Corporate social responsibility in market models

(1)

Corporate Social Responsibility in

Market Models

Author:

Margot van Moll Student Number: 6148883

Supervisor: dr. D´avid Kop´anyi Second reader: prof. dr. Jan Tuinstra

A thesis submitted to the University of Amsterdam in partial fulfilment of the requirements for the degree of Master of Science in Econometrics

March 24, 2017

University of Amsterdam Faculty of Economics and Business

Roetersstraat 11 1018 WB Amsterdam

(2)

Statement of Originality

This document is written by Student Margot van Moll who declares to take full respon-sibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

(3)

Abstract

This thesis regards the influence of corporate social responsibility in different market models. A model by Kopel and Brand [22] regarding a homogeneous Cournot duopoly is adjusted and modified to extend the analysis regarding the relationship between cor-porate social responsibility and profitability. In two different market models, results and conclusions will be drawn via numerical as well as formal analysis. In the Stackelberg model we find that it pays off to be socially responsible but not too socially responsible if the SR firm is follower. Furthermore we conclude there is a first mover advantage present for the PM firm and the SR firm. However, for the SR firm this advantage does depend on the level of social responsibility. Subsequently, in the model with multiple firms, it pays off to be socially concerned if the amount of SR firms in the market is not too large. If this condition holds, the profits of the SR firms are increasing until a certain level of social responsibility is reached, and decrease if this level is exceeded. Besides, we conclude that the overall welfare increases as the SR firms become more socially responsible if there are more SR firms active in the market. However, this is only true as long as the marginal costs of the SR firm do not exceed those of the PM firm, if they do, the welfare can decrease as more SR firms are active in the market and decrease if the SR firm puts more weight on social responsibility. All together we conclude the results by Kopel and Brand are confirmed in the analysis via different market models in this thesis.

(4)

Acknowledgements

I would like to express my thanks to my supervisor D´avid Kop´anyi for his patience and for his useful tips, remarks and commitment throughout the writing and learning process of this thesis.

(5)

Introduction

Corporate social responsibility (henceforth CSR) has become increasingly more impor-tant in the strategy of today’s companies. Currently, over 12000 of the largest companies all around the world have signed the UN Global Compact [32]. By signing this, they vow to follow the ten CSR principles this entails. The principles are carefully derived from several declarations, organizations and conventions [9] and will be briefly summarized next. The first two principles regard the field of human rights, by describing that firms should honor and back internationally declared human rights and that they should not to be associated with unethical activities regarding human rights offences. The next few principles regard the field of labour. The principles state that firms should support the freedom of association and that they should acknowledge the right to collective bargain-ing. Furthermore, the principles declare that firms should encourage the expulsion of forced labour, child labour and discrimination. Subsequently, several principles pay at-tention to the environment. They state that firms should support a prudent approach in solving environmental issues, firms should actively try to advertise greater environmental responsibility and that they should focus on advocating the development and spreading of environmentally friendly technologies. In conclusion there is a principle regarding the field of anti-corruption. It states that businesses should antagonize corruption in all forms.

Because there is a distinction between the sections of CSR, it is complicated to for-mulate a clear definition of CSR. The definition of CSR has been thoroughly reviewed and adjusted since the 1950s [5]. In the Netherlands, the definition of the European

(8)

CHAPTER 1. INTRODUCTION

Commission is used: “corporate social responsibility (CSR) refers to companies taking responsibility for their impact on society” [8]. However, in past literature considering CSR, a definition by Carroll (1979) is often used: “the social responsibility of business encompasses the economic, legal, ethical, and discretionary expectations that society has of organizations at a given point in time’ [Carroll, 1979 p. 500 [4]”. Because the definition by Carroll has been used in research concerning CSR over the past 25 years, the definition is a popular choice in current research about the subject [4].

Research about CSR is very important, because implementing some form of CSR in a company’s strategy has almost become obligatory since CSR has become mainstream. In the 2016 PwC Global CEO survey, it became apparent that the majority of CEOs are increasing their investments in CSR. In even 64% of the cases the CEOs said that “CSR is core to their business rather than being a stand-alone program” [29]. This might seem striking, however investing in CSR may be very smart. In a study by Nielsen, a firm specialized in consumer behaviour around the globe, it became clear that 55% of online customers across 60 countries are prepared to pay more for products and services supplied by firms dedicated to CSR.[27]

The link between corporate social responsibility (CSR) and corporate financial perfor-mance (CFP) has been thoroughly researched [25]. However, one conclusive answer has yet to be found. Therefore, the goal of this thesis is to explore this link further. This the-sis will build on and amplify the model suggested by Kopel and Brand [22], in which they try to define the link between CSR and CFP by analyzing a Cournot duopoly in which a profit maximizing (PM) firm competes against a social responsible (SR) firm. They do this by trying to answer the following two research questions: first, they ask themselves whether it pays off for a firm to be socially concerned, i.e. whether it can yield a com-petitive advantage to pursue goals different from profit maximization. Second, they try to find the impact of an increasing concern for consumer welfare on prices, quantities, industry profits and welfare. By analyzing the equilibrium results they conclude it can be profitable to be socially responsible, however not to be too socially responsible. This thesis will try to analyze whether the same conclusions will be reached, if in an extension different market models are being used.

In the first analysis, a Stackelberg model is used in which the duopoly consists of a leader and a follower. The questions that will be answered are:

(9)

CHAPTER 1. INTRODUCTION

1a. Does it pay off for a firm to be socially concerned, i.e. can it yield a competitive advantage to pursue goals different from profit maximization if one is leader or follower respectively?

1b. What is the impact of an increasing concern for consumer welfare on prices, quan-tities, industry profits and welfare and is there a first-mover or second-mover advantage present?

In the original model, both firms make their decisions simultaneously. However, a first-mover advantage or second-first-mover advantage are known in literature about sequential games [11] [23], [26]. An analysis will have to point out how these advantages are influ-enced by the type of company (PM or SR).

In the second analysis, a model in which multiple firms are active in the market is ana-lyzed. The following questions will be answered:

2a. Does it pay off for a firm to be socially concerned, i.e. can it yield a competitive advantage to pursue goals different from profit maximization and how does this depend on the proportion of SR firms in the market?

2b. What is the impact of an increasing concern for consumer welfare on prices, quanti-ties, industry profits and welfare and how does this depend on the proportion of SR firms in the market?

In the original model, a duopoly is investigated. Nonetheless, a duopoly is not nec-essarily a realistic view of the market as there might be more than two active firms in the market. Therefore I consider a model with N firms and I will look at what happens to the results if one fixes the types of firms in the market, but varies the market compe-tition.

In the analysis regarding both models, formal as well as numerical analyses will be used. By using standard optimization methods, we find that the use of different market models did not necessarily change the conclusions of the research by Kopel and Brand. In the next chapter, a broader view of past literature on CSR is reviewed. Then the original model by Kopel and Brand is explained and summarized. In the third and fourth chap-ters, the Stackelberg model and the model with multiple firms are analyzed, followed by the overall conclusion. Subsequently, formal derivations are presented in the Appendix.

(10)

Chapter 2

CSR and earlier research

2.1 Brief history of CSR

The history and development of CSR have been studied by many, including Carroll [3] and Cochran [7]. Carroll analyzed and presented an overall history of CSR while Cochran focused his paper on the evolution of CSR. Their view and main findings are presented next.

The concept and definition of CSR as we know it today, has been developing since the 1950s [3]. However, in the 1950s, CSR was not common yet and people were talking about it more than actually acting with respect to it. It was not until the 1960s that CSR started to get more popular, when there was an increase in the number of boycotts of companies doing business in South Africa [7]. The amount of investments in social responsibilities increased, and the phenomenon of corporate philantrophy was born. Cor-porate philantrophy consisted mainly of firms making donations to universities, hospitals or any other social cause to increase the overall well being of society [7]. It can be stated that social responsibility was yet to be linked to the firm’s business plan and that it was mainly focused on improving the image of the firm. Some even say it was frowned upon if the firm were to make profits of the donations it did.

