**3. Theoretical approach**

The classical libertarian free-market viewpoint sees that firms should not engage in charitable work with stockholders' money and should leave public goods to the public sector. Even if investors have the option of contributing to the public good via corporate giving in addition to their personal giving, the private channel is more efficient. Moreover, in a perfectly competitive environment, there is no room for charity which reduces profits. This is a well-known argument by Friedman [27]. What lies under the classical viewpoint are the assumptions of perfect competition and that consumer demand for products is independent of the public good. Friedman's viewpoint prevails under these conditions, and there is an ideal separation between the private and the public sectors. If firms, however, do benefit from acts of charity, in the form of increased sales, profits and share price, assumptions for a perfectly competition market must not hold.

boycott and threaten the firm into a settlement for more contribution toward the environment. This situation is studied as an extensive form game in Ref. [13]. The firm can link amount of giving *g* to per unit of output. The firm faces an inverse demand *P*(*q*, *g*), which is a function of quantity *q* and corporate giving. Corporate giving has a positive effect on the inverse demand,

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103

*π* = (*P*(*q*, *g*) − *MC* − *g*)*q*. (1)

In equilibrium, when the firm has a better market opportunity or the pollution level is high in the environment, the activist will spend more efforts and make a higher initial demand. In this type of model, consumer behavior is limited to one product by one firm. The multiple market interactions are assumed away. The part of firm value due to corporate giving is sup-

This type of models have a production technology jointly producing a public good (or alleviating a public bad) along with the product, called a green product. There is a cost advantage for such joint production over separate production if it requires less input to produce the same combination of product and public good. Consumers are making purchasing decisions maximizing utility. A representative consumer can allocate resources endowment into a private product or an environmental public good [14]. When the joint production of public good is engaged, this is called a green market. When there is a cost advantage in joint production, introducing the green market or improving the green technology may discourage private provision of public good. When the joint production is a simple bundling of the private and the public goods, such as direct donations with a unit of product, the equilibrium outcome is

Consumers have diverse preferences. It is realistic to consider two types of consumers: one type care about the public good and the other type do not [15]. Consumers have linear indi-

*V*(*p*, *g*) = *b* − *p* + γ*f*(*g*). (2)

The first term *b* is a constant, *f*(*g*) is the utility from public good, and γ is a 0/1 indicator for neutral and responsible consumers, respectively. Each consumer demands only one unit of the product and each firm produces one unit as well. Firms have constant returns to scale technology and constant marginal costs. Giving to public is committed with each unit of output. Firms compete in the market by announcing the pairs of product price and the amount of public good produced jointly with their products. Firms' strategies constitute a sorting Nash equilibrium that separates consumer types. There are two pairs of equilibrium price and social quality indicator for two groups of consumers. Responsible firms contribute to the public good and charge a high price, which is the marginal cost plus a premium. Increase in corporate giving induces short-run profits, and the value of a firm rises while the market adjusts to equilibrium. We can compare three modes of public good provision in this setting: corporate social responsibility, private voluntary contribution, and government provision. There is a crowding out

and shows up as an addition to the marginal cost *MC* in the profit function:

the same as consumer voluntary contribution without joint production.

rect utility *V*(*p*, *g*) from the product price *p* and public good *g*:

ported by the threat of boycotts.

**3.2. Joint production and consumer choice**

Therefore, to incorporate corporate social responsibility into profit-maximizing behavior, there need to be demand increases for firms' products associated with more public good. Moreover, firms operate in a variety of imperfectly competitive market structures. A successful model of corporate social responsibility needs to incorporate imperfect competition and demand shifts by the public good. The literature takes on a few different modeling strategies. We discuss these strands of models comparing the differences in their market structure, production technologies, and components of consumer utility, and how these increase the value of a firm. Firms may engage in socially responsible actions due to external pressure from activists for the fear of boycotts, or responding to incentives internal to the market. Socially responsible actions can take the form of donations to the public, joint production of the public good with products, or a better quality of products. The decision of engaging in socially responsible products may be made by managers in the firm, by investor through holding shares, or by consumers purchasing the products.

