Shadow Prices as Marginal Resource Values

In LP duality, each primal constraint has an associated dual variable whose optimal value $y_i^*$ is called the shadow price of that constraint. The shadow price measures the marginal value of the corresponding resource: it is the rate at which the optimal objective value $z^*$ changes per unit increase in the right-hand side $b_i,$

$$y_i^* = \dfrac{\partial z^*}{\partial b_i}.$$

For a small change $\Delta b_i$ in the right-hand side (within the range over which the current optimal basis remains optimal), the new optimal objective is

$$z^*_{\text{new}} = z^*_{\text{old}} + y_i^* \cdot \Delta b_i.$$

For example, suppose a workshop has optimal weekly profit z^* = \800$and the shadow price on labor is$y_1^* = $5$per hour. If$3$ additional labor-hours become available, then

$$z^*_{\text{new}} = 800 + 5 \cdot 3 = \$815.$$