measuring hospital performance in the presence of quasi-fixed inputs: an analysis of québec...

Journal of Productivity Analysis, 21, 183–199, 2004

# 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Measuring Hospital Performance in the Presence of

Quasi-Fixed Inputs: An Analysis of Quebec Hospitals

DANIEL BILODEAU

Departement des sciences economiques, Universite du Quebec a Montreal, Montreal, Canada

PIERRE-YVES CREMIEUX* [email protected]


Analysis Group, 111 Huntington, Boston, MA 02199, Cambridge, USA

BRIGITTE JAUMARD

Departement de genie industriel et de mathematiques appliquees, Ecole Polytechnique de Montreal,

Montreal, Canada

PIERRE OUELLETTE


TSEVI VOVOR

Departement de genie industriel et de mathematiques appliquees, Ecole Polytechnique de Montreal,

Montreal, Canada

Abstract

This research proposes an approach to measure hospital performance based on ageneralization of Banker and Morey (1986) and Førsund (1996). This approachconsiders quasi-fixed inputs explicitly, calculates their implicit cost, and quantifiesreturns to scale. The performance measure is decomposed into allocative andtechnical inefficiencies.Based on a very complete data set of Quebec hospitals, we find that significant

inefficiencies of up to 17% ($700 CAN million) could have been saved throughimproved performance. Postestimation analyses that include qualitative measures ofcare suggest that differences in performance are attributable to differences inmanagement or unobservable quality of care rather than patient case mix.

JEL Classification: I11, I18

Keywords: hospital performance, returns to scale, implicit price, quasi-fixed inputs, allocative

inefficiencies, technical inefficiencies

* Corresponding author.

1. Introduction

In this paper, we propose a non-parametric method to study the health careproduction process by hospitals in the Province of Quebec based on a very completedata set covering all short-term acute care hospitals over a 12-year period. This studyimproves on previous work by relying on data that include all variable inputs, alltypes of quasi-fixed inputs explicitly, including physicians, equipment, and buildings,and all the types of outputs including inpatient care, outpatient care, food, laundry,laboratory services, and teaching. The explicit inclusion of all the types of outputsand all variable and quasi-fixed inputs reduces the likelihood of biases resulting fromomitted variables.Evidence based on U.S. data suggests that hospitals’ production of health care is

rather inefficient (Eakin and Kniesner, 1988, 1992; Cowing and Holtman, 1983;Cowing et al., 1983; Byrnes and Valdmanis, 1994). This could result in part fromthird party payers, citizens’ perception that health care is desirable at any cost, and,in Quebec, a centralized financing and decision-making structure.Understanding the performance of the hospital industry is crucial in the context of

a centralized and socialized health care system. In the absence of competitivepressures driving to the usual adjustments, one might expect significant andpersistent inefficiencies. Ranking hospitals based on their relative efficiency (relativecosts all else equal) and assessing their distance from the envelope yield theprovincial cost of inefficiencies and help identify institutions most in need ofrestructuring. This study offers an approach to do so that minimizes potential biases.

2. Hospital Performance in a Non-Competitive Environment

Although hospitals in the United States typically bill their patients or their patients’payers and enjoy increased revenues as a result of increased activity, hospital servicesdo not operate in a truly competitive market. Hospitals enjoy local market power,benefit from third party payments and asymmetric information about theircustomers’ needs, and enjoy high barriers to entry. Nevertheless, higher profitsand the very survival of the institution still depend on cost minimization efforts.The hospital industry structure is quite different in Quebec where the government

determines budgets based on historical data. Increased activity typically leads toincreased revenue allocations from the government but overall budget constraints,geographical, or political considerations significantly affect the allocation mechan-ism. Through ownership or monopsony power, the Quebec government largelyimposes uniform management rules, salary structure, job descriptions andaccounting rules. While this does not imply that hospitals are equally efficient,they do operate within a similar framework. The very distribution of hospitals overthe territory is determined as much by political and language considerations as it isby demand for services. Furthermore, institution survival is rarely compromised byhigh costs. In fact, over the period studied here, not a single hospital was closed forfinancial reasons. This could lead to significant and persistent inefficiencies since

