Abstract
The two sides of envy, destructive and constructive, give rise to qualitatively different equilibria, depending on the economic, institutional, and cultural environment. If investment opportunities are scarce, inequality is high, property rights are not secure, and social comparisons are strong, society is likely to be in the “fear equilibrium,” in which better endowed agents underinvest in order to avoid destructive envy of the relatively poor. Otherwise, the standard “keeping up with the Joneses” competition arises, and envy is satisfied through suboptimally high efforts. Economic growth expands the production possibilities frontier and triggers an endogenous transition from one equilibrium to the other causing a qualitative shift in the relationship between envy and economic performance: envyavoidance behavior with its adverse effect on investment paves the way to creative emulation. From a welfare perspective, better institutions and wealth redistribution that move the society away from the lowoutput fear equilibrium need not be Pareto improving in the short run, as they unleash the negative consumption externality. In the long run, such policies contribute to an increase in social welfare due to enhanced productivity growth.
This is a preview of subscription content, access via your institution.
Notes
The term “culture” refers to features of preferences, beliefs, and social norms (Fernández 2011).
For a stateoftheart overview see various chapters in Benhabib et al. (2011). A different strand of literature looks at the endogenous formation of preferences in the context of longrun economic development, with recent examples including Doepke and Zilibotti (2008) on patience and work ethic and Galor and Michalopoulos (2012) on risk aversion.
A number of evolutionary theoretic explanations have been proposed for why people care about relative standing, see Hopkins (2008, Sect. 3) and Robson and Samuelson (2011, Sect. 4.2). Evidence documenting people’s concern for relative standing is abundant and comes from empirical happiness research (Luttmer 2005), job satisfaction studies (Card et al. 2012), experimental economics (Zizzo 2003; Rustichini 2008), neuroscience (Fliessbach et al. 2007), and surveys (Solnick and Hemenway 2005; Clark and Senik 2010); see Clark et al. (2008, Sect. 3) and Frank and Heffetz (2011, Sect. 3) for an overview.
Allowing for transfers replaces destructive activities of the poor with “voluntary” sharing of the rich, reflecting the evidence on the fearofenvymotivated redistribution in peasant societies of Latin America (Cancian 1965), Southeast Asia (Scott 1976), and Africa (Platteau 2000). This possibility is examined in Gershman (2012).
Certain psychological approaches treat “benign” and “malicious” envy as two separate emotions (van de Ven et al. 2009). The present theory is crucially different in that the emotion, envy, is the same, but its manifestation is an equilibrium outcome.
Second round, 2002–2003; raw data available at http://www.afrobarometer.org.
The rational fear of envy becomes curiously embedded in cultural beliefs. The term “institutionalized envy” coined by Wolf (1955) summarizes the set of cultural control mechanisms related to the fear of envy including gossip, the fear of witchcraft, and the evil eye belief. See Gershman (2014) for more details.
Some have argued that, in addition to its stimulating effect on labor supply, “wasteful” conspicuous consumption may hurt savings and, hence, economic growth (Frank 2007). Cozzi (2004) argues instead that status competition can lead to increased capital accumulation. Corneo and Jeanne (1998) show that the effect of status competition on savings depends crucially on the assumption about the timing of “contests for status” over the life cycle. Moav and Neeman (2012) develop a model in which both human capital and conspicuous consumption serve as signals for income and show that in this setup concern for status can generate a poverty trap due to increasing marginal propensity to save.
This is in line with evidence cited by Moav and Neeman (2012) and others that even in poor economies some people engage in conspicuous consumption. The theory that follows identifies the conditions that facilitate a transition from the fear of envy to peaceful status competition in developing societies.
For a similar perspective on envy, the rise of consumer culture, and the transition from envyavoidance to “envyprovocation” see Belk (1995, Chap. 1).
An extension with bequest dynamics is presented in the Supplementary Online Material.
Linearity in \(A_t\) and \(K_{it}\) is inessential. Having exponents on these terms would just add more parameters to the model. Linearity in \(L_{it}\) allows to obtain closedform solutions, but is not crucial for any of the qualitative results.
