To examine the duality concept, the idea of implicit values or shadow prices must be introduced. In the primal linear programming problem discussed previously, the values QX and QY maximize the firm's profit subject to constraints imposed by limitations of input factors A, B, and C. Duality theory indicates that an identical operating decision would result if one had instead chosen to minimize the costs of resources employed in producing QX and QY, subject to an output constraint.
The key to this duality is that relevant costs are not the acquisition costs of inputs but, rather, the economic costs of using them. For a resource that is available in a fixed amount, this cost is not acquisition cost but opportunity cost. Consider, for example, a skilled labor force employed by a firm. If workers are fully utilized producing valuable products, a reduction in skilled labor will reduce valuable output, and an increase in skilled labor will increase the production of valuable output. If some labor is shifted from the production of one product to another, the cost of using skilled labor in this new activity is the value of the original product that can no longer be produced. The marginal cost of a constrained resource that is fully utilized is its opportunity cost as measured by the value of foregone production. If a limited resource such as skilled labor is not fully utilized, then at least the last unit of that resource is not productive and its marginal value is zero. Acquiring additional excess resources does not increase valuable output. The firm would incur a zero opportunity cost if it applied currently unused resources in some different activity.
The economic value, or opportunity cost, of a constrained resource depends on the extent to which it is utilized. When a limited resource is fully utilized, its marginal value in use is positive. When a constrained resource is not fully utilized, its marginal value in use is zero. Minimizing the value of limited resources used to produce valuable output is nothing more than minimizing the opportunity cost of employing those resources. Minimization of opportunity costs is equivalent to maximizing the value of output produced with those resources.
Because the economic value of constrained resources is determined by their value in use rather than by historical acquisition costs, such amounts are called implicit values or shadow prices. The term shadow price is used because it represents the price that a manager would be willing to pay for additional units of a constrained resource. Comparing the shadow price of a resource with its acquisition price indicates whether the firm has an incentive to increase or decrease the amount acquired during future production periods. If shadow prices exceed acquisition prices, the resource's marginal value exceeds marginal cost and the firm has an incentive to expand employment. If acquisition cost exceeds the shadow price, there is an incentive to reduce employment. These relations and the importance of duality can be clarified by relating the dual to the linear programming problem discussed previously.
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