R&D investments in a duopoly model

R&D investments in a duopoly model lberto. Pinto 1, runo M. P. M. Oliveira 1,2, Fernanda. Ferreira 1,3 and Miguel Ferreira 1 1 Departamento de Matemática Pura, Faculdade de Ciências da Universidade do Porto Rua do Campo legre, 7, 19-7 Porto, Portugal aapinto@fc.up.pt, miguel.ferreira@fc.up.pt 2 Faculdade de Ciências da Nutrição e limentação da Universidade do Porto R. Dr. Roberto Frias, 25-5 Porto, Portugal bmpmo@fcna.up.pt 3 Departamento de Matemática, ESEIG - Instituto Politécnico do Porto Rua D. Sancho I, 91, -7 Vila do Conde, Portugal fernandaamelia@eseig.ipp.pt Summary. We present deterministic dynamics on the production costs of Cournot competitions, based on perfect Nash equilibria of nonlinear R&D investment strategies to reduce the production costs of the firms at every period of the game. We analyse the effects that the R&D investment strategies can have in the profits of the firms along the time. We show that small changes in the initial production costs or small changes in the parameters that determine the efficiency of the R&D programs or of the firms can produce strong economic effects in the long run of the profits of the firms. Keywords: Game Theory; Industrial Organization; Cournot model; R&D investments. JEL classification: C72; C73; L. Presenter: lberto. Pinto

2 lberto. Pinto et al. 1 Introduction We present deterministic dynamics on the production costs of Cournot competitions, based on R&D investment strategies of the firms. t every period of time, the firms involved in a Cournot competition invest in R&D projects to reduce their production costs. This competition is modeled by a two stages game for which the perfect Nash equilibria are computed. y deciding, at every period, to use the perfect Nash investment equilibria of the R&D strategies, the firms give rise to deterministic dynamics on the production costs characterizing the duopoly competition. We observe that small changes in the initial production costs or small changes in the parameters that determine the efficiency of the firms or of the corresponding R&D programs, the firms can produce drastic economic effects in the long run of the profits of the firms. For instance, we show that a firm, F 1, with a disadvantage in the initial production costs but with a higher quality of R&D programs is able to recover along the time to a level of production costs lower than the other firm, F 2. However, for small changes in the initial production costs of the firms, or in the efficiency of the R&D investment programs, the firm F 1 is not able anymore to recover. 2 R&D investments on costs The Cournot competition with R&D investment programs consists of two subgames in one period of time. The first subgame is an R&D investment program, where both firms have initial production costs and choose, simultaneously, the R&D investment strategies to obtain new production costs. The second subgame, for simplicity of the model, is a Cournot competition with production costs equal to the reduced cost determined by the R&D investment program. s it is well known, the second subgame has a unique perfect Nash equilibrium. In some parameter region of our model, the game presents a unique Perfect Nash equilibrium, except for initial costs far away of the minimum attainable reduced production cost where the uniqueness of the equilibrium is broken. We are going to find the perfect Nash equilibria of the Cournot competition with R&D investment programs by first discussing the Cournot competition and then the R&D investment strategies. Let us consider an economy with a monopolistic sector with two firms, F 1 and F 2, each one producing a differentiated good. s considered by Singh and Vives [], we take the representative consumer preferences described by the following utility function U(q 1, q 2 ) = α 1 q 1 + α 2 q 2 ( β 1 q 2 1 + 2q 1 q 2 + β 2 q 2 2) /2, where q i is the amount of good produced by the firm F i, and α i, β i >, for i {1, 2}. The inverse demands are linear and, letting p i be the price of the good produced by the firm F i, they are given by

R&D investments in a duopoly model 3 p 1 = α 1 β 1 q 1 q 2, p 2 = α 2 q 1 β 2 q 2, in the region of quantity space where prices are positive. The goods are substitutes, independent, or complements according to whether >, =, or <, respectively. Demand for good i is always downward sloping in its own price and increases (decreases) the price of the competitor, if the goods are substitutes (complements). When α 1 = α 2, the ratio 2 /β 1 β 2 expresses the degree of product differentiation ranging from zero, when the goods are independent, to one, when the goods are perfect substitutes. When > and 2 /β 1 β 2 approaches to one, we are close to a homogeneous market. c i c i i /2 i /2 new cost new cost i i λ i investment λ i investment Fig. 1. New production costs as a function of the investment. (): The maximum reduction in the production costs is i, obtained for an infinite investment v i = +. () (zoom of the left part of ()): When the investment is v i = λ i the reduction in the production costs is equal to i /2. The firm F i invests an amount v i in an R&D program that reduces the production costs to v i a i = c i ɛ i (c i c Li ), λ i + v i where the parameter c il > is the minimum attainable production cost for the firm F i, and c i is the firm F i s unitary production cost at the beginning of the period satisfying c il c i < α i (See Figure 1). The maximum reduction i = ɛ i (c i c il ) of the production cost is a percentage < ɛ i < 1 of the difference between the current cost, c i, and the lowest possible production cost, c il. The parameter λ i > can be seen as a measure of the inverse of the quality of the R&D program of the firm F i, since a smaller λ i will result in a bigger reduction of the production costs for the same investment. In particular, c i a i (λ i ) gives half i /2 of the maximum possible reduction i of the production cost for firm F i.

