In this paper we present a mixed duopoly model of supply function competition under uncertainty with product differentiation. We find that, regardless the nature of product heterogeneity, the best response of the private firm always arises as strategic complement. Contrary to this, state-owned firm's best response arises either as strategic complement or substitute depending on the product heterogeneity. As a result of the ex post realization of the demand uncertainty, different equilibria are reached

Gutiérrez-Hita, Carlos

Vicente-Pérez, José

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveOn supply function competition in amixed oligopolyGutiérrez-Hita, Carlos and Vicente-Pérez, JoséUniversity of Alicante7 January 2018Online at https://mpra.ub.uni-muenchen.de/83792/MPRA Paper No. 83792, posted 10 Jan 2018 12:04 UTCOn supply function equilibria in a mixed duopolyCarlos Gutiérrez-Hitaa,∗, José Vicente-PérezaaDepartamento de Fundamentos del Análisis Económico, University of Alicante, 03071 Alicante, SpainAbstractIn this paper we present a mixed duopoly model of supply function competition under uncertainty withproduct differentiation. We find that, regardless the nature of product heterogeneity, the best response ofthe private firm always arises as strategic complement. Contrary to this, state-owned firm’s best responsearises either as strategic complement or substitute depending on the product heterogeneity. As a result ofthe ex post realization of the demand uncertainty, different equilibria are reached.Keywords: Supply Function Equilibria, Mixed oligopoly, Differentiated products.JEL classification: D43, H42, L131. IntroductionWe investigate the endogenous choice of a supply function as firms’ strategic variable in a mixed duopolywhere a state-owned firm competes against a profit-maximizing firm. In a differentiated product settingwith a numeraire good (Singh and Vives, 1984), uncertainty in the demand side is introduced (Kemplererand Meyer, 1986). Firms’ strategies arise either as strategic complements or strategic subsitutes (Bulowet al., 1985; Akgün, 2004) as a result of product differentiation, yielding the following findings. Overall, itis found that social welfare is higher as product differentiation approaches perfect complementarity. At thesame time, prices and quantities increase. The contrary holds as products become substitutes. Moreover,no matter the nature of product differentiation, the state-owned firm always behaves more aggressive thanthe private firm producing a higher output and thus, enhancing welfare.State-owned firms are very common in many markets such as airlines, telecommunications, railways,electricity, and banking, among other sectors. Concerning the study of mixed oligopolies, classical marketstructures like Cournot and Bertrand can be found. In the strand of differentiated duopolies with a state-owned firm, Haraguchi and Matsumura (2016) have compared quantity and price competition. They alsoendogenize the strategic variable (either quantity or price) extending previous findings by Matsumura andOgawa (2012). In a more general setting Ghosh and Mitra (2010) also provide interesting results in theCournot and Bertrand comparison.To the best of our knowledge, the assumption of supply function competition (SFC, hereinafter) withheterogeneous goods in a context of a mixed oligopoly has not been yet studied. We introduce uncertaintyin the demand side, which is a more realistic market environment. Indeed, SFC is a natural way to endog-enize demand uncertainty because firms may offer different price-quantity pairs as a result of each demandrealization. As Kemplerer and Meyer (1989) have shown, in a symmetric industry the equilibrium in supplyfunctions is linear and unique. They also state that, under product differentiation with linear demand, theexistence of a unique supply function equilibrium is guaranteed provided that the demand shock has fullsupport.Our results differ form Cournot and Bertrand competition because in these settings strategic interactiontakes place only in one dimension, whereas in our framework firms manage a two-dimensional strategy.