Lecture 3

Then uyi uxi for all i and uyi uxi for some i

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Unformatted text preview: → R. Let xi ∈ RN be the consumption vector (a + + vector that specifies how much of each good an agent consumes) of agent i. Let e ∈ RN be the total amount of good in the economy (called the aggregate + I endowment ). An allocation {xi }i=1 ⊂ RN is feasible if + I xi ≤ e. i=1 (For N -vectors x = (x1 , . . . , xN ) and y = (y1 , . . . , yN ), we write x ≤ y if xn ≤ yn for all n; x < y if x ≤ y and x ̸= y ; and x ≪ y if xn < yn for all n.) An allocation {yi } is said to Pareto dominate {xi } if u(yi ) ≥ u(xi ) for all i and u(yi ) > u(xi ) for some i. A feasible allocation is said to be Pareto efficient, or just efficient, if it is not Pareto dominated by any other feasible allocation. Intuitively, an allocation is efficient when there is no other feasible allocation that makes every agent at least as well off and one agent strictly better off. The following proposition shows that an efficient allocation exists. Proposition 8. Let λ ∈ RI and define W : RN I → R by ++ + I λi ui (xi ). W (x1 , . . . , xI ) = i=1 Then (i) W attains the maximum over all feasible allocations. (ii) An allocation that achieves the maximum of W is Pareto efficient. 2014W Econ 172B Operations Research (B) Alexis Akira Toda I Proof. Let F = (x1 , . . . , xI ) ∈ RN I + i=1 xi ≤ e b e the set of feasible allocations. Clearly F is nonempty and compact. Since ui is continuous, so is W . Therefore by the Bolzano-Weierstrass theorem W has a maximum. Suppose that {xi } attains the maximum of W . If {xi } is inefficient, there exists a feasible allocation {yi } that Pareto dominates {xi }. Then u(yi ) ≥ u(xi ) for all i and u(yi ) > u(xi ) for some i. Since λi > 0 for all i, we get I I λi ui (yi ) = W (y1 , . . . , yI ), λi ui (xi ) < W (x1 , . . . , xI ) = i=1 i=1 which contradicts the maximality of {xi }. Hence {xi } is efficient. Clearly Proposition 8 holds with the weaker assumption that each ui is upper semi-continuous. When the utility function ui is concave, the following partial converse holds (continuity is not necessary). Proposition 9. Suppose that each ui is concave and the feasible allocation {xi } ¯ is efficient. Then there exists λ ∈ RI such that {xi } is the maximizer of ¯ + I λi ui (xi ). W (x1 , . . . , xI ) = i=1 Proof. Define V = v = (v1 , . . . , vI ) ∈ RI (∃ {xi } ∈ F )(∀i)vi ≤ ui (xi ) , the set of utilities that are (weakly) dominated by some feasible allocation. Since {xi } ∈ V , V is nonempty. Since each ui is concave, V is convex (exercise). Let ¯ u = (u(¯1 ), . . . , uI (¯I ) and D = u + RI , the set of utilities that (strictly) ¯ x x ¯ ++ dominate {xi }. Then D is nonempty and convex. Since {xi } is e...
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This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.

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