vitaliano health econ

Journal of Health Economics 6 (1987) 305-318. North-Holland

ON THE ESTIMATION OF HOSPITAL COST FUNCTIONS

Donald F. VITALIANO* Rensselaer Polytechnic Institute, Troy, NY 12181, USA

Received March 1987, final version received May 1987

Data from 166 general hospitals in New York State (1981) is used to estimate a quadratic and logarithmic long-run cost function. Both equations lit the data very well but give very different results. The quadratic appears to confirm the commonly-held view of a shallow U-shaped ave. age cost curve, whereas the log function indicates significant economies of scale: a total and avr;rage cost elasticity of 0.9 and -0.10, respectively (using beds or patient days to measure output). Ramseys RESET test is used to discriminate between the two models and the quadratic is clearly rejected as a misspecilication. Scale economies thus exist even where the usual quadratic suggests otherwise.

1. Introduction

Over the past 20 years a number of empirical studies of hospital cost functions and scale economies have appeared. A representative list might include the papers by Carr and Feldstein (1967), Culver et al. (1978) (U.K.), Evans (1971) (Canada), Feldstein (1968), Lave and iave (1970) and Sloan and Steinwald (1980). Among the most recent studies is that by Grannemann et al. (1986), which employs a multiple-output type of cost function. While it is hazardous (and perhaps foolhardy) to suggest the existence of anythinp approaching a consensus among economists, the most recent edition of a leading textbook on health economics summarizes its survey of the literature as follows: The shape of the average cost curve is shallow [U-shaped], that is it does not fall sharply nor is its minimum point much below that of hospitals on the ends of the curve [Feldstein (1983, p. 212)].

This paper first replicates the U-shaped cost function and then goes on to show that the same data set is capable of generating significantly decreasing unit costs with respect to size. The role of functional form misspecification and estimation technique in causing this apparent contradiction is formally explored. Statistical testing permits the decisive rejection of the U-shape in

*This paper was written while the author was visiting the Institute for Research in the Social Sciences and the Department of Economics and Related Studies at York University, Engl-nd. The valuable comments and suggestions of Professor J.P. Hutton are gratefully acknowledged. Professor A.J. Culyer and Dr. Keith Hartley have also made useful comments. All remaining errors are, of course, the sole responsibility of the author.

0167-6296/87/%3.50 @ 1987, Elsevier Science ?ublishers B.V. (North-Holland

306 D.F. Vitaliano, Estimation of hospital cost functions

favour of significant economies of scale. Even those who reject the U-shaped consensus will welcome the strong evidence here about scale economies which, as Berki (1972, p. 115) remarked, ought to exist.

Partly owing to the consensus just noted, but also because of a policy- driven concern with the cost of individual hospital services, it has become fashionable to estimate multiple-output cost functions. This new approach is useful in its own way and welcome, but it would be ill-advised to completely abandon the estimation of single output functions [see Hombrook and Monheit (1985)]. For example, the efficient planning choice between a single 500 bed hospital and two 250 bed units with a similar complement of facilities requires knowledge of conventionally measured scale economies in order to balance changes in treatment costs against changes in the travel costs of patients, visitors and staff as size varies. Moreover, the multiple output models typically suffer from difficulty in interpreting their results and from multicollinearity among the output measures which causes unreliable parameter estimates, thus weakening much of their policy usefulness.

In what follows, I present a brief view of selected past cost studies. There follows a description of the model and data base employed here. The next section presents the econometric results. A concluding part assesses the findings and attempts an interpretation of their significance.

2. Prior studies

Typically, past studies have relied upon single equation, cross-section multiple regression to estimate hospital cost functions [Carr and Feldstein (1967)]. A quadratic total cost function is fitted, with much attention devoted to attempting to control for the quantity and quality of output (e.g., case- mix or facilities available). A positive but small coefficient on the squared term in the quadratrc leading to the inference of a saucer shaped average cost curve [Carr and Feldstein (1967), Sloan and Steinwald (1980), Bays (1980)]. Francisco (1970) fitted an average cost function and reached a similar conclusion. Estimation of a total rather than an average cost function is preferred on econometric grounds because the latter introduces the size or output variable on both sides of the equation. This may cause bias in parameter estimates of as much as 10% and of unknown direction [Hough (1985)]. .

