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Competitive effects of informative advertising in distribution channels

Abstract

Informative advertising and persuasive advertising play quite different roles in the efficiency of distribution channels. A recent study observes that a direct channel structure outperforms an indirect one in the presence of advertising competition between manufacturers, assuming that advertising is persuasive. On the contrary, taking the informative view of advertising, this paper demonstrates an opposite result: Disintegration does make competing manufacturers better off for highly substitutable goods. We further discuss the optimum control of informative advertising under the indirect channel scenario, showing that retailers are more effective than manufacturers as advertisers.

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Notes

  1. The pure informative role of advertising assumption applies to, for example, the new product sale market where search costs are prohibitively high (Esteves 2009).

  2. The determination of channel structure is quite a long-term strategy, whereas the advertising and pricing decisions can be tailored for a specific good. It seems appropriate to assume that the channel structure has been pre-determined (Wang et al. 2011). Furthermore, empirical findings show that a mixed channel structure tends to become a pure one and only pure structures are stable over the long run (Coughlan 1985).

  3. In “Appendix 4,” we discuss the case when the market is locally covered. We thank the anonymous reviewers for suggesting this discussion.

  4. Throughout this paper, we drop the subscripts “A” and “B” to refer to the symmetric equilibrium results and use the superscripts “D”, “M,” and “R” to denote the equilibrium results under DC, IM, and IR, respectively.

  5. See Eqs. 37, 38, 49, and 50 in “Appendix 2” for details.

  6. See Eqs. 37, 38, 45, and 46 in “Appendix 2” for details.

  7. Note that this conclusion is different from conventional wisdom which suggests that persuasive advertising is anti-competitive because Wang et al. (2011) assume that advertising only increases consumer valuation without affecting the cross-price elasticity between products.

  8. The second-order conditions also hold, since \(\frac{\partial^2\Pi_i}{\partial p_i^2} = -\frac{\phi_i\phi_j}{t}<0\) and \(\frac{\partial^2\Pi_i}{\partial \phi_i^2}=-\alpha<0\).

  9. The second-order condition also holds, since \(\frac{\partial^2\Pi_{ri}}{\partial p_i^2} = -\frac{\phi_i\phi_j}{t}<0\).

  10. The second-order conditions also hold, since \(\frac{\partial^2\Pi_{mi}}{\partial w_i^2} = \frac{\phi_i\phi_j}{2t}\frac{\partial p_j^{M}}{\partial w_i}-\frac{\phi_i\phi_j}{2t}\frac{\partial p_i^{M}}{\partial w_i}-\frac{\phi_i\phi_j}{6t}=-\frac{\phi_i\phi_j}{3t}<0\) and \(\frac{\partial^2\Pi_{mi}}{\partial \phi_i^2} = \frac{w_i\phi_j}{2t}\frac{\partial p_j^{M}}{\partial \phi_i}-\frac{w_i\phi_j}{2t}\frac{\partial p_i^{M}}{\partial \phi_i}+\frac{w_i\phi_j}{3\phi^2}-\alpha=-\alpha<0\).

  11. The second-order conditions also hold, since \(\frac{\partial^2\Pi_{ri}}{\partial p_i^2} = -\frac{\phi_i\phi_j}{t}<0\) and \(\frac{\partial^2\Pi_{ri}}{\partial \phi_i^2} = -\alpha<0\).

  12. The second-order condition also holds, since \(\frac{\partial^2\Pi_{mi}}{\partial w_i^2} = \frac{\phi_i\phi_j}{2t}\frac{\partial p_j^{R}}{\partial w_i}-\frac{\phi_i\phi_j}{2t}\frac{\partial p_i^{R}}{\partial w_i}-\frac{\phi_i\phi_j}{6t}=-\frac{\phi_i\phi_j}{3t}<0\).

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Acknowledgements

The authors wish to thank the co-editor and the anonymous referees for their constructive comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 71071033).

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Correspondence to Jianqiang Zhang.

Appendices

Appendix 1: Derivation of Assumption 1

In this appendix, we derive the parameter restrictions that ensure that the market is fully covered and equilibrium advertising reach is less than 1. Following Soberman (2004), Simbanegavi (2009), and Ghosh and Stock (2010), we need to ensure that under all scenarios, (a) all prices above p = v − t are strictly dominated in equilibrium, (b) the symmetric equilibrium does not involve a corner solution p = v − t, (c) the equilibrium specified above is unique, and (d) ϕ < 1 in equilibrium.

