Nonlinear dynamics in a duopoly with price competition and vertical differentiation

Transcription

1 Nonlinear dynamics in a duopoly with price competition and vertical differentiation uciano Fanti, uca Gori and Mauro Sodini Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi, 10, I 5614 Pisa (PI), Italy Department of aw, University of Genoa, Via Balbi, 30/19, I 1616 Genoa (GE), Italy Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi, 10, I 5614 Pisa (PI), Italy This paper aims at studying the dynamics of a nonlinear duopoly with price competition (Bertrand) and different quality of products (vertical differentiation). Players are assumed to have limited information regarding profits no knowledge of the market (Bischi et al., 1998). They follow an adjustment process based on local estimates of the marginal profit in the current period (Bischi and Naimzada, 000). There exists a burgeoning literature that deals with nonlinear duopolies with quantity competition (Cournot) aiming at studying local and/or global dynamics in a context with homogeneous products (Bischi et al., 1998, 1999; Tramontana, 010) or horizontal differentiation (Fanti and Gori, 01) substitutability or complementarity between two products.

2 . Fanti,. Gori, M. Sodini Differently, the object of this paper is the analysis of local and global dynamic events in a nonlinear Bertrand duopoly with vertical differentiation (high and low quality). With regard to the case of covered markets, we find that local and global dynamics depends on the extent of consumer s heterogeneity. It is observed that firms often supply differentiated products on the market, so that consumers face a large domain of varieties, which can sometimes unambiguously be ranked along some quality ladders. There exists an established literature that deals with problems of horizontal and vertical differentiation in static oligopoly games, essentially to rank equilibrium outcomes in both Cournot and Bertrand competition models. Studies that deal with the former type of product differentiation date back at least to the works by Dixit (1979), Singh and Vives (1984) and Vives (1985), while examples of the latter can be found in Gabszewicz and Thisse (1979), Shaked and Sutton (198), Motta (1993), Wauthy (1996), äckner (000) and Correa-ópez and Naylor (004). The findings of this literature represent a cornerstone of the oligopoly theory. The model (covered market) Following Tarola et al. (011), we assume that: (1) there exist two firms ( and ) providing goods and services of different quality to the customers; () it is unanimously believed that products of firm are of a higher quality than products of firm ; (3) the average cost of production is not affected by quality, and it is set to zero without loss of generality.

3 Nonlinear dynamics in a duopoly with price competition and vertical differentiation Consumers are identified by the parameter φ [ a, b], where 0 a < b (that can be interpreted by following an established literature (Tirole, 1988; Motta, 1993) as the marginal rate of substitution between income and quality (Motta, 1993, p. 115). Therefore, φ measures the taste for quality of consumers, and it is assumed to be uniformly distributed with unit density (Motta, 1993; iao, 008), while the parameters a and b capture the extent of population (consumers) heterogeneity. The larger the difference between a and b, the higher the degree of heterogeneity amongst consumers. Preferences (U ) of consumer of type φ are described by the following expected utility function: Ui ( φ, pi ) = φ ui pi, i = {, }, (1) where u and u ( u > u) are two indexes that capture the different quality (of products of firms and ) perceived by consumers, and p i is the price that consumers pay to buy product i. Following Wauthy (1996), quality indexes are exogenous. Moreover, in order to guarantee that both firms set strictly positive prices at an interior equilibrium, we assume that: 0, b a. () et φ be an index that identifies the consumer indifferent between purchasing products of high or low quality from firms and at the price p and p, respectively. This is obtain by equating U ( φ, p ) = U ( φ, p). Then, by solving φ u p = φ u p, (3) for φ we get: p u φ =, (4) which depends on both the price differential and quality differential. p u 3

