Profit sharing in a unionized differentiated duopoly

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1 Proft sharng n a unonzed dfferentated duopoly Emmanuel Petraks Panagots Skartados May 2016 Work under progress. Please do not quote. Abstract Proft sharng, wth one form or the other, has been n wde use all over the world. Unonzaton does survve and plays a major role n wage settng n most OECD countres. We have endogenze unons decson of mergng, and frms decson of gvng proft sharng, n a multstage Cournot unonzed duopoly. In equlbrum, unons wll always merge, and multple equlbra arse: proft sharng mght be used by none, by one frm, or by both frms. JEL Classfcaton: J30; J40; J20 Keywords: unon olgopoly; barganng; proft sharng; Petraks E.(correspondng author) and Skartados P.: Department of Economcs, Unversty of Crete, Unv. Campus at Gallos, Rethymnon 74132, Greece, e-mal: petraks@uoc.gr. We would lke to thanks semnar partcpants of ASSET 2015 and ICMAIF 2016 for ther useful comments. We would lke to thanks Stamatopoulos G., and Vlasss M. for ther useful nsghts. Full responsblty for all shortcomngs s ours. 1

2 1 Introducton Proft sharng scheme s a type of remuneraton scheme, n whch addtonally to the fxed wage receved by the worker, part of a worker s salary s based on the profts made by the frm. Proft sharng schemes can be pad n cash, stocks or other, and can be pad annually, sem-annually or be kept by the frm, and be gven to workers at the form of a penson. In practce, proft sharng s qute complex, and contans a set of dfferent elements (OECD 1995). Nevertheless, proft sharng, wth one form or the other, has been n wde use all over the world (Dhllon and Petraks 2001). A survey of the largest 1250 global corporatons (Weeden et al. 1998) found that 33% of them offered some short of proft sharng schemes to all employees, and another 11% had plans to put n place a broad-based proft sharng scheme. Lberal economc reforms and deregulatons of 1970 s and 1980 s n USA, UK and elsewhere greatly affected unonzaton. But, emprcal studes (Vsser 2006) and OECD statstcs (stats.oecd.org) shows that unonzaton s not dead; s n effect today, but wth strong varatons from a country to another. Unons mght be naton-wde (central grand unon of workers) or frm-specfc (decentralzed dedcated unon); n any case they can metamorphose the remuneraton scheme bargan agenda, and strongly affect product market competton. Product market competton, unonzaton form and remuneraton schemes, as we wll show, are nterdependent. A strong unon affects remuneraton scheme, whch can affect frm s labor cost, whch can affect product prces, consumer s welfare, and fnally competton. The drvng force that moved us to do research over proft sharng s the generally accepted dea that proft sharng can lead to hgher frm profts and hgher worker earnngs, but not always do so. We consder: a) a decentralzed envronment wth two separate and dedcated unons U and U j, and two competng frms F and F j, or b) a central envronment wth one grand (naton-wde?) unon U, and two competng frms F and F j. The frms use a homogeneous and sngle qualty labor force suppled by the unon(s), as the only nput to produce a dfferentated product. Frms compete n product market à la Cournot (quantty competton). Usng a Leontef-style producton functon for both frms (constant returns to scale), 2

3 frms also decde for the amount of labor force they wll rent from ther respectve unon(s). In ths envronment, the choce of horzontal mergng of unons, allows the nternalzaton of an nsttutonal choce often found n labor economcs: f a central grand unon can delegate better n a bargan, n contrast to multple frm-specfc unons. Furthermore, the choce of contract type from frms (fxed wage or proft scheme), nternalzes the choce of the optmum remuneraton scheme over dfferent states of unons. All these are backed from a competton stage, n whch frms compete over quanttes or prce, formng a rght to manage framework, where the amount of labor force used by frms, s a frm-specfc choce, but at the same tme, the total compensaton s a matter of bargan between frm and unon. We have chosen the so called rght to manage delegaton model because we beleve that t s more realstc than a monopoly wage settng from unons (often used n lterature). In our defense, Horn and Wolnsky (1988) crtcze those who smply assume that unons are settng wages unlaterally, rather than partcpatng n some bargan process wth the frm. Our contrbuton to lterature s mult-fold. a) Frst, we hghlght the sgnfcance of remuneraton schemes and unonzaton for product market competton, consumer s welfare and employment. b) Second, we extent exstng lterature on Cournot competton by usng product dfferentaton 0 < γ < 1, and central coordnated bargans between one grand unon and two separate frms. Both of them are absent from exstng lterature. c) Thrd, we prove that mx stuatons are possble. Wth mx stuatons we mean one frm usng fxed wage scheme, whle th other usng proft sharng scheme. And wth possble we mean that ths mx stuaton s sub perfect pure strategy Nash equlbrum. The rest of the paper s as follows: second secton makes a lterature revew; thrd secton descrbes the model; fourth secton makes the equlbrum analyss, solves the game and shows the results; ffth secton states the results of ths research; sxth secton makes some conclusons and suggest further research. Fnally, follows references. 3

