Own price influences in a Stackelberg leadership with demand uncertainty

Transcription

1 R E B E Own price influences in a Stackelberg leadership with dem uncertainty Abstract: We consider a Stackelberg duopoly model with dem uncertainty only for the first mover We study the advantages of leadership flexibility with the variation of the dem uncertainty also with the variation of the own price effect We compute in terms of the dem uncertainty of the own price effect parameter the probability of the second firm to have higher profit than the leading firm We prove that even in presence of low uncertainty the expected value of the profit of the second firm increases to higher values than the ones of the leading firm with the increase of the own price effect Key words: Perfect Bayesian equilibrium Stackelberg duopoly uncertainty JEL Classification: C72 D43 L Ferna A Ferreira Flávio Ferreira Alberto A Pinto 2 ESEIG Instituto Politécnico do Porto Rua D Sancho I Vila do Conde Portugal fernaamelia@eseig ipppt flavioferreira@eseigipppt Fax: Departamento de Matemática Universidade do Minho Braga Portugal E- mail: aapinto@gmailcom R Bras Eco de Emp 28; 8(): 29-38

2 3 Own price influences in a Stackelberg leadership with dem uncertainty Introduction The model introduced by Stackelberg (934) is one of the most widely used models in industrial organization for analyzing firms behavior in a competitive environment It studies the strategic situation where firms sequentially choose their output levels in a market The question we ask is: Do first movers really have strategic advantage in practice? The belief of first-mover advantage was widely held among entrepreneurs venture capitalists but is now questioned by numerous practitioners We indicate some examples of successful pioneering firms (as described in Liu (25)): Dell was the first to introduce the direct-sale business model into the PC market it achieved great success growing from Mr Dell s small-dorm business into a giant in the PC market However we can find many counterexamples During the dot-com booming era Petscom Webvan Garden com etoys were all first movers in their respective market segments but they all ended up burning through their investment capital before attracting enough customers to sustain a business (see Stalter (22)) Why do these pioneering firms get very different results? The probability of success of pioneering in a market clearly depends on many factors including technology marketing strategy market dem product differentiation Several research papers focus their attention in giving answers to such question In this paper we extend Liu s results by focus not only on the effects of the market dem uncertainty but also on the own price effect to explain the advantages disadvantages of being the leading firm Usually the followers in markets get more market information than first movers before sinking their investments In some industries that we consider to have fairly stable predictable market dem the pioneering firm tends to be the biggest player However if a market has a high degree of uncertainty the followers can wait see the customers response to the new product introduced by the first movers as well as move along the differentiation curve of innovation As in the model studied by Liu (25) we consider that only the first mover (leading firm) faces dem uncertainty The dem uncertainty is given by a rom variable uniformly distributed with mean µ stard deviation σ characterizing the dem uncertainty parameter By the time the second mover chooses its output level that uncertainty is resolved Therefore the leading firm possesses first-mover advantage but the second mover enjoys an informational advantage because it can adjust the production level after observing the realized dem (flexibility) We study the advantages of flexibility over leadership as the own price effect parameter b changes We find explicit functions I b J b in terms of the own price effect parameter b characterizing the dem uncertainty parameter θ for which the leading firm looses its advantage for some realizations of the dem rom variable We show that the leading firm looses its advantage for high values of the dem intercept if the dem uncertainty parameter θ is greater than I b for low values of the dem intercept if the dem uncertainty parameter θ is greater than J b (see Figure 7) Hence for high values of the dem uncertainty parameter (θ > J b ) only in an intermediate zone of the realized dem does the first mover preserve its advantage We observe that for own price effect equal to cross price effect (β = ) the functions I b J b coincide ie I = J for own price effect greater than cross price effect (β > ) we have that I b > J b We make an ex-ante analysis by computing the expected value with respect to the dem realization of the profits of both firms in terms of the dem uncertainty parameter θ of the own price effect parameter b (see Figure 2) In particular we prove that even in the presence of low uncertainty the expected value of the profit of the second firm increases to higher values than the ones of the leading firm with the increase of the `own price effect (see Figures 3 4) Moreover we show that there is a value θ such that if the uncertainty parameter θ is greater than θ then the expected profit of the follower firm is always greater than the expected profit of the leading firm We also make an ex-post analysis by computing comparing the firms profits after the dem uncertainty has been resolved We also compute the R Bras Eco de Emp 28; 8(): 29-38

