Are big hospitals the best medicine.

1.0 Introduction

The health care issue has exploded to the top of the American political agenda. In the process, attention has increasingly focused on health care providers, especially hospitals, as centerpieces in the supply of medical services. However, the hospital industry scarcely fits the model of the traditional textbook market: Due to the preponderance of third-party payers, the typical consumer receives a good for which he or she pays only a small part of the total price. Moreover, the consumer has little or no expertise for evaluating the service being purchased.

Transcending these anomalies is a hot debate concerning the most efficient market structure for delivering hospital services. On one side, some contend that smaller hospital size and more competitive market structures in hospital services are less efficient, and serve to substantially and artificially inflate hospital costs and prices. On the other side it is argued that smaller size and competitively structured markets are essential to ensure that consumers and society are not harmed in traditionally monopolistic ways.

Which of these opposing views is correct has dramatically significant implications for policy makers: If greater size engenders lower prices and more efficiency then policy makers should encourage consolidation in order to achieve these benefits. If, however, smaller hospitals are found to be more efficient, and therefore more advantageous to consumers, then vigorous enforcement of antitrust laws by the Federal Trade Commission (FTC) and the Department of Justice (DOJ) would be in order to combat hospital combinations.

The idea of size is one that is easily blurred. Both relative and absolute size would seem to play a role in the cost structure of hospitals. Market concentration is the number of firms providing hospital services within a defined market area, which therefore gives an idea as to relative size; while firm size is the size of an individual firm within the market, that is, absolute size. It is conceivable to have a highly concentrated market, i.e. relatively few firms, which in fact has rather small hospitals. While both issues are of interest in the debate over the hospital industry, they should not be seen as redundant. The material covered herein deals with individual hospital size as its main area of inquiry.

2.0 Theory

The theoretical debate can be easily classified into two distinct divisions. On one side, is the increasing monopoly theory (Salkever 1978). This theory indicates that the lower the concentration and the representative size of a hospital in a particular hospital market, the higher will be the prices.

In justifying this, Salkever cites that in most cases, hospitals do not pay physicians. Rather they must attract physicians to their facility via attractive arrangements. Such arrangements might be high technology equipment, or increased staff. As hospitals compete with each other to woo the best physicians, they become entrapped in a type of "medical arms race." This drives up costs and in turn drives up the prices charged to the consumer.

Salkever argues that the consumer is made worse off by this competition. In essence, his argument is that hospital services are a natural monopoly, that is "competition is unworkable, unsustainable, and socially undesirable."(1) Nett (1992) also argues this point, modeling medical services under both the monopoly case and the competitive case. He concludes that any more than one provider in a market is inefficient.

On the opposite side is the traditional oligopoly approach which argues that there is a direct relationship between market concentration and hospital size and the prices they charge. More concentration leads to higher prices as hospitals no longer must compete as strongly with each other. A common, though not certain, by-product of increased concentration is larger hospitals. Moreover, a market dominated by larger-hospitals leads to higher prices as the bigger institutions are not efficient. For advocates of this position, the competitive free market is the force that ensures that consumers are not gouged and that society is protected.

In the case of hospitals, there are unique guidelines followed by the regulatory agencies as they evaluate proposed mergers:(2) 1) Entry Conditions: How likely is it that within two years a new hospital would enter the market, or a fringe hospital would expand its capacity; 2) Current Role of Merger Participants: Mavericks prone to shaking up the market versus fringe competitors with different areas of specialization; 3) Consumer Perceptions: Serve as "reality check" on analysis; 4) Efficiencies: Potential cost savings and other consumer benefits, if any, that would directly result from the merger, i.e. economies of scale.

This last point is intriguing for analysis. These efficiencies are the crux of what will be discussed here. Without them, any argument for a merger seems difficult to imagine. Moreover, in the presence of obvious efficiencies, even the most devout critic of size and concentration would be hard pressed to denounce collaboration.

