Shadow price
Shadow price is the increase in the optimal objective (typically total contribution) from relaxing a binding constraint by one unit. It shows the maximum extra amount a firm should pay to obtain one additional unit of a scarce resource.
FrameworkLinear programming
See it move
A binding constraint — say machine hours fully used — has a shadow price equal to the rise in total contribution from relaxing it by one unit. If that shadow price is €12 per hour, the firm should pay up to, but never more than, €12 for an extra hour of machine time through overtime or hire. The figure only holds within the constraint's sensitivity range; beyond it, a new optimal plan applies.
The formula
Variables
- increase in the optimal value of the objective function (e.g. total contribution) when binding constraint j is relaxed by one unit (€)
- one-unit increase in the right-hand side of binding constraint j (e.g. one extra machine-hour or kilogram of material) (units of scarce resource)
The shadow price equals the maximum additional amount a firm should pay to obtain one extra unit of a scarce resource. It holds only within the sensitivity range of the constraint; outside that range a new LP solution is required.
Check yourself
An LP solution for a furniture maker identifies machine hours as a binding constraint with a shadow price of €18 per hour. Current monthly capacity is 2,000 hours. A contractor offers overtime that would add 80 machine hours next month at an extra cost of €1,600. Should the offer be accepted?