Corporate social responsibility in a game-theoretic context

Using a simple Cournot duopoly model with differentiated products, this work studies the firms’ strategic choice of whether to adopt Corporate Social Responsibility (CSR) rules. The proposed game-theoretic approach shows that, depending on the degree of product differentiation and firms’ social concern, different equilibria arise: (1) all firms in the industry follow CSR rules, (2) all firms are profit-maximising (PM), (3) asymmetric equilibria are present (one CSR firm, one PM firm), and (4) multiple symmetric equilibria are present. Under Bertrand competition, universal PM is the unique equilibrium in which the social welfare is the least desirable: this implies that, in contrast to the conventional wisdom, Bertrand may appear as welfare dominated by Cournot. This work can help to explain the widely observed phenomenon, in the real world, of different industries in which firms’ CSR behaviours are more or less commonly widespread.

JEL Classification D21 Á L13 Á L14 Á M14 1 Introduction between morally motivated CSR and CSR induced by social pressure, and thus they tend to target their campaign against morally managed firms, because they are more vulnerable to the campaign than the self-interested firms. On the other hand, from the view of Benabou and Tirole (2010), CSR is a pro-social behaviour resulting from several interacting motivations, from pure altruism to material motivation, social and self-esteem concerns. Moreover, as argued by Reinhardt and Stavins (2010), among others, CSR may be a strategy like any other form of product differentiation if there are customers-identifiable by firms-willing to pay for ethical goods which may be profitable: the opportunity arises, for instance, because of the presence of asymmetric information, economies of scale, intellectual-property protection, and the successful defense of the resultant niche against imitators, according to basic economics. More interestingly, García-Gallego and Georgantzís (2009) consider the role of the ''pressure'' exerted by heterogeneous consumers in a vertically differentiated duopoly with firms' strategic choice of their degree of corporate social responsibility. Those authors study the effects of an increase in consumers' willingness to pay for products sold by socially responsible manufacturers. The consumers' preferences towards CSR behaviours increase the demand; however, a trade-off exists because the adoption of CSR rules implies a quadratic cost. They find that shifting towards socially responsible consumer preferences is, in most cases, welfare-improving. Conversely, it can be welfare-reducing when such an increase changes the market structure. Therefore, the above-mentioned literature, and in particular García-Gallego and Georgantzís (2009), examine the demand-side (the consumers' or NGOs) CSR preferences while the current work focuses on the supply-side (the firms') CSR preferences in the sense that supplied quantities are chosen by firms to maximise also consumers' interests. Moreover, in a recent paper to appear in this journal, Fanti and Buccella (2016) investigate the effect that the incumbent's CSR adoption can have on a potential entrant decision. On the other hand, the appearance of universal adoption of CSR behaviours in an industry, leading to a ''prisoner's dilemma'' situation, has been identified by Kopel and Brand (2012), which seems to be the unique paper dealing with this strategical analysis, to the best of the authors' knowledge.
In the current paper, we examine if a correct game-theoretic approach to this issue may explain the above-mentioned stylised facts; that is, the emergence of different equilibria in different industries (i.e. all firm adopt CSR rules, all firms do not follow CSR rules or asymmetric equilibria). Making use of a simple Cournot duopoly model with identical firms and differentiated products, we show that the strategic nature of the adoption of CSR rules may explain the occurrence in the equilibrium of situations in which (1) both firms follows CSR rules, (2) both firms do not follow CSR rules, (3) asymmetric equilibria in which one firm adopts CSR while the rival does not, and (4) multiple symmetric equilibria in which both firms may decide whether to adopt CSR, depending on the degree of product differentiation.
On the other hand, in the case of a Bertrand duopoly (analysed in the ''Appendix''), we find that profit maximization is the dominant strategy for firms. In other words, unilateral deviation from the profit maximisation rule is always unprofitable for firms. The rationale for this result is rather intuitive: given that under Bertrand the nature of competition is in strategic complements, the unilateral adoption of CSR behaviours (taking care of consumers' surplus) leads the firm to produce a larger amount of output than the rival, with a negative impact on the firm's price and profits. However, in the presence of strategic complements with complement goods, if the social concern is not too strong, there is an area in the parameter space in which the overall adoption of CSR rules payoff-dominates the profit maximisation equilibrium; that is, firms are cast into a classical prisoner's dilemma game.
