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.



Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?

  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.)

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.