Eco 600E: Advanced Microeconomics I (Spring, 1st, 2012)
Final Exam
1. True or False (10 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) Suppose that a concave function u represents preferences %. Then, % MUST be convex.
(b) Let xi be Marshallian demand and pi be the price of good i. Then, for any pair of goods a and b, ∂xa
∂pb
= ∂xb
∂pa
ALWAYS hold.
(c) If a firm’s technology shows increasing return to scale, then the marginal prod- uct (with respect to any input) CANNOT be decreasing.
(d) If an agent is risk neutral, then her certainty equivalent of a lottery MUST take the same value of the expectation of the lottery.
(e) The set of Pareto efficient allocations ALWAYS contains the core (if it exists). 2. Optimization (12 points)
A consumer gets utility from 2 sources: drinking (measured in liters x) and time spent on the phone (measured in hours y). Each liter of drink costs $4 and each hour on the phone costs $4. She has a total of $240 available for spending. Her utility function is given by:
u(x, y) = xy
(a) Does the consumer have convex preference? Explain why. (b) Solve the utility maximization problem.
(c) The health authorities are putting up a program to cut down alcohol consump- tion. They propose a quota that allows to consume a maximum of 16 liters. What are the optimal choices under this new scenario?
(d) Suppose the price of alcohol is reduced from $4 to $3. Then, what are her optimal choices if the quota is not imposed?
(e) What are the optimal choices when the price of alcohol is $3 but the quota (of 16 liters) is imposed?
(f) Find the amount of quota q that would make the consumer indifferent between the scenario (b) (no quota, price is $4) and (e) (quota of q, price is $3).
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3. Consumer Theory (8 points)
(a) Let xi(p, ω) denote the Marshallian demand function of good i and v(p, ω) denote the indirect utility function. Then, state the Roy’s identity.
(b) Let xhi(p, u) denote the Hicksian demand function of good i and e(p, u) denote the expenditure function. Then, state the Shephard’s lemma.
(c) Using envelope theorem, derive either (a) Roy’s identity, or (b) Shephard’s lemma. You can assume that the first order conditions guarantee the optimal solution, i.e., ignore the second order conditions.
4. Exchange Economy (12 points)
Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices and allocations in each case.
(a) Two agents, 1 and 2, have the following indirect utility functions: v1(p1, p2, ω) = ln ω − a ln p1− (1 − a) ln p2
v2(p1, p2, ω) = ln ω − b ln p1− (1 − b) ln p2
where 0 < a, b < 1 and initial endowments:
e1 = (2, 0), e2 = (0, 2).
(b) Two agents, a and b, have the following utility functions: ua(x1, x2) = x1 + x2
ub(x1, x2) = max{x1, x2} and initial endowments
e1 = e2 = (1, 1). 5. Expected Utility (8 points)
Suppose that the decision maker’s preferences under uncertainty are described by the vNM utility function, u(x) =√x.
(a) Is the decision maker risk-averse, risk-neutral, or risk-loving? Explain why. (b) Calculate the absolute risk aversion and the relative risk aversion, respectively.
(c) Consider the following lottery (gamble): x becomes $900 or $2500 with prob- ability 0.5 each. Derive the certainty equivalent of this lottery.
(d) If this decision maker participates the gamble in St Petersburg Paradox, (i.e., Receiving $2k when head shows kth trial), how much is she willing to pay?
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