Within-year variation in hospital utilization and its implications for hospital costs

Journal of Health Economics 23 (2004) 191–211

Within-year variation in hospital utilization and its implications for hospital costs Laurence C. Baker a,b,∗ , Ciaran S. Phibbs c , Cassandra Guarino d , Dylan Supina e , James L. Reynolds f a

Department of Health Research and Policy, Stanford University, HRP Redwood Building, Rm 253, Stanford, CA 94305-5405, USA b National Bureau of Economic Research, Cambridge, MA, USA c The Veterans Affairs Palo Alto Health Care System and Stanford University, Stanford, CA 94305-5405, USA d The Rand Corporation, 1700 Main St., Santa Monica, CA, USA e Bayer Pharmaceutical, USA f Columbia University, 2960 Broadway, New York, NY, USA Received 1 March 2002; accepted 1 September 2003

Abstract Variability in demand for hospital services may have important effects on hospital costs, but this has been difficult to examine because data on within-year variations in hospital use have not been available for large samples of hospitals. We measure daily occupancy in California hospitals and examine variation in hospital utilization at the daily level. We find substantial day-to-day variation in hospital utilization, and noticeable differences between hospitals in the amount of day-to-day variation in utilization. We examine the impact of variation on hospital costs, showing that increases in variance are associated with increases in hospital expenditures, but that the effects are qualitatively modest. © 2003 Elsevier B.V. All rights reserved. JEL classification: I1 Keywords: Hospital capacity; Hospital utilization; Hospital demand; Hospital costs; Mergers

1. Introduction Health care cannot be stored, and demand for health care is often unpredictable. The resulting importance of demand variability as a determinant of hospital capacity needs has ∗ Corresponding author. Tel.: +1-650-723-4098; fax: +1-650-723-3786. E-mail address: [email protected] (L.C. Baker).

0167-6296/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jhealeco.2003.09.005

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long been recognized (e.g., MacStravic, 1979, 1981; Long and Feldstein, 1967; Carey, 1998), as have the related effects of demand variability on hospital costs, particularly the costs incurred because hospitals that wish to meet peak demand must purchase and maintain standby capacity that is costly but rarely utilized (e.g., Gaynor and Anderson, 1995; Joskow, 1980; Pauly and Wilson, 1986; Lynk, 1995). One area in which the relationship between demand variability and hospital costs has policy relevance is hospital mergers. An argument commonly advanced in merger discussions is that larger hospitals are more efficient. Reductions in demand variability are one way such efficiencies could come about. As emphasized by Lynk (1995), merged hospitals, across their combined facilities, would likely face lower variability in demand relative to the mean, which may lead to lower costs. This paper explores the relationship between demand variability and hospital costs. In the next section, we highlight two ways that demand variability can influence hospital costs. First, we follow others (e.g., Gaynor and Anderson, 1995; Joskow, 1980; Lynk, 1995) in noting that if hospitals operate with the goal of keeping the probability of having to turn away patients below some threshold level, the capacity a given hospital requires will be a function of demand variability. High variability hospitals would need more capacity on hand to meet high demand peaks than low variability hospitals. A second way demand variability can increase costs, which has received less attention in previous work, is by increasing costs associated with input purchasing. Many health care inputs, like nursing labor or supplies can be purchased either through long-term contracts or on shorter term spot markets. As we discuss in the next section, hospitals with highly variable demand cannot take advantage of long-term contracts as easily as hospitals with lower variance. If spot-market prices are higher than prices that can be obtained through long-term contracts, overall costs tend to rise with demand variability. After reviewing these relationships, we turn to an empirical analysis of daily variability in hospital occupancy and its effects on costs. A number of previous papers have empirically examined aspects of the relationship between demand variability and hospital costs. Gaynor and Anderson (1995) estimated a model of hospital costs which included a demand variability term derived from forecast errors in annual hospital utilization data. This clever solution permits empirical analysis of the variance–cost relationship to the extent that it is captured in the annual data, but misses day-to-day and month-to-month variability. Lynk (1995) used daily utilization data from four hospitals to examine demand patterns facing hospitals under various merger strategies, concentrating on variation at the level of individual units and on the impacts of demand variation on staffing. His data support the argument that larger hospitals should have lower coefficients of variation (i.e., the standard deviation divided by the mean) than smaller hospitals, and that this should translate into lower costs, but his analysis lacked the cost data that would have been needed to directly assess the effects of demand variability on costs. Joskow (1980) and MacStravic (1979) both present data on within-year variability in demand for a small number of hospitals but do not directly assess the relationship between variability and costs. Other work (e.g., Green, 2003) looks at capacity needs of hospitals or hospital units. Finally, several papers focus on the related issue of excess capacity on hospital costs and the cost of empty hospital beds (e.g., Keeler and Ying, 1996; Friedman and Pauly, 1981, 1983; Pauly and Wilson, 1986; Schwartz and Joskow, 1980).

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Our analysis remedies important data limitations in previous work by examining daily utilization data for a large number of hospitals over a relatively long time period. Using detailed discharge records, we computed the midnight hospital census for every hospital in California for every day between January 1, 1983 and December 31, 1995. We use these data to characterize day-to-day variation in hospital utilization. We show that there is substantial variation in hospital utilization on a daily basis, and that patterns of utilization differ across hospitals. We then directly investigate the effects of utilization variance on costs, estimating hospital cost functions that include measures of variance. We find statistically significant but small effects of daily utilization variability on costs.

