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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 BolzanoWeierstrass 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
semicontinuous.
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.
 Winter '08
 Foster,C

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