3 Ferna A Ferreira Flávio Ferreira Alberto A Pinto 3 probability of the second firm to have higher profit than the leading firm in terms of the dem uncertainty parameter of the own price effect parameter (see Figure 9) We show that the probability of the follower firm to have higher profit than the leading firm increases with the degree of the dem uncertainty with the own price effect Furthermore when the own price effect is equal to the cross price effect (b = ) the probability of the leading firm to have higher profit than the follower firm is greater than the probability of the opposite situation; however when the own price effect is sufficiently greater than the cross price effect for sufficiently high level of uncertainty the probability of the follower firm to have higher profit than the leading firm is greater than the probability of the opposite situation 2 The model the perfect Bayesian equilibrium We start by describing the Stackelberg duopoly model The dem for simplicity is linear namely () with > b where p i is the price q i the amount produced of good by the firm F i for i { 2} We note that since b cross effects are dominated by own effects Firms have the same constant marginal cost c We consider from now on prices net of marginal costs This is without loss of generality since if marginal cost is positive we may replace by c We consider that the dem intercept is a rom variable uniformly distributed in the interval [ ] with > > We note that in this case the dem uncertainty parameter θ is equal to the ratio / The distribution of is of common knowledge Profit p i of firm F i is given by p i = p i q i = ( bq i q j ) q i (2) As already described in the Introduction the timing of the game is as follows: i) Firm F chooses a quantity level q without knowing the value of the dem realization; ii) Firm F 2 first observes the dem realization observes q then chooses a quantity level q 2 In the next theorem we show that this game has a uni\que perfect Bayesian equilibrium Let A = 6b 2 (b 2 ) + 6b + B = 6b 2 (2b 2 ) C = 2b (8b 3 3) denote by respectively the expressions (b a a ) = 4b 2 a 2(2b 2 ) a + (3) We observe that for all θ = / > so Σ are welldefined We also note that = a Σ Theorem Consider a Stackelberg duopoly model facing the dem system where the parameter a is uniformly distributed in the interval [a a ] Then there is a unique perfect Bayesian equilibrium as follows: (i) If θ K b then ; (ii) If θ K b then given Theorem is proved in the Appendix In the next corollary we compare the quantities produced by each firm Corollary Let I b = (4b 2 ) / (4b 2 3) a) If θ < I b then for all a [a a ] the leading firm produces more than firm F 2 b) If I b θ K b then there exists such that (i) if a R the leading firm produces more than firm F 2 ie ; R Bras Eco de Emp 28; 8(): 29-38

4 32 Own price influences in a Stackelberg leadership with dem uncertainty (ii) if a R firm F 2 produces more than the leading firm ie c) If θ K b then there exists such that (i) if a R 2 the leading firm produces 5 a more than firm F 2 ie ; (ii) if a R 2 firm F 2 produces more than the leading firm ie Corollary is proved in the Appendix The different situations described in this corollary are illustrated in Figure for some values of the parameters 4 b q q Quantities 2 q 2 Quantities q c R 4 q Quantities q 2 R 5 2 Figure : the quantities produced by each firm when the intervals of the uniform distribution of the parameter a are such that the ratio θ between its endpoints is as in each situation of Corollary for b = 2 (a) [a a ] = [ 5]; (b) [a a ] = [ 4]; (c) [a a ] = [5] In the next theorem we present an exante analysis by giving the profits that the firms can expect before the knowledge of the dem realization a For simplicity of notation we denote by the profit for i { 2} Theorem 2 While the dem realization is unknown for both firms their expected profits are given by where P = {(θ b) : ( < θ < 3 b ) (θ > 3 b b ) (θ = 3 b = )} R Bras Eco de Emp 28; 8(): 29-38

