Technological advance, social fragmentation and welfare

Bosworth, Steven J.; Snower, Dennis J. Article — Published Version Technological advance, social fragmentation and welfare Social Choice and Welfare Provided in Cooperation with: Springer Nature Suggested Citation: Bosworth, Steven J.; Snower, Dennis J. (2023) : Technological advance, social fragmentation and welfare, Social Choice and Welfare, ISSN 1432-217X, Springer, Berlin, Heidelberg, Vol. 62, Iss. 2, pp. 197-232, https://doi.org/10.1007/s00355-023-01484-0 This Version is available at: https://hdl.handle.net/10419/316991 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Social Choice and Welfare (2024) 62:197–232 https://doi.org/10.1007/s00355-023-01484-0 1 3 ORIGINAL PAPER Technological advance, social fragmentation andwelfare StevenJ.Bosworth1 · DennisJ.Snower2,3,4 Received: 11 August 2019 / Accepted: 4 August 2023 / Published online: 4 October 2023 © The Author(s) 2023 Abstract This paper models the welfare consequences of social fragmentation arising from technological advance. We start from the premise that technological progress falls primarily on market-traded commodities rather than prosocial relationships, since the latter intrinsically require the expenditure of time and thus are less amenable to productivity increases. Since prosocial relationships require individuals to identify with others in their social group whereas marketable commodities are commonly the objects of social status comparisons, a tradeoff arises between in-group affiliation and inter-group status comparisons. People consequently narrow the bounds of their social groups, reducing their prosocial relationships and extending their statusseeking activities. As prosocial relationships generate positive externalities whereas status-seeking activities generate negative preference externalities, technological advance may lead to a particular type of “decoupling” of social welfare from material prosperity. Once the share of status goods in total production exceeds a crucial threshold, technological advance is shown to be welfare-reducing. 1 Introduction This paper explores how productivity-enhancing economic forces–by increasing material prosperity–can give rise to social fragmentation and how this affects social welfare. People are assumed to be more cooperative within a social group (such as a family, a friendship circle or a workplace team) than between groups. The reason is that people commonly identify their wellbeing with other members of their We are deeply grateful to Paul Collier for his valuable insights.This manuscript benefitted enormously from the scrutiny of two anonymous referees. * Steven J. Bosworth s.j.boswor[email protected] 1 University ofReading, Reading, UK 2 Institute forNew Economic Thinking, University ofOxford, Oxford, UK 3 CEPR, London, UK 4 IZA, Bonn, Germany 198 S.J.Bosworth, D.J.Snower 1 3 social group, but do not do so with regard to out-group members. We investigate how productivity increases that fall on marketable goods and services–rather than on prosocial relationships within social groups–can reduce the size of social groups by raising the return to positional comparisons across groups and thereby influence social welfare. Productivity increases therefore not only raise people’s material standard of living, but also increase social fragmentation as measured by the size of social groups within the economy. We examine the conditions under which the second effect dominates the first, whereupon the productivity increases become welfare-reducing. Our paper seeks to capture a phenomenon that is receiving growing attention in the public debate, but is largely ignored in conventional economic analysis, namely, that around the world–in both developed and emerging market economies–we are witnessing how economic growth can be destructive of local, regional and national communities. In particular, we focus on the decline in people’s close relationships documented by McPherson etal. (2006). Material progress may shrink the scope of our social ties and thus have an ambiguous influence on social welfare–raising welfare by promoting the production of more goods and services for a given set of factor inputs, while reducing welfare through the disintegration of social relationships. For this purpose, we need to extend macroeconomic analysis beyond individualistic microfoundations to recognize to broad categories of economic activities that characterise humans as social creatures: positionally competitive activities (satisfying status-seeking motives, for which one’s welfare is assessed relative to the welfare of others) and cooperative activities within prosocial relationships (in which one’s welfare depends positively on the welfare of others). The three activities differ in terms of their preference externalities: individualistic activities are associated with no such externalities; positionally competitive activities have negative externalities; and prosocial relationships have positive externalities. Our analysis of how productivity growth affects on social fragmentation and welfare rests on two simplifying premises. First, the productivity growth from technological advance falls more on market-traded commodities associated with individualistic and positionally competitive activities than on prosocial relationships. Though prosocial relationships often benefit from technological innovations, their goals tend to be less closely associated with market commodities than are the goals of individualistic and positionally competitive activities. The reason is that these socially cooperative relationships typically, often intrinsically, require time spent in supportive social interactions and this time input cannot be substantially reduced through technological advance. The second premise is that prosocial relationships are more common for the relations within social groups than across social groups. Though many prosocial activities occur across social groups, prosocial relationships occur preferentially within social groups defined by a “we” (Akerlof 2016). The choice of whom to extend “we” rather than “you” and “I” has two natural implications: it defines the relevant group within which one is able to most easily overcome cooperation problems, and without which social comparisons become more salient. Under these two premises, we analyse how productivity growth promotes individualistic and positionally competitive activities at the expense of prosocial activities. We examine how these incentives reduce the size of social groups, thereby 199 1 3 Technological advance, social fragmentation andwelfare generating social fragmentation. Consequently, productivity