In the early 1970s the first investigations regarding CSR were conducted. One survey by Eilbert and Parket [12] resulted in the mapping of the most important CSR activities firms were engaging in. Examples of some of these activities were hiring of minorities, investing in environmental causes and contributions to arts. Since there still was no clear definition of CSR, mapping the CSR activities provided some insight into what

(11)

CHAPTER 2. CSR AND EARLIER RESEARCH

CSR primarly implied to firms. It was not until 1979, when Archie B. Carroll presented his definition of CSR: “The social responsibility of business encompasses the economic, legal, ethical, and discretionary expectations that society has of organizations at a given point in time” [4] that the concept of CSR as we know it today started to get shape. In the 1980s new and purified definitions of CSR were constructed and this inspired and boosted the research regarding CSR substantially. This trend was also amplified by the fact that the public’s awareness regarding CSR increased due to several ethical scandals caused by managerial and corporate malpractices [3]. The focus of the research about CSR shifted in the 1990s, when more research tried to define the link between CSR and corporate financial performance (CFP).

In 2002, Porter and Kramer [28] published an article reporting that the social and eco-nomic returns of an investment were often overlapping. They implied that ecoeco-nomic investments frequently resulted in social returns and the other way around. They pro-posed it made more sense to merge the two investments and look for investments that could result in financial as well as social returns. Other research in the beginning of the 21st century was mainly dedicated to exploring the link between CSR and CSP by investigating consequences resulting from the decision making regarding CSR on a firms’ financial performance [3].

However, a clear link is yet to be found, since the literature about the influence of CSR on CFP is conflicting. Although an unequivocal link is yet to be found, it is clear that companies invest enormous amounts of money in CSR. It was reported that in the US and UK alone, the Fortune Global 500 companies spent more than $15bn on CSR in 2014 [30].

2.2 Pro’s and con’s of CSR

But why do companies invest so much in CSR? One important reason, stated by a PwC survey (2016) [29], could be that correct implementation of CSR in a firm’s strategy contributes to the building of trust with consumers, partners, governments and their employees. This trust is important for the sales and image of a firm. Next to trust, there are also other factors that motivate CEOs to implement CSR in the firm’s strategy. Sprinkle and Maines (2010) [31] state there are at least six motivations for a firm to invest in CSR. First, there are the altruistic intentions, the management of a firm could

(12)

genuinely feel the need to do something back for the society, second there is the ’window dressing’. This window dressing refers to the fact that on paper it looks good to invest in CSR which in return contributes to a good image of a firm. Third, it is believed that CSR attracts talent and motivates employees. Fourth, it could increase sales because consumers buy a product because of the CSR investments of a firm. Fifth, investing in environmental social responsibility could reduce taxes which results in cost savings in the supply chain. Sixth, the article states that CSR investments reduce the risk of a company by reducing the likelihood of incidents occurring [31].

Carroll and Shabana (2010) [6] amplify these by adding other positive aspects of investing in CSR. First, they name the belief that it is in the best interest of the business itself to invest in CSR. This, because they state if firms want to have a healthy climate to function in, they do not have any other choice than to invest in CSR, ensuring long-term viability. Furthermore, it is pointed out that investing in CSR is the right thing to do because they state that pro-acting is better than reacting [6]. They refer to the fact that preventing and anticipating (pro-acting) is often cheaper and more convenient opposed to reacting to a social problem once this has occurred. Besides they cite an article by Bernstein [2] that declares businesses should invest in CSR because the public strongly backs it. The public’s view of corporate responsibility has extended over the years. Nowadays, safe and healthy working conditions, content stakeholders and a happy community are mildly considered to be amongst the responsibilities of a firm [2]. It is therefore believed that the aim of a firm should be to prioritize these over the aim of pure profit maximization.

However, not only advantages of CSR are known in practice. Sceptics believe CSR can also be of negative influence for a firm. Reason named for this could be that managers are forced to involve themselves in areas beyond their expertise [14]. Others believe it is not the responsibility of a firm to be socially responsible, but that the problem should be resolved by the mechanism of the free market system. If the free market system fails to solve the social issues, it is stated that it is the responsibility of government and legislation [4] to solve these and not the responsibility of the firm. Friedman (1970) is a firm believer of this view, stating that in his opinion, ”there is one and only one social responsibility of business - to use it resources and engage in activities designed to increase its profits so long as it stays within the rules of the game, which is to say, engages in open and free competition without deception or fraud.” [15].

(13)

Besides, Carroll at al. found no proof suggesting that CSR had a positive influence on a firms profits [21]. This suggests that if firms do invest in CSR, it puts them at a financial disadvantage compared to other firms that in fact do not invest in CSR. Besides, Davis [10] states that businesses already have enough power, so it would be a mistake to grant them even more power in the social field. Therefore it should be preferred not to expect firms to invest in CSR. [4].

Because of the conflicting literature, many economists try to capture the effect of CSR in an economic model. One of these models is analyzed in this thesis. We extend this model to see whether the results and conclusions remain valid in other market models too.

2.3 Models of CSR

Berger et al. [19] use 97 interviews with managers and consultants who were closely engaged in CSR initiatives to attain a view regarding the internal dynamics of an orga-nization engaging in CSR. Their findings show that a firm’s approach of CSR regarding distinct orientations or profiles can be categorized into three models, namely the Busi-ness Case Model, the Social Values-Led Model and the Syncretic Stewardship Model. The first two models contain CSR narratives already known to literature about CSR, but the third model appears to contain a new approach. In this model, social responsibility is arranged in a broad manner, making it possible to simultaneously satisfy different types of stakeholders. The most important aspects and implications of each model will be presented next.

2.3.1 The Business Case Model

In this first model, Berger et al. [19] state that investing in CSR is not altruistic, but that it is viewed as a business case. They declare that the main objective for firms fitting this model, is to invest in CSR is because it is good for the business. In practice, this means that of all changes resulting from CSR implementation, only the economic results are of importance to the managers and stakeholders. CSR is therefore interesting only if it can provide the firm with a competitive advantage. CSR investments of firms fitting this model are often based on data about the magnitude of the CSR market sector. If the statistics and consumer returns are in favor of CSR, it is implemented in the firm’s

(14)

strategy. However, if the statistics are opposed to favor an implementation of CSR, it is not. Besides data and consumer returns there are also other components that provide a foundation in favor of the business case. In the paper they point out examples like governmental regulation, protests aimed at certain sectors or the solution to a social concern can also motivate CSR investments. Carroll and Shabana [6] present a general business case for corporate social responsibility. In their paper, arguments opposed to the business case are weighted against arguments supporting the business case. They conclude that there is a growing support in favor of the business case, but that firms should comprehend the nature of the CSR investments. They state it is convenient and smart for firms to focus on those CSR activities that overlay the firm’s economic activities to ensure the best results.

2.3.2 The Social Values-Led Model

Berger et al. [19] name the second model the Social Values-Led Model. In this model, the objective and image of the firm are both focused on one specific social cause or issue. Firms fitting this model, prioritize CSR over everything in their decision mak-ing. Consequently, this implies that CSR is integrated in the very essence of the firm. However, this does not always work in favor of the firm. This is shown by the fact that the firm could go bankrupt due to its lack of focus on profits and its survival in the market. Because the economic advantages or motives are clearly not prioritized, Berger et al. state that the sector in which the Social Values-Led Model is categorised, represents the consolidation between the for-profit and the nonprofit sector. For firms fitting this model, it presents a challenge to survive in the market by competing against companies with other morals and prospects. However, competition between companies fitting this model is in contrast very gentle and not harsh at all. From the interviews the conclusion is drawn that to managers it is more important to solve the social issue or achieve the social goal, than it is to survive in the market. Even if this means to help the competition at your own cost. Firms fitting this model are therefore described by words as altruistic or marvelous.

