Three Approaches to the Analysis of Cost Function in Health Care

Transcription

1 Ekonomcký časops, 55, 2007, č. 1, s Three Approaches to the Analyss of Cost Functon n Health Care Martn DLOUHÝ Lenka FLUSSEROVÁ* Abstract In ths paper, we descrbe three methods of cost-functon analyss n health care: classcal econometrc analyss, fronter analyss, and survvor analyss. The strength of classcal econometrc analyss s a hghly developed methodology of hypothess testng; the weakness s that t s necessary to deal wth many problems related to estmaton technque, such as multcollnearty, autocorrelaton, heteroscedastcty, etc. The strength of stochastc fronter analyss s that t ncorporates random shocks n effcency evaluaton; on the other hand, strong assumptons about the dstrbuton of effcency have to be made. The advantages of survvor analyss are ts smplcty and the possblty to nclude the factors that are otherwse hard to measure; the dsadvantages are the applcaton only n the long-term studes, and the provson of no specfc nformaton on the character of cost functon n the studed ndustry. Dfferent methods have strengths and weaknesses and the choce of the approprate method depends on the objectve of the study. Keywords: cost functon, econometrc analyss, fronter analyss, survvor analyss, health care JEL Classfcaton: C10, D24, I10 1. Introducton A cost functon descrbes the relatonshp between output (total product) and cost; t shows the mnmal cost of producng the gven level of output. In health care, output can be measured n the number of outpatent consultatons, the number of admssons, the number of npatent days and the lke. The total cost functon * Martn DLOUHÝ Lenka FLUSSEROVÁ, Vysoká škola ekonomcká v Praze, Katedra ekonometre, nám. W. Churchlla 4, Praha 3, Česká republka; e-mal:dlouhy@vse.cz Acknowledgement: The paper was supported by the Grant Agency of the Czech Republc, project No. 402/06/0150 Quanttatve Models for the Analyss of Economc Effcency on Imperfect Markets.

2 70 can be wrtten as TC = c( y), where y s the level of output. The cost of producton can be also related to nputs (the number of beds, physcans or nurses) and prces of these nputs. In ths case, the total cost functon s expressed as TC = p1x1 + p2x pkxk (1) where x j (j = 1, 2,, k) are the quanttes of nputs (factors of producton) and p j (j = 1, 2,, k) are the prces of these nputs. In emprcal studes, researchers also relate cost wth external factors, such as the locaton of hosptal, the type of ownershp, or the chan afflaton. Estmaton of the cost functon offers: (1) to determne margnal cost, (2) to explore the exstence of scale economes, (3) to explore the exstence of economes of scope, (4) to evaluate the cost effcency of producton unt. Margnal cost s the cost related to producton of the addtonal unt of output. The margnal cost s the frst dervatve of the total cost functon. Economes of scale descrbe the stuaton n whch the long-term average cost (AC) of producton declnes wth ncreasng output y. The opposte of economes of scale are dseconomes of scale, whch descrbe stuaton n whch AC(y) < AC(y + d); d > 0. Ths rse of average cost s explaned by neffcences n the management of large organzatons. Economes of scope are possble only for the mult-product case. Hosptals that produce both npatent and outpatent care have a choce whether to provde a partcular servce on the outpatent or npatent bass. Accordng to economes of scope, t s more effcent to produce two products together than separately. It looks obvous n theory, but t s less obvous n practce. Concentraton of servces n health centers and hosptals may lead to excessve utlzaton of accessble servces. If ths s the case, the average costs of servces are lower, but the average cost per patent could actually ncrease. Thus, under some crcumstances, t may be a really dffcult queston whether the concentraton of servces means a hgher qualty or just more expensve medcne. Evaluaton of cost effcency: techncal effcency means that a frm does the best possble job, wthout any waste. If nputs are substtutes, dfferent combnatons of nputs can produce the same output. In ths case, a manager should choose the least costly alternatve of producton. There s a drect lnk between a producton functon and a cost functon: techncal effcency s a necessary condton for cost effcency. 2. Classcal Econometrc Analyss The cost functon s correctly specfed f all mportant varables are ncluded and f the rght functonal form s chosen. The strength of econometrc analyss s a hghly developed methodology of hypothess testng both the selecton of

