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Harvard. General Exams in Microeconomic and Macroeconomic Theory. Spring, 1992

 

The following general examinations for microeconomic and macroeconomic theory (Spring 1992) have 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. Economics in the Rear-view Mirror is most grateful for her generosity in sharing this valuable material.

The “Wozniak collection” is over 90 pages long, so it will take some time for all the exams to get transcribed.

Transcriptions are also available for: 

Spring 1991. General Examinations in Microeconomic Theory and Macroeconomic Theory.

Fall 1992. Microeconomic Theory and Macroeconomic Theory.

___________________________

HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

Economics 2010b: FINAL EXAMINATION and
GENERAL EXAMINATION IN MICROECONOMIC THEORY

Spring Term 1992

For those taking the GENERAL EXAM in microeconomic theory:

  1. You have FOUR
  2. Answer a total of FIVE questions subject to the following constraints:
    *at least ONE from part I;
    *at least TWO from Part II;
    *EXACTLY ONE from Part III.

For those taking the FINAL EXAMINATION in Economics 2010b (not the General Examination):

  1. You have THREE HOURS
  2. Answer a total of four questions subject to the following constraint:
    *DO NOT ANSWER ANY questions from Part I;
    *at least TWO from Part II;
    *at least ONE from Part III.

 

PLEASE USE A SEPARATE BLUE BOOK FOR EACH QUESTION

PLEASE PUT YOUR NAME (OR EXAM NUMBER) ON EACH BOOK

 

PART I (Questions 1 and 2)

QUESTION 1

A consumer in a three good economy (goods denoted x1, x2 and x3; prices by p1, p2, p3) with wealth level w>0 has demand functions for commodities 1 and 2 given by:

\begin{array}{l}{{x}_{1}}=100-5\,\,\frac{{{p}_{1}}}{{{p}_{3}}}+\beta \,\,\frac{{{p}_{2}}}{{{p}_{3}}}+\delta \,\,\frac{w}{{{p}_{3}}}\\{{x}_{2}}=\alpha -\beta \,\,\frac{{{p}_{1}}}{{{p}_{3}}}+\gamma \,\,\frac{{{p}_{2}}}{{{p}_{3}}}+\delta \,\,\frac{w}{{{p}_{3}}}\end{array}

Where Greek letters are non-zero constants.

i) Indicate (but don’t actually do it!) how to calculate the demand for the third good, good 3.

ii) Are the demand functions for x1 and x2 appropriately homogeneous?

iii) Calculate the numerical values of \alpha ,  \beta , \gamma . (Hint: what are the various restrictions on the consumer’s demand function and on the relationships between various demand functions? Have you made use yet of all of them?)

iv) Given your results above, draw for a fixed level of x3, the consumer’s indifference curve in the x, y plane.

v) What does your answer to (iv) imply about the form of the consumer’s utility function u(x1, x2, x3)?

 

QUESTION 2

Consider a Cournot duopoly. The inverse demand function is

p=100 – (q1 +q2)

where qi, i= 1,2, is the production of firm i (Note: if q1+q2 ≥100 then the price is zero). Marginal cost for the two firms is constant and equal to zero.

  1. Show that given the production qi≤100 of firm i the optimal reaction of firm j (j \ne i) is qj = ½ (100-qi). Use this to compute the Cournot-Nash equilibrium. Draw also the reaction function.
  2. Argue that it can never be a best response for any firm to supply more than x1=50. Argue then that if no firm will ever supply more than 50 then no firm will ever supply less than x2=25.
  3. Continue the above recursion and determine the limit of xt. Interpret in terms of the concept of rationalizability (define terms).

 

PART II (Questions 3, 4 and 5)

QUESTIONS 3

Consider the following 2 person, 2 commodity general equilibrium model. Individual 1 is risk neutral and has a utility function

u1(x,q) = x + y

Individual 2 is risk averse and has von Neumann-Morgenstern utility function

u2(x,y) = x½ + y.

Individual 1’s endowment of good y is {\bar{Y}}, he has no x. There are three states of nature and the endowment of individual 2 depends on the state. The levels of endowment of x are x1, x2 and x3. He has no y.

  1. Suppose the subjective beliefs are that each state is equally likely. Find a complete-market Arrow-Debreu equilibrium.
  2. Suppose there is a market for the delivery of x and y only uncontingently. Find the equilibrium. Is it efficient?
  3. Suppose that before any trade takes place, all individuals will be told whether or not state 1 will arise. They are not able to distinguish between states 2 and 3 at this stage. Find the equilibrium as it depends on the information disseminated.
  4. Calculate the expected utility attained in the equilibrium of part c), ex ante (i.e. before any announcement is known), using the fact that the probability that the individuals will be told that the state is 1 is 1/3, and that it is not 1 is 2/3. Under what conditions (on x1, x2 and x3) is the information socially valuable. Comment.

