Incorporating Concern for Relative Wealth Into Economic Models

This article develops a simple model that captures a concern for relative standing, or status. This concern is instrumental in the sense that individuals do not get utility directly from their relative standing, but, rather, the concern is induced because their relative standing affects their consumption of standard commodities. The article investigates the consequences of a concern for relative wealth in models in which individuals are making labor/leisure choice decisions. The analysis shows how individuals’ decisions are affected by the aggregate income distribution and how the concern for relative wealth can generate behavior that can be interpreted as conspicuous consumption when wealth is not directly observable. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. A standard, often implicit, assumption in economics is that people only accumulate wealth to fund consumption by themselves and their families. In this article, we will argue that, in many circumstances, people have other motivations for wealth acquisition. In particular, we will argue that people acquire wealth in order to be wealthier than other people. Moreover, while this desire to be wealthier than other people appears to capture a concern for relative status, it can be justified on narrow economic grounds. This desire to be relatively wealthy is similar to the social motivations for wealth acquisition mentioned by a number of prominent early economists. Broadly speaking, they argued that society views wealthy individuals positively and, furthermore, that this positive light serves as an important motivation for the acquisition of wealth. Adam Smith (1759, pp. 108–10) wrote For to what purpose is all the toil and bustle of this world? what is the end of avarice and ambition, of the pursuit of wealth, of power, and preheminence [sic]? Is it to supply the necessities of nature? The wages of the meanest labourer can supply them . . . . From whence, then, arises that emulation which runs through all the different ranks of men, and what are the advantages which we propose by that great purpose of human life which we call bettering our condition? To be observed, to be attended to, to be taken notice of with sympathy, complacency, and approbation, are all the advantages which we can propose to derive from it. It is the vanity, not the ease, or the pleasure, which interests us. But vanity is always founded upon the belief of our being the object of attention and approbation. Veblen (1899) argued that there developed within societies a belief about the level of conspicuous consumption that is appropriate to a particular rank within a society, that this consumption level increases with one’s rank, and, further, that as the society becomes richer, the appropriate level for any given rank rises. Veblen also argued that since the primary purpose of these conspicuous consumptions is to signal one’s success, they must be of a publicly observable nature or at least produce a publicly observable product. The pervasive assumption in current economic models that people are not concerned with relative wealth stems not from a belief in its descriptive accuracy, but rather from methodological considerations. Economics has been successful as a discipline because of the restrictions imposed by the assumptions of the models employed. A model can have predictive power only to the extent that some kinds of behavior are inconsistent with the assumptions of that model. Foremost among the assumptions that underlie economic models is that agents are rational: agents choose from the actions available that action which yields the highest utility. The assumption that agents maximize utility, however, puts no restrictions on behavior in the absence of restrictions on the nature of the utility function. Any observed pattern of behavior can be rationalized as utility-maximizing if utility functions can change arbitrarily through time. The force of the rationalagent assumption in economic models comes from the concurrent restrictions on the utility function, for example, the requirement that the utility function be either unchanging through time or changing in a well-defined way. Similarly, economists can assume that many variables affect individuals, but only at the cost of weakening the conclusions that can be drawn from the analysis. Typically, economists have restricted agents’ utility functions to depend only on consumption for this reason: allowing agents’ decisions to be affected by such things as feelings of competition, envy, or rivalry admits models that have no predictive power. We are interested in developing models that accommodate a concern for relative wealth in reduced-form models while maintaining the standard economic assumption that individuals ultimately care only about consumption. In these models, an agent’s concern for relative wealth is instrumental: he or she cares about relative wealth only because final consumption is related not just to wealth, but additionally to relative wealth. In Cole, Mailath, and Postlewaite 1992, we presented a model in which agents care about relative wealth because relative wealth affects mating. That model deals with an environment in which there is a succession of generations of men and women who match and jointly make a consumption/saving decision. The members of each sex differ only in their endowments. An immediate consequence of the assumption that consumption is joint is that each individual prefers to be matched with the richest member of the opposite sex, all other things being equal. If the matching in a particular period has no effect on how future generations will match, voluntary matching will be positively assortative on wealth; that is, the wealthiest men will match with the wealthiest women, and so on. When matching is positively assortative on wealth, individuals who are higher in the wealth distribution for their sex will end up with better matches (that is, richer mates). Thus individuals care about relative wealth, but in the instrumental way described above: they care about relative wealth because it leads to wealthier mates, which results in higher consumption. The purpose of the current article is twofold. First, we provide a simple exposition of the basic ideas contained in our earlier work and discuss in more depth the interaction between