Supplement to Corporate social responsibility in a gametheoretic context
1. The endogenous choice of k
In the case of the firms’ endogenous strategic choice of the level of “social
concern”, the level of ki , i  1,2 is totally under the control of the firms' owners
and does not depend on the stakeholders.
Let us assume as in the main text that in the asymmetric case firm 1 follows
CSR and firm 2 is profit maximizing. The optimal value of the parameter k1 is
obtained maximising eq. (11) in the main text, i.e. selecting that value of k1 such
that the gain of being social responsible rather than following PM, given that the
rival is PM, is maximal. Differentiation of (11) leads to the value
(S.1)
lower than the threshold value
(S.2)
to satisfy the non-negativity constraint on profits. Direct substitutions into the
profit expressions in (11) and (12) yield
On the other hand, in the strategic profile CSR/CSR, the firms’ objective
functions are
,
Maximisation leads to the following reaction functions
1
i  1,2 , i  j
(S.3)
Solving the system in (S.3), the firms’ output as function of the CSR level are
Substitution of the quantities into the profit functions and subsequent
maximization w.r.t. ki yield the CSR level reaction functions, given by
i  1,2 , i  j
(S.4)
Solving the system in (S.4), the unique relevant solution leading to the
equilibrium level of CSR is
(S.5)
Therefore, substituting back (S.5) into the profit functions, these are
Finally, in the case of both firms following PM, profits equal
2
Figure S.1: profits with Cournot competition
and endogenous choice of k
Given these payoffs, it is straight forward to construct the following Figure S.1.
A deeper analytical and graphical inspection reveals the following result.
Result S.1. 1) For   0 (substitute goods) given that  1  0 ,  2  0 , there
exists a unique SPNE, that is (CSR,CSR) and since (PM,PM) payoff-dominates
(CSR,CSR), it is Pareto inefficient, so a standard prisoner’s dilemma emerges;
2) For   0 (complement goods), given that  1  0 ,  2  0 there are two
pure-strategy asymmetric Nash equilibria, that is (CSR,PM) and (PM,CSR), and
the game is a coordination game.
Let us briefly discuss the welfare effects of the model. As in the main text, the
social welfare function (the sum of the industry profits and consumer surplus) is:
SW   i  CS , i  1,2 .
Substituting the relevant values of (S.1) and (S.5) in (6), (9) and (10) in the main
text, and making use of equations (2) and (3), one can easily obtain the relevant
expressions of the three configurations CSR/CSR, PM/PM, and asymmetric
regimes with the endogenous choice of ki , i  1,2 :
3
, SWPM 
a 2 (3   )
,
(2   )2
A direct analytical inspection reveals the following result.
Result S.2. A duopoly in which both firms adopt CSR rules leads to the optimal
social welfare outcome for   (0.767,1] while in   (.99, 0.767] the asymmetric
regime is the most welfare desirable outcome.
Proof: Simple comparison of the social welfare expressions reveals that
SWCSR  SWAsy  SWPM
in   (.767,1] , while in SWAsy  SWCSR  SWPM in
  (.99, .767] , as Figure S.2 shows.
Given Result S.1, Figure S.2 shows that for substitute goods, the firms’ strategic
interaction leads to the socially optimal outcome, although in contrast to the
firms’ interests. On the other hand, in the presence of complement goods, for
  (.767, 0] the firms’ strategic interaction generates asymmetric equilibria
which lead the economy to a second best welfare outcome; however, when the
goods are “very” complement, i.e.   (.99, .767] , those asymmetric equilibria
generate the attainment of the most desirable social welfare.
Figure S.2 Social Welfare with the endogenous choice of k
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