Midterm Exam: Solutions
Eco600E: Advanced Microeconomics I (Fall, 1st, 2014)
Nov. 4th 2014
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) The union of convex sets is convex.
(b) If ≿ is convex, then every consumer problem has at most one solution.
(c) A function u(·) : X → R is concave if u(αx+(1−α)y)≥Min{u(x), u(y)} for all x, y ∈ X and α ∈ (0, 1).
2 Consumer Theory (18 points)
(a) Let xi(p, w) denote the Marshallian demand function of good i and v(p, w) denote the indirect utility function. Then, state the Roy’s identity.
(b) Let xih(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 indentity, or (b) Shephard’s lemma. You can assume that the first order condition guarantee the optimal solution, i.e., ignore the second order conditions.
3 Duality (24 points)
Consumer a has expenditure function ea(p1, p2, ua) = ua√p1p2 and consumer b has utility function ub(x1, x2) = 6x13x23.
(a) Derive the Marshallian demand functions for x1 and x2 by consumer a (x1
a and x2 a). Denote the income of consumer a by wa and the income of consumer b by wb.
1
(b) Derive the Marshallian demand functions for x1 and x2 by consumer b (x1b and x2b).
4 Demand Function (43 points)
Assume utility function u(x1, x2) = kx1αx21−α for some α ∈ (0, 1) and k > 0.
(a) Denote the prices and income as p and w, respectively. Then, state utility maximization problem and solve it.
(b) Derive the (Marshallian) demand function for each good.
(c) Show the derived demand function has homogeneity of degree 0 (in prices and income). (d) Show the corresponding preference relation ≿ is strictly convex.
2
