Sot. Sci. Med. Vol. 24. No. IO. pp. 843-850. 1987 Printed in Great Britain. All rights resened

0277-9536!87 53.00 + 0.00 Copyright C 1987 Pergamon Journals Ltd

TECHNICAL PRODUCTION

AND ALLOCATIVE EFFICIENCY IN OF OUTPATIENT MENTAL HEALTH

CLINIC SERVICES

RICHARD G. FRANK’ and CARL A. TAUBE?

‘Health Services Research and Development Center, Johns Hopkins University, School of Hygiene & Public Health, 624 N. Broadway. Baltimore. MD 21205 and ‘Division of Biometry and Epidemiolon,

National Institute of Mental Health. Rockville, MD 20857, U.S.A.

Abstract-This paper presents an analysis of production of ambulatory mental health services in free standing outpatient clinics. The study empirically addresses several issues including: the nature of returns to scale, the impact of differing organizational forms on the volume of service produced and the efficiency of staffing patterns used by psychiatric clinics. An appraisal of two popular production functions is offered based on predictive performance. The results suggest the existence of decreasing returns to scale; input hiring decisions that depart from cost minimization; and the potential important of a decentralized clinic organization for expansion of access to mental health services.

Key words-production function, allocative efficiency, staffing

IFiTRODU<JTION

Ambulatory care in mental health is’provided in three major settings: office-based practice, hospital out- patient departments, and free-standing outpatient clinics. Clinics have a long history of development going back to the early 1900s. Child guidance clinics were initiated and grew after World War I. The National Mental Health Act of 1946 provided for- mula grants-in-aid to States to develop community- based psychiatric services. Many States passed com- munity mental health acts which provided State financing for local government mental health boards to operate outpatient clinics. In addition, States developed State hospital aftercare and outpatient services to complement community care [I]. In the 6Os, Federal support of Community Mental Health Centers (CMHCs) provided a further impetus to the development of organized ambulatory health facili- ties outside of hospitals. It is these facilities that are the object of the study reported here.

Since the initiation of the CMHC program in 1963, important changes have occurred in the composition of organized outpatient mental health settings. Be- tween 1971 and 1977, the number of general hospital outpatient psychiatric services dropped. The number of admissions to these services dropped 23% for public general hospitals and 14% for nonpublic. State mental hospitals and free standing clinics also experienced sharp declines in admission during the 1970s. Nonprofit and proprietary clinics on the other hand continued growing albeit at a much slower rate than previously [2]. In 1980, free-standing outpatient psychiatric clinics numbered 1053, accounting for 43% of the outpatient services and 31% of the admissions to such services. Federally funded CMHCs account for an additional 691 outpatient services, 29% of the total of such settings, and the remainder were outpatient services of genera1 hospi- tals or psychiatric hospitals. Such organized settings in 1980 had 2.6 million admissions [3]. An estimated

4.6 million persons had one or more visits in the office-based practice of psychiatrists or psychologists 141.

As markets for ambulatory mental health care have emerged questions have arisen as to the appro- priate market structure for the market. Specifically, whether mental health services should be region- alized? Advocates of regionalization suggest that the technology for delivering ambulatory mental health care is not consistent with a competitive market structure.

In this paper we attempt to shed some light on the factors related to the provision of mental health care in free standing outpatient clinics. We do so by estimating production functions for free standing outpatient mental health clinics. In addition, we evaluate the performance of two popular production functions in describing the technologies present in outpatient mental health clinics. The paper is organ- ized as follows: the second section takes up some theoretical issues needed for interpretation of pro- duction function estimates. The third section describes the empirical implementation of the two theoretical production functions. The fourth section presents results and the paper ends with a discussion of the key findings.

