Corporate social responsibility in a supply chain and competition from a vertically integrated firm

Abstract

This paper extends the analysis of the effects of corporate social responsibility in a bilateral monopoly to the case where a manufacturer and a retailer engaged in a supply chain face competition from a vertically integrated firm. The paper finds that when the manufacturer and the retailer non-cooperatively select their degrees of social concern, they choose to pursue pure profit maximization, irrespective of the order in which they make their choices. These choices put them at a disadvantage with respect to their vertically integrated competitor, who produces more output than they do and obtains higher profits than their joint profits. The paper then shows that when they cooperatively choose their degrees of social concern, they both choose positive degrees of social concern and, therefore, deviate from pure profit maximization. These choices give them a competitive edge over their vertically integrated rival: they produce a higher output and obtain higher joint profits than their competitor. In comparison with the non-cooperative outcome, the positive degrees of social concern cooperatively chosen by the manufacturer and the retailer imply higher output and profits for the retailer and the manufacturer and lower output and profits for their competitor. They also imply lower prices and higher consumer surplus and social welfare.

Appendix

1.1 Equilibrium outcome for exogenous degrees of social concern

For any given degrees of social concern $\theta_{r}$ and $\theta_{m}$, the equilibrium values are as follows:

$$\begin{aligned} & w^{*} = \frac{{ - \left( {a - c} \right)\theta_{r}^{2} + \left( {2a - 6c} \right)\theta_{r} - \left( {2a - c} \right)\theta_{m} + 9c + 3a}}{{12 - 4\theta_{r} - \theta_{m} }},\quad \\ &p^{*} = \frac{{ - \left( {c + 3a} \right)\theta_{r} - a\theta_{m} + 7c + 5a}}{{12 - 4\theta_{r} - \theta_{m} }} \end{aligned}$$

$$q_{r}^{*} = \frac{{\left( {a - c} \right)\left( {2\theta_{r} + \theta_{m} + 2} \right)}}{{12 - 4\theta_{r} - \theta_{m} }},\quad q_{v}^{*} = \frac{{\left( {a - c} \right)\left( { - 3\theta_{r} - \theta_{m} + 5} \right)}}{{12 - 4\theta_{r} - \theta_{m} }},\quad CS^{*} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)^{2} }}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }}$$

$$\begin{aligned} & \Pi_{m}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( { - \theta_{r}^{2} + 2\theta_{r} - 2\theta_{m} + 3} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }},\quad\\ & \Pi_{r}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( {\theta_{r}^{2} - 5\theta_{r} + \theta_{m} + 2} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} \end{aligned}$$

$$\begin{aligned} \Pi_{v}^{*} &= \frac{{\left( {a - c} \right)^{2} \left( {5 - 3\theta_{r} - \theta_{m} } \right)^{2} }}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }},\;\\ V_{r}^{*} &= \frac{{\left( {a - c} \right)^{2} \left( {5\theta_{r}^{3} + 2\theta_{m} \theta_{r}^{2} - 30\theta_{r}^{2} - 6\theta_{m} \theta_{r} + 37\theta_{r} + 2\theta_{m}^{2} + 8\theta_{m} + 8} \right)}}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} \end{aligned}$$

$$V_{m}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {\theta_{r}^{2} + 2\theta_{r} + 4\theta_{m} + 1} \right)}}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)}},\;SW^{*} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( {17 - 7\theta_{r} - 2\theta_{m} } \right)}}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }}$$

Proof of Proposition 1

The derivatives of the equilibrium variables are as follows:

$$\begin{aligned} &\frac{{\partial w^{*} }}{{\partial \theta_{m} }} = \frac{{ - \left( {a - c} \right)\left( {7 - \theta_{r} } \right)\left( {3 - \theta_{r} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} < 0,\;\\ &\frac{{\partial w^{*} }}{{\partial \theta_{r} }} = \frac{{2\left( {a - c} \right)\left( {2\theta_{r}^{2} + \theta_{m} \theta_{r} - 12\theta_{r} - 5\theta_{m} + 18} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} > 0, \end{aligned}$$