The formulation with pure destruction allows to focus on envy as the only motive for disruptive behavior. In a setup with theft, envy is an additional force contributing to appropriation. The implications of protection in a model of appropriation (without envy) were examined by Grossman and Kim (1995, 1996). As will become clear, the setup with time allocation makes the model scalefree: optimal destruction intensity will depend on the inequality (but not the scale) of firststage outcomes, which captures the essence of destructive envy.
Assume for simplicity that \(U_{it}=\infty \) whenever \(C_{it}\leqslant \theta C_{jt}\). Under the assumption (10) below, this will never be the case in equilibrium. The assumption on the elasticity of marginal utility with respect to relative consumption, \(\sigma \), is a convenient regularity condition that guarantees concavity of equilibrium outputs in endowments (see Sect. 3.3). Linearity in effort is assumed for analytical tractability. For simplicity, we also abstract from leisure in stage 2.
Additive comparison was assumed, among others, by Knell (1999) and Ljungqvist and Uhlig (2000). Boskin and Sheshinski (1978) and Carroll et al. (1997) are examples of models with ratio comparison. Clark and Oswald (1998) examine the properties of both formulations. The model in Gershman (2014) shows that the qualitative results of this section can be obtained in a framework with ratio comparison.
It is not uncommon in political economy literature to model group interaction in the context of twoagent games (Grossman and Kim 1995). The implicit assumption is that groups are able to solve the collective action problem and act in a coordinated way.
Dupor and Liu (2003) make a distinction between jealousy (envy) and KUJ behavior (emulation). The former is defined as \(\partial U_{it}/\partial C_{jt}<0\), while the latter is defined as \(\partial ^2 U_{it}/\partial C_{jt}\partial C_{it}>0\). Interestingly, for a class of utility functions including (5) these two notions are equivalent.
In the extension of the model available in the Supplementary Online Material generations are linked through bequests.
Aghion et al. (1999) assume that \(\alpha =1\). Here, \(\alpha <1\) in order to capture the possibility of a low productivity steady state, as will become clear from the analysis of Sect. 4. If \(\alpha =1\) or \(A_t\) evolves according to a Romerstyle “ideas production function,” the society never gets stuck in a “bad” longrun equilibrium, but all the main qualitative results still carry through. We also assume for simplicity that the new knowledge is only available to the next generation, that is, there is no contemporary spillover effect.
As shown in the proofs of the main results available online, the assumptions of the model guarantee that \(C_i\theta C_j>0\) in equilibrium for \(d_i^{*}<1\). Hence, \(d_i^{*}=1\) is never optimal and this case is not considered.
This assumption rules out the case in which the agent with higher investment outcome engages in destructive activities to improve his relative position even further. While such behavior is a theoretical possibility, the case considered here is more intuitive and consistent with the anecdotal evidence on destructive envy in Sect. 2.
Technically, these consumptionbased best responses correspond to the best responses in terms of firststage investment outcomes adjusted for the secondstage destructive activity. Such transformation focuses on final outputs and makes it easier to analyze the destructive equilibria.
The former assumption is made to cover all possible configurations of the bestresponse function. The latter imposes a lower bound on productivity which simplifies derivations.
The values of thresholds \(\widehat{C}_{11},\, \widehat{C}_{12},\, \widehat{C}_{13},\,\widehat{C}_{14}\), and \(\widehat{C}_{15}\) shown in Fig. 2 are specified in the detailed form of Lemma 2 in the Appendix.
In fact, for levels of \(C_1\) between \(\widehat{C}_{15}\) and \(\widehat{C}_{14}\) Agent 2 would be willing to produce more than \(AK_2\) to improve his relative position, but is constrained by the available resources.
The values of thresholds \(\widehat{C}_{21}\), \(\widehat{C}_{22}\), \(\widehat{C}_{23}\), and \(\widehat{C}_{24}\) shown in Fig. 3 are specified in the detailed version of Lemma 3 in the Appendix.
In particular, such split is typical under two assumptions: \(\underline{k}<\hat{k}\) and \(\widehat{A}_{22}>\widehat{A}_{23}\), where \(\hat{k}\) is defined in the detailed form of Proposition 1 in the Appendix. Both assumptions are maintained to achieve the highest possible variety of equilibria, with alternatives yielding only a subset of cases depicted in Fig. 5a. The blackened southwestern corner of the sector is not considered due to an earlier technical assumption that \(AK_2>\widehat{A}_{23}\).