THEORETICAL CONSIDERATIONS

The point of departure for the empirical work presented below is the economic theory of the firm. The free standing outpatient mental health clinic (here after ‘free standing clinic’) is viewed as a firm: which combines inputs, such as labor and capital, in order to provide mental health treatments measured by group and individual visits. The production func- tion for the firm summarizes the technical re- lationship between inputs and outputs (or services in this case). This means that it represents the trans- formation of inputs into services. In a simple world of a firm that produces a single output (Q) by

843

844 Rmwm G. FRANK and Cti~ A. TACBE

combining two variable inputs (X,, X2) with one or more fixed inputs we can represent the production function in the following manner:

Q =f(X,, ‘5) (1)

Q is assumed to be continuous with nonzero first and second order partial derivatives.

In this study we are concerned with the structure of production in free standing clinics. We do not assume that all types of producers use the same technology. We are interested in variation in output according to organizational form and clinic own- ership holding constant input levels. We study the issue by estimating the effect on output of clinic organization. This is operationalized by estimating shifts in the production function by organization and ownership. We also test for interaction effects of ownership and organization with input levels. As we shall show, strong inferences concerning ownership are difficult to draw in the mental health sector.

A second central concern is with staffing patterns. Economic theory makes very specific predictions with respect to input hiring decisions for cost minimizing firms. That is, if firms are input price takers (supply curves are perfectly elastic) the following condition will hold:

-=~,,w1=fi w, w, f, f2 w,f: (2)

where W, is the price of the ith input, and f; is the marginal product of the ith input.

This means that a firm’s actual hiring decision can be assessed relative to the cost minimizing conditions by estimating departure from the equalities [5] in equation (2). We will make use of these conditions to evaluate the cost minimizing behavior of free standing outpatient clinics.

The empirical approach to evaluation of condition (2) among other hypotheses will be to estimate pro- duction functions. One reason to examine production functions rather than cost functions is that cost function analysis imposes behavioral restrictions such as cost minimization on the model. Since this is a proposition that is testable, we estimate production functions. A second assumption necessary for use of a production function is that demand is exogenous to the individual clinic. Because of the strong role of third party payors, this assumption is reasonable. We use two functional forms to estimate the production function: the Cobb-Douglas and transcendental models.

The Cobb-Douglas model is a simple model that is rather easily estimated but which imposes im- portant restrictions on the form of the observed technology. The Cobb-Douglas model takes the form:

lnQ=lnA-t~InX,+BlnX~. (3)

This model implies that the ratio of the marginal products is:

fi zx2 -=-- f? BX,

which equals 2 (4) 2

for cost minimizing firms. This condition implies that the cost minimizing input levels do not depend on the level of output [6]. In addition, the Cobb-Douglas

model imposes strong restrictions on the degree of substitutability between inputs. One advantage of the model is that returns to scale are easily calculated. Returns to scale are defined as the percentage change in output resulting from, say, a 1% increase in all inputs. The measure of the returns to scale offer insight into the cost structure of the firm. In the case of the Cobb-Douglas model the returns to scale are reflected by calculating the elasticity of scale which is

of the following form:

z + B = elasticity of scale (ES).

We will also estimate a more fle.xible functional form which takes the following form.

InQ=A+r InX,+B,XL+BzX’. (5)

This model is known as the transcendental pro- duction function [7, 81. The ratio of the marginal products in this case is given by:

fi 3 = ZB, B;XJ’

(6)

Note that the cost minimizing input combination depends on output and the absolute Iev-els of inputs. In addition, it is possible to produce positive output with only one input (X,). The transcendental model is therefore a more general model than the Cobb-Douglas.

The calculation of the elasticity of scale with the transcendental model is considerably more involved than in the case of the Cobb--Douglas model.

ES=~----= 7 2 In X,

r + (B, + 28, X:)X,. (7)

If ES = I returns to scale are constant; if ES < 1 returns to scale are declining and if ES < 1 returns to scale are increasing. If more inputs are added to the production function the expression in (7) becomes substantially more complicated.

