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Harvard. General Examination in Microeconomic Theory. Spring, 1991

 

 

The following general examination in microeconomic theory (Spring 1991) comes from a collection of nine years’ worth of general exams at Harvard from the last decade of the 20th century shared by Abigail Waggoner Wozniak (Harvard economics Ph.D., 2005). Abigail Wozniak was an associate professor of economics at Notre Dame before she was appointed senior research economist and the first director of the Federal Reserve Bank of Minneapolis’ Opportunity & Inclusive Growth Institute. Economics in the Rear-view Mirror is grateful for her generosity in having a copy sent here for eventual transcription. 

The “Wozniak collection” is over 90 pages long, so it will take some time for all the exams to appear. But for now I can at least promise that the Spring 1991 macroeconomics examination will be posted soon.

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HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS
GENERAL EXAMINATION IN MICROECONOMIC THEORY
SPRING, 1991

Instructions:

For those taking the generals in microeconomic theory:

  1. You have FOUR hours.
  2. Answer a total of five questions subject to the following constraints:
    at least two from Part A;
    at least one from Part B;
    exactly one from Part C.

For those taking the final exam in Economics 2010B, but not the generals:

  1. You have three hours and ten minutes
  2. Answer a total of four questions subject to the following constraints:
    at least two from Part A;
    do not answer any questions from Part B;
    at least one from Part C.

Please use a separate blue book for each question, and please put your name (or number) on each book.

Unless otherwise specified, the parts within each question will be equally weighted.

 

PART A (questions 1, 2,3)

  1. Consider an economy composed of a large number of consumers who differ in their tastes and endowment. The preferences of each individual are described by u={{x}^{\alpha }}{{y}^{\beta }}{{z}^{\left( 1-\alpha -\beta \right)}}, where \alpha ,\beta >0,\,\,\,\,\,\alpha +\beta <1 and endowments are \omega =\left( {{\omega }_{x}},{{\omega }_{y}},{{\omega }_{z}} \right)
    1. Write the excess demand function for good x and y as a function of the prices of the goods p= (px, py, pz), using a knowledge of the statistics of the distribution F.
    2. Find an expression for the equilibrium price system.
    3. Show that it is unique.
    4. Suppose that the price system p is out of equilibrium at time t = 0 and that for each commodity price adjusts proportionately to excess demand. The constant of proportionality dk>0 for k= x, y, z is known to be positive but is not known to you. Can you nevertheless be sure that the price system will converge to the equilibrium found in b)? Explain
  2. Consider a firm with production function f(x) which is uncertain about the price of its product p. This uncertainty is summarized in the distribution function G(p). The firm wants to maximize its expected profits. The workers of this firm, represented by their union, want to make a contract with the firm that will guarantee them a certain level of expected utility. The union’s von Neumann-Morgenstern utility function is u(c,x), where c is the total payment received by the union and x is the quantity of labor provided by the union to the firm.
    1. Show that a contract in which c and x are specified in advance of the firm’s learning p is worse than one in which c and x can be chosen after p.
    2. Assume that the realization of p from the distribution G is observable to both the firm and the union and that contracts c(p), x(p) specifying the payment and employment as a function of p are possible. Describe an optimal contract mathematically as it depends on f, u and G.
    3. Now suppose that only the firm can observe p and that it must propose a contract c(x) which gives the compensation level as a function of labor demanded, and that the firm retains the right to choose x (and hence c) after the value of p is known. Write the problem of finding an optimal contract. What are the constraints?
    4. Now suppose that there are only two possible prices p, call them pH,pL. Moreover, assume that x is a normal good in the union’s utility function. Show that the constraints found in part c are binding. Relative to an efficient situation, compare the value of the marginal product of labor to the marginal rate of substitution between x and c in the union’s utility function.
  3. Consider a region consisting of three towns in which a jail must be built. The towns are configured as shown below:

1

2

3
    1. The towns do not want the jail located in their borders. Moreover they do not want it in the town adjacent to themselves. Assume that utilities are quasi-linear, so that we can speak of the willingness to pay to avoid having the jail in or near a given town in units of money, which is transferable among the towns. Each town has a willingness to pay for avoiding having the jail in its borders of 10. Their willingness to pay for having it in an adjacent town are:
Town Willingness to pay to avoid jail in a neighbor
1 5
2 3
3 0

Where should the jail be located on efficiency grounds?  (.15)

