How Many Beds Should a Hospital Department Serve?
by Dianne E. Thomas and Houston H. Stokes Departmental cost functions are constructed for selected hospital departments, using total number of beds in the hospital served as a proxy output measure. Calculation of maxima or minima for the resulting cost functions reveals that, on average, diferent departments have extremes in their cost functions at different levels of output. A relative cost index is constructed,
using parameters of the departmental cost functions, and departmental costs are compared across regions. The significance of departmental differences in optimum output is discussed with regard to sharing of services and modified system design.
Most research into hospital cost functions has concentrated either on the hospital as a unit or on the efficiency with which hospitals coordinate various units . These approaches are based on the assumption that the hospital as presently constituted will continue to exist. As a consequence, such analyses are limited to determining the optimum size of the hospital as a whole. Some of the most important research in the area of optimum size has been done by P. Feldstein , who attempted to measure a hospital cost function by using a weighted sum of various hospital operating aggregates. His study made some attempt to measure department cost functions, but in general the functions he estimated were not suitable for the calculation of maxima or minima. Other research on the problem has been conducted by Carr and P. Feldstein  for the United States, by M. Feldstein  for the United Kingdom, and by Lave and Lave  for the United States. Important contributors include Newhouse , who addressed himself to the question of determining the relationship between a hospitars nonprofit status and its economic efficiency, and Francisco , who attempted to measure hospital cost functions of the general form of the equations used in the present study. Francisco's regressions placed average cost per patient day on the right side of the equation. Francisco's findings suggest economies of scale for small hospitals and constant costs for large hospitals. Cohen  conducted a similar study and found that the minimum point of the hospital cost function was for hospitals in the size range of 540-790 beds. Quite recently, Pulley Address communications and requests for reprints to Dianne E. Thomas, J. Lloyd Johnson Associates, 778 Frontage Road, Northfield, IL 60093.
and Fulmer  published results from a study based on the 1971 and 1972 Duke Endowment data for North and South Carolina. Although they claim to have "pinpointed" the optimal hospital size, their regression results indicate that the optimal hospital size rose from 279.35 beds in 1971 to 345.92 beds in 1972. Such an extreme variation does not appear reasonable. Only six of their 15 variables were significant (at the 95 percent level or better) for the 1971 data, whereas for 1972 only seven were significant. While studies such as these have their value, they are limited in considering only the usual mode of hospital organization, and this implicit assumption throws away much that is of interest in the present-day context. Increasing emphasis is being placed on the components of health care rather than on the total institution. This trend suggests that it is appropriate to look at a hospital as a collection of modules, each a different department or service with a separate cost function. There is nothing in theory to imply that the cost function of each module has the same shape and location on the price/volume graph-that is, that each has a maximum or minimum point at the same scale of operation. A modular "micro" view of the hospital suggests that it may be more cost effective to have some departments serve more than one hospital or less than one hospital. The modular approach taken in this study has a number of advantages. It avoids some of the problems of aggregation of costs by investigating the costs of each module (department) separately. Moreover, a major problem in previous studies has been the lack of a suitable output measure for an entire hospital, as Mann and Yett point out in their review . Although this problem is not solved, the modular approach makes it less significant: one can construct useful cost functions for the separate departments without having to make assumptions about the ultimate hospital output. Our departmental output measure is the size (number of beds) of the hospital served by the department. Although the output of all acute care hospitals is not homogeneous, we have increased the sample homogeneity somewhat by excluding teaching hospitals. Homogeneity was further improved by using data sorted by size class and region. Nevertheless, use of hospital size as a proxy for a measure of output of the cost centers carries the implicit assumptions that the utilization of different size hospitals is uniform and that length of stay is generally uniform across hospital sizes and regions. It would be advantageous to be able to control for differences in hospital output over size and region, but in this preliminary study the available data should nevertheless provide some useful insights into the modular hospital concept.
Data The data for our study were obtained from the semiannual Hospital Administrative Services (HAS) reports of the American Hospital Association (AHA), covering 1972 and 1973 [11-14]. These reports represent approximately 43 percent of AHA members and about 55 percent of member beds.
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How MANY BEs?
