Question

Consider a money demand function that takes the form (M/P)^d = Y/3i, where M is the quantity of money, P is the price lelve, Y is real output, and i is the nominal interest rate (measured in percentage points.)

a) If the economy is growing g% per year and velocity is growing twice as fast what are the growth rates in real money balances and the interest rate?

b) Supposing the money supply is growing at 3% per year, what values of g indicate inflation? Deflation?

I am struggling with these questions. Could you please answer the following questions. Thank you

Answer 1

Real money demand:

$\small (\frac{M}{P})^d = \frac{Y}{3i}$$\small = \frac{Y}{v}$ where v = 3i = velocity of money

$\small ln(\frac{M}{P})^d = ln\frac{Y}{v} = ln Y - ln v$

Differentiating both sides:

$\small \Rightarrow \frac{(\Delta \frac{M}{P})^d}{( \frac{M}{P})^d} = \frac{\Delta Y}{Y} - \frac{\Delta v}{v}$

If the economy is growing g% per year and velocity is growing twice as fast:

$\small \frac{\Delta Y}{Y} = g\%$ and $\small \frac{\Delta v}{v} = 200\%$

$\small \Rightarrow \frac{(\Delta \frac{M}{P})^d}{( \frac{M}{P})^d} = \frac{\Delta Y}{Y} - \frac{\Delta v}{v}$$\small = (g - 200)\%$

Also

$\small 3i = v \Rightarrow i = \frac{v}{3}$

Thus,

$\small \frac{\Delta i}{i} = \frac{1}{3} \frac{\Delta v}{v} = 200/3 = 66.67\%$

b)

If money supply grows at 3% per year,

$\small P = MV/Y$

$\small ln P = ln (MV/Y) = ln M + ln V - ln Y$

Differentiating both sides we get:

$\small \frac{\Delta P}{P} = \frac{\Delta M}{M}+\frac{\Delta V}{V}-\frac{\Delta Y}{Y}$

$\small \Rightarrow \frac{\Delta P}{P} = (3+203-g)\% = (203-g)\%$

For inflation, $\small \frac{\Delta P}{P} > 0 \Rightarrow (203-g) > 0 \Rightarrow g < 203$

For deflation, $\small \frac{\Delta P}{P} < 0 \Rightarrow (203-g) < 0 \Rightarrow g > 203$