Eco 600E: Advanced Microeconomics I (Fall, 1st, 2013)
Final Exam: December 10 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 is complete and transitive, her demand behaviors always satisfy the weak axiom of revealed preference.
(b) Even if a firm’s technology shows increasing return to scale, the marginal product (with respect to some input) can be decreasing.
(c) When an indirect utility function takes Gorman form, its original utility func- tion must be quasi-linear.
(d) A firm’s cost function is homogeneous of degree 0 in the input price vectors. (e) If an allocation is Pareto efficient, it must be in the core.
2. Preferences (25 points)
Suppose % is a preference relation on X, i.e., % satisfies completeness and transi- tivity. Then, show the followings, given that ∼ and ≻ are defined as follows:
a∼ b ⇔ a % b and b % a, a≻ b ⇔ a % b and not b % a. (a) Reflexive: For any x ∈ X, x ∼ x.
(b) Transitive 1: For any x, y, z ∈ X, if x ≻ y and y ≻ z, then x ≻ z. (c) Transitive 2: For any x, y, z ∈ X, if x ∼ y and y ∼ z, then x ∼ z. (d) Transitive 3: For any x, y, z ∈ X, if x ∼ y and y % z, then x % z.
(e) Transitive 4: For any x, y, z ∈ X, if x ≻ y and y % z, then x ≻ z. 3. Production (20 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.
(a) Sketch the isoquant for this technology.
Hint: Isoquant is the combination of inputs that achieves a given level of output y. (similar to “indifference curve” in consumer theory.)
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(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). Hint: Note that solutions may change for different relative costs.
(d) Let p > 0 be a unit price of output y. Describe the profit maximization problem and solve it.
4. Exchange Economy (30 points)
Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices (price ratio) and allocations in each case.
(a) Two agents, a and b, have the following indirect utility functions: va(p1, p2, ω) = ln ω − α ln p1− (1 − a) ln p2
vb(p1, p2, ω) = ln ω − β ln p1− (1 − b) ln p2 where 0 < α, β < 1 and initial endowments:
ea= (2, 0), eb = (0, 2).
(b) Two agents, a and b, have the following utility functions: ua(x1, x2) = x1+ x2
ub(x1, x2) = min{x1, x2} and initial endowments
ea = eb = (1, 1).
(c) Two agents have the symmetric utility function and initial endowments: u(x1, x2) = x21+ x22, e = (1, 1).
5. Proofs (30 points)
(a) Let c(w, y) be a firm’s cost function. Show that it is concave in w.
(b) Let y = f (x1, x2) be a constant returns-to-scale production function. Show that if the average product of x1
= f (xx1,x2)
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is increasing in x1, the marginal product of x2 must be negative.
(c) Let (x, p) be a competitive equilibrium. Suppose ui(yi) > ui(xi) for some bundle yi. Then show that p · yi >p · xi. Does this depend on whether utility function is increasing or not?
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