Modeling brand advertising with heterogeneous consumer response: channel implications

Abstract

The paper explores the implications of heterogeneous consumer response to advertising for the equilibrium strategies chosen by firms in a distribution channel. We solve a simple game theoretic model using a variation of consumers’ Hotelling utility model for a decentralized and a coordinated channel. The key findings show that heterogeneity considerably affects the value of channel coordination. Overlooking heterogeneous responses to advertising can lead to either undercutting or overestimating channel coordination benefits. In particular, our results indicate that, in contrast to previous findings in the literature, channel coordination might result in higher consumer prices especially when the average response to brand advertising exceeds consumers’ marginal disutility cost.

Appendices

Appendix: The decentralized channel

Equilibrium advertising, prices and profits are obtained by solving the three-stage game backwards. We start by solving the retailer’s pricing problem and maximize the retailer’s profits with respect to the retail price (p) by solving the first-order condition ($\frac{\partial \Pi _\mathrm{R} }{\partial \mathrm{p}}=0$) in (p). We obtain the price reaction to advertising and to the wholesale price in Eq. (3).

Then, we substitute p by Eq. (3) into the manufacturer’s profit function and maximize the obtained expression w.r.t. the wholesale price (w). Solving the first-order condition ($\frac{\partial \Pi _\mathrm{M} }{\partial \mathrm{w}}=0$) in (w), we get the result in Eq. (4). Finally, substitute (w) back in the manufacturer’s profits and solve its first-order condition ($\frac{\partial \Pi _\mathrm{M} }{\partial \mathrm{a}}=0$) in (a) to obtain the equilibrium advertising for the game denoted by ($\mathrm{a}^\mathrm{d}$) and which expression is in Table 1.²

The equilibrium wholesale price ($\mathrm{w}^\mathrm{d}$) is then obtained by substituting advertising by ($\mathrm{a}^\mathrm{d}$) in Eq. (4). The equilibrium retail price is found by substituting (w) by ($\mathrm{w}^\mathrm{d}$) and (a) by ($\mathrm{a}^\mathrm{d}$) back in Eq. (3). We finally replace the equilibrium in the profit functions and in $\mathrm{x}_\mathrm{I}$ to obtain equilibrium profits and demands such as in Table 1.

Proof of Proposition 2

Results are obtained by computing the derivatives of the variables ($\mathrm{a}^\mathrm{d}, \mathrm{p}^\mathrm{d}, \Pi _\mathrm{R}^\mathrm{d} , \Pi _\mathrm{M}^\mathrm{d}$) w.r.t. the parameters ($\mathrm{s}_\mathrm{H} , \mathrm{s}_\mathrm{L} , \upalpha$), which gives

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }&= \mathrm{v}\upalpha \frac{4\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }&= \mathrm{v}\left( {1-\upalpha } \right) \frac{4\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \upalpha }&= \mathrm{v}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \frac{4\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}\right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }&= 6\mathrm{tv}\upalpha \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }&= 6\mathrm{tv}\left( {1-\upalpha } \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \upalpha }&= 6\mathrm{tv}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) }{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }&= \mathrm{v}\left( {2\upalpha -1} \right) \frac{4\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}, \end{aligned}$$

therefore

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }\Leftrightarrow \upalpha >1/2. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }=6\mathrm{tv}\left( {2\upalpha -1} \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}\right] ^{2}}, \end{aligned}$$

therefore,

$$\begin{aligned} \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }\Leftrightarrow \upalpha >1/2. \end{aligned}$$

Finally, from the equilibrium solution in Table 1, it is easy to see that the expressions of $\frac{\partial \mathrm{w}^\mathrm{d}}{\partial \mathrm{y}}, \frac{\partial \Pi _\mathrm{R}^\mathrm{d} }{\partial \mathrm{y}}, \frac{\partial \Pi _\mathrm{M}^\mathrm{d} }{\partial \mathrm{y}}, \frac{\partial \mathrm{D}^\mathrm{d}}{\partial \mathrm{y}}$ would have the same sign as $\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{y}},\forall \mathrm{y}\in \left( {\mathrm{s}_\mathrm{H} , \mathrm{s}_\mathrm{L} , \upalpha } \right)$ and are hence positive.