5 Ferna A Ferreira Flávio Ferreira Alberto A Pinto 33 with Q = {(θ b) : θ 3 b b } theorem we describe three regions for the dem uncertainty parameter q = a / a corresponding to three distinct profits relations between the leading the follower firms (see Figure 5) The lowmedium uncertainty boundary value is Theorem 2 is proved in the Appendix In Figure 2 we plot firms expected profits as functions of the dem uncertainty parameter θ of the own price effect parameter b Taking a = we see that there is a value θ (approximately equal to 73) such that if the uncertainty parameter θ is greater than θ then the expected profit of the follower firm is always greater than the expected profit of the leading firm Figure 3 illustrates cross sections of Figure 2 at b = 2 (Figure 3a) b = 5 (Figure 3b) Figure 4 illustrates cross sections of Figure 2 at the dem uncertainty parameter s values q = 2 (Figure 4a) q = 8 (Figure 4b) Figure 2: firms expected profits varying with the dem uncertainty parameter q with the own price effect parameter b by taking a = the medium-high uncertainty boundary value is Figure 5: three regions for the dem uncertainty parameter q corresponding to three distinct profits relations between the leading the follower firms The functions I b J b characterize the dem uncertainty parameter q for which the leading firm looses its advantage for some realizations of the dem rom variable In fact in the next theorem we will show that the leading firm looses its advantage for high values of the dem intercept if the dem uncertainty parameter q is greater than I b for low values of the dem intercept if the dem uncertainty parameter q is greater than J b Hence for high values of the dem uncertainty parameter (q > J b ) only in an intermediate zone of the realized dem does the first mover preserve its advantage We observe that for own price effect equal to cross price effect (b = ) the functions I b J b coincide ie I = J for own price effect greater than cross price effect (b > ) we have that I b < J b In Figure 6 we show the plots of I b J b K b as functions of the own price effect parameter b The value θ is given by Figure 3: cross sections of Figure 2 at (a) b = 2 (b) b = 5 a b where 3 3 Expected profits E(p ) 2 E(p 2 ) Expected profits 2 E(p ) E(p 2 ) with 3 b 5 b Figure 4: cross sections of Figure 2 at the uncertainty parameter s values (a) q = 2 (b) θ = 8 Now we are going to analyze the profits that the firms obtain after the observation of the dem realization In the next Figure 6: plots of the functions I b J b K b Theorem 3 a) If q < I b then for all a [a a ] the leading firm profits more than firm F 2 ie We observe that θ 73 R Bras Eco de Emp 28; 8(): 29-38

6 34 Own price influences in a Stackelberg leadership with dem uncertainty b) If I b q < J b then there exists 6 c π 2 such that i) if a R the leading firm profits more than firm F 2 ie ; ii) if a R firm F 2 profits more than the leading firm ie c) If q J b then there exist Firms profits Firms profits 2 L R 35 6 c π 2 π π such that L R [a a ] i) if L a R the leading firm profits more than firm F 2 ie ; ii) if either a L or a R firm F 2 profits more than the leading firm ie Theorem 3 is proved in the Appendix The different situations described in this theorem are illustrated in Figure 7 In Figure 7a we consider q < I b ; in Figure 7b we consider I b q < J b ; in Figure 7c we consider J b q < K b ; in Figure 7c 2 we consider q K b In Figure 8 we show the plots of L R as functions of the dem uncertainty parameter q profits Firms Firms profits 24 a b 5 8 π π 2 π 2 π 65 /3 L 2 R 2 Figure 7: the profits of both firms when the intervals of the uniform distribution of the parameter a are such that the ratio q between its endpoints is as in each situation of Theorem 3 for q = 2 (a) [a a ] = [ 5]; (b) [a a ] = [ 25] ; (c ) [a a ] = [ 35]; (c 2 ) [a a ] = [ 65] 5 R( θ) L( θ) I 5 =J =K θ 5 a b L(2 θ) R(2 θ) I 2 J 2 K 2 5 θ Figure 8: plots of the functions L L (b a a ) R R (b a a ) by considering a = in the cases of (a) b = ; (b) b = 2 Now we are going to compute in terms of the dem uncertainty parameter q of the own price effect parameter b the probability of the second firm to have higher profit than the leading firm Using the results presented in Theorem 3 we get the following corollary Corollary 2 a) If q < I b then b) If I b q J b then 5 25 R R Bras Eco de Emp 28; 8(): 29-38

