Due March 3 Economics 2010d Spring 2014

Problem Set 4

1. Suppose a representative household maximizes a standard intertemporal utility function:

0

ln lnst t s t ss

E C H H

, and faces the budget constraint: 1 1 1 1t t t t t t t t t tK B C K R K W H r B . The households assets consist of capital, K, and riskless private consumption bonds, B. R is the rental rate of capital (what firms pay per unit of K each period), r is the return on bonds, and W is the wage. The household satisfies standard transversality and no-Ponzi-game conditions. A competitive representative firm produces all output using the technology 1( )t t t tY K Z H

. The firm pays R and W to rent capital and labor. There is no government, so in equilibrium t t tY C I . A. Suppose that at the beginning of period t, the household is informed that next periods

technology, Zt+1, will be lower than expected. However, todays Zt does not change. In response to this new information, what happens to the ratio ( ) / ( )?t t tC H H H

How can this information be used to figure out the comovement of C and H in response to the shock?

B. Suppose households expect next periods consumption, Ct+1, to be low due to next periods

bad shock. Suppose further that the real interest rate between t and t+1, rt+1, falls when the news about Zt+1 is received. Can you say what happens to todays levels of Yt, Ct, Ht and It when the news about future Z is received? Make an intuitive argument; you do not have to solve the full dynamic model. Be sure to note if any result is ambiguous, and explain the economic reasoning.

C. Suppose households expect next periods consumption, Ct+1, to be low due to next periods bad shock. Suppose further that the real interest rate between t and t+1, rt+1, rises when the news about Zt+1 is received. Can you say what happens to todays levels of Yt, Ct, Ht and It when the news about future Z is received? Make an intuitive argument; you do not have to solve the full dynamic model. Be sure to note if any result is ambiguous, and explain the economic reasoning.

Due March 3 Economics 2010d Spring 2014 2. Suppose that the money demand function is

tit tt

M Y eP

where M is nominal money, P is the price level, Y is output, it = rt + Ett is the nominal interest rate, and > 0. (e is the mathematical constant 2.71828 .) Assume that Y and r are constant, and pick units such that 1rYe . A. Take the natural log of the money demand function under the assumptions above, using the

definition that 1 1 ,t t t t tE E p p where ln( ).t tp P Rearrange the result to solve for pt in terms of expected p and m. (Use lower-case letters to represent natural logs.)

B. Eliminate future prices from the equation in part A. Solve for p in terms of current and

expected future m (assume there is no bubble or non-fundamental solution). Now suppose that nominal money grows at a constant rate . Thus, 1 st s tM M . C. Solve for pt as a function of mt, , and . (You can leave the answer in the form of a

summation. Assume that all the relevant infinite sums converge to finite quantities.)

D. Suppose the monetary authority increases permanently. What happens to p and to m p? Explain the economics.

E. Now return to Part B, and show that in addition to the fundamental solution we have been discussing, there is also a bubble solution. Solve for the bubble term. Show that it implies that the price level can grow much faster than the money stock. Interpret the economics behind the bubble solution.

F. Suppose we embed this model of money demand into a model with dynamically optimizing consumers. Do you think that the assumption of consumer optimization would rule out bubbles? Discuss the cases of positive and negative bubbles separately.

Romer, Advanced Macro (4th ed.), problem 10.10