∗Corresponding authorEmail addresses: carlosghita@ua.es (Carlos Gutiérrez-Hita), jose.vicente@ua.es (José Vicente-Pérez)Preprint submitted to Economics Letters January 8, 2018Indeed, Cournot and Bertrand approaches may arise as limit cases from SFC (see, for instance, Martin,2002, pp. 68–69).2. The modelWe consider an industry with two firms simultaneously producing a heterogeneous product. Firm 0is a welfare-maximizing public firm (De Fraja and Delbono, 1990) that produces q0, whereas firm 1 is aprofit-maximizing private firm producing a quantity q1. Both firms bear a quadratic cost function given byCi(qi) = (c/2)q2i , with c ≥ 11. The representative consumer maximizes its surplus U(q0, q1)− (p0q0 + p1q1),where pi and qi denote firm i’s price and quantity, respectively, for i = 0, 1. The utility function U(q0, q1) =α(q0 + q1)− (q20 + 2γq0q1 + q21)/2 is assumed to be strictly concave (see Singh and Vives, 1984) with α > 0and 1 − γ2 > 0. Industry inverse demand functions follow and are given by pi = α − qi − γqj (i, j = 0, 1,i 6= j). Product differentiation is captured by γ, where γ ∈ Γ := (−1, 0) ∪ (0, 1). If γ is negative (positive)the products are complements (substitutes). Let us assume that firms are unable to know ex ante thevalue of α. Hence, from firms’ point of view, the intercept of the inverse market demand is subject to anex ante unobservable realization of α which has strictly positive density f(α) everywhere on the support(α, α) ⊂ R+, with E(α) = µ and V (α) = σ2. Firms simultaneously compete in supply functions as inKemplerer and Meyer (1989) by offering quantity-price pairs according with the linear2 function qi = βipi.Ex ante market clearing conditions yield pricespi(βi, βj) =α(1 + (1− γ)βj)1 + βi + βj + βiβj(1− γ2). (1)A strategy for each firm is to choose the slope βi ≥ 0 which determines prices and quantities. Taking (1), wedenote a firm’s supply function by qi(β0, β1) = βipi(β0, β1). The state-owned firm maximizes the expectedsocial welfare (consumer surplus and firms’ profits),maxβ0∫ αα(U(q0(β0, β1), q1(β0, β1))−1∑i=0Ci(qi(β0, β1)))f(α)dα,whereas the private firm maximizes its own expected profits,maxβ1∫ ααπ1(β0, β1)f(α)dα,where πi(β0, β1) = pi(β0, β1)qi(β0, β1)−Ci(qi(β0, β1)). For the sake of simplicity we assume hereinafter thatc = 1. First order conditions for the state-owned and private firms provide best response functions,β0(β1) =1 + 2β1(1− γ) + β211 + β1(2− γ2) + β21(1− 2γ + γ2), (2a)β1(β0) =1 + β02 + β0(2− γ2), (2b)which yield the optimal strategies reported in the following proposition.1As stated in De Fraja and Delbono (1990), if each firm’s marginal cost is constant the public firm will impose the rule ofpricing at marginal cost. This is true independently of the relative efficiency of private and public firms. We abstract from thisissue by considering increasing marginal costs.2It is possible to specify a more general setting where supply functions are defined as qi = υi+βipi. However, when marginalcost have zero intercept a supply function equilibrium of the form qi = βipi exists (Kemplerer and Meyer, 1989). Moreover,as Delbono and Lambertini (2015) states, it is possible to define a linear supply function equilibrium by choosing as strategicvariable βi leaving as a parameter υi. For the sake of simplicity and without loss of generality we take υi = 0 as done inCiarreta and Gutiérrez-Hita (2006).2Proposition 1. The optimal supply functions for the state-owned and private firms areβ∗0(γ) =−1 + 2ξ1− ξ(2− γ2),β∗1(γ) = ξ,where ξ is the unique real root of the degree 3 polynomial on t, Tγ(t) := (2−γ)2t3+(2γ3− 5γ2− 2γ+6)t2+2γt− 2.Proof. By combining (2a) and (2b) we get the above optimal supply functions β∗0(γ) and β∗1(γ). Thissolution is well-defined and unique. To show this, we