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Dynamic Games and Applications
Published in: Dynamic Games and Applications 1/2017

02-02-2016

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

Authors: Fabien Ngendakuriyo, Sihem Taboubi

Published in: Dynamic Games and Applications | Issue 1/2017

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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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Appendix
Available only for authorised users
Footnotes
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 B, R2M refer to bilateral monopoly and One-retailer–two-manufacturers vertical channel, respectively.
 
17
This result is in line with [24].
 
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Metadata
Title
Pricing Strategies of Complementary Products in Distribution Channels: A Dynamic Approach
Authors
Fabien Ngendakuriyo
Sihem Taboubi
Publication date
02-02-2016
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 1/2017
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-016-0181-7

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