Delayed Dynamics in Heterogeneous Competition with Product Differentiation

Delayed Dynamics in Heterogeneous Competition with Product Differentiation Akio Matsumoto Department of Economics Chuo University Hachioji, Tokyo, 19-0393, Japan akiom@tamacc.chuo-u.ac.jp Ferenc Szidarovszky Systems Industrial Engineering Department University of Arizona Tucson, Arizona, 8571-000, USA szidar@sie.arizona.edu August 5, 006 Abstract Heterogeneous duopolies with product differentiation isoelastic price functions are examined in which one firm is quantity setter the other is price setter. The reaction functions the Cournot- Bertr (CB) equilibrium are first determined. It is shown that the best response dynamics with continuous time scales without time delays is always locally asymptotically stable. This stability can be however lost in the presence of time delays. Both fixed continuously distributed time delays are examined, stability conditions derived the stability regions determined illustrated. The results are compared to Cournot-Cournot (CC) Bertr-Bertr (BB) dynamics. It turns out that continuously distributed lags have smaller instabilizing effect on the equilibria than fixedlags,both homogeneous (CC BB) competitions are more stable than the heterogeneous competitions. 1

1 Introduction The effect of information delay in dynamic economic models has been introduced by Invernizzi Medio(1991), its application to dynamic oligopolies has been examined by Chiarella Khomin(1996) Chiarella Szidarovszky(001). In investigating concave quantity adjusting oligopolies, they have shown that the presence of information delay has an instabilizing effect on the equilibrium. Furthermore a complete stability analysis was performed in the case of continuously distributed time lags, the occurrence of Hopf bifurcation was verified giving the possibility of the birth of limit cycles around the equilibrium. No such study was given for price adjusting oligopolies, for heterogeneous competitions for markets with isoelastic dem functions. In this paper, heterogeneous duopolies will be examined with product differentiation isoelastic inverse dem functions. The paper develops as follows. In Section the best responses of the firms will be determined the Nash equilibrium computed. In Section 3, the stability of the equilibrium will be examined without time delays. Dynamics with time delays will be studied, the one with fixed time delays in Section 4 then the one with continuously distributed delays in Section 5. In Section 6, we will also compare these two approaches. In Section 7 our results on heterogeneous duopolies will be compared to the homogeneous cases of quantity price adjusting duopolies, finally, concluding remarks will be given in Section 8. Cournot-Bertr Equilibrium A duopoly is considered with product differentiation isoelastic dem functions. The inverse dem functions of firms 1 are assumed to have the following forms: 1 p 1 = (1) x 1 + θ 1 x 1 p =, () θ x 1 + x where θ 1 θ denote the degrees of production differentiation fulfill Assumption 1 below. Assumption 1. 0 < θ i < 1.

In the following, we consider the Cournot-Bertr (CB henceforth) competition in which firm 1 is a quantity-setter firm is a price-setter. By interchanging the firms we can have the identical Bertr-Cournot (BC) competition. Let c 1 c denote the constant marginal costs. First we will find the best responses of the firms in terms of their decision variables x 1 p. From (1) (), x = 1 θ x 1 p x 1 = 1 p 1 θ 1 x, so the profit offirm 1 can be written as π 1 = (p 1 c 1 )x 1 = 1 ³ c 1 x 1 1 x 1 + θ 1 p θ x 1 = p x 1 θ 1 + x 1 p (1 θ 1 θ ) c 1x 1. Assuming interior optimum, the first order condition implies that π 1 x 1 = p θ 1 (θ 1 + x 1 p (1 θ 1 θ )) c 1 =0. Therefore the best response of firm 1 is the following: Ãs! 1 θ 1 x 1 = θ 1. (3) 1 θ 1 θ p c 1 p The profit offirm has the form with first order condition π = (p c )x µ 1 = (p c ) θ 1 x p 1 = 1 c p p θ x 1 + c θ x 1 π = c θ p p x 1 =0, 3

