Before we begin our examination of the various aspects of an efficient general equilibrium with a discussion of exchange efficiency, we introduce an extremely handy tool for analyzing market efficiency, the Edgeworth box. The box is named after the Irish economist Francis Edgeworth, and it illustrates an economy with two consumers and two goods (or, alternatively, two producers and two inputs). This simple setup allows us to demonstrate almost all the concepts we need to understand market efficiency.

Suppose there are two consumers—we’ll call them Jerry and Elaine—each of whom has his or her own preference for two goods, bowls of cereal and pancakes. Let’s also suppose that there are a total of 10 bowls of cereal and 8 pancakes that can be split between them. We want to figure out in what Pareto-efficient ways we can distribute these goods between Jerry and Elaine.

To do this analysis, we use an Edgeworth box. An Edgeworth box utilizes the fact that when there is a fixed total number of goods to be split between two individuals, giving one more unit of a good to one person necessarily means the other person gets one less unit. In our case, for example, if Elaine gets one more bowl of cereal, Jerry must get one less. A point within or on the sides of the Edgeworth box shows the distribution of two goods (like cereal and pancakes) between two people (like Jerry and Elaine).

Figure 15.5 illustrates how the Edgeworth box works in our example. The horizontal sides of the box measure 10 bowls of cereal; the vertical sides measure 8 pancakes. The lower-left-hand corner represents one consumer (we’ll say Elaine here, though we could just as easily have picked Jerry) who receives 0 units of both goods. If we give Elaine one more bowl of cereal, we move her allocation 1 unit to the right. If we give her one more pancake, we move Elaine’s allocation 1 unit up.

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The upper-right-hand corner represents Jerry when he receives 0 units of both goods. Note that if Jerry receives no goods, Elaine receives all of both goods—10 bowls of cereal and 8 pancakes. If we give Jerry one more bowl of cereal, his allocation moves 1 unit to the left (one less bowl for Elaine). If we give him one more pancake, we move Jerry’s allocation down 1 unit (one less pancake for Elaine).

To get accustomed to working with the Edgeworth box, let’s do some quick examples. Consider an initial allocation at point A in Figure 15.5. At point A, Elaine has 3 bowls of cereal and 6 pancakes, and Jerry has the rest of the goods, 7 bowls of cereal and 2 pancakes. If we change the allocation by giving Elaine another bowl of cereal and taking one from Jerry, we are at point B; Elaine has 4 bowls of cereal and 6 pancakes and Jerry has 6 bowls of cereal and 2 pancakes. If we change the allocation from point B by taking away a pancake from Elaine and giving it to Jerry, we move to point C (Elaine has 4 bowls of cereal and 5 pancakes and Jerry has 6 bowls of cereal and 3 pancakes). Any change in the allocation that moves one of the consumers in a certain direction will move the other consumer by the same amount but in the opposite direction. As a result, the Edgeworth box allows us to see simultaneously the effects of changes in goods allocations on both consumers. Think of it as almost a game board where each person’s side is the opposite of the other.

To analyze whether allocations are efficient, we need to know consumers’ preferences. In Chapter 4, we learned that indifference curves show consumers’ preferences. To see the gains from trade, we add Elaine’s and Jerry’s indifference curves to the Edgeworth box (Figure 15.6).

Elaine’s preferences for bowls of cereal and pancakes are represented by indifference curves U_E1, U_E2, U_E3, and so on. Each shows combinations of cereal and pancakes that make Elaine equally well off. Indifference curves farther away from Elaine’s origin depict more of each good, and so represent higher utility levels. Jerry’s preferences are represented by indifference curves U_J1, U_J2, U_J3, and so on. Jerry’s indifference curves may look odd, but when you remember that Jerry’s origin is in the upper right corner, then they look like all other indifference curves: bowed toward the origin. Indifference curves farther away from Jerry’s origin show higher utility levels.

