Interpreting Reduced Costs as Opportunity Costs

Reduced costs measure the per-unit gap between a product's profit and the opportunity cost of the resources it consumes, priced at the shadow prices. This lesson interprets reduced costs as opportunity-cost differentials and uses them to evaluate whether a non-basic product should enter the production plan, the profit loss from forcing production, and the break-even profit at which a product becomes attractive.

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Reduced Cost as Opportunity Cost

A firm solves a linear program to maximize profit subject to resource constraints. At the optimum, each resource $i$ has a shadow price $y_i$ — the marginal value of one more unit of that resource.

Suppose product $j$ uses $a_{ij}$ units of resource $i$ per unit produced and earns $c_j$ in profit per unit. The resources consumed by one unit of product $j$ have a total shadow-price value of

z_j = \sum_{i} y_i\, a_{ij}.

This is the opportunity cost of one unit of product $j$ : the value of the resources it ties up, measured at their best-alternative use.

The reduced cost of product $j$ is its profit per unit minus its opportunity cost:

\bar{c}_j \;=\; c_j - \sum_{i} y_i\, a_{ij} \;=\; c_j - z_j.

It measures whether one unit of product $j$ earns more profit than the value of the resources it would consume.

Illustration. Suppose two resources have shadow prices $y_1 = \$ 3 $and$ y_2 = $4 $. Product$ Q $uses$ a_{1Q}=2 $units of resource 1 and$ a_{2Q}=1 $unit of resource 2, with profit$ c_Q = $11$. Then

z_Q = 3\cdot 2 + 4\cdot 1 = 10, \qquad \bar{c}_Q = 11 - 10 = 1.

Product $Q$ earns $1 more per unit than the resources it consumes are worth elsewhere.