$$\frac{{\partial q_{v}^{*} }}{{\partial \theta_{m} }} = \frac{{ - \left( {a - c} \right)\left( {7 - \theta_{r} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} < 0,\;\frac{{\partial q_{v}^{*} }}{{\partial \theta_{r} }} = \frac{{ - \left( {a - c} \right)\left( {16 + \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} < 0,\;\frac{{\partial q_{r}^{*} }}{{\partial \theta_{m} }} = \frac{{2\left( {a - c} \right)\left( {7 - \theta_{r} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} > 0,$$

$$\frac{{\partial q_{r}^{*} }}{{\partial \theta_{r} }} = \frac{{2\left( {a - c} \right)\left( {16 + \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} > 0,\;\frac{{\partial p^{*} }}{{\partial \theta_{m} }} = \frac{{ - \left( {a - c} \right)\left( {7 - \theta_{r} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} < 0,\;\frac{{\partial p^{*} }}{{\partial \theta_{r} }} = \frac{{ - \left( {a - c} \right)\left( {16 + \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} < 0,$$

$$\frac{{\partial CS^{*} }}{{\partial \theta_{m} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)^{2} }}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,\;\frac{{\partial CS^{*} }}{{\partial \theta_{r} }} = \frac{{\left( {a - c} \right)^{2} \left( {16 + \theta_{m} } \right)\left( {7 - \theta_{r} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0$$

$$\frac{{\partial V_{m}^{*} }}{{\partial \theta_{m} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)^{2} }}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} > 0,\;\frac{{\partial V_{m}^{*} }}{{\partial \theta_{r} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( {2\theta_{r} + \theta_{m} + 2} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} > 0$$

$$\frac{{\partial V_{r}^{*} }}{{\partial \theta_{m} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( { - \theta_{r}^{2} - \theta_{m} \theta_{r} - \theta_{r} + 4\theta_{m} + 8} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,$$

$$\frac{{\partial V_{r}^{*} }}{{\partial \theta_{r} }} = \frac{{\left( {a - c} \right)^{2} \left( { - 20\theta_{r}^{3} - 15\theta_{m} \theta_{r}^{2} + 180\theta_{r}^{2} - 4\theta_{m}^{2} \theta_{r} + 84\theta_{m} \theta_{r} - 572\theta_{r} + 22\theta_{m}^{2} - 45\theta_{m} + 508} \right)}}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,$$

To see that $\frac{{\partial V_{r}^{*} }}{{\partial \theta_{r} }} > 0$ notice that the denominator is positive and the numerator is also positive because it is decreasing in $\theta_{r}$ and it is positive even when $\theta_{r} = 1$ ($\frac{{\partial V_{r}^{*} }}{{\partial \theta_{r} }} > \frac{{\left( {a - c} \right)^{2} \left( {18\theta_{m}^{2} + 24\theta_{m} + 96} \right)}}{{2\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0$)

$$\frac{{\partial SW^{*} }}{{\partial \theta_{m} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( {5 - 3\theta_{r} - \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,\;\frac{{\partial SW^{*} }}{{\partial \theta_{r} }} = \frac{{\left( {a - c} \right)^{2} \left( {16 + \theta_{m} } \right)\left( {5 - 3\theta_{r} - \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0$$

It is also of interest to note that:

$$\frac{{\partial \Pi_{v}^{*} }}{{\partial \theta_{m} }} = \frac{{ - 2\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( {5 - 3\theta_{r} - \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} < 0,\;\frac{{\partial \Pi_{v}^{*} }}{{\partial \theta_{r} }} = \frac{{ - 2\left( {a - c} \right)^{2} \left( {16 + \theta_{m} } \right)\left( {5 - 3\theta_{r} - \theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} < 0$$

$$\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{r} }} = \frac{{2\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( { - 4\theta_{r}^{2} - 3\theta_{m} \theta_{r} + 8\theta_{r} - \theta_{m}^{2} - 3\theta_{m} + 12} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,$$

$$\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{m} }} = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)\left( { - \theta_{m} \theta_{r} - 8\theta_{r} + 4\theta_{m} + 8} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} > 0,$$

Proof of Proposition 2

Assume first simultaneous choice of the degrees of social concern.

