Cash-back rewards versus equity-based electronic loyalty programs in e-commerce

Abstract

Loyalty is a crucial part of today’s business because retaining a customer is generally less expensive than attracting a new one. This relationship also holds true in e-commerce. Most of the e-loyalty programs available on the Internet utilize cash-back rewards. A new type of e-loyalty program in which customers are offered a fraction of merchant firm’s equity is emerging recently. The profitability of this approach versus cash-back reward programs is still an open question. In this paper, we first survey current e-loyalty programs, and then develop a two-period duopoly model in which one of the firms gives customers a small fraction of its equity and the other offers cash-back reward for a purchase. We derive analytical conditions to compare the total profits generated through each loyalty program. In particular, we find that equity-based e-loyalty programs provide higher total profits than those of cash-back programs in markets where it is difficult for customers to switch between firms.

Appendix (Proofs of Lemmas and Propositions)

Proof of Lemma 1: Firm A issues ɛ _A2 (Q _AA + Q _AB ) number of new stocks in the second period in addition to its d _A1 number of stocks in the first period. The dilution effect states that firm A’s total market value in the first period, d _A1 S _A1, should be equal to its total market value in the second period, [d _A1 + ɛ _A2 (Q _AA + Q _AB )]S _A2. Thus,

$$ \begin{aligned} d_{A1} S_{A1} &= {\left[{d_{A1} + \varepsilon_{A2} {\left({Q_{AA} + Q_{AB}} \right)}} \right]}S_{A2}\\ \Rightarrow S_{A2} &= \frac{{d_{A1} S_{A1}}}{{d_{A1} + \varepsilon_{A2} {\left({Q_{AA} + Q_{AB}} \right)}}}\\ \end{aligned} $$

In addition to the dilution effect, firm A’s second-period stock price is affected by the changes in its market share between the two periods. Denoting the ratio of change in market share by γ, we obtain:

$$ \gamma = \frac{Firm\,A's\,2nd\,period\,market\,share - Firm\,A's\,1st\,period\, market\,share}{Firm\,A's\,1st\,period\,market\,share} $$

$$ \gamma = \frac{\frac{Q_{AA} + Q_{AB}}{N} - \alpha}{\alpha} = \frac{Q_{AA} + Q_{AB} - \alpha N}{\alpha N} $$

Hence, the updated value of S _A2 above is calculated as:

$$ S_{A2} = \frac{{d_{A1} S_{A1}}}{{d_{A1} + \varepsilon_{A2} {\left({Q_{AA} + Q_{AB}} \right)}}}{\left({1 + \gamma} \right)} = \frac{{d_{A1} S_{A1}}}{{d_{A1} + \varepsilon_{A2} {\left({Q_{AA} + Q_{AB}} \right)}}}{\left({\frac{{Q_{AA} + Q_{AB}}}{{\alpha N}}} \right)} = \frac{{d_{A1} S_{A1} {\left({Q_{AA} + Q_{AB}} \right)}}}{{\alpha N{\left[{d_{A1} + \varepsilon_{A2} {\left({Q_{AA} + Q_{AB}} \right)}} \right]}}} $$

Proof of Lemma 2: Using the information presented in Fig. 2, we calculate the total numbers of firm A’s loyal and switching customers in the second period, Q _AA and Q _BA , as follows:

$$\begin{aligned} Q_{AA} &= \alpha N{\int\limits_{p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z}^\theta \frac{1}{\theta}} \, ds\\ Q_{AA} &= \frac{{\alpha N}}{\theta}{\left[{\theta - {\left({p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z} \right)}} \right]}\\ & = \alpha N{\left({1 - \frac{{p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z}}{\theta}} \right)}\\ Q_{BA} &= \alpha N{\int\limits_0^{p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z} {\frac{1}{\theta}}} ds = \frac{{\alpha N}}{\theta}{\left({p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z} \right)}\\ \end{aligned}$$

