Chapter 1 Preferences and Utility 1. Fundamental Assumptions The rst fundamental assumption that we make about people is that they know what they like: they know their preferences among the set of things. If a person is given a choice between x and y he can say (one and only one sentence is true): 1. He prefers x to y 2. He prefers y to x 3. He is indifferent between the two. This is the axiom of completeness. It seems reasonable enough. The second fundamental assumption is the axiom of transitivity. The assumpştion has four parts: 1. If a person prefers x to y and prefers y to z, then he prefers x to z. 2. If a person prefers x to y and is indifferent between y and z, then he prefers x to z. 3. If a person is indifferent between x and y and prefers y to z, then he prefers x to z. 4. If a person is indifferent between x and y, and is indifferent between y and z, then he is indifferent between x and z. 2. Best Alternatives and Utility Functions Asking "Would you prefer x to y" will never get you a measure of utility with well-de ned units, a zero, and other nice mathematical properties. But it will allow you to nd alternatives that are at least as good as all others, and, remarkably, it will allow you to construct a numerical measure to re ect tastes. The determination of best alternatives and the construction of a measure of satisfaction are both made possible by the completeness and transitivity assumptions on preferences. Therefore, the theory of preferences, with those two assumptions, is connected to and is a generalization of, the old-fashioned nineteenth-century theory of utility. 3. The Formal Model of Preferences Before we can proceed, we need to introduce some notation. Let x and y be two alternatives. We consider a group of people who are numbered 1, 2, 3, and so on. To symbolize the preferences of the ith person we write xRiy for "i prefers x is at least as good as y"; xPiy for "i prefers x to y"; and xliy for "i is indifferent between x and y.” xPiy if xRiy and not yRix xliy if xRiy and yRix In words: Person i prefers x to y if he thinks x is at least as good as y but he does not think y is at least as good as x. And i is indifferent between x and y if he thinks x is at least as good as y and he thinks y is at least as good as x. Now our fundamental axioms of completeness and transitivity are formally put this way: Completeness. For any pair of alternatives x and y, either xRiy or yRiX. fi fi fl fi Transivity. For any three alternatives x,y, and z, if xRiy and yRiz., then xRiz. 3. Decisions under Uncertainty and Expected Utility . . . Chapter 2 Barter Exchange 1. Introduction This chapter will be about such swapping, or barter exchange. In order to analyze barter exchange, we will construct a model in which a group of people exchange bundles of goods among themselves. 2. Allocations The set of alternatives is now a set of distributions or allocations of goods in an economy. ✦ There are n people, numbered 1,2,…,n. Usually a person is indexed with the letter i. Often we’ll let n = 2, in which case we are talking about what happens when there are only two people (like Adam and Eve exchanging fruit in the Garden of Eden). ✦ There are m different goods. Typically we index a good with the letter j, so the goods are numbered j = 1,2,…,m. This is the entire list of goods. In some context, there might be only one (m = 1) or two (m = 2) goods. some more important notation; ✦ We let xij be person i’s quantity of good j. ✦ We let xi person i’s bundle or vector of goods xi. =(xi1, xi2,…, xim); so xi shows m things, i’s quantity of good 1, i’s quantity of good 2, …, i’s quantity of good m. ✦ We de ne x = (x1, x2, …, xn). Now x shows person 1’s bundle of goods, person 2’s bundle of goods, …., person n’s bundle of goods. In the theory of exchange there is no production; wha goods are available are ther ein the beginning. We let wij be person i’s starting or initial quantity of good j. Similarly, wi is his initial bundle, and w is the initial list of bundles of goods. The total of hood j available must be w1j + w2j + w3j + ... + wnj = The symbol n “ ∑ i=1 n ∑ i=1 wij . wij” Is short-hand for “summation of the wij’s, where i ranges from 1 to n.” An allocation is a list of bundles of goods x with totals of goods consistent with the totals initially available. This means n ∑ i=1 xij must be equal to n ∑ i=1 wij fi for every j , that is, every good. We call the set of allocations a, and a is formally written as follows: n a = { x | xij a ≥ for all i,j and ∑ i=1 xij = n ∑ i=1 wij for all j } . The set of alternatives in the theory of exchange is a. One