4 2 Lterature Revew Proft sharng schemes ganed a lot of attenton due to Professor Wetzman s work. Wetzman (1983, 1984, 1985, 1987), a poneer of proft sharng schemes, states that proft sharng makes the cost of labor completely flexble and gves frms the ncentve to hre as many workers as are wllng to take jobs. Ths leads to an economy of proft sharng frms wth lower levels of unemployment and greater macroeconomc stablty. Lterature on the motve of ntroducton of proft sharng (OECD 1995) suggests that several groups of varables affects the ntroducton of proft sharng, such as: frm sze, nternal organzatonal structure, ndustral relatons, labor and legal nsttutons, and the external envronment. Sesl, Kroumova, Blas et al. (2002), studed 229 US major New Technology Frms (pharmaceutcals, semconductors, software, telecommuncatons & hgh-technology manufacturng), all offerng broad-based proft sharng plans. Usng multvarate analyss wth panel data they found that, n contrast wth ther non-proft sharng counterparts, proft sharng frms productvty ncreased 4%, total shareholder returns ncrease by 2%, and proft levels jump by about 14%. These gans are after dluton effect s taken nto account. Arrvng at smlar results, Kruse (1992), uses data from almost 3000 US frms, from 1971 to 1985, and shows that the ntroducton of a proft sharng scheme s (statstcally sgnfcant) assocated wth a productvty ncrease of 2.8% to 3.5% for manufacturng frms, and 2.5% to 4.2% for non-manufacturng frms. Kruse suggests that only the most proftable and most productve frms ntroduce proft sharng schemes, n order to algn frm and workers nterests, and through ths algnment to reach new, hgher levels of proftablty and market share. In Germany, Kraft and Ugarkovc (2005), usng panel data from more than 2000 German frms from 1998 to 2002, show that the ntroducton of a proft sharng system mproves proftablty. Bensaïd and Gary-Bobo (1991) are the frst to vew proft sharng schemes not just as an nternal ncentve system, but as a strategc commtment. They prove that the adopton of a proft sharng scheme by a frm and ts unon shfts market equlbrum and s a Pareto mprovement for both partes. They have shown that the choce of a proft sharng scheme by each frm s an equlbrum, and n the case of Cournot competton, the game has the structure of the prsoners dlemma. Last, they argue that proft sharng 4

5 schemes seems to be credble n short and mddle run, but not n the long run, because of nsttutonal and legal nfluences n contracts. Sorensen (1992) sets a three stage game, n whch two frms producng a sngle type homogeneous product, and two unons, one for each frm, as a sngle labor suppler for that frm. In stage one, frms decde the remuneraton system (fxed wage versus proft share), n stage two there s a determnaton of wages and proft share va barganng among frms and unons, and n stage three there s a Cournot style competton among frms, whch determnes output, prces, and employment levels. He shows that unon s bargan power s of crtcal mportance n frm s choce of ntroducng a proft sharng scheme. Nevertheless, the ntroducton of a proft sharng scheme seems to ncrease employment, and decrease prces. Fung (1989a) examnes proft sharng n three dfferent competton regmes: monopoly, perfect competton and Cournot olgopoly. He found that proft sharng can lower wages n all three dfferent types of competton. Ths wll not lower rents extracted from the unon, because there wll be a postve proft share percentage taken as well. Also, states that the jont rents of the unon and the frm wll ncrease wth proft sharng. Fnally, he shows that, under some restrctons, unon s wage demand wll be smaller under proft sharng than under fxed wage regme. In an artcle related wth the prevous, Fung (1989b) sets up a two-stage duopoly game n order to explore the effects of proft sharng n unemployment rates. In frst stage unons set wages as Bertrand duopolsts (maxmzng the economc rent extracted from labor), whle n the second stage frms compete à la Cournot. Under the assumptons of hs model, Fung fnds out that the effects of proft sharng can be decomposed n two parts; frst, an ndustry wde effect n whch proft sharng causes a wage reducton, whch leads to a lower product prce and thus n bgger total quanttes produced. Ths causes employment to rse. Second, a frm specfc effect n whch frms usng proft sharng schemes gan a bgger market share, and have hgher employment wth lower wage rates. These benefcal effects gve to the proft sharng frm a strategc advantage over the non proft sharng counterparts. An argument n ths paper s that because n many Japanese ndustry sectors, proft sharng s common, ths mght be a possble explanaton for the low unemployment rates and economy s success n ths country. Stewart (1989) proves that whle for a monopolst, the ntroducton of a proft sharng 5

6 scheme t s not a Pareto mprovement, for a frm n an olgopolstc sector there are ncentves for such an ntroducton. He sets up a two stage n-opoly game, usng standard assumptons. In frst stage, one of the homogeneous n-frms commt to proft sharng scheme, whle n second stage all n-frms compete à la Cournot. Holdng workers ncome constant (subsdzng equally a part of fxed wage wth a proft share), there s an ncrease n frm s profts. Ths creates ncentves to more frms enterng a proft sharng scheme, even f enterng all n-frms n proft sharng schemes creates profts below the level obtaned under the wage system. Goeddeke (2010) analyzes the emergence of proft sharng schemes, when wages are negotated n two extremes: a blateral monopoly way or decentralzed way. She s based upon Sorensen (1992) homogeneous product model, but t extends t, by usng an n opoly structure. She founds that n n opoly, under decentralzed bargans, frms have ncentves to replace fxed base wage schemes wth proft sharng schemes. But when the majorty of frms (depends on magntude of n) adopts proft sharng schemes, t s n the collectve nterest of both frms and unons to move back to fxed base wages. Also, she shows that the exstence of proft sharng scheme creates ncentves for frms to bargan ndependently, and not centrally. Negotatons between frm and unon over the proft share are not common n all ndustral countres. Poole (1988), surveys Brtsh frms, and shows that the decson about the level (percentage) of proft shares made by the managers n 97.7% of the cases studed, whle only 0.7% was made under bargan. Ths assumpton s behnd the research made by Pemberton (1991); he assumes that the very exstence and extent of proft sharng cannot be a process of bargan between frm and unon, but n real world s part of frm s proft maxmzaton strategy. He proves that n the absence of product subsdes, proft sharng schemes wll not be proftable for the frm, but f frms facng heterogeneous demand, and have dfferent technologes, there are ncentves for frms to apply a proft share. In France, snce 1967, there s a legal oblgaton for frms employng more than 50 workers, to apply some short of proft share, calculated on a bass of predetermned proft sharng formula. In 2011 ths legal oblgaton was renforced and became wder, requrng frms to also pay a socal dvdend, f ther net profts were hgher than last year. In a recent study (Fang, 2016) there has been a revew of emprcal studes showng 6

