Collusion under bequest motives

Collusion under bequest motives

Author:

Jakob Hinger

Supervisors:

Maarten Pieter Schinkel,

Leonard Treuren

A thesis submitted for the degree of Master of Science in Economics

Faculty of Economics and Business

Abstract

In this paper I analyze whether a cartel can achieve a better outcome for consumers than ordinary competition if consumers have bequest motives for future generations. The paper can be seen as an extension to the papers ”Can collusion promote sustain-able consumption and production” by Schinkel and Spiegel [44] and ”Can collusion promote sustainable consumption and production? Not beneficially beyond duopoly” by Treuren and Schinkel [49]. It adds a bequest parameter to their models and as-sumes that consumers are willing to pay more for a sustainably produced good if they know that their descendants benefit from it. Based on regulation by the European Commission and research on bequeathing, it is justified to include such a bequest pa-rameter and to base policy recommendations on the derived results. I estimate prices, quantities, investments in sustainability and consumer surplus in both collusionary and competitive scenarios. I find that, even when considering bequest motives, the only case where a production cartel yields the bigger consumer surplus is in a market with two firms, sufficiently homogenous products and cheap sustainability investments for firms. For a market with more than two firms competition always yields the biggest consumer surplus. In conclusion, this suggests that antitrust and regulatory agencies should be reluctant to rely on cartels but rather use traditional regulatory instruments to reach a certain sustainability level.

Acknowledgements

I would like to express my thanks to my supervisors Leonard Treuren and Maarten Pieter Schinkel for offering prompt support during my work and for sharing helpful ideas.

Contents

1 Introduction 3

2 Judicial framework 4

3 Literature 9

3.1 Bequest motives . . . 9

3.2 Recent merger assessment . . . 11

4 Consumers’ utility function 13 5 The model for n firms, δ2 = 0 14 5.1 Quantities and sustainability contributions in the different settings . . . 16

5.2 Welfare effects . . . 19

5.3 Price comparison with the case without bequest motives . . . 23

5.4 The socially optimal output . . . 24

5.5 Compensation by colluding firms, simplified for n=2 . . . 25

6 Analysis for δ2 >0 27 7 Conclusion 31 8 Appendix 34 8.1 Bertrand competition for n=2 firms . . . 34

8.2 Analysis for a multiplicative utility function . . . 38

1

Introduction

Awareness for the scarcity of natural resources and for the importance of a sustainable lifestyle to tackle environmental threats and climate change [27] is constantly increasing, and the need to preserve nature becomes more and more common sense [37]. It is therefore in societies’ interest that relevant rules and regulations are implemented to achieve this goal. This paper specifically deals with corporates’ investments in sustainability and the question whether colluding companies may achieve a better outcome for society than competition, whilst considering not just the living conditions now, but also the living conditions of future generations.

In their paper ”Can collusion promote sustainable consumption and production” [44] Schinkel and Spiegel dealt with the general question under which circumstances cartel agree-ments can increase consumer surplus by considering positive effects on utility through sus-tainability investments. Their finding was remarkable in especially two ways. Firstly, they found that sustainability investments are always highest under a production cartel and sec-ondly, that under specific circumstances, with respect to the parameter spaces of investment costs and substitutability within goods, a production cartel can bring about the biggest consumer surplus.

Recently Treuren and Schinkel published their paper ”Can collusion promote sustainable consumption and production? Not beneficially beyond duopoly” [49], which can be seen as an extension to Schinkel and Spiegel [44]. It extended the above-mentioned findings to the n-firm case with either full or partial monopoly. They concluded that for a market with at least 3 firms, consumer surplus is always bigger under competition than under a full cartel. What is more, the only setting where consumer surplus is not the biggest under competition, is when 2 out of 3 firms in a market form a production cartel, sustainability investment is cheap, and products are sufficiently homogenous.

My paper is an extension to the paper by Treuren and Schinkel [49] and adds a parameter for bequest motives within consumers to their model. It deals with the fact that when making the decision to purchase a product, consumers do not just have their own benefit in mind but also the impact on their descendants or future generations in general and current consumers’

utility increases with the living conditions of their descendants. This feature does not just change their purchasing decision, but through their demanded quantity also the companies’ profit function and their motivation to invest in sustainability. As investment costs are the same, companies in general are likely to invest more in sustainability compared to the findings in Treuren and Schinkel. It is part of my thesis to analyse under which circumstances such an inclusion of bequest motives changes the welfare effects of different cartels and especially under which circumstances cartels may be justified, as they maximize consumer surplus.

Section 2 of this paper shows the judicial framework for my thesis. It deals with Article 101 TFEU (ex Article 81 TEC), which states under which circumstances a cartel may be exempted from cartel regulation, and with previous cases regarding such exemptions, both in the Netherlands and in Europe in general. Also, the consequences of this regulation on my model are covered in this section. Section 3 shows previous research on bequest motives and why it is justified to include them in consumers’ utility. Moreover, I show the similarities between recent merger assessments and assessing competition in light of different levels of sustainability contributions including bequest motives. Section 4 examines the utility function I will use and its properties. Section 5 gives a welfare analysis for the case where the only impact on consumers by higher sustainability contributions is through a higher quantity demanded. It shows the impact of competition as well as of different collusionary scenarios on welfare, on sustainability contributions and on prices. Section 6 gives a similar analysis for the case where sustainability contributions, apart from increasing quantity demanded, also have a direct effect on consumers’ utility. Section 7 concludes and proposes potential future research and antitrust policies.

2

Judicial framework

Article 10 TFEU(ex Article 81 TEC) states that investments in green or in sustainability, which benefit consumers, can be exempted from cartel law. It says that cartel agreements between companies shall not be prohibited if they

”promote technical or economic progress, while allowing consumers a fair share of the resulting benefit, and which does not:

a) impose on the undertakings concerned restrictions which are not indispensable to the attainment of these objectives

b) afford such undertakings the possibility of eliminating competition in respect of a substantial part of the products in question.” [48]

Regarding the first point, the EC stated that such improvements can be either in quality through innovations and reduced pollution or in price reductions. Consumers do have to benefit, and cost savings incurred by colluding companies have to be passed on to consumers [19].

Based on Article 101(3) TFEU, the ACM defined clearly for what reasons a sustainability agreement may be exempted from cartel prohibition:

”1. The arrangement contributes to improving the production or distribution of goods or to promoting technical or economic progress;

2. Consumers receive a fair share of the resulting benefit;

3. The arrangement is necessary to achieve these benefits and does not go beyond what is necessary;

4. The arrangement does not lead to competition being eliminated in a substan-tial part of the market. The arrangement must leave enough room for competi-tion.” [5]

The ACM concluded that price fixing for instance is hardly ever exempted from cartel prohibition. One of the rare cases, where price fixing was exempted anyways, was the Assur-pol case [16], which was about a co-reinsurance pool, in which the participating companies fixed the risk premiums they had to pay for environmental pollution. On the other hand, notable examples when the EC did not grant cartel exemption were the VOTOB case [15], in which independent tank storage companies wanted to agree on a surcharge on produc-ers, which would have offset environmental costs incurred because of polluting producproduc-ers, and the Melkdubbeltje case [2], in which supermarket chains wanted to agree on a fixed price surcharge on milk bottles because of a foot-and-mouth-disease outbreak. Also, pass-on

costs agreements are mostly not seen as justifiable for exemption. On the other hand, stan-dardization and qualitative restriction of supply are more likely to be exempted from cartel prohibition [5].

Within Dutch Regulation, the most famous examples for cartel exemption or denial of exemption were the ”Chicken of tomorrow” case [6], the SER Energieakkoord case [4] (both denied) and the CECED case [17] (exempted).