Evans (1971) found evidence of a U-shaped average cost curve for hospitals in Ontario.

The recent paper by Grannemann et al. (1986) typifies the multiple-output approach in which total cost is regressed on a vector of output measures in order to estimate separate marginal costs. Using single equation least- squares, they find important scale economies for emergency room activity but not for other outpatient activity. Because of multicollinearity among the

D.F. Vitaliano, Estimation of h al cost functions 307

outputs, 30 of 64 reported coefXcients wer wide confidence intervals around the es hospital outputs.

t statistically significant, with es of marginal cost for all

One problem with the standard appro that firm data often exhibit non-constant error variance (heterosceda y) which can cause statistical tests and goodness of fit measures to incorrect, leading to incorrect inferences about coefficients and functio form [Pindyck and Rubinfeld (1981, pp. 14&152)]. Further problems se with the results that have appeared in the literature when it is realiz that the use of a single equation model implies either a reduced form ation or only one endogenous variable. Price, cost and output are ty lly determined jointly. In such instances, the interpretation of the c nts of a single equation is particularly hazardous as they may re a complex of unspecified or unidentified coefficients from a larger mode ost authors have been content to merely point out the problem of en y before proceeding to single equation estimation.

3. A data-consistent model of hospital cost

The data set employed in this study is especially well-suited to avoid the cndogeneity problem of single equation estimation. New York State (U.S.A.) has, since 1969, engaged in a comprehensive and compulsory system of prospective (i.e., ex-ante) hospital rate regulation of all its non-federal hospitals. In other words, the prices are set in advance for each hospital by the State Health Department, much as electricity and telephone rates are set by regulatory boards [Health Care Financing Administration (1981)]. Prospective reimbursement rates are based on complex formulae designed to measure efficient levels of cost. Hospitals are grouped by bedsize, service-mix and teaching status. Costs are classified as routine or ancillary, with the former on a per diem basis and the latter per admission. A group mean is calculated after excluding hospitals above 125% and below 750/, of the raw mean. The baseline costs thus derived are adjusted to allow for expected inflation to form the prospective reimbursement rate, which is also subject to adjustment and penalties for deviation from target occupancy and utilization levels. This implies that prices are exogenous to the hospital, much as to a perfectly competitive firm. In addition, hospitals are similar to electric utilities in that output is non-storable and supplied on demand to customer-patients. Together these two factors imply that total revenue is exogenous and thus use of a single equation, cost-minimizing cost function is appropriate [Wallis (1979)]. Cost minimization is much more general than profit maximization and consistent with a variety of non-profit maximizing models [Cowing et al. (1983, p. 263)]. Although almost all the hospitals included in this study are not-for-profit, the regulators allow them to keep


any profit and impose financial penalties for cost overruns. Thus it is not unreasonable to assume least-cost behaviour on the part of the hospitals.

The rationale advanced here for the use of a single equation model is similar to that employed by Nerlove (6963) in his classic study of returns to scale in the electric power industry. Following Nerlove, let us define the following generalized Cobb-Douglas production function:

Q=A Xdfl X$2X$3, (1)

where Q is output, the XS input, A a constant and the ors the output elasticities. The least cost equation derived from (1) under the assumption of cost minimization is [Nerlove (1963, p. 54)]

1 lnC=lnk+-lnQ+%nPI+%nP,+olflnP,,

V V V V 12)

where C is total cost, P the unit prices of the inputs and k- v(Aailti

D.F. Vitaliano, Estimation of hospital cost functions 309

cQ2 - a-dX zO.

e=a+bQ+cQ2+dX

Increasing, constant and decreasing costs obtain according as e$O. We shall employ e later in the paper when comparing (3) and (4) since the elasticity of total cost e = e + 1 = l/v.

4. Data base and choice of variables

American Hospital Association (1981) and data provided me by the New York State Health Department are the basic data sources employed. On& hundred and sixty-six general medical-surgical (or acute care) hospitals located in upstate, New York State (U.S.A.) are analyzed. These hospitals represent virtually the entire universe of acute care hospitals in the upstate region. The 166 units consist of 139 private, not-for-profit hospitals, 23 publicly-owned and operated hospitals and only four for-profit units. Not included in the data set are specialty hospitals such as mental institutions and military base and veterans hospitals. Included are 32 medical school affiliated hospitals.

