Affiliated marketing

Abstract

In information-intensive environments some firms may be sending their customers to other firms’ or affiliates’ websites in order to generate additional sales for their affiliates. Although this may provide a choice for the customers, from a firm’s point of view such strategies have the potential to cannibalize own sales. Hence, when customers purchase from other firms’ websites, the firm may loose its own customers. This study analyzes the optimality of such strategies using an analytical framework. The findings show that a firm may increase its prices and profit when its own customers purchase from other firm websites. An analysis of customer surplus and total welfare show that such strategies may have adverse effects. The results show that customer surplus and total welfare may decrease as a result of affiliations.

Appendix

Notations are given in Table 2.

Table 2 Notation table

1.1 Proofs of Propositions 1 and 2

First, we find the location of the indifferent customer. By solving v−xt−(1−δ)p ₁−δ((1−x)t + p ₂) = v−(1−x) t−p ₂, we find $x=\frac{{(1-\delta)(-p_1+p_2+t)}}{{t(2-\delta)}} .$ The location of the indifferent customer helps us to identify the respective firm demands. By multiplying the demands with the prices we find profits. Due to affiliation, firm 1 may sent δ fraction of its customers to firm 2 and may charge a commission fee of c. Then, the firm profits are given by π₁ = (1−δ)xp ₁ + δx c and π₂ = (1−x)p ₂ + δx (p ₂−c). We solve the game using subgame perfect Nash. First, firms simultaneously set their prices. Note that the second order conditions hold since the second derivatives of profits with respect to prices equal $-2\frac{{(1-\delta)^{2}}}{t(2-\delta)}.$ When we set the first derivatives of profits with respect to prices to zero we have the following equations:

$$ \begin{aligned} 2\frac{{\left(\delta-1 \right) ^{2}}}{{t \left( \delta-2 \right)}} -\frac{{\left( 1-\delta \right) \left( \delta-1 \right) {\mit p_1}}}{{t \left( \delta-2 \right)}}+\frac{{\left( 1-\delta \right) \left(\delta-1 \right) \left( -{\mit p_1}+t+{\mit p2} \right)}}{{t \left( \delta-2 \right)}}-\frac{{\delta \left(\delta-1 \right) c}}{{t \left( \delta-2 \right)}}=&0\\ -\frac{{\left( \delta-1 \right) {\mit p_2}}}{{t \left( \delta-2 \right)}}+1-\frac{{\left( \delta-1 \right) \left(-{\mit p_1}+t+{\mit p_2} \right) }}{{t \left( \delta-2 \right)}}+\frac{{\delta \left( \delta -1 \right) \left({\mit p_2}-c \right) }}{{t \left( \delta-2 \right)}}+\frac{{\delta \left( \delta-1 \right) \left( -{\mit p_1}+t+{\mit p_2} \right)}}{{t \left( \delta-2 \right)}}=&0 \end{aligned} $$

By simultaneously solving the above equations we find the prices: $p_1=1/3 \frac{{-3 \delta t+3 t+{\delta}^{2}t+3 {\delta}^{2}c-3 \delta c}}{{\left( \delta-1 \right) ^{2}}}$ and $p_2=1/3 \frac{{3{\delta}^{2}c-{\delta}^{2}t-3 \delta c+3 t}}{{\left( \delta-1 \right) ^{2}}}$. When we insert these prices in the profit functions we find $\pi_1=-1/9 \frac{{\left( -3 \delta+3+{\delta}^{2} \right) ^{2}t}}{{\left( \delta-1 \right) ^{2} \left(\delta-2 \right) }}$ and $\pi_2=-1/9 \frac{{{\delta}^{4}t-9 {\delta}^{3}c-6 {\delta}^{2}t+27 {\delta}^{2}c-18 \delta c+9 t}}{{\left( \delta-1 \right) ^{2} \left( \delta-2 \right)}} .$ Firm 1 profit with respect to δ increases since the derivative equals to $1/9 \frac{{\left( 3-\delta\right) \left(3-3 \delta+{\delta}^{ 2} \right) \left(1+\delta-{\delta}^{2} \right) t}}{{\left(2- \delta \right) ^{2} \left(1- \delta \right) ^{3}}} > 0.$ Hence, firm 1 finds it beneficial to set δ to the maximum value. By inserting the optimal prices, we see that the indifferent customer is located at $x=1/3 \frac{{-3 \delta+3+{\delta}^{2}}}{{\left( \delta-2 \right) \left( \delta-1 \right)}}.$ Since $x < 1,$ we need to have δ < 0.63. We also see that firm 1 profit does not increase with c, the firm weakly prefers setting c to zero. In order to see the profits without affiliation, we set δ to zero. This provides profits of t/2. However, we have already showed that a higher δ increases profits. Hence, both firms prefer to be affiliated.