7 that proft sharng s benefcal for employees through hgher earnngs and employment stablty, and at the same tme s benefcal for employers through hgher workspace productvty, and thus hgher frm proftablty. Proft sharng reduces supervson costs and s aganst shrkng behavor, whle at the same tme creates a bgger flexblty n wages. The author concludes that labor unons may want to work collaboratvely wth management to enhance the mutual benefts of proft sharng. A short lterature revew over unonzaton could reveal a great varaton: unonzaton s stll strong n some countres, whle n other s not. Unon densty 1 n Vsser (2006) and OECD Statstcs onlne 2, shows that durng the frst decade of 2000, n some countres (e.g. Iceland, Belgum, Fnland, Denmark, Norway, Sweden) unonzaton s way above 50%, whle n some other countres (e.g. France, Korea, USA, Japan, Span, Turkey, Netherlands) unonzaton s below 20%. Many countres le n between 20% and 50% (Canada, Italy, Ireland, Greece, Austra, Luxembourg). These charts shows a great varaton n unonzaton, but nevertheless unonzaton s n effect and wth stable percentages durng 2000 s. Labor market nsttutons show hgh varablty as of the level of wage negotatons. In Unted States, Unted Kngdom, Australa, Canada, and Japan, wages are usually barganed on a decentralzed frm level between unons and frms. In contrast, n Germany, Austra, Belgum, Greece and Scandnavan countres, bargan take place ether on an sector-wde level or even at a natonal level (Goeddeke 2010). 3 The model 3.1 Market structure and contract types We consder an ndustry, consstng of two competng frms, denoted F and F j respectvely. Each frm has ether: a) a separate, dedcated workers unon (decentralzed verson) as ts sole suppler of labor, denoted U and U j respectvely, or b) both frms face the same grand unon of workers (central coordnated verson), denoted U. We assume that: ) the two frms F and F j 1 Rato of employed unon members to all employed cvlans. 2 OECD Trade Unon Densty charts could be found at use a two-factor Leontef producton 7

8 functon q = Mn[L, K ], where L =Labor and K =Captal, ) produce under constant returns to scale, ) the amount of captal K s fxed n the short run, and large enough not to nduce zero margnal product of labor(l << K ). From these three assumptons we get that quantty equals employment: q = L, = 1, 2. Furthermore, we assume that each frm produces a sngle type of a dfferentated good. Frm F faces the followng nverse demand functon: P = α q γ q j, = 1, 2, where γ (0, 1) measures the dfferentaton between the two products 3. Total quantty sold by frms equals Q = q + q j. Because of the assumpton q = L, total quantty Q equals total employment Q = L = L + L j. We assume that market s bg enough (α s bg enough) to consume all quantty produced and sold, and thus no stock s kept by the frms. We assume a constant margnal cost per product, beng the sum of a (non-labor) cost: average cost = margnal cost = c per quantty q produced, where 0 < c < α, and a labor cost (wage rate) w per unt of labor L = q. Frms assumed to have no other costs or ncome, so they have a (gross 4 ) proft functon equal to: Π = P q (c + w ) q = (α c w q γq j ) q, = 1, 2 & j (1) Frm s decson makers (management) are assumed to be rsk-neutral, thus ther utlty functon equals (net) profts, whch are: a) wth proft share π = (1 a ) Π, = 1, 2, and b) wthout proft share: π = Π, = 1, 2. Unon s leaders are also assumed to be rsk-neutral, thus ther utlty functon can be modeled as: a) wth proft share UW P and b) wthout proft share UW W = [((w w 0 ) + a Π L )L ] = [(w w 0 )q + a Π ], = (w w 0 )L = (w w 0 )q where w s the fxed base wage, and w 0 0 s the (exogenous, and assumed constant) unemployment beneft 5. We use the unon s utlty functon of Pemberton (1988), who shows that a rentmaxmzng unon s equvalent to a manageral unon wth unon leaders who are n- 3 As n Sngh and Vves (1984), ths demand functon comes from a quadratc and strctly concave utlty functon of a contnuous of representatve consumers. Quadratc mean that consumers prefers more to less, and strctly concave mean consumers are rsk averse 4 For frms we use Π = for gross profts.e. profts before proft share (f applcable), and π =: net profts.e. profts after profts share deducton(f applcable). In case of usng a fxed base wage as remuneraton scheme, then gross and net profts are the same π = Π. For unons we use UW : unon s welfare, whch s calculated always after the addton of proft share percentage. 5 Unemployment benefts exsts all over the world. They protect workers durng unemployment aganst major ncome loses or acceptng jobs below ther qualfcatons. They ought to be below wages, else they could rase unemployment rates. 8

9 terested n employment, and unon members who are nterested n excess wages (.e. the amount (w w 0 )). 3.2 Sequence of events Frms and unons engage n a four stage game wth observable actons. Game tmng reflects the dea that long run decsons, such as the formng of grand coaltons of workers, may have consderable strategc effects n short run decsons, such as the employment decson made by the frm. Ths tmng mples that some varables (such as employment) are easer to change and are greatly affected by other varables, whch may be much more dffcult to alter. Ths tmng s standard n lterature, and allows us to capture the contract forms commtment value. Stage 1 Merger Stage Stage 2 Remuneraton Stage Stage 3 Bargan Stage Stage 4 Competton Stage Unons decde to merge or not. Frms decde about the remuneraton scheme. Bargans between frms and unons. Product competton between frms. Stage 1 The merger stage. Ths s a unon s decson stage. The unons decde whether they wll merge or not horzontally, formng a grand unon of workers. We use two dfferent types of bargan: central coordnated and decentralzed. Stage 2 The remuneraton scheme stage. Ths s a frm s decson stage. Frm F decdes whether to offer a fxed wage w, or a proft sharng scheme made from a fxed wage plus a proft share percentage (w, F ). Ths s a take t or leave t offer n a rght to manage bargan framework 6. There are four possble outcomes: a)both frms decde to gve a proft sharng scheme, b)both frms decde to gve fxed wage, and c) & d)a double symmetrcal case, where one frm apples a proft sharng scheme, whle the other apples a fxed wage scheme. Stage 3 The barganng stage. In stage 3, two separate unons U & U j, or one grand unon U from one sde, and two frms F & F j on the other sde, bargan over the remuneraton scheme. The soluton concept used here s the sub perfect pure strategy 6 There s a strong debate between rght to manage and Effcent Bargan frameworks. We have chosen rght to manage because we feel that no unon can enforce a certan amount of employment to a frm; employment has to do wth producton, and producton has to do wth competton. 9

