Dynamics in Delay Cournot Duopoly

Transcription

1 Dnamics in Dela Cournot Duopol Akio Matsumoto Chuo Universit Ferenc Szidarovszk Universit of Arizona Hirouki Yoshida Nihon Universit Abstract Dnamic linear oligopolies are eamined with continuous time scales and information delas. Sstems dnamics are compared with fied and continuousl distributed time lags. We first show that b assuming constant speeds of adjustment stabilit is preserved if the firms use instantaneous information on their outputs and have delas in the outputs of the competitors, and the stabilit might be lost if time delas are present in the firms own outputs. Similar results are obtained if each firm adjusts its growth rate proportionall to a change in its profit. In the case of stabilit loss Hopf bifurcation occurs giving the probabilit of the birth of limit ccles around the stationar state. 1 Introduction Oligopol models pla a central role in the literature of mathematical economics. Since the pioneering work of Cournot (1838), a large number of researchers discussed and eamined the classical Cournot model and its variants and etensions. The eistence and uniqueness of the equilibrium was the main focus of studies in the earl stages and later the research turned to the dnamic analsis of oligopolistic markets. A comprehensive summar of earlier results can be found in Okuguchi (1976), and their multiproduct generalization with several case studies are discussed in Okuguchi and Szidarovszk (1999). The attention was focused on linear and linearized models at the beginning which provided the local asmptotic properties of the equilibria. During the last two decades however an increasing attention has been given to the analsis of global dnamics. A surve of the newer results can be found in Bischi et al. (29) which This paper was prepared when the first author visited the Department of Sstems and Industrial Engineering of the Universit of Arizona. He appreciated its hospitalit over his sta. The usual disclaimer applies. Department of Economics, 742-1, Higasi-Nakano, Hachioji, Toko, , Japan. akiom@tamacc.chuo-u.ac.jp Department of Sstems and Industrial Engineering, Tucson, Arizona, , USA. szidar@sie.arizona.edu College of Economics, 1-3-2, Misaki, Chioda, Toko, , Japan. oshida.hirouki@nihon-u.ac.jp 1

2 contains models with both discrete and continuous time scales. The most common dnamic processes are based either on the gradients of the profit functions or on best responses. In both cases usuall constant speeds of adjustment are assumed and ver few studies consider output-dependent speeds of adjustment. In the case of discrete time scales we mention the works of Bischi and Naimuzada (1999) and Bischi and Lamantia (22). In this paper continuous time scales will be considered. In the cases of most models discussed earlier in the literature, it was assumed that each firm has instantaneous information about its own output and also on the outputs of the competitors. This assumption has mathematical convenience, however it is unrealistic in real economics, since there are alwas time delas due to determining and implementing decisions. In addition to these facts, in fast changing industries the firms do not want to follow sudden market changes, the rather want to react to averaged past information. Hence there are alwas time delas between the times when information is obtained and the times when the decisions are implemented. Howrod and Russel (1984) constructed a linear continuous dnamic oligopol model in which the outputs were adaptivel adjusted and the adjustments were subject to fied time delas. Their conclusions were ver clear: stabilit is not affected b the information lags about the rivals outputs while it is affected b the information lags about the firms own outputs. There are man economic situations in which the lags are uncertain or the firms are reacting to averaged past information. In such situations, continuousl distributed time lags are useful. In this stud, we have two main purposes: we first eamine whether or not the Howrod-Russel results still hold in a dnamic oligopol model with continuousl distributed time lags; second, we reconstruct the model such that the growth rate of the outputs are adjusted with the gradient method and show the eistence of comple dnamics when a stationar state loses stabilit. The paper is organized as follows. Section 2 constructs a basic duopol model with linear price and cost functions. Section 3 assumes constant speed of adjustment and introduces information time delas into the basic model. Section 4 discusses a nonlinear etension of the basic model b adopting the growth rate adjustment process. Section 5 concludes the paper. 2 The Basic Model Dnamics in a classical duopol model is considered. Let and be the quantities produced b firm and firm with linear production costs where the marginal costs are denoted b c and c. The price function is also assumed to be linear, p = a b( + ) with a> and b>. The profit functions are given b π (, ) =(a b( + )) c 2

