In an economy in which firms exercise market power in the markets for consumption goods and inputs (labor), we show that a merger to monopoly is Pareto improving when the number of firms is below a threshold. This threshold is larger the larger is the elasticity of labor supply and the smaller is the consumers'preference for goods variety. Consequently, market concentration may have non-monotonic general equilibrium effects on wage mark downs, employment and welfare.Moreno acknowledges financial support of the MCI (Spain), grant PGC2018-098510-B-I00, and from the Comunidad de Madrid, grant H2019/HUM-5891. Petrakis acknowledges financial support from UC3M-Santander 
chairs of Excellence

Moreno, Diego

Petrakis, Emmanuel

English

Universidad Carlos III de Madrid e-Archivo

                                  2022-08 Working paper. Economics ISSN 2340-5031  ON THE GENERAL EQUILIBRIUM EFFECTS OF MARKET POWER  Diego Moreno and Emmanuel Petrakis                                                          Serie disponible en http://hdl.handle.net/10016/11  Web: http://economia.uc3m.es/ Correo electrónico: departamento.economia@eco.uc3m.es Creative Commons Reconocimiento-NoComercial- SinObraDerivada 3.0 España  (CC BY-NC-ND 3.0 ES) On the General Equilibrium Effects of Market Power∗Diego Moreno† Emmanuel Petrakis‡July 2022AbstractIn an economy in which firms exercise market power in the markets for consump-tion goods and inputs (labor), we show that a merger to monopoly is Pareto im-proving when the number of firms is below a threshold. This threshold is larger thelarger is the elasticity of labor supply and the smaller is the consumers’preferencefor goods variety. Consequently, market concentration may have non-monotonicgeneral equilibrium effects on wage mark downs, employment and welfare.Keywords: General Equilibrium, Market Power; Market Effi ciency; Mergers.JEL Classification: D4, D5, D6, L1, L4.∗Moreno acknowledges financial support of the MCI (Spain), grant PGC2018-098510-B-I00, and fromthe Comunidad de Madrid, grant H2019/HUM-5891. Petrakis acknowledges financial support fromUC3M-Santander Chairs of Excellence. The usual disclaimer applies.†Departamento de Economía, Universidad Carlos III de Madrid.‡Department of Economics, University of Crete1 IntroductionCommon wisdom suggests that mergers leading to monopolies create important dead-weight losses, unless they generate large production effi ciencies. Well known exceptions in-clude (impracticable) perfectly price discriminating monopolies, markets for goods whoseproduction generates negative externalities such as pollution (see e.g. Schoonbeek andde Vries, 2009), or vertically differentiated markets in which there are status and envyeffects (Skartados and Zacharias, 2022). We study this issue from a general equilibriumperspective and find results opposing common wisdom.Specifically, we consider a simple market economy in which there are two consumptiongoods which firms produce using labor as input. There are two types of consumers, thefirms’owners (the capitalists) who use their share of the firms’returns to buy consumptiongoods, and workers, who supply labor and use their labor income to buy consumptiongoods. There are neither public goods nor production or consumption externalities. Thus,perfect competition generates Pareto optimal outcomes.Under imperfect competition, firms exercise market power both as good suppliers andlabor users, i.e., a firm internalizes the impact of its output and labor decisions on itsprofits via their effect on the prices of goods and on the wage. In the extreme case of amonopolistic economy, a single firm exercises monopoly power in the goods markets andmonopsony power in the labor market.We identify the Cournot-Walras and monopoly equilibria of the economy and showthat the monopoly equilibrium Pareto dominates the Cournot-Walras equilibrium whenthe number of firms is below a threshold. If the number of firms is below this threshold,then relative to the Cournot competitors the monopoly both, pays a greater real wage(and hence warrants a larger