The derivatives of the profits of the manufacturer and the retailer with respect to their own degrees of social concern are given by:

$$\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} = \frac{{ - \left( {a - c} \right)^{2} \theta_{m} \left( {7 - \theta_{r} } \right)^{2} }}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} \le 0,$$

$$\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }} = \frac{{\left( {a - c} \right)^{2} \left( {\left( { - 8\theta_{r}^{2} - 6\theta_{m} \theta_{r} + 72\theta_{r} - 2\theta_{m}^{2} + 28\theta_{m} - 216} \right)\theta_{r} + 11\theta_{m}^{2} + 2\theta_{m} - 40} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{3} }} < 0$$

Thus, it is optimal for the manufacturer to set $\theta_{m} = 0$ because, for any $\theta_{r} \epsilon \left[ {0,1} \right]$, $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} = 0$ for $\theta_{m} =$ 0 while $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} < 0$ whenever $\theta_{m} > 0$ and, reciprocally, it is optimal for the retailer to set $\theta_{r} = 0$ because $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }} < 0$ for all $\theta_{m} \epsilon \left[ {0,1} \right]$ and $\theta_{r} \epsilon \left[ {0,1} \right]$.

Assume now sequential choice of the degrees of social concern with the retailer as the leader.

The manufacturer chooses $\theta_{m}$ knowing the retailer’s choice. The manufacturer’s reaction function is to set $\theta_{m} = 0$ for all $\theta_{r} \epsilon \left[ {0,1} \right]$ because, again, irrespective of the value of $\theta_{r}$, $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} = 0$ for $\theta_{m} =$ 0 and $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} < 0$ whenever $\theta_{m} > 0$. Replacing the previous manufacturer’s reaction function $\theta_{m} = 0$ into the retailer’s profits, the retailer chooses $\theta_{r} = 0$ because $\Pi_{r}^{*}$ is always strictly decreasing in $\theta_{r}$.

Assume finally sequential choices of the degrees of social concern with the manufacturer as the leader.

The retailer chooses $\theta_{r}$ knowing the manufacturer’s choice $\theta_{m}$. The retailer chooses $\theta_{r} = 0$ irrespective of the value of $\theta_{m}$ because $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }} < 0$ for all $\theta_{m} \epsilon \left[ {0,1} \right]$ and $\theta_{r} \epsilon \left[ {0,1} \right]$. Thus, the retailer’s reaction function is to set $\theta_{r} = 0$ for all $\theta_{m} \epsilon \left[ {0,1} \right]$. Replacing this reaction function into the manufacturer’s profits, the manufacturer chooses $\theta_{m} = 0$ because it continues to hold that $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} = 0$ for $\theta_{m} =$ 0 and $\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} < 0$ whenever $\theta_{m} > 0$.

Proof of Lemma 1

Consider the problem of choosing $\theta_{m} \epsilon \left[ {0,1} \right]$, $\theta_{r} \epsilon \left[ {0,1} \right]$ to solve:

$${\text{Max}}\,\Pi_{m}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( { - \theta_{r}^{2} + 2\theta_{r} - 2\theta_{m} + 3} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }}$$

$${\text{s}} . {\text{t}}.\,\Pi_{r}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( {\theta_{r}^{2} - 5\theta_{r} + \theta_{m} + 2} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} = \bar{\Pi }_{r}$$

Notice that the constraint implicitly defines $\theta_{r}$ as a function of $\theta_{m}$ with $\frac{{{\text{d}}\theta_{r} }}{{{\text{d}}\theta_{m} }} = \frac{{ - \frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{m} }}}}{{\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }}}} > 0$ because $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{m} }} > 0$ and $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }} < 0$. One can thus write $\theta_{r}$ as a function of $\theta_{m}$ in the objective function to simplify the maximization problem into one that has $\theta_{m}$ as the only decision variable. The first-order condition of this simplified problem is:

$$\frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{m} }} + \frac{{\partial \Pi_{m}^{*} }}{{\partial \theta_{r} }}\left( {\frac{{ - \frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{m} }}}}{{\frac{{\partial \pi_{r}^{*} }}{{\partial \theta_{r} }}}}} \right) = \frac{{\left( {a - c} \right)^{2} \left( {7 - \theta_{r} } \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( {8 - 8\theta_{r} - 3\theta_{m} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} D}} = 0$$

with

$$D = \left( {8\theta_{r}^{2} + 6\theta_{m} \theta_{r} - 72\theta_{r} + 2\theta_{m}^{2} - 28\theta_{m} + 216} \right)\theta_{r} - 11\theta_{m}^{2} - 2\theta_{m} + 40 > 0$$

Notice now that when $\theta_{m} = 1$ and $\theta_{r} = 5/8$, the retailer profits are $\Pi_{r}^{*} = \frac{{\left( {a - c} \right)^{2} }}{64}$ and consider two different cases:

• i. Assume that $\bar{\Pi }_{r} \le \frac{{\left( {a - c} \right)^{2} }}{64}$. In this case, there is a unique pair $\theta_{m} \epsilon \left[ {0,1} \right]$, $\theta_{r} \epsilon \left[ {0,1} \right]$ that satisfies both the first-order condition, which amounts to $8 - 8\theta_{r} - 3\theta_{m} = 0,$ and the constraint $\Pi_{r}^{*} = \frac{{\left( {a - c} \right)^{2} \left( {2\theta_{r} + \theta_{m} + 2} \right)\left( {\theta_{r}^{2} - 5\theta_{r} + \theta_{m} + 2} \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} } \right)^{2} }} = \bar{\Pi }_{r}$. To see this, notice that replacing $\theta_{m} = \frac{{8 - 8\theta_{r} }}{3}$ from the first-order condition into $\Pi_{r}^{*}$ one can write the constraint as $\frac{{\left( {a - c} \right)^{2} \left( {2 - 3\theta_{r} } \right)}}{8} = \bar{\Pi }_{r}$. Solving this equation for $\theta_{r}$, one obtains $\theta_{r} = \frac{{2\left( {a - c} \right)^{2} - 8\bar{\Pi }_{r} }}{{3\left( {a - c} \right)^{2} }}$, and replacing this back into the first-order condition, one obtains $\theta_{m} = \frac{{8\left( {a - c} \right)^{2} + 64\bar{\Pi }_{r} }}{{9\left( {a - c} \right)^{2} }}$. When $\bar{\Pi }_{r} \le \frac{{\left( {a - c} \right)^{2} }}{64}$, these two values satisfy $\theta_{m} \epsilon \left[ {0,1} \right]$, $\theta_{r} \epsilon \left[ {0,1} \right]$.

• ii. $\bar{\Pi }_{r} > \frac{{\left( {a - c} \right)^{2} }}{64}$ implies $\theta_{r} < 5/8$, because $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{m} }} > 0$ and $\frac{{\partial \Pi_{r}^{*} }}{{\partial \theta_{r} }} < 0$. But $\theta_{r} < 5/8$ implies that the first-order condition is never satisfied because it implies $8 - 8\theta_{r} - 3\theta_{m} > 3 - 3\theta_{m} \ge 0$ and so the derivative of the objective function expressed as a function of $\theta_{m}$, the only decision variable, is always positive. Thus, one has a corner solution with $\theta_{m} = 1$

Proof of Proposition 4

$$w^{C} - w^{N} = - 0.0752\left( {a - c} \right),\;q_{v}^{C} - q_{v}^{N} = - 0.1191\left( {a - c} \right),\;q_{r}^{C} - q_{r}^{N} = 0.2382\left( {a - c} \right)$$

$$w^{C} - w^{N} = - 0.0752\left( {a - c} \right),\;p^{C} - p^{N} = - 0.1191\left( {a - c} \right)$$

$$CS^{C} - CS^{N} = 0.0766\left( {a - c} \right)^{2} ,\;V_{m}^{C} - V_{m}^{N} = 0.2758\left( {a - c} \right)^{2} ,\;V_{r}^{C} - V_{r}^{N} = 0.1210\left( {a - c} \right)^{2}$$

