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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?

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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.

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