Harvard. Graduate General Exams in Micro- and Macro-Theory. Fall 1993

The following general examination for macroeconomic theory (Spring 1993) has been transcribed from a collection of general exams at Harvard from the 1990s provided to Economics in the Rear-view Mirror by Abigail Waggoner Wozniak (Harvard economics Ph.D., 2005). Abigail Wozniak was an associate professor of economics at Notre Dame before being appointed a senior research economist and the first director of the Federal Reserve Bank of Minneapolis’ Opportunity & Inclusive Growth Institute. She is currently Vice-President of the Federal Reserve Bank of Minneapolis.

Economics in the Rear-view Mirror is most grateful for her generosity in sharing this valuable material.

Because the “Wozniak collection” is over 90 pages long, it will take time for all the exams to get transcribed. To date the following general examinations have been transcribed and posted:

Spring 1991
Microeconomics; Macroeconomics

Spring 1992
Micro- and Macroeconomics

Fall 1992
Micro- and Macroeconomics

Spring 1993
Microeconomics; Macroeconomics

_________________________

HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

GENERAL EXAMINATION
IN MICROECONOMIC THEORY
FALL 1993

Instructions:

You have FOUR hours.

Answer a total of FOUR questions, subject to the following constraints:

You must answer questions 1, 2, and 3. You must answer either question 4 or 5 (but not both).

Please use a separate blue book for each question.

Please put only your exam number on the blue book.

Good luck!

1. Indirect Utility (20 points)

This is a question on the indifference map of indirect utility functions.

There are two commodities. The utility function is $U(x_1, x_2)$ . Wealth is normalized to $w=1$ , so that the indirect utility function has the form $V(p_1, p_2)$ .

Assume that $U(x_1, x_2)$ is continuous and increasing in both arguments.

a) Draw an indifference map of the indirect utility function. Be sure that the axes are properly labelled, that the indifference curves have the proper shape and that the direction of increasing preference is clearly marked. (5 points)

b) Argue that if the utility function $U(x_1, x_2)$ is not quasi-concave then some indirect indifference curve will exhibit a kink. (5 points)

c) Suppose that an indifference curve of the indirect utility function has a piece that is a straight line. What does this correspond to in the direct utility function? Why? (5 points)

d) Consider the slope of an indifference curve of the indirect utility function. Interpret it in terms of the demand function. Justify your answer. (5 points)

2. (20 points)

Consider an industry in which there are two firms: an incumbent and a potential entrant. In each of two periods, total inverse demand in the industry is given by

$p = 5 - Q,$

where $Q$ is total output. The incumbent’s marginal cost of production $c_I$ can either be “high” ( $c_I = 2$ ) or “low” ( $c_I = 1$ ). The incumbent knows the value of $c_I$ , but the potential entrant knows only that the two values each have a 50% probability ex ante. In the first period, the incumbent chooses an output level $q_{I1}$ . The entrant can observe $q_{I1}$ (but not the incumbent’s profit). In the second period the two firms choose quantities $q_{I2}$ and $q_{E2}$ simultaneously. The entrant’s marginal cost $c_E$ is 2. In addition, if the entrant chooses $q_{E2} > 0$ , it incurs an “entry” cost of $\frac{1}{2}$ . The two firms each maximize their (expected) profits (the incumbent maximizes the expected sum of its first- and second-period profits). The model and all its parameters are common knowledge (the value of $c_I$ is not common knowledge, but the fifty-fifty prior distribution on 1 and 2 is common knowledge).

A. Suppose that the entrant concludes, after observing $q_{I1}$ , that the incumbent’s costs are low (i.e., the entrant attaches posterior probability 1 to $c_I = 1$ ). Show that, in this case, the entrant will not enter. That is, show that $q_{E2} = 0$ in the unique continuation equilibrium starting in period 2. What are the incumbent’s second-period output and profit in this continuation equilibrium?

B. Suppose that the entrant concludes, after observing $q_{I1}$ , that the incumbent’s costs are high ( $c_I = 2$ ). Show that, in this case, the entrant will enter, i.e., choose $q_{E2} > 0$ , in equilibrium. Find the second-period continuation equilibrium outputs and profits for both firms.