In the ”Chicken of tomorrow” case broiler chicken meat producers agreed on different practices to improve chickens’ living conditions and to reduce their environmental impact. Examples of such agreements were a reduction in the occupancy rate of the chicken barns, a reduction in antibiotics use, a reduction of certain emissions to reduce the CO2 footprint

or the use of just sustainably grown soy as animal feed. The ACM concluded that such agreements restrict competition, as future consumers would find it harder to buy regularly produced chicken meat, as all the large supermarket chains in the Netherlands were part of the agreement [6]. What is more, they concluded that it is in conflict with the free movement of goods within the EU, as the agreement would have cross-border effects. To check whether consumers would benefit from such an agreement, although the costs of the sustainability investment would be passed on to them, the ACM made a survey estimating consumers’ willingness to pay for better produced broiler chicken. Although consumers were found to be willing to pay more for better produced chicken, it was less than the expected passed on costs, so in total they would have been worse off than before. The ACM therefore decided to not grant exemption from cartel prohibition [6].

Moreover, the ACM assessed whether the agreement on coal power plants closure, the ”SER Energieakkoord”, in the Netherlands is in contrast with Article 101 TFEU. In this case energy providers agreed to close down coal power plants, which are highly pollutant. This would in turn obviously lead to less capacity to produce energy. The ACM concluded that electricity consumers would have to pay a higher price for electricity due to the agreement but would benefit from reduced emissions and a decrease in costs incurred to reduce emissions afterwards. Nevertheless, they concluded that the latter benefit can be assumed to be relatively small, as Dutch consumers just form a small part of the ETS area, on which the cost reduction would be distributed. All in all, also in this case, the ACM decided that the

benefits would not offset the higher prices and the cartel agreement should not be exempted [4].

Regarding the CECED case, the EC exempted a cartel of washing machine producers, who agreed to stop producing the least energy efficient washing machines. The EC reasoned that consumers benefit twofold. Firstly, they profit from lower energy costs, and secondly, they benefit from reduced CO2 emissions and hence, less pollution [17].

For an increased transparency in the assessment, the ACM issued certain principles, which it will follow when assessing sustainability agreements and it says it will only exempt them if they ”enjoy broad social support if all parties involved such as the government, citi-zen representatives, and businesses are positive about the arrangements” [7].

During the model analysis, I will use a consumer surplus perspective to assess the Wel-fare effects of the different cartels. Albeit this decision is ambivalent, there are certain reasons, additional to the statements issued by the EC and the ACM, why this approach is justified. As explained in Motta [34], in their heterogeneity and mass, consumers are different to producers and their willingness for lobbying and complaints are smaller. As individual consumers regularly purchase only a small amount of each product, the effect on them by, for example, a price increase is mostly small. Producers, on the other hand, may profit immensely from a price increase as their profits are normally distributed over a small number of them. For this reason, producers are much more likely to intervene for regulatory exemptions.

Besanko and Spulber [10] proposed a model, in which they show the advantages of collud-ing firms through an information asymmetry on their cost structure. They concluded that in order to offset this information asymmetry, antitrust authorities should assign greater weight to consumer surplus than to producer surplus. Neven and R¨oeller [35] argued similarly and formed a model including different lobbying activities incurred by companies on politicians and antitrust regulators. Like Besanko and Spulber [10] they argued that the weight on consumer surplus shall be higher to counterbalance those lobbying efforts.

Nevertheless, as nowadays the ownership structure of companies has changed and many of them are partly or fully owned by consumers, using a total welfare standard can be justified

as consumers not just benefit through consumer surplus but also through the profits made by companies, in which they have a stake, hence an increased producer surplus. Also, it is theoretically possible to redistribute producer surplus to consumers and therefore a maximum in total welfare would be welfare maximizing for consumers as well. Pittman [39], however, argued that it is ”not very satisfying (...) to note that (...) income redistribution policies could make everyone better off (...) — if in fact they do not. The compensation principle does not pay the rent.”

In line with the standard applied by the EC, I decided to base my analysis on consumer welfare. As will be shown later, calculating consumer welfare is also easier and results hand-ier to show.

Regarding my model addition to include bequests, in 2016 the EC issued a revised pol-icy rule on how to apply Article 101 TFEU, stating

”With this approach, the benefits both to the current consumer in the future, as well to future consumers of the product or service concerned are taken into account: it is about a longer term than right here, right now, and others that do not themselves consume the product.” [22]

Through this statement, the EC specifically stated that also future generations and the impact a certain policy has on them should be considered when assessing specific scenarios and deciding over different cases. In its vision document on competition and sustainability, the ACM highlighted the importance of future consumers and the need for both a static and a dynamic perspective regarding sustainability agreements. They even noted that it can be justified to decrease benefits for current consumers if this leads to a higher benefit for future generations [5]. For example, if overfishing is tackled now and quotas are introduced, this would lead to a decline in supply and most likely higher prices [40]. Nevertheless, it may be the only possibility to ensure sufficient fishing stock for future generations.

With respect to Article 101 TFEU a), which says that all the restrictions made on com-petition and on the behaviour of colluding companies, cannot be avoided to achieve the

sustainability gains, I assume that this prerequisite is satisfied. Unnecessary agreements harming consumers would result directly in higher prices without having a positive effect on consumer surplus through increased sustainability investments. Whether all colluding actions are necessary to achieve higher investments must therefore be dealt with on a case-by-case basis.

Moreover, regarding point b) in the specification, stating that competition must remain, I only compare the effect within a market of either two firms or with n firms that all participate in the cartel agreement. I especially do not consider fringe competitors in my thesis as this is computationally cumbersome and results hard to interpret.

3

Literature

Academic research on exemption from Article 101 TFEU [48] is still very limited. The two papers mentioned at the beginning by Schinkel and Spiegel [44] and Treuren and Schinkel [49] serve as a great starting point into the topic.

In the next section, I give a brief overview on research on bequest motives and recent academic merger assessment.

3.1

Bequest motives

Studies on household behaviour found that people care about bequests they make to future generations. Unfortunately, most of the surveys cover only personal bequests in cash or assets and literature on bequest motives through general living conditions is rather limited. Kopczuk and Lupton [31] found that the saving behaviour among people often differs substantially and that this fact stems from bequest motives. People with a strong bequest motive spend roughly one quarter less on consumption. What is more, elderly people, es-pecially those living in a single household, pass about 80% of their net wealth to future generations. About half of it can be attributed to a bequest motive.

Horioka et al [26] researched on the different models of household behaviour and checked which one of these is the most applicable one in the United States and Japan. They thereby categorized household behaviour as either following the life cycle model, the altruism model

or the dynasty model. The life cycle model, as probably the most famous one, was established by Modigliani and Brumberg [33] and claims that people are per se selfish and do not care about future generations but just themselves. Although they save, they do not do so for bequeathing, but for retirement and for precautionary reasons. In the altruism model by Barro and Becker [8], ”utility of parents depends on their own consumption, their fertility, and the utility of each of their children”. In short, utility is maximized if also consumption by future generations is considered, which is guaranteed by making bequests. In the dynasty model by Weil [54], on the other hand, families also care about future generations and leave bequests, but they do so for dynastic reasons to maintain the family line. Their bequests are therefore often unequally distributed, and mostly bequeathed to the first-born descendant. Horioka et al concluded that the model fitting the most in both the US and Japan is the life cycle model, although it is much more relevant for Japan. The altruism model is of minor importance in Japan, albeit being pretty popular in the US. The dynasty model is of negligible relevance in both countries, although applicable more in Japan. To sum up their conclusions, bequest motives do play a strong role, especially in the US.

Bernheim et al [9] studied strategic bequest motives, which means the possibility that parents make bequests not just motivated by the wellbeing of their descendants but also as a threat and reward to promote good behaviour towards them. This makes receiving bequests conditional on children’s actions before. They find that strategic bequest motives indeed play a major role in practice and are widespread within American families.