growth has an ambiguous influence on social welfare, since it promotes negative preference externalities (associated with positionally competitive activities) at the expense of positive preference externalities (associated with prosocial relationships). On the one hand, productivity growth promotes the production of individualistically want-satisfying commodities (thereby raising welfare); on the other, it promotes activities in which one person’s welfare gain is another’s welfare loss and discourages activities in which people gain from one another’s welfare. In this context, we derive a condition under which productivity growth reduces aggregate welfare.1 In these respects, this paper draws on and significantly extends the analysis of Snower and Bosworth (2016), which does not derive conditions for welfare-reducing technological advance. We also assess the empirical plausibility of this condition. In particular, we provide a rough calibration of our model for the United Kingdom, which indicates that welfare-reducing growth is indeed an empirical possibility, worthy of further examination. In this light, technological advance and globalisation can be associated with a well-known aspect of rising individualism (as described, for example, by Putnam (2000) and McPherson etal. (2006)), manifested through declining willingness to engage in civic activities, to contribute to public goods and to make contributions to social allegiances. The technologically-driven rise in social fragmentation can lead to a “decoupling” of social welfare from material progress. Our analysis points to the need for further investigation of the consequences of productivity growth for social communities and the need to bring macroeconomic policy and innovation policy into closer association with social policy. As indicated below, the possibility of welfare-reducing growth is not an argument for stopping technological advance and structural economic change, but rather for designing public policies and business strategies that sustain and nourish social communities. The rest of our paper is organised as follows. Section2 summarises the motivational foundations of decision making in our analysis. Section3 presents our analysis of comparative, individualistic and cooperative activities. Section4 describes the general equilibrium. Section5 derives the effect of productivity growth on aggregate production, social fragmentation and welfare. Section6 calibrates the parameters of the model to existing stylised facts. Section7 derives additional welfare implications when the proportion of positionally competitive activities rises in response to productivity growth and when there are diminishing returns to the production of market goods. Section8 concludes. 1 Our paper is certainly not the only to introduce a model wherein growth can be welfare-reducing. See Peng (2008) for a model in which envy can outstrip consumption utility. Our focus is rather specifically on the phenomenon of social fragmentation, and our results hold for an arbitrarily small disutility from envy. 200 S.J.Bosworth, D.J.Snower 1 3 2 Motivational foundations ofdecision making The individualistic, comparative and prosocial activities in our analysis are generally recognised to be driven by distinct human motives: • Self-interested wanting, whereby an individual’s utility depends exclusively on her own payoff, • Positional competition,2 whereby her utility depends on her payoff relative to her relevant reference group, and • Prosociality, whereby her utility depends positively on the utility of her in-group. 2.1 Motives ineconomic decision making The underlying insight is taken from motivation psychology,3 namely, that people have access to multiple “motives”, which are psychological forces that give direction and energy to one’s behaviour. Different motives can be associated with different utility functions. Which motives are active at any point in time depends on one’s social context. Prosocial motives engender group cohesion, whereas positionally competitive motives delineate and secure the individual’s place within social hierarchies. The self-interested wanting motive drives the satisfaction of wants that pertain to oneself, without reference to any social relations. All three motives are common in practice. Prosociality generates the desire to promote the wellbeing of others and to alleviate their suffering. It includes acts of benevolence, altruism, sympathy, as well as the need to be liked and the need for interpersonal relatedness. It occurs naturally among kin and is frequently extended to friends and other non-kin groups with whom one identifies. Positional competition takes a wide variety of forms in market economies, including concern with one’s wealth, physical appearance, possessions, political clout, business success, intellectual prowess, sports achievements, etc. relative to the other members of one’s reference group. It is manifested as ostentatious consumption, keeping up with the Jones’s, tournament contracts in the labour market, rankings of fund managers, tennis seeds, football leagues, and much more. Our analysis focuses on positional competition and prosociality since these motives exemplify two common, yet contrasting economic objectives. Under status-seeking, one’s payoff is diminished by the payoff of one’s competitors; whereas under prosociality, one’s payoff is enhanced by the payoff of the members of one’s reference group. Non-positional activities arise when we satisfy our basic needs for food, shelter, clothing, and other essentials for the maintenance of life. Except for people living in extreme poverty, most of our consumption activities satisfy “wants” rather than “needs,” and many of these wants arise from positional battles in social settings. The prevalence of such positional battles is clarified through evolution-based theories 2 For example, Heckhausen (1989, 2000); Heckhausen and Heckhausen (2010). 3 Heckhausen and Heckhausen (2010) provide an excellent survey. 201 1 3 Technological advance, social fragmentation andwelfare describing how survival and procreation depends on one’s social ranking. Prosociality is common within families; no child would survive without it. Much of the evolutionary success of homo sapiens is due to our ability to extend prosociality to non-kin groups. 