Question

11. Consider an exchange economy consisting of two people, A and B, endowed with two goods, 1 and 2. Person A is initially endowed with ω^A= (0,3) and person B is initially endowed with ω^B= (6,0). They have identical preferences, which are given by U^A(x1, x2) =U^B(x1, x2) =x₁²x₂.

(a) Write the equation of the contract curve (express x₂^Aas a function of x₁^A1).

(b) Let p²= 1. Find the competitive equilibrium price,p1, and allocations,x^A= (x_1^A, x_2^A) and x^B= (x_1^B, x_2^B).

(c) Now suppose that before A and B ever get to trade, some of B’s endowment of good 1 is destroyed,so she instead has an initial endowment of ω^B= (3,0) (and everything else is the same as in part(a), including A’s endowment and both players’ preferences. Calculate the new equilibrium price,p1, and the new allocations,x^A= (x^A₁, x₂^A) and x^B= (x₁^B, x₂^B) (You may assume that p₂= 1.)

For part C), the answer for x^B is x^B= (2,2) but I keep getting something else. Can anyone show the steps to solve for x^B in part C

Answer 1

a)

For both consumer A and B. For interior solutions, MRS_A= MRS_B

Since,

  

The Contract Curve is

b) Max U for consumer A

s.t.

Since p₂=1 and initial endowment income = 3

Substituting in the budget constraint we get,

We also know from these two equations,

and

  and  

Similarly,

Max U for consumer B

s.t.

and so

We also know from this equation,

and so  

c) The new contract curve is

  

For consumer 1,

s.t.

Since p₂=1 and initial endowment income = 3

Substituting in the budget constraint we get,

We also know from these two equations,

   and

  and  

Max U for consumer B

s.t.

and so

We also know from this equation,

and

  and from above