3 and π (, ) =(a b( + )) c. Firm determines its output to maimize its profit withrespectto and so does firm with respect to. Assuming interior optimal points and solving the first-order conditions for the outputs ield the best repl function for firm, R () = a c b 2b and the best repl function for firm, R () = a c b. 2b A Cournot point is an intersection of these best repl functions and its coordinates are c = a 2c + c 3b and c = a 2c + c. 3b In this paper we assume that the firms continuousl adjust their outputs proportionall to the change in their profits (i.e., gradient dnamics), ẋ = α() π and ẏ = β() π (1) where α() and β() are positive adjustment functions of firm and firm. 3 Dela Linear Duopolies In this section, we start with a simple case and assume constant adjustment coefficients: Assumption 1. α() =α > and β() =β >. The continuous dnamic duopol model is ẋ(t) =α(a c 2b(t) b(t)) ẏ(t) =β(a c b(t) 2b(t)). Using the best repl functions, the dnamic sstem can be rewritten as ẋ =ᾱ (R () ) ẏ = β (R () ) (2) (3) 3

4 where ᾱ =2bα and β =2bβ. In sstem (3), each firm adaptivel adjusts its output in such a wa that the adjustment rate of the output is proportional to the difference between the profit maimizing output and the current output. That is, each firm adjusts its output into the direction toward its best repl. The transformation from (2) to (3) or vice versa implies that for the firms, the gradient adjustment of the output is the same as the adaptive adjustment toward best repl. To eamine the stabilit of sstem (2), we consider its coefficient matri, J = 2bα bα bβ 2bβ with trace trj = 2b(α + β) < and determinant detj =3b 2 αβ >. Since (2) is linear, and the above conditions impl local asmptotical stabilit, we confirm the following well-known result: Theorem 1 The continuous dnamic duopol model (2) is globall asmptoticall stable. Howrod and Russel (1986) introduced the fied dela adjustment process in a general n-firm oligopol model and showed two main conclusions: first, the information delas in the competitors outputs are harmless to the stabilit of the models and second, stabilit is affected if all information available to the firms is subject to a dela. A duopol version of their model is presented b ẋ(t) =α(a c 2b(t S ) b(t T )) (4) ẏ(t) =β(a c b(t T ) 2b(t S )) where the firms adjust their current outputs based on delaed information at some preceding times t S i and t T i for i =,. The consider the situation basedoninformationinwhicheachfirm eperiences a time lag T i in obtaining information about the rival s output and a time lag S i in implementing information about its own output. The characteristic equation of sstem (4) can be obtained b looking for the solutions as (t) =e λt u and (t) =e λt v and substituting them into the corresponding homogeneous equations: λe λt u = α( 2be λ(t S) u be λ(t T) v), λe λt v = β( be λ(t T) u 2be λ(t S) v). 4

