On manufacturers complementing the traditional retail channel with a direct online channel

Abstract

With the explosion of the Internet and the reach that it affords, many manufacturers have complemented their existing retail channels with an online channel, which allows them to sell directly to their consumers. Interestingly, there is a significant variation within product categories in manufacturer's use of the Internet as a direct distribution channel. The main objective of this study is to examine the strategic forces that may influence the manufacturer's decision to complement the retail channel with a direct online channel. In particular, we are interested in answering the following questions:

• (I) Why is it that in some markets only a few firms find it optimal to complement their retail channels with a direct Internet channel while other firms do not?

• (II) What strategic role (if any), does the direct Internet channel serve and how do market characteristics impact this role?

To address these issues we develop a model with a single strategic manufacturer serving a market through a single strategic retailer. In addition to the focal manufacturer's product the retailer carries products of competing manufacturers. Consumers in this market are one of two types. They are either brand loyal or store loyal. The retailer sets the retail price and the level of retail support, which impact the demand for the manufacturer's product. The retailer's decisions in turn depend on the wholesale price as well as the Internet price of the product if the manufacturer decides to complement the retail channel with an online channel.

Our analysis reveals that the optimality of complementing the retail channel with an online channel and the role served by the latter depends critically upon the level of support that the retailer allocates to the manufacturer's product in the absence of the online channel. The level of support allocated by the retailer, in the absence of the online channel, depends upon the retail margins on the manufacturer's product relative to that on rival products in the product category. When the size of the brand loyal segment is small relative to the size of the store loyal segment then in the absence of the online channel, the manufacturer can lower wholesale price and enhance retail support, especially when the retail margins on the rival products are low. In contrast, when the size of the loyal segment is large and the retail margins on rival products are high the manufacturer will find it more profitable to charge a high wholesale price even if that induces the retailer to extend low levels of support. If the manufacturer decides to complement the retail channel with an online channel, some consumers who would have purchased from the retailer might prefer to purchase online. Our analysis reveals that when consumers' sensitivity to price differences across the competing channels exceeds a certain threshold it is not optimal for the manufacturer to complement the retail channel with an online channel. However, this price sensitivity threshold itself depends upon product/market characteristics, suggesting that manufacturers seeking to complement their retail channels with an online channel should look beyond the nature of threat the online channel poses to the retail channel in devising their optimal distribution strategies. When the retail margins on rival products are sufficiently small, complementing the retail channel with an online channel when optimal allows the manufacturer to price discriminate and enhance profits. In contrast when retail margins on rival products are sufficiently high, complementing the retail channel with an online channel serves to enhance retail support. We also identify market conditions under which profits of both the manufacturer and the retailer are greater with the online channel than that without it. This is particularly interesting since the online channel competes with the retail channel.

 

Lemma 1.

• (a) If k > k*, the manufacturer charges wholesale price w*=r and the retailer provides low level of service: s*=s _l .

• (b) if k > k*, the manufacturer charges wholesale price w*=r −k and the retailer provides high level of service: s*=s _h , where $k^ * = \frac{{\alpha _r (s_h - s_l )r}}{{\alpha _r s_h + \alpha _m }}$.

Proof:

Consider the retailer's optimal strategies given wholesale price w, The retailer's profit Π _r as a function of retail price p _r and service level s, is given by

$$\displaylines{ \Pi _r (p_r ,s) = (\alpha _r s + \alpha _m )(p_r - w) + k\alpha _r (\bar s - s)\quad\cr {\rm and,}\quad \frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }} = \alpha _r s + \alpha _m > 0.\quad {\rm Hence},\,p_r^ * = r. }$$

Given wholesale price w, the retailer provides high level of service if and only if

$$ \Pi _r (p_r ,s_h ) > \Pi _r (p_r ,s_l ). $$

Or,

$$ (\alpha _r s_h + \alpha _m )(r - w) + \alpha _r s_l k > (\alpha _r s_l + \alpha _m )(r - w) + \alpha _r s_h k, $$

which holds if w < r −k. Hence, to get high level of service the manufacturer cannot set w higher than r −k. Now, we consider the manufacturer's optimal strategies:

The manufacturer's profit Π _m is given by

$$ \Pi _m (w|p^ * (w),\;s^ * (w)) = (\alpha _r s + \alpha _m )w $$

Similarly, $\frac{{\partial \Pi _m (w)}}{{\partial w}} = \alpha _r s + \alpha _m > 0$, so that the manufacturer's profit is increasing with wholesale price w. This in turn implies that w* should be as high as possible.

