The Corporate Social Responsibility Is just a Twist in a Möbius Strip

In recent years economic agents and systems have become more and more interactive and juxtaposed, therefore the social sciences need to rely on the studies of physical sciences to analyze this complexity in the relationships. According to this point of view, the authors rely on the geometrical model of the Möbius strip used in electromagnetism, which analyzes the movements of the electrons that produce energy. They use a similar model in a Corporate Social Responsibility (CSR) context to devise a new cost-benefit model in order to take into account three positive effects on the efficiency of a socially responsible company: 1) cooperation among stakeholders in the same sector; 2) cooperation among similar stakeholders in different sectors; and 3) the stakeholders’ loyalty toward the company. By applying this model to a firm’s decision problem the authors find that investing in CSR activities is always convenient, depending on the number of sectors, the stakeholders’ sensitivity to these investments and the decay rate to alienation. Their work suggests a new method of analysis which should be developed not only at a theoretical level, but also at an empirical level. JEL L13 D21 Z1


Introduction
In recent years, in particular from the beginning of the 21st century, the social sciences started to strongly rely on the discoveries of physics of complexity to analyze complicated relations between models and social phenomena (Urry, 2003). For instance this is just the research eld of the econophysics which studies the applications of theories and methods developed by Physics in order to solve problems in Economics (for more details see Rosser, 2008). As in the studies of many physical systems, also in the social sciences there is a growing attention to go behind the traditional notions treating various agents as separated and distinct essences (Urry, 2003;Giddens,1984). Currently they are instead conceived as juxtaposed entities related trough a nonlinear mechanism where causes and eects are copresent and strongly integrated 1 .
In an even more globalized world very complex interactions characterize social and economic relationships. Therefore we need models taking into account this complexity and nonlinearity in the connections. Such links involve multiple positive and negative feedback loops making systems interdependent and interacting dissipatively with their environment.
In Economics this interdependence among systems and among agents is just the core of the models of Corporate Social Responsibility (since now on CSR), which consider the global integration between rms and stakeholders, including workers, customers and the full environment (see Becchetti et al., 2014). The CSR implies a move from the maximization of the shareholders wealth to the satisfaction of a more complex objective function in which interests of the other stakeholders are taken into account. On turn this creates also benets for the business. For instance Becchetti et al.(2014) show that since more and more prot maximizing rms are adopting CSR practices there must be pecuniary benets arising from them. The authors also document that the CSR has the potential to generate several values increasing eects by attracting better employees, enhancing their intrinsec motivation and loyalty, reducing turnover rates, improving the eciency and by reducing operating costs.
Nevertheless CSR improves boosting sales revenues, increases rivals costs and attracts more 1 " No party to a relation is therefore a monadic or molar entity. Each is instead a mutable function or the character of the mode-of-being related and its capacity for relationality" (Dillon, 2000) ethical consumers, so that the rm can benet from increases in her demand share.
All the above mentioned advantages can be seen as a sort of ethical capital accumulated trough the CSR practices, which also requires the payment of additional costs. Becchetti et al. (2014) underline, by using a dynamic model, the conditions implying that such benets overrun the costs. These advantages can also be considered as the result of the synergy which relates each subsystem's and each agent's performance.
Thanks to this synergy net benets from the relationships across to the stakeholders by the virtue of their connections to the rm and the net transactional benets across to the business system by the virtue of the intra-organizational cooperation.
There is also a large literature on the benets that in general arise by investing in CSR on stakeholders and in particular on workers. To this aim many analyses use the standard taxonomy of CSR criteria provided by Kinder, Lydenberg and Domini Research and Analytics, Inc. (KLD). They include the following eight wide-ranging categories into the Domini 400 index: i) community; ii) corporate governance; iii) diversity; iv) employee relations; v) environment; vi) human rights; vii) product quality; and viii) controversial business issues.