lberto. Pinto et al. The profit π i (q i, q j ) of the firm F i is given by π i (q i, q j ) = q i (α i β i q i q j a i ) v i, (1) for i, j {1, 2} and i j. The optimal output level q i = q i (q j) of the firm F i in response to the output level q j of the firm F j is given by qi = arg max q i (α i β i q i q j a i ) v i. q i α i/β i Hence, the optimal output level of firm F i is given by { qi = max, α } i a i q j. 2β i Let R = R 1 = 2β 2α 1 α 2 2β 2 a 1 + a 2 β 1 β 2 2. The Nash equilibrium output (q1, q2) is given by, if R < q1 2β = 2α 1 α 2 2β 2a 1+a 2 β 1β 2, if R < α2 a2 2 α 1 a 1 2β 1, if R α2 a2 q2 = α 2 a 2 2β 2, if R < 2β 1 α 2 α 1 2β 1 a 2 +a 1 β 1 β 2, if R < α 2 a 2 2, if R α 2 a 2 Thus, firm F 1 has profit v 1, if R < π1 (q1, q2) β 1 (2β 2 (α 1 a 1 ) (α 2 a 2 )) = 2 v (β 1β 2 2 ) 2 1, if R < α 2 a 2., and F 2 has profit π2 (q1, q2) = (α 1 a 1) 2 β 1 v 1, if R α 2 a 2 (α 2 a 2 ) 2 β 2 v 2, if R < β 2(2β 1(α 2 a 2) (α 1 a 1)) 2 v (β 1 β 2 2 ) 2 2, if R < α2 a2 v 2, if R α 2 a 2.

R&D investments in a duopoly model 5 The perfect Nash investment equilibrium (v 1, v 2) of the first subgame is given by v 1 = arg max v 1 π 1 and v 2 = arg max v 2 π 2. In Figure 2, we present the perfect Nash equilibrium investment (v 1, v 2) for non-identical firms. In Figure 2, we fix the production cost of the firm F 1 equal to 5.3 and we observe that the firm F 1 starts investing when the production cost of firm F 2 are close to 3, and increases its investment with the production costs of firm F 2. Firm F 2 starts investing when its production costs are close to the production cost of firm F 1, and it is not able to invest when its production costs are too large comparing with the production cost of the firm F 1. investment 2 1 1 prod. cost 1 prod. cost 2 1 NIE 12 1 2 1 2 3 5 7 initial production cost of Fig. 2. The effect of the production costs on the Nash investment equilibrium. Identical firms, producing homogeneous goods, with parameters α i = 1, β i =.13, c il =, ɛ i =.1 and λ i =.1. In Figure () we show a transversal cut of Figure (), taking an initial production costs of firm F 1 equal to c 1 = 5.3. In Figure 3 we present the profits (π 1, π 2) for identical firms when they choose the perfect Nash investment equilibrium as a function of the production costs of both firms. Figure 3 is a cross section of Figure 3 at the production costs c 1 of firm F 1 equal to 5.3. In Figure 3 we observe that with the increase of the production costs c 2 of firm F 2, the profits π 2 of firm F 2 decrease and the profits π 1 of firm F 1 increase.

lberto. Pinto et al. 15 1 profit profit 2 5 prod. cost 1 prod. cost 2 2 initial production cost of Fig. 3. The effect of the production costs in the profits. Identical firms producing homogeneous goods, with parameters α i = 1, β i =.13, c il =, ɛ i =.1 and λ i =.1. In Figure () we show a transversal cut of Figure (), taking an initial production costs of firm F 1 equal to c 1 = 5.3. 3 Deterministic Dynamics The deterministic dynamics on the production costs of the duopoly competition appear from the firms deciding to play the perfect Nash equilibrium in the Cournot competition with R&D investment programs, period after period. We consider that firm F 1 has a better R&D quality than firm F 2 (ɛ 1 > ɛ 2 ) but that firm F 1 has a higher initial production cost than firm F 2. We see that the advantage of the higher quality of the R&D program of firm F 1 allows along the time firm F 1 to decrease its production costs to a level lower than firm F 2 and like that recovering from the initial advantage of firm F 2 (see Figure ). small increase in the R&D quality of firm F 2 provokes that firm F 1 is not able anymore to recover along the time the production costs (see Figure 5). Moreover, the gap between the production costs of the two firms increases along the time. small decrease in the initial production costs of firm F 2 provokes that firm F 1 is not able anymore to recover along the time the production costs (see Figure ). Moreover, the gap between the production costs of the two firms increases along the time. Hence, in this case, a small decrease in the initial production costs of firm F 2 has similar effects in the long run of the profits of the firms to a small increase in the R&D quality of firm F 2.

