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ECON7230 Monetary Economics Summer 2024
Homework 2
Due Date: July 2 (Tuesday) (before the end of the day, submit through Moodle)
1. Completing the island model – We have derived the aggregate supply curve yt = a + bθ(pt − p(*)t ) in class, and we have also talked about the demand side but have not got into the details. The aggregate demand curve is yt + pt = xt , where xt is a normal random variable with zero mean and variance of σx(2) that moves the demand curve around. At time t you do not know xt but you know its previous values. Now we can find the rational expectations equilibrium for the whole economy.
a) Guess a solution for price in the form of pt = Co + C1xt. Under this solution, what is p(*)t?
b) Use the equilibrium condition and match the coefficients to find the unknown constants Co, C1 .
c) Suppose the variance of xt increases, how does it affect the prior belief pt ~N(p(*)t, σp(2))? How does it affect θ?
Explain your reasoning carefully.
2. A Simple Dynamic Model with Excel
We want to know the effects of reducing the money growth rate gmt permanently. In period 0, the economy is at the medium-run equilibrium, where un = ̅(g)y = 3% and gmo = 6%. Also, α = 1 and β = 0.4.
From period 1 onwards, the money growth rate is lowered to 5%.
a) Using Excel (or any similar program), calculate u, π , and gy from period 1 to period 50. Plot your answers as three separate curves from period 0 to period 50.
b) Do the same exercise, but this time we have β = 0.1 instead. How does that change your answers?
(Hint: To use Excel, you cannot use the three equations directly. You need to first rewrite the equations such that ut does not depend on πt and πt does not depend on ut.)
3. In the Fischer model, we change the aggregate supply and demand curves to
yts = pt − Et 一1 pt + ut
yt = Mt − pt
That is, there is no demand shock but there is a supply shock ut. The shock is an AR(1) process ut = put 一1 + et , where et is a zero-mean white noise. Everything else of the model is the same as the one discussed in class.
a) For the case of a one-period contract, show that money has no effect on output.
b) For the case of a two-period contract, show that money can make output less volatile.
c) How does your answer change if the supply shock is white noise instead of an AR(1) process?
4. Consider a specific version of the Barro-Gordon model:
ut = u
− 2(πt − πt(e))
u
= utn一1 + Et
zt = (Ut − 0.5U
)2 + (πt )2
The term Et is a zero-mean white noise. The policymaker aims at minimizing a discounted sum of zt.
a) Derive the equilibrium under discretion. How do inflation and unemployment rate behave over time (sketch your answer in a graph with time on the x-axis)?
b) Derive the equilibrium under rule. How do inflation and unemployment rate behave over time (sketch your answer in a graph with time on the x-axis)?
5. Bank Run Model: Consider the model about bank runs we covered in class and use the same set of
a) Calculate the expected utility at T = 0 when there is no bank and people have to invest in the project themselves.
b) Use the optimality condition UI (r1 ) = 2UI (r2 ) and the constraint 75r2 = 2(100 − 25r1 ) to solve for the best deal r1 and r2. Notice that U′ is the marginal utility (which in this case is 
), and U′ (r1 ) means you use r1 to replace C in the marginal utility.
c) Calculate the expected utility at T = 0 when there is a bank, based on the answer for b).
d) Using the shape of the utility function, explain intuitively why the optimal r1 and r2 here are different from the version we have covered in class? Without doing the calculation, describe what will happen to the optimal
r1 and r2 when the utility function is changed to U = 1 − √c/1.
6. Delegated Monitoring Model: Suppose the central bank has increased the interest rate and depositors now require a 10% return instead. All other parameters are the same as in the notes. Calculate the expected profits for the borrowers and the bank, and intuitively explain how they are different from the case when the required return is 5%.