# Need a math help for the Cagan's model in macroeconomics

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?

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