Thus, for high level of service, w=r −k. For low level of service, w=r

The manufacturer wants to get high level of service if and only if

$$ \Pi _m (r - k|p_r ,s_h ) > \Pi_ m(r|p_r ,s_l ), {\rm or}$$

$$ (\alpha _r s_h + \alpha _m )(r - k) > (\alpha _r s_l + \alpha _m )r, $$

which holds if $k < \frac{{\alpha _r (s_h - s_l )r}}{{\alpha _r s_h + \alpha _m }}$

Therefore, when $k < \frac{{\alpha _r (s_h - s_l )r}}{{\alpha _r s_h + \alpha _m }}$, w*=r −k, $p_r^ * = r$, s*=s _h . $\Pi _m = (\alpha _r s_h + \alpha _m )(r - k)$ When $k > \frac{{\alpha _r (s_h - s_l )r}}{{\alpha _r s_h + \alpha _m }}$, w*=r, $p_r^ * = r$, s*=s _l . $\Pi _m = (\alpha _r s_l + \alpha _m )r$.

Lemma 2.

The manufacturer's optimal strategy that will induce the retailer to extend low level of support, s*=s _l , is $w^* = p_m^* = r$

Proof:

If the retailer provides low level of service when the manufacturer decides to sell online the manufacturer's profits are:

$$ \Pi _m (w,p_m |p_r ,s_l ) = (\alpha _r s_l + \alpha _m (0.5 + \beta (p_m - p_r )))w + \alpha _m (0.5 + \beta (p_r - p_m ))p_m $$

Since $\frac{{\partial \Pi _m (w,p_m |p_r ,s_l )}}{{\partial w}} = \alpha _r s + \alpha _m (0.5 + \beta (p_m - p_r )) > 0$, w*=r. It follows that $p_m^ * = r$, s*=s _l and $p_r^ * = r$.

Lemma 3.

The optimal strategies of the manufacturer that induce the retailer to extend high level of support and the corresponding strategies for the retailer are:

• (a) ${\it If} k <k_1,\, {\it then}\, w^* = r - k,p_m^* = r,s^* = s_h$ and $p_r^* = r$.

• (b) If k ₁ $< k < k_2$ , then $w^* = r - k,w^*<p_m^* < r,s^* = s_h$ and $p_r^* = r$.

• (c) If $k_2 < k < k_3$ , then $w^* < p_m^* < r,s^* = s_h$ and $p_r^* = r$.

where $\displaystyle k_1 = \frac{1}{{2\beta }},\quad k_2 = \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }},\quad k_3 = \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }}$

Proof:

The manufacturer sets wholesale price recognizing its impact on the retailer's optimal strategies. The retailer's optimal price, $p_r^ *$ however, could be a corner solution or an interior. Consequently, we derive the manufacturer's optimal strategies under the following three scenarios respectively:

• 1. p _r is the corner solution or $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$.

• 2. p _r is the interior solution or $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } < 0$

After characterizing the equilibrium strategies, we identify the conditions, which delineate each of these three regions.

Consider, the first scenario:

1. p _r is the corner solution or $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0.$

There are two possibilities in this scenario:

• (a) $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$ and $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0$ or

• (b) $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$ and $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$

Case 1 (a):

$\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$ and $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0$.

In this case, the optimal retail price, $p_r^ * = r$.

The optimal manufacturer Internet price, $p_m^ *$, then can be obtained by solving the first order condition $\frac{{\partial \Pi _m (p_m ,w)}}{{\partial p_m }}\left| {_{p_r = r} } \right. = 0$

$$ p_m = \frac{{w + r}}{2} + \frac{1}{{4\beta }} $$

(1)

The retailer provides high level of support if

$$ \Pi _r \left( {p_r^ * \left| {_{s =\, s_h } } \right. =r,s_h } \right)> \Pi _r \left( {p_r^ * \left| {_{s = s_l } } \right. = r,s_l } \right). $$

Or,

$$\displaylines{ (\alpha _r s_h + \alpha _m (0.5 + \beta (p_m - p_r )))(r - w) + \alpha _r s_l k\cr > (\alpha _r s_l + \alpha _m (0.5 + \beta (p_m - p_r )))(r - w)+ \alpha _r s_h k}$$

which holds if w < r −k.

Now let us derive the optimal wholesale price that induces the retailer to offer high level of support.

Substituting (1 and $P_r^\ast$=r into the manufacturer's profit function and taking derivative of it with respect to w, we get

$$ \frac{{\partial \Pi _m (p_m ,w)}}{{\partial w}} = \frac{1}{4}(4\alpha _r s_h + \alpha _m (3 - 2\beta r + 2\beta w)) $$

We can see that at w = r −k, $\frac{{\partial \Pi _m (p_m ,w)}}{{\partial w}}\left| {_{w = r - k} } \right. = \frac{1}{4}\alpha _m (3 - 2\beta k) + \alpha _r s_h > 0$ if $k < \frac{3}{{2\beta }} + \frac{{2\alpha _r s_h }}{{\alpha _m \beta }}$.