3 71 varables and the selecton of functonal form can be tested (Gujarat, 2003). The cost functon s usually specfed as a multple regresson equaton, wth the dependent varable beng ether the total cost or the average cost. In many studes, the recurrent cost s only used because of problems wth the captal cost, whch s measured n hstorcal prces. Dependng on the objectve of the study and the data avalablty, the emprcal econometrc studes nclude a great varety of explanatory varables. It s possble to dvde these explanatory varables nto three man categores: a) Input-related explanatory varables For example, the number of departments, the number of beds, the number of physcans, the number of nurses, the number of outpatent departments. b) Output-related explanatory varables For example, the number of outpatent consultatons, the number of admssons (dscharges), the number of npatent days, the case mx (dagnoss, age, race), the ndcators of qualty of servces (hosptal mortalty, readmssons). c) Other explanatory varables For example, the type of ownershp (prvate or publc), the for-proft or non-for- -proft status, the market compettveness, the afflaton wth a mult-hosptal system, the afflaton wth a medcal school, locaton (urban or rural), and so forth. In hosptal economcs, the relaton between the number of beds and cost s especally studed. Ths s probably so for two reasons, frst, the relaton between the hosptal sze and the cost s a key relaton of hosptal economcs ndcatng scale of actvty, and second, the data on hosptal beds are usually the best avalable data on hosptals at all. The ntal, sngle-equaton model of hosptal cost s wrtten as Average Cost = β 0 + β 1 (Number of Beds) + β 2 (Number of Beds) 2 + u (2) A nonlnear relatonshp n the second cost functon (2), whch was ntroduced by the square of the number of beds, enables to study economes of scale. If larger hosptals are able to acheve scale economes, the average cost curve n relaton to scale of actvty s L-shaped. An alternatve assumpton s that, at some pont, the average cost begns to grow due to the neffcent management control over a large organzaton. Such average cost curve s U-shaped and mplctly assumes that there exsts an optmum sze of a hosptal. However, t seems that the number of beds s not as good ndcator of hosptal sze and cost as t was n the past. Wth the development of modern (hgh-cost) medcal technology, the outpatent servces are now a far more mportant part of a modern hosptal than decades ago. In order to assess the qualty of methods and results of the studes of hosptal cost functons, Posnett (Posnett, 2002) suggests that three mportant consderatons are: (a) an approprate unt of measurement, (b) adjustment for case mx, and (c) adjustment for nput prces.

4 72 Unt of measurement. As a measure of effcency, cost per case s superor to cost per day. A hosptal that mproved ts effcency by reducng the average length of stay may have hgher average cost per day because costs are typcally hgher n the frst few days after admsson. Adjustment for case mx. A dfference n case mx s one of the most obvous determnants n cost per case. The studes that do not adequately adjust for dfferences n resource ntensty between hosptals are dffcult to nterpret, especally n relaton to economes of scale. If larger hosptals attract a more resource-ntensve case mx, unt costs may be hgher even n the presence of economes of scale. Adjustment for nput prces. Costs are a functon of the nput mx and the prce of ndvdual nputs. If dfferences n the costs of nputs faced by a hosptal are not adjusted for, they may confound any true underlyng relatonshps between sze and effcency or other varables. 3. Fronter Analyss If there s no recognton of statstcal error, the level of effcency s calculated as the dstance between the observaton and the regresson lne. If the resdual s postve, the unt s relatvely effcent, and vce versa, f the resdual s negatve, the producton unt s relatvely neffcent. The shortcomng of such approach s that t concentrates on the estmaton of average behavor, not on the best performance. The classcal regresson analyss was, therefore, extended to the determnstc fronter analyss and stochastc fronter analyss (Kumbhakar and Lovell, 2000). Both methods can be appled to producton and cost functons. Estmaton of the nput-orented cost effcency s even a more nterestng case than estmaton of the output-orented techncal effcency because cost effcency can be decomposed nto nput-orented techncal neffcency and nput allocatve neffcency. Because techncal effcency s a necessary condton for cost effcency, a cost effcent unt has to be techncally effcent. Let us defne that E s the total expendture of producton unt, y s the vector of outputs produced by unt, x s the vector of nputs, w s the vector of nput prces faced by unt, and β s the vector of unknown technology parameters to be estmated. Let us suppose that outputs and nputs are nonnegatve and prces are postve. The total expendture of th unt s E = w T x. The cost fronter c(y,w ; β) s common for all producton unts. The cost effcency of th unt, CE, may be expressed as the rato of mnmum feasble cost to expendture: c( y, w ; β ) CE = (3) E

