Economies of scale and scope in mental health care

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Economies of scale and scope in mental health care

J.A. Wilschut*

Delft University of Technology, The Netherlands, j.a.wilschut@tudelft.nl

B.L. van Hulst

Delft University of Technology, The Netherlands, b.l.vanhulst@tudelft.nl

J.L.T Blank

Delft University of Technology, The Netherlands, j.l.t.blank@tudelft.nl

Abstract

Economies of scale and scope are usually derived under the assumption that the set of production possibilities are shared by all firms in an industry irrespective of whether they specialize in a single output or not. Mental health care institutions in the Netherlands vary substantially in the scale and the number of outputs. Estimation of one cost function therefore seems very restrictive and requires the allowance of zero-values. We used a translog cost function model with dummy-variables for different types of institutions, to allow for different technologies. We found evidence for differences in technologies between institutions specialized in counseling and integrated institutions that also performed other activities, expressed by the number of days in the hospital or permanent care, number of treatments in daycare or number of day activities. The marginal costs of counseling were lower for the integrated institutions than for the specialized institutions.

Keywords: economies of scale, economies of scope, stochastic frontier analysis, mental health care

Introduction

Governments seek tools to control the growth of healthcare spending. Producing at the optimal scale and full use of economies of scope can help reducing the healthcare spending. However it is important that the economies of scale and scope are correctly derived. In this study, we will determine economies of scale and scope of mental health care institutions in the Netherlands. The mental healthcare in the Netherlands is particularly interesting because of the increasing costs in mental health care due to an increasing number of patients. Furthermore, the size of the institutions and the combinations of activities offered varies widely between mental healthcare providers. If economies of

*

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scale and scope exist potential cost savings can be realized by restructuring the sector.

Economies of scale and scope are generally derived under the assumption that all firms in a certain industry operate under the same production possibilities, irrespective of whether they specialize in a single output or not (Baumol et al., 1988). In this paper we test that the assumption of the same technology for all for the mental health in the Netherlands. Mental health care institutions in the Netherlands vary widely in scale and in the number and kind of services they offer. Basically there are two types of firms; integrated firms that offer a wide range of care and ambulant firms that offer only ambulant care. Given the huge differences between institutions the possibility of a different cost function for the different types of institutions seems very appropriate. We therefore apply a cost model to the Dutch mental health care institutions that allows for such differences, in order to estimate economies of scale and scope as tools to increase the productivity level. Allowing for differences between the technologies of the two types of institutions seems a more realistic approach. Determination of economies of scale and scope starts with the estimation of a cost function. The assumption that all firms operate under the same production possibilities implies the estimation of one cost function for all firms. Economies of scope therefore only depend on differences in cost levels and do not account for differences in cost functions between specialized and integrated firms. Moreover, estimation of the frequently used translog function, introduced by (Christensen et al., 1973), requires the handling of substantial amount of zeros for the specialized institutions in the outputs. It has been suggested to estimate separate translog cost functions (Weninger, 2003). Here, we estimate a cost function where we allow for different parameters for the specialized and integrated institutions.

Methods Model

Institutions vary widely with respect to scale and type of treatment they offer. A substantial part of the institutions only perform ambulant care (counseling), and are likely to vary substantially from integrated institutions that not only offer counseling but also offer residence to their patients for example. We therefore estimated a cost function that allows for different technologies between different types of institutions by including dummy variables. We divided the institutions into two groups, the first group consisted of institutions that had counseling as the only output, and the second group was all other institutions.

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Under the assumption of a translog form (Christensen et al., 1973), we estimated the following cost function:

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= = = = = = = = = = = = = = = = T aa W Y ae W W ac Y Y ab W ac Y ab aa D T a W Y e W W c Y Y b W c Y b a D C n i n j m i n j j i ij j i ij m i m j j i ij n i i i m i i i amb n i n j m i n j j i ij j i ij m i m j j i ij n i i i m i i i g 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 int ln ln ln ln 2 1 ln ln 2 1 ln ln ln ln ln ln 2 1 ln ln 2 1 ln ln ln (1) With: C = Total costs;

Damb = Dummy variable for institutions that performed only counseling

Dintg = Dummy variable for integrated institutions

Yi = output i (i = 1,.., m); Wi = Price of input i (i = 1,.., n); T = year; , , , , , , , , , , , , _{i i ij ij ij o i i ij ij ij} o b c b c e aa ab ac ab ac ae a parameters to be estimated.

