A Dynamic Advertising Game with Market Growth

Abstract

The paper considers a market in which two firms compete on advertising over time. Each firm can use three types of advertising: offensive advertising which attempts to attract customers from the rival firm, defensive advertising which aims at protecting a firm’s customer base from the rival’s attacks, and generic advertising to make industry sales grow. We address questions like: How should a strategy for the simultaneous use of the three types of advertising be designed? How would the resulting time paths of sales look like? The paper studies a differential game played over an infinite time horizon and provides closed-form expressions for equilibrium advertising strategies and sales rate trajectories.

Appendix

Nonnegativity of Attraction Rates If a _i  > 0, d _j  > 0, attraction rates are given by

$$\displaystyle{ f_{i}\left (a_{i},d_{j}\right ) =\beta a_{i} -\lambda d_{j} = \left ( \frac{\beta ^{2}} {c_{a}} - \frac{\lambda ^{2}} {c_{d}}\right )\sqrt{S_{j}}\left (\varphi -\psi \right ) }$$

and invoking (6) we have $f_{i}\left (a_{i},d_{j}\right )> 0$ if φ −ψ > 0. Lemma 1 shows that given (6), the difference φ −ψ is positive and hence attraction rates are positive. Q.E.D.

Proof of Lemma 1 Using (11) provides

$$\displaystyle{ \varphi -\psi = -\frac{\sqrt{4C\left (C_{2 } - C_{1 } \right ) + 1} - 1} {2\left (C_{1} - C_{2}\right )}. }$$

Our assumption $\beta ^{2}/c_{a}>\lambda ^{2}/c_{d}$ implies C ₁ − C ₂ < 0 and hence φ −ψ > 0. Q.E.D.

Proof of Proposition 1 In Case 1, ψ > 0 follows from (10). Using ψ > 0 and φ −ψ > 0 shows that φ > 0 and then φ +ψ > 0. In Case 2, ψ < 0 follows from (10). The result in (12) is established by using (10) and the fact that B > b. Offensive and defensive advertising rates are positive in both cases since φ > ψ. The generic advertising rate g is positive in Case 1. In Case 2, use (11) to see that g is positive if C > B, zero if C ≤ B. Q.E.D.

Fig. 1

Case A1: equilibrium sales paths

Fig. 2

Case A2: equilibrium sales paths

Fig. 3

Case B: equilibrium sales paths