Over the two-year period the number of hospitals represented by the data ranged from 1,854 to 2,134. In the classification by size and region, the smallest cell represented six hospitals. Hospitals not reporting every month for the period of the report were excluded from our sample; as mentioned previously, teaching hospitals were also excluded. Six-month medians of direct cost figures for each type of department in hospitals in nine regions and eight size classifications were taken from the four semiannual HAS reports for 1972 and 1973; these reports provided four discrete time series values for each department-region-size observation, for a total of 288 observations of direct cost for each of eight department types. The department types and their cost measures were: nursing, cost per patient day; laundry, cost per 100 lb; laboratory, cost per test; radiology, cost per procedure; plant, maintenance cost per 1,000 sq ft; housekeeping, cost per 1,000 sq ft; linen, cost per patient day; and dietary, cost per meal. The nine regions were New England (NE), Middle Atlantic (MA), South Atlantic (SA), East North Central (ENC), East South Central (ESC), West North Central (WNC), West South Central (WSC), Mountain (M), and Pacific (P). Values of size shown below and used in regressions were computed as midpoints of the category ranges and adjusted to obtain a more uniform progression of the size intervals. Size category Size value used (beds) (beds) 400
Methodology The general plan of the study was to construct a series of cost functions containing dummy variables to control for regional variations and a time term to control for cost variations over the two years of data. The equation yielding the minimum standard error of estimate for a given module (department) was taken as the best fit . Once the appropriate function was identified, it was differentiated to allow calculation of the minimum or maximum of that module's cost curve. Further, given the regional coefficients of the dummy variables and the value of the intercept for a given module, a theoretical price index (discussed in more detail below) was calculated. This index allowed interregional and interdepartmental comparisons of relative costs.
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Two equations fulfilled our criterion of minimum standard error for one another module; between them, these two equations provided acceptable cost functions for all the modules examined:
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RijD + yjSj + Y3jSj3 + u,
where, for both equations,
Cj = average cost for module j (j = 1,... ,8) Aj = constant for module j (j = 1,... ,8) Aj = coefficient of time for module j(j1, . t
Rij= coefficient for region i dummy variables for module j = 1,. .,9) Di = regional dummy (i = 1, ... ,9) Ykj = =
coefficient on order k size variable for module j hospital size (beds)
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error terms were tested for normal assure that estimates of significance
distribution, using a modified x2 of the coefficients were unbiased. The confidence limits for this test, and the coefficients and their associated t-values, are shown for each module in Table 1. It must be noted that only direct costs were reflected in the data available. Because indirect costs for such things as electricity, heat, and maintenance generally rise with size (though not necessarily proportionally), the cost functions shown in Table 1 are a little below the true average cost curves, and approximately parallel. If overhead costs did not rise with size, the functions shown would approach the true average cost curves asymptotically. Even though there are nine regions, only eight dummy variables enter the regression; in each regression, one of the Ri. is zero and the constant serves as the intercept for a base region dummy. Since region 4, East North Central, falls in the middle of the cost spread, we took it as the base region for calculation of relative price (cost) indexes. The price indexes Pij for each module j in each region i were calculated using the coefficients RPj and the constants Aj in the following equation: pii R,j + Aj test, to
The indexes so estimated differ from the conventional Laspeyre price indexes. Whereas the Laspeyre index is a measure of the relative prices in the different regions at each region's point of operation on the cost function, the index from Eq. 3 is a measure of the relative position of the minimumcost point of the function in each region. If hospitals in all regions operated
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How MANY BEDs?
Table 2. Hospital Size of Extreme Points of Average Departmental Cost Function Models Department Nursing ...........
Laundry ........... Laboratory ......... Radiology .......... Plant .............. Housekeeping ...... Linen ............. Dietary ............
Model Eq. 1 Eq. 2 Eq. 1 Eq. 2 Eq. 2 Eq. 1 Eq. 1 Eq. 1
Maximum Minimum Minimum Maximum Maximum Maximum Maximum Minimum
497.97 518.00 421.81 584.47 393.94 380.81 457.26 260.78
at the same position relative to the extreme point of the cost curve, then the index values calculated from Eq. 3 would be the same as the Laspeyre values. If two hospitals differed only in their relative positions on the cost function, the conventional price index would show different values whereas the indexes from Eq. 3 would show no difference.