The basic production function models given by equations (3) and (5) are transformed into hybrid production functions so as to test hypotheses on the effect of differing ownership and organizational forms on production. Of particular interest are differences in output levels by type of organizational form. Specifically, we consider whether clinics with satellite facilities are more productive than clinics that operate using a single central facility. Multi- facility clinics may produce higher levels of output for a given level of inputs because of an ability to specialize and centralize management functions. This might be reflected in lower manager to clinician ratios. We empirically test the proposition that clinics with satellite facilities provide more care holding constant all labor and capital inputs.

Economic theory suggests that private ownership in general and the profit motive in particular should lead to greater efficiency than found in publicly owned enterprises. This might be reflected by cost minimizing input mixes, lower levels of organiza- tional slack and X-inefficiency [9]. Nonprofit enter- prises, are often thought to generate various inefficiencies. However, they are not insulated from the market place by public budgets as are state and county owned clinics. Thus, holding all inputs con- stant, privately owned clinics are expected to produce

Production of outpatient mental health services 843

Table I

Variable Definition

Linear form MD Physician hours includes both

psychiatrists and others NURSES Nurse hours includes LPNs

and RNs OTH Hours of combination of other

treatment staff such as counselors

ADM Administrative personnel hours

CAP Capital expenses such as depreciation measured in dollars

Log form

MD Log of MD hours

130

(157)

(I::) 333

(505) 410

(492) 120.57 I

(3387.00)

4.22 il.361

NURSES

0-l-H

ADM

SWPM

Log of nurse hours

Log of other hours

Log of administrative hours

Log of social worker psychologists

4.34 (2.19) 5.40

(2.35) 5.33

(1.47) 5.43

(1.28)

higher levels of output than do public clinics. We examine this aspect of technical efficiency related to ownership but as will be seen below inferences are complicated by measurement issues.

The performance of the two specifications of the production functions outlined above are also evalu- ated. This is accomplished by measuring the fore- casting performance of the two models using a ran- dom sample of clinics excluded from the sample used for parameter estimation. One systematic measure of forecast accuracy is Theil’s inequality coefficient

u = + /‘PI - A,)% \I A:in

where P, is the model’s predicted output value for the ith clinic and Ai is the actual output value for the ith clinic and n is the number of observed clinics. Clearly if P = A then U = 0 and the model gives perfect forecasts. As U increases in value the forecast deteriorates. We also report the simple correlation between P and A [IO].

E,MPIRICAL IMPLEMENTATION

The data used in this analysis is based on a survey of all known freestanding outpatient psychiatric clin- ics operating as of January 1982. The survey was conducted by the National Institute of Mental Health in conjunction with State Mental Health Authorities. Of the 1473 free-standing mental health clinics, 1107 responded to the survey.* Of these 1107,755 reported all the data required for the Cobb-Douglas and transcendental models.

Table 1 presents definitions of key inputs. All inputs, except for capital, are measured in hours. Capital is measured in dollars since medial equipment and floor space are not measured directly. The MD

*The abrupt increase between 1980 and 1982 is due to a The visit measure we use does appear to be reclassification of federally funded CMHCs to their homogeneous with respect to duration. We examined original facility designation. data on visits to psychiatrists and found duration of

variable included all physicians delivering serv+ces in a clinic regardless of specialty. The nursing variable NURSE include both LPN and RN level nurses. The SWPM include the sum of masters and doctoral level social workers and psychologists. The OTH included all clinical staff not listed above. These include a variety of workers such as counselors, alcohol treat- ment specialist, rehabilitation workers and so on. A copy of the survey instrument used is available from the Division of Biometry and Applied Sciences of the NIMH.

Spectjication

As is common in most studies of organized care settings, we do not measure the final output of the free standing outpatient clinic; mental health, instead we observe intermediate outputs. Two intermediate outputs are considered: individual clinic visits and group visits. Since previous work on state mental hospitals has not supported the presence of econ- omies of scope we transform all visits into individual visit equivalents using the relative prices of group versus individual visits [ 1 I].

