A dynamic marketing model with best reply and inertia
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1 A dynamic marketing model with best reply and inertia Gian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) URL: Urbino, September 18, 14 Urbino, September 18, 14 1 /
2 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i a j x j (t) Urbino, September 18, 14 2 /
3 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i Urbino, September 18, 14 2 /
4 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i market share: s i (t) = A i(t) n j=1 A j (t) Urbino, September 18, 14 2 /
5 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i market share: s i (t) = A i(t) n j=1 A j (t) profits: Π i (t) = Bs i (t) x i (t) Urbino, September 18, 14 2 /
6 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Urbino, September 18, 14 3 /
7 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Each firm maximizes its own profit function computing its gradient Π i (t + 1) x i = 0 Urbino, September 18, 14 3 /
8 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Each firm maximizes its own profit function computing its gradient Π i (t + 1) x i = 0 Setting β i = 1, this leads to x i (t + 1) = j =i a j x (e) j (t + 1) a i a j x (e) j (t + 1) j =i Urbino, September 18, 14 3 /
9 The Farris Market Model Assuming naïve expectations, x (e) j (t + 1) := x j (t), Farris et al. derive the following Best Response dynamic model with adaptive adjustment: x i (t + 1) = (1 λ i )x i (t) + λ i j =i a j x j (t) a i a j x j (t) j =i Urbino, September 18, 14 4 /
10 The Farris Market Model Assuming naïve expectations, x (e) j (t + 1) := x j (t), Farris et al. derive the following Best Response dynamic model with adaptive adjustment: x i (t + 1) = (1 λ i )x i (t) + λ i j =i a j x j (t) a i a j x j (t) j =i Setting N = 2 and the new (rescaled) variables x = a 1 a 2 x 1, y = a 1 a 2 x 2 we have x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x Urbino, September 18, 14 4 /
11 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 Urbino, September 18, 14 5 /
12 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Urbino, September 18, 14 5 /
13 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) Urbino, September 18, 14 5 /
14 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x Urbino, September 18, 14 5 /
15 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x We obtain the Puu from the Farris setting a 2 = 1/a 1 Urbino, September 18, 14 5 /
16 Equilibria The equilibria abscissas follows from the fourth order algebric equation: [ ] (1 a 1 a 2 ) 2 η η (1 a 1 a 2 ) η 2 + a 2 (a 1 + 1) η a 2 = 0 a 1 a 2 where η := x. An analogous equation for ζ := y holds. From Cardano s formula, the number of real solutions is given provided the sign of the discriminant: > 0 1 real solution D(a 1, a 2 ) : = 0 1 real solution and 2 coincident < 0 3 distinct real solutions Urbino, September 18, 14 6 /
17 Equilibria Section of the function D(a 1, a 2 ) with the plane (a 1, a 2, 0) Urbino, September 18, 14 7 /
18 Equilibria Section of the function D(a 1, a 2 ) with the plane (x, y, 0) Urbino, September 18, 14 8 /
19 The General case a 1 = a 2 Proposition Besides E 0 = (0, 0) a non vanishing fixed point always exists in the region S = (0, 1) (0, 1). If a 1 a 2 = 1 then two further distinct fixed points exist in the region S if the following inequality holds D(a 1, a 2 ) = a1 2a4 [ a1 108 (1 a 1 a 2 ) 6 a 2 (4a 1 + 4a 2 18) a1a ] < 0 and if D(a 1, a 2 ) = 0 these two further fixed points are merging, i.e. there are two real coincident solutions of the cubic ( equation. In the ) particular 1 1 case a 1 a 2 = 1 the unique fixed point E =, is get. (a 1 +1) 2 (a 2 +1) 2 Urbino, September 18, 14 9 /
20 Bifurcation path: a 2 = a 1 /9 + 13/ ,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /
21 Bifurcation path: a 2 = a ,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /
22 Bifurcation path: a 2 = a , a 2 = a 1 (gray) 1 6 0,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /
23 Typical scenario (after the final bifurcation) Urbino, September 18, /
24 The symmetric case: a = a 1 = a 2 Fixed poits are: E 1 = ( a 2 (1 + a) 2, a 2 ) (1 + a) 2 = { (x, y) R 2 x = y } Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
25 The symmetric case: a = a 1 = a 2 Fixed poits are: E 1 = ( a 2 (1 + a) 2, a 2 ) (1 + a) 2 = { (x, y) R 2 x = y } For a 3 the two further fixed points in symmetric positions with respect to : E 2 = E 3 = a 2 2 (a 1) 2 (a + 1) a 2 2 (a 1) 2 (a + 1) ( a 1 + (a + 1) (a 3), a 1 ) (a + 1) (a 3) ( a 1 (a + 1) (a 3), a 1 + ) (a + 1) (a 3) Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
26 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Urbino, September 18, /
27 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Transverse instability (saddle) via pichfork, merging of two further fixed points E 2 and E 3, for a a p Urbino, September 18, /
28 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Transverse instability (saddle) via pichfork, merging of two further fixed points E 2 and E 3, for a a p Unstability via flip for ( 1 a a f = ) λ 1 λ 2 λ 1 λ 2 Urbino, September 18, /
29 Fixed points and their basin of attraction (a) Codim. 2 bifurcation (b) Just after the pichfork (c) Just after the flip (d) Just after the subcritical flip Urbino, September 18, /
30 The symmetric case: stability of E 2 and E 3 Local asymptotic stability for ( a < a h = ) ( ) 2 16 λ 1 λ 2 λ 1 λ ( ) ( ) 2 16 λ 1 λ 2 λ 1 λ 2 27 Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
31 The symmetric case: stability of E 2 and E 3 Local asymptotic stability for ( a < a h = ) ( ) 2 16 λ 1 λ 2 λ 1 λ ( ) ( ) 2 16 λ 1 λ 2 λ 1 λ 2 27 Unstability via Hopf for a a h Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
32 Profits Computing the profits we get: Π 1 > Π 2 (x y)(x + y a 2 ) < 0 Urbino, September 18, /
33 Conclusions: effect of eterogeneities Farris Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
34 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
35 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:
36 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Urbino, September 18, /
37 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Urbino, September 18, /
38 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) Urbino, September 18, /
39 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) A rich dynamical scenario is observed Urbino, September 18, /
40 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) A rich dynamical scenario is observed Eterogeneities and initial conditions: asymptotic behavour of the system Urbino, September 18, /
41 Bibliography Bischi, G.I., Gardini, L. and Kopel, M Analysis of Global Bifurcations in a Market Share Attraction Model, Journal of Economic Dynamics and Control, 24 (00) Bischi, G.I., Cerboni Baiardi L. Fallacies of composition in nonlinear marketing models, Communications in Nonlinear Science and Numerical Simulation, DOI: /j.cnsns (14, in press) Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition Marketing - Journal of Research and Management, 1 (1) (05) Puu T. Chaos in Duopoly Pricing Chaos, Solitons & Fractals, 1 (6) (1991) Urbino, September 18, 14 /
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