Shadow Price of Health

Shadow Price of Health in a Fixed Health Budget

In health economic evaluation, the shadow price of health is the amount of health that one extra unit of money would buy for a fixed health budget at the margin, once the budget is already being used to produce as much health as it can. Turned the other way up, as money per QALY, it is the supply-side cost-effectiveness threshold against which the extra cost of a new technology is judged. Health is not traded in a market, so a health service has to infer its value from what the budget would otherwise achieve. The idea explains why net health benefit subtracts health displaced from health gained and why the value used in appraisal depends on which services are actually cut. The phrase has a second, separate meaning in the Grossman model of the demand for health.

Where the idea comes from in health care allocation

The term applies the general notion of a shadow price, the implicit value of a constrained resource with no observed market price, to a budget whose objective is health, usually counted in QALYs. The second ISPOR report on constrained optimisation methods, by Crown and colleagues, states that under a strict set of assumptions the threshold represents the inverse of the shadow price of the budget constraint, the shadow price being how much the objective in QALYs would increase for a one-unit increase in the budget. The report cites Weinstein and Zeckhauser's 1973 paper on critical ratios for that result.

Claxton and colleagues, in their 2015 NIHR report on the NICE threshold, write that formally the threshold can be seen as the shadow price of the budget constraint, which is the same quantity expressed in money per QALY. Culyer's bookshelf guide to thresholds takes the ratio of health gain to expenditure of the least effective intervention that should be included in a public insurance plan as the benchmark and its reciprocal as the first-best threshold.

The shadow price as a Lagrange multiplier on the budget

The cleanest statement comes from writing health maximisation as a constrained optimisation, in which a payer chooses how much of each programme to provide. The problem is as follows.

$$\max_{x} \; H = \sum_{i=1}^{J} h_i x_i \quad \text{subject to} \quad \sum_{i=1}^{J} c_i x_i \le B, \quad 0 \le x_i \le 1$$

where $H$ is total health gained in QALYs, $J$ is the number of programmes, $h_i$ is the health gain from providing programme $i$ in full, $c_i$ is its full cost, $x_i$ is the share provided and $B$ is the budget for the period. The multiplier on the budget constraint at the optimum is the shadow price of health.

$$\mu = \frac{\partial H^{*}}{\partial B}, \qquad k = \frac{1}{\mu}$$

where $H^{*}$ is the largest total health the budget can buy, $\mu$ is the shadow price in QALYs per unit of money, and $k$ is the threshold in money per QALY. Programmes with $h_i / c_i$ above $\mu$ are provided in full, those below it are not provided, and the marginal programme is provided in part. The budget constraint page works through this with four programmes and shows the threshold moving with the budget.

Converting extra cost into health displaced

Once $\mu$ or $k$ is known, the extra cost of a new technology can be turned into the health it is expected to displace. Claxton and colleagues express the decision as an analysis of net health benefit, as follows.

$$NHB = \Delta h - \frac{\Delta C_h}{k} = \Delta h - \mu \, \Delta C_h$$

where $NHB$ is net health benefit in QALYs, $\Delta h$ is the change in health generated by the new technology, $\Delta C_h$ is the additional health care cost imposed on the budget, and $k$ and $\mu$ are as above. A positive value means the health gained exceeds the health given up elsewhere, and for a technology that adds both cost and health, comparing its ICER with $k$ gives the same answer.

The NICE manual (PMG36) uses the same reasoning. Section 6.3.1 states that, given the fixed budget of the NHS, the appropriate maximum acceptable ICER is that of the opportunity cost of programmes displaced by new, more costly technologies, and that NICE lacks the information to define it precisely, so sections 6.3.4 to 6.3.8 apply a range of GBP 25,000 to GBP 35,000 per QALY gained.

The shadow price and the consumption value of health

A second value of health is often confused with the shadow price. The consumption value of health, written $v$, is the rate at which people are willing to give up other consumption for health, the quantity behind a demand-side willingness-to-pay threshold. Claxton and colleagues show that applying $v$ to both health gained and health forgone leaves the decision unchanged, as follows.

$$NHB_v = v \, \Delta h - v \, \frac{\Delta C_h}{k}$$

where $NHB_v$ is net health benefit valued in consumption terms and the other symbols are as before. For any positive $v$ the sign of $NHB_v$ is the sign of $NHB$. They conclude that the consumption value may guide decisions such as the level of the overall NHS budget but does not inform how a fixed budget is allocated, and that the size of $k$ is an empirical question, not a value judgement.

