A Note on Price and Quantity Competition in Differentiated Oligopolies

Transcription

1 August 17, 1999 A Note o Price ad Quatity Competitio i Differetiated Oligopolies Joas Häcer Departmet of Ecoomics, Stocholm Uiversity S Stocholm, Swede Phoe: Fax: joas.hacer@e.su.se Abstract I this ote we show that the results developed i Sigh ad Vives (1984) are sesitive to the duopoly assumptio (Rad Joural of Ecoomics 15, ). If there are more tha two firms, prices may be higher uder price competitio tha uder quatity competitio. This will be the case if quality differeces are large ad goods are complemets. If goods are substitutes, high-quality firms may ear higher profits uder price competitio tha uder quatity competitio. Hece, it is ot evidet which id of competitio is more efficiet. JEL Classificatio: D43, L13 Keywords: Product Differetiatio, Oligopoly, Courot, Bertrad

2 1 1. Itroductio I a classic cotributio to oligopoly theory Sigh ad Vives (1984) discuss the ature of competitio i Bertrad ad Courot marets usig the duopoly framewor developed i Dixit (1979). The coclusios are strog ad clear-cut. Courot competitio always yields higher prices ad lower welfare compared to Bertrad competitio. Whe goods are substitutes, firm profits are higher uder Courot competitio while if goods are complemets Bertrad competitio is more profitable. Cosiderig the theoretical ad practical importace of these results it seems importat to test their robustess with respect to alterative maret structures. We therefore exted the Dixit (1979) model to allow for a arbitrary umber of firms. The discussio is limited to prices ad profit levels sice the welfare issue becomes too complex i a geeral settig. For simplicity, we allow oly for two dimesios of firm heterogeeity, vertical product 1 differetiatio ad substitutability. Moreover, it is assumed that the margial cost of productio is equal across firms. The Dixit (1979) model geeralizes icely despite the asymmetry that stems from vertical product differetiatio. The liear demad structure esures that firm demad depeds oly o the average quality of the competitors products while it is uaffected by chages i the exact distributio of product qualities across firms. The duopoly results do ot to geeralize to the -firm settig if quality differeces are large. Whe goods are complemets (ad quality differeces large) low-quality firms will charge prices that are higher uder Bertrad competitio tha uder Courot competitio. If goods are substitutes, high-quality firms may ear higher profits uder Bertrad competitio tha uder Courot competitio. 2. The Model The utility fuctio i Sigh ad Vives (1984) is of the followig type 1 Whe goods are substitutes, the degree of substitutability could be iterpreted i terms of horizotal product differetiatio.

3 2 U(q 1, q 2, I) ' " 1 q 1 % " 2 q ($ 1 q 2 1 %$ 2 q 2 2 %2(q 1 q 2 ) % I. For simplicity let us assume that $ = $ = 1. Thus, utility is quadratic i the cosumptio of q- 1 2 goods ad liear i the cosumptio of other goods, I. The parameter (0[-1, 1] measures the substitutability betwee the products. If ( = 0, each firm has moopolistic maret power, while if ( = 1, the products are perfect substitutes. A egative ( implies that the goods are complemetary. Fially, " measures quality i a vertical sese. Other thigs equal, a icrease i i " icreases the margial utility of cosumig good i. i It is straightforward to geeralize the utility fuctio to allow for firms producig oe product variety each. U(q, I) ' j q i " i & 1 2 ( j q 2 i %2( j q i ) % I i j Cosumers maximize utility subject to the budget costrait 3piq i + I # m, where m deotes icome ad the price of the composite good is ormalized to oe. The first-order coditio determiig the optimal cosumptio of good is MU Mq ' " & q & ( jj & p ' 0. (1) Courot Competitio Firm s iverse demad fuctio ca be solved for directly from expressio 1. p (q, q &i ) ' " & q & ( jj Firms set quatities to maximize profits, B, taig the other firms quatities as give. If costs are ormalized to zero, firm s reactio fuctio equals

4 3 q (q & ) ' " & ( jj 2. Summig over all firms ad otig that j q i ' q % jj j " i ' " % jj (2) we ca solve for demad ad price i equilibrium q C ' p C ' " [((&2)%2]&( jj (2&()[((&1)%2] where superscript C idicates Courot equilibrium. Thus, firm s equilibrium price ad quatity deped o the average quality if its competitors products but are idepedet of the exact distributio of product qualities across firms. Bertrad Competitio Summig over all firms, expressio 1 ca be writte j " i & j q i & ((&1) j q i & j p i ' 0. (3) Expressios 1, 2 ad 3 the yield firm s demad fuctio. q (p, p & ) ' (" &p )[((&2)%1]&( jj ( &p j ) (1&()[((&1)%1]. (4)