(15)

2.3.3 Syncretic Stewardship Model

The third and last model Berger et al. [19] define is called the Syncretic Stewardship Model. Firms fitting this model, have a much broader and varying group of partners and shareholders. The group includes members, partners and organizations that are directly linked to the profitability of the firm but it also includes activists, charities and non profit organizations who are linked to the social image of the firm. Each representative of the group has a weighted vote regarding negotiations and integrations between the stakeholders. Nijhof et al. [1] describe firms fitting the Syncretic Stewardship Model as firms with a view on CSR in which their own organisation is part of the contribution to the steady solution of a relevant societal issue. They state that firms fitting the Synretic Stewardship Model are therefore often using an open system approach, in which communicating and interacting with the diverse group of stakeholders is incorporated in the entire production process [1]. Because the group of stakeholders is so diverse, making decisions presents quite a challenge, since one has to take different views from different stakeholders into account. Trying to deal with conflicting views on matters while compromising on others appears to be a daily task for companies fitting this model [19]. The view of the market by the Syncretic Stewardship Model can be seen as a merger between the Social Values-Led Model and the Business-Case Model. This, because Syncretic Stewerds are accepting the fact that CSR investments should be viewed from an economic objective, but they also acknowledge that the market is more diverse and complex than in the view of the Business Case Model. Carroll et al. state that the Syncretic Stewardship Model contains a broad view of CSR, since it acknowledges direct and indirect links between CSR and a firm’s performance [6]. This provides the Stewards with a great challenge in simultaneously striving economic and non-economic objectives in their CSR investments. It may not come easy, but implementing CSR in a company in the manner that the Syncretic Stewardship Model suggests to do, provides firms with a competitive advantage with respect to employees. In particular, it helps them to attract and motivate talent. Besides, Cunningham et al. [19] state that acknowledging the value and importance of CSR also attributes to the honesty and commitment of employees. The model by Kopel and Brand (2007) [22] that is explored and amplified in this thesis is inspired by a Syncretic Stewardship Model.

(16)

Model by Kopel and Brand

In the original model by Kopel and Brand [22] they analyze a duopoly in which a profit maximizing (PM) firm is competing against a social responsible (SR) firm in a homogeneous-product market. The SR firm embraces economic as well as non-economic goals, and can therefore be categorised under the Syncretic Stewardship Model. In the duopoly firms are distinguished by their objective functions. Both firms have the same linear inverse demand function p = max{a − b(qSR+ qP M), 0} and a linear cost function ck(qk) = ckqk with k ∈ {P M, SR} and ck < a. The PM firm is maximizing its profits: πP M = (a − b(qSR+ qP M) − cP M)qP M while the SR firm is optimizing the objective function VSR= (a − b(qSR+ qP M) − cSR)qSR+ θ

b(qSR_+qP M₎2

2

, consisting of its profits and a share of the Consumers Surplus (CS). In the SR objective function, the consumer surplus b(qSR+q₂P M)2 represents the investment in social responsibility. The parameter θ ∈ [0, 1] signifies the weight the SR firm puts on the CS. Subsequently, the authors investigate the following research questions:

RQ1: Does it pay off for a firm to be socially concerned, i.e. can it yield a competi-tive advantage to pursue goals different from profit maximization?

RQ2: What is the impact of an increasing concern for consumer welfare on prices, quan-tities, industry profits and welfare?

Kopel and Brand also have an extensive analysis about whether a firm should hire a manager or not, however this is outside of the scope of this thesis and therefore dis-regarded for the moment. They use backward induction and standard optimization techniques to find the equilibrium values and analyze how they depend on the weight put on CSR.

Most important results

In the results they make the distinction between the situation in which both firms have equal marginal costs, and the one in which the firms have different marginal costs. When the firms have equal marginal costs (cSR = cP M), the authors find that in the equilib-rium, the SR firm produces more (qSR> qP M) and therefore had a larger market share. This in its turn provides the SR firm with a larger profit than PM firm: πSR > πP M

(17)

if θ was exogenously given. They state that this result is in line with earlier research by Fershtman [13] and Kelsey and Milne [20]. Kopel and Brand join the conclusion that by adding a non-profit component in the objective function, firms could obtain an advantage in a situation with strategic interaction [22]. The authors also analyze the relationship between θ and VSR and θ and πSR. This, because even though θ was exoge-nously presented, its value can be endogeexoge-nously affected by the choice of stakeholders and the arrangement of the board. They find that VSRis increasing in θ and monotonic. However, the relationship between θ and πSR proves not to be monotonic, since πSR in-creases until a certain θ∗, but decreases again if θ > θ∗. They therefore conclude that can pay off for a firm to pursue non profit goals alongside the goal of profit maximization.

In the case where the firms had different marginal costs (cSR 6= cP M_{), they}

subde-vised the situation in two parts. In the first part, the SR firm had lower marginal costs than the PM firm (cSR _{< c}P M_{) and in the second part, this was the other way around}

(cSR _{> c}P M_{). By analyzing the first part (c}SR_{< c}P M_{), it appears complicated to draw}

one simple conclusion from the results, since these often depend on how big the differ-ence in marginal costs is. However, they did find that if this differdiffer-ence is substantially large (cSR << cP M) a monopoly appears because the PM firm will not enter the market. They also find that the overall welfare (welf are = πP M + πSR+ CS) is increasing in θ for cSR < cP M, but that it is decreasing in θ for cSR<< cP M. They do not provide an explanation for this fact. The second part of the analysis consists of the PM firm having the cost advantage (cSR > cP M). They do not extend the analysis to a broad level, but do mention that the overall welfare is increasing in θ. To summarize, a firm with non-profit motives can actually obtain higher profits than a PM firm given that it does not put too much weight on the consumers’ interest in the objective function (in the case cSR = cP M). Furthermore, a monopoly market appears when there is a cost-advantage of the SR firm as θ increases in which only the SR firm is active.

(18)

2.4 Modeling CSR in this thesis

Kopel and Brand [22] present some ideas for future research. This thesis presents two new market models. As Kopel and Brand state, they did not consider timing issues in their setting. However, it could be interesting to explore this and see if there will be a first- or second-mover advantage in the mixed market. Subsequently, as in the original model only a duopoly is considered, in this thesis a model with multiple firms (N=100) will be presented. These N=100 firms will consist of m SR firms and (N-m) PM firms. In the next chapter, the Stackelberg model will be discussed and in the fourth chapter the model with multiple firms will be discussed.

(19)

Chapter 3

Stackelberg Model

3.1 Theoretical background

The first model that is used in this thesis is the Stackelberg model. The classical von Stackelberg model consists of a duopoly in which two players make decisions regarding their quantities sequentially. One player is assigned with the role of leader, while the other player is assigned with the role of follower. The leader can make decisions about his quantity first. The follower will want to answer in the best way possible and play its best response to the quantity choice of the leader. So for every quantity choice of the leader, the follower will have a matching response. All these response points together constitute the reaction function of the follower. The leader is assumed to know this reaction curve, and choose the quantity matching the point on the curve that yields him the highest payoff. It is therefore of importance that the leader in the game knows ex-ante that the follower will observe its choice and will base its quantity upon the leader’s action. The chosen quantities constitute a von Stackelberg equilibrium.

3.1.1 von Stackelberg equilibrium

The formal definition of such equilibrium is formulated by Furth and Tuinstra (2012) [16] in the following way: Consider a game in which two players are active, one being the leader (L) and one being the follower (F). Each player has its own strategy space Xi, i ∈ {L, F }, a connected interval of R+. Each player has a payoff function πi. Player

(20)

func-CHAPTER 3. STACKELBERG MODEL

tion ϕ : XL XF. That is, when the leader chooses xL ∈ XL, the follower chooses ϕ(xL) ∈ XF. Payoffs are πL(xL; ϕ(xL)) to the leader and πF(ϕ(xL); xL) to the follower.

Definition 3.1: A von Stackelberg equilibrium of the above described sequential game, is a pair (x∗, ϕ∗) with x∗ ∈ X_L, ϕ∗ : XL XF and the following properties:

• πF(ϕ∗(xL); xL) ≥ πF(xF; xL) for all xL∈ XLand xF ∈ XF;

• π_L(x∗; ϕ∗(x∗)) ≥ πL(xL; ϕ∗(xL)) for all xL∈ XL.