10 Nash equlbrum of smultaneous generalzed Nash Barganng problems. Unon(s) has bargan power 7 equal to β (0, 1) whle frm has bargan power equal to (1 β). We assume that unon(s) leadershp engage n wage barganng wth frm s management on behalf of ther (same skll/homogeneous) afflated workers. Frms and unon(s) negotate smultaneously, and t s assumed that at the end they wll reach an agreement. The man goal of solvng ths stage s to fnd, under all possble remuneraton schemes and under all possble forms of centralzed/decentralzed bargan, Nash equlbra values of wages w and of proft sharng (f applcable) a for = 1, 2, as functons of exogenous varables. Stage 4 The competton stage. Frms compete n Cournot style competton, where frms maxmze profts over quantty. Ths stage determnes employment, snce q = L. 3.3 Barganng Framework We model the bargan between one grand unon U or two separate unons U 1 and U 2, and two separate frms F and F j as to be done smultaneously and separately. Ths assumpton captures the fact that each par of negotatons (U & F vs U & F j ) has ncentves to behave opportunstcally and to reach a mutually favorable agreement that enhances frm s compettve poston n expense to the other rval frm. We obtan a unque equlbrum by mposng parwse proofness on the equlbrum contracts. That s, we requre that the contract between U and F s mmune to a blateral devaton of U wth the rval frm F j, holdng the contract wth F constant. We also mpose exclusvty n frm-unon relatons. Each frm supples t s labor force from a sngle dedcated unon of workers, whle a sngle unon of workers supply t s labor force to a sngle frm. If bargan between them fals, then both gets zero profts. Two addtonal key assumptons of our model are passve belefs and non-contngent contracts. Passve belefs state that frm F wll handle any out-of-equlbrum offer from U as a tremble, uncorrelated wth any offer from U to rval frm F j. F beleves that under any offer receved from U, the par U & F j has reached an equlbrum outcome. Dfferent belefs wll gve rse to dfferent equlbra, as stated n lterature (McAfee and Schwartz 7 Not many thnks are mentoned n lterature about bargan power β. It s generally accepted beng an exogenous varable of the game, manly affected from legal framework, frm s nternal organzaton, team sprt, unon s ablty to strke, frm s costs of hrng, tranng, and frng, unemployment rates and/or dffcultes to match frm s needs wth sklled workers, wage dscrmnaton, and many more. 10

11 1994). Non-contngent contracts state that any breakdown n bargans between F and U wll be non-permanent and non-rrevocable, and ths s common knowledge (Horn and Wolnsky 1988). Ths wll lead the rval par F j and U to bargan n a blateral monopoly fashon, wth F j employng and sellng monopoly quantty, but usng the same wage w j and the same proft share percentage a j as n duopoly (because of the fact that any breakdown n bargans between F and U wll be non-permanent). 4 Frms competton and barganng outcomes 4.1 Stage 4: Cournot Competton In the last stage of the game, frms F & F j engage n quantty (Cournot) competton. Under Cournot competton, frms maxmze ther profts Π over quantty q : max[π (q, q j, w )] = max[(p (q, q j ) c w ) q ] = max[(α q γq j c w ) q ] (2) q q q The frst order condtons gve rse to the followng reacton functon: R (q j, w ) = 1 2 (α c γq j w ) (3) A decrease n the wage w faced by F shfts ts reacton functon upwards, and turns F nto a more aggressve compettor n product market. Solvng the system of reacton functons for = 1, 2 & j, we obtan the equlbrum quanttes and ( gross ) profts for gven levels of wages: q (w, w j ) = (2 γ)(α c) 2w + γw j 4 γ 2, Π (w, w j ) = (q (w, w j )) 2 (4) 4.2 Stage 3: Barganng Stage In Stage 3 frms and unon(s) bargan over wages and proft shares. Snce the barganng game n case of one grand central unon s dfferent from the case of two decentralzed unons, we wll analyze the two cases separately. 11

12 4.2.1 Stage 3a: Decentralzed unons When unons are separate and dedcated, then two pars (U, F ) & (U j, F j ) exsts. Each par has to choose between a fxed wage scheme (w ), or a proft sharng scheme (w, a ). Due to symmetry, there are three possble subgames: I. Fxed wage case: both frms use fxed wage schemes, II. Mx case: one frm use fxed wage, whle the other use proft sharng, and III. Proft sharng case: both frms use proft sharng schemes. We mpose an exclusvty n frm-unon relatons; each frm gets labor force from ts own unon, whle each unon supply only one frm. If bargans fal, then both frm and unon gets zero profts. I. Fxed wage case Both frms use fxed wage schemes. Unons are separate and dedcated to respectve frms. Frms and unons bargan over the followng Nash bargan product: NBP DF (w, w j ) = [Π (w, w j )] 1 β [(w w 0 ) q (w, w j )] β (5) where superscrpt DF stands for: D=decentralzed bargans, and F=fxed wage remuneraton schemes for both frms. Substtutng Π (w, w j ) and q (w, w j ) from Equaton 4, and maxmzng NBP DF (w, w j ) over wage w we get: max w [NBP DF ] w DF (w j ) = 1 4 (4w 0 + 2β(α c w 0 ) βγ(α c w j )) (6) Notce that wdf (w j ) w j > 0, that s: an ncrease n one frm s wage, wll cause the other frm to ncrease wages as well. Imposng symmetry n equlbrum: w DF (w j ) = wj DF (w ) = w DF, we derve the equlbrum wage: w DF = w 0 + β(2 γ)ã 4 βγ (7) w DF γ We have set: 0 < α c w 0 = ã < α. Clearly: w DF > w 0, whle wdf β > 0 and < 0. Equlbrum wage ncreases, as unon s bargan power ncreases, and decreases as product dfferentaton decreases (γ 1). Nash equlbrum quanttes, prces, and 12

