Economic and Emotional Rationality: An Application to Wealth Concentration

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1 Theoetical Economics Lettes, 203, 3, doi:0.4236/tel Published Online August 203 ( Economic and Emotional Rationality: An Application to Wealth Concentation Jose Rigobeto Paada-Daza, Miguel Ignacio Paada-Contzen 2 Univesidad de Concepcion, Concepcion, Chile 2 Technische Univesität Belin (TUB), Belin, Gemany paada@udec.cl, miguelpaadacontzen@gmail.com Received May 29, 203; evised June 29, 203; accepted July 7, 203 Copyight 203 Jose Rigobeto Paada-Daza, Miguel Ignacio Paada-Contzen. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. ABSTRACT This Pape pesents a theoetical outline egading the Emotional Well-being (EW) function as an extension of the economic utility function. EW includes habitual factos that ae always pesent in eveyday decision making. Fistly, an analytical-mathematical conceptualization of EW is caied out, followed by a study of the concept of emotional secuity, in ode to define a new idea of emotional ationality as a complement to economic ationality. An explanation is put foth, as an application, of the concentation of wealth phenomenon accoding to the focus on economic and emotional ationality. The conclusion is that EW is a theoetical appoach which can claify the undestanding of the decision making pocess in economics activities. Keywods: Utility Function; Emotional Well-Being; Economic Rationality; Emotional Rationality; Concentation of Wealth. Intoduction The elation between levels of wealth and utility can be explained conceptually, fom an exclusively economic point of view, by adopting the economic utility function concept, based on the theoy of ational investment decisions measued in utilitaian units. This is the foundation of the agument fo financial and economic decision making theoy. Howeve, this vision can change if vaiables ae incopoated that ae not completely explained by ationality of economic man when the intepetation of a paticula economic fact is attempted. Sometimes it is not completely convincing that the peson o pesons behave solely as economic agents when making economic decisions. As such, following the economic utility function, new vaiables can be incopoated into the analysis in ode to explain the elation between wealth and utility. In this pespective, Paada-Daza [], a function is developed that incopoates the economic ationale adding an element to intepet othe vaiables that influence in making decisions duing an economic act. This new facto includes factos such as: ethics, social esponsibility and othe chaacteistics elated to people. This function has been titled the Emotional Well-being Function. The EW function has been used in liteatue to evaluate copoate social esponsibility Paada-Daza [2], the sustainable development of small and medium entepises, Bilius [3] and to evaluate the economic and non-economic behavio of vaious counties, Paada-Daza [4]. In this pape analytically distinct chaacteistics of the EW function ae developed. Emphasis is placed on two new concepts; emotional ationality and emotional secuity to popose an application of the Emotional Well Being function that allows undestanding, fom a diffeent pespective, what motivates the concentation of wealth and the peception that the actions of ich and poo citizens must be sepaated. 2. Pio Definitions and Liteay Analysis To substantiate the popositions of this pape, some concepts must be defined. These ae extapolated fom consulted liteatue and new concepts incopoated hee. 2.. Emotional Well-Being Pesonal Emotional Well-being (EW) is undestood as the level of satisfaction that is felt when undetaking any act of daily life. This includes economic and global emotional satisfaction. This is evaluated by emotional pleas-

2 234 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN ue. Highe the level of emotional pleasue, highe the level of emotional wellbeing. It is consideed hee that EW is a vaiable that is dependent on an individuals wealth Relative Wealth Let w a,i be the wealth of individual i, valued in absolute monetay units a, in an economy compised of n individuals and W the total wealth in the economy in the following fom: n i w ai, W Thus, a vaiable is defined as wi, 0, which epesents the elative wealth of an individual i as: w n ai, wi,, such that wi,. W i The elative wealth of the individual i pemits the gouping of people as moe wealthy and less wealthy. Theefoe if an individual i has a wealth elative to 0.9 it indicates that, of the entie wealth of a society of n individuals, this individual i has accumulated 90% and the est of the n individuals, possess 0% of the total wealth Utility Function and Its Liteay Discussion Economic Utility In economic decision making theoy, the utility function is used as the basis of analysis fo explanation unde isky conditions. A set of axioms has been developed to explain the choice o decision theoy, Copeland and Weston [5] established a quadatic utility function as well as the logaithmic utility function that has been widely used in liteatue. Theoies have been developed based on this type of function that eflect the behavio of people fom an economic ationality standpoint, giving undestanding as to why people ae essentially maximizes and only move