184 BILODEAU ET AL.

hospital survival is never at stake. As such, performance measures allow thegovernment to identify hospital characteristics associated with better overallperformance and to single out hospitals that systematically show poor performance.The analysis of the population of Quebec short-term hospitals rather than asubsample is particularly relevant from a policy point of view because centralizedfunding decisions imply that the performance and funding of any given hospitalaffect the funding of all other establishments.In this analysis, we rely on a DEA analysis which requires fewer hypotheses on the

functional form and stochastic structure of the technology than would be required in atraditional cost function estimation (Grosskopf and Valdmanis, 1987; Kooreman,1994a, b; Fare et al., 1993;Grosskopf et al., 1995). However, it also greatly complicatesthe use of qualitative variables measuring differences in quality or complexity of care.Specifically, qualitative variables cannot be directly incorporated into DEA analysisunless such variables are monotonously related to cost or efficiency. Alternatively,homogenous subsamples could be created but would obviously limit the number ofobservations available. Here, we rely on a post-DEA analysis to assess whetherdifferences in case mix or patient population are determinants of performance(unfortunately, no index of quality of care is available). This is done by regressingoverall efficiency on potential determinants of inter-hospital performance differences.

3. Method

To examine the efficiency of Quebec hospitals, we adopt a two-step process based onBanker and Morey (1986) and Byrnes and Valdmanis (1994). Banker and Morey firstintroduced the distinction between discretionary and nondiscretionary (quasi-fixed)inputs, the latter being out of the firm’s control in the short-run. This analysisintroduces this distinction between inputs in the Byrnes and Valdmanis model.Hence, this model accounts for quasi-fixed inputs into the production process and isa generalization of the standard DEA model.1

The first step is to define a production frontier based on the set of hospitals notdominated by any other hospital. Then, for each hospital, we chose, based on observedhospital specific variable input prices, the combination of variable inputs that yields thehospital’s level of production at minimum variable cost given the observed quasi-fixedinputs. The distance between the observed variable cost and the minimal variable costusing the optimal set of inputs is the overall efficiency. This overall efficiency is theproduct of the technical and allocation efficiencies (Farrell, 1957).

3.1. Determining the Production Frontier and the Cost of Technical Inefficiencies

The following minimization problem must be solved to determine the productionfrontier for hospital h (for h ¼ 1; . . . ;H):

min yh under the constraint that f ðyh; yhxh; khÞ � 0 and yh > 0;

HOSPITAL PERFORMANCE IN THE PRESENCE OF QUASI-FIXED INPUTS 185

where yh:ðyh1; . . . ; yhMÞ is the M-vector of outputs, xh:ðxh1; . . . ; xhNÞ, is the N-vectorof variable inputs, kh:ðkh1; . . . ; khLÞ is the L-vector of quasi-fixed inputs, yh, a scalar,is a measure of efficiency, and h ¼ 1; . . . ;H is a hospital index. This minimizationproblem identifies whether the level of variable input can be reduced given theavailable level of quasi-fixed factors within the same production possibility space.This production frontier can be approximated using the following linear program:2

TEh ¼ miny;l

fyhjly � yh; lk � kh; lx � xhyh;Shlh ¼ 1; lh � 0; for all hg; ð1Þ

where x ¼ ½x1 x2 . . . xh . . . xH � is the matrix made up of the H column vector ofvariable inputs, xh; y ¼ ½y1 y2 . . . yh . . . yH � is the matrix made up of the H columnvector of outputs, yh; k ¼ ½k1 k2 . . . kh . . . kH � is the matrix made up of the H columnvector of quasi-fixed inputs, kh.TE h is the measure of technical efficiency for hospital h in year t. It is the ratio of

the cost of a technically efficient variable input bundle over observed variable cost.The measure of technical efficiency is therefore expressed in terms of variable inputlevels.Variable input prices yield the cost of technical inefficiency measured by

ð1� TEhÞ. This cost is:

ð1� TEhÞ � Ch ¼ ð1� TEhÞ � Snwhnx

hn;

where Ch is the observed variable cost of hospital h and whi is the market price of the

ith variable input. The efficient (but not necessarily optimal) cost can be calculated asTEh � Ch.