The comparison still says something about the budget. One unit of money spent at the margin buys $\mu$ QALYs, worth $v \mu$ in consumption, so where $v$ exceeds $k$ the marginal health is worth more than the money it costs, which is an argument for a larger health budget. Culyer notes that the demand-side and supply-side approaches are equivalent only in a health-maximising equilibrium.

When the services displaced are not the least cost-effective

The textbook reading assumes that new spending is found by cutting the least cost-effective activity and that existing provision is efficient. Claxton and colleagues list constant returns to scale and perfect divisibility of programmes among the simplifying assumptions behind comparing an ICER with a threshold, and Stinnett and Paltiel showed that mathematical programming can relax them, a topic the integer programming page covers.

Eckermann and Pekarsky derive a shadow price of the health effects of a new technology for a health system that is not allocatively efficient. They distinguish $d$, the ICER of the service actually displaced, $m$, the ICER of the least cost-effective existing service that could be contracted, and $n$, the ICER of the most cost-effective existing service that could be expanded. Their shadow price is as follows.

$$\beta_c = \left( \frac{1}{n} + \frac{1}{d} - \frac{1}{m} \right)^{-1}$$

where $\beta_c$ is the critical ICER for the new technology in money per QALY and $n$, $d$ and $m$ are the three ICERs just defined. Funding a technology with ICER $r$ by displacing the service at $d$ yields $\frac{1}{r} - \frac{1}{d}$ QALYs per unit of money, while optimal reallocation, expanding the best existing service and contracting the worst, yields $\frac{1}{n} - \frac{1}{m}$. The technology is the better use of the money only when $r$ is below $\beta_c$. If provision were completely efficient, $n$, $d$ and $m$ would be equal and $\beta_c$ would reduce to that common value, the textbook threshold.

Worked example with imperfect displacement

All figures in this example are illustrative. A health service could expand an existing service at GBP 12,000 per QALY ($n$) or contract its least cost-effective service at GBP 30,000 per QALY ($m$), and in practice finds new money by cutting a service at GBP 20,000 per QALY ($d$). A new technology costs GBP 1.5 million a year and gains 100 QALYs, an ICER of 1,500,000 / 100 = GBP 15,000 per QALY.

1. Judge the technology at the least cost-effective service. Displacing activity at $m$ would remove 1,500,000 / 30,000 = 50 QALYs, a net gain of 100 − 50 = 50 QALYs.

2. Allow for what is actually displaced. Cutting the service at $d$ removes 1,500,000 / 20,000 = 75 QALYs, a net gain of 100 − 75 = 25 QALYs.

3. Price the alternative use of the same money. Contracting the service at $m$ by GBP 1.5 million loses 50 QALYs, and expanding the service at $n$ with that sum gains 1,500,000 / 12,000 = 125 QALYs, a net gain of 125 − 50 = 75 QALYs.

4. Compute the shadow price. The reciprocals of $n$, $d$ and $m$ are 0.00008333, 0.00005 and 0.00003333 QALYs per pound, so the first two less the third is 0.00008333 + 0.00005 − 0.00003333 = 0.0001 and $\beta_c$ = 1 / 0.0001 = GBP 10,000 per QALY.

5. Check the result. At an ICER of exactly GBP 10,000 the technology would gain 1,500,000 / 10,000 = 150 QALYs and, after the 75 QALYs displaced, net 75 QALYs, equal to the gain from reallocation.

6. Interpretation. The technology adds health even after allowing for what is actually cut, yet it is a worse use of GBP 1.5 million than moving money between existing services, and its critical ICER is lower than any of the three ICERs in the system. The calculation assumes the reallocation could actually be carried out. Where it cannot, the comparison that remains is the one in step 2.

Estimating and updating the shadow price

Because $k$ is an empirical quantity, it can be estimated from the observed relationship between spending and health outcomes. Claxton and colleagues concluded that the appropriate threshold should be lower for technologies with a greater impact on NHS costs. McCabe, Claxton and Culyer argued similarly that interventions with a large budgetary impact may need a lower threshold because they are likely to displace more than the marginal activities. Later estimates are covered on the opportunity cost page.