C. Using the results from parts (A) and (B), show that there exists a “separating” perfect Bayesian equilibrium for the two-period model in which (i) the incumbent chooses a “small” value of $q_{I1}$ if $c_I = 2$ and a “big” value if $c_I = 1$ , and (ii) the entrant enters if and only if $q_{I1}$ was small. In particular, show that $q_{I1}$ is at the high-cost monopoly level if $c_I = 2$ but strictly above the low-cost monopoly level if $c_I = 1$ .

3. Gross Substitute Excess Demand (20 points)

Consider an excess demand system $(f_1(p_1,\ldots,p_{L-1}),\ldots,f_{L-1}(p_1,\ldots,p_{L-1}))$ . There is an Lth commodity for which price is 1 and for which demand is not explicitly recorded.

a) How would you determine the demand for the Lth commodity? (5 points)

b) Suppose that the excess demand system for all L commodities satisfies the gross substitute property (when the price of a single commodity goes up the excess demand of each of the other commodities increases). Argue that the competitive equilibrium is then unique. (5 points)

c) Suppose that the excess demand functions for all L commodities are differentiable and that as in (b) the gross substitute property holds. What is the sign pattern induced in the Jacobian matrix of price effects? (5 points)

d) Suppose that the excess demand functions for all L commodities are differentiable and satisfy the gross substitute property. Suppose now that there is a shock (if this is helpful you can limit yourself to a small shock) the effect of which is to decrease the demand for commodities 1 to L−1 at every vector of prices. What should be the effect of this shock on the equilibrium prices of commodities 1 to L−1? Give a justification of your answer which is as rigorous as possible. (5 points)

Part IV:
Answer ONE of the following two questions:

4. Suppose there is another energy shock and the world-market price of oil goes from its present price of under $20 per barrel to $100. Analyze the short and long term effects on growth and distribution from neoclassical, Marxian, and Keynesian perspectives on the simplifying assumption that all energy is imported at the world-market price.

5. Milton Friedman has argued (“The Methodology of Positive Economics”) that a theory should be judged not on the realism of its assumptions, but on its predictive power. What are the principal criticisms of Friedman’s view? How might Friedman respond to the criticisms?

_________________________

HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

GENERAL EXAMINATION
IN MACROECONOMIC THEORY
FALL 1993

Instructions:

You have FOUR hours.

You will answer a total of SIX questions from four sections of the exam:

In section I, you must answer the one question given.
In section II, answer both questions.
In section III, answer one of the two questions.
In section IV, answer two of the three questions.

Please use a separate blue book for each question.

Please put only your exam number on the blue book.

Good luck!

PART I:

1. Suppose that each household-producer has the production function,

$Y_i = AK_i \cdot (G/Y)^{\alpha},$

where $A>0$ , $0<\alpha<1$ . $Y_i$ and $K_i$ are the ith producer’s output and capital stock, respectively, G is the total of government purchases and public services, and Y is aggregate output. The output of all producers is identical and can be used on a one-for-one basis for consumption, investment, or government services. Each household seeks to maximize utility, given by the standard Ramsey expression,

$\displaystyle U = \int_0^{\infty} \left(\frac{c^{1-\theta}-1}{1-\theta}\right) e^{-\rho t}\,dt,$

where $\theta>0$ , $\rho>0$ , and c is consumption per person. The economy is closed, depreciation and population growth are zero, and no technological change occurs. The government finances its purchases by a proportional tax on output at the rate $\tau$ :

$G = \tau Y,$

where $\tau$ is constant over time.

a. What is the growth rate of $Y$ ? Does the model exhibit a convergence property? Why or why not?

b. How does the growth rate depend on $\tau$ ? What condition holds if $\tau$ is chosen to maximize the growth rate?

c. Are the outcomes Pareto optimal if $\tau$ is set to maximize the growth rate? How does this result relate to the usual view that lump-sum taxation is superior to proportional taxation?

PART II:
Answer BOTH questions:

2.—

a.) Under what assumptions does the permanent income hypothesis predict that consumption should follow a random walk?

b.) Derive explicitly the random walk of result (a).

c.) Discuss briefly what you know about the empirical evidence on the random walk.

d.) Consider a pre-announced future temporary tax cut. What are the effects on consumption of this policy before, during, and after the tax cut? Consider the cases of: (1) the life cycle model; (2) the life cycle model with bequests, (3) the Keynesian consumption function; (4) a life cycle model with liquidity constraints.