With Shephard’s lemma we obtain share equations :

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= = = = (2) With :

Sj = Cost share equation of input j (j = 1,.., n)

The following restrictions were imposed on the parameters to impose linear homogeneity in input prices and symmetry:

ji ij ji ij ji ij ji ij b c c ab ab ac ac b = ; = ; = ; =

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We first estimated the cost function under the assumption of the same technology for different types of institutions by assuming the same parameters for the ambulant institutions as for the integrated institutions. For the specialized institutions we multiplied the parameters of the zero-outputs with a dummy variable to make sure they were not estimated for that observation. The obvious restrictions that we impose in case of the assumption of same technology: ij ij ij ij ij ij i i i i o o a ab b ac c ab b ac c ae e aa = , = , = , = , = , =

Next, we estimated the model under the assumption of different technologies. We tested the hypothesis of a same technology using a loglikelihood ratio test.

The models were estimated with maximum likelihood. Moreover, we used a thick frontier approach with the first estimation over all observations and the second estimation over the 50% most efficient observations (according to the first estimation). We also tested the required monotonicity and concavity of the model.

The estimated cost functions were used to derive economies of scale and economies of scope. The economies of scale were represented by the cost flexibility which described the increase in cost relative to the increase in output (Baumol et al., 1988), depending on the type of institution. The cost flexibility is described by the following formula:

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Economies of scope were deducted from the marginal cost of the overlapping outputs between specialized and integrated institutions. We determined the marginal costs at varying levels of output to account for differences in scale between institutions. The marginal costs were derived as follows:

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= = + + = ¶ ¶ = ¶ ¶ = Data

The mental health care institutions in the Netherlands report to the Ministry of Health in the Netherlands. The yearly data collected for this purpose were used in this analysis over the years 2008-2010. We selected the institutions that dealt with mental health care only, so departments of psychiatry as part of general hospitals for example were not included. We selected those institutions that had valid and plausible values for all variables. In total 201 observations (institutions per year) remained (59 in 2008, 73 in 2009 and 69 in 2010).

We included four measures of treatment as output variables: the number of counsels, the number of days in residence, the number of part-time treatments and the number of day activities (Table 1). Of these institutions, 32 were specialized in the sense that they only performed counseling. The other institutions did counseling and at least one of the other treatment activities. None of the institutions was specialized in one of the other activities. We therefore used two groups to which we refer to as specialized and integrated institutions. We used two inputs, personnel and material and capital. The latter two were added and used as one variable. Costs of the inputs were available and we used the number of full time jobs to calculate the price of personnel. The prices of materiel and capital were based on an index constructed from the Consumer Price Index and a Price Index for investments of fixed activa of the government (Statistics Netherlands, www.cbs.nl).

Results and discussions

We estimated the cost function under the assumption of a common technology for both specialized and integrated institutions and under the assumption of different technologies (Table2). The hypothesis of the same technology for both groups was rejected by the log likelihood ratio test (p< 0.001).

The parameters of the models are very different. Moreover, under the assumption of a common technology the signs of two of the outputs become negative. The parameter of output 1 (counseling) had a value that was close to

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the value of the specialized institutions under the assumption of different technologies, but then overestimated the total costs of the integrated institutions. Therefore, the parameters of the other outputs were given values that decrease the total costs of the integrated institutions. Under the assumption of different technologies, the costs of the specialized institutions were slightly more influenced by costs of materiel and capital than by personnel, compared to the integrated institutions. No significant change was found over time. The model fulfilled the criteria of monotonicity and concavity.

Since the model with the assumption of the same technology for both types of institutions does not result in plausible estimates, we report economies of scale and scope only for the model with different technologies. The integrated institutions operate under diseconomies of scale on average (cost flexibility 1.05). This particularly applies to the relatively small integrated institutions (Table 3). The larger institutions operate under increasing economies of scale. The specialized institutions operate under economies of scale on average (0.72). However, the cost flexibility rapidly increases for these institutions and the larger ones operate under diseconomies of scale. All the specialized institutions are smaller than 0.5 of the average size institution.

The economies of scope follow from the marginal costs of counseling in a specialized or integrated institution with the same scale (Table 4). The marginal costs of counseling are higher for the specialized institutions than for the integrated institutions, except for the very small institutions. Moreover, costs increase with scale for the specialized institutions, and decrease with scale for the integrated institutions. The marginal costs of all other products also decrease with scale, reflecting the economies of scale.

Conclusions

Specialized and integrated mental health care institutions in the Netherlands vary in the way they operate, as shown by the different technology assumption. We also found indications that increasing the scale of the institutions and that integrating counseling in institutions with other types of treatment could increase the productivity of the sector.

