Consider the same consumers A and B that live for two periods and suppose now !1 = 6 and owned by A. This is the entire crop of this economy at date 1. Suppose also that !2 = 12 and owned by B. This is the entire crop of this economy at date 2. The good is perishable and cannot store. (a) Suppose A and B invented a way to trade over times 1 and 2. Formulate the consumer choice problems for A and B when such trade is possible. (b) What is the equilibrium? (c) Verify that trading over time allows to achieve Pareto e¢ ciency.

(a) Consumer Choice Problems:

For consumer A, the problem is to maximize utility over the two periods subject to the budget constraint. Let's denote the amount of good A consumes in period 1 as x1A and in period 2 as x2A. The utility function for A is U(x1A, x2A). The budget constraint is x1A + x2A/(1+r) <= 6, where r is the interest rate.

Similarly, for consumer B, the problem is to maximize utility over the two periods subject to the budget constraint. Let's denote the amount of good B consumes in period 1 as x1B and in period 2 as x2B. The utility function for B is U(x1B, x2B). The budget constraint is x1B + x2B/(1+r) <= 12.

(b) Equilibrium:

The equilibrium is determined by the market clearing condition, which requires that the total amount of the good consumed in each period equals the total amount of the good available in that period. In other words, x1A + x1B = 6 and x2A + x2B = 12.

Given the utility functions and budget constraints of A and B, we can solve these equations to find the equilibrium quantities x1A, x2A, x1B, and x2B, and the equilibrium interest rate r.

(c) Pareto Efficiency:

Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off.

In this case, if the trade between A and B is voluntary, it must be the case that both A and B are better off after the trade than before. Therefore, the trade leads to a Pareto improvement.

Furthermore, once the trade has taken place, there is no other possible trade that could make either A or B better off without making the other worse off. Therefore, the outcome after the trade is Pareto efficient.

To verify this, we can check that the marginal rate of substitution (MRS) between the two goods is the same for A and B in the equilibrium. The MRS is the rate at which a consumer is willing to trade off one good for another while maintaining the same level of utility. If the MRS is the same for A and B, it means that there is no other possible trade that could make either A or B better off without making the other worse off. Therefore, the equilibrium is Pareto efficient.