Similarly, we calculate the total numbers of firm B’s loyal and switching customers in the second period, Q _BB and Q _AB , as follows:

$$ Q_{BB} = {\left({1 - \alpha} \right)}N{\left({1 - \frac{{p_{B2} - p_{A2} + \varepsilon_{A2} S_{A2} - m_{B2}}}{\theta}} \right)} $$

$$ Q_{AB} = \frac{{{\left({1 - \alpha} \right)}N}}{\theta}{\left({p_{B2} - p_{A2} + \varepsilon_{A2} S_{A2} - m_{B2}} \right)} $$

Proof of Lemma 3: We first insert the values of S _A2, Q _AA , Q _BA , Q _BB , and Q _AB from the Eqs. (1), (4), (5), (6), and (7) into the following maximization problems:

$$ {\mathop {\Pi^{*}_{A2} = {\mathop {Max}\limits_{\varepsilon_{A2}}}}\limits} \Pi_{A2} = {\left({p_{A2} - \varepsilon_{A2} S_{A2} - c} \right)}{\left({Q_{AA} + Q_{AB}} \right)} $$

$$ {\mathop {\Pi^{*}_{B2} = {\mathop {Max}\limits_{m_{B2}}}}\limits} \Pi_{B2} = {\left({p_{B2} - m_{B2} - c} \right)}{\left({Q_{BB} + Q_{BA}} \right)} $$

We then solve the following equations together to find the optimum values of ɛ _A2 and m _B2.

$$ \frac{{\partial \Pi_{A2}}}{{\partial \varepsilon_{A2}}} = 0\quad{\hbox{and}}\quad\frac{{\partial \Pi_{B2}}}{{\partial m_{B2}}} =0 $$

Simultaneous solution of the two equations above gives us m ^* _B2 as follows:

$$ m^{*}_{B2} = p_{B2} - c - \frac{1}{3}{\left[{\alpha Z + {\left({2 - \alpha} \right)}\theta} \right]} $$

(A.1)

The solution for ɛ^* _A2 requires further work since it appears as a function of Q _AA and Q _AB :

$$ \varepsilon^{*}_{A2} = \frac{{\alpha Nd_{A1} {\left[{3{\left({p_{A2} - c} \right)} + \alpha Z - {\left({1 + \alpha} \right)}\theta} \right]}}}{{{\left({Q_{AA} + Q_{AB}} \right)}{\left[{3d_{A1} S_{A1} - \alpha N{\left({3{\left({p_{A2} - c} \right)} + \alpha Z - {\left({1 + \alpha} \right)}\theta} \right)}} \right]}}} $$

(A.2)

To solve for ɛ^* _A2 independent from Q _AA and Q _AB , we first find the optimum values of Q _AA and Q _AB by inserting Eqs. (1), (A.1) and (A.2) into the following equations:

$$\begin{aligned} Q_{AA} &= \alpha N{\left({1 - \frac{{p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z}}{\theta}} \right)}\quad{\hbox{and}}\\Q_{AB} &= \frac{{{\left({1 - \alpha} \right)}N}}{\theta}{\left({p_{B2} - p_{A2} + \varepsilon_{A2} S_{A2} - m_{B2}} \right)} \end{aligned}$$

After simplifications of algebra, we obtain:

$$ Q^{*}_{AA} = \frac{{\alpha N}}{{3\theta}}{\left[{4\theta - 3Z + 2\alpha {\left({Z - \theta} \right)}} \right]}\;{\hbox{and}}\; Q^{*}_{AB} = \frac{{2{\left({1 - \alpha} \right)}N}}{{3\theta}}{\left[\!\frac{\theta}{2} + {\alpha {\left({Z - \theta} \right)}} \!\right]} $$

(A.3)

Inserting these values back into Eq. (A.2) gives us the value of ɛ^* _A2:

$$ \varepsilon^{*}_{A2} = \frac{{3\alpha \theta d_{A1} {\left[{3{\left({p_{A2} - c} \right)} + \alpha Z - {\left({1 + \alpha} \right)}\theta} \right]}}}{{{\left[{\alpha Z - {\left({1 + \alpha} \right)}\theta} \right]}{\left[{\alpha N{\left({3{\left({p_{A2} - c} \right)} + \alpha Z - {\left({1 + \alpha} \right)}\theta} \right)} - 3d_{A1} S_{A1}} \right]}}} $$