employ analytical tools for cubic equations (see, forinstance, Press et al., 1992, Section 5.6) in order to guarantee that, for each γ ∈ Γ, the polynomial Tγ hasa unique real root ξ. Moreover, one has Tγ(0) = −2 < 0 and limt→+∞Tγ(t) = +∞ since (2 − γ)2 > 0, whichshow that β∗1(γ) = ξ > 0 in virtue of the well-known Bolzano’s Theorem. Besides that, one can check thatTγ(12 ) < 0 < Tγ(12−γ2 ) for all γ ∈ Γ, which implies12 < ξ <12−γ2 for all γ ∈ Γ3, and so β∗0(γ) > 0.We point out that ξ depends on γ although, in the interest of simplicity, we do not make explicit thisdependence and we just write ξ instead of ξ(γ). Furthermore, as illustrated in Figure 1, one has that thestate-owned firm always behave more aggressive than the private firm, i.e., β∗0(γ)− β∗1(γ) > 0 for all γ ∈ Γ.This fact shows that, as the state-owned firm is social welfare-maximizer, it has an additional incentive toincrease output by cutting prices and thus, the state-owned firm’s optimal supply function is always abovethe one chosen by the private firm.Figure 1: Optimal supply functions β∗0(γ) and β∗1(γ).3. Results and discussionOnce optimal supply functions β∗i (γ) have been characterized in Proposition 1, we present in the nextproposition the main findings from the best response functions in (2). The core findings of our model follows.Proposition 2. Consider a mixed duopoly with product differentiation and SFC. Then,(i) For β0 > 0, the best response function of the private firm operates under strategic complementarity nomatter products are complement or substitutes, i.e., ∂β1(β0)∂β0> 0 for all γ ∈ Γ.3Following the same reasoning, one can provide tight lower and upper bounds for ξ. Thus, 12−0.5γ< ξ < 12−0.6γwheneverγ ∈ (−1, 0), and 12−0.4γ< ξ < 12−0.5γwhenever γ ∈ (0, 1).3(ii) For β1 >12 , the best response function of the state-owned firm operates under strategic substitutabil-ity (complementarity, respectively) when products are complements (substitutes, respectively), i.e.,∂β0(β1)∂β1< 0 for all γ ∈ (−1, 0) (∂β0(β1)∂β1> 0 for all γ ∈ (0, 1), respectively).Proof. Consider the best responses in (2a) and (2b). On the one hand, it follows that∂β1(β0)∂β0=γ2(2 + β0(2− γ2))2 > 0for all γ ∈ Γ. On the other hand, one can see that∂β0(β1)∂β1=γ(2− γ)((3− 2γ)β21 + 2β1 − 1)(1 + β1(2− γ2) + β21(1− 2γ + γ2))2 ,whose sign solely depends on γ since the denominator is positive, the factor 2− γ is positive for all γ ∈ Γ,and the factor (3− 2γ)β21 + 2β1 − 1 is positive for all γ ∈ Γ too. To see the last assertion, we consider thepolynomial Sγ(t) := (3 − 2γ)t2 + 2t − 1 and observe that 3 − 2γ > 0 for all γ ∈ Γ. Then, Sγ(t) > 0 if andonly if either t < sγ :=−1−√4−2γ3−2γ or t > sγ :=−1+√4−2γ3−2γ . It can be checked that sγ < 0 and12 > sγ for allγ ∈ Γ. Hence, as β1 >12 > sγ , one gets Sγ(β1) > 0. Thus, the conclusion follows.Observe that the assumption β1 >12 in Proposition 2 (ii) is not restrictive since, according to (i), onehas β1(0) =12 < β1(β0) <12−γ2 = limβ0→+∞β1(β0) for all β0 > 0.Proposition 2 states that in the linear equilibrium of a SFC game, private firm’s best response is upwardsloping, i.e., as the state-owned firm becomes more aggressive, the optimal response of the private firm is tobehave aggressive as well. Notice that the contrary holds in a Cournot game where the optimal response isto restrict output as the rival firm increases it, yielding to the well known result that quantities are strategicsubstitutes. In this sense, in our framework the private firm mimics Bertrand behavior. Nevertheless, thebest response of the state-owned firm is upward sloping only when products are substitutes whereas it isa decreasing function under product complementarity. The intuition behind is that when products aresubstitutes both firms have the incentive to capture market share stealing business to the rival; in thissetting, the best response of the state-owned firm is to reduce prices as long as the private firm behavesmore aggressive. However, when products