2.2 Motives pertaining tosocial groups Both positional competition and prosociality take place with respect to pre-existing reference groups, defined by our social identities. For the purposes of our analysis, we restrict our conception of social identity to the formation of social class groups. Specifically, each identity describes an in-group, the payoff of whose members we seek to promote, and a “competing out-group,” the payoff of whose members we seek to surpass.4 People are assumed to be motivated by prosociality toward their in-group and by positional competition toward their out-group. These assumptions are admittedly drastic simplifications of people’s actual relationships, but they provide a simple analytical framework for exploring something important, which has received little if any attention in traditional economic analysis. In particular, the Care and Affiliation motives generate positive externalities, whereas the positional competition motive generates negative externalities. This turns out to have potentially important implications for the influence of productivity growth on social welfare. There is substantial psychological evidence that positionally competitive and prosocial motives are in fundamental conflict due to their opposing internalisations of others’ welfare. This conflict is mediated by identification: other people are categorised as “us”, with whom we affiliate or “them”, with whom we differentiate (Akerlof 2016). Aron etal. (1991) characterise close relationships as featuring a high degree of overlap between conceptions of the self and the other person.5 Galinski etal. (2005) show that this self-other overlap explains why close relationships foster social cooperation (prosocial motives). McFarland etal. (2001) find muted affective responses to social comparisons with close others. Gardner etal. (2002) experimentally prime interdependent self-constual (close identification with others) and find that unfavourable social comparisons become cause for celebration rather than envy, and favourable social comparisons cease to be cause for pride. Chen and Li (2009) induce group identity and measure social preferences using a number of strategic economic games, finding that in-group members display greater altruism and lower envy toward one another. Similarly, Oveis etal. (2010) show that both traitand state-induced compassion is associated with increased perceived self-other similarity, while pride is associated with a decreased sense of similarity to weak 4 In practice, people also have “non-competing out-groups,” the payoff of whose members is irrelevant to their decisions. For analytical simplicity, however, we ignore this category in our analysis. Genicot and Ray (2017) for example study the motivating effects of social comparisons with those of very close incomes. Our analysis is consistent with the view that social comparisons with out-group members of similar income are most important since our model’s results hinge on optimisation with respect to who the marginal in-group member is. 5 Gächter, Starmer and Tufano (2015) review an experimentally tractable and validated measure of perceived self-other closeness. 202 S.J.Bosworth, D.J.Snower 1 3 others. Our assumption that there is more prosociality within groups and more positional competition between groups is therefore well founded. 2.3 Technological market bias Our analysis rests on the hypothesis that productivity growth arising from technological advance falls more on market activities than on non-market, prosocial relationships – what we shall call the “technological market-bias hypothesis”. The reason underlying this hypothesis akin to the “Baumol effect.”6 The amount of time input required by social relationships powered by prosociality – such as socially supportive relationships with one’s spouse and children – has changed much less over the past century than the huge technology-driven productivity improvements in the production of goods and services. To be a good friend or good relative generally calls for substantial unmediated personal exchanges. We argue that though these social interactions can be promoted through technological advances, the latitude for doing so is far more limited than for goods and services devoted to the purposes of positional competition and materialistic consumption. Though goods and services can serve many goals – comparative, individualistic and socially supportive relationships – we claim that the prosocial relationships invariably require much time to be spent together and technological advance cannot significantly reduce this time input without degrading the relationships. Goods and services are often consumed in the process of conducting socially supportive relationships and although these goods and services are complementary to these relationships, technological advances in the production of these goods and services do not significantly reduce people’s time spent in tending to the relationships, at least in comparison to the effect of technology on positionally competitive and individualistic pursuits. For example, advances in computer technologies have given rise to vast productivity improvements in the production of positional goods such as automobiles and jets, but we still require much the same amount of time to give socially supportive care to friends, children and the elderly. Maintaining socially cooperative relationships may be aided by technological developments – such as advances in communication technology – but these are incidental to the relationships themselves and must combine with time and attention devoted to others. This latter ingredient by its nature can hardly be economised on.7 Dealing specifically with a technology complementary to social relationships, 6 Baumol’s “cost disease of the services” refers to service sector jobs that experience wage growth though they do not benefit from technological progress. These service sector jobs are market activities, to be distinguished from non-market relationships. Like many services, the labour productivity of nonmarket relationships – such as playing tag with one’s children, dancing with loved ones, playing tennis with friends – cannot be raised significantly through technological progress, since the time input of the participants is central to these activities. Unlike Baumol’s phenomenon however, this productivity difference between socially cooperative relationships and competitive and individualistic activities does not arise from the distinction between goods and services. Our distinction is rather between goods and services that meet competitive and individualistic goals versus those that meet socially cooperative goals. 7 This holds intrinsically, since the non-market, prosocial relationships rest centrally on the expenditure of time with others. 