Economies of scale arise in instances where the nature of the industry is such that as the size of the firm increases, ceteris paribus, average costs decrease. That is, resources are used more efficiently when they are used to a greater extent. Those mergers that demonstrate greater efficiencies are not as likely to receive FTC and DOJ challenges (Wagner, 1992 p. 11).

Some in the hospital industry argue for relaxation of antitrust enforcement. Greater flexibility in merging and collaborating, would ostensibly reduce health care costs (Wagner, 1992 p. 11). In a Q&A report to member hospitals, the American Hospital Association explains that the possibility of challenge is high in those instances where there are few competing hospitals, unless one of the participants is in danger of failing, or significant efficiencies counterbalance the potential for increased market power (AHA 1992, p. 4).

A number of approaches have been used to attempt to discern economies of scale in the hospital industry. A majority of these have centered on estimating a cost function for hospitals, and from that calculating the presence or absence of economies of scale. There is no consensus in the literature on the nature of the cost function to be estimated, and perhaps more importantly, there is no uniformity of results among these studies.

3.0 Model

There are a variety of approaches in constructing a cost function to test for the presence of economies of scale. Perhaps the most fundamental is the question of a structural vs. a behavioral cost function. A structural cost function is one that is derived exclusively from economic theory and only involves the relationships between costs, output levels, and input prices. Conversely, a behavioral cost function includes variables not dictated by economic theory, those variables which capture the "real-world" differences between hospitals (Folland, Goodman, and Stano p. 335). Because of their subjective nature, behavioral cost functions are ad hoc, making interpretation of coefficient estimates difficult. There is no theoretical justification upon which to base expectations as to a given variable's impact on the model.

A structural cost function is the preferred approach here, because its derivation can be shown through theory rather than some ad hoc approach.

Given the form of the cost function, the next question is the time horizon. Economic theory designates the short run as the time period in which one of the factor inputs to the production function (i.e. labor or capital) is fixed; in the long run all inputs are variable. For purposes of discerning economies of scale, it is necessary to estimate a long run function because by definition, "a firm is said to experience economies of scale when its long-run average cost curve is declining in output," (Folland, Goodman, and Stano p. 331, emphasis added). Or, put another way, economies of scale occur when increasing all inputs some percentage, increases output more than that percentage; the short run requirement that at least one input be fixed would not allow this.

Given that the cost function will be a structural long run cost function, we can derive it from economic theory below. We assume that a hospital's production function reflects Cobb-Douglas technology of the form:

y = Q [W.sup.a][K.sup.b]

y equals output produced by inputs labor (W) and capital (K). Q is some constant, and a and b are the elasticities of production for each input respectively (Vitaliano, p. 308, 1987).

Solving for the respective factor demand equations leaves the cost equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where C is total cost, and w and k are the respective input prices. The exponent, I/a+b, reflects the elasticity of total cost with respect to output. By taking natural logs we get:

In C = ln [Alpha] + [Beta] ln w + [Gamma] ln k + [Lambda] ln y

Taking natural logs allows us to estimate the nonlinear cost function as a linear function. We are able to estimate 1/a + b (the elasticity of total cost with respect to output) with [Lambda]. A [Lambda] [is greater than] 1 indicates decreasing returns to scale, while [Lambda] [is less than] 1 signifies increasing returns to scale. In the graph below, the region along the curve from a to b would be increasing returns to scale; at point b, returns would be constant; and from point b to point c would be decreasing returns to scale (Folland, Goodman, and Stano, p. 334).

For purposes of estimation we estimate the total cost function, because estimation of the average cost function would necessitate placing the output variable on both sides of the equation (because to arrive at the average cost function both sides of the total cost function would be divided by output, y). This would potentially bias parameter estimates (Vitaliano, p. 306, 1987).