The assumption of the same technology did not give plausible estimates because of the negative parameter values of two of the outputs. Instead of using dummy variables, we also tried to replace the zero-values with the minima of the non-zero values and estimate the model under the assumption of the same technology for all institutions. The estimates were very similar then to the estimates of the model that allows for different technologies, as derived for the integrated institutions alone. This approach also led to the rejection of the hypothesis of equal technologies.

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The technology can differ between institutions for several reasons. These differences can be caused by the kind and severity of the disorders that are treated, and for example by the average age of the patients. We had data on the types of treatments as expressed by the outputs we used, but did not have any information about the patients or the disorders that were treated. A Norwegian study showed a substantial impact of case mix on productivity growth estimates but did not differentiate between types of institutions (Halsteinli et al., 2010). Given the data, we could only differentiate between institutions that had counseling as a single output, and all other institutions. No further differentiation was possible for example of institutions that only performed one of the other outputs.

We used the marginal costs as a proxy for the economies of scope. Due to the lack of observations we could not estimate a separate cost function without counseling. Moreover, the costs cannot be estimated by a translog function in case of a zero output. Direct estimation of the economies of scope was therefore not possible. We compared the marginal costs of specialized and integrated institutions of the same size. The differences in costs could be overestimated because the size of the integrated institution that relates to counseling is smaller and the marginal costs of counseling increase with a decreasing scale for these institutions.

We used 201 observations of mental health care institutions over a 3 year period to compare the costs of the institutions relative to their production. Unfortunately, we had to exclude another 346 observations because of missing or implausible values. A substantial part of the excluded institutions had high costs relative to the number of outputs. Most of the excluded institutions perform other activities which are not counted in the outputs we used, like for example reintegration activities or parental support. We did not have usable data to systematically exclude institutions for this reason. It is therefore likely that some of the institutions with other activities are included in the analysis. Because of the high costs relative to the output, this would underestimate the cost efficiency. Due to a lack of data of the costs of mental health care activities alone, we were not able to include 137 health care institutions with more than just mental health care, like for example the general hospitals. Summarizing, this analysis covers a substantial but selective part of the mental health care sector.

Measuring the output of health care institutions is subject of debate (Hollingsworth & Street, 2006). In this study we only used the number of treatments and not the number of patients or the effectiveness of the treatment. We used quality of care measurements from interviews to evaluate the relation between the efficiency score of the institutions and quality. We found a negative

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relation with patient perception of the effectiveness of the treatment and a non-significant correlation with all other quality measures (results not shown).

The plausibility of the same technology assumption between firms or institutions has been largely unexplored. The couple of studies we found came to the same conclusion that different cost functions are required for specialized and diversified firms. Differences in cost functions were for example found between general and specialty hospitals in Vietnam (Weaver & Deolalikar, 2004), between firms that provided both freight and passenger railway services in the US and firms that offered primarily freight services (Weninger, 2003) and between the water and sewerage firms in England and the water only companies (Bottasso et al., 2011).

In conclusion, specialized and integrated mental health care institutions in the Netherlands operate under different cost functions. The issue of different technologies deserves more attention. Drawing the wrong conclusions from the possibly incorrect same technology assumption can have major policy implications on scale and scope.

References

[1] Baumol, J., Panzar, J.C., & Willig, R.D. (1988). Contestable markets and the theory of industry structure Sydney: Marcourt Brace Jovanovich.

[2] Bottasso, A., Conti, M., Piacenz, M., & Vannoni, D. (2011). The appropriateness of the poolability assumption for multiproduct technologies: Evidence from the English water and sewerage utilities. International Journal of Production Economics, 130(1), 112-117.

[3] Christensen, L.R., Jorgenson, D.W., & Lau, L.J. (1973). Transcendental Logarithmic Production Frontiers. The Review of Economics and Statistics, 55(1), 28-45.

[4] Halsteinli, V., Kittelsen, S.A., & Magnussen, J. (2010). Productivity growth in outpatient child and adolescent mental health services: the impact of case-mix adjustment. Soc Sci Med, 70(3), 439-446. [5] Hollingsworth, B., & Street, A. (2006). The market for efficiency analysis of health care organisations.

Health Econ, 15(10), 1055-1059.

[6] Weaver, M., & Deolalikar, A. (2004). Economies of scale and scope in Vietnamese hospitals. Soc Sci Med, 59(1), 199-208.

[7] Weninger, Q. (2003). Estimating multiproduct costs when some outputs are not produced. Empirical Economics, 28(4), 753-765.