J¨uerges [29] examined life cycle savings with and without a bequest motive and the main reason for saving in Germany. He found that it is not possible to infer a bequest motive from having children and that bequest motives effect the saving behaviour of households. Nevertheless, neither the life cycle model with bequests nor the respective one without, can be concluded to be the prevalent one, as both models give insignificant results after running both OLS and IV regressions using cross-section data from the German Socioeconomic Panel. Fujiu and Yano [24] studied the importance of altruism while making bequests. They estimated the possibility of children being more altruistic towards their parents than the other way round, which they called ”backward altruism”. They found that altruism by children can be a reason for bequests as in their ”setting, parents give transfers to children

so as to enlarge repayments from children” [24]. They tested this with an intergenerational model for an infinite horizon. In the end, they concluded that by taking time inconsistency into consideration, there exists a ”stationary rational expectations equilibrium (SREE)”, in which parents make bequests to their altruistic children, even if altruism exists just from children to parents and not the other way around.

As already mentioned above, literature on bequest motives in a setting extended from personal bequests made to descendants is very limited. Brechet and Lambrecht [12] studied privately owned national resources, such as timber forests and the impact of bequest motives on their exploitation. Their main finding is a trade-off between the bequest motive and the resource’s substitutability with capital. They found that for a low substitutability with cap-ital the resource will never be used up, no matter whether owners care for future generations or not at all. On the other hand, if the resource has a strong substitutability with capital, owners have to have a strong bequest motive in order not to use up the resource.

To conclude, as shown in the articles mentioned above, bequest motives play a major role in humans’ actions and generally people care about future generations and preservation of sustainable living conditions for their descendants. Although bequest motives may not be as strong as intuitively expected, studies found that for a large part of society, making bequests is important, may the reasoning behind this attitude be less altruistic but rather strategic. For this reason, it seems justified to incorporate bequest motives in my model and to include a bequest parameter in peoples’ utility function. Nevertheless, one should be careful in this setting as Brechet and Lambrecht [12] found that resources with a low substitutability with capital will be preserved whether bequest motives exist or not. For simplicity, I abstracted from this case in my model and I analyse the utility function based on resources with a strong substitutability with capital.

3.2

Recent merger assessment

The assessment of mergers and especially research joint ventures so far has mostly excluded the effects on future generations and sustainability. Nevertheless, the reasoning if including bequests is similar and is in line with recent analysis. So far, the main argument by companies of why to allow mergers and research joint ventures was an efficiency defence, which means

that the benefits of a merger outweigh the extra costs.

Gugler and Siebert [25] assessed research joint ventures in the semiconductor industry and estimated the effects they bring about, especially regarding an efficiency defence. They found that market shares of participating companies normally increase due to the joint research agreement, a fact that is in contrast with Salop’s circular city model [43], which predicts falling market shares, which in turn can just be offset by immense efficiency gains. They explained their finding with the introduction of new products by the colluding firms, which attract consumers away from their competitors. What is more, they concluded that especially consumer benefits through research joint ventures outweigh potential negative effects, a finding which is less clear in ordinary production cartels. Especially their finding of new introduced products and its importance raise the necessity of a dynamic perspective which includes future generations.

On the other hand, Porter argued that for assessing long term benefits one should look at productivity growth as ”it is the single most important determinant of long term consumer welfare and a nations standard of living” and not profitability growth or price increases [41]. His argument is in line with previous publications by Solow [47], Schumpeter [46] and Abramowitz [1]. He therefore argued that also for assessing merger effects, the immediate possible price increase should be valued less than the possible innovative potential. The long-term view combined with innovative potential is in line with my model, as bequests and long-term growth can be seen as his ”holy grail”. Similar analysis can be found in Carlton [13] and Pittmann [39]. Also, the US Horizontal Merger Guidelines [50] allow for including the benefits of future consumers and the US Department of Justice comments that mergers can be allowed even if ”they cannot be expected to result in direct, short-term, procompetitive price effects because consumers may benefit from them over the longer term” [14].

A different conclusion was reached by Oldale [36]. She approached the consumer benefit analysis from a behavioural perspective and found, based on Iyengar and Lepper [28], that consumers dislike choice. She concluded that consumers could therefore benefit form mergers that raise prices, but also reduce the number of options on the market. This finding is in stark contrast to standard research, but also allows for including future generations in an

analysis on cartels. Although the main assumption on profitability is different, if assuming future generations to have the same preferences on choice, the impact on them by a merger should be similar.

The EC itself has given a higher weight on future generations in their latest merger assess-ment including a new theory of harm. When assessing recent mergers in the pharmaceutical and agro industry such as Dow/DuPont [23], they were concerned on the potential negative effect this merger could have on innovation. Based on their analysis the potential merger effects on innovation are twofold. On the one hand, it leads to an increase in companies’ market share, but on the other hand it leads to cannibalization of their own sales. Only if the first effect is sufficiently higher than the latter one, the firm has incentives to innovate.

All in all, as both official policies and statements by European and US institutions as well as analysis by academics and merger and cartel experts allow for adapting for bequest motives, it seems justified to include a bequest perspective and give future generations a stake when assessing welfare effects of production or sustainability cartels.

4

Consumers’ utility function

Consumers’ utility function is taken from ”Can collusion promote sustainable consumption and production” [44] and adapted to include bequest motives:

ui = n X i=1 (a + ∞ X t=0 δtvi)qi− 1 2 n X i=1 q_i2− γ n X i=1 n X j6=i qiqj − n X i=1 piqi+ ∞ X t=0 n X i=1 δt₂vi = (1) n X i=1 (a + vib)qi− 1 2 n X i=1 q_i2− γ n X i=1 n X j6=i qiqj− n X i=1 vib2 , (2) where P∞ t=0δ t ₌ 1 1−δ = b and P∞ t=0δ t 2 = 1 1−δ2 = b2 .

This is the utility function of a representative consumer with symmetric preferences over n goods. a > 0 is a utility parameter, γ represents a factor for substitutability within two goods, which is bounded between 0 and 1, with 0 meaning no substitutability and 1 meaning the two products being perfect substitutes, q=(q1, ..., qn) are quantities produced,

con-tributions and δ, δ2 < 1 are future discount factors. The utility function neither controls

for non-consumers nor for spillovers effects of the sustainability investment. This strong simplification, nevertheless, is in line with Article 101 TFEU.

Abstracting to ignore non-consumers of a good is justified as Article 101 TFEU explicitly says that just consumers should have a fair share of the benefit, which means that the impact on people not buying does not have to be controlled for here [48].

Basis for adapting the utility function was Blanchard’s intertemporal utility function [11] with a constant discount factor δ. The second to last term on the right means that sustain-ability investments result in higher consumer welfare through a higher quantity demanded and the last one means that the sustainability investments have a pure effect on utility.

An issue with considering future generations is that they are not present, and their needs cannot yet be included in the model. Whether future generations will consume the product at all, have different preferences or will face a change in product variety, can just be assumed and is complicated to address in a utility function set in the present. It is for this reason that bequest motives are included as a proxy for future generations’ utility and needs. By making the buying decision of the current generation dependent on future generations, future generations have an immediate impact on the quantity demanded and the willingness to pay of their ascendants. As shown above, literature and studies in the specific field indeed show that the willingness to leave bequests is part of peoples’ everyday decisions.

Moreover, in this utility function consumers are willing to pay a higher price for higher quantities, higher sustainability contributions and if they have higher bequest motives or care more about future generations. Consumers’ utility is strictly increasing in quantities, sus-tainability contributions and bequest motives as δui(q,v,b)

δqi > 0,

δui(q,v,b)

δvi > 0 and

δui(q,v,b)

δb > 0.

5

The model for n firms, δ

= 0

In the first part of the analysis I will assume δ2 = 0, which means sustainability contributions

Companies’ profit function is

πi = piqi− kqi−

rvi

2 , (3)

with k being variable costs per unit and r a proxy for investment costs for sustainability. Setting P∞

t=0δt = 1

1−δ = b, consumers’ utility function yields an inverse demand function,

after setting δu1

δqi = 0, as pi = a − qi− γ n X j6=i qj + vib , (4)

and, after plugging (4) into (3) and setting a − k = A, companies’ profit function as

πi = (A − qi− γ n X j6=i qj + vib)qi− rv2 i 2 . (5)

Companies in this model compete in two stages. In the first stage they choose their sustainability investments and in the second stage they compete in quantities based on the sustainability contributions set in the first stage, which means they compete in Cournot. In the appendix an analysis is presented for the same utility function with companies competing in prices, which means in Bertrand.

