Microeconomics contains a theoretically based framework that describes how an individual business enterprise chooses to optimize production and cost efficiency, given existing technologies and prices of inputs. Within this supply side structure, the production function models the relationship between outputs produced and inputs used in the process, and the cost function models the relationship between the production cost of different levels of output accounting for input prices. The two functions are related in the sense that the production function shows the various ways of combining inputs to produce outputs, given the state of technology, and the cost function shows how to do it at minimum cost. Given certain basic mathematical properties, a duality or one-to-one correspondence exists between a set of production possibilities and the respective minimum cost function. In modeling the provision of health care services, economists often prefer the cost function to the production function because input prices are plausibly assumed to be determined outside of the model of firm behavior, whereas the selection of inputs in the production process are not.
The cost function is a powerful tool in the econometric application of the theory of production. In health economics, the preponderance of cost function estimation studies have focused on the hospital, which lies at the nexus of health care services and is the foremost component of health care spending. A number of issues involved in cost function estimation in health care have been addressed in empirical studies of US hospital costs. The remainder of this article will highlight the key issues involved in cost function estimation largely in that context.
Approaches To Cost Function Estimation
Short-Run Versus Long-Run Cost Functions
In any cost function estimation, a fundamental determination facing the researcher is whether to adopt the short-run or the long-run perspective. The distinction lies in assumptions regarding the state of equilibrium, or whether the firm has set all its inputs at their cost-minimizing levels. A variable cost function assumes the short-run scenario in which a firm’s capital costs are fixed, whereas a total cost function takes the long-run perspective, in which all costs are variable and inputs have been chosen such that total costs are minimized. In the short-run variable cost function specification, the dependent variable measuring costs does not include capital costs; however, the fixed measure of capital is included as an explanatory variable. In the long-run total cost function, the dependent variable includes capital costs.
The appropriate choice of the short-run versus the long run approach draws on both theoretical and practical considerations. If the firms are believed to be employing all inputs at the cost minimizing levels, then the long-run total cost function is indicated by theory. However, if firms cannot adjust their capital stock quickly in response to changing output levels or input prices, a short-run variable cost function is the preferable specification. From a practical perspective, estimation of a long-run cost function requires a measure of capital costs, which are often difficult to observe. In addition, the long-run cost function should include measures of all input prices including those of capital, which in most applications can be achieved only as rough approximations.
In hospital studies, it is generally agreed that capital stocks are adjusted over time horizons exceeding the periods of study included in most datasets. Moreover, the industry has experienced considerable organizational, regulatory, and demand side changes over recent decades. These factors, together with the challenges of measuring capital costs and capital input prices, generally have led economists to estimate short-run variable cost functions for hospital studies. This specification does require reliable measures of fixed inputs. It also assumes that those inputs are exogenous, or that hospitals do not have the opportunity to significantly adjust their physical plant size.
Structural Versus Behavioral Cost Functions
In pure theoretical form, costs are modeled solely as a function of output levels and prices of inputs, controlling for fixed inputs or capital in the case of the short-run variable cost function. However, in empirical applications, cost functions generally incorporate other observable factors that have been found both conceptually and empirically to account for significant variation in the costs of producing specific products or services. This is particularly important in the health services literature where such cost estimations are alternatively referred to as behavioral cost functions or hybrid cost functions as opposed to structural or pure theoretic cost functions.
In the hospital literature, variables included in behavioral cost functions may not have a particular role in the microeconomic theory of the firm, but they incorporate real world differences in hospitals and reflect patterns of variation found in actual hospital cost data. Typically, hospital cost functions contain a primary measure of output such as number of admissions, one or more measures of input prices, and a measure of fixed capital such as the number of beds or the amount of total fixed assets. Admissions alone do not capture variation in hospital output. Other product descriptor variables commonly included are average length of inpatient stay, a case-mix index that is usually based on the relative costliness of the diagnosis-related groups assigned to admitted Medicare patients, and the number of hospital outpatient visits.
