Jurnal f CENTRU Cathedra Vlume 4, Issue 2, 20 26-223 JCC Jurnal f CENTRU Cathedra Calitin Frmatin and Data Envelpment Analysis Rlf Färe Oregn State University, Crvallis, OR, USA Shawna Grsspf Oregn State University, Crvallis, OR, USA Dimitris argaritis * University f Aucland Business Schl, Aucland, New Zealand Abstract The intrductin f a framewr fr ptimal calitin frmatin using data envelpment analysis (DEA) methds is the fcus f this paper. Simple eamples illustrate hw DEA is useful in frmulating calitin mdels and deriving ptimal slutins. In particular, the paper shws the relevance f the prpsed framewr in the cntet f analyzing hw cmpanies may reach decisins t acquire ptential partners. eywrds: DEA (data envelpment analysis), calitin frmatin, cre thery, mergers and acquisitins JEL Classificatin cdes: C6, D70 We draw n the thery f the cre, a tpic ften taught in cnnectin with general equilibrium thery, t study calitin frmatin using data envelpment analysis (DEA) methds. In the thery f the cre, there is a set {I} f individuals endwed with preferences and initial allcatins f resurces. A grup f {I}, say a subset S I, can imprve upn its members if each member may increase his r her utility by being a member. The cre then cnsists f the allcatin f resurces upn which n calitin can imprve it. Decisin-maing units (DUs) may enter a calitin with ther DUs t imprve their perfrmance (e.g., revenue efficiency r any ther apprpriate perfrmance measure). 2 Deviating frm the standard thery f the cre, ne can frmulate the agents (DUs) by means f DEA r activity analysis mdels. Thus, data prvided, ne can estimate the pssible gains f frming calitins. Having nly a finite number f agents, ne can estimate the best cnditin(s) fr calitin frmatin. Nte that this paper des nt address the allcatin f gains amng participants in a calitin.
Calitin Frmatin and Data Envelpment Analysis 27 First Eample Fr the sae f cnvenience, a simple illustratin will aid in intrducing the tpic f calitin frmatin and DEA. Cnsider three DUs ( =,2,3) each using ne input ( ) t prduce a single utput (y ). The inputs and utputs are hmgeneus, s their sum is well defined, and 3 3, y are ttal input and utput, = = respectively. Table shws hw the three DUs mae up the DEA technlgy. Table DU Inputs and Outputs DU DU 2 DU 3 Output y y 2 y 3 Input 2 3 Intrducing intensity variables, ne fr each DU, z 0 ( =,2,3), allws fr frmulatin f a DEA r activity analysis mdel. In terms f utput sets, ne culd write such a mdel as fllws: 3 3 P ( ) = y: zy y, z, z 0, =, 2,3 = =. One may prve that this mdel has strngly dispsable input and utput (the first tw inequalities) and ehibits cnstant returns t scale, that is P( λ) = λp ( ), λ > 0. The maimal utput that DU ( =, 2, 3) can prduce is estimated as 3 3 F ( ' ) = a zy : z ', z 0, =, 2,3. = = Its efficiency is the rati y' / F( ' ), =, 2, 3. Allwing DU and DU 2 t frm a calitin wuld raise the questin f hw much utput they culd jintly prduce using their cmbined amunt f input, 2 + 2=. The utput wuld be 3 3 3 3 2 2 2 F( 2) = a zy+ zy : z, z 2, + 2 2, z 0, z 0, =, 2,3 = = = =. Frmulating a calitin between DU and DU 2 wuld be beneficial if F( ) > F( ) + F( ). 2 2 Alternatively, ne culd frmulate a weaer cnditin fr calitin frmatin in this case as F( 2) > y + y2, where F( ) and F( 2) are the maimal utput each DU can prduce nt being a member f any calitin, and y and y 2 are the bserved utputs. Similarly, ne culd frm calitins between DUs and 3, between DUs 2 and 3, r amng DUs, 2, and 3. Determining the best calitin wuld invlve cmparing all the alternatives as fllws: F( 2) vs. F( 3) vs. F( 23) vs. F( 23).
