The explicit solution of the profit maximization problem with box-constrained inputs

In this paper we study the profit-maximization problem, considering maximum constraints for the general case of m-inputs and using the Cobb–Douglas model for the production function. To do so, we previously study the firm’s cost minimization problem, proposing an equivalent infimal convolution problem for exponential-type functions. This study provides an analytical expression of the production cost function, which is found to be a piece-wise potential. Moreover, we prove that this solution belongs to class C1. Using this cost function, we obtain the explicit expression of maximum profit. Finally, we illustrate the results obtained in this paper with an example.