Calculate the shadow price of a linear programming constraint from the optimal objective value before and after raising its right-hand side by one unit.

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Shadow Price Formula

Shadow price measures the change in the objective function when a constrained resource is increased by one unit. This calculator evaluates a one-unit finite difference. It equals the local shadow price only when that +1 RHS change remains within an unchanged linear sensitivity range.

SP = Vf + 1 - Vf
  • SP = shadow price
  • Vf+1 = value of the objective function with one additional unit of the resource
  • Vf = original value of the objective function

This value is commonly used in optimization, operations research, economics, and business planning to estimate the marginal value of scarce resources such as labor hours, machine time, storage space, budget, or raw materials.

What the Shadow Price Tells You

A shadow price answers a practical question: How much better or worse does the objective become if I get one more unit of a limited resource? In a maximization problem, a positive shadow price usually means that the extra unit increases profit, revenue, or output. In a minimization problem, the sign must be interpreted relative to the objective, since an extra unit may reduce total cost.

  • Positive value: the objective increases; that improves a maximization objective but worsens a minimization objective. Interpret the RHS change with the constraint direction.
  • Zero shadow price: the additional unit does not change the objective, often because the constraint is not currently binding.
  • Larger magnitude: the objective is more sensitive per specified resource unit; this alone does not establish economic value or improvement.

How to Calculate Shadow Price

  1. Determine the original objective function value.
  2. Determine the objective function value after increasing the constrained input by one unit.
  3. Subtract the original value from the new value.
SP = New Objective Value - Original Objective Value

If the result is 25, then one extra unit of that resource changes the objective by 25 units of value. If the result is 0, the added unit has no marginal benefit under the current solution.

Example

Assume a companyโ€™s profit is 1,250 under the current resource limit. If adding one more unit of the constrained resource raises profit to 1,310, then the shadow price is:

SP = 1310 - 1250 = 60

The observed one-unit change is 60 in objective value. It represents the local shadow price only if that interval stays inside the applicable sensitivity range.

Why Shadow Price Matters

  • Capacity planning: shows whether buying more machine hours, labor time, or storage is likely to be worthwhile.
  • Budget allocation: helps identify which constrained input creates the greatest return if expanded.
  • Production optimization: reveals which bottlenecks are limiting profit or throughput.
  • Negotiation and purchasing: provides a rational ceiling for what an extra unit of a resource may be worth.
  • Sensitivity analysis: helps measure how sensitive the optimal solution is to changes in constraint limits.

Interpreting Results Correctly

Shadow price is a marginal measure, not a blanket value for unlimited expansion. It is usually valid only within the range where the current optimization structure remains unchanged. Once the resource increase is large enough to change the active constraints or the optimal mix, the shadow price may also change.

  • Use it for small, incremental changes.
  • Do not assume the same value applies to every additional unit indefinitely.
  • If the constraint is non-binding, the shadow price is often zero.
  • Compare shadow prices only after accounting for resource units, objective sense, constraint direction and applicable sensitivity ranges.

Common Uses

  • Determining the value of one more labor hour
  • Estimating the benefit of additional inventory capacity
  • Evaluating extra production time on constrained equipment
  • Measuring the impact of more advertising budget or shelf space
  • Prioritizing scarce materials across competing products

Shadow Price vs. Market Price

Shadow price is not necessarily the same as an actual market price. Market price is what you pay in a transaction, while shadow price is the internal economic value of relaxing a constraint by one unit inside a model or decision system. A resource can have a low purchase price but a high shadow price if it is severely limiting operations.

Quick Reference

  • Use this calculator when: you know the original objective value and the value after adding one unit.
  • Main output: the one-unit objective difference, with conditions for interpreting it as a shadow price.
  • Best interpretation: the worth of relaxing a bottleneck by one unit under current conditions.