Corporate Social Responsibility, Environmental Emissions and Time-Consistent Taxation

Abstract

We formally model a Cournot duopoly market in which a corporate socially responsible (CSR) firm interacts with a profit-maximizing firm and where the market is regulated with an emission tax. We consider three different kinds of CSR firm behaviors: (i) consumer-friendly; (ii) environmentally-friendly; and (iii) consumer-environmentally friendly. Unlike most theoretical works within this literature, which typically use specific functional forms, we use general structures for the inverse demand function, the cost function, and for emission levels and damage functions. In terms of modeling strategy, we use two game-theoretic approaches: (i) a simultaneous game and (ii) a sequential three-stage ex-post game, in which decisions are time consistent. We found that the optimal emissions taxation rule is modified when considering different CSR motivations. We show that depending upon the CSR motivation and the price elasticity of demand in some cases we can obtain optimal emission tax rates higher, lower, or equal to marginal external emission. Finally, we also found that firms adopting consumer-friendly CSR behavior are more effective in improving the environment compared to environmentally friendly firms.

Appendices

Appendices

Alternative CSR-Firm’s Objective Functions Used in the Literature

Authors	Title	Year	CSR-firm’s objective function
CSR firms as consumer-friendly firms
Seung-Leul Kim, Sang-Ho Lee. and Tosliihiro Matsumura	Corporate social responsibility and privatization policy in a mixed oligopoly	2019	${U}_{{{i}}} = \pi _{{{i}}} + \alpha _{i}CS$. where $\alpha _{i}$ ($0< \alpha _{i} < 1$) represents the CSR level, which is exogenously given. That is, CSR implies the private firm is interested in consumers’ surplus in addition to its profit. CS stands for consumer surplus.
Arturo Garcia, Mariel Leal and Sang-Ho Lee	Time-inconsistent environmental policies with a consumer-friendly firm: Tradable permits versus emission tax	2018	$V_{0} = \pi _{0} + {\theta }CS$. where the parameter $\theta \in [0,1]$ measures the degree of concern on consumer surplus that the consumer-friendly firm has. which is exogenously given. CS stands for consumer surplus.
Lili Xn and Sang-Ho Lee	Corporate Social Responsibility and Environmental Taxation with Endogenous Entry	2018	$G=\pi _{0}+\alpha CS$. where $\alpha \in [0,1]$, CS is consumer surplus. They assume that CSR. initiative includes both profitability and consumer surplus, as a proxy of its own concern on consumers, and thus the objective of the CSR-firm is a combination of consumers surplus and its own profit.
Luciano Fanti and Domenico Buccella	Corporate social responsibility, profits and welfare with managerial firms	2017	$W_{i} = \pi _{i} + kCS$. where $k \in [0,1]$ denotes the weight that CSR firms assign to consumer surplus. CS stands for consumer surplus.
Luca Lambertini and Alessandro Tampieri	Incentives, performance and desirability of socially responsible firms in a Cournot oligopoly	2015	$O_{csr}=\pi _{csr}- gq_{csr} + z\frac{Q^{2}}{2}$, where $O_{csr}$ represents the objective function of a firm adopting a CSR statute, gqcsr represents environmental damage and $z \in [0, 1]$ denotes the weight that the firm assigns to consumer surplus.
Tosliihiro Matsumura and Akira Ogawa	Corporate Social Responsibility or Payoff Asymmetry? A Study of an Endogenous Timing Game	2014	$V_{i} = \theta _{i}SW + (1 - \theta _{i})\pi _{i}$, where $\theta _{i} \in [0,1)$. SW is the total social surplus (sum of the firms’ profits and consumer surplus), and 7rt is firm i’s profit.
Gregory E. Goering	The Profit-Maximizing Case for Corporate Social Responsibility in a Bilateral Monopoly	2014	$\lambda _{r} = \pi _{r} + {\gamma }CS$ where $\pi _{r}$ represents profits plus a given fraction ($\gamma >0$) of the consumer surplus (CS) of its customers’ (stakeholders’).
Bjorn Brand and Michael Grothe	Social responsibility in a bilateral monopoly	2014	$v_{i} = \pi _{i} + \theta _{i}CS$, where $\theta _{i}$ indicates the weight put on consumer surplus. CS stands for consumer surplus. For $\theta _{i} = 0$ the firm operates like a profit maximizer while for $\theta _{i} = 1$ the whole consumer surplus is considered in the firm’s objective function.
Michael Kopel and Bjorn Brand	Socially responsible firms and endogenous choice of strategic incentives	2012	$U_{SR} = \pi _{SR} + {\theta }CS + {\gamma }SR (CS - \pi _{SR})$. the compensation contract gives incentives to contribute to socially responsible (SR) firm’s objective including the non-profit motives, but corrects for differences between the two components consumer surplus CS and profit $\pi _{SR}$. For ${\gamma }_{SR} = 0$ the SR. manager’s goal coincides with the firm’s objective.
CSR firms as environment-friendly firms
Juan Carlos Barcena-Ruiz, Amagoia Sagasta	International trade and environmental corporate social responsibility	2022	$V_{i} = \pi _{i} - {\alpha }ED_{i},i \ne j; \hbox {i,j} = 1,2$. where aEDt can be interpreted as measuring the cost of factoring environmental considerations into all business activities. Parameter $\alpha$. which is assumed equal for both firms, denotes the weight that firm i places on environmental damage in addition to its profits and thus represents the degree of ECSR. Hence, $\alpha = 0$ means that the owner of firm i is only concerned about its profit and the higher parameter $\alpha$ is, the greater the concern of firm i for environmental damage is. The weight attached to environmental damage by firm i, $\alpha$. is exogenous, with $\alpha \in [0,1/2]$.
Lili Xu. Yuyan Chen and Sang-Ho Lee	Emission tax and strategic’ environmental corporate social responsibility in a Cournot Bertrand comparison	2022	$V_{\i } = \pi _{i} - \beta _{i}ED$. where $\beta _{i} \in [0,1]$ represents the degree of ECSR (environmental corporate social responsibility) and ED represents environmental damage. Note that $\beta _{i} = 0$ indicates that firm i is a private firm pursuing absolute profits.
Katsufumi Fukuda and Yasunori Ouchidab	Corporate social responsibility (CSR) and the environment: Does CSR increase emissions?	2022	$V = \pi + \theta \langle CS - D(E)\rangle$ where the exogenous parameter $\theta \in [0,1]$ presents the degree of CSR. A higher value of $\theta$ denotes a higher degree of CSR. $\theta \langle CS - D(E)\rangle$ is called social concern. When $\theta =0$, then the firm maximizes only its own profit. Conversely, when $\theta = 1$, then the monopolist behaves as the most socially responsible firm. CS stands for consumer surplus and D(E) for environmental damage.

