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Pricing Strategies of Complementary Products in Distribution Channels: A Dynamic Approach

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Abstract

This paper investigates the dynamic pricing strategies of firms selling complementary products in a marketing channel. The problem is modelled as a non-cooperative differential game that takes place between decisions makers controlling transfer and retail prices. We computed and compared prices and sales rates of channel members under two scenarios: (i) The first involves a single retailer that sells a unique brand produced by a monopolist manufacturer and (ii) in the second, a complementary product is introduced by an additional manufacturer. We found that in both scenarios, transfer and retail prices decrease over time, but prices decrease faster when the complementary product is introduced into the market. Furthermore, the entry of the complementary product onto the market boosts the sales rate of the existing product. Finally, we found that the retailer in the second scenario always has a non-negative retail margin, meaning that practicing a loss-leadership strategy is not optimal.

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Notes

  1. The interested reader could read [13] for a complete survey of non-competitive models on this topic.

  2. See [14] for a complete survey on the topic.

  3. Studies that modelled the dynamic price effects via the reference price are: [11, 12] and [22]. Only [19, 20] and [2] examined this issue in a vertical channel context.

  4. See [1].

  5. The author built a general model and then examined different subclasses of it. In the particular case where the diffusion rate was technically similar to the model used in our paper, the author found that the monopolistic firm fixed a price that decreased monotonically. This result was not affected by the discount rate level or the presence of cost learning.

  6. Competition is attributed either to the presence of substitute products offered by competitors in an oligopoly, or to the introduction of new generations of technologies by the same firm.

  7. The only exception is Dockner and Gaunersdorfer [8], in which the authors used a feedback information structure. But here again, the authors examined the case of a duopoly where two substitute products were sold.

  8. A common assumption in this body of literature is that the number of adopters is equal to the number of units sold. Hence, the cumulative number of adopters at the end of the planning horizon is equal to the market potential.

  9. We assume that both manufacturers face similar constant unit production costs c.

  10. The steady state is obtained by replacing the pricing strategy \(p_{R}(t)\) with its expression from (11) in the state Eq. (1) and solving \(\overset{.}{x}(t)=0\).

  11. This expression represents the sales rate when \(x(t)=0\) and when the product is sold at cost.

  12. Since the objective functionals are quadratic in the state and the control variables, and the state dynamics is linear in these variables, we have a linear-quadratic differential game.

  13. Another result not reported here is \(\alpha _{0}=2\beta _{0}\).

  14. If these expressions are positive.

  15. The differential game studied under this second scenario is particular in that the Ricatti Eqs. (44)–(46) and (50)–(52) given in “Appendix B” are highly nonlinear. We solve them numerically with MATLAB’s fsolve routine to obtain the value function coefficients \(h_{11},h_{12},h_{13},h_{21},h_{22}\), \(h_{23}\) and \(R_{1},R_{2},R_{3}\), respectively. Once these coefficients are known, we compute the values function coefficients \(h_{14},h_{15},h_{24}\), \(h_{25}\) and \(R_{4},R_{5}\) from (47)–(48) and (53)–(54). Finally, we solve the Eqs. (49) and (55) for \(h_{16},h_{26}\) and \(R_{6}\).

  16. Superscripts BR2M refer to bilateral monopoly and One-retailer–two-manufacturers vertical channel, respectively.

  17. This result is in line with [24].

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Authors and Affiliations

Authors

Corresponding author

Correspondence to Sihem Taboubi.

Additional information

F. Ngendakuriyo: The project was initiated during the post-doctorate at GERAD.

Research supported by NSERC, Canada.

The authors wish to thank the two anonymous Reviewers for their constructive comments.

Appendix

Appendix

1.1 Appendix A

The sufficient condition for a stationary feedback Stackelberg equilibrium requires us to find bounded and continuously differentiable functions, \(V_{R}(x)\) and \(V_{M}(x)\) for the retailer and the manufacturer, respectively, which satisfy for all \(x(t)\geqslant 0\), the HJB equations obtained after the substitution of (10) and (8) into Eqs (7) and (9).