Thus, when $k < \frac{3}{{2\beta }} + \frac{{2\alpha _r s_h }}{{\alpha _m \beta }}$, the optimal wholesale price to induce the retailer to offer high level of support is w*=r −k.

Substituting w*=r −k into (1), we get $p_m = r - \frac{k}{2} + \frac{1}{{4\beta }}$.

Since p _m can not exceed r, $p_m^ * = \min \{ {r,r - \frac{k}{2} + \frac{1}{{4\beta }}} \}$.

In other words, if $k < \frac{1}{{2\beta }}$, $p_r^ * = r$. If $k > \frac{1}{{2\beta }}$, $p_m^* = r - \frac{k}{2} + \frac{1}{{4\beta }}$

Now we identify the conditions under which case (a) will arise i.e. $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$ and $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0 }$.

The retailer's profit Π _r (p _r , s _h ), given wholesale price w* and manufacturer Internet price $p_m^ *$, is

$$ \Pi _r (p_r ,s) = ( {\alpha _r s + \alpha _m ( {0.5 + \beta ( {p_m^* - p_r })})})(p_r - w^* ) - k\alpha _r s $$

As we have shown, if $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$, $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0$ and $k < \frac{1}{{2\beta }}$, then $p_m^ * = r$,

$$ \frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }}\left| {_{w = r - k,p_m = r,p_r = r} } \right. = \frac{{\alpha _m }}{2} + \alpha _r s - \alpha _m \beta k $$

It is easily seen that $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$ and $\smash{\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0 }$ require that $k < \frac{1}{{2\beta }} + \frac{{\alpha _r s_l }}{{\alpha _m \beta }}$.

Therefore when $\smash{k < \frac{1}{{2\beta }}}$, the optimal strategy of the manufacturer that induces the retailer to offer high service is w*=r −k, when $\smash{\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$, $\smash{\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0 }$. The corresponding strategy of the retailer is s*=s _h and $p_r^ * = r$. The corresponding optimal Internet price is $p_m^ * = r$. We denote this as strategy 1.

Recall that, when $\smash{ k > \frac{1}{{2\beta }} }$, p _m would be the interior solution when the optimal retail strategy is $\smash{p_r^* | {_{s = s_h } } = r }$ and $\smash{ p_r^* | {_{s = s_l } } = r }$. We will now identify the condition under which $\smash{ p_r^*| {_{s = s_h } } = r }$, $\smash{ p_r^* | {_{s = s_l } } = r }$ when p _m is the interior solution.

Suppose $\smash{ k > \frac{1}{{2\beta }} }$, then w*=r −k and $\smash{ p_m^ * = r - \frac{k}{2} + \frac{1}{{4\beta }} }$

$$ \frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }}\left| {_{w = r - k,p_m = r - \frac{k}{2} + \frac{1}{{4\beta }},p_r = r} } \right. = \frac{{3\alpha _m }}{4} + \alpha _r s - \frac{3}{2}\alpha _m \beta k, $$

Thus $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$ and $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0 }$ require $\smash{ k < \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }} }$.

Therefore, when $\smash{ \frac{1}{{2\beta }} < k < \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }} }$, the optimal strategies when the $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$, $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } \ge 0 }$ are $p_m^ * = r - \frac{k}{2} + \frac{1}{{4\beta }}$, w*=r −k, and s*=s _h $p_r^ * = r$. Denote this as strategy 2.

Now , we check whether $p_m^ * - w^ * > 0$

$$ p_m^ * - w^ * = \frac{k}{2} - \frac{1}{{4\beta }} $$

Since $k > \frac{1}{{2\beta }}$, $p_m^ * - w^ * > 0$.

Case 1 (b):

$\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$ but $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0 }$.

Suppose $\smash{ \partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0 }$ but $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$. The retailer charges $p_r^ * = r$ when providing high level of support. The manufacturer would charge an Internet price such that the following first order condition is satisfied

$$ \begin{array}{*{20}c}\displaystyle{\frac{{\partial \Pi _m (p_m ,w)}}{{\partial p_m }}\left|\displaystyle {_{p_r = r,s = s_h } } \right. = 0} \hfill \\{{\rm It}\,{\rm gives}\,p_m^* = \displaystyle\frac{{r + w}}{2} + \frac{1}{{4\beta }}} \hfill \\ \end{array} $$

(2)