13 ( gross = net ) profts are: q DF = 2(2 β)ã (γ + 2)(4 βγ), ΠDF = π DF = (q DF ) 2, P DF 2(2 β)(γ + 1)ã = α (γ + 2)(4 βγ) (8) It s qdf β < 0 and qdf γ < 0, so equlbrum wage, and equlbrum profts are decreasng as unon s bargan power ncrease, and as product dfferentaton decrease. Furthermore, equlbrum prce ncrease as unon s bargan power ncrease, and decrease as product dfferentaton decrease. Overall, a strong unon wll demand hgher wages. Frms wll bypass ths hgher labor cost to consumers through hgher prces. Lesser product dfferentaton wll enhance product market competton, leadng to prce cut, and so to lower frm s profts. Unon welfare and Consumer Surplus are: UW DF = UW DF j = 2(2 β)β(2 γ)ã2 (γ + 2)(4 βγ) 2, CS DF = 8(2 β) 2 ã 2 (γ + 2) 2 (4 βγ) 2 (9) Unon welfare s ncreasng wth ts own bargan power β, and decreasng as product dfferentaton decreases. Consumer surplus ( 1 2 Q2 ) s decreasng when unon s bargan power ncrease, and when product dfferentaton decrease. II. Mx case In ths mx case scenaro one frm use fxed wage scheme, whle the other use proft sharng scheme. Unons are separate and dedcated to respectve frms. Due to dfferent remuneraton schemes, the two barganng pars (U, F ) & (U j, F j ) bargan over dfferent Nash bargan products: NBP DM (w, w j, a ) = [(1 α)π (w, w j )] 1 β [(w w 0 )q (w, w j ) + απ (w, w j )] β (10a) NBP DM j (w, w j ) = [Π j (w, w j )] 1 β [(w w 0 ) q j (w, w j )] β (10b) where superscrpt DM stands for: D=decentralzed bargans, and M=mx case. Maxmzng NBP DM (w, w j, a ) over both w and a, and maxmzng NBPj DM (w, w j ) 13

14 over w j only, we get: max[nbp DM (w, w j, a )] w DM (w j ) & a DM (w, w j ) (11a) w,a max w j [NBP DM j (w, w j )] w DM j (w ) (11b) Due to the exstence of proft share percentage 0 < a < 1, par s U & F remuneraton contract s blaterally effcent, that s, t maxmzes the jont surplus of the barganng par, gven the rval par s barganng outcome. In contrast, a fxed wage remuneraton scheme contract s not blaterally effcent, snce t does not nclude a proft share percentage. max[nbp DM (w, w j, a )] a NBP DM (w, w j, a ) a = 0 a M (w, w j ) = β [Π (w, w j )] (1 β) [(w w 0 )q (w, w j )] Π (w, w j ) (12) The reason because a s dvded by Π (w, w j ) s the fact that a s a percentage of frm s profts, so t has to be normalzed to 0%-100%. The use of a s to splt the pe of jont profts n two peces, n accordance wth each party s bargan power. If we set Equaton 12 to Equaton 11, then we get: NBP DM (w, w j ) a M (w,w j ) = (1 β) 1 β β β JP DM (w, w j ) (13) where: JP DM (w, w j ) = Π (w, w j ) + (w w 0 ) q (w, w j ) are the jont profts of bargan par F & U. That s, maxmzng Nash bargan product s about maxmzng jont profts. Solvng the system of these three equatons over (w DM, w DM j, a DM ) we get: w DM q DM = = w 0 (2 γ)(βγ + 4)γ2 ã, a DM 32 + βγ 4 16γ 2 = β + 1(1 2 β)γ2 (14a) β(2 γ)(γ + 2)(4 γ(γ + 2))ã w DM j q DM j = = w βγ 4 16γ 2 (14b) 2(2 γ)(βγ + 4)ã, π DM 32 + βγ 4 16γ 2 = (1 a DM ) (q DM ) 2 (14c) 2(2 β)(γ(γ + 2) 4)ã, π βγ 4 16γ 2 j DM = (qj DM ) (14d) We can easly check that: w DM < w 0 whle w DM j 14 > w 0. The frm F that uses a proft

15 sharng scheme s able to subsdze fxed wage wth proft share percentage, effectvely lowerng fxed wage below unemployment beneft. Total worker s compensaton (both proft share and fxed wage) s stll above unemployment beneft, but the fact that part of fxed labor cost transformed nto proft share (varable cost) gave to frm F a strategc advantage over frm F j whch use a fxed wage scheme only. Furthermore, wdm β both wages (w DM w DM a DM β fxed wage. < 0 and wdm γ < 0, whle wdm j β > 0 and wdm j γ < 0. That s, whle and wj DM ) lower wth product dfferentaton (as γ goes to 1), wage lowers wth unon s bargan power β whle w DM j s ascendng. Consder also that > 0, we get the ntuton that a stronger unon wll prefer more proft share and less Unons welfare and consumer surplus are: UW DM = 2β(2 γ)2 (2 γ 2 )(βγ + 4) 2 ã 2, UW DM (βγ 4 16γ ) 2 j = 2(2 β)β(4 γ2 )(4 γ(γ + 2)) 2 ã 2 (βγ 4 16γ ) 2 CS DM = 8(2β(γ 1) γ(γ + 4) + 8)2 ã 2 (βγ 4 16γ ) 2 (15a) (15b) Both unons U & U j have ther welfare decrease as product dfferentaton decrease, and U has ts welfare ncrease as bargan power ncrease. For U j, nterestngly, thngs are a bt more complcated: UW j > 0 only for β(γ ) < 32 16(1 β)γ 2. So, there s an area n (β, γ)-plane where UW j < 0. Especally for β = γ = 1 ths holds true. In mx case, when product s homogeneous, monopolstc bargan power wll harm the unon wth fxed wage. III. Proft sharng case Both frms use proft sharng schemes. Unons are separate and dedcated to respectve frms. Frms and unons bargan over the followng Nash bargan product: NBP DP (w, w j, a ) = [(1 a )Π (w, w j )] 1 β [(w w 0 )q (w, w j ) + a Π (w, w j )] β (16) where superscrpt DP stands for: D=decentralzed bargans, and P=proft sharng remuneraton schemes for both frms. Substtutng Π (w, w j ) and q (w, w j ) from Equaton 4, and maxmzng NBP DP (w, w j, a ) over wage w, and over proft share percentage a, we get: 15

