On The Application Of Weighted Am-Gm Inequality To Profit Maximization Problem In The Case Of The Cobb-Douglas Production Function With Two Input Factors

Profit maximization in the case of the Cobb-Douglas production function is one of the fundamental problems in the microeconomic analysis of a company. In the literature, the problem of profit maximization with Cobb-Douglas technology is almost always solved by applying a differential calculus. Therefore, this paper aims to show how by applying the weighted arithmetic mean - geometric mean inequality (weighted AM-GM inequality) the problem of profit maximization with Cobb-Douglas production function with two input factors can be solved in an alternative, new way, without derivatives. Compared to the differential calculus, the application of the weighted AM-GM inequality bypasses a non-trivial check of the necessary and sufficient conditions for the optimal solution of the problem. The elegance of the new way of calculating the maximum profit originates from the direct application of the weighted AM-GM inequality and the very definition of a strict global maximum. However, it should be noted that the application of weighted AM-GM inequality, which belongs to the tools of elementary mathematics, should by no means be understood as superior, but as a complementary way to differential calculus in solving and better understanding this microeconomic problem.