If the retailer were to offer low level of support, the optimal retail prices and Internet price can be obtained by solving the following first order conditions simultaneously:

$$ \begin{array}{*{20}c}\displaystyle{\frac{{\partial \Pi _m (p_m ,w)}}{{\partial p_m }}\left| {_{s = s_l } } \right. = \alpha _m \left( \displaystyle{\frac{1}{2} + \beta (p_r + w - 2p_m )} \right) = 0} \hfill \\\displaystyle{\frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }}\left| {_{s = s_l } } \right. = \alpha _r s_l + \alpha _m \left( \displaystyle{\frac{1}{2} + \beta (p_m - 2p_r + w)} \right) = 0} \hfill \\\end{array} $$

Which gives,

$$ \begin{array}{*{20}c}\displaystyle{p_r^* = \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }} + w} \hfill \\ \displaystyle{p_m^* = \frac{1}{{2\beta }} + \frac{{\alpha _r s_l }}{{3\alpha _m \beta }} + w} \hfill \\\end{array} $$

(3)

The retailer provides high level of support if

$$ \Pi _r \left( {p_r^* |_{s = s_h } = r,s_h } \right) \ge \Pi _r \left( {p_r^* |_{s = s_l } < r,s_l } \right) $$

Substituting (2) into $\smash{ \Pi _r ( {p_r^* |_{s = s_h } = r,s_h }) }$ and (3) into $\smash{ \Pi _r( {p_r^* |_{s = s_l } < r,s_l }) }$ we get

$$\displaylines{ \Pi _r ( {p_r^* |_{s\! =\! s_h }\! =\! r,s_h } ) - \Pi _r ( {p_r^* |_{s\! =\! s_l } < r,s_l })\cr =\left( {\begin{array}{*{20}c}{\alpha _r ks_l - \displaystyle\frac{{9\alpha _m^2 + 36\alpha _m \beta \alpha _r s_h k + 24\alpha _m \alpha _r s_l + 16\alpha _r^2 s_l^2 }}{{36\alpha _m \beta }}} \\{ + \displaystyle\frac{1}{4}\left( {r - w} \right)\left( {4\alpha _r s_h + 3\alpha _m - 2\alpha _m \beta \left( {r - w} \right)} \right)} \\ \end{array}} \right)} $$

(4)

In order to get high level of support, the manufacturer must set w such that (4>0, which implies that

$$ r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} - \frac{{\sqrt A }}{{12\alpha _m \beta }} < w < r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }} $$

where $A = 9\alpha _m^2 - 24\alpha _m \alpha _r \left( {12\beta ks_h - 9s_h + 8s_l - 12\beta ks_l } \right) + 16\alpha _r^2 ( {9s_h^2 - 8s_l^2 })$

Now let us derive the optimal wholesale price that induces the retailer to offer high level of support.

Substituting (2) and $p_r^ * = r$ into the manufacturer's profit function and taking derivative of it with respect to w, we get

$$ \frac{{\partial \Pi _m (p_m ,w)}}{{\partial w}} = \frac{1}{4}(4\alpha _r s_h + \alpha _m (3 - 2\beta r + 2\beta w)) $$

At $w = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }}$, it is easily seen that $\frac{{\partial \Pi _m (p_m ,w)}}{{\partial w}}\left| {_{w = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }}} } \right. = \frac{1}{{24}}\left( {9\alpha _m + 12\alpha _r s_h + \sqrt A } \right) > 0$Therefore, the optimal wholesale price to induce the retailer to offer high level of support is

$$ w^* = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }} $$

(5)

Substituting (5) into (2), we get

$$ p_m^* = r - \frac{1}{{8\beta }} - \frac{{\alpha _r s_h }}{{2\alpha _m \beta }} + \frac{{\sqrt A }}{{24\alpha _m \beta }} $$

(6)

Now we derive the conditions under which $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$ but $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$.

$$ \frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }}\left| {_{p_r = r,s = s_h ,w = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }},p_m = r - \frac{1}{{8\beta }} - \frac{{\alpha _r s_h }}{{2\alpha _m \beta }} + \frac{{\sqrt A }}{{24\alpha _m \beta }}} } \right. \!= \!\frac{1}{8}( { \!-\! 3\alpha _m \!-\! 4\alpha _r s_h \!+\! \sqrt A } ) $$

$A - (3\alpha _m + 4\alpha _r s_h )^2 > 0$ implies that $32\alpha _r (s_h - s_l )(\alpha _m (6 - 9\beta k) + 4\alpha _r (s_h + s_l )) > 0$, or $k < \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }}$, $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$ requires that $p_r^ * = r$ ( $p_r^ *$ as specified in(3) given w* as specified in (5)).

