Product cannibalization and the effect of a service strategy

Abstract

Product cannibalization can push some consumers to shift their purchasing preferences from new to used products. This is a costly issue for manufacturers, who have to adjust their pricing strategies accordingly to mitigate the negative effect of cannibalization. In this paper, we characterize an atypical channel to examine the effect of product cannibalization within the DellReconnect project. In particular, we investigate how the presence of a Goodwill agency in a second-hand market impacts the business of a manufacturer (e.g., Dell) in a new market through cannibalization, and how the manufacturer reacts to mitigate its effects. We show that even if the manufacturer adjusts its price to decrease the negative effects of cannibalization, this effect is so severe that it always loses some profits. Nevertheless, when the manufacturer provides some additional services to new consumers, the negative effects of cannibalization can be partially overcome.

Appendix 1: Analytical developments

Proof of Proposition 1

Since both players choose their strategies simultaneously, we will characterize the non-cooperative Nash equilibrium solution. Taking the derivatives of the profit functions in Eqs. (3) and (4) with respect to $p_{M}$ and $p_{C},$ respectively, and setting them equal to zero, we can derive the following reaction functions:

$$ p_{M}(p_{C})=\frac{\alpha _{M}+c_{P}(\beta _{M}+\gamma )-H_{M}\beta _{M}+\gamma p_{C}}{2(\beta _{M}+\gamma )}\quad p_{C}({p_{M}})=\frac{ \alpha _{C}-H_{C}\beta _{C}+\gamma p_{M}}{2(\beta _{C}+\gamma )}$$

These reaction functions are upward sloping because $\frac{\partial p_{M}}{\partial p_{C}}=\frac{\gamma }{2(\beta _{M}+\gamma )}>0$ and $\frac{\partial p_{C}}{\partial p_{M}}=\frac{\gamma }{2(\beta _{C}+\gamma )}>0.$ Furthermore, the slope of the manufacturer’s reaction function is strictly greater than the slope of the collector’s reaction function. To see this, note that $\frac{2(\beta _{M}+\gamma )}{\gamma }>\frac{\gamma }{2(\beta _{C}+\gamma )}$, implies that $4(\beta _{M}+\gamma )(\beta _{C}+\gamma )-\gamma ^{2}>0$, which is always true in our model. This implies that the two reaction functions intersect only once and hence the optimal solution is unique.

Solving simultaneously the reaction functions yields the optimal prices; $p_{M}^{\mathcal {B}}$ and $p_{C}^{\mathcal {B}}$ are given in Eqs. (5) and (6) in Proposition 1. In addition, the Nash equilibrium prices of the manufacturer and the collector are stable, because if any of the players were to deviate from the optimal solution, based on the reaction functions, $p_{M}(p_C)$ and $p_C(p_M)$, the solution would converge back to $p_{M}^{\mathcal {B}}$ and $p_{C}^{\mathcal {B}}$.

The expressions for optimal profit of the manufacturer and the collector are

$$ \Pi _{M}^{\mathcal {B}}=\frac{\left\{ \begin{array}{c} \Omega _{2}\left[ \Omega _{1}\left( \alpha _{M}+c_{p}\left( \gamma +\beta _{M}\right) -H_{M}\beta _{M}\right) -\Omega _{2}^{{}}\left( \gamma +\beta _{M}\right) \right] \\ +\Omega _{3}\left( \gamma \Omega _{2}-\Omega _{1}\left( \gamma c_{p}+H_{M}\beta _{C}\right) \right) +\Omega _{1}^{2}\left( H_{M}\left( \alpha _{C}+\alpha _{M}\right) -c_{p}\alpha _{M}\right) \end{array}\right\} }{\left[ 4(\beta _{M}+\gamma )(\beta _{C}+\gamma )-\gamma ^{2} \right] ^{2}} $$