Direct channel

Domination of prices above v − t

When p > v − t, manufacturer i’s profit maximization problem changes from Eq. 1 into

$$ \mathop{\max} \limits_ {p_i, \phi_i}\ \hat{\Pi}_i=p_i\Big(\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}\Big)-\frac{\alpha\phi_i^2}{2}, $$
(6)

where the mark “^” refers to the case when the market for partially informed consumers is not fully covered. Differentiating \(\hat{\Pi}_i\) w.r.t. p i yields

$$\label{A2} \frac {\partial\hat{\Pi}_i} {\partial p_i}=\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}-\frac{p_i\phi_i(2-\phi_j)}{2t}, $$
(7)

from which we have \(\frac{\partial\Pi}{\partial p}^+\Big|_{p=v-t}=\frac{\phi}{2t}(2t-v)(2-\phi)\) at a symmetric equilibrium. Clearly, \(\hat{\Pi}_i\) is a concave function of p i since \(\frac{\partial^2\hat{\Pi}_i}{\partial p_i^2}\!=\!-\frac{\phi_i(1-\phi_j)}{t}-\frac{\phi_i\phi_j}{2t}-\frac{\phi_i(2-\phi_j)}{2t}\!<\!0\). To ensure that all prices above p = v − t are strictly dominated in equilibrium, we need to have \(\frac{\partial\Pi}{\partial p}^+\Big|_{p=v-t}<0\), i.e.,

$$ t<v/2. $$
(8)

No corner solution

Next, we should give conditions under which the symmetric equilibrium cannot involve a corner solution p = v − t. It is easily shown that \(\frac {\partial^2\Pi_i} {\partial p_i^2}<0\) when p i  ≤ v − t (see “Direct channel” in “Appendix 2”). If there exists a corner equilibrium at p = v − t, it must be satisfied that \(\frac{\partial\Pi}{\partial p}^-\Big|_{p=v-t}>0\) and \(\frac{\partial\Pi}{\partial p}^+\Big|_{p=v-t}<0\), where the second inequality does hold due to Eq. 8. To exclude the possibility of a corner equilibrium, we need to ensure that \(\frac{\partial\Pi}{\partial p}^-\Big|_{p=v-t}<0\). From Eq. 1, we have \(\frac{\partial\Pi}{\partial p}^-\Big|_{p=v-t}=\frac{\phi}{2t}(2t-v\phi)\). Plugging p = v − t into Eq. 1, we obtain firm i’s profit maximization problem w.r.t. ϕ i :

$$\mathop{\max} \limits_ {\phi_i}\ \Pi_i=(v-t)\phi_i\Big(1-\frac{\phi_j}{2}\Big)-\frac{\alpha\phi_i^2}{2}, $$
(9)

whose first- and second-order conditions are \(\frac{\partial\Pi_i}{\partial\phi_i}=(v-t)\big(1-\frac{\phi_j}{2}\big)-\alpha\phi_i=0\) and \(\frac{\partial^2\Pi_i}{\partial\phi_i^2}=-\alpha<0\), respectively. Thus, when there exists a corner equilibrium at p = v − t, the advertising reach is solved to be \(\phi=\frac{2(v-t)}{v-t+2\alpha}\), substituting which into \(\frac{\partial\Pi}{\partial p}^-\Big|_{p=v-t}=\frac{\phi}{2t}(2t-v\phi)<0\) yields

$$ \alpha<\frac{(v-t)^2}{2t}. $$
(10)

Uniqueness of equilibrium

Seeing Table 1 for p D and ϕ D, it is easily shown that the pure strategy equilibrium is stable to defections by either firm to p = v − t, since \(\Pi_i\left(\phi^D, p^D\right)>\Pi_i\left(\phi^D, p_i=v-t, p_j=p^D\right)\Leftrightarrow \frac{2\alpha}{\left(1+\sqrt{2\alpha/t}\right)^2}\,>\,(v\,-\,t)\,\left(\frac{2}{1\,+\,\sqrt{2\alpha/t}}\hfill\right.\) \(\left.{\kern-4pt}\left(1-\frac{2}{1+\sqrt{2\alpha/t}}\right) +\left(\frac{2}{1+\sqrt{2\alpha/t}}\right)^2\cdot\frac{t-v+t+\sqrt{2\alpha t}}{2t}\right)-\frac{2\alpha}{\left(1+\sqrt{2\alpha/t}\right)^2}\!\Leftrightarrow\! 2\alpha>(v\!-\!t)\left(\sqrt{\frac{2\alpha}{t}}\!-\!1+\right.\) \(\left.{\kern-2pt}\frac{2t-v+\sqrt{2\alpha t}}{t}\right)\Leftrightarrow 2\sqrt{\frac{2\alpha}{t}}-\frac{v-t}{t}<\frac{2\alpha}{v-t}\Leftrightarrow \frac{8\alpha}{t}<\left(\frac{2\alpha}{v-t}+\frac{v-t}{t}\right)^2\Leftrightarrow \left(\frac{2\alpha}{v-t}-\frac{v-t}{t}\right)^2>0\).