though this type of function. Using the concept of elative wealth, fo the puposes of this pape a utility function is defined as: U :0, and w U Uw. Thus, U is diffeentiable up to at least second gade and is stictly inceasing. i.e. du 0, w 0, In this appoach, mathematically, utility is a function of wealth and it is supposed that the geate the wealth, the geate the utility. Each point shows the value that coesponds to each level of wealth. Economic man moves ove the combination of these points. In egads to the peviously mentioned it is assumed that the utility function is both inceasing and limited. This implies that the geate the wealth, the geate the degee of economic satisfaction. The fact that it is limited means that cetain levels exist whee people conside a cetain amount of satisfaction is acceptable. Anothe chaacteistic of this function is that thee is a diminishing maginal utility. That is, compaed to an incease of wealth, an augment in utility will be gadually deceasing. Mathematically this is epesented by a negative value of the second deivative of the utility function of which coesponds gaphically to a convex function: 2 d U 0, w 0, 2 Some utility functions commonly used in economic and financial liteatue ae: logaithmic function, quadatic function, exponential negative function and potential function, Main & Rubio [6]. In continuation, the two most common functions used in the economic liteatue ae detailed Logaithmic Function ln, with, 0, U w a w c a c () D. Benoulli (730-73) applied the logaithmic utility function, i.e. U(w) = ln(w). Given that this pape utilizes elative wealth, which is defined between zeo and one, the use of a logaithmic function is not possible because the function would take negative values. To avoid this poblem, the agument of the logaithmic function is augmented in one unity as shown in Equation (). The logaithmic function is used because it is inceasing thoughout its domain, is convex and has no elative maximum point o inflection. Figue shows the gaph of this function. The dotted line epesents the utility function value when w = Quadatic Function 2 U w a w a, with a 0 (2) Figue. Function U(w ) = ln(w + ) + c.

3 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN 235 The quadatic function has been used by seveal authos such as R. Meton [7], Copeland-Weston [5], R. Jaow [8]. W. Shape [9], Also epesents the utility function by a quadatic function. The main featue of this is that it pesents a maximum point of utility fo a level of wealth. Beyond this point the function gaph deceases and, theefoe, the utility deceases with inceasing wealth. Regading this, W. Shape [9] said: this is clealy unacceptable. This is explained following the concept of economic ationality which is an intesection between a nomative and a positive appoach, as thee ae people in eal life who behave accoding to this standad howeve othes will not. Thus thei eveyday economic actions ae not fully explained by the assumption of economic ationality and the ethic that this standad implies. Figue 2 shows the gaph of this function Discussion Thus, pesonal economic pefomance educes when maximum utility is obtained. This is methodologically explained though the mathematic maximization of the desied utility of an event vaious altenative popositions. The expected utility implies that, faced with two possibilities of obtaining compensation fo a decision that is made, that which has a highe expected utility is appopiate. The afoementioned is that which is defined as the expected utility hypothesis which epesents the ational behavio of a peson faced with uncetainty. With espect to this, Lafont [0] poses the following: a) The definition of the utility function with these nomative assumptions is a woking hypothesis and theefoe it is necessay to deduce veifiable empiical implications. If the empiical study doesn t dismiss, it can be concluded that people act as if they will maximize the expected utility. b) The utility function is a nomative intepetation that consists of demonstating that ational agents should maximize thei expected utility. The concept of ational behavio ests within the peviously mentioned postulations and is defined as the abil- Figue 2. Function U(w ) = a(w + ) 2 a. ity to choose as if it wee a lottey chaacteized by vaious paths of etibution. This intepetation is an economic definition and the ational concept should not be undestood as a synonym fo tems such as: easonable, pudent, just and fai amongst othes. Thus, a being is ational only if they behave accoding to the economic ule o standad, which fom a Theoy of Knowledge point of view, is based on a ational and empiical model. These two aspects of egulation and maximization ae essential fo undestanding and easoning what, unde these suppositions, implies the concept of ational economic man. Caoll [] follows Max Webe in The Potestant Ethic and the Spiit of Capitalism, which explains that people s seach fo wealth is fo thei own use and possession, and that this is the main cause of both the system and of individuals. Robinson [2] indicates that the maximization of pofit is a metaphoical concept of impenetable ciculaity. Debeu [3] shows that thee ae continuous and non-continuous utility functions. Makowitz [4] based his appoach on the utility function. Patt and Aow [5] establish ways of measuing isk ewad. Poweful utility functions have been developed with complex mathematical fomulas unde the same nomative pinciples as indicated in Ait-Sahalia & Bandt [6], Ang [7], Meha & Pescott [8], Fiend & Blume [9]. Hwang and Satchell [20] elaboate fom the Utility function, how much should be paid in ode to acquie infomation. The utility function has been used to explain donations and signal that these have a simila behavio to luxuy goods. Inhabe and Caoll [2] signal that luxuy goods ae geneally associated with assets such as at, jewely, spoting equipment, etc. Which ae always goods in an economic sense and theefoe, assimilable to any othe economic good. Caoll [] indicates that the love of wealth as motivation is assuedly an exteme position; thee ae othe types of motivation such as: job satisfaction, status, philanthopic ambition, powe, etc. Consideing the afoementioned, the utility function theoy has a conceptual and philosophical foundation but is also egulative, focing the undestanding of people s behavio exclusively as ational economic entities. In this way, the utility function explains any othe type of noneconomic motivation which would be well epesented unde its assumptions. Fom a Theoy of Knowledge stand point, the utility function theoy is a mix between ationalism and empiicism, whee an intesection can be made between the nomative and the positive. The ationalism and empiicism appoach is pesent in financial theoy. The intepetation of an economic act fom a solely ational economic point of view and its methodological epesentation though the utility function can at times give patial esults. Fo which, these acts should be ana-

4 236 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN lyzed fom a view point that encompasses both ational economic standads and thei implicit ethics, as with othe ethic views that can fulminate distinct decisions to those that would suggest an analysis fom a puely economic view point. 3. Emotional Well-Being Function 3.. Definition Fom the postulations of Shape (Op.Ci.), who indicates that no investo is located on the descending pat of the utility function cuve, a moe global function of economic utility is discussed hee. This new function explains the pefomance of people and businesses simultaneously incopoating both the facets of man with economic ationality and the view of people and businesses that also act motivated by othe ethical values, Paada []. This same model has been used to explain the economic cisis in small and medium businesses and thei sustainable politics, Bilius [3], in ode to assess the social esponsibility of the business, Paada-Daza [2], and by this same autho in ode to evaluate the economic behaviou of people in diffeent counties, Paada-Daza [4]. An Emotional Well-being (EW) function consides that thee ae economic sacifices in exchange fo benefits povoked by diffeent easons and exclusively economic motives. This is tanslated mathematically in that the function is not necessaily stictly inceasing thoughout its domain. U :0, (3) w U U w U w, f e whee U (w ) coesponds to a classic utility function as peviously descibed. The tem U e (w, f) joins all cultual, ethical, social, emotional o pactical chaacteistics of a distinct natue to the stictly economic, that can cause the individual to be sepaated fom the behavio of an economic man. Unlike the oiginal aticle of Paada-Daza [], an innovation is intoduced hee, making a genealization of both components and incopoating the f paamete to synthesize the non-economic chaacteistics mentioned. Fo analysis, the following function is consideed because the conclusions deived fom it ae easily genealizable to the types of functions descibed above. EW : 0, EW w U w aln w bsin 2f w With abc,,, f paametes of the function and a + b =. Paametes a and b ae intepeted as the elative impotance that people o oganizations submit to economic c (4) ationality and the espective emotional component. In effect, if a =, then b = 0, a classic utility function is obtained as peviously expessed. The paamete c epesents the minimum satisfaction that is independent fom the level of wealth of each peson o business. This could be zeo, in which case it is implied that emotional wellbeing only depends on wealth. The coefficient c is intepeted as a Satisfaction of Belonging fo being an integal pat of a business o society that povides emotional satisfaction to the peson, independent to thei wealth. This enjoyment can be explained by such factos as: social and business pestige, business and social histoy and tadition, social and business cultue and othe specific factos and chaacteistics of each peson o business. The paamete f is intepeted as the fequency with which the peson o oganization allows the avesion of stictly economic ationality. Distinct values of f involve distinct types of behavio. While highe the value of f, highe will be the oscillation of the emotional vaiable with espect to a puely economic component. The appoach of (4) is an extension of the Emotional Well-being function intoduced by Paada-Daza [], because in this pape, the paamete f is consideed pemanently in f = /2 and, given that it was woking with absolute wealth (in geneal, w > ), the displacement of the algoithmic component was not consideed. Howeve, in this pape