Guided by the model’s linear-quadratic structure, we conjecture that the functions \(V_{R}(x)\) and \(V_{M}(x)\) are quadratic and given by the expressions (13) and (14) in the proposition. The coefficients \(\beta _{1},\beta _{2},\beta _{0},\alpha _{1},\alpha _{2}\) and \(\alpha _{0}\) are obtained by identification after replacing \(V_{R}(x)\) and \(V_{M}(x)\) as well as their first derivatives into the HJB equations.

Finally, plugging the derivatives of these values functions into the expressions (10) and (8) provides the channel members’ pricing strategies at the equilibrium displayed in the proposition.

The terminal conditions

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }e^{-\rho t}V_{R}(x(t))&=0, \end{aligned}$$
(39)
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }e^{-\rho t}V_{M}(x(t))&=0. \end{aligned}$$
(40)

are sufficient conditions guaranteeing that the expressions (13) and (14) are the retailer’s and manufacturer’s value functions and that (11)–(12) are the pricing strategies.

1.2 Appendix B

As in the previous scenario, after computing the retailer’s reaction functions, we move to the manufacturer’s problems. We consider that both manufacturers play à la Nash. In other words, they solve their optimization problems simultaneously. The manufacturer i’s HJB equations are:

$$\begin{aligned} \rho V_{Mi}(x_{1},x_{2})= & {} \max \limits _{p_{Mi}}\left\{ \left( p_{Mi}-c+\frac{\partial V_{Mi}}{\partial x_{i}}\right) [a_{i}+bx_{i}-\kappa p_{Ri}\left( t\right) +\gamma p_{Rj}(t)]\right. \nonumber \\&\left. +\frac{\partial V_{Mi}}{\partial x_{j}}[a_{j}+bx_{j}-\kappa p_{Rj}(t)+\gamma p_{Ri}(t)]\right\} . \end{aligned}$$
(41)

Substituting (24) and (25) into (41), the HJB equations of the leaders, and performing the maximization of the right-hand side provide the strategies:

$$\begin{aligned} p_{M1}&=\frac{\lambda _{1}+bx_{2}\gamma +2bx_{1}\kappa +\gamma ^{2}\left( \frac{\partial V_{M2}}{\partial x_{1}}-\frac{\partial V_{R}}{\partial x_{1}}\right) +2\kappa ^{2}\left( \frac{\partial V_{R}}{\partial x_{1}}-\frac{\partial V_{M1}}{\partial x_{1}}\right) +\gamma \kappa \left( c-\frac{\partial V_{R}}{\partial x_{2}}+2\frac{\partial V_{M1}}{\partial x_{2}}-\frac{\partial V_{M2}}{\partial x_{2}}\right) }{4\kappa ^{2}-\gamma ^{2}}, \end{aligned}$$
(42)
$$\begin{aligned} p_{M2}&=\frac{\lambda _{2}+bx_{1}\gamma +2bx_{2}\kappa \gamma ^{2}\left( \frac{\partial V_{M1}}{\partial x_{2}}-\frac{\partial V_{R}}{\partial x_{2}}\right) +2\kappa ^{2}\left( \frac{\partial V_{R}}{\partial x_{2}}-\frac{\partial V_{M2}}{\partial x_{2}}\right) +\gamma \kappa \left( c-\frac{\partial V_{R}}{\partial x_{1}}+2\frac{\partial V_{M2}}{\partial x_{1}}-\frac{\partial V_{M1}}{\partial x_{1}}\right) }{4\kappa ^{2}-\gamma ^{2}}. \end{aligned}$$
(43)

where \(\lambda _{1}=a_{2}\gamma +2\kappa (a_{1}+c\kappa )\) and \(\lambda _{2}=a_{1}\gamma +2\kappa (a_{2}+c\kappa ).\)

Substituting \(p_{Ri}\) and \(p_{Mi}\), namely (24), (25), (42), 43), into the HJB Eqs. (23) and (41) leads to conjecture the quadratic value functions in Eqs. (30) and (31).

