Strategic corporate social responsibility, imperfect competition, and market concentration

Abstract

We examine the strategic use of corporate social responsibility (CSR) in imperfectly competitive markets. Before firms decide upon supply, they choose a level of CSR which determines the weight they put on consumer surplus in their objective function. First, we consider Cournot competition and show that the endogenous level of CSR is positive for any given number of firms. However, positive CSR levels imply smaller equilibrium profits. Second, we find that an incumbent monopolist can use CSR as an entry deterrent. Both results indicate that CSR may increase market concentration. Finally, we show that CSR levels decrease as the degree of product heterogeneity increases in Cournot competition and are zero in Bertrand Competition.

Appendices

Appendix

A Proof of Proposition 3 and Corollary 5

Given the inverse demands (14), consumer surplus can be written as follows:

$$\begin{aligned} CS = \frac{1}{2} \cdot \left[ \gamma (q_1+q_2)^2 + (1-\gamma ) (q_1^2+q_2^2)\right] = q_1^2 + q_2^2 + 2 \gamma q_1q_2. \end{aligned}$$

(15)

In the second stage of the game, firm $i \in \{1,2\}$ chooses $q_i$ in order to maximize $V_i=\pi _i + \theta _i CS$. For $i,j \in \{1,2\}$, $i \not = j$, the first-order conditions $\frac{\partial V_i}{\partial q_i} = 0$ imply the reaction functions¹⁸

$$\begin{aligned} q_i = \frac{1-\gamma (1-\theta _i)q_j}{2-\theta _i}. \end{aligned}$$

(16)

Solving the system of equations (16) yields

$$\begin{aligned} q_i = \frac{(1-\gamma )+(1-\theta _j)+\gamma \theta _i}{2+(2-\gamma ^2)(1-\theta _j) -[1+(1-\gamma ^2)(1-\theta _j)]\theta _i} \end{aligned}$$

(17)

for $i,j \in \{1,2\}$, $i \not = j$. We now use (14) and (17) to compute the corresponding prices

$$\begin{aligned} p_i = \frac{(1-\gamma )+(1-\theta _j)-[1+(1-\gamma ^2)(1-\theta _j)]\theta _i}{2+(2-\gamma ^2)(1-\theta _j)-[1+(1-\gamma ^2)(1-\theta _j)]\theta _i}. \end{aligned}$$

(18)

In the first stage of the game, firm $i \in \{1,2\}$ chooses $\theta _i$ in order to maximize $\pi _i=p_iq_i$. Using (17) and (18), the first-order conditions $\frac{\partial \pi _i}{\partial \theta _i} = 0$ imply the reaction functions

$$\begin{aligned} \theta _i = \frac{\gamma ^2(1-\theta _j)(2-\theta _j-\gamma )}{[2-\theta _j-(1-\theta _j) \gamma ^2](2-\theta _j+\gamma )} \end{aligned}$$

(19)

for $i,j \in \{1,2\}$, $i \not = j$. Solving the system of equations (19) for $i,j \in \{1,2\}$, $i \not = j$ yields a unique feasible solution:¹⁹

$$\begin{aligned} \theta _i = \frac{2(1+\gamma )+\gamma ^2-\sqrt{[2(1+\gamma )]^2+\gamma ^4}}{2(1+\gamma )} =: \theta _C^*(\gamma ) > 0. \end{aligned}$$

(20)

Using (20), it is straightforward to show that $\frac{d\theta _C^*(\gamma )}{d\gamma } > 0$ for $\gamma >0$, and $\frac{d\theta _C^*(\gamma )}{d\gamma } < 0$ for $\gamma <0$. This proves Proposition 3.

Finally, to prove Corollary 5, we use (20) and compute firm i’s equilibrium profit

$$\begin{aligned} \pi _i^* = \frac{2\left( \sqrt{[2(1+\gamma )]^2+\gamma ^4}-\gamma ^2-2\gamma )\right) }{\left( \sqrt{[2(1+\gamma )]^2+\gamma ^4}-\gamma ^2+2\right) ^2} \end{aligned}$$

whereas the regular Cournot profit without strategic CSR ($\theta _1=\theta _2=0$) equals $\pi _i^C=\frac{1}{(2+\gamma )^2}$. A comparison shows that, in the relevant range, $\pi _i^* > \pi _i^C$ if and only if $-1< \gamma <0$.