it has been decided to open the spectal analysis to include the paamete f and with it explain why the valo f = /2 has been peviously used. Theefoe, in contay to the Paada-Daza [] pape, in this contibution the elative wealth concept has been adopted, with w 0,, and the oiginal appoach has been modified adding to the agument of the logaithmic function, in such a way to adjust it in ode to wok without the EW function assuming negative values, Mao [22] Coodinates of Emotional Well-Being: Enveloping Functions The family of cuves (4), defined by the paamete f, includes supeio and infeio enveloping functions. This is explained by the sinusoidal component of the EW function that has a bounded ange in [,], independent of its agument. A cuve is said to be enveloping a family of plane cuves if it is tangent with all the lines of said family, additionally each point of the enveloping function has contact with some of the lines of the family that is examined, Demidovich [23]. Fo a family of plane cuves depending on the paamete α that complies with the following equation: f x, y, 0. The enveloping equation is detemined by way of the,, 0 f x, y, 0. equation systems: f x y and

5 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN 237 By eliminating the paamete α, a disciminating cuve (Φ) is obtained that contains the envelopment of the family studied, stemming fom the following elation: Φ(x,y) = 0. In the geneic case of the EW function poposed in (3), paamete α coesponds to f and x w, y U, then: f : U U wuew, f 0 U (5) e f : 0 f Consideing, as in (4), an emotional component of sinusoidal chaacteistics, U w, f bsin 2f w, e fom the second equation of the system (5) it is concluded that: U e 0 bsin2 f w f f 0 2bw cos2f w0 2f w 2 f 4w Replacing these values in the fist equation of the system (5), the following disciminating function is obtained: : U U w bsin Given that sin 2 2 0, two enveloping cuves ae obtained which coespond to: U wu w b U w U w b Note that no matte the type of stictly economic function that is used (logaithmic, quadatic o othe), fo an emotional component of the sinusoidal chaacteistic, the enveloping cuves ae of the same fom as the economic utility function. These ae intepeted as the maximum and minimum value of the descibed Emotional Wellbeing function. U + coesponds to the behavio of an economic man as a whole while U is intepeted as the minimum accepted level of secuity, which is pesonal and can be epesented by a minimum equiement. In othe wods U + indicates the economic expectations and aspiations of an agent and U the minimum equiements. In ealie papes, Paada-Daza [] the existence of enveloping functions exclusively fo a function with a ational logaithmic pat. In this section, said poof has been boadened to include any type of utility function (U ) that will give oigin to enveloping functions with the same chaacteistics as U. In the paticula case of the equation poposed in (4), U w aln w c is given. Theefoe, eplacing the expessions ealie deduced, the envelopments ae: U w aln w bc (6) U w aln w bc 3.3. Emotional Well-Being Function Gaph Figue 3 pesents a gaph of the EW function accoding to that which has been outlined peviously, the same as both enveloping functions. The Emotional Well-being function (continuous cuve) is contained between both enveloping functions (segmented lines). This chaacteistic allows the affimation that enveloping functions have tendency to incease, although, a as a poduct of the emotional component, clealy identifiable intevals exist whee the cuve is deceasing. With this the following question aises: Is it valid to be located above the deceasing inteval of the cuve? If conventional economic ationality necessaily locates economic agents above the ascending pat of the utility function, what would justify this behavio contay to that habitually studied in economic theoy? Obseve in Figue 3 that the Emotional Well-being function is always above the minimum established by the infeio envelopment (U ). That is to say that thee is always a level of economic satisfaction limit that the individual, society o oganization is not willing to compomise. On the othe hand, it can be obseved that thee is an inceasing level of economic aspiations (U + ) that coincides with the behaviou of any agent in eal life. The diffeence that exists between this nomative model and the actions of a eal economic agent is contasted by the diffeences between the minimum level of secuity established and the eal actions to explain the making of detemined decisions. In othe wods, The EW model encompasses complex agents, with a much wide philosophical undestanding than an exclusively economic entity. In addition to the utilitaian chaacteistics which ae taken as a nomative assumption, othe ethical concepts ae added (such as modeation, bavey, justice o libety) that pemit a moe complete analysis of human tasks. These chaacteistics ae intended to be eflected though the paamete ƒ. Lage values of this paamete indicate a tendency to ecede epeatedly fom the imposed nomative model denoting cetain emotional instability. In tun, values that ae too small, demonstate excessive pudence that diminishes in the long un. In fact, when ƒ = 0, then the EW cuve coincides with the aveage between both limits of behavio (fine segmented line Figue 3).

