The effect of within-group inequality in a conflict against a unitary threat

The e¤ect of within-group inequality in a con‡ict against a unitary threat Maria CubelSantiago Sanchez-Pagesy July 27, 2012 Abstract A group of agents must defend their individual income from an external threat by pooling their e¤orts against it. The winner of this confrontation is determined by a contest success function where members’ e¤orts may display di¤erent degrees of complementarity. Individual e¤ort is costly and follows a convex isoelastic function. We investigate how the success of the group in the con‡ict and its members’utilities vary with the degree of within-group inequality. We show that there is a natural relationship between the group’s probability of victory and the Atkinson index of inequality. If members’e¤orts are complementary or the cost function convex enough, more egalitarianism within the group increases the likelihood of victory against the external threat. The opposite holds when members’e¤orts are substitutes and the cost linear enough. Finally, we obtain conditions under which richer members of the group are willing to make transfers to poorer members in order to enhance their …nal payo¤. Keywords: Con‡ict, Inequality, Atkinson index, Redistribution. JEL codes: D31, D63, D72, D74. University of Barcelona, Dept of Public Economics, and IEB. Email: [email protected]. yUniversity of Barcelona, Dept of Economic Theory, and University of Edinburgh, School of Economics. E-mail: [email protected]. URL: http://www.homepages.ed.ac.uk/ssanchez/. 1 1 Introduction Societies and communities often have to defend from or compete with hostile out-groups. Cities and villages often su¤ered raids from barbarians, pirates or bandits. Ordinary citizens need to protect themselves from the appropriation e¤orts by criminal networks. States and empires often clash over the control of natural resources or have to resist the attack of rival nations. Suppose that these communities, while remaining in confrontation with an out-group, have solved any possible con‡ict of interests within themselves. They accept the current distribution of income or hold binding agreements on how to share the value of the object they are …ghting for against the outgroup. We then ask the following question: Are more egalitarian societies more or less likely to prevail in such confrontations? Apart from its intrinsic interest, this question is important because the interplay between chances of success and within-group inequality opens the door to income redistribution. If, for instance, a more egalitarian distribution of income within the community enhances its prospects of victory, members of that society may voluntarily transfer part of their income to poorer members. In short, the presence of external con‡icts may provide a rationale for the redistribution of income we observe in societies. In this paper we show that the answers to these two questions, whether egalitarianism enhances the chances of victory of society and whether a society may want to engage in income redistribution as a result, depend on the technologies of con‡ict. If the e¤orts of the members are substitutes, more inequality is better because inequality increases the incentives to contribute of richer members, who are the ones who have most to gain from victory. If, on the contrary, e¤orts are complements, more egalitarian societies fare better in the confrontation. This is because all members must contribute for society to be successful in the con‡ict, implying that members with the lowest incentives to contribute are key. These members are the poorer members since they are the ones with the lowest stake in the …ght. So the richer they are, that is, the more egalitarian is the distribution of income within that society, the more they contribute to defeat the out-group. The cost of con‡ict contributions plays also a crucial role. If the marginal cost of e¤orts increases rapidly, this will deter richer members from contributing substantially. In that case, more egalitarianism makes the group more e¤ective in the confrontation. Con‡ict thus generates two types of redistribution. First, con‡ict shapes the income distribution within society because its members contribute to the success of the group. The resulting distribution of income could be more or 2 less egalitarian than the initial one. We show that as e¤orts become more complementary the distribution of contributions becomes more egalitarian so con‡ict is in e¤ect implementing a regressive tax scheme. As a result, the …nal distribution of income is more unequal than the initial one. We