Productive Efficiency

Productive Efficiency in Health Care Production

Productive efficiency asks whether a hospital, clinic or programme could deliver the same care, or the same health gain, for less money. A provider meets the test only when it wastes no inputs and also combines staff, drugs, equipment and beds in the proportions that cost least at the prices it actually pays. The idea matters because a unit can look lean on a staffing ratio yet still spend more than necessary through a costly skill mix, and the remedies for the two problems differ. Hospital frontier studies measure it as cost efficiency, and economic evaluation uses the same logic for interventions with directly comparable outcomes. Its limit is that it treats the output as given, so it cannot say whether that output was worth producing.

Two readings of productive efficiency in health economics

Health economists use the term in two closely linked ways. Palmer and Torgerson, in a short BMJ note on definitions of efficiency, describe productive efficiency as the maximisation of health outcome for a given cost, or the minimisation of cost for a given outcome. Their example is Down's syndrome screening, where replacing maternal age screening with biochemical screening uses fewer amniocenteses but adds biochemical testing. Because the two programmes use different combinations of inputs, the choice turns on the relative costs of those inputs. If the biochemical programme costs no more and its outcomes are equal or better, it is productively efficient relative to maternal age screening.

The production-frontier literature uses the same idea for providers. Farrell's 1957 framework, as set out by Førsund and Sarafoglou, divides the performance of a production unit into technical efficiency, price (or allocative) efficiency and overall efficiency, and the overall measure is the one that corresponds to producing at least cost. Jacobs, writing on hospital cost frontiers, notes that inefficiency on the cost side is often called cost inefficiency, the total of technical and allocative inefficiency. Cost efficiency, overall efficiency and productive efficiency thus name one target, and allocative here means the input mix, not the programme mix of the allocative efficiency page.

Palmer and Torgerson also set out how the concepts nest. Productive efficiency implies technical efficiency and allocative efficiency, in their programme-mix sense, implies productive efficiency, but none of the converse implications necessarily holds. Faced with limited resources, the productive test rules out some input combinations that are technically efficient, because they cost more than another combination that produces the same result.

Farrell's split into technical and price efficiency

Farrell's measures compare an observed unit with a reference point on the best-practice frontier reached by scaling all its inputs down in proportion, so the observed input ratios are kept. On a diagram with one output and two inputs, P is the unit's observed input bundle, Q is the point on the frontier along the ray from the origin through P, and R is the point on that ray whose cost equals the minimum cost of producing the output at observed input prices.

$$OE = \frac{OR}{OP} = \frac{OQ}{OP} \times \frac{OR}{OQ} = TE \times PE$$

where $OE$ is overall (productive or cost) efficiency, $TE = OQ/OP$ is technical efficiency, the share of observed inputs needed at best practice to produce the observed output, and $PE = OR/OQ$ is price efficiency, the minimum cost relative to the cost of the technically efficient bundle Q. Each ratio lies above 0 and at most 1, and a value of 1 means no saving is available from that source.

Least cost for a given output

The benchmark for productive efficiency is the lowest cost at which the observed output can be produced with the available technology and the prices the provider faces. Written as a minimisation, it is the problem that defines a cost function.

$$C(w, y) = \min_{x} \, w^{\top} x \quad \text{subject to } x \text{ being able to produce } y$$

where $x$ is the vector of input quantities (for example nurse hours, consultant hours and theatre sessions), $w$ is the vector of their unit prices, $y$ is the output, and $C(w, y)$ is the minimum attainable cost. A unit with observed inputs $x^{o}$ has cost efficiency

$$CE = \frac{C(w, y)}{w^{\top} x^{o}}$$

where $w^{\top} x^{o}$ is its actual spending on inputs at prices $w$, and $CE$ equals Farrell's overall efficiency. The minimising bundle depends on $w$, so the least-cost mix changes when relative prices change even if the technology and the output do not.

Worked example of a day-case cataract unit

All figures in this example are illustrative. Three day-case units each perform 1,000 cataract operations a year, with comparable case mix and outcomes, using nurse hours and surgeon hours. Unit B uses 6,000 nurse hours and 1,500 surgeon hours, Unit C uses 3,000 nurse hours and 3,000 surgeon hours, and both lie on the best-practice frontier, which runs along the straight line between them. Unit A uses 5,400 nurse hours and 2,700 surgeon hours, a ratio of 2 to 1.