The New York State rate-setting scheme under which all these hospitals operate imposes on them a uniform system of cost accounting. This helps to minimize a source of error common to cross-section studies arising from non-uniform treatment of costs (e.g., depreciation) across observations. This is a particular weakness of prior hospital cost studies that used a sample drawn from various states.

The choice of an appropriate index of hospital size is the next issue to be addressed. In this paper I use beds as the measure of size, and thus the proxy for size-related output. Carr and Feldstein employed average daily census and Francisco used patient days to measure hospital size. Culyer et al., Bays and Sloan and Steinwald used beds as the size variable. Since the general conclusion of all these authors about costs is similar, the choice of size index does not appear crucial. Nevertheless, Carr and Feldbieins (1967, p. 231) argument against use of bed size is worthy of note: available beds might be defined differently across hospitals and smaller hospitals are required to keep a larger fraction of beds vacant in order to assure a given probability of bed availability. The latter, they allege, distorts the true cost picture. In our view the latter phenomenon represents a genuine scale diseconomy, analogous to inventory costs as between small and large shops. Because all New York hospitals are required to use uniform procedures to calculate costs and other

*Upstate in the present corztext refers to the area of the state outside New York City and Long Island. The latter locations are excluded because the rate-setting procedure varies between upstate and down, and because this paper is part of a larger project on hospital location in the upstate region.


hospital statistics, the probability of a varying definition of bed capacity across hospitals is low. Another reason to choose beds as a size measure is that admissions, census and patient-day measures all reflect short-run output volume variations and short-run costs that may lie off the long-run locus of interest to us. Hospital beds better reflect installed capacity, and planning/ location decisions are usually made in terms of bed capacity. However, as indicated below (see footnote 4), the results of this paper are not sensitive to the choice between beds and patient days.

The data base includes hospitals ranging from 23 beds to 1,070 beds, with a mean and standard deviation of 214 and 175, respectively. Thus, a very wide variation in hospital size is included.

Besides measurement of size, it is necessary to allow for non-size related influences on cost. Among these, three in particular warrant attention: factor prices, facilities and me t. cal school affiliation.

The most likely source of factor price variation in the present situation is between hospitals located in rural areas versus locations in urban areas. An index of the size of the metropolitan area in which the hospital is located is utilized to capture this effect. It varies from 0 (rural area) to 6 (population in excess of 2.5 million). The mean value of this index is 2.6, between 100,000 and 500,000 population.

With regard to service-mix, an unweighted index of 11 available hospital facilities is employed. (The included facilities are shown in an appendix.) Francisco (1970) has shown that such an index is a good proxy for the service-output mix. 2 The mean value of this index is 3.4. The choice of a facilities or services index in preference to a case-mix variable reflects the important distinction between the two made by Tatchell (1983). The former reflects the supply-side, the latter demand influences. The service-mix approach may better reflect reality in countries such as the U.S. . . . The extent, degree of sophistication and quality of facilities and services offered plays an important role in attracting doctors, and hence their patients, to particular hospitals (p. 880).

A dummy variable to represent the presence of a medical school affiliation is used. The teaching-related costs of a medical school are likely to exert an upshift, ceteris paribus, on hospital costs [Culyer et al. (1978)]. Normally, the expected sign of the medical school variable would be positive.

Finally, a variable designed to capture any market-power related effects on hospital costs is included. A monopsony or oligr >psony hospital may be able to exert downward pressure on factor prices. (I%:sumably regulation of price bits monopoly exploitation.) Market power iG measured by the share of

2Francisco (1970) showed that an unweighted facilities _ .-dex performed equally well in a multiple regression e service item.

uation 8s a more complex &es of dummy variable,s for each individual


beds in its county represented by each hospital. The mean value of bed share is 0.28.