Advertising reach less than one

Finally, we restrict ϕ such that ϕ < 1. Seeing Table 1 for ϕ D, we have \(\frac{2}{1+\sqrt{2\alpha/t}}<1\), i.e.,

$$ \alpha>\frac{t}{2}. $$
(11)

Inequalities Eqs. 8, 10, and 11 together provide the restrictions of parameters α, v, and t under the direct channel scenario, which can be simply written as

$$ \alpha\in \bigg(\frac{t}{2}, \frac{(v-t)^2}{2t}\bigg). $$
(12)

Indirect channel and manufacturer advertises

No corner solution

We first derive conditions under which p = v − t cannot be a corner equilibrium under the full market coverage assumption. As the Stackelberg follower, each retailer determines the retail price given the decisions of w and ϕ. Therefore, as long as p M < v − t, p = v − t cannot be a corner equilibrium. Seeing Table 1 for p M, we have \(t+\sqrt{t^2+16\alpha t}<v-t\), i.e.,

$$\alpha< \frac{(v-t)(v-3t)}{16t}. $$
(13)

Advertising reach less than one

From \(\phi^{M}=\frac{2}{1+\frac{1}{4}(1+\sqrt{1+16\alpha/t})}<1\), we have

$$\alpha>\frac{t}{2}. $$
(14)

Furthermore, it must be satisfied that \(\frac{t}{2}<\frac{(v-t)(v-3t)}{16t}\), i.e.,

$$ t<v/5. $$
(15)

Domination of prices above v − t

Next, we derive conditions under which all prices above p = v − t are strictly dominated in equilibrium. When p > v − t, retailer i’s profit maximization problem changes from Eq. 2 into

$$ \mathop{\max} \limits_ {p_i}\ \hat{\Pi}_{ri}=(p_i-w_i)\Big(\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}\Big). $$
(16)

Differentiating \(\hat{\Pi}_{ri}\) w.r.t. p i yields

$$ \frac {\partial\hat{\Pi}_{ri}} {\partial p_i} =\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}-\frac{\phi_i(2-\phi_j)(p_i-w_i)}{2t}, $$
(17)

from which we have \(\frac{\partial\Pi_r}{\partial p}^+\Big|_{p=v-t}=\frac{\phi}{2t}(2t-v+w)(2-\phi)\) at a symmetric equilibrium. Because \(\frac{\partial^2\hat{\Pi}_{ri}}{\partial p_i^2}<0\), \(\hat{\Pi}_{ri}\) is a concave function of p i . To ensure that all prices above p = v − t are strictly dominated in equilibrium, it must be satisfied that \(\frac{\partial\Pi_r}{\partial p}^+\Big|_{p=v-t}<0\), i.e.,

$$ t<\frac{v-\hat{w}}{2}. $$
(18)

Note that we use \(\hat{w}\), \(\hat{p}\), and \(\hat{\phi}\) to denote the equilibrium results when p > v − t.

Next, we solve \(\hat{w}\) when p > v − t. Evaluating at the symmetric equilibrium, from Eq. 17, we have

$$ \hat{p}=\frac{2v-2v\hat{\phi}+\hat{\phi}t+2\hat{w}-\hat{w}\hat{\phi}}{4-3\hat{\phi}}. $$
(19)

Due to the symmetry between p i and p j , seeing Eq. 17, we have

$$ p_i=\frac{(4\phi_i+6\phi_j-2\phi_i\phi_j-8)v+(\phi_i-4)\phi_jt+\big(4(\phi_i+\phi_j)-2\phi_i\phi_j-8\big)w_i+\phi_j(\phi_i-2)w_j} {8(\phi_i+\phi_j)-3\phi_i\phi_j-16}. $$
(20)

Therefore,

$$ \frac{\partial p_i}{\partial w_i}=\frac{4(\phi_i+\phi_j)-2\phi_i\phi_j-8}{8(\phi_i+\phi_j)-3\phi_i\phi_j-16}, $$
(21)
$$ \frac{\partial p_j}{\partial w_i}=\frac{\phi_i\phi_j-2\phi_i}{8(\phi_i+\phi_j)-3\phi_i\phi_j-16}. $$
(22)

Manufacturer i’s profit maximization problem when p > v − t is

$$ \mathop{\max} \limits_ {w_i, \phi_i}\ \hat{\Pi}_{mi}=w_i\Big(\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}\Big)-\frac{\alpha\phi_i^2}{2}. $$
(23)

Differentiating \(\hat{\Pi}_{mi}\) w.r.t. w i yields

$$ \frac{\partial \hat{\Pi}_{mi}}{\partial w_i}=\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t} -\frac{w_i\phi_i(2-\phi_j)}{2t}\frac{\partial p_i}{\partial w_i} +\frac{w_i\phi_i\phi_j}{2t}\frac{\partial p_j}{\partial w_i}. $$
(24)

Substituting Eqs. 19, 21, and 22 into \(\frac{\partial \hat{\Pi}_{mi}}{\partial w_i}=0\), the symmetric equilibrium \(\hat{w}\) is derived to be

$$ \hat{w}=\frac{2v-2v\hat{\phi}+\hat{\phi}t}{3(2-\hat{\phi})}. $$
(25)

Substituting Eq. 25 into Eq. 18 yields \(\frac{2v-2v\hat{\phi}+\hat{\phi}t}{3(2-\hat{\phi})}<v-2t\), which can be simplified as

$$ 4v-12t>\hat{\phi}(v-5t). $$
(26)

That is to say, Eq. 18 is equivalent to Eq. 26. From Eq. 15, we know v > 5t. Thus, 4v − 12t > v − 5t. Since \(\hat{\phi}\leq1\), Eq. 26 holds necessarily. Therefore, all prices above p = v − t are strictly dominated in equilibrium as long as Eqs. 13 and 14 are satisfied.

