Provides conditions for a parametric optimization problem to have continuous solutions
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959.[1] The theorem is primarily used in mathematical economics and optimal control.
Statement of theorem
Maximum Theorem.[2][3][4][5]
Let and be topological spaces, be a continuous function on the product , and be a compact-valued correspondence such that for all . Define the marginal function (or value function) by
and the set of maximizers by
- .
If is continuous (i.e. both upper and lower hemicontinuous) at , then the value function is continuous, and the set of maximizers is upper-hemicontinuous with nonempty and compact values. As a consequence, the may be replaced by .
Variants
The maximum theorem can be used for minimization by considering the function instead.
Proof
Throughout this proof we will use the term neighborhood to refer to an open set containing a particular point. We preface with a preliminary lemma, which is a general fact in the calculus of correspondences. Recall that a correspondence is closed if its graph is closed.
Lemma.[6][7][8]
If are correspondences, is upper hemicontinuous and compact-valued, and is closed, then defined by is upper hemicontinuous.
Proof
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Let , and suppose is an open set containing . If , then the result follows immediately. Otherwise, observe that for each we have , and since is closed there is a neighborhood of in which whenever . The collection of sets forms an open cover of the compact set , which allows us to extract a finite subcover . By upper hemicontinuity, there is a neighborhood of such that . Then whenever , we have , and so . This completes the proof.
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The continuity of in the maximum theorem is the result of combining two independent theorems together.
Theorem 1.[9][10][11]
If is upper semicontinuous and is upper hemicontinuous, nonempty and compact-valued, then is upper semicontinuous.
Proof of Theorem 1
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Fix , and let be arbitrary. For each , there exists a neighborhood of such that whenever , we have . The set of neighborhoods covers , which is compact, so suffice. Furthermore, since is upper hemicontinuous, there exists a neighborhood of such that whenever it follows that . Let . Then for all , we have for each , as for some . It follows that
which was desired.
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Theorem 2.[12][13][14]
If is lower semicontinuous and is lower hemicontinuous, then is lower semicontinuous.
Proof of Theorem 2
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Fix , and let be arbitrary.
By definition of , there exists such that .
Now, since is lower semicontinuous, there exists a neighborhood of such that whenever we have . Observe that (in particular, ). Therefore, since is lower hemicontinuous, there exists a neighborhood such that whenever there exists .
Let .
Then whenever there exists , which implies
which was desired.
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Under the hypotheses of the Maximum theorem, is continuous. It remains to verify that is an upper hemicontinuous correspondence with compact values. Let . To see that is nonempty, observe that the function by is continuous on the compact set . The Extreme Value theorem implies that is nonempty. In addition, since is continuous, it follows that a closed subset of the compact set , which implies is compact. Finally, let be defined by . Since is a continuous function, is a closed correspondence. Moreover, since , the preliminary Lemma implies that is upper hemicontinuous.
Examples
Consider a utility maximization problem where a consumer makes a choice from their budget set. Translating from the notation above to the standard consumer theory notation,
- is the space of all bundles of commodities,
- represents the price vector of the commodities and the consumer's wealth ,
- is the consumer's utility function, and
- is the consumer's budget set.
Then,
Proofs in general equilibrium theory often apply the Brouwer or Kakutani fixed-point theorems to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so.