$$\Pi_{v}^{C} - \Pi_{v}^{N} = - 0.0851\left( {a - c} \right)^{2} ,\;\Pi_{m}^{C} - \Pi_{m}^{N} = 0.0291\left( {a - c} \right)^{2} ,\;\Pi_{r}^{C} - \Pi_{r}^{N} = 0.0219\left( {a - c} \right)^{2}$$

$$SW^{C} - SW^{N} = 0.0425\left( {a - c} \right)^{2} .$$

Proposition A1

When the degrees of social concern are allowed to be higher than one, then cooperative choice of these degrees results in $\theta_{m}^{H} = \frac{104}{81}$, $\theta_{r}^{H} = \frac{14}{27}$. These choices lead to:

$q_{v}^{H} = \frac{a - c}{4}$, $q_{r}^{H} = \frac{a - c}{2}$, $w^{H} = \frac{31c + 5a}{36}$, $p^{H} = \frac{3c + a}{4}$, $CS^{H} = \frac{{9\left( {a - c} \right)^{2} }}{32}$, $\Pi_{v}^{H} = \frac{{\left( {a - c} \right)^{2} }}{16}$, $\Pi_{m}^{H} = \frac{{5\left( {a - c} \right)^{2} }}{72}$, $\Pi_{r}^{H} = \frac{{\left( {a - c} \right)^{2} }}{18}$, $SW^{H} = \frac{{15\left( {a - c} \right)^{2} }}{32}$, $V_{m}^{H} = \frac{{31\left( {a - c} \right)^{2} }}{72}$, $V_{r}^{H} = \frac{{29\left( {c - a } \right)^{2} }}{144}$

Proof

Replacing $\theta_{m} = \frac{{8 - 8\theta_{r} }}{3}$ from Eq. (11) into $\Pi_{m}^{*}$ and $\Pi_{r}^{*}$, one can write¹⁷ the Nash Product P given in (13) as $P = \frac{{\left( {a - c} \right)^{4} \left( {9\theta_{r} - 4} \right)\left( {16 - 27\theta_{r} } \right)}}{1728}$ which is maximized when $\theta_{r} = 14/27$. Replacing this value back in Eq. (11) yields $\theta_{m} = \frac{104}{81}$. These values can then be inserted into the equilibrium outcomes given in Proposition 1.□

The differences between the equilibrium values with and without cooperation when the degrees of social concern can be higher than unity are:

$$q_{v}^{H} - q_{v}^{N} = - \frac{a - c}{6} < 0,\;q_{r}^{H} - q_{r}^{N} = \frac{a - c}{3} > 0,\;w^{H} - w^{N} = - \frac{a - c}{9} < 0,$$

$$p^{H} - p^{N} = - \frac{a - c}{6} < 0,\;CS^{H} - CS^{N} = \frac{{\left( {a - c} \right)^{2} }}{9} > 0,\;\Pi_{v}^{H} - \Pi_{v}^{N} = - \frac{{\left( {a - c} \right)^{2} }}{9} < 0$$

$$\Pi_{m}^{H} - \Pi_{m}^{N} = \frac{{\left( {a - c} \right)^{2} }}{36} > 0,\;\Pi_{r}^{H} - \Pi_{r}^{N} = \frac{{\left( {a - c} \right)^{2} }}{36} > 0,\;SW^{H} - SW^{N} = \frac{{\left( {a - c} \right)^{2} }}{18} > 0$$

$$V_{m}^{H} - V_{m}^{N} = \frac{{7\left( {a - c} \right)^{2} }}{18} > 0,\;V_{r}^{H} - V_{r}^{N} = \frac{{25\left( {a - c} \right)^{2} }}{144} > 0$$

Proof of Proposition 5

$$w^{**} = \frac{{\left( {c - a} \right)\theta_{r}^{2} + \left( {2a - 6c + 2c\theta_{v} } \right)\theta_{r} - \left( {2a - c} \right)\theta_{m} + \left( {a + c} \right)\theta_{v}^{2} - \left( {4a + 6c} \right)\theta_{v} + 9c + 3a}}{{12 + \left( {2\theta_{v} - 4} \right)\theta_{r} - \theta_{m} + 2\theta_{v}^{2} - 10\theta_{v} }}$$