203 1 3 Technological advance, social fragmentation andwelfare Rotondi etal. (2017) show that smartphone adoption degrades the overall quality of one’s social interactions and resulting wellbeing. Furthermore, socially cooperative relationships cannot typically be re-framed into material transactions without significantly diminishing the nature of the exchange.8 The quest for positional status on the other hand, is very much tied in with material plenty. Showing others that one commands plentiful material resources generally promotes one’s place in a social hierarchy. Conspicuous consumption is a prime example of a market activity, whose productivity is strongly affected by technological progress. But the domain of positionally competitive activities amenable to technological progress is far wider than this, because the benefits of technological progress fall more on high-earners than on low-earners and high earnings are a common source of positional status. In our analysis, market-traded goods are divided into positional and non-positional consumption. For parsimony, we first assume that this fraction remains constant as society becomes more prosperous. This is a conservative assumption, as diminishing marginal utility for non-positional consumption implies that income growth is most likely to be spent on positional consumption at the margin. People first satisfy their basic needs for nutrition, clothing, shelter and transportation, and only then seek out artisanal food, designer clothing, large houses for their possessions, and luxury cars.9 In this context, our analysis shows how productivity growth has an ambiguous influence on social welfare. This influence may be decomposed into a firstand second-order effect. In the first-order effect, productivity growth raises welfare by enabling the production of more non-positional commodities with given factor inputs, but it reduces welfare by reducing the scope of people’s in-group identification, thereby promoting positionally competitive relationships (which are zero-sum) at the expense of prosocial relationships (which are positive-sum). Whether this first-order effect is positive or negative depends on the relative strength of these two forces. The second-order effect depends on preference and production changes that occur once positional competition has increased at the expense of prosociality. More positional competition may be expected to give rise to increased sensitivity to the gains from positional competition and diminishing returns in the production of positional and non-positional goods. Each of these effects further reduces the social welfare generated by productivity growth. 2.4 Positional competition andindividualism There is a large literature on the rise of individualism, particularly in the West (e.g. Rahn and Transue 1998; Putnam 2000; McPherson etal. 2006). Of particular concern for us is the time series evidence showing a narrowing of social relations in terms of socioeconomic heterogeneity. Paxton (1999) documents a decline 8 For example, we do not show our appreciation for a friend’s dinner party by paying the friend at the end of the party. 9 We consider this extension in Sect.7, where our quantitative conclusions are strengthened while our qualitative results remain unchanged. The rebalancing of consumption towards more positional goods exacerbates, but is not a necessary condition for, the welfare-reducing effects of growth. 204 S.J.Bosworth, D.J.Snower 1 3 in evenings spent with neighbours over a 20 year period in the United States, with some substitution towards other friends. Li etal. (2003) document increasing class polarisation of friendship networks in the United Kingdom from 1972 to 1998. This corroborates McPherson etal. (2006) who find that the number of people with whom General Social Survey respondents in the United States discuss personal matters has shrunk between 1985 and 2004, and that the average educational heterogeneity of these close friendship networks has also fallen. McPherson etal. also show that the reason why time spent with close ones has not fallen by as much is that people socialise more intensely with a narrower range of people (pp. 361). There is also evidence that these trends are associated with rising levels of economic growth. Panel regressions show that even though interpersonal trust promotes growth (Algan and Cahuc 2010), growth degrades interpersonal trust (Roth 2009; see also Mahdavi and Azizmohammadlou (2013)). The implications of individualism for well-being have also been studied extensively, with much evidence indicating that a decline in social ties is inversely associated with self-reported happiness and various objective measures of well-being (e.g. Ogihara and Uchida 2014). Bartolini and Bilancini (2010) track changes in socialisation and income across a panel of countries and find that income per capita predicts modest increases in subjective wellbeing, but only when controlling for the quality of people’s social relations. A straightforward application of omitted variable bias means that these changes in income are correlated with drops in sociality. The reasons adduced for why individualism can reduce well-being are diverse: an erosion of trust, a decline in the sense of connectedness to others, and a rise in narcissism (e.g. Bosson etal. 2008; Putnam 2000; Twenge 2006; Twenge and Campbell 2010). There is much evidence that well-being depends significantly and substantially on personal relationships, starting with psychologists’ recognition of such relationships as a basic human need (e.g. Baumeister and Leary 1995; Kasser and Ryan 1999; Ryff and Singer 2000; Deci and Ryan 2001) and proceeding to economists’ studies on the correlation between self-reported happiness and personal relationships (e.g. Uhlaner 1989; Gui 2000; Frey and Stutzer 2002; Helliwell 2002; Bruni and Stanca 2008; Becchetti etal. 2008, 2009; Gui and Stanca 2010). The importance of positional competition in market economies has received substantial empirical attention. For example, on the basis of social surveys and contingent choice studies, Easterlin (1974); Kahneman etal. (1999) and others have found that people’s subjective well-being and life satisfaction were more closely associated with their relative material status than their absolute income. These findings are consonant with survey evidence that people voluntarily accept reductions in their absolute incomes in return for improvements in their rank within the income distribution (e.g. Solnick and Hemenway 1998). The first major investigation of how economic growth is associated with a proportional growth of positional goods relative to non-positional goods was conducted by Hirsch (1976). He argued that rising affluence is associated with a rising proportion 211 1 3 Technological advance, social fragmentation andwelfare By increasing