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Eco 600E: Advanced Microeconomics I (Fall, 1st, 2013)

Midterm Exam 1. True or False (9 points)

Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.

(a) Suppose % is represented by utility function u(·). Then, u(·) is quasi-concave IF AND ONLY IF % is convex.

(b) Marshallian demand function is ALWAYS weakly decreasing in its own price. (c) Lagrange’s method ALWAYS derives optimal solutions for any optimization

problem with equality constraints. 2. Consumer Theory (30 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 $120 available for spending. Her utility function is given by:

u(x, y) = xy (a) Set up the utility maximization problem. (b) Solve this 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 8 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 8 liters) is imposed?

(f) Find the amount of quota q that would make the consumer indifferent between the scenario (a) (no quota, price is $4) and (e) (quota of q, price is $3).

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3. Kuhn-Tucker Condition (21 points)

Consider the following maximization problem with two inequality constraints,

x,y>0maxx

1/2

y

s.t. x+ y ≤ a and y ≤ 4 where 0 < a < 6.

(a) Explain whether the objective function is (i) concave, and (ii) quasi-concave, respectively.

Hint: You can graphically show the claims if you prefer to do so. (b) Derive the Kuhn-Tucker conditions.

(c) Derive optimal solutions (you can assume that second order conditions are satisfied), and the maximum value function.

4. Duality (20 points)

An (minimum) expenditure function of some consumer is given by

e(p1, p2, u) = u × pα1p12−α.

(a) State the Shephard’s lemma, and derive the Hicksian demand for good 1. (b) Explain why the Hicksian demand is always weakly decreasing in its own price.

(c) Using duality, derive the indirect utility function.

(d) State the Roy’s identity, and derive the Marshallian demand for good 1.

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