3.2. Identifying Allocative Inefficiencies

The cost of technical inefficiency results from a choice of inputs that is below theproduction frontier. Other costs might result because all choices on the productionfrontier are not equivalent. Given the price ðwhÞ of variable input factors ðxhÞ, somepoints on the frontier will yield a minimal production cost. To determine thesepoints, the following minimization problem must be solved:

minx

Snwhnx

hn j f ð yh; xh; khÞ � 0

� �:

It can be approximated with the following linear program:

Cminh ¼ Chðw; k; yÞ

¼ minXE ;lfwh � xE j ly � yh; lk � kh; lx � xh; lx� xE ¼ 0;

Shlh ¼ 1; lh � 0; for all hg; ð2Þ

where xE is the solution to the cost minimization problem. The cost of allocativeinefficiencies is the difference between the efficient cost calculated in the previous

186 BILODEAU ET AL.

section and the minimal cost:

TEh � Ch � Chmin:

The allocative inefficiency expressed as a percentage ðAEhÞ of the efficient cost is:

AEh ¼ Chmin

TEh � Ch:

Added together, the allocative and technical inefficiencies yield the total cost surplus.Similarly, the product of the allocative and technical inefficiencies yields the globalinefficiency as a percentage of the observed cost:

OEh ¼ AEh � TEh

¼ Chmin

Ch:

4. Technology Measurements

This section is a straightforward generalization of Førsund (1996) that includes anevaluation of implicit prices with an input-oriented inefficiency and allows for quasi-fixed inputs since hospitals cannot constantly set buildings, equipment and medicalstaff at their optimal levels.The Lagrangian function for the linear program described by (1) is:

Lh ¼ yh þ Smuhm yhm � Sh

j lhj ymj

� �þ Snv

hn SH

j lhj xnj � yhxhn

� �þ Slrhl SH

j lhj klj � khl

� �þ uh SH

j lhj � 1

� �for all h ¼ 1; . . . ;H;

where uhm; vhn; r

hl and uh are Lagrange multipliers associated with the constraints. The

dual problem is:

maxfSmuhmy

hm � Slrhl k

hl þ uh jSnv

hnx

hn ¼ 1;Smu

hmymj � Snv

hnxnj � Slrhl klj � 0

for all j ¼ 1; . . . ;Hg:

Using the definition of yh, we get:

Smuhmymj � Snv

hnxnjy

h � Slrhl klj ¼ 0 for all j ¼ 1; . . . ;H;

which corresponds to the transformation function Fðyh; xh; khÞ ¼ f ðyhÞ� gðxh; khÞ ¼ 0. Using this result yields the partial derivatives and implicit pricesfor quasi-fixed factors:

qFqyhm

¼ uhm;qF

qðxhnyhÞ¼ � vhn;

qFqkhl

¼ � rhl : ð3Þ


4.1. Implicit Prices

The implicit price, zl, calculated for each quasi-fixed input l can be calculated basedon the behavior of the efficient cost-minimizing firm:

Cðw; k; yÞ ¼ min xfSnwnxn jFðy; x; kÞ ¼ 0g:

The Envelope theorem implies:

zl ¼ � qCqkl

¼ gqFqkl

;

where g is the Lagrange multiplier associated with the technology constraint.This implicit price represents the maximum acceptable price for an additional unit

of quasi-fixed factor, kl. It is equal to the reduction in variable cost resulting from thefirm’s use of that additional unit of capital. It is the product of the Lagrangemultiplier, g, associated with the technology constraint and kl’s marginalproductivity ðqF=qklÞ calculated in the previous section (equation (3)). The Lagrangemultiplier, g, can be calculated from the first-order conditions on the costminimization problem: wn ¼ gqF=qxn or g ¼ wn=ðqF=qxnÞ ¼ �wn=v

hn. Together,

these two results yield the implicit price relative to the nth variable input price, wn:

zl ¼ gqFqkl

¼ wn

vhn

� ��rhl :

Alternatively, if the variable input prices are normalized such that Snwn ¼ 1, then

g ¼ 1

SnqFh=qxn¼ �1

Snvhn; and zl ¼

rhlSnvhn

:

Implicit prices compared with market prices will determine whether the hospital isover or under-capitalized. The telltale sign of undercapitalization would be implicitprices greater than the market price while overcapitalization would be characterizedby the opposite.

