Production function studies in health economics have taken three divergent approaches. Some of these studies focus on the production function for general and regress health (such as reduced mortality) against a variety of factors. Another strand examines the technological relationship between medical care and the inputs that are used to produce medical care. A third approach examines medical care production efficiency more specifically through stochastic frontier or data envelopment analysis techniques. This article describes all three approaches with a primary focus on production functions and on efficiency analysis.
The production of health approach involves a more general specification of the production process, including a variety of societal factors as inputs, such as consumption of medical care, technology, demographics, and personal health habits. Findings in this literature include a positive relationship between medical care and health; demographics (such as education and income) and health; and avoiding risky behaviors (such as smoking and other substance abuse) and health. One recent example of this approach is production function estimation for health in the Organization for Economic Co-operation and Development (OECD) countries, which postulates that life expectancy at the age of 65 years depends on health expenditures, medical technology, and lifestyle. Health expenditures significantly affect life expectancy at the age of 65 years in OECD countries.
In contrast, two other major strands of production function estimation have examined the technology and efficiency associated with production of medical care, which is the primary focus of this article. One approach examines the technological relationship between inputs (such as employment of different types of health care workers, physical capital, and possibly other inputs) and output or outputs, which can include client counts (admissions or discharges), relative value units, or others. More recently, interest has focused on the relationship between inputs and the quality of output, but this is an area deserving much greater attention. The outputs have centered around a variety of different health care services, ranging from hospitals, to physicians and specialty care treatment centers.
Hospital Production Functions
The literature on hospital production functions is quite extensive. In the general hospitals literature on production functions, researchers examine individual hospitals (or other medical entities such as practices) which maximize utility rather than minimize costs, where utility may be defined as a function of effort and leisure. Effort is defined as the number of discharges or admissions. The hospitals maximize their utility, given a budget, labor market conditions, and a production function. The production function for each hospital describes the technological relationship between the capital and labor inputs, and the process by which capital and labor are translated into output.
A typical hospital production function can be estimated by least squares regression techniques, after adding an ‘error’ term.
It is noteworthy that ‘output’ can be represented by admissions, discharges, and/or relative value units; the stock of physical capital can be measured by beds, and/or value of equipment and structures; the supply of labor can include either full-time equivalents, number of total employment hours, or one of these measures for several separate labor categories such as physicians, nurses, etc. The labor variable may include one type of labor and focus on aggregate hours or full-time equivalents, whereas alternatively separate labor variables can be included for different types of hospital workers (i.e., physicians, nurses, clinicians, clerical workers, etc.). The latter can be advantageous in assessing the substitutability of different types of workers. Often a vector of client mix or client demographic variables that affect the position of the production function is included in the statistical estimation.
The derivative of output with respect to each input is denoted as the marginal product of the input; it describes how output changes when there is a small change in the amount of one input, while holding constant all other factors of production. Specifically, in the hospital context, the marginal product of physicians is the additional patients who can be treated when there is a slight increase in the number of physicians. This marginal product is required to be positive, and the value of output should be zero in the presence of zero inputs (i.e., if there is only one labor input, hospital full-time equivalents, then zero hospital full-time equivalents implies zero patients treated). If there are several labor inputs (i.e., physicians, nurses, clinicians, clerical workers, etc.), it is permissible that some (but not all) of these labor inputs equal zero. Also, the production function should increase at a decreasing rate – in other words, the marginal product decreases as more physicians (or nurses) are added.
Functional Forms
Before estimating the medical care production function with regression analysis, a functional form must be specified. There are several common functional form assumptions that have been used in the literature, including Cobb–Douglas, translog, and generalized Leontief. A Cobb–Douglas production function is perhaps the most straightforward because of its linear structure in logarithms. A convenient feature of the Cobb–Douglas is that the regression parameter estimates are also elasticities. In assessing the marginal product of labor, for instance, this would be the elasticity of output with respect to labor, times the output level divided by the employment level.
Although the Cobb–Douglas has some advantages due to computational simplicity, and it diminishes the potential for multicollinearity because of a lack of interaction terms, a disadvantage is that it does not allow the elasticities of substitution among different types of hospital workers (or among a particular type of hospital employee and number of beds, for instance) to be different from unity. In other words, a hospital production function can generate information on how the facilities are able to substitute capital for labor by examining the elasticity of substitution, but the Cobb–Douglas production function assumes this elasticity is constant and equal to one at all levels of input use. For most hospitals, this assumption is quite restrictive and unrealistic. So, some researchers have considered an alternative that is more flexible, known as the translog. The translog production function is a generalization of the Cobb–Douglas – in other words, it builds on the Cobb–Douglas by adding interaction terms (in logarithms) for all of the possible combinations of inputs.