We find that the adoption of CSR rules always implies a more intensive product market competition, under both the Cournot and Bertrand conjectures. However, the role of the product differentiation is crucial. Under Cournot competition, with substitute goods, the reaction functions have a standard downward sloping shape: goods are strategic substitutes. In the presence of complement goods, the reaction functions become upward sloping, and goods behave as when strategic complements. In both cases, the increasing weight attached to the stakeholder-consumers' interests leads the socially concerned firms to an aggressive market behaviour via output expansion, which is stronger with substitute than with complement products. On the other hand, under Bertrand competition, with substitute goods, the reaction functions are upward sloping (strategic complements), while with complement goods, the reaction functions become downward sloping, and goods behave as when strategic substitutes. Also in these cases, an increase of the social concern level yields aggressive product market behaviour of the CSR firm through a price reduction, which is more intense with complements than substitute products. Thus, these results may be of great interest for the current policy debate on firms' social concern.
Concerning social welfare, under Cournot competition, the firms' strategic interaction leads to the most desirable welfare outcome 1 , when the goods are sufficiently substitutes and sufficiently complements and, in particular, if the level of social concerns are adequately low. However, while the former case is in contrast to firms' interests, in the latter case the firms' and social interests coincide. Therefore, when the emergent equilibrium is the best achievable and the interests of all the agents (both firms, consumers and social welfare as a whole) coincide, the best achievable outcome can be also defined as the Pareto-superior outcome. Nonetheless, there is a wide area of the relevant parameter space characterised by low-medium degrees of goods' substitutability/complementarity in which universal profit maximisation arises as endogenous equilibrium, driving the economy to the third best achievable social welfare outcome.
On the other hand, in the case of Bertrand competition, product market interactions always determine the rise of universal profit maximisation as endogenous equilibrium, and this outcome generates the third best achievable outcome in terms of welfare for a wide set of the parameter space.
The remainder of the paper is organized as follows. Section 2 presents the basic ingredients of the duopoly model, with firms deciding whether to adopt CSR rules. Section 3 describes the analysis of the game equilibria. Section 4 discusses the implications in terms of social welfare. Finally, Sect. 5 concludes with a brief discussion of the main results and an outline of the future research directions.

The model
We assume that firms produce heterogeneous goods. As usual, the standard linear inverse market demand is as follows where p denotes price, q i and q j are the firms' output levels for i, j = 1, 2 and i = j and À1\c 1. Note that, if c\0ð [ 0Þ, then products are complements (substitutes) (if c ¼ 0, goods are completely differentiated and each firm is a monopolist for its own product). For tractability, we assume that both firms have zero production costs. 2

Both firms follow CSR rules
Following the recent established literature (e.g. Goering 2007Goering , 2008Lambertini and Tampieri 2010), we consider in our model that all social concerns can be interpreted as part of consumer surplus; thus, the feature of a CSR firm is to be sensitive to it. Therefore, we suppose that the firm, in its objective, wishes to maximize profits plus a fraction of the market consumer surplus k, the firm's social concern or care for consumer outcomes in the market. Therefore, the stakeholders' participation in governance applies regarding market decisions. That is, stakeholders exercise ''pressure'' with regard to the firm's objective, while private owners make the company's strategic choices. As a consequence, in the current model, the firms' strategic choice is whether to adopt CSR rules. On the other hand, the stakeholders, once involved in the firm's governance, fix the level of social engagement. Therefore, when a firm decides to be socially responsible, the level of social concern k is taken as exogenously given by the private owners. In other words, the owners of the firms can make a binary choice: either k ¼ 0 in the case of profit maximisation or a given positive k in the case of CSR rules adoption (where k is exogenously given, for instance, by the customary toughness of the stakeholders incorporated in the firm's objective function). This is coherent with the results of the empirical study of Spitzeck and Hansen (2010), who find that the stakeholder engagement mechanism is mainly confined to ''dialogue and issues'' advisory, while the structural strategic choices (i.e. different from the output choices) of the company are outside their sphere. Moreover, given that the firms compete for the same clients in the industry, it can be reasonably assumed that the level of social engagement stakeholders request is identical for both companies. It follows that the objective function of the firm following CSR rules can be specified as a simple parameterised combination of profits and consumer surplus. Profits and consumer surplus are, respectively: Thus, the CSR objective function (W) is where k 2 ½0; 1 denotes the weight that CSR firm assigns to consumer surplus. 3 The analysis is carried as usual through the maximisation of (4) with respect to the quantity, which leads to the two reaction functions for i, j = 1, 2, and i = j. Solving the system in (5), one gets the equilibrium output Substituting (6) backwards, we obtain the equilibrium profits 4,5 Note that the satisfaction of the non-negativity constraint on profits requires ultimately 6 that k 1 1þc k T (that is, the firm's interest for the consumer's welfare has not to be too high, especially if there is a fierce product competition). This upper bound, however, is not relevant in the case of c\0: in fact, when goods are complements, profits are positive for any k 2 ½0; 1. We assume that this inequality (which is also the most stringent) holds true for the rest of the paper.