Some results in the literature may be driven by modeling features. Consumers consume and firms produce indivisible products; competing firms produce identical products, or the public good is jointly produced with a private product at a fixed ratio. Firms' roles are suppressed; either they are not making production decisions or their actions are limited by indivisibility and linearity. Hence, there is the equivalence result and that corporate giving crowds out investors' personal giving. Firms, however, should have the full range of price or quality decisions and also choice of contribution levels. Discussions on the benefits from altruistic business actions and the different ways in which firms execute them can be found in Ref. [9]. Other model features in the literature include, for example, the warm glow effect, which is an extra assumption that appeals to personal emotions of giving in addition to public consumption. Portfolio choice models assume fixed profits or linear profit functions, which are not based on market foundations. It would be fruitful if the interlinked relationships among the public and private goods, being complementary or substitutive, can be further explored. A model that exhibits different degrees of complementarity and substitutability among different goods would be an alternative approach [28, 29].

#### **3.1. External activists**

A firm can expand some output to improve the environment, and such efforts toward the environment will be rewarded by more sales of its product. An activist may also launch a boycott and threaten the firm into a settlement for more contribution toward the environment. This situation is studied as an extensive form game in Ref. [13]. The firm can link amount of giving *g* to per unit of output. The firm faces an inverse demand *P*(*q*, *g*), which is a function of quantity *q* and corporate giving. Corporate giving has a positive effect on the inverse demand, and shows up as an addition to the marginal cost *MC* in the profit function:

$$
\pi = \langle P(q, \mathbf{g}) - \mathbf{M}\mathbf{C} - \mathbf{g} \rangle q. \tag{1}
$$

In equilibrium, when the firm has a better market opportunity or the pollution level is high in the environment, the activist will spend more efforts and make a higher initial demand. In this type of model, consumer behavior is limited to one product by one firm. The multiple market interactions are assumed away. The part of firm value due to corporate giving is supported by the threat of boycotts.

#### **3.2. Joint production and consumer choice**

**3. Theoretical approach**

102 Firm Value - Theory and Empirical Evidence

shares, or by consumers purchasing the products.

**3.1. External activists**

The classical libertarian free-market viewpoint sees that firms should not engage in charitable work with stockholders' money and should leave public goods to the public sector. Even if investors have the option of contributing to the public good via corporate giving in addition to their personal giving, the private channel is more efficient. Moreover, in a perfectly competitive environment, there is no room for charity which reduces profits. This is a well-known argument by Friedman [27]. What lies under the classical viewpoint are the assumptions of perfect competition and that consumer demand for products is independent of the public good. Friedman's viewpoint prevails under these conditions, and there is an ideal separation between the private and the public sectors. If firms, however, do benefit from acts of charity, in the form of increased sales, profits and share price, assumptions for a perfectly competition market must not hold.

Therefore, to incorporate corporate social responsibility into profit-maximizing behavior, there need to be demand increases for firms' products associated with more public good. Moreover, firms operate in a variety of imperfectly competitive market structures. A successful model of corporate social responsibility needs to incorporate imperfect competition and demand shifts by the public good. The literature takes on a few different modeling strategies. We discuss these strands of models comparing the differences in their market structure, production technologies, and components of consumer utility, and how these increase the value of a firm. Firms may engage in socially responsible actions due to external pressure from activists for the fear of boycotts, or responding to incentives internal to the market. Socially responsible actions can take the form of donations to the public, joint production of the public good with products, or a better quality of products. The decision of engaging in socially responsible products may be made by managers in the firm, by investor through holding

Some results in the literature may be driven by modeling features. Consumers consume and firms produce indivisible products; competing firms produce identical products, or the public good is jointly produced with a private product at a fixed ratio. Firms' roles are suppressed; either they are not making production decisions or their actions are limited by indivisibility and linearity. Hence, there is the equivalence result and that corporate giving crowds out investors' personal giving. Firms, however, should have the full range of price or quality decisions and also choice of contribution levels. Discussions on the benefits from altruistic business actions and the different ways in which firms execute them can be found in Ref. [9]. Other model features in the literature include, for example, the warm glow effect, which is an extra assumption that appeals to personal emotions of giving in addition to public consumption. Portfolio choice models assume fixed profits or linear profit functions, which are not based on market foundations. It would be fruitful if the interlinked relationships among the public and private goods, being complementary or substitutive, can be further explored. A model that exhibits different degrees of complementarity and

substitutability among different goods would be an alternative approach [28, 29].