Other key variables that have been demonstrated to account for variation in hospital costs and are often included as controls in the cost function are measures of local market competition, ownership status (for-profit, not-for-profit, or government), and the presence of a teaching mission. Market competition is often measured using a Herfindahl–Hirschman index of market concentration. The index, calculated as the sum of the squared market shares of individual firms competing in the same market, is a function of the number of competitors and the distribution of their relative market shares. Its values fall in the range of 0–1 where lower measures signify many hospitals competing within the market and higher measures indicate fewer hospitals. Teaching hospitals are more costly because of the extra resources involved in performing an educational, in addition to a therapeutic, mission. These costs are sometimes captured by a binary variable such as membership in the Council of Teaching Hospitals or alternatively by a continuous variable measuring the number of medical residents affiliated with the hospital.
Challenges In Cost Function Estimation
Measuring Output: The Multiproduct Cost Function
Health care provision is highly complex, and measuring the output of a firm that supplies health care services is often complicated. For example, a typical general hospital treats patients with a large number of diverse conditions using thousands of different medical procedures. Resource utilization for surgical inpatients is greater than for medical inpatients, and inpatients are more resource intensive than outpatients. In physician practices, office visits for established patients have cost implications that are unlike those driven by visits with new patients, emergency room visits, or hospital visits. Nursing homes provide distinct levels of care for their residents, and skilled nursing patient days have different cost implications than intermediate care or other patient days.
Most health care cost estimations rely on the multiproduct cost function (also referred to as the multiple output cost function), which defines the cost of producing more than one type of output assuming that all inputs are used efficiently. Incorporating more than a single output into the cost function adds realism to the model. The multiple output specification also allows for a richer set of theoretical constructs useful in applications of cost function results. However, greater output complexity also introduces additional challenges in capturing unit costs of production. These issues are discussed in further detail in the section on Average Costs.
Controlling For Quality
Microeconomic theory assumes that the firm minimizes cost in choosing inputs to the production process to produce outputs at a given level of quality. Although measurement of firm cost is generally straightforward and measures of output are usually feasible, the quality of health care service provision is multidimensional and difficult to quantify. Yet, it has long been established that if quality of service is not controlled in a cost function, biases result.
Variation in quality levels also complicates the theoretical modeling of health care cost. In the case of hospitals, high nurse staffing ratios, the extra resources required by teaching hospitals, sophisticated information systems, and/or innovative high technology services are cost increasing features that have been found to be associated with higher observed hospital quality. Yet, low quality also can be cost increasing if it is related to lapses leading to preventable adverse events or postoperative complications that require additional services. These dynamics are interrelated. For instance, higher nurse staffing levels and/or sophisticated information systems not only have a direct and positive impact on costs but also reduce the probability of expensive adverse events, thereby simultaneously having an indirect effect that is cost reducing. Overall, the theoretical relationship between costs and quality is complex, consisting of the joint effects of many different factors operating simultaneously.
Quality of health care also has presented repeated problems of measurement and data availability. Consequently, many cost function studies have not included explicit quality measures, confounding the impact of cost containment policies. In the absence of observed measures, some hospital cost functions have incorporated unobserved quality by building on the economic theory or by exploiting the structure of the error term in regression models. Studies that have included observed quality controls have relied heavily on structural measures of hospital quality such as teaching activity. There is widespread agreement that quality of care tends to be higher in teaching hospitals, which have access to the newest technologies. Yet, patient satisfaction and continuity of care are often worse in teaching hospitals and reports of resident exhaustion not uncommon. Teaching per se also represents the specific hospital output of medical education so that teaching is at best a proxy variable for hospital quality. Other structural measures include the presence of high-technology services, board certification of staff, hospital accreditation, and registered nurses as a percentage of full-time nursing staff. Finally, process measures such as outpatient follow-up to inpatient care, or outcome measures such as readmission, mortality, or adverse event rates have been used as quality controls.
The Profit Maximization Assumption In Health Care
The empirically estimated cost function derives from a theoretical framework, which assumes that the firm’s fundamental goal is profit maximization. However, it generally is agreed that producers of health care services are often motivated by other objectives. For-profit enterprises constitute a minority of general hospitals in developed countries, and a large percentage of nursing homes are nonprofit organizations. Although a number of theoretical models have been developed in order to explain the objectives of nonprofit firms in the health sector, the empirical cost function literature on hospitals does not find that ownership drives cost differences. Growing competition in the hospital industry may force nonprofit hospitals to behave much like for-profit hospitals to remain viable.
