Healthcare Production Function

Healthcare Production Function, Marginal Product and the Cost of Care

Written as a function of staff, capital, drugs and technology, the healthcare production function underpins provider efficiency analysis and the marginal cost of activity. That is the extra cost of one more consultation or episode, which, when only staff time changes, equals the cost of the added staff time divided by its marginal product. The function differs from the health production function, in which health itself is the output.

From staff, beds and equipment to visits and episodes

Economists treat the transformation of inputs into valued outputs as a production function giving the maximum feasible output for given inputs, so any shortfall signals inefficiency. The idea applies from a single physician's office to an entire health system. In health care the output is usually activity, such as consultations, procedures or episodes of hospital care. Health care is an intermediate product, a means to the end of better health, so this function stops one step short of health itself.

In general form the function can be written as follows.

$$Q = f(L, K, M; T)$$

where $Q$ is maximum output in a period (such as consultations or case-mix-adjusted episodes), $L$ is labour, often split by skill type, $K$ is capital such as beds and equipment, $M$ is drugs and consumables, and $T$ is the technology and organisation of care.

A framework chapter in a European Observatory on Health Systems and Policies volume describes hospital production as a chain of stages; the stage studied decides what counts as input and output, and cost per QALY summarises the whole chain.

Technology and organisation shift the whole function rather than moving along it. In a study of 2.8 million Medicare heart attack patients that measured hospital output as patient survival, hospitals quickly adopting cost-effective treatments such as beta blockers and aspirin had substantially better outcomes, while extra spending with adoption held constant brought relatively modest returns.

Marginal product, average product and diminishing returns

The marginal product is the extra activity from one more unit of an input, and the average product is activity per unit of the input. Both are read off the production function. For labour, they are defined as follows.

$$MP_L = \frac{\partial Q}{\partial L}, \qquad AP_L = \frac{Q}{L}$$

where $MP_L$ is the marginal product of labour with other inputs held constant, measured in tables as the change in output divided by the change in labour, and $AP_L$ is the average product of labour.

Diminishing marginal returns arise when one input increases while others stay fixed. Marginal product may rise at first, as staff divide tasks, but eventually each extra unit adds less than the one before because it shares fixed rooms, beds or equipment. This is one reason marginal analysis looks at the effect of a small change instead of at averages.

Returns to scale are a long-run idea about a proportionate increase in all inputs together. They are constant when output rises in the same proportion, decreasing when it rises less and increasing when it rises more. A provider can face diminishing returns to extra nurses in a fixed building yet constant returns to scale when everything expands together.

The Cobb-Douglas form used in hospital studies

Hospital studies often estimate a Cobb-Douglas production function, sometimes alongside the more flexible translog form. Its parameters read directly as elasticities. With two inputs, it takes the form below.

$$Q = A L^{\alpha} K^{\beta}$$

where $A$ is a productivity parameter and $\alpha$ and $\beta$ are the elasticities of output with respect to labour and capital. The marginal product of labour is $\alpha Q / L$, the sum $\alpha + \beta$ measures returns to scale, the marginal rate of technical substitution is the ratio of marginal products, and the elasticity of substitution is fixed at 1.

A 2022 study of 94 Polish county hospitals, with 2018 patient-days as the output, estimated elasticities of 0.48 for electricity costs and 0.2 for nurse full-time equivalents. A 1% rise in both inputs would therefore raise patient-days by less than 1%, a sign of decreasing returns to scale.

From the production function to the cost function and marginal cost

A cost function gives the lowest cost of producing each level of output. Reading the production function backwards gives the inputs needed for each output level, and multiplying them by their prices gives cost; under suitable conditions the cost function is the dual of the production function.

With labour as the only variable input, paid a wage $w$ per unit, the per-unit costs follow directly from the products of labour, as shown below.

$$MC = \frac{w}{MP_L}, \qquad AVC = \frac{w}{AP_L}$$

where $MC$ is marginal cost, the extra cost of one more unit of output, $AVC$ is average variable cost per unit of output, and $MP_L$ and $AP_L$ are as defined above. Average cost is total cost, including fixed costs, divided by output.

As marginal product falls, each extra unit of activity needs more labour, so marginal cost rises. Average cost first falls as fixed costs are spread, then rises once marginal cost exceeds it. The marginal cost of healthcare therefore depends on where the provider sits on its production function.

Worked example of adding GP sessions to a practice of fixed size

The figures are illustrative. A practice has fixed premises, reception and equipment costing GBP 6,000 a week, and each full-time equivalent GP costs GBP 3,000 a week. The table shows weekly consultations with one to five GPs and the products and costs per consultation derived from them.

1. Products of the fourth GP. Marginal product is the change in consultations and average product is consultations per GP, as follows.

$$MP_4 = 560 - 460 = 100, \qquad AP_4 = \frac{560}{4} = 140$$

where $MP_4$ is the marginal product of the fourth GP and $AP_4$ the average product with four GPs. Marginal product rises to 170 with the second GP, then falls to 140, 100 and 60 once diminishing returns set in from the third GP because rooms and support staff are fixed.

2. Average cost with four GPs. Total cost, the fixed cost plus four GPs, is divided by consultations.

$$AC_4 = \frac{6{,}000 + 4 \times 3{,}000}{560} = \frac{18{,}000}{560} = 32.14$$

where $AC_4$ is the average cost per consultation in GBP with four GPs. Of this, average fixed cost is $AFC_4 = 6{,}000/560 = 10.71$ and average variable cost is $AVC_4 = 12{,}000/560 = 21.43 = 3{,}000/140$, as $w/AP_L$ predicts.