 

QUESTION 4

[This problem concerns a firm which is not explicitly optimizing anything. We just want to study the stability of its profitability and debt-equity structure.]

At each instant of time, the firm employs only a single factor, capital, in an amount K. Gross profits are f(K), concave and increasing in K. It has a certain level of borrowing B on which it owes rB in interest costs; r is fixed. Net profits therefore are

\pi = f(K) – rB

Some of the profit is distributed in the form of dividends, D. The rest of it is held as retained earnings R. Suppose the firm follows the “dividend policy”

\begin{array}{l}R=\alpha \pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,0\le \alpha \le 1\\D=\left( 1-\alpha  \right)\pi \end{array}

The change in firm’s stock capital is equal to the sum of its level of retained earnings R and new borrowing B minus depreciation \delta K (\delta > 0)

\dot{K}=R+\dot{B}-\delta K

Its level of new borrowing responds positively to the excess of current net profits above a target level \pi *, according to

\dot{B}=\beta \left( \pi -\pi * \right)

\beta \ge 0.

In an equilibrium, K, B, \pi , R and D are all constant.

1) Assume we have an equilibrium for fixed parameters \alpha and \beta . For what values is it locally stable?

2) Equity in the firm is, by definition, K – B. Find the steady-state debt-equity ratio, in terms of the parameters and the production function.

Assuming that the equilibrium is stable,

3) What is the effect of a small increase in \pi * on the equilibrium debt-equity ratio?

 

QUESTION 5

There are n men and n women. Each man has an ordering over the set of women, and each woman has an ordering over the set of men. (indifference is not allowed)

  1. Show that in any Pareto optimal system of pairing men and (i.e. a pairing that cannot be dominated by another system), someone must get his/her first choice.
  2. Consider the following system: Each man proposes to the woman whom he most prefers. Women receiving more than one proposal accept the one they like the best. (Women receiving exactly one proposal accept it.)
    Next, all men whose proposals were rejected, propose to the woman they most prefer from among those who have not yet accepted proposals. The rules for acceptance are as above. This continues until all pairings have been completed.
    Is the result a Pareto optimum?
    Is it in the core of this game, suitably defined? (supply your own definition here).
  3. Show by example how the system could be manipulated by a man who knew the preferences of all other men and who could profess false preferences.

 

PART III (ANSWER EXACTLY 1 QUESTION) (Questions 6 and 7)

QUESTION 6

What does “competition” mean in the context of neoclassical theory? To what extent are neo-Marxian and neo-Keynesian theories compatible with the neoclassical idea of competitive markets? What alternatives to neoclassical adjustment processes characterize Marxian and Keynesian theories?

 

QUESTION 7

Paul Samuelson once urged us “(r)emember that in a perfectly competitive market it really doesn’t matter who hires whom; so have labor hire ‘capital’”. (“Wages and Interest: a Modern Dissection of Marxian Economic Models.” American Economic Review, 1957) Does it really not matter who hires whom in neoclassical theory? Is the same true in a Marxian framework? A Keynesian framework?

 

Source: Department of Economics, Harvard University. Past General Exams Spring 1991-Spring 1999, pp. 60-66. Copy provided to Economics in the Rear-view Mirror by Abigail Wozniak.

___________________________

HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

Economics 2010d: FINAL EXAMINATION
and
GENERAL EXAMINATION IN MACRO ECONOMIC THEORY

Spring Term 1992

FINAL EXAMINATION

Instructions for all Economics Department graduate students:

The examination will last four hours.

Answer all three parts of the examination (Parts I, II and III). Within each part, be sure to answer two questions (so that, in all, you will answer six questions).

Use a separate bluebook for each question. Clearly indicate the question number and your identification number on the front of each bluebook.

Do not indicate your name on any bluebook you submit.

Instructions for all other students:

The examination will last three hours.

Answer Parts II and III only. Within each part, be sure to answer two questions (so that, in all, you will answer four questions).

Use a separate bluebook for each question. Clearly indicate the question number and your name on the front of each bluebook.

 

PART I:

  1.  Consider the following q-theory of investment:

(1) I=I\left( q \right)\,\,\,\,\,\,I\acute{\ }>0

(2) \dot{K}=I-\delta K

(3) r={\left( \dot{q}+\pi  \right)}/{q}\;

(4) \pi =\pi \left( K \right)\,\,\,\,\,\,\,\,\,\pi \acute{\ }

Where I is investment, q is Tobin’s q, K is the capital stock, \delta is the depreciation rate, r is the required rate of return, and \pi is the profit from owning one unit of capital.

