Harvard. General Examination in Microeconomic Theory. Spring, 1993

Economics in the Rear-view Mirror has been provided a copy of nearly all the 1990s general exams in micro- and macroeconomic theory from Harvard through the collegial generosity of Minneapolis Fed economist Abigail Wozniak. With this post you now have the Spring 1993 graduate general exams in microeconomic theory.

While these exams lie outside of my personal comfort zone as a historian of economics (1870-1970), for fledgling historians of economics of today and tomorrow these are indeed legitimate historical artifacts definitely worth transcription. I am rather slow in digitizing them because transcription of mathematics for this blog requires latex inserts. Latex expressions involve considerably longer roundabout production than the application of my talents for touch-typing/OCR to non-mathematical text. Patience! The Rest is Yet to Come!

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Previously Transcribed Harvard Graduate General Exams

Spring 1989: Economic Theory

Spring 1991: Microeconomics; Macroeconomics

Spring 1992: Micro- and Macroeconomics

Fall 1992: Micro- and Macroeconomics

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Graduate Microeconomic Theory Sequence, 1992-93

Economics 2010a. Economic Theory

Michael D. Whinston and Eric S. Maskin

Covers the theory of individual behavior including the following topics: constrained maximization, duality, theory of the consumer, theory of the producer, behavior under uncertainty, consumer choice of financial assets, externalities, monopolistic distortions, game theory, oligopolistic behavior, asymmetric information.

Prerequisite: Economics 2030 or equivalent; can be taken concurrently.
Half course (fall term). Tu., Th., 10-11:30.

Economics 2010b. Economic Theory

Andreu Mas-Colell and Stephen A. Marglin

General equilibrium, stability, pure and applied welfare economics, uncertainty, descriptive and optimal growth theory, income distribution, methodology.

Prerequisite: Economics 2010a.
Half course (spring term). Tu., Th., 10-11:30.

Source: Harvard University, Faculty of Arts and Sciences. Courses of Instruction 1992-1993, p. 248.

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HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

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

Spring Term 1993

For those taking the GENERAL EXAM in microeconomic theory:

You have FOUR hours.
Answer a total 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):

You have THREE HOURS
Answer a total of four questions subject to the following constraints:

— 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 EXAM NUMBER ON EACH BOOK

Part I (Questions 1 and 2)

Suppose that there are J firms producing good differentiable cost function c(w,q) where w is a vector of input prices and q is the firm’s output level. The differentiable aggregate demand function for good is x(p), where p is good ’s price. Assume c(w,q) is strictly convex in q and that x´(p)≤0. Also assume that partial equilibrium analysis is justified.
1. Suppose that all factor inputs can be adjusted in the long-run, but that input k cannot be adjusted in the short-run. Suppose that we are initially at an equilibrium where all inputs are optimally adjusted to the equilibrium level of output $\bar{q}$ and factor prices $\bar{w}$ so that, letting $z_{k}\left( \bar{w} ,\bar{q} \right)$ be the conditional factor demand function for factor k, we have $z_{k}=z_{k}\left( \bar{w} ,\bar{q} \right)$ . What can be said about the short-run versus long-run output response of the firm to a differential change in the price of good $\ell$ ? What does this imply about the short-run versus long-run equilibrium response of p to a differential exogeneous shift in the demand function (hold the number of firms fixed in both cases)? (Hint: Define a short-run cost function $c_{s}\left( w,q,z_{k}\right)$ giving the minimized cost of producing output q given factor prices w when factor k is fixed at level $z_{k}$ ).
2. Now suppose that all factor inputs can be freely adjusted. Give the weakest possible sufficient condition, stated in terms of marginal and average costs and/or their derivatives, that guarantees that if the price of input $k\left( w_{k}\right)$ marginally increases, then firms’ equilibrium profits decline for any demand function $x\left( \cdot \right)$ with $x^{\prime }\left( \cdot \right) \leq 0$ . Show that if your condition is not satisfied, then there exist demand functions such that profits increase when the price of input k increases. What does your condition imply about the firm’s conditional factor demand for input k?
A. Consider a one-shot two-player game in which player 1 has a set of possible moves M₁ (with n₁ elements) and player 2 has a set of possible moves M₂ (with n₂ elements). Players move simultaneously. How many strategies does each player have?

B. Now suppose that the game is changed so that player 1 moves before 2, and 2 observes 1’s move, but that the game is otherwise the same as before. That is, the sets of moves are still M₁ and M₂, and player 1’s and 2’s payoffs as functions of moves $\psi_{1} \left( m_{1},m_{2}\right) \text{ and } \psi_{2} \left( m_{1},m_{2}\right)$ , respectively, are unchanged. How many strategies does each player have in the altered game?

C. The game of part B may have multiple subgame-perfect equilibria. Show, however, that, if this is the case, there exist two pairs of moves $\left( m_{1},m_{2}\right)\text{ and } \left( m^{\prime }_{1},m^{\prime }_{2}\right)$ (where either $m_{1}\neq m^{\prime }_{1}\text{ or } m_{2}\neq m^{\prime }_{2}$ ) such that either

(*) $\psi_{1} \left( m_{1},m_{2}\right) =\psi_{1} \left( m^{\prime }_{1},m^{\prime }_{2}\right)$
or
(**) $\psi_{2} \left( m_{1},m_{2}\right) =\psi_{2} \left( m^{\prime }_{1},m^{\prime }_{2}\right)$ .