In terms of the specific variable definitions, y is the measure of the level of hospital output and corresponds to the product of overall admissions and the hospital's case mix index.(3) This draws from Lave, Lave, and Silverman who show the need to account for differences in hospital case mix as well as Vita who also makes the distinction. Not accounting for dissimilarities in individual hospital output could lead to biased result (Folland, Goodman, and Stano, p. 336). For example, in the graph below, the long run average cost curve for four hospitals (1-4) treating different severities of illness (intersection points a-d) are shown. Note that each has a flat curve (i.e. constant returns to scale); however if the severity of illness is not accounted for, (under the reasonable assumption that more severe illnesses are more costly) a researcher would perceive that there are diseconomies of scale (Folland, Goodman, and Stano, p. 337).

w represents the index for hospital wages.(4) This fulfills the need for a measure of labor prices which is dictated by production theory. k(5) is the Geographic Adjustment Factor, which is used by HCFA in evaluating differences in the cost of capital among hospitals. Like the wage index it is calculated by Metropolitan Statistical Area (MSA), and therefore reflects the cost faced by the hospitals in a given market.

Demand is assumed to be exogenous. If it were not, then the output variable could be endogenous, and therefore correlated with the disturbance term (Pindyck and Rubenfeld, p. 303, 1991), making OLS estimates biased and inconsistent. To justify exogeniety of output we assume that hospital administrators must provide services sought by physicians on demand; the administrator must provide factor inputs which are combined with physician services to provide necessary treatments. Hospitals do not pay for physician services, so it is in composition of the other inputs that hospital costs are generated; therefore the demand for services is exogenous to the administrator and the hospital (Conrad and Strauss, p. 341, 1983). This assumption is further enforced by restricting the study to within state boundaries. As Conrad and Strauss point out, "restricting the study to one state (North Carolina) was desirable because it ensures a single institutional and regulatory environment," (p. 345).

In our case, the two states chosen for consideration are Florida and Ohio. Florida has a highly regulated hospital industry where the hospitals are assigned a Maximum Allowable Rate of Increase (MARI) in gross revenue. The MARI is set by the state government. If a hospital exceeds its MARI, it is fined, and if it comes in under its MARI, it is given an award. This would seem to further enforce the assumption of exogenous output. The situation in Florida appears very similar to that in New York at the time of Vitaliano's study, in that both markets are subject to a high degree of government intervention, strengthening the assumption of the exogeneity of demand. Ohio has no such governmental controls; however, the previous justification for exogeneity seems applicable.

The specific cost function (with appropriate variable names) is:

TEXP = [Alpha] + [[Beta].sup.1]RAD + [[Beta].sup.2]WIND + [[Beta].sup.3]K + u

where:

TEXP = natural log of total hospital

expenditures

RAD = natural log of number of

admissions deflated by the

case mix index

WIND = natural log of wage index

K = natural log of capital Geographic

Adjustment Factor

4.0 Data

Data on total expenditures and admissions, were obtained from the 1991 American Hospital Association Guide to the Health Care Field. The Guide publishes financial information annually on all hospitals in the United States.

For purposes of this study, Florida and Ohio were selected for consideration. They were selected first, because both were relatively large and densely populated so as to provide enough observations for analysis. In addition, it was thought that because Florida has a noticeably older population, this might have a different effect on health care costs in general and hospital costs in particular.

The study was constrained to short term general medical hospitals. Short term hospitals are those where the average length of stay for all patients is less than 30 days.(6) General medical hospitals are those where the main purpose of the facility is to perform therapeutic and diagnostic patient services.(7)

The wage index, the case mix index, and capital index were obtained from the Health Care Financing Administration (HCFA). The case mix index represents the relative patient mix of an individual hospital. The wage index represents the relative wage rate faced by hospitals within a given Metropolitan Statistical Area (MSA). The capital index provides the relative cost of capital faced by the hospital. These would seem to be excellent input measures, as they reflect the market environment.

