Signature
Q = A * L^alpha * K^beta; RTS = alpha + beta; Q_s = A * (s * L)^alpha * (s * K)^beta; g_s = Q_s / Q
| Inputs | Definition | Unit |
|---|---|---|
A | Activity obtained from one unit of each input, the level of the function, which shifts with technology and organisation and is above zero | units of activity per period |
L | Quantity of staff time used, such as nurse full-time equivalents, above zero | full-time equivalents |
alpha | Percentage change in Q from a 1% rise in L with K fixed, above zero and the same at every level of the inputs | none |
K | Quantity of capital or of another non-labour input, such as beds, or the electricity costs that Sielska and Nojszewska used beside nurses, above zero | as measured, such as beds or currency per period |
beta | Percentage change in Q from a 1% rise in K with L fixed, above zero and the same at every level of the inputs | none |
s | Multiple applied to both L and K, such as 1.01 for a 1% rise or 2 for doubling, above zero | none |
Q | Maximum activity the provider obtains in the period from labour L and capital K, such as patient-days or consultations | units of activity per period |
|---|---|---|
RTS | Sum of the output elasticities, the percentage change in Q when L and K both rise by 1%, equal to 1 under constant returns to scale | none |
Q_s | Activity from s times L and s times K with A unchanged | units of activity per period |
g_s | Q_s divided by Q, equal to s to the power RTS (derived) | ratio, without unit |
Function
Healthcare production function mapping staff, capital, consumables and technology to provider activity
Maps a provider's labour L, capital K, drugs and consumables M and technology and organisation T to the maximum activity Q it can produce in a period, such as consultations or case-mix-adjusted episodes. It is the provider counterpart of the household function HE-FN-HPF-001, in which health itself is the output. The quantities read off the function are held in live records and are cited here, not restated. The Cobb-Douglas form with exponents summing to one is HE-FM-HPF-001 with A = exp(rho_H E), alpha = 1-K and beta = K, K there being the own time share, and the marginal products, each exponent times output over the input, and their ratio, the marginal rate of technical substitution, take the form of those in HE-FM-HPF-002, which computes them only for exponents summing to one; the same form holds for any exponents (derived). Marginal cost as the price of an input over its marginal product, MC = w / MP_L when labour is the only variable input, is Grossman's condition held in HE-SR-HPF-003 and tested in HE-VT-HPF-006, and average variable cost w / AP_L is the unit labour cost w / a_h of HE-FM-BAUM-001. A discrete marginal product from a production schedule is a change in output over a change in input, the same difference-quotient form as HE-FM-ICER-001, and an average product is a partial productivity ratio such as the bed turnover rate of HE-FM-BTO-001. Distance from an estimated frontier is measured by HE-FM-EFF-001 to HE-FM-EFF-003. Average cost, cost functions and the marginal cost of a change in scale belong to the Average Cost, Cost Function and Marginal Analysis pages. This package adds the degree of returns to scale of a Cobb-Douglas function whose output elasticities are estimated freely (HE-FM-HCPF-001). Notation follows the Healthcare Production Function article.
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Implementations
Excel
Cobb-Douglas hospital output and returns to scale from named cells
With the productivity parameter in CDProductivity, the inputs in LabourInput and CapitalInput, the elasticities in LabourElasticity and CapitalElasticity and the scale factor in ScaleFactor, the four formulas fill CDOutput, ReturnsToScale, ScaledOutput and ScaleOutputRatio in turn. ScaledOutput recomputes output from the scaled inputs, so ScaleOutputRatio does not use ReturnsToScale.
=CDProductivity*LabourInput^LabourElasticity*CapitalInput^CapitalElasticity; =LabourElasticity+CapitalElasticity; =CDProductivity*(ScaleFactor*LabourInput)^LabourElasticity*(ScaleFactor*CapitalInput)^CapitalElasticity; =ScaledOutput/CDOutput
Assumptions
Constant output elasticities in the Cobb-Douglas hospital production function
Each output elasticity is a constant that does not depend on how much of each input is used, Sielska and Nojszewska's equation (10), so RTS is the same for a small and a large provider, and the elasticity of substitution is fixed at 1 (their equation (13); HE-AS-HPF-001 states the same for the household function). A provider whose returns to scale differ with its size cannot be described by one set of Cobb-Douglas elasticities. A more flexible form, such as the translog function that Sielska and Nojszewska also estimated, is needed for that.
Positive inputs and elasticities with technology held fixed in the hospital scale formula
L, K, A and s are above zero, and alpha and beta are above zero, the condition Sielska and Nojszewska give for positive marginal products. The scale change moves the provider along one function with A unchanged; a change in technology or organisation shifts the whole function, as the article describes, and appears as a change in A rather than in RTS.
Returns to scale measured over the inputs included in the estimated hospital function
RTS describes the response of activity to the inputs that appear in the function. Sielska and Nojszewska estimated two-factor and three-factor functions, each using some of beds, doctors, nurses, materials, electricity and outsourced services. A sum below 1 for two included inputs therefore describes scaling those two, not every resource the hospital uses (derived).
