Linear Programming and Resource Allocation Modeling

11Parametric Programming and the Theory of the Firm

11.1 The Supply Function for the Output of an Activity (or for an Individual Product)

Let us determine how the firm reacts to a variation in the price of one of its outputs by generating a supply function for a particular product. Specifically, we want to determine the ceteris paribus amounts supplied at each possible price (the implication of the ceteris paribus assumption is that the firm’s technology is taken as given or unchanging as are the prices of its other products and variable inputs and the quantities of its fixed inputs). It is further assumed that at each possible price the firm supplies an amount of the product that maximizes its gross profit so that, as required by our parametric analysis of the objective function, we move from one optimal basic feasible solution to the next as the price of the product under consideration varies continuously.

From (7.25.1) our problem is to initially

Once the price p_j (the jth component of P) of one of the firm’s output changes, it is replaced in the objective function by itself is replaced by P^* = P + θe_j, where e_j denotes the jth unit column vector, j = 1, …, p. Thus, the firm now desires to

If the problem presented in Example 7.3 is interpreted as an optimum product mix ...