2. The effects of demand variability on hospital costs There have been numerous studies of hospital costs, most of which assume that demand is known to the hospital (see Breyer, 1987; Vitaliano, 1987; Cowing et al., 1983; Vita, 1990; and Ellis, 1991 for reviews). We wish to explicitly incorporate demand variability, so we begin from a model incorporating variance, originally developed by Gaynor and Anderson (1995) based on work by Duncan (1990) and Friedman and Pauly (1981). We assume that hospitals make capacity and staffing decisions in two parts, and that the potential for variability in demand to influence costs exists in both. At the first stage, we assume that hospitals make decisions about the provision of fixed or “quasi-fixed” infrastructure, like buildings and beds, that cannot be changed in the short run. Following previous literature, we assume that each hospital does this by setting a “turnaway probability,” the probability that it will have to turn away new patients or discharge some patients early because its facilities are full, which then determines the level of capacity the hospital requires.1 To illustrate, suppose that on any given day the hospital faces demand z, where z is a random variable with distribution function G(z) and corresponding probability density function g(z). We assume for simplicity that G is known to the hospital.2 Acting to satisfy its goals for service to its community or perhaps other requirements, the hospital sets some turnaway probability, which we will denote as α. Choosing α defines the maximum level of demand it has to be prepared to face: z¯ = G−1 (1 − α), and the hospital needs then provide buildings, beds, and other fixed or quasi-fixed factors sufficient to meet that level of demand. Increases in the variance of G increase the standby capacity that hospitals require to meet peak demand but which will go unused at other times. The distribution of demand for hospital care is often characterized as being approximately normal, and we assume for our discussion here that G is normal. We later provide empirical evidence to support this assumption. For the normal distribution (and other sufficiently similar distributions), increases in the variance of G will increase z¯ . If hospitals must invest in infrastructure to meet this level of demand, costs will increase with the variance of G. At the second stage, taking as given the presence of a certain level of fixed factors, hospitals must adopt a strategy for providing variable inputs like staffing and supplies. For 1 2

As in Gaynor and Anderson, 1995; Joskow, 1980; Mulligan, 1985; and Shonick, 1970, 1972, among others. Modeling the distribution of demand conditional on past realizations of demand would produce similar results.

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example, assume that hospitals can buy labor and supplies using either long-term contracts or spot markets. Input amounts agreed under long-term contracts must be purchased regardless of the actual realization of demand at any given point in time, but inputs purchased in spot markets may be varied instantaneously to match realized demand.3 Prices are often higher in spot markets. Perhaps the classic example is the market for nursing, where hospitals hire full-time nurses with some responsibility to pay salaries even when demand is low. Hospitals can also pay overtime to their regular nurses and hire temporary nurses when demand is high, but at rates typically substantially higher than in the normal nursing labor market. By deciding how much of the variable inputs to purchase with long-term contracts, the hospital defines its strategy for providing the variable inputs. Even with optimal planning, increases in demand variability will tend to increase costs by decreasing the ability of hospitals to use long-term contracts and increasing the extent to which hospitals must rely on typically more expensive spot markets. Such an effect can be formally demonstrated,4 but for a simple illustration consider the effects of two demand structures that could face a given hospital. In the first instance, suppose the hospital has completely known demand of z∗ on every day. Such a hospital could provide for all of its variable input needs through long-term contracts sufficient to meet z∗ demand. In the second instance, suppose that on any given day the hospital has a 50% chance of facing each of two levels of demand, z− = z∗ − a and z+ = z∗ + a, with 0 < a < z∗ , and it cannot predict which. The hospital could write long-term contracts for an amount of variable inputs to meet demand levels on the interval (z∗ , z+ ), which by definition would cost more than the contracts for z∗ required when there was no variability. It could also write long-term contracts sufficient to cover demand levels on the interval (z− , z∗ ). Here, since average demand in this scenario is still z∗ , the hospital would pay for variable inputs on the more expensive spot market to meet the difference between the amount of demand covered in the long-term contracts and z∗ , which would also cost more than in the no-variance case because spot market prices are higher. Writing a long-term contract for exactly z∗ would also end up more expensive since the hospital would need to use the spot market on days when z+ was realized. Embedding these decisions in a cost function framework produces a cost function of the general form (see Gaynor and Anderson, 1995; Appendix A): C = C(y, α, µ, σ, ws , wl , k)

(1)

where C is the hospital costs, y the output, µ the average daily demand facing the hospital, σ the standard deviation of demand facing the hospital, ws and wl are the price per unit that a hospital pays for inputs purchased with long-term and spot market contracts, respectively, and k denotes the previously determined level of fixed capacity. The presence of α, µ, and 3 This structure departs somewhat from the framework used by Gaynor and Anderson (1995), where hospitals are assumed to be able to adjust their variable input purchases up or down after demand is realized. Our characterization of long-term contracts limits hospitals’ ability to adjust their variable input purchases down. This might be more realistic in many cases, where, for example, salaries of personnel must be paid regardless of the actual realization of demand. 4 We demonstrate this result formally in an appendix available from the authors upon request, or at http://www.herc.research.med.va.gov/journalarticles.htm.

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σ in (1) capture both effects that they have through standby capacity and effects through decisions made about the use of long-term contracts and spot markets for variable inputs. We will estimate a function of this form in Section 4. 3. Daily variation in hospital utilization We now turn to an examination of data on hospital demand variability and its effects on hospital costs. 3.1. Data To compute estimates of variation in hospital utilization, we obtained (with IRB approval) confidential discharge abstracts for each patient discharged from a California hospital between January 1, 1983 and December 31, 1995 from the California Office of Statewide Health Planning and Development (OSHPD). This amounted to 46.9 million records. We excluded discharges from non-acute facilities and discharges associated with lengths of stay greater than 1 year, which together made up 3.8% of the original records. Each record indicated the admission and discharge date. We used this information to compute the “daily census” for each hospital each day. Following custom, we defined this as the number of patients in a given hospital at the midnight that ends each day. For example, a person admitted on January 1 and discharged on January 2 would be included in the census count on January 1. Individuals admitted and discharged on the same day are not included in the daily census counts since they were not in the hospital over any midnight. We compiled separate data on the number of individuals admitted and discharged on the same day. There are relatively few “same-day” discharges. For example, in 1994 there were 111,013 total same-day discharges reflected in the California discharge data, relative to 14.6 million overnight patient days. We should note that our data do not capture people who are seen in a hospital but never admitted. Some hospitals have “short stay” units that function like inpatient units in many ways, except that patients never stay longer than 24 hours and are not formally admitted to the hospital. Over time, the number of such units has grown, but we are unable to track this type of hospital utilization here. The use of these kinds of units has grown most substantially in recent years, so that use of these kinds of arrangements was almost certainly less widespread during the time period covered by this study than it is now. Our data include only those patients discharged during the sample period. Patients in the hospital during the study period but discharged after December 31, 1995 were, therefore, not counted. This will cause us to undercount daily census in the days leading up to the end of 1995. We estimate that less than 1% of patients in California had a length of stay of 31 days or more. We thus eliminated December 1995 from all subsequent calculations about utilization patterns. For our analyses of costs, we eliminate all of 1995. These data represent about 500 hospitals per year. Consistent with general industry trends (AHA, 1999), the number of hospitals in the samples declines over time. For example, the sample contains 572 hospitals in 1984 but only 471 hospitals in 1994.

