The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers as a parameter changes. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics.
Statement of theorem
Let and be metric spaces, be a function jointly continuous in its two arguments, and be a compact-valued correspondence.
If is continuous (i.e. both upper and lower hemicontinuous) at some , then is continuous at and is non-empty, compact-valued, and upper hemicontinuous at .
The theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter. In this case, is the parameter space, is the function to be maximized, and gives the constraint set that is maximized over. Then, is the maximized value of the function and is the set of points that maximize .
The result is that if the elements of an optimization problem are sufficiently continuous, then some, but not all, of that continuity is preserved in the solutions.
The proof relies primarily on the sequential definitions of upper and lower hemicontinuity.
Because is compact-valued and is continuous, the extreme value theorem guarantees the constrained maximum of is well-defined and is non-empty for all in . Then, let be a sequence converging to and be a sequence in . Since is upper hemicontinuous, there exists a convergent subsequence .
which would simultaneously prove the continuity of and the upper hemicontinuity of .
Suppose to the contrary that , i.e. there exists an such that . Because is lower hemicontinuous, there is a further subsequence of such that and . By the continuity of and the contradiction hypothesis,
But this implies that for sufficiently large ,
which would mean is not a maximizer, a contradiction of . This establishes the continuity of and the upper hemicontinuity of .
Because and is compact, it is sufficient to show is closed-valued for it to be compact-valued. This can be done by contradiction using sequences similar to above.
If in addition to the conditions above, is quasiconcave in for each and is convex-valued, then is also convex-valued. If is strictly quasiconcave in for each and is convex-valued, then is single-valued, and thus is a continuous function rather than a correspondence.
If is concave and has a convex graph, then is concave and is convex-valued. Similarly to above, if is strictly concave, then is a continuous function.
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.
- is the indirect utility function and
- is the Marshallian demand.
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.
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