5 Nontrivial solution eists if and onl if the determinant of the coefficient matri J F = λ +2bαe λs bαe λt (5) bβe λt λ +2bβe λs is zero, which provides a mied eponential-polnomial equation for λ: λ +2bαe λs λ +2bβe λs b 2 αβe λt e λt =. Multipling both sides b e λs e λs e λt e λt, we get λe λs +2bα λe λs +2bβ e λt e λt b 2 αβe λs e λs =. (6) To clarif the effects caused b the time lags on dnamics, we first eamine the case in which the firms adjust their outputs using correct information on their own outputs and have uncertaint about their rival s output (i.e., S = S =while T > and T > ) and then consider the opposite case in which the firms have lags on their outputs (i.e., T = T =while S > and S > ). Substituting S = S =, T > and T > into (6), we obtain the characteristic equation (λ +2bα)(λ +2bβ) b 2 αβe λ(t +T ) =. Notice first that λ =cannot be a solution. So assume that λ 6= and R(λ) where λ = μ + iξ. Then we have λ +2bα λ +2bβ 4b 2 αβ and b 2 αβe λ(t+t) = b 2 αβ e μ(t+t) e iξ(t+t) b 2 αβ The last two inequalities impl that no λ with R(λ) can solve the characteristic equation. Hence an solution of the characteristic equation must have negative real parts. The information lags on the rival s outputs are harmless and do not affect the stabilit of (4). Taking T = T =, S > and S > and repeating the same procedure ield the characteristic equation of the form λ 2 b 2 αβ +2bαλe λs +2bβλe λs +4b 2 αβe λ(s+s) =. (7) In the absence of information lags, the linear sstem (4) is locall asmptoticall stable. The characteristic equation implies that λ =is not a solution of (7). B continuit, all eigenvalues of (7) have negative real parts for sufficientl small S + S >. Freedman and Rao (1986) estimated the range of S + S for which the Cournot point remains asmptoticall stable. In particular, according 5

6 to their theorem (i.e., (ivβ) of Theorem 3.1), the following estimates on S and S impl asmptotical stabilit: ma[s,s ] Γ, S + S < η 4b 2 αβ, η Cos[Γv + ]= 2b(α + β), where < η < 2b(α + β) and v + = b n 2(α + β)+ p o (α + β) αβ. Figure 1 illustrates two time trajectories of when α = β =1, b =1and η =4. Under these parameter specifications, the Cournot point is asmptoticall stable for ma[s,s ].262 and S + S < 1. We take S =.24 and S =.26 in Figure 1(A) and S =.7 and S =.8 in Figure 1(B). It can be seen that the large information lags on the own outputs of the firms have an instabilizing effect. Wethushaveconfirmed the results of Howrod and Russel (1986) in the duopol case. (A) Converging trajector (B) Diverging trajector Figure 1. Time trajectories of (t) generated b (4) In real economic situations, the lags are usuall uncertain and can be considered to be fied onl under special circumstances. Therefore, we will model time lags in a continuousl distributed manner. Given Assumption 1, the continuous dnamic duopol model with continuousl distributed time lags is ẋ(t) =α(a c 2b ε (t) b e (t)), ẏ(t) =β(a c b e (t) 2b ε (t)), (8) 6

7 with epectations on its own outputs ε (t) = ω(t s, S,n )(s)ds and ε (t) = and epectations on its rival s outputs e (t) = ω(t s, T,m )(s)ds and e (t) = ω(t s, S,n )(s)ds ω(t s, T,m )(s)ds. The weighting function ω is assumed to have the form 1 t s e Γ Γ if ` =, ω(t s, Γ,`)= `+1 1 ` (t s)`e `(t s) Γ `! Γ if ` 1. Here we assume that Γ > and ` is a nonnegative integer. Substituting (9) into the epectation formations defined above, and the resulting epressions of the epectations into (8) ield a sstem of integro-differential equations. In order to analze the dnamic behavior of the sstem, we consider the corresponding homogeneous sstem. Letting δ and δ denote the deviations of and from their Cournot levels of outputs, c and c, the homogeneous sstem can be formulated as follows: ẋ δ = α n 2b R t ω(t s, S,n ) δ (s)ds b R t ω(t s, T,m ) δ (s)ds n ẏ δ = β b R t ω(t s, T,m ) δ (s)ds 2b R o t ω(t s, S,n ) δ (s)ds. (1) We seek the solutions in the eponential form δ (t) =e λt u and δ (t) =e λt v. Substituting these solutions into equation (1) and arranging terms ield ³ λ +2bα R t ω(t s, S,n )e λ(t s) ds u + bα R t ω(t s, T,m )e λ(t s) ds v =, bβ R ³ t ω(t s, T,m )e λ(t s) ds u + λ +2bβ R t ω(t s, S,n )e λ(t s) ds v =. Introducing a new variable z = t s, we can simplif the integral terms b noticing that Allowing t, we have ω(t s, Γ,`)e λ(t s) ds = lim t ω(z, Γ,`)e λz dz = 7 ω(z,γ,`)e λz dz. 1+ λγ q (`+1) o, (9)

