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Unformatted text preview: Section 7 Pareto Efficiency in an Exchange Economy Todd Sarver Northwestern University Winter 2011 1 An Exchange Economy In this lecture, we will be studying an exchange economy. An exchange economy is an economy in which there is no production, but individuals have endowments of goods. In previous examples, we have already examined Pareto efficiency in the context of an exchange economy with a single good. We now consider the (more interesting) problem of determining which allocations are Pareto efficient in an economy with multiple goods. Although we will pay special attention to the case of 2 goods, the main ideas of this lecture generalize to the case of many goods. Formally, the environment we are considering is an economy with two consumers, 1 and 2, and two goods, x and y . Let x i denote the quantity of good x allocated to consumer i , and let y i denote the quantity of good y allocated to consumer i . Thus, ( x 1 ,y 1 ) denotes the final allocation to consumer 1. ( x 2 ,y 2 ) denotes the final allocation to consumer 2. But how much of each good is available to allocate between these consumers? Although there is no production in this economy, we assume that each consumer has an initial endowment of each good which can then be redistributed between the two consumers. Let e x i denote consumer i s initial endowment of good x , and define e y i similarly. ( e x 1 ,e y 1 ) denotes the endowment allocation of consumer 1. ( e x 2 ,e y 2 ) denotes the endowment allocation of consumer 2. e x = e x 1 + e x 2 is the aggregate amount of good x available in the economy. e y = e y 1 + e y 2 is the aggregate amount of good y available in the economy. At this point, we have not said anything about the mechanism for distributing the two goods and how the final allocation depends on the initial endowments and preferences of the agents (or if it depends on these at all). In later lectures, we will study particular mechanisms for redistributing Section 7 Pareto Efficiency in an Exchange Economy 2 endowments, such as equilibrium trade between the agents. However, for the moment we are only interested in whether the final allocation is Pareto optimal, not how this allocation comes about. Each consumer i has a complete and transitive preference i over the allocations of the goods, represented by a utility function u i . For now, we assume that each consumer cares only about his own consumption, so i and u i depend only on ( x i ,y i ). Therefore, we will simplify notation and write u i ( x i ,y i ) instead of u i ( x 1 ,y 1 ,x 2 ,y 2 ). We will also assume that x and y are goods in the sense that both consumers strictly prefer more of each good to less. This standard assumption is known as the monotonicity axiom: Axiom 1 (Monotonicity) If x i x i and y i y i , then ( x i ,y i ) i ( x i , y i ) . If, in addition, x i > x i or y i > y i , then ( x i ,y i ) i ( x i , y i ) ....
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