We replace \(V_{Mi}(x_{1},x_{2})\) as well as their first derivatives in the HJB equations. To identify the value function coefficients \(h_{ij}\) for \(i=1,2\) and \(j=1,2,...,6\) for manufacturers, we need to solve the twelve Ricatti equations (not printed here). Their canonical forms are the following, for \(i=1,2\); \(\forall i\ne j\):

$$\begin{aligned} f_{1} (h_{i1}, h_{i3}, h_{j1}, h_{j3}, R_{1},R_{3})&= 0, \end{aligned}$$
(44)
$$\begin{aligned} f_{2} (h_{i2}, h_{i3}, h_{j2}, h_{j3}, R_{2},R_{3})&= 0, \end{aligned}$$
(45)
$$\begin{aligned} f_{3} (h_{i1}, h_{i2}, h_{i3}, h_{j1}, h_{j2}, h_{j3}, R_{1},R_{2},R_{3})&= 0, \end{aligned}$$
(46)
$$\begin{aligned} f_{4}(h_{i1},h_{i3},h_{i4},h_{i5},h_{j1}, h_{j3}, h_{j4}, h_{j5}, R_{1}, R_{3},R_{4}, R_{5})&= 0 , \end{aligned}$$
(47)
$$\begin{aligned} f_{5} (h_{i2}, h_{i3}, h_{i4}, h_{i5}, h_{j2}, h_{j3}, h_{j4}, h_{j5}, R_{2},R_{3}, R_{4}, R_{5})&= 0, \end{aligned}$$
(48)
$$\begin{aligned} f_{6} (h_{i4}, h_{i5}, h_{i6}, h_{j4}, h_{j5}, h_{j6}, R_{4},R_{5})&= 0. \end{aligned}$$
(49)

The value function coefficients for the retailer, \(R_{i}\) for \(i=1,2,....,6\), are identified through the six Ricatti equations below.

$$\begin{aligned} s_{1}(R_{1},h_{11},h_{21},h_{13},h_{23},R_{3})&=0, \end{aligned}$$
(50)
$$\begin{aligned} s_{2}(R_{2},h_{12},h_{22},h_{13},h_{23},R_{3})&=0, \end{aligned}$$
(51)
$$\begin{aligned} s_{3}(R_{3},h_{11},h_{12},h_{21},h_{22},h_{13},h_{23},R_{1},R_{3})&=0, \end{aligned}$$
(52)
$$\begin{aligned} s_{4}(R_{4},h_{11},h_{13},h_{14},h_{15},h_{21},h_{23},h_{24},h_{25},R_{1},R_{3},R_{5})&=0, \end{aligned}$$
(53)
$$\begin{aligned} s_{5}(R_{5},h_{12},h_{13},h_{14},h_{15},h_{22},h_{23},h_{24},h_{25},R_{2},R_{3},R_{4})&=0, \end{aligned}$$
(54)
$$\begin{aligned} s_{6}(R_{6},h_{14},h_{15},h_{16},h_{24},h_{25},h_{26},R_{4},R_{5})&=0. \end{aligned}$$
(55)