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L.C. Baker et al. / Journal of Health Economics 23 (2004) 191–211 115 Mean Daily Census

110 105 83 89 92 95

100 95 90 85 80 Dec

Nov

Oct

Sep

Aug

Jul

Jun

May

Apr

Mar

Feb

Jan

75

Fig. 1. Average daily census by month for California Hospitals, 1983, 1989, 1992, and 1995.

3.2. Summarizing variation in occupancy We use our data to summarize occupancy in several ways. First, we compute average daily census across hospitals by month and year. There is considerable variation in daily census by year and by month. Fig. 1 plots mean daily census by month for selected years. Consistent with previous literature using annual data, there are notable declines in average daily census in the early- to mid-1990s. Mean daily census is highest in January and February and falls over the course of the year, reaching a minimum in the late summer and fall. We do not plot the data here, but analyses by week of the year reveal similar patterns, and also show drops in daily census in the weeks near the Thanksgiving, Christmas, and New Years holidays. Fig. 2 plots mean daily census values for each day of the week. Daily census is highest during the week and lowest on weekends. These data are consistent with the types of variations one would expect, given that at least some hospital utilization is elective and short-term, and can be scheduled to coincide with convenient times for patients and physicians. Variation across months and days of the week accounts for a relatively small share of total occupancy—the difference between the mean Sunday census and the mean Thursday census is only about 10% of the average daily

Mean Daily Census, 1983-1995

105 100 95 90 85 80 Sun

Mon

Tue

Wed

Thu

Fri

Sat

Fig. 2. Average daily census by day of the week, California Hospitals, 1983–1995. The values shown are averages across all years.

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Fig. 3. 1990 Daily census distributions for four selected California Hospitals.

census. Nonetheless, even without accounting for unexpected shocks in demand, variations in average census across months of the year and days of the week will force hospitals to have beds that will not be filled at all times. To meet Thursday’s demand, the average hospital needs about 10 more beds than it will need on Sunday. For any given hospital, the distribution of daily census over days of the year appears approximately normal. Fig. 3 plots the distribution of daily census with superimposed normal distribution for each of four example hospitals, one randomly selected from each of four bed size groups. In all four of these cases, the daily census distribution has a characteristic normal shape.5 Generally, similar patterns are observed in other hospitals we have examined. The amount of variation in daily census itself varies across hospitals. Table 1 reports summary statistics for the variance and standard deviation of daily census across California hospitals for some sample years. The first row of each section contains the average variance and the average standard deviation across hospitals in 1984, 1990, and 1994. That is, for each hospital, we computed the variance and the standard deviation across days of the 5 All four of the distributions fail formal statistical tests for normality. At least part of the failure appears to result from relatively small departures from normality in the tails of the distribution.

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Table 1 Summary statistics of the variation in daily census in California Hospitals, 1984, 1990, and 1994 Daily census

Daily census variance

Daily census S.D.

Daily census coefficient of variation

Daily census 75th pctl.–25th pctl.a

Daily census 90th pctl.–10th pctl.a

R2 from regression on day of week and week of year

1984 Mean across hospitals 10th percentile across hospitals 50th percentile across hospitals 90th percentile across hospitals

522 522 522 522

100.4 10.9 60.8 231.3

259.6 14.5 102.9 591.9

12.8 3.8 10.1 24.3

25.1 8.9 16.5 37.2

17.5 5 14 33

32.9 10 26 63

0.73 0.58 0.74 0.87

1990 Mean across hospitals 10th percentile across hospitals 50th percentile across hospitals 90th percentile across hospitals

468 468 468 468

100.7 7.8 64.2 235.5

230.1 9.9 103.0 505.5

12.2 3.2 10.1 22.5

27.7 8.6 15.5 47.7

16.5 4 14 31

31.0 8 26 57

0.69 0.54 0.70 0.84

1994 Mean across hospitals 10th percentile across hospitals 50th percentile across hospitals 90th percentile across hospitals

465 465 465 465

86.2 5.9 57.8 206.5

205.1 9.4 98.0 434.0

11.5 3.1 9.9 20.8

37.7 9.8 17.0 51.0

15.2 4 13 27

28.8 6 25 53

0.68 0.52 0.68 0.83

Note: Hospitals that had zero total census for all days in the specified year are excluded. Percentiles are computed by year, so that, for example, the 10th percentile hospital in 1984 need not be the same as the 10th percentile hospital in 1990. a These figure are based on a calculation that first computes the difference between the 75th and 25th or between the 90th and 10th percentile of daily census for each hospital within a year. The figures in the table are derived from the distribution of this figure across hospitals.