2.2
The asymmetric case: one firm follows CSR rules, while the other is profit-maximising Let us consider the case that firm i maximises the social objective function (CSR), while the rival j maximises profits (PM). Hence, firm i maximises (4) with respect to the quantity, obtaining the usual reaction function (5). On the other hand, firm j maximises its profit function (2), and thus the FOCs for firm j determine the following reaction function: Solving the system composed by (5) and (8) for the firms' quantities, we obtain firm i's output and firm j's output, respectively: By substituting (9) and (10) in (2), we find the following equilibrium profits of the mixed case: Now we are in a position to derive which rule (CSR or profit-maximising) endogenously emerges as SPNE for both firms.

The role of CSR on firms' reaction functions
At this point, in the spirit of Bulow et al. (1985), a short digression may provide further details on the strategic effects that the adoption of CSR rules has on firms' reaction functions. 7 7 As known, according to the pioneering work of Bulow et al. (1985), decisions are strategic substitutes under Cournot competition and strategic complements under Bertrand competition. Formally, let a i denote firm i's action such as quantity or price decisions, and let p i ða i ; a j Þ be the profit function of firm i when it takes action a i and the rival, firm j, takes action a j . Then, strategic substitutes are defined as It is easy to see from the reaction functions (5) and (8) that a socially concerned firm has an incentive to offer a level of output larger than a profit-maximising firm. Irrespective of the value of c, as the intensity of the social concerns (k) rises, the output of the CSR firm expands as well. In fact, the differentiation of (5) and (8) shows that oq i ok [ 0, no matter goods are substitutes (c [ 0) or complements (c\0). Figure 1 depicts this result. The left box shows the case of substitute goods: the reaction functions are downward sloping; i.e. we are in the presence of goods which are strategic substitutes. On the other hand, the right box depicts the case of complement goods: the reaction functions are upward sloping and the goods behave as when the strategies are complements. In both cases, the adoption of CSR rules and the increasing weight attached to the stakeholder-consumers' interests lead the firms to demonstrate more aggressive market behaviour through output expansion, even though the relative aggressiveness is stronger when goods are substitutes than complements (the output expansion for a given level of k is larger when c [ 0 than when c\0). As a consequence, the effect of following CSR behaviours is that of accelerating product market competition, in the vein of the delegation games' logic with managers (e.g. Fershtman and Judd 1987).

The game analysis
Let us define the following profit differentials: For instance, CSR represents a commitment on the decision variables which plays a different strategic role depending on the competition mode, as in the case of forward market: with regard to Cournot competition, see Allaz and Vila (1993), while for Bertrand, see Mahenc and Salanié (2004). ð14Þ Dp 1 describes the gain or loss of being social responsible rather than PM, given that the rival is PM. On the other hand, Dp 2 defines the gain or loss of being socially responsible rather than PM, given that the rival adopts CSR rules 8 . Finally, Dp 3 describes the gain or loss of the overall PM rule in the industry rather than the universal adoption of social responsible behaviours. First, we analyse the usual case of homogeneous product.
Proof The proof is directly derived from (1), profit differentials (13), (14), and (15), and (2) the two pure-strategy asymmetric Nash equilibria follow from the firms' payoff structure in k 2 ½k Ã ; k T : p , is shown by direct comparison.
