Final answer:
To find the contract curve, we calculate the marginal rates of substitution for each trader from their utility functions, set them equal to each other, and solve under the constraints of the total amounts of goods G and H available.
Step-by-step explanation:
To solve for the contract curve in a pure exchange economy with two traders with Cobb-Douglas utility functions, we first need to determine the marginal rates of substitution (MRS) for each trader, since the contract curve is where the MRS of the two traders is equal. Tony's utility function is U_t = G_t * H_t, and Margaret's utility function is U_m = G_m * (H_m)^2. We start by calculating the marginal utilities:
The MRS for each trader is the ratio of their marginal utilities of good G to good H. So we have:
Tony's MRS: MRS_t = MU_{Gt} / MU_{Ht} = H_t / G_t
Margaret's MRS: MRS_m = MU_{Gm} / MU_{Hm} = (H_m)^2 / (2 * G_m * H_m) = H_m / (2 * G_m)
For the contract curve, we equate the MRS of Tony to the MRS of Margaret:
H_t / G_t = H_m / (2 * G_m)
Since they own 100 units of G and 50 units of H between them, we have the constraints:
G_t + G_m = 100
H_t + H_m = 50
Substituting the constraints into the MRS equations and solving them simultaneously gives us the points on the contract curve.