Eco 600E: Advanced Microeconomics I (Fall, 1st, 2014)

Final Exam: December 1

1. Consumer Theory (20 points)

A consumer has a utility function u(x, y, z) = e^{min}^{{x,y}}e^{z}. The prices of the three
goods are given by (p, q, 1) and the consumer’s wealth is given by ω.

(a) Formulate the utility maximization problem of this consumer.

(b) Note that this consumer’s preference can be expressed in the form of U (x, y, z) = V(x, y) + z. Derive V (x, y).

(c) What are the demand functions for each of the three goods? (d) What is the indirect utility function?

2. Production (25 points)

Suppose that a production function takes the following form,
y = (αx^{1}+ x^{2})^{β}

where α > 0 and 0 < β < 1. Let w^{1}, w2 > 0 be the prices for inputs x^{1} and x^{2}
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.)

(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(w^{1}, w2_{, y).}

Hint: Note that solutions may change for different relative costs.

(d) Let p > 0 be the price of output y. Describe the profit maximization problem in two different ways, one-step and two-step procedures.

(e) Solve the profit maximization problem you derived in (d) (in either procedure).

1

3. Partial Equilibrium (10 points)

Consider the following partial equilibrium analysis. Let CS(p) and P S(p) be the consumer surplus and producer surplus (for a given market price p), respectively. Show that the competitive price minimizes the total surplus, i.e., CS(p) + P S(p). Why does the equilibrium price minimize rather than maximize the welfare? 4. Exchange Economy (20 points)

Consider an economy with two consumers and two commodities. Consumer 1 has an endowment of three units of good 1 and none of good 2. Consumer 2 has an endowment of one unit of good 1 and four units of good 2. Each consumer has the utility function

u_{(x}1, x2_{) = β ln x}1+ (1 − β) ln x^{2}^{,}
where β lies strictly between 0 and 1.

(a) Find all of the Pareto efficient allocations for this economy. (b) Formulate the utility maximization problem for each consumer.

(c) Solving the consumer problem you derived in (b), find the competitive equi- librium allocation.

(d) Assume that the government tries to achieve the equitable allocation, i.e., each consumer receives two units of both goods, through the market mechanism. How much lump-sum transfer of good 2 is needed to achieve this goal?

5. Production Economy (25 points)

Consider an economy with two firms and two consumers. Firm 1 is entirely owned by consumer 1; it produces good A from input X via the production function a= 2x. Firm 2 is entirely owned by consumer 2; it produces good B from input X via the production function b = 3x. Each consumer owns 10 units of X. Consumers’ preferences are given by the following utility functions:

u^{1}(a, b) = a^{2}^{/5}b^{3}^{/5},
u^{2}(a, b) = 10 + ^{1}

2^{ln a +}
1
2^{ln b.}

A (unit) price of input X is normalized to be 1, while prices of goods A and B are denoted by p and q, respectively.

(a) Formulate the profit maximization problem for each firm. (b) Formulate the utility maximization problem for each consumer.

(c) Derive the competitive equilibrium prices of goods A and B, the equilibrium output for each firm, and the equilibrium allocation for each consumer.

2