We include the fulltime investment case (18) in the group of KUJtype equilibria, since it does not feature either destructive envy or the fear of it. One caveat, however, is that at very low levels of productivity working full time is unrelated to KUJtype incentives. With this in mind, by default we refer to case (18) as the one in which it is the catchingup behavior that is limited by the available resources.
Note that (16) would always be the unique equilibrium of the envy game in the absence of destructive technology (under perfect property rights protection) and the resource constraint.
Mui (1995) constructs a theoretical framework in which (costless) technological innovation may not be adopted in anticipation of envious retaliation. The intuition of the fear equilibrium in the present theory is similar, except that here the fear of envy operates on the intensive margin by discouraging (costly) investment. Furthermore, in Mui’s framework retaliation reduces envy directly by assumption rather than through the improvement of the relative standing of the envier. Finally, his paper ignores constructive envy and thus, focuses on one side of the big picture.
This is due to the assumption that \(\underline{k}<\hat{k}\). If, on the other hand, \(\underline{k}>\tilde{k}\), a high enough level of productivity guarantees a peaceful KUJ equilibrium. The thresholds \(\tilde{k}\) and \(\hat{k}\) are given by (28).
A shift towards this sector from higher starting levels of inequality may happen endogenously due to bequest dynamics, see the Online Supplementary Material.
This “opportunityenhancing” effect of redistribution under diminishing returns to individual endowments and imperfect capital markets is wellknown in the literature, see, for example, Aghion et al (1999, Sect. 2.2).
Another reason for why the economy might start in the feartype equilibrium is if it has redistributive mechanisms in place. A variation of the basic model in Gershman (2012) shows how in the presence of preemptive transfers destructive equilibrium can be replaced by a “fear equilibrium with transfers.”
The lower bound on initial productivity is \(\bar{A}\equiv [\tau \theta /((1\tau \theta ^2)K_1)]^{\sigma /(\sigma 1)}\cdot [(1+\theta ^2)K_2/2]^{1/(\sigma 1)}\).
Specifically, as follows directly from Proposition 1, expost inequality is equal to \(C_1/C_2=\tau \theta \) in the fear region, then monotonically decreases to \((k^{1/\sigma }+\theta )/(1+\theta k^{1/\sigma })\) and stays at that level thereafter.
All major world religions denounce envy. In JudeoChristian tradition envy is one of the deadly sins and features prominently in the tenth commandment. Schoeck (1969, p. 160) goes as far as to say that “a society from which all cause of envy had disappeared would not need the moral message of Christianity.”
Figure 8 ignores for simplicity that for low enough \(A\) the dynamics is governed by the outcomes of destructive equilibria.
An alternative, but similar way to think about it would be to consider a benevolent utilitarian social planner maximizing the (discounted) welfare of current and all future generations of the society. A more involved option would be to incorporate Barrostyle dynastic preferences in the analysis. In any case the crucial point will be the differential welfare effect on current versus future generations.
Clearly, an increase in \(C_1\) and \(C_2\) by the same factor due to rising productivity makes everyone better off since own consumption is valued more than reference consumption \((\theta <1)\).
Note also that in the short run a Pareto improvement is more likely to happen in an unequal society, while in the long run it is the other way round. The reason is that in the former case it is the rich agent who is critical and his preference is to maintain higher inequality, while in the latter case the poor agent is critical and he prefers equality. See the proof of Proposition 4 for details.
References
Aghion, P., Caroli, E., & GarcíaPeñalosa, C. (1999). Inequality and economic growth: The perspective of the new growth theories. Journal of Economic Literature, 37(4), 1615–1660.
Banerjee, A. (1990). Envy. In B. Dutta, S. Gangopadhyay, D. Ray, & D. Mookherjee (Eds.), Economic theory and policy: Essays in honour of Dipak Banerjee (pp. 91–111). New York: Oxford University Press.
Barnett, R. C., Bhattacharya, J., & Bunzel, H. (2010). Choosing to keep up with the Joneses and income inequality. Economic Theory, 45(3), 469–496.
Belk, R. W. (1995). Collecting in a consumer society. London: Routledge.
Benhabib, J., Bisin, A., & Jackson, M. (Eds.). (2011). Handbook of social economics (Vol. 1A). Amsterdam: Elsevier.
