Harvard. General Examination in Economic Theory, Spring 1989

Trawling the Zvi Griliches papers at the Harvard archives recently, I was able to retrieve the following copy of the economic theory general examination (transcribed below) for Harvard Ph.D. students from 1989. The Berlin Wall was still standing and the exam took place close to the time of the Tiananmen Square demonstrations. We see that it was business as usual at Harvard Square with respect to economic theory. Students were discouraged from answering more than one question (i.e., 25% of the exam) dealing with the history of economic theories.

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Some later general examinations

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HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

GENERAL EXAMINATION IN ECONOMIC THEORY
SPRING 1989

You have four hours. Spend approximately one hour on each question as all questions will receive equal weight in grading.

Start each question in a fresh blue book. Write your number and the number of the question answered on the cover of each blue book. DO NOT INDICATE YOUR NAME.

PART A
ANSWER BOTH QUESTIONS.

Question 1

Consider an investor with von Neumann-Morgenstern utility

${u_{i}=-e^{-a_{i}w_{i}}}$

where a_i > 0 is a constant and w_i is his wealth. Suppose this investor can choose to invest his initial wealth ${\bar{w}_{i} }$ in a set of n assets. One (n = 1) is a safe asset with return unity, and the others are risky. Let their returns be denoted (r₂,..,r_n) which are jointly distributed according to F(r₂,…., r_n). Take the prices of these assets, in units of the safe asset, to be (p₂,..,p_n).

Show that the optimal portfolio puts a share of initial wealth in each of the risky assets that is proportional to initial wealth.
Suppose that the risky assets are in fixed supply—for example, they are issued inelastically by firms in exchange for the investors’ initial wealth. The safe asset is elastically supplied.
Write the conditions on asset prices that determine an equilibrium on the asset markets.
Now suppose that some additional investors are added to the economy, each with positive initial wealth and the same form of utility function. Show that the asset prices change in such a way that the mean return to the portfolio of all risky assets decreases.
Take the limiting case in which the number of investors and their total wealth becomes very large in comparison with the fixed asset supplies. Comment on your results.

Question 2

Empirical evidence indicates that money and output are positively associated over business cycles. What are the strong points and shortcomings of each of the following three approaches to explaining this observation.

The Keynesian model.
The new classical model with incomplete information.
Models with “endogenous money”.

Part B
ANSWER TWO OF THE SIX QUESTIONS.
YOU SHOULD NOT ANSWER BOTH QUESTION 7 & 8.

Question 3

A government is concerned with providing disability insurance for its citizens. Each individual has a probability of ${\pi }$ of being disabled. If disabled, the individual has utility function v(c) where c is the individual’s consumption of society’s sole consumption good. If able, the individual has utility level u(c) – h where c is consumption (of society’s sole consumption good) and h is the number of hours worked by the individual. All jobs require H hours of work and one hour of work produces 1 unit of the consumption good.

Suppose the government can observe whether an individual is disabled or not. Suppose, also, that there are enough individuals in society so that exactly a fraction ${\pi }$ end up disabled. Set up and solve the government’s problem of finding the optimal disability insurance program. (HINT: Think of the government as directly picking a consumption level for the disabled c_d, and a consumption level for the able c_a, to maximize a representative individual’s expected utility subject to the overall societal resource constraint. Assume that able individuals always work.] What do consumption levels in the optimal program look like? Draw a picture in (c_a, c_d) space illustrating the solution.
Suppose now that the government cannot observe whether an individual is disabled or not. Thus, an individual can pretend to be disabled by choosing not to work.

(i) Set up the government’s problem under the assumption that it is desirable to have all able individuals choose to work.

(ii) Characterize the solution under the assumption that for any (c_a,c_d) such that
u(c_a) – H ≥ v(c_d) we have u'(c_a) < v'(c_d).
Draw a picture in (c_a,c_d) space illustrating the solution.

Suppose that disability is unobservable but that now there are two types of individuals. They differ only in their probability of being disabled. Suppose the fraction of high probability types is ${\lambda_{H}}$ , and that the government maximizes a utilitarian social welfare function. What do consumption levels with the optimal disability insurance program look like?
Draw a picture in (c_a, c_d) space illustrating the solution.

Question 4

Suppose that the representative household/producer attempts to maximize

${U=\int^{\infty }_{0} u\left( c\right) e^{-\rho t}dt }$

where c is consumption per household, ${\rho >0}$ , ${u\left( c\right) =\left( c^{1-\sigma }-1\right) /\left( 1-\sigma \right) }$ , and ${\sigma >0}$ . There is a constant number of these immortal households.

Production is ${y=c+\dot{k} =Ak}$ , where y is output per household, k is the capital stock per household, and ${A>\rho }$ . The capital stock, which does not depreciate, begins at time zero at the quantity k(0).

