FINAL EXAMINATION

ADVANCED PRICE THEORY

Instructions: ALL SECTIONS ARE EQUALLY WEIGHTED. If a question has multiple parts, indicate exactly where you answer each part. GOOD LUCK!!

VERY SHORT ANSWERS:

ANSWER ALL OF THESE. Carefully define the following terms. Whenever possible, give both mathematical and verbal definition.

Quasi-Convex	Risk Premium
Time Separable Utility	Risk Aversion
Bellman's Equation	Substitution Axiom
Moral Hazard	Archimedean Axiom
Independence of Irrelevant Alternatives	Anonymity Principle

SHORT ANSWERS:

DO ANY TWO (2) OF THE FOLLOWING QUESTIONS. ALL QUESTIONS ARE EQUALLY WEIGHTED.

1. Define the weak Pareto criterion. Is it an ordering? (Show.) Is it a good collective choice rule? (Discuss.)
2. Consider Robin, who attends a lot of parties. S/he loves cake, but s/he is “polite”: s/he never takes the largest piece available, but s/he takes the biggest piece subject to that normative constraint. Is Robin rational? Is s/he a maximizer? Does s/he obey Sen's alpha and beta? (If not, which is violated?)
3. Use expected utility theory to analyze the optimal response of potential tax-cheats to the risk of audit. You can assume the IRS wishes to recoup its losses on average.
4. Prove the first fundamental theorem of welfare economics algebraically and interpret your proof.

LONGER ANSWER:

ALL STUDENTS MUST ANSWER ONE (1) OF THE FOLLOWING QUESTIONS:

1. Give a precise statement of Arrow's “impossibility theorem” and discuss its implications as you see them. Prove Arrow's “impossibility theorem”.
2. Hall (1978, JPE) considered a simple consumption problem: max E₀Sum_t=0^infty beta^tU(c_t) subject to the budget constraint A_t+1=R(A_t+y-c_t) with U(c_t)=-(cbar-c_t)²/2.
   • Set up the associated Bellman equation and interpret.
   • Derive the first order and envelope conditions and interpret. (An “interpretation” gives the economic reasoning behind these conditions; do not just restate the equations in words.)
   • Derive Hall's result that consumption follows a martingale with drift. Be sure to provide economic interpretations at each step of your derivation. (What are the implications of this result for empirical research on the consumption function?)
3. Give an intuitive discussion of the principal agent problem along with some concrete, real world situations where it might arise. Suggest a general procedure by which the principal might derive an optimal contract with the agent, and indicate precisely the constraints which the principal must incorporate in this procedure. Present and interpret the first order conditions, with careful attention to the sign and magnitude of the multipliers. (Good intuition is as important to your answer as algebra.)