Income inequality and saving in a class society: the role of ordinal status

This paper examines the impact of income growth and income inequality on household saving rates and payoffs in a non-cooperative game where each player’s payoff depends on her present and future consumption and her rank in the present consumption distribution. The setting is a pooling equilibrium with three clusters of successive income groups, each cluster having its own present-consumption standard and rank in the present-consumption distribution. In this way the analysis addresses the saving behaviour and welfare of three social classes: the lower, middle and upper class. Explanations are found for the Easterlin paradox and the Kuznets consumption puzzle and it is concluded that rank concerns tend to strengthen the case for more income equality and weaken the standard effect of inequality on aggregate saving. (Published in Special Issue The economics of social status) JEL C72 D31 D62 E21 Z10 D71


Introduction
In recent decades, a new literature on consumer behaviour has emerged that moves away from the traditional notion that a person's consumption and labour supply are completely independent of what others do. A major line of research builds on the assumption that people care about how their choices compare with those of others in the consumption and income hierarchy. In evaluating their relative position, people tend to be upward-looking and particularly envy those who, in some relevant dimension, are near to them (Frank, 1985a, Chapter 2;Elster, 1991). One theme in this research programme considers the consequences of social comparison for aggregate saving and another studies the consequences for happiness. For instance, social comparison can explain the observed positive correlation between saving rate and household income, which is hard to reconcile with the life cycle and permanent-income hypotheses (Duesenberry, 1949;Frank, 1985b;Dynan et al., 2004). Social comparison can also explain the well-known Easterlin paradox: the observation of strong growth of real per capita income in Western countries since World War II without any corresponding rise in self-reported happiness (Easterlin, 1974;Hirsch, 1976;Layard, 2005, Clark et al., 2008. This paper contributes to these themes by analysing the impact of income growth and income inequality on household saving rates and payo s in a non-cooperative game where each player's payo depends on her present and future consumption and her rank in the presentconsumption distribution. The setting is a speci c pooling equilibrium with three clusters of successive income groups, each cluster having its own present-consumption standard and rank in the present-consumption distribution. Within each cluster, their concern with rank induces the members of the lower income groups to consume at the level set by the highest income group, and consequently to neglect saving for future consumption. In this way the analysis aims to address the saving behaviour and welfare of three social classes: the lower, middle and upper class (just three classes for illustrative purposes). In particular, we examine how the social-comparison motive alters the standard analysis of two questions: (1) does across-the-board income growth make everyone better o and also raise the aggregate saving rate? (2) does reducing income inequality by creating a larger middle class favour the poor and increase overall payo and aggregate saving?
A person's rank in the present-consumption distribution is given by the fraction of people who consume the same as or less than that person. By relating individual choices to rank rather than distance to some average consumption level, the paper follows the seminal article of Frank (1985b) and more recent contributions, including Kornienko (2004, 2009 1 The paper particularly builds on the ordinal status game of Haagsma and van Mouche (2010), henceforth HM (2010), which stands out by assuming a nite number of agents instead of an uncountably in nite number. The discretization assumption seems appropriate, since positional concerns typically play a role in small local environments, i.e. where the size of a person's relevant reference group is limited. 2 An important implication of the above de nition of rank in the case of a nite number of consumers is that in their competition for higher position, consumers also have a tendency to conform. Because a rst place shared with others yields the same rank as a unique rst place, people do not want to fall behind the highest consumption level in their reference group, nor do they want to go ahead of this standard if it is costly to do so. It is this conformist element that creates the possibility of pooling equilibria (see HM, 2010). 3 The number of consumption standards { and thus the social class structure { is an endogenous variable, however, and ultimately depends on the shape of the underlying income distribution. A similar structure of social classes characterizes also the equilibria of the status game studied by Immorlica et al. (2017), where the players are embedded in a network (for other economic explanations of class structure, see e.g. Bernheim, 1994, Akerlof, 1997, Oxoby, 2004. Once we have linked the standard model of intertemporal consumption and saving to the ordinal status game of HM (2010), the analysis is relatively straightforward and yields the following main results. For each social class, we nd that by matching the consumption standard of the highest income group of their class, lower income groups consume too much in the present and save too little for later. This results in lower payo s as compared with the situation where individual rank is xed and determined by social class. The saving rate of a lower income group is decreasing in the consumption standard and increasing in income, because higher income relieves the burden of complying with the standard. These results are pretty much in line with the relative income hypothesis of Duesenberry (1949) and the work of Frank (1985aFrank ( , 1985b, that revived the interest in relative income, and also re ect more recent empirical work on consumption and saving, including Dynan et al. (2004), Alvarez-Cuadrado and El-Attar (2012), and Bertrand and Morse (2016), although none of these studies formalizes class or reference-group structure as such.