4.2. Returns to Scale

A hospital exhibits increasing returns to scale if:

RTSh ¼� SnðqF=qðxhny

hÞÞðxhnyhÞ þ SlðqF=qkhl Þkhl

� �SmðqF=qyhmÞyhm� � > 1:

Returns to scale are measured at efficient points ðyh ¼ 1Þ or on the radial projectionon the frontier ðyh < 1Þ.3 Following Førsund (1996), we substitute the aboverelations in the definition of RTS to obtain a generalization of RTS including

188 BILODEAU ET AL.

quasi-fixed inputs:

RTSh ¼ Snvhnx

hny

h þ Slrhl khl

Smuhmyhm

> 1:

4.3. Economies of Scope

In the two outputs case, economies of scope are present if the cost of producing twooutputs ðy1; y2Þ in one firm is lower than the cost of producing one output in eachfirm (Panzar and Willing, 1980). This definition can be generalized in the multi-output case. Of course, the multi-output definition must distinguish betweenproportional and non-proportional increases in each output. In a centralized healthsystem, the government typically chooses between marginal increases in thespecialization of each hospital and an increase in all services at all hospitalssimultaneously. This contrasts with the strict Panzar and Willig approach in whichthe government would make a dramatic choice between one large general hospital ormultiple specialized hospitals. In reality, the decision is marginal rather thandichotomous.The inequality below illustrates the marginal nature of the government’s

opportunity to exploit economies of scope in any given year:

Cðy1 þ Dy1; y2 þ Dy2; . . . ; yM þ DyMÞ < Cðy1 þ Dy1; y2; . . . ; yMÞþ Cðy1; y2 þ Dy2; y3; . . . ; yMÞ þ � � � þ Cðy1; y2; . . . ; yl�1; yM þ DyM Þ:

Each of the elements is calculated using a dichotomous search Cðy1; y2; . . . ;yj�1; yj þ Dyj; yjþ1; . . . ; yMÞ: We search for the smallest value Dyj (if an acceptableone exists) that will affect C. The search interval varies from one yj to another. Thevalues of the Dyj from each of the dichotomous searches are then incorporated intoboth sides of the inequality.

5. Data4

All Quebec hospitals are required to file yearly detailed financial (AS-471) andactivity (AS-477) reports with the department of health (DOH) of Quebec (Ministeredes affaires sociales, various years; Ministere de la sante et des services sociaux,various years). Because the DOH pays all hospital expenses through the nationalhealth insurance system, the reports are quite detailed. They are designed by theDOH to allow careful control of the hospitals’ activity. Similarly, the Regie del’Assurance-Maladie du Quebec (RAMQ) keeps detailed data on physician activitysince it is the government body that reimburses physicians for the care they provide(Regie de l’assurance-maladie du Quebec, 1993). Our analysis is based on data thatspan all 121 short-term hospitals in Quebec over the 1981–82 through 1992–93


periods that were still in operation in 1992.5 Remaining data on prices were availablefrom Statistics Canada (various years). Table A1 provides a list of all the variablesused in the analysis.There are no omitted inputs or outputs in the financial and output reports.

Outputs include not only traditional measures of inpatient days and outpatient clinicvisits but also laboratory exams performed for pay (for non-hospitalized patients),laundry and cafeteria services (again, for non-hospitalized patients) and teaching.Similarly, inputs available from the reports are comprehensive and include the totalnumber of hours and expenses on labor, expenditures on supplies (medical,administrative, maintenance, security, and capital maintenance), food and mealsprepared for inpatients, drugs (total expenditures only), energy (oil, gas, electricity,steam, other sources), and an ‘‘other’’ category including all remaining variableinputs. For quasi-fixed inputs, the reports include equipment, building andphysicians by specialty (from the RAMQ reports). The introduction of physiciansas a quasi-fixed inputs is discussed in Jensen and Morrisey (1986a,b) and Bilodeau etal. (2000). In the province of Quebec, the number of physicians is not determined byhospital administrators. Physicians request access privileges from the Physician,Dentist and Pharmacist Council that must accommodate them if possible. Suchprivileges have no fixed durations and are not under the administrative control of thehospital. The variables’ descriptive statistics are reported in Table A1.The inclusion of all inputs and all types of outputs in the cost function reduces

