Find the contract curve as a function and graph it

Consider the setting of the exchange economy is as follows:

• There are only two goods.

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• 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 1 and 2.

Assume that utility functions are Cobb-Douglas form, that is:

u(q1, q2)=(q1^α)(q2^(1-α))

v(z1, z2)=(z1^β)(z2^(1-β))

To make the analysis easier, assume that 1 = 2 = 1

(1) Find the contract curve as a function of q1 vs q2 and graph it on the q1 − q2 plane when α= β = 1/2.

(2) Similarly, find the contract curve and graph it when α = 2/3 and β= 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.

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