28 Calitin Frmatin and Data Envelpment Analysis One culd als mae weaer cmparisns in relatin t bserved utputs as shwn abve. The net sectin invlves generalizing these ideas int multi-utput multi-input technlgies with finite. The General Case Revenue aimizatin N Let inputs R +, utputs y R +, and assume there are =,..., DUs (r firms). The cnstant returns t scale technlgy may be mdeled via DEA r activity analysis as P ( ) = y: zy m ym, m=,...,, z n n, n=,..., N, z 0, =,..., = = where, =,..., are the intensity variables frming the cnve cne f the bservatins (,y ), =,...,. Thin f as the initial endwment belnging t DU, and assume that sme f the inputs may be reallcated amng the DUs, say inputs n =,..., N*. The rest are nn-allcable and stay with their DU. Nte that ne may have N* = N. Althugh here each DU shares the same technlgy, generated by the data (,y ), =,...,, the DUs utput set may differ because they may have different initial endwments, (e.g., n, n ). In the multi-utput frmulatin, ne cannt maimize utputs, s selectin f a methd that allws fr maimizatin is required. Assuming that utput prices ( p ) are nwn, ne may maimize revenue by maimizing m= p y m m subject t a technlgical cnstraint. The maimum revenue fr DU is ' R (, p) = a pm ym : zy m ym, m=,...,, z n n, n=,..., N, z 0, =,..., m= = =. One may estimate the revenue efficiency fr DU as the rati f bserved revenue ' ' ' t maimum revenue R (, p ), that is, ( Ry )/ R (, p. ) m= py m m ' = Ry ' ( ) Net, estimate the revenue efficiency gain DU can mae by frming a calitin with, say, DU 2. Their jint revenue, R(, 2, p ), is estimated as fllws: 2 m m m m m= m= R(, 2, p) a p y + p y = = = zy y, m=,...,, m m z, n=,..., N, * n n = z, n= N+,..., N, * n n z 0, =,...,,
Calitin Frmatin and Data Envelpment Analysis 29 = = z y y, m=,...,, 2 2 m m z, n=,..., N, 2 2 * n n = 2 z, n = N+,..., N, 2 * n 2n z 0, =,...,, + = +, n=,..., N. 2 2 * n n n 2n Perhaps sme further eplanatin is necessary at this pint. Nte the fllwing:. Each DU has its wn intensity variables, z, z, =,...,. 2 2. Bth DUs face the same utput prices; this case can be generalized t 3. One may reallcate the first n =,..., N* inputs t maimize the jint revenue. ' p, =, 2. 2 4. One can cmpare the calitin s revenue, R(, 2, p ), t individual firm revenue, R(,2, p) Ry (, p) + Ry (, p), t determine whether a calitin is beneficial. Evaluating the best calitin ptin fr DU requires a cmparisn with all ther DUs, such as DU 3 thrugh ; fr eample, (, 2, 3), (, 2, 4), and s n. A best calitin eists with being finite althugh it need nt be unique. The General Case Distance Functins aimizatin When data n utput prices are nt available, ne culd add directinal distance functins. The functins are independent f measurement units and, hence, can be aggregated. They als generalize the first eample f adding (scalar) utputs. The apprach is a generalizatin f Jhansen s (972) industry prductin mdel (see Färe & Grsspf, 2004) and can be viewed as an applicatin f benefit thery due t Luenberger (995). First, let P ( ) be an utput set, P( ) = { y : can prduce y }, and g R+, g 0, a directinal vectr. The directinal utput distance functin is defined as fllws: D (, ; ) sup : ( ) ( ) yg = y+ g P { β β }. The directinal distance functin measures the distance, in the directin f frm y t the bundary f the utput set; and is a generalizatin f Shephard s (970) utput distance functin, { θ θ } D (, ) inf : ( / ) ( ) y = y P, where the relatin between the distance functins, fr g = y, is given by DT ( yy, ; ) =. D( y, )
220 Calitin Frmatin and Data Envelpment Analysis One may estimate the directinal utput distance functin using the DEA r activity analysis frmulatin f the utput set P ( ) as D y g β ' ' (,, ) = sup = zy y + β g, m=,...,, m 'm m = z, n=,..., N, n ' n z 0, =,...,. Net, use a distance functin criterin t evaluate the benefits f frming a calitin. Paralleling the revenue maimizatin case, ne may calculate the jint directinal distance functin in the event that DU and DU 2 frm a calitin as fllws: D (, 2; g) = aβ + β = = m m m 2 zy y + β g, m=,...,, z, n=,..., N, * n n = z, n= N+,..., N, * n n z 0, =,...,, = = z y y + β g, m=,...,, 2 2 m m 2 m z, n=,..., N, 2 2 * n n = 2 z, n= N+,..., N, 2 * n 2n z 0, =,...,, + = +, n=,..., N. 2 2 * n n n 2n Emplying the jint distance functin, D (, 2; g), fr the tw individual DU distance functins, (, D ; ) 2 2 y g and D (, ; ) y g, shws whether a calitin between DU and DU2 wuld be beneficial. Again, by evaluating all pssible calitins, ne can find the best gruping amng DUs.
