On the Modelling of Price Effects in the Diffusion of Optional Contingent Products

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.

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:

$$\displaystyle \begin{aligned} \int_{0}^{\infty }e^{-rt}(p_{1}(t)-c_{1})\dot{x}_{1}(t)\,dt \end{aligned}$$

taking into account (6).

In order to find the first-order conditions necessary for optimality, we construct the current value Hamiltonian:

$$\displaystyle \begin{aligned} H^{1}(x_{1},p_{1},\lambda _{1})=(p_{1}-c_{1})\dot{x}_{1}+\lambda _{1}\dot{x} _{1}=(p_{1}-c_{1}+\lambda _{1})\dot{x}_{1}, \end{aligned}$$

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:

$$\displaystyle \begin{aligned} p_1=\frac{\alpha_1 + (c_1-\lambda_1) \beta_1+ b_1 x_1}{2\beta_1}. {} \end{aligned} $$

(9)

The Maximum Principle optimality conditions also include

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot \lambda_1&\displaystyle =&\displaystyle r \lambda_1-\frac{\partial H^1}{\partial x_1}, \quad \lim_{t\to \infty} \lambda_1 (t) x_1(t) e^{-rt}=0, \\ \dot x_1&\displaystyle =&\displaystyle (\alpha _{1}-\beta _{1}p_{1}+b_{1}x_{1})(M-x_{1}), \quad x_1(0)=x_{10}. \end{array} \end{aligned} $$

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:

$$\displaystyle \begin{aligned} \int_{0}^{\infty }e^{-rt}(p_{2}(t)-c_{2})\dot{x}_{2}(t)\,dt \end{aligned}$$

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:

$$\displaystyle \begin{aligned} H^{2}(x_{2},p_{2},\lambda _{2})=(p_{2}-c_{2})\dot{x}_{2}+\lambda _{2}\dot{x} _{2}=(p_{2}-c_{2}+\lambda _{2})\dot{x}_{2}, \end{aligned}$$

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:

$$\displaystyle \begin{aligned} p_2=\frac{\alpha_2 + (c_2-\lambda_2) \beta_2+ b_2 x_2}{2\beta_2}. {} \end{aligned} $$

(10)

The Maximum Principle optimality conditions also include

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot \lambda_2&\displaystyle =&\displaystyle r \lambda_2-\frac{\partial H^2}{\partial x_2}, \quad \lim_{t\to \infty} \lambda_2 (t) x_2(t) e^{-rt}=0, \\ \dot{x}_{2}&\displaystyle =&\displaystyle (\alpha _{2}-\beta _{2}p_{2}+b_{2}x_{2})(x_{1}-x_{2}),\quad x_{2}(0)=x_{20}. \end{array} \end{aligned} $$

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:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle (x_{1ss}^{(1)}, \lambda_{1ss}^{(1)}, x_{2ss}^{(1)}, \lambda_{2ss}^{(1)})=(M,\lambda_1^{(1)} , M, \lambda_2^{(1)}), \\ &\displaystyle &\displaystyle (x_{1ss}^{(2)}, \lambda_{1ss}^{(2)}, x_{2ss}^{(2)}, \lambda_{2ss}^{(2)})=(M,\lambda_1^{(1)} , M, \lambda_2^{(2)}), \\ &\displaystyle &\displaystyle (x_{1ss}^{(3)}, \lambda_{1ss}^{(3)}, x_{2ss}^{(3)}, \lambda_{2ss}^{(3)})=(M,\lambda_1^{(2)} , M, \lambda_2^{(1)}), \\ &\displaystyle &\displaystyle (x_{1ss}^{(4)}, \lambda_{1ss}^{(4)}, x_{2ss}^{(4)}, \lambda_{2ss}^{(4)})=(M,\lambda_1^{(2)} , M, \lambda_2^{(2)}), \end{array} \end{aligned} $$

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda_i^{(1)}&\displaystyle =&\displaystyle -\frac{1}{\beta_i}\left(\alpha_i -c_i \beta_i+ 2 r + b_i M+ 2\sqrt{r(\alpha_i -c_i \beta_i+ r + b_i M)}\right), \\ \lambda_i^{(2)}&\displaystyle =&\displaystyle -\frac{1}{\beta_i}\left(\alpha_i -c_i \beta_i+ 2 r + b_i M- 2\sqrt{r(\alpha_i -c_i \beta_i+ r + b_i M)}\right), \quad i=1, 2. \end{array} \end{aligned} $$

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

$$\displaystyle \begin{aligned} \mu_i=r-\sqrt{r(\alpha_i -c_i \beta_i+ r + b_i M)}, i=1, 2. \end{aligned}$$

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:

$$\displaystyle \begin{aligned} \begin{array}{rcl} x_1(t)&\displaystyle =&\displaystyle (x_{10}-M) e^{\mu t} + M, \\ \lambda_1(t)&\displaystyle =&\displaystyle w_1^{(2)} e^{\mu t}+ \lambda_{1ss}^{(4)}, \\ x_2(t)&\displaystyle =&\displaystyle (x_{20}-M) e^{\mu t} +w_2^{(3)} t e^{\mu t} + M, \\ \lambda_2(t)&\displaystyle =&\displaystyle w_1^{(4)}e^{\mu t}+w_2^{(4)} t e^{\mu t}+ \lambda_{1ss}^{(4)}, \end{array} \end{aligned} $$

where $w_i^{(k)}$ is the k-th component of vector $\bar w_i$, with

$$\displaystyle \begin{aligned} \bar w_1= \varphi \bar v_1 + \eta \bar v_2,\quad \bar w_2= (\Omega-\mu I_4) \bar w_1, \end{aligned}$$

and

$$\displaystyle \begin{aligned} \varphi=\frac{1}{v_1^{(1)}}(x_{10}-M-(x_{20}-M)v_2^{(1)}), \quad \eta=x_{20}-M. \end{aligned}$$