(10)

$$ \Pi _{C}^{\mathcal {B}}=\frac{\Omega _{3}\left[ \Omega _{1}\left( \alpha _{C}-H_{C}\beta _{C}\right) -\left( \gamma +\beta _{C}\right) \Omega _{3} \right] +\Omega _{2}\left( \gamma \Omega _{3}-\Omega _{1}H_{C}\beta _{M}\right) +\Omega _{1}^{2}H_{C}\left( \alpha _{C}+\alpha _{M}\right) }{ \left[ 4(\beta _{M}+\gamma )(\beta _{C}+\gamma )-\gamma ^{2}\right] ^{2}} $$

(11)

with $\Omega _{1}={\rm DEN}\{ p_{M}^{\mathcal {B}}\}, \Omega _{2}={\rm NUM}\{ p_{C}^{\mathcal {B}}\}$ and $\Omega _{3}={\rm NUM}\{ p_{M}^{\mathcal {B}}\} \square$.

Proof of Proposition 2

We compute the derivatives of firms’ prices with respect to the cannibalization effect, $\gamma ,$ to show that:

• $\frac{\partial p_{M}^{\mathcal {B}}}{\partial \gamma }=-\frac{2(\alpha _{M}-H_{M}\beta _{M}) ( 4\beta _{C}^{2}+6\beta _{C}\gamma +3\gamma ^{2}) +( \alpha _{C}-H_{C}\beta _{C}) ( 3\gamma ^{2}-4\beta _{C}\beta _{M}) +2c_{p}\gamma ( \beta _{M}\gamma +\beta _{C}( 2\beta _{M}+\gamma ) ) }{[ 4(\beta _{M}+\gamma )(\beta _{C}+\gamma )-\gamma ^{2}] ^{2}}<0,\forall \gamma .$

• the sign of $\frac{\partial p_{C}^{\mathcal {B}}}{\partial \gamma }= \frac{[ 2( \alpha _{C}-H_{C}\beta _{C}) +K_{1}] [ 4( \beta _{C}+\gamma ) ( \beta _{M}+\gamma ) -\gamma ^{2}] -2[ 2( \beta _{M}+\beta _{C}) +3\gamma ] [ 2( \alpha _{C}+H_{C}\beta _{C}) ( \beta _{M}+\gamma ) +\gamma K_{1}] }{[ 4(\beta _{M}+\gamma )(\beta _{C}+\gamma )-\gamma ^{2}] ^{2}}$ depends on $\gamma ,$ where $K_{1}=\alpha _{M}-H_{M}\beta _{M}+c_{p}( \beta _{M}+\gamma ) >0.$ Solving that derivative with respect to $\gamma$ gives two roots, one positive and one negative; indeed, only one of those is feasible $({\rm e.g.},\,\gamma \in (0,1))$ and results in $\gamma ^{ \mathcal {B}*}=\frac{2\beta _{M}K_{2}-\sqrt{( 2\beta _{M}K_{2}^{{}}) ^{2}+16\beta _{M}K_{3}[ \alpha _{M}\beta _{C}+\beta _{M}( ( 2H_{C}-H_{M}+c_{p}) \beta _{C}-2\alpha _{C}) ] }}{2K_{3}}$, where $K_{2}=2[ 3( \alpha _{C}-H_{C}\beta _{C}) -2c_{p}\beta _{C}] >0$ and $K_{3}=-( K_{2}+3( \alpha _{M}-H_{M}\beta _{M}) -c_{p}\beta _{M}) <0$. Consequently, $\frac{\partial p_{C}^{\mathcal {B}}}{\partial \gamma } \ge 0,\forall \gamma \in (0,\gamma ^{\mathcal {B}*}]$ and $\frac{ \partial p_{C}^{{\mathcal {B}}}}{\partial \gamma }<0\forall \gamma \in (\gamma ^{\mathcal {B}*},1].$ Intuitively, because $p_{M}^{\mathcal {B} }>p_{C}^{\mathcal {B}}, \frac{\partial ( p_{M}^{\mathcal {B}}-p_{C}^{ \mathcal {B}}) }{\partial \gamma }<0,\forall \gamma .$ $\square$