5.0 Econometric Results

Table 1 shows the regression results of the above model for Florida and Ohio. Note that the variable measuring cost of capital has been dropped to deal with multicollinearity in the model.(8) The White test was used to test for the presence of heteroskedasticity. The White test is advantageous because it does not require any assumptions about the form of the heteroskedasticity, as do other tests. For example, the Goldfeld-Quandt test calls for dividing the data into two arbitrary groups and making an assumption about the nature of the heteroskedasticity. The results of the White test indicated the presence of heteroskedasticity.

TABLE 1 OLS Results

            FLORIDA        OHIO

Intercept   2.3227629(*)   1.9839096
            0.2219276      0.4796526
            0.00000001     0.00006

RAD         0.9207175(*)   0.960139
            0.0234591      0 0471604
            0.0009567      0.3993989

WIND        0.7447904(*)   0.8139255(*)
            0.2023027      0.4413209
            0.0003377      0.0672096

n              134         146

Note: Parameter estimate/standard error (White corrected)/prob value (Beta = 1 for RAD; otherwise Beta = 0)

(*) significant at 10%

To correct for heteroskedasticity the White procedure was used. Note that coefficient estimates are consistent in the presence of heteroskedasticity; however, the standard errors will be wrong leading to incorrect hypothesis tests if they are not corrected. The White procedure is chosen, again, because it requires no arbitrary assumptions be made. Weighted Least Squares, another popular remedy for heteroskedasticity, requires employing some ad hoc weight to divide through the model. The standard errors in table 1 reflect those derived from the White procedure.

The value of the coefficient estimate on RAD is significantly different from one in Florida and not so in Ohio (a value of one indicating constant returns to scale). This indicates that there are some economies of scale in the hospital industry in Florida.

Because of the nature of the data, we must test to see if the independent variables were measured with error. If they were, then the independent variables would be correlated with the disturbance term, making our estimates biased and inconsistent (Pindyck and Rubenfeld, p. 160, 1991). We can test for this via the Hausman test. The Hausman test first requires the estimation of a companion model using instrumental variable estimation. For this an instrumental variable, i.e. one that is correlated with the independent variable, but not with the disturbance term is needed:

[[Beta].sup.iv] = [(Z`X).sup.-1] (Z`Y)

The intuition behind the Hausman test is that the probability limit of [[Beta].sup.iv] will always equal [Beta]. However, the probability limit of [[Beta].sup.ols] will only equal [Beta] if the error is not correlated to the independent variables. Therefore, we test if [[Beta].sup.iv] = [[Beta].sup.ols] if this is true, then the error and the independent variables are believed to be uncorrelated.

Using Durbin's method we use the variables relative rank as the instrument, as this would be highly correlated to the explanatory variable in question, without being correlated to the error. X is the initial matrix of explanatory variables; Z is the matrix of explanatory variables with the variable rank in place of the variable being tested. The results of this are seen in table 2. The Hausman test works under the null hypothesis that the OLS estimator is consistent i.e. the OLS estimator is equal to the instrumental variable estimator. In our case, the null cannot be rejected in either state; indicating no correlation with the disturbance term.

TABLE 2 Hausman Test

            FLORIDA                       OHIO
            RAD inst.      WIND inst.     RAD inst.     WIND inst.

Intercept   2.2793379(*)   2.3117864(*)   1.910003(*)   2.0103827
            0.3575399      0.3198847      0.5746734     6.1956284
            1.14E-07       0.00000001     0.0047815     0.1557636

RAD         0.9254533(*)   0.9214586(*)   0.9679436     0.9578705
            0.0150213      0.1554333      0.0201054     0.7804484
            0.0000354      0.0300994      0.1979007     0.7864749

WIND        0.7358485(*)   0.6710035(*)   0.7742507(*)  0.8707284(*)
            0.2708481      0.2012265      0.4114058     0.5408007
            0.0040177      0.0057281      0.0023819     0.0034321

Wald        0.0769268      0.0993424      0.1042352     0.0433787
p.v. Wald   0.999279       0.99988065     0.9986882     0.9997682
n              134                           146

Note: parameter estimate/standard error (White corrected)/prob value (Beta = 1 for RAD; otherwise Beta = 0)

(*) significant at = 10%

This test is also interesting because it seems to give further credence to our assumption of the exogeneity of demand. If demand were endogenous then the output variable would be related to the error term; however, the Hausman test indicates that is not the case.

