Allocative inefficiency and its cost:: The case of Spanish public hospitals

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Int. J. Production Economics 92 (2004) 99–111

*Correspondin

985-10-48-71.

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(A. Rodr!ıguez- !A

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doi:10.1016/j.ijpe

Allocative inefficiency and its cost:The case of Spanish public hospitals

Ana Rodr!ıguez- !Alvareza,*, V!ıctor Fern!andez-Blancoa, C.A. Knox Lovellb

a Departamento de Econom!ıa, Universidad de Oviedo, Campus del Cristo, 33071 Oviedo, Spainb Department of Economics, Terry College of Business, University of Georgia, Athens, GA 30602, USA

Received 2 October 2002; accepted 21 August 2003

Abstract

In this paper, we present a new empirical model based on an input distance function, from which estimates of

allocative inefficiency and its cost can be obtained. The model avoids the ‘‘Greene problem,’’ which refers to the

difficulty in practice of using a cost system to separate economic inefficiency into its technical and allocative

components. We also develop a procedure to calculate the cost of allocative inefficiency. Using a panel of Spanish

public hospitals to apply our methodology, we find evidence of systematic allocative inefficiency in the employment of

variable inputs. Since inputs are generally poor substitutes, the cost of this allocative inefficiency is high.

r 2003 Elsevier B.V. All rights reserved.

Keywords: Allocative inefficiency cost; Spanish public hospitals; Duality theory; Greene problem; Parametric input distance function

1. Introduction

In this paper, we develop and estimate a modelto analyse the inefficiency of resource allocation inSpanish public hospitals and the cost associatedwith this inefficiency. We test the hypothesis that,given technology and prices, hospital inputs arenot optimally allocated in the sense that costs arenot minimised. To do this, we develop a modelthat allows the unbiased estimation of allocativeinefficiency of input use in two ways: an errorcomponents approach and a parametric approach.In both approaches the general procedure involves

g author. Tel.: +34-985-10-48-84; fax: +34-

ss: [email protected]

lvarez).

front matter r 2003 Elsevier B.V. All rights reserve

.2003.08.012

the estimation of a system of equations formed byan input distance function and the associated inputcost share equations. We allow allocative ineffi-ciency to be systematic, and we incorporate thispossibility into our empirical model. This specifi-cation distinguishes our study from other studiesthat concern themselves with the estimation ofallocative inefficiency.

Using duality theory, we propose a methodol-ogy based on a parametric (Shephard, 1953) inputdistance function, which has certain advantagesover production and cost functions. Unlike aproduction function, a distance function allows forseveral outputs. In contrast to a cost function, itdoes not require the assumption of cost minimisingbehaviour. These advantages are especially im-portant when we consider some characteristics ofthe Spanish public hospital sector: (a) it provides a

d.

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A. Rodr!ıguez- !Alvarez et al. / Int. J. Production Economics 92 (2004) 99–111100

wide range of services and (b) public ownershipand a lack of competition mean that cost control isnot a survival condition. Thus, the absence of aprofit motive weakens incentives to act in a cost-minimising manner thereby allowing for a persis-tent and costly misallocation of resources in theprovision of health care. In this study, we developa model that enables us to analyse this misalloca-tion and its cost.

The paper is organised as follows. In Section 2we outline a model that allows us to determine ifthere exists a costly misallocation of resources inthe provision of health care. In Section 3 wediscuss the two econometric specifications and anumber of econometric issues related to thespecifications. In Section 4 we present and discussour empirical findings. Briefly, we find a pattern ofsystematic input misallocation that has increasedcost by approximately 14% over minimum cost. Inthe final section we summarise our findings.

2. The empirical model

We present two methods that permit thecalculation of allocative efficiency. Atkinson andCornwell (1994) identify the two methods as anerror components approach and a parametricapproach. We demonstrate that replacing theusual cost frontier with an input distance functioncan overcome the main drawbacks of eachapproach.

2.1. The error components approach

This approach is based on the cost frontiersystem of equations:

ln C ¼ ln Cðy;wÞ þ v þ u; ð1Þ

xiwi

C¼

q ln Cðy;wÞq ln wi

þ vi þ Ai; ð2Þ

where C is the cost, y is the output vector, x is theinput vector, w is the input prices vector andCðy;wÞ is a minimum cost frontier. The errorcomponents v and vi represent statistical noise, andare assumed to be distributed as multivariatenormal with zero mean and constant variance.

The error component uX0 represents the cost ofinefficiency (technical as well as allocative). Inprinciple the error component u can be decom-posed into two terms, uAX0 capturing the cost ofallocative inefficiency, and uTX0 capturing thecost of technical inefficiency. In practice this hasproved difficult. The error components Aix0; i ¼1;y; n; represent allocative inefficiencies, and souA cannot be independent of the Ai: The difficultywith the error components approach, denoted byBauer (1990) as the ‘‘Greene problem,’’ is specify-ing a tractable relationship between uA and the Ai:Kumbhakar (1997) used a translog functionalform to derive an exact (and extremely compli-cated) relationship between uA and the Ai; but hisformulation remains to be implemented empiri-cally.

