Customer Acquisition and Customer Retention in a Competitive Industry

Abstract

We study optimal customer acquisition and retention strategies in an infinite-horizon model of dynamic competition. We find that acquisition expenditures constitute the larger share of the marketing budget, when the customer profit margin is either low or large, but for intermediate profit margin values, firms spend more resources for customer retention. If customer profit margins rise for exogenous reasons, we find that the share of customer acquisition expenditures in the marketing budget increases in markets with high profit margins, whereas it decreases in markets with low profit margins. The impact of entry of new firms in the market on the optimal strategy depends on the effect of the entry on profit margins and absolute levels of profit margins. A similar phenomenon may also appear in a single segmented market: the impact of higher competition on the luxury and mass-consumption segments of a market would be different.

Appendices

Appendix 1: Characterization of the Equilibrium of the Firm

We write the Hamiltonian function as follows: H ⁱ = exp[−ρ t]{(m − e ⁱ)l ⁱ + λ ⁱ{−β l ⁱ + N ^p s ⁱ}, where λ ⁱ is a co-state variable. As dN ^p/dl ⁰ = 0 and ds ⁱ/dA ^j = 0, the first-order conditions are as follows:

$$ {\lambda}^i\left(-\frac{\mathrm{d}{\beta}^i}{\mathrm{d}{e}^i}\right)= \exp \left[-\rho t\right], $$

(12)

$$ {\lambda}^i{N}^p\left(\frac{\mathrm{d}{s}^i}{\mathrm{d}{A}^i}\right)= \exp \left[-\rho t\right], $$

(13)

$$ \frac{\mathrm{d}{l}^i}{\mathrm{d}t}=-{\beta}^i{l}^i+{N}^p{s}^i, $$

(14)

$$ \frac{\mathrm{d}{\lambda}^i}{\mathrm{d}t}=- \exp \left[-\rho t\right]\left(m-{e}^i\right)+{\lambda}^i{\beta}^i. $$

(15)

The solution paths must also satisfy the transversality condition, lim _t→∞ λ ⁱ = 0. Equations (12) and (13) combined give the condition:

$$ {N}^p\left(\mathrm{d}{s}^i/\mathrm{d}{A}^i\right)=-\left({\beta}^i\right)\hbox{'}. $$

(16)

Differentiating equation (12) with respect to t and substituting the result into (15), we obtain the expression:

$$ \frac{\mathrm{d}{e}^i}{\mathrm{d}t}=-\left[{\left({\beta}^i\right)}^{\prime}\left(m-{e}^i\right)+\left({\beta}^i+\rho \right)\right]\frac{{\left({\beta}^i\right)}^{\prime }}{{\left({\beta}^i\right)}^{{\prime\prime} }}. $$

(17)

For the symmetric long-run equilibrium, (1), (14), (16), and (17) provide (7)–(9). It may be shown that the function H ⁱ is quasi-concave jointly in (e ⁱ, A ⁱ, l ⁱ). Hence, a path that satisfies the first-order necessary conditions gives the optimal solution.

Appendix 2: Existence and Uniqueness of the Long-Run Equilibrium

Considering the optimal path of e determined by (17), we define $G(e)\equiv \beta \hbox{'}(e)\left(m-e\right)+\beta (e)+\rho$. It can be seen that G(0) < 0, G(e) monotonically increases in e on [0, m], G(m) > 0 and G(e) is positive and decreases in e for all e > m. Therefore, for e ∊ [0, m], there exists the unique e ^*, such that G(e ^*) = 0. Thus, e(t) = e ^* is the unique stationary solution of de/dt = 0.

For paths that start at e > e ^*, we have de/dt > 0, which implies e(t) → const > m, as t → ∞. These paths cannot be equilibrium, as firms get negative per-period profit in the long run. For paths starting at e < e ^*, we have de/dt < 0, so they lead to e = 0. These paths cannot be equilibrium paths: marginal retention expenditures at unit marginal cost would bring the positive next-period gain −β′(0) m > 1 and would therefore increase the profit.