While it would be ideal to have an all inclusive measure of the mental health improvements of patients treated in free standing clinics few such measures exist and none are routinely collected for a national sample of clinics. Focusing on visits can, however, provide insights on factors that determine access to ambulatory mental health care. Concerns over access to mental health care are increasingly being raised in public policy debates over reductions in public funding of mental health care [ 121. For this reason analyses using visits as an ‘intermediate out- put’ may generate results of interest to the public policy arena.

The theoretically appropriate measure of visits would take into account the duration of the visit and the diagnosis and severity of the illness for which the visit was made. The visit measure used here falls short of that ideal. Our casemix measure depends on information on whether particular groups of patients are disproportinately represented among the clinic population. These groups are specified as indepen- dent dummy variables equal to one if they are reported as ‘primary problem’ groups, and zero otherwise. There are five groups identified. Alcoholics (ALCOHOL), drug abusers (DRUG), the mentally retarded (MR), children (CHILD) and the adult mentally ill as the reference group. Results from analyses of inpatient mental health care suggest that case mix adjustment made on the basis of diagnosis does little to homogenize hospital use [13, 141. Sever- ity indices in psychiatry are in their infancy. Work by Horn [15] has pointed to some potential for devel- oping severity measures in mental health. However, one would need an enormous amount of information on the clinical circumstances of individual patients to construct such measures. Moreover, Horn’s work has been criticized as tautological because utilization measures are used in the severity index which is then used to explain utilization.

846 RICHARD G. FRASK and CARL A. TAcBf

visits to be similar across practice settings and over time. The basic visit is a 15 min ‘hour’ of psycho- therapy. Variation by diagnosis ranged from 30 min for organic brain syndrome to 46 min for neuroses. Clinic visits had similar patterns to office visits [16, 171. iMore detailed information on individual clinics was not avaiiable.

In addition, to the casemix variables in the model the ownership and regional dummy variables are likely to capture some variation in severity of illness. Recent work by Manderscheid er al. [18] shows that there appears to be systematic sorting of patients across types of mental health facilities (ownership). Thus, one might also expect ambulatory patients to be sorted by clinic ownership. Three dichotomous dummy variables describe ownership status: state ownership (ST.4TE), county ownership (COUNTY) and private nonprofit and for-profit comprise the reference category. The possibility of the ownership dummies capturing severity effects clouds our ability to draw inferences on the impact of ownership on output levels.

Finally, we specify dummy variables for the four census regions of the county where west is the reference category (Rl-R3).

The mismeasurement of casemix and product mix has been analyzed by Feldstein [19]. In comparing hospital analyses that excluded casemix measures with those that incorporated such measures Feldstein found that exclusion of casemix measures resulted in underestimates of returns to scale. The underestimates were, however, small in magnitude. Thus, we would expect some bias in our coefficient estimates due to measurement error in our casemix variables. Yet the bias could reasonably be expected to be small. More- over, if the bias were substantial that should be reflected in our evaluation of the models forecast performances.

For both the Cobb-Douglas and transcendental production models we estimate specifications with either five or six input variables. The core inputs consist of the number of physicians (MD), the num- ber of social workers and psychologists (SWPM), the number of nurses (either RN or LPN; NURSE), other patient care staff (OTH) and administrative staff (ADM). These variables can be found in linear, logarithmic and quadratic form depending on which model is being estimated. Capital (CAP) is included in one specification of each model and excluded from one. Medical equipment, and floor space are not measured directly. As a proxy we measure a variable that includes depreciation and indirect expenses. Since this is a rather crude proxy for physical units of capital we experiment with use of the variable. We view our choice as being one between omitted variable bias and bias due to measurement error.

Both the Cobb-Douglas and transcendental models are estimated as single equation models using ordinary least squares. This has been a point of some concern in the econometrics literature due to the joint determination of inputs and outputs. Reinhardt [I, among others, has argued that in practice the bias is small if factor and product prices vary substantially across the sample observations [20]. Examination of visit fees and nurses’ wages across states suggests substantial variation in both sets of prices. Thus, we

follow convention and estimate single equation production functions.

