Capital Rationing

Capital Rationing in Health Care Investment Appraisal

Capital rationing is a budget constraint on investment spending that binds when the projects passing an appraisal cost more in total than the funds set aside for them. In a health system the usual case is an NHS trust holding a fixed capital allocation while several worthwhile schemes, such as new scanners and refurbished theatres, compete for the same money. Once funds are rationed, accepting every project with a positive net present value is no longer possible, and the analyst must find the combination that adds the most value within the limit. The same reasoning lies behind the cost-effectiveness threshold, since a fixed health budget makes every funding decision a rationing problem, so the idea matters for cost-benefit analysis of hospital investment and for cost-effectiveness analysis.

Why a spending limit changes the net present value rule

Without a spending limit, an appraisal funds every independent project whose discounted benefits exceed its discounted costs. When capital falls short of the total cost of those projects, the objective becomes the largest sum of net present values the funds can buy. Brealey, Myers and Allen write this as a mathematical programme in which each project is accepted in full, in part or not at all, subject to a spending limit in each period.

$$\max_{x_1, \ldots, x_n} \sum_{j=1}^{n} x_j V_j \quad \text{subject to} \quad \sum_{j=1}^{n} x_j K_{jt} \le B_t \ \text{for each period } t, \qquad 0 \le x_j \le 1$$

where $x_j$ is the share of project $j$ that is funded, $V_j$ is the net present value of project $j$, $K_{jt}$ is the net cash outflow of project $j$ in period $t$, negative when the project pays cash back, $B_t$ is the capital available in period $t$, and $n$ is the number of candidate projects. When a project cannot be split, such as a single building, $x_j$ is restricted to 0 or 1.

The HM Treasury Green Book (2026) describes public appraisal in the same terms. It states that the public sector generally aims to maximise social benefits and minimise social costs subject to a public sector budget constraint, and offers the return on public sector cost to show how well an option uses that money. In health terms, $V_j$ is the net present social value of a capital scheme and $B_t$ the capital a trust can spend that year.

Ranking by return per pound and where ranking fails

When there is a single constrained period and projects can be scaled, the programme has a simple solution. Each project is scored by the present value it returns per pound of constrained capital, and projects are funded in descending order of that score until the money runs out.

$$PI_j = \frac{PV_j}{K_j} = 1 + \frac{V_j}{K_j}$$

where $PI_j$ is the profitability index of project $j$, $PV_j$ is the present value of the benefits that follow the outlay, $K_j$ is the present value of the capital outlay, and $V_j$ is the net present value, so that $PV_j$ equals $K_j$ plus $V_j$. In public appraisal the matching ratio is the benefit-cost ratio, or the return on public sector cost when part of the cost falls outside the constrained budget.

Cost-effectiveness analysis uses the same device with health in place of money. Karlsson and Johannesson show how a fixed budget can serve as a decision rule, funding programmes in order of incremental cost-effectiveness ratio, which is the logic of a cost-effectiveness league table.

Ranking breaks down once the problem has more structure than one limit and divisible projects. Brealey, Myers and Allen give four projects facing a limit of 10 million in each of two periods. Ranking by profitability index selects projects B and C, with a combined net present value of 28 million. Funding project A instead releases cash in the second period, which pays for project D, and A with D gives 34 million. They note that ranking also fails when projects are mutually exclusive, when one project depends on another, and when a non-financial resource such as design staff is limited.

Health care shares these features. Stinnett and Paltiel showed that a general mathematical programming framework can handle returns to scale, partial and complete indivisibility and interdependence between programmes, and can add ethical constraints with an explicit cost of equity. The integer programming and linear programming pages show how such models are built. Van Baal, Morton and Severens add that a shortage of skilled staff acts as a second constraint, and that applying the default threshold rule when such a constraint binds leads to suboptimal decisions unless the cost categories are reweighted.

Capital rationing in the NHS and public appraisal

UK central government budgets for capital apart from day-to-day spending, and the Green Book asks the financial case to show public sector costs in resource or capital Departmental Expenditure Limits or in Annually Managed Expenditure. The Green Book records public sector financial costs in the financial dimension of the business case, split into capital and revenue components. The capital budget page describes the NHS capital regime.

Williams, Allen and Plahe interviewed senior finance professionals in 30 NHS provider organisations across England about how they respond to capital limits. Their findings suggest a pervasive sense of impending crisis, with capital restrictions limiting investment in buildings and equipment. Using the forms of rationing framework, the authors found selection, dilution and delay to be widespread, while denial and termination were comparatively rare. They also report little tailored support for priority setting in capital spending and a perception that decisions are often reactive and short term.

The Green Book limits how far its summary metrics can carry such choices. It treats them as an initial ranking of options that meet the same objectives, so ranking unrelated schemes across a whole capital programme by ratio goes beyond the use the guidance makes of them.

The shadow price of a rationed budget

A binding limit gives the funds a value above their face value. In the programme above, the multiplier on the budget constraint is the extra net present value one more pound of capital would buy, a shadow price of the limit. A project acceptable at the market price of capital may then be rejected, because the opportunity cost of the funds is the return on the best project they would displace.

Van Baal, Morton and Severens derive the threshold for a fixed health care budget the same way. In their model the multiplier is the shadow price of the health care budget, the QALYs that one more unit of budget would buy, and its reciprocal is the cost-effectiveness threshold. The budget constraint page develops that reading.

The NICE manual (PMG36, section 6.3.1) uses the same reasoning in words. It 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. Because NICE lacks full information on the costs and QALYs of all competing programmes, it uses a range of GBP 25,000 to GBP 35,000 per QALY (sections 6.3.4 to 6.3.8). Claxton and colleagues estimated the central NHS threshold at GBP 12,936 per QALY using 2008 expenditure data, well below the NICE range.