Useful Constructs
The magnitudes of coefficients on independent variables generated by the cost function are not in themselves meaningful. However, a number of constructs fundamental to the theory of the firm can be determined using the cost function estimates. Key measures include marginal cost, average cost, economies of scale, and economies of scope. These represent a highly constructive set of tools that frequently are used in cost function applications to research and policy.
Marginal Costs
Marginal cost is the increment in cost that occurs when the output produced is increased by one unit. More formally, it is the derivative of the total cost function with respect to output. Marginal costs are important because economic decisions are made at the margin. For example, the economic decision of a physician practice to expand or reduce a particular service in response to a change in fixed payment rates will depend on the marginal cost of producing that service.
Average Costs
Average cost is defined as the total cost of production divided by the number of output units. Although a conceptually simple construct, calculation of average costs is complicated in health care cost functions. Because of the multiproduct nature of production, it is difficult to describe output in a single utilization measure. The American Hospital Association Annual Survey Database contains measures of ‘adjusted’ discharges and patient days where these outputs are inflated by the ratio of total (inpatient plus outpatient) revenues to inpatient revenues. These measures are widely accepted and used in hospital cost function estimations; however, it is recognized that they are biased to the extent that hospitals cross subsidize across inpatient and outpatient services. Although the ratio of costs rather than revenues would be a more accurate economic adjustment, separation of costs in this way is not generally available in hospital accounting systems.
Economies Of Scale
Economies of scale refer to the notion that average cost falls as the firm expands. Conversely, diseconomies of scale occur when expansion incurs increasing average costs. From a technical standpoint, a measure of economies of scale is equivalent to the ratio of marginal to average costs. This is because if cost at the margin is lower than average cost, then average cost will fall with increased output.
In the multiproduct context, there are two distinct economies of scale concepts. Product specific economies of scale characterize the cost effects of expanding each output separately while holding production levels of other outputs constant. The alternative adaptation is ray scale economies, which assumes a proportional increase in cost resulting from a simultaneous proportional increase in all outputs. Either construct may be appropriate; the choice depends on the context involved in the specific analysis.
Economies Of Scope
The nature of multiproduct cost functions also gives rise to the related concept of economies of scope. Typically, a health care enterprise will produce more than one product because sharing of resources generally means that it is cheaper to produce products together than to produce them separately. Economies of scope refer to the savings incurred as a result of joint production.
Functional Form Of The Cost Function
The cost function is not derived from a specific production technology; hence, no particular functional form is called for in estimation. Yet, because the functional form of the minimum cost function is unknown to the researcher, there is a risk of misspecification, in which the model may yield poor or even erroneous predictions. Some judgment is called for in selecting a functional form for the cost function, and the econometrician practices a degree of art as well as science in formulating the econometric model.
A variety of specifications are employed in practice. The most commonly used in the health industries is the translog, a ‘flexible functional form,’ which represents a local second-order Taylor approximation to any true differential function. The translog involves logarithmic transformation of the dependent and independent variables and includes squared terms as well as interactions among outputs as independent variables. An important drawback to the translog that estimates a large number of parameters is the problem of multicollinearity among its many terms so that some precision of the estimates is sacrificed for functional flexibility, a trade-off that may or may not be warranted depending on the size of the dataset being used and the objectives of the particular research question. The problem is exacerbated in multiproduct cost functions and increases with finer disaggregation of outputs.
An alternative to the translog that often has been adopted in hospital and nursing home studies is a model that is logarithmic in costs with cubic polynomials on output. Although less flexible than the translog, the cubic specification is consistent with the classic U-shaped average cost function. It is particularly useful when the focus of the research is on marginal effects. There are other functional forms that have been used to estimate hospital cost functions. Of particular mention are the generalized translog, which often is used for multiproduct cost functions in cases where an output takes a value of 0 for some firms, and the generalized Leontief, which is useful in studies where the determination of input substitutability is of particular interest.