of the simplest examples of an exchange economy involves only two people and one good; so n = 2 and m = 1. Let the total quantity of the good initially available,w11 + w21, be equal to 1. Then a is the set of all pairs (x11 + x21) such that x11 + x21 ≥ 0 and x11 + x21 = 1. The set of allocations in this economy can be easily diagrammed. To picture a, draw a line segment one unit long. Choose a point x on the line segment, and let the distance from the lefthand end of the line to x represent the person i's quantity of the good x11, and let the distance from the righthand end of line x represent person 2's quantity of the good x21. Now x11 + x21 ≥ 0 and x11 + x21 = 1, so every such point represents an allocation, and, conversely, every allocation can be represented by such a point or division of the line. 3. The Edgeworth Box Diagram The most useful example of an exchange economy is one in which there are two people and two goods. This economy's set of allocations can be illustrated in an Edgeworth box diagram. 4. Pareto Optimal Allocations and the Core An allocation x is not Pareto optimal if there is another allocation y such that ui(y) ≥ ui(x) for all i = 1,2,...,n ui(y) ≥ ui(x) for at least one i . If there is no such alternative, x is a Pareto optimal, or ef cient allocation. fi Let's consider an illustration in an Edgeworth box diagram. When there are just two traders and two goods, and both traders have convex- shaped smooth indifference curves like the curves in Figure 2.3, the Pareto optimal allocations are the points, like x, y, and z, at which indifference curves of the two people are tangent. At these points, it is impossible to make one party better off without hurting the other. Point X, however, is blocked by person 1, since u1(w1) > u1(x1). Similarly, z is blocked by 2. The core is the locus of Pareto optimal points, such as y, lying on or within the lens-shaped area bounded by the indifference curves passing through w. 5. Algebraic Examples How to calculate the Pareto optimal and core allocations in an exchange economy. u1 = x11x12 u2 = x21 + x22 . In other words, person I's utility level is the product of the quantities of the two goods he has, person 2's utility level is equal to the amount of good 1 he has plus twice the amount of good 2 he has. Let the initial allocation be w1 = (1/2,1/2), w2 = (1/2,1/2). Each starts with 1/2 unit of each good. In this case, person 1's indifference curves are hyperbolic (because x11 x12 = constant is the formula for a hyperbola) and person 2's indifference curves are straight lines. In order to proceed, we need to nd expressions for the marginal rates of substitution or the absolute values of the slopes of the two people's indifference curves. The loss in utility equals the gain in utility, in absolute value, ▵ xi1 . MU of good 1 for i = ▵ xi2 . MU of good 2 for i From which it follows that ▵ xi2 ▵ xi1 MRS for person i = = MU of good 1 for i . MU of good 2 for i MRS person 1 = MRS person 2. This gives x12 1 1 = or x12 = x11 x11 2 2 fi Graphically, this is the straight line from person i’s origin to point z. This straight line segment gives part of the set of Pareto optimal allocations since any move from a point on it (like x or y) must make someone worse off. But there are Pareto optimal allocations other than these tangency points. Consider, for instance, the point w. In order to make person 1 better off, starting at w, we would have to move above the hyperbolic indifference curve for person 1 going to w. This would make person 2 worse off. That is, there is no way to make everyone as well off and at least one better off. In fact, all the points on the right-hand side of the box above point z are nontangency Pareto optimal allocations. Where are the core allocations in this example? Any core allocation must also be Pareto optimal, so the core allocations lie somewhere on the lines we've already identi ed as Pareto optimal. But a core allocation must not be blocked by person 1, or by person 2. Person 1 would block any allocation that gives him less utility than w1, or any allocation below his hyperbolic indifference curve through w. Similarly, person 2 would block any allocation that gives him less utility than w1, or any allocation below (with respect to his origin) his straight-line indifference curves through w. Consequently, the core is the locus of points on the straight line between points x and y, including the endpoints x and y. fi In this example, then, we would expect barter exchange between persons 1 and 2 to move the economy from w to a point on the line segment from x to y.