Calitin Frmatin and Data Envelpment Analysis 22 Secnd Eample The secnd eample relates t a case invlving strategic chices f cmpanies. In particular, the prpsed framewr is useful in analyzing hw cmpanies may reach decisins t acquire ptential partners. Because cmpanies eperience increasing difficulty in achieving and sustaining grwth, ften they resrt t frming strategic alliances (e.g., airlines) r acquiring ther cmpanies (e.g., the massive waves f mergers and acquisitins activity in the late 990s). A hypthetical case invlving three bans will aid in investigating the issue further. Assume that the bans use tw inputs, (persnnel) and (capital), t prduce a single utput, (lans and ther investments), as evident in Table 2. Table 2 Ban Input and Output Data Ban A B C y 2 2 2 2 2 The net questin is with which f the ther tw bans, B r C, Ban A shuld frm a partnership. In this case, allw bth inputs t be reallcated. Befre cmmitting t a strategy, Ban A must assess the amunt f redundant resurces that will be a burden shuld it decide t team up with either Ban B r Ban C. The ban culd use surplus resurces t achieve ecnmies f scale r alternatively cut csts by eliminating thse resurces (Dyer, ale, & Singh, 2004). T answer the questin, ne needs t slve tw linear prgramming prblems: LP Prblem : a y + y 2 z + z + z y Ban A 2 3 z 2+ z + z 2 2 3 z + z 2+ z 2 2 3 2 z 0, =, 2,3 z + z + z y Ban B 2 2 2 2 3 2 z 2+ z + z 2 2 2 2 2 3 2 z + z 2+ z 2 2 2 2 2 3 22 z 0, =, 2,3 2 + 3, + 3 2 2 22
222 Calitin Frmatin and Data Envelpment Analysis LP Prblem 2: a y + y 3 z + z + z y Ban A 2 3 z 2+ z + z 2 2 3 z + z 2+ z 2 2 3 2 z 0, =, 2,3 z + z + z y Ban C 3 3 3 2 3 3 z 2+ z + z 2 3 3 3 2 3 3 z + z 2+ z 2 3 3 3 2 3 32 z 0, =, 2,3 3 + 4, + 3 3 2 32 The results shw that Ban A shuld frm a partnership with Ban C, nt with Ban B. In this case, the ttal utput frm Bans A and C is 2.5 units, which is greater than the bserved utput sum f 2 units prduced by any tw ther bans individually r than the maimum jint utput resulting frm a ptential calitin between Bans A and B, which is als equal t 2 units. The requirement in this eample is weaer than in earlier sectins, but it illustrates the pint. Summary The fcus f this paper was t prpse a framewr and present eamples demnstrating hw ne can frmulate and estimate ptimal calitins using DEA methds. At the center f such analyses may be cases invlving strategic chices f cmpanies (e.g., frming alliances r pursuing taevers in the interest f bsting sales revenue and prfits and maimizing sharehlder wealth). Given that crprate histry is fraught with a myriad f failed acquisitins and alliances while taever activity has remained strng as cmpanies eperience even mre difficulty achieving and sustaining grwth, there is strng interest in develping analytical tls t assist cmpanies in maing better deals. The prpsed framewr ffers sme insights int and tls fr helping cmpanies decide whether they shuld acquire ptential partners.
Calitin Frmatin and Data Envelpment Analysis 223 Ftntes In essence, the cre is a generalizatin f the idea f the Paret set. If an allcatin is in the cre, every grup f agents gets sme gain frm trade, and n grup has an incentive t defect (Varian, 992). 2 Eamples f such calitins may include crprate alliances and crprate taevers (Dyer, ale, & Singh, 2004; artynva & Rennebg, 2008; Sudarsanam, 2003). References Dyer, J. H., ale, P., & Singh, H. (2004). When t ally and when t acquire. Harvard Business Review, 82, 08-5. Färe, R., & Grsspf, S. (2004). New directins: Efficiency and prductivity. Bstn, A: luwer. Jhansen, L. (972). Prductin functins. Amsterdam, Netherlands: Nrth-Hlland. Luenberger, D. G. (995). icrecnmic thery. New Yr, NY: cgraw-hill. artynva,., & Rennebg, L. (2008). A century f crprate taevers: What have we learned and where d we stand? Jurnal f Baning and Finance, 32, 248-277. Shephard, R. W. (970). Thery f cst and prductin functins. New Jersey: Princetn University Press. Sudarsanam, S. (2003). Creating value frm mergers and acquisitins: The challenges. Harlw, England: Prentice Hall. Varian, H. R. (992). icrecnmic analysis (3 rd ed.). New Yr, NY: W.W. Nrtn. Authrs Nte Rlf Färe, Department f Agricultural and Resurce Ecnmics and Department f Ecnmics, Oregn State University, Crvallis, OR, USA. Shawna Grsspf, Department f Ecnmics, Oregn State University, Crvallis, OR, USA. Dimitris argaritis, Department f Accunting and Finance, University f Aucland Business Schl, Aucland, New Zealand. Crrespndence cncerning this article shuld be addressed t Dimitris argaritis, Email: d.margaritis@aucland.ac.nz