Every category has its strengths and weaknesses identied and analyzed within the index, as well as the suggestion of corporate activities compliant with each specic category. For instance, by using the KLD index, Becchetti et al.(2016) show that the CSR rms which take into account the workers well-being are less exposed to business risks and prot volatility. Other authors analyse the eects of increased productivity of the individual workers, see Rob and al. (2000). The authors show how the specic investiments in Corporate Social Responsibility can be seen as the optimal incentives that foster the employees to allocate greater eorts to cooperative tasks because their derive utility from cooperation. In the meta-analysis devised by Harter et al (2003) positive workplace perceptions and feelings are associated with higher business-unit customer loyalty, higher protability, higher productivity and lower rates of turnover. In Gond et al (2010) it is explained how employees' perceptions of CSR trigger attitudes and behavior in the workplace which aect organiza- Therefore according to the CSR point of view rms and stakeholders can be depicted not as two distinct and unconnected systems, but they are a cross-system where transfers occur in a such a way that a business becomes a stakeholders' interest and conversely stakeholders well-being becomes part of the business. In this crossed-system the output of each part is transferred across them to become the others' input, so that these subsystems are strongly overloaded and linked inextricably together.
According to our point of view the best metaphor, suggested by the physical sciences, to approximate and represent this new conceptualization of links in economics systems and between agents is the Möbius strip. This is a topological enigma independently documented in 1858 by two mathematicians A.
F. Möbius and J.B. Listing. It is a bend of paper given a 180 degree twist prior to having its two ends connected. The rst use of the Möbius strip as a metaphor in the business relationships, on our knowledge, is that of Litz (2008), who discusses an alternative approach to business family and family business relationships.
In this work we aim to extend this approach to the CSR analysis by extensively relying on the recently discoveries in the electromagnetism. We assimilate rm and stakeholders' contributes to the action of electrons travelling on a Möbius strip which, unlike a regular bend, return to a mirror reality in each count. In particular we strictly follow the model of Yacubo et al. (2003) who show that the electrons travelling on a Möbius strip produce energy of higher intensity or equivalently there is a lower energy dissipation thanks to the decreased resistance by virtue of the twist in the bend. We analyze how contributions of the economic agents in a CSR context, thanks to the eect to the ethical capital, produce higher benets and a lower dissipation of the costs thanks the augmented cooperation.
The paper is divided into four sections (including introduction and conclusions). In the sec-ond section we describe the building of the geometrical model for the electrons travelling in a Möbius strip. In the third section we investigate how to apply this model to the behavior of rms and economics agents in a CSR context. We dene a new cost function that show the convenience to invest in social responsible activities thanks to three positive crossed effects on the eciency: i)cooperation among stakeholders in the same sector; ii)cooperation among similar stakeholders in dierent sectors and iii)the stakeholders' loyalty towards the company. We provide an example of a rm's decisional problem which decides whether to invest in social responsibility. Our analytical results show that this is ever the optimal choice depending on the number of sectors, the stakeholders' sensitivity to these investments and the decay rate to alienation. In the fourth section we discuss our conclusions.

How to build a geometrical model for the electrons travelling in a Möbius strip
The Möbius strip is a bi-dimensional manifold with only one face. It can be built from a strip of paper by joining together its both ends after having twisted one of them a half turn (see Figure 1).

Figure 1: How to build a Möbius strip
The Möbius strip has one side and a single border and if we move along the centre line, the meridian, of the strip we need to go through the circle twice in order to return to the original position. This behavior is similar to that of the electrons generating a ux periodicity of persistent currents in a Möbius strip in Yacubo et al. (2003), who describe it by using the Hubbard model (1963). This last is the simplest model of interacting particles (electrons) in a lattice and consists of a Hamiltonian with only two terms: a kinetic term which represents the kinetic energy of electrons hopping between atoms and a potential term consisting of an on-site interaction which represents the potential energy arising from the charges on the electrons. Therefore the Hamiltonian is a sum of potential and kinetic energy and is applied to describe how the one kind of energy repeatedly changes into the other one over time.
If we assume that there are N sites then we'll say that if an electron tunnels from lattice site j to site l, its energy changes by an amount −t jl . This tunneling eect is equivalent of annihilating the electron at site j and creating it again at site l, so the portion of the Hamiltonian, the kinetic term, dealing with tunneling can be written as where a † l , a j are the fermion (since electrons are fermions) creation and annihilation operators. For many practical purposes it suces to assume that t jl is none-zero, only when j and l are the nearest neighbors in which case it is usually approximated by a constant t.