welfare to workers), and generates larger real returns (andhence warrants a larger welfare to capitalists). The threshold on the number of firmsleading to this result is larger the larger is the elasticity of labor supply and the smalleris the consumers’preference for goods variety.Interestingly, this result arises even though the monopoly generates no productioneffi ciencies. Simply, when the number of firms is below the threshold, the monopolist’sincentives are better aligned with those of society than those of the Cournot competi-tors. As the number of firms increases above the threshold then employment and realwage increase above those of the monopoly equilibrium, and workers become better off.Unsurprisingly, as the number of firms becomes arbitrarily large the Cournot-Walrasequilibrium approaches the competitive equilibrium.12 TheEconomyConsideramarketeconomyàlaAzarandVives(2021),inwhichtherearetwoconsump-tiongoods,c1andc2.Bothgoodsareproducedusinglabor,l,asinput. ThereareNfirmsproducingthegoodc1,andanotherN firmsproducingthegoodc2.Eachfirm’sproductionfunctionisF(l)=lα,whereα∈(0,1]. Thefirms’owners,towhomwereferascapitalists,buygoodsusingtheirshareofthefirms’returnsastheirexclusivesourceofincome.Inaddition,thereisacontinuumofworkers(whosemeasureisnormalizedto2)whosupplylaborandusetheirlaborincometobuygoods.Bothworkersandcapitalistscareabouttheirconsumptionofacompositegoodc,givenbyc=v(c1,c2):=121θ 1cθ 1θ1 +cθ 1θ2θθ 1.Theparameterθ∈(1,∞)istheelasticityofsubstitutionbetweenc1andc2. Thelargerisθ,themoreworkersandcapitalistsvaluevariety.Inaddition,workerscareabouttheirleisure(oritscounterpart,labor)andtheirpreferencesarerepresentedbytheutilityfunctionu(l,c)=c1−σ1−σ−l1+ξ1+ξ,wherelistheworker’slaborsupply,σ∈(0,1)andξ>0.Fori∈{1,2}letusdenotebypithepriceofgoodci.Bysolvingtheproblemmin(c1,c2)∈R2+p1c1+p2c2,subjectto:v(c1,c2)=c,wegetc∗i=1212p1−θ1 +p1−θ2θ1 θp−θi cfori∈{1,2}.Hencep1c∗1+p2c∗2=pc,wherep=12p1−θ1 +p1−θ211 θ. (1)istheeffectivepriceofc,bestknownasthe Dixit-Stiglitzpriceindex. Denotingbyρi:=pi/ptherealpriceofgoodi∈{1,2},wemaywriteci(ρ1,ρ2)=12ρ−θi c(ρ1,ρ2). (2)2Letusdenotebywthenominalwageandbyω:=w/ptherealwage.Solvingtheproblemmax(l,c)∈R2+u(l,c),subjectto:c=ωl.wegetaworker’slaborsupplyanddemandofthecompositeconsumptiongood,ls(ω)=ωη,cw(ω)=ωη+1,whereη:=(1−σ)/(ξ+σ)>0istheelasticityoflaborsupply. Notethatηisdecreasinginbothσandξ.Theworkers’aggregatesupplyoflaborisLS(ω)=2ωη. (3)Notethat LS(ω)isstrictlyincreasing. Also,LS(ω)isconvex(concave)ifη >1(η <1);i.e., whenηislarge,asmalincreasein wageleadstoalargeincreaseinemploymentlevel.Workers’(indirect)utilityasafunctionoftherealwageisU(ω):=u(cw(ω),ls(ω))=σ+ξ(1−σ)(1+ξ)ω(1 σ)(1+ξ)σ+ξ =ωη(1+ξ)η(1+ξ),whichisincreasing.Asthecapitalistscareonlyabouttheirconsumptionofthecompositegood,itisnaturaltoassumethatafirm’sobjectiveisto maximizerealreturns. Thecapitalists’aggregatedemandofthecompositegoodisjustthefirms’aggregaterealreturns,whichwedenotebyΠ.3 CompetitiveEquilibriumInacompetitiveeconomyafirmproducinggoodi∈{1,2}choosesitslabortosolvemaxl∈R+ρilα−ωlIfα<1,thenitslabordemandisl(ρi,ω)=αρiω11 α.3Inoursymmetricsettingacompetitiveequilibriumsatisfiesp1=p2=p,andhenceρ1=ρ2=1.Thus,theaggregatelabordemandisLD(ω)=2Nαω11 α,andthelabormarketclearingconditionisLS(ω)=LD(ω)⇔ 2ωη=2Nαω11 α.HencetheequilibriumrealwageisωCE = N1−ααβ,whereβ:=11+(1−α)η.Notingthat(1−β)/(1−α)=ηβ,weseethatinequilibriumeachfirmuseslaborlCE :=l(1,ωCE)=αηNβ.Theworkers’aggregateconsumptionofthecompositegoodisCwCE =2ωη+1CEandcapitalists’aggregateconsumptionofthecompositegoodisΠCE =2N(lαCE −ωCElCE).Ifα=1,theninthecompetitiveequilibriumρ1=ρ2=ω,andtheformulaeaboveidentifyingtheworkers’andcapitalists’aggregateconsumptionofthecompositegoodremainsvalid.Interestingly,theworkers’shareoftheoutputofthecompositegood,whichwe mayrefertoasthelaborshareoftheeconomy’srealGDPisCwCECwCE +ΠCE=α.Thus,thereal wage,andhencethe workers’welfare,andthelaborshareofthe4economy’srealGDParelargerthelargerarethefirms’returnstoscale. Further,whenfirmshaveconstantreturnstoscale,i.e.,α=1,theworkerscapturetheentireeconomy’srealGDP.4 