Dual Constraints

In the primal problem, the constraints stated that the total units of each input used to produce X and Y must be equal to or less than the available quantity of input. In the dual, the constraints state that the total value of inputs used to produce one unit of X or one unit of Y must not be less than the profit contribution provided by a unit of these products. In other words, the shadow prices of A, B, and C times the amount of each of the inputs needed to produce a unit of X or Y must be equal to or greater than the unit profit contribution of X or of Y. Because resources have value only when used to produce output, they can never have an implicit value, or opportunity cost, that is less than the value of output.

In the example, unit profit is defined as the excess of price over variable cost, price and variable cost are both constant, and profit per unit for X is $12 and for Y is $9. As shown in Table 9.1, each unit of X requires 4 units of A, 1 unit of B, and 0 units of C. The total implicit value of resources used to produce X is 4 VA + 1VB. The constraint requiring that the implicit cost of producing X be equal to or greater than the profit contribution of X is

Because 2 units of A, 1 unit of B, and 3 units of C are required to produce each unit of Y, the second dual constraint is

Because the firm produces only two products, the dual problem has only two constraint equations.

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