Further, we nd that economy-wide income growth raises or lowers the aggregate saving rate, depending on whether the highest income groups of each social class see present consumption as a necessity or luxury. For instance, in the case of a luxury good (the example we elaborate on), income growth raises consumption standards more than proportionally, lowering the saving rates of all income groups of a social class. In the case of unitary income elasticity, consumption standards rise at the same pace as income, so that saving rates remain constant. Hence, this case provides a solution to the Kuznets consumption puzzle: the observation that saving rates increase with income in cross-section data but are constant in time series (Kuznets, 1942). Our result echoes Duesenberry's proposed solution to the puzzle, which was quickly overshadowed by the life-cycle hypothesis of Modigliani and Brumberg (1954) and the permanent-income hypothesis of Friedman (1957). Note that we obtain this solution under the usual assumption of homothetic preferences (and so unitary income elasticities) of life cycle/permanent-income models.
Although individual payo is increasing in income, the impact of economy-wide income growth on payo s is ambiguous. Higher incomes across the board raise the payo s of those who are in the top income and nearby income groups, but we nd that it may hurt consumers at the bottom of a social class. Granted that present consumption is a normal good, the top income groups of the social classes may raise consumption standards to such an extent that bottom income groups, in spite of their higher income, see their payo s reduced. These results can explain the observation that average happiness scores tend to change more slowly than average income. Thus the analysis o ers another illustration of how social comparison can explain the Easterlin paradox (for similar approaches, see Kornienko, 2004, Clark et al., 2008).
Establishing more income equality by expanding the middle class particularly alters social ranks and thereby consumption standards. The out ow of people from the lower class to the middle class decreases the social rank of those who stay behind in the lower class, while the out ow of people from the upper class raises the social rank of everyone in the middle class. We prove that this makes those who migrate from the lower class better o as well as those who already were in the middle class. However, people who stay behind in the lower class are worse o . Their social rank has dropped and they have to spend more of their income to conform to the consumption standard of their class, because this has been raised by their peers in response to the lower rank. Evaluating the e ect that runs through changes in social rank on overall payo , we nd it can go both ways. That is, it remains an open question whether this e ect mitigates or strengthens the standard impact of income redistribution that works through changing class sizes. Nevertheless, the exposition o ers some improvement over the analysis by Hopkins and Kornienko (2004), which can only assess the e ects on individual payo s for given income levels. Whereas their analysis concludes that the poor are worse o under more income equality, we show that this only holds for those who stay behind in the lower class; those who move to a middle-class income group are always better o (for empirical work, see Ravina, 2007, Oishi et al., 2011). 4 Since more income equality changes consumption standards, it also changes the average saving rates of the social classes. Because the consumption standard goes up in the lower class and down in the middle class, the saving rate of the former falls and that of the latter rises. These e ects appear in addition to the standard e ect of income redistribution on aggregate saving that arises from changing class sizes. The theoretical literature is not unambiguous on the sign of the standard e ect (for overviews, see Schmidt-Hebbel and Serv en, 2000, Bovinger and Scheuermeyer, 2016). If the marginal propensity to save strictly increases with income, as found by Dynan et al. (2004) for the US, more equality would reduce aggregate saving. Our analysis shows that in this case the social-rank e ect of redistributing income tends to mitigate the standard e ect. This is in line with a number of recent empirical papers with di erent modelling of upward-looking comparisons that nd that more income equality tends to reduce peer pressures on people's consumption and thus promote aggregate saving (Alvarez-Cuadrado and El-Attar, 2012, 2016, Frank et al., 2014, Bertrand andMorse, 2016).
The remainder of the paper is organized as follows. Section 2 constructs the basic model and speci es the particular pooling equilibrium with three social classes. Section 3 examines the impact of across-the-board income growth on individual payo s and aggregate saving. Sections 4-6 study the impact of income inequality on individual payo s, overall average payo , and aggregate saving, respectively. Section 7 concludes. A number of appendices support the link between the basic model and the ordinal status game of HM (2010) and also derive su cient conditions for the existence of the pooling equilibrium.

Basic model
We start by incorporating social rank in a standard intertemporal model with saving and then link this to the ordinal status game of HM (2010). To increase structure, two additional conditions are introduced, resulting in each Nash equilibrium showing weakly positive sorting, in this case: an increasing relation between income and present consumption. Next, we specify a particular pooling equilibrium with three levels of present consumption and discuss its properties for a simple concrete utility function. This concrete setting also forms the baseline for the remaining sections.