Using this wholesale price, one obtains the equilibrium profits:

$$\Pi_{m}^{**} = \frac{{\left( {a - c} \right)^{2} \left( {2 + 2\theta_{r} + \theta_{m} - 3\theta_{v} - \theta_{r} \theta_{v} + \theta_{v}^{2} } \right)\left( {3 + 2\theta_{r} - \theta_{r}^{2} - 2\theta_{m} - 4\theta_{v} + \theta_{v}^{2} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} - 10\theta_{v} + 2\theta_{r} \theta_{v} + 2\theta_{v}^{2} } \right)^{2} }}$$

$$\Pi_{r}^{**} = \frac{{\left( {a - c} \right)^{2} \left( {2 + 2\theta_{r} + \theta_{m} - 3\theta_{v} - \theta_{r} \theta_{v} + \theta_{v}^{2} } \right)\left( {2 - 5\theta_{r} + \theta_{m} + \theta_{r}^{2} - 3\theta_{v} + 2\theta_{r} \theta_{v} + \theta_{v}^{2} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} - 10\theta_{v} + 2\theta_{r} \theta_{v} + 2\theta_{v}^{2} } \right)^{2} }}$$

$$\Pi_{v}^{**} = \frac{{\left( {a - c} \right)^{2} \left( {5 - 3\theta_{r} - \theta_{m} + \theta_{r} \theta_{v} - \theta_{v}^{2} } \right)\left( {5 - 3\theta_{r} - \theta_{m} - 7\theta_{v} + 2\theta_{r} \theta_{v} + 2\theta_{v}^{2} } \right)}}{{\left( {12 - 4\theta_{r} - \theta_{m} - 10\theta_{v} + 2\theta_{v}^{2} + 2\theta_{v} \theta_{r} } \right)^{2} }}$$

and using these profit functions one obtains the results in the proposition:

• i. given any $\theta_{m}$ and $\theta_{v}$, it is optimal for the retailer to set $\theta_{r} = 0$ because $\Pi_{r}^{**}$ is strictly decreasing in $\theta_{r}$: $\frac{{\partial \Pi_{r}^{**} }}{{\partial \theta_{r} }} = - \frac{{\left( {a - c} \right)^{2} N}}{{D^{3} }}$ with

$D = 2\theta_{v} \theta_{r} - 4\theta_{r} - \theta_{m} + 2\theta_{v}^{2} - 10\theta_{v} + 12 > 0$ because it is decreasing in $\theta_{v}$ and positive when $\theta_{v} = 1$, and

$$N = \left( \begin{gathered} 2\theta _{v} ^{2} \theta _{r} ^{3} - 8\theta _{v} \theta _{r} ^{3} + 8\theta _{r} ^{3} - 3\theta _{v} \theta _{m} \theta _{r} ^{2} + 6\theta _{m} \theta _{r} ^{2} + 6\theta _{v} ^{3} \theta _{r} ^{2} - 42\theta _{v} ^{2} \theta _{r} ^{2} + \\ 96\theta _{v} \theta _{r} ^{2} - 72\theta _{r} ^{2} + 2\theta _{m} ^{2} \theta _{r} - 4\theta _{v} ^{2} \theta _{m} \theta _{r} + 22\theta _{v} \theta _{m} \theta _{r} - 28\theta _{m} \theta _{r} + 6\theta _{v} ^{4} \theta _{r} - 60\theta _{v} ^{3} \theta _{r} + \\ 222\theta _{v} ^{2} \theta _{r} - 360\theta _{v} \theta _{r} + 216\theta _{r} + 5\theta _{v} \theta _{m} ^{2} - 11\theta _{m} ^{2} + 7\theta _{v} ^{3} \theta _{m} - 30\theta _{v} ^{2} \theta _{m} + 33\theta _{v} \theta _{m} - \\ 2\theta _{m} + 2\theta _{v} ^{5} - 10\theta _{v} ^{4} + 6\theta _{v} ^{3} + 42\theta _{v} ^{2} - 80\theta _{v} + 40 \\ \end{gathered} \right) > 0\,{\text{because}}:$$