the productivity of engaging in positional competition, technological advance and globalisation induce individuals to substitute status relationships for socially cooperative relationships, which explains the decline in group size. Furthermore, the increase in productivity leads to a rise in the production of commodities xi . There is a direct effect on material welfare (via the rise in market good production for a given amount of effort) and an indirect effect that operates via the rise in social fragmentation): The direct effect is denoted by the first term ( 1 2+ai ) and the indirect effect is denoted by the the second term −( 𝜆𝜕N∗ 𝜕𝛽 ). Since both effects are positive, note that the rise in social fragmentation augments the production-enhancing effect of the initial productivity stimulus from technological advance. Next, we consider the welfare implications of productivity growth, accompanied by a growing quest for positional status, whereby people can gain only at each other’s expense. These welfare implications may be assessed in terms of the following social welfare function i.e. the sum of the utilities of all social groups. The economy contains K+1≡⌈1∕N∗⌉ social groups, with the upper K=⌊1∕N∗⌋ groups16 having equilibrium size N∗ and a smaller “rump group,” of size size 1−KN∗ at the bottom of the ability distribution, that is left over once the highest-ranking members of all the other groups have made their choices of group members. The welfare effect of productivity growth is the sum of a direct effect 𝜕W 𝜕𝛽 (holding group size constant) and an indirect effect 𝜕N∗ 𝜕𝛽 dW dN∗ (via the change in group size N∗ ): The direct effect (by Eq. (5)) is The indirect effect represents the influence of a rise in productivity 𝛽 on group size N∗ and thereby on the three components of welfare: Uc from socially cooperative relationships, Un from non-positional commodities, and Us from positional commodities. (18) dUn i d𝛽 =𝛾 ( 1 2 +ai ) − ( 𝛾𝜆𝜕N∗ 𝜕𝛽 ) =𝛾 ( 1 2 +ai ) +𝛾𝜆(𝛼−𝜆) 𝛽 2 𝜋(1−𝛾) > 0 (19) W = K+1 ∑ k=1 ∫ak a k Uidai , (20) dW d𝛽 = 𝜕W 𝜕𝛽 + 𝜕N∗ 𝜕𝛽 ⋅ dW dN ∗ . 𝜕W 𝜕𝛽 =𝛄 . 16 In the comparative static exercises to follow we will treat K as fixed. Note that this restricts our attention to small changes in group alignments, from which the effects of larger changes may be approximated. 212 S.J.Bosworth, D.J.Snower 1 3 We begin by calculating the effect of a rise in group size on positional utility: dUs∕dN∗ . We first consider discrete changes in group size, and then take a limit to derive the differential effect on welfare. The process of individualisation leads to a cascade of social demotions down the ladder of positional status, starting with a shrinking top-status group and rippling down to the progressively shrinking lowerstatus groups. Each step in the individualisation process generates “demotees” (who are relegated to the next-lower social position) and remaining “incumbents” (who maintain their previous social position). In our analysis, each social group is of equal size, comprising the incumbents and demotees from a higher-status group. This implies however that groups’ lower membership boundaries will shift by more than their upper membership boundaries, and in fact the lower down the social stratum, the more demotees relative to incumbents there will be. Figure2 illustrates. The highest-status group 1 shrinks by Δa1 . The next-highest-status group both shrinks in size by Δa1 but also shifts to incorporate all the demotees from the first group. Therefore the lower membership boundary for this second group shifts by Δa2=2Δa1 . Likewise Δa3=3Δa1 . In general, dak∕dak=k∕k+1<1 . As noted, people are envious of higher-status groups and proud regarding lowerstatus groups, but they experience neither pride nor envy regarding members of their own social group. Suppose that the group size changes by ΔN∗ and that this implies changes in group boundaries by Δak , ak+1 by Δak+1 , and so on. Then the change in the aggregate positional status-driven utility Us may be expressed where the first term represents the change in utility of the people who have not switched groups, and the second term represents the change in utility of all those who have switched groups (i.e. those, for positive Δak , who were members of group k but are now members of group k+1 ). Taking the limit of ΔUs∕ΔN∗ as ΔN∗ approaches zero, we derive the effect of group size on welfare from positional commodities17: (21) Δ Us= ∑ k∫ a k ak+Δak ΔUs idai ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ incumbents +∫ a k +Δa k ak ΔUs idai ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ demotees 17 A full derivation may be found in the attached workings. Fig. 2 Visualising the cascade of social demotions 213 1 3 Technological advance, social fragmentation andwelfare On this basis, the indirect effect may be derived as follows. By Eq. (17), the effect of productivity growth on group size is negative. Furthermore, it can be shown that the effect of group size on welfare is positive18 Intuitively, only the highest-ability member of each group has a marginal utility from prosocial relationships equal to the marginal utility from commodity production. For all other members of the group, the marginal utility of prosocial relationships is greater than the marginal utility from commodity production. Thus for the group as a whole, welfare falls as group size falls. Unlike in the case of non-positional commodities, we suppose there is no aggregate direct effect of productivity growth on status utility Us . More specifically we assume tha U s by definition satisfies and that there are only indirect effects of productivity 𝛽 on positional utility Us through its impact on equilibrium group size N∗ , dN∗∕d𝛽 . Were U s not subtracted out of status utility we would observe scaling effects from increasing 𝛽 which would amplify people’s experienced pride or envy. We feel that from a welfare perspective it would not be appropriate to count this because status is inherently relative and its aggregate quantity cannot increase when the objects of status competition become more abundant. Thus the effect of productivity growth on social welfare may be expressed as follows: (22) dUs dN ∗= 𝛽𝜀 2 (1−𝛾)K ( N∗2−(1−KN∗)2 ) −(1−𝛾)𝜆 . (23) dW dN ∗=𝛼−𝜆+ 𝛽𝜀 2 (1−𝛾)K ( N∗2−(1−KN∗)2 ) > 0. (24) 𝜕 𝜕𝛽 K+1 ∑ k=1 ∫ak a k Us idai= 0; (25) dW d𝛽=𝛾 ⏟⏟⏟ direct effect + 𝛾𝜆(𝛼−𝜆) 𝛽2𝜋(1−𝛾) ⏟⏞⏞⏞⏟⏞⏞⏞⏟ effort effect ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ increased non-positional commodities − 𝛼(𝛼−𝜆) 𝛽2𝜋(1−𝛾) ⏟⏞⏞⏞⏟⏞⏞⏞⏟ lost prosocial relationships −(𝛼−𝜆)𝜀K(N∗2−(1−KN∗)2) 2𝛽𝜋 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ excess envy +(1−𝛾)𝜆(𝛼−𝜆) 𝛽2𝜋(1−𝛾) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ effort effect ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ increased positional commodities . 