Proof Proposition 1

Proof

Solving for (6) we have: $\frac{\partial v_{0}(q_{0},w_{0})}{\partial q_{0}}=f(Q^{*})+q_{0}^{*}\frac{\partial f(Q^{*})}{\partial q_{0}}-\frac{\partial c_{0}^{*}}{\partial q_{0}}-t\frac{\partial d_{0}^{*}}{\partial q_{0}}+\theta \left( f(Q^{*})\frac{\partial Q^{*}}{\partial q_{0}}-f(Q^{*})\frac{\partial Q^{*}}{\partial q_{0}}-\frac{\partial f(Q^{*})}{\partial q_{0}}Q^{*}\right) -\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial q_{0}}=0$

$\Longrightarrow f(Q^{*})+q_{0}^{*}\frac{\partial f(Q^{*})}{\partial q_{0}}-\frac{\partial c_{0}^{*}}{\partial q_{0}}-t\frac{\partial d_{0}^{*}}{\partial q_{0}}+\theta f(Q^{*})\frac{\partial Q^{*}}{\partial q_{0}}-\theta f(Q^{*})\frac{\partial Q^{*}}{\partial q_{0}}-\theta \frac{\partial f(Q^{*})}{\partial q_{0}}Q^{*}-\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial q_{0}}=0$.

Cancelling out the term $\theta f(Q^{*})\frac{\partial Q^{*}}{\partial q_{0}}$, we get (i).