5 73 For a cost effcent unt, the observed expendture E equals to the mnmum feasble cost c(y,w ;β), and therefore the cost effcency CE = 1. If the observed expendture E s hgher than the mnmum feasble cost, the producton unt s not cost effcent and CE < 1. Ths formulaton (3) s a determnstc cost fronter, whch gnores random shocks and attrbutes the hgher expendture of the unt to cost neffcency. Notce that cost effcency of the unt can be estmated wthout observng the nput vector x. A stochastc cost fronter s formulated as [c(y, w ; β) exp{v }], where c(y, w ; β) s the determnstc part and exp{v } s the unt-specfc stochastc part of the fronter. The nput-orented cost effcency s then gven by the rato CE c( y, w ; β ) exp{ v} = (4) E The composed error term ε n the stochastc cost fronter model s defned as v + u, where v s the two-sded random-nose component, and u s the nonnegatve cost neffcency component. The composed error ε s asymmetrc and postvely skewed because u 0. The cost fronter must be lnearly homogeneous n nput prces: c(y, λw ; β) = λc(y, w ; β) for λ > 0. One soluton s the restrcton that the sum of the technology parameters β j equals one, or another soluton s that the cost fronter model s reformulated. Let us assume that the stochastc cost fronter takes the Cobb-Douglas functonal form. The reformulated stochastc cost fronter s then wrtten as k 1 E w j β0 β y y β j v wk j = 1 wk ln = ln + ln u (5) where w j s the prce of jth nput faced by th unt, and k s the number of nputs. The frst part of equaton (5) measures the relaton between the expendture E and output y, and the second part of equaton (5) measures the relaton between the expendtures and nput prces faced by unt. The measure of cost effcency for the th producton unt s calculated as CE = exp{ u } (6) The estmates of cost effcency can be obtaned by the mean or the mode pont estmators E(u ε ) and M(u ε ). They are gven by E( u ε ) ϕ( ε λ / σ ) ε λ = σ * 1 Φ( ελ / σ) σ (7) and by

6 74 M ( u ε ) = 2 σ u ε 0 f ε 0, 2 σ otherwse (8) where ε s the composed error term, σ v 2 and σ u 2 are dstrbuton parameters of v and u, Φ( ) s the cumulatve densty functon of standard normal dstrbuton, ϕ( ) s the densty functon of standard normal dstrbuton, σ = (σ v 2 + σ u 2 ) 1/2, λ = σ u /σ v, and σ * = (σ v 2 σ u 2 )/σ 2. When the pont estmates of u are obtaned, the cost effcency of each producer (6) s estmated by exp{ u }. The man advantage of stochastc fronter analyss s that t s able to ncorporate random shocks n effcency evaluaton. On the other hand, strong assumptons about the dstrbuton of effcency have to be made (e.g., exponental dstrbuton of cost effcency s assumed). 4. Survvor Analyss Several complcated ssues must be consdered by a researcher n estmatng cost functons measurng of case mx, handlng dfference n qualty, and dstngushng between the short-run and long-run costs. A survvor analyss, developed by Stgler (1958), s recommended by health economcs textbooks as a smple alternatve to estmatng cost functons (Fedsten, 1998; Folland, Goodman and Stano, 2001). The dea of the method s straghtforward: those categores that grow relatve to the rest of the ndustry are assumed to have some advantage over the other ones. In the long-run, the dstrbuton of provders should tend toward an optmum, whch s, by the analyss, dentfed as the category (-es) wth the fastest growth. Categores may be defned by the sze of hosptal or of group practce (when estmatng economes of scale), by the specalty, by the type of ownershp, by locaton, and so forth. For example, f scale economes exst then, n the long-term, hosptals that are too small or too large wll ext the market and there wll be only those of the optmum sze. An advantage of classcal, unvarate survvor analyss s that the method ncludes both the factor to be nvestgated and all other factors. The analyss thus ncludes factors that are hard to measure n econometrc studes of cost functon. On the other hand, a lmtaton of the survvor analyss s that t s not able to solate the effects of those factors. Ths lmtaton can be moderated by takng an explct account of such factors n an expanded, multvarate survvor analyss. The lnear verson of the multvarate survvor analyss takes the form s = β 0 + β 1 x 1 + β 2 x β k x k + u, (9)