7 Ferna A Ferreira Flávio Ferreira Alberto A Pinto 35 c) If I b q K b then d) If q > K b then where is defined by (3) The economic interpretations of this result include: (i) given the own price effect parameter the probability of the follower firm to have higher profit than the leading firm increases with the degree of the dem uncertainty; (ii) given the dem uncertainty the probability of the follower firm to have higher profit than the leading firm increases with the own price effect Furthermore when the own price effect is equal to the cross price effect the probability of the leading firm to have higher profit than the follower firm is greater than the probability of the opposite situation; however when the own price effect is sufficiently greater than the cross price effect for sufficiently high level of uncertainty the probability of the follower firm to have higher profit than the leading firm is greater than the probability of the opposite situation (see Figure 9) Prob(π 2 > π ) β=5 β=2 β= θ Figure 9: the probability of the second firm to have higher profit than the leading firm as a function of the dem uncertainty parameter θ for different values of the own price effect parameter b by taking a = 3 Conclusions For the Stackelberg model considered (i) we observed that in the case of dem certainty the leading firm has advantage over the follower one; (ii) we showed that in the case of dem uncertainty the leading firm does not necessarily have advantage over the one that follows In order to analyze the leadership flexibility advantages we had to consider three different situations that depend upon the size of the ratio q = a / a between the endpoints of the interval [a a ] in which the parameter a were uniformly distributed That is we found two functions I b J b that depend upon the own price effect parameter b such that (i) if q < I b then the leading firm profits more than the follower; (ii) if I b q < J b then when the realized dem is very high the leading firm profits less than the follower otherwise the leading firm profits more than the follower; (iii) if q J b then when the realized dem is either very low or very high the leading firm profits less than the follower when the realized dem is in an intermediate region the leading firm profits more than the follower We observed that for own price effect equal to cross price effect (b = ) the functions I J coincide ie I = J b for own b price effect greater than cross price effect (b > ) we have that I b < J b We showed that when the own price effect is equal to the cross price effect the probability of the leading firm to have higher profit than the follower firm is greater than the probability of the opposite situation; however when the own price effect is sufficiently greater than the cross price effect for sufficiently high level of uncertainty the probability of the follower firm to have higher profit than the leading firm is greater than the probability of the opposite situation References VAN DAMME E & HURKENS S Endogenous Stackelberg leadership Games Economic Behavior N p 5-29 GAL-OR E First mover second mover advantages International Economic Review 3 N p GAL-OR E First-mover disadvantages with private information Review of Economic Studies 2 N p R Bras Eco de Emp 28; 8(): 29-38

