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

Econ 816

Spring, 2000

**HW#4 Answers**

** **

**Part 1: Mathematical Problems**

1. Romer, problem 9.8.

a. Since
this is an RE model without exogenous uncertainty, the fact that expected
inflation p^{e}=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(M_{t}/P_{t})=a-bi+ln Y_{t},
so:

M_{t}/P_{t}=e^{a-bi}Y_{t}

i=r+p^{e}

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

p=g_{M}-g_{Y}

_{ }

i=r+g_{M}-g_{Y}

Plugging into the equation for real balances:

M_{t}/P_{t}=exp(a)exp(-b[r+g_{M}-g_{Y}])Yt

Seigniorage equals the change in real balances:

_{}

Plugging in:

S=g_{M}*exp(a)exp(-b[r+g_{M}-g_{Y}])Yt

Then just maximize S wrt g_{M}, getting:

[exp(a)exp(-b[r+g_{M}-g_{Y}])Yt]-bg_{M}[exp(a)exp(-b[r+g_{M}-g_{Y}])Yt]=0

Thus:

[exp(a)exp(-b[r+g_{M}-g_{Y}])Yt][1-bg_{M}]=0

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

[1-bg_{M}]=0

Implying g_{M}*=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.