implying that the best response of firm is r c p =. (4) θ x 1 To characterize the CB equilibrium it is convenient to re-define these reaction functions in the quantity space (x 1,x ): θ 1 (θ x 1 + x )=c 1 (x 1 + θ 1 x ) θ x 1 = c (θ x 1 + x ), which are simple consequences of relations (), (3) (4). Dividing the first equation by the second introducing new variables z = x x 1 c = c c1 give, after arrangements, c θ µ 1 1+θ1 z (z + θ )=. (5) θ θ + z The intersection of the right h side the left h side functions determines the CB equilibrium ratio of outputs. Let us denote the left h side by f(z), f(z) =c θ 1 (z + θ ), θ the right h side by g(z), µ 1+θ1 z g(z) =. θ + z Then it is easy to see that g(0) = 1 θ > 0, g 0 (z) = (1 θ 1θ )(1 + θ 1 z) (z + θ ) 3 < 0, g 00 (z) = (1 θ 1θ )(3 θ 1 θ +θ 1 z) (z + θ ) 4 > 0. Clearly, f(z) strictly increases, linear f(0) = cθ 1. The graph of functions f(z) g(z) are illustrated in Figure 1. A unique positive intersection α = α(c, θ 1, θ ) exists if only if the following condition holds: 4

Assumption. c< 1 θ 1 θ. Insert Figure 1 Here It is also easy to see, as shown in Figure 1, that α < 1 cθ 1θ. cθ 1 θ The equilibrium of the CB duopoly is clearly given by x 1 = θ c (α + θ ), p = c (α + θ ) θ. 3 Continuous Dynamics without Time Delays The best response CB dynamic system has the form à Ãs!! 1 θ 1 ẋ 1 = k 1 θ 1 x 1, 1 θ 1 θ p c 1 p µr c ṗ = k p. θ x 1 Its Jacobian is k 1 k 1 γ 1 J =, k γ k where γ i is the derivative of firm i 0 s reaction function at the CB equilibrium. Simple calculation shows that s θ 1 θ γ 1 = 1 1 c(α + θ ) (1 θ 1 θ )c (α + θ ) θ 1 θ γ = c (α + θ ) 3 θ, 5

furthermore by using (5), we have The characteristic equation is γ 1 γ = (1 θ 1θ ) αθ 1 θ 1 θ. 4(1 θ 1 θ ) λ +(k 1 + k )λ + k 1 k (1 γ 1 γ )=0. Since γ 1 γ = (1 θ 1θ ) αθ 1 θ 1 θ < 1 4(1 θ 1 θ ) 4, (6) both coefficients are positive, therefore the real parts of the eigenvalues are confirmed to be negative. Therefore the CB dynamic system is always locally asymptotically stable under Assumptions 1. 4 ContinuousDynamicswithFixedTimeDelays Letusdenotethereactionfunctionsofthefirms by R 1 (p ) R (x 1 ) assume that each firm i has a fixed time delay T i on its competitor s variable. The delayed dynamic system is ẋ 1 (t) = k 1 {R 1 (p (t T 1 )) x 1 (t)}, (7) ṗ (t) = k {R (x 1 (t T )) p (t)}. By linearizing these equations,we have a linear system: ẋ 1δ (t) = k 1 γ 1 p δ (t T 1 ) k 1 x 1δ (t), (8) ṗ δ (t) = k γ x 1δ (t T ) k p δ (t), where x 1δ p δ denote the deviations of x 1 p from their equilibrium levels. By looking for the solution in the usual exponential forms x 1δ = e λt u p δ = e λt v, substituting these functions into equations (8), we have k 1 λ k 1 γ 1 e λt 1 u 0 =. k γ e λt k λ v 0 6