When the best response rF : XL XF of the follower is a function, clearly ϕ

∗_(x L) =

rF(xL). When it is not a function, but a correspondence rF : XL P (XF) one should have ϕ∗(xL) ∈ rF(xL), with P(XF) the set of all subsets of XL[16].

3.1.2 Literature review

There has been some past research regarding the first-mover or second-mover advantage in Stackelberg models. It can not be concluded that it will always be favorable for each firm to be in the attributed roles. If one extends the game in a way that players can choose their role it is not the case that both players will always want to be leader [11]. In a paper by Dowrick [11], the conditions under which firms will agree regarding the choice of leader or follower as a role in the von Stackelberg duopoly model are analyzed. He found that firms will generally will not end up in the Stackelberg equilibrium if the firms have similar cost and demand functions. He states that the Stackelberg equilibrium will not be a Nash equilibrium if firms can choose the role of leader and follower themselves. The equilibrium will only be viable if a firm can benefit from acting as a follower, responding to the leader. Dowrick [11] formulatea some conditions under which this is the case. These conditions ensure the viability of the equilibrium, even when firms are given the chance to pick the roles of leader and follower respectively themselves. He finds that if firms both had positive reaction functions and there was an adequate inequality in cost or demand functions for one firm to prefer to be in leading role, the choice of roles will also constitute a Nash equilibrium. This results in the viability of the Stackelberg equilibrium, even if firms can make decisions regarding their roles themselves. It will be interesting to see what happens to the equilibrium functions in the model that is being

(21)

CHAPTER 3. STACKELBERG MODEL

discussed in this section if one creates an asymmetry in cost functions and under which conditions the choice of role could also constitute a Nash equilibrium.

Others that investigated the effect of the distribution of roles and the advantages that result from this were Kopel and L¨offler [23]. In their research, they try to find the circumstances under which a firm with a first-mover advantage gets surpassed by a follower. They assume there is a leader choosing a quantity and a follower responding to this chosen quantity. Furthermore they give the firms the opportunity to make decisions regarding their internal organization and their production technology. They establish this by providing each firm with the choice between hiring a manager or not, and the choice between investing in Research & Development (R&D) or not. They assume that an investment in R&D results in a reduction of production costs per unit. They find that in the equilibrium, both firms are investing in R&D and that only the follower delegates his decision. They also find that the follower can conquer the first-mover advantage of the leader and make a higher profit. An explanation for this is that even though the leader also has the option to delegate, he can not make use of this because this option becomes worthless if you are the first-mover in the market. Therefore one can conclude that in their model there is a second-mover advantage present. It will be interesting to see if this is also the case in the model considered next, and if it is present, under which conditions it holds.

Another research that is interesting, is by Goering [17]. He reviews a bilateral monopoly model that consists of an up-stream manufacturer and a down-stream retailer. One of these, either the up-stream or the down-stream firm, is socially concerned and pays attention to the end-customers’ welfare next to its profits. They assume that in both situations the up-stream firm is the Stackelberg leader and the down-stream firm is the follower. In their analysis, they find that the optimal solution can be highly altered by CSR implementations of either the up-stream (leader) or the down-stream (follower) firm. They also find little to no correlation between CSR and profits. In the first model in this thesis, one will see if these results will be confirmed.

The last research that is being discussed is one by Matsumara and Ogawa [26]. In their research, they try to find and define the effect of CSR if firms are being placed in an endogenous timing game. In this duopoly, each player is placed in leader or follower position and can subsequently choose their quantities and level of social responsibility. They find that when neither firm invests in CSR, the only equilibrium is that both

(22)

players want to be in leader position. They are therefore not ending up in the Stackelberg equilibrium. When one firm chooses to invest in CSR while the other does not, there are two equilibria: one for each sequential game. It will be interesting to see what happens if one looks at the sequential equilibria in a more extensive way, and look at the differences between the role distribution regarding leader and follower respectively.

3.2 Stackelberg competition between SR and PM firms

Both situations where the social responsible firm moves first as well as the situation in which the profit maximizing moves first will be analyzed.

In this thesis the same linear demand function is used as the one by Kopel and Brand [22], p = max{a − b(q_iSR+ q_jP M), 0} in which i, j ∈ {L, F } and i 6= j. Furthermore a, b > 0 also holds. The production costs of either firm are constituted by: cSR(q_iSR) = cSR_i qSR_i and cP M(q_jP M) = cP M_j q_jP M, with again i, j ∈ {L, F } and i 6= j another condition is that cSR_i , cP M_j < a. If one puts these functions and conditions together, one obtains the following functions: π_jP M(qSR_i , q_jP M) = (a − b(q_iSR+ qP M_j ) − cP M)q_jP M π_iSR(qSR_i , q_jP M) = (a − b(q_iSR+ qP M_j ) − cSR)q_iSR V_iSR(qSR_i , q_jP M) = (a − b(q_iSR+ qP M_j ) − cSR)q_iSR+ θb(q SR i + qP Mj )2 2

in which i,j ∈ {L, F } and i 6= j. The SR firm maximizes its objective function V_iSR, in which b(q SR i +q P M j ) 2

2 constitutes the consumer surplus, i.e. the part that represents the

social responsibility in the objective function and θ displays the amount of consumer surplus that is accounted for in the objective function. The PM firm maximizes its profits π_jP M, with i ∈ {L, F } and does not take the consumer surplus into account. The game is as follows: First the firm that is assigned with the role of leader decides its quantity. Subsequently, the follower selects his quantity. We first take a look at the situation in which the PM firm is leader, and the SR firm is follower. Thereafter we analyze the situation in which the SR firm is leader and the PM firm is follower. From each of these situations, we construct optimal equilibrium functions regarding quantities, profits and welfare. In the section that follows after this, an extensive analysis regarding these functions is conducted.

(23)

3.2.1 PM firm leader and SR firm follower

The optimization problem is solved via backward induction in three steps. If the PM firm is the leader and the SR firm is the follower, the first step is to take a look at the optimization problem of the SR firm.

Step 1: The social responsible firm solves: max qSR F V_FSR = max qSR F (a − b(q_FSR+ qP M_L ) − cSR)q_FSR+ θb(q SR F + qLP M)2 2

By solving the first order condition, this results in the best response function of the social responsible firm:

ϕSR_F (q_LP M) = −(a − c

SR_{+ q}P M

L (bθ − b))

b(θ − 2) Step 2: The profit maximizing firm solves:

max qP M L π_LP M(ϕSR_F (q_LP M), qP M_L ) = max qP M L −qP M L (a(1 − θ) + cSR+ cP M(θ − 2) − bqP ML ) θ − 2

From this objective function and the first order condition, the optimal quantity of the profit maximizing firm as leader is constructed. Subsequently, by filling in this equi-librium quantity qP M ∗ in the best response function of the social responsible firm, the equilibrium quantity of the SR firm as follower is obtained:

qP M ∗_L (θ) = a(−θ + 1) + c SR_{+ c}P M_{(θ − 2)} 2b qSR∗_F (θ) = ϕSR_F (q_LP M ∗) = a(θ 2_{− 2θ − 1) + c}SR_{(3 − θ) + c}P M_(−θ2_{+ 3θ − 2)} 2b(θ − 2)

To obtain some insight regarding the relationship between the optimal quantities and θ, derivatives are calculated:

∂q_LP M ∗ ∂θ = 2b(cP M − a) 4b2 < 0 ∂qSR∗_F ∂θ = 5a − cSR− 4cP M_{+ (θ}2_{− 4θ)(a − c}P M₎ 2b(θ − 2)2 > 0 if 5a − 4cP M − cSR a − cP M < 4θ − θ 2

We know the quantities of the PM firm are decreasing in θ because cP M _{− a < 0 and}

4b2 > 0. We know the quantities of the SR firm are increasing in θ if 5a − cSR− 4cP M₊

(24)

since 2b(θ − 2)2 > 0.

