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31. Exchange - Solutions

31. Exchange - Solutions - Chapter 31 NAME Exchange...

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Chapter 31 NAME Exchange Introduction. The Edgeworth box is a thing of beauty. An amazing amount of information is displayed with a few lines, points and curves. In fact one can use an Edgeworth box to show just about everything there is to say about the case of two traders dealing in two commodities. Economists know that the real world has more than two people and more than two commodities. But it turns out that the insights gained from this model extend nicely to the case of many traders and many commodities. So for the purpose of introducing the subject of exchange equilibrium, the Edgeworth box is exactly the right tool. We will start you out with an example of two gardeners engaged in trade. You will get most out of this example if you fill in the box as you read along. Example: Alice and Byron consume two goods, camelias and dahlias. Alice has 16 camelias and 4 dahlias. Byron has 8 camelias and 8 dahlias. They consume no other goods, and they trade only with each other. To describe the possible allocations of flowers, we first draw a box whose width is the total number of camelias and whose height is the total number of dahlias that Alice and Byron have between them. The width of the box is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12. Dahlias Byron 12 6 0 6 12 18 24 Alice Camelias Any feasible allocation of flowers between Alice and Byron is fully described by a single point in the box. Consider, for example, the alloca- tion where Alice gets the bundle (15 , 9) and Byron gets the bundle (9 , 3). This allocation is represented by the point A = (15 , 9) in the Edgeworth box, which you should draw in. The distance 15 from A to the left side of the box is the number of camelias for Alice and the distance 9 from A to the bottom of the box is the number of dahlias for Alice. This point also determines Byron’s consumption of camelias and dahlias. The distance 9 from A to the right side of the box is the total number of camelias consumed by Byron, and the distance from A to the top of the box is the number of dahlias consumed by Byron. Since the width of the box is the total supply of camelias and the height of the box is the total supply of
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384 EXCHANGE (Ch. 31) dahlias, these conventions ensure that any point in the box represents a feasible allocation of the total supply of camelias and dahlias. It is also useful to mark the initial allocation in the Edgeworth box, which, in this case, is the point E = (16 , 4). Now suppose that Alice’s utility function is U ( c, d ) = c +2 d and Byron’s utility funtion is U ( c, d ) = cd . Alice’s indifference curves will be straight lines with slope 1 / 2. The indifference curve that passes through her initial endowment, for example, will be a line that runs from the point (24 , 0) to the point (0 , 12). Since Byron has Cobb-Douglas utility, his indifference curves will be rectangular hyperbolas, but since quantities for Byron are measured from the upper right corner of the box, these indifference curves will be flipped over as in the Edgeworth box diagrams in your textbook.
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