6 238 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN quency f with which emotional behavio is accepted. The index is intepeted as the elation between that which is not eaned by not being an economic man with espect to the benefit obtained by ove the minimum level of secuity. In ode to coespond as close to eality as possible, this loss cannot be highe than the emotional benefit of situating ove the minimum ( The Secuity Cushion ). As a consequence, a pudent fom of behavio would yield a numeic value less than fo the ESI. Figue 3. Emotional well-being utility function. 4. Measuing Emotional Rationality 4.. Emotional Secuity Index (ESI) Given the EW function, the Emotional Secuity Index (ESI) as the ation between that which is lost fom economic eanings by paying attention to an emotion othe than an economic one, divided by the excess eaning ( Secuity cushion ) with espect to the minimum desied pofit fo any point of elative wealth w. That is to say, the numeato U w EW w coesponds to lost eanings at a point w by paying attention to values diffeent to denominato EW w U w, in tun, coesponds to excess eaning ( Secuity cushion ) with espect to minimum equiements at this point w. The index is the following: ESI w EW w U w U w w EW In the paticula case of Equation (4), when b = 0, the index cannot be defined by the numeato as the denominato in the expession (7) would have a value equal to zeo. Replacing in (7), the expessions (4) and (6) and simplifying, fo b 0, the index takes the following fom: ESI w (7) sin 2f w (8) sin 2 f w With sin 2f w 0. Note that always ESIw 0, due to the numeato and denominato ae always positive. Futhemoe, ESI doesn t depend on paametes a and b but only of f. That is to say, it doesn t depend on how much impotance is given to emotionality by each individual, oganization o economic system, only the fe Emotional Rationality A decision is defined as emotionally ational if the economic loss though emotional behavio is less than the benefits with espect to the minimum toleated level. That is to say when: ESI w (9) In the context of (4), given that fo any value of 2f w, both the ESI numeato and denominato ae positive, thus: f w f w ESI w sin 2 sin 2f w 0sin 2 02f w 0 f w 2 As w 0,, in ode to maintain emotional ationality on evey level of wealth the following should be obseved: 0 f 2. Fo subsequent analysis, f = /2 is chosen which epesents an exteme case. Thus, maximum fequency ensues emotionally ational behavio accoding to that which is defined in (9), fo evey level of wealth. Note that any infeio value would make the EW function seem simila to a pue utility function. In paticula, if f = 0, then the EW function coincides exactly with aveage between both enveloping functions, that is: w a w c 0 0.5U w U w U w EW ln With this, fom this point onwads, Emotional Wellbeing and ESI functions will be implemented in the following fom: w a w b w sinw w sin w EW ln sin c ESI (0) EW Function Popeties unde the Supposition of Emotional Rationality ) Point of tangency with U +

7 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN 239 The EW function is tangent with U + in w = /2. Indeed, in ode to have: By (6) and (0) then: w U w EW ln aln w bsin 2f w c a w bc Reducing, the following is obtained: sin 2f w. This complies with the function domain only if: 2f w 2. Additionally, f = /2, thus: w 2. That is, the EW function eaches a supeio level (U + ) at a solitay point within the domain when the emotional component eaches its maximum value. Additionally, at this point, the maginal gowth ate fo both functions is the same: d EW w w 2 w 2 w 2 a bcosw w a w d U w w 2 w 2 These gowth ates ae intepeted as maginal emotional and economic wellbeing. That is, fo this point, the maginal emotional wellbeing is exactly the same as the maginal emotional wellbeing fo an economic man. 2) Detemining the maximum level of emotional wellbeing Deiving EW with espect to elative wealth, the following is obtained: dew a 2f bcos2f w w On the othe hand, the second deivative of the EW function coesponds to: f bsin 2f w 2 dew a 2 2 w The deived functions can be disassembled into two pats u y u 2 such as: dew u u 2 () a With: u and u 2f bcos2f w 2 w The fist expession of (), u, is intepeted as the ate of change in EW as a poduct of the emotional component of the behavio. Note that this change is always stictly positive. That is to say, the economic component of the behavio is stictly inceasing in all the domain and that in the behavio of individuals, oganizations o economies that appoach the paadigm of economic ationality (i.e. when a and b 0), the EW function will have a maximum level only in the oute lying eaches of its domain when w. Moeove, the second component of (), u 2, coesponds to the EW ate of