Setting the stage
Consider a standard intertemporal two-period setting with only income in the rst period, where individuals have preferences over current and future consumption (we will ignore any bequest motive). Second-period consumption, enjoyed after retirement, is limited by the accumulated saving in the rst period plus interest. The quest for a higher social position relates to consumption in the rst period. Work done by development psychologists and sociologists suggests that interpersonal comparisons are especially important early in life, when people are busy building a career and setting up a family (see e.g. Frank 1985a, Ch. 8, 1985b, Alvarez-Cuadrado and El-Attar, 2012). For a given integer N 2, let N := f1; : : : ; N g be the set of consumers. An individual i 2 N chooses a combination of consumption in the two periods, with quantities c i (1) and c i (2) , to maximize utility: given a social production function (speci ed below): and subject to a budget constraint: Here r i refers to her social status and c{ (1) is the vector of rst-period consumption levels of all other consumers. The utility function U : R 2 + [0; 1] ! R is such that, for each r i 2 [0; 1], U ( ; ; r i ) is continuous, strictly increasing, and strictly quasi-concave on all budget lines (thus allowing for e.g. a Cobb-Douglas speci cation). 5 Moreover, U is strictly increasing in the third variable, the individual's social status. Further, income w i > 0 for all i 2N and interest rate 0. Importantly, given our topic, consumers may di er only with respect to income. We assume that social status is produced in an`ordinal' way. Striving for a higher position is then like racing: one only has to be faster than the others; nothing is gained by increasing the lead. An ordinal measure is close to the sociological literature, where social status is connected to rank-ordered relationships among people, as illustrated by`social ladder' (see e.g. Ridgeway and Walker, 1995). Studies that model positional concerns in terms of ordinal rank typically relate individual actions to the cumulative distribution of other people's actions. 6 We follow this approach, in particular HM (2010), by assuming that the social rank of individual i depends positively on the fraction of consumers with strictly lower or equal levels of rst-period consumption: where # means`the number of elements of'. Another way of seeing this is that, since leaving more people behind means fewer of them in front, a person's rank depends negatively on the fraction of people with strictly higher consumption levels. This agrees with the nding that people tend to look upward when making comparisons, as suggested by, for example, the concept of (egoistic) relative deprivation (Runciman, 1966) and the welfare-economic notion of envy (Varian, 1974) (see also Frank, 1985a, Elster, 1991, Stark and Wang, 2005. For a further discussion of (4), see HM (2010). As usual in this literature, each individual chooses her utility-maximizing combination of 5`S trictly increasing' means that for all a 1 ; a 2 ; b 1 ; b 2 2 R + , we have U (a 2 ; b 2 ; r i ) U (a 1 ; b 1 ; r i ) whenever a 2 a 1 and b 2 b 1 and the inequality is strict whenever a 2 > a 1 and b 2 > b 1 . consumption, given the choices of all the others. Individuals do so simultaneously and independently, thus the above describes a game in strategic form. Imposing one more restriction on the shape of U : max w i g, the game boils down to the non-cooperative ordinal status game de ned and studied by HM (2010). Restriction (5) just implies that spending all income on current consumption can never be a best reply, which avoids trivial corner solutions. The connection with HM (2010) clearly allows us to apply some key results derived in that study. Their game has only a single action variable, but by using the budget constraint U can be expressed in terms of rst-period consumption only (see Appendix A).
Finally, a little more structure completes our baseline model by creating, as shown by the proposition below, a positive equilibrium relation between income groups and social classes (ordered with respect to rank). Two conditions are critical, though not far-fetched (see Appendix B for their formal statement). One is that the utility-maximizing quantity of rst-period consumption at a given rank di ers for consumers with unequal incomes. The other essentially states that if the change in payo s from an increase in rst-period consumption (given the consumption levels of the others) is positive for a speci c consumer, then it is also positive for any other consumer whose income is not lower. These conditions are satis ed by assuming from now on that the optimal consumption quantity is an interior solution for all income groups and that the function U also is twice continuously di erentiable on the interior of its domain with partial derivatives U 11 ; U 12 ; U 23 0 and U 22 < 0 (see Appendix B). First-period consumption then is a normal good. 7 In the next section we will work with a simple concrete utility function that has these properties.