  1. What is Tobin’s q?
  2. Interpret each equation.
  3. Write the model in terms of two variables and two laws of motion.
  4. What is the state (non-jumping) variable, and what is the costate (jumping) variable?
  5. Draw the phase diagram for this model, including arrows showing the dynamics, the steady state, and the stable path.
  6. Because of an increase in acid rain, the depreciation of capital (\delta ) suddenly increases. Assume the change is permanent. Compare the old and new steady states.
  7. Describe the transition between the old and the new steady states.
  8. Suppose now that (because of strong environmental policies) the increase in \delta is only temporary. Describe the effects.

 

Question 2 (Answer both Parts)

  1. (20 minutes)
    The Solow model assumes a constant gross saving rate, whereas the neoclassical growth model determines the saving rate through household optimization.

    1. Explain the forces in the neoclassical growth model that make the saving rate rise or fall as an economy develops.
    2. How does this behavior of the saving rate influence the convergence rate—that is, the per capita growth rate of a poor country relative to a rich country? (Assume here that all countries are closed and have the same underlying parameters of technology and preferences.)
  2. (20 minutes)
    “The imperfection of private credit markets implies that a cut in lump-sum taxes, financed by a budget deficit, affects the economy. In particular, in a full-employment setting,

    1. a larger deficit crowds out private investment, and
    2. a budget deficit is a “bad idea.”
      Discuss and comment.

 

PART II

Answer any two of the following three questions. Be sure to use a separate bluebook for each answer.

  1. A familiar suggestion to central banks is to conduct monetary policy by “targeting” nominal income. Distinguish (a) the argument for targeting nominal income on the ground that this is the measure of economic activity the central bank should ultimately care about affecting, from (b) the argument for targeting nominal income on the ground that doing so will best enable the central bank to affect some other aspect(s) of economic activity in an optimal way. In your analysis of argument (a), be specific about the preference that this notion of nominal income targeting implies with respect to real income and prices separately, and say whether you think these preferences are sensible (and why). In your analysis of (b), this way would lead to a better outcome than focusing monetary policy directly on those aspects of economic activity that the central bank ultimately seeks to affect.
  2. “When money, interest-bearing government debt and productive capital are imperfect substitutes in investors’ portfolios, the economy can achieve a stable growth equilibrium only if the part of the government’s primary deficit (that is, its deficit exclusive of interest payments) that it finances by issuing new debt is equal to some particular value in relation to the economy’s size, where the deficit/income ratio is uniquely determined by specific parameters describing economic behavior. If the government attempts to run a larger debt-financed primary deficit, its growing debt will crowd out private capital, which in turn will raise the real interest rate, which will cause the government’s interest payments to rise, which will make its outstanding debt grow even faster—all in an explosive way—and vice versa if the deficit/income ratio is too small.” Do you agree or disagree with this statement? Why? Be specific about the assumptions underlying your answer.
  3. Under what conditions would the government’s choice among alternative forms of interest-bearing debt, to finance a given non-monetized deficit, affect (a) the aggregate level of real economic activity, and/or (b) the composition of real economic activity, and/or (c) the price level? Why? Under what conditions would this choice affect none of (a), (b) or (c)? Be explicit about the assumptions you make in describing both sets of conditions.

 

PART III

QUESTION 6

Consider an economy with overlapping generations of identical two-period lived consumers who work in the first period of their life and consume in the second period. The utility function of a consumer representative of the generation born at time t is

{{U}_{t}}={{c}_{t+1}}-\frac{1}{2}a_{t}^{2},

Where ct+1 denotes the consumption taking place at t +1, of a consumer born at t and a her labor supply at time t. Output is produced with labor using a linear production function: one unit of labor yields one unit of output. Output is non-storable. Population is constant.

  1. Show that the competitive equilibrium of this economy is Pareto sub-optimal. Explain why.
  2. Propose two schemes to eliminate this inefficiency. Show precisely how to implement them and compare the resulting welfare levels to the welfare level achieved by the economy described in the previous question. [Restrict yourself, for simplicity, to schemes yielding constant consumption and employment over time.]
  3. What is the channel through which your two proposed schemes affect employment? How does this channel compare with the transmission mechanism of i) real business cycle models, and ii) Keynesian models?

QUESTION 7

You are asked to testify in the Senate about the advisability of phasing out the current U.S. social security scheme. Sketch the argument that you would make in favor of or against this phasing out, knowing that the Senators whom you are addressing have no tolerance for mathematics.

QUESTION 8

In a large open economy, what is the equilibrium effect on national saving and investment of a shift in desired domestic saving (originating, say, from a change in how patient domestic consumers are) or in desired domestic investment (coming, for instance, from a permanent productivity shock)? Does your answer shed any light on the debate about how well international capital markets operate?

 

Source: Department of Economics, Harvard University. Past General Exams Spring 1991-Spring 1999, pp. 67-71. Copy provided to Economics in the Rear-view Mirror by Abigail Wozniak.