change poduced by the emotional component of the behavio. This change can be both positive o negative, so it can be pesumed that within the defined domain, w [0,], the EW function can have a maximum o minimum point in w* when: dew w w* 0 u w* u w* 2 Note that the value of w* depends on the paametes of the function a, b = ( a) and f. this equation does not have a diect analytical solution, fo which an analysis of all possible paamete values is not caied out. Futhemoe, when w 0 the EW deivative is positive and theefoe, the EW function is inceasing in a vicinity of 0. Also, unde the supposition of emotional ationality stated ealie (f = /2), the second deivative is negative thoughout the EW domain, thus the EW function gaph is convex. In othe wods, these popeties ensue that thee is a point whee the function eaches its maximum value within the domain [0, ]. Theefoe the EW function will be inceasing in the inteval [0, w*] and deceasing in the inteval [w*, ]. In which case with the supposition of emotional ationality the w* point complies with: a acosw* w* The solution to this equation depends exclusively on the value of paamete a. An exteme case is given when w* =. In this situation, the function will be stictly inceasing thoughout its domain. If the solution to the equation is such that w* >, theefoe the maximum value of EW will coespond to EW max = EW (w = ), since the function would be inceasing thoughout the entie domain. Fom the latte it can be deduced that a value limit exists fo the paamete a fo which a maximum wellbeing is eached fo total wealth (w = ). Indeed, making w* = fom the last expession is obtained

8 240 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN 2 that: a This means that, unde the supposition of emotional ationality, fo values of a supeio to the EW function will be stictly inceasing thoughout its domain and will each maximum in w* =. 3) EW function gaph Figue 4 shows an EW function unde supposed emotional ationality function (continuous blue line) togethe with both enveloping functions (continuous geen lines). Note the tangent point in w = /2 and that the function have a maximum value close to w = ESI Popeties unde Supposed Emotional Rationality Note that the popeties expessed in continuation ae valid when f = /2. ) Inceasing and deceasing intevals Consideing that w 0,, the function is deceasing when: d ESIw 0 2cosw 0 2 sinw 2cosw 0 0 w 2 0 w 2 Analogous to the pevious, the function is inceasing when: d ESI w 0 2 w 2) Maximum and minimum points The w = /2 point is a singula point, in effect: d ESI w 0 w 2 Taking this into consideation as well as the inceasing and deceasing intevals peviously calculated, the maximum and minimum points of the function ae given by: w w w Maximum : w 0 ESI Minimum : w 2 ESI 0 Maximum : w ESI Fom this, it can be deduced that the ESI ange coe- RangeESI w 0,. 3) Symmety of the ESI sponds to Figue 4. EW unde supposed emotional ationality. The ESI shows symmetic behavio in elation to the w = /2 point. This is demonstated in continuation. Accoding to the definition of symmety in diffeential calculus, a function f(x) is symmetic with espect to x 0 when f x0 x f x0 x. Let δ > 0, thus ESI 2 sin 2 sin 2 sin 2 cos sin cos 2 sin 2 cos sin cos 2 cos cos ESI 2 sin 2 sin 2 sin 2 cos sin cos 2 sin 2 cos sin cos 2 cos cos cos cos Theefoe ESI 2 ESI 2 and the function is symmetic with espect to w = /2. Pecisely fo w = /2, the EW function is tangent to its supeio limit o utility function. That is to say, the loss poduced by distancing fom economic behavio is minimized at this point. Note that if points ae defined as w, y w,2, these two points ae symmetic (that is, w, = /2 δ and w,2 = /2 + δ) if and only if w, + w,2 =. In othe wods, two diffeent levels of wealth exist fo which the ESI eaches the same value. These levels ae such that togethe they epesent all economic wealth, as it was peviously den fined that wi,. i 4) ESI Gaph

9 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN 24 It is necessay to note that the index is a elative measue and not absolute. In symmetic intevals, who has moe wealth (w ) has thei EW balanced the same as those with less wealth (w 0), howeve with diffeent levels of economic demands and expectancies. These levels of demands ae given by U and exclusively economic expectancies by U +. This implies that, even if they behave in diffeent wealth dimension, both agents ae equally satisfied because they shae the same elationship between economic compomise and secuity cushion. In ode to explain the pevious, the following example is used. Suppose that two individuals exist. One individual has 20% of the total wealth; on the contay, the othe individual has 80%. Fo symmety, both individuals have the same level of emotional