Let us denote a Nash equilibrium of rst-period consumption levels by x = (x 1 ; :::; x N ). The analysis will build on the following fundamental insights resulting from these two conditions: (A proof is in Appendix B). Suppose there are e U income groups, each containing one or more consumers with the same amount of income. The rst result of the proposition directly implies that the number of di erent levels of rst-period consumption in a Nash equilibrium are at most e U . Hence, only equilibria with e U or with fewer than e U consumption levels can exist. Equilibria with e U levels, or separating equilibria, show a one-to-one correspondence between consumption level and income group. Equilibria with fewer than e U levels, or pooling equilibria, show two or more income groups whose members have the same consumption level. The second and third results indicate that, in each Nash equilibrium, the distribution of rst-period consumption is positively related to the income distribution. That is, in a separating equilibrium, higher consumption levels correspond to higher income groups. In a pooling equilibrium, at least one quantity of consumption is chosen by two or more successive income groups. 8 Note that the possibility of a pooling equilibrium increases if income di erences become smaller (see HM, 2010). This paper studies a particular pooling equilibrium. To characterize this equilibrium, let us rst deal with the following question: if the members of a set of two (or more) successive income groups consume the same amount in a Nash equilibrium, what can we say about their consumption level? So suppose two members i and j with w i < w j and x i = x j , and suppose that w j equals the highest income level of this set of successive income groups. Because consumption levels are the same, both have the same social rank: r i = r j . This rank is r i = r := R(x i ; x{). Now letĉ (1) (r; w) denote the unique maximizer of the function U (c (1) ; (1 + )(w c (1) ); r). It is the utility-maximizing quantity of rst-period consumption if the individual cannot change her rank r. Applying a basic result in HM (2010, Proposition 6), we know that, in any Nash equilibrium, each individual h has a consumption level x h equal to or larger than this quantity at the attained rank R( . It follows that the consumption level of the two individuals in the pooling equilibrium is at leastĉ (1) (r ; w j ), that is, at least equal to the utility-maximizing quantity at given rank r of members of the highest income group of the cluster. This also illustrates the ine ciency of status seeking. Person i tries to`catch up with the Joneses' by matching the consumption of person j. The former indulges in overconsumption, because she consumes more than if her rank were exogenously xed at r (for the Pareto e ciency of separating and pooling equilibria, see HM, 2010).
We will study the impact of income growth and redistribution for a pooling equilibrium with three clusters of successive income groups. Each cluster has its own consumption standard, which equals the utility-maximizing quantity of its highest income group. 9 Since members of the same cluster share the same rank in equilibrium, the clusters are referred to as social classes. Thus a distinction is drawn between the`lower class', the`middle class', and the`upper class'. Many sociologists suggest ve social classes (distinguishing also between upper-and lower middle class, and between working class and underclass), but we restrict the divisions to three classes for the sake of clarity. 8 The succession property boils down to: if w i < w j < w k and x i = x k , then x j = x i . We prove this by contradiction, using the second result of the proposition. So suppose x j < x i . Then w j < w i , which is a contradiction. Suppose x j > x i , so also x j > x k . Then w j > w k , which is also a contradiction. 9 Given the three clusters of income groups, there generall exists a family of pooling equilibria with three consumption levels (see Appendix C). The selected equilibrium is the Pareto-dominating member of this family (see HM, 2010, Proposition 18).
Speci cally, for large enough N , x three integers e L , e M , and e U such that 1 e L < e M < e U . Let w k denote the xed income level of income group k, and assume w 1 < < w e U . The set of individuals of income group k is W k := fi 2 N j w i = w k g and their number is n k := # W k . The lower class is the set of individuals L := fi 2 N j w 1 w i w e L g and their number is N L := #L = P e L k=1 n k . Similarly, the middle class is given by M := fi 2 N j w e L +1 w i w e M g with number N M := #M = P e M k=e L +1 n k , and the upper class by U : and with (noting (4)) In Appendix C we derive su cient conditions for the existence of such a pooling equilibrium. The conditions essentially require that members of the lowest income group of a social class are not better o by choosing some lower consumption than the standard of their class (conditions (44)-(46)) and members of the highest income group of a social class are not better o by choosing the standard of a higher social class (conditions (47)-(49)). These requirements can be ful lled by an appropriate shape of the underlying income distribution. Income growth and redistribution may clearly upset the pooling equilibrium. Hereafter we consider the implications for a pooling equilibrium where income groups e L , e M , and e U still set the consumption standard of their social class (though the standards may be di erent than before). Su cient conditions for the existence of the new pooling equilibrium can be readily constructed using conditions (44)-(49) in Appendix C.

A concrete baseline
As baseline, consider a pooling equilibrium with three consumption standards where individual utility is given by with 0 < < 1 and > 0. Parameter measures sensitivity to rank. The optimal quantity of present consumption if individual i could not change her rank is To guarantee an interior solution for consumers of all income groups, it is assumed So optimal present consumption is then increasing in income and decreasing in social rank. It is even a luxury good, which accords with its status-signalling function. The negative relation with rank does not necessarily follow from our general assumptions in the previous section, but it is plausible. For example, consider a person's response to an exogenous event that causes the incomes of all other people to rise, and thus to increase their present consumption. Since her own income has not risen while her (exogenous) rank has fallen, the person su ers a decline in utility.
