Consider the setting of the exchange economy is as follows:
• There are only two goods.
• There are only two individuals (economic agents).
• No production: only exchange between these two people is possible; goods are consumed.
• The first agent has the utility function u(q1, q2)
• The second agent has the utility function v(z1, z2)
• The total endowments of these goods are W1 and W2.
Assume that utility functions are Cobb-Douglas form, that is:
To make the analysis easier, assume that W1 = W2 = 1
(1) Find the contract curve as a function of q1 vs q2 and graph it on the q1 − q2 plane when a= b = 1/2.
(2) Similarly, find the contract curve and graph it when a = 2/3 and b= 1/2.
Hint: Recall that on the contract curve, marginal rates of substitution (MRSs) must be equalized: MRS1=MRS2
Also, recall that MRS=marginal utility for good 1/marginal utility for good 2.
Use the relations: z1 = W1 − q1, z2 = W2 − q2.