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Section-12-CE-Welfare-Theorems-slides

# Section-12-CE-Welfare-Theorems-slides - Competitive...

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Competitive Equilibrium Welfare Theorems Todd Sarver Northwestern University Econ 310-2 – Fall 2011 Todd Sarver (Northwestern University) Competitive Equilibrium Welfare Theorems Econ 310-2 – Fall 2011 1 / 16

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Setting: An Exchange Economy In this lecture, we will focus on the environment of an exchange economy introduced in the previous set of slides. However, these results also hold in economies with production. Todd Sarver (Northwestern University) Competitive Equilibrium Welfare Theorems Econ 310-2 – Fall 2011 2 / 16
Setting: An Exchange Economy In this lecture, we will focus on the environment of an exchange economy introduced in the previous set of slides. However, these results also hold in economies with production. Reminder of the setting: Suppose there are n consumers and two goods, x and y . (For simplicity, we will often focus on the case of 2 consumers.) Use ( x i , y i ) to denote an allocation for consumer i . Each consumer has a utility function u i ( x i , y i ) . Endowment economy (i.e., no production): Each consumer has an initial endowment of each good and can buy and sell the goods at the going market prices p x and p y . Let ( e x i , e y i ) denote consumer i ’s initial endowments of x and y . Todd Sarver (Northwestern University) Competitive Equilibrium Welfare Theorems Econ 310-2 – Fall 2011 2 / 16

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First Theorem of Welfare Economics Recall the monotonicity axiom: 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) Competitive Equilibrium Welfare Theorems Econ 310-2 – Fall 2011 3 / 16
First Theorem of Welfare Economics Recall the monotonicity axiom: 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 ) . So long as this basic property of preferences is satisfied, we get the following important result for any competitive equilibrium: Theorem (First Theorem of Welfare Economics) Given that consumers’ preferences satisfy the monotonicity axiom, any competitive equilibrium allocation is Pareto efficient. Todd Sarver (Northwestern University) Competitive Equilibrium Welfare Theorems Econ 310-2 – Fall 2011 3 / 16

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First Theorem of Welfare Economics Recall the monotonicity axiom: 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 ) . So long as this basic property of preferences is satisfied, we get the following important result for any competitive equilibrium: Theorem (First Theorem of Welfare Economics) Given that consumers’ preferences satisfy the monotonicity axiom, any competitive equilibrium allocation is Pareto efficient.
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