The fall in rank raises the marginal payo from present consumption and lowers that of future consumption, however, so she can reduce the decline in utility by saving less and increasing her present consumption. 10 The consumption standards of the lower, middle, and upper class follow as with ranks r L and r M given by (8). The trendsetters of each social class (i 2 fW e L ; W e M ; W e U g) consume their optimal amount (by assumption), but their followers consume too much. A follower of the lower class, for example, consumes more than her optimal quantity at the prevailing rank, and her overconsumption is higher, the lower her income: In particular, a follower has a higher payo than any trendsetter with a lower income since the former is able to match the consumption level of the latter and at the same time save more (in Appendix D we prove that payo increases by social class). Overconsumption is accompanied by undersaving. For example, the saving rate s i of a mem-ber of the lower class is (using (12)). Hence, the saving rate of a follower is decreasing in the consumption standard and it is increasing in income because higher income relieves the burden of complying with the standard. The saving rate of a trendsetter, however, is decreasing in income if present consumption is a luxury good, which is the case here. The relationship between income and saving rate for the three social classes typically describes a saw-like curve, as sketched in Figure 1(B). Note that a higher sensitivity to rank ( ) raises saving rates of both followers and trendsetters, because this lowers the marginal payo from present consumption and thus also from consumption standards.
Hereafter, the impact of income growth and redistribution is studied for individual payo s and the aggregate saving rate, denoted by s. The latter is a weighted sum of the average saving rates of the three social classes: where 3 Is income growth for everyone bene cial for everyone?
Suppose everyone's income rises with the same percentage. Then clearly everyone would be better o if consumption standards c L , c M , and c U stayed put. However, as indicated by (12), the trendsetters of the social classes can gain even more by increasing the consumption standards. While the trendsetters, then, are always better o , this is not immediately clear for the followers, typically the majority of the consumers. Therefore, suppose the income of each consumer i (i 2 N ) rises from w i to w i0 according to but with z not too large to preserve our type of pooling equilibrium, where income groups e L , e M , and e U set the consumption standards. Again taking a member of the lower class, her payo becomes Su cient for being strictly better o is that the increase in income ((z 1)w i ) covers the extra expenditure due to the higher standard (c L 0 c L ), so that while rst-period consumption increases, her saving does not fall. This comes down to w i (1 )w e L , which holds for consumers with incomes close to those of trendsetters but not necessarily for poor consumers of the lower class.
By di erentiating (18) with respect to z using (9) and evaluating at z = 1, we nd a necessary and su cient condition for being strictly better o : (note that the ratio is less than 1). The condition may not hold for consumers at the bottom of the lower class. For this category, adhering to the increased consumption standard may be accompanied by such a large decline in saving, and thus also future consumption, that, in spite of their higher income, their payo will fall. Let us see how concerns for rank can explain the Easterlin paradox. To do this, we examine the relationship between the growth rate of income and the growth rate of individual payo (or happiness, see Introduction). Just for now it is convenient to measure time as a continuous variable and consider a restricted time path that preserves the particular type of pooling equilibrium. Let g denote the income growth rate and v i L the payo growth rate of individual i in the lower class. 11 Then the latter is simply proportional to the former: (using (9)). Consider rst the members of the highest income group, i.e. with w i = w e L . Their payo growth rate follows as v i L = g h we L w e L + r L i (i 2 W e L ), which is less than the growth rate of their income. Further, because the second ratio of (20) is smaller than the rst ratio if w i < w e L , the payo growth rate of lower income groups is lower than that of the highest income group. Indeed, it is easily veri ed that the lower the income, the lower the growth rate of payo . Hence, the lowest payo growth rate occurs in the lowest income group. Payo growth rates at the bottom of the social class are even zero or negative if condition (19) fails to hold (z approaching 1 mimics continuous time). It is clear that similar results are obtained for the middle and upper class. Taken together, this implies that the change in the average payo of a social class can seriously lag behind universal income growth. The analysis thus supports the empirical observation, rst made by Easterlin (1974), that average happiness scores seem to move more slowly than average income.
Though high-income groups of a social class are able to save more when income rises, saving 11 Let t denote time and write w i (t) and we L (t). Then v i L is de ned as the growth rate of U (c L (t); (1+ )(w i (t) c L (t)); r L ) with c L (t) := (1 )we L (t) r L (i 2 L).
rates fall for everyone. The saving rate of a member of the lower class becomes (using (12), (17) and (18)). Hence, economy-wide income growth induces such higher consumption standards that it decreases the aggregate saving rate. Note that this result critically hinges on the property that trendsetters consider present consumption to be a luxury good. More generally (i.e., ignoring (9)), it is easily veri ed that the aggregate saving rate increases or decreases, depending on whether the trendsetting income groups see present consumption as a necessity or a luxury, and stays constant in the case of unitary income elasticity. The latter case provides a solution to the Kuznets consumption puzzle (see Introduction). Speci cally, we already found that saving rates are increasing in income for followers (arguably the majority of consumers, see (14)) and now we know that, under unitary income elasticity, saving rates stay constant if income changes across the board.
