Maximum theorem

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 $X$ and $\Theta$ be metric spaces, $f:X\times\Theta\to\mathbb{R}$ be a function jointly continuous in its two arguments, and $C:\Theta\twoheadrightarrow X$ be a compact-valued correspondence.

For $x$ in $X$ and $\theta$ in $\Theta$, let

$f^*(\theta)=\max\{f(x,\theta)|x\in C(\theta)\}$ and

$C^*(\theta)=\mathrm{arg}\max\{f(x,\theta)|x\in C(\theta)\}=\{x\in C(\theta)\,|\,f(x,\theta)=f^*(\theta)\}$.

If $C$ is continuous (i.e. both upper and lower hemicontinuous) at some $\theta$, then $f^*$ is continuous at $\theta$ and $C^*$ is non-empty, compact-valued, and upper hemicontinuous at $\theta$.

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, $\Theta$ is the parameter space, $f(x,\theta)$ is the function to be maximized, and $C(\theta)$ gives the constraint set that $f$ is maximized over. Then, $f^*(\theta)$ is the maximized value of the function and $C^*$ is the set of points that maximize $f$.

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 hemicontinuity.

Because $C$ is compact-valued and $f$ is continuous, the extreme value theorem guarantees the constrained maximum of $f$ is well-defined and $C^*(\theta)$ is non-empty for all $\theta$ in $\Theta$. Then, let $\theta_n$ be a sequence converging to $\theta$ and $x_n \in C^*(\theta_n)$ be a sequence in $X$. Since $C$ is upper hemicontinuous, there exists a convergent subsequence $x_{n_k}\to x \in C(\theta)$.

If it is shown that $x \in C^*(\theta)$, then

$\lim_{k \to \infty} f^*(\theta_{n_k}) = \lim_{k \to \infty} f(x_{n_k},\theta_{n_k}) = f(x,\theta) = f^*(\theta)$

which would simultaneously prove the continuity of $f^*$ and the upper hemicontinuity of $C^*$.

Suppose to the contrary that $x \not\in C^*(\theta)$, i.e. there exists an $\hat{x}\in C(\theta)$ such that $f(\hat{x},\theta) > f(x,\theta)$. Because $C$ is lower hemicontinuous, there is a further subsequence of $n_k$ such that $\hat{x}_{n_j} \in C(\theta_{n_j})$ and $\hat{x}_{n_j} \to \hat{x}$. By the continuity of $f$ and the contradiction hypothesis,

$\lim_{j \to \infty} f(\hat{x}_{n_j},\theta_{n_j}) = f(\hat{x},\theta) > f(x,\theta) = \lim_{j \to \infty} f(x_{n_j},\theta_{n_j})$.

But this implies that for sufficiently large $j$,

$f(\hat{x}_{n_j},\theta_{n_j}) > f(x_{n_j},\theta_{n_j})$

which would mean $x_{n_j}$ is not a maximizer, a contradiction of $x_n \in C^*(\theta_n)$. This establishes the continuity of $f^*$ and the upper hemicontinuity of $C^*$.

Because $C^*(\theta) \subset C(\theta)$ and $C(\theta)$ is compact, it is sufficient to show $C^*$ is closed-valued for it to be compact-valued. This can be done by contradiction using sequences similar to above.

Variants and generalizations

If in addition to the conditions above, $f$ is quasiconcave in $x$ for each $\theta$ and $C$ is convex-valued, then $C^*$ is also convex-valued. If $f$ is strictly quasiconcave in $x$ for each $\theta$ and $C$ is convex-valued, then $C^*$ is single-valued, and thus is a continuous function rather than a correspondence.

If $f$ is concave and $C$ has a convex graph, then $f^*$ is concave and $C^*$ is convex-valued. Similarly to above, if $f$ is strictly concave, then $C^*$ is a continuous function.

It is also possible to generalize Berge's theorem to non-compact set-valued correspondences if the objective function is K-inf-compact.

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,

• $X=\mathbb{R}_+^l$ is the space of all bundles of $l$ commodities,
• $\Theta=\mathbb{R}_{++}^l \times \mathbb{R}_{++}$ represents the price vector of the commodities $p$ and the consumer's wealth $w$,
• $f(x,\theta)=u(x)$ is the consumer's utility function, and
• $C(\theta)=B(p,w)=\{x \,|\, px \leq w\}$ is the consumer's budget set.

Then,

• $f^*(\theta)=v(p,w)$ is the indirect utility function and
• $C^*(\theta)=x(p,w)$ 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.

See also

• Envelope theorem