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.
is continuous (i.e. both upper and lower hemicontinuous) at
is upper-hemicontinuous with nonempty and compact values.
The maximum theorem can be used for minimization by considering the function
The theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter.
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.
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.
is upper hemicontinuous and compact-valued, and
forms an open cover of the compact set
, which allows us to extract a finite subcover
By upper hemicontinuity, there is a neighborhood
is upper hemicontinuous, nonempty and compact-valued, then
is upper hemicontinuous, there exists a neighborhood
is lower semicontinuous, there exists a neighborhood
is lower hemicontinuous, there exists a neighborhood
is an upper hemicontinuous correspondence with compact values.
is nonempty, observe that the function
The Extreme Value theorem implies that
a closed subset of the compact set
, the preliminary Lemma implies that
A natural generalization from the above results gives sufficient local conditions for
to be nonempty, compact-valued, and upper semi-continuous.
is single-valued, and thus is a continuous function rather than a correspondence.
[15] It is also possible to generalize Berge's theorem to non-compact correspondences if the objective function is K-inf-compact.
[16] 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, 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.