MonopolyEquilibriumLetuscalculatetheequilibriumofthiseconomyassumingthatasingleagentcontrolsthe2N firms,behavingasa monopolyinthegoods marketsandasa monopsonyinthelabormarket.Ifthe monopolyusesthelaborprofile =(l11,..l1N,l21,..l2N),thenusingthelaborsupply(3)wemaycalculatetherealwagethatclearsthelabormarket,2ωη=Nj=1l1j+Nj=1l2j⇔ ω()=Nj=1l1j+Nj=1l2j21η. (4)Moreover,ifthe monopolysuppliesitsoutputs,then marketclearingrequiresthattheaggregatedemandofgoodi,Ci,satisfyCi=Nj=1lαij=:Ci(). (5)HencetheaggregatedemandofthecompositegoodisC()=121θ 1C1()θ 1θ +C2()θ 1θθθ 1. (6)Usingequation(2),wederivetherealpriceofgoodi∈{1,2},Ci()=ρ−θiC()2⇔ ρi()=C()2Ci()1θ. (7)Thus,themonopolysolves:max∈R2N+ρ1()C1()+ρ2()C2()−ω()Nj=1l1j+Nj=1l2j5The unique solution to this problem isl∗ij = lM =(1N(αη1 + η)η)β, for i ∈ {1, 2}, j ∈ {1, ..., N}.Hence the equilibrium real wage isωM =(N1−ααη1 + η)β,and the real prices of the goods are equal to 1.The workers’aggregate consumption of the composite good isCwM = 2ωη+1M ,and capitalists’aggregate consumption of the composite good isΠM = 2N (lαM − ωM lM) .The workers’share of the monopolistic economy’s output of the composite good isCwMCwM + ΠM=αη1 + η.Thus, also in a monopolistic economy the real wage, and hence the workers’welfare,and the labor share of the economy’s real GDP are larger the larger are the firms’returnsto scale. However, even when firms have constant returns to scale, i.e., α = 1, thecapitalists continue to capture a positive share of the economy’s real GDP.5 Cournot-Walras EquilibriumLet us next calculate the equilibrium of the economy assuming that the 2N firms competeà la Cournot by choosing the amount of labor they use. Note that firms exercise marketpower in both the goods markets and the labor market.Let ` = (l11, ...l1N , l21, ...l2N) be the profile identifying the labor used by each of thefirms. Then the real return of firm ij ∈ {1, 2} × {1, ..., N} isπij(`) = ρi(`)lαij − ω(`)lij,6where ρi(`) and ω(`) are given in equations (7) and (4), respectively. It is easily verifiedthat the economy has a unique Cournot-Walras equilibrium, which is symmetric. Inequilibrium the labor used by each firm isl∗ij = lCW =[1N((2Nθ − 1)αη(2Nη + 1) θ)η]β.Hence the goods’real prices are equal to 1 and the real wage isωCW =(N1−α (2Nθ − 1)αη(2Nη + 1) θ)β.The workers’aggregate consumption of the composite good isCwCW = 2ωη+1CW ,and capitalists’aggregate consumption of the composite good isΠCW = 2N (lαCW − ωCW lCW ) .The workers’share of the oligopolistic economy’s output of the composite good isCwCWCwCW + ΠCW=(2Nθ − 1)αη(2Nη + 1) θ.Thus, the labor share of the economy’s real GDP increases with the number of firms,N, the firms’returns to scale, α, the elasticity of labor supply, η, and the agents’preferencefor variety, θ. As expected, as N approaches infinity, the labor share of the economy’sreal GDP approaches α.6 DiscussionClearly, the capitalists are better off in the monopoly equilibrium than in the Cournot-Walras equilibrium, and in the later than in the competitive equilibrium, i.e.,ΠM > ΠCW > ΠCE,Notably, the comparison between employment, real wages and labor share of the econ-omy’s real GDP across equilibria is not as straightforward. The comparison of employ-7ment in competitive and monopolistic economies is clear,lCE = N−βαηβ > N−βαηβ(η1 + η)ηβ= lM ,as it is the comparison between competitive and oligopolistic economies,lCE = N−βαηβ > N−βαηβ((2Nθ − 1) η(2Nη + 1) θ)ηβ= lCW ,where the last inequality follows since(2Nθ − 1) η − (2Nη + 1) θ = −(θ + η) < 0.Consequently, real wages, satisfywCE > max{wM , wCW}.As for the labor share of the economy’s real GDP, it is easy to see thatCwCECwCE + ΠCE> max{CwMCwM + ΠM,CwCWCwCW + ΠCW}.The obvious implication of these inequalities is that workers are better off in the compet-itive equilibrium than in either the monopoly equilibrium or the Cournot-Walras equilib-rium.However, the comparisons of employment, real wage, and welfare in the monopolyequilibrium and in the Cournot-Walras equilibrium is more cumbersome. Specifically, itis easy to show thatlM ≥ lCW ⇔ N ≤ N̄(η, θ)wM ≥ wCW ⇔ N ≤ N̄(η, θ)CwMCwM + ΠM≥ CwCWCwCW + ΠCW⇔ N ≤ N̄(η, θ),whereN̄(η, θ) :=1 + η2θ+12.Thus, while capitalists are better off in the monopoly equilibrium than in the Cournot-Walras equilibrium, whether workers’are better or worse off depends on the number of8firms and the parameters η and θ. The bound N̄(η, θ) decreases with the preference forvariety, θ, and increases with the elasticity of labor supply, η. If the goods are nearlyperfect substitutes (i.e., θ is near 1), for example, then N̄(η, θ) > 1, that is, a monopolyproduces a Pareto superior outcome to that of a differentiated product Cournot duopoly(i.e., N = 1). It also increases the labor share of the economy’s real GDP. Note thatwhen N < N̄(η, θ) a merger to monopoly turns out to be pro-competitive and welfareenhancing.Proposition 1 In an oligopolistic economy in which N < N̄(η, θ), a merger to monopolyleads to an increase of the employment and the real wage, and produces a Pareto superiorallocation. Thus, market power may have non-monotonic effects on employment, realwage mark downs and welfare.To understand the intuition for this result, let us consider the case N = 1, for whichthe premise of Proposition 1 requires the inequality θ < 1 + η to hold. In the Cournot-Walras equilibrium of this duopolistic economy, each competitor sets its output (i.e.,labor) to equate its marginal real revenue and its marginal real cost. It is easy to identifythe qualitative conditions under which a merger would lead the resulting monopoly toincrease the output of both goods, generating an equilibrium with a larger employmentand a larger real wage: A marginal increase of the output of good 1, for example, willlead to an increase of the monopoly real revenue, as well as to an increase of the realcost (via the increase in the real wage), in both markets. The effect on the monopolist’sreal returns in the market for good 1 (evaluated at the Cournot-Walras equilibrium) iszero. Yet, the sign of the effect on the monopolist’s real returns in the market for good2 —an effect that is not a concern of a Cournot competitor —depends on the differencebetween the increase of its real revenue due to the increase of the real price of good 2,ρ2, and the increase of its real cost due to the increase of the real wage, ω. The effect onρ2 is larger the smaller is θ, while the effect on ω is smaller the larger is η. Thus, when θis suffi ciently small and η is suffi ciently large (specifically, when θ < 1 + η), the effect onthe real returns in the market for good 2 is positive. When this is the case, the monopolyequilibrium entails a larger output on both markets as well as a larger employment andreal wage.When N > 1, a marginal increase of the monopolist output in any of its plantsgenerates an additional effect that reduces its real revenue: Since the increase of output9decreases the real price of the good, it also reduces the real revenue in other plantsproducing that good. This negative effect is larger the larger is N , and dominates thepositive effect on the real revenue generated in the market for the other good when(2N − 1)θ < η + 1.References[1] Azar, J. and X. Vives: General Equilibrium Oligopoly and Ownership Structure.Econometrica 89 (2021): 999-1048.[2] Eeckhout, J.: The Profit Paradox. How Thriving Firms Threaten the Future of Work.Princeton University Press, 2021.[3] Farrell, G. and C. Shapiro: Asset Ownership and Market Structure in Oligopoly.RAND J Econ 21 (1990): 275-292.[4] De Loecker, J., J. Eeckhout, and G. Unger: The Rise of Market Power and theMacroeconomic Implications. Q J Econ 135 (2020): 561-644.[5] Skartados, P. and E. Zacharias: Merge to monopoly in a vertically differentiatedmarket with positional effects. Mimeo, Athens University of Economics and Business(2022).[6] Schoonbeek, L., and F.P. de Vries: Environmental taxes and industry monopolization.J of Reg Econ 36 (2009): 94-106.10

On the General Equilibrium Effects of Market Power

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On the General Equilibrium Effects of Market Power

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