Uniqueness of equilibrium

Seeing Table 1 for p M and ϕ M where we use Λ to denote \(\sqrt{t^2+16\alpha t}\) for simplicity, we show that no retailer has an incentive to deviate from the equilibrium retail price to p = v − t, since \(\Pi_{ri}(\phi^{M}, p^{M})>\Pi_{ri}(\phi^{M}, p_i=v-t, p_j=p^{M}) \Leftrightarrow 2t\big(\frac{t+\Lambda}{5t+\Lambda}\big)^2\!>\! \big(v-t-\frac{3}{4}(t+\Lambda)\big)\big(\frac{8t}{5t+\Lambda}\big(1-\frac{8t}{5t+\Lambda}\big)+ \big(\frac{8t}{5t+\Lambda}\big)^2\cdot\,\frac{t-v+t+t+\Lambda}{2t}\big)\Leftrightarrow (t+\Lambda)^2>\big(v-t-\frac{3}{4}(t+\Lambda)\big)(36t-16v+20\Lambda) \Leftrightarrow (\Lambda+2t-v)^2>0\).

Inequalities 13 and 14 together provide the restrictions of parameters α, v, and t under IM, which can be simply written as

$$ \alpha\in \bigg(\frac{t}{2}, \frac{(v-t)(v-3t)}{16t}\bigg). $$
(27)

Indirect channel and retailer advertises

No corner solution

We first derive conditions under which p = v − t cannot be a corner equilibrium under the full market coverage assumption. Plugging p i  = p j  = v − t into Eq. 4, we obtain retailer i’s profit maximization problem w.r.t. ϕ i :

$$ \mathop{\max} \limits_ {\phi_i}\ \Pi_{ri}=(v-t-w_i)\phi_i\Big(1-\frac{\phi_j}{2}\Big)-\frac{\alpha\phi_i^2}{2}. $$
(28)

Evaluating at the symmetric equilibrium, the equilibrium advertising level equals \(\phi=\frac{2(v-t-w)}{v-t-w+2\alpha}\). From Eq. 4, we have \(\frac{\partial\Pi_r}{\partial p}^-\Big|_{p=v-t}=\frac{\phi}{2}\big(2t-\phi(v-w)\big)\). Plugging \(\phi=\frac{2(v-t-w)}{v-t-w+2\alpha}\) into \(\frac{\partial\Pi_r}{\partial p}^-\Big|_{p=v-t}<0\) yields \(\alpha<\frac{(v-t-w)^2}{2t}\). When p = v − t, solving manufacturers’ problems (follow the same process shown in “Indirect channel and retailer advertises” in “Appendix 2”), we have \(w=\frac{v-t+\sqrt{(v-t)^2-24\alpha t}}{2}\), substituting which into \(\alpha<\frac{(v-t-w)^2}{2t}\) yields

$$ \alpha<\frac{(v-t)^2}{32t}. $$
(29)

Advertising reach less than one

From \(\phi^{R}=\frac{2}{1+\sqrt{2\alpha/t}}<1\), we obtain

$$ \alpha>\frac{t}{2}. $$
(30)

From \(\frac{t}{2}<\frac{(v-t)^2}{32t}\), i.e., (v + 3t)(v − 5t) > 0, we have

$$ t<v/5. $$
(31)

Domination of prices above v − t

Next, we show that all prices above p = v − t are strictly dominated in equilibrium provided that Eqs. 29 and 30 are satisfied. When p > v − t, retailer i’s profit maximization problem changes from Eq. 4 into

$$ \mathop{\max} \limits_ {p_i, \phi_i}\ \hat{\Pi}_{ri}=(p_i-w_i)\Big(\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}\Big)-\frac{\alpha\phi_i^2}{2}, $$
(32)

and manufacturer i’s profit maximization problem change from Eq. 5 into

$$ \mathop{\max} \limits_ {w_i}\ \hat{\Pi}_{mi}=w_i\Big(\phi_i(1-\phi_j)\frac{v-p_i}{t}+\phi_i\phi_j\frac{t-p_i+p_j}{2t}\Big). $$
(33)

Since \(\frac{\partial\hat{\Pi}_{ri}}{\partial p_i}\) and \(\frac{\partial\hat{\Pi}_{mi}}{\partial w_i}\) are the same as Eqs. 17 and 24, similar to the derivation processes of Eqs. 1826, we know that all prices above p = v − t are strictly dominated in equilibrium as long as \(4v-12t>\hat{\phi}(v-5t)\), which definitely holds due to Eq. 31.

