The chapter starts by formulating the standard problem in the theory of the firm: namely, to minimise combined capital and labour costs while producing a certain output. A general formulation of a constrained optimisation problem (with one constraint) is given and it is explained how to solve such problems by the method of Lagrange multipliers. This leads to a method to calculate the cost function of firms given their production functions and capital and labour unit costs. This enables us to derive the supply sets for efficient small firms with Cobb--Douglas production functions.
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