D. Suppose that, for any two pairs of moves $\left( m_{1},m_{2}\right)\text{ and } \left( m^{\prime }_{1},m^{\prime }_{2}\right)$ such that $m_{1}\neq m^{\prime }_{1}\text{ or } m_{2}\neq m^{\prime }_{2}$ , (**) is violated, i.e., $\psi_{2} \left( m_{1},m_{2}\right) \neq \psi_{2} \left( m^{\prime }_{1},m^{\prime }_{2}\right)$ . In other words, player 2 is never indifferent between pairs of moves. Suppose that there exists a pure-strategy equilibrium in the game of part A in which $\pi_{1}$ is player 1’s payoff. Show that in any subgame-perfect equilibrium of part B, player 1’s payoff is at least $\pi_{1}$ . Would this conclusion necessarily hold for any Nash equilibrium of part B? Why or why not?

E. Show, by example, that the conclusion of part D may fail if either

(a) $\psi_{2} \left( m_{1},m_{2}\right) =\psi_{2} \left( m^{\prime }_{1},m^{\prime }_{2}\right)$ holds for some pair $\left( m_{1},m_{2}\right) ,\left( m^{\prime }_{1},m^{\prime }_{2}\right)$ with $m_{1}=m^{\prime }_{1}\text{ and } m_{2}=m^{\prime }_{2}$ ; or

(b) we replace the phrase “pure-strategy equilibrium” with “mixed-strategy equilibrium.”

Part II (Questions 3, 4, & 5)

QUESTION 3 (General Equilibrium with Gorman Preferences)
(20 points)

Suppose you have a population of consumers i = 1,….,I. Ever consumer i has an endowment vector of commodities $\omega_{i} \in R^{I}$ and preferences expressed by an indirect utility function $v_{i}\left( p,w_{i}\right)$ .

(a) (5 points)

Let $\left(\bar{x}_{1},\cdots,\bar{x}_{I}\right)$ be a Pareto optimal allocation. The utility levels of this allocation are $\left(\bar{u}_{1},\cdots,\bar{u}_{I}\right)$ . The second welfare theorem asserts the existence of a price vector $\bar{p}$ and wealth levels $\left(\bar{w}_{1},\cdots,\bar{w}_{I}\right)$ supporting the allocation. What does this mean? Express $\left(\bar{u}_{1},\cdots,\bar{u}_{I}\right)$ in terms of the indirect utility functions.

Assume for the next two parts of this question (b and c) that the indirect utility functions take the (Gorman) form $v_{i}\left( p,w_{i}\right) =a_{i}\left( p\right) +b\left( p\right) w_{i}$ . Note that $b\left(\cdot\right)$ does not depend on i. In the following, neglect always boundary allocations. Use of pictures is permissible and helpful.

(b) (5 points)

Show that for the above family of utility functions all the Pareto optimal allocations are supported by the same price vector.

Use the conclusion of part (b) to argue that the Walrasian equilibrium allocation is unique. (Assume preferences are strictly convex.)

For the last part of the question (d) assume that indirect utilities are of the form $v_{i}\left( p,w_{i}\right)=b_{i}\left(p_{i}\right)w$ , that is, the preferences on commodity bundles are homothetic (but possibly different across consumers).

(d) (5 points)

Argue by means of an Edgeworth box example (or in any other way you wish!) that the multiplicity of Walrasian equilibria is possible even if preferences are restricted to be homothetic.

QUESTION 4 (Revelation of Information Through Prices)
(20 Points)

Suppose there are two equally likely states $s_{1},s_{2}$ and two traders. In each state there is a spot market where a good is exchanged against numeraire. The utilities of the two traders are (the second good is the numeraire):

	STATE 1	STATE 2
TRADER 1	2 ln x₁₁ – x₂₁	4 ln x₁₁ + x₂₁
TRADER 2	4 ln x₁₂ – x₂₂	2 ln x₁₂ + x₂₂

The total endowment of the first good equals 6 in the first state and $6+\varepsilon$ in the second state. All the endowments of this good are received by the second trader. Assume that the endowments of numeraire for the two traders are sufficient for us to neglect the possibility of boundary equilibria. The price of the numeraire is fixed to 1 in the two states. The prices of the non-numeraire good in the two states are denoted $\left( p_{1},p_{2}\right)$ .

(a) (5 points)

Suppose that when uncertainty is resolved both traders know which state of the world has occurred. Determine the spot equilibrium prices $\left(\hat{p}_{1}\left(\varepsilon\right) ,\hat{p}_{2}\left(\varepsilon\right) \right)$ in the two states (as function of the parameter $\varepsilon$ ).