In light of the difficulties associated with the useof a cost frontier system to estimate allocativeefficiency, we formulate a dual input distancefunction system as

ln 1 ¼ ln DI ðy;xÞ þ v þ u; ð3Þ

wixi

C¼

q ln DI ðy;xÞq ln xi

þ vi þ Ai; ð4Þ

where the input distance function: DI ðy; xÞ ¼maxdfdX1 : x=dALðyÞg; where LðyÞ ¼ fxARþ

n : x

can produce yARþmg: For xALðyÞ; DI ðy; xÞX1;

with DI ðy;xÞ ¼ 13x is technically (but notnecessarily allocatively) efficient for y:

As in the cost frontier system (1) and (2), theerror components v and vi represent statisticalnoise, and the error components Ai‘0 representallocative inefficiencies, represented by the differ-ence between actual and stochastic input costshares. However in contrast to (1), the errorcomponent u in (3) represents the magnitude oftechnical inefficiency, rather than the cost oftechnical and allocative inefficiency. Thus, thegreat advantage of estimating the input distancefunction system (3) and (4) instead of the costfrontier system (1) and (2) is that the errorcomponent u in (3) does not include the cost ofallocative inefficiency, and so u and the Ai areinherently independent, and the ‘‘Greene pro-blem’’ disappears.

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2.2. The parametric approach

In this approach, firms are assumed to minimisethe shadow cost of producing a given output vectorfor some shadow input price vector ws; and so

Cðy;wsÞ ¼ minxfwsx : xALðyÞg: ð5Þ

Firms minimise shadow cost by equating marginalrates of substitution with shadow input priceratios, which may diverge from actual input priceratios. The estimation of these shadow price ratios,and their comparison with the market price ratios,enables the calculation of the magnitudes and costof allocative inefficiency.

Starting with Toda (1976), a line of research hasbeen developed in which these shadow price ratioshave been calculated. Initially these studies werebased on the estimation of a cost-based system ofequations from which expressions for actual costand actual input cost shares are obtained fromshadow cost and shadow input cost shares. Thissystem of equations had the property of establish-ing a relationship between shadow input priceratios and market input price ratios, using para-metric corrections kij‘1 to market input priceratios sufficient to satisfy the cost minimisationconditions.

Ideally the parametric corrections to marketinput price ratios would be input- and firm-specific. However, if only cross-section or timeseries data are available, this method must assumetechnical efficiency, and it is unable to estimatespecific kijs for each observation without endo-genising the kijs: The availability of panel dataallows us to obtain firm-specific estimates of bothtechnical and allocative efficiency. This approach,used by Atkinson and Cornwell (1994), does notcompletely solve the problem, however, as it isassumed that technical efficiency and the kijs aretime-invariant for each firm. It is possible to relaxthis assumption by normalising on a reference firm(Balk and Van Leeuwen, 1999), although para-meter estimates are not invariant to the choice ofthe reference firm.1

1 For a comparison of the approaches of Atkinson and

Cornwell (1994) and Balk and Van Leeuwen (1999), who

normalised on a reference firm, see Maietta (2002).

F.are and Grosskopf (1990) modified this ap-proach by using an input distance function, ratherthan a shadow cost function, to represent technol-ogy.2 This makes it possible to estimate kij for eachfirm in each time period. Thus, in a shadow pricemodel (5) where firms are assumed to minimiseshadow cost, the dual Shephard’s lemma

qDI ðy; xÞqxi

¼ wsi ðy; xÞ ¼

wsi

Cðy;wsÞ; ð6Þ

yields the shadow price ratios:

qDI ðy;xÞ=qxi

qDI ðy; xÞ=qxj

¼ws

i

wsj

: ð7Þ

If the cost minimisation assumption is satisfied,these shadow price ratios coincide with actualprice ratios. However, if expense preferencebehaviour causes allocative inefficiency, the twoprice ratios differ. To study the magnitude anddirection of such deviations, a relationship be-tween the shadow price ratios in (7) and the actualprice ratios is introduced by means of theparametric price ratio corrections:

wsi

wsj

¼ kij

wi

wj

: ð8Þ

The degree to which shadow price ratios differfrom actual price ratios is calculated from (8). Ifkij ¼ 1; there is allocative efficiency; if kij > 1; inputi is under-utilised relative to input j; if kijo1; inputi is over-utilised relative to input j:

Numerous studies have used input distancefunctions to estimate the extent of allocativeinefficiency. However, it is preferable to estimatethe input distance function jointly with the shadowinput share equations, thereby improving effi-ciency in estimation. To do this, one estimatessystem (3)–(4) and then obtains estimates of thekijs using (7) and (8) (Grosskopf and Hayes, 1993;Atkinson and Primont, 2002).

However, in system (3)–(4) the Ais are typicallyassumed to have zero means, which has theimportant drawback of introducing the assump-tion that allocative inefficiency is random ratherthan systematic. As the objective of this research is

2 On the other hand, it is possible to calculate output shadow

prices using an output distance function (see for example Reig-

Mart!ınez et al., 2001).

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precisely to determine whether allocative ineffi-ciency is systematic in the Spanish hospital sector,this issue is of particular relevance. In the nextsection, we modify system (3)–(4) to allow forsystematic allocative inefficiency, and we use twoapproaches, the error components approach andthe parametric approach.