8 with 1 if ` =, q = ` if ` 1. Then equations (1) can be simplified as à λ +2bα 1+ λs! (n +1) u + bα 1+ λt (m +1) v = q q bβ 1+ λt à (m+1) u + λ +2bβ 1+ λs! (n+1) v =. q q which can be rewritten in the matri form: A (λ) B (λ) u B (λ) A (λ) v = where " A (λ) = λ 1+ λs n +1 +2bα q " A (λ) = λ # 1+ λt q 1+ λs # n+1 +2bβ 1+ λt q q B (λ) =ba 1+ λs n +1 q and B (λ) =bβ 1+ λs n+1. q A non-trivial solution eists if and onl if m +1, m+1, A (λ)a (λ) B (λ)b (λ) =. (11) We denote the left hand side b ϕ(λ) and call it the characteristic polnomial of sstem (1). There is an interesting relation between the characteristic polnomials (6) and (11) of the sstems with fied and continuousl distributed time lags. Assume that n,n,m and m converge to infinit. Since q i = n i 1 and q i = m i 1 for i =,, we have the limits A (λ) λe λs +2bα e λt, A (λ) λe λs +2bβ e λt, B (λ) bαe λs 8

9 and B (λ) bβe λs, therefore in the limiting case, the characteristic equation of the continuousl distributed dela model converges to the characteristic equation of the fied dela model. 3.1 Information lags about the rival s output In this section we confine our analsis to the case in which firms eperience no information lags on their own outputs: Assumption 2. S =and S =. Under Assumption 2, the characteristic equation has the form (λ +2bα)(λ +2bβ) 1+ λt m+1 1+ λt m+1 b 2 αβ = (12) q q We will prove that all roots of equation (12) have negative real parts impling the global asmptotical stabilit of the Cournot point. Assume in contrar that there is a root λ = α + iβ with α. Notice firstthatwithanrealpositive numbers A and B, A + Bλ 2 = (A + Bα)+i(Bβ) 2 = (A + Bα) 2 +(Bβ) 2 (A + Bα) 2 A 2. Therefore the two terms of (12) have to be different, since +2bα)(λ +2bβ) 1+ (λ λt m λt m +1 q q 4b2 αβ >b 2 αβ. Consequentl λ cannot satisf equation (12). This observation implies that the stabilit part of the result of Howrod and Russel (1986) remains true in the case of continuousl distributed lags. 3.2 Information lags on the own outputs In this section we confine our analsis to the case in which the firms have no information lags on their rivals outputs. Assumption 3. T =and T =. 9

10 Under Assumption 3, the characteristic equation has the form à λ 1+ λs!ã n+1 +2bα λ 1+ λs! n+1 +2bβ b 2 αβ q q Since it is difficult to check whether the real parts of the roots are negative or positive in general, we will show some simple special cases in which analtical results can be obtained. Case 1. S > with n =and S =. As a special case, we assume first that firm has information lag on its own output, however, firm uses instantaneous information about its own output, that is, S > and S =. We also assume eponential weighting function with n =. Then equation (13) reduces to the following cubic equation: where the coefficients are defined as Furthermore, we have a λ 3 + a 1 λ 2 + a 2 λ + a 3 = a = S >, a 1 =1+2bβS >, a 2 =2b(α + β) b 2 αβs R, a 3 =3b 2 αβ >. a 1 a 2 a a 3 =2b b 2 αβ 2 S 2 +2bβ 2 S +(α + β) ª. (14) If a 2 > and a 1 a 2 a a 3 > are confirmed, then the Routh-Hurwitz stabilit theorem implies that the sstem (1) is globall asmptoticall stable. It suffices for our purpose to show that a 1 a 2 a a 3 > in this case since a >, a 1 > and a 3 > impl that a 2 >. Let f(s ) be the right hand side of (14). Since f() > and f () >, equation f(s )=has one positive root, 1+ λs n+1 1+ λs n+1 = q q (13) S = β + p α 2 + αβ + β 2 bαβ >. Hence we have a 2 > and a 1 a 2 a a 3 > for S <S. We summarize this result as follows: Theorem 2 If S > with n =and S =under Assumption 3, then the dela dnamic sstem (8) is globall asmptoticall stable when <S <S and it is unstable when S >S where S = β + p α 2 + αβ + β 2. bαβ 1