Proof of Proposition 3

$\frac{{\rm d}\Pi _{M}^{\mathcal {B}}}{{\rm d}\gamma }= \frac{\partial \Pi _{M}^{\mathcal {B}}}{\partial p_{C}^{\mathcal {B}}}\frac{ \partial p_{C}^{\mathcal {B}}}{\partial p_{M}^{\mathcal {B}}}\frac{\partial p_{M}^{\mathcal {B}}}{\partial \gamma }-[p_{M}^{\mathcal {B}}-p_{C}^{\mathcal {B }}][p_{M}^{\mathcal {B}}-c_{P}]<0$ as $\frac{\partial \Pi _{M}^{\mathcal {B}}}{ \partial p_{C}}>0, \frac{\partial p_{C}^{\mathcal {B}}}{\partial p_{M}^{ \mathcal {B}}}>0$, and $\frac{\partial p_{M}^{\mathcal {B}}}{\partial \gamma } <0$ by Proposition 2. Similarly, $\frac{{\rm d}\Pi _{C}^{\mathcal {B}}}{{\rm d}\gamma }= \frac{\partial \Pi _{C}^{\mathcal {B}}}{\partial p_{M}^{\mathcal {B}}}\frac{ \partial p_{M}^{\mathcal {B}}}{\partial p_{C}^{\mathcal {B}}}\frac{\partial p_{C}^{\mathcal {B}}}{\partial \gamma }+[p_{M}^{\mathcal {B}}-p_{C}^{\mathcal {B }}]p_{C}>0,$ as $\frac{\partial \Pi _{C}^{\mathcal {B}}}{\partial p_{M}^{ \mathcal {B}}}>0, \frac{\partial p_{M}^{\mathcal {B}}}{\partial p_{C}^{ \mathcal {B}}}>0$, and $\frac{\partial p_{C}^{\mathcal {B}}}{\partial \gamma } >0$ by Proposition 2. $\square$

Proof of Proposition 4

Since M and C choose their strategies simultaneously, we characterize first the non-cooperative Nash equilibrium solution. Using the given demand functions and the firms’ profit functions, we compute the derivatives of Eq. (3) with respect to $p_{M}$ and A, and the derivative of Eq. (4) with respect to $p_{C}$. Putting these derivatives equal to zero, the first-order necessary conditions lead to the following reaction functions:

$$ p_{M}(p_{C},A)= \frac{nA+\alpha _{M}+\gamma p_{C}+c_{p}(\beta _{M}+\gamma )-H_{M}\beta _{M}}{2(\beta _{M}+\gamma )} $$

(12)

$$ p_{C}(p_{M},A)= \frac{\alpha _{C}+\gamma p_{M}-H_{C}\beta _{C}}{2(\beta _{C}+\gamma )}\quad A(p_{M})=\frac{n(p_{M}-c_{p}+H_{M})}{l} $$

(13)

From (12) and (13), we can see that the reaction curves for M and C are upward sloping because $\frac{\partial p_{M}}{ \partial p_{C}}=\frac{\gamma }{2(\beta _{M}+\gamma )}>0,\frac{\partial p_{M} }{\partial A}=\frac{n}{2(\beta _{M}+\gamma )}>0,\frac{\partial p_{C}}{\partial p_{M}}=\frac{\gamma }{2(\beta _{C}+\gamma )}>0$, and $\frac{ \partial A}{\partial p_{M}}=\frac{n}{l}>0$. Solving the reaction curves yields the optimal prices chosen by the firms, $p_{M}^{\mathcal {A}}$ and $p_{C}^{\mathcal {A}}$ as well as the optimal level of service efforts, A, as given in Proposition 4. The Nash equilibrium solution is unique and stable. To see this note that substituting $A(p_{M})=\frac{ n(p_{M}-c_{p}+H_{M})}{l}$ into (12), we get $\frac{\partial p_{M}(p_{C})}{\partial p_{C}}=\frac{\gamma l}{2(\beta _{M}+\gamma )l-n^{2}}$. Now, the reaction functions will intersect only once if $\frac{2(\beta _{M}+\gamma )l-n^{2}}{\gamma l}>\frac{\gamma }{2(\beta _{C}+\gamma )}$, implying that $4l(\beta _{M}+\gamma )(\beta _{C}+\gamma )-2n^{2}(\beta _{C}+\gamma )-l\gamma ^{2}>0$, which is always true in our model.