3. Econometric specification

We now consider how to estimate system (3)–(4), how to deal with a number of econometricissues.

3.1. Functional form

We have chosen a flexible functional form, atranslog short run multiproduct input distancefunction. This functional form has supplanted theCobb–Douglas functional form used in earlyhospital research (Feldstein, 1967). The short-runinput distance function is specified as

ln 1 ¼B0 þXm

r¼1

ar ln yrht þ1

2

Xm

r¼1

Xm

s¼1

ars ln yrht ln ysht

þXn

i¼1

bi ln xiht þ1

2

Xn

i¼1

Xn

j¼1

bij ln xiht ln xjht

þ xf ln xfht þ1

2xff ln xf 2

ht

þXm

r¼1

Xn

i¼1

rri ln yrht ln xiht

þXn

i¼1

xfi ln xfht ln xiht þXm

r¼1

xfr ln xfht ln yrht

þXT¼1

t¼1

gtT þ eht: ð9Þ

The associated variable input cost share equationsare

xihtwiht

Cht

¼ bi þXn

j¼1

bij ln xjht þXm

r¼1

rri ln yrht

þ xfi ln xfht þ miht; ð10Þ

where y ¼ ðy1;y; ymÞ is an output vector, x ¼ðx1;y;xnÞ is a variable input vector, xf is a quasi-

fixed input, T is a time dummy, h ¼ 1;y;Hdenotes hospitals, and eht and miht are disturbanceterms. Homogeneity of degree +1 of the inputdistance function in variable inputs is enforced byimposing the restrictions

Pni¼1 bi ¼ 1;

Pnj¼1 bij ¼

0;Pn

i¼1 rri ¼ 0 ð8r ¼ 1;y;mÞ;Pn

i¼1 xfi ¼ 0: Wealso impose the symmetry conditions ars ¼ asr;bij ¼ bji; xfi ¼ xif ; xfr ¼ xrf ; rri ¼ rir:

3.2. The error structure

We assume that the disturbance term in (9) hasthe structure

eht ¼ Zht þ dh; h ¼ 1;y;H; t ¼ 1;y;T ; ð11Þ

where ZhtBiid Nð0;s2Þ is a random disturbanceterm, and the dh are hospital-specific disturbancesthat capture unobserved heterogeneity. Sincehospital technology is highly complex, it is unlikelythat an econometric distance function will fullyencompass all the elements that affect it. Ifhospital-specific differences exist and are notexplicitly picked up in the model, a problem ofomitted variables exists and the estimated coeffi-cients of the included variables are biased. AsCarey (1997) points out, the main advantage ofusing panel data when dealing with the hospitalsector is that it is possible to capture unobservedsystematic differences among hospitals (for exam-ple, quality of services and severity of illnesses).

The fixed effects dh are traditionally interpretedas indices of the technical inefficiency of each firm.However, these fixed effects capture not justvariation in technical efficiency but also theinfluence of other variables that have not beenfully incorporated into the model and that do notchange over the sample period, such as servicequality and the geographic location of eachhospital. This will have an effect on the indicesof technical efficiency, which are picked up in thesefixed effects.

Turning to the disturbance term in (10), weassume it has the structure

miht ¼ Ziht þ Ai; i ¼ 1;y;N; ð12Þ

where Ziht is normally distributed with zero mean.The existence of contemporaneous correlationamong the Zs is made possible by the estimation

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3 To test the validity of using IV estimation, we calculate the

Hausman (1978) test of exogeneity. The value of the test

statistic corresponding to the null hypothesis bGLS ¼ bIV is

1113, which exceeds the critical value of the w2 distribution for

46 degrees of freedom at any reasonable significance level (the

critical value at 0.01 level is 71.2). The result of this test

confirms the appropriateness of an IV approach.4 A likelihood ratio test of the Cobb–Douglas restriction on

the translog functional form yields a test statistic of 259.56,

which exceeds the critical value of the w2 distribution for 66

degrees of freedom at the usual levels of significance.5 The autoregressive parameter estimate in the distance

function is �0.0147 with a t-statistic of �0.696. Since it is not

significantly different from zero, we set this autocorrelation

coefficient to zero.


of a system of equations, and it constitutes anadvantage over the estimation of a single equation,since it allows us to assume that stochastic factorsexist that can affect the disturbance terms of thedifferent equations in a period of time.

The Ai terms in (12) are interpreted as measuresof allocative inefficiency, which could be systema-tic in the Spanish public hospital sector. In otherwords, the Ais could have non-zero means. Wetherefore assume that the Ais have means ai; andwe propose the following transformation of (10),inspired by Ferrier and Lovell (1990):

xihtwiht

Cht

¼ ðbi þ aiÞ þXn

j¼1

bij ln xjht þXm

r¼1

rri ln yrht

þ xfi ln xfht þ ðAi � aiÞ þ Ziht; ð13Þ

where the transformed error terms ðmiht � aiÞ havezero means.

3.3. Endogeneity of variable inputs

Given the special characteristics of Spanishpublic hospitals, we assume that hospital manage-ment may have incentives to choose hospitalinputs following criteria other than cost-minimisa-tion (Lee, 1971; Spicer, 1982). If this is so, thevariable input regressors could be endogenous,and for this reason we use an instrumentalvariables (IV) approach. Each variable input isregressed on a vector of variables that is expectedto be correlated with the variable inputs butuncorrelated with the error term. The predictedvalues are used in the estimation of (9)–(13).