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TABLE 2 SUMMARY DATA OF INCLUDED INSTITUTIONS

Average stddev min max

Output (x 1.000)

Counseling (contacts) 137.7 170.8 1.1 888.8

Residence (day) 112.8 125.9 0.0 503.5

Part-time treatment 11.8 16.3 0.0 71.2

Day-activity 22.5 46.3 0.0 258.9

Costs (x 1 million euro)

Personnel 34.1 37.6 0.2 153.9

Kapital 2.1 2.8 0.0 14.9

Material 12.3 12.8 0.1 54.2

Total 48.5 52.7 0.6 215.7

Total # FTE 609.6 646.7 10.8 2602.9

104 TABLE 3 PARAMETER ESTIMATES

Shared technology Different technologies

Integrated institutions Specialized institutions Variable Paramete r Estima te Er ro r t-statisti c Estim ate Err or t-statisti c Estim ate Err or t-statisti c Constant A0 -0.59 0.0 9 -6.88 0.30 0.05 6.03 -1.47 0.16 -9.11 Counselin g (y1) B1 1.35 0.0 4 31.17 0.25 0.06 4.31 1.40 0.06 22.30 Day in residence (y2) B2 0.22 0.1 0 2.18 0.34 0.05 7.20 Part-time treatment (y3) B3 -0.28 0.0 9 -3.27 0.30 0.06 5.38 Dayactivit y (y4) B4 -0.17 0.0 5 -3.84 0.01 0.02 0.63 Personnel (w1) C1 0.71 0.0 1 65.71 0.70 0.01 61.28 0.55 0.04 15.38 Materiel & capital (w2) C2 0.29 0.0 1 27.39 0.30 0.01 26.10 0.45 0.04 12.46 Time A1 -0.03 0.0 3 -0.95 -0.02 0.02 -1.20 0.05 0.05 1.02 y1 x y1 B11 0.06 0.0 5 1.38 0.26 0.05 5.41 0.35 0.03 12.03 y1 x y2 B12 -0.34 0.0 7 -5.17 -0.15 0.03 -4.88 y1 x y3 B13 0.21 0.0 4 5.85 -0.17 0.04 -4.46 y1 x y4 B14 0.19 0.0 3 6.82 0.00 0.01 0.23 y2 x y2 B22 0.16 0.0 7 2.35 0.05 0.04 1.35 y2 x y3 B23 0.05 0.0 1.10 0.03 0.02 1.45

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Shared technology Different technologies

4 y2 x y4 B24 -0.03 0.0 1 -2.00 -0.01 0.01 -1.19 y3 x y3 B33 0.04 0.0 7 0.48 0.17 0.04 4.07 y3 x y4 B34 -0.15 0.0 3 -5.89 -0.02 0.01 -1.82 y4 x y4 B44 -0.01 0.0 1 -0.66 0.00 0.01 0.63 w1 x w1 C11 0.13 0.0 5 2.55 0.12 0.07 1.63 0.08 0.08 1.01 w2 x w2 C22 0.13 0.0 5 2.55 0.12 0.07 1.63 0.08 0.08 1.01 w1 x w2 C12 -0.13 0.0 5 -2.55 -0.12 0.07 -1.63 -0.08 0.08 -1.01 y1 x w1 E11 -0.01 0.0 1 -1.41 0.01 0.01 0.82 -0.08 0.02 -5.00 y2 x w1 E21 -0.02 0.0 1 -1.85 -0.01 0.01 -1.46 y3 x w1 E31 0.03 0.0 1 3.46 0.02 0.01 2.82 y4 x w1 E41 0.00 0.0 0 -0.73 -0.01 0.00 -1.84 y1 x w2 E12 0.01 0.0 1 1.41 -0.01 0.01 -0.82 0.08 0.02 5.00 y2 x w2 E22 0.02 0.0 1 1.85 0.01 0.01 1.46 y3 x w2 E32 -0.03 0.0 1 -3.46 -0.02 0.01 -2.82 y4 x w2 E42 0.00 0.0 0 0.73 0.01 0.00 1.84 R2 0.98 0.99 Loglikelih ood 123 206

106 TABLE 4 ECONOMIES OF SCALE

Total costs* Specialized Integrated

0.2 1.32 1.14

0.5 1.54 1.04

1.0 0.94

2.0 0.83

*Standardized costs: the costs of an average size institution are equal to 1

TABLE 5 MARGINAL COSTS BY TYPE AND SCALE OF THE INSTITUTION

Total Costs * Specialized Integrated

Counsel Counsel Days in residence Part-time treatment Dayactivities

0.2 113 142 183 779 81

0.5 172 134 172 854 60

1.0 117 149 873 32

2.0 92 114 827 3