• ii. N is strictly increasing in $\theta_{r}$:

$\frac{\partial N}{{\partial \theta_{r} }} = 2\left( {3\theta_{v}^{2} \theta_{r}^{2} - 12\theta_{v} \theta_{r}^{2} + 12\theta_{r}^{2} - 3\theta_{v} \theta_{m} \theta_{r} + 6\theta_{m} \theta_{r} + 6\theta_{v}^{3} \theta_{r} - 42\theta_{v}^{2} \theta_{r} + 96\theta_{v} \theta_{r} - 72\theta_{r} + \theta_{m}^{2} - 2\theta_{v}^{2} \theta_{m} + 11\theta_{v} \theta_{m} - 14\theta_{m} + 3\theta_{v}^{4} - 30\theta_{v}^{3} + 111\theta_{v}^{2} - 180\theta_{v} + 108} \right) > 0$ because it is decreasing in $\theta_{m}$ ($\frac{{\partial^{2} N}}{{\partial \theta_{m} \partial \theta_{r} }} = - 6\theta_{v} \theta_{r} + 12\theta_{r} + 4\theta_{m} - 4\theta_{v}^{2} + 22\theta_{v} - 28 < 0$ since it increases in $\theta_{v}$ and it is negative even when $\theta_{v} = 1$) and it is positive even when $\theta_{m} = 1$ (When $\theta_{m} = 1$, $\frac{\partial N}{{\partial \theta_{r} }} = 2\left( {3\theta_{v}^{2} \theta_{r}^{2} - 12\theta_{v} \theta_{r}^{2} + 12\theta_{r}^{2} + 6\theta_{v}^{3} \theta_{r} - 42\theta_{v}^{2} \theta_{r} + 93\theta_{v} \theta_{r} - 66\theta_{r} + 3\theta_{v}^{4} - 30\theta_{v}^{3} + 109\theta_{v}^{2} - 169\theta_{v} + 95} \right) > 0$ because it is decreasing in $\theta_{r}$ ($\frac{{\partial^{2} N}}{{\partial \theta_{r}^{2} }} = 6\left( {\theta_{v} - 2} \right)\left( {2\theta_{v} \theta_{r} - 4\theta_{r} + 2\theta_{v}^{2} - 10\theta_{v} + 11} \right) < 0 )$ and it is positive even when $\theta_{r} = 1$: when $\theta_{m} = 1$ and $\theta_{r} = 1$,$\frac{\partial N}{{\partial \theta_{r} }} = 6\theta_{v}^{4} - 48\theta_{v}^{3} + 140\theta_{v}^{2} - 176\theta_{v} + 82 > 0$)

• iii. $N > 0$ when $\theta_{r} = 0$, because then: $N = \left( {\theta_{m} + \theta_{v}^{2} - 3\theta_{v} + 2} \right)\left( {5\theta_{v} \theta_{m} - 11\theta_{m} + 2\theta_{v}^{3} - 4\theta_{v}^{2} - 10\theta_{v} + 20} \right)$ is the product of two positive terms, since each term is decreasing in $\theta_{v}$ and positive when $\theta_{v} = 1$.

• iv. given any $\theta_{r}$ and $\theta_{v}$, it is optimal for the manufacturer to set $\theta_{m} = 0$ because $\Pi_{m}^{**}$ is decreasing in $\theta_{m}$:$\frac{{\partial \Pi_{m}^{**} }}{{\partial \theta_{m} }}$ = $- \frac{{\left( {a - c} \right)^{2} \theta_{m} \left( {\theta_{r} + 3\theta_{v} - 7} \right)^{2} }}{{\left( {2\theta_{v} \theta_{r} - 4\theta_{r} - \theta_{m} + 2\theta_{v}^{2} - 10\theta_{v} + 12} \right)^{3} }}$ < 0 for $\theta_{m} > 0$ (To see that the denominator is positive, notice that it is positive when $\theta_{v} = 1$ and it is decreasing in $\theta_{v}$)