16 max[nbp DP ] w DP (w j ) & a DP (w, w j ) (17) w,a Imposng symmetry n equlbrum: w DP (w j ) = wj DP (w ) = w DP and a DP (w, w j ) = a DP j (w, w j ) = a DP, we derve the equlbrum wage and proft share: w DP γ Notce that: w DP < 0 and wdp β w DP = w 0 γ 2 ã 4 + γ(2 γ) & adp = β (1 β)γ2 (18) < w 0 whch s n contrast wth other versons of the game, whle = 0. The last nequalty s dfferent from other versons of the game, showng that a varaton n unon s bargan power wll not affect wages. Nash equlbrum quanttes, prces, and ( gross and net ) profts are: q DP = 2ã 4 + γ(2 γ), πdp = (1 a DP )(q DP ) 2, P DP 2(1 + γ)ã = α 4 + γ(2 γ) (19) The nterestng concluson when both frms usng proft sharng, s that equlbrum wage, quantty, profts and prce are ndependent of unon s bargan power β. Ths has a straghtforward economc ntuton: under proft sharng, unon and frm maxmze jont profts, are to dstrbute them they use proft share percentage and not wage. Thus, bargan power affects only proft sharng and not any other varable of the game. A decrease n product dfferentaton (γ 1) means lower equlbrum wage, lower equlbrum quantty, profts and prce, but hgher proft share (n favor of the unon). Unon welfare and consumer surplus are: UW DP = UW DP j = 2β(2 γ2 )ã 2 (4 + (2 γ)γ) 2, CSDP = 8ã 2 (4 + (2 γ)γ) 2 (20) Unon welfare s decreasng as product dfferentaton decreases, and ncreasng as unon s bargan power ncreases. Interestngly, n ths verson of the game, consumer surplus s ndependent of unon s bargan power β. Mathematcally ths happens because equlbrum quantty s ndependent of β. An economc ntuton, though, says that through proft sharng unon and frm maxmze jont profts, and face consumers rather lke a blateral monopoly, so unon s bargan power has nothng to do wth quantty produced or consumer surplus; s part of an nternal procedure of splttng jont profts nto two parts, weghted by each party s bargan 16

17 power Stage 3b: Central grand unon We wll analyze the case of one grand unon of workers U, a monopolst of labor to both frms F and F j. Ths sngle monopolst of labor creates consderable dfferent ncentves to frms. To llustrate the dfference wth decentralzed bargans, consder the followng example: f n decentralzed bargans, a frm-unon par, say U & F, fals to reach an agreement over wages, then they both get a zero proft. In dfferent words, the dsagreement payoff of both frms and unons was zero. Thngs are dfferent n central bargans. If a grand unon of workers U (a monopolst of labor supply to both frms) fal to reach an agreement wth one frm, say frm F, then there s always the other frm F j to bargan wth. Ths creates the so called outsde opton (or dsagreement payoff) n favor of the grand unon. Whle frm F stll gets zero profts f an agreement s not set, a grand unon has a mnmum securty net of profts, whch holds ts profts way above zero. In Horn and Wolnsky (1988), we encounter a report over dfferent types of outsde optons. In ther paper, they use a non-contngent duopoly outsde opton: f the grand unon U fals to bargan wth frm F (so F gets zero labor force and thus produce zero quantty and have zero profts), then the other frm F j wll fal to notce, and par U & F j wll bargan over duopoly quantty and prces. We choose to use a non-contngent monopoly outsde opton: f the grand unon U fals to bargan wth frm F (agan zero quantty and profts for frm F ), then frm F j wll notce that, and thus par U & F j wll bargan over monopoly quantty and prces. We have chosen ths type of outsde opton because t s more reasonable to suppose that the falure n negotatons between the par U & F wll be notced by the other frm F j, whch wll adjust ts behavor from duopolst to monopolst, knowng that t s the last and only chance the grand unon has to sell ts workforce. Havng these n mnd, we model outsde opton as: OutOpt CF (w j ) = (w j w 0 ) qj Mon (w j ) = (w j w 0 ) 1(α c w 2 j) (21) for fxed wage remuneraton schemes, and: 17

18 OutOpt CP (w j ) = (w j w 0 )qj Mon (w j )+a j Π Mon j (w j ) = (w j w 0 ) 1(α c w 2 )+a j [ 1(α c w 2 )] 2 for proft sharng schemes. Also, profts of grand unon U are now profts made from supplyng labor to both frms, nstead of just one frm n decentralzed verson of the game, so: (22) GUW CF (w, w j ) = (w w 0 ) q (w, w j ) + (w j w 0 ) q j (w, w j ) (23) for fxed wage remuneraton schemes, and: GUW CP (w, w j ) = (w w 0 )q (w, w j ) + (w j w 0 )q j (w, w j ) + a Π (w, w j ) + a j Π j (w, w j ) for proft sharng schemes. (24) I. Fxed wage case Both frms use fxed wage schemes. There s only one grand unon for both frms. Frms and grand unon bargan over the followng Nash bargan product: NBP CF (w, w j ) = [Π (w, w j )] 1 β [UW CF (w, w j ) OutOpt CF (w j )] β (25) where the superscrpt CF denotes: C=central coordnated bargans, and F=fxed wage. We wll substtute q (w, w j ) and Π (w, w j ) from Equaton 4, and we wll maxmze NBP CF (w, w j ) over wage w. We wll do the same for par U & F j, and we wll mpose symmetry n equlbrum w CF equlbrum quanttes, profts, and wages: (w j ) = wj CF (w ) = w CF. After all these steps, we wll get w CF = w βã, qcf = (2 β)ã 2(γ + 2), πcf = Π CF = (q CF ) 2, P CF (2 β)(γ + 1)ã = α 2(γ + 2) (26) Wage w CF s hgher than unemployment beneft w CF > w 0, and becomes hgher as unon s bargan power rses wcf β rses qcf β < 0. > 0. Moreover, quantty falls as unon s bargan power 18

