Allocative Efficiency

Allocative Efficiency and the Best Mix of Health Services

Allocative efficiency in health care concerns which services a system funds and in what quantities; how well each service is run is a separate question. In practice the test is made at the margin, where the last pound in every funded use should buy roughly the same benefit. Palmer and Torgerson place it at the top of three nested ideas of efficiency, defined as achieving the right mixture of healthcare programmes to maximise the health of society.

Technical, productive and allocative efficiency

Palmer and Torgerson distinguish three concepts. Technical efficiency is the maximum improvement in outcome from a set of inputs; their example is a 10 mg daily dose of alendronate found as effective as 20 mg. Productive efficiency maximises outcome for a given cost, or minimises cost for a given outcome, so it can compare interventions using different input mixes, but not a shift of resources from geriatric care to mental illness, whose outcomes are incommensurate.

The three are nested. Productive efficiency implies technical efficiency and allocative efficiency implies productive efficiency, but none of the converse implications necessarily holds. Cylus, Papanicolas and Smith add that allocative efficiency has an output side, whether resources produce the right mix of outputs given their relative value, and an input side, whether the input mix, such as skill mix, minimises cost at given prices. Health economics usage mostly means the output side. A clinical team can be efficient at its own task inside a system that misallocates between preventive and curative care.

What counts as the best mix under Pareto, welfarism and extra-welfarism

Allocative efficiency is relative to an objective, and choosing it is a value judgement. Arrow's statement of the first optimality theorem gives the classical benchmark. A competitive equilibrium in which everything relevant to costs or utilities is priced is optimal in Pareto's sense, so no other allocation makes all participants better off. Brouwer and colleagues stress that there may be many such optima, and the Pareto efficiency criterion alone cannot choose between them.

Palmer and Torgerson note that strict adherence to the Pareto criterion has proved impossible in practice, so the working rule is to maximise the welfare of the community. In welfare economics that rule has two readings. Welfarism, in Brouwer and colleagues' account, judges a situation solely by the utility levels individuals attain, with gains and losses compared in money terms under the potential Pareto criterion. Extra-welfarism permits outcomes other than utility, sources of value other than the affected individuals, non-preference weights and interpersonal comparisons; the QALY lets analysts serve decision makers who have chosen health maximisation, so allocative efficiency becomes the most health from a fixed budget. The same authors warn that forgone health cannot be the only opportunity cost of public resources, even if it is the principal one inside the health sector. Equity weights change which allocation counts as efficient, as distributional cost-effectiveness analysis makes explicit.

The equal-margins condition

With a fixed budget and a health objective, allocation is a constrained maximisation. Spending is spread until no transfer raises the total.

$$\max_{x_1,\ldots,x_J} \sum_{j=1}^{J} H_j(x_j) \quad \text{subject to} \quad \sum_{j=1}^{J} x_j \le B$$

where $x_j$ is spending on programme $j$, $H_j(x_j)$ the health (in QALYs) it produces, $J$ the number of programmes and $B$ the budget. If each $H_j$ is smooth with diminishing returns and every programme is funded, the solution satisfies

$$H_1'(x_1) = H_2'(x_2) = \cdots = H_J'(x_J) = \lambda$$

where $H_j'(x_j)$ is the extra health from one more pound in programme $j$ and $\lambda$ is the health produced by the last pound of budget, so $\frac{1}{\lambda}$ is the cost per QALY at the margin. Mitton and Donaldson describe the same point, at which no further reallocation increases total benefit, and in theory marginal cost to marginal benefit ratios are equal across services. The condition concerns the last unit of each programme, not its average; indivisible services, fixed capacity and links between programmes can stop the margins equalising.

Thresholds as the price of displaced health

Cylus and colleagues link this to health technology assessment. Funding treatments in order of incremental cost per QALY until the budget runs out maximises health, and an equivalent rule accepts only treatments below a threshold set by the size of the budget. A cost-effectiveness threshold is therefore allocatively meaningful only if it approximates $\frac{1}{\lambda}$, the cost of health given up elsewhere.

Claxton and colleagues define the threshold as the extra cost the system must bear to forgo one QALY through displacement, with a central estimate of £12,936 per QALY (2008 expenditure, 2008 to 2010 mortality) and substantial uncertainty. Lomas, Martin and Claxton, who present such estimates as a way to improve allocative efficiency, found that an additional QALY cost the English NHS between £5,000 and £15,000 from 2003 to 2012, in current prices. The NICE manual (PMG36) requires committees to consider programmes displaced by a new technology, including those NICE has not evaluated, and has committees refer explicitly to listed factors as the incremental cost-effectiveness ratio (ICER) rises from £25,000 to £35,000 per QALY. Where a decision threshold sits above the marginal productivity of the budget, a technology with an ICER between the two displaces more health than it adds. Net health benefit states the comparison directly.

$$\text{NHB} = \Delta E - \frac{\Delta C}{k}$$

where $\Delta E$ is the QALY gain of the new option, $\Delta C$ its incremental cost to the budget and $k$ the cost per QALY of the spending it displaces.

