Question

2. a) Starting with the product rule for derivatives, show that the monetary exchange equation may be written as where AM dM etc. b) As interest payments made on the debt are extinguished from the money supply, more money must be created. Agreater portion of that money also goes towards making debt interest payments. Using the monetary exchange equation to explain what happens to prices. Assume the velocity of money remains constant.

Answer 1

2) a)The Quantity theory of money (QTM)  equation after modification is given by

M*V=P*Y

Here, M= Quantity of Money

V= Income Velocity of Money

P= Price of one unit of output

Y= Output/ Total Income

Now if we take log on both sides of the above equation

ln(M*V) = ln(P*Y)

ln M + ln V = ln P+ ln Y ( because ln (a*b) = ln a+ ln b)

Now differentiating on both sides, ( Implicit differentiation)

$\frac{1}{M}dM +\frac{1}{V}dV = \frac{1}{P}dP + \frac{1}{Y}dY$  

$\Delta M+ \Delta V= \Delta P + \Delta Y$ ( Given $\frac{1}{M}dM = \Delta M$ )

Hence Proved

b) From a) part we have

$\Delta M+ \Delta V= \Delta P + \Delta Y$

But since V constant. $\Delta V= 0 %u2191 \Delta P + \Delta Y$

As per the question, more quantity of money is required for interest payment on the debt.

Therefore in L.H.S., there is an increase in $\Delta M$ . Now R.H.S. needs to increase to match L.HS. so that equation is satisfied.

i.e. $%u2191 \Delta M + \Delta V = %u2191 \Delta P + \Delta Y$ ↑$\Delta M = %u2191 \Delta P + \Delta Y$ ↑ $\Delta P + \Delta Y$

That is, if the velocity is fixed, the quantity of money determines the dollar value of the economy’s output. And if output is also fixed then increase in money causes inflation.