We compute the pricing strategies of all channel members by substituting the derivatives of the value functions into expressions (42), (43), (24) and (25). where

$$\begin{aligned}&\Delta _{1} = \frac{\kappa \left( -2 b+\gamma \left( -2 h_{13}+h_{23}+R_3\right) +2 h_{11} \kappa \right) -h_{21} \gamma ^2+R_1 \left( \gamma ^2-2 \kappa ^2\right) }{\gamma ^2-4 \kappa ^2}, \end{aligned}$$
(56)
$$\begin{aligned}&\Delta _{2} = \frac{-\gamma \left( b+\gamma \left( h_{23}-R_3\right) \right) +\gamma \kappa \left( -2 h_{12}+h_{22}+R_2\right) +2 \kappa ^2 \left( h_{13}-R_3\right) }{\gamma ^2-4 \kappa ^2}, \end{aligned}$$
(57)
$$\begin{aligned}&\Delta _{3} = \frac{-\gamma \left( a_2+\gamma \left( h_{24}-R_4\right) \right) -2 a_1 \kappa +\gamma \kappa \left( -c-2 h_{15}+h_{25}+R_5\right) -2 \kappa ^2 \left( c-h_{14}+R_4\right) }{\gamma ^2-4 \kappa ^2}, \end{aligned}$$
(58)
$$\begin{aligned}&\Delta _{4} = \frac{-\gamma \left( b+\gamma \left( h_{13}-R_3\right) \right) +\gamma \kappa \left( h_{11}-2 h_{21}+R_1\right) +2 \kappa ^2 \left( h_{23}-R_3\right) }{\gamma ^2-4 \kappa ^2}, \end{aligned}$$
(59)
$$\begin{aligned}&\Delta _{5} = \frac{-2 b \kappa +\gamma ^2 \left( R_2-h_{12}\right) +\gamma \kappa \left( h_{13}-2 h_{23}+R_3\right) +2 \kappa ^2 \left( h_{22}-R_2\right) }{\gamma ^2-4 \kappa ^2}, \end{aligned}$$
(60)
$$\begin{aligned}&\Delta _{6} = \frac{-\gamma \left( a_1+\gamma \left( h_{15}-R_5\right) \right) -2 a_2 \kappa +\gamma \kappa \left( -c+h_{14}-2 h_{24}+R_4\right) -2 \kappa ^2 \left( c-h_{25}+R_5\right) }{\gamma ^2-4 \kappa ^2} , \end{aligned}$$
(61)
$$\begin{aligned}&\Delta _{7} = \frac{\kappa \left( \left( \kappa -\gamma \right) \left( \gamma +\kappa \right) \left( -\gamma \left( h_{23}+R_3\right) -2 \kappa \left( h_{11}+R_1\right) +2 h_{13} \gamma \right) -3 b \left( \gamma ^2-2 \kappa ^2 \right) \right) +h_{21} \gamma ^2 \left( \kappa ^2-\gamma ^2\right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) }, \end{aligned}$$
(62)
$$\begin{aligned}&\Delta _{8} = \frac{b \left( 5 \gamma \kappa ^2-2 \gamma ^3\right) -\left( \gamma -\kappa \right) \left( \gamma +\kappa \right) \left( h_{23} \gamma ^2-\kappa \left( \gamma \left( -2 h_{12}+h_{22}+R_2\right) +2 \kappa \left( h_{13}+R_3\right) \right) \right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) } , \end{aligned}$$
(63)
$$\begin{aligned}&\Delta _{9} = \frac{\kappa \left( -3 a_1 \left( \gamma ^2-2 \kappa ^2\right) -\left( \gamma -\kappa \right) \left( \gamma +\kappa \right) \left( c \left( \gamma +2 \kappa \right) -\gamma \left( h_{25}+R_5\right) -2 \kappa \left( h_{14}+R_4\right) +2 h_{15} \gamma \right) \right) +a_2 \left( 5 \gamma \kappa ^2-2 \gamma ^3\right) +h_{24} \gamma ^2 \left( \kappa ^2-\gamma ^2\right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) },\nonumber \\ \end{aligned}$$
(64)
$$\begin{aligned}&\Delta _{10} = \frac{b \left( 5 \gamma \kappa ^2-2 \gamma ^3\right) -\left( \gamma -\kappa \right) \left( \gamma +\kappa \right) \left( \kappa \left( \gamma \left( -h_{11}+2 h_{21}-R_1\right) -2 \kappa \left( h_{23}+R_3\right) \right) +h_{13} \gamma ^2\right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) }, \end{aligned}$$
(65)
$$\begin{aligned}&\Delta _{11} = \frac{\kappa \left( \left( \gamma -\kappa \right) \left( \gamma +\kappa \right) \left( \gamma \left( h_{13}-2 h_{23}+R_3\right) +2 \kappa \left( h_{22}+R_2\right) \right) -3 b \left( \gamma ^2-2 \kappa ^2 \right) \right) +h_{12} \gamma ^2 \left( \kappa ^2-\gamma ^2\right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) } , \end{aligned}$$
(66)
$$\begin{aligned}&\Delta _{12} = \frac{\kappa \left( \left( \gamma -\kappa \right) \left( \gamma +\kappa \right) \left( \gamma \left( -c+h_{14}-2 h_{24}+R_4\right) +2 \kappa \left( -c+h_{25}+R_5\right) \right) -3 a_2 \left( \gamma ^2-2 \kappa ^2\right) \right) +a_1 \left( 5 \gamma \kappa ^2-2 \gamma ^3\right) +h_{15} \gamma ^2 \left( \kappa ^2-\gamma ^2\right) }{2 \left( \gamma ^4-5 \gamma ^2 \kappa ^2+4 \kappa ^4\right) }.\nonumber \\ \end{aligned}$$
(67)