Substituting (3) and (5) into $r - p_r^ *$, we get

$$ \begin{array}{*{20}c}{r - p_r^* = \frac{1}{{12\alpha _m \beta }}( {3\alpha _m + 4\alpha _r (3s_h - 2s_l ) - \sqrt A })} \hfill \\{(3\alpha _m + 4\alpha _r (3s_h - 2s_l ))^2 - A = - 48\alpha _r (s_h - s_l )(\alpha _m (3 - 6\beta k) + 4\alpha _r s_l )} \hfill \\\end{array} $$

When $k > \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }}$, condition $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$ is satisfied given w* as specified in (5).

Therefore, when $\smash{ \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }} < k < \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }} }$, the optimal strategies when the $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } > 0$ but $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_l } < 0$ are $p_m^* = r - \frac{1}{{8\beta }} - \frac{{\alpha _r s_h }}{{2\alpha _m \beta }} + \frac{{\sqrt A }}{{24\alpha _m \beta }}$, $w^* = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }}$, and s*=s _h , $p_r^ * = r$. Denote this as strategy 3.

Now let us check whether $\smash{ p_m^ * - w^ * > 0 }$

$$ \begin{array}{*{20}c}{p_m^* - w^* = \displaystyle\frac{1}{{24\alpha _m \beta }}( {15\alpha _m + 12\alpha _r s_h - \sqrt A })} \hfill \\{(15\alpha _m + 12\alpha _r s_h )^2 - A = 8( {27\alpha _m^2 + 16\alpha _r^2 s_l^2 + 6\alpha _m \alpha _r (3s_h + 4s_l + 6\beta k(s_h - s_l ))})} \hfill \\\end{array} $$

Obviously, $p_m^ * - w^ * > 0$.

Then we check whether $p_m^ * < r$

$$ \begin{array}{*{20}c}{r - p_m^* = \displaystyle\frac{1}{{24\alpha _m \beta }}(3\alpha _m + 12\alpha _r s_h - \sqrt A )} \hfill \\{(3\alpha _m + 12\alpha _r s_h )^2 - A = 16\alpha _r ( {8\alpha _r s_l^2 + 3\alpha _m ( - 3s_h + 4s_l + 6\beta k(s_h - s_l ))})} \hfill \\\end{array} $$

Since $k > \frac{1}{{2\beta }} + \frac{{2\alpha _r s_l }}{{3\alpha _m \beta }}$, $(3\alpha _s + 12\alpha _u s_h )^2 - A > 0$. So $p_m^ * < r$.

Case 2.

p _r is the interior solution or $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } < 0$.

Suppose $\partial \Pi _r /\partial p_r |_{p_r = r,s = s_h } < 0$. For any given w, the optimal p _r and p _m to induce the retailer to provides high level of support can be obtained by solving the following first order conditions

$$ \begin{array}{*{20}c}{\displaystyle\frac{{\partial \Pi (p_m ,w)}}{{\partial p_m }}\bigg| {_{s = s_h } } = \alpha _m \bigg( {\displaystyle\frac{1}{2} + \beta (p_r + w - 2p_m )} \bigg) = 0} \hfill \\{\displaystyle\frac{{\partial \Pi _r (p_r ,s)}}{{\partial p_r }}\bigg| {_{s = s_h } } = \alpha _r s_h + \alpha _m \bigg( {\displaystyle\frac{1}{2} + \beta (p_m - 2p_r + w)} \bigg) = 0} \hfill \\\end{array} $$

which gives

$$ \begin{array}{*{20}c}{p_r^* = \displaystyle\frac{1}{{2\beta }} + \displaystyle\frac{{2\alpha _r s_h }}{{3\alpha _m \beta }} + w} \hfill \\{p_m^* = \displaystyle\frac{1}{{2\beta }} + \displaystyle\frac{{\alpha _r s_h }}{{3\alpha _m \beta }} + w} \hfill \\\end{array} $$

(7)

Anticipating the optimal retail price and Internet price as specified in (7), the manufacturer would set its wholesale price w as high as possible.

Since $p_m^ * < p_r^ * \le r$, the maximum w the manufacturer could charge is such that $p_r^ * = r$. Thus,

$$ w^* = r - \frac{1}{{2\beta }} - \frac{{2\alpha _r s_h }}{{3\alpha _m \beta }} $$

Next, we check the conditions under which the retailer would offer high level of support.

The retailer would offer high level of support if and only if

$$ \Pi ( {p_r^* ,s_h }) - \Pi( {p_r^* ,s_l } ) \ge 0 $$

Substituting (7) into $\Pi( {p_r^* ,s_h })$ and (3) into $\Pi ( {p_r^* ,s_l })$, we get

$$ \Pi ( {p_r^* ,s_h }) - \Pi ( {p_r^* ,s_l } ) = \frac{{\alpha _r (s_h - s_l )(\alpha _m (6 - 9\beta k) + 4\alpha _r (s_h + s_l ))}}{{9\alpha _m \beta }} $$

Thus, $\smash{ \Pi ( {p_r^* ,s_h }) - \Pi ( {p_r^* ,s_l } ) \ge 0 }$, if and only if $\smash{k < \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }} }$. In other words, when $\smash{ k > \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }} }$, the retailer would not offer high level of support to the manufacturer.