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Now that we’ve laid out Jerry and Elaine’s preferences, how can we use them to determine what an efficient allocation of the two goods would be? We start at an arbitrary allocation, point A, and ask if we can reshuffle who gets what and make both Elaine and Jerry better off by doing so. (Remember the definition of Pareto efficiency: There is no possible way to reallocate who gets what without making at least one individual worse off than before.) At allocation A, such a reshuffling is possible. Let’s see why.

Elaine and Jerry each have indifference curves that pass through point A, U_E3 and U_J3. We know from our analysis in Chapter 4 that any allocation giving Elaine quantities of cereal and pancakes that are above and to the right of U_E3 will give her a higher utility than at point A. Similarly, any allocation that raises Jerry’s utility will be below and to the left of U_J3. Knowing this, we can figure out the allocations that make both Elaine and Jerry better off, or at least make one of them better off without making the other worse off. These are the allocations in the shaded area in Figure 15.6. Any distribution of goods in this area must be on indifference curves (not drawn to keep the graph uncluttered) that correspond to higher utility levels for both Elaine and Jerry. Any change that moves the allocation of goods from A to somewhere in the shaded area makes both Elaine and Jerry better off. For example, if we gave one of Jerry’s bowls of cereal to Elaine and one of Elaine’s pancakes to Jerry, we’d be at point C inside the shaded area, and Jerry and Elaine would both have higher utility levels than at the initial allocation (point A).

Remember that a Pareto-efficient allocation is one that, if changed in any way, would make at least one person worse off. Because there is a set of allocations that would make both Jerry and Elaine better off (the shaded area), point A cannot be Pareto-efficient. (Reallocations that make at least one person better off without making anyone worse off are sometimes referred to as Pareto-improving reallocations or just Pareto improvements.) Jerry and Elaine, if given allocation A, would both be willing to make the trade: Elaine could give a pancake to Jerry in exchange for a bowl of cereal.8

If the existence of a shaded area like that in Figure 15.6 means an allocation isn’t Pareto-efficient, you might think that only allocations without shaded areas are Pareto-efficient, and you’d be right. To see what this would look like, let’s think about another allocation that is inside the shaded area, point C.

Although C gives both Jerry and Elaine higher utility levels than A, there are still mutually beneficial trades that can be made. These can be seen in Figure 15.7. For example, indifference curves U_E4 and U_J4 are Elaine and Jerry’s indifference curves that pass through C. The area that U_E4 and U_J4 enclose are cereal and pancake allocations that would make both Elaine and Jerry better off than they are with allocation C. Therefore, allocation C can’t be Pareto-efficient either. But, notice that this area is smaller than the corresponding area for allocation A. We seem to be closing in on Pareto efficiency.

To find a Pareto-efficient allocation, we need to locate one where there is no area between Jerry and Elaine’s indifference curves. When will this hold? When we reach indifference curves for Elaine and Jerry that meet at a single point, point D in Figure 15.7. Notice that Elaine and Jerry’s indifference curves that pass through this point, U_E5 and U_J5, are tangent to each other. Also notice that we can’t change the allocation even slightly without making either Jerry or Elaine worse off by going below and to the left of U_E5, or above and to the right of U_J5. Allocation D, therefore, is Pareto-efficient.

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We’ve shown that exchange efficiency is achieved when two consumers’ indifference curves are tangent. Only at such a tangency are there no mutually beneficial gains from trade. This tangency condition offers an interpretation for what must be true for exchange efficiency to hold. Recall from Chapter 4 that the slope of an indifference curve at any point reflects the marginal rate of substitution (MRS) between goods at that point. That means at a common tangency like point D, the slopes of both Jerry and Elaine’s indifference curves are the same. (Remember, the slope of a curve at a particular point equals the slope of a line tangent to the curve at that point.) Thus, at point D, Jerry and Elaine have the same MRS between cereal and pancakes. At any point that isn’t a Pareto-efficient allocation, consumers have different MRS for the goods. At points A and C in Figure 15.7, the difference in the slopes of Elaine and Jerry’s indifference curves is clear.