11 Case 2. S > with n =and S > with n =. In the second special case, we assume that both firms have information lags about their own outputs and have eponentiall declining weighting functions. Then equation (12) reduces to the following fourth degree polnomial equation: where the coefficients are defined as a λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = a = S S >, a 1 = S + S >, a 2 =1+2b(αS + βs ) b 2 αβs S R, a 3 =2b(α + β) b 2 αβ(s + S ) R, a 4 =3b 2 αβ >. The Routh-Hurwitz stabilit theorem implies that the roots of the characteristic equation have negative real parts if and onl if all coefficients are positive and the following determinants are positive: Notice that J 2 = a1 a a 3 a 2 > and J 3 = a 1 a a 3 a 2 a 1 a 4 a 3 J 2 = S + S +2b(αS 2 + βs 2 ) > >. and J 3 = a 3 J 2 a 2 1a 4. We can easil show that the single condition J 3 > implies that all roots have negative real parts. Since J 2 and a 1 and a 4 are positive, J 3 > implies a 3 >. Then relation J 2 = a 1 a 2 a a 3 > implies that a 2 also have to be positive. Theorem 3 Given Assumption 2, if S > with n =and S > with n =, then the dela dnamic sstem (8) is globall asmptoticall stable for (S,S ) below the partition line, J 3 =, and unstable for (S,S ) above the line where J 3 = b S + S +2b(αS 2 + βs) 2 [2(α + β) bαβ(s + S )] 3bαβ(S + S ) 2ª It is possible to obtain the analtic form of the partition line with larger values of n and n. It becomes, however, much more complicated as shown in Theorem 3. Therefore we check numericall the shapes of the partition lines. Taking α = β =1, b =1and repeating the above procedure with increasing values of n = n = 1, 2, 3, 4, we obtain the four partition lines illustrated in Figure 2 in which P i means the partition line when n = n = i. Since the partition line divides the parameter space into the stable and the unstable regions, stabilit of the dnamic sstem (4) is clearl affected b the information 11

12 lags about the firms s own outputs, similarl to the case of fied time lags as shown b Howrod and Russel (1986). Figure 2 indicates that the stable region becomes larger as the partition line shifts upward with increasing values of n and n and converge to the stable region with fied time dela when n and n tend to infinit. Figure 2. Partition lines 4 Dela Nonlinear Duopolies In this section we consider the case in which the adjustment speeds are positive functions of the outputs of the firms. In particular, we adopt the linear dependenc: Assumption 4. α() =α with α > and β() =β with β >. The continuous dnamic sstem (1) under Assumption 4 can be written as ẋ(t) (t) = α(a c 2b(t) b(t)), ẏ(t) (t) = β(a c 2b(t) b(t)). (15) This implies that each firm adjusts its growth rate of the outputs proportionall to a change in the profit. As we have done before, sstem (15) can be also epressed in terms of the best repl functions, ẋ(t) (t) =ᾱ(r ((t)) (t)), ẏ(t) (t) = β(r ((t)) (t)), (16) 12