Result 2 For whatever values of k, there exists a unique SPNE, that is (CSR, CSR), and since (CSR, CSR) payoff-dominates (PM, PM), it is also Pareto-efficient for both firms.
In the general case of heterogeneous products (i.e. À1\c\1), the following holds: Result 3 Given Eqs. (13)-(15) and the condition p i ! 0, it is possible to construct Fig. 2 in the ðc; kÞ-space, which has the following characteristics: 1. In region A, the unique SPNE is PM/PM. 2. In regions B and E, there are two asymmetric SPNE: PM/CSR and CSR/PM. 3. In regions C and D, the unique SPNE is CSR/CSR. 4. In region F, the unique SPNE is PM/PM.

Proof
The proof is derived straightforwardly from the following inequalities: 9 region A: Dp 1 \0; Dp 2 [ 0; Dp 3 [ 0; region B: Dp 1 [ 0; Dp 2 [ 0; Dp 3 [ 0; region C: Dp 1 [ 0; Dp 2 \0; Dp 3 [ 0; region D: Dp 1 [ 0; Dp 2 \0; Dp 3 \0; region E: Dp 1 [ 0; Dp 2 [ 0; Dp 3 \0; region F: Dp 1 \0; Dp 2 [ 0; Dp 3 \0.  (15)] in the ðc; kÞ-space. The graphs are drawn for a ¼ 1, The comments to Results 1-3 are simple. First, the rich gamma of equilibria is determined by a varying degree of product competition. In essence, when goods are substitutes, the higher (resp. lower) the product competition is, i.e. the higher (the lower) the value of c, the more likely CSR (resp. PM or even asymmetric and multiple choices) is the SPNE. The rationale for this result is as follows: from Sect. 2.3 it is recognized that the higher the level of social engagement, the larger the output expansion of the firm and, thus, the larger the market share of the CSR firm with respect to the PM firm. Therefore, with close substitute goods, sufficiently low levels of CSR engagement favour the emergence of CSR as the SPNE, because an increasing social concern makes the unilateral deviation from PM increasingly profitable for a firm; consequently, both firms opt for the adoption of CSR rules. However, when the value of k is large enough, the firm choosing CSR expands output at a level such that the rival reduces production to sustain prices and prefers PM which ensures profits higher than a CSR duopoly; thus, asymmetric equilibria arise. Finally, combinations of more independent goods and high values of k lead firms to choose PM. On the other hand, with close complement goods, given the strategic complementarity, both firms mainly prefer the adoption of CSR rules.
The empirical predictions are that, in industries (1) with very differentiated products CSR behaviours should be rarely observed, (2) with intermediate values of the differentiation degree the case of one firm following CSR rules and the other profit-maximising should be more often found, and (3) at the two extremes of the substitutes/close complements, the presence of CSR firms should be more often observed.
While Result 3 provides all of the SPNEs of the game, the following result provides the welfare properties of such SPNEs: Result 4 (1) In region A, the SPNE PM/PM is Pareto-efficient. (2) In region D, the SPNE CSR/CSR is Pareto-efficient. (3) In region C, the SPNE CSR/CSR is Paretoinefficient, and the game is a ''prisoner's dilemma''. (4) In region B, the asymmetric equilibrium (CSR,PM) payoff-dominates (CSR,CSR), while the asymmetric equilibrium (PM,CSR) is payoff-dominated by (PM,PM). (5) In region E, the asymmetric equilibrium (PM,CSR) payoff-dominates (PM,PM), while the asymmetric equilibrium (CSR,PM) is payoff-dominated by (CSR,CSR). (6) In region F, the SPNE PM/PM is Pareto-inefficient, and the game is a ''prisoner's dilemma''.
Proof The proof derives directly from Results 1-3 and the following firms' payoff structure in region E: p The intuition is that, following the interest of stakeholder-consumers means a more aggressive behaviour on the market; such aggressiveness, provided that it is not too intense and that the competition is fierce, may be convenient for strategic reasons for each firm. However, this leads to a typical prisoner's dilemma situation.