Proof of Proposition 3

Results are obtained by computing the derivatives of the variables ($\mathrm{a}^\mathrm{d}, \mathrm{p}^\mathrm{d}, \Pi _\mathrm{R}^\mathrm{d} , \Pi _\mathrm{M}^\mathrm{d}$) w.r.t. the parameter (t), which gives;

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{t}}&= -4\mathrm{v}\frac{\mathrm{s}_\mathrm{L} +\upalpha \left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) }{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}<0,\\ \frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{t}}&= -3\mathrm{v}\frac{\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}<0. \end{aligned}$$

Finally, from the equilibrium solution in Table 1, it is easy to see that the expressions of $\frac{\partial \mathrm{w}^\mathrm{d}}{\partial \mathrm{t}}, \frac{\partial \Pi _\mathrm{R}^\mathrm{d} }{\partial \mathrm{t}}, \frac{\partial \Pi _\mathrm{M}^\mathrm{d} }{\partial \mathrm{t}}, \frac{\partial \mathrm{D}^\mathrm{d}}{\partial \mathrm{t}}$ would have the same sign as $\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{t}}$ and are hence negative.

The coordinated channel

For the coordinated channel, the problem is to maximize total channel profits ($\Pi$) with respect to consumer price and advertising such as;

$$\begin{aligned} \mathop {\max \Pi }\limits _\mathrm{a, p} = \hbox {pD}-\frac{1}{2}\mathrm{a}^{2}. \end{aligned}$$

The equilibrium advertising and price are obtained by solving the two-stage game backwards. We start by solving the pricing problem and maximize the total channel profits ($\Pi$) with respect to the retail price (p) by solving the first-order condition ($\frac{\partial \Pi }{\partial \mathrm{p}}=0$) in (p). We obtain the following price reaction to advertising

$$\begin{aligned} \mathrm{p}=\frac{1}{2}\left( {\mathrm{v}+\hbox {as}_\mathrm{L} +\mathrm{a}\upalpha \mathrm{s}_\mathrm{H} -\mathrm{a}\upalpha \mathrm{s}_\mathrm{L} } \right) . \end{aligned}$$

(6)

In a second stage, we substitute (p) back in the channel’s profits and solve the first-order condition ($\frac{\partial \Pi }{\partial \mathrm{a}}=0$) in (a) to obtain the equilibrium advertising for the coordinated game

$$\begin{aligned} \mathrm{a}^\mathrm{c}=\mathrm{v}\frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}. \end{aligned}$$

The equilibrium price is found by substituting and (a) by ($\mathrm{a}^\mathrm{c}$) back in (6), which gives

$$\begin{aligned} \mathrm{p}^\mathrm{c}=\frac{\mathrm{tv}}{2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}. \end{aligned}$$

We finally replace the equilibrium in the profit functions and in ($\mathrm{x}_\mathrm{I}$) to obtain equilibrium profits and demands such as in Table 1.

Proof of Proposition 4

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }&= \mathrm{v}\upalpha \frac{2\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }&= \mathrm{v}\left( {1-\upalpha } \right) \frac{2\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}\right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \upalpha }&= \mathrm{v}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \frac{2\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}\right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }&= 2\mathrm{tv}\upalpha \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }&= 2\mathrm{tv}\left( {1-\upalpha } \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \upalpha }&= 2\mathrm{tv}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}>0,\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }&= \mathrm{v}\left( {2\upalpha -1} \right) \frac{2\mathrm{t}+\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}, \end{aligned}$$

therefore,

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }\Leftrightarrow \upalpha >1/2. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }=2\mathrm{tv}\left( {2\upalpha -1} \right) \frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}\right] ^{2}}, \end{aligned}$$

therefore,

$$\begin{aligned} \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }\Leftrightarrow \upalpha >1/2. \end{aligned}$$

Compute the derivatives of equilibrium advertising and pricing w.r.t. the cost parameter (t) to obtain;

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{t}}&= -2\mathrm{v}\frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}<0,\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{t}}&= -\mathrm{v}\frac{\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}<0. \end{aligned}$$

Finally, from the equilibrium solution in Table 1, it is easy to see that the expressions of $\frac{\partial \Pi ^\mathrm{c}}{\partial \mathrm{y}}, \frac{\partial \mathrm{D}^\mathrm{c}}{\partial \mathrm{y}}$ would have the same sign as $\frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{y}},\forall \mathrm{y}\in \left( {\mathrm{s}_\mathrm{H} , \mathrm{s}_\mathrm{L} , \upalpha , t} \right)$.

We can then conclude that ($\frac{\partial \mathrm{n}^\mathrm{c}}{\partial \mathrm{y}}$) and ($\frac{\partial \mathrm{n}^\mathrm{d}}{\partial \mathrm{y}}$) have the same sign, $\forall \mathrm{n}\in \left( {\mathrm{a, p, D}, \Pi } \right)$ and $\forall \mathrm{y}\in \left( {\mathrm{s}_\mathrm{H} , \mathrm{s}_\mathrm{L} , \upalpha , \mathrm{t}} \right)$.