3. Consider a monopolistic trade union, that can unilaterally set the nominal wage. The union’s utility function is:

$\displaystyle U = \left(\frac{w}{p}\right)^{\frac{1}{2}} + \beta N$

Where: w is the nominal wage
p is the price level
N is employment
and $\beta>0$ is a parameter.

$\displaystyle N = n_0 - \alpha\frac{w}{p},$

where $n_0>0$ and $\alpha>0$ are parameters

1) Suppose first that p is fixed, and compute what is the nominal wage chosen by the union.

2) Suppose that a monetary authority can influence the price level with monetary policy, and suppose that the union can change the nominal wage at any time but with a small cost; that is, any change in the nominal wage has a very small administrative cost for the union. Discuss the effects of monetary policy in this model.

PART III:
Answer ONE of the following two questions:

4. Consider the case of a central bank conducting monetary policy in a standard fractional reserve banking system. The central bank can set either the short-term interest rate or the quantity of bank reserves (but not both simultaneously) to whatever level is chosen. What factors determine the central bank’s optimal choice of which one to set, under each of the following sets of circumstances?

(a) The central bank is trying to come as close as possible to hitting a target for the money stock (defined as public holdings of bank deposits).

(b) The central bank is trying to come as close as possible to hitting a target for real output.

In answering each part, show explicitly what difference (if any) it makes whether the economy is subject to aggregate supply shocks or whether demand-type disturbances are the only ones the economy faces.

5. Under what conditions can a central bank’s prior announcements of its policy actions influence the macroeconomic effect of those actions? If the central bank is free to conduct a politically independent monetary policy, what determines whether it will then actually implement the actions it has announced? Be explicit about the assumptions on which your answer relies.

PART IV:
Answer TWO of the following THREE questions:

6. Consider an unemployed worker searching for work. Each period she draws a wage offer w from a distribution $F(w)$ . Assume that there are no quits or layoffs so once the worker accepts a job, she works at it forever. Assume that wages in future periods are discounted by $\beta$ .

(a) Write a Bellman equation that characterizes the value of an optimal search policy given a current offer of w.

(b) The optimal policy is to set a reservation wage. Explain what this means and write an equation that implicitly defines the reservation wage.

(c) Suppose that the worker only receives a wage offer with probability $\phi$ each period. Explain in words or equations what happens to the reservation wage and why?

(d) Suppose that the worker receives two wage offers each period. Explain in words or equations what happens to the reservation wage and why?

7. Consider the following open economy macroeconomic model

(1) $Y = f(K, L)$
(2) $W/P = f_L$
(3) $Y = C(Y) + G + NX(e, Y)$
(4) $e = EP^*/P$
(5) $M/P = L(Y)$

Where asterisks denote foreign variables, NX is net exports, E is the nominal exchange rate and all other variables have their usual interpretations.

(a) What is the sign of $\partial NX/\partial e$ ? Explain.

(b) Suppose that workers and firms set nominal wage contracts:

$W = \bar{W}$

What is the effect of a change in $M$ or in $G$ on $Y$ ?

$W = \bar{w}P^{\alpha}(EP^*)^{1-\alpha}$

What is the effect of a change in $M$ or in $G$ on $Y$ ?

8. Suppose that price in a market is initially equal to zero, but is expected to rise linearly with time:

$p(t) = \alpha t.$

Suppose that you have a machine that costs k to install and once installed produces a unit flow of output. Time is continuous and the date zero present value of a dollar at date t is $e^{-rt}$ .

(a) At what date t should you have the machine installed in order to maximize profits?

(b) What is the present value of profits at the optimal time of investment?

(c) What is the value of Tobin’s Q at the optimal time of investment? What did Tobin argue should be the relationship between Q and investment? Speculate about what is special about the case considered here.

Source: Department of Economics, Harvard University. Past General Exams, Spring 1991-Spring 1999, pp. 84-88. Private copy of Abigail Waggoner Wozniak.

Image Source: Harvard University Economics PhD graduates from May 2025.

GENERAL EXAMINATION IN MICROECONOMIC THEORY FALL 1993

GENERAL EXAMINATION IN MACROECONOMIC THEORY FALL 1993

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GENERAL EXAMINATION
IN MICROECONOMIC THEORY
FALL 1993

GENERAL EXAMINATION
IN MACROECONOMIC THEORY
FALL 1993