Results and Discussion Differentiation of the appropriate equation permitted calculation of the point on the hospital size axis where each cost function showed a maximum or minimum cost. These extreme points are shown in Table 2. A maximum cost for nursing occurs at the relatively large size of about 498 beds. This indicates the least economical hospital size under current modes of organization of nursing: costs associated with the nursing department would be lower in both smaller and larger hospitals. This finding suggests that hospitals near this size could achieve economies of scale by reorganizing nursing departments into smaller or larger modules. There are two variables which, if they could be included in the analysis, might affect this result: the division of labor and the intensity of care in various size hospitals. With the first, it is possible that results were influenced by varied organization of nursing units. Below a certain hospital size, nurses may be more likely to be "jacks of all trades," whereas in larger hospitals it may be more common to employ nursing personnel with a greater variety of skill levels and use a substantial division of labor. Further research is indicated in this area, with primary emphasis on the ratios of different skill levels among personnel in hospitals. In the case of the second variable, the types and quality of care given in different hospitals might be so diverse that the relatively large hospitals, where the most intensive care is likely given, may have notably higher costs. We have excluded teaching hospitals to reduce diversity in the types of patient care; the data do not permit further control. The findings for laundry (Table 2) show that substantial economies of scale can be achieved by large-scale operations serving hospitals of at least
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Table 3. Relative Departmental Costs in Nine Regions Department
Nursing ........ 1.239 1.015 0.829 Laundry ....... 0.981 1.091 0.940 Laboratory ..........1.005 0.979 0.930 Radiology ...... 1.114 1.152 1.010 Plant .......... 0.964 1.044 0.914 1.164 1.149 1.066 Housekeeping Linen ......... 1.967 0.929 0.961 Dietary ........ 1.191 1.024 0.911 * See text for region code.
Region* ENC ESC WNC
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.886 0.848 0.877 1.132 0.777 0.908 0.249 0.858
1.099 0.953 0.992 1.235 0.752 0.842 1.435 0.937
1.678 1.017 1.205 1.660 0.995 1.296 2.514 1.201
0.700 0.912 1.012 0.970 0.807 0.981 0.496 0.791
0.846 0.910 0.954 1.010 0.752 0.806 0.407 0.869
500 beds. Smaller hospitals may capture these economies of scale by joining together with other hospitals in a shared laundry effort or by purchasing services. Many hospitals across the country are, in fact, sharing their laundry facilities to permit their operation at a more economical level. The relative success of such ventures is generally dependent on the degree of cost effectiveness of the organization and the management employed. The result for laboratory shows a minimum at a level of output consistent with a hospital size near 422 beds. At this size, the volume of laboratory determinations performed is sufficiently large that effective use may be made of automated instrumentation, decreasing the amount of relatively expensive labor necessary. This finding strongly suggests that smaller hospitals in an area could effectively share laboratory facilities to reduce cost. The existence of this minimum point is impressive when one considers that number of tests performed per 100 beds generally rises with hospital size and that more expensive tests, such as radioimmunoassay, are usually included in the repertoire in larger institutions. For radiology, costs rise to the maximum at 584 beds and then fall off. As hospital size increases, the radiology service generally performs a wider range of procedures, such as angiocardiography and lymphangiography, requiring more expensive equipment and more specialized personnel. Although more detailed data are desirable, our findings suggest that radiology departments could obtain economies of scale if the operation were commensurate with the demand of any hospital size other than 584 beds. Perhaps the more specialized procedures could be centralized in a larger hospital that would make those procedures available to smaller hospitals. This would allow both large and small hospitals to capture available scale economies: the large hospital would operate at a point on the average cost function to the right of the maximum point, and the smaller hospital would operate even further left of the maximum. Problems of patient transportation and scheduling of procedures might arise, but the appearance of the maximum indicates that detailed examination of alternatives to present modes of operation is worthwhile. 248
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How MANY BEDS?
The results for plant and housekeeping show cost maxima for hospitals of about 394 and 381 beds, respectively. Since physical plant costs include power, heat, and other items that might be analyzed as indirect costs to other departments, it is difficult to know how the finding for plant should be interpreted. For housekeeping, alternative modes of organization or sharing of services might be devised. At the very least the appearance of these maxima suggests that large hospitals should look for ways to economize on these services. Results for the linen service also show a maximum, at about 457 beds. This indicates that linen costs per patient day increase with hospital size over a considerable range, a result that is hard to explain. However, since a vast preponderance of hospitals have fewer than 400 beds, this indicates that most hospitals operate well to the left of this unexpected peak. Very large hospitals might profit by reorganizing their linen departments or examining their operations in detail. Our findings for the dietary department are based on coefficients having less significance than those used in calculating extremes for the other cost centers and are therefore less reliable. The minimum cost per meal near 261 beds suggests some diseconomies in large hospitals and in smaller ones.