secuity. This is epesented diectly in Figue 5. In this case ESI0.2 ESI Howeve, the individual who possesses 80% of the wealth has an EW value supeio to the individual with 20% as can be obseved in Figue 4. This is additionally accompanied by a highe level of minimum demand (U ) and an economic expectancy of (U + ) Concentation of Wealth Diffeential Gowth In ode to undestand the way in which people behave between the two utility functions, U + y U, it is necessay to analyze the maginal gowth of both cuves fo each given inteval of wealth. By maginal gowth (decline) of a function f(x) in the inteval [x 0, x f ], is undestood the positive (negative) value calculated though the expession: f xf f x0 f x0, xf (2) f x Symmetic Inteval Application Fo the emotional Wellbeing Functions and thei highe Figue 5. ESI(w ) function unde supposed emotional ationality. 0 and lowe levels, maginal gowth can be calculated at a symmetic inteval defined between w and ( w ), with w 2 w, as: U w aln w bc wew w EW w ln 2 ln U w aln w bc U w U w w,w U EW w,w EW aln 2 w aln w a w a w aln w bsin w c U w U w w,w U aln 2 w aln w The numeato of the second expession (Δ EW ) is obtained though simplification using the popety: sin x sin cosxsin xcos sin x sin x Note that: sinw aln w bcaln w bsin wc aln w bc aln w bc aln w bsin wc aln w bc aln 2 waln w aln w bc aln 2 waln w aln w bsin wc aln 2 waln w aln w bc U w, w w, w w, w EW U (3) Expession (3) implies that the incease in demand U is supeio to the incease of EW (Δ EW ) and supeio to the incease of economic expectations U though analysis of symmetic intevals. This implication

10 242 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN is necessay in ode to explain the following point Distibution of Wealth Suppose an economy whee only two agents exist who distibute wealth in an unequal manne. In such a way that the fist agent has a elative wealth of w given by w y and the second by w 2 = w. Fom symmety, both agents have an equivalent ESI. That is, both have the same fom of coveing thei economic expectations with espect to thei level of secuity. It is assumed that the agent with a wealth of w is less wealthy than the second agent with a wealth of w 2, ego w < w 2. Howeve, the second agent has a highe level of EW. In effect, if: EW w EW w2 Replacing and solving you have: 0 EWw EWw Using (0): sin w 0 aln 2w bsin w aln w b Reducing 2 w : 0 ln w 2 w The following is obtained: w 2 w This necessay in ode to fulfill the pemise of inequality between both agents. Additionally, in ode to achieve this supeio level of emotional wellbeing, the second agent has a highe level of demands (U ) than the fist agent. Futhemoe, thei economic expectations (U + ) ae also highe. The symmetic inteval analysis of maginal gowth in the EW function and its two enveloping Equation (3), concludes that the incease in U is highe than the incease in U +. Thus, the quantitative diffeence between the ich agent and the poo agent is popotionally highe in the case of minimum demands (U ) than in the case of economic expectations. This implies that as wealth inceases, the iche agent becomes moe demanding in elation to thei economic expectations. The latte is consistent with that which is obseved in pactice when individuals incease thei wealth and, pesumably fo social easons, thei tastes and minimum standads become moe demanding and efined. That is the case, even though the incease in economic expectations is not significantly as high as the incease in demand. The concentation in wealth is explained though a numeical analysis in continuation. Take an economy of two individuals govened by an EW function with the following paametes: a = 0.7, b = 0.3, c =.4 and f = 0.5. Figues 6 and 7 show gaphs of each function of the associated ESI. Supposing that the fist economic agent concentates 20% of thei wealth and the second agent 80%, the peviously mentioned figues mak the points that each the EW and ESI in said concentations of wealth. Numeically, BE(0.2) =.704 < BE(0.8) =.988 and ESI(0.2) = ESI(0.8) = It is clea that, even in the emotional ationality limit scenaio (f = /2), the individual that concentates the geate elative wealth has a highe level of EW, although both shae the same emotional index secuity value. That is to say that, if both economic agents behave in a ational emotional way, then the accumulation of wealth will ewad with a highe level of emotional wellbeing with at least the same level of emotional secuity fo both. On the othe hand, if in addition to the pevious, maginal vaiations ae calculated fo the EW function and its envelopments, the following ae obtained: 0.2, % ; 0.2, % and U EW Figue 6. EW concentation of wealth. Using: sin x sin cos x sin x cos sin xsin x Figue 7. ESI concentation of wealth.