We summarize the result in the following graph. The optimal strategies of the manufacturer that induce the retailer to extend high level of support and the corresponding strategies for the retailer are depicted in the following graph.

• Strategy 1: $p_m^ * = r$, w*=r −k, $p_r^ * = r$.

• Strategy 2: $p_m^* = r - \frac{k}{2} + \frac{1}{{4\beta }}$, w*=r −k, $p_r^ * = r$

• Strategy 3: $p_m^* = r - \frac{1}{{8\beta }} - \frac{{\alpha _r s_h }}{{2\alpha _m \beta }} + \frac{{\sqrt A }}{{24\alpha _m \beta }}$, $w^* = r - \frac{3}{{4\beta }} - \frac{{\alpha _r s_h }}{{\alpha _m \beta }} + \frac{{\sqrt A }}{{12\alpha _m \beta }}$, $p_r^ * = r$.

Where $A = 9\alpha _m^2 - 24\alpha _m \alpha _r \left( {12\beta ks_h - 9s_h + 8s_l - 12\beta ks_l } \right) + 16\alpha _r^2 \left( {9s_h^2 - 8s_l^2 } \right)$

$$ \begin{array}{*{20}c}{k_1 :\frac{1}{{2\beta }}} \hfill \\{k_2 :\displaystyle\frac{1}{{2\beta }} + \displaystyle\frac{{2\alpha _r s_l }}{{3\alpha _m \beta }}} \hfill \\{k_3 :\displaystyle\frac{2}{{3\beta }} + \displaystyle\frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \displaystyle\frac{{4\alpha _r s_l }}{{9\alpha _m \beta }}} \hfill \\\end{array} $$

Proposition 1.

If $\beta > {\hat \beta} \hbox{ or equivalently } k^* > k_3 \hbox{ then }\forall k \in [k_3, k^*]$ the manufacturer strictly prefers not to complement the traditional retail channel with a direct online channel and is indifferent for all $k> k^* > k_3$.

Proof:

Let $\smash{ \Pi _{mh} ^c }$ and $\smash{ \Pi _{ml} ^c }$ denote the profits of the manufacturer when he gets high and low support respectively, with the online channel. Let $\smash{ \Pi _{mh} ^{nc} }$ and $\smash{ \Pi _{ml} ^{nc} }$ denote the profits of the manufacturer when he gets high and low support respectively, in the absence of the online channel.

Recall from Lemma 3, when k > k ₃, the retailer offers low level of support to the manufacturer with the online channel. With Lemma 2, It is easy to see that $\Pi _{ml} ^{nc} = \Pi _{ml} ^c$. In Lemma 1, we have shown that when k > k*, $\Pi _{ml} ^{nc} < \Pi _{mh} ^c$ in the absence of an online channel. Thus, when k ₃ < k <k*, $\Pi _{ml} ^c < \Pi _{mh} ^{nc}$. So the manufacturer will prefer not to complement its traditional retail channel with a direct online channel.

$$ {\rm The}\,{\rm condition}\,k^ * > k_3 \,{\rm implie}\,{\rm that}\,\frac{{\alpha _r (s_h - s_l )r}}{{\alpha _r s_h + \alpha _m }} > \frac{2}{{3\beta }} + \frac{{4\alpha _r s_h }}{{9\alpha _m \beta }} + \frac{{4\alpha _r s_l }}{{9\alpha _m \beta }} $$

$$ {\rm Which}\,{\rm yields},\,\beta > \frac{{(6\alpha _m + 4\alpha _r s_h + 4\alpha _r s_l )(\alpha _r s_h + \alpha _m )}}{{9\alpha _m k\alpha _r (s_h - s_l )r}} $$

$$ {\rm Define}\,\mathord{\over \beta} = \frac{{(6\alpha _m + 4\alpha _r s_h + 4\alpha _r s_l )(\alpha _r s_h + \alpha _m )}}{{9\alpha _m k\alpha _r (s_h - s_l )r}} $$

Therefore, If $\beta\,\, >\,\, \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \beta }$ or equivalently k*<k ₃, then $\forall k \in [k_3 ,k^* ]$, the maufacturer prefers not to complement the traditional retail channel with a direct online channel.