relative standing and economic behavior. Second, we apply these ideas to two economic problems of independent interest. We first develop an effort model with complete information and show how the concern about relative wealth affects individuals’ effort decisions. We then develop a second model which extends the analysis to include private information about income, which induces signaling that can be interpreted as conspicuous consumption. We should emphasize that the direct implications of these models in which agents care about relative wealth do not necessarily differ from those that would obtain if relative wealth were put directly into the utility function. There are, however, advantages to our approach. First, an agent’s concern for relative wealth in reduced-form preferences is induced by the fundamentals of the environment. Changes in the fundamentals of that environment will lead to predictable changes in reduced-form preferences. Here, unlike the case in which relative wealth is put directly into the utility function, testable implications can be derived about the relationship between fundamentals and reducedform preferences. The dependence of reduced-form preferences on the fundamentals provides for additional scope in explaining why seemingly similar agents behave differently. An Effort Model With Complete Information Consider a one-period model in which there are two types of agents, men and women. There exist a continuum of men indexed by i ∈ [0,1] and a continuum of women indexed by j ∈ [0,1]. Male i is exogenously endowed with i units of good x, while female j can produce good y by expending effort. There is no trade; each agent seeks to match with an agent of the opposite sex in order to consume both goods. By assumption, both goods are jointly consumed by two matched individuals. All agents have identical utility functions over the joint consumption of a matched pair’s bundle given by u(y) + x. We assume that the female agent has a disutility for effort given by –v(l), where l denotes labor effort. Female output of good y is given by a( j)l, where the productivity function a( j) gives female j’s productivity per unit of effort. We allow the productivity levels, denoted by a( j), to differ across females. We assume that the females are ordered so that a( j) is increasing in j, the index or names of the females. Matching is voluntary and, in this section, based on complete information. A given matching is voluntary if no two unmatched agents mutually strictly prefer each other to their current matches. Since all consumption is joint by assumption, agents desire to be matched with as wealthy a mate as possible. Consequently, in any voluntary matching, the wealthiest male will match with the wealthiest female and, more generally, the kth-percentile male in the wealth distribution of men will be matched with the kthpercentile woman in the female wealth distribution. Since the distribution of good x is fixed exogenously, women’s effort decisions determine the matching of men and women, along with the consumption levels in the matches. In equilibrium, each female takes as given other women’s effort decisions, and hence the endowment level of her equilibrium match is given by her rank in the distribution of y. For a particular choice of outputs by women, we summarize the relationship between an individual female’s output and her mate’s endowment by the matching function m(y), which indicates the endowment of the man who will match with a woman with wealth y. If female j produces y units of output, while half of the other females produce less than y and half more, then female j will be matched with the male with the median endowment, or an endowment of one-half. If female j’s output is such that exactly three-quarters of the females produce less than she, then she will be matched in equilibrium with the male whose endowment is three-quarters. In other words, m is the distribution function of female output. If female output (wealth) is strictly increasing in j, then the matching function is simply the inverse of the output function: if female j produces y = y( j) units of output, then the index (and so endowment) of her mate is given by m(y) = y(y( j)) = j. Given a matching function m(·), female j’s optimal effort level will be the solution to the following problem: (1) maxl u(a( j)l) – v(l) + m(a( j)l). A female’s total utility is determined by her direct utility from consumption of her own output, her disutility from effort, and the utility she derives from consuming her mate’s endowment of x. It is not difficult to establish that in equilibrium a female’s output level is increasing in her productivity. We establish this result in Proposition 1 in the Appendix. The first-order condition which, under certain conditions, characterizes the solution to the problem (1) is (2) a( j)[u′(a( j)l) + m′(a( j)l)] – v′(l) = 0. We denote the value of effort l that solves (1) by l( j). The first-order condition indicates how the impact of the equilibrium match quality affects a woman’s effort decision. The concern about her relative output level induced by the tournament for males, reflected by m′ in the first-order condition, leads to an increase in the effort level. When m′ is relatively large, there is an incentive to work harder since the resulting increase in output has a greater impact on the quality of the resulting match. Since m(y) is the fraction of females whose output level is below y, if the distribution of females’ output is tight, m′ is large; that is, a small change in an individual female’s output can have a large effect on her rank; conversely, if females’ outputs are disperse, the opposite is true. A female is concerned about her output rank only to the extent that males differ in the levels of the male good. If male j’s endowment level was given by γ j, then, if the effort levels of the other females are held fixed, the new matching function would be given by γm(a( j)l( j)), and the impact on female j’s effort decision would be larger or smaller as γ was greater or less than one. Note that in the extreme case where γ = 0, matching would be irrelevant from the females’ perspective, and they would choose their effort levels so that a( j)u′(a( j)l) – v′(l) = 0. An equilibrium, then, is an effort function l: [0,1] → + and a matching function m: + → [0,1] such that (3) l( j) maximizes u(a( j)l) – v(l) + m(a( j)l)