Allocative Efficiency Measurement with Endogenous Prices

Transcription

1 Allocatve Effcency Measurement wth Endogenous Prces Andrew L. Johnson Texas A&M Unversty John Ruggero Unversty of Dayton December 29, 200 Abstract In the nonparametrc measurement of allocatve effcency, output prces are fxed. If prces are endogenous, the overall output n the market determnes the allocatve effcent pont. We develop an alternatve sem-nonparametrc model that allows prces to be endogenously determned. Keywords: nonparametrc fronters; allocatve effcency JEL classfcaton codes: C6; D2 Correspondng author. aohnson@tamu.edu; Department of Industral and Systems Engneerng, Texas A&M Unversty, 237K Zachry Engneerng Center, College Staton, Texas ; Telephone: + (979) ; Fax: + (979)

2 page 2. Introducton The semnal work of Farrell (957) ntroduced both the concepts of techncal effcency and allocatve effcency. Allocatve effcency assumes fxed market prces for the nputs used n the producton process and for the outputs (products) generated from the producton process. Much of the ntal work related to producton effcency focused on the agrcultural ndustry, where fxed prces may be a reasonable assumpton due to the compettve nature of the ndustry and the relatvely small mpact any ndvdual farmer has on prces, Farrell and Feldhouse (962), Boles (966). However, as measurng productvty and effcency have become common n other types of producton settngs, such as manufacturng and servces, the same methods for estmatng techncal and allocatve effcency have been used, Lovell (993). A wdely accepted prncple n mcroeconomcs s that frms face downward slopng demand curves; when competton s not perfect, frms output levels nfluence prce, Chamberln (933). As researchers extend the standard nonparametrc techncal and allocatve effcency measurement methods to ndustres that are not perfectly compettve, current methods wll requre adustment to consder the effects of changng output levels on prces for those outputs. Ths s a crtcal consderaton, because a frm that strves to become techncally effcent may actually reduce overall profts by ncreasng the supply of a partcular product and thus reducng the market clearng prce. In ths paper, we ntroduce an approach to model the dependency of prce on output level when estmatng allocatve effcency. Ths modelng approach s more approprate n markets characterzed by monopolstc competton than the standard nonparametrc effcency estmaton models that assume perfect competton. Secton 2 of ths paper ntroduces the notaton and the standard models for estmatng techncal and allocatve effcency. Secton 3 gves an example of the dfferent results of modelng the dependency of prce on output. Fnally, secton 4 concludes.

3 page 3 2. onparametrc Fronters and Effcency Measurement 2 We assume that frms produce a vector of S outputs Y = ( y,..., y S ) usng M nputs X x x M = (,..., ). We further defne the observed output vector for frm for =,..., as Y = ( y,..., ys ) and the observed nput vector as X = ( x,..., xm ). Then, followng Banker et al. (984), the output-orented techncal effcency ( TE ) of frm s measured, assumng varable returns to scale, wth the followng lnear program: TE = max θ λθ, s. t. λ y θ y =,..., S, l l, λ x x k =,..., M, l lk k, λ =, λ 0,...,. l () Ths measure dentfes the radal dstance to the fronter. Techncally effcent frms may be output allocatvely neffcent f the observed mx of outputs does not maxmze total revenue for gven nput usage. Followng Färe et al. (994), we assume that frm faces output prces p = ( p,..., pm ). The maxmum revenue R that can be obtaned relatve to the producton technology s gven by: 2 In ths paper, we focus on output expanson and revenues, holdng nputs fxed. The extenson to an nput orentaton and mnmzng costs s straghtforward.

4 page 4 R S = max p y y, λ = s. t. λ y y =,..., S, l l, λ x x k =,..., M, l lk k, λ =, l λ 0,...,, l (2) where each y s obtaned n the soluton of (2). Fnally, gven observed revenue of S = p y for = R frm, we can measure output revenue effcency as ORE R =. Färe et al. (994) provde a R complete decomposton of output revenue neffcency nto techncal, allocatve, and scale neffcences. 3 The optmal revenue obtaned va (2) assumes exogenous output prces, whch s consstent wth perfect competton but no other market structure. If n fact the frm faces a downward slopng demand curve, the results obtaned n (2) wll over-estmate the revenue level. 3. Example A smple example llustrates ths pont. Suppose fve frms ( A E ) are observed producng dfferng levels of two outputs y and y2 wth the dentcal level of one nput x =. Data are presented n Table. As shown, only frm E s techncally neffcent; relatve to frm B, frm E could expand both outputs by.5 wthout ncreasng ts nput level. 3 For our purposes, we focus on the mplcatons of endogenous output prces wthout regard to the decomposton. Extendng our results to ths decomposton s straghtforward.

5 page 5 Table : Illustratve Data Frm y y 2 A 5 B 3 4 C 4 3 D 5 E We assume frm E faces demand curves pe= 0 0.5yE and pe 2 = 2 ye 2 for outputs y and y2, respectvely. Wth current producton, p E = 9 and p E 2 = Based on observed prces and output quanttes, frm E s revenue s $42.9. Solvng (2) usng these fxed prces, we obtan y = 3, y2 = 4 and R E = $ Accordng to ths model, f E became techncally effcent, ts revenue would ncrease by $2.44. However, ths gnores that ncreased outputs lead to lower output prces. Thus, based on the gven demand curves, prces would decrease to p E = 8. 5 and p E 2 = 8. If frm E became techncally effcent by producng y = 3 and y2 = 4, the resultng revenue would be $57.50, a more modest ncrease than suggested by (2). Of course, ths occurs because of the endogenous prce decreases. However, producng y = 3 and y2 = 4 s not optmal, because the output prce rato changes. In general, a frm wll be able to change ts output relatve to (2) to accommodate the prce-rato change. We note that the soluton of (2) assumng p = 8. E 5 and p = E 2 8 s y = 4, y = 3 2 and R E = 58, a reflecton that the optmal mx s ndeed dfferent when prces are based on techncally effcent producton. Because the mx has changed wth these prces, we need to solve the followng nonlnear program to properly dentfy output revenue effcency:

6 page 6 S = max ( ) λ, y = R p y y s. t. λ y y =,..., S, l l, λ x x k =,..., M, l lk k, λ =, λ 0,...,, l (3) where p ( y ) s the demand functon that depends on output y. 4 Returnng to our example, we fnd the soluton of (3) for frm E s: p p y y R E= 8, E2 = 9, = 4, 2 = 3 and E = $59. The fve-observaton, two-output problem shown here s only for llustratve purposes. The data requrements dscussed n the effcency and productvty lterature for nonparametrc models s mxed. For example Cooper et al. (2007) argue the mnmum data requrements are n max { m s,3( m+ s)} based on degrees of freedom. However, Smar and Wlson (2008) argue that the asymptotc convergence rate of nonparametrc estmators s much slower than those of ther parametrc counterparts and s nfluenced by the dmensonalty of the estmaton space. Thus, more data s necessary to estmate parameters wth a smlar level of confdence n nonparametrc models. We recommend followng the data requrements by Smar and Wlson n practcal applcatons. We have focused on a frm competng under monopolstc competton where the output prce s endogenous. An alternatve consders the olgopoly market structure, where the output and prce decsons of each frm strategcally depend on the decsons of all other frms. In ths case, for example, 4 ote the model proposed n (3) s sem-nonparametrc because the functonal form of the demand functon s assumed. However, f a nonparametrc characterzaton of the demand functon s used, the model wll be fully nonparametrc, see McMllan et al. (989). Johnson and Kuosmanen (2009) dscuss ntegratng multple optmzaton problems wthn an effcency settng.

7 page 7 the demand functon wll depend on the producton levels of all frms. In the long-run, changes n the scale of operaton would also the other frms mpact prcng and output decsons. 5 The olgopoly market structure strengthens our argument that usng fxed prces for outputs s napproprate, but makes the modelng more complex. The fronter benchmark whch all frms would try to acheve (n the classcal Farrell framework) would be techncally and allocatvely effcent (n the producton sense), leadng to a cost-mnmzng mx of nputs and producton at the most productve scale Conclusons Ths paper extends the models avalable for measurng techncal and allocatve effcency to settngs of mperfect competton where output prces are endogenously determned. Thus, a frm attempts to maxmze ts own revenue and the resultng output levels are consstent wth proft maxmzaton, yet are not socally optmal. In our monopolstc competton example, frms use market power to restrct ther output levels, leadng to deadweght loss. The new model descrbed captures the relatonshp between prce and output level when measurng allocatve effcency. Current effcency studes tend to focus on a statc estmaton of cross-sectonal effcency. However, analysts often use these results to gve advce regardng techncal and allocatve effcency strateges. Ths paper mples that movng towards ths statc allocatvely effcent benchmark when prces are endogenous s not approprate. Rather, changes n output prces should be taken nto account when dentfyng an allocatve and techncally effcent benchmark. We suggest that future research could consder a game theoretc model wth both nterdependent and endogenous prces. References Banker, R. D., A. Charnes and W. W. Cooper (984). Some Models for Estmatng Techncal and Scale Ineffcences n Data Envelopment Analyss. Management Scence 30(9): Boles, J.. (966). Effcency squared- effcent computaton of effcency ndexes. Western Farm Economc Assocaton, Annual Meetng, Los Angles, Pullman. 5 We thank an anonymous referee for makng ths pont. 6 Ths would lkely be the Cournot n-frm ash equlbrum soluton. However, a sgnfcant extenson of Cournot competton to multple outputs would be necessary. We leave ths extenson for future research.

8 page 8 Chamberln, E. (933). The theory of monopolstc competton. Cambrdge,, Harvard Unversty Press. Cooper, W.W., L.M. Seford, and K. Tone (2007). Data Envelopment Analyss: a comprehensve text wth models, applcatons, references and DEA-solver software. ew York, Y, USA, Sprnger. Färe, R., S. Grosskopf and C. A. K. Lovell (994). Producton Fronters. Cambrdge [England] ; ew York, Y, USA, Cambrdge Unversty Press. Farrell, M. J. (957). The measurement of productve effcency. Journal of the Royal Statstcal Socety. Seres A. General 20: Farrell, M. J. and M. Feldhouse (962). Estmatong effcent productons functons under ncreasng returns to scale. Journal of the Royal Statstcal Socety. Seres A. General 25: Johnson, A.L. and Kuosmanen, T. (2009), "How Operatonal Condtons and Practces Affect Productve Performance? Effcent Sem-Parametrc One-Stage Estmators." Avalable at SSR: Lovell, C. A. K. (993). Producton fronters and productve effcency. H.O. Fred, C.A.K. Lovell and S.S. Schmdt eds. The Measurement of Productvty and Effcency. ew York, Oxford Unversty Press. pp McMllan, J., A. Ullah, and H.D. Vnod (989). Estmaton of the shape of the demand curve by nonparametrc kernel methods, Advances n Econometrcs and Modelng. Dordrecht: Kluwer, pp Smar, L. and P.W. Wlson (2008). Statstcal nference n nonparametrc fronter models: Recent developments and perspectves. H.O. Fred, C.A.K. Lovell and S.S. Schmdt eds. The Measurement of Productve Effcency and Productvty Growth ew York, Oxford Unversty Press. pp