4 Does more income equality favour the poor?
Suppose a more equal income distribution is contemplated that expands the size of the middle class by reducing the numbers of people in the lower and upper class. While traditional analysis points at the bene ts of such a policy for people in the lower income brackets, the picture is more di use now, because redistribution of income may alter social ranks and consumption standards. Speci cally, the out ow of people to the middle class lowers the social rank of those who stay behind in the lower class, thereby inducing a higher consumption standard. In contrast, the in ow of people from the upper class raises the social rank of everyone in the middle class, which causes a fall in the consumption standard. Whether an egalitarian income policy has the intended e ects, therefore, remains to be seen. Consider the following redistribution scheme. Suppose a random draw of members of the lower class and a random draw of members of the upper class are randomly allocated to the income groups of the middle class, keeping aggregate income constant. Fix the proportion of the lower social class that ows out, denoted by (0 < 1). Then the redistribution is such that the size of an income group becomes an integer n 0 k with n 0 k := where is the proportion of the upper class that ows out and the growth rate of the middle class. The value of follows from the condition that the total income gain of those who leave the lower class must be equal to the total income loss of those who leave the upper class. If w L is the average income of the lower class ( w L := 1 N L X i2L w i ) and w M and w U the average incomes of the middle and upper class, follows from (it is assumed that is small enough to yield < 1). Finally, is implied by Note that the degree of income redistribution is entirely determined by the value of . As before, we consider changes in that induce a pooling equilibrium where income groups e L , e M , and e U still set the consumption standards.
Using primes for ex post variables, the scheme changes the ranks of the social classes as follows: So, while nothing happens with the social rank of the upper class, the rank of the lower class falls by N L N 1 , whereas the rank of the middle class rises by N U N 1 . For the new consumption standards, we nd accordingly These adjustments follow from the property that standards are decreasing in rank.
Let us now determine the welfare e ects of the redistribution scheme. To shorten notation, de ne for i 2 W k (k = 1; :::; e U ) and note that V k ( ; r i ) is downward-sloping if c i >ĉ (1) (r i ; w k ). We consider the change in payo for ve groups of individuals: Those who stay in the lower class are worse o , due to both the lower rank and the higher standard.
Formally, we have V k (c L 0 ; r L 0 ) < V k (c L 0 ; r L ) (k = 1; :::; e L ). Because c L < c L 0 , it holdŝ Those who stay in the middle class are better o , because of both the higher rank and the lower standard.
Those who stay in the upper class are una ected, since both rank and standard remain the same.
Those who leave the lower class are better o for two reasons: as shown above, middle-class consumers are better o than before, and payo always increases by social class. In sum, returning to the question of whether a more equal income distribution favours the poor, the answer is yes, and no. The policy results in both lower-class consumers receiving a higher income and middle-class consumers being better o . However, lower-class consumers who do not receive a higher income are worse o . Their rank in the social hierarchy drops and they have to spend more of their income to conform to the consumption standard of their class, because this has been raised by their peers in response to the lower rank.

Does more income equality increase happiness?
Let us now take a utilitarian approach and use the results in the previous section to explore whether a more equal income distribution increases overall average payo (`mean happiness'). Particularly, we are interested in how the social-comparison component alters the standard e ect of income equality on aggregate payo . Therefore, let V denote overall average payo and de ne where is the average payo of the lower class, and similar de nitions apply to those of the middle and upper class, V M and V U . Overall average payo after income redistribution ( V 0 ) can be written as where Now recall that the payo s of individuals who stay in the upper class are una ected, so V 0 U = V U . Then, using (24), the induced change in overall average payo can be split up into these two terms: The rst term is the standard e ect of redistributing income. It compares the gain of those who ow from the lower class into the middle class with the loss of those who arrive from the upper class. The second term arises because income redistribution changes the ranks of the social classes. The two e ects are examined further below.
Regarding the standard e ect, let us eliminate by de ning Note that is a strictly positive parameter. It measures the relative income gap between lower and middle class, as compared with the income gap between middle and upper class. Then the rst term of (34) can be written as (using (23)). For example, if = 1 (income gaps between social classes are the same), the standard e ect is positive if average payo increases by social class at a decreasing rate. This re ects the Benthamite proposal for reducing income inequality.