We can solve this model with backward induction. First, we let companies maximize their profits with respect to quantities and then with respect to sustainability contributions, keeping in mind the quantities that will be chosen afterwards, conditional on the sustainabil-ity contributions in the first stage. In this model there are four possible different competitive or collusive scenarios. If companies compete in both stages, they face full competition, if they compete in the first stage in sustainability, but collude in quantities produced, they form a production cartel, if they collude in sustainability, but not in quantities, they form a sustainability cartel and if they collude in both stages, they form a full cartel.

Similar to the calculations in Treuren and Schinkel, companies maximize their profits in the different settings the following way:

In full competition: n X j6=i δπi δqj δqj,• δvi + δπi δvi = 0 , (6)

in a production cartel: δπi δqi δqi,coll δvi + n X j6=i δπi δqj δqj,coll δvi + δπi δvi = 0 , (7) in a sustainability cartel: n X j6=i δπi δqj δqj,• δvi + δπi δvi + n X j6=i ( n X j6=i6=l δπj δql δql,• δvi +δπj δqi δqi,• δvi +δπj δvi ) = 0 , (8)

and in full collusion:

δπi δvi + n X j6=i δπj δvi = 0 , (9)

where ”•” stands for competition and ”coll” for collusion in the quantity setting stage. Define β1 = γ(n − 1) + 2, β2 = γ(n − 2) + 2 and β3 = γ(n − 3) + 2.

5.1

Quantities and sustainability contributions in the different

settings

Companies competing in production maximize their profits (5) through setting δπi

δqi = 0. This yields quantities as qi,• = A + vib − γ Pn j6=iqi 2 . (10)

Symmetry in quantities between firms yields

qi,• =

(A + vib)β1− γPn_i6=j(A + vjb)

(2 − γ)β1

. (11)

Companies forming a production cartel maximize their profits (5) through setting

δPn i=1πi

δqi = 0. This yields quantities as

qi,coll =

A + vib − 2γ

Pn

j6=iqi

2 . (12)

Symmetry in quantities between firms yields

qi,coll = (β2− 1)(A + vib) − γ Pn i6=j(A + vjb) 2(1 − γ)(β1 − 1) . (13)

Companies, which compete in sustainability, maximize their profit (5) with the quantities (11) or (13), based on whether they compete in the second stage, with setting δπi

δvi = 0. Due

to symmetry, this yields the following contributions:

vi,∗ = 2Aβ2b rβ2 1(2 − γ) − 2β2b2 , (14) vi,pc = Aβ3b 4r(1 − γ)(β1 − 1) − β3b2 , (15)

where ”*” stands for competition in both stages and ”pc” for a production cartel.

If companies form a sustainability cartel, they maximize their profits (5) with the quan-tities (11) or (13), based on whether they compete in the second stage, with setting

δPn i=1πi

δvi = 0, which yield due to symmetry:

vi,sc = 2Ab rβ2 1 − 2b2 , (16) vi,f c = Ab 2r(β1− 1) − b2 , (17)

where ”sc” stands for a sustainability cartel ”fc” for full collusion (=collusion in both stages). In symmetric equilibrium quantities simplify to:

qi,• = A + vi,∗,scb β1 , (18) and qi,coll = A + vi,pc,f cb 2(β1− 1) . (19)

Proposition 1: Similar to the findings in Schinkel and Spiegel [44], the production cartel always yields the highest sustainability contribution, followed by competition and full collusion. Paradoxically, a sustainability cartel yields the lowest sustainability contribution. The order of the contributions is:

Proof 1: vi,pc− vi,∗ = Ab(rγ2_{(n − 1)(4n + 2(n − 1)(n − 2)γ) − (n − 1)(n − 3)γ}2₎ (4r(1 − γ)(β1− 1) − β3b2)(rβ12(2 − γ) − 2β2b2) > 0 , (21) vi,∗− vi,f c= Abrγ(1 − n)(4 + γ(n − 1)(2 + γ)) (2r(β1− 1) − b2)(rβ12(2 − γ) − 2β2b2) > 0 , (22) vi,f c− vi,sc = Abr(β1− 2)2 (rβ2 1 − 2b2)(2r(β1− 1) − b2) > 0 . (23) The ranking of the sustainability contributions can also be explained intuitively when looking at the profit maximizing functions in (6), (7), (8) and (9), as in Treuren and Schinkel [49]. As the profit maximizing functions are the same in my case, the intuitive explanation for the order of sustainability contributions stays the same as well and is not affected by the added bequest parameter.

Proposition 2: Both sustainability contributions and produced quantities are higher when considering bequest motives than without doing so.

Proof 2: vi,∗,b>1− vi,∗,b=1 = 2β2(b − 1)(β12r(2 − γ) + 2β2b) (β2 1r(2 − γ) − 2β2)(β12r(2 − γ) − 2β2b2) > 0 , (24) vi,sc,b>1− vi,sc,b=1 = 2(b − 1)(β2 1r + 2b) (β2 1r − 2)(β12r − 2b2) > 0 , (25) vi,pc,b>1− vi,pc,b=1 = β3(b − 1)(4r(1 − γ)(β1 − 1) + β3b) (4r(1 − γ)(β1− 1) − β3)(4r(1 − γ)(β1− 1) − β3b2) > 0 , (26) vi,f c,b>1− vi,f c,b=1 = (b − 1)(2r(β1− 1) + b) (2r(β1− 1) − 1)(2r(β1− 1) − b2) > 0 , (27) qi,∗,b>1 − qi,∗,b=1 = (b − 1)v∗,b>1+ (v∗,b>1− v∗,b=1) β1 > 0 , (28)

qi,sc,b>1− qi,sc,b=1 = (b − 1)vsc,b>1+ (vsc,b>1− vsc,b=1) β1 > 0 , (29) qi,pc,b>1− qi,pc,b=1 = (b − 1)vpc,b>1+ (vpc,b>1− vpc,b=1) 2(β1− 1) > 0 , (30) qi,f c,b>1− qi,f c,b=1 = (b − 1)vf c,b>1+ (vf c,b>1− vf c,b=1) 2(β1− 1) > 0 . (31)

This general finding can be explained intuitively. Consumers derive a higher utility on any given quantity and sustainability contribution if they have bequest motives. For companies, on the other hand, sustainability costs stay the same. As higher prices can be charged due to an increased utility for higher qi and vi, companies are willing to invest and produce more

and profit from these higher prices.

5.2

Welfare effects

After putting quantities (4) into the utility function (1), consumer surplus in equilibrium can be written as

CS = n(β1− 1)q

2 i

2 . (32)

It can be seen that the order of consumer surpluses just depends on the order of quantities.

Proposition 3: For a narrow parameter space, a production cartel can yield the biggest consumer surplus. In all other cases, the highest welfare is obtained under competition. The order of consumer surpluses is

(i) CSpc > CS∗ > CSsc > CSf c if

r < b

2_{(4 − 2γ + γ}2₎

(ii) CS∗ > CSpc> CSsc > CSf c if r < b 2_(β 3+ 2) 2(1 − γ)β1 , n ≥ 3 , (34) b2_{(4 − 2γ + γ}2₎ 2(1 − γ)(4 − γ2₎ < r < b2_(β 3+ 2) 2(1 − γ)β1 , n = 2 , (35) (iii) CS∗ > CSsc > CSpc > CSf c if r > b 2_(β 3+ 2) 2(1 − γ)β1 . (36) Proof 3: qi,∗− qi,sc = b β1 (v∗− vsc) > 0 , (37) qi,pc− qi,f c= b 2(β1 − 1) (vpc− vf c) > 0 , (38) qi,sc− qi,f c= 2Arγβ1(n − 1)(β1− 1)(rβ1− b2) (2r(β1− 1) − b2)(rβ12− 2b2) > 0 , (39) qi,∗− qi,pc= Arγ(1 − n)(2rβ1(γ − 2)(γ − 1) + (γ(γ − 2)(n − 3) − 4)b2) (4r(1 − γ)(β1− 1) − β3b2)(rβ12(2 − γ) − 2β2b2) R 0 , (40) qi,pc− qi,sc = Arγ(1 − n)(2rβ1(γ − 1) + (β3+ 2)b2) (4r(γ − 1)(β1− 1) + β3b2)(rβ12− 2b2) R 0 . (41)

The latter two expressions can be either bigger or smaller than 0, depending on the value of r with respect to the estimated thresholds in (33), (34), (35) and (36), with (40) being strictly greater than 0 for n≥3.