Nontrivial solution exists only if the determinant of the coefficient matrix is zero. Therefore the characteristic equation is λ +(k 1 + k )λ + k 1 k k 1 k γe λτ =0, (9) where we introduce two new variables, τ denoting the sum of the two time lags γ denoting the product of the derivatives of the reaction functions, τ = T 1 + T γ = γ 1 γ. Firstweshowthatλ =0cannot be an eigenvalue. Suppose λ =0, then the characteristic function becomes k 1 k (1 γ) > 0, which implies that λ =0cannot be a solution. In order to underst the stability switches of the delayed dynamic system, it is crucial to determine the critical values of time lag at which the characteristic equation may have a pair of conjugate pure imaginary roots. Suppose now that λ = iω, with ω > 0, is a solution of the characteristic equation. Introducing a new variable θ = ωτ, we have ω + k 1 k k 1 k γ cos θ =0 (10) (k 1 + k )ω + k 1 k γ sin θ =0. (11) Thus, by moving ω + k 1 k (k 1 + k )ω to the right h sides, squaring adding the resulted equations, we obtain Hence (k 1 k γ) =( ω + k 1 k ) +(k 1 + k ) ω. ω 4 +(k 1 + k )ω +(k 1 k ) (1 γ )=0. (1) Introducing z = ω makes the left h side a quadratic polynomial in terms of z, ϕ(z) =z +(k 1 + k )z +(k 1 k ) (1 γ ). (13) Notice that ϕ(0) = (k 1 k ) (1 γ ) ϕ 0 (0) = (k 1 + k ) > 0, 7

furthermore the discriminant of the quadratic polynomial (13) is always positive. For γ [ 1, 1), ϕ(0) > 0 with 4 ϕ0 (0) > 0 implying that ϕ(z) =0has real solutions, both are negative. Thus there are no imaginary solutions such as λ = iω with ω > 0. In order to prove that the equilibrium is always locally asymptotically stable, we will show that with sufficiently small γ, this is the case. Let λ = a + ib be a solution of equation (9) with a 0. Substituting this solution into the equation we have a b +abi +(k 1 + k )(a + bi)+k 1 k 1 γe aτ (cos bτ i sin bτ) =0. The imaginary part of this equation shows that which can be written as ab +(k 1 + k )b + k 1 k γe aτ sin bτ =0, a = (k 1 + k ) C where C = k 1 k γ e aτ τ sin bτ bτ k 1k τ γ. So, if γ < k 1+k k 1 k, then a<0, which completes the proof. Hence we have: τ Theorem 1 If 1 5 γ < 1, no stability switching occurs the delayed 4 system is always locally asymptotically stable. Consider next the case where γ < 1. In this case, ϕ(0) < 0 ϕ 0 (0) > 0, therefore, ϕ(z) =0has one positive one negative root, z ± = ω ± = (k 1 + k) ± p (k1 + k) 4(k 1 k ) (1 γ ). (14) In this case, only one imaginary root, λ = iω +, exists with ω + > 0. We select τ as the bifurcation parameter. The critical level of time lag for which stability switching may occur is determined by τ = θ ω +, where cos θ = k 1k ω + k 1 k γ 8

sin θ = (k 1 + k )ω + > 0 k 1 k γ as consequences of the definition of θ relations (10) (11). We have k 1 k ω + = (k 1 + k ) p (k1 + k) 4(k 1 k ) (1 γ ). By comparing the squares of the two terms of the numerator, it is easy to see that sign ª nh k 1 k ω + = sign (k 1 + k ) p k 1 k γ ih (k 1 + k )+ p io k 1 k γ. The first factor of the right h side is always positive. Define next γ = k 1 + k. (15) k1 k It is easy to see that γ < 1. The second factor is negative if only if γ < γ. Hence 0 if γ γ < 1, k 1 k ω + < 0 otherwise. Notice in addition that by using equation (1), we have (k 1 k ω +) = k 1k k 1 k ω + + ω 4 + = k 1k γ (k 1 k ) ω + <k1k γ, so k 1 k ω + < k 1 k γ, therefore θ can be uniquely defined for all γ < 1. If k 1 k ω + 0, then θ [ π, π) if k 1k ω + < 0, then θ (0, π). In order to observe stability switching, we need to determine the sign of the derivative of the real part of the purely imaginary root. We can think of the roots of the characteristic equation as continuous functions in terms of the delay τ. By implicit differentiationofequation(9)wehave λ +(k1 + k )+k 1 k γτe λτª dλ dτ = k 1k γλe λτ. For convenience, we study (dλ/dτ) 1 instead of dλ/dτ. We have µ 1 dλ = λ +(k 1 + k )+k 1 k γτe λτ dτ k 1 k γλe λτ 9