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year. We then average these figures to arrive at the average variance and average standard deviation across hospitals. Below the average across hospitals are the 10th, 50th, and 90th percentiles of the distribution of variance across hospitals. In 1990, the average variance was 228. There is wide variation across hospitals: in 1990 the 10th percentile of variance was 10, and the 90th percentile was 503. Variance is scale dependent. To adjust for this, we computed coefficients of variation, which express the amount of variation as a percent of the mean. The CVs in column 4 of Table 1 indicate that, on average, the standard deviation is about one fourth of the mean. The CVs also indicate substantial differences across hospitals in the degree of variation in daily census, even after normalizing by the means. To the extent that the distributions of daily census values are non-normal, mean and variance may not fully characterize the distributions. Percentile distributions provide a non-parametric way of summarizing variation. Table 1 reports summary statistics for the difference between the 75th and 25th percentile of daily demand and the difference between the 90th and 10th percentile of daily demand for hospitals. That is, for each hospital, we compute the difference between the 75th and 25th percentile and the 90th and 10th percentile of daily demand, and then compute means and percentiles of these figures across hospitals. As above, these indicate sizable variations in daily census within-years and substantial differences across hospitals. Daily census variance declined over time. Fig. 4 plots the average daily census variance and various percentiles of daily census variance across hospitals for California hospitals by year for 1983–1995. At all levels, variance declines between 1983 and 1995, with the average variance across hospitals falling by about 30 percent. Much of the decline in variances observed between 1983 and 1995 is associated with declines in average daily census. Fig. 5 plots average coefficients of variation. No decline in the coefficient of variation is evident between 1983 and 1995, although sizable differences remain across hospitals. Hospital size is associated with variation. Larger hospitals have higher variances than smaller hospitals, closely related to their larger mean utilization. However, larger hospitals have smaller coefficients of variation than smaller hospitals. For example, the average CV 700 600

mean

Variance

500

10th percentile 400

25th percentile median

300

75th percentile 200

90th percentile

100 0 1983

1985

1987

1989

1991

1993

1995

Fig. 4. Means and selected percentiles of daily census variance for California Hospitals by year, 1983–1995.

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60

Coefficient of Variation

50 40

mean 10th percentile

30

25th percentile median

20

75th percentile 90th percentile

10 0 1983

1985

1987

1989

1991

1993

1995

Fig. 5. Mean and selected percentiles of the daily census coefficient of variation for California Hospitals by year, 1983–1995.

in a hospital with more than 225 beds was 11.1, while the average CV among hospitals with 151–225 beds was 15.3, the average for those with 76–150 beds was 19.4, and, among hospitals with 75 beds or fewer, the average CV was 37.8. 3.3. Predictable and unpredictable variance Some portion of the variation in daily census might reasonably be anticipated by hospital administrators. It may be that variation that can be anticipated has a different impact on costs than variation that cannot be anticipated. Anticipated variance can lead to increases in costs due to the need for standby capacity but may not increase costs associated with input purchasing if contracts can be written to provide for expected variations. Variations in utilization by day of the week and week of the year seem likely to be predictable. To estimate the amount of variation that these account for, we estimated regressions of the following form for each hospital each year: Ch,y,d = β0 + β1 DAY OF WEEK + β2 WEEK OF YEAR + εh,y,d ,

(2)

where subscript h denotes hospital h, subscript y denotes year y, subscript d denotes day (d = 1, . . . , Dy where Dy is the number of days in year y) and Ch,y,d is the patient census in hospital h in year y on day d. DAY OF WEEK is a set of six dummy variables accounting for variation across days of week Sunday through Saturday, and WEEK OF YEAR is a set of 51 dummy variables accounting for variation across the 52 weeks of the year. For each hospital each year, the R2 value from the regression measures the amount of variation explained by day of the week and week of the year patterns, which could be anticipated by an administrator familiar with patterns of time fluctuations in his or her hospital.6 (This will, of course, underestimate the total amount of variation that might be anticipated by an 6 We also experimented with a model where day of week effects were allowed to vary by quarter of the year, and found very similar patterns.

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administrator aware of additional sources of variation, such as a high volume surgeon going on vacation.)7 The rightmost column of Table 1 presents the results from this exercise for 1984, 1990, and 1994. Across hospitals, the average R2 from these models was about 0.70, suggesting that about 70% of the variation could be explained by variations in day of week and week of year in the average hospital and might well be predictable by savvy administrators. Notably, however, there was substantial variation across hospitals in the sample, with variations of about 0.3 between the 10th and the 90th percentile hospitals in any given year.

4. Relationship between occupancy variability and hospital costs In this section, we test the relationship between occupancy variation and costs using financial data on California hospitals from the California OSHPD, which requires all non-federal hospitals in California, with some minor exceptions, to submit quarterly and annual financial reports.8 These reports contain information on total operating costs, bed size, wages, ownership, teaching status, and a number of other variables. Our main measure of costs is total inpatient operating costs, which was obtained by summing inpatient operating costs for each quarter from the quarterly financial filings of each hospital. Our original sample contained 6357 hospital-year observations. We included in the analysis hospitals that reported providing short-term general care and for which there were no missing financial and other necessary data elements. To eliminate clearly problematic cases, we excluded hospitals that reported zero inpatient operating expenditures or zero beds in any year. In the end, the data set we use contains 4900 hospital-year observations. There are 482 unique hospitals included in the panel, with an average of 10.2 observations each. The panel is unbalanced, with some hospitals appearing a different number of times than others. We use these data to estimate short-run hospital cost functions of the general form of Eq. (3). Theory does not provide guidance as to the specific form of the cost function, so we follow much of the previous work in this area and adopt a hybrid form of the trans-log cost function used by Gaynor and Anderson (1995), including measures of variance and controls for average census, beds, wages, and other cost shifters. Trans-log models have the usual properties of cost functions, allow for flexible substitution, and locally approximate 7 This could also overstate predictable variance because it assumes that administrators are familiar with current patterns, and in reality they may only be able to anticipate variations that follow previously observed patterns. We experimented with models that predict current utilization as a function of only patterns in previous years, and found similar results with regard to the portion of variance explained by seemingly observable time patterns. 8 The annual reports filed by hospitals pertain to each particular hospital’s fiscal year. The starting and ending points of fiscal years vary from hospital to hospital, and some cases hospitals change their fiscal years over time. To allow for standardized analysis, we constructed data pertaining to calendar years (January 1 through December 31) from the annual reports. The quarterly reports are all filed for calendar quarters, so conversion to consistent dates was not necessary for the quarterly data.

