Edgeworth Box and Contract Curve

LEARNING GOALS:

V.B.1: Determine the conditions under which two consumers will have gains from trade.
V.B.2: Solve for the competitive equilibrium of an exchange economy and graphically depict it in an Edgeworth box.
V.B.4: Define and solve for the lens, contract curve, and core of an exchange economy.

Consider the following Edgeworth box economy. There are two consumers, Aria and Brixton, and two goods, 1 and 2. The two consumers have identical preferences represented by the following utility functions:

$u^A(x^A_1, x^A_2) = (x^A_1)^\frac{1}{2}(x^A_2)^\frac{1}{2} \; \textrm{and} \; u^B(x^B_1, x^B_2) = (x^B_1)^\frac{1}{2}(x^B_2)^\frac{1}{2}$

Full screen version

In the above graph, drag the endowment point to the place where Aria (origin bottom-left) has $x_1^A = 120$ units of good 1 and $x_2^A = 20$ units of good 2. How much of each good does Brixton have?
Check the “show indifference curves” box. This will show the indifference curves for Aria and Brixton that go through the endowment point. Who is better off at this point?
Describe in words the region of the graph where both Aria and Brixton are better off than they are at the endowment point. What is this region called?
Check the “show potential trade” box and drag the point, X, to a place where you think Aria and Brixton could end up if they were allowed to trade with each other. In words, what must be true of the indifference curves at this equilibrium point?
Now check the “show contract curve” box. Is the point you chose on the contract curve? Should it be? Explain the relationship between Pareto efficiency and the contract curve.
Use 3 equations to characterize the set of Pareto efficient allocations.
Set up the utility maximization problems for Aria and Brixton, write down the Lagrangians, and derive their first order conditions.
In equilibrium, the resource constraints must hold, and Aria and Brixton face the same prices. Set the price of good 1 to 1, and solve for the equilibrium price of good 2 and the equilibrium allocations of the goods.
Is your equilibrium allocation on the contract curve? Compare the equilibrium allocation you found to the X you chose in (d).