Worked examples
Sielska and Nojszewska elasticities with a 1% rise in nurses and electricity
With the output elasticities that Sielska and Nojszewska estimated for 94 Polish county hospitals, 0.2 for nurse full-time equivalents and 0.48 for electricity costs, RTS is 0.68, so a 1% rise in both inputs raises patient-days by 0.68%, less than 1%, as the article states. The productivity parameter of 2,000, the 100 nurse FTEs and the electricity input of 500 are illustrative and not taken from the study, and g_s does not depend on them.
A = 2000; L = 100; K = 500; alpha = 0.2; beta = 0.48; s = 1.01; Q = 99205; RTS = 0.68; Q_s = 99879; g_s = 1.006789
Doubling nurses and electricity with returns to scale of 0.68
With the same illustrative hospital, doubling both inputs multiplies patient-days by 2 to the power 0.68, 1.6021, a rise of 60.2% rather than 100% (computed here for illustration).
A = 2000; L = 100; K = 500; alpha = 0.2; beta = 0.48; s = 2; Q = 99205; RTS = 0.68; Q_s = 158941; g_s = 1.6021
Constant returns limiting case of the hospital scale formula
With elasticities of 0.5 each, the inputs of HE-EX-HPF-001 read as 10 units of labour and 40 of capital and A of 1, output is 20, the value HE-FM-HPF-001 gives, RTS is 1 and doubling both inputs doubles output, a limiting case (illustrative).
A = 1; L = 10; K = 40; alpha = 0.5; beta = 0.5; s = 2; Q = 20; RTS = 1; Q_s = 40; g_s = 2
Common errors
Output elasticities multiplied instead of added for hospital returns to scale
The degree of returns to scale is the sum of the output elasticities. Multiplying Sielska and Nojszewska's 0.2 and 0.48 gives 0.096 and implies that a 1% rise in nurses and electricity raises patient-days by about 0.096%, about a seventh of the 0.68% that the sum gives (computed here for illustration).
Diminishing marginal returns to staff read as decreasing returns to scale
Diminishing returns concern one input rising with the others fixed, returns to scale all inputs rising together. With elasticities of 0.5 each, 40 units of capital and A of 1, raising labour from 10 to 20 lowers its marginal product alpha Q / L (HE-FM-HPF-002) from 1 to 0.7071, yet doubling both inputs doubles output because RTS is 1 (computed here for illustration). A falling marginal product of nurses in a fixed building says nothing on its own about the returns to expanding the whole hospital.
Returns to scale of two included inputs read as the returns of the whole hospital
If an illustrative hospital's full function also had doctors with an elasticity of 0.32, its returns to scale would be 1, while the elasticities of nurses and electricity in that function, 0.2 and 0.48, sum to 0.68. A 1% rise in nurses and electricity alone, with doctors fixed, then raises output by 0.68%, and a 1% rise in all three by 1% (derived, computed here for illustration). A sum below 1 from a function that leaves out inputs does not by itself show decreasing returns to scale for the hospital.
Linear reading of returns to scale for a large change in hospital inputs
RTS is the percentage response to a small proportional change. For a large change the exact ratio is s to the power RTS. Doubling both inputs with RTS of 0.68 raises output by 60.2%, while multiplying the 100% rise by 0.68 gives 68% (computed here for illustration).
Sources
Sielska and Nojszewska on Cobb-Douglas elasticities and economies of scale in county hospitals
Sielska A, Nojszewska E. Production function for modeling hospital activities. The case of Polish county hospitals. PLoS One. 2022;17(5):e0268350. The Methods section defines economies of scale as the sum of the elasticities of production with respect to each factor, showing by how many percent production changes when the employment of all factors rises by 1%, in an unnumbered expression before equation (5), which gives this elasticity of scale as a limit. Equations (8) to (10) give the Cobb-Douglas function as alpha_0 times the product of each factor raised to its power alpha_i, its logarithmic form and the elasticity of production of each factor, equal to alpha_i and not dependent on the employment of factors, and the section states that the functions were estimated by least squares in R. The Discussion reports, for the two-factor Cobb-Douglas function with electricity and nurses, that a 1% rise in electricity costs raises patient-days by 0.48% and a 1% rise in nurse FTEs by 0.2%, and that economies of scale do not exceed 1, so a 1% rise in both inputs gives less than 1% more patient service capacity. The text after equation (11) describes Cobb-Douglas economies of scale as the sum of the elasticities but prints the expression with a product sign; the records follow the sum. The sum 0.68 and the ratio s to the power RTS are computed here and are not printed in the paper.
Observatory glossary definitions of constant, decreasing and increasing returns to scale
Cylus J, Papanicolas I, Smith PC, editors. Health system efficiency: how to make measurement matter for policy and management. Copenhagen: European Observatory on Health Systems and Policies; 2016. The Glossary, page xvi, defines decreasing returns to scale as a proportionate increase in all inputs leading to a lower proportionate increase in outputs, increasing returns to scale as one leading to a greater proportionate increase in outputs, and constant returns to scale as a given increase in inputs always giving the same increase in outputs, regardless of the scale of operations.
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