A firm can expand some output to improve the environment, and such efforts toward the environment will be rewarded by more sales of its product. An activist may also launch a This type of models have a production technology jointly producing a public good (or alleviating a public bad) along with the product, called a green product. There is a cost advantage for such joint production over separate production if it requires less input to produce the same combination of product and public good. Consumers are making purchasing decisions maximizing utility. A representative consumer can allocate resources endowment into a private product or an environmental public good [14]. When the joint production of public good is engaged, this is called a green market. When there is a cost advantage in joint production, introducing the green market or improving the green technology may discourage private provision of public good. When the joint production is a simple bundling of the private and the public goods, such as direct donations with a unit of product, the equilibrium outcome is the same as consumer voluntary contribution without joint production.

Consumers have diverse preferences. It is realistic to consider two types of consumers: one type care about the public good and the other type do not [15]. Consumers have linear indirect utility *V*(*p*, *g*) from the product price *p* and public good *g*:

$$V(p, \emptyset) = |b - p + \gamma f(\emptyset). \tag{2}$$

The first term *b* is a constant, *f*(*g*) is the utility from public good, and γ is a 0/1 indicator for neutral and responsible consumers, respectively. Each consumer demands only one unit of the product and each firm produces one unit as well. Firms have constant returns to scale technology and constant marginal costs. Giving to public is committed with each unit of output. Firms compete in the market by announcing the pairs of product price and the amount of public good produced jointly with their products. Firms' strategies constitute a sorting Nash equilibrium that separates consumer types. There are two pairs of equilibrium price and social quality indicator for two groups of consumers. Responsible firms contribute to the public good and charge a high price, which is the marginal cost plus a premium. Increase in corporate giving induces short-run profits, and the value of a firm rises while the market adjusts to equilibrium.

We can compare three modes of public good provision in this setting: corporate social responsibility, private voluntary contribution, and government provision. There is a crowding out effect on government provision from the other two modes. Corporation social responsibility will produce public goods at exactly the same level as predicted by the standard voluntary contribution equilibrium by individuals. Yet, corporate provision has an advantage when public good is naturally bundled together with the private good in production.

#### **3.3. Oligopolistic competition and linked products**

This type of models compare corporate giving in oligopolistic markets following Cournot type and Bertrand type of competition [16]. Firms produce identical products. In Cournot (Bertrand, respectively) competition, firms decide their output quantities (product prices) and leave the price (quantities) to be determined in the market. Firms can link a contribution to the public good with one unit of their products. When linked, a portion of sales is donated to a charitable cause. Both versions of the products, linked and unlinked, are available in the market. Consumers demand only one unit of product, either linked or not. They are heterogeneous in the willingness to pay for private and public goods. All consumers enjoy the public good, and there is a warm glow effect [30] associated with purchasing the linked product. They have an additive utility function containing nonlinked product *x*, linked product *y*, and public good *g*:

$$\mathsf{LI}(\mathsf{x}\_{\mathsf{t}}\mathsf{y}\_{\mathsf{t}}\mathsf{g}).\tag{3}$$

The investor maximizes utility over the portfolio of shares and direct giving. When the model parameters satisfy a certain condition, shares of the responsible firm trade at a lower price than the neutral firm. When there are heterogeneous investors in the market and some strictly prefer corporate giving to direct giving, the responsible firm will adopt the socially responsible policy of a positive amount of charitable giving in order to maximize share price.