19 Grand Unon s welfare and consumer surplus are: GUW CF = (2 β)βã2 2(γ + 2), CS CF = (2 β)2 ã 2 2(γ + 2) 2 (27) Grand unon s welfare decreases as product dfferentaton decrease, and ncrease wth grand unon s bargan power. On the other hand, consumer surplus decreases wth both product dfferentaton decrease, and wth grand unon s bargan power ncrease. II. Mx case In ths mx case, one frm, say frm F, use a proft sharng scheme (w, a ), whle the rval frm F j use a fxed wage scheme (w j ). Both frms face the same grand unon of workers U, whch has an outsde opton of not gvng labor force to one frm, makng the other frm a monopolst n product market. Nash bargan products of these pars (U, F ) & (U, F j ) are: NBP CM (w, w j, a ) = [(1 a )Π (w, w j )] 1 β [(w w 0 )q (w, w j ) + (w j w 0 )q j (w, w j )+ + a Π (w, w j ) (w j w 0 )q Mon j (w j )] β (28a) NBP CM j (w, w j, a ) = [Π j (w, w j )] 1 β [(w w 0 )q (w, w j ) + (w j w 0 )q j (w, w j )+ + a Π (w, w j ) (w w 0 )q Mon (w ) a Π Mon (w )] β (28b) where the superscrpt CM denotes: C=central coordnated bargans, and M=mxed case. Substtutng q (w, w j ) and Π (w, w j ) from Equaton 4, and solvng the bargan problem for par U & F we get: max[nbp CM (w, w j, a )] w,a NBP CM (w, w j, a ) w = 0& NBP CM (w, w j, a ) a = 0 (29) From these two frst order condtons we wll get two expressons: one for wages w CM (w j, a ), and one for proft share a CM (w, wj). We have proven before that the exstence of proft share percentage 0 < a < 1 makes the remuneraton contract blaterally effcent, that s t maxmzes (excess) jont profts of par U & F. Solvng the bargan problem of the other par U & F j we get: 19

20 max[nbpj CM (w, w j, a )] w j NBP CM j (w, w j, a ) w j = 0 (30) Ths frst order condton wll gve us an expresson of wage w CM j (w, a ). Solvng the system of three equatons we derve equlbrum expressons of wages, profts, proft shares, quanttes, and prces. We have closed form solutons, whch wll not state, n order to save some space. What we could state are the followng: w CM j and qcm = 0. β > w 0, wcm j > 0, γ Grand unon s welfare s a very bg expresson to state here (expressons avalable upon request), whle consumer surplus s: CS CM = 1 8 (2 γ)2 ã 2 2(β 2)(β(γ 2 2)γ (γ + 2)γ 2 + 4) ( (γ 2 2)((β 1) 2 γ 4 2((β 1)β + 2)γ ) γ 2 )2 (31) Grand unon s welfare s always ncreasng wth grand unon s bargan power β, whle GUW CM γ β and γ. has no stable sgn. On the other hand, consumer surplus s decreasng wth both III. Proft sharng case Both frms F and F j have decded to gve proft sharng schemes, whle unons have merged nto one grand unon U. Par U & F bargans over the followng Nash bargan product: NBP CP (w, w j, a, a j ) = [(1 a )Π (w, w j )] 1 β [(w w 0 )q (w, w j )+ + (w j w 0 )q j (w, w j ) + a Π (w, w j ) + a j Π j (w, w j ) (w j w 0 )q Mon j (w j ) a j Π Mon (w j )] β where superscrpt CP denotes: C=central coordnated bargans, and P=proft sharng scheme for both frms. Maxmzaton of Nash bargan product NBP CP (w, w j, a, a j ) over proft sharng percentage a leads to excess jont profts maxmzaton, where excess jont profts are: j (32) EJP CP (w, w j, a j ) = (w w 0 )q (w, w j ) + (w j w 0 )q j (w, w j ) + Π (w, w j )+ + a j Π j (w, w j ) (w j w 0 )qj Mon (w j ) a j Π Mon j (w j ) (33) 20

21 So: max w [NBP CP (w, w j, a, a j )] max w [EJP CP (w, w j, a j )] w CP (w j, a j ) (34) And: a CP max a [NBP CP Imposng symmetry n equlbrum: w CP (w, w j, a, a j )] a CP (w, w j, a j ) (35) (w j, a j ) = wj CP (w, a ) = w CP (w, a), and (w, w j, a j ) = a CP j (w, w j, a ) = a CP (w, w, a), and solvng the system of equaton we get equlbrum expressons: w CP = w 0 + γ(β(γ2 (γ + 4) 8) γ(γ(γ + 4) 4))ã 2γ((γ + 2)((β 1)γ 2 2β) + 6γ) 16 (36a) a CP 4(β + 2) = (β 1)γ (36b) q CP = (γ 2)((β 1)γ 2 + 4)ã 2γ((γ + 2)((β 1)γ 2 2β) + 6γ) 16 π CP = (1 a CP ) Π CP = (1 a CP ) (q CP ) 2 P CP = α (γ 2)(γ + 1)((β 1)γ 2 + 4)ã 2γ((γ + 2)((β 1)γ 2 2β) + 6γ) 16 Grand Unon s welfare and consumer surplus are: (36c) (36d) (36e) GUW CP = (γ 2)((β 1)γ2 + 4)(β(γ(γ(γ(γ + 2) + 4) 4) 8) γ 3 (γ + 2))ã 2 2(γ((γ + 2)((β 1)γ 2 2β) + 6γ) 8) 2 CS CP = (2 γ) 2 ((β 1)γ 2 + 4) 2 ã 2 2(γ((γ + 2)((β 1)γ 2 2β) + 6γ) 8) 2 (37a) (37b) Grand unon welfare s ncreasng wth β, whle surplus s decreasng wth both γ and β. GUW CM γ has no stable sgn. Consumer 4.3 Stage 2: Remuneraton stage In ths stage, frms separately and smultaneously decde over the remuneraton scheme. They have two optons: a) to gve a fxed wage, or b) to gve a proft sharng scheme. 21