On the other hand, the output of the CSR firm larger than the PM one has a different effect on the market prices of the two companies. In fact, the negative impact of the aggressive behaviour of the CSR firm on its own price is higher than on that of the PM firm, and the degree of product differentiation attenuates this effect on the latter. Therefore, the presence of areas B and E in Fig. 1 seems to Econ Polit Ind (2017) 44:371-390 381 suggest that, in those portions of the ðc; kÞ-space the combination of the parameters is such that one firm finds profitable not to adopt CSR rules, because it may charge a relatively high price without drastically lowering output levels with respect to the CSR firm. Conversely, when competition is low and the interest for consumers is too high, no firm may find profitable to follow CSR rules. 10

Welfare considerations
Let us consider the welfare effects of the model. The social welfare function (the sum of the industry profits and consumer surplus) is given by: Using Eqs. (2) and (3), the relevant expressions of the three configurations CSR/ CSR, PM/PM, and asymmetric regimes are SW CSR ¼ a 2 ½3þcÀ2kð1þcÞ ½2þcÀkð1þcÞ 2 , SW PM ¼ a 2 ð3þcÞ ð2þcÞ 2 , SW Asy ¼ a 2 ½2ð1ÀkÞ 2 c 3 Àð2À6kþk 2 Þc 2 À2ð8À7kþ2k 2 Þcþ3k 2 À20kþ24 2½2ð2ÀkÞÀc 2 ð1ÀkÞ 2 . A direct analytical inspection reveals the following result.
Result 5 A duopoly in which both firms adopt CSR rules always leads to the best achievable social welfare outcome.
Proof Simple comparison of the social welfare expressions reveals that, in the feasible set of the ðc; kÞ-space, SW CSR [ SW Asy [ SW PM . Given Result 5, Fig. 3 shows the relation between the firms' game equilibria and social welfare. A straightforward graphical inspection reveals that universal CSR arises as the SPNE in regions C and D, and this outcome also represents the best achievable outcome in terms of social welfare. While in region C universal CSR is Pareto-inefficient, in region D it is Pareto-efficient for firms. In the former case, the strategic interactions in the product market drive the result against the firms' interests; however, the socially best achievable outcome is attained. On the other hand, in the latter case, when goods are ''very'' complements, the firms' strategic 10 In the supplement, we examine the case of the firms' non-cooperative endogenous choice of the level of social concern (i.e. the level of k is totally under the control of the firms' owners and does not depend on the stakeholders). We find that, in the case of substitute goods, there exists a unique SPNE in which both firms adopt CSR and, given that PM payoff-dominates CSR, this equilibrium is Pareto-inefficient, so that a standard prisoner's dilemma emerges. On the other hand, in the case of complement goods, there are two pure-strategy asymmetric Nash equilibria, that is (CSR,PM) and (PM,CSR), and the game becomes a coordination game. In addition, in the case in which the endogenous choice of k depends on the maximisation of the objective function W, the results are in a certain sense rather trivial. In fact, the value of the consumers' surplus dominates that of profits in the function W (a well-known result in the IO literature in the presence of standard linear demand and technology). This means that the objective function W is always increasing in the weight of the consumer surplus, and it reaches a maximum compatible with the non-negativity constraint condition of profits. As a consequence, not only would it be a convenient an unilateral introduction, but of CSR rules: all firms, in equilibrium, choose to introduce CSR with zero profits. interaction leads to the most desirable welfare outcome, and this is in line with the firms' interests.
Regions A and F, in which PM/PM emerges as endogenous equilibrium, represent the most problematic areas, because product market interactions lead the economy to the third best achievable social welfare outcome. In fact, the strategic interactions induce duopolists to select PM; nonetheless, this equilibrium is Paretoinefficient, because the universal adoption of CSR would make both firms better off, and this would be in line with the objective of a social welfare-maximising government.
These results offer the policy implication that decision makers should pay attention both to the degree of product differentiation and the level of social engagement to design the appropriate intervention tools (e.g. mandatory introduction of quality standards). However, given that the universal presence of socially concerned firms facilitates the attainment of the highest welfare level the governments should encourage the overall adoption of CSR rules. 11 Fig. 3 Endogenous equilibria and social welfare in the ðc; kÞ-space. The graphs are drawn for a ¼ 1 11 In the case of the non-cooperative endogenous choice of k, the firms' strategic interaction with substitute goods leads to the socially best achievable outcome given by universal CSR, although this is in contrast to the firms' interests because universal PM payoff-dominates universal CSR. On the other hand, with complement goods, the firms' strategic interaction generates asymmetric equilibria which lead the economy to a second best achievable welfare outcome for a sufficiently large range of c; however, when the goods are ''very'' complement, those asymmetric equilibria leads to the most desirable social welfare. These results are reported in detail in the supplement.