The expressions for optimal profits of the manufacturer and the collector are:

$$\begin{aligned} \Pi _{M}^{\mathcal {A}} &= \frac{\left\{ \begin{array}{c} 2\Psi _{2}\left( n\Psi _{4}+\gamma \Psi _{3}\right) -2\Psi _{1}\left[ n\left( c_{p}-H_{M}\right) \Psi _{4}+\left( \gamma c_{p}+H_{M}\beta _{C}\right) \Psi _{3}\right] -2\gamma \Psi _{2}^{2} \\ +2\Psi _{1}\left[ \Psi _{2}\left( \alpha _{M}-H_{M}\beta _{M}+\left( \gamma +\beta _{M}\right) c_{p}\right) -\Psi _{1}\left( c_{p}\alpha _{M}-H_{M}\left( \alpha _{M}+\alpha _{C}\right) \right) \right] \end{array} \right\} }{\Psi _{1}^{2}}\\ \Pi _{C}^{\mathcal {A}} &= \frac{\left( \gamma \Psi _{3}-H_{C}\Psi _{1}\beta _{M}\right) \Psi _{2}+\Psi _{1}\left( nH_{C}\Psi _{4}+H_{C}\Psi _{1}\left( \alpha _{C}+\alpha _{M}\right) \right) +\Psi _{3}\left( \Psi _{1}\left( \alpha _{C}-H_{C}\beta _{C}\right) -\Psi _{3}\left( \gamma +\beta _{C}\right) \right) }{\Psi _{1}^{2}} \end{aligned}$$

where $\Psi _{1}={\rm DEN}\{ p_{M}^{\mathcal {A}}\},\; \Psi _{2}={\rm NUM}\{ p_{M}^{\mathcal {A}}\}, \Psi _{3}={\rm NUM}\{ p_{C}^{ \mathcal {A}}\}$ and $\Psi _{4}={\rm NUM}\{ A\}$. $\square$

Proof of Proposition 5

Consider the set of constant terms $L_{k}>0,k=1,\ldots ,5$.⁵ We compute the derivatives of the firms’ strategies with respect to $\gamma$ to show that:

• $\frac{\partial p_{M}^{\mathcal {A}}}{\partial \gamma }=-\frac{ \begin{array}{c}l(2n^{2}((\alpha _{C}-(2H_{M}+H_{C})\beta _{C})\beta _{C}+(3H_{M}-cp)L_{1})+(\alpha _{C}-H_{C}\beta _{C})L_{2}\\ +2(H_{M}\beta _{M}-\alpha _{M})L_{3})-l(2c_{p}\gamma (\beta _{M}\gamma +\beta _{C}(2\beta _{M}+\gamma )))\end{array}}{( 2( \beta _{C}+\gamma ) ( 2l( \beta _{M}+\gamma ) -n^{2}) -l\gamma ^{2}) ^{2}}<0,\quad\forall \gamma \in [ 0,1]$,

• $\frac{\partial A}{\partial \gamma }=-\frac{\begin{array}{c}n(2n^{2}(\alpha _{C}\beta _{C}-H_{C}\beta _{C}^{2}+H_{M}L_{3}-cpL_{1})+l(( \alpha _{C}-H_{C}\beta _{C}) L_{2}-2c_{p}\gamma (\beta _{M}\gamma\\ +\beta _{C}(2\beta _{M}+\gamma ))+2(\alpha _{M}-H_{M}\beta _{M})L_{3}))\end{array}}{( 2( \beta _{C}+\gamma ) ( 2l( \beta _{M}+\gamma ) -n^{2}) -l\gamma ^{2}) ^{2}}<0,\quad \forall \gamma \in [ 0,1],$