potential biases resulting from omitted variables. Of the 1,452 potential observations(121 hospitals over 12 years), 105 to 116 hospitals are included each year ðH ¼105 to 116Þ for a total of 1,359 observations (94% of the population). Each hospitalhas six variable inputs ðn ¼ 6Þ, up to five outputs ðm ¼ 5Þ, three quasi-fixed inputsðl ¼ 3Þ and is observed for up to 12 years ðt ¼ 1; . . . ; 12Þ. The minimum cost iscalculated for each hospital h in each year t. Of course, hospitals produce such alarge number of outputs using a large number of variable and quasi-fixed inputs thatDEA analysis requires some aggregation. Outputs and inputs are aggregated withinDMUs but not across DMUs (see Blackorby and Russell, 1999). Therefore, we donot aggregate efficiency scores. Instead, we are aggregating different quantities usinghomogeneous output aggregations (e.g., outpatients, inpatients, laboratories,teaching) and inputs (e.g., energy, labor).Unlike in many prior studies, physicians are included as a quasi-fixed factor. This

eliminates an important potential source of bias (Bays, 1980; Jensen and Morrisey,1986a,b; Cremieux and Ouellette, 2001). Price indices for specialists and generalpractice physicians were generated using the RAMQ data set by regressing totalspending on physicians (total salaries) on the number of specialists, the number ofgeneral practitioners, and a dichotomous variable capturing whether there wereresidents in the hospital.Another common source of bias is the omission of the stock of equipment as a quasi-

fixed factor. Here, the stock is calculated using the perpetual inventory methodEtþ1 ¼ Etð1� dÞ þ It, where E is the quantity of equipment, d(¼ 0:18,according to Statistics Canada) is the rate of depreciation, and I is the level ofinvestment in furniture and equipment reported in theAS-471. The hospital is assumed

190 BILODEAU ET AL.

to be in long-run equilibrium the first year ðE1981 ¼ I1981=dÞ and the depreciation isbased on the average life span of furniture investments reported by StatisticsCanada.One innovation of this analysis is to include all five types of output. Of these,

inpatient and outpatient days stand out as the primary measures of a hospital’sactivity but the production of food, laundry and laboratory services for visitors orother hospitals as well as teaching of residents are a growing dimension of hospitalproduction. Clearly, the primary measures of activity (inpatient and outpatient care)are incomplete since the nature of the underlying interventions might well varyacross hospitals. Unfortunately, within the DEA context, such case-mix measurescannot be incorporated. Nevertheless, such differences are considered in a post DEAanalysis based on two additional indicators of activity. The first is simply the numberof specialities offered (for inpatient care) or the number of speciality clinics (foroutpatients). The second is a measure of the relative importance of each specialityweighted by its complexity (proxied by relative costs across all hospitals in thesample). It is calculated using the formula below:

X9

i¼1

P121h¼1 total costihP121

h¼1 number of patient daysih� number of patient daysis � 100

total patient dayss

!;

where i indexes specialities and h indexes hospitals. The outpatient complexitymeasure replaces patient days with clinic visits.6

A NIRRU index, which is akin to the diagnostic related groups (DRG) forQuebec, was also introduced to benchmark complexity.

6. Results

6.1. Allocation Efficiency, Technical Efficiency, Overall Efficiency

Table 1 indicates that over the 1981 through 1993 period, 37% of hospitals areefficient. This percentage varies from 29% in 1982–1983 to 45% in 1992–1993. Ofcourse, even these hospitals could improve their performance by radically changingtheir practice but, given observed hospital production processes and their choice ofinputs and outputs, they are on the production frontier. This measure of inefficiencyis rather forgiving since it only considers efficiency gains from incremental changes inthe hospital’s management rather than fundamental changes in operation not yetincorporated in any of the hospitals in Quebec.Inefficiency results in large part from technical rather than allocative inefficiency

(Table 2). This suggests that inefficient hospitals combine their inputs efficiently butuse more resources to achieve similar production levels. Table 3 shows thatinefficiencies lead to overall excess costs of $500–700 million per year over 105(1981–1982) to 116 (1992–1993) hospitals. This represents roughly 15% of the $3.3billion (1981–1982) to $4.4 billion (1992–1993) total hospital budgets.


These higher costs result from a suboptimal allocation of inputs and an overreliance on quasi-fixed inputs. Table 4 shows that variable input levels are typically15 to 20% above their optimal level and that, as mentioned earlier, overcapitalization is generally present.The 20 worst performing hospitals out of 119 over the period studied accounted

for nearly 50% of total inefficiencies while the top 20 hospitals accounted for lessthan 1% of inefficiencies. Costs incurred as a result of inefficient managementcould therefore, be significantly reduced by focusing on a small subset ofhospitals.

Table 1. Number of efficient hospitals.