are complements the private firm exerts higher market power (asproducts are used according to a bundle consumers’ willingness to pay is potentially high). Hence, whenthe private firm increases the price (cutting output), the best response of the state-owned firm is to behavemore aggressive reducing the price and thus, increasing output.Now we are in position to discuss more in detail the implications that firms’ optimal supply functionshave in prices and quantities. By combining (1) and Proposition 1, we obtain the following optimal pricesand quantities,p∗0(γ) =α(1 + ξ(1− γ))(1− ξ(2− γ2))γ2ξ(2− ξ), q∗0(γ) =α(1 + ξ(1− γ))(−1 + 2ξ)γ2ξ(2− ξ),p∗1(γ) =α(1− ξ(2− γ))γξ(2− ξ), q∗1(γ) =α(1− ξ(2− γ))γ(2− ξ),which have been represented in Figure 2 below for the case α = 1. Equilibrium prices and quantities aboveenjoy the following properties: for any γ ∈ Γ,(i)∂p∗i (γ)∂γ< 0,∂q∗i (γ)∂γ< 0,(ii) p∗1(γ)− p∗0(γ) > 0, q∗0(γ)− q∗1(γ) > 0.4Hence, as complementarity (substitutability) decreases (increases), prices and quantities decrease. In ad-dition, private firm prices are over the state-owned firm prices, and the contrary holds for quantities (seeFigure 2). These facts can be explained from the above reasoning on the best response functions features.SFC allows both private and state-owned firms manage at the same time price and quantities. Hence, whenproduct complementarity takes place, there is weaker competition and the private firm fixes higher prices(restricting output). Accordingly, the state-owned firm behaves more aggressive by offering lower prices(and hence increasing output) aimed to partially offset the market power exerted by the private firm andthus, enhancing social welfare. As long as products become substitutes, competitiveness increases in order tocapture market share and then, the private firm cuts prices. The state-owned firm reacts by cutting pricesas well, although less aggressively than the private firm. As a result, prices are lower than those underproduct complementarity, reaching to be equal as γ tends to one. Besides, quantity traded also decreases.The latter comes from the increasing marginal cost. Indeed, as product substitutability provides lower pricesas a result of a fierce competition, firms decrease output because the marginal income from an extra unitproduced is lower than its marginal cost of production. In other words, when SFC takes place, firms mayadjust price and quantity by choosing the slope of the supply function in order to maximize profits (in thecase of the private firm) and social welfare (in the case of the state-owned firm).Figure 2: Optimal prices and quantities as a function of γ.Finally, the resulting expected social welfare and private firm’s expected profit can be characterized asfollows,E(SW ∗(γ)) = (σ2 + µ2)(U(q∗0(γ), q∗1(γ))−1∑i=0Ci(q∗i (γ))),E(π∗1(γ)) = (σ2 + µ2)(p∗1(γ)−q∗1(γ)2)q∗1(γ),which are unique for each possible realization of the market demand on the support (α, α) ⊂ R+. Accordinglywith the evolution of prices and quantities, as the degree of product differentiation changes, it can bechecked that ∂E(SW∗(γ))∂γ< 0 and∂E(π∗1(γ))∂γ< 0 (see Figure 3, where it is assumed without loss of generalityσ2 + µ2 = 1). The highest level of expected social welfare is attained when products are almost perfectcomplements (γ → −1) with the private firm also obtaining higher expected profits. This relationshipis monotonically decreasing as product becomes susbstitutes. However, expected consumer surplus (theevaluation of the utility function at the equilibrium minus the expenditure) is maximized at γ → 1 (seeFigure 3). This fact comes from the product heterogeneity, as we explain below.Firstly, under product complementarity market power is large. The state-owned firm tries to preserveconsumer surplus (by increasing output above than produced by the private firm, and also fixing the pricebelow than the private firm). The intuition behind is that when products are