3.4. Corrections for autocorrelation and

heteroscedasticity

We introduce intra-equation intertemporal ef-fects by permitting the error terms to follow first-order autoregressive processes. Although equalacross firms, we allow the autoregressive para-meter in the distance function disturbance term todiffer from that in the share equation disturbanceterms. To ensure consistency in summation,however, we impose equality of share equationautoregressive parameters across shares (Berndt

et al., 1993). Heteroscedasticity is corrected usingWhite’s (1980) method.

4. Empirical results

To estimate system (9)–(13) we use an iterativeseemingly unrelated regressions (ITSUR) proce-dure, with instruments for the endogenous vari-ables. In Appendix A we describe our data set usedin the estimation.3 Because the cost shares sum tounity, one of the share equations is deleted.Results are invariant to the choice of equation tobe deleted.

The variables are divided by their geometricmeans, so that the estimated distance function is aTaylor series approximation to the true butunknown distance function at the mean of thedata. The estimated distance function satisfies therequisite regularity conditions at the sample mean:it is non-decreasing and concave in inputs anddecreasing in outputs.4 The AR1 parameter thatwe introduce in each share equation to correct forfirst-order autocorrelation is statistically signifi-cant.5 The parameters estimated from the systemof equations are presented in Appendix B.

4.1. The error components approach

The indices of allocative efficiency estimatedusing the error components approach are pre-sented in Table 1. These parameters indicate thesystematic allocative inefficiency arising from theuse of the corresponding input in a non-cost

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Table 1

Mean values of systematic allocative inefficiencies

Variable Coefficient t-statistic

agraduates 0.0880 2.3463��

atechnicians �0.1031 �2.1884��

asupplies 0.1394 3.3018��

aotherpersonnel �0.1243 �2.2884��

��Statistically significant from zero at 5%.

Table 2

Time effects

Period TCa t-statistic

88–89 �0.0424 �3.8708��

89–90 �0.0811 �7.6770��

90–91 �0.0711 �7.6899��

91–92 �0.0143 �1.4876

92–93 0.0061 0.6806

93–94 0.0143 1.6372�

a Evaluated at the means of the data using parameter

estimates of (9)–(13).�Statistically significant from zero at 10%.��Statistically significant from zero at 5%.


minimising mix. All parameter estimates differsignificantly from zero, which implies that, at thesample mean, the input mix is systematically

inefficient. The proportion in which care graduatesand supplies are used is above optimum, while thatof care technicians and other personnel is belowoptimum. Thus too much is spent on caregraduates and supplies, and too little is spent oncare technicians and other personnel.

The coefficients of the time dummies show theeffect on the distance function of unobservedvariables that, in their evolution through time,affect all hospitals equally. The influence of thepassage of time is given by the expression

TCtþ1;t ¼ gtþ1 � gt: ð14Þ

A positive (negative) value of TCtþ1;t indicates anupward (downward) shift in the distance function,which is typically associated with technical change.The indices obtained from (14) are presented inTable 2. From 1988 through 1992, TCtþ1;to0;indicating that hospital performance deterioratedduring this period. This trend began to reversebeginning in 1992–93. This pattern may reflect theincreased control over hospital administratorscreated by the implementation of the managementcontracts in INSALUD hospitals beginning in1992.6 Gonz!alez and Barber (1996) find similarresults.

4.2. The parametric approach

We now analyse allocative efficiency using theparametric approach. Although it is possible toapply this approach by estimating only the inputdistance function, we opt for its joint estimation

6 Prior to 1992 the hospitals run by INSALUD gesti !on-

directa had retrospective budgets with little or no delegation of

responsibilities. The resulting lack of effective control mechan-

isms provided an incentive to increase expenses. In order to

improve the delegation of responsibilities, the Contratos-

Programa (Management Contracts) were designed to replace

the retrospective budgets. Whereas the latter were based on

historical expenses, the Contratos-Programa were based on

budgets by objectives, with these objectives quantified in

advance, in both activity and financial terms. The new

budgetary system began in 1992, and in 1993 the Contratos-

Programa began to be applied to the INSALUD gesti !on-directa

hospitals.

with the cost share equations to improve efficiencyin estimation. Applying the parameters estimatedin (9)–(13) to (8), we obtain indices of allocativeinefficiency for each observation according to theexpression

kij ¼wjhtxjht

#bi þPn

j¼1#bij ln xjht þ

Pmr¼1 #rri ln yrht þ #xfi ln xffht

h i

wihtxiht#bj þ

Pni¼1

#bij ln xiht þPm

r¼1 #rrj ln yrht þ #xfj ln xffht

h i:

ð15Þ

It is important to distinguish these measures ofallocative inefficiency from those obtained fromthe error components approach, in which the aisrepresent the systematic allocative inefficiency for

each input. In the parametric approach the kijsindicate the allocative inefficiency for each pair of

inputs. Moreover, in the error components ap-proach, behind the ais lies the assumption of anadditive relationship between wi and ws

i : In theparametric approach a multiplicative relationshipbetween wi and ws

i is specified, yielding thecoefficients kij as indexes of allocative inefficiency.

