Eco 600E: Advanced Microeconomics I (Spring, 1st, 2013)
Final Exam: June 6 1. True or False (15 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) If a consumer’s preference satisfies completeness and transitivity, her prefer- ence can be ALWAYS represented by some utility function.
(b) It is POSSIBLE that an expenditure function is a convex function of prices. (c) If the utility function is quasi-linear, the compensating variation is ALWAYS
equal to the equivalent variation.
(d) vNM functions are invariant with respect to ANY monotone transformation. (e) An allocation in the core MUST BE Pareto efficient.
2. Consumer Theory (15 points)
A consumer has a utility function u(x, y, z) = min{x, y}+z. The prices of the three goods are given by (px, py, pz) and the consumer’s wealth is given by ω.
(a) Note that the utility function u can be written in the form of U (V (x, y), z). Derive the functions V (x, y) and U (V, z).
(b) What are the demand functions for the three goods? (c) What is the indirect utility function?
3. Duality (10 points)
An (minimum) expenditure function of some consumer is given by e(p1, p2, u) = u × pα1p1−α2 .
(a) What are the Hicksian demand functions for x1 and x2? (b) What are the Marshallian demand functions x1 and x2? 4. Production (15 points)
Suppose that a production function takes the following form, y = (αx1+ βx2)1/2
where α, β > 0. Let w1, w2 > 0 be the prices for inputs x1 and x2 respectively. Then, answer the following questions.
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(a) Sketch the isoquant for this technology.
Hint: Isoquant is the combination of inputs that achieves a given given level of output y. (corresponds to “indifference curve” in consumer theory.)
(b) Does this production function display increasing, constant, or decreasing re- turns to scale? Explain why.
(c) Formulate the cost minimization problem (you may denote a target output level by y). Then, solve it and derive the (minimum) cost function, c(w1, w2, y). 5. Risk Aversion (15 points)
Suppose that a division maker has the vNM utility function, u(x) = ln 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 aversions, respectively.
(c) Consider the following lottery (gamble): x becomes $100 or $400 with proba- bility 0.5 each. Calculate the certainty equivalence of this lottery.
6. General Equilibrium (30 points)
Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x1 and a consumption good x2. Ann and Bob have equal en- dowments of time (= ω1) to be allocated between leisure and work, so the total endowment of time is 2ω1 units. There is no (initial) endowment of consumption good. Each individual has a common utility function U (x) = ln x1+ 2a ln x2. Sup- pose that only Ann owns the firm and its production function is y2 = √z1, where y2 is the output of consumption good and z1 is the input of (total) labor. Let the price of x2 be normalized by 1 and the price of labor, i.e., wage, be denoted by w.
(a) Solve the profit maximization problem of this firm.
(b) Solve the utility maximization problem of each individual.
(c) Derive the competitive equilibrium (both price w∗ and allocation x∗).
(d) Now consider an exchange economy with n consumers and k goods. We de- note the bundle of total endowments by ω = (ω1, . . . , ωk). Suppose that all consumers have identical (strictly) convex preferences. Then, show that equal division of total endowments, i.e., xi = ω/n for all consumer i, is always a Pareto efficient allocation.
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