Year

Number of

Efficient Hospitals

Total Number

of Hospitals

1981–1982 42 105

1982–1983 31 107

1983–1984 33 109

1984–1985 56 110

1985–1986 45 111

1986–1987 45 116

1987–1988 36 116

1988–1989 39 116

1989–1990 38 117

1990–1991 39 118

1991–1992 50 118

1992–1993 52 116

1981–1993 504 1,359

Table 2. The extent of inefficiencies.

Year

Overall

Efficiency

Technical

Efficiency

Allocative

Efficiency

1981–1982 0.78 0.84 0.92

1982–1983 0.75 0.81 0.93

1983–1984 0.73 0.79 0.91

1984–1985 0.87 0.92 0.94

1985–1986 0.85 0.91 0.94

1986–1987 0.84 0.88 0.94

1987–1988 0.83 0.89 0.93

1988–1989 0.84 0.89 0.95

1989–1990 0.83 0.87 0.95

1990–1991 0.83 0.88 0.94

1991–1992 0.84 0.89 0.94

1992–1993 0.85 0.89 0.95

1981–1993 0.82 0.87 0.94

192 BILODEAU ET AL.

6.2. Technology Measurements

. Economies of scale and scopeTable 5 indicates that most hospitals have grown beyond their optimal size andreached decreasing returns to scale. Furthermore, since 1987–1988, thepercentage of hospitals exhibiting decreasing returns to scale has grown from69 to 84% in 1992–1993. Changes in the hospitals system seem to lead morehospitals to become inefficiently large structures. Not surprisingly, Table 6indicates that these hospitals are two to three times larger than hospitals withconstant returns to scale.

Table 3. The costs of inefficiencies (in million of dollars).

Year

Average Observed

Cost [1]

Average Minimum

Cost [2]

Average Excess

Cost [3] ¼ [1]� [2]

Total Excess Cost

[3]*# of Hospitals

1981–1982 31.1 24.3 6.8 715

1982–1983 29.9 22.6 7.3 779

1983–1984 30.1 21.9 8.2 893

1984–1985 30.2 26.7 3.5 387

1985–1986 30.8 26.6 4.2 466

1986–1987 29.2 24.6 4.5 524

1987–1988 31.6 26.6 5.0 576

1988–1989 32.2 27.8 4.4 511

1989–1990 33.1 28.2 4.8 566

1990–1991 34.2 29.3 5.0 584

1991–1992 34.7 30.4 4.3 508

1992–1993 36.2 31.7 4.4 516

1981–1993 32.0 26.8 5.2 7,025

Table 4. Percentage difference between observed and optimal input levels (drugs, food, furniture, energy,

labor, other).

Year Drugs (%) Food (%) Furniture (%) Energy (%) Labor (%) Other (%)

1981–1982 � 22 � 23 � 14 � 23 � 22 � 24

1982–1983 � 26 � 24 � 21 � 24 � 24 � 28

1983–1984 � 25 � 23 � 23 � 29 � 26 � 30

1984–1985 � 12 � 14 � 11 � 14 � 13 � 13

1985–1986 � 14 � 16 � 16 � 16 � 14 � 14

1986–1987 � 18 � 16 � 17 � 18 � 16 � 14

1987–1988 � 17 � 17 � 17 � 22 � 16 � 14

1988–1989 � 16 � 14 � 18 � 20 � 15 � 15

1989–1990 � 18 � 16 � 20 � 20 � 16 � 19

1990–1991 � 16 � 13 � 21 � 17 � 15 � 15

1991–1992 � 17 � 8 � 17 � 18 � 15 � 16

1992–1993 � 18 � 8 � 18 � 17 � 14 � 17

1981–1993 � 18 � 16 � 18 � 20 � 17 � 18


Increasing returns to scale experienced by other hospitals suggest a smallerthan optimal size. These hospitals are roughly half the size of hospitals withconstant returns to scale. In this case, however, non-economic considerationssuch as the need for regional hospitals serving a sparsely populated area mightexplain and justify the suboptimal size of the establishments. In fact, 35% of ruralhospitals would benefit from increased size versus only 26% in urban areas.Nevertheless, even in rural areas, 65% of hospitals seem too large. A densityvariable to capture any systematic differences between rural and urban hospitalsfails to unveil any statistically significant differences. Over the 11-year periodstudied here, 72 hospitals account for the 593 hospital-years during whichdecreasing returns to scale are observed. Similarly, 25 hospitals account for the122-hospital years for which increasing returns to scale are observed. Again, thissuggests great stability over time but also that targeting a small number ofinstitutions might lead to significant gains in overall efficiency not withstandingregional requirements.