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any arbitrary function. Specifically, our models take the form: ln Yh,y = β0 + β1 lnVh,y + β2 (ln Vh,y )2 + β3 ln Dh,y + β4 (ln Dh,y )2 + β5 ln Wh,y + β6 (ln Wh,y )2 + β7 ln Bh,y + β8 (ln Bh,y )2 + β9 ln Dh,y ∗ ln Wh,y + β10 ln Dh,y ∗ ln Bh,y + β11 ln Wh,y ∗ ln Bh,y + β12 Xh,y + β13 YEARy + β14 HOSPITALh + ε

(3)

where Y is the hospital costs, V the variance of daily census, D the patient days (i.e., the sum of daily census), W the measure of wages, B the available beds, X the vector of other covariates expected to shift the cost function, YEAR the vector of year dummies, HOSPITAL the vector of hospital dummies, and ε is the error term. We force the log of the variance in daily census ln(V) to enter without interactions in this specification for parsimony and clarity in the presentation of results. More complex specifications with interactions terms in ln(V) produced qualitatively similar results. Our measure of costs is annual inpatient operating costs, as reported by the hospitals to OSHPD. This is a comprehensive measure that should capture the majority of the costs hospitals incur to operate their inpatient enterprise. As a commonly reported measure, it is generally consistently reported across institutions, and is similar to cost measures used in other studies. We concentrate on inpatient operating costs rather than total operating costs since total operating costs for hospitals can include spending on a wide range of other enterprises (like outpatient clinics) that hospitals also run and for which we have limited ability to control. One question that arises is the extent to which measures of operating expenditures will capture effects of variance on costs through both standby capacity and through increases in costs for inputs. Operating expenditure measures are not designed to capture capital costs associated with new construction, and so their ability to capture standby capacity costs is limited. Nonetheless, they will capture the incremental operating costs associated with maintaining additional available space like utilities costs and maintenance costs. Of course, increased costs that arise in the procurement of inputs should be well-captured. We control for bed size, a proxy for capital stock and fixed costs, using the number of available beds in the hospital as of January 1 each year. Available beds are defined as beds that exist and are available for patient use, but need not be staffed. The wage variable we use is the average wage for six categories of labor: registered nurses, licensed vocational nurses, aids and orderlies, management and supervisory employees, technicians and specialists, and clerical workers. The wage variable is weighted by the total hours worked by each category. We compute wage measures at the level of the California “Health Facility Planning Area” in which each hospital is located. Results from models that use hospital-specific wage measures are similar. The inclusion of other less common categories of labor (e.g., advanced practice nurses, housestaff) did not affect the conclusions, nor did using wages for the various categories of labor separately. This wage variable captures spending on regular hospital staff, and so is probably best conceptualized as a long-term contract price rather than a spot-market price. Covariates expected to shift the cost function include case mix (i.e., the average DRG weight per discharge), the percent of discharges for which an HMO was the expected source

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Table 2 Means and standard deviations (S.D.) of cost function variables Variable

Mean

S.D.

Inpatient operating expenses ln(inpatient operating expenses) Variance in daily census ln(variance in daily census) Mean daily census ln(mean daily census) Available beds ln(available beds) Mean wage ln(mean wage) Case mix index ln(case mix index) % Discharges HMO No teaching/low teaching hospital Intermediate teaching hospital Academic medical center Church owned Private non-profit For profit Government owned Year = 1983 Year = 1984 Year = 1985 Year = 1986 Year = 1987 Year = 1988 Year = 1989 Year = 1990 Year = 1991 Year = 1992 Year = 1993 Year = 1994 Available beds ≤75 Available beds 76–150 Available beds 151–225 Available beds >225 unexplained variance ln(unexplained variance) explained variance ln(explained variance) N

35737704 16.748 234.025 4.565 101.986 4.061 191.851 4.936 14.808 2.659 0.985 −0.030 0.097 0.946 0.037 0.017 0.096 0.368 0.299 0.209 0.092 0.091 0.089 0.087 0.083 0.084 0.082 0.080 0.080 0.079 0.078 0.075 0.233 0.278 0.172 0.317 60.398 3.446 173.627 4.101 4900

46938252 1.224 549.899 1.435 113.943 1.209 161.406 0.846 3.998 0.269 0.177 0.175 0.139 0.226 0.188 0.131 0.295 0.482 0.458 0.407 0.289 0.287 0.285 0.282 0.275 0.278 0.275 0.272 0.271 0.269 0.267 0.263 0.423 0.448 0.377 0.465 85.513 1.271 484.026 1.572

Note: Intermediate teaching hospitals are COTH members that are not academic medical centers.

of payment, a vector of dummies for teaching status indicating academic medical centers as well as other hospitals with residency programs (“intermediate teaching hospitals”), and a vector of dummies for ownership type indicating church-owned, private non-profit, for-profit, and government hospitals (including district hospitals). Means and standard deviations of the variables used in the analysis are shown in Table 2.