The second term sums two expressions. The rst one is negative since V 0 L < V L , and refers to the loss in payo for those who stay behind in the lower class. The second expression is positive since V 0 M > V M , and includes a gain for those who were already in the middle class (N M ) and what could be seen as a bonus for the new arrivals ( N M ). Those who arrived from the lower class receive more than the initial payo of the middle class ( V M ) and those who arrived from the upper class su er less than in the case of the initial payo of the middle class. Without further assumptions, however, the second term cannot be signed. Hence, whether the e ect due to social comparison mitigates or strengthens the standard e ect of income redistribution remains an open question.
6 Does more income equality increase aggregate saving?
In the neoclassical case with homothetic preferences over present and future consumption, saving rates do not depend on income. Redistributing income then has no impact on the aggregate saving rate. With more general preferences where saving rates do depend on income, redistributing income typically alters the aggregate saving rate through its impact on the weights given by the relative sizes of the income groups, or, under the above redistribution scheme, the relative sizes of the social classes. In the case of social comparison, however, there is an additional e ect. Since the scheme changes social ranks and thus consumption standards, it also a ects the average saving rates of the social classes.
The aggregate saving rate after income redistribution ( s 0 ) can be written as where (see Section 4 and (15) and (16)). Now note that, since the consumption standard of the upper class is una ected by redistribution, the saving rate of this class stays the same: s 0 U = s U . Then the induced change is given by the sum of two terms: Just as in the previous section, the rst term is a standard e ect that occurs through the change in the relative sizes of the social classes. Redistribution expands the middle class by in ows of N L consumers from the lower class and N U consumers from the upper class. The second term is the additional e ect due to social comparison, which causes changes in the saving rates of the lower and middle class.
Using (23) and (35), the rst term of (39) can be written as This shows that the direction of the standard e ect is independent of the degree of income redistribution ( ). If = 1 (income gaps between social classes are the same), the standard e ect is negative if the average saving rate rises by social class at an increasing rate: more income equality then reduces aggregate saving. The opposite holds if the average saving rate rises at a decreasing rate.
As for the additional e ect, a little calculation shows that using (12), (25) and (26). Hence, because the consumption standard increases in the lower class and decreases in the middle class, the saving rate of the former falls and that of the latter rises. This already suggests that the direction of the additional e ect is ambiguous. Using (22)-(25), the second term of (38) can be expressed as The bracketed term cannot be signed a priori, but it is seen that a positive outcome becomes more likely as the degree of redistribution increases. Also, if the relative income gap between lower and middle class ( ) is small, redistribution tends to have a negative additional e ect. If the relative income gap ( ) is large, it is just the opposite: redistribution tends to have a positive additional e ect. Let us draw a conclusion for the plausible case where saving rates rise by social class, so s L < s M < s U . If the relative income gap between lower and middle class ( ) is small, redistribution tends to have a positive standard e ect (it increases aggregate saving) and a negative additional e ect. If the relative income gap ( ) is large, redistribution tends to have a negative standard e ect (it decreases aggregate saving) and a positive additional e ect. Our conclusion then is that income redistribution in the case of upward-looking comparisons is likely to mitigate the standard e ect of income redistribution on aggregate saving.

Conclusion
Above we analysed how social-rank concerns alter the usual impact of income growth and redistribution on individual payo s and saving rates. After linking the standard model of intertemporal consumption and saving to the ordinal status game of Haagsma and van Mouche (2010), the analysis yielded, among other things, explanations for the Easterlin paradox and the Kuznets consumption puzzle and the insight that rank concerns tend to weaken the standard e ect of inequality on aggregate saving.
Assuming a nite number of consumers di ering only in income, an individual's social rank was de ned as the fraction of consumers who spend the same as or less than her on present consumption. The resulting interdependency among consumers can give rise to two types of Nash equilibria: separating equilibria, where each income group has its own consumption standard, and pooling equilibria, where at least one consumption standard is shared by two or more income groups. Whereas the literature focuses on separating equilibria with a continuum of agents, the paper shows that it is the possibility of pooling equilibria that o ers another step towards a more realistic account of the phenomenon of status seeking. Perhaps this is indeed the typical manifestation of status seeking: people not only raising their spending but actually matching the consumption expenditure of those in slightly higher income groups. In any case, it accords with basic sociological notions that social interdependence promotes uniform behaviour. The possibility of pooling equilibria can also illustrate the phenomenon of class structure. While we distinguished three social classes, the number of classes is an endogenous variable ultimately determined by the shape of the underlying income distribution.

