# Prof. Bryan Caplan

bcaplan@gmu.edu

http://www.gmu.edu/departments/economics/bcaplan

Econ 816

Spring, 2000

Part 1: Mathematical Problems

1.  Romer, problem 9.8.

a.  Since this is an RE model without exogenous uncertainty, the fact that expected inflation pe=b/a implies that actual inflation p=b/a.  Therefore,  .

b.  The AS function in the intial period will be:

The 1-period OF is therefore:

Differentiating:

, so

Plugging into the lifetime OF yields:

Whereas if the policymaker did not defect, lifetime OF would be:

c.  Find when :

Cancelling and simplifying:

This leaves us with a quadratic equation in .  Solve using the quadratic formula, it can be seen that the above inequality holds if:

Simplifying:

The two critical values are thus:  and .

Cooperation will then be possible so long as:

.

If , it cannot be larger than b/a, so equilibrium will exist so long as:

, implying that .

2.  Romer, problem 9.9.

In all three problems, the cooperation utility remains:

a.  Defection will only pay if one can defect one period, take the punishment for one period, and be better off than if one had cooperated.  So compare:

to:

Simplifying:

Again solving the quadratic equation for the critical values (and simplifying!):

, so the critical values are  and .

Cooperation will then be possible so long as:

.

b. This implies a two-period cost of defection.  For convenience, define .  Then compare:

to:

So in equilibrium:

Simplifying:

Applying the quadratic formula yields the following mess:

Which shows that with extreme punishments, you can get cooperation in more general circumstances than you can otherwise.

c.  This is always an equilibrium; in each period, the policy-maker is optimizing relative to the public's actions, and vice versa.

3.  Romer, problem 9.16.

ln(Mt/Pt)=a-bi+ln Yt, so:

Mt/Pt=ea-biYt

i=r+pe

The percent change in the money supply plus the inflation rate must equal the real rate of growth, so:

p=gM-gY

i=r+gM-gY

Plugging into the equation for real balances:

Mt/Pt=exp(a)exp(-b[r+gM-gY])Yt

Seigniorage equals the change in real balances:

Plugging in:

S=gM*exp(a)exp(-b[r+gM-gY])Yt

Then just maximize S wrt gM, getting:

[exp(a)exp(-b[r+gM-gY])Yt]-bgM[exp(a)exp(-b[r+gM-gY])Yt]=0

Thus:

[exp(a)exp(-b[r+gM-gY])Yt][1-bgM]=0

The first term in brackets is always positive, so the only solution is given by:

[1-bgM]=0

Implying gM*=1/b

4.  Romer, problem 9.17.

a. As in the last problem, .

Define  and .  Then

Next, note that inflation is just the difference between nominal money growth and real money growth:

Let G be the amount of seigniorage the government spends:

, so

The money demand equation can be used to compute :

Plugging into the equation for inflation:

Now plug the above into the equation for the change in expected inflation:

, so:

Re-arranging terms:

b.  In this case, you essentially get a hyperinflation - every period to have to make inflation higher to get more than S* in revenue.

c.  Now in the steady state, expected inflation equals actual inflation equals the rate of money growth.