The transversality condition

$$\begin{aligned} \lim \limits _{t\rightarrow \infty } e^{- \rho t} V_{Mi}(x_{1}(t), x_{2}(t))= & {} 0 ; \\ \lim \limits _{t\rightarrow \infty } e^{- \rho t} V_{R}(x_{1}(t), x_{2}(t))= & {} 0 \end{aligned}$$

guarantees that the expressions (30), (31); and (26), (27), (28) and (29) are the manufacturer i’s and retailer’s value functions, and the pricing strategies. Notice that \((x_{1}(t),x_{2}(t)) \) is the solution of the closed-loop dynamics resulting in substitution of (26), (27), (28) and (29) into the dynamic demand (4). We have to solve the following dynamical system:

$$\begin{aligned} \dot{x_{1}}(t)&=g_{1}x_{1}+g_{2}x_{2}+g_{0}, \end{aligned}$$
(68)
$$\begin{aligned} \dot{x_{2}}(t)&=r_{1}x_{1}+r_{2}x_{2}+r_{0}. \end{aligned}$$
(69)

where

$$\begin{aligned} g_{1}&=b-\kappa \Delta _{7}+\gamma \Delta _{10}, \end{aligned}$$
(70)
$$\begin{aligned} g_{2}&=b-\kappa \Delta _{8}+\gamma \Delta _{11}, \end{aligned}$$
(71)
$$\begin{aligned} g_{0}&=a_{1}-\kappa \Delta _{9}+\gamma \Delta _{12}, \end{aligned}$$
(72)
$$\begin{aligned} r_{1}&=b-\kappa \Delta _{10}+\gamma \Delta _{7}, \end{aligned}$$
(73)
$$\begin{aligned} r_{2}&=b-\kappa \Delta _{11}+\gamma \Delta _{8}, \end{aligned}$$
(74)
$$\begin{aligned} r_{0}&=a_{2}-\kappa \Delta _{12}+\gamma \Delta _{9}, \end{aligned}$$
(75)

The solution gives the expressions for both products’ cumulative sales rate trajectories \(x_{1}(t)\) and \(x_{2}(t)\) given in the proposition. The paths for the pricing strategies are obtained after replacing \(x_{1}(t)\) and \(x_{2}(t)\) by their expressions (32) and (33) in (26), (27), (28) and (29).

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Ngendakuriyo, F., Taboubi, S. Pricing Strategies of Complementary Products in Distribution Channels: A Dynamic Approach. Dyn Games Appl 7, 48–66 (2017). https://doi.org/10.1007/s13235-016-0181-7

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