The dependent variable is total hospital costs, including payroll an payroll costs (inclusive of interest and historic depreciation). All the variables described thus far are as defined in the American Hospital Association

, survey. We may summarize the hypothesis thus far advanced,

C = F(B, F, M, u, s), (61 where

C = total hospital costs, B =number of beds available, F = index of hospital facilities, M = medical school affiliation (a dummy), U =index of urban area size, S =market share, the hospitals share of total beds in the county.

In the next section we use multiple regression to estimate (6) using alternate functional forms (3) and (4) and estimating techniques, with attention particularly focused on the effect of bed size on C. The role of occupancy rates and length of stay is also discussed in section 5.

5. Econometric results

We begin by first estimating the total cost function using the quadratic functional form typically found in the literature. Eq. (7) presents that result (t ratios in parenthesis),

Total cost = - 3,073,130* + 57,477 Beds* + 18.27 Beds2 * * (-2.46) (5.99) (1.84)

+ 6,733,225 Med. schl. + 1,202,13 1 Facil.* (3.98) (4.78)

+ 682,736 Urban* - 20,07 1 Share, (2.73) (-1.13)

R2 =0.89, F(6, 159) =214.4,

* = signitrl dnt at 95% level, ** = significant at 90% level.

Many researchers would find (7) quite an acceptable result: most of the coeficients have the expected signs and are significant at the 9 the R2 is very high for cross-section data. The coefficients on th


is not significant at the 95% level but the null hypothesis of a zero coefficient can be rejected at the 900/ level. Overall, eq. (7) broadly conforms with the results widely reported in the literature. Only the market share variable Share, which was somewhat speculative to start with, is clearly insignificant. Dropping this variable from the regression has no effect on the remaining coefficients.

The Goldfeld-Quandt test of (7) indicates the existence of heteroscedasticity associated with hospital bed size: a test statistic in excess of 70 versus a critical value of about 1.5 (95% level). The Park-Glejser test allows one to estimate the appropriate weight to use in a weighted least squares regression, the most common technique for allowing for heteroscedasticity [Pindyck and Rubinfeld (1981, pp. M-152)]. The test involves regressing the residuals from (7) on bed size and using the resultant coefficient as the appropriate weight.3 In the present instance that weight is 1.26; thus eq. (7) is rerun using weighted least square and shown below (t ratios in parentheses),

Total cost = - 1,317&U* +64,303 Beds* + 24.57 Beds** (-2.49) (10.22) (2.30)

+ 5,194,528 Med. schZ.* + 206,373 Urban** + 470,134 Facil? (3.85) (1.73) (3.03)

- 9,686 Share, (0.99)

R=0.92, F(6, 159) = 371.0, .

* = significant at 95% level, ** = significant at 90/, level. (8)

In eq. (8) the coefficient on Beds* is now significant and a shallow b-shaped average cost curve seems clearly confirmed. The quotes around the R* indicates that it is a pseudo correlation coeflficient, consisting of the simple correlation between C and fitted C using (8). This measure is used because there is no entirely satisfactory measure of R* when weighted least squares is employed [Pindyck and Rubinfeld (1981, p. 146)]. The high R* in (8) and (9) combined with only one insignificant coeficient suggests the absence of any serious multicollinearity among the explanatory variables.

It should be noted that the negative constant term is not a problem because inclusion of typical values of Facil., which represents a further dimen- sion of output, will render the intercept positive [Klein (1962, p. 120)].

3The result is In e2 = 22.7 + 1.26 In Beds, R2 =0.14,

(18.0) (5.15) where e2 is the squared residuals from (7).


Utilizing (8) and the elasticity of average cost (S), along with the mean value of Facil. (= 3.4), the average cost falls up until bed size of 107 is reached and rises thereafter. At the mean bed size of 214, the average cost elasticity e = 0.028 and total cost elasticity e = 1.028.

