Question

Pure Exchange Model 1. Consider a Pure Exchange Economy with two agents A and B and two goods X and Y in which each agent acts competitively. Their preferences are given by the following utility function U(X,Y)=X13*Y23 Their initial endowments are as follows W=(5,20) w- (25,10) a) Calculate the demand functions for Good X and Good Y for each agent. b) State the equilibrium conditions for this economy. c) Using these conditions and the demand functions found in part a) calculate the general equilibrium for this economy show it on an Edgeworth Box (i.e., P. XA. YAX8*, and YB), call this point E. What is the value of Mr. A's initial endowment in terms of good X and good Y at P*? What is the value of Ms. B's initial endowment in terms of good X and good Y at P*? Is this allocation Pareto Optimal? Explain d) Is allocation E Pareto superior to allocation W (the initial endowment)? Explain Is the movement from allocation W to E a Pareto improvement? Explain e) Find the contract curve and show it on your Edgeworth box. Are any points along the contract curve you found Pareto noncomparable? Explain

Answer 1

1. a. Normalizing the price of good 2 to be equal to 1.
Consumer 1's problem:

max x1/2/3 s.t. 5p +20 = px + y

At equilibrium, marginal rate of substitution is equal to the price ratio:

MRS - MU, - 23x1737-173 - x = P + y = 2px MRS. MU, 1/3x2312/3

Substituting this into the consumer's budget equation:

10p+ 40 5p+20 3px = 5p +20 + X= 3py=-

Consumer 2's problem:

max x1/2/3 s.t. 25p+ 10 = px + y

Substituting the optimality condition in the budget equation:

25p+10 - 3px 25p+10 = 50p+ 20 y= 3

b. At equilibrium, the total demand for x will be equal to the total endowment of x according to Walras Law.

c. Applying Walras Law on the market for good y:

10p+40 50p+20 + 3 = 30 p =

The equilibrium price vector is:
$\small p^*=(\frac12,1)$
The competitive equilibrium allocation for consumer 1 is:

(xi. Yi) = (15,15)

The competitive equilibrium allocation for consumer 2 is:

(XS, ES) = ( 15.15)

Value of consumer 1's initial endowment in terms of the equilibrium price vector is 22.5, for consumer two it is 22.5.
At this allocation, the marginal rates of substitution for the two individuals are:

y 151 y21 RS1 = 2x - 30 = 3 = MRS2 = 2

Since the marginal rate of substitution is equal for both consumers, the competitive equilibrium allocation is Pareto Optimal. This follows from the first fundamental theorem of welfare economics. When preferences are increasing, continuous, and quasi concave, every competitive equilibrium allocation is also Pareto Optimal.

d. Since the allocation is Pareto Optimal, it is Pareto Superior to the endowment as at the endowment, the marginal rates of substitution is not equal for both consumers.

MRS1 = 2x = 20 = 3 MRS2 = 2* = 20 = 4 MRS. Y 51. y2 25 5


41(5, 10) = 12.56 < 41(15, 15) = 15, Auz(25, 10) = 13.57 < u2(15,15) = 15

Since both individuals are better off, a movement from the endowment vector to the competitive equilibrium vector is a Pareto Improvement.