Midterm Exam
Date: April 28, 2010
Subject: Advanced Microeconomics I (ECO600E) Professor: Yosuke YASUDA
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) Even if the consumer demand is rationalized by some preference relation, the weak axiom of revealed preference can SOMETIMES be violated.
(c) Lagrange’s method ALWAYS derives optimal solutions for any optimization problem with equality constraints.
2. Revealed Preference (10 points)
Consider the following choice problem. There are 4 feasible elements, and we denote the set of all elements as X = fa; b; c; dg. Suppose individual choice behaviors are described by two di¤erent choice functions, f1 and f2.
f1(A) = a; f1(B) = b; f1(C) = c; f1(D) = a
f2(A) = b; f2(B) = b; f2(C) = c; f2(D) = a
where A = fa; bg; B = fb; cg; C = fa; cg; D = fa; b; dg. (a) Derive A \ B and C [ D respectively.
(b) Does each choice function satisfy the weak axiom of revealed preference? You should also explain why.
Hint: The weak axiom (in general choice problems) is violated if and only if there exist two sets X and Y such that
fx; yg X; Y, and f(X) = x and f (Y ) = y.
(c) Show that, for each choice function, there exists no preference relation that can rationalize it.
Hint: A preference relation % is said to “rationalize” a choice function f ( ), if for any choice set X, f (X) is the unique % best alternative within X.
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3. Lagrange’s Method (16 points)
Consider the following maximization problem with an equality constraint,
x;y>0maxx+ 2y s:t: x2 + y2 = a where a > 0.
(a) Explain whether the objective function is (i) homogeneous of degree 0, and (ii) quasi-concave, respectively.
Hint: You can graphically show the claims if you prefer to do so.
(b) Derive the critical points (i.e., the combinations satisfying the …rst order con- ditions) of this maximization problem by using Lagrange’s method.
(c) What is the (maximum) value function? Is it strictly increasing in a?
(d) Derive the bordered Hessian and verify that your solution is a global maximum. Hint: You can assume that the sign of is the same as what you derive in (b).
4. Kuhn-Tucker Condition (15 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 satis…ed), and the maximum value function.
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