代做ECON7230 Monetary Economics Summer 2024 Homework 2代写C/C++编程

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

parameters: the project pays 1 unit if aborted at T  = 1 and pays 2 units if completed at T  = 2, and that there are 100 people and 25 of them are the urgent type, and so on. The only difference now is that the utility function is U = 1 − c2/1

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%.