The shadow price also belongs to a particular budget at a particular date. Culyer states that the first-best threshold rises or falls as the health budget rises or falls, other things equal, and that distinct and separable health budgets imply multiple thresholds. Claxton, Paulden, Gravelle, Brouwer and Culyer showed in 2011 that, if the budget is fixed and decisions are based on ICERs, discounting costs and health gains at the same rate is correct only if the threshold remains constant, so an expected change in the shadow price also bears on the choice of discount rates.

The other meaning in the Grossman model

In Grossman's model of the demand for health, health is a durable capital stock that yields healthy time and can be increased by investment, with medical care as one input among many. Grossman explains that in this framework the shadow price of health depends on many variables besides the price of medical care, and shows that it rises with age if the rate of depreciation rises over the life cycle and falls with education if more educated people are more efficient producers of health. This individual price, part of the wider Grossman model, has no direct connection with a payer's budget constraint.

Boundaries with nearby concepts

Opportunity cost is the wider principle, and the shadow price of health is the rate that converts extra spending into health opportunity cost. The cost-effectiveness threshold is the number a body actually uses, which need not equal an estimate of $k$, while a willingness-to-pay threshold reflects $v$ and answers a different question.

Limitations and common misreadings

Several errors recur when the idea is applied. Most come from forgetting that the shadow price is marginal, empirical and specific to one budget.

1. Treating the shadow price as a constant. It depends on the budget, the efficiency of current provision and the date, so estimates do not transfer automatically.
2. Reading it as a value judgement. On the supply-side view it describes what current spending achieves at the margin, and what a society would pay for health is a separate quantity.
3. Assuming displacement at the margin. Real cuts may fall on services more cost-effective than the least cost-effective one, and large budget impacts reach past the margin, both of which lower the relevant threshold.
4. Overlooking the evidence gap. Gafni and Birch argue that threshold values in use lack justification, including their relationship with the opportunity cost of marginal resources.
5. Mixing the two meanings. The payer's shadow price of health and the individual's shadow price in the Grossman model share a name only.

Sources

• Claxton K, Martin S, Soares M, Rice N, Spackman E, Hinde S, Devlin N, Smith PC, Sculpher M. Methods for the estimation of the National Institute for Health and Care Excellence cost-effectiveness threshold. Health Technology Assessment. 2015;19(14):1-503, v-vi. doi:10.3310/hta19140
• Claxton K, Paulden M, Gravelle H, Brouwer W, Culyer AJ. Discounting and decision making in the economic evaluation of health-care technologies. Health Economics. 2011;20(1):2-15. doi:10.1002/hec.1612
• Crown W, Buyukkaramikli N, Sir MY, Thokala P, Morton A, Marshall DA, Tosh JC, Ijzerman MJ, Padula WV, Pasupathy KS. Application of constrained optimization methods in health services research: report 2 of the ISPOR Optimization Methods Emerging Good Practices Task Force. Value in Health. 2018;21(9):1019-1028. doi:10.1016/j.jval.2018.05.003
• Culyer AJ. Cost-effectiveness thresholds in health care: a bookshelf guide to their meaning and use. Health Economics, Policy and Law. 2016;11(4):415-432. doi:10.1017/S1744133116000049
• Eckermann S, Pekarsky B. Can the real opportunity cost stand up: displaced services, the straw man outside the room. PharmacoEconomics. 2014;32(4):319-325. doi:10.1007/s40273-014-0140-3
• Gafni A, Birch S. Incremental cost-effectiveness ratios (ICERs): the silence of the lambda. Social Science and Medicine. 2006;62(9):2091-2100. doi:10.1016/j.socscimed.2005.10.023
• Grossman M. The human capital model. In Culyer AJ, Newhouse JP, eds. Handbook of Health Economics, volume 1. Amsterdam: Elsevier; 2000. Chapter 7, 347-408.
• McCabe C, Claxton K, Culyer AJ. The NICE cost-effectiveness threshold: what it is and what that means. PharmacoEconomics. 2008;26(9):733-744. doi:10.2165/00019053-200826090-00004
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 6.3.1 and 6.3.4 to 6.3.8. https://www.nice.org.uk/process/pmg36
• Stinnett AA, Paltiel AD. Mathematical programming for the efficient allocation of health care resources. Journal of Health Economics. 1996;15(5):641-653. doi:10.1016/S0167-6296(96)00493-6