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Theory suggests that the turnaway probability adopted by each hospital will influence its costs, but we have no measure of this probability. Since we expect that hospital objectives with respect to the level of care they will provide for their community are relatively constant over time, we expect the use of hospital fixed effects to control for hospital turnaway probabilities, among other fixed characteristics of hospitals and their communities that might influence costs. Similarly, theory suggests that spot-market prices will influence costs, but we do not measure these directly. We rely on the assumption that variations in these are captured by the hospital and year fixed effects and other hospital controls. Because spot markets for nursing and related services often function over wide geographic areas, we expect the amount of variation in spot market prices across hospitals, conditional on the hospital characteristics for which we control, to be very small. Theory also suggests that variation in demand should drive the need for standby capacity, but our measures capture variation in occupancy. Demand and occupancy will be closely related except possibly when hospitals are nearly full, when they may turn away some new patients or make efforts to discharge some patients who might otherwise have stayed. We expect that occupancy variation provides a reasonable approximation to demand variance since most of the hospitals in our sample are rarely full. For the median hospital-year in our sample, the maximum occupancy observed during the year is 70% of available beds. Of the hospital-years in our sample, 10% have a maximum daily census that is 93% or more of the number of available beds, and in only a very small number of hospital-years does the maximum daily census equal the reported number of available beds. Results from estimation of Eq. (3) are presented in Table 3. Most of the variables have the expected signs and are statistically significant. We focus on the variance terms, which are positive and significant in all specifications. We begin in column 1 with a basic model that includes only the variance terms, average daily census terms, and year dummies, excluding the hospital fixed effects and other covariates. The variance coefficients are statistically significant individually and in a joint test for the significance of both the linear and quadratic terms simultaneously, shown at the bottom of Table 3. Here, the linear ln(V) term is 0.202, and the quadratic term is −0.012, suggesting some concavity in the effect of demand variability. Evaluated at the mean level of ln(V), 4.565, this suggests that the elasticity of inpatient operating expenditures with respect to daily census variance is 0.092—a one percent increase in variance would lead to a 0.092 percent increase in inpatient operating expenditures. The second column adds controls for other hospital characteristics that are likely to be important, and cuts the estimated elasticity (evaluated at the mean) to 0.066. Adding in hospital dummies to control for other unobservables across hospitals further reduces the estimated elasticity to 0.043 (column 3). Although there is a significant relationship between variability and costs, the effect of an increase in variability is modest in absolute terms. With an elasticity of 0.043, the average hospital reporting a little over US$ 35 million in inpatient operating expenditures, would see a 1% increase in variance increase inpatient operating expenditures by about US$ 15,400. The specifications in columns 2 and 3 include a control for the number of available beds, as (among other things) a proxy for capital stock and fixed costs. Available beds is interpretable, at least in part, as a measure of standby capacity. Thus, an interpretation of the effects of variance in these models is that they reflect, to at least a large extent, increases in the costs associated with input purchasing rather than costs associated with the

Table 3 Main cost model regression results dependent variable: ln(inpatient operating expenditures) 2

3

4

0.202 (0.067) −0.012 (0.006) 0.355 (0.087) 0.067 (0.010) – – – – – – – – – – – – – – – 0.098 (0.016) 0.176 (0.016) 0.251 (0.016) 0.315 (0.017) 0.403 (0.017) 0.478 (0.017) 0.596 (0.018) 0.692 (0.018) 0.802 (0.020) 0.858 (0.021) 0.896 (0.020) 13.000 (0.067) No 4900 0.95 26.28 <0.001

0.167 (0.049) −0.011 (0.004) 0.436 (0.211) 0.151 (0.012) 0.236 (0.253) 0.206 (0.024) −1.254 (0.990) 0.380 (0.201) −0.330 (0.030) 0.251 (0.087) −0.257 (0.097) 0.534 (0.025) 0.147 (0.036) 0.122 (0.016) 0.245 (0.023) −0.027 (0.022) −0.066 (0.021) −0.133 (0.022) −0.042 (0.022) 0.071 (0.013) 0.131 (0.014) 0.203 (0.014) 0.254 (0.014) 0.336 (0.014) 0.406 (0.014) 0.504 (0.015) 0.575 (0.015) 0.654 (0.016) 0.693 (0.017) 0.730 (0.017) 13.324 (1.278) No 4900 0.97 18.92 <0.001

0.098 (0.042) −0.006 (0.004) 0.675 (0.462) 0.127 (0.018) −0.078 (0.346) 0.247 (0.032) −18.693 (18.160) 3.469 (3.362) −0.359 (0.044) 0.249 (0.178) −0.232 (0.114) 0.443 (0.038) 0.162 (0.025) 0.783 (0.175) 0.663 (0.123) −0.001 (0.019) 0.010 (0.014) −0.020 (0.029) −0.071 (0.028) 0.059 (0.009) 0.113 (0.009) 0.185 (0.009) 0.237 (0.010) 0.313 (0.009) 0.377 (0.010) 0.467 (0.010) 0.536 (0.011) 0.608 (0.011) 0.637 (0.012) 0.669 (0.013) 38.233 (24.400) Yes 4900 0.99 10.04 <0.001

0.109 (0.048) −0.007 (0.004) 0.429 (0.532) 0.055 (0.012) – – −22.150 (21.113) 4.191 (3.941) – −0.097 (0.211) – 0.414 (0.042) 0.193 (0.028) 1.009 (0.208) 0.442 (0.144) −0.010 (0.024) 0.005 (0.015) −0.005 (0.032) −0.030 (0.034) 0.055 (0.010) 0.109 (0.010) 0.180 (0.010) 0.235 (0.010) 0.311 (0.010) 0.377 (0.011) 0.473 (0.011) 0.546 (0.012) 0.632 (0.013) 0.668 (0.014) 0.696 (0.014) 43.493 (28.170) Yes 4900 0.99 10.77 <0.001

Note: Robust standard errors in parentheses. F-statistic is from the joint test that the linear and quadratic ln(daily census variance) terms are equal to 0.

