For a given utility function derive the contact curve

Question

Write a brief explanation of the following concepts using mathematics and graphs when necessary and discuss their use and/or significance in microeconomic theory.

1. The Edgeworth-Bowley box

2. Walras' law

3. The numeraire

4. The Auctioneer

ANALYTICAL EXERCISES-

1. Consider an Edgeworth box pure exchange economy with two individuals (A and B) and two goods (x₁ and x₂). The consumers' utility functions are u^A[x₁, x₂] and u^B[x₁, x₂]. Assume that both utility functions are continuous and quasi-concave. The initial endowments of the two consumers are ω^A = (ω₁^A, ω₂^A) and ω^B = (ω₁^B, ω₂^B) and total resources are ω¯₁ = ω₁^A + ω₁^B and ω¯₂ = ω₂^A + ω₂^B.

a) For a given utility function derive the contact curve equation in terms of ω¯₁ and ω¯₂.

b) What are the Pareto-efficient bounds of the efficient contract locus in terms of ω₁^A, ω₂^A, ω₁^B, ω₂^B?

c) What is the purely equitable Pareto-efficient equilibrium?

d) Sketch and label an Edgeworth box for this economy.

2. Consider an economy with 2 commodities and a large number, N , of consumers, each of which has the same Cobb-Douglas utility function:

 u[x₁, x₂] = x₁^1/2 x₂^1/2        with

The total endowment of the economy for each good is

ω¯₁ = _i=1Σ^N ω_i1 = 100

ω¯₂ = _i=1Σ^Nω_i2 = 50

Consumer i 's budget constraint is p₁x₁ + p₂x₂ = w = p₁ω_i1 + p₂ω_i2 for i = 1, ..., N . The equilibrium condition for good l is _i=1Σ^Nx^∗_il = ω¯_l for l = 1, 2.

a) What are the competitive equilibrium prices of this economy? (Hint: simplify us- ing a monotonic transformation of the utility function and first solve for the Marshallian demands for each commodity. Use the price of good 1 as the numeraire and use the equilibrium condition to solve.)

b) Derive the excess demand function for this economy. (Hint: the solution should only be a function of prices and endowments for all consumers.)

3. Consider an economy with two individuals (A and B) and two goods (x₁ and x₂). The consumers' utility functions are

u^A[x₁, x₂] = (x₁^A)^α(x₂^A)^1-α              

u^B[x₁, x₂] = (x₁^B)^β(x₂^B)^1-β

The initial endowments of the two consumers are ω^A = (1, 0) and ω^B = (0, 1).

a) Derive the excess demand function.

b) Prove that it satisfies Walras' law.

c) With α = β = 0.5 derive the equilibrium price vector. (Hint: use the price of good 2 as the numeraire.)

d) Make sure you understand the general solution to this problem, i.e. that you are able to solve it for a variety of utility functions.

ESSAY (CHOOSE ONE)

Please use mathematics and graphs when necessary.

4. In a short essay discuss the notions of trading equilibrium and Walrasian equilibrium. Discuss why a trading equilibrium is more general than Walrasian equilibrium. Make sure to touch upon issues of existence, uniqueness and stability in each case.

5. In a short essay discuss the fundamental theorems of welfare economics. Explain both theorems and discuss the ways in which they are an apologia for the capitalist mode of production. Make sure to touch upon the underlying assumptions of the theorems and what the theorems say about coordination problems and the institutional conundrum.