complements, the state-ownedfirm is aggressive in order to relax the market power exerted by the private firm. As a consequence, the5Figure 3: Expected social welfare, consumer surplus and firm’s profits as a function of γ.state-owned firm puts in the market a large amount of output which obliges to the private firm to increaseoutput as well, enhancing the expected social welfare. Nevertheless, firms’ expected profits have a highershare in social welfare.Secondly, as product substitutability increases, expected consumer surplus benefits from lower prices,although expected social welfare and private firm’s expected profits decrease. Indeed, the state-owned firmtakes relatively more care about the own profits and private firm profits (by decreasing output at a higherpace than the private firm). This result arises again as a consequence of the SFC. As we pointed outabove, although competitiveness is enhanced as product becomes substitutes, marginal cost increases withquantities, so firms adjust production as prices decrease in order to save costs. However, expected consumersurplus has a higher share in the expected social welfare.To summarize, it can be said that when SFC takes place, the strategic effect induced by product comple-mentarity is higher than the welfare effect that the state-owned firm has in the market. As a consequence,although the expected social welfare is high, the expected consumer surplus is under firms’ profits whenproduct complementarity is large. Contrary to this, the strategic effect induced by product substitutabilityis lower than the welfare effect that the state-owned firm has in the market. As a result, although theexpected social welfare is reduced by the effect of competitiveness (but at a lower rate than it was a pureduopoly market) the expected consumer surplus is enhanced and, in fact, it is larger than private firms’profits.AcknowledgementsC. Gutiérrez-Hita was partially supported by MINECO of Spain, Grant MTM2014-54199-P, and J. Vicente-Pérez waspartially supported by MINECO of Spain and ERDF of EU, Grants MTM2014-59179-C2-1-P and ECO2016-77200-P.ReferencesU. Akgün. Mergers with supply function. The Journal of Industrial Economics, LII(4):535–546, 2004.J. Bulow, J. Geanakoplos, and P. Klemperer. Multimarket oligopoly: strategic substitutes and complements. Journal ofPolitical Economy, 93(3):488–511, 1985.A. Ciarreta and C. Gutiérrez-Hita. Supply function vs quantity competition in supergames. International Journal of IndustrialOrganization, 24(4):773–783, 2006.G. De Fraja and F. Delbono. Game theoretic models of mixed oligopoly. Journal of Economics Surveys, 4(1):1–17, 1990.F. Delbono and L. Lambertini. On the properties of linear supply functions in oligopoly. Economics Letters, 135:22–24, 2015.A. Ghosh and M. Mitra. Comparing bertrand and cournot in mixed markets. Economics Letters, 109(2):72–74, 2010.J. Haraguchi and T. Matsumura. Cournot–bertrand comparison in a mixed oligopoly. Journal of Economics, 117(2):117–136,2016.P. Kemplerer and M. Meyer. Price competition vs. quantity competition: the role of uncertainty. The RAND Journal ofEconomics, 17(4):618–638, 1986.P. Kemplerer and M. Meyer. Supply function equilibria in oligopoly under uncertainty. Econometrica, 57(6):1243–1277, 1989.S. Martin. Advanced Industrial Economics. Blackwell Publishers, 2nd edition, 2002.6T. Matsumura and A. Ogawa. Price versus quantity in a mixed duopoly. Economics Letters, 116(2):174–177, 2012.W. Press, S. Teukolsky, W. Vetterling, and B. Flannery. Numerical Recipes in C: The Art of Scientific Computing. CambridgeUniversity Press, 2nd edition, 1992.N. Singh and X. Vives. Price and quantity competition in a differentiated duopoly. The RAND Journal of Economics, 15(4):546–554, 1984.7

On supply function competition in a mixed oligopoly

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On supply function competition in a mixed oligopoly

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