3. Marginal cost. The consultations added by the fourth and fifth GPs cost the amounts below.

$$MC_4 = \frac{3{,}000}{100} = 30.00, \qquad MC_5 = \frac{3{,}000}{60} = 50.00$$

where $MC_4$ and $MC_5$ are the marginal costs in GBP per extra consultation.

4. Why average cost turns upwards. The fourth GP's consultations cost 30.00 each at the margin, below the average of 32.61 with three GPs, so average cost falls to 32.14. The fifth GP's consultations cost 50.00 each, above 32.14, so average cost rises to $AC_5 = 21{,}000/620 = 33.87$, where $AC_5$ is average cost with five GPs.

5. What the numbers show. Four GPs give the lowest average cost, but average cost is a poor guide to the cost of expansion. Paying GBP 32.14 for each of the 60 extra consultations a fifth GP provides would cover less than two thirds of their marginal cost of GBP 50.00. The example holds case mix and consultation quality constant.

Measuring efficiency against the production frontier

Because the true production function is never observed, efficiency analysis compares each provider with a frontier estimated from similar providers. Technical efficiency means producing the maximum output from given inputs, whatever the value of that output. Productive efficiency brings in input prices, so that the chosen output is produced at the lowest possible cost, which needs technical efficiency and the least-cost mix of inputs, so a practice can be technically efficient yet productively inefficient if a cheaper skill mix would deliver the same care. Terminology varies. Frontier studies often call this cost efficiency, some sources call the input-mix part allocative efficiency on the input side, and Palmer and Torgerson frame productive efficiency in terms of health outcomes. On the output side, allocative efficiency asks whether the right outputs are being produced at all, given what society values, which is where cost per QALY analysis and health technology assessment come in.

Data envelopment analysis (DEA) uses linear programming to build the frontier from the best performers. Stochastic frontier analysis (SFA) uses regression and splits the error term into inefficiency and random noise; because an SFA production frontier needs all outputs combined into one measure, analysts often estimate cost frontiers instead. Patients are not inputs transformed into a product, so one common approach to case mix is to adjust outputs, for example with diagnosis-related group weights as in a case mix index.

Choices that change the estimated function

Estimates embody choices that can move results as much as real differences in performance. Activity counts are easy to observe but ignore quality and health gain, while outcomes are closer to what is valued but hard to attribute to the provider. Labour can be split by skill type or combined using relative wages, and capital is often proxied crudely by beds or floor space. Population, disease severity, geography and local primary care affect what a hospital can achieve, and can be handled by comparing similar providers, by regression or by risk adjustment.

Healthcare production compared with health production

The two functions share a name and a mathematical form but answer different questions. The healthcare production function concerns providers, whose output is activity. The health production function, formalised in Grossman's 1972 Grossman model of the demand for health, concerns individuals. Its output is health, and its inputs include medical care alongside own time, diet, exercise and housing, with education affecting how efficiently health is produced. The two connect through demand. In Grossman's framework the demand for medical care is a derived demand arising from the demand for health, so the output of the healthcare production function becomes one input into the health production function.

Use in costing and service planning

The production function seldom appears directly in an economic evaluation, but it sits behind cost inputs and workforce plans. Valuing additional activity at an average unit cost assumes that marginal cost equals average cost, which generally holds only at the lowest point of the average cost curve; the worked example shows how far the two can diverge. Marginal products and substitution rates show whether one staff group can take on work from another, so substitution has to be demonstrated, not assumed.

Common misreadings of healthcare production estimates

Most errors come from treating an estimate as more precise or general than it is. Each reflects one sample of providers, one output measure and one functional form.

• Confusing activity with health. A provider can look productive on activity while adding little health gain.
• Over-reading benchmarks and rankings. Frontier estimates show which providers produce less than their peers, not whether the outputs are worth producing. Alternative specifications within accepted practice have given similar mean inefficiency but substantially different rankings of individual hospitals.
• Ignoring case mix and context. A provider treating sicker patients, or in a remote area, can appear inefficient for reasons outside its control.

Sources

• Cylus J, Papanicolas I, Smith PC. A framework for thinking about health system efficiency. In: Cylus J, Papanicolas I, Smith PC, eds. Health system efficiency: how to make measurement matter for policy and management. Copenhagen: European Observatory on Health Systems and Policies; 2016. Chapter 1 and Glossary.
• Folland ST, Hofler RA. How reliable are hospital efficiency estimates? Exploiting the dual to homothetic production. Health Economics. 2001;10(8):683-698. doi:10.1002/hec.600
• Greenlaw SA, Shapiro D, MacDonald D. Principles of Economics. 3rd ed. Houston: OpenStax; 2022. Sections 7.2 and 7.3.
• Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. New York: National Bureau of Economic Research; 1972. Introduction and Chapter 1.
• Hollingsworth B. Health system efficiency: measurement and policy. In: Cylus J, Papanicolas I, Smith PC, eds. Health system efficiency: how to make measurement matter for policy and management. Copenhagen: European Observatory on Health Systems and Policies; 2016. Chapter 5.
• Palmer S, Torgerson DJ. Definitions of efficiency. BMJ. 1999;318(7191):1136. doi:10.1136/bmj.318.7191.1136
• Sielska A, Nojszewska E. Production function for modeling hospital activities. The case of Polish county hospitals. PLoS One. 2022;17(5):e0268350. doi:10.1371/journal.pone.0268350
• Skinner J, Staiger D. Technology diffusion and productivity growth in health care. Review of Economics and Statistics. 2015;97(5):951-964. doi:10.1162/REST_a_00535