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:

$$\displaystyle \begin{aligned} \int_{0}^{\infty }e^{-rt}\left[ (p_{1}(t)-c_{1})\dot{x} _{1}(t)+(p_{2}(t)-c_{2})\dot{x}_{2}(t)\right] \,dt \end{aligned}$$

taking into account the differential equations (6) and (7 ).

The current-value Hamiltonian reads¹⁶:

$$\displaystyle \begin{aligned} \begin{array}{rcl} H^m(x_1, p_1, x_2, p_2, \lambda_1^m, \lambda_2^m)&\displaystyle =&\displaystyle (p_{1}-c_{1})\dot{x} _{1}+(p_{2}-c_{2})\dot{x}_{2}+ \lambda_1^m \dot{x}_{1}+ \lambda_2^m \dot{x} _{2} \\ &\displaystyle =&\displaystyle (p_{1}-c_{1}+ \lambda_1^m)\dot{x}_{1}+ (p_{2}-c_{2}+ \lambda_2^m)\dot{x} _{2}, \end{array} \end{aligned} $$

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:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial H^m}{\partial p_i}&\displaystyle =&\displaystyle 0, \quad i=1,2, \\ \dot \lambda_i^m&\displaystyle =&\displaystyle r\lambda_i^m-\frac{\partial H^m}{\partial x}, \quad \lim_{t\to \infty} \lambda_i^m(t) x_i(t) e^{-rt}=0\quad i=1,2, \\ \dot{x}_{1}&\displaystyle =&\displaystyle (\alpha _{1}-\beta _{1}p_{1}+b_{1}x_{1})(M-x_{1}),\quad x_{1}(0)=x_{10}, \\ \dot{x}_{2}&\displaystyle =&\displaystyle (\alpha _{2}-\beta _{2}p_{2}+b_{2}x_{2})(x_{1}-x_{2}),\quad x_{2}(0)=x_{20}. \end{array} \end{aligned} $$

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:

$$\displaystyle \begin{aligned} p_i=\frac{1}{2\beta_i}(b_i x_i+\alpha_i+ \beta_i(c_i-\lambda_i)), \quad i=1, 2. \end{aligned}$$

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

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda _{1ss}^{(m)} &\displaystyle =&\displaystyle -\frac{1}{\beta _{1}\beta _{2}}\left( \alpha _{1}-c_{1}\beta _{1}+2r+b_{1}M-2\sqrt{r\Gamma }\right) , \\ \lambda _{2ss}^{(m)} &\displaystyle =&\displaystyle -\frac{1}{\beta _{2}}\left( \alpha _{2}-c_{2}\beta _{2}+2r+b_{2}M-2\sqrt{r(\alpha _{2}-c_{2}\beta _{2}+r+b_{2}M)}\right) , \end{array} \end{aligned} $$

with Γ given by:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Gamma &\displaystyle =&\displaystyle \beta _{2}\left[ (\alpha _{2}-c_{2}\beta _{2}+2r+b_{2}M-2\sqrt{ r(\alpha _{2}-c_{2}\beta _{2}+r+b_{2}M)})\beta _{1}\right. \\ &\displaystyle &\displaystyle \left. +(\alpha _{1}-c_{1}\beta _{1}+r+b_{1}M)\beta _{2}\right] . \end{array} \end{aligned} $$

The eigenvalues of the Jacobian matrix evaluated at this steady-state are

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mu _{1} &\displaystyle =&\displaystyle r-\sqrt{r(\alpha _{2}-c_{2}\beta _{2}+r+b_{2}M)}, \\ \mu _{2} &\displaystyle =&\displaystyle r-\frac{1}{\beta _{2}}\sqrt{r\Gamma }. \end{array} \end{aligned} $$

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:

$$\displaystyle \begin{aligned} \begin{array}{rcl} x_{1}(t) &\displaystyle =&\displaystyle (x_{10}-M)e^{\mu _{2}t}+M, \\ \lambda _{1}(t) &\displaystyle =&\displaystyle \xi v_{1}^{(m2)}e^{\mu _{1}t}+\psi v_{2}^{(m2)}e^{\mu _{2}t}+\lambda _{1ss}^{(m)}, \\ x_{2}(t) &\displaystyle =&\displaystyle \xi v_{1}^{(m3)}e^{\mu _{1}t}+\psi v_{2}^{(m3)}e^{\mu _{2}t}+M, \\ \lambda _{2}(t) &\displaystyle =&\displaystyle \xi e^{\mu _{1}t}+\psi e^{\mu _{2}t}+\lambda _{2ss}^{(m)}, \end{array} \end{aligned} $$

where

$$\displaystyle \begin{aligned} \xi =\frac{(x_{20}-M)v_{2}^{(m1)}-(x_{10}-M)v_{2}^{(m3)}}{ v_{1}^{(m3)}v_{2}^{(m1)}},\quad \psi =\frac{x_{10}-M}{v_{1}^{(m3)}}. \end{aligned}$$