Economics 407: Topics in Macroeconomics

Economics 407: Topics in Macroeconomics

Homework #4

This homework is due on Tuesday, March 5th.

1.Consider the Friedman-Lucas “Worker Misperception” model in which workers observe their nominal wage perfectly but not their real wage. Suppose the central bank wishes to reduce the price level and announces (ahead of time) that it will reduce the money supply to accomplish this.

a. Suppose that workers do not believe/understand the central bank. What happens to real aggregate variables (unemployment rate, real GDP, employed labor) and the price level when the central bank reduces the money supply?

When the Fed reduces the money supply, aggregate demand falls. Firms, upon realizing fewer people are purchasing their products, reduce prices and simultaneously reduce their employees’ nominal wages by the same amount as prices fell thereby leaving the real wage unchanged. Employees, not observing their real wage, believe the nominal wage reduction is tantamount to a reduction in their real wage, reduce their labor supply. A lower labor supply results in less employment and reduced output. Interestingly, because markets still clear and the labor supply reduction was voluntary, the unemployment rate remains the unchanged.

Eventually, workers will realize businesses are offering an unchanged real wage. When they do, workers will respond by increasing their labor supply back to its original level thereby increasing real GDP back to its initial level.

b. Alternatively, suppose that workers believes and understands the central bank. What happens to real variables and the price level when the central bank reduces the money supply?

Again, the Fed reduces the money supply, aggregate demand falls, and firms lower both prices and wages proportionally. In this case, workers understand that the reduction in nominal wages has left their real wage unchanged. Workers do not reduce their labor supply and no changes in output occur. All that happens is that the price level falls.

c. Compare your results in parts a and b. In which case does the price level change more for a given reduction in money supply? What do you think a central bank can do to make its policy announcements more credible?

In the long run, the decrease in money causes the same sized reduction in prices regardless of if workers initially understand or fail to understand the Fed’s actions. In the long run, workers are not fooled into reducing their labor supply so aggregate supply remains unchanged relative to the perfect information case.

In the short run, the fact that workers reduce labor supply and thereby reduce output means that products become more scarce under situation a. Since products become more scarce, prices fall less than they do when workers maintain their initial level of production. In short, one can see this by realizing that part a suggests a positively sloped aggregate supply curve while part b suggests a vertical aggregate supply curve. For any decline in aggregate demand, prices will fall less when the aggregate supply curve is positively sloped relative to one that is vertical (in other words, prices will fall less when there is a supply response to a decline in demand).

2.Consider the Lucas “Imperfect Information” model where producers observe pi but not p (and hence not ri). Consider two economies, one in which the money supply is very stable and one in which the money supply is highly variable. Show how the aggregate supply curves of these economies differ. Explain why.

The Lucas surprise supply curve is given by the equation: . In country a, 2p is large—the highly variable money supply results in a highly variable price level which makes it difficult for the firm to identify the difference between a relative and absolute price change. If it is difficult to observe this, then when a firm observes a higher price for their own good, the firm will likely conclude that the reason their price is higher is because the price level rose and hence be less likely to change production. Mathematically, as 2p grows, the impact of p on y declines—the AS curve because more steep. Thus, in country b the AS curve is flatter and there is more output response to a given surprise price shock.

If the money supply (or AD change) is not a surprise, neither country’s producers react and total output does not change.

3.Consider a consumer who lives for two periods and has the following intertemporal utility function:

>0

where is the discount rate. A larger value of  implies that this consumer values future consumption less than consumption today. The consumer receives income of Yt and Yt+1 and pays taxes of Tt and Tt+1 in each period as described in class. The consumer is not constrained in either their borrowing or lending and for each unit borrowed or lent, the interest rate r applies.

b. What is ? Is it less than or greater than one? Comment on this derivative in light of consumption being less variable than income.

Because this derivative is less than one, consumption rises less than one-for-one with income. In other words, an increase in income leads to higher consumption by a lesser amount (hence, some savings occurs).

c. What is where Y = Yt + Yt+1? Compare this derivative to that which you found in (b) and comment on it in light of the Permanent Income Hypothesis.

A rise in permanent income (as opposed to a one time change in income) leads to a greater consumption response than what occurs given an equal sized change in income during just one period.

For the remainder of this problem consider a case where individuals pay a higher interest rate on any borrowing it partakes than it would receive on savings. To be specific, when the consumer saves, she gets an interest rate of rs but when she borrowers she has to pay rb on her loan where rbrs.

d. What is the budget constraint faced by the consumer in each period? Distinguish between the cases in which she is a net saver or borrower. Write down the intertemporal budget constraint and draw it in space (Ct, Ct+1).

and draw it in space (Ct, Ct+1).

If saver:

If borrower:

e. Using the above utility function, solve the maximization problem and determine the consumption level in the first period of life when the consumer is a net saver and when the consumer is a net borrower. [Hint: The way to solve this problem is to consider solutions to two distinct problems with different interest rates and then think about when each solution applies.

if Ctyt – τt

if Ctyt – τt

Substituting the solutions for Ct into the conditional equations gives:

if

if