Turning next to alternate functional forms of the estimating equation, the log form equation (3) is used to re-estimate the total cost function (t ratios in parentheses),

In Total cost = 11.2485 + 0.9035 In Beds* (79.6) (24.59)

+ 0.2734 Med. schl.* + 0.02 1 Urban* + 0.059 1 Facil.* (2.42) - - (1.97) - (4.49)

+ 0.001 Share (1.16)

R2=0.92, F(5, 160)= 15,871,

* = significant at 95% level. (9)

Eq. (9) was fitted using weighted least squares and the same weight employed in (8). An unweighted version gave quite similar results but with somewhat lower t ratios on the size and intercept terms. It clearly indicates significant economies of scale and decreasing costs, with an elasticity of total cost of 0.90. The null hypothesis that l/v=e= 1 is rejected at the 95% and 99% confidence levels (one-tail test). The coefficients on medical school affiliation, urban size and facilities are all positive, as a priori reasoning would suggest. In both the log and quadratic form of the cost function the market share variable S is not significantly different from zero. The results suggest that perhaps researchers have not gone far enough in exploring alternative specifications when estimating hospital cost functions. A given data set that apparently suggests a conventional U-shaped average cost curve is shown instead to imply significantly decreasing costs with respect to hospital bed size.

Differences in capacity utilization and/or average length of patient stay in hospital might introduce error into the analysis because, for example, larger hospitals might display lower average cost per bed because patients stay for longer periods and the later part of the hospitalization episode consists mostly of hotel costs. Systematic variations in occupancy rates, with larger hospitals displaying higher rates, may lead to greater ease in spreading fixed costs as hospitals grow in size. While a genuine source of scale economies, one would be interested in clearly identifying and measuring it. In the event, inclusion of average length of stay or occupancy rate into eqs. (8) and (9) resulted in insignificant coefficients on those variables with virtually no

314 D.F. Vita!iano, Estimation of hospital cost functions

impact on the other coefficients. The reason for the failure of length of stay (LOS) or occupancy (OCC) to exert any size-related effect on hospital costs is undoubtedly because the New York State regulatory system imposes financial penalties on hospitals that fail to adhere to predetermined LOS and OCC guidelines [Health Care Financing Administration (19Sl)].

Although only cost and beds are in log form in eq. (9), estimation with all right-hand variables (except the medical school dummy) in logs gives virtually the same results in terms of cost elasticity. Some economists seem to prefer patient days to beds as an output measure in cost function estimation in spite of the reasons for choosing beds discussed above. Consequently, eq. (9) was run using adjusted patient days (American Hospital Association definition) instead of beds. Happily, the results are virtually identical? In the following section an attempt is made to discriminate between the quadratic and log-form cost functions.

One of the most powerful tests for detecting functional form misspecification is RESET (regression specification error test) developed by Ramsey (1969j; its properties and power were explored by Thursby and Schmidt (1977).

Consider the standard linear regression model y=/IiXi + u, where y is the dependent variable, X the explanatory variable (s), u the disturbance term and /? the coefficients to be estimated. The null hypothesis of no misspecification involves E(u 1 X) = 0, whereas the alternative hypothesis E(u 1 X) = A # 0 involves one or more of the following misspecification errors: omitted variables, incorrect functional form, and correlation between X and u mursby and Schmidt (1977, p. 635)]. Assuming ilk 42, the RESET test involves forming the augmented regression y= /IX + #Z+ u and testing whether 4=0 in the usual way. This corresponds to the null hypothesis E(uIX)=O.

Using Monte Carlo methods, Thursby and Schmidt explored various candidates for 2 and found, among other things, that powers of fitted values of the dependent variable (i.e., j2,jj3) performed very well: a lOOo/, detection rate (95% confidence level) when a known *true functional form was tested against an alternative model.

It is a straightforward matter to rerun eqs. (8) and (9) adding a c2 term to

?he alternative regression (with t ratios in parentheses) is In Total cost = 6.385* + 0.873 In Patient days* + 0.03 Facil.*

(17.16) (23.6) (2.83) +0.319 Med schf.* +0.046 Urban size*, (10) (4.50) (4.17)

R2 =0.91, N = 166, RESET test =0.53*, is significant at 95% level. The Park-Glejser test indicates no heteroscedasticity in (lo), and the null of equal cost elasticity in (10) and (9) is not rejected (95% level of confidence). The coincidence between the two estimates may suggest the existence of long-run equilibrium.


each, where e is the Wed value of total cost estimated from (8) and (9), respectively. The results are presented below (t ratios in parentheses),

Total cost = - 6,486,148 i- 154,683 Beds - 526 Beds2 (- 5.01) (8.01) (5.43)