Step 3: If we fill in these equilibrium quantities, it results in the following equilib-rium profit functions and equilibequilib-rium utility function:

π_LP M(qSR∗_F , qP M ∗_L ) = −(a(−θ + 1) + c SR_{+ c}P M_{(θ − 2))}2 4b(θ − 2) πSR_F (qSR∗_F , qP M ∗_L ) = a(−θ + 1) + c SR_{(2θ − 3) + c}P M_{(θ + 2)} 4b(θ − 2)2 ×(a(−θ2+ 2θ + 1) + cSR(θ − 3) + cP M(θ2− 3θ + 2)) V_FSR(qSR∗_F , qP M ∗_L ) = 1 8b(θ − 2)(−(a 2_(−3θ2_{+ 6θ + 1) + (c}SR₎2_{(−4θ + 9) + ac}SR_(4θ2_{− 6θ − 6)} +cSRcP M(4θ2+ 14θ − 12) + acP M(2θ2− 6θ + 4) + (cP M)2(θ2− 4θ + 4))) I tried to obtain insight regarding the relationship between the profits, utility func-tion and θ. However, formulating an analytical solufunc-tion appears too complicated. The relationship between the functions and θ will be analyzed graphically in section 3.3.

3.2.2 SR firm leader and PM firm follower

The same three steps will be applied to the model in which the SR firm is the leader and the PM firm is the follower.

Step 1: the profit maximizing firm solves: max qP M F πP M_F = max qP M F (a − b(q_LSR+ q_FP M) − cP M)q_FP M

By solving the first order condition, this results in the best response function of the profit maximizing firm:

ϕP M_F (qSR_L ) = −(c

P M _{− a + bq}SR L )

2b Step 2: The social responsible firm solves:

max qSR L V_LSR(qSR_L , ϕP M_F (qSR_L )) = q SR L (a − cP M − 2cSR− bqSRL ) 2 + θ(a − cP M + bq_LSR)2 8b

Solving this optimization problem, results in the optimal quantity of the SR firm qSR∗ and by filling in qSR∗ in the best response function of the PM firm, one obtains the

(25)

following equilibrium quantities: q_LSR∗= a(2 + θ) − 4c SR_{+ c}P M_{(2 − θ)} b(4 − θ) qP M ∗_F = ϕP M_F (q_LSR∗) = a(1 − θ) + 2c SR_{+ c}P M_{(θ − 3)} b(4 − θ)

We will again try to acquire some awareness regarding the relationship between θ and the optimal quantities by taking derivatives:

∂qP M ∗_F ∂θ = cP M − 3a + 2cSR b(θ − 4)2 < 0 ∂qSR∗ L ∂θ = −2(cP M_{− 3a + 2c}SR₎ b(θ − 4)2 > 0

We can see the quantities of the PM firm are always decreasing in θ and the quantities of the SR firm are always increasing because b(θ − 4)2> 0 and cP M − 3a + 2cSR _{< 0}

Step 3: to obtain the equilibrium profit functions and the equilibrium utility function, one fills in the optimal quantities:

π_FP M(qSR∗_L , qP M ∗_F ) = (a(−θ + 1) + 2c SR_{+ c}P M_{(θ − 3))}2 b(θ − 4)2 πSR_L (qSR∗_L , qP M ∗_F ) = 1 b(θ − 4)2(a 2_(−θ2_{+ 4θ + 2) + (c}SR₎2_{(−4θ + 8) + ac}SR_(θ2_{− 4θ − 8)} +cSRcP M(−θ2+ 4θ − 8)) V_LSR(qSR∗_L , qP M ∗_F ) = 1 2b(θ − 4)(a 2_{(−2θ − 1) − 4(c}SR₎2_{+ ac}SR_{(2θ + 4) + c}SR_cP M_{(−2θ + 4)} +acP M(2θ + 2) − (cP M)2)

To obtain insight regarding the relationship between the optimal quantities, utility and θ we take the first derivatives:

∂πP M ∗_F ∂θ = −2(cP M _{− 3a + 2c}SR_{)(a − 3c}P M_{+ 2c}SR_{− aθ + c}P M_θ) b(θ − 4)3 < 0 ∂πSR∗_L ∂θ = θ(cP M − 3a + 2cSR₎2 b(θ − 4)3 ≤ 0

3.3 Results

Now that the equilibrium have been calculated, an analysis of the results will be con-ducted. The analysis will be done regarding two separate cases, namely the situation in

(26)

which cSR= cP M, and the situation in which cSR6= cP M_.

3.3.1 First case: cSR_{= c}P M _{= c}

In the case of equal marginal costs, the functions change to the following: qP M ∗_L = (a − c)(1 − θ) 2b q_FSR∗= (a − c)(θ 2_{− 2θ − 1)} 2b(θ − 2) πP M_L (q_FSR∗, q_LP M ∗) = (a − c) 2_{(1 − θ)}2 4b(θ − 2) π_FSR(q_FSR∗, q_LP M ∗) = (a − c) 2_(θ3_{− 3θ}2_{+ θ + 1)} 4b(θ − 2)2 V_FSR(q_FSR∗, q_LP M ∗) = (a − c) 2_(3θ2_{− 6θ − 1)} 8b(θ − 2) q_LSR∗= −(a − c)(θ + 2) b(θ − 4) qP M ∗_F = (a − c)(θ − 1) b(θ − 4) πP M_F (q_LSR∗, q_FP M ∗) = (a − c) 2_{(θ − 1)}2 b(θ − 4)2 π_LSR(q_LSR∗, q_FP M ∗) = −(a − c) 2_(θ2_{+ θ − 2)} b(θ − 4)2 V_LSR(q_LSR∗, q_FP M ∗) = −(a − c) 2_{(2θ + 1)} 2b(θ − 4)

One knows, that in the case of equal marginal costs, the firm that produces the most will also have the highest profits. It is therefore useful to see for which parameter values and value of θ holds that qSR∗ > qP M ∗and the other way around. If one calculates this in the case in which the PM firm is leader, and the SR firm is follower, it appears that q_LSR= qP M_F holds if θ = 5₄ −1

4

√

17. Therefore one can conclude the following:        πP M L > πFSR for 0 ≤ θ < 54 − 1 4 √ 17 πP M L = πFSR for θ = 54− 1 4 √ 17 π_LP M > π_FSR for 5₄ −1₄√17 < θ ≤ 1

If one calculates this in the case in which the SR firm is leader, and the PM firm is follower, it appears that θ > −1₂ is the condition under which qSR∗ > qP M ∗. Because

(27)

this is always the case, one can conclude that in the case of SR leader and PM follower, the SR will always have higher profits than the PM firm : π_LSR> π_FP M ∀θ ∈ [0, 1]. Furthermore we can conclude that the PM firm always has decreasing quantities in θ, since ∂qLP M ∗ ∂θ = − (a−c) 2b < 0 and ∂qP M ∗ F ∂θ = − 3(a−c)

b(θ−4)2 < 0. In addition, the SR firm has

increasing quantities in θ, because ∂qLSR∗

∂θ = 6(a−c) b(θ−4)2 > 0 and ∂qSR∗ F ∂θ = (a−c)(θ2_−4θ+5) 2b(θ−2)2 > 0, because θ ∈ [0, 1] and a − c > 0.