18 The positive effect follows from three conditions: (i) Eq. (12), (ii) the rump group is smaller than the other groups: (K+1)N∗>1 (for otherwise the rump group would have formed as another social group), and (iii) the number of people in the rump group is positive: KN∗<1 . For a formal proof, see Workings: in the supplementary materials. 214 S.J.Bosworth, D.J.Snower 1 3 As this equation shows, technology-driven growth affects social welfare via three channels: 1. Non-positional commodities: The productivity increase raises the production of non-positional commodities (i.e. the ones captured in conventional utility functions). This effect can be decomposed into a direct effect (more non-positional commodities produced for the same amount of effort) and effort-related effect (more effort is devoted to non-positional commodities, at the expense of prosocial relationships). (a) Direct effect (first term): productivity growth permits the production of non-positional commodities for the same amount of effort input. This is the effect in the absence of a change in effort on non-positional production and on prosocial relationships. In other words, it can be thought of as the traditional “manna from heaven” portrayal of productivity growth: people gain additional consumption at the margin from the effort they were already putting in. The resulting social welfare effect is, not surprisingly, unambiguously positive. The magnitude of this effect depends on 𝛾 , the proportion of non-positional commodities relative to GDP. (b) Effort-related effect (second term): productivity growth also leads people to substitute more time into market activities, away from socially cooperative relationships. This generates both more positional and non-positional commodities, on account of the greater labour input and the increased productivity of this input.19 2. Socially cooperative relationships (third term): Productivity growth favours market activities relative to the non-market prosocial ones. Thereby it leads to increased individualisation, in the form of smaller social groups, which hurts socially cooperative relationships since these relationships are club goods. This resulting social welfare effect is unambiguously negative: − 𝛼 2 𝛽𝜋(1−𝛾) < 0. Note that the standard microeconomic result that an increase in the productivity of one private good relative to another has substitution effects which sum to zero20 does not obtain here, due to the club-good nature of prosocial relationships.21 3. Positional commodities: The productivity increase raises the production of positional commodities. This effect can be decomposed into an excess envy effect (individuals at the bottom of the social hierarchy have more people to feel envious 19 If individuals were not allowed to change their effort, or if there were no tradeoff between goods consumption and caring relationships (when the consumption substitutability parameter is 𝜆=0 ), this term is zero. 20 This would be justified by an application of the envelope theorem to U in the case of private goods. Note that here only a measure-zero subset of agents have their first-order conditions satisfied. 21 The substitution effect away from caring activities may be greater or less than the substitution effect towards non-positional commodities, depending on the parameters of the model, including the consumption substitutability parameter 𝜆 . 215 1 3 Technological advance, social fragmentation andwelfare of) and effort-related effect (more time is devoted to positional commodities, at the expense of prosocial relationships). 1. Excess envy (fourth term): the formation of smaller social groups leads to a rise in positionally competitive activities. When 𝜀>𝜋 (Boyce etal. 2010 provide empirical support for this claim) increased positional competition has an unambiguously negative effect on social welfare. However, even under the assumption 𝜋>𝜀 , the increased pride utility and effort-related goods production will not on net exceed the lost utility from socially cooperative relationships. This follows from the result in Eq.23. While it is true that for every person who gains from a relative rise in positional status, there is another person who loses from a relative loss in status,22 this does not mean that status seeking is socially neutral. The reason is that increased individualisation leaves the the worst-off group worse off than it was before (i.e. there is a rump group which gets bigger). 2. Effort-related effect (fifth term): productivity growth also leads people to substitute more time into market activities, away from socially cooperative relationships. This generates greater consumption of positional commodities, on account of the greater labour input and the increased productivity of this input. The “welfare implications of growth” equation has implications given in the following: 5.1 Proposition When the proportion 𝛾 of non-positional goods is lower than 𝛾 , then productivity growth unambiguously reduces social welfare, where the proportion of non-positional goods is approximately In general there is not a closed-form solution for 𝛾 since N∗ depends on the share (1−𝛾) of positional goods in consumption. We can however use the edge cases K=1∕N∗ (population exactly partitioned into equal size groups, so that there is no rump group) as an approximation of 𝛾 . In these cases, N∗ drops out of the expression for W𝛽 . By implication, if productivity growth is generating a higher proportion of positional goods than 𝛾 , then the welfare effects of growth must be negative. We consider this possibility empirically plausible (See Sect.6 below for a rough calibration). (26) 𝛾 (𝛼,𝛽,𝜆,𝜀,𝜋)≃1 2+ √ √ √ √ √ 1 4− ( 𝜋+𝜀 2 ) (𝛼−𝜆)2 𝛽2𝜋2 . 22 Recall that the total amount of status in society must remain constant, as indicated through the normalisation of status utility (subtracting U s from Us i,j ) in Eq (6): This means there is no direct effect from the increased productivity of status production. 