R&D investments in a duopoly model 7 production costs 7 5 NIE 2 2 1 5 C 2 1 profit 3 2 1 2 1 Fig.. Firms with different R&D investment programs. () Production costs. () Nash Investment Equilibrium. (C) Profit: (i) The lines correspond to the evolution of the profits determined by the stochastic dynamics; (ii) The dashes determine the profits of the firms in the case where both firms do not invest in that period; (iii) The dots determine the profits of the firms in the case where both firms never invest; (iv) The dash-dot is the profit in the case where both firms play the Nash equilibrium of the Cournot game with the minimum attainable production cost c Li. For initial production costs c 11 =, c 21 =.3 and parameters α i = 1, β i =.13, c il =, ɛ 1 =.2, ɛ 2 =.1 and λ i = 1. Conclusions We presented deterministic dynamics on the production costs of Cournot competitions, based on perfect Nash equilibria of R&D investment strategies of the firms at every period of the game. The following conclusions are valid in some parameter region of our model. The R&D investment programm presented a unique perfect Nash equilibrium, except for initial costs far away of the minimum attainable reduced production cost where the uniqueness of the equilibrium is broken. We illustrated the transients and the asymptotic limits of the deterministic dynamics on the production costs of the duopoly competition. We have shown strong long term economic effects resulting from small changes in the initial production costs of the firms, or in the efficiency of

lberto. Pinto et al. 5 production costs 7.5 7.5 NIE 3 2 1 5.5 5 1 15 2 25 3 5 1 15 2 25 3 35 C 3 25 profit 2 15 1 5 5 1 15 2 25 3 Fig. 5. Firms with different R&D investment programs. () Production costs. () Nash Investment Equilibrium. (C) Profit: (i) The lines correspond to the evolution of the profits determined by the stochastic dynamics; (ii) The dashes determine the profits of the firms in the case where both firms do not invest in that period; (iii) The dots determine the profits of the firms in the case where both firms never invest; (iv) The dash-dot is the profit in the case where both firms play the Nash equilibrium of the Cournot game with the minimum attainable production cost c Li. For initial production costs c 11 =, c 21 =.3 and parameters α i = 1, β i =.13, c il =, ɛ 1 =.2, ɛ 2 =.13 and λ i = 1.

R&D investments in a duopoly model 9 5 production costs 7.5 7.5 NIE 3 2 1 5.5 5 1 15 2 25 3 5 1 15 2 25 3 35 C profit 3 25 2 15 1 5 5 1 15 2 25 3 Fig.. Firms with different R&D investment programs. () Production costs. () Nash Investment Equilibrium. (C) Profit. (i) The lines correspond to the evolution of the profits determined by the stochastic dynamics; (ii) The dashes determine the profits of the firms in the case where both firms do not invest in that period; (iii) The dots determine the profits of the firms in the case where both firms never invest; (iv) The dash-dot is the profit in the case where both firms play the Nash equilibrium of the Cournot game with the minimum attainable production cost c Li. For initial production costs c 11 =, c 21 =.2 and parameters α i = 1, β i =.13, c il =, ɛ 1 =.2, ɛ 2 =.1 and λ i = 1. the R&D investment programs. For instance, we have shown that a firm, F 1, with a disadvantage in the initial production costs but with a higher quality of R&D programs is able to recover along the time to a level of production costs lower than the other firm, F 2. However, for small changes in the initial production costs of the firms, or in the efficiency of the R&D investment programs, the firm F 1 is not able anymore to recover.

1 lberto. Pinto et al. cknowledgments We thank the Programs POCTI and POSI by FCT and Ministério da Ciência, Tecnologia e do Ensino Superior, and Centro de Matemática da Universidade do Porto for their financial support. Fernanda Ferreira and runo Oliveira gratefully acknowledge financial support from PRODEP III by FSE and EU. Fernanda Ferreira also acknowledges financial support from ESEIG/IPP. References 1. mir R, Evstigneev I, Wooders J (21) Noncooperative versus cooperative R&D with endogenous spillover rates. Core Discussion Paper 21/5, Louvainla-Neuve, elgium 2. ischi GI, Gallegati M, Naimzada (1999) Symmetry-breaking bifurcations and representative firm in dynamic duopoly games. nnals of Operations Research 9:253-272 3. rander J, Spencer J (193) Strategic commitment with R&D: the symmetric case. The ell Journal of Economics 1:225-235.. Cournot (13) Recherches sur les Principes Mathématiques de la Théorie des Richesses, Paris. English edition: (197) Researches into the Mathematical Principles of the Theory of Wealth. Edited by N. acon, New York, Macmillan 5. Qiu DL (1997) On the dynamic efficiency of ertrand and Cournot equilibria. Journal of Economic Theory 75:213-229. Singh N, Vives X (19) Price and quantity competition in a differentiated duopoly. RND Journal of Economics 15:5-55