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1

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ln(daily census variance) ln(daily census variance)2 ln(average daily census) ln(average daily census)2 ln(available beds) ln(available beds)2 ln(mean wage) ln(mean wage)2 ln(average daily census) × ln(available beds) ln(available daily census) × ln(mean wage) ln(available beds) × ln(mean wage) ln(case mix index) % Discharges HMO Intermediate teaching hosp Academic medical center Church owned Private non-profit For profit Government owned Year = 1984 Year = 1985 Year = 1986 Year = 1987 Year = 1988 Year = 1989 Year = 1990 Year = 1991 Year = 1992 Year = 1993 Year = 1994 Constant Hospital fixed effects Observations R2 F-statistic P-value

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Table 4 Effects of variance by size of hospital dependent variable: ln(inpatient operating expenditures)

ln(daily census variance) ln(daily census variance) × beds 76–150 ln(daily census variance) × beds 151–225 ln(daily census variance) × beds >225 ln(unexplained variance) ln(explained variance) Hospital fixed effects Observations R2

1

2

3

0.069 (0.011) −0.011 (0.004) −0.013 (0.005) −0.013 (0.005) – – No 4900 0.97

0.036 (0.010) −0.004 (0.005) −0.001 (0.006) −0.001 (0.007) – – Yes 4900 0.99

– – – – 0.106 (0.015) 0.013 (0.006) Yes 4900 0.99

Note: Robust standard errors in parentheses. Models also contain the same controls shown in Table 3, column 3.

infrastructure associated with standby capacity.9 To allow the model to more fully capture effects through standby capacity as well, column 4 drops the controls for available beds. This increases the estimated elasticity only a small amount, to 0.045. (Note that this may still be an understatement of the full effect of variance since our dependent variable, inpatient operating expenditures, will not capture capital spending on things like new construction.) Hospital size may be related to the elasticity of inpatient operating expenditures with respect to variance. This could result because hospitals differ in their ability to effectively manage demand shocks depending on their size. It is also possible that some scale-dependence in variance persists, despite the inclusion of controls for hospital size and hospital fixed effects, and influences the estimates. To investigate these effects, we estimated models that interact log variance with dummies for hospitals with 75–150, 151–225, and more than 225 beds. These points are approximately the 25th, 50th, and 75th percentiles of the bed distribution observed in our data. For simplicity in presentation, we report results from a model that included only a linear term in variance; more complex specifications produce similar results. These results are shown in Table 4. In the model without fixed effects, we observe a relationship between hospital size and the effect of variance, with larger hospitals having smaller effects. Most of this effect is observed in the move from below 75 beds to above 75 beds, with only very small incremental changes at higher bed sizes. Relative to hospitals with 75 or fewer beds, hospitals with 76–150 beds have a ln(variance) coefficient about 16% lower. When we include hospital fixed effects, we observe no change in the effect of variance with changes in bed size, though with limited variation in bed size over time within hospitals in the sample, the power of the fixed effects models to detect an effect is also limited. Predictable and unpredictable variability in demand may influence costs to different degrees. Predictable variability would increase costs to the extent that hospitals have to provide for high levels of demand with resources that will expectedly go unused at times 9 This interpretation is less clear if one interprets “standby capacity” to include only beds set up and staffed. In this sense, writing long-term contracts to provide staffing for some number of beds is a key part of determining standby capacity. In this sense, including controls for beds may capture some effects on costs via standby capacity needs.

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of low demand. But, unpredictable variation may increase costs even more, by forcing the use of more expensive spot-market variable inputs as well. To investigate this, we used results from Eq. (2) to divide the total variance into the portion that is explained by day of week and week of year, and the portion that is not. We then include the explained, perhaps “predictable,” and unexplained, perhaps “unpredictable,” variances in the model separately. Column 3 of Table 4 shows results (again from a model that includes only linear terms in variance; models using quadratic specifications produce similar conclusions). Both explained and unexplained variance increase costs, but the effect of unexplained variance is much stronger. The model in column 3 controls for beds, thus possibly absorbing some effect of variance through standby capacity costs. If we drop the beds variables, we observe only a very small increase in the effect of explained variance, consistent with the view that changes in the share of predictable variance have a more important effect on input purchasing costs than on standby capacity costs. We experimented with further decomposing variance into portions explained by day of the week, explained by week of the year, and unexplained, because of the potential for hospitals to incorporate variation across day of the week into labor contracts more easily than variation across week of the year. The results, however, showed no significant differences between variation attributable to day of the week and to week of the year. We tested our results for robustness using a variety of measures of demand variability, including the standard deviation of daily census, the CV of daily census, the difference between the 90th and 10th percentiles of daily census, and the difference between maximum and minimum levels of daily census. All of these results (not shown) are consistent with the conclusion that increases in variability are associated with small increases in costs.

5. The expected effects of hospital mergers on variance and costs Forecasting the effects of mergers on costs as mediated through changes in demand or occupancy variance is difficult because mergers change many hospital characteristics that may influence costs, making the effects of variance changes difficult to isolate. We can, nonetheless, use our data to provide information about the expected impacts of mergers on measures of variability and discuss the effects that these changes could have on spending. As noted by Lynk (1995), increases in size of hospitals via mergers may be expected to reduce variation in demand relative to mean demand. Others (e.g., Stigler, 1964) have long since pointed out that firms that control larger shares of markets face lower variability in demand relative to the mean. To illustrate, consider a merger between two hospitals facing normally distributed daily demand xa and xb , with means µa and µb and variances σa2 and σb2 . If these hospitals were to merge, they would together face normally distributed daily demand 2 with mean µa+b = µa + µb and variance σa+b = σa2 + σb2 + 2 cov(xa , xb ). One measure of the amount of the total amount of demand variability facing the two hospitals before any merger is (σa +σb )/(µa +µb ), which we will term the “joint CV”. One interpretation of this statistic is as an index of the maximum census the two hospitals would have to be prepared to face. After a merger, this could be compared to the CV facing the merged institution, σa+b /µa+b , and it is not hard to show that σa+b /µa+b ≤ (σa + σb )/(µa + µb ). These