Again from Lemma 3, the retailer would offer low level of support to the manufacturer with the online channel when k<k ₃. From Lemma 1, when k<k*, $\Pi _{ml} ^{nc} > \Pi _{mh} ^{nc}$, in other words, the manufacturer would get low level of support from the retailer. Thus when k > k* > k ₃, the manufacturer is indifferent.

Proposition 2.

If $k < K_3$ then the manufacturer prefers to complement the traditional retail channel with a direct online channel.

Proof:

From Lemma 3, when k ≤ k ₂, the optimal strategies are w*=r −k, $r - k < p_m^ * \le r$ and $p_r^ * = r$ if the manufacturer wants to get high level of support when going online. Let M denote the proportion of committed consumers who purchase from the online channel, 0 > M > 1. Then $\Pi _{mh} ^c$, the manufacturer's profits from getting high level of support in the presence of the online channel, is given by

$$\displaylines{ \Pi _{mh}^c= (\alpha _r s_h + \alpha _m (1 - M))w + \alpha _m Mp_m\cr \Pi _{mh}^c= (\alpha _r s_h + \alpha _m )w^* + \alpha _m M ( {p_m^* - w^* })\cr= (\alpha _r s_h + \alpha _m )(r - k) + \alpha _m M( {p_m^* - w^* })\cr\quad \ge (\alpha _r s_h + \alpha _m )(r - k) }$$

Since $\Pi _{mh}^{nc} = (\alpha _r s_h + \alpha _m )(r - k)$, $\Pi _{mh}^c > \Pi _{mh}^{nc}$ if k ≤ k ₂.

Now consider the case when k ₂ < k ≤ k ₃.

Define $DF(k) = \Pi _{mh}^c - \Pi _{mh}^{nc}$. Differentiating DF with respect to k, we get

$$\displaylines{ \frac{{dDF(k)}}{{dk}} = \frac{1}{2}\cr\left( \!\!{2\alpha _m \!+\! \alpha _r (s_h \!+\! s_l ) \!-\! \frac{{3\alpha _r (3\alpha _m \!+\! 4\alpha _r s_h )(s_h \!-\! s_l )}}{{\sqrt {9\alpha _m^2 \!+\! 24\alpha _m \alpha _r (9s_h \!-\! 8s_l \!-\! 12\beta k(s_h - s_l )) \!+\! 16\alpha _r^2 \left( {9s_h^2 \!-\! 8s_l^2 } \right)} }}} \right) }$$

Note that $\frac{{dDF(k)}}{{dk}}$ decreases with k. In other words, DF(k) is concave in k in the interval k ∈ [k ₂, k ₃]. Thus, if DF is positive at both ends of the interval, then DF is positive over the entire interval. We know that DF(k ₂)<0 by the above argument. We now check whether DF(k ₃)<0

$$ DF(k_3 ) = \frac{{15\alpha _m^2 - 4\alpha _r^2 s_h^2 + 16\alpha _r^2 s_h s_l + 10\alpha _m \alpha _r s_h + 16\alpha _m \alpha _r s_l }}{{36\alpha _m \beta }} $$

Since α _m <2α _r s _h /3, DF(k ₃)<0. Thus DF(k)<0 over the interval k ∈ [k ₂, k ₃]

Note that we are discussing the case in which the manufacturer's optimal strategy is to get high level of support in the absence of an online channel. In other words, k < k*.

Recall from Lemma 1, if k<k*, then $\Pi _{mh} ^{nc} > \Pi _{ml} ^{nc}$. Therefore when k ≤ k ₃ and k<k*, $\Pi _{mh} ^c <\Pi _{mh} ^{nc} > \Pi _{ml} ^{nc}$, and the manufacturer prefers to complement the existing retail outlet with an online channel.

Proposition 3.

If $\beta < \hat \beta$, then there must exist an interval $[k^* ,k^* + \varsigma ],\,\varsigma > 0$, such that when k is within the interval, the manufacturer can increase profits by complementing the retail channel with a direct online channel.

Proof:

Recall from Lemma 1, when k ≥ k*, the manufacturer prefers low level of support in the absence of the online channel, or $\Pi _{ml} ^{nc} \ge \Pi _{mh} ^{nc}$. In Lemma 2, we have shown that $\Pi _{ml} ^{nc} = \Pi _{ml} ^c$. Thus, the manufacturer can make more profits by selling on the Internet only if his optimal strategy when selling online is to get high level of support or $\Pi _{mh} ^c > \Pi _{ml} ^{nc} = \Pi _{ml} ^c$. Following Lemma 3, the manufacturer will get high level of support when selling online only if k < k ₃.