One advantage of the translog, compared with the Cobb– Douglas production function, is that the translog allows for the possibility of elasticities of substitution between physicians and nurses to be different from unity, and these elasticities can vary across hospitals. This is a desirable feature of the translog because it provides valuable information that can be useful in policy recommendations. But a potential problem with the translog arises when there is a zero in one or more of the inputs for some hospitals. For instance, if the labor inputs in the model include physicians, nurses, clinicians, and clerical workers, and if some hospitals have no clinicians, then this will be problematic because the log of zero is undefined. As an alternative, researchers have considered a generalized Leontief production function. Typically, the generalized Leontief hospital production function models admissions and/or discharges as a function of total labor and capital inputs and other shift variables. Alternatively, several types of labor can be included along with one type of capital and several shift factors. The generalized Leontief allows for interaction of the square roots of each variable (i.e., every type of labor, capital, and other shift variables).
Interpretation Of Production Function Estimates
Average And Marginal Products
There are a couple of possible scenarios that may be evident with the production function estimates. First, hiring additional workers may provide resources for clerical workers to perform more administrative tasks while allowing physicians and clinicians to specialize in treating patients, leading to higher average numbers of patients treated per employee. Alternatively, there may be a sufficiently large number of employees at clinics so that having additional workers might lead to office overcrowding, physicians and clerical workers getting in each others’ way, and possibly more difficulty in getting reimbursements for treatment because of additional bureaucratic layers within these organizations. If the regression estimates of the production function support this second scenario, having fewer employees at clinics would be expected to result in higher average number of patients treated per employee. The marginal product of labor is the additional clients that can be treated when an additional worker is hired, while the average number of patients treated by workers is the average product of labor. If the marginal product of labor is greater (less) than the average product, then the average product of labor rises (falls) as more workers are hired.
Elasticities Of Complementarity And Substitutability
Clinicians or physicians may work better when they have access to additional equipment, such as computers that may help with record keeping and billing. If so, the additional physical capital may allow the physicians to focus on the tasks they are trained to perform (that is, treating patients), while physical capital can be used for other tasks. This scenario is called q-substitutability; else, would physicians be able to treat greater numbers of patients if the hospitals were to hire additional support workers (such as nurses, clerical staff, or other employees)? This scenario is called q-complementarity. In other words, through the production function regressions one can address the question of whether workers and physical capital or two individual types of labor, are q-complements or q-substitutes.
The technological relationships between any two factor inputs can be assessed by examining the production function for q-complementarity. Here, capital and labor will be q-complements (q-substitutes) if an increase in capital increases (decreases) the marginal product of labor. More generally, the Hicks elasticity is defined as the product of output and the change in marginal product of labor resulting from a change in capital, all divided by the product of the marginal product of capital and the marginal product of labor. The Hicks elasticity measures the relative ease by which one factor can be substituted for another, while keeping admissions or discharges constant, so if the Hicks elasticity is positive (negative), this implies capital and labor are q-complements (q-substitutes). Nonphysician labor has been found to significantly impact physician productivity in the context of hospitals and a translog production function. Other translog production function analysis explains the relationships between physicians and other employees in health care settings. A common finding is that physicians and nonphysician employees are q-complements. In the context of hospital efficiency with a generalized Leontief production function, capital and physicians have been found to be q-complements; capital and technicians/aides have been found to be q-complements as well.
Stochastic Frontier Estimation And Data Envelopment Analysis
The third type of health production function studies focus on efficiency analysis. These studies, which also estimate production functions for medical care, are based on stochastic frontier models, and data envelopment analysis.
