Prof. Bryan Caplan

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 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:





, 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:







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:








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:





So in equilibrium:






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:






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






Plugging into the equation for real balances:




Seigniorage equals the change in real balances:



Plugging in:




Then just maximize S wrt gM, getting:








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




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.