Uniqueness of equilibrium

Seeing Table 1 for p R and ϕ R, we show that no retailer has an incentive to deviate from the equilibrium retail price to p = v − t, since \(\Pi_{ri}(\phi^{R}, p^{R})>\Pi_{ri}(\phi^{R}\!, p_i\!=\!v-t, p_j\!=\!p^{R})\!\Leftrightarrow\!\frac{2\alpha}{(1+\sqrt{2\alpha/t})^2}\!>\!\big(v-t-3\sqrt{2\alpha t}\big)\!\Big(\!\frac{2}{1+\sqrt{2\alpha/t}}\Big(\!1\!-\!\frac{2}{1+\sqrt{2\alpha/t}}\!\Big) +\Big(\frac{2}{1+\sqrt{2\alpha/t}}\Big)^2\cdot\frac{t-v+t+4\sqrt{2\alpha t}}{2t}\Big)-\frac{2\alpha}{(1+\sqrt{2\alpha/t})^2}\!\Leftrightarrow\!4\alpha\!>\!\big(v-t-3\sqrt{2\alpha t}\big)\big(5\sqrt{2\alpha/t}-\frac{v-t}{t}\big)\!\Leftrightarrow 4\alpha\!>\!5(v-t)\sqrt{2\alpha/t}+3\sqrt{2\alpha t}\frac{v-t}{t}-\frac{(v-t)^2}{t}-30\alpha\!\Leftrightarrow\! 8\sqrt{2\alpha/t}-\frac{v-t}{t}<\frac{34\alpha}{v-t}\!\Leftrightarrow\! \frac{128\alpha}{t}\!< \big(\frac{34\alpha}{v-t}+\frac{v-t}{t}\big)^2\Leftrightarrow \big(\frac{34\alpha}{v-t}-\frac{v-t}{t}\big)^2+\frac{8\alpha}{t}>0\).

Inequalities 29 and 30 together provide the restrictions of parameters α, v, and t under IR, which can be simply written as

$$\alpha\in \bigg(\frac{t}{2}, \frac{(v-t)^2}{32t}\bigg). $$
(34)

As the last step of this appendix, we compare the intervals shown by Eqs. 12, 27, and 34. It is easily seen that \(\frac{(v-t)^2}{2t}>\frac{(v-t)(v-3t)}{16t}\). Moreover, we have \(\frac{(v-t)(v-3t)}{16t}>\frac{(v-t)^2}{32t}\) since v > 5t. This establishes Assumption 1, namely \(\alpha\in\big(\frac{t}{2}, \frac{(v-t)^2}{32t}\big)\).

Appendix 2: Derivation of Table 1

Direct channel

Seeing Eq. 1, the partial derivatives of Π i w.r.t. p i and ϕ i are, respectively,

$$ \frac {\partial\Pi_i} {\partial p_i}=\phi_i(1-\phi_j)+\phi_i\phi_j\frac{t-p_i+p_j}{2t}-\frac{p_i\phi_i\phi_j}{2t}, $$
(35)
$$ \frac {\partial\Pi_i} {\partial \phi_i} = p_i\Big(1-\phi_j+\phi_j\frac{t-p_i+p_j}{2t}\Big)-\alpha\phi_i. $$
(36)

Assuming symmetry, we have the first-order conditionsFootnote 8

$$ \phi\Big(1-\frac{\phi}{2}\Big)-\frac{p\phi^2}{2t}=0, $$
(37)
$$ p\Big(1-\frac{\phi}{2}\Big)-\alpha\phi=0, $$
(38)

through which we obtain the equilibrium decisions, \(p^{D}=\sqrt{2\alpha t}\) and \(\phi^{D}=\frac{2}{1+\sqrt{2\alpha/t}}\). Consequently, the equilibrium profit is \(\Pi^{D}=\frac{2\alpha}{(1+\sqrt{2\alpha/t})^2}\) (see also in Tirole 1988).

Indirect channel and manufacturer advertises

Seeing Eq. 2, the derivative of Π ri w.r.t. p i is

$$ \frac {\partial\Pi_{ri}} {\partial p_i} = \phi_i(1-\phi_j)+\phi_i\phi_j\frac{t-p_i+p_j}{2t}-\frac{(p_i-w_i)\phi_i\phi_j}{2t}. $$
(39)

Due to the symmetry between p i and p j , we have the first-order conditionsFootnote 9

$$ 1-\phi_j+\phi_j\frac{t-p_i+p_j}{2t}-\frac{(p_i-w_i)\phi_j}{2t}=0, $$
(40)
$$ 1-\phi_i+\phi_i\frac{t-p_j+p_i}{2t}-\frac{(p_j-w_j)\phi_i}{2t}=0, $$
(41)

from which retailer i’s decision on retail price is derived as

$$ p_i^{M}=\frac{2}{3}w_i+\frac{1}{3}w_j+\frac{2t}{3\phi_i}+\frac{4t}{3\phi_j}-t, $$
(42)

where w i , w j , ϕ i , and ϕ j are determined by manufacturers.

