文档介绍：Math Review for . Students
ECO 1011H (Fall, 2002)
by Shouyong Shi
Lecture II: Sets, Metric Spaces, and Existence
Reading materials:
. Dixit (2nd edition), chapters 6 and 7.
Stokey, Lucas with Prescott, -49, 55-65, 516-525.
1. Motivation
(1) Why do we need more general methods for existence?
The constraints G(x) c may not have the \nice" properties, ., di®erentiability;
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Cumbersome to verify the second-order conditions;
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Local versus global optimum;
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The problem may plicated objects, ., distributions.
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(2) Basic intuition for the general method for existence:
Suppose that x¤ is the optimum, achieving a value v¤. Then,
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{ Choices that satisfy the constraints G(x) c cannot achieve higher than v¤;
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{ Choices that achieve higher than v¤ must violate G(x) c.
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That is, the optimum has the separation property: the set of choices satisfying the
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constraints (., the feasibility set) and the set of choices achieving higher than v¤ are
separated from each other.
Illustration.
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This separation property does not rely on strong requirements such as di®erentiability.
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Contour sets: For any function F(x), Rn R, and v R,
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upper contour set for v: x : F (x) v ;
f ¸ g
lower contour set for v: x : F(x) v .
f · g
The separation property says that the upper contour set of U for v¤ and the lower
² contour set of G for c are separated from each other, except for the boundary points.
Signi¯cance:
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Some of the boundary points are the optimum;
The two sets are separated by a \linear" line, which can often be interpreted as the
price line.
For the above reason, the separation result indicates that petitive economy
achieves the optimum (in a large class of economies).
2. Formal statements of results
(1) The Weierstrass Theorem
Consider the optimization problem
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sup U(x) st. x ¡,
2
where ¡ is the feasible set. If the set ¡ is \compact" and U is \upper semicontinuous",
the