Concept Architecture
Concept
Theoretically, a Shadow Price is the implicit economic value of an additional unit of a constrained resource, representing the marginal change in the objective function resulting from a one-unit relaxation of that constraint. It is founded on constrained optimisation theory and mathematical programming and reflects the opportunity cost of scarce resources rather than observed market prices. In health economics, shadow prices are used to value constrained healthcare resources, estimate opportunity costs and support efficient resource allocation.
Mathematically, the Shadow Price is represented by the Lagrange multiplier associated with a binding constraint in an optimisation problem. It measures the marginal improvement in the objective function resulting from a one-unit increase in the availability of the constrained resource. In linear programming, the shadow price is valid over the allowable range of the constraint.
In practice, Shadow Prices are estimated using linear programming, mathematical optimisation models and economic evaluation. They are applied to estimate the opportunity cost of limited healthcare resources such as hospital beds, operating theatre time, healthcare personnel and fixed budgets, particularly where market prices do not accurately reflect economic value.
Purpose
Used to estimate the marginal economic value of scarce healthcare resources and to support efficient resource allocation under binding constraints.
Mathematical Formulae
Primary Formula
For a constrained optimisation problem:
max? f(x)
subject to
g(x) � b
the shadow price is
? = ?f* / ?b
where:
- ? = shadow price
- f* = optimal value of the objective function
- b = resource constraint
Supporting Formulae
Lagrangian function:
L(x, ?) = f(x) + ?[b ? g(x)]
First-order condition:
?L / ?x = 0
Related Mathematical Methods
- Linear Programming
- Mathematical Optimisation
- Lagrange Multipliers
- Opportunity Cost Analysis
- Resource Allocation Modelling
Example
A hospital optimisation model maximises total health gain subject to an annual operating theatre capacity of 5,000 hours. The optimisation solution reports a shadow price of 0.08 QALYs per additional operating hour.
Increasing available theatre capacity by one hour would therefore increase the maximum achievable health benefit by approximately:
0.08 QALYs
provided the constraint remains within its allowable range.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Solver | Solver Sensitivity Report | Calculates shadow prices for binding resource constraints in optimisation models |
| SUMPRODUCT | =SUMPRODUCT(B2:B20,C2:C20) | Calculates objective function values in optimisation problems |
| SUM | =SUM(D2:D20) | Calculates total resource utilisation against capacity constraints |
| IF | =IF(E2>Capacity,""Constraint Binding"",""Slack Available"") | Identifies whether resource constraints are binding |
VBA (Optional)
Automate optimisation analyses and extract shadow prices from Solver sensitivity reports for healthcare resource allocation studies.
Sources
- Dantzig GB. Linear Programming and Extensions. Princeton University Press.
- Winston WL. Operations Research: Applications and Algorithms.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Bazaraa MS, Jarvis JJ, Sherali HD. Linear Programming and Network Flows.
Related Concepts (3)
Library
Publications
1
Conjoint Analysis Applications in Health — A Checklist: A Report of the ISPOR Good Research Practices for Conjoint Analysis Task Force — Bridges, Hauber, Marshall, Lloyd, Prosser, Regier, Johnson & Mauskopf, Vol. 14, No. 4 ed., 2011 (Value in Health)
The ISPOR good-practice checklist for conjoint analysis and discrete-choice experiments in health — the stated-preference methods used to elicit patient and public preferences over treatment attributes for value assessment and priority-setting.
Journal ArticleView source →
Frequently Asked Questions (6)
What is a shadow price?
The implicit value of a resource with no observable market price, reflecting the opportunity cost of using it in a particular way.
Source: Weinstein & Zeckhauser 1973
When is a shadow price needed?
Where a resource is genuinely consumed but has no market price, or where the observed price does not reflect what the resource is worth in its alternative use. Volunteer time, informal care provided by relatives, donated goods and facilities occupied at nil charge all fall into the first category. Administered prices, monopoly prices and internal transfer prices fall into the second. In both cases using the observed figure, or treating the resource as costless, misstates what is being given up. The requirement follows from the definition of cost as opportunity forgone, so a resource with no price is not thereby without cost.
Source: Weinstein & Zeckhauser 1973
How is a shadow price estimated?
By identifying what the resource would have produced in its best alternative use and valuing that. For volunteer or carer time, the usual approaches are the earnings forgone, the market price of purchasing the equivalent service, or what the person would need in compensation. For a distorted market price, the correction removes the element attributable to the distortion, such as a tax that transfers rather than consumes resources. Each approach requires a judgement that should be stated. Where several methods are defensible, reporting the result under each is preferable to selecting one, since the choice frequently moves the total materially.
Source: Drummond et al. 2015
Where do shadow prices arise in health economics?
In valuing informal care, which is frequently large enough to change a conclusion for conditions requiring prolonged support at home. In valuing patient and carer time spent obtaining treatment. In adjusting for taxes and subsidies, which move resources between parties rather than consuming them. And in constrained systems, where the shadow price of the budget itself expresses the health forgone by committing a further unit of expenditure. Shadow pricing also arises in valuing capital already owned, where the relevant cost is what the asset could otherwise be used for rather than what was paid for it.
Source: Weinstein & Zeckhauser 1973
Why does omitting shadow prices bias an analysis?
Because resources treated as costless are consumed without appearing in the total, which systematically favours interventions relying on them. An intervention shifting care from paid staff to family members appears to save money while transferring the burden rather than removing it. The bias operates in one direction and is invisible in the results, which makes it more serious than random imprecision and is the principal reason methods guidance recommends including these categories. The same reasoning applies to interventions relying on volunteer effort, which appear cheap precisely because a real resource is being valued at nothing.
Source: van den Berg, Brouwer & Koopmanschap 2004
What should be reported when a shadow price is used?
The resource valued, the method used to derive the value, the assumptions behind it, and the result with and without that component included. Because the choice of method can change the figure substantially, sensitivity analysis across plausible alternatives is more informative than a single point estimate. Where a resource was known to be consumed but no defensible value could be attached, saying so is preferable to omitting it silently. Reporting the quantity of the resource alongside its assumed value allows a reader to substitute their own valuation without repeating the analysis.
Source: Drummond et al. 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 4 Aug 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/shadow-price
- Term code
- HE-EE-CBA-046
Stable URI · Machine-readable · Resolvable · CC BY 4.0