Coalition Formation and Data Envelopment Analysis

Transcription

1 Jurnal f CENTRU Cathedra Vlume 4, Issue 2, 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.

2 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 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).

3 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, =,...,,

4 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, =,..., 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, =, 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, )

5 220 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.

6 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 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 z + z 2+ z z 0, =, 2,3 z + z + z y Ban B z 2+ z + z z + z 2+ z z 0, =, 2, ,

7 222 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 z + z 2+ z z 0, =, 2,3 z + z + z y Ban C z 2+ z + z z + z 2+ z z 0, =, 2, , 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.

8 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, 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, 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, d.margaritis@aucland.ac.nz