    1. Assume that the towns could freely bargain about the location of the jail, and that they can make deals involving monetary compensation among themselves. Describe the set of such arrangements that are robust against defection or recontracting. (.60)
    2. Now assume that each town is populated by an identical number of citizens with identical utility functions such that their individual willingnesses to pay sum up to the town willingness to pay as given in part a. Could some type of competitive market be arranged to produce an efficient outcome? How would you organize it? (.25)

 

PART B (questions 4 and 5)

  1. There are two firms, an incumbent and a potential entrant. To produce at all, a firm must install at least ko units of capacity. The cost of capacity is q (>0) per unit. A firm that installs k units of capacity (k ≧ko) can produce up to k units of output. The marginal cost of output is c. Inverse demand is given by p = a – bx, where x is total output (the sum of the two firms’ outputs). The incumbent moves in period 1 and selects its capacity level kI. The entrant then moves in period 2 and either chooses not to enter or else selects a capacity kE. Finally, the two firms select output levels simultaneously in period 3. Subgame-perfect equilibrium is the solution concept.
    1. What level of capacity must the incumbent install in order to deter entry? (Note: you need just set up the equation; it is not necessary to solve it explicitly). Will the incumbent ever choose to install capacity that it does not use in equilibrium?
    2. If the incumbent chooses to accommodate entry, how much capacity will it install? Again, just set up the maximization).
    3. Under what conditions on the parameter values will the incumbent act to deter entry rather than accommodate it?
    4. Now suppose that demand is random and that the uncertainty is not resolved until the beginning of period 3.
      Specifically, suppose that inverse demand is a-bx+\varepsilon  or  a-bx-\varepsilon with equal likelihood where \varepsilon is “small”. Assume that firms are risk neutral. Does the uncertainty increase or decrease the capacity needed to deter entry? (It should not be necessary to perform any computations to answer this question).
  2. Suppose we are in a three commodity market. Good 3 is a numeraire and the demand functions for the other two goods are:

x1(p,w)= a1+ b1p1+ c1p2+ d1p1p2
x2(p,w)= a2+ b2p1+ c2p2+ d2p1p2

    1. Note that the demand for goods x1, x2 does not depend on wealth. Write down the most general class of utility functions whose demand has this property.
    2. Argue that if the above demand functions are generated from utility maximization then the values of the parameters cannot be arbitrary. Write down as exhaustive a list as you can of the restrictions implied by utility maximization. Justify your answer.
    3. Suppose that the conditions identified in (ii) hold. The initial price situation is
      p= (p1, p2) and we consider a change to p´= (p´1, p´2). Define the concept of consumer surplus generated in going from p to p´.
    4. Let the value of the parameters be
      a1= a2= ½ , b1= c2= -1, c1= b2= ½, d1= d2= 0. Suppose the initial price situation is p= (1,1). Compute the consumer surplus for a move to p´ for each of the following three cases: (1) p´= (2,1), (2) p´= (1,2), (3) p´= (2,2). Denote by CS1, CS2, CS3 the respective answers. Under which condition will you have CS3= CS1+ CS2. Discuss.

 

PART C  (questions 6 and 7)

  1. In both neo-Marxian and neo-Keynesian theories there is, to paraphrase Schumpeter, no growth without profit and no profit without growth. But the interaction between profit and growth is different in the two theories. What are the most important differences?
  2. Consider the problem of predicting the shots made by an expert billiard player. It seems not at all unreasonable that excellent predictions would be yielded by the hypothesis that the billiard player made his shots as if he knew the complicated mathematical formulas that would give the optimum directions of travel, could estimate accurately by eye the angles, etc., describing the location of the balls, could make lightning calculations from the formulas, and could then make the balls travel in the direction indicated by the formulas.
    It is only a short step from these examples to the economic hypothesis that under a wide range of circumstances individual firms behave as if they were seeking rationally to maximize their expect returns (generally if misleading called “profits”) and had full knowledge of the data needed to succeed in this attempt; as if, that is, they knew the relevant cost and demand functions, calculated marginal costs and marginal revenue from all actions open to them, and pushed each line of action to the point at which the relevant marginal coast and marginal revenue were equal. (Milton Friedman, “The Methodology of Positive Economics,” in Essays in Positive Economics, pp. 21-22).

What are the most important criticisms of Friedman’s position? What would be lost for economics, normative as well as positive, if the maximization hypothesis were abandoned?

 

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

Image Source: Abigail Wozniak webpage at the University of Notre Dame.