Abstract
In this chapter, we study the pricing strategies of firms in a multi-product diffusion model where we use a new formalization of the price effects. More particularly, we introduce the impact of prices on one of the factors that affect the diffusion of new products: the innovation coefficient. By doing so, we relax one of the hypotheses in the existing literature stating that this rate is constant. In order to assess the impact of this functional form on the pricing policies of firms selling optional contingent products, we use our model to study two scenarios already investigated in the multiplicative form model suggested by Mahajan and Muller (M&M).
We follow a “logical experimentation” perspective by computing and comparing the results of three models: (1) The M&M model, (2) a modified version of M&M where the planning horizon is infinite, and (3) our model, where the new formalization of the innovation effect is introduced. This perspective allows us to attribute the differences in results to either the length of the planning horizon or to our model’s formalization. Besides its contribution to the literature on pricing and diffusion, this paper highlights the sensitivity of results to the hypothesis used in product diffusion modelling and could explain the mixed results obtained in the empirical validations of diffusion models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The number of non-adopters depends also on the state variable. It captures the remaining potential market at a particular time period. It is computed as the difference, in each time period, between the total market potential and the cumulative number of adopters at that time period.
- 2.
In the M&M model, the authors study additional scenarios involving the two cases of contingency described above (i.e., optional and captive). In order to focus on the main objective of our study, we restrict our analysis to investigate only the case of optional contingent products.
- 3.
The state dynamics of the cumulative number of adopters when the innovation rate parameter a equals zero corresponds to the well-known logistic equation usually described by \( \dot {P}=rP(1-P/k)\), where P is the population size, r a constant that defines the growth rate, and k is the carrying capacity.
- 4.
For a detailed analysis of these limits, see Mahajan et al. (1990).
- 5.
I.e., GMB states for the General Bass Model. The GBM is an extension of the Bass model that incorporates, in a multiplicative way, the effect of the marketing variables. According to this model, diffusion is described by the following differential equation:
$$\displaystyle \begin{aligned} \dot{x}\left( t\right) =\left( a+bx\left( t\right) \right) \left( M-x\left( t\right) \right) g(V), \end{aligned}$$with g(V ) representing a function capturing the impact of firms’ decision variables (e.g. advertising, price, etc.)
- 6.
The effect here is found to be multiplicative, as in the GBM, meaning that advertising affects equally the innovation and the imitation rates.
- 7.
See Peres et al. (2010) for a more recent review of diffusion models.
- 8.
Hence, the maximum number of consumers who could buy the contingent product should not exceed the maximum number of consumers who already bought the base product. (i.e. the market potential M).
- 9.
We use the superscript m to denote the case of an integrated monopolist (i.e., a single firm selling both products).
- 10.
Remark: In the case of captive contingency, the price effect is introduced by considering that each one of the product diffusion processes is affected by not only its own price, but also by the price of the other product.
- 11.
This result indicates that \(p_{1}^{m}\) can be greater or lower than \( p_{2}^{m}\) for some time periods because, as mentioned above, M&M consider the symmetric scenario with respect to the parameters, including the symmetry in production costs (i.e., c 1 = c 2).
- 12.
Initially M&M assume x 20 = 0, but in their numerical simulations it seems that they consider other initial values for variable x 2 positive but lower than the initial value for the variable x 1.
- 13.
Similar figures have been computed for the following cases:
-
x 20 = 3000, 6000;
-
M = 40, 000, 70, 000, 80, 000;
-
ε 1 = ε 2 = 0.02.
showed qualitatively similar results.
-
- 14.
However, the transitional dynamics, that is, the transition towards these steady states could be different.
- 15.
Qualitatively similar figures have been obtained for the following cases:
-
x 20 = 3000, 6000;
-
M = 40, 000, 80, 000.
-
- 16.
The superscript stands for “monopolistic scenario.”
References
Bass, F. M. (1969). A new product growth for model consumer durables. Management Science, 15(5), 215–227.
Bass, F. M., Krishnan, T. V., & Jain, D. C. (1994). Why the bass model fits without decision variables. Marketing Science, 13(3), 203.
Dockner, E., & Jorgensen, S. (1988). Optimal advertising policies for diffusion models of new product innovation in monopolistic situations. Management Science, 34(1), 119–130.
Horsky, D. (1990). A diffusion model incorporating product benefits, price, income and information. Marketing Science, 9(4), 279–365.
Horsky, D., & Simon, L. S. (1983). Advertising and the diffusion of new products. Marketing Science, 2(1), 1–17.
Jones, J. M., & Ritz, C. J. (1991). Incorporating distribution into new product diffusion models. International Journal of Research in Marketing, 8(2), 91–112.