13 which can be interpreted as each firm adjusts its growth rate proportionall to the difference between its profit maimizing output and its actual output. The discrete versions of the duopol dnamic model (15) or (16) are considered b Huang (22) adopting a feedback controlling method to stabilize a discrete sstem and also b Hassen (24) showing that the stabilit region of the Nash equilibrium can become larger in a delaed duopol model. As in the case of linear duopol, we first introduce fied information lags on the own behavior and rewrite (15) as ẋ(t) =α(t)(a c 2b(t S ) b(t)), ẏ(t) =β(t)(a c b(t) 2b(t S )) which has the same dnamic structure as the Lotka-Volterra tpe models. Appling the results obtained b Shibata and Saito (198), we can demonstrate that this fied dela Cournot model displas various dnamic behavior ranging from periodic solutions to chaotic solutions. Figure 3 illustrates emergence of chaotic oscillation under α = β =1,c = c =1, a =3,S =1.6 and S =.9. Figure 3. Chaotic solution of the fied dela Cournot model We now introduce continuousl distributed lags into (15), ẋ(t) =α(t)(a c 2b ε (t) b e (t)), ẏ(t) =β(t)(a c b e (t) 2b ε (t)), (17) where the epected values of the firms own outputs, the epected values of the competitors outputs and their weighting functions are given as in the previous section. To eamine local dnamics of the above sstem in a neighborhood of the equilibrium point ( c, c ) where c = e = ε and c = e = ε, we need to eamine the linearized version of sstem (17). B considering the right hand sides 13

14 as three-variable functions (depending on, ε, e and, e, ε, respectivel), the linearized sstem becomes ẋ δ (t) = α 2b c w(t s, S,n ) δ (s)ds b w(t s, T,m ) δ (s)ds, ẏ δ (t) = β c b w(t s, T,m ) δ (s)ds 2b w(t s, S,n ) δ (s)ds, which has eactl the same form as the linear sstem (1) if α and β are replaced b α c and β c. Thus the characteristic equation is given b where Ā (λ)ā(λ) B (λ) B (λ) = à Ā (λ) = λ 1+ λs!ã n +1 +2bα c 1+ λt! m +1, q q à Ā (λ) = λ 1+ λs!ã n +1 +2bβ c 1+ λt! m +1, q q B (λ) =bα 1+ c λs n+1, q B (λ) =bβ 1+ c λs n +1. q 4.1 Effects of the information lag about the own output Following the same procedure as in the previous section, we have the following result: Theorem 4 If S > with n =and S =under Assumption 3, then the dela dnamic sstem (17) is locall asmptoticall stable when <S <S and it is locall unstable when S >S where S = βc + p (α c ) 2 +(β c ) 2 + αβ c c bαβ c c. The a 1 a 2 a a 3 =line divides the parameter space into two parts: one in which the sstem is locall stable and the other in which it is locall unstable. We call it the partition line. Substituting a 3 = a 1a 2 a into the characteristic equation ϕ(λ) =and factorizing it ield (a 1 + a λ)(a 2 + a λ 2 )=which can be solved for λ. Two of the characteristic roots are purel imaginar and the third is real and negative: r λ 1,2 = ± a 2 = ±iξ a 14

15 and λ 3 = a 1 <. a We show the appearance of Hopf bifurcation giving the possibilit of the birth of limit ccles. To this end, we need to confirm whether the real part of the comple roots is sensitive to a change in a bifurcation parameter. We select S as the bifurcation parameter and suppose that λ is a function of S. Implicitl differentiating the characteristic equation gives dλ = λ3 +2bβ c λ 2 +( b 2 αβ c c )λ ds (3a λ 2. +2a 1 λ + a 2 ) Substituting λ = iξ with ξ 2 = a 2 a, rationalizing the right hand side and noticing that the terms with λ and λ 3 are imaginar and the constant and quadratic terms are real ield the following form of the real part of the derivative of λ with respect to the bifurcation parameter: dλ Re = a 2 4(bβ c ) 2 (bα c S 1) ds 2a (a 1 a 3 + a 2 2 ) >. We summarize this result as follows: Theorem 5 The stationar point ( c, c ) of dela sstem (17) is destabilized via a Hopf bifurcation when S increases from S. Figure 4 shows the birth of a limit ccle in this case. In the opposite case in which S =with n =and S =, wehavethesameresultsastheorems4 and 5 if S, c and α are replaced b S, c and β. Figure 4. Birth of a limit ccle Appling Theorem 3, we also have the following result: 15