Seeing Eq. 3, the partial derivatives of Π mi w.r.t. ϕ i and w i are, respectively,

$$ \frac {\partial\Pi_{mi}} {\partial \phi_i} = w_i\left(1-\phi_j+\phi_j\frac{t-p_i^{M}+p_j^{M}}{2t}\right)-\frac{w_i\phi_j}{3\phi_i}-\alpha\phi_i, $$
(43)
$$ \frac {\partial\Pi_{mi}} {\partial w_i} = \phi_i\left(1-\phi_j+\phi_j\frac{t-p_i^{M}+p_j^{M}}{2t}\right)-\frac{w_i\phi_i\phi_j}{6t}. $$
(44)

Considering the symmetric equilibrium, we obtain the first-order conditionsFootnote 10

$$ w\Big(1-\frac{\phi}{2}\Big)-\frac{w}{3}-\alpha\phi=0, $$
(45)
$$ \phi\Big(1-\frac{\phi}{2}\Big)-\frac{w\phi^2}{6t}=0, $$
(46)

through which the equilibrium wholesale price and the equilibrium advertising level are obtained: \(w^{M}=\frac{3}{4}(t+\sqrt{t^2+16\alpha t})\) and \(\phi^{M}=\frac{2}{1+\frac{1}{4}(1+\sqrt{1+16\alpha/t})}\), plugging which into Eq. 42 we have the equilibrium retail price: \(p^{M}=t+\sqrt{t^2+16\alpha t}\). Consequently, we obtain the equilibrium profits as follows: \(\Pi_{r}^{M}=2t\Big(\frac{t+\sqrt{t^2+16\alpha t}}{5t+\sqrt{t^2+16\alpha t}}\Big)^2\), \(\Pi_{m}^{M}=\frac{6t(t+\sqrt{t^2+16\alpha t})^2-32\alpha t^2}{(5t+\sqrt{t^2+16\alpha t})^2}\) and \(\Pi^{M}=\Pi_{r}^{M}+\Pi_{m}^{M}=\frac{8t(t+\sqrt{t^2+16\alpha t})^2-32\alpha t^2}{(5t+\sqrt{t^2+16\alpha t})^2}\).

Indirect channel and retailer advertises

Seeing Eq. 4, the partial derivatives of Π ri w.r.t. p i and ϕ i are, respectively,

$$ \frac {\partial\Pi_{ri}} {\partial p_i} = \phi_i(1-\phi_j)+\phi_i\phi_j\frac{t-p_i+p_j}{2t}-\frac{(p_i-w_i)\phi_i\phi_j}{2t}, $$
(47)
$$ \frac {\partial\Pi_{ri}} {\partial \phi_i} = (p_i-w_i)\Big(1-\phi_j+\phi_j\frac{t-p_i+p_j}{2t}\Big)-\alpha\phi_i. $$
(48)

Considering the symmetric equilibrium, we have the first-order conditionsFootnote 11

$$ \phi\left(1-\frac{\phi}{2}\right)-(p-w)\frac{\phi^2}{2t}=0, $$
(49)
$$ (p-w)\left(1-\frac{\phi}{2}\right)-\alpha\phi=0, $$
(50)

from which retailer i’s decisions are derived as

$$ p^{R}=w+ \sqrt{2\alpha t}, $$
(51)
$$ \phi^{R}=\frac{2}{1+\sqrt{2\alpha/t}}. $$
(52)

From Eq. 52, it can be seen that ϕ R is independent of manufacturers’ decisions at any symmetric equilibrium. From Eq. 47, we have \(p_i^{R}=\frac{2}{3}w_i+\frac{1}{3}w_j+\sqrt{2\alpha t}\). Thus, \(\frac{\partial p_i^{R}}{\partial w_i}=\frac{2}{3}\) and \(\frac{\partial p_j^{R}}{\partial w_i}=\frac{1}{3}\).

Seeing Eq. 5, the partial derivative of Π mi w.r.t. w i is

$$ \frac {\partial\Pi_{mi}} {\partial w_i} = \phi_i\left(1-\phi_j+\phi_j\frac{t-p_i^{R}+p_j^{R}}{2t}\right)-\frac{w_i\phi_i\phi_j}{6t}. $$
(53)

Considering the symmetric equilibrium, w is solved from the first-order condition:Footnote 12

$$w^{R}=\frac{6t}{\phi}-3t=3\sqrt{2\alpha t}, $$
(54)

substituting which into Eq. 51 we have \(p^{R}=4\sqrt{2\alpha t}\). Consequently, we obtain the equilibrium profits as follows: \(\Pi_{r}^{R}=\frac{2\alpha}{(1+\sqrt{2\alpha/t})^2}\), \(\Pi_{m}^{R}=\frac{12\alpha}{(1+\sqrt{2\alpha/t})^2}\) and \(\Pi^{R}=\Pi_{r}^{R}+\Pi_{m}^{R}=\frac{14\alpha}{(1+\sqrt{2\alpha/t})^2}\).