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A capital market with heterogeneous investors can be built on this model [18]. Firms have fixed profits and will distribute profits as financial returns. Besides two types of firms, there are also types of investors differentiated by a parameter *θ*, which indicates how strongly they feel about corporate giving. When corporate giving is a perfect substitute for personal giving, the former crowds out the latter and has no aggregate effect on the public good level. A critical level of *θ* separates investors into two groups. Investors with lower *θ* prefer personal giving and will not buy shares of the responsible firm, since corporate giving carries a higher cost. On the other hand, investors with higher *θ* prefer corporate giving, hence buying shares of the responsible firm.

The mechanism of managerial decision is added to this model in Refs. [31, 32]. Managerial contracts and personal utility induce managers to engage in socially responsible actions. The market value of the firm has a positive covariance with social returns. Firm's profit function *π*(*e*, *S*) is determined by managerial effort *e* and social expenditure *S*. There is a distribution of managers with differentiated ability levels, parameterized by *a*. Managers have utility function *u*(*I*, *e*, *S*), where *I* is the compensation specified by a managerial contract. The contract compensation *I*(*a*, *e*, *S*) is determined by a linear function of observed profit *π* and social expenditures. There are two parameters in the *I* function that set the profit incentive and social incentive for managers. Managers maximize utility over two policy variables, the effort *e* and social expenditure *S*. Investors who own shares of a firm receive a financial return equal to profit minus contract compensation to the manager. Parameter *θ* shows how strongly investors prefer corporate giving to personal giving. It separates investors into two groups, those with lower *θ* will give personally and buy no share of the responsible firm, those with higher

Some results in the approaches discussed above may be driven by their modeling features. Consumers consume and firms produce indivisible products; competing firms produce identical products; or the public good is jointly produced with a private product at a fixed ratio. The equivalence result between corporate giving and personal giving comes from these modeling features that suppresses the roles of firms. Either they are not making production decisions or their actions are limited by indivisibility and linearity. Firms in an ideal model, however, should have the full choice range of price, quantity, and also contribution levels. Discussions on the benefits from altruism and the ways in which it is executed in corporations are provided in Ref. [9]. Other model limitations include, for example, that the warm glow effect is an extra assumption that appeals to personal emotions of giving in addition to public consumption. Portfolio choice models assume fixed profits or arbitrary profit functions, which are not based on market foundations. We intro-

*θ* buy shares but will not give personally.

**3.5. Monopolistic competition and industry equilibrium**

duce two recent approaches that incorporate a market of many firms.

In equilibrium, two types of firms compete for socially responsible customers, and this can lead to overprovision of the public good. In this setting, both underprovision and overprovision of public good may occur. There is a tradeoff between efficient private good production and the efficiency of public good provision between these two modes. Namely, there is a higher level of public good under the Cournot competition which also has a higher product price.

#### **3.4. Portfolio choice and managerial decision**

The representative investor's utility function contains a private good and a public good. The private good is produced by two firms. One of them is a socially responsible firm that produces the public good together with the private good. Investors may earn financial returns from shares of these two firms. The public good is composed of corporate giving from the firm and personal giving from investors, which also has a warm glow effect on utility. An investor has a choice of giving to charity directly or buying shares of the socially responsible firm and, hence, engaging in altruistic investing. This is a model of corporate giving versus direct giving through portfolio choice [17]. The link between firm value and its giving is explicit in this type of model, reflected in share price. The limitation to this approach lies in the number of firms and competition among firms.

Upon buying *n* dollar worth of shares of the socially responsible firm, γ cents per dollar of return will be donated to the public good. Thus, private return is *q* = (1 − *γ*)*n*. If the investor gives *m* dollars to the public directly, she consumes a public good level *g*, together with private donation as warm glow. And

$$\mathcal{S} = \mathcal{N} + \mathcal{m}.\tag{4}$$

The investor maximizes utility over the portfolio of shares and direct giving. When the model parameters satisfy a certain condition, shares of the responsible firm trade at a lower price than the neutral firm. When there are heterogeneous investors in the market and some strictly prefer corporate giving to direct giving, the responsible firm will adopt the socially responsible policy of a positive amount of charitable giving in order to maximize share price.