16 Theorem 6 Given Assumption 3, if S > with n =and S > with n =, then the dela dnamic sstem (17) is locall asmptoticall stable for (S,S ) below the partition line, J 3 =, and locall unstable for (S,S ) above it where J 3 = b S + S +2b(α c S 2 + βc S 2 ) [2(α c + β c ) bαβ c c (S + S )] 3b 2 αβ c c (S + S ) 2. The curve of J 3 = a 1 a 2 a 3 (a a a 2 1a 4 )=is the partition line between the stable and unstable regions. Substituting a 4 = a 1a 2 a 3 a a 2 3 a 2 1 into the characteristic polnomial and factoring it ield (a 3 + a 1 λ 2 )(a 1 a 2 a a 3 + a 2 1λ + a a 1 λ 2 )= Two of the characteristic roots are purel imaginar, r λ 1,2 = ± a 3 = ±iξ, a 1 and the other roots are the solutions of the quadratic equation a a 1 λ 2 + a 2 1λ + a 1 a 2 a a 3 =, which have negative real parts, since all coefficients are positive. The appearance of Hopf bifurcation as the value of S crosses the partition line can be eamined similarl to the previous case. Case 3. S > with n =1and S > with n =1. In the third special case we assume that both firms have information lags about their outputs and the weighting function has a bell-shaped curve peaked around t s = S i for i =,. Substituting these values into the characteristic equation and arranging terms indicate that the eigenvalues are the roots of the following sith degree polnomial, ϕ(λ) =a λ 6 + a 1 λ 5 + a 2 λ 4 + a 3 λ 3 + a 4 λ 2 + a 5 λ + a 6, where the coefficients are defined as a =(S S ) 2 >, a 1 =2S S (S + S ) >, a 2 = S 2 +4S S + S 2 b 2 αβ c c S 2 S 2, a 3 =2(S + S )+2b(α c S 2 + β c S 2 ) 2b 2 αβ c c S S (S + S ), a 4 =1+4b(α c S + β c S ) b 2 αβ c c (S 2 +4S S + S 2 ), a 5 =2b(α c + β c ) 2b 2 αβ c c (S + S ), a 6 =3b 2 αβ c c >. 16

17 The Routh-Hurwitz determinants are defined as follows: J 2 = a 1 a 2 a a 3, a 1 a a 1 a a 1 a J 3 = a 3 a 2 a 1 a 5 a 4 a 3, J 4 = a 3 a 2 a 1 a a 3 a 2 a 1 a a 5 a 4 a 3 a 2 and J 5 = a 5 a 4 a 3 a 2 a 1. a 6 a 5 a 4 a 6 a 5 a 4 a 3 a 6 a 7 It is ver hard to analze the signs of these determinants analticall, therefore we perform numerical simulations. We take a =1,b=1, α = β =1and c = c =1and numericall confirm that the J 5 =curve is the partition line in Figure 5 where the curves, a 2 =,J 3 =and J 4 =are located outside the region shown in the figure. In the shaded region below the J 5 =curve, the Routh-Hurwitz stabilit conditions are satisfied. Figure 5. Tthe partition line and stable region We also numericall confirm that with the selection of S =.4 and S =.2 the dnamic sstem (17) generates comple dnamics. It is illustrated in Figure 6 where the time trajector is plotted in the (ln, ln ) space. It is well known that the weighting function (9) converges to the Dirac delta function when n and n tend to infinit. We also showed earlier that the characteristic polnomial of continuousl distributed dela models also converge to that of the fied dela, so in the limiting case, the model with continuousl distributed time lag becomes identical with the fied time dela model. Comparing Figure 3 with Figure 6 reveals the similarit between dnamics generated b the model with continuousl distributed time lag and dnamics b the fied time dela 17