From (6) we also obtain:

$\frac{\partial v_{0}(q_{0},w_{0})}{\partial w_{0}}=-\frac{\partial c_{0}^{*}}{\partial w_{0}}-t\frac{\partial d_{0}^{*}}{\partial w_{0}}+\theta \left( f(Q^{*})\frac{\partial Q^{*}}{\partial w_{0}}-f(Q^{*})\frac{\partial Q^{*}}{\partial w_{0}}-\frac{\partial f(Q^{*})}{\partial w_{0}}Q^{*}\right) -\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{ \partial w_{0}}=0$

$\Longrightarrow -\frac{\partial c_{0}^{*}}{\partial w_{0}}-t\frac{\partial d_{0}^{*}}{\partial w_{0}}+\theta f(Q^{*})\frac{\partial Q^{*}}{\partial w_{0}}-\theta f(Q^{*})\frac{\partial Q^{*}}{\partial w_{0}}-\theta \frac{\partial f(Q^{*})}{\partial w_{0}}Q^{*}-\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial w_{0}}=0$.

Cancelling out the term $\theta f(Q^{*})\frac{\partial Q^{*}}{\partial w_{0}}$, we get: $-\frac{\partial c_{0}^{*}}{\partial w_{0}}-t\frac{\partial d_{0}^{*}}{\partial w_{0}}-\theta \frac{\partial f(Q^{*})}{\partial w_{0}}Q^{*}-\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial w_{0}}=0$. Since $\frac{\partial f(Q^{*})}{\partial w_{0}}=0$, we get (ii).

Solving for (5) we have:

$\frac{\partial \pi _{1}(q_{1},w_{1})}{\partial q_{1}}=f(Q^{*})+q_{1}^{*}\frac{\partial f(Q^{*})}{\partial q_{1}}-\frac{\partial c_{1}^{*}}{\partial q_{1}}-t\frac{\partial d_{1}^{*}}{\partial q_{1}}=0$. Hence (iii).

From (5) we also obtain: $\frac{\partial \pi _{1}(q_{1},w_{1})}{\partial w_{1}}=-\frac{\partial c_{1}^{*}}{\partial w_{1}}-t\frac{\partial d_{1}^{*}}{\partial w_{1}}=0$. Thus (iv). $\square$

Proof Proposition 2

Proof

It can be noted that:$\frac{\partial d_{0}^{*}}{\partial t}=\frac{ \partial d_{0}}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}+\frac{\partial d_{0} }{\partial w_{0}}\frac{dw_{0}^{*}}{dt}$ and $\frac{\partial d_{1}^{*} }{\partial t}=\frac{\partial d_{1}}{\partial q_{1}}\frac{dq_{1}^{*}}{dt}+ \frac{\partial d_{1}}{\partial w_{1}}\frac{dw_{1}^{*}}{dt}$, which are the equilibrium emissions levels: $d_{\acute{\imath }}(q^{*}(t),w^{*}(t))$, (for $i=0,1)$, derived by totally differentiating emissions levels with respect to t.

Replacing these expressions into (8), we obtain:

$$\begin{aligned} f(Q^{*})\frac{\partial Q^{*}}{\partial t}-\left[ \frac{ \partial c_{0}^{*}}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}+\frac{ \partial c_{0}^{*}}{\partial w_{0}}\frac{dw_{0}^{*}}{dt}\right] - \left[ \frac{\partial c_{1}^{*}}{\partial q_{1}}\frac{dq_{1}^{*}}{dt} +\frac{\partial c_{1}^{*}}{\partial w_{1}}\frac{dw_{1}^{*}}{dt} \right] =\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{ \partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{ \partial t}. \end{aligned}$$

Distributing the negative sign in the LHS:

$$\begin{aligned} f(Q^{*})\frac{ \partial Q^{*}}{\partial t}-\frac{\partial c_{0}^{*}}{\partial q_{0}} \frac{dq_{0}^{*}}{dt}-\frac{\partial c_{0}^{*}}{\partial w_{0}}\frac{ dw_{0}^{*}}{dt}-\frac{\partial c_{1}^{*}}{\partial q_{1}}\frac{ dq_{1}^{*}}{dt}-\frac{\partial c_{1}^{*}}{\partial w_{1}}\frac{ dw_{1}^{*}}{dt}=\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}. \end{aligned}$$

