Name: Year π π e (a) (2 points) What is the cyclical unemployment rate in each of the four years?

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1 Name: Intermediate Macroeconomic Theory II, Winter 2008 Instructor: Dmytro Hryshko Problem Set 3 (36 points; +6 bonus points. Bonus will be graded only if you attempt to solve the entire problem!). Due Thursday, March 27, by 4PM. 1. (4 points) Consider an economy in long-run equilibrium with an inflation rate π of 12% (=0.12) per year and a natural unemployment rate u n of 6% (=0.06). The Phillips curve is π = π e 2(u u n ). Assume that Okun s law holds so that a 1% increase in the unemployment rate maintained for one year reduces GDP by 2% of full-employment output. Consider a four-year disinflation according to the following table: Year π π e (a) (2 points) What is the cyclical unemployment rate in each of the four years? (b) (1 points) By what percentage does output fall short of full-employment output each year? (c) (1 point) What is the sacrifice ratio for this disinflation? 1

2 2. (14 points) Assume that money demand is derived from the classical quantity equation of money. Assume also that the money demand is stable. These assumptions should affect your answers to all of the sub-questions to follow. (Hints: 1) The classical quantity equation of money is M V = P Y. 2) Stability of money demand implies that income velocity of money is constant. 3) Now, a very fat hint: you should be able to argue that the LM curve in this case is vertical.) (a) (2 points) Draw the LM curve for this case. Also, describe it in words. (b) (3 points) Suppose that the central bank targets the level of the real interest rate, i.e., adjusts the money supply when the real interest rate changes due to shocks in the economy. Assume also that the economy is currently at the equilibrium defined by the targeted level of the real interest rate, and the level of output below the long-run, fullemployment level of output. Draw the IS LM diagram and the full-employment LRAS curve. 2

3 (c) (3 points) Now suppose that policy advisers tell the government to cut taxes to reach this level of output faster. Draw the IS LM diagram and the full employment LRAS curve. What happens to the real interest rate, output, and investment? (d) (3 points) If the real interest deviates from its target because of the cut in taxes, what is the action of the central bank? Draw the IS LM diagram and the full employment LRAS curve. What happens to output, real interest rate, and investment due to this action of the central bank? (e) (3 points) Is fiscal policy effective in this case or not? aggregate output in this economy? What causes fluctuations in 3

4 3. (14 points) Consider a person deciding how to allocate her consumption over two periods. She has utility function U(C 1, C 2 ) = C 1/2 1 + β C 1/2 2, β 1. β is called the time discount factor, and measures how the person values current consumption versus future consumption, or, in other words, person s impatience. (The lower the β, the more impatient you are.) Assume that income in the first period is Y 1, income in the second period Y 2, and the real interest rate is r. (Hints: For this utility function MU 1 = C1, and MU 2 = 1 2 β 1 C2.) (a) (2 points) Write down an Euler equation for a consumer who is not borrowing constrained. (b) (1 point) Using the Euler equation only, write down the gross growth rate in consumption, i.e. C 2 C 1, in terms of the real interest rate and the time discount factor. (c) (1 point) If the impatience motive outweighs the speculative motive (i.e., if β (1 + r) < 1) will you observe an increasing/decreasing/or a constant consumption profile? (d) (1 point) If the impatience motive cancels out the speculative motive, i.e., when β (1 + r) = 1 will you observe an increasing/decreasing/or a constant consumption profile? 4

5 (e) (1 point) If the speculative motive outweighs the impatience motive, i.e., when β (1 + r) > 1 will you observe an increasing/decreasing/or a constant consumption profile? (f) (4 points) Now assume that the real interest rate is r = 1/20, and β = 1 1+r ; income in the first period is Y 1 = 81, and in the second period, Y 2 = 40. Solve for the optimal consumption in each period. Find the optimal savings in period 1. (g) (4 points) Now suppose that, instead, the agent can save at the rate r, but is unable to borrow at all. Solve for the optimal agent s consumption in each period. 5

6 4. (4 points) Indicate for each of the statements below whether it is true or false, or elaborate on a statement if it does not require the true/false judgment. Briefly explain. (a) (2 points) The Permanent Income Hypothesis (PIH) predicts that if one has a temporary, surprise, reduction in income, e.g., due to a temporary illness, consumption will be reduced by the magnitude of the income reduction. (b) (2 points) The PIH predicts that if one expects that his disposable income will increase in the next period (e.g., when the last mortgage payment is done) the adjustment of consumption between the current (t) and next periods (t + 1) will be sensitive to this predictable increase in the disposable income: i.e., C t+1 will be sensitive to a portion of Y t+1 due to the predicted increase in disposable income. 6

7 5. (BONUS OPTIONAL) (6 points) Consider a two period Fisher model. A person can borrow at the rate r b =1 (or 100% interest). A person can save at the rate of r s =0. The preferences are given by U(C 1, C 2 ) = ln(c 1 ) + ln(c 2 ). Borrowing or saving takes place only in period 1. (a) Write down an Euler equation of intertemporal utility maximization if the person is a saver. (Hint: an Euler equation links the marginal utility from consumption in period 1 and 2. Note that for this function MU 1 = 1 C 1, and MU 2 = 1 C 2.) (b) Write down an Euler equation of intertemporal utility maximization if the individual is a borrower. (c) Write down an intertemporal budget constraint for a saver with income in period one, Y 1, and income in period 2, Y 2. 7

8 (d) Write down an intertemporal budget constraint for a borrower with income in period one, Y 1, and income in period 2, Y 2. (e) Consider Mr. X with the income stream Y 1 = Y 2 = 32. What is his optimal first and second period consumption and his overall utility? Is he a borrower, a saver, or neither of them? (f) Consider Mr. Y with the income stream Y 1 = 0, and Y 2 = 64. What is his optimal first and second period consumption and his overall utility? (Hint: this consumer is a borrower since the utility from having consumption of zero in period 1 is prohibitively low: ln(0) = ). 8

9 (g) Consider Mr. Z with the income stream Y 1 = 24, and Y 2 = 40. What is his optimal first and second period consumption and his overall utility? Is he a borrower, a saver, or neither of them? (h) Compare lifetime utilities for Mr. X, Y, and Z. Who is better off and why? 9