from the characteristic equation we obtain Thus Therefore, ½ ¾ d(re λ) sign dτ e λτ = λ +(k 1 + k )λ + k 1 k. k 1 k γ µ 1 dλ λ +(k 1 + k ) = dτ λ(λ +(k 1 + k )λ + k 1 k ) τ λ. ( µ ) 1 dλ = sign Re dτ ½ µ ¾ λ +(k 1 + k ) = sign Re λ(λ +(k 1 + k )λ + k 1 k ) ½ µ ¾ (k 1 + k )+iω = sign Re (k 1 + k )ω i(k 1 k ω )ω ½ ¾ (k = sign 1 + k +ω ) > 0. (k 1 k γ) Consequently, the crossing of the imaginary axis is from left to right as τ increases thus resulting in the loss of stability. In summary, we have Theorem Given γ < 1, the delayed dynamic system with fixed time delays is locally asymptotically stable when τ < τ unstable when τ > τ where τ = θ /ω + with ω + = (k 1 + k)+ p (k1 + k) 4(k 1 k ) (1 γ ), sin θ = (k 1 + k )ω + > 0, k 1 k γ cos θ = k 1k ω +, k 1 k γ where θ [ π, π) if k 1k ω + 0, θ (0, π) otherwise. 5 Dynamics with Continuously Distributed Lags Matsumoto Szidarovszky(006) derived the stability conditions in the delayed Cournot competition with continuously distributed time lags. Instead 10

of assuming fixed time delays, they assumed that the lags were continuously distributed with exponential kernel functions. Similarly, in equations (7), p (t T 1 ) x 1 (t T ) are now replaced by their expectations: where p e (t) = x e 1(t) = Z t 0 Z t 0 w(t s, T 1 )p (s)ds w(t s, T )x 1 (s)ds, w(t s, T )= 1 T e t s T. So a system of Volterra-type integro-differential equations is obtained. As it has been shown in Matsumoto Szidarovszky(006), the characteristic equation of the system has the special form, (λ + k 1 )(λ + k )(1 + λt 1 )(1 + λt ) k 1 k γ =0, (16) where γ = γ 1 γ asbefore.thisisaquarticequationforλ, with coefficients a 0 λ 4 + a 1 λ 3 + a λ + a 3 λ + a 4 =0 a 0 = T 1 T, a 1 = T 1 + T + T 1 T (k 1 + k ), a = 1+T 1 T k 1 k +(k 1 + k )(T 1 + T ), a 3 = k 1 + k + k 1 k (T 1 + T ), a 4 = k 1 k (1 γ). Since all coefficients are positive, the Routh-Hurwitz criterion (see Szidarovszky Bahill(1998)) implies that all eigenvalues have negative real parts if only if a1 a 0 a 3 a > 0 a 1 a 0 0 a 3 a a 1 0 a 4 a 3 > 0. Simple algebra shows that the second order determinant is always positive. The sign of the third order determinant depends on the value of γ. Solving the inequality for γ gives the following result: 11

Theorem 3 For the dynamic system with continuously distributed time lags, the equilibrium is locally asymptotically stable if γ > (k 1 + k )(T 1 + T )(1 + k 1 T 1 )(1 + k T 1 )(1 + k 1 T )(1 + k T ) k 1 k (T 1 + T + T 1 T (k 1 + k )), is unstable if this relation is violated with strict opposite inequality. 6 Comparison of The Two Approaches We compared the effects caused by fixed continuously distributed lags on stability by determining illustrating the stability regions in the two cases. Figure shows the division of the (T 1,c) space in which the upper boldcurvethelowerboldcurvearetheboundariesbetweenthestable regionstheunstableregionsinthecasesoffixed lags continuously distributed lags, respectively. In particular, from Theorems, 3 relation (6), these boundaries are given by µ (T 1 + T )ω + =cos 1 k 1 k ω +, k 1 k γ CB (c, θ 1, θ ) where ω + is obtained from equation (14), γ CB (c, θ 1, θ )= (k 1 + k )(T 1 + T )(1 + k 1 T 1 )(1 + k T 1 )(1 + k 1 T )(1 + k T ) k 1 k (T 1 + T + T 1 T (k 1 + k )). In both cases γ CB (c, θ 1, θ ) endogenously determines the product of derivatives of the reaction functions is given as γ CB (c, θ 1, θ )= (1 θ 1θ ) α(c, θ 1, θ )θ 1 θ 1 θ. 4(1 θ 1 θ ) We set k 1 = k =0.8, θ 1 = θ =0.8 T =.5 in Figure in which D means distributed time lags F means fixed time lags. It is numerically confirmed that there are only small differences between the symmetric (k 1 = k θ 1 = θ ) nonsymmetric (k 1 6= k /or θ 1 6= θ )cases. It is also proved that the dynamics with fixedtimelagsisunstablebelow the upper bold curve stable above it. Similarly, the dynamics with continuously distributed time lags is unstable below the lower bold curve stable above the curve. As we have proved earlier, the CB dynamics without time delay is always locally asymptotically stable. By introducing time lags, this stability might be lost an instability region develops. Notice that in 1