Combining this equation with the ones highlighted in Proposition 1, we obtain:

$$\begin{aligned} \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}= & {} f(Q^{*})\frac{\partial Q^{*}}{\partial t}-\left( f(Q)+q_{0}\frac{\partial f(Q)}{\partial q_{0}}-t\frac{\partial d_{0}}{\partial q_{0}}-\theta Q\frac{\partial f(Q)}{\partial q_{0}}-\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial q_{0}}\right) \frac{dq_{0}^{*}}{dt}\\{} & {} +\left( t\frac{\partial d_{0}}{\partial w_{0}}+\gamma \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial w_{0}}\right) \frac{dw_{0}^{*}}{dt}-\left( f(Q)+q_{1}\frac{\partial f(Q)}{\partial q_{1}}-t\frac{\partial d_{1}}{\partial q_{1}}\right) \frac{dq_{1}^{*}}{dt}-\left( -t\frac{\partial d_{1}}{\partial w_{1}}\right) \frac{dw_{1}^{*}}{dt}. \end{aligned}$$

Collecting terms in the LHS we get:

$$\begin{aligned} \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}= & {} \frac{\partial d_{0}}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}\left( t+\gamma \frac{\partial D}{\partial d_{0}}\right) +\frac{\partial d_{0}}{\partial w_{0}}\frac{dw_{0}^{*}}{dt}\left( t+\gamma \frac{\partial D}{\partial d_{0}}\right) + t\left( \frac{\partial d_{1}}{\partial q_{1}}\frac{dq_{1}^{*}}{dt}+\frac{\partial d_{1}}{\partial w_{1}}\frac{dw_{1}^{*}}{dt}\right) \\{} & {} +\frac{\partial f(Q)}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}\left( -q_{0}+\theta Q\right) +f(Q^{*})\left( \frac{\partial Q^{*}}{\partial t}-\frac{dq_{0}^{*}}{dt}-\frac{dq_{1}^{*}}{dt}\right) -q_{1}\frac{\partial f(Q)}{\partial q_{1}}\frac{dq_{1}^{*} }{dt}. \end{aligned}$$

Given that $Q=q_{0}+q_{1}$, differentiating with respect to t we obtain:

$\frac{\partial Q^{*}}{\partial t}=\frac{dq_{0}^{*}}{dt}+\frac{dq_{1}^{*}}{dt}$ and so $\frac{\partial Q^{*}}{\partial t}-\frac{dq_{0}^{*}}{dt}-\frac{dq_{1}^{*}}{dt}=0$.

Collecting terms in the LHS we obtain: $\frac{\partial D}{\partial d_{0}} \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}} \frac{\partial d_{1}^{*}}{\partial t}=\left( t+\gamma \frac{\partial D}{\partial d_{0}}\right) \left( \frac{\partial d_{0} }{\partial q_{0}}\frac{dq_{0}^{*}}{dt}+\frac{\partial d_{0}}{\partial w_{0}}\frac{dw_{0}^{*}}{dt}\right) +t\left( \frac{\partial d_{1}}{ \partial q_{1}}\frac{dq_{1}^{*}}{dt}+\frac{\partial d_{1}}{\partial w_{1} }\frac{dw_{1}^{*}}{dt}\right) +\frac{\partial f(Q)}{\partial q_{0}}\frac{ dq_{0}^{*}}{dt}\left( -q_{0}+\theta Q\right) -q_{1}\frac{\partial f(Q)}{ \partial q_{1}}\frac{dq_{1}^{*}}{dt}$.

Replacing $\frac{\partial d_{0}^{*}}{\partial t}$and $\frac{\partial d_{1}^{*}}{\partial t}$ and rearranging terms, we get:

$$\begin{aligned} \left( t+\gamma \frac{\partial D}{\partial d_{0}}\right) \frac{\partial d_{0}^{*}}{\partial t}+t\frac{\partial d_{1}^{*}}{\partial t}-\frac{\partial D}{\partial d_{0}} \frac{\partial d_{0}^{*}}{\partial t}-\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}=\frac{\partial f(Q)}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}\left( q_{0}-\theta Q\right) +q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}. \end{aligned}$$