It can be seen that in the case for n>2 consumer surplus under full competition is always the biggest. This means that even when taking bequest motives into consideration, consumers will not benefit from a cartel in any stage if there are more than 2 firms involved.

Overall consumer welfare is biggest under the production cartel if n=2 and

r < b

2_{(4 + γ}2_{− 2γ)}

2(4 − γ2_{)(1 − γ)} . (42)

The right side is increasing in b, which increases the threshold for r for a larger bequest impact.

Figure 1: Areas for qpc> q∗

In Figure 1 it can be seen that the size of the area, where the production cartel yields the highest consumer surplus, is increasing in b. As the discount factor on future genera-tions increases, the chance that collusion yields higher consumer surplus than competition is getting bigger.

Additionally, the findings are closely related to the findings in Treuren and Schinkel and in Schinkel and Spiegel. As in Treuren and Schinkel, the production cartel can just be more profitable for consumers if there are two firms, although now the space for profitability is bigger due to the bequest motives.

Note that for estimating quantities, two different factors come into play. First, the sustainability investments, and then the quantities, based on the amount invested in sus-tainability.

For a given level of sustainability the difference in quantities produced for n firms is

n(qi,•(v0) − qi,coll(v0)) = (A + v0b) nγ(n − 1) 2β1(β1− 1) → limn→∞ = A + v0b 2γ . (43) It can be seen, that for n getting larger, the difference in hypothetical quantities produced will still be positive and approaches A+v_2γ0b. The impact of sustainability contributions on

quantities is given as δqi,• δvi = b β1 → limn→∞ = 0 , (44) δqi,coll δvi = b 2(β1− 1) → limn→∞ = 0 . (45)

From the latter two expressions it can be seen that the impact on quantities by sustain-ability contributions is decreasing for growing n and approximates 0 as n goes to infinity. Although the impact of sustainability contributions vanishes, the quantity reductions condi-tional on sustainability contributions remain present even for large n.

Proposition 4: Adding bequest motives to the utility function has the same effect on the order of sustainability contributions and quantities as cheaper investment, hence decreasing r. Proof 4: δ(vi,∗ vi,pc)b δb = −2 δ(vi,∗ vi,pc)r δr . (46)

r is bounded by being ≥1 and b is bounded insofar as to stay in the quasiconcave param-eter space necessary to guarantee for non-negativity of profits, sustainability contributions and quantities. Especially the relationship between r and b is relevant to ensure the latter condition as the denominator of sustainability contributions (14), (15), (16) and (17) is de-pendent on both of them and consists of the sum of r multiplied with a positive constant and b2 _{multiplied with a negative constant.}

As decreasing investment costs do not make the production cartel more profitable for consumers than competition (to ensure the quasiconcavity other parameters have to increase as well), increasing bequest motives cannot have this effect neither.

5.3

Price comparison with the case without bequest motives

Prices can be calculated by plugging sustainability contributions (14), (15), (16), (17) and quantities (18), (19) into the inverse demand function (4). This yields the following prices:

pi,∗ = a − r(2 − γ)(β1− 1)β12− 4b2(A + β1− 1) β1(r(2 − γ)β12− 4b2) , (47) pi,sc = a − r(2 + γ)2_(β 1− 1) + 2b2(A + β1− 1) β1(r(2 + γ)2 − 2b2) , (48) pi,pc= a − 4r(β1− 1)(1 − γ2) + b2(2 − γ)(A(β3 − 3) − β1+ 1) 2(1 + γ)(4r(1 − γ2_{) − b}2_{(2 − γ))} , (49) pi,f c = a − 2r(1 + γ)(β1− 1) + b2((β1− 1) − A(β3− 3)) 2(1 + γ)(2r(1 + γ) − b2₎ . (50)

Proposition 5: The prices when considering bequest motives are higher than the re-spective ones without bequest motives.

Proof 5: pi,∗,b>1− pi,∗,b=1 = 4A(b2 _{− 1)r(2 − γ)β} 1 (r(2 − γ)β2 1 − 4)(r(2 − γ)β12− 4b2) > 0 , (51) pi,sc,b>1 − pi,sc,b=1 = 2A(b2− 1)r(2 + γ)2 β1(r(2 + γ)2− 2)(r(2 + γ)2− 2b2) > 0 , (52) pi,pc,b>1− pi,pc,b=1 = 2A(b2_{− 1)r(β} 3− 3)(2 − γ)(1 − γ) (4r(1 − γ2_{) − b}2_{(2 − γ))(4r(1 − γ}2_{) − (2 − γ))} > 0 , (53) pi,f c,b>1− pi,f c,b=1 = A(b2_{− 1)r(β} 3 − 3) (2r(1 + γ) − b2_{)(2r(1 + γ) − 1)} > 0 . (54)

The reason, why higher sustainability contributions and higher quantities in the produc-tion cartel still do not yield a bigger overall consumer surplus than competiproduc-tion if bequest motives are considered, stems from the fact that a higher utility and a higher valuation of investment in sustainability lead to higher prices as well. Companies anticipate consumers’ higher utility and higher willingness to pay for sustainably produced goods if they include bequest motives and will charge a higher price accordingly to maximize their profits. The higher prices charged therefore undermine the positive effects by increased investment in sustainability.

5.4

The socially optimal output

To estimate the socially optimal output and the socially optimal sustainability contributions, overall welfare needs to be maximized. Overall welfare can be written as:

W F = n X i=1 πi+ CS = n X i=1 Aqi− 1 2 n X i=1 q2_i − γ n X i=1 n X j6=i qiqj+ ∞ X t=0 n X i=1 δtviqi− r 2 n X i=1 v2_i . (55) Maximizing overall Welfare through setting δW F_δq

i = 0 and δW F δvi = 0 yields qi,∗∗= Ar r(β1 − 1) − b2 , (56) vi,∗∗= Ab r(β1− 1) − b2 , (57)

where ”**” stands for the socially optimal amount.

Proposition 6: The sustainability contribution in both a production cartel and under competition can be bigger than the socially optimal one. The specific order of sustainability contributions is

(i) vi,∗∗> vi,pc> vi,∗ > vi,f c > vi,sc if

n < 2 − γ

2 , (58)

(ii) vi,pc > vi,∗∗> vi,∗ > vi,f c > vi,sc if

2 − γ

2 < n <

γ − γ2_{+ γ}3₊pγ2 ₂

+ 2γ2_{− γ}4

γ3 , (59)

(iii) vi,pc> vi,∗ > vi,∗∗> vi,f c > vi,sc if

n > γ − γ

2_{+ γ}3₊pγ2 ₂

+ 2γ2_{− γ}4

γ3 . (60)

Proof 6:

vi,∗∗− vi,f c = Abr(β1− 1) (r(β1− 1) − b2)(2r(β1− 1) − b2) > 0 , (62) vi,∗∗− vi,pc= Arb(β1− 1)(4(1 − γ) − β3) (r(β1− 1) − b2)(4r(1 − γ)(β1− 1) − β3b2) R 0 , (63) vi,∗∗− vi,∗ = Arb(β2 1(2 − γ) − 2β2(β1− 1)) (r(β1− 1) − b2)(r(2 − γ)β12− 2β2b2) R 0 . (64)

The latter two expressions are bigger than 0, depending on whether n is above or below the specific threshold calculated above in (58), (59) and (60).

