Calculation of simple elasticity problem

Question

I'm reading Advanced Macroeconomics by Romer and having trouble to understand very simple calculation provided in just one sentence.

Define real money market equilibrium condition as $(\frac{M}{P})^s=L(Y,i)=(\frac{M}{P})^d$
Then the price leve is $P=\frac{M}{L(Y,i)}$
If the interest elasticity of money demand is $-0.2$, we need almost $32$ times rise in interest rate to increase price level double.

I can't understand where "$32$ times" comes from.
From the definition of elasticity, and using the fact that
if we want to make price level double then $L(Y,i)$ should be reduced by half,
$\frac{\partial L}{\partial i}\frac{i}{L}=\frac{\partial L}{L}\frac{i}{\partial i}=-\frac{1}{2}\frac{i}{\partial i}=-0.2$ then $\frac{\partial i}{i}=2.3$
Can anyone tell me how to solve this problem?

Answer 1

If $L(Y,i)$ has a constant elasticity of $-0.2$ then you can write it as: $$ L(Y,i) = g(Y)i^{-0.2}. $$ Then for the period 0 and period 1 price levels you have: $$ P_0 = \frac{M}{g(Y)i_0^{-0.2}} \text{ and } P_1 = \frac{M}{g(Y)i_1^{-0.2}}. $$ We have that $P_1 = 2 P_0$ so: $$ 2 \frac{M}{g(y) i_0^{-0.2}} = \frac{M}{g(Y)i_1^{-0.2}}. $$ Simplifying gives: $$ 2 i_1^{-0.2} = i_0^{-0.2}. $$ This gives: $$ \begin{align*} \frac{i_1}{i_0} = 2^5 = 32 \end{align*} $$