Question

In an exchange economy there are two people (A and B), and two goods (x1 and x2). Their respective utility functions and endowments are: UA(xA1 ,xA2 ) = xA1 +2xA2 , UB(xB1 ,xB2 ) = 2xB1 +xB2 , ωA = (1,1), ωB = (1, 1). What are the minimum and maximum possible amounts of xA1 on the contract curve?

Answer 1

Finding the minimum and maximum amounts of xA1 on the contract curve involves analyzing the utility functions and MRS for two individuals in an exchange economy. In this case, the diagonal connecting individual endowments represents the contract curve where no further trade can improve both individuals' utilities.

Step-by-step explanation:

To find the minimum and maximum possible amounts of xA1 on the contract curve in an exchange economy, one would typically set up the utility maximization problem subject to a budget constraint. In this case, however, since the utility functions and endowments are clearly specified, and considering that the endowments are identical and the utility functions are linear, the contract curve can be derived from the fact that individuals will trade until the marginal rate of substitution (MRS) of each other's utility functions is equal. The MRS for person A is the rate at which they are willing to substitute good 2 for good 1 without changing their utility level, similarly for person B.

Since person A's utility (UA) is xA1 + 2xA2 and person B's utility (UB) is 2xB1 + xB2, the MRS for A is 2, and for B it is 0.5. The contract curve in this scenario will feature all allocations where MRS of A equals MRS of B (i.e., when 2 = 0.5), which is not plausible. Hence, the trade would result in a situation where there is no further scope for mutually beneficial trade. Given their endowments and utility functions, the contract curve will be the diagonal line connecting the two endowments as they will only exchange goods on a one-for-one basis.