A capital market with heterogeneous investors can be built on this model [18]. Firms have fixed profits and will distribute profits as financial returns. Besides two types of firms, there are also types of investors differentiated by a parameter *θ*, which indicates how strongly they feel about corporate giving. When corporate giving is a perfect substitute for personal giving, the former crowds out the latter and has no aggregate effect on the public good level. A critical level of *θ* separates investors into two groups. Investors with lower *θ* prefer personal giving and will not buy shares of the responsible firm, since corporate giving carries a higher cost. On the other hand, investors with higher *θ* prefer corporate giving, hence buying shares of the responsible firm.

The mechanism of managerial decision is added to this model in Refs. [31, 32]. Managerial contracts and personal utility induce managers to engage in socially responsible actions. The market value of the firm has a positive covariance with social returns. Firm's profit function *π*(*e*, *S*) is determined by managerial effort *e* and social expenditure *S*. There is a distribution of managers with differentiated ability levels, parameterized by *a*. Managers have utility function *u*(*I*, *e*, *S*), where *I* is the compensation specified by a managerial contract. The contract compensation *I*(*a*, *e*, *S*) is determined by a linear function of observed profit *π* and social expenditures. There are two parameters in the *I* function that set the profit incentive and social incentive for managers. Managers maximize utility over two policy variables, the effort *e* and social expenditure *S*. Investors who own shares of a firm receive a financial return equal to profit minus contract compensation to the manager. Parameter *θ* shows how strongly investors prefer corporate giving to personal giving. It separates investors into two groups, those with lower *θ* will give personally and buy no share of the responsible firm, those with higher *θ* buy shares but will not give personally.

#### **3.5. Monopolistic competition and industry equilibrium**

effect on government provision from the other two modes. Corporation social responsibility will produce public goods at exactly the same level as predicted by the standard voluntary contribution equilibrium by individuals. Yet, corporate provision has an advantage when

This type of models compare corporate giving in oligopolistic markets following Cournot type and Bertrand type of competition [16]. Firms produce identical products. In Cournot (Bertrand, respectively) competition, firms decide their output quantities (product prices) and leave the price (quantities) to be determined in the market. Firms can link a contribution to the public good with one unit of their products. When linked, a portion of sales is donated to a charitable cause. Both versions of the products, linked and unlinked, are available in the market. Consumers demand only one unit of product, either linked or not. They are heterogeneous in the willingness to pay for private and public goods. All consumers enjoy the public good, and there is a warm glow effect [30] associated with purchasing the linked product. They have an additive utility function containing nonlinked product *x*, linked product *y*, and public good *g*:

*U*(*x*, *y*, *g*). (3)

In equilibrium, two types of firms compete for socially responsible customers, and this can lead to overprovision of the public good. In this setting, both underprovision and overprovision of public good may occur. There is a tradeoff between efficient private good production and the efficiency of public good provision between these two modes. Namely, there is a higher level

The representative investor's utility function contains a private good and a public good. The private good is produced by two firms. One of them is a socially responsible firm that produces the public good together with the private good. Investors may earn financial returns from shares of these two firms. The public good is composed of corporate giving from the firm and personal giving from investors, which also has a warm glow effect on utility. An investor has a choice of giving to charity directly or buying shares of the socially responsible firm and, hence, engaging in altruistic investing. This is a model of corporate giving versus direct giving through portfolio choice [17]. The link between firm value and its giving is explicit in this type of model, reflected in share price. The limitation to this approach lies in the number

Upon buying *n* dollar worth of shares of the socially responsible firm, γ cents per dollar of return will be donated to the public good. Thus, private return is *q* = (1 − *γ*)*n*. If the investor gives *m* dollars to the public directly, she consumes a public good level *g*, together with pri-

*g* = *n* + *m*. (4)

of public good under the Cournot competition which also has a higher product price.

public good is naturally bundled together with the private good in production.

