# Analysis of a Diffusion Duopoly

• Gary M. Erickson
Part of the International Series in Quantitative Marketing book series (ISQM, volume 13)

## Abstract

A third model we analyze is the diffusion formulation with repeat purchase. With this model, we assume as before that the two competitors want to maximize discounted profits over an infinite horizon:
$$\begin{gathered} \mathop{{\max }}\limits_{{{A_1}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} \left( {{h_1}\frac{{{\gamma_1}}}{{{\gamma_1} + {\gamma_2}}}S - {A_1}} \right)dt \hfill \\ \mathop{{\max }}\limits_{{{A_2}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} \left( {{h_2}\frac{{{\gamma_2}}}{{{\gamma_1} + {\gamma_2}}}S - {A_2}} \right)dt \hfill \\ \end{gathered}$$
(5.1)
The dynamic constraint on industry sales S in the diffusion model is
$$\dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}}\varepsilon S)(N - S) - \delta S \dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}})(N - S) - \delta S$$
(5.2)
The diffusion model is similar to the Vidale-Wolfe structure analyzed in chapter 4, with the addition of the word-of-mouth parameter ε. As in the Vidale-Wolfe model, r is the common discount rate, h 1 and h 2 are unit contributions, α 1 and α 2 are advertising elasticities, β 1 and β 2 are advertising effectiveness parameters, δ is a decay factor, and N is the maximum market size.

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