We obtain insights regarding the relationship between the profits and θ by also tak-ing derivatives of each profit function with respect to θ. For the PM firm, ∂πP M ∗L

∂θ =

−(a−c)_4b(θ−2)2(θ2−4θ+3)2 ≤ 0, because (θ2 − 4θ + 3) ≥ 0, and

∂πP M ∗ F

∂θ = −

6(a−c)2_(θ−1)

b(θ−4)3 < 0, for

θ ∈ [0, 1] we can therefore conclude that the PM firm has decreasing profits in θ. For the SR firm, ∂πSR∗L ∂θ = 9θ(a−c)2 b(θ−4)3 ≤ 0, because b(θ − 4)3 < 0. For ∂πSR∗ F ∂θ = (a−c)2_(θ3_−6θ2_+11θ−4) 4b(θ−2)3

it is not easy to draw one simple conclusion. Because (a − c)2 > 0 and 4b(θ − 2)3 < 0 the sign of the derivative depends on the term θ3_{− 6θ}2_{+ 11θ − 4. To keep things orderly,}

we name this term X = θ3_{− 6θ}2_{+ 11θ − 4. If X < 0, the SR firm has increasing profits}

in θ, but if X > 0, it has decreasing profits in θ. By solving the equation X = 0 we find ˜ θ = 2 − (1 − √ 26√27 27 ) 1 3 − 1 3(1− √ 26√27 27 ) 1 3

≈ 0.4768, so the SR firm has increasing profits for θ < 0.4786 and decreasing profits for θ > 0.4768. Summarized, we can conclude the SR firm has decreasing profits in θ when he is leader. However, when the SR firm is follower, he has increasing profits in θ if θ < 0.4786 and he has decreasing profits if θ > 0.4786. A graphical illustration is presented in figure 3.1.

(28)

These results are summarized in the following propositions:

Proposition 3.1: If marginal costs are equal and the PM firm is the leader, it obtains a higher profit than the SR firm for θ < θ∗ = 5₄ +1₄√17. For θ > θ∗, the SR firm obtains the highest profit.

Proposition 3.2: If marginal costs are equal, the SR firm obtains a higher profit than the PM firm when it is in leader position.

Proposition 3.3: If marginal costs are equal, the PM firm always has decreasing quanti-ties and profits in θ.

Proposition 3.4: If marginal costs are equal, the SR firm always has increasing quantities in θ. When the SR firm is in leader position, its profits are decreasing in θ. When the SR firm is follower, it has increasing profits for θ < ˜θ ≈ 0.4768 and decreasing profits for θ > ˜θ.

We see, as the SR firm pays more attention to the CS, he will produce more. This has as a result that the prices will drop. Subsequently, the PM firm will produce less and will therefore end up with a lower profit.

If we take a look at the overall welf arei,j = πSR_i + πP M_j + CS with {i, j} ∈ {L, F }, we formulate welfare in the situation with SR leader and PM follower as

welf areL,F = πSR_L + π_FP M + b(q SR∗ L + qP M ∗F )2 2 = −3(2θ − 5)(a − c)2 2b(θ − 4)2

In the case with PM leader and SR follower, the welfare looks as follows: welf areF,L= πSR_F + π_LP M + b(q SR∗ F + qLP M ∗)2 2 = (a − c)2(3θ2− 14θ + 15) 8b(θ − 2)2

The first thing we can conclude is that the welfare is higher if the SR firm is leader, since welf areL,F − welf areF,L₌ −3(2θ−5)(a−c)2

2b(θ−4)2 −

(a−c)2_(3θ2_−14θ+15)

8b(θ−2)2 =

−θ(3θ−8)(a−c)2_(θ−1)2

8b(θ2_−6θ+8)2 ≥

0, because θ ∈ [0, 1] and squares are always positive. Furthermore we can conclude that the welf arei,j is always increasing in θ, since ∂welf are_∂θ L,F = 3(a−c)_b(θ−4)2(θ−1)3 > 0, and

∂welf areF,L

∂θ =

(a−c)2_(θ−1)

(29)

Figure 3.2: Welfare. cP M _{= c}SR_{= 20, a = 250 and b = 70}

The results are confirmed by Figure 3.2, because welfare is indeed increasing in θ and higher when the SR firm is leader . The results regarding welfare are summarized in the following proposition:

Proposition 3.5: In the case of equal marginal costs, the overall welfare is increasing in θ and higher in the situation with the SR firm as leader and the PM firm as follower.

3.3.2 Second case: cSR 6= cP M

We will have a look at two sections in this case, namely the case in which cSR< cP M and the case in which cSR> cP M. We have to do this, because as θ increases, the quantity of the SR firm also increases. This, in return, causes prices to drop. Because of this, profits could become negative or equal to zero. This causes a change in the structure of the model.

SR firm has lower marginal costs: cSR< cP M

The first case that will be discussed, is the case in which the SR firm has lower marginal costs, or as is stated in other literature, has a higher technical efficiency [18]. Because

∂qSR∗ L ∂θ = −2(cP M_−3a+2cSR₎ b(θ−4)2 > 0, and ∂qSR∗ F ∂θ = −(4cP M_+cSR_{−5a+θ(4−θ)(a−c}P M₎₎ 2b(θ−2)2 > 0 the

(30)

equi-CHAPTER 3. STACKELBERG MODEL

librium quantity qSR_i is always increasing in θ for i ∈ {L, F }. For the PM firm, this is different. Because ∂qP M ∗L ∂θ = −(a−cP M₎ 2b < 0 and ∂qP M ∗ F ∂θ = cP M_−3a+2cSR b(θ−4)2 < 0, we can

conclude that the PM firm has decreasing quantities in θ. This means that the SR firm produces higher amounts as θ increases. This causes the prices to drop. This could result in the case where the SR firm ends up with negative profits. In this thesis, it is assumed that the SR firm will have to choose its quantities given the condition of πSR ≥ 0, in order to survive in the market. The value of θ for which holds πSR(θ) = 0 is named θi,j₁ , in which i constitutes the role of the SR firm and j the role of the PM firm so i 6= j. So θL,F₁ is the value for which holds: π_LSR(θ) = 0. Because of the asymmetry in cost functions, the PM firm will have a different value of θ for which holds: πP M(θ) = 0, we name this value θi,j₂ . However, since the PM firm is maximizing profits, we can assume that for the value of θ₂i,j the PM firm is indifferent between producing that quantity and dropping out of the market. It is assumed here, that because qP M

i is decreasing in θ for

i ∈ {L, F }, qP M

i (θ) = 0 for values of θ ≥ θ i,j

2 , which implies that the PM firm will drop

out of the market, because both options yield him a profit of zero. Because cSR _{< c}P M_{, we know θ}i,j

2 < θ i,j

1 . Therefore we have to take into account that

the value of θ1 is calculated given the fact that qP M ∗= 0. In the case where the SR firm

has lower marginal costs, the expression for θ1 is found by solving πSR(qSR∗, 0) = 0. We

find that the equilibrium functions will entail four separate sections for the quantities with respect to θ. One in which both firms are active in the market, one in which only the SR firm is in the market, under the condition that the PM firm produces nothing, one in which the SR firm is a true monopolist with its quantities still increasing in θ, and one in which the quantity of the SR firm is constant and positive, which will be the maximal value of θ which is the value for which holds πSR= 0.

The maximization problems of the firms therefore look as follows, for {i, j} ∈ {L, F } and i 6= j:

for the SR firm:    max_qSR i V SR i (qiSR, qjP M) s.t. πSR≥ 0, qP Mj > 0 max_qSR i V SR i (qiSR, 0) s.t. πSR≥ 0, qP Mj ≤ 0

for the PM firm: {max_qP M j π

P M_(qSR

(31)

Optimal quantities:

First we find expressions for θi,j₁ and θ₂i,j. By solving πSR_F (qSR∗, 0) = 0 we obtain a value for θF,L₁ = 1 and solving q_LP M = 0 provides us the value of θ₂F,L = a−2c_a−cP MP M+cSR. It is

indeed the case that θ₁F,L > θ₂F,L. since 1 − θ₂F,L = cP M_a−c−cP MSR > 0. Given the stated

conditions and objective functions, one obtains the following quantities:

q_LP M ∗(θ)= (_a(−θ+1)+cSR_+cP M_(θ−2) 2b 0 ≤ θ < θ F,L 2 0 θF,L₂ ≤ θ ≤ 1 q_FSR∗(θ) =    a(θ2_{−2θ−1)+c}SR_(3−θ)+cP M_(−θ2_+3θ−2) 2b(θ−2) 0 ≤ θ < θ F,L 2 ϕSR_F (0) = c_b(θ−2)SR−a θ₂F,L≤ θ ≤ 1