**3.3. Oligopolistic competition and linked products**

104 Firm Value - Theory and Empirical Evidence

**3.4. Portfolio choice and managerial decision**

of firms and competition among firms.

vate donation as warm glow. And

Some results in the approaches discussed above may be driven by their modeling features. Consumers consume and firms produce indivisible products; competing firms produce identical products; or the public good is jointly produced with a private product at a fixed ratio. The equivalence result between corporate giving and personal giving comes from these modeling features that suppresses the roles of firms. Either they are not making production decisions or their actions are limited by indivisibility and linearity. Firms in an ideal model, however, should have the full choice range of price, quantity, and also contribution levels. Discussions on the benefits from altruism and the ways in which it is executed in corporations are provided in Ref. [9]. Other model limitations include, for example, that the warm glow effect is an extra assumption that appeals to personal emotions of giving in addition to public consumption. Portfolio choice models assume fixed profits or arbitrary profit functions, which are not based on market foundations. We introduce two recent approaches that incorporate a market of many firms.

Socially responsible actions by a firm can bring customer loyalty from those who care about the public; this leads to less elastic demand. With a lower demand elasticity, firm's profit is less sensitive to market fluctuations and provides a less risky stream of financial returns to investors. Thus, corporate social responsibility is a tool of risk management [19]. There are two types of products in the market. All products *c i* are labeled on the unit interval representing variety, responsible products distribute over *i* ∈ (0, *μ*) and regular products distribute over *<sup>i</sup>* <sup>∈</sup> (*μ*, 1). A responsible product has a lower elasticity of substitution *σ<sup>r</sup>* and a regular product has a higher elasticity of substitution *σ<sup>n</sup>* . The parameter *α* is the share of expenditure on responsible goods. Representative investor's utility is

$$\mathbf{C} = \left(\int\_{0}^{\mu} c\_{l}^{o} \, d\bar{t}\right)^{\underline{\sigma}} + \left(\int\_{\mu}^{1} c\_{l}^{o} \, d\bar{t}\right)^{\frac{1-\mu}{\underline{\sigma}}} \tag{5}$$

are not indivisible); corporate giving is a separate decision from production (do not need to

− <sup>∂</sup><sup>2</sup> \_\_\_\_\_*<sup>V</sup>* ∂*w*∂*g*

A firm chooses quantity as strategy, find corresponding prices on the demand curve, and then announce prices in the market. This is an approach advocated by Refs. [33, 34]. By analyzing the derivative of the profit function with respect to *g*, we found a cutoff point for the cross par-

with the public good. The equilibrium condition for corporate social responsibility and that for voluntary contribution are independent, and hence they are not perfect substitutes. More

value. In this setting, giving is a strategic market decision under competition with other firms.

In practice, companies engage in responsible activities for a few main reasons. First, the tax code provides incentives for companies to make charitable contributions as doing so lowers their taxable income. The highest US corporate tax rate is 35% and when combined with state and local taxes, the actual corporate tax rate is closer to 39%. Hence, for every \$1 contributed to charitable causes, the company can save about 39 cents in lower tax payments. Second, making charitable contributions improves the corporate image. In addition, these contributions support the communities in which their employees live making the community a better place to live. Corporate giving garners respect from the employees. Third, these contributions support the communities in which their employees live making the community a better place to live. Fourth, corporate giving garners respect from the employees. Klara Kozlov, head of corporate clients at the Charities Aid Foundation cites companies desire to "do good" as motivation for corporate gifts. Moreover, these contributions also serve to increase the popularity of the business which may increase consumer loyalty to the company. Fifth, companies involved with social corporate philanthropy receive media exposure and positive public attention/rec-

ognition. Hence providing the company with valuable advertising and marketing.

There are numerous examples of companies who are involved in corporate social responsibility. We provide some examples here, highlighting some of the companies that have recently been recognized for their generosity. In the United States, the Motley Fool in 2017 ranked the 12 most charitable US companies with health care, bank, and technology companies leading the list [35]. While there were two notable exceptions in Exxon and Walmart on the leading charitable company list, the remaining companies were comprised of health care, banking, and technology. The key component that drives corporate donations is company profitability.