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Table 3

Coefficients kij

Coefficients Meana t-statistic

k graduates, technicians 0.4414 9.3106��

(0.37–0.58)

k graduates, other personnel 0.4568 9.1760��

(0.36–0.57)

k graduates, supplies 1.2720 1.3317

(0.91–1.63)

k technicians, other personnel 1.0553 0.0756

(0.86–1.20)

k technicians, supplies 2.9465 2.5693��

(2.32–3.15)

k other personnel, supplies 2.9759 2.5005��

(2.43–3.43)

Note: Confidence intervals of kij obtained from bootstrapping

are in parentheses. To obtain it the percentile method has been

used. The reestimation of system (9)–(13) with the pseudo-data

generated was repeated 100 times.a Evaluated at the means of the data using parameter

estimates of (9)–(13).��Statistically significant, different from one at 5% level.


Finally, with the error components approach wecan only estimate allocative inefficiencies at thesample mean. The parametric approach avoidsthis drawback. In contrast to the ai coefficients,individual estimates of the kijs for each observa-tion can be identified. Once the parameters of thesystem are estimated, we can obtain firm- andtime-specific measures of allocative inefficiencyfrom the kijs; since these coefficients depend oninput quantities and input prices.

The mean values of the kijs obtained from (15),together with their t-statistics, are presented inTable 3.7 We use bootstrap techniques (Efron andTibshirani, 1986) to obtain confidence intervals forthe kijs: In this way, we can determine if thesecoefficients are robust to small changes of themodel specification. From the results we concludethat both care graduates and supplies are sig-nificantly over-utilised relative to care techniciansand other personnel, and that supplies are insig-nificantly over-utilised relative to care graduates.These findings are consistent with the typicalbureaucratic behaviour of preference for suppliesand highly qualified personnel (Lee, 1971; Spicer,1982).

We have used two different approaches todetermine the nature of allocative inefficiency inSpanish public hospitals. We have discussed thedifferences between the two approaches, but it isalso important to emphasise the relationship thatexists between them. As we have pointed out, ifsystematic inefficiency exists (that is, if the ais havemean values significantly different from zero),their inclusion is necessary if the estimatedparameters, and consequently the estimated kijs;are to be unbiased. This specification differentiatesthe present study from others that have analysedallocative inefficiency in a bureaucratic setting. Bynot allowing allocative inefficiency to be systema-tic, their estimated kijs may be biased. Our findingsprovide empirical evidence that systematic alloca-tive inefficiency exists in this sector. This in turn

7 We have analysed the pattern of the estimated kij

coefficients across hospitals of differing complexity and size,

and have found no important differences. We also have

analysed the time path of the kij and have not found important

trends.

supports the hypothesised bureaucratic model, andimplies that the inclusion of the parameters ai inthe empirical model is indispensable in order toobtain unbiased estimates of the kijs:

4.3. The cost of allocative inefficiency

The results indicate that the variable input mixis inefficient at the mean, and costs could thereforebe reduced through a more efficient allocation.The next logical step in the investigation is todetermine the extent to which the technologyallows substitution among variable inputs. Ifallocative inefficiency exists, it is not very costlyif variable inputs are good substitutes, but it is verycostly if they are poor substitutes. Since the inputdistance function describes the technology, we cananalyse the degree of substitutability among thevariable inputs by means of Morishima elasticitiesof substitution. These elasticities are defined as

Mijðy;xÞ

¼ �d ln½Diðy;x=xiÞ=Djðy;x=xjÞ=d ln½xi=xj

¼ xiDijðy; xÞ=Djðy; xÞ � xiDiiðy; xÞ=Diðy;xÞ

¼ Eijðy;xÞ � Eiiðy;xÞ; ð16Þ

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Table 4

Estimated Morishima substitution elasticities Mij

Meana t-statistic Meana t-statistic

Mgraduates;technicians �0.0155 �0.5838 Mtechnicians;graduates 0.2211 1.7851�

Mgraduates;otherpersonnel 0.1276 14.6371�� Motherpersonnel;graduates 0.6387 4.7887��

Mgraduates;supplies 0.0571 2.0979�� Msupplies;graduates 0.1451 0.6598

Mtechnicians;otherpersonnel 0.7345 3.6204�� Motherpersonnel;technicians 0.8206 3.7837��

Motherpersonnel;supplies 0.4244 1.6841� Msupplies;otherpersonnel 0.1427 0.5710

Mtechnicians;supplies 0.5233 2.5761�� Msupplies;technicians 0.1995 0.8373

a Evaluated at the means of the data using parameter estimates of (9)–(13).�Statistically significant from zero at 10%.��Statistically significant from zero at 5%.