Regional Cost Indexes Our estimates of the theoretical regional cost indexes from Eq. 3 are reported in Table 3. It will be remembered that the approach we have taken treats hospital cost modules separately and adjusts for any differences in scale of operation between regions. The theoretical price indexes thus allow exact interregional comparisons of the costs of various hospital modules, unobscured by differences in utilization. It is possible not only to compare the relative cost of, e.g., radiology across regions but also the relative cost of radiology and laboratory in different regions. It can be seen from Table 3 that cost patterns vary across the regions. For example, nursing costs are relatively high on the east and west coasts (regions NE and P), but costs for plant operation in these same regions are relatively low. The most important implication of the results, therefore, is that the relative cost pattern of one module across regions is likely not to be similar to the cost pattern of any other module across the same regions. Thus "optimum" hospitals in two different regions may require different relative sizes of the various modules. The variation in relative size of some of the modules across regions has a further implication. The spread of some modules is indicative of differing capital/labor cost ratios throughout the regions. The range of costs is greatest for such labor-intensive modules as nursing: the high-cost labor regions, such as the east and west coasts, have higher relative costs in comparison to the lower-cost regions, such as East South Central. In contrast, the relative spread of costs in the different regions is substantially less for a more capitalintensive module such as laundry.
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Regarding the module cost functions that have been described, it is clear that further investigation, using more detailed data and closer control of variables, would be worthwhile. The results we have presented must be regarded as tentative, but they indicate that sharing of many services and reorganization of hospital staffing structures could yield improvements in cost efficiency. Obviously it would be difficult, under the present modes of hospital organization and physical design, to alter some services: centralizing some portions of a set of radiology departments, for example, might be a reasonable alternative in an urban setting where several hospitals were close together but more difficult in a rural area; even in a city this would introduce new problems of communication and patient transport. For another example, the relatively small size indicated as optimum for a food service might represent an irreducible trade-off between economies of scale and increased spoilage and waste in bulk handling of food with present techniques. Nevertheless, if further study indicates that potential savings are substantial, the modular hospital concept could be a fertile source of inspiration for new techniques, physical designs, and organizational structures that could in the future benefit hospitals as organizations and health care as a more cost-effective function. Acknowledgments: Computer time was supplied by the University of Illinois at Chicago Circle Computer Center. The authors are indebted to Carson Bays, who made suggestions on an earlier draft. The authors are responsible for any errors or omissions.
1. Longest, B. Relationship between coordination, efficiency, and quality of care in general hospitals. Hosp Adm 19:65 Fall 1974. 2. Feldstein, P. J. An Empirical Investigation of the Marginal Cost of Hospital Services. Graduate Program in Hospital Administration, University of Chicago, 1961. 3. Carr, J. and P. Feldstein. The relationship of cost to hospital size. Inquiry 4:45 June 1967. 4. Feldstein, M. Economic Analysis for Health Service Efficiency. Amsterdam: NorthHolland, 1967. 5. Lave, J. and L. Lave. Hospital cost functions. Am Econ Rev 60:379 June 1970. 6. Newhouse, J. Toward a theory of non-profit institutions: An economic model of a hospital. Am Econ Rev 60:64 Mar. 1970. 7. Francisco, E. Analysis of Cost Variations Among Short-term General Hospitals. In H. Klarman (ed.), Empirical Studies on Health Economics-Proceedings of the Second Conference on Economics of Health, pp. 321-332. Baltimore, MD: Johns Hopkins Press, 1970. 8. Cohen, H. Hospital Cost Curves with Emphasis on Measuring Patient Care Output. In H. Klarman (ed.), Empirical Studies on Health Economics-Proceedings of the Second Conference on Economics of Health, pp. 279-293. Baltimore, MD: Johns Hopkins Press, 1970. 9. Pulley, A. V. and J. G. Fulmer. The optimal hospital size. Hosp Adm 20:16 Spring 1975. 10. Mann, J. and D. Yett. The analysis of hospital costs: A review article. J Bus 41:191 Apr. 1968. 11. Hospital Administrative Services. Six-Month National Data for Period Ending June 30, 1971. Chicago: American Hospital Association, 1971.
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How MANY BEDS? 12. Hospital Administrative Services. Six-Month National Data for Period Ending December 31, 1971. Chicago: American Hospital Association, 1972. 13. Hospital Administrative Services. Six-Month National Data for Period Ending June 30, 1972. Chicago: American Hospital Association, 1972. 14. Hospital Administrative Services. Six-Month National Data for Period Ending December 31, 1972. Chicago: American Hospital Association, 1973. 15. Johnston, J. Statistical Cost Analysis. New York: McCraw-Hill, 1960.