Boskin, M. J., & Sheshinski, E. (1978). Optimal redistributive taxation when individual welfare depends upon relative income. Quarterly Journal of Economics, 92(4), 589–601.
Bowles, S., & Park, Y. (2005). Emulation, inequality, and work hours: Was Thorsten Veblen right? Economic Journal, 115(507), F397–F412.
Cancian, F. (1965). Economics and prestige in a Maya community. Stanford, CA: Stanford University Press.
Card, D., Mas, A., Moretti, E., & Saez, E. (2012). Inequality at work: The effect of peer salaries on job satisfaction. American Economic Review, 102(6), 2981–3003.
Carroll, C. D., Overland, J. R., & Weil, D. N. (1997). Comparison utility in a growth model. Journal of Economic Growth, 2(4), 339–367.
Clanton, G. (2006). Jealousy and envy, Chapter 18. In J. E. Stets & J. H. Turner (Eds.), Handbook of the sociology of emotions (pp. 410–442). Berlin: Springer.
Clark, A. E., & Senik, C. (2010). Who compares to whom? The anatomy of income comparisons in Europe. Economic Journal, 120(544), 573–594.
Clark, A. E., Frijters, P., & Shields, M. (2008). Relative income, happiness and utility: An explanation for the Easterlin Paradox and other puzzles. Journal of Economic Literature, 46(1), 95–144.
Clark, A. E., & Oswald, A. J. (1998). Comparisonconcave utility and following behaviour in social and economic settings. Journal of Public Economics, 70(1), 133–155.
Corneo, G., & Jeanne, O. (1998). Social organization, status, and savings behavior. Journal of Public Economics, 70(1), 37–51.
Cozzi, G. (2004). Rat race, redistribution, and growth. Review of Economic Dynamics, 7(4), 900–915.
D’Arms, J., & Kerr, A. D. (2008). Envy in the philosophical tradition, Chapter 3. In R. H. Smith (Ed.), Envy: Theory and research (pp. 39–59). Oxford: Oxford University Press.
Demsetz, H. (1967). Toward a theory of property rights. American Economic Review, 57(2), 347–359.
Doepke, M., & Zilibotti, F. (2008). Occupational choice and the spirit of capitalism. Quarterly Journal of Economics, 123(2), 747–793.
Dow, J. (1981). The image of limited production: Envy and the domestic mode of production in peasant society. Human Organization, 40(4), 360–363.
Dupor, B., & Liu, W.F. (2003). Jealousy and equilibrium overconsumption. American Economic Review, 93(1), 423–428.
Elster, J. (1991). Envy in social life, Chapter 3. In R. J. Zeckhauser (Ed.), Strategy and choice (pp. 49–82). Cambridge, MA: MIT Press.
Falk, A., & Knell, M. (2004). Choosing the Joneses: Endogenous goals and reference standards. Scandinavian Journal of Economics, 106(3), 417–435.
Fernández de la Mora, G. (1987). Egalitarian envy: The political foundations of social justice. New York: Paragon House Publishers.
Fernández, R. (2011). Does culture matter?, Chapter 11. In J. Benhabib, A. Bisin, & M. Jackson (Eds.), Handbook of social economics (Vol. 1A, pp. 481–510). Amsterdam: Elsevier.
Fliessbach, K., Weber, B., Trautner, P., Dohmen, T., Sunde, U., Elger, C. E., et al. (2007). Social comparison affects rewardrelated brain activity in the human ventral striatum. Science, 318(5854), 1305–1308.
Foster, G. (1972). The anatomy of envy: A study in symbolic behavior. Current Anthropology, 13(2), 165–202.
Foster, G. (1979). Tzintzuntzan: Mexican peasants in a changing world. New York: Elsevier.
Frank, R. H. (1985). The demand for unobservable and other nonpositional goods. American Economic Review, 75(1), 101–116.
Frank, R. H. (2007). Falling behind: How rising inequality harms the middle class. Berkeley: University of California Press.
Frank, R. H., & Heffetz, O. (2011). Preferences for status: Evidence and economic implications, Chapter 3. In J. Benhabib, A. Bisin, & M. Jackson (Eds.), Handbook of social economics (Vol. 1A, pp. 69–91). Amsterdam: Elsevier.