say then that con‡ict is pro-rich. The opposite holds when e¤orts are substitutes. Con‡ict is pro-poor because richer members contribute a bigger share of their income than poorer members. Second, members of the community may be willing to engage in income redistribution as a result of the presence of the hostile out-group. The incentives to redistribute vary in their direction, from the rich to the poor or viceversa, depending on the technologies of con‡ict. When e¤orts are complementary enough or their cost is convex enough we show that richer members are willing to transfer voluntarily part of their income to poorer members. The opposite holds when e¤orts are substitutes or the cost is linear enough. This is because even though success of the group increases with inequality in that case, poorer members have less to gain from victory. They have little incentives to transfer their income to richer members in order to induce them to …ght harder. The relationship between egalitarianism and collective action has been subject to analysis for long now. Olson (1965) argued informally that more inequality favors collective action, public good provision for instance, since it maximizes the incentives of richer members to engage in it. Hirshleifer (1983) argued that this result rests critically on the assumption that the amount of public good provided depends on the sum of contributions. When contributions are perfect complements, i.e., the weakest-link technology of provision, inequality hinders public good porvision. Using examples, Cornes (1993) and Cornes and Sandler (1996) corroborated that Olson’s intuition does not hold in general. In the closest contribution to ours, Ray et al. (2007) characterize the relationship between the surplus generated by a joint project, inequality in the shares of the resulting output and the technology of production. These authors show, as we do, that egalitarianism can be welfare enhancing if contributions are complementary enough. We focus on the speci…c case of con‡ict against an out-group and consider the incentives to redistribute income that communities may have as a result. In the context of public goods, Vicary (1990) and Cornes (1993) explored this issue but only for the weakest-link technology. Both show that under this technology the "distribution neutrality" result by Bergstrom et al. (1986) no longer holds. In the literature on con‡ict, to the best of our knowledge, only Esteban and Ray´s (2011) model of ethnic con‡ict has analyzed the role of within-group inequality. These authors model a situation where members can contribute with their time or use money to increase the activism of 3 other members. Money and time are thus substitutes. They show that more within-group inequality makes groups more violent because the opportunity cost of time for poorer member decreases, so richer members …nd it easier to buy higher levels of activism from them. Our model also belongs to the literature that has explored the role of con‡ict in producing income redistribution. Hirshleifer (1991) pointed out that income redistribution is one of society’s responses to the threat of internal con‡ict. Individuals or social groups resort to con‡ict if by doing so they can improve their position relative to the current distribution of income. Within-society income redistribution thus becomes a way of avoiding internal con‡ict. Bevia and Corchon (2010) argued that a similar mechanism can help societies to avoid con‡ict against external agents. If a rich society transfers some of its income to an external group, that out-group becomes less interested in initiating a costly confrontation. In our case, we study how redistribution within a society can help to improve the society’s chances of victory in an external confrontation. The rest of the paper is organised as follows. In the next section we present the basic elements of the model. In Section 3, we characterize its equilibrium and perform comparative statics. Section 4 explores the implicit redistribution that con‡ict generates and the incentives of members of the group to engage in income transfers. Section 5 concludes. 2 The model 2.1 Con‡ict Let us consider a group formed n > 1members who di¤er in income. A member iowns a share i>0of the total income of the group (net of subsistence level) denoted by Ysuch that Pn i=1 i= 1:Let us index members increasingly by income so ii+1 for i= 1; :::; n 1: This group is subject to the threat of an external entity that we model as a unitary agent. We will simply refer to it as the threat. Both the group and the threat are in confrontation. If the group prevails its members are able to retain their individual income yi=iY. If the threat wins the con‡ict, it appropriates the entire income of the group and its members get nothing:Alternatively we could interpret the setting as a situation where the group and the threat are competing for a prize of value Y(a territory, a monopoly rent, a natural resource) so the vector = (1; :::; n)represents a binding agreement among group members on how to divide that prize in case of victory. 