Result significant for partial equilibrium analysis

Question

1. Suppose that consumer i’s preferences for L goods can be represented by the (Cobb-Douglas) utility function u_i[x_1i......x_Li] = Σ_l log[x_Li]

a) With a budget constraint _l=1Σ^L p_lx_l ≤ w

b) What is the wealth effect?

c) What happens to the wealth effect in the limit when the number of goods, L, increases?

d) Why is this result significant for partial equilibrium analysis?

2. Consider a two-good quasi-linear economy with one consumer and one firm (I = 1 and J = 1). The initial endowment of the numeraire is ω_m > 0, and the initial endowment of good l is 0. Let the consumer’s quasi-linear utility function be f [x]+m where f [x] = α+βlog[x] for some (α,β) >> 0. Let the firm’s cost function be c[q] = σq for some scalar σ > 0. Assume that the consumer receives all the profits of the firm. Both the firm and the consumer act as price takers. Normalize the price of good m to equal 1, and denote the price of good l by p.

a) Derive the consumer’s and the firm’s first-order conditions.

b) Derive the competitive equilibrium price and output of good l. How do these vary with α,β, and σ?

3. Assume that Robinson Crusoe can spend his time relaxing on the beach or hunting for oysters. Robinson has a Cobb-Douglas utility function of the form u[x,R] = αlog[x] + (1−α)log[R] where R is leisure time and x is oysters. He can procure oysters according to the CRS production function x = f [L] = αL. Further, Robinson is endowed with 24 hours in a day such that 24 = L + R. The time Robinson has spent in isolation on his island has driven him a bit mad. In wishing he was back in a market economy, Robinson devises a monetary scheme using shells as currency that he pays himself for his labor in the form of a wage w. As Robinson looses further touch with reality and his mind, he believes that the market price of oysters is p.

a) Derive Robinson’s profit maximizing condition, where his profit is π = px −wL

b) Derive Robinson’s utility maximizing condition, where his budget constraint is px +wR = Y .

c) Derive Robinson’s equilibrium demand for labor L* and equilibrium supply of x as functions of w,α, and p.

d) Derive the equilibrium wage w∗

e) What is Robinson’s profit at the equilibrium wage?

4. Consider an exchange economy in which two agents have quasi-linear utilities:

u^A [x₁, x₂] = x₁ + 2√x₂

u^B [x₁, x₂] = x₁ + √x₂

Agent 1 starts out holding amounts (2, 1) of the two goods while agent 2 has (2, 4)

a) An core exchange equilibrium is an allocation of the goods over the agents such that each agent is no worse off than the original allocation and such that there are no feasible trades that would make one agent strictly better off while leaving the other agent no worse off. Find the set of core trading equilibria for this economy and draw it in an Edgeworth box. (Hint: The set of core trading equilibria is a subset of the contract curve which should be horizontal.)

b) Identify the numerical bounds of this set where either A or B captures all utility gains while the other is not made any worse off.

c) Assuming both goods are fully divisible, find the equilibrium allocation that makes both A and B as better off as possible, i.e. the allocation that equally shares the benefits of exchange.


5. Consider an market economy in which two agents have utilities

u^A[x₁,x₂] = αlog[x₁]+(1−α)log[x₂]

u^B[x₁,x₂] = βlog[x₁]+(1−β)log[x₂]

Let the price of each good be p1,p2 respectively. Assume the total resources in the economy are ω ¯ 1 = 15 and ω ¯ 2 = 10 and that A’s initial endowments are (w₁^A = 15, w₂^A =1)

a) Find both consumer’s Marshallian demands for x₁ and x₂.

b) Find the equation for the contract curve.

c) Find the equilibrium price ratio (p1*/p2*) in terms of α and β.