Lecture12 - ECON 301 LECTURE#12 WELFARE THEOREMS STATEMENT...

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1 ECON 301 – LECTURE #12 WELFARE THEOREMS STATEMENT: “A competitive equilibrium is Pareto Optimal”. PROOF: (by contradiction) Suppose that an allocation bundle x (i.e. x = ( x 1 A , x 2 A , x 1 B , x 2 B )) is a competitive equilibrium that is not Pareto Optimal. Thus, there exists an allocation bundle y (i.e. y = ( y 1 A , y 2 A , y 1 B , y 2 B )) such that y is feasible… y 1 A + y 1 B = ! 1 A + ! 1 B (1) and U A ( y ) ! U A ( x ) but U B ( y ) > U B ( x ) y 2 A + y 2 B = ! 2 A + ! 2 B (2) However, since x was the allocation bundle chosen for utility maximization at the equilibrium price vector, we have… P 1 x 1 A + P 2 x 2 A = P 1 ! 1 A + P 2 ! 2 A Budget Constraints are satisfied. P 1 x 1 B + P 2 x 2 B = P 1 ! 1 B + P 2 ! 2 B meaning that for allocation bundle y to be “better” than allocation bundle x the following must be true… P 1 y 1 A + P 2 y 2 A " P 1 ! 1 A + P 2 ! 2 A P 1 y 1 B + P 2 y 2 B > P 1 ! 1 B + P 2 ! 2 B which implies, P 1 (y 1 A + y 1 B ) + P 2 (y 2 A + y 2 B ) > P 1 ( ! 1 A + ! 1 B ) + P 2 ( ! 2 A + ! 2 B ) (3) but the allocation bundle y must be feasible. ..subbing (1) and (2) into (3), we get P 1 ( ! 1 A + ! 1 B ) + P 2 ( ! 2 A + ! 2 B ) > P 1 ( ! 1 A + ! 1 B ) + P 2 ( ! 2 A + ! 2 B ) the above says that, at allocation bundle y , the value of the individual endowments for good 1 and good 2 exceeds the value of the individual endowments of goods 1 and 2 in the economy. So at this equilibrium price vector the allocation bundle y is not feasible…we have our contradiction!
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2 We derived this contradiction by assuming that the competitive equilibrium solution (allocation bundle x ) was not Pareto efficient. Therefore, this assumption must be wrong. THE FIRST FUNDAMENTAL THEOREM The First Fundamental Theorem of Welfare Economics provides the link which connects the Walrasian General Equilibrium Theory to the Pareto Optimal Theory. On one hand, Walrasian Theory deals with the various concepts of equilibrium. On the other hand, Pareto Theory deals with the various concepts of efficiency. The First Fundamental Theorem of Welfare Economics essentially says that “the Walrasian concept of general equilibrium is consistent with the Pareto concept of efficiency”. So what does this mean in the context of our Pure Exchange Economy? We have studied both the Walrasian general equilibrium pure exchange economy and the Pareto optimal pure exchange economy with two goods (X,Y) and two consumers (A,B). Let’s summarize what we know so far… WALRASIAN PURE EXCHANGE PARETO OPTIMAL PURE EXCHANGE The Walrasian pure exchange model The Pareto Optimal pure exchange model provides equilibrium prices (P X , P Y ) provides optimal allocations (X A , X B , Y A , and allocation (X A , X B , Y A , Y B ) such Y B ) such that that (a) no one can be made better off without (a) both consumers are in equilibrium making the other worse off (indifference (indifference curves are tangent to curves are tangent to each other) budget lines) (b) both goods are fully allocated (quantities
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Lecture12 - ECON 301 LECTURE#12 WELFARE THEOREMS STATEMENT...

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