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UCSB ECON 1 - PUBLIC ECONOMY

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Lecture 1–A Primitive Public EconomyTheodore Bergstrom, UCSBMarch 31, 2002c1998Chapter 1A Primitive Public EconomyAnne and Bruce are roommates. They are interested in only two things; thetemperature of their room and playing cribbage together. Each of them hasa different favorite combination of room temperature and games of cribbageper week. Anne’s preferred temperature may depend on the number ofgames of cribbage that she is allowed to play per week and her preferrednumber of games of cribbage may depend on the room temperature. Giventhe number of games of cribbage, the further the temperature deviates fromher favorite level, the less happy she is. Similarly, given the temperature, Anne is less happy the more the number of games of cribbage differs from herpreferred number. Bruce’s preferences have the same qualitative characteras Anne’s, but his favorite combination is different from hers.The landlord pays for the cost of heating their room and the cost of adeck of cards is negligible. Since there are no scarce resources in the usualsense, you might think that there is not much here for economists to study.Indeed if Anne lived alone and her only choices involved temperature andsolitaire, the economic analysis would be pretty trivial. She would pick herbliss point and that’s that.The tale of Anne and Bruce is economically more interesting because al-thought they may disagree about the best temperature and the best amountof cribbage-playing, each must live with the same room temperature and(since they are allowed no other game-partners) each must play the samenumber of games of cribbage as the other. Somehow they will have to set-tle on an outcome in the presence of conflicting interests. This situationturns out to be a useful prototype for a wide variety of problems in publiceconomics.We begin our study with an analysis of efficient conduct of the Anne–1Figure 1.1: Indifference Curves for Anne and BruceABZWYXVGames of CribbageTemperatureBruce household. A diagram will help us to understand how things are withAnne and Bruce. In Figure 1.1, the points A and B represent Anne’s andBruce’s favorite combinations of cribbage and temperature. These points areknown as Anne’s and Bruce’s bliss points, respectively. The closed curvesencircling A are indifference curves for Anne. She regards all points onsuch a curve as equally good, while she prefers points on the inside of herindifference curves to points on the outside. In similar fashion, the closedcurves encircling B are Bruce’s indifference curves.We shall speak of each combination of a room temperature and a numberof games of cribbage as a situation. If everybody likes situation α as well assituation β and someone likes α better, we say that α is Pareto superior toβ. A situation is said to be Pareto optimal if there are no possible situationsthat are Pareto superior to it. Thus if a situation is not Pareto optimal,it should be possible to obtain unanimous consent for a beneficial change.If the existing situation is Pareto optimal, then there is pure conflict ofinterest in the sense that any benefit to one person can come only at thecost of harming another.Our task is now to find the set of Pareto optimal situations, chez Anneand Bruce. Consider a point like X in Figure 1.1. This point is not Paretooptimal. Since each person prefers his inner indifference curves to his outer2ones, it should be clear that the situation Y is preferred by both Anne andBruce to X. Anne and Bruce each have exactly one indifference curve pass-ing through any point on the graph. At any point that is not on boundaryof the diagram, Anne’s and Bruce’s indifference curves through this pointeither cross each other or are tangent. If they cross at a point, then, by justthe sort of reasoning used for the point X, we see that this point can notbe Pareto optimal. Therefore Pareto optimal points must either be pointsat which Anne’s indifference curves are tangent to Bruce’s or they must beon the boundary of the diagram.In Figure 1.1, all of the Pareto optimal points are points of tangency be-tween Anne’s and Bruce’s indifference curves. Points Z and W are examplesof Pareto optima. In fact there are many more Pareto optima which couldbe found by drawing more indifference curves and finding their tangencies.The set of such Pareto optima is depicted by the line BA in Figure 1.1.Although every interior Pareto optimum must be a point of tangency, notevery interior point of tangency is a Pareto optimum. To see this, take alook at the point V on the diagram. This is a point of tangency between oneof Anne’s indifference curves and one of Bruce’s. But the situation V is notPareto optimal. For example, both Anne and Bruce prefer B to V . In ourlater discussion we will explain mathematical techniques that enable you todistinguish the “good” tangencies, like Z and W , from the “bad” ones, likeV .Let us define a person’s marginal rate of substitution between tempera-ture and cribbage in a given situation to be the slope of his indifference curveas it passes through that situation. From our discussion above, it shouldbe clear that at an interior Pareto optimum, Anne’s marginal rate of sub-stitution between temperature and cribbage must be the same as Bruce’s.If we compare a Pareto optimal tangency like the point Z in Figure 1.1with a non–optimal tangency like the point V , we notice a second necessarycondition for an interior Pareto optimum. At Z, Anne wants more cribbageand a lower temperature while Bruce wants less cribbage and a higher tem-perature. At V , although their marginal rates of substitution are the same, both want more cribbage and a lower temperature. Thus a more completenecessary condition for a Pareto optimum is that their marginal rates ofsubstitution be equal and their preferred directions of change be opposite.3The Utility Possibility Frontier and the ContractCurveWith the aid of Anne and Bruce we can introduce some further notions thatare important building blocks in the theory of public decisions.The Utility Possibility Set and the Utility Possibility FrontierSuppose that Anne and Bruce have utility functions UA(C, T ) and UB(C, T ),representing their preferences over games of cribbage and temperature. Wecan graph the possible distributions of utility between them. On the hori-zontal axis of Figure 1.2, we measure Anne’s utility and on the vertical axiswe measure Bruce’s


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