Appendix 3: Proof of propositions

Proof of Proposition 1

Since \(\frac{\alpha}{(1+\sqrt{2\alpha/t})^2}=\frac{1}{(\sqrt{1/\alpha}+\sqrt{2/t})^2}\) which is an increasing function of α, we have \(\frac{\partial\Pi^D}{\partial\alpha}>0\), \(\frac{\partial\Pi_r^{R}}{\partial\alpha}>0\), \(\frac{\partial\Pi_m^{R}}{\partial\alpha}>0\), and \(\frac{\partial\Pi^{R}}{\partial\alpha}>0\). Furthermore, by simple algebra, we obtain that \(\frac{\partial\Pi_r^{M}}{\partial\alpha}=\frac{32t^3(4t+4\sqrt{t^2+16\alpha t})}{\sqrt{t^2+16\alpha t}(5t+\sqrt{t^2+16\alpha t})^3}\), \(\frac{\partial\Pi_m^{M}}{\partial\alpha}=\frac{32t^3(11t+7\sqrt{t^2+16\alpha t})}{\sqrt{t^2+16\alpha t}(5t+\sqrt{t^2+16\alpha t})^3}\), and \(\frac{\partial\Pi^{M}}{\partial\alpha}=\frac{32t^3(15t+11\sqrt{t^2+16\alpha t})}{\sqrt{t^2+16\alpha t}(5t+\sqrt{t^2+16\alpha t})^3}\), which are all positive.□

Proof of Proposition 2

It is easily seen that p D < p M. Furthermore, \(p^{M}<p^{R}\Leftrightarrow t+\sqrt{t^2+16\alpha t}<4\sqrt{2\alpha t}\Leftrightarrow 2t^2+16\alpha t+2t\sqrt{t^2+16\alpha t}<32\alpha t\Leftrightarrow \sqrt{t^2+16\alpha t}<8\alpha-t\Leftrightarrow t^2+16\alpha t<t^2-16\alpha t+64\alpha^2\Leftrightarrow\alpha>\frac{t}{2}\). Therefore, we have p D < p M < p R.

Obviously, ϕ D = ϕ R. Moreover, \(\phi^{R}<\phi^{M}\Leftrightarrow \frac{2}{1+\sqrt{2\alpha/t}}<\frac{2}{1+\frac{1}{4}(1+\sqrt{1+16\alpha/t})}\Leftrightarrow 4\sqrt{2\alpha/t}>1\!+\!\sqrt{1+16\alpha/t}\!\Leftrightarrow\! 2+16\alpha/t\!+\!2\sqrt{1\!+\!16\alpha/t}<32\alpha/t\!\Leftrightarrow\! \sqrt{1+16\alpha/t}< 8\alpha/t-1\Leftrightarrow\alpha>t/2\). Therefore, we have ϕ D = ϕ R < ϕ M.

Since \(w^{M}=\frac{3}{4}(t+\sqrt{t^2+16\alpha t})>\frac{3}{4}t+\frac{3}{4}\sqrt{16\alpha t}=\frac{3}{4}t+3\sqrt{\alpha t}>\sqrt{2\alpha t}\), we have w M > p D. Obviously, \(p^{R}-w^{R}=\sqrt{2\alpha t}=p^D\).□

Proof of Proposition 3

We first prove that \(\Pi_r^{R}<\Pi_r^{M}\). Since \(\Pi_r^{M}\) can be expressed as an increasing function of p M, i.e., \(\Pi_r^{M}=2t(\frac{p^{M}}{4t+p^{M}})^2\) and \(p^{M}=t+\sqrt{t^2+16\alpha t}>4\sqrt{\alpha t}\), we have \(\Pi_r^{M}>2t\Big(\frac{4\sqrt{\alpha t}}{4t+4\sqrt{\alpha t}}\Big)^2=\frac{2\alpha}{(1+\sqrt{\alpha/t})^2}>\frac{2\alpha}{(1+\sqrt{2\alpha/t})^2}=\Pi_r^{R}\).

We then prove that ΠM < ΠR. Since ΠM can be expressed as an increasing function of p M, i.e., \(\Pi^{M}=\frac{8t(p^{M})^2-32\alpha t^2}{(4t+p^{M})^2}\) and \(p^{M}<p^{R}=4\sqrt{2\alpha t}\) (see Proposition 2), we have \(\Pi^{M}<\frac{8t(4\sqrt{2\alpha t})^2-32\alpha t^2}{(4t+4\sqrt{2\alpha t})^2}=\frac{14\alpha t^2}{(t+\sqrt{2\alpha t})^2}=\frac{14\alpha}{(1+\sqrt{2\alpha/t})^2}=\Pi^{R}\).