the middle shaded region surrounded by the bold curves, the CB equilibrium is stable under continuously distributed time lags unstable under fixed time lags. In other words, the instability region is larger when fixed time lags are assumed. Therefore it is demonstrated that the destabilizing effect of fixed time lags is larger than that in the case of continuously distributed time lags. Insert Figure Here 7 Heterogeneous vs Homogeneous Competition In Matsumoto Szidarovszky(006), the delayed continuous dynamic system is considered under the traditional Cournot competition (abbreviated as Cournot-Cournot or CC competition) in which both firms are quantitysetters. In this case, the firms have the best response functions r θ1 x R 1 (x )= θ 1 x c 1 Equation (5) has the form R (x 1 )= c θ 1 θ z = r θ x 1 c θ x 1. µ 1+θ1 z, (17) θ + z which has always a unique positive solution α C = α C (c, θ 1, θ ) if Assumption 1 is fulfilled. The CC equilibrium is given as x C 1 = θ c (θ + α C ) x C α C θ = c (θ + α C ). The Jacobian of the corresponding best response dynamics has the same form as in the CB duopoly case with the only difference that in the CC case, α = α C is the solution of (17), γ C := γ C 1 γ C = 1 4 ½ 1+θ 1 θ 13 µ α C θ 1 + 1 ¾ α θ C.

It can be proved that with appropriate values of α C, γ C 1 γ C can have any real value between 1 similarly to the CB case. Figure 3 illustrates the 4 stability regions in the CC competition case. The dotted upper lower curves have the same meanings as the upper lower bold curves in Figure. Thus it can be said that under CC competition, fixedtimelagshavethe larger destabilizing effect in comparison with continuously distributed time lags. Insert Figure 3 Here The numerical results of Figures 3 are repeated in a different way in Figure 4, where it can be seen that both dotted boundary curves of CC competition are located below the corresponding boundary curves of CB competition. In the middle shaded region between these boundary curves, the equilibrium is stable under CC competition unstable under CB competition. Consequently, the stability regions are much larger for CC dynamics than for CB dynamics showing that CC competition is more stable than CB competition. Insert Figure 4 Here In Matsumoto Szidarovszky(006), the case of Bertr-Bertr (BB) competition has been also discussed when both firms are price-setters. In this case the reaction functions of the two firms are r c1 p p 1 = r c p 1 p =. θ The equilibrium prices outputs are s s p B c 1 = 1c 3 θ,p B c = c 3 1 1θ θ, θ 1 s r x B 1 3 θ 1 = 1θ 1 sθ 3 1θ 3 1 θ 1 θ c 1 c c 1 c θ 1 14