Table 5. Distribution of hospitals by returns to scale (average ¼ 0.94).

Year

Returns to

Scale< 0.95

0.95<Returns

to Scale< 1.05

Returns to

Scale> 1.05

1981–1982 71 15 19

1982–1983 56 19 32

1983–1984 52 16 40

1984–1985 48 23 39

1985–1986 72 15 24

1986–1987 76 16 24

1987–1988 69 22 25

1988–1989 76 12 28

1989–1990 78 12 27

1990–1991 78 19 21

1991–1992 72 23 23

1992–1993 84 11 21

1981–1993 832 203 323

Table 6. Hospital output characteristics by returns to scale.

RTS< 0.95 0.95<RTS< 1.05 RTS> 1.05

Aver. number of patient days (1992–1993) 113,154 50,170 44,875

Aver. number of clinic visits (1992–1993) 158,498 60,324 24,472

Aver. number of patient days (1981–1993) 121,970 59,995 40,006

Aver. number of clinic visits (1981–1993) 128,144 57,015 21,575

Note: Similar relative output levels were observed in the three other dimensions (laboratory, laundry and

cafeteria, and teaching).

194 BILODEAU ET AL.

All hospitals included in the sample experienced economies of scope. Thisimplies not only that there are no potential efficiency gains from specializationbut also that a lower level of hospital specialization might lead to increasedefficiency.

Together the returns to scale and returns to scope results suggest that theQuebec government recent focus on creating ‘‘super hospitals’’ in the largemetropolitan areas might be ill advised unless the concentration of ultraspecialized units can counteract the losses in overall efficiency resulting fromdecreasing returns to scale and economies of scope. Our results provide evidenceon a hospital-by-hospital basis concerning the likely efficiency effect of mergers.Of course, DEA results provide insufficient guidance for public policy. However,DEA is helpful to identify hospitals likely to be successful candidates for mergersand, together with careful case studies, could help predict the likely consequencesof mergers in terms of economic efficiency. Together with issues such asconvenience for patients and fair access to care, economic efficiency is one aspectof mergers that should be kept in mind.

. Implicit quasi-fixed input pricesThe three quasi-fixed inputs are physicians, equipment, and building. Forphysicians and equipment, there is clear overcapitalization as evidenced by thezero implicit price in roughly 60% of observed cases (841/1,359). In other words,there is no evidence that increasing the number of physicians or the level ofequipment would decrease variable cost by even a dollar. While the market priceof capital and equipment is unknown, it is clearly positive. Similar results arefound for buildings. This seems to contradict the casual observation of regularshortages of physicians and space particularly in emergency services duringwinter months. However, our results do not imply that all departments within agiven hospital are over staffed at all time but rather that the overall level ofphysician staffing is inefficiently high. Shortages during peak demand or withincertain services might be observed and could be addressed while reducingovercapitalization by relying on part-time staff, increasing the level of supportstaff during the winter months, and reallocating space to emergency services atpeak time.

6.3. Determinants of Efficiency

Differences in performance levels could result from at least three different sources.First, differences in management could lead hospitals with identical case mix andquality of care to exhibit different costs. Second, hospitals with a more complex casemix might, despite equally good management and quality of care, also exhibithigher costs and therefore be considered under performers. Finally, hospitals thatoffer higher quality of care could, all else equal, experience higher costs. Resultsfrom the post-DEA analysis designed to examine the role of quality differences are


reported in Table 7. The regressions include case-mix and scope of services both forinpatients and outpatients, diagnosis related groups (DRG), average patient age anddensity.Since the dependent variable that measures efficiency is right truncated at 1, the

analysis is performed using a Tobit (technically, the efficiency measure is also lefttruncated at 0 but no observation takes this value (see Ferrier and Valdmanis,1996)).The DRG measure was not used as an index of case-mix in the DEA analysis

because it is not available for the early years. A significant coefficient on the DRGvariable would indicate that performance captures differences in case-mix ratherthan actual differences in managerial performance across hospitals with similar case-mix. The results for 1991 suggest that none of these variables have a statisticallysignificant effect except the diversity of services offered to outpatients. In particular,financial DRGs and complexity measures are insignificant. It is therefore unlikelythat performance captures differences in case-mix.This leaves differences in management practices and unmeasurable differences in

quality as potential sources of the observed differences in performance.