Finally, collecting terms in the LHS we get (9). $\square$

Proof Corollary 1

Proof

Rearranging (9) we obtain: $t^{*}\frac{\partial d_{0}^{*}}{\partial t}-\left( 1-\gamma \right) \frac{\partial D^{*}}{ \partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+t^{*}\frac{ \partial d_{1}^{*}}{\partial t}-\frac{\partial D^{*}}{\partial d_{1}} \frac{\partial d_{1}^{*}}{\partial t}=q_{0}^{*}\frac{dq_{0}^{*}}{ dt}\frac{\partial f(Q^{*})}{\partial q_{0}}-\theta Q^{*}\frac{ dq_{0}^{*}}{dt}\frac{\partial f(Q^{*})}{\partial q_{0}}+q_{1}^{*} \frac{dq_{1}^{*}}{dt}\frac{\partial f(Q^{*})}{\partial q_{1}}$, which we differentiate by $\gamma$ and get:$\frac{dt^{*}}{d\gamma }=- \frac{\frac{\partial D^{*}}{\partial d_{0}}\frac{\partial d_{0}^{*}}{ \partial t}}{\left( \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}\right) }$, which by assumptions 3, we know that $\frac{\partial D^{*}}{\partial d_{0}}>0$. Regardless of the sign of $\frac{\partial d_{i}^{*}}{\partial t}$, we get that: $\frac{dt^{*}}{d\gamma }<0$.

Now differentiating this expression by $\theta$ and rearranging terms we obtain:$\frac{dt^{*}}{d\theta }=- \frac{Q^{*}\frac{\partial q_{0}^{*}}{\partial t}\frac{\partial f(Q^{*})}{ \partial q_{0}}}{\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}$, differentiating Proposition 1, panel i with respect to t allow us conclude that $sign\left( \frac{\partial q_{0}^{*}}{\partial t}\right) =sign\left( \frac{\partial d_{0}^{*}}{\partial t}\right)$, which implies that $\frac{\frac{\partial q_{0}^{*}}{\partial t}}{\frac{\partial d_0^*}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}>0$, and then: $\frac{dt^{*}}{d\theta }>0$. $\square$

Proof Corollary 2

Proof

Case (i): ($\theta =0$ and $\gamma =0)$).

From (9) we obtain:$\left( t^{*}-\frac{\partial D}{\partial d_{0}}\right) \left( \frac{\partial d_{0}^{*}}{\partial t}\right) +\left( t^{*}-\frac{\partial D}{\partial d_{1}}\right) \left( \frac{\partial d_{1}^{*}}{\partial t}\right) =\frac{\partial f(Q)}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}\left( q_{0}\right) +q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

After distributing terms in the LHS we get: $t^{*}\frac{\partial d_{0}^{*}}{\partial t}-\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+t^{*}\frac{\partial d_{1}^{*}}{\partial t}-\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}=q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}\Longrightarrow t^{*}\left( \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}\right) = \frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

Finally, clearing for $t^{*}$, which we now denote by $t_{pm}^{*}$ we get: $t_{pm}^{*}=\frac{\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}}{\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}$.

Case (ii): ($\theta >0$ and $\gamma =0$).

From (9) we get:$\left( t^{*}-\frac{\partial D}{\partial d_{0}}\right) \left( \frac{\partial d_{0}^{*}}{\partial t}\right) +\left( t^{*}-\frac{\partial D}{\partial d_{1}}\right) \left( \frac{\partial d_{1}^{*}}{\partial t}\right) =\frac{\partial f(Q)}{\partial q_{0}}\frac{dq_{0}^{*}}{dt}\left( q_{0}-\theta Q\right) +q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

After distributing terms in the LHS we obtain: $t^{*}\frac{\partial d_{0}^{*}}{\partial t}-\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+t^{*}\frac{\partial d_{1}^{*}}{\partial t}-\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}=\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}\Longrightarrow t^{*}\left( \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}\right) =\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

Finally, clearing for $t^{*}$, which we now denote by $t_{cf}^{*}$ we get: $t_{cf}^{*}=\frac{\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}}{\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}$.

Case (iii): ($\theta =0$ and $\gamma >0$).