Stochastic frontier models (sometimes referred to as ‘frontier models’) are based on the assumption that the regression error term distribution does not follow a normal distribution. Because the production function models described in the section Hospital Production Functions are actually ideal production function models when there is no inefficiency, whenever the error term is nonzero there is inefficiency. In other words, technological inefficiency for any given hospital occurs when the error term is nonzero. Also, frontier models are assumed to have a typical production function functional form (such as Cobb–Douglas, translog, or generalized Leontief), as well as two error terms. One of these error terms is because of ‘technical efficiency’ and is assumed to be negative (because of the inability of a hospital to reach the frontier), and another error component is because of unobservables and other measurement difficulties. For any individual hospital, the frontier model can be written as a term that includes the production function as well as the inefficiency error term, and a separate error component from unobservables. Each hospital’s deviation from the mean efficiency can be calculated, to assess how inefficiently individual hospitals are operating. The model can be estimated by assuming the errors follow half of a standard normal distribution, and then using maximum likelihood estimation techniques.
Multiproduct Adjustments
When there are multiple outputs, such as different services provided in each hospital, a distance function stochastic frontier approach is appropriate. A stochastic frontier distance function to assess efficiency in Australian hospitals with several outputs has found a range of efficiency scores between 0.7 and 0.75, whereas there were a handful of outliers with efficiency scores that were close to 1. A distance function approach is crucial in the context of hospitals because most hospitals produce many different outputs, including inpatient and outpatient services. But a more disaggregated approach to address the efficiency of an array of many different hospital services necessitates a distance function approach. This can be done by specifying a ‘netput’ (or net input) transformation function, where a function of vectors of inputs and outputs are set equal to zero.
An alternative that can address the multiple outputs issue is a hospital cost function approach. Duality implies that profit maximization, through input choice for the production function, yields the ‘same’ result as cost minimization, where inputs are chosen to minimize costs. The optimal cost function depends on input prices and outputs, which can include several outputs. In this respect, a cost function approach to efficiency can be estimated using least squares regression techniques, to test for the presence of economies of scale; or, a cost function approach to represent technology can also be implemented as part of a stochastic frontier estimation.
Data envelopment analysis is a nonparametric approach to measure hospital efficiency, after considering several outputs and inputs. It uses the approach of linear programming, to estimate a model. Here, an ‘expansion factor’ for each hospital must be chosen so that the ‘expanded’ output of each type at any hospital must be no greater than the weighted average of all other firms’ output of that same type. At the same time, this hospital must use each input in such a manner that it is no less than the weighted average of all other firms’ input usage of that same input type. In this approach, which essentially estimates a production possibilities frontier, hospitals’ inefficiencies lead to deviations from the frontier. Some of the advantages of this approach are that no functional form needs to be imposed, and also it is possible that in some situations firms can produce outside the production possibilities frontier. In other words, if efficiency is defined as the inverse of the optimal value of this linear programming model, then the hospital is efficient relative to the other hospitals in the sample if the inverse of the optimal value equals 1. If the inverse of the optimal value is less than 1 for a given hospital, then that hospital is inefficient relative to the other hospitals in the sample. This measure of efficiency can be obtained by solving this optimization problem for each hospital in the sample.
Specific Applications: Specialty Care
Although there are few known applications of production function estimation for substance abuse treatment, there are more mental health applications. One application evaluates how changes to mental health workforce levels, composition, and degree of labor substitutability, affect practice output (measured as relative value unit’s) at US Department of Veterans Affairs mental health practices. This estimates the qcomplementarity/q-substitutability of mental health workers, using a generalized Leontief production function, examining many labor types, including residents, and then estimates the marginal product for each labor type as well as the substitutability and complementarity of physicians and other mental health workers. Among 28 unique labor–capital pairs, 17 are q-complements and 11 are q-substitutes. Complementarity among several labor types provides evidence of a team approach to mental health service provision at these providers.
Another application studies the efficiency of nursing homes in the state of Connecticut, USA, using a data envelopment analysis approach and finds that among the 140 nursing homes in the state, nearly 100 are more efficient than the mean. The mean efficiency score for Connecticut nursing homes is approximately 0.90, whereas similar studies of nursing homes in other locations have found a range of efficiency scores approximately from 0.57 to 0.93.
A general practice dentistry application estimates a translog production function for a sample of approximately 29 000 dentists in the US, and finds that dentists tend to use dental assistance in practices where they are ‘profitable’ in terms of their marginal products. Also, dentists in the age range of the mid-40s tend to be the most productive among all age ranges in the sample.
Conclusion
Since the 1970s, production functions in efficiency and productivity studies have been pervasive in a wide variety of applications. Future research should focus on how to assess quality of care in addition to quantity. This would be helpful to practitioners who might rely on marginal product estimates in compensation decisions and to governments in pay-for-performance calculations.
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