Table 5

Estimated cross and direct price elasticities Eij and Eii

Meana t-statistic Meana t-statistic

Egraduates;technicians �0.0682 �2.3266�� Etechnicians;graduates �0.1715 �3.4995��

Egraduates;otherpersonnel 0.0745 8.6833�� Eotherpersonnel;graduates 0.2201 3.9012��

Egraduates;supplies 0.0040 0.1458 Esupplies;graduates 0.0045 0.1417

Etechnicians;otherpersonnel 0.3419 3.2745�� Eotherpersonnel;technicians 0.4019 3.3260��

Eotherpersonnel;supplies 0.0057 0.0303 Esupplies;otherpersonnel 0.0022 0.0302

Etechnicians;supplies 0.1307 0.9342 Esupplies;technicians 0.0589 0.9075

Egraduates;graduates �0.0530 �22.7467�� Etechnicians;technicians �0.3926 �3.2475��

Eotherpersonnel;otherpersonnel �0.4187 �3.1733�� Esupplies;supplies �0.1405 �0.7016

a Evaluated at the means of the data using parameter estimates of (9)–(13).��Statistically significant from zero at 5%.

8 This approach, which is inspired by Kopp and Diewert

(1982), is also a parametric approach, but it is expressed in

terms of input quantities rather than input prices.


where the Eijs are cross shadow price elasticitiesindicating whether the input pairs are netsubstitutes or net complements, and the Eiis aredirect shadow price elasticities. Estimated Mor-ishima substitution elasticities Mij and their twocomponents Eij and Eii are provided in Tables 4and 5.

The estimated Morishima elasticities Mij suggestvery limited possibilities for substitution amongthe different pairs of inputs. All point estimates arenumerically small, and most are significantly lessthan unity. The same patterns hold for theestimated cross shadow price elasticities Eij : Thesefindings imply that the estimated allocative in-efficiency is likely to be very costly.

To estimate the cost of allocative inefficiency,we define an input vector x� that fulfils thecost minimisation conditions. Then, starting fromthe dual Shephard’s lemma (6), at the optimum

we have

qDI ðy; xÞqx�i

¼q ln DI ðy; xÞ

q ln x�i

DI ðy;xÞx�i

¼wi

Cðy;wÞð17Þ

and for any two inputs,

ðq ln DI ðy;xÞ=q ln x�i Þ=ðq ln DI ðy;xÞ=q ln x�j Þ

wi=wj

¼x�ix�j

: ð18Þ

If we define an input quantity correction zi suchthat x�i ¼ zixi; we can write (18) as8

ðq ln DI ðy;xÞ=q ln x�i Þ=ðq ln DI ðy;xÞ=q ln x�j Þ

wi=wj

¼zixi

zjxj

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Table 6

The cost of allocative inefficiency per hospital per year (thousands of euros)

Actual cost� Optimal cost� Allocative inefficiency cost� Allocative inefficiency cost� (%)

Graduates 6493.71 5890.45 603.27 9.3

Technicians 6701.21 6762.86 �61.65 �1

Other personnel 8029.70 7723.77 305.93 3.81

Supplies 10,162.01 6889.85 3272.17 32.2

Total 31,386.64 27,266.93 4119.72 14

�Cost per hospital and year.


)ðq ln DI ðy;xÞ=q ln x�i Þ=ðq ln DI ðy;xÞ=q ln x�j Þ

xiwi=xjwj

¼zi

zj

: ð19Þ

This is a simultaneous system of n � 1 non-linearequations in n variables z: However, since shareequations are homogeneous of degree zero ininputs, we can normalise by some arbitrarilychosen zjxj : In this way, we obtain zij parametersðzij ¼ zi=zjÞ such that x�i ¼ zijzjxi: From thecalculated values of zij it is possible to derive zj

directly by substituting the zij values into theestimated distance function to obtain

ln 1 ¼ ln DI ½ðy; ðzijzjxiÞ; i ¼ 1;y; n: ð20Þ

Finally, deriving the remaining zi parameters ðiajÞis easy, because zi ¼ zijzj :

Calculated parametric corrections to the opti-mal input quantities at the sample mean are:9

zsupplies ¼ 0:678; zgraduates ¼ 0:907; zotherpersonnel ¼0:962 and ztechnicians ¼ 1:01: Thus to correct allo-cative inefficiency of input quantities, it is neces-sary to decrease utilisation of supplies, graduatesand other personnel by 32.2%, 9.3% and 3.8%,respectively, and to increase utilisation of techni-cians by 1%. Once the optimal input quantitieshave been calculated, the effect of allocativeinefficiency on cost can be evaluated by comparingactual cost with minimum cost. The resultsreported in Table 6 indicate that allocativeinefficiency has increased cost by 14% at thesample mean. The primary source of cost ineffi-ciency is the excessive use of supplies.

9 To perform the optimisation we use the mathematical

program MATLAB. The initial values of zij chosen are (1,1,1).

5. Summary and conclusions

In this paper, we propose an empirical model toevaluate allocative inefficiency. It consists ofestimating an equation system, formed by aninput distance function and (n � 1) input costshare equations, that allows us to estimateallocative inefficiency in two ways: analysing theerror terms of the share cost equations and using aparametric approach. Both enable us to avoid the‘‘Greene problem’’, and to incorporate the possi-bility that the employment of an input in aproportion different from one that would minimisecost could be systematic. This specification allowsus to obtain unbiased estimates of allocativeinefficiency.

We provide an empirical application of thismodel to the study of allocative inefficiency inSpanish public hospitals sector, based on anunbalanced panel consisting of 67 general hospi-tals observed over the period 1987–94. This sectoris characterised by a lack of incentives on the partof the agents who manage public hospitals toadopt the criterion of cost minimisation.

