Question

Consider the following endowment economy. Mr. Peanutbutter and Princess Carolyn have the same utility function: u (c1, c2) = c₁^1/3 c₂^2/3 . Mr. Peanutbutter has endowment e_P = (4, 6) and Princess Carolyn has endowment e_C = (6, 4).

a. State the first and second welfare theorems.

b. Define Pareto efficient allocation for this economy.

c. Define the competitive equilibrium for this economy.

d. Solve for the competitive equilbrium

e. The government wants both Mr. Peanutbutter and Princess Carolyn to consume (5, 5) in equilibrium. Is (5, 5) Pareto efficient? If not, explain why not. If it is Pareto efficient, characterize the set of transfers on initial endowments such that (5, 5) is implemented in equilibrium.

Answer 1

a) first welfare theorem - competative eqm leads to set of pareto efficient allocation of resources.

second welfare - An efficient allocation can be attained by a competitive equilibrium, given the market mechanisms leading to redistribution.

b) for pareto optimali allocation

MRSA = MRSB

1/3c1^-2/3*c2^2/3 = 2/3*c2^-1/3*c1^1/3

c1^-1 = 2 c2-¹

1/c1 = 2/c2

c2 = 2c1

c) for competative eqm

c1_a + c1_b = 10

m1 =4p1 + 6p2

m2 = 6p1 + 4p2

c1_a = 1/3 * m1/p1 = 1/3* (4p1+6p2)/p1

c1_b =1/3*m2/p1 = 1/3*(6p1+4p2)/p1

1/3* (4p1+6p2)/p1 + 1/3*(6p1+4p2)/p1 = 10

10p1 + 10p2 / 3p1 = 10

10 p1 + 10p2 = 30 p1

10p2 = 20p1

p2 = 2p1

p1/p2 = 1/2