Conclusions
Using a simple two-stage game in which duopoly firms compete à la Cournot with differentiated products, this paper investigates the firms' strategic selection in adopting CSR rules. To the best of our knowledge, the game-theoretic approach proposed here is the unique work which shows that, depending on the degree of product differentiation and firms' social concerns, in equilibrium, there are situations in which the following arise: (1) universal adoption of CSR rules: (2) the presence of only PM firms; (3) the co-existence of a CSR with a PM firm; that is, an equilibrium in which identical firms engaged in different levels of CSR may arise as simple effect of the strategic context; and (4) multiple symmetric equilibria. Therefore, grounded on a game theoretical analysis, this work can help to shed light on the rationale for the emergence, widely observed in the real world, of various equilibria in different industries, related to firms' strategic moves, in which CSR behaviours are more or less frequently widespread. With regard to welfare, the firms' strategic interaction in the product market leads to the adoption of CSR rules both in the presence of sufficiently high levels of product substitutability/complementarity, especially when the levels of social concerns are adequately low, driving the economy to the best achievable welfare outcome. In the former case, this is in contrast to firms' interests, while, in the latter, the firms' and social interests coincide. However, there exists a wide area in the relevant parameter space defined by low-medium degrees of substitutability/complementarity among products in which universal profit maximisation rule emerges in equilibrium, leading the economy to the less-desirable social welfare level. Moreover, under Bertrand competition, the SPNE is always universal PM in which the social welfare is the least desirable: this implies that, in contrast to the conventional wisdom, Bertrand may appear as welfare dominated by Cournot.
These results contribute to the increasing policy debate on firms' social concerns, showing the thus-far not fully explored effects of such concerns in shaping the industry equilibrium. However, those results call for a robustness check under different model specifications such as the presence of network effects and the presence of asymmetric and endogenous costs (such as unionised labour costs). These extensions are left for further research.

Compliance with ethical standards
Funding This study was not funded by any Institution.

Conflict of interest
The authors declare that they have no conflict of interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Observing the reaction functions in (18), it can be noted that a socially concerned firm has an incentive to sell products at a price lower than a profit-maximising firm.
Regardless of the value of c, as k increases, the price of the socially concerned firm is reduced; the differentiation of (18) reveals that op i ok \0 both for substitute (c [ 0) and complement (c\0) goods. Figure 4 shows this result. The left box depicts the case of substitute goods: the reaction functions are upward sloping (strategic complements). The right box depicts the case of complement goods: the reaction functions are downward sloping. In this case, goods behave as when the strategies are substitutes. In both cases, the increasing weight attached to the stakeholder-consumers' interests by socially Fig. 4 Adoption of different levels of social engagement (CSR rules) and firms' reaction functions in Bertrand competition. Left box perfect substitutes (c ¼ 1). Right box almost perfect complements (c ¼ À:99). The graphs are drawn for a ¼ 1 concerned firms drives more aggressive market behaviour through price reduction; however, the relative aggressiveness is weaker when goods are substitutes than complements (the price reduction for a given level of k is smaller when c [ 0 than when c\0). Therefore, also in the case of Bertrand competition, the adoption of CSR rules implies a fiercer competition in the product market.
From (18), the equilibrium prices are Inserting Eq. (19) into Eq. (17), the equilibrium quantity in Bertrand competition is 13 thus leading to the equilibrium profits On the other hand, to get all the relevant expressions when both firms are profitmaximising, one evaluates Eqs. (17)-(21) at k ¼ 0. Thus, the firms' profits are Finally, in the asymmetric case, one firm follows CSR and the other is profitmaximising. As in the main text, let us assume that firm i maximises the social objective function in Eq. (4) (with p i ¼ p i q i ), while the rival j simply maximises profits (k ¼ 0, therefore p j ¼ p j q j ). Under these assumptions, after the application of the usual standard maximisation techniques, one gets the following equilibrium values: a½ð2 À c À c 2 Þð1 À kÞ ½2ð2 À kÞ À c 2 ð1 À kÞ ; p j ¼ a½ð2 þ k À c À c 2 ð1 À kÞ ½2ð2 À kÞ À c 2 ð1 À kÞ ; ð23Þ 13 Under Bertrand competition, it can be easily verified that the non-negativity conditions on prices and quantities are always satisfied in the relevant parameter space.