(A.4)

Finally, by inserting Eqs. (A.3) and (A.4) into (1), we can find S ^* _A2 as follows:

$$ S^{*}_{A2} = \frac{{{\left[{\alpha Z - {\left({1 + \alpha} \right)}\theta} \right]}{\left[{\alpha N{\left({3{\left({p_{A2} - c} \right)} + \alpha Z - {\left({1 + \alpha} \right)}\theta} \right)} - 3d_{A1} S_{A1}} \right]}}}{{9\alpha \theta d_{A1}}} $$

(A.5)

The values of Q ^* _BA and Q ^* _BB can be found similarly by inserting Eqs. (A.1), (A.4), and (A.5) into the following equations:

$$ \begin{aligned}Q_{BA} &= \frac{{\alpha N}}{\theta}{\left({p_{A2} - p_{B2} + m_{B2} - \varepsilon_{A2} S_{A2} + Z} \right)}\quad{\hbox{and}}\\Q_{BB} &= {\left({1 - \alpha} \right)}N{\left({1 - \frac{{p_{B2} - p_{A2} + \varepsilon_{A2} S_{A2} - m_{B2}}}{\theta}} \right)} \end{aligned}$$

which leads to:

$$ Q^{*}_{BA} = \frac{{\alpha N}}{{3\theta}}{\left[{3Z - \theta - 2\alpha {\left({Z - \theta} \right)}} \right]}\quad{\hbox{and}}\quad Q^{*}_{BB} = \frac{{2{\left({1 - \alpha} \right)}N}}{{3\theta}}{\left[{\theta - \alpha {\left({Z - \theta} \right)}} \right]} $$

(A.6)

Proof of Lemma 4: We can find the second-period optimum profit of firm A, Π^* _A2, by inserting Eqs. (A.3), (A.4), and (A.5) into:

$$ {\mathop {\Pi^{*}_{A2} =}\limits} {\mathop {Max}\limits_{\varepsilon_{A2}}} \Pi_{A2} = {\left({p_{A2} - \varepsilon_{A2} S_{A2} - c} \right)}{\left({Q_{AA} + Q_{AB}} \right)} $$

After simplifications, firm A’s optimum profit in the second period reduces to the following form:

$$ \Pi^{*}_{A2} = \frac{N}{{9\theta}}{\left[{{\left({1 + \alpha} \right)}\theta - \alpha Z} \right]}^{2} $$

Similarly, firm B’s second-period optimum profit is found by inserting (A.1) and (A.6) into:

$$ {\mathop {\Pi^{*}_{B2} = {\mathop {Max}\limits_{m_{B2}}}}\limits} \Pi_{B2} = {\left({p_{B2} - m_{B2} - c} \right)}{\left({Q_{BB} + Q_{BA}} \right)} $$

which results the required equation for Π^* _B2 :

$$ \Pi^{*}_{B2} = \frac{N}{{9\theta}}{\left[{{\left({2 - \alpha} \right)}\theta + \alpha Z} \right]}^{2} $$

Proof of Lemma 5: To find ɛ^* _A1 and m ^* _B1, we solve the following first-order conditions simultaneously by using Eqs. (20) and (21). Further simplifications lead to the expressions in (22) and (23).

$$ \frac{{\partial \Pi_{A}}}{{\partial \varepsilon_{A1}}} = 0\quad{\hbox{and}}\quad\frac{{\partial \Pi_{B}}}{{\partial m_{B1}}} = 0 $$

Proof of Proposition 1: The optimum total firm profits, Π^* _A and Π^* _B , are calculated by inserting Eqs. (22) and (23) into (20) and (21). Simplification of the resulting algebra leads to (24) and (25).

Proof of Proposition 2: Subtracting Eq. (25) from (24) gives us the difference between the optimum total firm profits, Π^* _Δ. Further simplification of the resulting algebra leads to the inequality in (26).