Comparing the sensitivity of equilibrium solution to changes in the model’s parameters across the coordinated and the decentralized channel structures, we obtain;

$$\begin{aligned} \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }&= \mathrm{v}\upalpha \mathrm{A},\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }-\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }&= \mathrm{v}\left( {1-\upalpha } \right) \mathrm{A},\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \upalpha }-\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \upalpha }&= \mathrm{v}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \mathrm{A},\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }-\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }&= 2\mathrm{tv}\upalpha \left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) \mathrm{B},\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }-\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }&= 2\mathrm{tv}\left( {1-\upalpha } \right) \left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H}-\upalpha \mathrm{s}_\mathrm{L} } \right) \mathrm{B},\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \upalpha }-\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \upalpha }&= 2\mathrm{tv}\left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \left( {\mathrm{s}_\mathrm{L}+\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) \mathrm{B},\\ \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{t}}-\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{t}}&= -2\mathrm{v}\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) \mathrm{E},\\ \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{t}}-\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{t}}&= -\mathrm{v}\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}\mathrm{B},\\ \end{aligned}$$

with

$$\begin{aligned} \mathrm{A}=2\mathrm{t}\frac{8\mathrm{t}^{2}+3\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}, \end{aligned}$$

hence $\mathrm{A}>0$, which implies that $\frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }, \; \frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }>\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }$ and $\frac{\partial \mathrm{a}^\mathrm{c}}{\partial \upalpha }>\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \upalpha }$.

$$\begin{aligned} \mathrm{B}=2\frac{2\mathrm{t}^{2}+\left( {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right) \left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}, \end{aligned}$$

hence $\mathrm{B}>0$, which implies that $\frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{H} }>\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{H} }, \; \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{s}_\mathrm{L} }>\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{s}_\mathrm{L} }, \; \frac{\partial \mathrm{p}^\mathrm{c}}{\partial \upalpha }>\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \upalpha }$ and $\frac{\partial \mathrm{p}^\mathrm{c}}{\partial \mathrm{t}}<\frac{\partial \mathrm{p}^\mathrm{d}}{\partial \mathrm{t}}$.

$$\begin{aligned} \mathrm{E}=\frac{8\mathrm{t}^{2}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{4}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}\left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] ^{2}}, \end{aligned}$$

hence $\mathrm{E}>0$, which implies that $\frac{\partial \mathrm{a}^\mathrm{c}}{\partial \mathrm{t}}<\frac{\partial \mathrm{a}^\mathrm{d}}{\partial \mathrm{t}}$.

Effect of channel coordination

Proof of Proposition 5

Comparison of equilibrium strategies and outputs from the coordinated and the uncoordinated channels gives;

$$\begin{aligned} \mathrm{p}^\mathrm{c}-\mathrm{p}^\mathrm{d}=-2\mathrm{tv}\frac{\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] \left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }, \end{aligned}$$

therefore

$$\begin{aligned} \mathrm{p}^\mathrm{c}>\mathrm{p}^\mathrm{d}&\Leftrightarrow \mathrm{t}<\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2},\\ \mathrm{a}^\mathrm{c}-\mathrm{a}^\mathrm{d}&= 2\mathrm{tv}\frac{\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] \left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }>0,\\ \Pi ^\mathrm{C}-(\Pi _\mathrm{R}^\mathrm{d} +\Pi _\mathrm{M}^\mathrm{d} )&= \frac{2\mathrm{t}^{2}\mathrm{v}^{2}}{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] \left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }>0,\\ \mathrm{x}_\mathrm{H}^\mathrm{c} -\mathrm{x}_\mathrm{H}^\mathrm{d}&= 2\mathrm{v}\frac{\mathrm{t}+\left( {1-\upalpha } \right) \left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] \left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }>0,\\ \mathrm{x}_\mathrm{L}^\mathrm{c} -\mathrm{x}_\mathrm{L}^\mathrm{d}&= 2\mathrm{v}\frac{\mathrm{t}-\upalpha \left( {\mathrm{s}_\mathrm{H} -\mathrm{s}_\mathrm{L} } \right) \left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) }{\left[ {2\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] \left[ {4\mathrm{t}-\left( {\mathrm{s}_\mathrm{L} +\upalpha \mathrm{s}_\mathrm{H} -\upalpha \mathrm{s}_\mathrm{L} } \right) ^{2}} \right] }>0,\\ \end{aligned}$$

and since $\mathrm{x}_\mathrm{H}^\mathrm{c} > \mathrm{x}_\mathrm{H}^\mathrm{d}$ and $\mathrm{x}_\mathrm{L}^\mathrm{c} > \mathrm{x}_\mathrm{L}^\mathrm{d}$, we get;

$$\begin{aligned} \mathrm{D}^\mathrm{c}- \mathrm{D}^\mathrm{d}>0. \end{aligned}$$