Given the outcomes in (21), (22), and (25) and defining the profit differentials as in the main text, one can directly verify that Dp 1 \0, Dp 2 [ 0 and Dp 3 (see also Fig. 5). 14 As a consequence, the following result holds.
Result A.1 Given that Dp 1 \0, Dp 2 [ 0 and Dp 3 [ \ 0 for k [ \ k T (1) simple profit maximisation is the dominant strategy for firms; therefore, (PM, PM) is the unique SPNE (2) if k ! k T ,(PM, PM) payoff-dominates (CSR, CSR), and the equilibrium is Pareto-efficient for firm, while, if k\k T , the opposite holds true: the Nash equilibrium is Pareto-inefficient, so that a standard prisoner's dilemma emerges.
Therefore, unilateral deviation from the profit maximisation rule is always unprofitable. The reason for this result is intuitive. Under Bertrand competition, firms compete in strategic complements. The unilateral adoption of CSR rules leads the firm to expand output, with negative effects on the price and profits. Nonetheless, when the degree of complementarity among products is adequately high and the level of the social concern is not too strong, there exists an area in the ðc; kÞ-space in which the (CSR, CSR) equilibrium payoff-dominates the (PM, PM) equilibrium: as such, firms face a classical prisoner's dilemma situation. Fig. 5 Plot of the profit differential Dp 3 ¼ p PM=PM À p CSR=CSR in the ðc; kÞ-space. The graph is drawn for a = 1, Dp 3 ¼ 0 , k T ¼ cðcÀ2Þ ð1ÀcÞ 2 (black solid line) With regard to welfare, given (19)-(25) and the definition of social welfare in (16), the expressions of the social welfare under the three configurations CSR/CSR, PM/PM, and asymmetric regimes are SW CSR ¼ a 2 ½3 À 2c À 2kð1 À cÞ ð1 þ cÞ½2 À k À cð1 À kÞ 2 ; SW PM ¼ a 2 ð3 À 2cÞ ð1 þ cÞð2 À cÞ 2 ; SW Asy ¼ a 2 ð2 À kÞð4 À 3kÞc À 2ð1 À kÞc 2 ð2 À kÞc þ ð5 À kÞ ½ þ 3k 2 À 20k þ 24 È É 2ð1 þ cÞ 2ð2 À kÞ À c 2 ð1 À kÞ ½ 2 An analytical inspection reveals the following result.
Result A.2 A duopoly in which both firms adopt CSR rules always leads to the best achievable social welfare outcome.
Proof Direct comparison of the social welfare expressions reveals that, in the ðc; kÞspace, SW CSR [ SW Asy [ SW PM for k k Ã ðcÞ ¼ 2ð1ÀcÞð2ÀcÞð2þcÞ 2 2c 3 ð1þcÞÀcð11cÀ4Þþ12 , while for k [ k Ã ðcÞ, the ranking is SW CSR [ SW PM [ SW Asy .
Given Result A.2, Fig. 6 depicts the relation between the firms' endogenous game equilibria and social welfare. The firms' strategic interactions in the product market lead to universal PM (see Result A.1), and this outcome represents the third Fig. 6 Endogenous equilibria and social welfare in the ðc; kÞ-space, Bertrand competition. The graphs is drawn for a = 1, Dp 3 ¼ 0 , k T ¼ cðcÀ2Þ ð1ÀcÞ 2 ; SW Asy ¼ SW PM , k ¼ k Ã ðcÞ best achievable result in terms of social welfare for a large area of the parameter set. Therefore, by comparing Figs. 3 and 6, we may observe that the best achievable welfare outcome is more likely under Cournot than Bertrand. Because CSR emerges as an equilibrium under Cournot but never under Bertrand, then, in contrast to the conventional wisdom, Bertrand may appear as welfare dominated by Cournot.