5 Profit maximizatio (ad zero costs) the implies the followig reactio fuctio 4 p (p & ) ' " 2 & ( jj ( &p j ) 2[((&2)%1]. (5) Summig this over all firms ad usig expressios 2 ad 5 the yields the equilibrium prices ad quatities for firm p B ' " [( 2 ( 2 &5%5)%3((&2)%2]&( jj [((&2)%1] [((&3)%2][((2&3)%2] q B ' " [( 2 ( 2 &5%5)%3((&2)%2]&( jj [((&2)%1] [((&2)%1] (1&()[((&3)%2][((&1)%1][((2&3)%2] where B deotes Bertrad equilibrium. Agai, firm s equilibrium price ad quatity deped o the average quality if its competitors products but are idepedet of the exact distributio. 3. Bertrad ad Courot Equilibria The first propositio i Sigh ad Vives (1984) deals with the welfare implicatios of price ad quatity competitio. Courot prices are foud to be higher tha Bertrad prices regardless of whether goods are complemets or substitutes. This meas that welfare is always higher uder price competitio. Moreover, Courot profits are show to be higher tha Bertrad profits whe goods are substitutes. If goods are complemets, the opposite relatio holds. If we tae the differece betwee the Courot price ad the Bertrad price of the geeralized model we arrive at the followig coditio. sig(p C & p B ) ' sig(" [(2 ( 2 &5%5)%4((&2)%4] & ( 2 (&2) jj ) Hece, uless = 2, the price differetial cosists of a positive ad a egative term. The sig will

6 5 be determied by the umber of firms, the degree of horizoal differetiatio ad the quality distributio betwee firm ad its competitors. Propositio 1: Assume that > 2. i) Whe goods are substitutes, p > p. Hece, welfare is uambiguously higher whe firms compete i prices. ii) Whe goods are complemets, ad quality differeces are large, p < p for low-quality firms. Proof: First cosider the case whe goods are substitutes. Let z be the ratio betwee the average quality offered by the rival firms, 3" /(-1), ad the quality offered by firm, ". If z <1 firm j produces a better product tha the average rival ad vice versa. Uless z is small eough, firm will be drive out of the maret i equilibrium. Specifically, uder Bertrad competitio B q > 0 if z < (2 ( 2 &5%5)%3((&2)%2 ((&1)(((&2)%1) / z B while the correspodig coditio uder Courot competitio is z < ((&2)%2 ((&1) / z C. The secod-order coditio for a iterior solutio uder Bertrad competitio is B C c > 1/(1-). This esures that z < z. Hece, it is ot meaigful to tal about a -firm maret B uless z < z. The price differetial is positive uless z is large. Specifically, p - p > 0 uless z > (2 ( 2 &5%5)%4((&2)%4 ( 2 (&2)(&1) / z(. It is straightforward to verify that there exists o z that ca satisfy the latter ad the former iequalities simultaeously if ( > 0. This proves the first part of the propositio. Assume ow that goods are complemets ad that the secod-order coditios are satisfied, i.e. (0[1/(1-), 0]. The, all firms face a positive demad i equilibrium regardless of the distributio of qualities. Cosequetly, z > z* is a sufficiet coditio for a egative price

7 differetial. It ca be checed that z* icreases mootoically i (. Moreover, z* > 0 whe 6 ( = 1/(1-) ad z* approaches +4 as ( approaches zero from below. Hece, for every ( i the iterval [1/(1-), 0] there exists a threshold value z* such that z > z* implies that p < p. Fially, z* decreases i ad approaches z* = 1 as approaches ifiity. Hece, oly firms producig qualities below average will charge higher prices uder Bertrad competitio tha uder Courot competitio. ~ A switch from Courot to Bertrad competitio will reduce the prices of high-quality products ad icrease equilibrium demad. If goods are complemets, this will icrease the demad for low-quality products which, i tur, may eable low-quality producers to raise their prices. B C Propositio 2: i) Whe goods are complemets, B > B. ii) Whe goods are substitutes, ad quality differeces are small, B > B. Whe quality differeces are large it may be the case that B C B > B for high-quality firms. Proof: First defie G(z ) = B /B ad let z ad z be defied as i the proof of Propositio 1. B If ( = 0 the G(z ) = 1. Whe goods are substitutes, G(z ) icreases i z. G(1) > 1 ad G(z ) = +4. Hece, oly high-quality firms (i.e. firms with a low z ) may have a profit ratio that is smaller tha oe. For istace, suppose that " = z ", œj. The, let z^ deote the lowest value of z for j which there is a positive demad for low-quality products i the Bertrad equilibrium. G(z^) < 1 as log as ( < ˆ(() / 4 &2 3 &5 2 %14&7% 2 &5%5 2( 2 &3%2). (^ is icreasig i. (^(2) = 0 while (^(4) = 1. Hece, for ay maret structure there exist a ( small ~ eough to mae G(z^) < 1. This, i tur, implies that there exists some positive iterval [z^, z] such ~ that G(z ) < 0 if z 0[z^, z]. Whe goods are complemets G(z ) decreases i z. Sice G(0) < 1 the profit ratio is always smaller tha oe.~ Whe quality differeces are large, high-quality firms become isulated from competitio from

8 7 the low-quality segmet. Therefore, price competitio may ot hurt firm profits more tha quatity competitio. 4. Cocludig Remars We may coclude that the results i Sigh ad Vives (1984) are sesitive to the duopoly assumptio. Although we have imposed more symmetry o the model, compared to the origial formulatio, the clear-cut dichotomy betwee Bertrad ad Courot competitio is lost i the - firm specificatio. Hece, it is ot evidet which type of competitio is more efficiet.

9 8 Refereces Dixit, A., 1979, A Model of Duopoly Suggestig a Theory of Etry Barriers, Bell Joural of Ecoomics 10, Sigh, N., ad X. Vives, 1984, Price ad Quatity Competitio i a Differetiated Duopoly, Rad Joural of Ecoomics 15,