Our results confirm that resource allocation isnot allocatively efficient. We find statisticallysignificant evidence of allocative inefficiency,which takes the form of systematic over-utilisationof supplies and care graduates relative to caretechnicians and other personnel. We also find verylimited possibilities for input substitution, whichimplies that input misallocation is costly. Weestimate the cost of the misallocation to be 14%of actual cost. These findings provide quantitativesupport for our initial hypothesis that cost-minimising behaviour does not characterise theoperation of Spanish public hospitals.

ARTICLE IN PRESS


Appendix A. The data

The data have been obtained from the SpanishMinistry of Health and Consumption. With theobjective of homogenising the sample, we haveexcluded specialised hospitals that cannot beclassified as general hospitals, and hospitals havingfewer than 100 beds (Carey, 1997). During thesample period several hospitals have merged, andwe have treated a merged hospital as a newhospital. As a result, the final sample is anunbalanced panel consisting of 318 observationson 67 general hospitals of the INSALUD gesti !on-

directa over the period 1987–94. To estimatemodel (9)–(13) we need data on variable inputs,quasi-fixed inputs, outputs and operating ex-penses. Table 7 provides definitions of these andother variables used in the analysis.

We classify hospital discharges into medicine(MED), surgery (SUR), obstetrics (OBS), paedia-trics (PED) and intensive care (UIC). Apart fromthese primary hospital activities, other activitiessuch as outpatient visits and emergencies arecarried out in most of the sample hospitals. Thesetwo complementary activities are merged into anaggregate variable (AM), each of the activitiesbeing weighted in accordance with its UPA

Table 7

Definitions of variables used in the analysis

Variable Type Description

MED Output Discharges in general medicine, psychi

SUR Output Discharges in surgery, paediatric and g

OBS Output Discharges in obstetrics.

PED Output Discharges in paediatric medicine and

UIC Output Discharges in units of intensive care, b

AM Output Weighted sum of first and successive v

G Input Care graduates (doctors, pharmacists a

T Input Care technicians (nurses, matrons and

RES Input Other personnel (directive personnel, a

S Input Deflated expenses on sanitary material

BED Quasi-fixed input Number of beds

EG Cost Expenses on assistant graduates

ETEC Cost Expenses on assistant technicians

ERES Cost Expenses on non-assistants and other p

ESU Cost Expenses on supplies

EDU Var. complexity Number of medical students

Dt Time Time dummy variable

classification. The UPA weights hospital activitiesaccording to resource consumption. The weightsare: first outpatient visit (0.25); successive out-patient visit (0.15) and emergencies (0.3).

The variable inputs include care graduates (G);care technicians (T); other non-assistant personnel(RES) and supplies (S). Since S is contaminated bythe effect of price variation, both temporal andspatial, we deflate this variable by the consumerprices index of the National Statistics Agency(INE), taking into account the group of goods towhich each supply type belongs and the differencesin prices that exist between the different regions.

We include as a quasi-fixed input the number ofbeds in the hospital (BED). Although this is clearlynot the most suitable measure, it is commonly usedsince the data relating to capital and its amortisa-tion are unreliable.

To account for expenses on personnel we usesalaries, wages and overtime reported in thedifferent categories. In this sense, the unit cost ofeach personnel category is considered. Expenseson supplies include purchases of disposablegoods.

We also include a teaching variable (EDU) in anattempt to capture the differences in complexityacross hospitals. This variable is defined as the

atry, tuberculosis, long stay, rehabilitation and others

ynaecological surgery

neonatology

urns and intensive neonatals

isits and emergencies

nd other graduates)

others)

dministration managers, qualified and non-qualified personnel)

, drugs, food, clothing, fuels and others

ersonnel

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number of students carrying out their studies in ahospital. It has become common to include thisvariable, since teaching hospitals are generally thelargest and most complex, as well as being locatedin metropolitan areas and having a relatively highinvestment in technology. These hospitals alsohave as an additional mission the professionaltraining of medical students.

Finally we include a dummy variable (Dt) foreach sample year, in an effort to reflect variablesthat, in their temporal evolution, affect allhospitals in the same way.