Because of the electron may tunnel also from lattice site l to site j, the Hamiltonian becomes where ε k represents the site energy and a † k , a k are the fermion creation and annihilation operators at the site k.
rections on 2M wires and transverse directions on N wires. Specically, starting from a rectangular lattice including N × 2M sites (see Figure 2), the rectangle is then twisted by 180 degrees and its two sides are connected, such that longitudinal wire 1 is attached to wire 2M, wire 2 is attached to wire 2M − 1 and so on (see Figure 3). The Möbius strip so constructed includes M longitudinal wires with 2N sites on each one.  According to the Hubbard model (1963) the Hamiltonian is then where a nm is the fermion operator at the site (n, m) with n = 1, 2, ..., 2N and m = 1, 2, ..., M ).
The quantity ε nm is the site energy so that represents the potential term.
The kinetic term is made up of three parts: Nevertheless we think that in a context of CSR this function does not take into account all the crossed eects that social responsible activities can generate in terms of productivity and costs saving (see Becchetti et al. 2014). In particular some of these eects concern the externalities due to the CSR benets on the stakeholders, which on turn are transferred into positive returns on the rm's traditional activities.According to this point of view, we consider a SR company with n = 1, 2, .., N stakeholders or cluster of stakeholders and are the specic activities devoted to the CSR. We denote by 0 ≤ a nm < 1 the contribution of the stakeholder n in the sector m measured as percentage per unit of a product. For instance if a 11 = 1 5 we say the stakeholder 1 is able to produce the 20 per cent of a unit in a working hour. Like in a Möbius strip also in a social responsible rm the eects of a twist may be considered as the returns due to the CSR activities on the stakeholders and rm production, which therefore amplify the crossed contributions of dierent stakeholders also operating in dierent sectors of the company (see Figure 4). contribution that the stakeholders 1 would give thanks to the social responsible activity 2M. The same for the other social responsible activities which are ordered in such a way that 2M is more relevant for the sector 1, 2M − 1 is more relevant for the sector 2, etc (for instance 2M could be seen as the social responsible activities dedicated to assure safety work condition in sector 1, 2M − 1 those to assure safety work condition in sector 2 and so on). Therefore in this work we propose the use of a new cost function for CSR companies suggested by (1), that in our case becomes: (1 − δ)a nm a n+1m , that we call the neighbouroud eciency term, measures the gains associated to the crossed contributions of n in the sector m with the nearest n+1 in the same sector. For instance if a 11 = 1 5 and a 21 = 1 7 , when the SR stakeholder 1 supports the stakeholder 2 helping him to produce his share 1 7 , the stakeholder 1 contributes with his ability of 1 5 to the production of 1+ 1 7 units of the good. Therefore his total contribution is now 1 5 1 + 1 7 . Obviously also the stakeholder 2 can support the stakeholder 1 and this would correspond to Hermitian conjugate of this term. In the rest of the paper, to avoid excessive complexity, we don't consider the hermitian conjugate of (2) because this doesn't aect our analysis. Moreover we assume that 0 < δ < 1 is the decay rate due to the possible eect of alienation (caused for instance by satiety, low free time, etc.). Finally t 1 represents the sensitivity of the stakeholders' contributions to the SR activities devoted to them; Therefore the order matters as investments and return are specic into the rm. Obviously we can imagine there are also externalities requiring no specic orders, but they are dicult to measure and not related to specic company's activities and investments while CSR measures are specic for sectors and stakeholders so implying specic returns. In particular the three above mentioned eects depend on the extremely strick and precise conditions of how CSR investments operate so that the twist is just a Mobius strip twist rather than some less well-ordered reshuing of cross-cutting eects across the stakeholders.
In that follows we aim to apply this function to a general decisional problem of a company which wants to minimize the costs taking into account these crossed benets due to the SR activities.
3.2 An application to a rm decisional problem with constant contributions and costs In this section we consider only one type of stakeholders and specically we assume that there are N workers in m = 1, 2, ..., M traditional sectors. We assume that the total production is equal to the sum of the contributions of these workers, which could be measured in term of pieces produced by worker in that sector in a working hour, which is constant for each worker and sector, a nm = a, with a ∈ R and 0 ≤ a < 1 for all n = 1, 2, ..., N and m = 1, 2, ..., M.