B Proof of Proposition 4

The inverse demand functions (14) imply the following direct demands

$$\begin{aligned} q_i = \frac{1-\gamma - p_i + \gamma p_j}{1-\gamma ^2} \end{aligned}$$

(21)

for $i,j \in \{1,2\}$, $i\ne j$. Using (15) and (21), consumer surplus can be expressed in terms of prices:

$$\begin{aligned} CS = \frac{2(1-\gamma )(1-p_1-p_2) - 2\gamma p_1p_2 + p_1^2 +p_2^2}{2(1-\gamma ^2)}. \end{aligned}$$

In the second stage of the game, firm $i \in \{1,2\}$ chooses $p_i$ in order to maximize $V_i=\pi _i + \theta _i CS$. For $i,j \in \{1,2\}$, $i \not = j$, the first-order conditions $\frac{\partial V_i}{\partial p_i} = 0$ imply the reaction functions²⁰

$$\begin{aligned} p_i = \frac{(1-\theta _i)(1 - \gamma + \gamma p_j)}{2-\theta _i}. \end{aligned}$$

(22)

Solving the system of equations (22) yields

$$\begin{aligned} p_i =\frac{(1-\gamma )(1-\theta _i)[2-\theta _j+\gamma (1-\theta _j)]}{(2-\theta _i)(2-\theta _j)-(1-\theta _i)(1-\theta _j)\gamma ^2} \end{aligned}$$

(23)

for $i,j \in \{1,2\}$, $i \not = j$. We now use (21) and (23) to compute the corresponding profits

$$\begin{aligned} \pi _i = p_iq_i = \frac{(1-\gamma )(1-\theta _i)[2-\theta _j+\gamma (1-\theta _j)]^2}{(1+\gamma ) [(2-\theta _i)(2-\theta _j)-(1-\theta _i)(1-\theta _j)\gamma ^2]^2}. \end{aligned}$$

(24)

In the first stage of the game, firm $i \in \{1,2\}$ chooses $\theta _i$ in order to maximize $\pi$. Using (24), it is straightforward to show that for $i,j \in \{1,2\}$, $i \not = j$

$$\begin{aligned} \frac{\partial \pi _i}{\partial \theta _i} = - \frac{(1-\gamma )[(2-\theta _j)\theta _i+(1-\theta _i)(1-\theta _j)\gamma ^2] [2-\theta _j+\gamma (1-\theta _j)]^2}{(1+\gamma )[(2-\theta _i)(2-\theta _j) -(1-\theta _i)(1-\theta _j)\gamma ^2]^3} < 0 \end{aligned}$$

(25)

for all $0<|\gamma |<1$. Consequently, each firm $i \in \{1,2\}$ will choose the lowest CSR level possible, i.e., $\theta _i=0=:\theta ^*_B(\gamma )$.

C Proof of Proposition 5

At the second stage, the objective functions of the two firms are given by

$$\begin{aligned} V_1&=p(q_1+q_2)q_1+\theta _1CS(q_1+q_2),\\ V_2&=p(q_1+q_2)q_2+\theta _2CS(q_1+q_2). \end{aligned}$$

The maximizing quantities satisfy the first-order conditions

$$\begin{aligned} \dfrac{\partial V_1}{\partial q_1}&=p'(q_1+q_2)q_1+p(q_1+q_2)+\theta _1CS'(q_1+q_2)=0, \end{aligned}$$

(26)

$$\begin{aligned} \dfrac{\partial V_2}{\partial q_2}&=p'(q_1+q_2)q_2+p(q_1+q_2)+\theta _2CS'(q_1+q_2)=0, \end{aligned}$$

(27)

as well as the second order conditions $\dfrac{\partial ^2 V_i}{\partial q_i^2} < 0$. Using $CS(q_1+q_2)=\int _0^{q_1+q_2}[p(q)-p(q_1+q_2)]dq$ and thus $CS'(q_1+q_2)=-(q_1+q_2)p'(q_1+q_2)$, we rewrite Eqs. (26) and (27):

$$\begin{aligned}{}[(1-\theta _1)q_1-\theta _1q_2]p'(q_1+q_2)+p(q_1+q_2)&=0, \end{aligned}$$

(28)

$$\begin{aligned} p'(q_1+q_2)+p(q_1+q_2)&=0. \end{aligned}$$

(29)

Denote the left-hand side of Eqs. (28) and (29) by $F_1(\theta _1,\theta _2,q_1,q_2)$ and $F_2(\theta _1,\theta _2, q_1,q_2)$, respectively.

