \ln\left(\frac{p_{t+1}}{p_t}\right)$$ (re-writing the expression to include logs)
$$RHS = -\gamma ln(1+\frac{\Delta p_{t+1}}{p_t}) \approx -\gamma \frac{p_{t+1}-p_t}{p_t} $$$$RHS = -\gamma \ln\left(1+\frac{\Delta p_{t+1}}{p_t}\right) \approx -\gamma \frac{p_{t+1}-p_t}{p_t} $$ $$\text{(using } ln(1+x) \approx x)$$$$\text{(using } \ln(1+x) \approx x)$$
Now sub this back to the first expression:
$$ln(\frac{m_t}{p_t}) \approx \gamma \frac{p_{t+1}-p_t}{p_t} \approx ln(1-\gamma \frac{p_{t+1}-p_t}{p_t}) $$$$\ln\left(\frac{m_t}{p_t}\right) \approx \gamma \frac{p_{t+1}-p_t}{p_t} \approx \ln\left(1-\gamma \frac{p_{t+1}-p_t}{p_t}\right) $$ $$ \text{using } \gamma x \approx \ln(1+ \gamma x) $$
$$\frac{m_t}{p_t} \approx 1 - \gamma \frac{p_{t+1}-p_t}{p_t}$$
Am I correct and/or on the right lines? Not sure how to go about it from here.