4 Both the group and the Threat can invest resources in order to prevail in this confrontation. The outcome of the con‡ict depends on the e¤orts spent by each of the two sides. Denote by x= (x1; :::; xn)the vector of con‡ict e¤orts made by the members of the group and denote by xothe e¤ort made by the threat. We assume that the group’s winning probability is p(x) = h(x;n) xo+h(x;n);(1) where the function h(x;n) = n[ n X i=1 1 nxi1]1 1;(2) is called the impact function of the group. The parameter 0represents the degree of complementarity between members’e¤orts. In the context of public good provision, similar functions have been used by Cornes (1993) and Ray et al. (2007). On the other hand, the group Contest Success Function (CSF) in (1) has been axiomatized by Münster (2009). It encompasses as particular cases the Tullock CSF (Tullock, 1967) when = 0 and the weakest-link technology (Hirshleifer, 1983) when ! 1: In particular, note that impact function (2) satis…es two important properties: Constant returns to scale: For all k > 0,h(kx; n) = kh(x; n): No group-size bias (Kolmar and Rommeswinkel, 2011): For any natural number k, it holds that h(x k; kn) = h(x;n): Constant returns to scale is an appealing property in this context because it implies that the relative success of a contender does not depend on the unit of measurement of e¤ort. Note that this property also implies that the CSF in (1) is homogeneous of degree zero (Münster, 2009). On the other hand, the No group-size bias property implies that the impact of two groups who have exerted the same total e¤ort should be the same regardless of their size. Many CSFs implicitly build in some group-size bias because this property is closely related to the degree of complementarity of e¤orts. Consider for instance the seemingly more natural impact function. g(x;n) = [ n X i=1 xi1]1 1: It is straightforward to show that this impact function presents positive group-size bias,i.e. h(x k; kn)> h(x;n);if and only if  < 1;and a negative 5 group-size bias,i.e. h(x k; kn)< h(x;n);if and only if  > 1:By assuming away any group-size bias, we avoid any confounding e¤ect from group size on our result and focus only on the e¤ect of internal inequality. We assume that the cost of e¤ort is iso-elastic and of the form c(xi) = 1 1 + x1+ i; where 0. Similarly for the threat. This functional form was …rst considered by Esteban and Ray (1999) and its properties studied in relation to the group-size paradox in Esteban and Ray (2001). The payo¤ function for a group member boils down to ui=pyic(xi) = h(x;n) xo+h(x;n)iY1 1 + x1+ i;(3) whereas for the threat it is just uo= (1 p)Yc(xo) = xo xo+h(x;n)Y1 1 + x1+ o:(4) 2.2 Inequality Consider that the income distribution in a society is given by the vector y= (y1; :::; yn):The measure of income inequality we will consider here was proposed by Atkinson (1970) and it is de…ned as A"(y) = 1 y" y; where yis society’s average income and y"is the Equally Distributed Equivalent Income (EDEI) which is given by y"= [ 1 n n X i=1 y1" i]1 1"=Y[1 n n X i=1 i1"]1 1":(5) The parameter "measures society’s attitude towards inequality. In our set up, the Atkinson index boils down to A"(y) = 8 < : 1n[1 nPn i=1 i1"]1 1"for "6= 1 1n n Q i=1  1 n ifor "= 1 :(6) 6 The parameter "embeds a normative judgment over income inequality.1 Observe that A"(0) = 0 and that lim "!1A"(y)=1n1;so the inequality index depends only on the income of the worst-o¤ member in the group. In the original formulation of the Atkinson index, it is always assumed that "0so incomes are socially evaluated according to a concave function, implying y"<y: Under that assumption, the index is consistent with the following principle: Principle of transfers (Pigou-Dalton principle): Take two vectors yand y0, where y0is obtained by adding >0to yiand substracting it from yjfor j > i and such that yj> yi+ :Then A"(y)> A"(y0). This principle states that when "0a rank-preserving income transfer from a richer individual to a poorer individual cannot increase inequality. However, the functional form of the Atkinson index does not preclude that society may have a preference for inequality, i.e. " < 0:The