Another concern is that some variables may have been omitted from the model. This is of particular interest in this case because we dropped one of the input price variables. Omitted variables result in biased coefficient estimates (Pindyck and Rubenfeld, p. 164, 1991). This bias could be positive or negative depending the nature of the omitted variable. To test for omitted variable bias, Ramsey's RESET test was employed. For this, the squared term of the independent variable RAD is included in a regression equation. The coefficient on RAD2 was tested under the null hypothesis that it equal zero; indicating no variable omission. The results are shown in table 3.

TABLE 3 RESET Test

            FLORIDA        OHIO

Intercept   5.9355959(*)   4.5689634(*)
            1.5536199      1.8873949
            0.0002054      0.0167503
RAD         0.1043272(*)   0.3566157
            0.335556       0.3891799
            0.0085723      0.1005035
WIND        0.8897088(*)   0.7563976(*)
            0.2303986      0.4081396
            0.0001768      0.0659173
RAD2        0.0456812(*)   0.0346401(*)
            0.0180891      0.020367
            0.01276        0.0911702
n             134             146

Note: parameter estimate/standard error (White corrected)/prob value (Beta = 1 for RAD; otherwise Beta = 0)

(*) significant at 10%

These results indicate that we may wish to temper the interpretation of the results generated from dropping one of the input price variables, as it is possible this omission may be affecting our results.

As one last inquiry in this model, behavioral variables were added to the structural model, as has been done in other studies. While there is no theoretical justification for these variables, as stated earlier, they may prove interesting in the study of hospital costs. Variables added were: PD, the population density in the hospital's county; MM and MF, the mean male and female age in the country, respectively (these were added separately, because tests indicated that they had a high correlation); and IPC, the income per capita for the county. These characteristics seemed intuitively to impact on hospital costs. Data on the mean population age for each gender, population density, and income per capita, all at the county level, were obtained from the 1991 Survey of Buying Power Demographics.

Table 4 shows the impact of including these variables in the model. It is interesting to observe that of the previously omitted variables, population density was significantly different from zero in Ohio, and in Florida (when mean female age was included) at the ten percent level; and both mean age levels were significant in Florida, while only the mean male age was significantly different from zero in Ohio. A joint test was used to see if all the "behavioral" variables were jointly significantly different from zero, which they were. These results make us cautious in accepting the results of our structural cost function. It is unclear if the apparent omissions are the result of dropping one of the factor input prices, or if perhaps the behavioral model is more appropriate in the case of hospitals.

TABLE 4 Behavioral Model

                       FLORIDA                            OHIO
                          MF                MM             MF

Intercept              2.5617847(*)     2.6011486(*)    1.7055957(*)
                       0.2746237        0.282711        0.335502
                       4.16E-16         8.60E-16        9.69E-07

PD                     0.0000399(*)     0.0000365       0.0001249(*)
                       0.0000221        0.000022        0.0000474
                       0.073913         0.105147        0.0091981

MF or MM              -0.008828(*)     -0.010347(*)      0.010708
                       0.0028334        0.0034067        6.99E-03
                       0.0022665        0.0028935        0.1272668

IPC                    8.18E-06         7.84E-06         1.63E-06
                       9.83E-06         9.88E-06         8.61E-06
                       0.4072006        0.4287493        0.8499337

RAD                    0.9153911(*)     0.9146599(*)     0.9230989(*)
                       0.0219996        0.0220698        0.0433366
                       0.0001884        0.0001746        0.0777693

