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Unformatted text preview: Pareto Efficiency in an Exchange Economy Todd Sarver Northwestern University Econ 3102 – Fall 2011 Todd Sarver (Northwestern University) PE in an Exchange Economy Econ 3102 – Fall 2011 1 / 22 Setting 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. Todd Sarver (Northwestern University) PE in an Exchange Economy Econ 3102 – Fall 2011 2 / 22 Setting 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. 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. Todd Sarver (Northwestern University) PE in an Exchange Economy Econ 3102 – Fall 2011 2 / 22 Preferences Utility functions: u i ( x i ,y i ) We will also assume that x and y are “goods” in the following sense: Axiom (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 ) . Todd Sarver (Northwestern University) PE in an Exchange Economy Econ 3102 – Fall 2011 3 / 22 Preferences Utility functions: u i ( x i ,y i ) We will also assume that x and y are “goods” in the following sense: Axiom (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 ) . Feasible Allocations In this economy, the set of feasible allocations (or alternatives) consists of all divisions of the goods between the individuals, that is, the set of all allocations (( x 1 ,y 1 ) , ( x 2 ,y 2 )) such that x 1 ,x 2 ,y 1 ,y 2 ≥ , x 1 + x 2 = e x , y 1 + y 2 = e y . Todd Sarver (Northwestern University) PE in an Exchange Economy Econ 3102 – Fall 2011 3 / 22 Edgeworth Box (Francis Edgeworth (1845–1926)) Forget about consumer 2 for a moment. Feasible allocations for consumer 1 satisfy ≤ x 1 ≤ e x and ≤ y 1 ≤ e y ....
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This note was uploaded on 01/03/2012 for the course ECON 3102 taught by Professor Sarver during the Spring '08 term at Northwestern.
 Spring '08
 SARVER
 Microeconomics

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