From Proposition 2, when k < k ₃, $\Pi _{mh} ^c > \Pi _{mh} ^{nc}$. From Lemma 1, when k = k*, $\Pi _{mh} ^{nc} = \Pi _{ml} ^{nc}$ thus $\Pi _{mh} ^c > \Pi _{ml} ^{nc}$. By continuity, there must exist an interval exist an interval [k*, k* + ς] such that when k is within the interval $\Pi _{mh} ^c > \Pi _{ml} ^{nc}$. Following proposition 1, condition k* < k ₃ requires that $\beta < \hat \beta$.

Thus, if $\beta < \hat \beta$, then there must exist an interval [k*, k* + ς], ς<0, such that when k is within the interval, the manufacturer can increase profits by complementing the retail channel with a direct online channel.

Proposition 4.

When the manufacturer finds it optimal to complement the retail channel with an online channel, compared to the profits in the absence of such a channel the retailer's profits with the online channel are:

• (a) Lower if k < k*

• (b) Higher if k>k*.

Proof:

• (a) Let $\Pi _t ^{nc}( {\Pi _t ^c })$ denote the total profits of the manufacturer and the retailer in the absence (presence) of the online channel. When k < k*,

  $$ \Pi _t ^{nc} = (\alpha _r s_h + \alpha _m )p_r - \alpha _r s_h k = (\alpha _r s_h + \alpha _m )r + \alpha _r s_l k $$

  Let M denote the proportion of brand loyal consumers who purchase online. The total profits of the manufacturer and the retailer, denoted by $\Pi _t ^c$ are

  $$ \Pi _t ^c = \alpha _r s_h p_r + \alpha _m Mp_m + \alpha _m (1 - M)p_r + \alpha _r s_l k $$

  Recall from Proposition 2, only if k ≤ k ₃ the manufacturer finds it optimal to complement the retail channel with an online channel when k ≤ k*. Recall from Lemma 3, for k ≤ k ₃ there are two cases: k ≤ k ₁ and k ₁ ≤ k ≤ k ₃. We consider them in turn.

  When k ≤ k ₁, p _r =p _m =r.

  $$ \Pi _t ^c \!=\! \alpha _r s_h p_r + \alpha _m Mp_m + \alpha _m (1 - M)p_r + \alpha _r s_l k \!=\! (\alpha _r s_h + \alpha _m )r + \alpha _r s_l k = \Pi _t ^{nc} $$

  Thus, the total profits are the same when the manufacturer does not sell on the Internet as when he sells on the Internet. Since the manufacturer's profits from selling on the Internet are higher, the retailer's profits must be lower compared to the case when the manufacturer does not sell on the Internet.

  When k ₁ ≤ k ≤ k ₃, p _r =r and p _m < r.

  $$\displaylines{ \Pi _t ^c= \alpha _r s_h p_r + \alpha _m Mp_m + \alpha _m (1 - M)p_r + \alpha _r s_l k < \alpha _r s_h r \cr\quad+ \alpha _m Mr + \alpha _m (1 - M)r + \alpha _r s_l k = \Pi _t ^{nc} }$$

  Now, the total profits when the manufacturer sells on the Internet are less than when he does not sell on the Internet. Since the manufacturer makes more profits when the manufacturer sells on the Internet, the retailer's profits must be lower.

  Therefore, if k < k*, the retailer makes less profits when the manufacturer sells on the Internet.

• (b) When k > k*, the manufacturer prefers low level of support when he does not sell on the Internet. In this case, p _r =r and w=r. The retailer's profits when the manufacturer does not sell on the Internet, denoted by $\Pi _r ^{nc}$, are:

  $$ \Pi _r ^{nc} = (\alpha _r s_l + \alpha_m)(p_r-w) + \alpha _r s_h k $$

  As noted before, the manufacturer prefers high level of service when he sells on the Internet. Then the retailer's profits when the manufacturer sells on the Internet, denoted by $\Pi _r ^c$, are given by

  $$ \Pi _r ^c = (\alpha _r s_h + (1 - M)\alpha _m )(p_r - w) + \alpha _r s_l k $$

  Let us consider the two cases: k ≤ k ₂ and k ₂ ≤ k ≤ k ₃. When k ≤ k ₂, p _r −w=k. So, $\Pi _r ^c = (\alpha _r s_h + (1 - M)\alpha _m )(p_r - w) + \alpha _r s_l k = \alpha_r s_h k + (1 - M)\alpha _m k +\alpha_r s_l k> \Pi _r ^{nc}$.

  When k ₂ ≤ k ≤ k ₃, $\Pi _r ^c = (\alpha _r s_h + (1 - M)\alpha _m )(p_r - w) + \alpha _r s_l k =\alpha _r s_h k + \break(1-M) \alpha_m k + \alpha_r s_l k > \Pi _r ^{nc}$

  Therefore, if k>k*, the retailer makes more profits when the manufacturer finds it optimal to sell on the internet.