The casemix, ownership and regional variables discussed above are included in all specifications. We also include a dicotomous variable to denote when a clinic operates stellite facilities (SATELLITE). This is equal to one if at least one satellite exists and zero otherwise. Table 1 reports descriptions and basic statistics for variables in the two models.

The models are estimated for 90% of the sample. We exclude a 10% random sample of clinics for the purpose of evaluating the performance of the two functional forms used for estimation. We used the 10% sample to analyze forcast error and to calculate Theil’s equality coefficient.

RESULTS

In this section we present results for both the Cobb-Douglas and transcendental forms of out- patient clinic production functions. In addition, we also present results of our evaluation of the relative forecast performance of the two models.

Cobb-Douglas model

Table 2 presents production function estimates for both the Cobb-Douglas and transcendental specification. The first two columns of Table 2 report the Cobb-Douglas estimates. The model as a whole had considerable explanatory power as evidenced by R2 statistics of 0.55 and 0.56. The calculation of the elasticity of scale indicates decreasing returns to scale. The elasticity of scale coefficients, calculated were 0.59 and 0.62 depending upon whether capital was included in the model. A doubling of all inputs would therefore lead to between a 59 and 62% increase in output. Thus, for the observed data (which is essen- tially the universe of such clinics) the marginal cost curve is upward sloping for the intermediate good, individual visit equivalents. This result suggests that the production technology is consistent with a competitive market structure.

Privately owned clinics produce higher levels of output than county owned clinics holding constant input levels. The coefficient estimate for the COUNTY dummy is negative and significantly different from zero at conventional levels (0.05). The magnitude of the coefficient indicates that a county owned clinic produces roughly 15!&fewer+&+than do clinics that are otherwise similar but held privately (either nonprofit or for-profit) [21]. Clinics owned and operated by the State government appear to produce similar levels of output to privately owned facilities other factors held constant. This analysis assumes that ownership differences are reflected by a simple shift in the intercept of the production function. Possible interactions between ownership and input levels were also estimated. Those results indicate that use of a simple intercept appears to be generally justified [22]. One exception should be noted. The interaction of COUNTY and NURSE was significant and positive. However, the inter- actions as a group were not significant using an incremental F test [23]. Thus, the intercepts differ by ownership type but the marginal products are the same.

Production of outpatient mental health services a37

Table 2. Production function estimates

C-D’ C-D Tram’ estimate estimate estimate

Tram estimate Variable

MD

SWPM

NURSE

OTH

ADM

CAP

MDSQ

NURSESQ

OTHSQ

ADMSQ

COUNTY

STATE

SATEL

ALCOHOL

DRUG

MR

CHILD

Rl

R2

R3

INTERCEPT

R? F N MSE

0.19. (7.73) 0.19*

(7.65) 0.06’

(3.65) 0.06.

(4.13) 0.09.

(378) -

0.17” (6.96) 0.19’

(6.88) 0.05.

(3.07) 0.05’

(3.42) 0.09.

(3.75) 0.07.

(2.14) -

O.M)3’

‘;:E’

(8.41; O.OW5’

(3.97) 0.OOQ4*

(2.91) 0.0004*

(2.58) -

- - - - - - - - - - - - - -

-0.15’ (2.16) 0.05

(0.57) 0.13.

(2.15) 0.18t

(1.93) 0.03

(0.36) -0.01

(0.25) 0.07 (0.75) 0.19’

(2.13) 0.05

(0.62) 0.02

(0.19) 6.11.

(43.19) 0.55

53.26’ 672

0.46 0.59

-

-0.14. (2.00) 0.11

(1.27) 0.12t

(1.95) 0.21.

(2.20) 0.0 I

(0.09) -0.04

(0.63) 0.07

(0.71) 0.20.