• $\frac{\partial p_{C}^{\mathcal {A}}}{\partial \gamma }=\frac{\left\{ \begin{array}{c} -2n^{4}L_{4}+ln^{2}\left[ 6L_{4}\beta _{M}+2(\alpha _{C}-H_{C}\beta _{C})(\beta _{M}+3\gamma )+(c_{p}-3H_{M})\gamma ^{2}+2\beta _{C}(2c_{p}\gamma +\alpha _{M})\right] \\ -l^{2}\left[ 2\left( \alpha _{M}-H_{M}\beta _{M}\right) L_{2}+2\left( \alpha _{C}-H_{C}\beta _{C}\right) L_{3}-c_{p}L_{5}\right] \end{array} \right\} }{( 2( \beta _{C}+\gamma ) ( 2l( \beta _{M}+\gamma ) -n^{2}) -l\gamma ^{2}) ^{2}}$ depends on the amplitude of $\gamma .$ By solving $\frac{\partial p_{C}^{\mathcal {A}}}{ \partial \gamma }$ with respect to $\gamma$ we find two roots and only one of these is feasible $({\rm e.g.},\,\gamma \in ( 0,1) )$ and turns out to be: $\gamma ^{\mathcal {A}*}=\frac{2ln^{2}((2c_{p}+3H_{C})\beta _{C}-3\alpha _{C})+\sqrt{\left\{ \begin{array}{c} 4l^{2}n^{4}((2c_{p}+3H_{C})\beta _{C}-3\alpha _{C})^{2}-4n^{2}l(c_{p}-3H_{M})[c_{p}l^{2}L_{5}-2L_{4}(n^{4}-3ln\beta _{M})\\ -l(l(2L_{3}(\alpha _{C}-H_{C}\beta _{C})+L_{2}(\alpha _{M}+H_{M}\beta _{M}))+2n^{2}(\alpha _{M}\beta _{C}-(\alpha _{C}-H_{C}\beta _{C})\beta _{M})] \end{array} \right\} }}{2ln^{2}(c_{p}-3H_{M})}.$ Thus $\frac{\partial p_{C}^{\mathcal {A}} }{\partial \gamma }\ge 0\Rightarrow \forall \gamma \le \gamma ^{\mathcal {A} *}. \square$

Proof of Proposition 5

$\frac{{\rm d}\Pi _{M}(p_{M}^{\mathcal {A} },p_{C}^{\mathcal {A}},A)}{{\rm d}\gamma }=\frac{\partial \Pi _{M}^{\mathcal {A}}}{ \partial p_{C}^{\mathcal {A}}}\frac{\partial p_{C}^{\mathcal {A}}}{\partial p_{M}^{\mathcal {A}}}\frac{\partial p_{M}^{\mathcal {A}}}{\partial \gamma } -[p_{M}^{\mathcal {A}}-p_{C}^{\mathcal {A}}][p_{M}^{\mathcal {A}}-c_{p}^{ \mathcal {A}}]<0$ as $\frac{\partial \Pi _{M}^{\mathcal {A}}}{\partial p_{C}^{ \mathcal {A}}}>0, \frac{\partial p_{C}^{\mathcal {A}}}{\partial p_{M}^{ \mathcal {A}}}>0$ (see Eq. 12), and $\frac{\partial p_{M}^{\mathcal {A }}}{\partial \gamma }<0$ by Proposition 5. Similarly, $\frac{{\rm d}\Pi _{C}^{ \mathcal {A}}(p_{M}^{\mathcal {A}},p_{C}^{\mathcal {A}},A)}{{\rm d}\gamma }=\frac{ \partial \Pi _{C}^{\mathcal {A}}}{\partial p_{M}^{\mathcal {A}}}\frac{\partial p_{M}^{\mathcal {A}}}{\partial p_{C}^{\mathcal {A}}}\frac{\partial p_{C}^{ \mathcal {A}}}{\partial \gamma ^{\mathcal {A}}}+p_{C}^{\mathcal {A}}[p_{M}^{ \mathcal {A}}-p_{C}^{\mathcal {A}}]>0, \frac{\partial \Pi _{M}^{\mathcal {A}} }{\partial p_{C}^{\mathcal {A}}}>0, \frac{\partial p_{M}^{\mathcal {A}}}{ \partial p_{C}^{\mathcal {A}}}>0$ (see Eq. 12), and $\frac{\partial p_{C}^{\mathcal {A}}}{\partial \gamma }>0$ by Proposition 5. $\square$