literature on inequality measurement never considers this case since this literature implicitly assumes that inequality is not socially desirable. When " < 0 it turns out that ye> y; and the index becomes non-positive, with higher absolute values corresponding to higher levels of inequality. Observe for instance that lim "!1A"(y) = 1 nn;so the index is negative unless the distribution of income is perfectly equal, i.e. i=1 nfor all members. In this case thus, the index satis…es the Reversed principle of transfers, that is, given two distributions yand y0de…ned as above A"(y)< A"(y0). It is easy to show that for any value of "; either positive or negative, the Atkinson index satis…es two particularly relevant properties (Lambert, 2001). Scale invariance: For all k > 0; A"(ky) = A"(y): Principle of population: For any natural number kdenote by yk the vector containing ktimes each and all of the elements in y:Then A"(yk) = A"(y): 1Atkinson (1970) proposes an equivalence between inequality aversion and risk aversion based on the idea that behind the veil of ignorance more risk-averse individuals would prefer more egalitarian distributions of income. Under that interpretation, the EDEI is the level of income that if equally distributed would give individuals a level of equality equal to the expected utility they would enjoy under the original distribution behind the veil of ignorance . 7 Scale invariance, i.e. homogeneity of degree zero, is an appealing property because it implies that inequality does not depend on the unit of measurement of income. On the other hand, the Principle of population implies that the Atkinson index remains invariant under replications of the population (and its incomes). These properties will be helpful later when characterizing the equilibrium of the con‡ict game. 3 The equilibrium 3.1 Existence Let us now characterize the equilibrium of this game. To do so we exploit the properties of the Atkinson index we have just outlined. A member iof the group seeks to maximize (3) taking as given the e¤ort of other members and the e¤ort made by the threat. Her optimal decision is characterized by the following expression @ui @xi =p(1 p) h(x;n) @h(x;n) @xi iYx i= 0 i= 1; :::; n: (7) From this it is possible to write the relation between the optimal e¤orts of any two members xi xj = ( i j )1 +:(8) This expression give us a …rst indication of how the impact function and the cost function a¤ect the distribution of e¤orts across members. Member’s e¤orts become more similar the more complementary e¤orts are and the more convex their cost, i.e. the higher +. Actually, expression (7) implies that whenever + > 0it cannot be a best response for a member to exert no e¤ort if another member is exerting a positive e¤ort. Hence, in any equilibrium, either all members or no member are contributing. There are, however, two exceptions to this result. Case 1: Tullock contest (Tullock, 1967) Consider the case where members’e¤orts are perfect substitutes and the cost is linear, i.e. += 0: In that case, observe that the …rst order condition (7) becomes @ui @xi =p(1 p) Pn i=1 xi iY1i= 1; :::; n; implying that only the member with the highest income, that is, member n; exerts positive e¤ort. This is a well-known result in the literature on contests (Baik, 1993). 8 Case 2: Weakest-link technology (Hirshleifer, 1983) When the impact function displays perfect complementarity between members’efforts, i.e. ! 1;it boils down to h(x;n) = nminfx1; :::; xng: In that case, it is clear that in any equilibrium all members will exert the same level of e¤ort. De…ne x1as the e¤ort choice of the poorest member that satis…es @ui @xi =xo (nx1+xo)21Y1 = 0; that is, x1is member 1 optimal choice under the assumption that all other members also exert x1:Then, it is quite straightforward to see that there exists a continuum of equilibria in which all members contribute the same amount of e¤ort, ranging from 0 to x1:Members of the group face thus a coordination problem. For the sake of exposition we assume that + > 0and focus on fully interior equilibria. On the other hand, the optimal e¤ort choice for the threat is given by the following FOC @ui @xo =p(1 p) xo Yx o= 0:(9) Combining (7) and (9) it is possible to express the relationship between the optimal e¤ort decisions of the group members and the threat in a compact way c(xi) c(xo)= ( xi xo )1+=ixi h(x;n) @h(x;n) @xi =i 1+ + Pn j=1 j 1 + (10) We are …nally in the position to state our …rst result Proposition 1 When + > 0, there exists a unique interior equilibrium e¤ort pro…le characterized by x1+ o=p(1 p)Y; x1+ i=p(1 p)tiyii= 1; :::; n; where