(2.22) 0.05

(0.55) 0.03

(0.37) 5.36’

(16.65) 0.56

50.90’ 648

0.44 0.62

-

-0.000001* (3.87)

-0.OOooO2’ (3.88)

-8.7e-0a’ (1.92)

-5.67e-“’ (4.02)

-0.09 (1.28) 0.08

(0.93) 0.18.

(2.99) 0.16t

(1.71) 0.01

(0.13) 0.03

(0.44) 0.07

(0.82) 0.19.

(2.16) 0.07

(0.79) 0.04

(0.51) 7.01.

(51.53) 0.56

43.01. 672

0.46

0.002*

(5.68) 0.20.

(7.30) 0.002*

(3.60) o.Mw3*

(2.36) 0.0003t

(1.95) 0.09’

(2.39) -0.OOOo01* (3.30)

-0.OOOoO2’ (3.57)

-7.13e-Mt (1.77)

4.49~~08 (1.50)

-0.09 (1.35) 0.14

(1.61) 0.16’

(2.71) O.lSt

(1.87) 0.001

(0.07) 0.01

(0.21) 0.09

(0.90) 0.20*

(2.28) 0.06

(0.77) 0.59

(0.72) 6.12.

(16.00) 0.56

40219 648

0.44 0.79 Elasticity’ of scale 0.72

‘r statistics in parentheses. %AP and SWPM are logged all other variables are in natural units.’ ‘Translog models evaluated at means. l Sig P < 0.05 2 tailed test. tSig P < 0.10 2 tailed test.

The estimate for the variable describing one aspect of clinic organization (SATELLITE) indicates that decentralized clinics, holding constant all inputs, produce about 12% more visits than do centralized clinics. The reasons for this finding are not entirely clear. We probed the result further by examining the doctoral level clinician to administrative staff ratio in centralized versus decentralized organizations. In clinics with no satellites the clinician to adminis- tration ratio (in hours) was 81/157 or 0.49. In con- trast for clinics with satellite the ratio was 145/510 or 0.28. Thus administrative staff is a larger portion of total staff in decentralized clinics. A second expla- nation may be that in the presence of decreasing returns to scale one can produce more output by organizing the clinic into a number of smaller organ- izationally linked facilities rather than in a larger single facility organization.

Table 3 reports examples of estimated ratios of marginal products evaluated at mean input levels and ratios of mean wages obtained from the National Council of Community Mental Health Centers. We focus on the ratio of physicians to other clinic staff. The results indicate that the ratios of marginal prod- ucts are substantially smaller than the wage ratios for all clinical personnel. This means that clinics, on average, tend to hire more physicians relative to clinical staff than would be dictated by cost mini- mization [24]. In contrast outpatient clinics appear to hire more administrators, relative to physicians, than is consistent with minimized cost. Reinhardt [7’j identified a tendency among physicians to use ‘too much’ of their own time relative to aide inputs. This means that physicians office based practices could lower costs without decreasing quantity by substi- tuting the time of aides for physician time. Goldman

818 RIcw G. FIWNK and CARL A. TAL’BE

Table 3. Marginal producrs-wages comparison

Cobb-Douglas

.MP,;.MP; w,, w,t

ZD, SWPM 1.66 2.75 MD. NURSE 1.41 3.23 MD, OTH 2.56 4.23 MD. ADM 6.65 2.20

Tnmcendenml MD. SWPM 2.50 2.75 NURSE. SWPM 2.45 0.89 OTH. SWPM 0.50 0.72 ADM, SWPM 0.40 1.31

*Data from Cobb-Douglas wrh CAP-evaluated as mean input levels.

+Data from 1983 Community Mental Health Center Salary Survey NCCMHC (national means).

Results were similar when regional wages were used along with regional MP calculations.

and Grossman [5] report that for some types of personnel (aides) too few are utilized relative to physician time, while for all other community health center workers too many are used relative to phys- icians. Unfortunately, since wage data is only re- ported as a national average of the clinics in our sample, we cannot probe our results further. They do, however, signal the departure from cost minimizing hiring decisions.