4 . Fanti,. Gori, M. Sodini Then, consumers identified by φ < φ < b (resp. a < φ < φ ) purchase products of high (resp. low) quality. Since the market is covered, D ( p, p) + D ( p, p) = 1, with D i( ) > 0 for i = {, } (Gabszewicz and Thisse, 1979; Wauthy, 1996). Then, the demand functions to firms and are respectively given by: D p p ( p, p) = b φ = b, (5.1) u u p p D ( p, p) = φ a = a. (5.) u u Π ( ) = p D (. Since average costs are zero, profits of the ith firm are i i i ) Therefore, Π p p ( p =, p) p b, (6.1) u u p p Π = a ( p, p) p. (6.) u u The maximisation of Eqs. (6.1) and (6.) with respect to p and the following marginal profits: Π p = b u Π p p = u u p u a p gives, (7.1). (7.) Therefore, the reaction- or best-reply functions of firms and are determined by equating Eqs. (7.1) and (7.) to zero and solving for p and p, respectively, that is: We now consider a dynamic version of the Bertrand duopoly with quality differentiation. Time is discrete and indexed by t = 0,1,,... We assume that both players have limited information regarding profits (no knowledge of the market). Each player uses local estimation where local means at the current state of production of the marginal value of its own objective function in order to follow the steepest local slope of that function. owever they follow an adjustment process based on local estimates of the marginal profits in the current period. We adopt the adjustment mechanism of prices over time proposed by Bischi and Naimzada (000) in a model with quantity competition and discrete time, that is: Πi pi ( t + 1) = pi + α pi, (8) i 4

5 Nonlinear dynamics in a duopoly with price competition and vertical differentiation where α > 0 and α p i (t) is the intensity of the reaction of every player. Therefore, firm i increases or decreases prices at time t + 1 depending on whether Π i / is positive or negative. i The two-dimensional system that characterises the dynamics of this economy is: Then p p p p ( t + 1) = p ( t + 1) = p ( t + 1) = p ( t + 1) = p + α p + α p ( t + 1) + α p ( t + 1) + α p Π p u Π ( t) p p b u u p u a. (9). (10) The non-negative fixed points are given by: b E = 0,0), E = ( u u ), 0, (11) and 0 ( E 3 3 = ( b a)( u u ), ( b a)( u u ). (1) ocal analysis: we find that the fixed point E loses stability exclusively through a flip bifurcation. From an economic point of view: Remark 1. Under the hypothesis of covered market, local stability around E the pure crucially depends on the size of the consumers type (the degree of population heterogeneity), as captured by the parameter a. The level of vertical differentiation (quality) between products and, as captured by the parameters u and u, does not matter. 5

6 . Fanti,. Gori, M. Sodini Fixing b = 1, α = 1. 85, u = 1 and u = 0. 5 only for illustrative purposes and letting a reduce, we present the following simulation results. Figure 1. Bifurcation diagram for a ( p ). As long as a reduces the fixed point E looses stability through a flip bifurcation. Then, the two-period cycle looses stability through a Neimark-Sacker bifurcation when a becomes lower. Moreover, we observe an increase in the complexity of the dynamics when a reduces further on, with some windows where the dynamics becomes more regular. 6

7 Nonlinear dynamics in a duopoly with price competition and vertical differentiation Figure. Bifurcation diagram for a ( p ). Enlargement. Figure 3. a =

8 . Fanti,. Gori, M. Sodini When a reduces we observe the (apparently) chaotic attractor. Figure 4. a = Figure 5. a =

9 Nonlinear dynamics in a duopoly with price competition and vertical differentiation Conclusions This study originates from the increasing interest in the study of nonlinear duopolies (Bischi et al., 010). The novelty of this paper is the analysis of a duopoly game with price competition, and vertical differentiation under the assumption of covered market. We find that a reduction in the degree of consumer s heterogeneity causes the loss of stability of the unique interior fixed point. Moreover, other interesting dynamic events are observed: the two-period cycle looses stability through a Neimark-Sacker bifurcation. Interesting phenomena of synchronisation can also be observed. This classification has to be deepened. Thank you! 9