In this section, we abstract from costs of CSR (Z = 0) and solve the game by backward induction for its subgame perfect equilibria (SPE). We focus on potential SPE in which 𝜃i ∈ [0,1] for i ∈{1,2}, i.e., no firm puts more weight on consumer surplus than on profits.
At the second stage of the game, the first-order conditions ∂Vi/∂qi = 0 imply the reaction functions
$$ \begin{array}{@{}rcl@{}} q_{1}(q_{2})&=& \frac{1-(1-\theta_{1})q_{2}}{2-\theta_{1}},\\ q_{2}(q_{1})&=& \frac{1-c-(1-\theta_{2})q_{1}}{2-\theta_{2}}, \end{array} $$
and thus, the second stage quantity choices as functions of the CSR levels:
$$ \begin{array}{@{}rcl@{}} q_{1}&=& \frac{1-\theta_{2}+\theta_{1}+c(1-\theta_{1})}{3-\theta_{1}-\theta_{2}}, \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} q_{2}&=& \frac{1-\theta_{1}+\theta_{2}-c(2-\theta_{1})}{3-\theta_{1}-\theta_{2}}. \end{array} $$
(2)
At the first stage, the firms anticipate these choices and maximize their respective profits
$$ \begin{array}{@{}rcl@{}} \pi_{1}&=& \frac{(1-\theta_{2}+c-\theta_{1})(1-\theta_{2}+c+(1-c)\theta_{1})}{(3-\theta_{1}-\theta_{2})^{2}}, \end{array} $$
(3)
$$ \begin{array}{@{}rcl@{}} \pi_{2}&=& \frac{(1-2c-(1-c)\theta_{1}-(1-c)\theta_{2})(1-2c-(1-c)\theta_{1}+\theta_{2})}{(3-\theta_{1}-\theta_{2})^{2}}, \end{array} $$
(4)
by the choice of their CSR levels. The first-order conditions ∂πi/∂𝜃i = 0 imply
$$ \begin{array}{@{}rcl@{}} \theta_{1}(\theta_{2})&=& \frac{(1-\theta_{2})^{2}+(1-\theta_{2})c}{3-\theta_{2}-c}, \end{array} $$
(5)
$$ \begin{array}{@{}rcl@{}} \theta_{2}(\theta_{1})&=& \frac{(1-\theta_{1})^{2}-(1-\theta_{1})(2-\theta_{1})c}{3-\theta_{1}-(2-\theta_{1})c}. \end{array} $$
(6)
It is straightforward to show that 0 ≤ 𝜃1(𝜃2) < 1 for all 0 < c < 1 and all 𝜃2 ∈ [0,1] as well as 𝜃2(𝜃1) < 1 for all 0 < c < 1 and all 𝜃1 ∈ [0,1]. Moreover, 0 < 𝜃2(𝜃1) for 0 < c < 1 and 𝜃1 ∈ [0,1] if and only if
$$ \theta_{1} < \frac{1-2c}{1-c}. $$
(7)
Consequently, for all 0 < c < 1 and 𝜃1,𝜃2 ∈ [0,1], the first stage best responses of the firms are given by the reaction functions r1(𝜃2) := 𝜃1(𝜃2) and \(r_{2}(\theta _{1}):=\max \limits \{\theta _{2}(\theta _{1}),0\}\), where 𝜃1(𝜃2) and 𝜃2(𝜃1) are defined by Eqs. 5 and 6, respectively.
Figure 1 illustrates the equilibrium CSR levels depicting the reaction functions r1 and r2 for the cost differentials c = 0, c = 1/4, and c = 1/3, respectively. Lemma 1 in the Appendix provides the comparative statics properties of the reaction functions. In particular, it shows that an increase in c increases r1 and decreases r2 wherever positive. For c = 1/3, we have r1(0) = 𝜃1(0) = 1/2 and r2(1/2) = 𝜃2(1/2) = 0 according to Eqs. 5 and 6, and thus, r1 and r2 intersect at (𝜃1,𝜃2) = (1/2,0). Lemma 1 then implies that, for any c ≥ 1/3, we always have 𝜃2 = 0 where r1 and r2 intersect. If 𝜃2 = 0, however, Eq. 5 implies the best response 𝜃1 = (1 + c)/(3 − c), and thus, q2 < 0 for all c > 1/3 by Eq. 2, i.e., the non-negativity constraint on the quantity of firm 2 will be violated. This proves
Proposition 1
If c ≥ 1/3, the less efficient firm will leave the market.
Notice that, without the strategic use of CSR (𝜃1 = 𝜃2 = 0), the threshold marginal cost above which the less efficient firm leaves the market is larger (c = 1/2). Strategic CSR may thus increase the market power of more efficient firms and foster market consolidation as well as the adaption of new technologies.
For smaller marginal costs, the intersection of the reaction functions r1 and r2 constitutes a SPE. We asterisk the corresponding equilibrium values.
Proposition 2
For all c ∈ (0,1/3), the two-stage game with strategic CSR and Cournot competition between two asymmetric firms has a SPE in which
-
(a)
The firm with the lower marginal costs chooses a higher CSR level, produces more output, and earns higher profits, i.e., \(\theta ^{*}_{1} > \theta ^{*}_{2} > 0\), \(q^{*}_{1}>q^{*}_{2} > 0\), and \(\pi ^{*}_{1}>\pi ^{*}_{2} > 0\) for all 0 < c < 1/3.
-
(b)
An increase in the cost differential increases the CSR level of the advantaged firm and decreases the CSR level of the disadvantaged firm, i.e., \(d\theta ^{*}_{1}/dc > 0\) and \(d\theta ^{*}_{2}/dc < 0\) for all c ∈ (0,1/3).
The proof can be found in the Appendix. For the intuition behind these results, note that in this model a higher CSR level (i.e., more weight on consumer surplus) represents a strategic commitment to a higher output. Since the more efficient firm faces lower costs of production, increasing its output is less costly for this firm. Therefore, it has stronger incentives to use this commitment device.Footnote 4 The model thus applies particularly well to environments in which CSR measures aim at a high market coverage, e.g., in the provision of pharmaceuticals in developing countries.
Proposition 2 is in line with several findings in the recent literature on strategic delegation. Straume (2006) and Fanti and Meccheri (2017) also find that the more efficient firm chooses a higher weight on the additional objective if managers maximize a weighted combination of profits and sales. For revenues as additional objective, Delbono et al. (2016) show that the more efficient firm earns higher equilibrium profits.Footnote 5 Moreover, Colombo (2019) finds that for sufficiently high cost differences the more efficient firm may even earn higher profits in the delegation equilibrium than if both firms abstained from delegation.