First, we compute the sign of $dq_1/d\theta _1$. Treating $\theta _2$ as fixed and applying the implicit function theorem yields

$$\begin{aligned} \dfrac{dq_1}{d\theta _1}= \dfrac{-\dfrac{\partial F_1}{\partial \theta _1}\dfrac{\partial F_2}{\partial q_2}}{\dfrac{\partial F_1}{\partial q_1}\dfrac{\partial F_2}{\partial q_2}-\dfrac{\partial F_1}{\partial q_2}\dfrac{\partial F_2}{\partial q_1}}. \end{aligned}$$

(30)

Notice that $\partial F_1/\partial \theta _1=-(q_1+q_2)p'(q_1+q_2)>0$. Moreover, $\partial F_i/\partial q_i=\partial ^2V_i/\partial q_i^2<0$ for $i \in \{1,2\}$, as implied by the second order conditions on the solution of the maximization problem. Thus the numerator of (30) is positive. Taking the respective derivatives, using the symmetry $\theta _1=\theta _2=\theta$ in equilibrium, writing $q_1+q_2=Q$, and simplifying terms, we compute the denominator

$$\begin{aligned} \dfrac{\partial F_1}{\partial q_1}\dfrac{\partial F_2}{\partial q_2}-\dfrac{\partial F_1}{\partial q_2}\dfrac{\partial F_2}{\partial q_1}=p'(Q)[(3-2\theta )p'(Q)+(1-2\theta )Qp''(Q)]. \end{aligned}$$

(31)

Due to $p'(Q)<0$, an increase in a firm’s CSR level will increase its output, i.e., $dq_1/d\theta _1>0$, if and only if

$$\begin{aligned} (3-2\theta )p'(Q)+(1-2\theta )Qp''(Q)<0. \end{aligned}$$

(32)

Next, we compute the sign of $d q_1/d \theta _2$. Treating $\theta _1$ as fixed now and applying the implicit function theorem yields

$$\begin{aligned} \dfrac{dq_1}{d\theta _2}= \dfrac{\dfrac{\partial F_2}{\partial \theta _2}\dfrac{\partial F_1}{\partial q_2}}{\dfrac{\partial F_1}{\partial q_1}\dfrac{\partial F_2}{\partial q_2}-\dfrac{\partial F_1}{\partial q_2}\dfrac{\partial F_2}{\partial q_1}}. \end{aligned}$$

(33)

The denominator of (33) equals that of (30). Again due to the symmetry in equilibrium, the numerator simplifies to

$$\begin{aligned} \dfrac{\partial F_2}{\partial \theta _2}\dfrac{\partial F_1}{\partial q_2}=-Qp'(Q)[(1-\theta )p'(Q)+(1-2\theta )Qp''(Q)]. \end{aligned}$$

(34)

Due to $-Qp'(Q)>0$, an increase in a firm’s CSR level will decrease its rival’s output, $dq_1/d\theta _2<0$, if and only if

$$\begin{aligned} (3-2\theta )p'(Q)+(1-2\theta )Qp''(Q) \end{aligned}$$

(35)

and

$$\begin{aligned} (1-\theta )p'(Q)+(1-2\theta )Qp''(Q) \end{aligned}$$

(36)

are either both positive or both negative, such that either the numerator of (33) is positive and the denominator of (33) is negative or vice versa. Note that (35) < (36) and thus we obtain the condition that $dq_1/d\theta _2<0$ if and only if either

$$\begin{aligned} (3-2\theta )p'(Q)+(1-2\theta )Qp''(Q)>0 \end{aligned}$$

(37)

or

$$\begin{aligned} (1-\theta )p'(Q)+(1-2\theta )Qp''(Q)<0. \end{aligned}$$

(38)

Finally, notice that (37) and (32) can never both be fulfilled at the same time and that (38) already implies (32).