Galor, O., & Michalopoulos, S. (2012). Evolution and the growth process: Natural selection of entrepreneurial traits. Journal of Economic Theory, 147(2), 759–780.
Gershman, B. (2012). Economic development, institutions, and culture through the lens of envy. PhD dissertation, Brown University.
Gershman, B. (2014). The economic origins of the evil eye belief. Working Paper, American University.
Graham, C. (2010). Happiness around the world: The paradox of happy peasants and miserable millionaires. Oxford: Oxford University Press.
Grossman, H. I., & Kim, M. (1995). Swords or plowshares? A theory of the security of claims to property. Journal of Political Economy, 103(6), 1275–1288.
Grossman, H. I., & Kim, M. (1996). Predation and production, Chapter 4. In M. R. Garfinkel & S. Skaperdas (Eds.), The political economy of conflict and appropriation (pp. 57–72). Cambridge, MA: Cambridge University Press.
Hopkins, E. (2008). Inequality, happiness and relative concerns: What actually is their relationship? Journal of Economic Inequality, 6(4), 351–372.
Hopkins, E., & Kornienko, T. (2004). Running to keep in the same place: Consumer choice as a game of status. American Economic Review, 94(4), 1085–1107.
Knell, M. (1999). Social comparisons, inequality, and growth. Journal of Institutional and Theoretical Economics, 155(4), 664–695.
Ljungqvist, L., & Uhlig, H. (2000). Tax policy and aggregate demand management under catching up with the Joneses. American Economic Review, 90(3), 356–366.
Luttmer, E. F. (2005). Neighbors as negatives: Relative earnings and wellbeing. Quarterly Journal of Economics, 120(3), 963–1002.
Matt, S. J. (2003). Keeping up with the Joneses: Envy in American consumer society, 1890–1930. Philadelphia: University of Pennsylvania Press.
Mitsopoulos, M. (2009). Envy, institutions and growth. Bulletin of Economic Research, 61(3), 201–222.
Moav, O., & Neeman, Z. (2012). Saving rates and poverty: The role of conspicuous consumption and human capital. Economic Journal, 122(563), 933–956.
Mui, V.L. (1995). The economics of envy. Journal of Economic Behavior & Organization, 26(3), 311–336.
Nash, J. (1970). In the eyes of the ancestors: Belief and behavior in a Maya community. New Haven: Yale University Press.
Neumark, D., & Postlewaite, A. (1998). Relative income concerns and the rise in married women’s employment. Journal of Public Economics, 70(1), 157–183.
Oswald, A. J. (1983). Altruism, jealousy and the theory of optimal nonlinear taxation. Journal of Public Economics, 20(1), 77–87.
Park, Y. (2010). The second paycheck to keep up with the Joneses: Relative income concerns and labor market decisions of married women. Eastern Economic Journal, 36(2), 255–276.
PérezAsenjo, E. (2011). If happiness is relative, against whom do we compare ourselves? Implications for labour supply. Journal of Population Economics, 24(4), 1411–1442.
Platteau, J.P. (2000). Institutions, social norms, and economic development. Amsterdam: Harwood Academic Publishers.
Robson, A. J., & Samuelson, L. (2011). The evolutionary foundations of preferences, Chapter 7. In J. Benhabib, A. Bisin, & M. Jackson (Eds.), Handbook of social economics (Vol. 1A, pp. 221–310). Amsterdam: Elsevier.
Rustichini, A. (2008). Dominance and competition. Journal of the European Economic Association, 6(2–3), 647–656.
Schoeck, H. (1969). Envy: A theory of social behavior. New York: Harcourt, Brace, and World.
Schor, J. B. (1991). The overworked American: The unexpected decline of leisure. New York: Basic Books.
Scott, J. C. (1976). The moral economy of the peasant: Rebellion and subsistence in Southeast Asia. New Haven, CT: Yale University Press.
Smith, R. H., & Kim, S. H. (2007). Comprehending envy. Psychological Bulletin, 133(1), 46–64.
Solnick, S. J., & Hemenway, D. (2005). Are positional concerns stronger in some domains than in others? American Economic Review Papers and Proceedings, 95(2), 147–151.
van de Ven, N., Zeelenberg, M., & Pieters, R. (2009). Leveling up and down: The experiences of benign and malicious envy. Emotion, 9(3), 419–429.