Worked example with four schemes and one capital allocation

All figures here are illustrative. Because the schemes serve different objectives, the ranking in step 2 shows only the arithmetic of rationing, and the Green Book would not use it as a decision rule. An NHS trust has GBP 10 million of capital for the year and four independent schemes, each appraised in present values with every cost falling on the capital allocation. Values are in GBP million, and NPSV is net present social value.

1. Without a limit. Every scheme has a positive NPSV, so all four would be funded for a total cost of 6.0 + 5.0 + 5.0 + 3.0 = GBP 19.0 million and a total NPSV of 12.0 + 9.0 + 8.5 + 3.0 = GBP 32.5 million. The GBP 10 million allocation rules this out.

2. Ranking by ratio. Funding in order of benefit-cost ratio takes X first, using GBP 6.0 million. Y and Z would each raise spending to GBP 11.0 million, so both are skipped, and W brings spending to GBP 9.0 million. The ranked choice of X and W gives an NPSV of 12.0 + 3.0 = GBP 15.0 million and leaves GBP 1.0 million unspent.

3. Best whole combination. Y and Z together cost exactly GBP 10.0 million and give 9.0 + 8.5 = GBP 17.5 million, the most any affordable set yields and GBP 2.5 million more than the ranked choice. That gap is the indivisibility loss that the integer programming page works through in full.

4. If schemes could be scaled. If each scheme could be funded in part, as with a phased equipment programme, the trust would fund all of X and 4.0 / 5.0 = 0.8 of Y, for an NPSV of 12.0 + 0.8 × 9.0 = GBP 19.2 million. The marginal scheme is Y, so one more pound of capital adds 9.0 / 5.0 = GBP 1.80 of NPSV, and a budget of GBP 11 million would give 19.2 + 1.8 = GBP 21.0 million, which is X and Y in full.

5. Reading the shadow price. In the divisible case each pound of rationed capital is worth GBP 1.80 of net value at the margin, or GBP 2.80 of benefits once the pound itself is counted. A new scheme competing for this allocation would need a benefit-cost ratio above 2.8 to displace part of Y, not merely above 1. For whole schemes the shadow price is less exact and the integer programme has to be solved directly. In a cost-effectiveness analysis the same steps apply with QALYs in place of NPSV and the threshold in place of the marginal ratio.

Welfare economics and the rationed health budget

In welfare economics, a budget limit is itself a choice that a full cost-benefit analysis could question. Cost-effectiveness analysis from a health care perspective usually takes the budget as given, and van Baal, Morton and Severens treat the health care budget as exogenous to the decision problem, the setting in which threshold rules work.

Birch and Gafni argued that the decision rules of cost-effectiveness and cost-utility analysis can serve managerial objectives in highly constrained environments but are inconsistent with welfare economic objectives. The argument is one strand of the debate between welfarist and extra-welfarist approaches to economic evaluation in health.

Capital rationing and neighbouring ideas

Capital rationing is a particular case of a budget constraint, one that applies to investment and binds because worthwhile projects exceed the funds. The capital budget is the allocation itself, while capital rationing names the situation in which that allocation is too small for the appraised schemes. Constrained optimisation supplies the methods for solving the problem. The social opportunity cost of capital, a related idea, asks what return public funds would earn elsewhere in the economy, a different question from the shadow price of an internal spending limit.

Common misreadings and limits

Most errors carry the unconstrained rule, or a simple ranking, into a setting with more structure than either can handle.

1. Funding every scheme with positive net present value. Under rationing that rule overspends the allocation, and the funded set must be chosen as a portfolio.
2. Treating a ratio ranking as optimal in every case. Ranking works only for a single limit and divisible projects. Whole schemes and limits over several years need a mathematical programme.
3. Applying a fixed ratio cut-off. The Green Book warns against rejecting proposals because their benefit-cost ratio falls below a set value, and the right cut-off under rationing is the ratio of the marginal project, which moves with the budget.
4. Ignoring limits other than money. Staff, theatre time and site capacity can bind before capital does, and van Baal, Morton and Severens show that the default threshold rule then needs adjusting.
5. Taking the limit as fixed for ever. A rationed budget reflects a policy choice, and its shadow price shows those who set it what relaxing the limit would be worth.

Sources

• Birch S, Gafni A. Cost effectiveness/utility analyses. Do current decision rules lead us to where we want to be? Journal of Health Economics. 1992;11(3):279-296. https://doi.org/10.1016/0167-6296(92)90004-K
• Brealey RA, Myers SC, Allen F. Capital rationing models. Online supplement to Principles of Corporate Finance, Global Edition, 11th ed. McGraw-Hill Education; 2014. https://highered.mheducation.com/sites/dl/free/0077151569/977788/c05c_Capital_rationing_models.pdf
• 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. https://doi.org/10.3310/hta19140
• HM Treasury. The Green Book (2026). Updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Karlsson G, Johannesson M. The decision rules of cost-effectiveness analysis. PharmacoEconomics. 1996;9(2):113-120. https://doi.org/10.2165/00019053-199609020-00003
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). 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. https://doi.org/10.1016/S0167-6296(96)00493-6
• van Baal P, Morton A, Severens JL. Health care input constraints and cost effectiveness analysis decision rules. Social Science & Medicine. 2018;200:59-64. https://doi.org/10.1016/j.socscimed.2018.01.026
• Williams I, Allen K, Plahe G. Reports of rationing from the neglected realm of capital investment: responses to resource constraint in the English National Health Service. Social Science & Medicine. 2019;225:1-8. https://doi.org/10.1016/j.socscimed.2019.02.015