From (9) we get: $\left( t^{*}-\frac{\partial D}{\partial d_{0}}\left( 1-\gamma \right) \right) \left( \frac{\partial d_{0}^{*}}{\partial t}\right) +\left( t^{*}-\frac{\partial D}{\partial d_{1}}\right) \left( \frac{\partial d_{1}^{*}}{\partial t}\right) =q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

After distributing terms in the LHS we obtain: $t^{*}\frac{\partial d_{0}^{*}}{\partial t}-\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+t^{*}\frac{\partial d_{1}^{*}}{\partial t}-\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}=q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}\Longrightarrow t^{*}\left( \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}\right) =\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

Finally, clearing for $t^{*}$, which we now denote by $t_{ef}^{*}$ we get we get:

$$\begin{aligned} t_{ef}^{*}=\frac{\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+q_{0}\frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}.}{\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}. \end{aligned}$$

Case (iv): ($\theta >0$ and $\gamma >0)$.

From (9) we get:$\left( t^{*}-\frac{\partial D}{\partial d_{0}}\left( 1-\gamma \right) \right) \left( \frac{\partial d_{0}^{*}}{\partial t}\right) +\left( t^{*}-\frac{\partial D}{\partial d_{1}}\right) \left( \frac{\partial d_{1}^{*}}{\partial t}\right) =\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

After distributing terms in the LHS we obtain: $t^{*}\frac{\partial d_{0}^{*}}{\partial t}-\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+t^{*}\frac{\partial d_{1}^{*}}{\partial t}-\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}=\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}\Longrightarrow t^{*}\left( \frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}\right) =\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+ \frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}$.

Finally, clearing for $t^{*}$, which we now denote by $t_{cef}^{*}$ we get we get:

$$\begin{aligned} t_{cef}^{*}=\frac{\left( 1-\gamma \right) \frac{\partial D}{\partial d_{0}}\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial D}{\partial d_{1}}\frac{\partial d_{1}^{*}}{\partial t}+\left( q_{0}-\theta Q\right) \frac{dq_{0}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{0}}+q_{1}\frac{dq_{1}^{*}}{dt}\frac{\partial f(Q)}{\partial q_{1}}}{\frac{\partial d_{0}^{*}}{\partial t}+\frac{\partial d_{1}^{*}}{\partial t}}. \end{aligned}$$

$\square$

Proof Proposition 4

Proof

Using the fact that from stage 2 we also get $c_0^{(2)}(w_0,w_1)$, $c_1^{(2)}(w_0,w_1)$, $d_0^{(2)}(w_0,w_1)$, $d_1^{(2)}(w_0,w_1)$ and $D^{(2)}(w_0,w_1)$ in Eqs. (17) and (18) we get:

$$\begin{aligned}{} & {} f(Q)= \frac{\frac{\partial c_1}{\partial w_1}+t\frac{\partial d_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}}{\frac{\partial q_1}{\partial w_1}}\\{} & {} f\left( Q\right) \frac{\partial q_0}{\partial w_0}= \frac{\frac{\partial c_0}{\partial w_0}+t\frac{\partial d_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\theta Q\frac{\partial f(Q)}{\partial w_0}+\gamma \frac{\partial D}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}}{\frac{\partial q_0}{\partial w_0}} \end{aligned}$$

Equating the equations above we get Proposition 4$\square$

Proof Corollary 3

Proof

Solving Proposition 4 for t, we have that:

$$\begin{aligned} t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\theta Q\frac{\partial f(Q)}{\partial w_0}+\gamma \frac{\partial D}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}} \end{aligned}$$

Differentiating with respect to $\theta$ we obtain:

$$\begin{aligned} \frac{\partial t^*}{\partial \theta }=\frac{Q\frac{\partial q_1}{\partial w_1}\frac{\partial f(Q)}{\partial w_0}}{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}. \end{aligned}$$

Differentiating with respect to $\gamma$ we get:

$$\begin{aligned} \frac{\partial t^*}{\partial \gamma }=\frac{\frac{\partial q_1}{\partial w_1}\frac{\partial D}{\partial w_0}}{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}. \end{aligned}$$

Since $\frac{\partial q_1}{\partial w_1}>0$ and $\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}>0$, it is clear that $\frac{\partial t^*}{\partial \gamma }<0$ and $\frac{\partial t^*}{\partial \theta }>0$ when $\frac{\partial f(Q)}{\partial w_0}>0$ $\square$

Proof Corollary 4

Proof

Solving Proposition 4 for t, we have that:

$$\begin{aligned} t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\theta Q\frac{\partial f(Q)}{\partial w_0}+\gamma \frac{\partial D}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}} \end{aligned}$$

It is straightforward to find each case. Just replace $\theta$ and $\gamma$ with the values of each case as follows.