216 S.J.Bosworth, D.J.Snower 1 3 Note that the condition 𝛾 < �𝛾 , under which economic growth (a rise in productivity level 𝛽 ) reduces welfare (W), is itself dependent on the current productivity 𝛽 . Figure3 illustrates how welfare depends on growth, under three scenarios. (i) When 𝛽 is small ( 𝛽<𝛽 1 ), there is no social fragmentation ( N∗=1 ) and thus growth in the level of productivity 𝛽 raises welfare, since it raises the consumption of non-positional goods without raising social fragmentation. However welfare does not rise as fast as output, since the share of non-positional consumption is 𝛾<1 . (ii) When 𝛽 is large ( 𝛽1≤𝛽<𝛽 2 ), increases in the level of productivity 𝛽 lead to increased social fragmentation (K rises as N∗ falls) and then correspondingly welfare falls, provided that the condition 𝛾 < �𝛾 is fulfilled. (iii) When 𝛽 is very large ( 𝛽≥𝛽2 ), there is hardly any social capital left to depreciate and then any rise in the level of productivity 𝛽 again leads to an increase in the consumption of non-positional goods without further raising social fragmentation. Thus welfare starts to rise again, with a limiting slope lim𝛽 → ∞dW∕d𝛽=𝛾 . This upward-sloping region has little if any practical relevance, since it describes an economy in which social groups have virtually disappeared. Since social belonging is a fundamental human need (otherwise solitary confinement in prison would not be punishment), such an economy would be psychologically unbearable, leading social upheaval, associated with a change in the other parameters of our model. Thus far, we have considered only the effect of productivity growth on social welfare, via reductions in the size of social groups (increased individualism). This of course is a comparative static analysis – assuming all other parameters remain constant. The model’s other parameters will not in practise remain fixed as 𝛽 increases. Recall that group size can be reduced even more through the consequences of the gains from increased positional competition (rises in 𝜋 ), and diminishing returns to the consumption of market goods relative to prosocial relationships (falls in 𝜆 ). Fig. 3 Effects of growth – Output vs. welfare for fixed 𝛾 217 1 3 Technological advance, social fragmentation andwelfare Obviously, in the presence of these changes, the lower bound on the proportion of non-positional goods ( 𝛾 ) is even lower than that given by Eq. (26). Furthermore since the limiting slope of the welfare function W is equal to the share of non-positional goods 𝛾 in total output, the evolution of this share has important implications for the dynamics of growth and welfare, as explored in Sect.7. 6 Calibration As indicated above, productivity growth becomes welfare-reducing once the proportion of non-positional goods falls beneath the threshold level 𝛾 . We now make a rough assessment of the empirical plausibility of reaching this threshold level with regard to key data from published research. For this purpose, we start with a simplifying assumption. We make the conservative assumption that the consumption substitution parameter is 𝜆=0 , i.e. increases in prosocial activities does not reduce the consumption and therefore production of market commodities. Under these conditions, by Equation (12), the equilibrium group size is N∗ = 𝛼 𝛽𝜋(1−𝛾) and the threshold proportion of non-positional goods 𝛾 simplifies to Our analysis indicates that if the proportion of non-positional goods falls beneath this threshold value 𝛾 , productivity growth becomes welfare-reducing. Note that the threshold proportion 𝛾 is the product of two terms: (i) the interaction-weighted “productivity ratio” (𝛼N∗∕𝛽) is, i.e. the ratio of prosocial output ( 𝛼N∗ ) to market productivity ( 𝛽 ) and (ii) the “envy-pride parameter” 𝜀∕𝜋 .23 The parameter 𝜀 can be normalised to 1. Boyce etal. (2010) suggest that 𝜋 is equal to 1/1.75. While 𝛼 is the productivity of an individual’s contribution to maintaining her social relationships, 𝛼N∗ is her total utility, which is the output of her prosocial relationships. Naturally, both individual productivity and group size matter for how much individuals choose to invest in public/club goods – individual productivity because people consider the opportunity cost of their investment, and group size because contributing to the public good benefits everyone in the group.24 In order to match the parameters with a moment from the data then, we need to know the total value that people place on their social relationships and set this equal to 𝛼N∗ . Wendner & Goulder (2008) suggest that positional consumption is at least 20% of total consumption,25 so that 𝛾 is at most 0.8. The median income in the United Kingdom in 2017 is £42,515. Social relationships may be valued along the following lines laid out by Powdthavee (2008): using data from the British Household Panel Survey, changes in life satisfaction arising from (27) 𝛾 = 𝛼N∗ 2𝛽 ⋅ ( 2+ 𝜀 𝜋) 23 Note d( 𝜀 𝜋) ∕d𝜀> 0 and d( 𝜀 𝜋) ∕d𝜋< 0 . 24 Weimann etal. (2018) provide evidence that both matter to experimental subjects. 25 Wendner and Goulder (2008) provide a range of estimates. 218 S.J.Bosworth, D.J.Snower 1 3 meeting with friends and family and speaking with neighbours are compared with the same changes arising from changes in income. Powdthavee assumes as his base category people who meet with their friends and relatives and speak to their neighbours less than once a month. Relative to these people, those who meet with friends or relatives once or twice a month (11% of the sample) experience an increase in life satisfaction equivalent to £57,500; those who meet with friends or relatives once or twice a week (40% of the sample) experience an increase in life satisfaction equivalent to £69,500; and those who meet with friends or relatives on most days (47% of the sample) experience an increase in life satisfaction equivalent to £85,000 of annual income (in 1996 pounds Sterling). Furthermore those who talk to their neighbours once or twice a week (40% of the sample) experience an increase in life satisfaction equivalent to £22,500; and those who talk to their neighbours on most days (36% of the sample) experience an increase in life satisfaction equivalent to £37,000 in annual income. We take these numbers to mean that the average value of each Briton’s social relations is equal to £172,019 in 2017 pounds Sterling. In the analysis above, we do not interpret the relative valuation of income and social relationships in monetary terms (refer to sec.3.2, footnote 11). The estimates above however are given in monetary terms. To transform this ratio back into utility terms, we make reference to the elasticity of social group size with respect to income. McPherson etal. (2006) document a 33% reduction in the extent of people’s close social groups over 1985–2004 in the United States. Real income per capita grew by 132% over this period however. Note that our model is equivalent to Cobb-Douglas utility and as such the elasticity of group size N∗ with respect to 𝛽 is This means we need to map a 132% growth in income into a 33% growth in consumption utility. The simplest way to do this is with an exponential consumption utility of money function: where m is the value of consumption at market prices with 0<𝜌<1 . We therefore set 𝜌=33∕132 and set the ratio of social relationship utility ( u(£172, 019) ) to mean income ( u(£42, 515) ),26 equal to Setting 𝜋=1∕1.75 , and 𝜀=1 , we obtain the condition 𝛾≤ 𝛼N ∗ 2𝛽 ⋅ ( 2+𝜀 𝜋) < 5.32 in order for productivity growth to be welfare-reducing. This exercise shows that the phenomenon of welfare-reducing growth is an empirically plausible possibility; and merits further investigation by empirical economists. dN∗ d𝛽 ⋅ 𝛽 N ∗ =− 1. xi=m𝜌 𝛼 N∗ 2𝛽=(172, 019) 33∕132 (42, 515) 33∕132 ≈ 1.42. 