To understand why the MRSs need to be equal, think for a moment about what it would mean if they weren’t. One consumer would have a higher marginal utility from consuming one of the goods than the other consumer would. For the other good, the order of the two consumers’ marginal utilities would be switched. When marginal utilities for the same good are unequal across consumers, each could give a unit of her low-marginal-utility good to the other. She would be getting rid of what is, for her, a relatively low-value item, but she would receive in return the good for which she has a higher marginal utility. The same would be true of the consumer receiving the unit, so both individuals would be better off. Only when marginal utility ratios are equal across the two consumers—that is, when exchange efficiency has been achieved—would there not be a mutual benefit from trade.

In allocation A in Figure 15.7, Elaine would be willing to trade 3 pancakes for 1 bowl of cereal. That is, her MRS_cp (marginal rate of substitution of cereal for pancakes) and the absolute value of the slope of her indifference curve is 3. Jerry would trade 3 bowls of cereal for 1 pancake, or equivalently, one-third of a pancake for a bowl of cereal. So, Jerry’s MRS_cp = 1/3. Elaine therefore has a relatively high marginal utility from cereal at allocation A; she’d be willing to give up 3 pancakes to get 1 bowl. Jerry feels the opposite. He would enjoy another pancake so much he would be willing to part with 3 bowls of cereal for it. Clearly, both would benefit from making a trade. Jerry could give Elaine a bowl of cereal (which he doesn’t value much, relatively speaking) in exchange for a pancake (which he does). Elaine would be happy to accept that trade, and they would end up at allocation C, which we know Elaine and Jerry like better than A. In fact, any trade where Jerry gives Elaine a bowl of cereal in exchange for anywhere between one-third and 3 pancakes will make Jerry and Elaine better off. Mutually beneficial trade between Elaine and Jerry could continue like this until allocation D is reached. Only at that point, when their marginal rates of substitution are equal, would neither want to trade. Further exchange would necessarily make one of them worse off than at D, where their MRSs are equal.

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Prices and the Allocation of Goods Up to this point, we’ve talked about the allocation of goods as if magical economists shuffle goods among consumers. In reality, consumers choose how much of each good to consume given the prices they face. In Chapter 4, we learned that a utility-maximizing individual consumes the product bundle at which her MRS between goods equals the ratio of those goods’ prices. (This optimal consumption bundle is located where the indifference curve is tangent to the budget line.)

In Elaine and Jerry’s case, then, we know that they will both consume amounts where . We also just saw that Elaine and Jerry’s MRS_cp will be equal in a Pareto-efficient allocation. Thus, it must be that Pareto efficiency in a market implies

In other words, an efficient market will result in the goods’ price ratio equaling consumers’ marginal rates of substitution for those goods.

If an initial allocation is not efficient, consumers will be willing to sell their allocated goods that (for them) have marginal utilities below the market price. They could then use the proceeds to buy high-marginal-utility goods. There will be individuals whose relative marginal utilities for those goods are the reverse and who will be willing to be on the other side of those transactions. In this way, we don’t need magical economists to reshuffle goods among people to get everyone’s MRS to match. Prices drive people to do it themselves.

The Consumption Contract Curve The key condition of exchange efficiency that consumers’ marginal rates of substitution are equal—that is, their indifference curves are tangent to one another—is true at more than one location in the Edgeworth box. In Figure 15.8, we’ve drawn a more extensive set of Elaine and Jerry’s indifference curves, showing several tangencies. If we connect these tangencies and all those in between (remember, there are indifference curves running through every point in the box), every allocation on the line is Pareto-efficient—they all meet the equal-MRS test. This line, called the consumption contract curve, shows all possible Pareto-efficient allocations between two consumers buying two goods.

Looking at the contract curve emphasizes the distinction between efficiency and equity that we mentioned earlier. While all allocations along the contract curve are efficient, they have very different implications for Jerry and Elaine’s respective utility levels. Some—those toward the lower-left corner of the Edgeworth box—imply low utility for Elaine and high utility for Jerry. Those toward the upper right, on the other hand, will be good for Elaine and bad for Jerry.9

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