In the case in which the PM firm is the leader and the SR firm is the follower. To calculate the quantities in the case with the PM firm follower and SR firm leader, we first calculate the values of θ₁L,F and θL,F₂ . We obtain θL,F₁ = 1 and θ₂L,F = a−3c_a−cP MP M+2cSR

and again 1 − θ₂L,F = 2(c_a−cP MP M−cSR₎ > 0. We obtain the following equilibrium quantities:

q_FP M ∗(θ)=    a(1−θ)+2cSR+cP M(θ−3) b(4−θ) 0 ≤ θ < θ L,F 2 0 θL,F₂ ≤ θ ≤ 1 q_LSR∗(θ) =        a(2+θ)−4cSR+cP M(2−θ) b(4−θ) 0 ≤ θ < θ L,F 2 a−cP M b θ L,F 2 ≤ θ < θ F,L 2 cSR_−a b(θ−2) θ F,L 2 ≤ θ ≤ 1

To get some insight into the implications this has, we will take a look at Figure 3.3. Looking at the graphs, we can conclude the following: q_iSR is indeed increasing in θ, and we see q_LSR ≥ qSR

F , proving this analytically did not provide a clear answer and

only complicated conditions can be formulated, it is chosen to not present those here. For the PM firm the decreasing quantities are also confirmed, as q_jP M is indeed de-creasing in θ. Furthermore it appears qP M_L ≥ qP M

F , however the computations were

again not decisive in sign and are not presented here. One can also see the values of θL,F₂ ≈ 0.8261 which is the case with the SR firm is leader and θF,L₂ ≈ 0.9130 when

(32)

Figure 3.3: Optimal quantities with unequal marginal costs. (cSR, cP M) is (0,20) in the left picture and (15,20) in the right picture, a = 250, b = 70

the SR firm is follower in the left figure and θ₂L,F ≈ 0.9565 if the SR firm is leader and θF,L₂ ≈ 0.9783 when the SR firm is follower in the right figure. As stated before, in all cases, the value of θi,j₁ = 1 with i, j ∈ {F, L} and i 6= j. It appears that as the difference in marginal costs decreases, the value θ₂i,j increases. This is confirmed if we look at the solutions of θ₂F,L= a−2c_a−cP MP M+cSR which increases as the difference in marginal

costs decreases and for θL,F₂ = a−3c_a−cP MP M+2cSR, holds the same. It also appears that θ

L,F 2

is smaller. This is confirmed by calculating the difference between the two values of θi,j₂ : θF,L₂ − θL,F₂ = a−2c_a−cP MP M+cSR −a−3c

P M_+2cSR

a−cP M = c

P M_−cSR

a−cP M > 0. One interesting aspect

of the SR quantities is that it remains constant for θL,F₂ ≤ θ ≤ θF,L₂ if it is the leader. This is the case, because of the condition qP M_F ≤ 0, prevents the SR to produce more for those values of θ, as the SR firm is not a true monopolist in the sense that it needs to keep the PM firm out of the market. You can also see this by the fact that optimal quantities for θ > θF,L₂ , which is the case with the SR firm follower, are equal to the optimal quantities with the SR leader for θ > θF,L₂ , in both cases they are c_b(θ−2)SR−a. Be-cause θF,L₂ > θL,F₂ , we have to keep the quantities of the SR firm constant for values of θL,F₂ ≤ θ < θ₂F,L, because otherwise the best response of the PM firm to the optimal SR quantity is producing a(−θ+1)+cSR_2b+cP M(θ−2). which is the optimal quantity of the PM firm when it is in leader position.

(33)

Equilibrium Profits:

The profits also depend on the different values of θ. They are as follows:

π_LP M ∗(θ) =    −(a(−θ+1)+cSR_+cP M_(θ−2))2 4b(θ−2) 0 ≤ θ < θ F,L 2 0 θF,L₂ ≤ θ ≤ 1 π_FSR∗(θ)=    (a(−θ+1)+cSR_(2θ−3)+cP M_{(θ−2))×(a(−θ}2_+2θ+1)+cSR_(θ−3)+cP M_(θ2_−3θ+2)) 4b(θ−2)2 0 ≤ θ < θ F,L 2 −(cSR−a)2(θ−1) b(θ−2)2 θ F,L 2 ≤ θ ≤ 1

If SR has the role of the follower and PM has role task of leader. When roles are the other way around, profits are given by:

π_FP M ∗(θ) =    (a(−θ+1)+2cSR_+cP M_(θ−3))2 b(θ−4)2 0 ≤ θ < θ L,F 2 0 θL,F₂ ≤ θ ≤ 1 π_LSR∗(θ) =        a2_(−θ2_+4θ+2)+(cSR₎2_{(−4θ+8)+ac}SR_(θ2_{−4θ−8)+c}SR_cP M_(−θ2_+4θ−8) b(θ−4)2 0 ≤ θ < θ L,F 2 (a−cP M_)(cP M_−cSR₎ b θ L,F 2 ≤ θ < θ F,L 2 −(cSR_b(θ−2)−a)2(θ−1)2 θ F,L 2 ≤ θ ≤ 1

Again, to obtain insight into these functions one takes a look at Figure 3.4.

If we take a look at the left graph in Figure 3.4, we can state the following. It appears π_jP M is decreasing in θ for j = L, F . Calculating this analytically, proved to contain many conditions so it is not presented here. It also appears π_LP M ≥ πP M

F . Again, finding

an analytical solution appeared to be too complicated. If we take a look at the profits of the SR firm, one can see that again, it appears that if it is the follower its profits will increase until a certain θ∗ and decrease after that and that its profits always decrease if it is the leader. One can also see that the SR firm will have constant profits for values of θL,F₂ ≤ θ < θF,L₂ . The PM firm will again have higher profits than the SR firm until a certain ˜θ when he is leader. Calculating this value ˜θ analytically appeared too com-plicated, but solving it with the given parameters gave the values ˜θ ≈ 0.0146 in the left figure and ˜θ ≈ 0.1637 in the right figure. Furthermore, the overall profits of the SR firm appear to be higher when his cost advantage is largest and the overall profits of the PM firm are lower when this is the case. In analyzing the welfare, we find that welfare is again increasing in θ, and that the overall value is higher in the case where the SR firm is leader.

(34)

Figure 3.4: Optimal profits with unequal marginal costs. (cSR, cP M) is (0,20) in the left picture and (15,20) in the right picture, a = 250, b = 70

PM firm has lower marginal costs: cP M < cSR

If the PM firm has lower marginal costs, the analysis is a bit less complicated. Because the SR firm has higher marginal costs, we can state that θ₁i,j < θi,j₂ , where θi,j₁ ∈ [0.1] is the smallest value of θ for which π_iSR∗(θ) = 0. Solving this, yields us with the expression for θF,L₁ = a+2c_a+cP MP M_−2c−3cSRSR and a value for θ

L,F 1 = a+c P M_−2cSR a−cSR . θ i,j 2 ∈ [0, 1] is the same

expression as before which is the solution to π_jP M ∗= 0. It is checked that θi,j₁ is indeed smaller than θ₂i,j: θ₁F,L− θF,L₂ = _(a−c2(a−cP MSR_)(a+c)(cP MP M−c_−2cSRSR) ₎ < 0 if cSR > cP M and the same

holds for θ₁L,F− θL,F₂ = −(cP M_(a−c−cSRP M)(c_)(a−cP M−3a+2cSR₎ SR) < 0. For values of θ > θ

i,j

1 the quantity

of the SR firm will remain constant, because it would get a negative profit if it still increased its quantities and this is not allowed. Because the PM firm has decreasing profits and quantities in θ (as stated before), the PM firm will not have to decrease its own quantities for higher values of θ, because the SR firm will not be able to act differently for values θ > θi,j₁ . It is therefore that the quantities and profits for the PM firm will will remain constant for θ > θi,j₁ . We will therefore not need θ₂i,j at all.