Proposition 7: If companies in competition are required to make at least the socially optimal sustainability investments, consumers can never be worse off than without this rule.

Proof 7: As quantities and hence, consumer surplus, are strictly increasing in sustain-ability contributions, the higher the sustainsustain-ability contribution, the higher the consumer surplus.

If vi,∗ ≥ vi,∗∗ is binding, companies under competition will invest more in sustainability

under the minimum requirement rule than without it. If it is not binding, they will invest the competitively optimal amount, which is then higher than the minimum requirement.

5.5

Compensation by colluding firms, simplified for n=2

Under a compensation scheme, firms are bounded to produce at least the competitive amount, so qi ≥ qi,•.

Proposition 8: If firms colluding in quantities are bounded by producing at least the competitive amount by qi ≥ qi,•, they will always do so and never produce more, as the most

profitable deviation strategy is to produce less than the competitive amount.

Proof 8: Deviation profit would be

πi,dev = (A + vib − qi− γ(n − 1)q•)qi−

rv2 i

where πi,dev means deviation profit, which yields deviation quantity and sustainability

con-tribution after setting δπi,dev

δqi = 0 and δπi,dev δvi = 0: qi,dev = 2Ar(4(4 − γ2_{) − 2b}2₎ (2r − b2_{)(r(4 − γ}2_{)(2 + γ) − 4b}2₎ , (66) vi,dev = 2Ab(4(4 − γ2) − 2b2) (2r − b2_{)(r(4 − γ}2_{)(2 + γ) − 4b}2₎ . (67) qi,dev < qi,• as qi,•− qi,dev = Ab2_rγ2 (2r − b2_{)(r(4 − γ}2_{)(2 + γ) − 4b}2₎ > 0 . (68)

Companies’ profit functions (5) become in symmetry, due to the constraint being binding,

πi = (A + vib − (β1− 1)q•)q• −

rv2 1

2 , (69)

which yields, after maximizing with respect to vi through setting δπ_δvii = 0

vi,comp =

Ab(4 − γ2)

r(2 + γ)2_{(2 − γ) − 4b}2 , (70)

where vi,comp is the sustainability contribution under the compensation scheme.

Proposition 9: If colluding firms are required to produce at least the competitive amount, their sustainability contribution will be lower than it would be under competition.

vi,comp < vi,∗ . (71)

Proof 9:

vi,∗− vi,comp=

Abγ2

6

Analysis for δ

>0

I now adapt the utility function used above to the case where δ2 >0 and I have:

ui = n X i=1 aqi− 1 2 n X i=1 q_i2− γ n X i=1 n X j6=i qiqj − n X i=1 piqi+ ∞ X t=0 n X i=1 δtviqi+ ∞ X t=0 n X i=1 δ₂tvi . (73)

Besides the previous effect on quantities demanded, sustainability contributions in this utility function also have a direct impact on current and future generations utility. Intuitively, as sustainability contributions are always highest under the production cartel, the last term, which is added in the consumer surplus evaluation, is also always highest in a production cartel and for a large enough discount factor δ2, the production cartel will always yield the

biggest consumer surplus.

If δ2 >0, the inverse demand function estimated after setting δu_δqii = 0 is the same as in

the case for δ2=0 estimated above. This leads to similar quantities and similar sustainability

contributions. After putting quantities (4) into the utility function (73), consumer surplus in equilibrium can be written, after setting P∞

t=0δ t ₌ 1 1−δ = b and P∞ t=0δ t 2 = 1 1−δ2 = b2, as CS = n 2(β1− 1)q 2 i + nb2vi . (74)

Evaluating the order of consumer surpluses is cumbersome and, all else equal, the only factors in which they change between the different settings is through vi and qi. Consumer

surplus is strictly increasing in both parameters. As vpc > v∗ > vf c > vsc, q∗,pc > qf c and

q∗ > qsc, it can be concluded that CS∗ > CSf c,sc and CSpc > CSf c. We get:

(i) CSpc > CS∗ if b2 > (β1− 1)r2( 4(1−γ) 2 (4(β1−1)r(1−γ)−b2β3)2 − β2 1(2−γ)2 β2 1r(2−γ)2−2β2b2) 2b( β3 4(β1−1)r(1−γ)−b2β3 − 2β2 β2 1r(2−γ)−2β2b2) , (75) (ii) CSpc > CSsc if b2 > (β1− 1)r2( 4(1−γ) 2 (4(β1−1)r(1−γ)−b2β3)2 − β2 1 β2 1r−2b2 ) 2b( β3 4(β1−1)r(1−γ)−b2β3 − 2 β2 1r−2b2 ) , (76) (iii) CSf c > CSsc if b2 > (β1− 1)r((2 + β1)b4− 4β12b2r + β12(3β1− 2)r2) 2b(β1− 2)(2r(β1− 1) − b2)(β12r − 2b2) . (77)

As these estimations for b2 are hard to understand I made Table 1, which shows for

different values of r, b, b2 and n the corresponding threshold for γ to make consumer surplus

under a production cartel bigger than under competition. For γ above the upper thresh-old, the model is not identified, for γ below the lower threshthresh-old, consumer surplus under competition is bigger than under a production cartel. It can be seen that for increasing b2,

the possible parameter space with having the biggest consumer surplus under a production cartel, is expanding quickly for γ. In general, the more homogeneous the goods are, the higher the chance that a production cartel yields the biggest consumer surplus. What is more, unfortunately the possible parameter space which satisfies the model assumptions of non-negativity of profits and sustainability contributions, is decreasing quickly with b and an increasing amount of firms present.

T able 1: When is a pro duction cartel the most profitable for consumers? r=50 r=8 r=1 b2 =10 b2 =3 b2 =1 b2 =10 b2 =3 b2 =1 b2 =10 b2 =3 b2 =1 b=1 n=2 0.963 > γ > 0.696 0.963 > γ > 0.870 0.963 > γ > 0.945 0.900 > γ > 0.287 0.900 > γ > 0.551 0.900 > γ > 0.752 0.614 > γ > 0.046 0.614 > γ > 0.129 0.614 > γ > 0.268 n=10 0.864 > γ > 0.401 0.864 > γ > 0.688 0.864 > γ > 0.862 0.683 > γ > 0.084 0.683 > γ > 0.250 0.683 > γ > 0.50 0.261 > γ > 0.010 0.261 > γ > 0.031 0.261 > γ > 0.082 n=80 0.642 > γ > 0.079 0.642 > γ > 0.232 0.642 > γ > 0.477 0.336 > γ > 0.011 0.336 > γ > 0.041 0.336 > γ > 0.119 0.054 > γ > 0.001 0.054 > γ > 0.004 0.054 > γ > 0.011 b=3 n=2 0.854 > γ > 0.441 0.854 > γ > 0.684 0.854 > γ > 0.819 – – – – – – n=10 0.602 > γ > 0.166 0.602 > γ > 0.405 – – – – – – – n=80 0.259 > γ > 0.024 0.259 > γ > 0.083 0.259 > γ > 0.217 – – – – – – b=10 n=2 – – – – – – – – – n=10 – – – – – – – – – n=80 – – – – – – – – –

To estimate the socially optimal output and the socially optimal sustainability contribu-tions, overall welfare needs to be maximized, which can be written as:

W Fb2>0 = n X i=1 πi + CS = n X i=1 Aqi− 1 2 n X i=1 q_i2− γ n X i=1 n X j6=i qiqj+ ∞ X t=0 n X i=1 δtviqi+ ∞ X t=0 n X i=1 δ₂tvi − r 2 n X i=1 v_i2 . (78)

Maximizing welfare with respect to quantities and sustainability contributions yields after setting δW Fb2>0 δqi = 0 and δW F_b2>0 δvi = 0 qi,∗∗= Ar + bb2 r(β1 − 1) − b2 , (79) vi,∗∗= Ab + b2(β1− 1) r(β1− 1) − b2 . (80)