What is the household’s first-order optimization condition for consumption over time? What is the transversality condition?
What are the steady-state growth and saving rates in this economy? How do growth and saving behave in the transition to the steady state?
What condition on A and ${\rho}$ ensures that the transversality condition is satisfied? Is utility per household bounded in this case?

Question 5

MATCHING.

Suppose there are two hospitals, X, Y and two doctors A, B out of medical school. Hospitals and doctors have to be matched up for residency purposes. A matching is a pairing of hospitals and doctors,
e.g. {(X, A), (Y, B)}.

Hospitals or doctors have preferences for the doctors, or hospitals, they are matched with. For example, Hospital X could have preferences A >_X B, similarly X >_A Y, Y >_B X, etc. For simplicity, suppose there is never strict indifference.

Define a concept of Pareto Optimality for matchings.
Define a matching as stable if there is no pair of hospital-resident that can block it. A pair can block if both would [be] better off leaving their current partners and pairing off. For example, for the preference: A >_X B, A >_Y B, Y >_A X, X >_B Y the matching {(X, A), (Y, B)} is not stable because (Y,A) can block (why?) Is it Pareto optimal? Exhibit a stable matching for the above preferences.
Suppose that interns are allocated to hospitals as follows: Hospital X chooses an intern and Hospital Y gets the remaining one (call this Allocation System I). Show that this system guarantees that the obtained matching is Pareto optimal but not necessarily stable. (Give an example to make the last point.)
Consider now a different allocation system (call it Allocation System II): Each hospital (the sequencing does not matter) makes an offer to some intern. If each intern receives an offer, this determines the matching. If an intern receives two offers, s/he chooses which one to accept, the rejected hospital then pairs off with the remaining intern. Which matching would this system yield for the preferences:
A >_X B, A >_Y B, Y >_A X, X >_B Y? Show that for any pattern of preference this system yields a stable matching.
Suppose you now consider a System III which is identical to System II except that the roles of hospitals and interns are reversed: The offers are now made by interns and accepted or rejected by the hospitals. Show by example that the stable matchings generated by System II and III can be different. Discuss this difference in terms of which side of the “market” is relatively favored.

Question 6

Assume that a consumer maximizes

${E_{0}\sum^{\infty }_{t=0} \beta^{t} \frac{exp\left( -\rho c_{t}\right) }{-\rho }}$

subject to

${a_{t+1}=R\left( a_{t}+y_{t}-c_{t}\right) ,}$

${y_{t+1}=y_{t}+\epsilon_{t+1} ,\ \epsilon_{t+1} \sim N\left( 0,\sigma^{2} \right) }$

and a transversality condition to rule out Ponzi schemes. Impose ${\beta \in \left( 0,1\right) }$ and ${\rho >0}$ , and define a_t = nonhuman wealth at t, and y_t = endowment at t. R is the (gross) safe interest rate, and E₀ denotes expectation conditional on information available at time 0. The ${\epsilon^{\prime } s}$ are assumed to be identically and independently distributed.

Write the first-order condition associated with an optimum consumption path.
Show that, if ${\beta R=1}$ , the optimum consumption path satisfies

${c_{t+1}=c_{t}+\frac{\rho \sigma^{2} }{2} +\epsilon_{t} }$

Explain the presence of the second term on the righthand-side of this expression. [Hint: It is reminded that if ${x\sim N\left( m,2s^{2}\right)}$ , then ${E\ exp\left( x\right) =exp\left( m+s^{2}\right)}$ .]

Suppose that our consumer is representative of all other consumers in the economy. Compute the equilibrium safe interest rate.
What are the factors conducive to a high equilibrium safe interest rate?

Question 7

Economists of different schools of thought have emphasized the symmetry of resource allocation and income distribution with respect to the organization of production. As Samuelson put it in “Wages and Interest: A Modern Dissection of Marxian Economic Models,” (AER, 1957), “Remember that in a perfectly competitive market it really doesn’t matter who hires whom; so have labor hire ‘capital.’” Yet one of the more striking facts of contemporary economic life from Chicago to Moscow is that labor is generally the hired factor rather than the hiring factor. Give accounts of this phenomenon which roughly fit within neoclassical, Marxian, and Keynesian frameworks, respectively. What are the strengths and weaknesses of each?

Question 8

“Economic science is, and must be, one of slow and continuous growth. Some of the best work of the present generation has indeed appeared at first sight to be antagonistic to that of earlier writers; but when it has had time to settle down into its proper place, and its rough edges have been worn away, it has been found to involve no real breach of continuity in the science.” (A. Marshall).

Critically evaluate this view of the history of economics.

Source: Harvard University Archives. Papers of Zvi Griliches, Box 125, Folder “General Examination in Economic Theory and its History, undated.”

Some later general examinations

Spring 1991

Spring 1992

Fall 1992

HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS

GENERAL EXAMINATION IN ECONOMIC THEORY SPRING 1989

PART A ANSWER BOTH QUESTIONS.