7. Conclusion

In a non-competitive environment, measuring performance to determine whichhospitals are efficient is an important management tool. Government planners needto identify poorly performing establishments since they cannot rely on the market toweed them out. In fact, given the complexity of the health care market, even apseudo-competitive system such as that observed in the United States might still

Table 7. Determinants of overall efficiency for 1991. (Tobit regression with efficiency measure truncated

below 0 and above 1.)

Variable Coefficient T-Statistic

Constant 1.201 4.45

Quebec financial DRG 0.021 0.20

Inpatient diversity index � 0.002 � 0.12

Inpatient complexity index � 0.00002 � 1.47

Outpatient diversity index 0.013 2.11

Outpatient complexity index � 0.00005 � 0.62

Percentage of patients over 65 � 0.145 � 0.47

Increase in patients over 65 since 1981 � 0.231 � 0.48

Density � 0.00002 � 1.04

Note: The coefficient of the variable sigma is 0.22 with a T-Statistic ¼ 9.37. No. of obs. ¼ 90 with 54

uncensored observations and 36 right-censored observations at overall efficiency� 1.

196 BILODEAU ET AL.

allow persistent inefficiencies. Once these inefficiencies are identified, it is importantto determine the source of the poor performance and remedy the situation.To evaluate the performance of the Quebec hospital system, we apply a

generalized version of Banker and Morey (1986) and Førsund (1996). We showhow all inputs including quasi-fixed inputs, and all types of outputs can be includedin the production function and find that the scope of inefficiencies among Quebechospitals is not trivial. Out of an overall hospital budget of $4.2 billion in 1992–1993,up to $700 million (17%) might have been saved had performance been improved. Inany given year over the 1981–1982 through 1992–1993 period, only 37% of hospitalsare on the production frontier while the remaining hospitals show various levels ofinefficiencies. These inefficiencies in part might be unavoidable if services are to beprovided in all areas of the Province including remote sparsely populated ones.However, diseconomies of scale even in remotely located hospitals suggestmanagerial and bureaucratic decisions as likely culprits rather than regionalimperatives. Overall, diseconomies of scale and overcapitalization in building,equipment and physicians appear the most likely sources of inefficiencies.To identify whether differences in performance might reflect differences in the

quality of care or the patient population rather than differences in management andstructure, we regressed hospital performance on differences in DRGs, averagepatient age, and other relevant variables. The results failed to identify significantdeterminants of performance. In particular, differences in case mix whethermeasured by a weighted services measure or DRGs do not seem to be driving theobserved differences in performance. This suggests that unobservable differences inmanagement methods or unmeasurable differences in the quality of care underlie thedifferences in observed performance.

Notes

1. Malmquist measures could provide an alternative approach that would allow technological change.

However, such an approach remains uncertain when returns to scale are not constant (see Grifell-Tatje

and Lovell, 1995). Since the approach followed here implicitely assumes a stable technology, we tested

this assumption using two subperiods (1981–1985 and 1986–1992). The results are qualitatively the

same.

2. For clarity, we omit the t indexes for the matrices and right-hand side elements of the linear program

below. The reference technology includes the entire time period.

3. See Førsund and Hjalmarsson (2002) for an in-depth discussion of issues associated with multiple

solutions when calculating returns to scale.

4. For a more detailed description of the data, see Bilodeau et al. (2000).

5. Results do not extend beyond 1992 because data on physicians were no longer available after 1992.

6. An extra step was necessary to create the complexity index because of the lack of data on total costs

for ambulatory care clinics or on the number of outpatient visits. First, the relative cost of each clinic

is based on a regression of total cost on the number of visits in each clinic or aggregation of clinics, the

complexity index of each care unit as well as its square. An iterated maximum likelihood method

yielded coefficients, which are the corresponding clinic’s marginal cost and are used as prices. We also

added ambulatory care prices and ambulatory psychiatric care prices obtained from the ratio of

spending to the number of visits. Based on prices generated for all the clinics through estimation or

directly from the data, we calculated the complexity indices as for the inpatient care.


Appendix

A.1. Descriptive statistics

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Numbers of residents Number 21.84 49.64 0 329

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