Jørgensen, S., & Zaccour, G. (2004). Differential games in marketing. International series in quantitative marketing. Boston: Kluwer Academic Publishers.
Kalish, S. (1985). A new product adoption model with price, advertising, and uncertainty. Management Science, 31(12), 1569–1585.
Kort, P. M., Taboubi, S., & Zaccour, G. (2018). Pricing decisions in marketing channels in the presence of optional contingent products. Central European Journal of Operations Research. https://doi.org/10.1007/s10100-018-0527-x
Kotler, P. (1988). Marketing management (6th ed.). Englewood Cliffs: Prentice Hall.
Mahajan, V., & Muller, E. (1991). Pricing and diffusion of primary and contingent products. Technological Forecasting and Social Change, 39, 291–307.
Mahajan, V., Muller, E., & Bass, F. M. (1990). New product diffusion models in marketing: A review and directions for research. Journal of Marketing, 54(1), 1–22.
Mahajan, V., Peterson, R. A., Jain, A. K., & Malhotra, N. (1979). A new product growth model with a dynamic market potential. Long Range Planning, 12(4), 51–58.
Mesak, H. I. (1996). Incorporating price, advertising and distribution in diffusion models of innovation: Some theoretical and empirical results. Computers & Operations Research, 23(10), 1007–1023.
Moorthy, K. S. (1993). Theoretical modeling in marketing. Journal of Marketing, 57(2), 92–106.
Peres, R., Muller, E., & Mahajan, V. (2010). Innovation diffusion and new product growth models: A critical review and research directions. International Journal of Research in Marketing, 27(2), 91–106.
Peterson, R. A., & Mahajan, V. (1978). Multi-product growth models. In S. Jagdish (Ed.), Research in marketing (pp. 201–231). London: JAI Press.
Robinson, B., & Lakhani, C. (1975). Dynamic price models for new product planning. Management Science, 21(10), 1113–1122.
Teng, J., & Thompson, G. L. (1983). Oligopoly models for optimal advertising when production costs obey a learning curve. Management Science, 29(9), 1087–1101.
Thompson, G. L., & Teng, J. (1984). Optimal pricing and advertising policies for new product oligopoly models. Marketing Science, 3(2), 148–168.
Acknowledgements
We are grateful to two anonymous reviewers for valuable comments and suggestions on an earlier draft of this paper. This research is partially supported by Spanish MINECO under projects ECO2014-52343-P and ECO2017-82227-P (AEI) and by Junta de Castilla y León under projects VA105G18 and VA024P17 co-financed by FEDER funds (EU).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix
Appendix
In the scenario where the independent producers control the pricing decisions of the primary and contingent products, the objective for the primary-product’s producer is to choose the price p 1 in order to maximize the following functional:
taking into account (6).
In order to find the first-order conditions necessary for optimality, we construct the current value Hamiltonian:
where λ 1 denotes the costate variable associated with x 1.
The maximization of H 1 with respect to p 1 yields ∂H 1∕∂p 1 = 0, and assuming x 1 is different from M, from this condition one gets:
The Maximum Principle optimality conditions also include
This boundary value problem taking into account expression (9) reads:
Function H 1 is concave with respect to p 1.
The objective for the contingent-product’s producer is to choose the price p 2 in order to maximize the following functional:
taking into account the differential equations describing the dynamics of the cumulative adoption of the primary and contingent products (6) and (7), respectively.
The current value Hamiltonian reads:
where λ 2 denotes the costate variable associated with x 2.
Assuming that x 1 is different from x 2, from the optimality condition ∂H 2∕∂p 2 = 0, one gets:
The Maximum Principle optimality conditions also include
Substituting the expression of p 2 given by (10) into this system of differential equations we get:
Function H 2 is concave with respect to p 2.
The characterization of the optimal time paths of the cumulative sales and prices of both products requires the solution of the differential equations for the state and costate variables x 1, x 2, λ 1 and λ 2. First of all, we focus on the characterization of the steady-state values and their asymptotically stability.
Because α i − β ip i + b ix i for i = 2, 2 are strictly positive, the unique steady-state value of the cumulative sales is given by x 1ss = M and x 2ss = x 1ss = M. Taking these values into account, we compute the steady-state values of the costate variables λ 1 and λ 2. It can be easily proved that the system of differential equations admits the following four different steady-state values:
where
In order to analyze the stability of the steady states we compute the eigenvalues and associated eigenvectors of the Jacobian matrix evaluated at each of the steady states.