s r x B 1 3 θ = θ 1 1 sθ 3 θ 3 1. 1 θ 1 θ c 1 c c c 1 Simple calculation shows that the form of the Jacobian of the corresponding dynamic system is the same as in the CB case with the only difference that in the BB case γ B = γ B 1 γ B = 1 4. Becauseofthisspecialvalueofγ B, thebbdynamicsisalwayslocallyasymptotically stable without also with time delays with both fixed continuously distributed time lags. Notice that the results of Section 3, Theorem 1 (with γ = 1 4 )Theorem3canbeappliedintheBBcasewithout any limitation. If we call both BB CC competitions homogeneous,in which both firms are either price-setters or quantity setters CB competition heterogeneous, in which behavioral specification of one firm is different from that of the other firm, then we can summarize our results of this section as follows: both homogeneous competitions are much more stable than the heterogeneous competition regardless of the forms of the time delays. 8 Concluding Remarks In this paper, heterogeneous duopolies were examined, where one firm is quantity-setter the other is price-setter. After the reaction functions of the firms were determined, the unique CB-equilibrium was computed. Continuous dynamics were investigated without time delays, with fixed continuously distributed time lags. The CB dynamics without time delay is always locally asymptotically stable. This stability can be lost in the presence of time delays. Complete stability analysis was presented for two types of time lags, the stability regions were illustrated. We have seen that fixed time lags have larger destabilizing effect on the dynamics than continuously distributed time lags. Similar conclusion could be reached in the case of CC duopolies, however the instability regions were much smaller in the case of CC dynamics than in CB dynamics. Therefore we can conclude that if any one of the firms of a CC competition changes from quantity setting behavior to price setting policy, then it has a destabilizing effect on the equilibrium. However, if the other firm also switches to price setting policy, then the corresponding dynamic system becomes always locally asymptotically stable with without time delays. Hence, homogeneous competitions in which 15

both firms are either quantity setters or price setters are more stable than the heterogeneous competitions in which one firm is quantity setter the other is price setter. In our study we considered only exponential weighting functions for continuously distributed time delays, more complex weighting functions will be examined in a future paper. Acknowledgement This paper was written while the second was visiting the Institute of Economic Reseach, Chuo University. The authors highly appreciate financial supports from the Japan Ministry of Education, Culture, Sports, Science Technology (Grant-in-Aid for Scientific Research (B) 15330037), the US Air Force Office of Scientific Research (MURI grant N00014-03-1-0510) Chuo University Grant for Special Research. The usual disclaimer applies. References [1] Chiarella, C. A. Khomin, "An Analysis of the Complex Dynamic Behavior of Nonlinear Oligopoly Models with Time Lags," Chaos, Solitons Fractals, vol.7, no.1, pp.049-065, 1996. [] Chiarella, C. F. Szidarovszky, "The Birth of Limit Cycles in Nonlinear Oligopolies with Continuously Distributed Information Lags," in Dror, M., P. L Ecuyer F. Szidarovszky (eds) Modeling Uncertainty, Kluwer Academic Publishers, Boston, 49-68, 001. [3] Invernizzi, S., A. Medio, "On Lags Chaos in Economic Dynamic Models," Journal of Mathematical Economics, vol.0, pp. 51-550, 1991. [4] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [5] Matsumoto, A. F. Szidarovszky, "Delayed Nonlinear Cournot Bertr Dynamics with Product Differentiation," Discussion Paper No.89 of the Institute of Economic Research, ChuoUniversity,Tokyo, Japan, 006. [6] Szidarovszky, F. T. A. Bahill, Linear Systems Theory, CRC Press, Boca Raton/London, 1998. 16

1êq ghzl fhzl fhal=ghal cq 1 q 1 a 1 - cq 1 q ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ cq 1 q z Figure 1: Determination of optimal output ratio. c 0.4 @D&F: StableD 0.3 0. @D: Stable; F: UnstableD 0.1 @D&F: UnstableD 1 3 4 5 T 1 Figure : Stability regions in CC competition. 17

c 0.1 @D&F: StableD 0.08 0.06 @D:Stable;F:UnstableD 0.04 0.0 @D&F: UnstableD 1 3 4 5 T 1 Figure 3: Stability regions in CB competition. c 0.4 0.3 Fixed Delays @CB&CC: StableD c 0.1 0.08 Distributed Delays @CB&CC: StableD 0. @CB: Unstable; CC: StableD 0.06 0.04 @CB: Unstable; CC: StableD 0.1 @CB&CC: UnstableD 0.0 @CB&CC: UnstableD 1 3 4 5 T 1 1 3 4 5 T 1 Figure 4: Stability regions in CC CB competitions. 18