8 36 Own price influences in a Stackelberg leadership with dem uncertainty GIBBONS R A Primer in Game Theory Pearson Prentice Hall: Harlow 992 HIROKAWA M Strategic choice of quantity stickiness Stackelberg leadership Bulletin of Economic Research N 53 2 p 9-34 HIROKAWA M & Dan Sasaki Endogeneous co-leadership when dem is uncertain Australian Economic Papers 3 N 39 2 p HIROKAWA M & DAN SASAKI Endogeneously asynchronous entries into an uncertain industry Journal of Economics Management Strategy 3 N 2 p HOPPE H The timing of new technology adoption: theoretical models empirical evidence Manchester School N 7 22 p HOPPE H & Lehmann-Grube U Second-mover advantages in dynamic quality competition Journal of Economics Management Strategy N 2 p HOPPE H & LEHMANN-GRUBE U Innovation timing games: a general framework with applications Journal of Economic Theory N 2 25 p 3-5 HUCK S; KONRAD K & MULLER W Big fish eat small fish: on merger in Stackelberg markets Economics Letters 2 N 73 2 p LIU Z Stackelberg leadership with dem uncertainty Managerial Decision Economics N p MAGGI G Endogenous leadership in a new market The RAND Journal of Economics N p PAL D Cournot duopoly with two production periods cost differentials Journal of Economic Theory N55 99 p PAL D & SARKAR J A Stackelberg oligopoly with nonidentical firms Bulletin of Economic Research 2 N53 2 p ROBSON A Stackelberg marshall American Economic Review N 8 99 p VON STACKELBERG H Marktform und Gleichgewicht Julius Springer: Vienna 934 STALTER K Moving first on a winning idea doesn t ensure first-place finish: pioneers find pitfalls Investor s Business Daily october 7 22 VIVES X Duopoly information equilibrium: Cournot Bertr Journal of Economic Theory N p 7-94 Appendix In this Appendix we give the proofs of the results presented throughout the paper Proof of Theorem Using backwardsinduction we first compute firm F 2 s reaction to an arbitrary quantity q fixed by firm F to the realized dem parameter a The quantity given by which yields Therefore firm F s problem in the first stage of the game amounts to determine is the cases: (I) a q (II) a q Note that the density function of a s distribution is / (a a ) Case I: a q (see Figure ) In this case a a q so Therefore Figure : a q (Case I) Then firm F s best quantity the equation Hence (4) solves (5) where E( ) is the expectation with respect to the dem intercept a We are going to study separately so (6) R Bras Eco de Emp 28; 8(): 29-38

9 Ferna A Ferreira Flávio Ferreira Alberto A Pinto 37 We observe that the value obtained in satisfies the hypothesis a q considered in Case I if only if q K b Case II: a q (see Figure ) In this case (i) if a q then ; (ii) if a q then Therefore if only if Figure : a q (Case II) Therefore (7) Then firm F s best quantity solves the equation Hence (8) so Since if only if q < I b I b < K b we get the statement a) Since q I b we get the statement b) if only if Now suppose that q K b By Theorem we get that if a < / 3 then which is positive; if a / 3 then (9) We observe that the value obtained in satisfies the hypothesis a q considered in Case II if only if q K b We note that in Case I (q K b ) we have that in Case II (q K b ) we have that Figure 2) E(?) (see Therefore if only if which implies the statement c) profit Proof of Theorem 2 Firm F s expected is obtained by (4) (7) Firm F 2 s expected profit is determined by E( ) q Figure 2: case I (q K b ): (q K b ): q q q ; case II Proof of Corollary First suppose that q K b By Theorem we get that with given respectively by (5) (6) in the case of q K b given respectively by (8) (9) in the case of q K b Finally we note that the values (q b) in the set P are the ones that satisfy q K b the values (q b) in the set Q are the ones that satisfy q K b Proof of Theorem 3 By Theorem in the case of q K b the profits at equilibrium are given by R Bras Eco de Emp 28; 8(): 29-38

10 38 Own price influences in a Stackelberg leadership with dem uncertainty a) For q < I b we have that L < a So we have that only if if Again by Theorem in the case of q K b the profits at equilibrium are given by R > a Therefore a [a a ] for all b) For I b q J b we have that L < a a R a Therefore if a < R then ; if a > R then c) For J b < q K b we have that a L R a Therefore if L a R then ; if either a L R a Therefore if a < L or a L R a Therefore if a > R then For q > K b we have that a L 2 R 2 < a Therefore if L 2 a R 2 then ; if either a < L 2 or a > R 2 then So we have that only if if R Bras Eco de Emp 28; 8(): 29-38