1.2 Proof of Proposition 3

The profits with no affiliation equal to t/2 based on the above proposition. The incremental profit with affiliation is the difference between the optimal profit with affiliation and t/2. That is $\pi_1-t/2=-1/18 \frac{{ \left( 2 \delta-3 \right) \left({\delta}^{2}-3\right) \delta t}}{{\left( \delta-1 \right) ^{2} \left( \delta-2\right)}}$ and $\pi_2-t/2=-1/18 \frac{{\left( 2 \delta-3 \right) \left( {\delta}^{2}+6 \delta-15 \right) \delta t}}{{\left( \delta-1 \right) ^{2} \left(\delta-2 \right).}}$ The respective derivatives with respect to t equal to $1/18 \frac{{\delta \left(3- 2 \delta \right) \left(3- {\delta}^{2} \right) }}{{\left(2- \delta \right) \left(1- \delta\right) ^{2}}}$ and $1/18 \frac{{\delta \left(3- 2 \delta \right) \left(15- {\delta}^{2}-6 \delta \right) }}{{\left(2- \delta \right) \left(1- \delta\right) ^{2}}}$ which are positive. Hence, as t increases the incremental profit also increases and the profit potential due to affiliations increase. We also see that the derivatives of prices with respect to t for firm 1 and firm 2 equal to $1/3\frac{{3+\delta^2-3\delta}}{{(1-\delta)^2}}$ and $1/3\frac{{3-\delta^2}}{{(1-\delta)^2}},$ which are positive.

1.3 Proof of Proposition 4

When firms are not affiliated, both firms obtain half the market and the indifferent customer is located at 1/2. When firms are affiliated we see that the indifferent customer is located at $x=1/3 \frac{{-3 \delta+3+{\delta}^{2}}}{{\left( \delta-2 \right) \left( \delta-1 \right)}}$ note that $x-1/2=1/6{ \frac{{\delta \left(3- \delta \right) }}{{\left(2- \delta \right) \left(1- \delta \right)}}} > 0.$ Hence more customers prefer to visit firm 1. Firms unit sales are given by δx. We see that $\delta x-1/2=1/6 \frac{{2 {\delta}^{3}-9 {\delta}^{2}+15 \delta-6}}{{\left( \delta-2 \right) \left(\delta-1 \right)}},$ which is positive when δ > 0.571. Similar arguments show that fewer customers initially want to visit firm 2. However, firm 2 unit sales is given by (1−δ)x + (1−x) which is higher than 1/2 when δ < 0.571.

1.4 Proof of Proposition 5

Customer surplus (CS) is found by integrating customer surplus over all customers. CS equals

$$ \int\limits_0^{x} ({v-(1-\delta)p_1-ty-\delta(p_2+t(1-y))} ){\rm d}y+\int\limits_{x}^{1} ({v-p_2-t(1-y)} ) {\rm d}y $$

This gives us $CS=1/18 {\frac {-t{\delta}^{4}+18 v{\delta}^{3}+3 t{\delta}^{3}-72 v{\delta}^{2}+9 {\delta}^{2}t+90 v\delta-45 \delta t+45 t-36v}{ \left( \delta-1 \right) ^{2} \left( \delta-2 \right) }}.$ The derivative of customer surplus with respect to δ equals to $-1/18 \frac{{t \left( \delta-3 \right) \left( {\delta}^{4}-4 { \delta}^{3}+17 {\delta}^{2}-48 \delta+45 \right) }}{{\left( \delta-1 \right) ^{3} \left( \delta-2 \right) ^{2}}},$ which is negative. Hence, customer surplus decreases when δ is greater than zero. Total welfare (TW) equals to CS + π₁ + π₂. The derivative of TW with respect to δ equals to $-1/18 \frac{{t \left( \delta-3 \right) \left( 5 {\delta}^{4}-20 {\delta}^{3}+31 {\delta}^{2}-24 \delta+9 \right)}}{{\left( \delta-1 \right) ^{3} \left( \delta-2 \right) ^{2}}}$ which is also negative. Hence, we see that total welfare also decreases with affiliations.