Pareto Efficiency

In the last two sections we saw how to construct the production possibilities set, the set that describes the feasible consumption bundles for the economy as a whole. Here we consider Pareto efficient ways to choose among the feasible consumption bundles.

We will indicate aggregate consumption bundles by (Xl,X2). This indicates that there are X1 units of good 1 and X2 units of good 2 that are available for consumption. In the Crusoe/Friday economy, the two goods are coconuts and fish, but we will use the (X1, X2) notation in order to emphasize the similarities with the analysis in Chapter 31. Once we know the total amount of each good, we can draw an Edgeworth box as in Figure 32.9.

Given (X1,^2), the set of Pareto efficient consumption bundles will be the same sort as those examined in the last chapter: the Pareto efficient consumption levels will lie along the Pareto set—the line of mutual tan-gencies of the indifference curves, as illustrated in Figure 32.9. These are the allocations in which each consumer's marginal rate of substitution—the rate at which he or she is just willing to trade—equals that of the other.

These allocations are Pareto efficient as far as the consumption decisions are concerned. If people can simply trade one good for another, the Pareto set describes the set of bundles that exhausts the gains from trade. But in an economy with production, there is another way to "exchange" one good for another—namely, to produce less of one good and more of another.

GOOD 2

Equilibrium consumption ^

Equilibrium consumption ^

GOOD

Production and the Edgeworth box. At each point on the production possibilities frontier, we can draw an Edgeworth box to illustrate the possible consumption allocations.

The Pareto set describes the set of Pareto efficient bundles given the amounts of goods 1 and 2 available, but in an economy with production those amounts can themselves be chosen out of the production possibilities set. Which choices from the production possibilities set will be Pareto efficient choices?

Let us think about the logic underlying the marginal rate of substitution condition. We argued that in a Pareto efficient allocation, the MRS of consumer A had to be equal to the MRS of consumer B: the rate at which consumer A would just be willing to trade one good for the other should be equal to the rate at which consumer B would just be willing to trade one good for the other. If this were not true, then there would be some trade that would make both consumers better off.

Recall that the marginal rate of transformation (MRT) measures the rate at which one good can be "transformed" into the other. Of course, one good really isn't being literally transformed into the other. Rather the factors of production are being moved around so as to produce less of one good and more of the other.

Suppose that the economy were operating at a position where the marginal rate of substitution of one of the consumers was not equal to the marginal rate of transformation between the two goods. Then such a position cannot be Pareto efficient. Why? Because at this point, the rate at which the consumer is willing to trade good 1 for good 2 is different from the rate at which good 1 can be transformed into good 2—there is a way to make the consumer better off by rearranging the pattern of production.

Suppose, for example, that the consumer's MRS is 1; the consumer is just willing to substitute good 1 for good 2 on a one-to-one basis. Suppose that the MRT is 2, which means that giving up one unit of good 1 will allow society to produce two units of good 2. Then clearly it makes sense to reduce the production of good 1 by one unit; this will generate two extra units of good 2. Since the consumer was just indifferent between giving up one unit of good 1 and getting one unit of the other good in exchange, he or she will now certainly be better off by getting two extra units of good 2.

The same argument can be made whenever one of the consumers has a MRS that is different from the MRT—there will always be a rearrangement of consumption and production that will make that consumer better off. We have already seen that for Pareto efficiency each consumer's MRS should be the same, and the argument given above implies that each consumer's MRS should in fact be equal to the MRT.

Figure 32.9 illustrates a Pareto efficient allocation. The MRSs of each consumer are the same, since their indifference curves are tangent in the Edge worth box. And each consumer's MRS is equal to the MRT—the slope of the production possibilities set.

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