ti=i 1 + Pn j=1 j 1 + (11) 9 of prevailing over the threat. To this end we can exploit the properties of the Atkinson index. Recall that the group winning probability pis decreasing in the Atkinson index of inequality. Because the properties of the index depend on the value of b", transfers have di¤erent e¤ects under di¤erent technologies. Proposition 6 The group’s equilibrium winning probability pincreases (i) With a progressive transfer if b"0; The increase is larger the poorer the two members involved in the transfer. (ii) With a regressive transfer if b" < 0:When b"2(1;0) this increase is larger the poorer the two members involved in the transfer. The opposite holds when b" 1: Proof. The proof of this Proposition rests on the properties of the Atkinson index. When "0;the index satis…es the Principle of transfers. In addition, the fact that w000(yi)0implies that a given transfer between poorer agents is more e¤ective in decreasing inequality than the same transfer made between richer individuals. This is called the Principle of diminishing transfers (Kolm, 1976). Things are slightly less straightforward when b" < 0:In that case, w000(yi) 0if and only if b" 1:Recall that in that case, the Atkinson index decreases only with regressive transfers. A non positive third derivative w000(yi)implies here that such transfer is more e¤ective in reducing the index when is made between two poorer individuals. To see this consider two individuals iand jsuch that yi=yj+  and an in…nitesimal transfer dmade from the latter to the former. The net increase in individuals utilities is just w= [w0(yi)w0(yj)]d= [w0(yj+ ) w0(yj)]d: For the impact on the index to be greater when the transfer is made between poorer it must be that wis decreasing yj;which requires w00(yj+ ) < w00(yj); and thus that w000(yi)<0:Finally, when the third derivative is non-negative the opposite property holds and transfers have a greater impact on the index if made between richer members. The Proposition suggests that members may be interested in redistributing income voluntarily before the con‡ict takes place in order to increase 16 their prospects of victory. When b" > 0;the group’s winning probability increases with progressive transfers so richer members may be interested in transferring some of their income to poorer members in order to incentivise them to exert more e¤ort. The opposite might happen when b" < 0since in that case impact and cost technologies tend to depress poor agents’e¤ort. They might be interested in transferring some income to richer and more active individuals within the group. The following proposition characterizes whether these types of voluntary transfers will take place or not. We have to restrict ourselves to particular (but relevant) cases since the analysis of general cases is not analytically tractable. Proposition 7 In a Tullock contest with both linear or quadratic costs, i.e. b"! 1 or b"= 0;no voluntary transfers take place. Under the weakestlink technology, i.e. b"= 2;there exists a income threshold such that any member with income share i>  is willing to make a transfer to the worst-o¤ member of the group: Proof. Let us start with the cases in which voluntary transfers do not take place. When b"! 1 the payo¤ of any member i6=nis ui=n 1 + n iY i = 1; :::; n 1: From here it is straightforward to check that no member has an incentive to transfer a share of its income to member n: Similarly for b"= 0;the expected payo¤ of any member is u i=1 2yi(1 i 2(1 + )); which does not depend in the income shares of other members and is always increasing in i:Hence no member has any incentive to transfer part of its income to another member. Finally, consider the case where b"= 2:The expected payo¤ for any member i6= 1 is just ui=n1 1 1+ 1 + n1 1 1+ Y(i1 1 +  1 1 + n1 1 1+ )i= 2; :::; n: Knowing that i= 11Pj6=1;i jthen the impact of an in…nitesimal transfer of income from ito member 1 on ispayo¤ is given by the derivative @ui @1 =p(1 p) 1 +  Y 1 (i1p 1 + 1)pY(1 + 1p 1 + (1 p 1 + )): 17 And this derivative is positive if and only if i>  1(1 + 1 +  1p+12p 1 + ): This Proposition shows that the incentives to redistribute income that an external confrontation creates have di¤erent strengths depending on the con‡ict and cost technologies. When b" < 0;a poorer member can increase the chance of victory of the group by transferring some of her income to a richer member, but her incentive to do so is rather small because she has a small stake in the confrontation. When b" > 0progressive transfers increase the group winning probability, and richer members can …nd pro…table to give away part of their income in order to enhance the con‡ict e¤ort of poorer members. For the case b"! 