Therefore if we denote by p the price of the nal good and by w the wages paid to workers, the rm's prot function is: We also assume that the company nances the social responsible activities with an expense c ≥ 0 equal for each sector and worker and proportional to their contributions, that is c nm = ca for all n = 1, 2, ..., N and m = 1, 2, ..., M. Notice that this assumptions constant expense c is not trivial and unrealistic. In fact, if we consider the same type of stakeholders, in order to avoid any discrimination the rm should invest, for each them, the same amount which is proportional only to the own contribution (meritocracy). Otherwise it might have counterproductive eects (like envy, frustration due to inequality, etc) instead of stimulating cooperation and eciency. In addition we suppose that the worker's sensitivities t 1 and t 2 are equal and are related to the investment in CSR through the function where k is a positive constant and β ∈ R.
Under these assumptions, the company, for given values p and w, wants to maximize the benets associated to the investment in CSR measured by the function (2) that in this case is subjected to Obviously the constraint (4) implies that the rm can't expend in CSR more than what she would earn without social responsible activities. 2 Simplifying (3) we get Therefore the company chooses the value of c that solves We can distinguish three cases: which is a feasible solution only if c * 1 < p − w. We can see that c * 1 increases for high values of δ. In fact, being convenient to enforce workers' sensitivity to SR to earn the high benets due to β > 1, the company should invest more c to counteract the ii) for β < 1 Obviously the above mentioned eects of δ, β and M on the optimal value of c are reversed when the workers are low sensitive to SR activities.
which is constant. Therefore, if it is ever convenient to invest in CSR and the company chooses the optimal value of c satisfying (4), as she can easily recover the costs from the proportional increase in t for k ≥ 1. This condition is more probably satised for high values of k and a.
Our ndings reveals that the convenience to invest in CSR mainly depends on the following factors: 1) the workers' sensitivity; 2) the alienation; 3) the number of sectors; 1. Being favorable to CSR inuences productivity, wages of eciency and intrinsic motivation (see Becchetti et al.,2012). In our decisional problem we nd that, other factors being equal, high workers' sensitivity to CSR practices makes absolutely convenient for the rm to support the related expenses because the workers' productivity increases. Nevertheless these expenses decrease as the sensitivity raises because the workers' marginal productivity for units of investment is higher and the rm can get the same eort even with lower costs.
2. The alienation eect, measured by delta, implies an higher workers' aversion to the task and the company or a greater preference for other activities, leisure or family.

Conclusions
In the ongoing times characterized by an even more globalized world, the reduction of distances thank to technologies make people and systems (economic, social, cultural, etc) strongly interrelated and juxtaposed. Therefore what happens somewhere inuences things happening elsewhere. From a theoretical point of view to study these more interacting systems the traditional economic models are improved also relying on the discoveries of the physical sciences to take into account the several crossed eects among the agents' actions.
In particular in a CSR context her related activities generate a sort of interlinked eects which should be adequately analyzed. In this work we extensively draw from the physical science and specically from the geometrical model of the Möbius strip where the electrons move in several directions to produce energy.
Similarly in a CSR context the social responsible activities have the eects going in several directions which can increase the stakeholders' productivity and eciency so reducing production costs. Therefore we devise a new cost-function where three crossed eects are at work:1) increases in the eciency in virtue of the augmented cooperation among the nearest stakeholders in the same sector; 2) increases in eciency in virtue of the augmented cooperation among stakeholders in the nearest sectors;3) increases in the eciency due to the augmented stakeholders loyalty towards the vision of the company (and also the management and the shareholders)and so towards her nal production.
We show how the benets of the CSR in terms of those three eects incentive the investment in CSR activities and we also provide an example on how this new cost-function can be used to analyze a simple SR rm's decisional problem. Our results show that investing in CSR activities can ever be convenient depending on the number of sectors, the stakeholders' sensitivity to these investments and the decay rate to alienation.
We think that this approach could make light on eects in productivity which not have been adequately taken into account and need to be more analyzed both at a theoretical and empirical level. In particular proceeding from our theoretical model new empirical measures on these crossed eects should be produced to translate our model into reality.