Some Applications
Health economists have used the cost function approach to address an extensive array of research questions. A description of the full range is beyond the scope of this narrative. However, this section highlights several notable issues that have been explored using cost function estimates. The purpose is to provide insight into the usefulness of the cost function approach in addressing important health policy concerns.
An economic question that lies at the core of the theory of the firm is optimization of firm size and the related issue of scale economies. The importance of economies of scale as a determinant of industry structure underlies economic arguments that have been put forth as justification for various forms of hospital regulation. A wave of hospital mergers in the 1980s and 1990s, for example, led the US federal antitrust authorities to develop guidelines for hospital mergers that allowed for demonstration of economic efficiency stemming from economies of scale. Economists have used the cost function to estimate the optimal hospital size, measured in patient days, or alternatively in number of beds. More recent policy concern has been over rapid growth of small physician- owned specialty hospitals. The economic cost function approach has been used to address the question of whether these hospitals are large enough to capture scale and scope efficiencies.
The cost function approach also has been applied to changes occurring in the internal organization of hospitals over the past two decades. Steep declines in the length of hospital inpatient stays began in the 1980s in response to insurer and government payer pressures on hospitals to absorb greater financial risk in their treatment decisions. The cost function has been used to examine the marginal cost of patient days over the course of a hospital stay. If the marginal cost of a patient day is relatively small, because the patient is in the recuperation stage and resource utilization is relatively low, then shortening the stay may or may not be an effective cost containment strategy.
An interesting policy question relating to the production of physician services is whether physician payments reflect marginal costs. For example, the Resource-Based Relative Value System through which US physicians are paid under the Medicare system was designed to reimburse at cost; however, the formulae used by Medicare is based on accounting cost systems that may not accurately reflect true production costs. A multiple output physician cost function is a tool that can more accurately reveal how marginal costs of production vary across different physician services that may be reimbursed at the same rate under administered pricing or privately negotiated rates.
The multiproduct cost function is well suited to empirical analysis of the US nursing home industry, which serves residents under explicitly distinct payment mechanisms: Rates received for Medicaid patients covered under various state programs for the poor are known to be considerably lower than those charged to self-paying patients. The cost function is a useful tool for exploring a number of policy questions. Are Medicaid rates paid by states to nursing homes for providing care for their poor elderly populations equal to the cost of treatment? Conversely, do higher rates charged to self-paying patients cross-subsidize Medicaid patients?
Stochastic Frontier Cost Function Estimation: Measuring Inefficiency
As expenditures on health care in developed countries have mounted in recent years, the goal of improving efficiency in health care provision has become a central objective for policy makers. At the same time, the demand for improved capability in measuring provider performance has stimulated the development of frontier analysis, which generates empirically based inefficiency measures at the provider level. Frontier studies define inefficiency as the extent to which an organization’s performance exceeds the optimum (or frontier) as predicted by either production function or cost function estimates.
Within this empirical framework, the stochastic frontier cost function is the principal econometric technique for identifying the cost inefficiency of an individual provider. In contrast to a typical cost function that fits the average level that best fits the data, the stochastic frontier cost function traces out the least cost locus econometrically for varying output levels and in that sense is more consistent with the theoretical concept of cost minimization. Inefficiency is inherently unobservable and assumed to be absorbed in the residual term. Allowing for unobserved firm-specific random shocks, the technique identifies an inefficiency term according to the deviation of the firm’s actual cost to the least possible cost as determined by the cost function. Focus on the inefficiency term in stochastic frontier cost function analysis differs from traditional cost function analysis, in which interest is centered on estimated coefficients. In examining the performance of hospitals over the past decade, stochastic frontier analysis has been more prevalent in the literature than traditional cost function estimation.
A particular challenge for stochastic frontier cost function estimation is the ongoing difficulty in adequately controlling for quality. In hospital studies, for example, if quality is cost increasing overall, failure to account for it will result in confounding the inefficiency measures because it is not possible to differentiate between higher residual costs resulting from unobserved superior quality and higher costs resulting from managerial inefficiency or slack.
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