Proposition 10: Both the production cartel and competition can yield a bigger sus-tainability contribution than the socially optimal amount. The specific order is

(i) vi,∗∗,b2>0 > vi,pc > vi,∗ > vi,f c> vi,sc if

b2 >

Arb(γ(n + 1) − 2) 4r(1 − γ)(β1− 1) − b2β3

, (81)

(ii) vi,pc > vi,∗∗,b2>0 > vi,∗> vi,f c> vi,sc if

Arb(4 − 2γ(n − 2) + 2γ2(n − 1) − γ3(n − 1)2) (β1− 1)(rβ12(2 − γ) − 2β2b2) < b2 < Arb(γ(n + 1) − 2) 4r(1 − γ)(β1− 1) − b2β3 , (82)

(iii) vi,pc> vi,∗ > vi,∗∗,b2>0 > vi,f c > vi,sc if

b2 < Arb(4 − 2γ(n − 2) + 2γ2_{(n − 1) − γ}3_{(n − 1)}2₎ (β1− 1)(rβ12(2 − γ) − 2β2b2) . (83) Proof 10: vi,∗∗,b2>0− vi,∗∗,b2=0 = b2(β1− 1) r(β1− 1) − b2 > 0 , (84)

vi,∗∗,b2=0 > vi,f c > vi,sc → (62) , (86)

vi,∗∗,b2>0− vi,∗ R 0 , (87)

vi,∗∗,b2>0− vi,pc R 0 . (88)

The last two expressions are either bigger or smaller than 0, depending whether b2 is

above or below the thresholds estimated in (81), (82) and (83).

To sum up, under both the production cartel and competition, the investment in sus-tainability can be higher than the socially optimal level.

Note that in this model, the parameter space for having a higher sustainability contribu-tion than the socially optimal amount in a produccontribu-tion cartel and competicontribu-tion is smaller than in the case discussed above with b2=0, as the sustainability contributions by companies are

independent of b2, but the socially optimal sustainability contribution is strictly increasing

in b2 as δvi,pc δb2 = δvi,∗ δb2 = 0 and δvi,∗∗ δb2 = β1−1 r(β1−1)−b2 > 0.

This specific utility function in turn is problematic for at least two reasons. Firstly, cal-culations and evaluations of the different scenarios are complicated as not just an additional parameter is added to the model but also consumer surplus is not just a function of quantities produced, a fact that simplified the calculating process immensely before.

What is more, as explained intuitively, for a large enough discount factor, a production cartel always yields the biggest consumer surplus. This is problematic as this would mean a justification for allowing any cartel as long as there is some contribution to sustainabil-ity involved. Therefore, also this utilsustainabil-ity function at best gives an idea of the additional complications and necessary considerations in the model and should not be applied directly.

7

Conclusion

As shown above, including bequest motives in the standard utility function (1) with δ2 = 0

can never be a justification for allowing a production cartel if there are more than 2 firms present in the market. Under the utility function I used, competition in both stages always yields the biggest consumer surplus. Although sustainability contributions and quantities

produced are higher when considering bequest motives than when not taking them into account, consumers are still not better off. Rational profit-maximizing companies will in-corporate the bigger utility through increased sustainability contributions and quantities and charge higher prices for their products. The gains in consumer surplus through higher sustainability investments are therefore undermined by the need to pay more. It therefore does not seem convincing to rely on companies to improve consumers’ utility and to increase welfare in general.

For the case of δ2 > 0, on the other hand, the likelihood of the production cartel yielding

the biggest consumer surplus, is increasing in δ2. A high δ2 means that future generations’

utility through a direct impact by sustainability contributions on consumer surplus is hardly discounted. If assuming a high enough δ2, a production cartel always yields the biggest

consumer surplus, no matter the value of the other parameters. Regarding the above used utility function, there still exist two significant shortcomings. Firstly, an ”everything goes” approach when choosing a sufficiently high discount factor, is hardly justifiable in light of cartels’ threats and their consequences [32]. Secondly, as can be seen in Table 1, the parameter space, under which the model is working, is decreasing with δ, b and n and for many combinations of parameter values, the assumption of non-negativity for profits and sustainability contributions is not satisfied.

In Article 101(3) TFEU the EC stated that cartels may be exempted from cartel reg-ulation if ”consumers receive a fair share of the benefit.” Although future generations are not explicitly included here, it is justifiable to include bequest motives in consumers’ util-ity function as the welfare of descendants directly effects the utilutil-ity of current consumers and people feel better if they know future generations are well off ( [8], [9], [26], [31], [54] ). Unfortunately, even when including bequest motives, the results hardly change and, albeit increasing consumer surplus, consumer welfare stays the biggest under competition, unless the parameters fall in a very narrow parameter space. This suggests that relying on com-panies to increase sustainability may not be effective. It seems more convincing to rely on standard regulatory practices. Instead of letting companies incorporate consumers’ increased utility and higher willingness to pay for improved sustainability (e.g. better living conditions for chicken leading to improvements in poultry keeping) regulatory agencies should rather

set minimum standards (for poultry keeping). If this minimum standard is set at the socially optimal amount, consumers can never be worse off under this regulation than without it (see Proposition 7).

Nevertheless, it is difficult to give unambiguous policy recommendations when keeping the above-mentioned shortcomings of the model in mind. Potential future research should try to tackle these shortcomings. Especially adapting the utility function to expand the parameter space is crucial to achieve a widely valid result. Unfortunately, the results obtained above remain incomplete insofar, as many possible parameter cases cannot be covered by this model.

A possible extension to the utility function used above adds a multiplicative element and is discussed briefly in the appendix.

8

Appendix

8.1

Bertrand competition for n=2 firms

As in Cournot, we have p1 = a + v1b − q1− γq2 , (89) p2 = a + v2b − q2− γq1 , (90) π1 = (p1− k)q1− rv2 1 2 , (91) π2 = (p2− k)q2− rv2 2 2 , (92) which yields q1 = a + v1b − p1− γq2 , (93) q2 = a + v2b − p2− γq1 . (94)

Plugging (94) into (93) and (93) into (94) yields

q1 = (a + v1b − p1) − γ(a + v2b − p2) 1 − γ2 , (95) q2 = (a + v2b − p2) − γ(a + v1b − p1) 1 − γ2 , (96)

and plugging (95) into (91) and (96) into (92) yields

π1 = (p1− k) (a + v1b − p1) − γ(a + v2b − p2) 1 − γ2 − rv2 1 2 , (97) π2 = (p2− k) (a + v2b − p2) − γ(a + v1b − p1) 1 − γ2 − rv2₂ 2 . (98)

Profit maximization in prices through setting δπ1 δp1 = 0 and δπ2 δp2 = 0 yields: p1,• = a + v1b + k − γ(a + v2b − p2) 2 , (99) p2,• = a + v2b + k − γ(a + v1b − p1) 2 , (100)

where ”•” stands for competition in the price setting stage. Plugging (100) into (99) and (99) into (100) yields

p1,• = a(2 − γ − γ2_{) + (2 + γ)k + (2 − γ}2_)v 1b − γv2b 4 − γ2 , (101) p2,• = a(2 − γ − γ2_{) + (2 + γ)k + (2 − γ}2_)v 2b − γv1b 4 − γ2 , (102)

and plugging (101) into (97) and (102) into (98) yields

π1,• = ( A(2 − γ − γ2) + (2 − γ2)v1b − γv2b 4 − γ2 ) 2 1 1 − γ2 − rv2₁ 2 , (103) π2,• = ( A(2 − γ − γ2_{) + (2 − γ}2_)v 2b − γv1b 4 − γ2 ) 2 1 1 − γ2 − rv2 2 2 . (104)

Companies competing in both stages choose their sustainability contribution through maximizing their own profit function and set δπ1,•

δv1 = 0,

δπ2,•

δv2 = 0. This yields in symmetry:

vi,∗ =

2Ab(2 − γ2₎

r(2 − γ)2_{(2 + γ)(1 + γ) − 2(2 − γ}2_)b2 . (105)