1. Technical efficiency. The frontier point with Unit A's 2 to 1 ratio lies halfway between B and C, at 4,500 nurse hours and 2,250 surgeon hours (the average of 6,000 and 3,000, and of 1,500 and 3,000). Technical efficiency is 4,500 / 5,400 = 0.833, and 2,250 / 2,700 gives the same value, so Unit A could cut both inputs by about 17% without losing output.
2. Costs at current prices. With a nurse hour costing GBP 40 and a surgeon hour GBP 100, Unit A spends 5,400 × 40 + 2,700 × 100 = GBP 486,000. The technically efficient bundle would cost 4,500 × 40 + 2,250 × 100 = GBP 405,000, and 405,000 / 486,000 = 0.833, matching the technical score.
3. Least-cost point. Along a straight frontier the cheapest bundle is one of its end points. Unit B's mix costs 6,000 × 40 + 1,500 × 100 = GBP 390,000 and Unit C's costs 3,000 × 40 + 3,000 × 100 = GBP 420,000, so the nurse-intensive mix is cheapest. Price efficiency is 390,000 / 405,000 = 0.963.
4. Productive (cost) efficiency. Overall, 390,000 / 486,000 = 0.802, and 0.833 × 0.963 = 0.802, as Farrell's decomposition requires. Unit A's cost per operation is GBP 486 against a least cost of GBP 390.
5. Where the saving lies. Of the GBP 96,000 gap, GBP 81,000 (486,000 less 405,000) comes from wasted hours and GBP 15,000 (405,000 less 390,000) from the input mix.
6. A change in relative prices. Units B and C cost the same when a surgeon hour costs twice a nurse hour (3,000 nurse hours against 1,500 surgeon hours), GBP 80 here. At GBP 70 a surgeon hour, B's mix costs 240,000 + 105,000 = GBP 345,000, C's costs 120,000 + 210,000 = GBP 330,000 and Unit A spends 216,000 + 189,000 = GBP 405,000. Technical efficiency stays at 0.833, the least-cost mix switches to Unit C's, price efficiency becomes 330,000 / 337,500 = 0.978 and cost efficiency 330,000 / 405,000 = 0.815.

Most of Unit A's excess is waste, which a technical score would reveal. A unit at the halfway point would be technically efficient yet spend GBP 15,000 a year more than Unit B at the first set of wages, and which skill mix is cheapest depends on what staff cost, which no physical frontier can show.

Where productive inefficiency arises in health services

Cylus, Papanicolas and Smith trace waste through each stage at which money becomes inputs and inputs become care. The first question, given the chosen inputs, is whether they are bought at minimum cost, for example whether an organisation uses branded rather than generic drugs or pays wage rates above local market rates. A cost efficiency score computed at the prices a unit itself pays does not detect overpayment of this kind. An average hourly wage adjusted for skill mix can flag this, and the authors note that without that adjustment the same index also picks up the allocative side of input choice, whether the right mix of doctors, other professionals and administrators is being deployed.

At the next stage, inputs become activities such as tests and procedures. The same chapter lists the use of highly skilled, and therefore costly, workers for activities that less specialised workers could do, and the use of excessive hours of labour or other inputs for a given activity. On the input side the authors also mention treatment in the wrong setting, such as an emergency department rather than primary care, and inappropriate inputs such as emergency ambulance transport for non-urgent care.

Assumptions behind a productive efficiency verdict

A judgement that a provider is or is not productively efficient holds only under the assumptions used to build the benchmark. The output must be held constant in both quantity and quality. Cylus and colleagues describe activity measures such as patients treated or operations undertaken as manifestly inadequate, because they miss variations in the effectiveness of care, yet analysts often have no alternative. A cheaper unit with worse outcomes has not shown productive efficiency.