Model extension to the case where the retailer also invests in local advertising

We extend the model in the paper to consider the retailer’s local advertising $n$. A consumer who belongs to segment $I\in (L,H)$ and is located at distance $x\in (0,1)$ from his/her ideal point, pays a price ($p$) for the product and derives a utility of $U_{I } (a, n, p, x)$ given by;

$$\begin{aligned} U_L&= v + s_{L} (a+n)-p-tx,\\ U_H&= v + s_{H} (a+n)-p-tx. \end{aligned}$$

For the decentralized channel, each channel members maximizes its profit function such as

$$\begin{aligned} \mathop {\max }\limits _{w,a} \Pi _{M}&= wD-{{a}^{2}}/{2,}\\ \mathop {\max }\limits _{p,n} \Pi _{R}&= (p-w)D-{n^{2}}/2. \end{aligned}$$

For the coordinated channel, the total channel profit is given by

$$\begin{aligned} \mathop {\max }\limits _{p,n,a} \Pi =pD-({a}^{2}+n^{2})/2. \end{aligned}$$

We solve the optimization problem for the coordinated channel to find the optimal strategies in p, n and a. For the decentralized channel, we solve a three-stage-game as in the previous sections, with the difference that the retailer now chooses simultaneously its price and local advertising effort.

Equilibrium solution is obtained using the backward induction method. We can easily verify that interior solutions are obtained for $t > 3 \delta ^{2}/4$ in the case of the decentralized channel and for $t > \delta ^{2}$ for the coordinated channel.

It is given in Table 3.

Table 3 Equilibrium solutions—model extension to include retail advertising (n)

Assuming interior equilibrium solutions for both the decentralized and the coordinated channels, we find, similarly to the paper that

$$\begin{aligned}&{a}^{c}>{a}^{d},n^{c}>n^{d},x_I^c >x_I^{d} ,\Pi ^{c}>\Pi ^{d},\\&{p}^{c}>{p}^{d}\Leftrightarrow t<2\delta ^{2}.\\ \end{aligned}$$

In fact;

$$\begin{aligned} a^{c}-a^{d}&= \;-\frac{1}{2}\;\frac{\delta \;\nu \;(\delta ^{2}-2\;t)}{(\delta ^{2}-t)\;(3\delta ^{2}-4t)}\;>\;0,\\ n^{c}-n^{d}&= \;-\frac{1}{2}\;\frac{\delta \;\nu \;(\delta ^{2}-2\;t)}{(\delta ^{2}-t)\;(3\delta ^{2}-4\;t)}\;>\;0,\\ x_L^c -x_L^d&= \;-\frac{1}{2}\;\frac{\nu \;(2\;\delta ^{2}-\;t-2\;\delta \;s_L )\;(\delta ^{2}-2\;t)}{t\;(\delta ^{2}-t)\;(3\;\delta ^{2}-4\;t)}\;>\;0,\\ x_H^c -x_H^d&= \;-\frac{1}{2}\;\frac{\nu \;(2\;\delta ^{2}-\;t-2\;\delta \;s_H )\;(\delta ^{2}-2\;t)}{t\;(\delta ^{2}-t)\;(3\;\delta ^{2}-4\;t)}\;>\;0,\\ \Pi ^{c}-\Pi ^{d}&= \;-\frac{1}{4}\;\frac{\nu ^{2}\;(\;\delta ^{2}-\;2\;t)^{2}}{(\delta ^{2}-t)\;(3\;\delta ^{2}-4\;t)^{2}}\;>\;0,\\ p^{c}-p^{d}&= \;-\frac{1}{2}\;\frac{\nu \;(\delta ^{2}-\;2t)\;(2\;\delta ^{2}-\;t)}{(\delta ^{2}-t)\;(3\;\delta ^{2}-4\;t)}\;,\;\hbox {hence};\\ p^{c}>p^{d}&\Leftrightarrow t<\delta ^{2}/2. \end{aligned}$$