Transcendental model

The two right hand side columns of Table 2 present coefficient estimates for the transcendental production function. The overall explanatory power of the model is similar to that of the Cobb-Douglas production function. The R’ statistics were 0.56 for both specifications of the transcendental model. The elasticities of scale were evaluated for the mean input levels. Those estimates are broadly consistent with the Cobb-Douglas results in that they indicate de- creasing returns to scale. The elasticity of scale esti- mates were 0.72 and 0.79. Goldman and Grossman [5], in their study of community health centers found constant returns to scale (elasticity = 1.0) when the transcendental model was evaluated at the input means [25].

With respect to the technical efficiency concerns of this study, the results for the efficiency impacts of organizational structure persisted. That is, the SATELLITE variable had estimated coefficients of 0.18 and 0.16 both ,of which were significantly different from zero at the 0.05 level. A decentralized organization produced between 16 and 18% more output, other factors held constant, than do central- ized clinics. Differences by ownership essentially vanished. While the direction of the ownership effects remained, the significance fell below conventional confidence levels.

As in the case of the Cobb-Douglas model our casemix controls had little effect on productivity. One exception, is the alcohol variable, whose coefficient was consistently positive and significant at at least the 0.10 level. The ALCOHOL coefficient was positive and varied between 0.16 and 0.21 depending upon the detailed specification. This indicates that clinics where alcoholics are heavily represented in the patient population are able to produce more visits.

This may be due to mismeasurement in the output variable, because group therapy for alcoholics tend to have more patients per group than other mental disorders.

Staffing decisions for the transcendental model are illustrated by comparisons of wage ratios to ratios of marginal products. Some illustrative comparisons are reported in Table 3. Qualitatively the results are similar to those found in the analysis of the Cobb-Douglas model. For ease of computation the social worker/psychologist category of workers is used as a reference group. The results indicate con- siderable departure from cost minimizing stuffing patterns. In particular there appears to be too few nurses relative to social workers, too many adminis- trators and other treatment staff and slightly more physicians than are needed to minimize costs. There appear to be substantial gains possible from reallocation of personnel.

Relatice performance of‘ production _fUnction models

In this section we provide evidence on the relative forecast performance of the Cobb-Douglas and tran- scendental production function models. We use three basic criteria in reviewing model performance. They are: conformance of coefficient estimates with theory. mean square error of estimated models, and forecast error.

Both the Cobb-Douglas and transcendental models have coefficient estimates which conform with theoretical expectation. By ‘conforming with theory’ we mean that the isoquant between two inputs. implied by the estimates, is convex to the origin (761. The coefficient estimates for the inputs in the two models are all statistically significant using one tailed tests of significance at the 0.05 level. Both models perform well along our initial criteria.

The second criteria involves evaluation of the mean square error for estimated production function models. Table 4 presents estimates of the mean squared errors for the two models. As is immediately evident the models performed identically with respect to the mean square error criteria.

Our third criteria, forecast performance is based on the notion that there is a trade-off between global satisfaction of regularity conditions on the part of the Cobb-Douglas model and Rexibility to model com- plex technologies but failure to meet global regularity conditions [27]. Moreover, since computation is more difficult using the flexible functional form our desire is to understand how well these models approximate technologies existing in the health care sector. We accomplish this via two measures of forecast per- formance, the correlation between actual and pre- dicted value for our 10% validation subsample (‘AJ) and calculation of Theil’s inequality coefficient (281. Table 4 presents estimates of both ‘,d.P and Theil’s inequality coefficient. Both results indicate that the

Table 4. Production function model performance

Model

Indicator Cobb-Douglas Transcendental

‘A,P 0.78 0.75

Theil’s C. 0.071 0.079

Mean square error 0.4-l 0.44

Production of outpatient mental health services 849

two models perform well in that ‘d,P is relatively large (0.75 or greater) and Theil’s Li is close to zero. In neither case are the two statistics substantially different. However, the Cob&Douglas model per- forms somewhat better along both criteria. The ‘A,P statistic is closer to 1 for the Cobb-Douglas model and Theil’s U is slightly closer to zero. However, one must conclude that the two models’ performances are essentially identical. This means that the simpler more restrictive but globally well behaved Cobb- Douglas model performs at least as well as the transcendental model in terms of forecasting ability.