The prices matter as much as the technology. The least-cost mix in the example moved when one wage changed, so a verdict depends on which prices are used and whether they reflect the opportunity cost of the resources. Cylus and colleagues add that using total cost as the single input assumes that organisations are free to deploy inputs efficiently, taking account of relative prices. In practice parts of the input mix are fixed in the short term, the stock of capital in particular, so an analysis may need to keep inputs separate to reflect the mixes organisations have inherited.

The frontier itself is estimated from the units observed. A best-practice frontier built from a sample shows the best achieved in that sample, so units that all share a costly practice can each appear productively efficient.

Measuring productive efficiency with frontier methods

The two main empirical tools, data envelopment analysis and stochastic frontier analysis, have their own pages, and only their treatment of cost matters here. Jacobs observes that, by letting each hospital choose its own output weights, data envelopment analysis addresses technical efficiency. Hollingsworth notes that other forms of efficiency, allocative efficiency among them, can also be measured with it. Separating the price component requires data on input prices as well as quantities.

A stochastic cost frontier estimates the minimum cost for each hospital's output and reads the one-sided part of the residual as inefficiency. In Jacobs' account the cost efficiency of a hospital is its expected cost given its inefficiency divided by its expected cost with no inefficiency, a value between one and infinity, so the scores have to be inverted before they can be compared with data envelopment analysis scores on the Farrell scale. If allocative efficiency is assumed, the one-sided term relates closely to the cost of technical inefficiency. Otherwise it may contain both technical and allocative inefficiency, and without further assumptions the method cannot separate them. Jacobs stresses that the distinction matters because the two require different policy responses.

Productive efficiency in economic evaluation

In economic evaluation, productive efficiency is the question answered when the outcome is fixed and only the cost of reaching it varies. Palmer and Torgerson note that it allows the relative value for money of interventions with directly comparable outcomes to be assessed. Cost-minimisation analysis is the most direct case, since once outcomes are shown to be equivalent the choice reduces to the lowest cost.

Their note is equally clear about the limit. Productive efficiency cannot judge a broader reallocation, such as moving resources out of geriatric care and into mental illness services, because the health outcomes are incommensurate. That wider question belongs to allocative efficiency, which needs a common measure of value across programmes, such as the QALY, and a view of what each programme displaces. The wider family of concepts is on the efficiency page.

Common misreadings of productive efficiency

Several errors recur in reports and teaching material. The first is to present a technical efficiency score, such as staff needed at best practice divided by staff used, as a measure of productive efficiency. A score of that kind ignores input prices and so cannot detect a costly skill mix.

A lower unit cost is also not proof of productive efficiency unless output, quality and case mix are comparable. Nor is the least-cost mix a fixed property of a service, since it moves with relative prices, as step 6 of the example shows. Finally, an inefficiency score from a stochastic cost frontier should not be read as pure waste, because without further assumptions it combines waste with a poor input mix.

Sources

• Cylus J, Papanicolas I, Smith PC. A framework for thinking about health system efficiency. In: Cylus J, Papanicolas I, Smith PC, editors. Health System Efficiency: How to Make Measurement Matter for Policy and Management. Health Policy Series 46. Copenhagen: World Health Organization on behalf of the European Observatory on Health Systems and Policies; 2016. p. 1-20. https://iris.who.int/server/api/core/bitstreams/2794c497-bcca-4a89-8161-b65f1b5cb998/content
• Førsund FR, Sarafoglou N. On the origins of data envelopment analysis. Memorandum 24/2000. Oslo: Department of Economics, University of Oslo; 2000. https://www.sv.uio.no/econ/english/research/publications/memoranda/pdf-files/2000/Memo-24-2000.pdf
• Hollingsworth B. Health system efficiency: measurement and policy. In: Cylus J, Papanicolas I, Smith PC, editors. Health System Efficiency (as above); 2016. p. 99-114.
• Jacobs R. Alternative Methods to Examine Hospital Efficiency: Data Envelopment Analysis and Stochastic Frontier Analysis. CHE Discussion Paper 177. York: Centre for Health Economics, University of York; 2000. https://www.york.ac.uk/media/che/documents/papers/discussionpapers/CHE%20Discussion%20Paper%20177.pdf
• Palmer S, Torgerson DJ. Definitions of efficiency. BMJ. 1999;318(7191):1136. doi:10.1136/bmj.318.7191.1136