Veblen, T. B. (1891). Some neglected points in the theory of socialism. Annals of the American Academy of Political and Social Science, 2(3), 57–74.
Wolf, E. R. (1955). Types of Latin American peasantry: A preliminary discussion. American Anthropologist, 57(3, Part 1), 452–471.
Zizzo, D. J. (2003). Money burning and rank egalitarianism with random dictators. Economics Letters, 81(2), 263–266.
Zizzo, D. J. (2008). The cognitive and behavioral economics of envy, Chapter 11. In R. H. Smith (Ed.), Envy: Theory and research (pp. 190–210). Oxford: Oxford University Press.
Acknowledgments
I am grateful to the Editor, Oded Galor, and two anonymous referees for their advice. Quamrul Ashraf, Pedro Dal Bó, CarlJohan Dalgaard, Geoffroy de Clippel, Peter Howitt, Mark Koyama, Nippe Lagerlöf, Ross Levine, Glenn Loury, Stelios Michalopoulos, Michael Ostrovsky, JeanPhilippe Platteau, Louis Putterman, Eytan Sheshinski, Enrico Spolaore, Ilya Strebulaev, Holger Strulik, David Weil, and Peyton Young provided valuable comments. I also thank seminar and conference participants at American University, Brown University, George Mason University, Gettysburg College, Higher School of Economics, Fall 2011 Midwest economic theory meetings at Vanderbilt University, Moscow State University, 2011 NEUDC conference at Yale University, 2013 ASREC conference in Arlington, 2013 EEAESEM congress in Gothenburg, New Economic School, SEA 80th annual conference in Atlanta, University of Copenhagen, University of Namur, University of Oxford, University of South Carolina, and Williams College.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendix
Appendix
Detailed Form of Lemma 2 The detailed version of Eq. (12) is:

1.
If \(AK_2\geqslant \widehat{A}_{21}\), then
$$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{11}; \\ \theta C_1+(AK_2)^{1/\sigma }, &{} \hbox {if} \quad \widehat{C}_{12}\leqslant C_1<\widehat{C}_{11}; \\ C_1\cdot \frac{1}{\tau \theta }, &{} \hbox {if} \quad \widehat{C}_{13}\leqslant C_1<\widehat{C}_{12}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{13}, \end{array}\right. } \end{aligned}$$where
$$\begin{aligned} \widehat{C}_{11}\equiv \frac{AK_2(AK_2)^{1/\sigma }}{\theta },\! \quad \widehat{C}_{12}\equiv \frac{\tau \theta (AK_2)^{1/\sigma }}{1\tau \theta ^2}, \quad \!\widehat{C}_{13}\equiv \frac{\tau \theta }{1\tau \theta ^2}\left( \frac{1+\theta ^2}{2}AK_2\right) ^{1/\sigma }, \end{aligned}$$and \(C_2^d(C_1)\) is implicitly given by
$$\begin{aligned} C_2\theta C_1=\phi \cdot \left( \frac{C_1+\theta C_2}{C_2}\right) ^{1/\sigma }, \quad \phi \equiv \left( \frac{1+\theta ^2}{2\theta (1+\tau )}AK_2\right) ^{1/\sigma }. \end{aligned}$$(26) 
2.
If \(AK_2\in [\widehat{A}_{22},\widehat{A}_{21})\), then
$$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{14}; \\ C_1\cdot \frac{1}{\tau \theta }, &{} \hbox {if} \quad \widehat{C}_{13}\leqslant C_1<\widehat{C}_{14}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{13}, \end{array}\right. } \end{aligned}$$where \(\widehat{C}_{14}\equiv \tau \theta AK_2\).

3.
If \(AK_2\in (\widehat{A}_{23}, \widehat{A}_{22})\), then
$$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{14}; \\ \widetilde{C}_2^{d}(C_1), &{} \hbox {if} \quad \widehat{C}_{15}\leqslant C_1<\widehat{C}_{14}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{15}, \end{array}\right. } \end{aligned}$$where \(\widetilde{C}_2^{d}(C_1)\) is implicitly given by
$$\begin{aligned} C_1=\theta C_2\cdot \left( \frac{1+\tau }{AK_2}\cdot C_21\right) \end{aligned}$$and \(\widehat{C}_{15}\) solves \(C_2^d(\widehat{C}_{15})=\widetilde{C}_2^d(\widehat{C}_{15})\).
