Maximum Theorem. 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 by and the set of maximizers by If is continuous at , then is continuous and is upper hemicontinuous with nonempty and compact values. As a consequence, the may be replaced by and the by.
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
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. If are correspondences, is upper hemicontinuous and compact-valued, and is closed, then defined by is upper hemicontinuous. 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. Then whenever, we have, and so. This completes the proof. The continuity of in the maximum theorem is the result of combining two independent theorems together. Theorem 1. If is upper semicontinuous and is upper hemicontinuous, nonempty and compact-valued, then is upper semicontinuous. 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. Theorem 2. If is lower semicontinuous and is lower hemicontinuous, then is lower semicontinuous. 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 . Therefore, since is lower hemicontinuous, there exists a neighborhood such that whenever there exists. Let. Then whenever there exists, which implies which was desired. 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.
Variants and generalizations
A natural generalization from the above results gives sufficient local conditions for to be continuous and to be nonempty, compact-valued, and upper semi-continuous. 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. It is also possible to generalize Berge's theorem to non-compact set-valued correspondences if the objective function is K-inf-compact.
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.