1;su¢ ciently rich members are willing to transfer part of their income to poorest member in order to fuel her con‡ict e¤ort. Note that such transfers are Pareto improving since all members bene…t from them. Note also that as these transfers are made, the identity of the worst-o¤ member changes and that changes the threshold as well. 5 Conclusion The relation between inequality and social con‡ict has been subject to intense theoretical and empirical scrutiny in the last few years. Less attention however has been given to the relationship between external con‡ict and internal inequality. In this paper we have shown that the technology of con‡ict -protection or appropriationplays a crucial role in that relation. Egalitarian societies are more likely to prevail in a confrontation against an external group if the e¤ort of their members are complementary. This technology encompasses cases such as military secrecy, defence of forti…cations or modern armies where strength is very related to the lowest e¤ort made by members of society. Given that technology, egalitarianism increases the chances of the group prevailing in the con‡ict because it increases the stakes of the poorer members. On the other hand, unequal societies are more e¤ective in con‡ict when e¤orts of their members are substitutes. This scenario encompasses cases such as lobbying or the use of small mercenary armies. In this case, the community is more successful as richer members get richer because they have bigger stakes in the con‡ict. We characterize the relationship between within-group inequality and con‡ict expenditures as a function of the Atkinson index of inequality. This 18 allows us to express equilibrium variables as a relationship of inequality in a transparent manner and allows us to exploit the properties of the index when performing comparative statics. For a given distribution of income, more complementarity of e¤orts is bad for the community. Victory depends increasingly in the lowest e¤ort, which is made by the poorest members of the group, which in turn are the ones with the smallest stakes in the con‡ict. Hence, more complementarity makes the success of the group rest on those individuals who have the lowest incentives to contribute. On the other hand, society can obtain an advantage in con‡ict as it becomes bigger provided that the cost of con‡ict is convex enough. This is in line with previous results (Esteban and Ray, 2001) showing that convexity of the cost is key for groups to overcome the collective action problem pointed out by Olson (1965). More importantly, we show that members of society have incentives to engage in voluntary redistribution. This redistribution is aimed at increasing the incentives of other members to contribute more to the con‡ict e¤ort. This incentives are asymmetric, however. We show that when more egalitarianism within the community makes it more likely to prevail, richer agents have incentives to transfer part of their income to poorer members. These transfers can be Pareto improving for the group. However, when more inequality makes the group more likely to prevail, poorer members do not have incentives to make transfers to richer members. This is because poorer members do not have much to gain from victory. Hence, we conclude that external threat constitutes a force that explains the progressive income redistribution that we observe in modern societies. There are two limitations in our analysis that deserve further exploration. One is the assumption that the external threat is unitary. In that sense, con‡ict e¤orts could also be interpreted as e¤orts aimed at mitigating the e¤ects of a natural catastrophe or at avoiding an epidemic. It would be interesting to explore the role of relative inequality between the threat and the group in the equilibrium. The incentives of a society to engage in income redistribution should also vary depending on the inequality of the out-group it is facing. The second limitation has to do with the completely decentralized nature of interactions within the group. Members contribute e¤orts voluntarily and, if willing to, transfer part of their income to other members. In reality, states have traditionally requested these e¤orts from its subjects, often using coercion. Taxation has arisen as an institutional mechanism aimed at redistributing income in order to help societies to wage war. We intend to explore more centralized mechanisms in our future research. 19 References [1] Atkinson, AB. (1970). 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