18 model even when n = n =1. Figure 6. Trajector in the (ln, ln ) space 4.2 Effects of the information lag about the rival s output In this section we assume again that there are no information lags about the own outputs of the firms, similarl to the model introduced in Section 3.1, however we assume the nonlinear adjustment process (15). In this case the characteristic equation (11) has the form (λ+2bα c )(λ+2bβ c ) 1+ λt m λt m +1 b 2 αβ c c =. (18) q q This is the same equation as (12) when α and β are replace b α c and β c, respectivel. In Section 3.1 we have proved that all roots of equation (12) have negative real parts, which also holds for equation (18) showing that the stationar state is alwas locall asmptoticall stable. 5 Conclusion Dnamic duopolies were eamined with linear price and cost functions. We assumed firstconstantspeedsofadjustmentina gradient adjustment process. It is well-known that the stationar state is alwas globall asmptoticall stable. If the firms have fied or continuousl distributed time delas onl on the outputs of the rivals, then stabilit is preserved. The stabilit might be lost if the firms have delas in their own outputs. Similar results were shown in the case when the firms adjust their growth rates of the outputs proportionall to the changes in their profits. The resulting dnamicsstemsarenonlinear,andwehaveshownthatsimilarresultshold 18

19 about the local asmptotical stabilit of the stead state. In the case of stabilit loss Hopf bifurcation occurs giving the possibilit of the birth of limit ccles around the stationar state. 19

20 References [1] Ahmed, A., H. Agiza and S. Hassen, "On Modifications of Puu s Dnamical Duopol," Chaos, Solitions and Fractals, 11(2), [2] Bischi, G. I., C. Chiarella, M. Kopel and F. Szidarovszk, Nonlinear Oligopolies: Stabilit and Bifurcations, 29, New York, Springer- Verlag. [3] Bischi, G. I. and F. Lamatia, "Chaos Snchronization and Intermittenc in a Duopol Game with Spillover Effects," in Oligopol Dnamics ed. T. Puu and I. Sushko, , 22, Berlin/Herdleberg/New York, Springer-Verlag. [4] Bischi, G. I. and A. Naimzada, "Global Analsis of a Duopol Game with Bounded Rationalit," Advnces in Dnamic Games and Applications, 5(1999), , Birkhuser. [5] Chiarella, C., and F. Szidarovszk, "Birth of Limit Ccles in Nonlinear Oligopolies with Continuousl Distributed Information Lags," in Modeling Uncertaint ed. M. Dror, P. LEcuer and F. Szidarovszk, , 22, Boston/Dordrecht/London, Kluwer Academic Publishers. [6] Cournot, A., Recherches sur les principes mathematiques de la theorie de richessess, 1838, Paris: Hachette. (English translation (196): Researches into the Mathematical Principles of the Theor of Wealth, New York: Kell). [7] Freedman, H. and V. Rao, "Stabilit Criteria for a Sstem Involving Two Time Delas," SIAM Journal of Applied Mathematics, 46(1986), [8] Hassen, S., "On Delaed Dnamical Duopol," Applied Mathematics and Computation, 151(24), [9] Howrod, T. and A. Russel, "Cournot Oligopol Models with Time Delas," Journal of Mathematical Economics, 13(1984), [1] Huang, W., "Controlling Chaos through Growth Rate Adjustment," Discrete Dnamics in Nature and Societ, 7(22), [11] Okuguchi, K., Epectations and Stabilit in Oligopol Models, 1976, New York, Springer-Verlag. [12] Okuguchi, K., and F. Szidarovszk, The Theor of Oligopol with Multi- Product Firms (2nd ed.), 1999, New York, Springer-Verlag. [13] Russell, A., J. Rickard and T. Howrod, "The effects of Delas on the Stabilit and Rate of Convergence to Equilibrium of Oligopolies," Economic Records, 62(1986), [14] Shibata, A. and N. Saito, "Time Delas and Chaos in Two Competing Species," Mathematical Biosciences, 51(198),