• v. given $\theta_{m} = \theta_{r} = 0$, $\Pi_{v}^{**}$ is concave, and the first-order condition $\frac{{\partial \Pi_{v}^{**} }}{{\partial \theta_{v} }} = \frac{{\left( {a - c} \right)^{2} \left( {13\theta_{v}^{4} - 93\theta_{v}^{3} + 231\theta_{v}^{2} - 215\theta_{v} + 40} \right)}}{{4\left( {\theta_{v} - 3} \right)^{3} \left( {\theta_{v} - 2} \right)^{3} }} = 0$ yields $\theta_{v}^{NV} = 0.2439$.

Using the equilibrium degrees of CSR, one can obtain the equilibrium values for the rest of the variables. The inequalities in the proposition hold because:

$$q_{v}^{NV} - q_{v}^{N} = 0.0937\left( {a - c} \right),\;q_{r}^{NV} - q_{r}^{N} = - 0.0295\left( {a - c} \right), \;\Pi_{v}^{NV} - \Pi_{v}^{N} = 0.0063\left( {a - c} \right)^{2}$$

$$\Pi_{m}^{NV} - \Pi_{m}^{N} = - 0.0121\left( {a - c} \right)^{2} ,\;\Pi_{r}^{NV} - \Pi_{r}^{N} = - 0.0090\left( {a - c} \right)^{2}$$

$$w^{NV} - w^{N} = - 0.0347\left( {a - c} \right),\;p^{NV} - p^{N} = - 0.0642\left( {a - c} \right)$$

$$CS^{NV} - CS^{N} = 0.0395\left( {a - c} \right)^{2} ,\;SW^{NV} - SW^{N} = 0.0247\left( {a - c} \right)^{2} .$$

Proof of Proposition 6

When $\theta_{v} = 0$ and $\theta_{m} = 1$, the Nash Product $P^{V}$ can be written as:$P^{V} = - \frac{{\left( {a - c} \right)^{4} AB}}{C}$, with

$A = \left( {143825692\theta_{r}^{3} - 37939774\theta_{r}^{2} - 762154664\theta_{r} + 41182819} \right)$, $B = \left( {1820162036\theta_{r}^{3} - 6644541894\theta_{r}^{2} - 6683867938\theta_{r} + 6118794979} \right)$, $C = 65446516094957228\left( {4\theta_{r} - 11} \right)^{4}$ and it is maximized at the only point where its derivative vanishes, $\theta_{r} = 0.4008$.

Reciprocally, given that $\theta_{m} = 1$ and $\theta_{r} = 0.4008$, it can be checked that $\frac{{\partial \prod_{v}^{**} }}{{\partial \theta_{v} }} = - 0.0102 < 0$ when $\theta_{v} = 0$ and that $\frac{{\partial \prod_{v}^{**} }}{{\partial \theta_{v} }}$ has no roots in the interval $\theta_{v} \in \left[ {0,1} \right]$ and thus, it is optimal for the vertically integrated firm to set $\theta_{v} = 0$.

The inequalities in the proposition hold because:

$$q_{v}^{CV} - q_{v}^{NV} = - 0.2127\left( {a - c} \right),\;q_{r}^{CV} - q_{r}^{NV} = 0.2674\left( {a - c} \right),$$

$$\Pi_{v}^{CV} - \Pi_{v}^{NV} = - 0.0913\left( {a - c} \right)^{2} ,\;\Pi_{m}^{CV} - \Pi_{m}^{NV} = 0.0411\left( {a - c} \right)^{2} ,$$

$$\Pi_{r}^{CV} - \Pi_{r}^{NV} = 0.0310\left( {a - c} \right)^{2} ,\;w^{CV} - w^{NV} = - 0.0406\left( {a - c} \right)$$

$$p^{CV} - p^{NV} = - 0.0547\left( {a - c} \right),\;CS^{CV} - CS^{NV} = 0.0369\left( {a - c} \right)^{2}$$

$$SW^{CV} - SW^{NV} = 0.0178\left( {a - c} \right)^{2} .$$