WIND                   0.6308486(*)     0.638669(*)      0.0061755
                       0.2691017        0.2699049        0.194938
                       0.0206035        0.0194687        0.9747648

Joint "Behavioral"   223.56684        298.26376       1117.7957
prob value             0.000000001      0.000000001      0.000000001
n                         134                               146

                        MM

Intercept               1.6663824(*)
                        0.3284023
                        1.01E-06

PD                      0.0001295(*)
                        0.0000483
                        0.0080768

MF or MM                0.0136896(*)
                        0.0076664
                        0.0759366

IPC                     3.03E-07
                        8.17E-06
                        0.9704659

RAD                     0.9213975(*)
                        0.0436046
                        0.0732194

WIND                    0.0048503
                        0.1946754
                        0.9801521

Joint "Behavioral"   1388.7004
prob value              0.000000001
n

Note: parameter estimate/standard error (White corrected)/prob value (Beta = 1 for RAD; otherwise Beta = 0)

(*) significant at 10%

There is no apparent effect on economies of scale in the Florida model, where admissions continue to be different from one, and the wage index is significant and positive. In Ohio, however, admissions are now significantly different from one, a decided change from the earlier model where they had never come out as statistically different from one. Also, the wage index coefficient estimate is much lower than previously, and no longer significantly different from zero.

6.0 Conclusions

These results indicate that there are economies of scale in the hospital industry in the state of Florida, where the coefficient on RAD was consistently significantly different from one; on the other hand in Ohio there was no evidence from the theoretical model that the coefficient on RAD was different from one. However, when including extra variables to account for any omissions, the coefficient on RAD was significantly different from one at the ten percent level in Ohio.

These results are somewhat different from earlier studies. Conrad and Strauss (1983), Eakin and Kniesner (1988), and Vita (1990) all use some form of the multiple output translog cost function and find no evidence of economies of scale. On the other hand, Fournier and Mitchell (1992) also use a multiple output translog cost function and do find evidence of economies of scale. Vitaliano (1987) estimates both a quadratic and a logarithmic behavioral single output function, and finds definite economies of scale.

Our use of the "weighted" output, where admissions are multiplied by a case mix index would seem to account for the multiple output nature of hospitals, while allowing us to eschew the use of a multiple output functional form. The results of the RESET test and the subsequent estimation of the behavioral model suggest that there is some problem with the use of only one of the input price indices. A better measure of hospital input prices would therefore be advantageous, as it would allow for the full estimation of the theoretical model without the collinear problems affecting this model.

Based on our findings, it would seem that there may be economies of scale to be exploited by hospitals in Ohio and Florida. Again, because of the problems with multicollinearity, followed by the possibility of omitted variables, it is wise to look upon these results with some circumspection.

Notes

(1.) Adams and Brock (1986).

(2.) Horoschak (1992).

(3.) This case mix index is calculated by the Health Care Financing Administration (HCFA) to differentiate between hospitals in terms of case mix severity. This figure is used by the federal government to account for differences in hospital case mix for purposes of Medicare reimbursement.

(4.) Again, as calculated by HCFA.

(5.) As calculated by HCFA.

(6.) As defined in the AHA Guide, p. A8.

(7.) As defined in the AHA Guide, p A5.

(8.) When the model was run with both labor and cost of capital (WIND and K), the sign on the wage index was negative, contradicting economic theory. This suggested the presence of multicollinearity in the model. Running pairwise regressions and calculating a condition number confirmed this. The model was then run once including WIND only and once including K only. The results were indifferent to which input price variable was dropped, both models had the appropriate sign on the input price. Only the model including WIND is presented here. The fact that both WIND and K are indices which measure input price of a hospital within a given MSA would probably explain their collinearity.

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F. Patrick Hallagan, Adapted from an Exit Paper, completed May 1995, for the Master of Arts in Economics Degree from Miami University, Oxford, Ohio 45056. I am thankful for the assistance and advice of Professors Antonio Ligeralde James Brock, and Thomas Hall.