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quantities would be equal only if corr(xa , xb ) = 1. As long as the correlation is less than one, which would be expected in many cases, mergers reduce the CV. The variance itself, and the ratio of the variance to the mean, need not be reduced by mergers and, in fact, are likely to increase. Our data bear out these patterns. We used our daily utilization data to simulate 1000 mergers of California hospitals. To simulate a merger, we first randomly chose a year and a pair of hospitals in the same county operating in that year.10 It is possible that the same pair of hospitals appears more than once in the analysis in different years. For each simulated merged entity, we computed predicted daily demand as the sum of the daily demands from each of the pre-merger hospitals. We then examined measures of demand variability for hospitals before and after a simulated merger. Table 5 reports results. Across all 1000 simulated mergers, the average mean daily census of the hospitals increases linearly as expected. Variance increases more than proportionally, as would be expected given the nonlinear nature of variance. The average ratio of variance to mean daily census (MDC) in the pre-merger hospitals is 2.61. The average ratio in the simulated merged hospital is 2.90. The coefficients of variation fall. Average CVs in the pre-merger hospitals are just over 0.20, and the average joint CV is 0.15. The average CV of the simulated merged entities is 0.12. The average ratio of joint CV to merged CV is 0.81. Across simulated mergers, the 10th percentile ratio of joint CV to merged CV is 0.72, and the 90th percentile is 0.90. To investigate whether these effects vary by the sizes of the merging hospitals, we computed measures for two subsets of our simulated mergers: those where the two hospitals both had MDC over 70, and those where both hospitals had MDC under 70. For mergers of small hospitals, the reductions in variance are more noticeable. Variance increased with the merger, but by a lower factor than in the overall case. The ratio of variance to MDC fell, averaging 2.8 in the pre-merger hospitals and 2.6 after the simulated merger. The average ratio of joint to merged CV here is 0.77. Mergers of larger hospitals show less pronounced variation reducing effects. The ratio of variance to mean was notably higher after the simulated merger, going from an average of 2.6 to 3.5. The average ratio of joint to merged CV is 0.83. Identifying the effects of changes in variation on the levels of spending in the context of mergers with any precision is difficult. One approach is to use our regression results to compute levels of predicted spending for the pre-merger and simulated merger hospitals, but we found this unsatisfactory. We regress log spending on log variance in a context with high spending and variance levels, so even small imprecisions in estimation can produce very large swings in predicted levels of spending. Another difficulty is that mergers can influence many characteristics of hospitals, including bed size and other parameters in our models. Rather than attempt to forecast spending using our model, we present some more general results derived from expectations about changes in variation applied to the estimated variance elasticities. To do this, one must evaluate the amount of overall variation across the two 10 If area population characteristics are the source of at least some variation in demand, and hospitals in the same county serve populations that overlap to at least some extent, then we would expect cov(xa , xb ) to be positive in these cases, and possibly lead to smaller estimates of the reduction in variability than would be calculated for mergers between more distant hospitals serving more distinct populations and thus having lower covariance.

N mergers

All mergers Both <70 MDC Both >70 MDC

1000 223 266

Note: MDC is mean daily census.

MDC

Variance

Coefficient of variation

Hospital 1

Hospital 2

Merged

Hospital 1

Hospital 2

Merged

Hospital 1

Hospital 2

Joint

Merged

120 34 212

122 38 194

241 72 405

378 94 802

316 72 525

828 181 1688

0.22 0.35 0.12

0.21 0.3 0.12

0.15 0.25 0.11

0.12 0.19 0.09

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Table 5 Changes in demand variability associated with simulated mergers

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pre-merger hospitals before the merger. It is not clear that there is a precise way to do this. As one estimate, we computed the average pre-merger “joint” variance as (σa + σb )2 where σ a and σ b are the observed standard deviations for the two hospitals. We then compared this to the actual observed variance in the simulated merger. Across our 1000 simulated mergers, the mean pre-merger “joint” variance was 1144. The mean simulated merger variance was 828, a reduction of 28%. Applying an elasticity of 0.043, a 28% reduction in variance would correspond to a reduction in inpatient operating expenditures of 1.2%, or about US$ 430,000 for a hospital with sample mean annual inpatient operating expenditures of about US$ 35 million. Various non-linearities in the specification of the model and in variances could distort the predicted effects. As a check, we also re-estimated the model shown in Table 3, column 3, using the log of CV instead of the log of variance. Evaluated at the sample average log CV (2.827), the elasticity for CV is 0.065. Our simulations suggest about a 20% reduction from the joint CV to the merged CV on average, which would produce a 1.3% reduction in inpatient operating expenditures. Comparing the larger declines between the individual hospital CVs before the merger and the simulated merged CV would yield a decline of about 40%, which would imply a correspondingly larger spending reduction. These estimates suggest that, while savings could likely be realized through mergers that reduce variance, these savings may not be large. At the same time, these results are only rough approximations, and may not apply in every case. Some mergers may realize smaller reductions in variance than we discuss, and thus also correspondingly smaller reductions in spending. Others may have larger effects, particularly mergers between smaller hospitals.

6. Conclusions This paper uses data from California hospitals to examine daily census variability. Variation in demand is an important component of hospital cost modeling, but data on within-year variations has not previously been available for a large sample of hospitals. We demonstrate that there is substantial within-year variation, and that there is also considerable variability in utilization variation across hospitals. Some of this variation should be predictable by hospital managers, but it is unlikely that all of it is. We then examined the impact of variation in daily census on hospital costs, showing that increases in variance are associated with statistically significant but small increases in total inpatient operating expenditures. This confirms previous work that argued the importance of demand variability as a factor in hospital cost functions. A common argument in merger discussions is that merging hospitals will tend to lower the amount of demand variability facing the merged entity relative to the amount facing the unmerged hospitals. These results suggest that such effects are likely to be modest in size, and that very large reductions in variability would have to be realized to generate significant savings.

Acknowledgements We gratefully acknowledge helpful comments from the editor, two anonymous referees, and seminar participants at the National Bureau of Economic Research. We are also

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