7 75 where s s the change n market share of group, and x 1, x 2,, x k are the explanatory varables (factors). An alternatve s a bnary growth model, but n ths type of model, the nformaton s lost n convertng a contnuous varable nto a bnary one. The unvarate survvor analyss has one advantage over other methods: t s ts smplcty. But both the unvarate and multvarate survvor analyses, lke the other methods of cost analyss, are not able to overcome the fact that the governments, not the market forces play a sgnfcant role n the health sector. Changes n the structure of the health care market may rather demonstrate planned governmental nterventons than a real economc struggle of provders for ther survval. However, the survvor analyss of governmental polcy seems to be also an nterestng applcaton. Accordng to Koutsoyanns (1979), the survvor analyss suffers from serous lmtatons. The survvor analyss assumes that: (a) the frms pursue the same objectves (hosptals are both prvate and publc); (b) the frms operate n smlar envronments so that they do not have locatonal or other ad-vantages (e.g., publc hosptals receve subsdes); (c) prces of factors and technology are not changng (e.g., prces of health technology grow rapdly); (d) the frms operate n a very compettve market structure, that s, there are no barrers to entry or collusve agreements, snce under such condtons neffcent (hgh-cost) frms would probably survve for long perods of tme (hosptals are local monopoles, the entry s regulated). Another shortcomng of the survval analyss s that t s not able to explan cases where the sze dstrbuton of frms remans constant over tme. If the share of the varous plant szes does not change over tme, ths does not mply that all scales of plant are equally effcent. Koutsoyanns (1979) also argues that the survvor analyss ndcates only the broad shape of the long-run cost curve, but t does not show the actual magntude of economes of scale. 5. Applcatons Although the methodology of cost functon can be appled to a varety of producton unts n health care (physcan, department, hosptal, communty health center, health system), the health care applcatons are domnated by the studes where the producton unt s a hosptal. The analyss of cost functon wth methods of quanttatve economc analyss are relatvely rare n the Czech health system, therefore, we also present llustratve examples from other countres. In ther study on hosptal effcency, Dlouhý and Strnad (1999) estmated hosptal total cost functon on a sample of 40 Czech acute-care hosptals, the 1997 data. The ndependent varables were: the number of beds, the number of beds n ntensve care unts, and the number of outpatent departments. All parameters were

8 76 statstcally sgnfcant. As the model assumed the constant average cost, the exstence of economes of scale could not be tested, whch was a weakness of the study. Frech and Mobley (1995) tred to resolve the problem of hosptal economes of scale by the multvarate survval analyss. They analyzed hosptal data from Calforna for years 1983 and In the contnuous verson of ther survval model, the dependent varable was defned as a change n market share, measured n npatent days, for a gven group of hosptals. The explanatory varables ncluded the level of output (average daly census), the chan afflaton, the Herfndahl ndex of market concentraton, the change n local market-level proporton of hosptal revenues under dscount contracts, and some other varables. The study found the exstence of scale economes possbly up to a sze as large as 220 bed-days (370 beds). Over the through perods, Blodeau, Crémeux et al. (2004) studed all short-term hosptals n the Provnce of Québec, Canada. They used two-step analyss: frst, they estmated techncal effcency; second, the costs of techncal neffcency were calculated. Blodeau et al. found that sgnfcant neffcences up to 17% could have been saved through mproved performance. Post-estmaton analyss that ncluded quanttatve measures of care suggested that dfferences n performance were attrbutable to dfferences n management and unobservable qualty of care rather than patent case mx. The nterestng result was that the 20 worst performng hosptals out of 119 over the perod studed had accounted for nearly 50% of total neffcences. Mark (1996) emprcally examned whether an ownershp affects the qualty of psychatrc care. He estmated a qualty devaton functon and a fronter cost functon usng data on psychatrc hosptals n Calforna for the years The estmaton of parameters was obtaned by two-stage least squares. The analyss found the evdence that nonproft psychatrc hosptals experenced fewer complants and volaton than for-proft psychatrc hosptals. There was no evdence that nonproft hosptals were more neffcent than for-proft hosptals. Those fndngs supported the vew that descrbes nonproft hosptals as offerng advantages n markets characterzed by the asymmetrc nformaton. Posnett (2002) revewed the lterature on economes of scale and found that larger hosptals dd not have to have necessarly lower average costs and better outcomes. The research lterature suggests that f economes on scale exst, they appear to be fully exploted at a relatvely low scale, somewhere n the range of beds. Ths sze s lkely a mnmum sze of acute general hosptal that s able to provde all complementary medcal, dagnostc and support servces. For hosptals wth 200 and more beds, we may observe a roughly constant average cost. Somewhere n the range of beds the average cost may be

