Maximum theorem
Encyclopedia
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
.
and 
be metric spaces, 
be a function jointly continuous in its two arguments, and 
be a compact-valued correspondence
.
For
in 
and 
in 
, let
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 
.
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.
.
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 
.
If it is shown that
, then
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.
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.
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 theorem
s to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so.
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
of an optimized
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
function and the set of its maximizers as a parameter changes. The statement was first proven by Claude Berge
Claude Berge
Claude Berge was a French mathematician, recognized as one of the modern founders of combinatorics and graph theory. He is particularly remembered for his famous conjectures on perfect graphs and for Berge's lemma, which states that a matching M in a graph G is maximum if and only if there is in...
in 1959. The theorem is primarily used in mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
.
Statement of theorem
Let and be metric spaces, be a function jointly continuous in its two arguments, and be a compact-valued correspondenceMultivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
.
For
in and in , let
-
and
-
.
If
is continuous (i.e. both upper and lower hemicontinuousHemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extendible to multi-valued mappings or correspondences. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extention...
) at some
, then is continuous at and is non-empty, compact-valued, and upper hemicontinuous at .Interpretation
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.
Proof
The proof relies primarily on the sequential definitions of upper and lower hemicontinuityHemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extendible to multi-valued mappings or correspondences. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extention...
.
Because
is compact-valued and is continuous, the extreme value theoremExtreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
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 .If it is shown that
, then
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.Variants
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 concaveConcave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
and
has a convexConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
graph, then
is concave and is convex-valued. Similarly to above, if is strictly concave, then is a continuous function.Examples
Consider a utility maximization problemUtility maximization problem
In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...
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 setBudget setA budget set or opportunity set includes all possible consumption bundles that someone can afford given the prices of goods and the person's income level...
.
Then,
-
is the indirect utility functionIndirect utility functionIn economics, a consumer's indirect utility functionv gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over market conditions....
and -
is the Marshallian demand.
Proofs in general equilibrium theory often apply the Brouwer
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
or Kakutani fixed point theorem
Kakutani fixed point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing...
s to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so.