DISCLSSION

The two models of the production technology of freestanding outpatient mental health clinics reveal a variety of important features relating to the structure of production. We take up aspects of technical efficiency first.

In both the Cobb-Douglas and the transcendental models of production the estimated coefficients im- plied decreasing returns to scale. Thus, the marginal cost curve is upward sloping. This means there are likely to be few benefits to regionalization. The result indicates that a competitive market is sustainable. A second related result is that clinics with satellite facilities tend to produce 12-18% more visits, holding input levels constant, than do single site clinics. The implication here is that decentralized organizations run more efficiently. An important consequence of this finding relates to equity-efficiency trade-offs. It is often suggested that health care delivery organ- izations should be decentralized so that care is more available to those ‘in need’. This position was sup- ported by the work of Acton [29] which showed that time and distance posed substantial barriers to treatment among poverty populations in New York. Our rest&s indicate that organization along lines associated with decentralization would not lead to higher costs of production. Thus one may be able to improve access without raising treatment costs appre- ciably. The mechanism underlying this result remains unclear.

The results concerning the impact of ownership on clinic output are difficult to interpret. Because patients may be sorted among clinics according to their severity of illness ownership may reflect casemix as well as difference in productivity arising from motivations to minimize costs. The finding that counties appear to produce fewer visits for given amounts of inputs than either state owned or pti- vately owned clinics may reflect the fact that county clinics often serve as the provider of last resort. This might mean they treat more severely ill indigent patients than do other providers.

The results pertaining to staffing patterns suggest that hiring decisions do not seem to meet the condi- tions necessary for cost minimization. We do not have sufficient information to probe this result in great detail. However, several possible explanations may underlie the result. First, most directors of clinics tend to be clinicians (psychiatrists, psychol- ogists or social workers). Eighty percent of commu- nity mental health centers are directed by a member of one of the three ‘core’ clinical professions [30]. This

may lead to staffing patterns that reflect the pursuit of other goals such as ‘quality of care’. Since we do not measure factors of such a quality we cannot in- vestigate this proposition further. However, our re- sults do indicate a tendency of clinics to hire ‘too much’ physician time relative to other clinical staff. This might in fact reflect quality concerns. Altema- tively, most states require certain minimal levels of physician staff for a clinic to be licensed. These regulatory standards may conflict with choosing a cost minimizing staffing pattern. Finally, most of these clinics rely on fee for service reimbursement which does not encourage cost minimizing behavior on the part of providers. This is especially true when fee schedules in part depend on staffing patterns of such clinics.

The analysis of performance of the econometric models used for estimation leads us to several con- clusions. First, the production technology in free- standing outpatient mental health clinics appears to be rather simple. We believe that the strong per- formance of the Cobb-Douglas model in fitting data as well as forecasting of our validation subsample points to this. A second conclusion is that the CobbDouglas model performs at least as well as the transcendental production function. Using any of the criteria specified the Cobb-Douglas model did as well as the transcendental. Therefore, we conclude that for purposes of modelling simple technologies in the health field such as clinics and office based practices one gives up little by choosing the computational simplicity of the Cobb-Douglas model.

Acknowledgements-Frank’s research effort was supported by order number 84M05841020lD from the National Institute of Mental Health. We are grateful to David Salkever, Gordon DeFreise and an anonymous referee for comments on an earlier draft of this paper.

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850 RICHARD G. F~USK and CML A. TALBE

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