At the first steady state \((x_{1ss}^{(1)}, \lambda _{1ss}^{(1)}, x_{2ss}^{(1)}, \lambda _{2ss}^{(1)})\) the Jacobian matrix has two negative eigenvalues and it can be proved that there exists a bi-parametric family of solutions converging to this steady state. This family of solutions imposes that x 1(t) = M, x 2(t) = M for all t.
At the second steady state \((x_{1ss}^{(2)}, \lambda _{1ss}^{(2)}, x_{2ss}^{(2)}, \lambda _{2ss}^{(2)})\) the Jacobian matrix has two negative eigenvalues and it can be proved that there exists a one-parametric family of solutions converging to this steady state. This family of solutions imposes that x 1(t) = M for all t.
At the third steady state \((x_{1ss}^{(3)}, \lambda _{1ss}^{(3)}, x_{2ss}^{(3)}, \lambda _{2ss}^{(3)})\) the Jacobian matrix has two negative eigenvalues and it can be proved that a relationship among the initial values of the state variables, x 10 and x 20 is needed to ensure the convergence of the optimal paths to this steady state.
At the fourth steady state \((x_{1ss}^{(4)}, \lambda _{1ss}^{(4)}, x_{2ss}^{(4)}, \lambda _{2ss}^{(4)})\) the Jacobian matrix has two negative eigenvalues and it can be proved that there exists a unique optimal path converging to this steady state.
The numerical simulations carried out focus on this fourth scenario. In this case, the two negative eigenvalues are given by
Following M&M, the values of the model parameters used in the numerical simulations are assumed to be completely symmetric. Consequently, under this assumption there is a double negative eigenvalue, μ = μ i, i = 1, 2. We have computed the associated generalized eigenvectors denoted by \(\bar v_1=(v_1^{(1)}, v_1^{(2)}, 0, 1)\) and \(\bar v_2=(v_2^{(1)}, v_2^{(2)}, 1, 0)\) , with \(v_i^{(j)}\) the j-th component of the i-th eigenvector (omitted for brevity). The solution of the system of differential equations read:
where \(w_i^{(k)}\) is the k-th component of vector \(\bar w_i\), with
and
Matrices Ω and I 4 denote the Jacobian matrix associated with the system of differential equations evaluated at the steady state \( (x_{1ss}^{(4)}, \lambda _{1ss}^{(4)}, x_{2ss}^{(4)}, \lambda _{2ss}^{(4)})\) and the fourth-order identity matrix, respectively.
The characterization of the optimal time-paths of the prices and cumulative adoption of both products in the case of the integrated monopolist follows the same steps as previously described for the scenario of two independent producers.
The objective in the case of the integrated monopolist is to choose the prices, p 1 and p 2, in order to maximize the following functional:
taking into account the differential equations (6) and (7 ).
The current-value Hamiltonian readsFootnote 16:
where \(\lambda _1^m\) and \(\lambda _2^m\) denote the costate variables associated with x 1 and x 2, respectively.
The first-order optimality conditions for an interior solution read:
Assuming that x 1 and x 2 are different from M and x 1, respectively, from the two first optimality conditions the following expressions from the prices can be derived:
Substituting these expressions in the differential equations describing the time evolution of the state and costate variables, these equations read:
The characterization of the steady-states and the analysis of their stability follow the same steps as the analysis developed in the case of the independent producers. Four steady states can be characterized and the numerical simulations focus on the only steady state for which there is a unique optimal path converging to this steady state. This steady-state reads \((x_{1ss}^{(m)},\lambda _{1ss}^{(m)},x_{2ss}^{(m)},\lambda _{2ss}^{(m)})\) with \(x_{1ss}^{(m)}=x_{2ss}^{(m)}=M\), and
with Γ given by:
The eigenvalues of the Jacobian matrix evaluated at this steady-state are
The eigenvectors associated are \(\bar {v} _{1}^{m}=(0,v_{1}^{(m2)},v_{1}^{(m3)},1)\) and \(\bar {v} _{2}^{m}=(v_{2}^{(m1)},v_{2}^{(m2)},v_{2}^{(m3)},1)\), with \(v_{i}^{(mj)}\) the j-th component of vector \(\bar {v}_{i}^{m}\) (omitted for brevity). The solution of the system of differential equations read:
where
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Martín-Herrán, G., Taboubi, S. (2020). On the Modelling of Price Effects in the Diffusion of Optional Contingent Products. In: Pineau, PO., Sigué, S., Taboubi, S. (eds) Games in Management Science. International Series in Operations Research & Management Science, vol 280. Springer, Cham. https://doi.org/10.1007/978-3-030-19107-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-19107-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19106-1
Online ISBN: 978-3-030-19107-8
eBook Packages: Business and ManagementBusiness and Management (R0)