11 J. R. PARADA-DAZA, M. I. PARADA-CONTZEN , % U This indicates that the change fom a position of povety to a position of elative wealth is accompanied by an incease of minimum demands by the agent. In this way indicating that the ich agent is compaatively moe demanding that the poo agent and that continuing to concentate wealth povides geate comfot. 5. Conclusions This pape has pesented a theoetic outline in efeence to the Emotional Wellbeing function as an extension of the economic utility function. Afte analytical-mathematical conceptualization of the new function, a concept of emotional secuity has been developed which, in tun, is used to define a new idea of emotional ationality as a complement to economic ationality. This concept allows fo the explanation of a moe integated vision with espect to economic decisions whethe they ae elated to business o othe eveyday acts. On the othe hand, an analytical explication has been put foth of the concentation of wealth phenomenon though a function of ational emotional wellbeing. In the intoduction it was stated that the economic Utility function may epesent a biased appoach when tying to descibe the behavio, fo example, of a manage in the decision making pocess. Using a utility function can lead to an incomplete desciption of the pocess of economic decision. An Emotional welfae function, howeve, includes factos usually neglected by the economic ationalist appoach, but always pesent in any business with o without a pofitable outcome. Theefoe it has been poposed that the ole of Emotional Well-being may epesent phenomenon elated to both economic and non-economic satisfaction of agents who make decisions. In conclusion, this pape is an analytical attempt of integation between a ational economic appoach and a management appoach to explaining decision making in diffeent daily acts, whethe they ae fo business o othe activities. Indeed, this pape uses an analytical language, typical of the utility function, to show that mathematical models ae applicable to human behavio, and that they can collect vaious influences of ethical schools distinct to economic maximizing utilitaianism. In paticula, the studied function of EW suggests that thee ae at least two ways to conside the influence of these othe schools of thought on human behavio. The fist is the elationship between the a and b coefficients of the function. It is shown that though the consideation of these elements, a geate (o lowe) elative impotance an agent o company gives to both the economic maximizing component as well as the emotional component of thei behavio. Futhemoe, it is concluded that the paamete f epesents the type of emotionality that the economic agent values in thei decisions. This paamete is closely elated to the concepts of emotional secuity and emotional ationality. These aspects wee synthesized though the inclusion and development of an emotional secuity index (ESI). Regadless of the values of a and b, this is an indicato of economic sacifice in exchange fo emotional satisfaction obtained fo each level of wealth. Thus, this indicato is a peiodic function that depends only on f. That is, the type of emotionality of the individual. This indicato coesponds to a theoetical novelty as it had not appeaed befoe in liteay efeences. This is an impotant finding because fom the index develops the concept of emotional ationality, which allows fo a moe convincing justification of the stictly non-maximizing behavio of economic agents. This new concept of emotional ationality complements the economic ationality appoach and allows an integated view of the explanation of decisions to be geneated. A conclusion that is also elevant, thanks to the concept of elative wealth intoduced in this pape, is that an application on the concentation of wealth is pesented. Indeed, it is concluded as the poposed EW function helps to explain the phenomenon of concentation of wealth though an emotional incentive to do so. This will complement the analysis caied out to show the pactical and conceptual validity of the poposition of EW. This pape concludes that the function of EW has a new application, not thought of at fist, that joins with othe applications mentioned in liteatue. Anothe conclusion of the analytical wok, that whateve the fom adopted by utility functions most commonly used in economic liteatue (quadatic, logaithmic, powe, negative exponential o othe), they ae also envelopments of an emotional wellbeing function. Fom this it can be concluded that the utility functions ae an explanation of edge (o limit) behavio of people duing thei economic acts. Lastly, a sinusoidal chaacteistic of the emotional pat of the EW function is poposed hee. Howeve, the definition of EW function that has been given opens the option to use othe featues. In paticula, it is only equied that the emotional chaacteistic is a bounded function on its ange to ensue the existence of enveloping functions. Thus it is possible to define othe chaacteistics (Fouie seies, stochastic functions, etc.) that can bette epesent a paticula decision making pocess. REFERENCES [] J. R. Paada-Daza, The Utility Function and the Emotional Well-Being Function, Electonic Jounal of Business Ethics and Oganization Studies, Vol. 9, No. 2, 2004, pp [2] J. R. Paada-Daza, A Valuation Model fo Copoate

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