If companies form a sustainability cartel, they jointly maximize their profits with respect to v, and set δ(π1,•+π2,•) δv1 = 0 and δ(π1,•+π2,•) δv2 = 0. This yields: vi,sc = 2Ab(1 − γ) r(γ + 1)(2 − γ)2_{− 2(1 − γ)b}2 . (106)

Companies forming a production cartel jointly maximize their profits (97) in prices and set δ(π1,+π2)

δp1 = 0 and

δ(π1,+π2)

δp2 = 0. This yields prices as

p1,coll =

(a + k)(1 − γ − γ2_{) + v}

1b(1 − γ2) − v2bγ + 2p2γ

p2,coll =

(a + k)(1 − γ − γ2) + v2b(1 − γ2) − v1bγ + 2p1γ

2(1 − γ2₎ , (108)

where ”coll” stands for collusion in the price setting stage, which simplifies in equilibrium to

pi,coll =

a + k + vib

2 , (109)

which yields after plugging (109) into (97)

π1,coll = ( a + v1b − k 2 )( a + v1b − k − γ(a + v2b − k) 2(1 − γ2₎ ) − rv₁2 2 , (110) π2,coll = ( a + v2b − k 2 )( a + v2b − k − γ(a + v1b − k) 2(1 − γ2₎ ) − rv2 2 2 . (111)

Firms competing in sustainability maximize their profits and set δπ1,coll

δv1 = 0,

δπ2,coll

δv2 = 0,

which yields in symmetry

vi,pc=

A(2 − γ)b

4r(1 − γ2_{) − (2 − γ)b}2 . (112)

Firms forming a cartel in both stages maximize their profits jointly and set δ(π1,coll+π2,coll)

δv1 =

0 and δ(π1,coll+π2,coll)

δv2 = 0, which yields in equilibrium:

vi,f c =

Ab

2(1 + γ) − b2 . (113)

The latter two sustainability contributions are both similar to the Cournot case.

Proposition 11: The order of the sustainability contributions is the same as in Cournot with

vi,pc> vi,∗ > vi,f c > vi,sc . (114)

Proof 11: vi,pc− vi,∗ = Abrγ2_{(8 + 4γ − 5γ}2_{− γ}3₎ (4r(1 − γ2_{) − b}2_{(2 − γ))(r(2 − γ)}2_{(2 + 3γ + γ}2_{) − 2b}2_{(2 − γ}2₎₎ > 0 , (115) vi,∗− vi,f c= Abrγ(4 + 2γ − 3γ2− γ3₎ (2r(1 + γ) − b2_{)(r(2 − γ)}2_{(2 + 3γ + γ}2_{) − 2b}2_{(2 − γ}2₎₎ > 0 , (116)

vi,f c− vi,sc =

Abrγ2(1 + γ)

(2r(1 + γ) − b2_{)(r(2 − γ)}2_{(1 + γ) − 2b}2_{(1 − γ))} > 0 . (117)

For the same reasons as in Cournot, bequest motives do not change the order of the sustainability contributions.

After plugging (101), (102), (105), (106) and (109), (112), (113) into (95) and (96), quantities can be written as:

qi,• = A + v∗,scb (2 − γ)(1 + γ) , (118) qi,coll = A + vpc,f cb 2(1 + γ) . (119)

As in Cournot, consumer surplus can be written as

CS = n(β1− 1)q

2 i

2 . (120)

Proposition 12: Like in Cournot, the production cartel can yield the biggest consumer surplus under the right parameters. In all other cases consumer surplus is biggest under competition. The specific order is:

(i) CSpc > CS∗ > CSsc > CSf c if r < b 2_{(4 + 2γ − 3γ}2₎ 8 − 10γ2_{+ 2γ}4 , (121) (ii) CS∗ > CSpc> CSsc > CSf c b2(4 + 2γ − 3γ2) 8 − 10γ2_{+ 2γ}4 < r < b2(4 − 3γ) 4 − 2γ − 4γ2_{+ 2γ}3 , (122) (iii) CS∗ > CSsc > CSpc > CSf c if r > b 2_{(4 − 3γ)} 4 − 2γ − 4γ2_{+ 2γ}3 . (123) Proof 12: qi,∗− qi,sc = Ab (γ + 1)(2 − γ)(v∗− vsc) > 0 , (124)

qi,pc− qi,f c= Ab 2(γ + 1)(vpc− vf c) > 0 , (125) qi,sc− qi,f c = Arγ(r(2 + γ − γ2_{) − b}2₎ (2r(1 + γ) − b2_)(2b2_{(−1 + γ) + r(−2 + γ)}2_{(1 + γ))} > 0 , (126) qi,∗− qi,pc = Arγ(2r(1 − γ2_{)(4 − γ}2_{) − b}2_{(4 + 2γ − 3γ}2₎₎ (r(2 − γ)2_{(2 + γ)(1 + γ) − 2(2 − γ}2_)b2_{)(4r(1 − γ}2_{) − (2 − γ)b}2₎ R 0 , (127) qi,pc− qi,sc = Arγ(2r(γ − 1)(2 + γ + γ2_{) + b}2_{(4 − 3γ))} (r(2 − γ)2_{(1 + γ) − 2(1 − γ)b}2_{)(4r(1 − γ}2_{) − (2 − γ)b}2₎ R 0 . (128)

As in Cournot, the latter two expressions can be either positive or negative, depending on r with respect to its thresholds given in (121), (122) and (123).

8.2

Analysis for a multiplicative utility function

A multiplicative utility function could look the following way:

ui,m = vi2( n X i=1 (a + ∞ X t=0 δtvi)qi− 1 2 n X i=1 q_i2− γ n X i=1 n X j6=i qiqj − n X i=1 piqi) . (129)

It multiplies the previous utility function with the square of the respective sustainability contribution and through that assigns a bigger weight on investment in sustainability.

As for the case with w>0, the inverse demand function estimated after setting δui,m

δqi = 0

is the same as in the case for w=0 estimated above, which in turn leads to similar quantities and similar sustainability contributions. After putting quantities (4) into the utility function (129), consumer surplus in equilibrium can be written as

CSm =

n(β1− 1)

2 (qivi)

2

. (130)

Now consumer surplus is the product of the quantity and the sustainability contribution squared, multiplied with a constant. As vi,pc > vi,∗ > vi,f c > vi,sc, qi,∗,pc,sc > qi,f c and

We also get that CSpc> CSsc as CSpc− CSsc = 2br( (1 − γ)β3 (b2_β 3− 4r(1 − γ)(β1− 1))2 − β1 (rβ2 1 − 2b2)2 ) > 0 . (131)

Estimating the rest of the orders in a general notation is hardly possible and to highlight the difference I simplify here for the case n=3.

CS∗ > CSpc if r > b 2_{(4 − 5γ}2_{+ γ}4_{+ γ}p36 + 24γ − 41γ2 _{2 − 30γ3} + 4γ4_{+ 6γ}5_{+ γ}6₎ 2(4 − 2γ − 18γ2_{+ 3γ}3_{+ 17γ}4_{− γ}5_{− 3γ}6₎ , (132) and CSsc > CSf c if r > b 2_{(1 + γ + γ}√2 _{1 + γ)} 2(1 + 2γ − γ3₎ . (133)

For n≥4, with the right parameters, a production cartel can be more profitable for con-sumers than competition, although calculating the parameter spaces becomes complex and has to be done on a case by case basis.

Glossary

ACM Autoriteit Consument en Markt (Dutch Market Authority, since 2013). 5–8, 40, 41

EC European Commission. 5, 7, 8, 13, 32, 41, 42 ETS European system for emissions trading. 6 EU European Union. 6

IV Instrumental variable. 10

NMa Nederlandse Mededingingsautoriteit (Dutch Market Authority, until 2013). 40

OLS Ordinary least squares. 10

TEC Treaty Establishing the European Community. 4

TFEU Treaty on the Functioning of the European Union. 4–6, 8, 9, 14, 32

9

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