*g*

*g*

, *x*2

, …,*<sup>g</sup>*). With wealth *w* and indirect utility *V*, the demand

Corporate Social Responsibility and Firm Value: Recent Developments

> 0. (6)

increases more strongly with the pub-

*g*

will increase demand and its

decreases

107

, …) and a public good *g*. The price

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be joint production). There is a profile of private goods (*<sup>x</sup>*<sup>1</sup>

, *x*2

*<sup>g</sup>* \_\_\_ *pi*

lic good and the firm contributes more. And demand for a product with a smaller *Uxi*

increases with the public good, if

. Demand for a product with larger *Uxi*

corporate giving from a firm whose product has a higher *Uxi*

. Consumer utility is *U*(*x*<sup>1</sup>

*Uxi*

*g*

of each *xi*

is *pi*

for a private good *xi*

tial derivative *Uxi*

**4. Practice**

A firm can choose to invest in a production technology for a product among the continuous variety of products. It takes a fixed cost investing in one of these technologies. The fixed cost of socially responsible technology follows a distribution with a lower bound that is smaller than the fixed cost of the regular products. After acquiring the technology, production has constant returns to scale. Investors are endowed with stocks and cash. They allocate endowment into consumption, stock holdings, and bonds. In period one, investment decisions are made and there is an aggregate consumption good which is not differentiated. It is found that responsible products sell at a premium to regular products. Shares of responsible firms trade on average higher than those of regular firms.

Another approach explores the interlinked relationships among the public and private goods, being complementary or substitutive. Consumer utility contains multiple private goods that exhibit different degrees of complementarity and substitutivity with the public good [28, 29]. There is no cost advantage in public good production tied with any product. The public good has differential effects on private products; it may be complementary to one and substitutive to another. For example, roads will increase the marginal utility of automobiles; this is a public good complementary to private products. On the other hand, national defense and police force will decrease the marginal utility of privately owned firearms; this is a public good substitutive to private products. PBS programs will increase the marginal utility of television sets and at the same time decrease the marginal utility of television programs. Without assumptions like cost advantage in joint production or indivisibility, complementarity is enough to explain the endogenous demand increase caused by a public good. When there are products that are complementary or substitutive to the public good in various degrees, it is apparent that firms whose products that are more complementary to the public good will face demand increases with a higher public good. Thus, there are incentives to contribute to the public. Firms whose products that are more substitutive to the public good suffer a demand decrease with a higher public good level.

A model of monopolistic competition with differentiated products and a public good is presented in Ref. [20]. Individuals and firms contribute at the same time but for different reasons. Individuals are looking to enjoy the public good directly, while firms contribute to induce demand increases. Consumers and firms can choose quantities freely in the market (products are not indivisible); corporate giving is a separate decision from production (do not need to be joint production). There is a profile of private goods (*<sup>x</sup>*<sup>1</sup> , *x*2 , …) and a public good *g*. The price of each *xi* is *pi* . Consumer utility is *U*(*x*<sup>1</sup> , *x*2 , …,*<sup>g</sup>*). With wealth *w* and indirect utility *V*, the demand for a private good *xi* increases with the public good, if

$$\frac{\mathcal{U}\_{\varepsilon\mu}}{p\_i} - \frac{\partial^2 V}{\partial w \, \partial g} > 0. \tag{6}$$

A firm chooses quantity as strategy, find corresponding prices on the demand curve, and then announce prices in the market. This is an approach advocated by Refs. [33, 34]. By analyzing the derivative of the profit function with respect to *g*, we found a cutoff point for the cross partial derivative *Uxi g* . Demand for a product with larger *Uxi g* increases more strongly with the public good and the firm contributes more. And demand for a product with a smaller *Uxi g* decreases with the public good. The equilibrium condition for corporate social responsibility and that for voluntary contribution are independent, and hence they are not perfect substitutes. More corporate giving from a firm whose product has a higher *Uxi g* will increase demand and its value. In this setting, giving is a strategic market decision under competition with other firms.