9 77 expected to rse. It s mportant, however, to bear n mnd that optmum hosptal sze s a drect functon of the health care needs of the populaton that t s desgned to serve. Concluson In ths paper, we descrbed three methods of cost-functon analyss n health care: classcal econometrc analyss, fronter analyss, and survvor analyss. The strength of classcal econometrc analyss s a hghly developed methodology of hypothess testng both the selecton of varables and the selecton of functonal form can be tested. The weakness of econometrc analyss s that t s necessary to deal wth many problems related to estmaton technque, such as multcollnearty, autocorrelaton, heteroscedastcty, etc. The strength of stochastc fronter analyss s that t ncorporates random shocks n effcency evaluaton. On the other hand, strong assumptons about the dstrbuton of effcency have to be made. The advantages of survvor analyss are the smplcty and the possblty to nclude the factors that are otherwse hard to measure. The dsadvantages of survvor analyss are: the method can be appled only n the long-term studes (not applcable to cross-sectonal data); t does not provde specfc nformaton on the character of cost functon n the studed ndustry. Hence results of survvor analyss are not controlled for possble confoundng factors, such as case mx and so on. No method of cost analyss s able to overcome the fact that the governments, not the market forces play a sgnfcant role n the health sector. Changes n the structure of the health care market may rather demonstrate governmental nterventons than a struggle of health provders for ther economc survval. Health care s an applcaton area wth very specfc context and characterstcs. But ths complexty s the reason why health care as an applcaton area s so challengng for researchers. The man topc of ths study was showng how quanttatve economc analyss of cost functon could be useful n evaluatng effcency of resources n the health system. Dfferent approaches have strengths and weaknesses and the choce of the approprate method depends, of course, on the objectve of the study. References [1] BILODEAU, D. CRÉMIEUX, P.-Y. JAUMARD, B. OUELLETTE, P. VOVOR, T. (2004): Measurng Hosptal Performance n the Presence of Quas-fxed Inputs: An Analyss of Québec Hosptals. Journal of Productvty Analyss, 21, pp [2] DLOUHÝ, M. STRNAD, L. (1999): Nemocnce kvalta, efektvta, fnance [A Hosptal: Qualty, Effcency, Fnancng]. Praha: Grantová agentura Mnsterstva zdravotnctví ČR. <

10 78 [3] FELDSTEIN, P. J. (1998): Health Care Economcs. Ffth Edton. Albany, NY: Delmar Publshers. [4] FOLLAND, S. GOODMAN, A. C. STANO, M. (2001): The Economcs of Health and Health Care. Thrd Edton. Upper Saddle Rver: Prentce Hall. [5] FRECH, H. E. III MOBLEY, L. R. (1995): Resolvng the Impasse on Hosptal Scale Economes: A New Approach. Appled Economcs, 27, pp [6] GUJARATI, D. N. (2003): Basc Econometrcs. Fourth Edton. New York: McGraw-Hll. [7] KOUTSOYIANNIS, A. (1979): Modern Mcroeconomcs. Second Edton. London and Basngstoke: The Macmllan Press. [8] KUMBHAKAR, S. C. LOVELL, C. A. K. (2000): Stochastc Fronter Analyss. Cambrdge: Cambrdge Unversty Press. [9] MARK, T. L. (1996): Psychatrc Hosptal Ownershp and Performance: Do Nonproft Organzatons offer Advantages n Market Characterzed by Asymmetrc Informaton? The Journal of Human Resources, 31, pp [10] POSNETT, J. (2002): Are Bgger Hosptals Better? In: M. McKee and J. Healy (eds.): Hosptals n Changng Europe. Buckngham: Open Unversty Press. [11] STIGLER, G. J. (1958): The Economcs of Scale. The Journal of Law and Economcs, 1,