Obviously, \(\Pi_m^{M}<\Pi_m^{R}\) since \(\Pi_m^{M}=\Pi^{M}-\Pi_r^{M}\) and \(\Pi_m^{R}=\Pi^{R}-\Pi_r^{R}\).

Since \(\Pi_m^{M}>\Pi_r^{M}\Leftrightarrow 4t(t+\sqrt{t^2+16\alpha t})^2-32\alpha t^2>0\Leftrightarrow (t+\sqrt{t^2+16\alpha t})^2>8\alpha t\Leftrightarrow 2t^2+16\alpha t+2t\sqrt{t^2+16\alpha t}>8\alpha t\), we get \(\Pi_m^{M}>\Pi_r^{M}>\Pi_r^{R}=\Pi^D\).

Finally, ΠD < ΠM since \(\Pi^D=\Pi_r^{R}<\Pi_r^{M}<\Pi^{M}\). □

Appendix 4: Local coverage

Suppose that t is large enough such that the market is locally covered. Under local monopoly, a consumer located at the midpoint of the Hotelling line does not purchase either product, even when fully informed, i.e., \(v-p-\frac{t}{2}<0\) at any equilibrium (Hamilton 2009).

Under this assumption, a consumer who receives product A’s advertising message will buy it if and only if v − p A  − tx > 0; a consumer who receives product B’s message will buy it if and only if v − p B  − t(1 − x) > 0. Hence, demand facing product i equals \(\phi_i\frac{v-p_i}{t}\).

Direct channel

Manufacturer i’s profit is given by \(\Pi_i=p_i\phi_i\frac{v-p_i}{t}-\frac{\alpha\phi_i^2}{2}\). The optimal decisions are solved to be \(p^D=\frac{v}{2}\) and \(\phi^D=\frac{v^2}{4\alpha t}\), plugging which back into the profit function we obtain \(\Pi^D=\frac{v^4}{16\alpha t^2}\). Note that local coverage assumption implies \(v-p^D-\frac{t}{2}<0\), i.e., t > v.

Indirect channel and manufacturer advertises

Retailer i’s profit is \(\Pi_{ri}=(p_i-w_i)\phi_i\frac{v-p_i}{t}\), which is maximized at \(p_i=\frac{v+w_i}{2}\). Manufacturer i’s profit is \(\Pi_{mi}=w_i\phi_i\frac{v-p_i}{t}-\frac{\alpha\phi_i^2}{2}=w_i\phi_i\frac{v-w_i}{2t}-\frac{\alpha\phi_i^2}{2}\), which is maximized at \(w^{M}=\frac{v}{2}\) and \(\phi^{M}=\frac{v^2}{8\alpha t}\). Thus, \(p^{M}=\frac{3}{4}v\). The equilibrium profits are \(\Pi_r^{M}=\Pi_m^{M}=\frac{v^4}{128\alpha t^2}\) and \(\Pi^{M}=\frac{v^4}{64\alpha t^2}\). Since t > v, we ensure that \(v-p^{M}-\frac{t}{2}<0\), i.e., the market is locally covered.

Indirect channel and retailer advertises

Retailer i’s profit is \(\Pi_{ri}=(p_i-w_i)\phi_i\frac{v-p_i}{t}-\frac{\alpha\phi_i^2}{2}\). The optimal decisions are \(\phi_i=\frac{(v-w_i)^2}{4\alpha t}\) and \(p_i=\frac{v+w_i}{2}\). Manufacturer i’s profit is \(\Pi_{mi}=w_i\phi_i\frac{v-p_i}{t}=\frac{w_i(v-w_i)^3}{8\alpha t^2}\), which is maximized at \(w^{R}=\frac{v}{4}\). Thus, we have \(\phi^{R}=\frac{9v^2}{64\alpha t}\) and \(p^{R}=\frac{5}{8}v\). The equilibrium profits are \(\Pi_r^{R}=\frac{81v^4}{8192\alpha t^2}\), \(\Pi_m^{R}=\frac{27v^4}{2048\alpha t^2}\), and \(\Pi^{R}=\frac{189v^4}{8192\alpha t^2}\). Since t > v, we ensure that \(v-p^{R}-\frac{t}{2}<0\), i.e., the market is locally covered.

To sum up, under local monopoly (t > v), we have \(\Pi^D=\frac{v^4}{16\alpha t^2}\), \(\Pi^{M}=\frac{v^4}{64\alpha t^2}\), and \(\Pi^{R}=\frac{189v^4}{8192\alpha t^2}\). Clearly, ΠD > ΠR > ΠM.

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Zhang, J., Zhong, W. & Mei, S. Competitive effects of informative advertising in distribution channels. Mark Lett 23, 561–584 (2012). https://doi.org/10.1007/s11002-011-9161-2

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Keywords

  • Informative advertising
  • Channel competition
  • Distribution channel