Question

Consider the exchange economy. It consists of two individuals, A and B.

The only two goods available for consumption are red wine and choco-

late. The endowment of the economy is 64 units of red wine and 16 units

of chocolate. A’s endowment is w_a ? and c_a ? ; B’s endowment is w_b ? and

c_b ? . The utility functions for each individual are U = 2(w_a c_a )1/2 and

UB = 10(wBcB)1/2, where wj and cj are the amounts of wine and choco- late consumed by individual j.

(a) Identify (solve for) ALL Pareto efficient allocations in this economy and show them in the graph.

(b) Draw the the utilities possibility frontier. What is the equation of the utility possibilities frontier?

(c) What does the slope of the utilities possibility indicate?

(d) Suppose that a social dictator decides to maximize the social welfare function S = ?AUA + ?BUB. What is the equation for an iso-welfare curve? What is the slope of an iso-welfare curve? What does the slope indicate?

(e) Ifthedictatordecidesthattheweightsfrombothindividualsshouldbe equal, what is the highest attainable iso-welfare contour? Why would A prefer that the ratio ?A/?B > 5?

(f) Suppose, instead, that the social dictator has determined that the best point on the utility possibilities frontier is UA = 16, UB = 240. How much of each good is consumed by each individual at this allocation? What would the price of wine have to be to support this allocation as competitive equilibrium? How much income would each individual require at this competitive equilibrium? How much of the numeraire good, chocolate, would have to be redistributed and in what direc- tion, in order to generate this social optimum by means of competitive trade? What important economic proposition has just been verified? Locate the location of this social optimum, the price line, and the en- dowment point after redistribution in your Edgeworth box.

Answer 1

1. Identify (solve for) ALL Pareto efficient allocations in this economy and show them in the graph.

In the given problem:

Consumers: A & B

Goods: Red Wine (R) & Chocolate (C)

Endowments (w) = 64 units of R and 16 units of C

A endowment = wa + ca

B endowment = wb + cb

Total endowment of goods in the economy:

Good R = wa + wb

Good C = ca + cb

Demand X = (xa, xb) = (64, 16) = ((xa1 + xa2), (xb1 + xb2))

Demands should be feasible:

Good R = xa1 + xb1 = wa + ca

Good C = xa2 + xb2 = ca + cb

For this equilibrium analysis we use Edgeworth box:

Here,

w = endowment = area under AI, AL for country A

x = demand = area under AH, AK for Country A

Now,

w = endowment = area under BD, BG for country B

x = demand = area under BC, BF for country B

Country A buys HI units of good C and sells KL of R

1. Draw the the utilities possibility frontier. What is the equation of the utility possibilities frontier?