Case i: $\theta =\gamma =0$ $\Longrightarrow$ $t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}$

Case ii: $\theta >0$ and $\gamma =0$ $\Longrightarrow$ $t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\theta Q\frac{\partial f(Q)}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}$

Case iii: $\theta =0$ and $\gamma >0$ $\Longrightarrow$ $t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\gamma \frac{\partial D}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}$

Case iv: $\theta >0$ and $\gamma >0$ $\Longrightarrow$ $t^*=\frac{\frac{\partial q_1}{\partial w_1}\left( \frac{\partial c_0}{\partial w_0}+d_0\frac{\partial t}{\partial w_0}+\theta Q\frac{\partial f(Q)}{\partial w_0}+\gamma \frac{\partial D}{\partial w_0}-q_0\frac{\partial f(Q)}{\partial w_0}\right) -\frac{\partial q_0}{\partial w_0}\left( \frac{\partial c_1}{\partial w_1}+d_1\frac{\partial t}{\partial w_1}-q_1\frac{\partial f(Q)}{\partial w_1}\right) }{\frac{\partial q_0}{\partial w_0}\frac{\partial d_1}{\partial w_1}-\frac{\partial q_1}{\partial w_1}\frac{\partial d_0}{\partial w_0}}$ $\square$

Output of a CSR Firm is Less than the Output of a Profit Maximizing Firm

Since we know that $q_0=\frac{\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)+\gamma \frac{\partial D}{\partial d_0}}{(1-\theta )\frac{\partial f(Q)}{\partial q_0}}+q_1\frac{\theta }{1-\theta }$, let’s show that production is greater when $\theta =\gamma =0$ than when $\theta >0$ and $\gamma >0$. Then, it should be the case that:

$$\begin{aligned} \frac{\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)}{\frac{\partial f(Q)}{\partial q_0}}>\frac{\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)+\gamma \frac{\partial D}{\partial d_0}}{(1-\theta )\frac{\partial f(Q)}{\partial q_0}}+q_1\frac{\theta }{1-\theta } \end{aligned}$$

Proof

$$\begin{aligned}{} & {} (1-\theta )\frac{\partial c_0}{\partial q_0}+t(1-\theta )\frac{\partial d_0}{\partial q_0}-(1-\theta )f(Q)>\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)+\gamma \frac{\partial D}{\partial d_0}+\theta q_1\frac{\partial f(Q)}{\partial q_0}\\{} & {} \Rightarrow f(Q)-\frac{\partial c_0}{\partial q_0}-t\frac{\partial d_0}{\partial q_0}>q_1\frac{\partial f(Q)}{\partial q_0} \end{aligned}$$

Since f(Q) is de inverse demand function, $\frac{\partial f(Q)}{\partial q_0}<0$ (assuming normal goods), and therefore the right-hand side of the inequality is negative. However, we know that the reaction function of a firm without CSR motivations is $q_i=\frac{\frac{\partial c_i}{\partial q_i}+t\frac{\partial d_i}{\partial q_i}-f(Q)}{\frac{\partial f(Q)}{\partial q_i}}>0$ (if not the case, the firm doesn’t produce). Then, because $\frac{\partial f(Q)}{\partial q_0}<0$, it may be the case that $\frac{\partial c_i}{\partial q_i}+t\frac{\partial d_i}{\partial q_i}-f(Q)<0$ $\Leftrightarrow$ $f(Q)-\frac{\partial c_i}{\partial q_i}-t\frac{\partial d_i}{\partial q_i}>0$, and therefore the left-hand side of the inequality is positive, then:

$$\begin{aligned} \frac{\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)}{\frac{\partial f(Q)}{\partial q_0}}>\frac{\frac{\partial c_0}{\partial q_0}+t\frac{\partial d_0}{\partial q_0}-f(Q)+\gamma \frac{\partial D}{\partial d_0}}{(1-\theta )\frac{\partial f(Q)}{\partial q_0}}+q_1\frac{\theta }{1-\theta } \end{aligned}$$

$\square$