Question

Consider a pure exchange economy with two consumers, Ann (A) and Bob (B), and two commodities, 1 and 2, denoted by (x^A_1, x^A_2) and (x^B_1, x^B_2). Ann’s initial endowment consists of 20 units of good 1 and 5 units of good 2. Bob’s initial endowment consists of 0 unit of good 1 and 5 units of good 2. The consumers’ preferences are represented by the following Cobb-Douglas utility functions:U^A(x^A_1, x^A_2) = (x^A_1)^2(x^A_2)^2 and U^B=√x&B1√x^B2. Denote by p1 and p2 the price of good 1 and good 2, respectively.

1. Draw the Edgeworth box for this economy. Illustrate the initial allocation. Label it with sizeof the box, consumers’ initial endowments. (Depict Ann’s consumption from the lower leftcorner and Bob’s consumption from the upper right corner.)

2. For each consumer, derive the equation for the indifference curve passing through the initialendowment allocation.

3. Derive the equation of the contract curve and draw it in the Edgeworth box.

4. Using the contract curve, determine whether the initial allocation is Pareto Efficient or not.What is a definition of Pareto efficient allocations?

5. Determine Ann’s and Bob’s demand functions for all possible prices p1 and p2.

6. Derive the Walrasian equilibrium (i.e., the equilibrium allocation and the equilibrium prices)for the pure exchange economy. Normalize the price of good 1 to be 1,p1= 1. Illustrate theequilibrium situation in the Edgeworth box.

7. Using the contract curve, determine whether the equilibrium allocation is Pareto Efficient or not. Can we conclude that the first theorem of welfare economics is satisfied?

Answer 1

7)

First welfare theorem says that if preferences are strictly convex , no market welfare exists,

Then all Competitive eqm CE are pareto effecient

So as CE lies on Cobtract curve, so CE is pareto optimal, thus First welfare theorem holds