- 586,854 Med. schl. + 325,249 Urban + 302,373 Facil. (-2.16) (1.37) (1.08)

- 9,006 Share + 0. (-0.55) (5.65)

R2 =0.91, I (8 1

In Total cost = 11.20 + 0.933 In Beds + 0.212 Med. schl. (3.17) (0.97) (0.60)

+ 0.061 Urban + 0.0638 Facil. + 0.001 Share (2.30) (0.93) (0.75)

-o.7419e2, ( -0.002)

R2 = 0.89. I (9)

The implications of the RESET test are clear: the coefficient of e2 in the quadratic form is significantly different (99% level of confidence) from zero and the null hypothesis of no misspecification is rejected. [The small size of 4 in (8) is due to the dimensions of e: millions squared.] In contrast the 4 coefficient in (9) is highly insignificant and the null of no misspecification cannot be rejected. Thus, it seems reasonable to conclude that general surgical-medical hospitals exhibit significantly decreasing costs pr patient bed as bed size increases.

The next thing to be discussed are potential errors or bias other than functional form that might vitiate the inference of decreasing costs.

Use of a facilities index is an approximation to some ideal measure designed to control for the composition of hospital output. Because larger hospitals will generally deal with more complex and costly cases, the efFect of failing to fully allow for case-mix is to ouerstate the costs of larger hospitals. Any bias thereby introduced thus works to strengthen the conclusion that decreasing costs obtain in hospitals. In addition to using the Facil. index, Facile2 was tried in order to capture any non-linear effects, as well as a bed size facilities interaction term. he FaciL2 usually performed somewhat better than Fad.


in the sense of having a smaller standard error, but did not affect the bed size coefficients in any meaningful way in either the quadratic or log form of the cost function. The facilities-bed size interaction was not sign&an different from zero in either functional form.

It seems safe to conclude that the results presented in eq. (9) are robust both as to functional form and errors in variables and that pure size-related decreasing costs exist in the 166 hospitals herein analyzed. A total cost elasticity of 0.9 and average cost elasticity of -0.10 is found, which in turn implies a returns to scale coefficient v = 1.10. Given the small standard error (0.0367) on the bed size variable in eq. (9), it appears that the case for rejecting the consensus hypothesis of a U-shaped hospital average cost curve is strong. Returns to scale of 1.1 are quite significant and lie at the upper end of the range reported in studies of other industries [Mansfield (1985, pp. 223-225), Walters (1961)].

6. Conclusion

Using cross-section weighted least squares to estimate a total cost function for 166 acute care hospitals in New York State in 1981, an elasticity of total cost with respect to bed size of 0.9 and average cost of -0.10 is found.

The reason for overturning the U-shaped long-run average cost curve in preference for significantly decreasing average costs lies with the choice of the cost function that is estimated. We have shown that our own data set is capable of replicating the standard result very nicely. Rut use of a log form cost function that is fully consistent with economic theory and the insti- tutional arrangements in the hospital industry produce results that are far more satisfactory from an econometric point of view and which are consistent with the great body of cost studies covering other sectors of the economy - including regulated and public utility industries similar to the New York hospitals examined here.

It is worthwhile pondering briefly the possible sources of the scale economies uncovered in this paper. The most commonly accepted reason for falling unit costs is indivisibilities of labour and capital. A hospital is composed of highly specialized personnel and equipment. To the extent these specialized components reach minimum costs at different activity levels, average costs for the hospital as a whole will decline as the scale of activity grows. In addition, the purely hotel aspects of a hospital (e.g., catering and cleaning) are probably subject to scale economies, if one is to judge by the ever-growing size of newly-built commercial hotels. Average costs are thought to eventually turn up when the preceding forces are overwhelmed by increased control and decision-making costs as the unit grows. But use of computers and modern management methods may have the effect of delaying the point of upturn very considerably.


q dixt List of items in facilities index

1. Electroencephalography (EEG) 2. CT. scanner (head unit) 3. C.T. scanner (body unit) 4. Diagnostic radioisotope facility 5. Open heart surgery 6. Ultrasound 7. Megavolt radiation therapy 8. Therapeutic radioisotope facility 9. X-ray radiation therapy

10. Ambulatory surgical services 11. Kidney transplant

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