APPENDIX 12
Appendix A We show that the game is an`ordinal status game' as de ned and studied by HM (2010). Since, for each r i 2 [0; 1], U ( ; ; r i ) is strictly increasing, the budget constraint will hold with strict equality at any maximizer and equilibrium, so we can substitute for c i (2) in U and write With X i := [0; L i ] := [0; w i ] the domain of the action variable x i and Q := fq 1 ; :::; q N g with q k := k 1 N 1 (k 2 N ) the domain of the rank variable r i (note that Q [0; 1]), we de ne the function u i : Then using (4) we arrive at the payo function v i : X 1 X N ! R as de ned by HM (2010): The assumed shape of U ensures that function u i is continuous in the rst variable, strictly quasi-concave in the rst variable, and strictly increasing in the second variable. Restriction (5) ensures that u i also satis es the so-called relevance condition mentioned by HM (2010). Hence, the game indeed is an ordinal status game.

Now this inequality can be rewritten as
First we will show that u i (c 0 ; r 0 ) u i (c; r 0 ) u j (c 0 ; r 0 ) u j (c; r 0 ) and then u i (c; r 0 ) u i (c; r) u j (c; r 0 ) u j (c; r).
For the proof of Proposition 1 in the main text, we use the two conditions and the following result: Lemma 4 Let x be a Nash equilibrium. Then for all i; j 2 N , with r i := R(x i ; x{) and r j := R(x j ; x|), it holds Proof. As x i < x j and (4) holds, we have . Because x is a Nash equilibrium and x j w i , u i (x i ; r i ) = u i (x i ; R(x i ; x{)) u i (x j ; R(x j ; x{)). Now, u i (x j ; R(x j ; x{)) = u i (x j ; R(x j ; x|)) = u i (x j ; r j ). So the rst inequality holds. Similarly, u j (x j ; r j ) = u j (x j ; R(x j ; x|)) u j (x i ; R(x i ; x|)) = u j (x i ; R(x i ; x{) + 1 N 1 ) > u j (x i ; r i ). So the second inequality also holds.
Proof of Proposition 1.
(1 ) We can apply Theorem 7 in HM (2010) if we can prove that two consumers i; j 2 N are`homogeneous', as de ned by HM (2010), if and only if w i = w j . Well,`if' is obvious. As for`only if', suppose i and j are homogeneous. Then, by Theorem 5 in HM (2010),ĉ (1) (0; w i ) =ĉ (1) (0; w j ). Imposing Condition 1, this requires w i = w j .
(2 ) By contradiction. Suppose w i < w j and x i > x j . Then we have x i ; x j 2 [0; w i ] and, with r i := R(x i ; x{) and r j := R(x j ; x|), also r j < r i . Now, by Lemma 4, So we have u i (x i ; r i ) u i (x j ; r j ) > 0. According to Condition 2, this implies u j (x i ; r i ) u j (x j ; r j ) > 0. But this contradicts Lemma 4.
(3 ) Because of the rst and second statement, it is su cient to prove that w i < w j ) x i < x j . So suppose w i < w j . Then i and j are not homogeneous players (see under (1 )). Because x is a separating equilibrium, it follows that x i 6 = x j . So x i < x j or x i > x j . But x i > x j is impossible because of the second statement.

Appendix C
We derive su cient conditions for the existence of the particular pooling equilibrium by applying Theorem 11 in HM (2010). To connect to this theorem we rst introduce two auxiliary functions and change notation a bit.
Letĉ i (r) :=ĉ (1) (r; w i ) (i 2 N ) and note that, by assumption, (iii) c i + is strictly increasing in its rst variable, strictly decreasing in its second variable, and strictly increasing in its third variable; (iv) c i Proof. See Lemmas 20 and 21 in HM (2010).
We also need to know how the two functions depend on income. For this recall that U is twice continuously di erentiable with partial derivatives U 11 ; U 12 ; U 23 0 and U 22 < 0.
Lemma 6 For each pair of consumers i; j 2 N , Proof. Because c i (b; a) >ĉ i (a) and b > a, this inequality indeed holds if both U 21 (1 + )U 22 > 0 and U 23 0. This is so by assumption. ( Again the implicit theorem can be applied, implying that c i + (b; a; d) is a di erentiable function of w i . Di erentiation of the above expression wrt. w i yields The denominator is strictly negative because c i + (b; a; d) >ĉ i (b). The numerator is strictly negative if and only if U 2 (d; (1 + )(w d); a) < U 2 ( c + ; (1 + )(w c + ); b).
Because c i + (b; a; d) > d and b > a, the above inequality indeed holds if both U 21 (1 + )U 22 > 0 and U 23 0. This is so by assumption.

Appendix D
The following result implies that payo increases by social class.
Proposition 7 Let x be a Nash equilibrium. Then for all i; j 2 N with w i < w j , and writing r i := R(x i ; x{) and r j := R(x j ; x|), it holds x i < x j ) u i (x i ; r i ) < u j (x j ; r j ): Proof. Because x i < x j and given (4), we have r j = R(x j ; x|) = R(x j ; x{) and r i = R(x i ; x{) = R(x i ; x|) 1 N 1 .

Appendix E
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