Analysis of a Vidale-Wolfe Duopoly

Abstract

In this chapter, we analyze the Vidale-Wolfe model version introduced in chapter 2. We assume the objectives

$$\begin{array}{*{20}{c}} {\mathop{{\max }}\limits_{{{{A}_{1}}}} \int\limits_{0}^{\infty } {{{e}^{{ - rt}}}\left( {{{h}_{1}}\frac{{{{\gamma }_{1}}}}{{{{\gamma }_{1}} + {{\gamma }_{2}}}}S - {{A}_{1}}} \right)} dt} \hfill \\ {\mathop{{\max }}\limits_{{{{A}_{2}}}} \int\limits_{0}^{\infty } {{{e}^{{ - rt}}}\left( {{{h}_{2}}\frac{{{{\gamma }_{2}}}}{{{{\gamma }_{1}} + {{\gamma }_{2}}}}S - {{A}_{2}}} \right)} dt} \hfill \\ \end{array}$$

(4.1)

and that total industry sales level S is subject to the dynamic constraint

$$\dot{S} = \left( {{{\beta }_{1}}A_{1}^{{{{\alpha }_{1}}}} + {{\beta }_{2}}A_{2}^{{{{\alpha }_{2}}}}} \right)\left( {N - S} \right) - \delta S$$

(1)

In the Vidale-Wolfe model we analyze, each competitor’s advertising is used to attract industry sales S, which are divided between the two competitors according to brand-strength parameters γ₁, γ₂. The industry sales level changes dynamically according to (4.2). The maximum sales level is N, and the difference between the maximum and current sales levels is attractable through advertising. In addition to the buildup in sales through advertising, there is a decay at the rate of δ times the current level of sales. The parameters h ₁, h ₂, are the unit contributions of the two competitors, and r is the common discount rate.