22 Ths create a world wth four possble outcomes. Due to mposed symmetry, these four outcomes collapse nt three: ) both frms gve fxed wage schemes, ) both frms gve proft sharng schemes, and ) one frm gve fxed wage whle the other gve proft sharng scheme. The decson over whch remuneraton scheme to provde, s clearly a part of frm s proft maxmzaton procedure. It s clear by now that ths decson s hghly depended on values of product dfferentaton γ, and unon s bargan power β, and t s a dfferent decson when unons are separate than when they are merged. To sum up, we state the followng proposton β I III β I III 0.2 II (a) (β, γ) plane n decentralzed verson γ 0.2 II IV (b) (β, γ) plane n central verson γ Fgure 1 Proposton 4.1. equlbra: When unons are separate (decentralzed verson), we have two Both frms gve Fxed wage, when β, γ are n Area I of Fgure 1a, Both frms gve Proft Sharng, when β, γ are n Area II of Fgure 1a. Area III of Fgure 1a, s a double equlbrum area. Pareto-ranked equlbrum: both frms to gve fxed wage. In decentralzed verson, mx case s never an equlbrum. When unons merge (central coordnated verson), we ave three equlbra: Both frms gve Fxed wage, when β, γ are n Area I of Fgure 1b, Both frms gve Proft Sharng, when β, γ are n Area II of Fgure 1b, 22

23 Mx case s an equlbrum, when β, γ are n Area III of Fgure 1b. Area III of Fgure 1b, s a double equlbrum area. Pareto-ranked equlbrum: both frms to gve fxed wage. Proof. Usng the expressons we have derved for frm s profts under fxed wage, and under mx case, t s straghtforward to verfy that (devatons reasonng): A) For decentralzed unons: () π DF > π DM for β, γ n Area I and III of Fgure 1a. ()π DP > π DM j for β, γ n Area II and III of Fgure 1a. () Followng system of nequaltes never hold: π DF < π DM & π DP < πj DM, so mx case can never be an equlbrum n decentralzed verson of the game. (v) In Area III of Fgure 1a, where both equlbra exsts, we can Pareto-rank them because for β, γ n ths are the followng hold: π DF > π DP. So f a frm can choose between two possble equlbra, t wll choose the one wth the hghest profts. ()π CP π CF B) For a central unon: () π CF > π CM for β, γ n Area I and III of Fgure 1b. > π CM j for β, γ n Area II and III of Fgure 1b. () System of nequaltes: < π CM & π CP < πj CM hold for β, γ n Area III of Fgure 1b, so mx case can be an equlbrum n central coordnated verson of the game. (v) In Area III of Fgure 1b, where both equlbra exsts, we can Pareto-rank them because for β, γ n ths are the followng hold: π CF > π CP. So f a frm can choose between two possble equlbra, t wll choose the one wth the hghest profts. 4.4 Stage 1: Merger stage In ths stage, unons decde whether to bargan separately, or to merge and form a grand unon. The followng proposton clarfes ths decson. Proposton 4.2. Unons always have ncentves to merge. Central coordnated verson s the only equlbrum verson for any β (0, 1) and γ (0, 1). Proof. A) When both frms gve fxed wage, the followng hold under any value of β (0, 1) and γ (0, 1): UW DF + UW DF j < GUW CF. B) When both frms gve fxed wage, the followng hold under any value of β (0, 1) and γ (0, 1): UW DP + UW DP j < GUW CP. C) When one frm gve fxed wage and the other frm gve proft sharng scheme, the followng hold under any value of β (0, 1) and γ (0, 1): UW DM 23 +UW DM j < GUW CM.

24 D) There are spots n β, γ-plane, where frms mght choose a dfferent remuneraton scheme under decentralzed verson, or central coordnated verson. It s straghtforward to prove that these cross nequaltes hold for any β (0, 1) and γ (0, 1): UW DF + UW DF j < GUW CP, UW DP +UWj DP < GUW CMP, and UW DF +UWj DF < GUW CM. 5 Conclusons In Introducton and n Lterature Revew we have seen emprcal evdence and stylzed facts that support proft sharng as a wde spread remuneraton scheme. We have seen that many researchers agree that proft sharng can ncrease frm s profts and can ncrease employment, by alterng fxed labor costs nto varable costs per quantty produced. Furthermore, we have encounter OECD contemporary statstcs that show that unonzaton stll exsts, but wth great varaton from country to country. In the man body of our paper we have endogenze two decsons: 1)unons decson to merge or not, and 2)frms decson for gvng proft sharng or fxed wage. Ths s somethng unque n exstng lterature. We went even further, usng a bargan framework, and product dfferentaton. Wth a devatons reasonng, and wth Pareto rankng we have proven that unons wll always want to merge and form a grand unon. Under a grand unon, and for dfferent values of β (0, 1) and γ (0, 1), all three cases can be equlbra: both frms mght choose to gve fxed wage, both frms mght choose proft sharng, and for a rather small are of (β, γ)-plane one frm could choose fxed wage whle the other could choose proft sharng scheme. Frms net profts are always hgher when use fxed wage, but a prsoner s dlemma stuaton lead a frm to follow the rval and ntroduce proft sharng scheme (except for the small area of (β, γ)-plane mentoned just before). 24

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