26 Per Eq.4 the mean income in this economy is 𝛽 , therefore 𝛽= 42,515. 219 1 3 Technological advance, social fragmentation andwelfare 7 Further welfare effects ofproductivity growth In Sect.5, we have seen how productivity growth leads to a reduction in the size of social groups, thereby promoting people’s positionally competitive activities with regard to those outside their social groups and reducing prosocial relationships within their social groups. Since the positionally competitive activities are associated with negative preference externalities whereas the prosocial relationships are associated with positive preference externalities, productivity growth leads to a “decoupling” of social welfare from GDP (the sum of all market production). This decoupling phenomenon can be reinforced through the effect of productivity growth on the following phenomena. 7.1 Rising proportion ofpositionally competitive activities Productivity growth increases GDP per capita and may thereby raise the share of positional goods in total production. The reason is that while basic individual material needs may be satisfied with finite resources, positional status needs are inherently insatiable, since one individual’s status needs must always be satisfied relative to those of others.27 In the context of our model, a rise in the share of positional goods reduces the size of social groups: The associated welfare effect is also negative: In accordance with our hypothesis that productivity growth raises the share of positional goods, we now assume that the proportion of non-positional goods 𝛾(𝛽) is inversely related to the productivity parameter 𝛽 : and for 𝛾( ⋅ ) continuous on [0, +∞) . Figure4a provides an example. These assumptions formalise the hypothesis that positional consumption rises in importance as people’s basic material needs become increasingly satisfied. (28) dN∗ d (1−𝛾)=− 𝛼−𝜆 𝜋𝛽(1−𝛾) 2< 0 dW d (1−𝛾)=−𝛽− 𝛼−𝜆 𝜋𝛽(1−𝛾) 2 ⋅ dW dN∗< 0. (see above) 𝛾(0)=1 lim 𝛽 → +∞ 𝛾(𝛽)= 0 d𝛾 d𝛽≡ 𝛾𝛽 ≤0 27 Hopkins and Kornienko (2004) provide a theory for how this might arise endogenously. 220 S.J.Bosworth, D.J.Snower 1 3 Firstly, we re-express the aggregate marginal utility of growth (i.e. the welfare effects of increasing 𝛽 holding group size fixed) as Note that, in comparison with the base case, there are effects on both the direct and effort-related effects of growth on non-positional consumption. The direct effect becomes 𝛾+𝛾𝛽𝛽≤𝛾 , meaning that each additional £/€/$ of production will consist of | | | 𝛾𝛽 | | | ⋅ 𝛽 fewer non-positional goods. Secondly however, the effort-related substitution effect increases because the tradeoff between group size and goods production becomes steeper. As before we then express the total welfare implications of technology-driven economic growth by using the expression for the total derivative: now taking into account that knock-on effects from changes in 𝛾 : As above, technology-driven growth affects social welfare via three channels. We compare the differences with the baseline model below: (29) 𝜕W 𝜕𝛽 =𝛾+𝛾𝛽𝛽 . dW d𝛽 = 𝜕W 𝜕𝛽 + 𝜕N∗ 𝜕𝛽 ⋅ dW dN ∗ , (30) dW d𝛽=𝛾+𝛾𝛽𝛽 ⏟⏟⏟ direct effect +𝜆(𝛼−𝜆) ( 1−𝛾−𝛾𝛽𝛽 ) 𝛽2𝜋(1−𝛾)2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ effort effect −𝛼 ( 1−𝛾−𝛽𝛾𝛽 ) (𝛼−𝜆) 𝛽2𝜋(1−𝛾)2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ lost prosocial relationships −(𝛼−𝜆)(1−𝛾−𝛽𝛾𝛽)𝜀K(N∗2−(1−KN∗)2) 2𝛽𝜋(1−𝛾) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ excess envy . Fig. 4 Effects of growth–diminishing 𝛾 (a) and its effects on welfare (b) 227 1 3 Technological advance, social fragmentation andwelfare Finally, we can take the limit of the total change in status utility above as ΔN∗→0 at N0=N∗ : Proof that dW∕dN∗>0 We must show that given the equilibrium group size condition N∗=(𝛼−𝜆)∕𝛽𝜋(1−𝛾) and the definition of the number of groups K, with KN∗≤1 and (K+1)N∗>1 . We know that the first term 𝛼=N∗𝛽𝜋(1−𝛾)+𝜆 , so making this substitution and collecting terms we have Note from the condition (K+1)N∗>1 that N∗≥1−KN∗ and thusly N ∗2 −(1−KN ∗ )2≥0 . Since the expression for dW∕dN∗ above contains only nonnegative elements, we therefore know that dW∕dN∗>0 . Derivation of   Recall that the expression dW∕d𝛽 involved an expression for the number of groups K+1 which depends on 𝛾 in a non-linear fashion. For this reason, we evaluate this expression at the edge case where K=1∕N∗ : = K ∑ k=1 ( 𝛽 2(1−𝛾) ( −ΔN∗ ( N0+kΔN∗ )( 𝜋kN0+𝜀(k−1) ( N0+ΔN∗ )) +kΔN∗(𝜀N0(kΔN∗+N0)+𝜋(N0+ΔN∗)(N0+(k+1)ΔN∗)) )) −𝛽𝜀 2 (1−𝛾)KΔN∗ ( 1−K ( N0+ΔN∗ ))( 1−KN0 ) −(1−𝛾)𝜆ΔN dU s dN ∗ ≡ lim ΔN∗→0 ΔUs ΔN∗= K ∑ k=1 ( 𝛽𝜀 2(1−𝛾)N∗2 ) +𝛽𝜀 2(1−𝛾)K(1−KN∗)2−(1−𝛾)𝜆 =𝛽𝜀 2 (1−𝛾)K ( N∗2−(1−KN∗)2 ) −(1−𝛾)𝜆 . dW dN ∗=𝛼−𝜆+ 𝛽𝜀 2 (1−𝛾)K ( N∗2−(1−KN∗)2 ) > 0 dW dN ∗=N∗𝛽𝜋(1−𝛾)+𝜆−𝜆+ 𝛽𝜀 2(1−𝛾)K ( N∗2−(1−KN∗)2 ) =(1−𝛾) ( 𝛽 ( 𝜋N∗+𝜀K 2( N∗2−(1−KN∗)2 ))) . 228 S.J.Bosworth, D.J.Snower 1 3 Setting dW ∕d𝛽 | | | |K=1∕N ∗ = 0 and solving for 𝛾 gives us An application of the quadratic formula gives us Solving for   inthecalibration We have, from Eq.26 that dW d𝛽 | |||K=1∕N∗ =𝛾+ 𝜆(𝛼−𝜆) 𝛽2𝜋(1−𝛾)− 𝛼(𝛼−𝜆) 𝛽2𝜋(1−𝛾)− (𝛼−𝜆)𝜀N∗ 2𝛽𝜋 . =𝛾+𝜆 𝛽 N∗−𝛼 𝛽 N∗−(𝛼−𝜆)𝜀N∗ 2𝛽𝜋 =𝛾−(𝛼−𝜆 𝛽)N∗−(𝛼−𝜆) 𝛽𝜋 ⋅ 𝜀N∗ 2 =𝛾−(𝛼−𝜆 𝛽)N∗(1+𝜀 2𝜋) =𝛾−(𝛼−𝜆 𝛽)( 𝛼−𝜆 𝛽𝜋(1−𝛾))(1+𝜀 2𝜋) 𝛾−(𝜋+𝜀 2)(𝛼−𝜆)2 𝛽 2 𝜋 2 (1−𝛾) 𝛾 = ( 𝜋+𝜀 2 ) (𝛼−𝜆)2 𝛽2𝜋2(1−𝛾 ) → 𝛾 (1−𝛾 )=(𝜋+𝜀 2)(𝛼−𝜆)2 𝛽2𝜋2 →𝛾 −𝛾 2=(𝜋+𝜀 2)(𝛼−𝜆)2 𝛽2𝜋2 →𝛾 2−𝛾 +(𝜋+𝜀 2)(𝛼−𝜆)2 𝛽 2 𝜋 2= 0. 𝛾 = 1+ √ 1−4⋅ ( 𝜋+𝜀 2 ) (𝛼−𝜆)2 𝛽2𝜋2 2 =1 2+√ √ √ √ √ 1 4−(𝜋+𝜀 2)(𝛼−𝜆)2 𝛽2𝜋2 . 229 1 3 Technological advance, social fragmentation andwelfare and from Eq.12 that So we must solve Subtracting 1/2 from both sides and squaring we get Proof that d2 W∕dˇd ˇ ≥ 0 Recall the cross-partial derivative of welfare with respect to technological progress 𝛽 and the gradient of the share of status goods with respect to technological progress 𝛾𝛽 was 𝛾 =1 2+ √ 1 4−𝛼2(𝜀+2𝜋) 2𝛽2𝜋2 𝛼 = √𝛽( 1− 𝛾)𝜋 ⋅ 𝛼 N∗ . 𝛾 =1 2+ � � � � � � � � 1 4−⎛⎜⎜⎜⎝ � 𝛽�1−𝛾 �𝜋⋅𝛼N∗ 𝛽⎞⎟⎟⎟⎠ 2 (𝜀+2𝜋) 2𝜋2 =1 2+� � � � 1 4+� � 1−𝛾 � ⋅𝛼N∗ 𝛽�(𝜀+2𝜋) 2𝜋. 𝛾 2−𝛾 +1 4=1 4+ (( 1−𝛾 ) ⋅𝛼N∗ 𝛽 ) (𝜀+2𝜋 ) 2𝜋 𝛾 (1−𝛾 )=((1−𝛾 )⋅𝛼N∗ 𝛽)(𝜀+2𝜋) 2𝜋 𝛾 = ( 𝛼N∗ 2𝛽)( 2+𝜀 𝜋) . 230 S.J.Bosworth, D.J.Snower 1 3 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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