The maximization problems of the firms therefore look as follows, for {i, j} ∈ {L, F } and i 6= j:

for the SR firm: {max_qSR i V

SR

(35)

for the PM firm:    max_qP M j π P M j (qiSR, qjP M) s.t. πSR> 0, qjP M ≥ 0 max_qP M j π P M j (qiSR, qjP M) s.t. πSR= 0, qjP M ≥ 0 Optimal quantities

We obtain the following quantities:

q_LP M ∗(θ)=    a(−θ+1)+cSR_+cP M_(θ−2) 2b 0 ≤ θ < θ F,L 1 a(−θF,L₁ +1)+cSR_+cP M_(θF,L 1 −2) 2b θ F,L 1 ≤ θ ≤ 1 q_FSR∗(θ) =    a(θ2_{−2θ−1)+c}SR_(3−θ)+cP M_(−θ2_+3θ−2) 2b(θ−2) 0 ≤ θ < θ F,L 1 a((θ₁F,L)2_−2θF,L 1 −1)+cSR(3−θ F,L 1 )+cP M(−(θ F,L 1 )2+3θ F,L 1 −2) 2b(θF,L₁ −2) θ F,L 1 ≤ θ ≤ 1

In the case in which the PM firm is the leader and the SR firm is the follower, and:

q_FP M ∗(θ)=    a(1−θ)+2cSR_+cP M_(θ−3) b(4−θ) 0 ≤ θ < θ L,F 1 a(1−θ₁L,F)+2cSR_+cP M_(θL,F 1 −3) b(4−θL,F₁ ) θ L,F 1 ≤ θ ≤ 1 q_LSR∗(θ) is:    a(2+θ)−4cSR_+cP M_(2−θ) b(4−θ) 0 ≤ θ < θ L,F 1 a(2+θ₁L,F)−4cSR_+cP M_(2−θL,F 1 ) b(4−θL,F₁ ) θ L,F 1 ≤ θ ≤ 1

The optimal quantities are plotted in figure 3.5. We can conclude from the figure that the situation appears to be quite similar to the previous case in which the SR firms had the lowest marginal costs. In the figure, with the stated parameters, the quantities of both firms are constant for θ > θ₁L,F = 21₂₃ if the SR firm is leader and θ > θ₁F,L= 19₂₁ if the SR firm is follower. The constant quantities for θ > θL,F₁ = 45₄₆ and θ > θ₁F,L= 44₄₅ with the stated parameters are shown in the right figure. Furthermore, it appears that the PM firm has decreasing quantities and the SR firm appears to have increasing quantities in θ. As stated before, an analytical confirmation of these statements is too complicated. We can also see that it appears that again, leaders produce more than followers.

Equilibrium profits

(36)

Figure 3.5: Optimal quantities with unequal marginal costs. (cSR, cP M) is (20,0) in the left picture and (20,15) in the right picture, a = 250, b = 70

π_LP M ∗(θ) =    −(a(−θ+1)+cSR_+cP M_(θ−2))2 4b(θ−2) 0 ≤ θ < θ F,L 1 −(a(−θF,L₁ +1)+cSR_+cP M_(θF,L 1 −2))2 4b(θF,L₁ −2) θ F,L 1 ≤ θ ≤ 1 πSR∗ F (θ)=    (a(−θ+1)+cSR_(2θ−3)+cP M_{(θ−2))×(a(−θ}2_+2θ+1)+cSR_(θ−3)+cP M_(θ2_−3θ+2)) 4b(θ−2)2 0 ≤ θ < θ F,L 1 0 θ₁F,L≤ θ ≤ 1

If SR has the role of the follower and PM has the role of leader. If it is the other way around it looks like this:

π_FP M ∗(θ) =    (a(−θ+1)+2cSR_+cP M_(θ−3))2 b(θ−4)2 0 ≤ θ < θ L,F 1 (a(−θL,F₁ +1)+2cSR_+cP M_(θL,F 1 −3))2 b(θ₁L,F−4)2 θ L,F 1 ≤ θ ≤ 1 π_LSR∗(θ) =    a2_(−θ2_+4θ+2)+(cSR₎2_{(−4θ+8)+ac}SR_(θ2_{−4θ−8)+c}SR_cP M_(−θ2_+4θ−8) b(θ−4)2 0 ≤ θ < θ L,F 1 0 θL,F₁ ≤ θ ≤ 1

By analyzing these equilibrium profits with the same parameters as before, we can con-clude that the same patterns as before become present, as the profits of the SR firm are decreasing in θ as leader, and increasing until a certain θ∗ where after it was decreasing again when it is the follower. Furthermore, if the PM firms’ costs are lower, its profits

(37)

are higher.

Welfare:

The overall welfare is analyzed graphically next, because analytical solutions are very complicated. In figure 3.6, it becomes clear that the relationship between the welfare and θ depends greatly on the (difference in) marginal costs. If the cost advantage of the PM firm is not too large, welfare will be increasing in θ with the stated parameters. However, it appears that if the PM firm has much lower marginal costs, and he is leader, the welfare can be decreasing in θ. The reason for this is graphically shown in figure 3.7. Here we see the components that constitute the welfare. As we can see, if the marginal costs of the PM firm are way lower than those of the SR firm, the ratio between the increase in CS and the decrease in PM is just failing to create an overall increase in welfare. We can see that the slope of the CS in the case when the PM firm is leader is just not steep enough to moderate the decrease in the total profits. It is therefore difficult to draw a simple conclusion regarding the relationship between welfare and θ.

Figure 3.6: Welfare with unequal marginal costs. (cSR, cP M) is (20,0) in the left picture and (20,15) in the right picture, a = 250, b = 70

(38)

Figure 3.7: CS and sum of profits with unequal marginal costs. (cSR, cP M) is (20,0) in a = 250, b = 70

We will conclude this section with unequal marginal costs by formulating the following propositions:

Proposition 3.6: If the SR firm has lower marginal costs, the PM firm will drop out of the market for values of θ > θ₂i,j with i, j ∈ L, F and i 6= j.

Proposition 3.7: If the SR firm has a marginal cost advantage over the PM firm and the SR firm is leader, there will be a section in which he has constant quantities and profits for θL,F₂ ≤ θ < θ₂F,L to keep the PM firm out of the market

Proposition 3.8: If the SR firm has a marginal cost advantage over the PM firm the welfare is increasing in θ and higher when the SR firm is leader.

Proposition 3.9: If the PM firm has lower marginal costs, the PM firm will always have constant positive profits that are higher than those of the SR firm for θ > θi,j₁

Proposition 3.10: If the PM firm has a marginal cost advantage over the SR firm, the welfare can be increasing, as well as decreasing in θ, depending on the choice of parameters.

(39)

Chapter 4

Model with Multiple firms

4.1 Theoretical background

The second model that is being considered in this thesis is a model similar to the model by Kopel and Brand, but instead of a duopoly, a model homogeneous oligopoly with N firms is analyzed. These N firms consist of m SR firms and (N-m) PM firms. The firms make their decisions about quantities simultaneously and the price is endogenously determined according to the inverse demand function. Because the firms compete in quantities opposed to prices, this type of model is called a Cournot oligopoly. In an oligopoly multiple firms are active in the market. We will ultimately try to find the Nash equilibrium for homogeneous Cournot oligopolies. The definition that is being presented next, is the one formulated by Furth and Tuinstra (2002) [16]

4.1.1 Cournot-Nash equilibrium

Definition 4.1: A profile of strategies (actions) q∗ = (q∗₁, ..., q∗_N) is a Nash equilibrium when for all i ∈ N and all q ≥ 0

πi(qi∗, q−i∗ ) ≥ πi(q; q−i)

In the Nash equilibrium, every player gets, given the choices of the other players, the highest payoff. This ensures the Nash equilibrium to be sustainable, since not one single player will have an incentive to deviate. Each player will play a strategy qi, when the

Corporate social responsibility in market models