Table 8

Variable Coefficienta t-statistic

System of equations estimated parameters

L(MED) �0.1542 �5.3477��

L(SUR) �0.0488 �1.6671�

L(OBS) �0.0262 �1.8475�

L(PED) �0.0558 �2.7190��

L(UIC) �0.0049 �1.0734

L(AM) �0.0385 �2.0527��

L(G) 0.1350 3.6200��

L(T) 0.3262 6.9419��

L(RES) 0.3833 7.0838��

L(S) 0.1553 5.0237��

L(BED) 0.1161 2.3272��

L(EDU) �0.0166 �3.8721��

L(MED).L(MED) 0.2285 3.1937��

L(SUR).L(SUR) 0.2304 1.6526�

L(OBS).L(OBS) 0.0055 1.0369

L(PED).L(PED) �0.0002 �0.0254

L(UIC).L(UIC) �0.0019 �0.8067

L(AM).L(AM) �0.0558 �0.9885

L(G).L(G) 0.1099 3.7168��

L(G).L(T) �0.0662 �2.9103��

L(G).L(RES) �0.0232 �0.7834

L(G).L(S) �0.0203 �1.0777

L(G).L(BED) 0.0044 0.3430

L(T) �L(T) 0.0920 2.3420��

L(T) �L(RES) 0.0057 0.1621

L(T).L(S) �0.0314 �1.6950�

L(RES).L(S) �0.0587 �2.3929��

L(RES).L(RES) 0.0762 1.5453

L(S).L(S) 0.1105 4.5014��

L(BED).L(BED) 0.0952 0.6391

L(MED).L(SUR) 0.1655 2.6105��

L(MED).L(OBS) 0.0028 0.1001

L(MED).L(PED) �0.0030 �0.0845

L(MED).L(UIC) 0.0033 0.4017

L(MED).L(AM) �0.1895 �3.1729��

L(MED).L(G) 0.0124 1.4051

L(MED).L(T) 0.0179 2.3935��

As instruments, we employ the exogenousvariables of the model and the following threevariables that we also consider to be exogenous:childbirths, childbirths where the child weighs lessthan 2.5 kg, and the endowment of X-ray rooms ofeach hospital.

Appendix B

The system of equations estimated is presentedin Table 8.

Variable Coefficienta t-statistic

L(OBS).L(PED) 0.0235 1.5317

L(OBS).L(UIC) 0.0081 2.1984��

L(OBS).L(AM) 0.0071 0.3039

G(OBS).L(G) �0.0015 �0.8080

L(OBS).L(T) 0.0004 0.2653

L(OBS).L(RES) 0.0039 1.6820�

L(OBS).L(S) �0.0028 �1.3965

L(OBS).L(BED) �0.0874 �2.5544��

L(PED).L(UIC) �0.0072 �1.8208�

L(PED).L(AM) �0.0204 �0.6921

L(PED).L(G) 0.0045 1.6891�

L(PED).L(T) 0.0006 0.2969

L(PED).L(RES) �0.0023 �0.7113

L(PED).L(S) �0.0029 �0.9881

L(PED).L(BED) �0.0264 0.6297

L(UIC).L(AM) 0.0064 1.2038

L(UIC).L(G) �0.0005 �0.4034

L(UIC).L(T) �0.0005 �0.4775

L(UIC).L(RES) 0.0008 0.5173

L(UIC).L(S) 0.0002 0.1628

L(UIC).L(BED) �0.0329 �3.2697��

L(AM).L(G) �0.0023 �0.2197

L(AM).L(T) 0.0144 1.5405

L(AM).L(RES) 0.0115 0.8392

L(AM).L(S) �0.0235 �2.0122��

L(AM).L(BED) 0.3368 4.7580��

L(T).L(BED) �0.0211 �1.9629��

L(RES).L(BED) 0.0001 0.0068

L(S).L(BED) 0.0165 1.2165

L(EDU).L(EDU) �0.0007 �0.2370

L(MED).L(EDU) �0.0483 �3.8651��

L(SUR).L(EDU) �0.0435 �2.4635��

L(0BS).L(EDU) �0.0014 �0.3026

L(PED).L(EDU) �0.0015 �0.2969

L(UIC).L(EDU) 0.0061 4.1820��

L(AM).L(EDU) �0.0030 �0.3127

L(G).L(EDU) �0.0024 �1.1355

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Table 8 (continued)

Variable Coefficienta t-statistic Variable Coefficienta t-statistic

L(MED).L(RES) �0.0072 �0.6688 L(T).L(EDU) �0.0023 �1.3326

L(MED).L(S) �0.0231 �2.3784�� L(RES).L(EDU) �0.0002 �0.0855

L(MED).L(BED) �0.0632 �0.7497 L(S).L(EDU) 0.0049 2.1730��

L(SUR).L(OBS) 0.0540 3.0476�� L(BED).L(EDU) 0.0784 4.4009��

L(SUR).L(PED) 0.0023 0.0774 AR1 0.4396 16.0545��

L(SUR).L(UIC) �0.0001 �0.0233 g89 �0.0424 �3.8708��

L(SUR).L(AM) �0.0448 �0.6458 g90 �0.1235 �9.9131��

L(SUR).L(G) �0.0252 �2.0202�� g91 �0.1947 �14.5330��

L(SUR.L(T) �0.0109 �1.0387 g92 �0.2090 �13.4983��

L(SUR).L(RES) 0.0025 0.1658 g93 �0.2029 �12.4191��

L(SUR).L(S) 0.0336 2.4493�� g94 �0.1885 �10.6124��

L(SUR).L(BED) �0.3298 �3.2306��

Equation R2 DW S.E. regression

Statistics of the model

Distance function — 1.7454 0.0403

Share graduates 0.4124 1.5282 0.0392

Share technicians 0.3321 1.6253 0.0330

Share other personnel 0.3612 1.3909 0.0476

Share supplies 0.4813 1.6227 0.0435

Number of observations=318.

Number of hospitals=67.a Standard error estimates employ the White (1980) heteroscedasticity-robust computation.�Statistically significant at 10%.��Statistically significant at 5%.


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Allocative inefficiency and its cost:: The case of Spanish public hospitals