(b) (5 points)

We assume now when a state occurs Trader 2 knows it while Trader 1 remains uninformed (i.e. s/he must keep thinking of the two states or equally likely). Under this information set up determine the spot equilibrium prices $\left( \bar{p}_{1}\left(\varepsilon\right) ,\bar{p}_{2}\left(\varepsilon \right)\right)$ in the two states.

We are as in (b), except that now we allow Trader 1 to deduce the state of the world from prices. That is, if $p_{1}\neq p_{2}$ then Trader 1 is actually informed while if $p_{1}=p_{2}$ , s/he is not informed. A system of equilibrium spot prices $\left( p^{\ast }_{1}\left(\varepsilon\right) ,p^{\ast }_{2}\left(\varepsilon\right) \right)$ is a rational expectation equilibrium if at the equilibrium Trader 1 derives information from $\left( p^{\ast }_{1}\left(\varepsilon\right) ,p^{\ast }_{2}\left(\varepsilon\right) \right)$ in the manner described. Let $\varepsilon \neq 0$ . Exhibit a rational expectations equilibrium. Comment.

(d) (5 points)

Show that if $\varepsilon = 0$ then there is no rational expectations equilibrium.

QUESTION 5 (20 Points)

There are three participants in a public good decision problem with two outcomes. If the public good project is not carried out then the utility is zero for everybody. If it is carried out then the utility is 3 for the “project-lovers” and -1 for the “project-haters.” The cost of the project is zero.

We consider first the following decision mechanism. People are asked if they are PL (project-lovers) or PH (project haters). If at least one participant announces PL the project is carried out and the self-declared PH are exactly compensated for their loss. The resources for the compensation comes from a tax imposed on the self-declared PL (equal across them).

(a) (5 points)

Show that the above mechanism is not straightforward. Define your terms.

(b) (5 points)

Suppose now that participants know each others characteristics (i.e. if they are project-lovers or project-haters). Consider the situation where everybody self-declares truthfully. Argue that this is an equilibrium (i.e. it does not pay to any participant to deviate) if there is one but not if there are two PLs. Which are the equilibrium situations in the latter case?

We now change the set-up somewhat. Suppose that the designer knows how many PLs there are and that the participants know that the designer knows (or, simply you can assume that both designer and participants have this information). Say that the number of PLs is $\alpha \in \left( 1,2,3\right)$ . (Hence there is at least one PL.) Then the decision mechanism is as above except that for the project to be carried out it is now required that at least $\alpha$ self-declare as PL.

Show that for this mechanism it does not always pay to self-declare truthfully (that is, the truth is not a dominant strategy).

(d) (5 points)

Suppose that it is understood (Precisely, it is common knowledge) that no participant will ever use a dominated announcement. Show then that it cannot hurt to self-declare truthfully (technically, the truth is dominant after one round of deletion of dominated strategies. There is a subtle point here—that you may want to discuss—namely, if “dominated” should be understood as “weakly dominated” or “strongly dominated.” The distinction does not matter for the case $\alpha =1$ but it does for the case $\alpha =2$ .)

Part III (Questions 6 and 7)

(a) How does the following idea (or vision, in Schumpeter’s sense of the term) get reflected in the neo-Keynesian model presented in this course?

…there is a subtle reason drawn from economic analysis why…faith may work. For if we act consistently on the optimistic premise, this hypothesis will tend to be realized; whilst by acting on the pessimistic premise, we keep ourselves for ever in the pit of want. (Keynes, Essays in Persuasion, pp. vii-viii)

(b) Why does Knight’s dictum [following] fail to characterize the neo-Keynesian model?

…competition among even a very few [entrepreneurs]will raise the rate of contractual returns [wages] and lower the residual share [profits], if they know their own powers. If they do not, the size of their profits will again depend on their “optimism,” varying inversely with the latter. (Knight, Risk, Uncertainty, and Profit, p. 285.)

(c) Is it true, as Joan Robinson once wrote, that in a neo-Keynesian conception of the world businessmen are free to make the rate of profit anything they wish?

(d) More generally, how can investment demand be exogenous in a model where income and expenditure must be equal as a condition of equilibrium? What features of the theory allow investors’ preferences and investment demand to play a role in neo-Keynesian theory which differs from the role played by consumers preferences and consumption demand in neoclassical theory?

Economic theories are, among other things, theories of knowledge—implicitly if not explicitly. What is the neoclassical theory of knowledge? Which do you regard as the more serious of the many objections to this theory of knowledge? Why in your view has the theory been able to survive the objections?

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

Previously Transcribed Harvard Graduate General Exams

Spring 1989: Economic Theory

Spring 1991: Microeconomics; Macroeconomics

Spring 1992: Micro- and Macroeconomics

Fall 1992: Micro- and Macroeconomics

Graduate Microeconomic Theory Sequence, 1992-93

Economics 2010a. Economic Theory

Economics 2010b. Economic Theory

HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS

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

Spring Term 1993

Part I (Questions 1 and 2)

Part II (Questions 3, 4, & 5)

Part III (Questions 6 and 7)

Related

HARVARD UNIVERSITY
DEPARTMENT OF ECONOMICS

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