Worked example of moving £100,000 between two programmes

The figures are illustrative. A commissioner splits £1,000,000 into £100,000 tranches for falls prevention (A) and cardiac rehabilitation (B). Each further tranche yields fewer QALYs; A now has six tranches and B four.

1. Current allocation. A produces 30 + 22 + 15 + 10 + 6 + 3 = 86 QALYs and B produces 25 + 20 + 16 + 12 = 73, a total of 159.
2. Average against margin. A's average cost is £600,000 / 86 = £6,977 per QALY, but its sixth tranche costs £100,000 / 3 = £33,333 per QALY. B's next tranche would cost £100,000 / 8 = £12,500 per QALY.
3. Move one tranche. Cutting A's sixth tranche loses 3 QALYs and funding B's fifth gains 8, a net gain of 5. A now gives 83, B gives 81 and the total is 164 from the same budget.
4. Test further moves. Moving A's fifth tranche (6 QALYs) to B's sixth (5) loses 1 QALY; moving B's fifth (8) back to A's sixth (3) loses 5. No transfer helps, so five tranches each is efficient, although the last yields (6 and 8) differ because spending moves in lumps.
5. Implied threshold. The cheapest health to give up is A's fifth tranche, so $k$ = £100,000 / 6 = £16,667 per QALY. A new technology costing £100,000 for 5 QALYs has an ICER of £20,000, below £25,000, yet displaces 6 QALYs.

$$\text{NHB} = \Delta E - \frac{\Delta C}{k} = 5 - 6 = -1$$

where 5 is the technology's QALY gain and 6 is $\Delta C/k$, the QALYs lost by cutting A's fifth tranche. Adopting it would cost one QALY overall, more if disinvestment fell on a more productive service.

Market failure and allocative distortion

An allocative distortion is a departure from the efficient mix, usually through market failure, regulation or mispricing. In such cases the marginal value of a use departs from its marginal cost or, in a budgeted system, a pound buys more health in one use than another. Arrow argued that when the competitive preconditions fail, welfare falls below what existing resources and technology could achieve. Thomas and Chalkidou add that, without perfectly competitive markets and with information asymmetry between providers and patients, a market-driven allocation of health care is likely to be inefficient.

Several market failures open such gaps. Arrow's example of an externality is communicable disease, where a person who is not immunised risks the health of others as well as their own. Insurance creates moral hazard; Feldstein and Gruber show that a person with a 20 per cent coinsurance rate uses care until the last dollar of services is worth only 20 cents to them, an 80-cent welfare loss; summed over the extra use, this is a deadweight loss. Regulation can distort too. Arrow notes that licensing restricts entry to the medical profession and raises the cost of care, defended as a guarantee of minimum quality. Cylus and colleagues observe that inappropriate care may persist because the financial regime still rewards it, in which case responsibility may lie with policymakers rather than providers.

Programme budgeting and marginal analysis

Programme budgeting and marginal analysis (PBMA) applies the equal-margins rule to local priority setting. Mitton and Donaldson build it on opportunity cost and the margin. With a flat budget, the question is whether some areas should be cut back so that others can expand. They identify releasing resources through real service reductions as the harder task. Marginal analysis covers the steps in detail.

Measuring allocative efficiency

Two of the most common methods for measuring health care efficiency use frontier analysis, which Hollingsworth describes as assessing how a unit such as a hospital uses its own inputs to produce outputs, against similar units. Data envelopment analysis and stochastic frontier analysis therefore mainly measure technical efficiency, which Cylus and colleagues define as minimising inputs for chosen outputs regardless of the value placed on those outputs. The allocative measure in this literature is the input-side one.

$$TE = \frac{OA}{OC}, \qquad AE = \frac{OB}{OA}$$

where, on a diagram of labour and capital for one level of output, C is the hospital's actual input use, A is the point on the frontier with the same input mix, and B lies on the least-cost line at prevailing input prices along that ray. Hollingsworth reads AE as the excess cost of using inputs in inappropriate proportions, so it needs input prices and says nothing about whether those outputs should be produced. Because AE is measured here separately from TE, a hospital can score 1 on AE while technically inefficient; the product OB/OC is overall cost efficiency, which is productive efficiency in Palmer and Torgerson's sense. The earlier nesting applies to the programme-mix sense.

Output-side allocative efficiency is harder to measure. Thomas and Chalkidou note that the optimal mix must be known before any deviation from it can be estimated, which at system level may mean comparing all programmes against one another. Indicators such as avoidable admissions or excessive antibiotic prescribing can signal misallocation, once case mix and workforce or capital constraints are checked.

Common misreadings

A lower unit cost or a better frontier score shows technical or productive gains; where the released resources go is a further allocative question. One beneficial transfer, like the one above, shows an improvement and not a global optimum. A Pareto-efficient or health-maximising allocation can still be very unequal; Arrow called the judgement behind the Pareto criterion a very weak value judgement. An ICER below the decision threshold shows allocative efficiency only if the threshold reflects the health actually displaced.

Sources

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