The threshold \(\widehat{C}_1\) from Lemma 2 is defined as \(\widehat{C}_1\equiv \min \{\widehat{C}_{13},\hat{C}_{14}\}\).
Detailed Form of Lemma 3 The detailed version of Eq. (15) is:

1.
If \(AK_1\geqslant \widehat{A}_{11}\), then
$$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{22}; \\ C_1^d(C_2), &{} \hbox {if} \quad \widehat{C}_{21}\leqslant C_2<\widehat{C}_{22}; \\ \theta C_2+(AK_1)^{1/\sigma }, &{} \hbox {if} \quad C_2<\widehat{C}_{21}, \end{array}\right. } \end{aligned}$$where
$$\begin{aligned} \widetilde{C}_1^d(C_2)\equiv \frac{1+\tau }{\tau }AK_1\theta C_2, \quad \widehat{C}_{21}\equiv \frac{(AK_1)^{1/\sigma }}{\theta (\tau 1)}, \end{aligned}$$the function \(C_1^d(C_2)\) is implicitly given by
$$\begin{aligned} C_1\theta C_2=\psi \cdot \left( \frac{C_1}{C_1+\theta C_2}\right) ^{1/\sigma }, \quad \psi \equiv \left( \frac{1+\tau }{\tau }AK_1\right) ^{1/\sigma }, \end{aligned}$$(27)and \(\widehat{C}_{22}\) solves \(C_1^d(\widehat{C}_{22})=\widetilde{C}_1^d(\widehat{C}_{22})\).

2.
If \(AK_1\in [\widehat{A}_{12},\widehat{A}_{11})\), then
$$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{24}; \\ AK_1, &{} \hbox {if} \quad \widehat{C}_{23}\leqslant C_2<\widehat{C}_{24}; \\ \theta C_2+(AK_1)^{1/\sigma }, &{} \hbox {if} \quad C_2<\widehat{C}_{23}, \end{array}\right. } \end{aligned}$$where
$$\begin{aligned} \widehat{C}_{23}\equiv \frac{AK_1(AK_1)^{1/\sigma }}{\theta }, \quad \widehat{C}_{24}\equiv \frac{AK_1}{\tau \theta }. \end{aligned}$$ 
3.
If \(AK_1<\widehat{A}_{12}\), then
$$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{24}; \\ AK_1, &{} \hbox {if} \quad C_2< \widehat{C}_{24}. \end{array}\right. } \end{aligned}$$
The threshold \(\widehat{C}_2\) from Lemma 3 is defined as \(\widehat{C}_2\equiv \min \{\widehat{C}_{21},\hat{C}_{24}\}\).
Detailed Form of Proposition 1 The unique subgame perfect equilibrium \((C_1^{*},C_2^{*})\) of the envy game is determined as follows.

1.
If \(AK_1\geqslant \widehat{A}_{11}\) and \(AK_2\geqslant \widehat{A}_{21}\), then:

(a)
If \(k\geqslant \tilde{k}\), it is the KUJ equilibrium (16);

(b)
If \(\hat{k}\leqslant k<\tilde{k}\), it is the fear equilibrium (19);

(c)
If \(k<\hat{k}\), it is the destructive equilibrium implicitly defined by
$$\begin{aligned} {\left\{ \begin{array}{ll} C_1^{*}=\min \{C_1^d(C_2^{*}),\widetilde{C}_1^d(C_2^{*})\};\\ C_2^{*}=C_2^d(C_1^{*}). \end{array}\right. } \end{aligned}$$
The threshold values of \(k\) are given by
$$\begin{aligned} \tilde{k}\equiv \left[ \frac{\theta (\tau 1)}{1\tau \theta ^2}\right] ^{\sigma }, \quad \hat{k}\equiv \tilde{k}\cdot \frac{(1+\theta ^2)}{2}. \end{aligned}$$(28) 
(a)

2.
If \(\widehat{A}_{12}\leqslant AK_1<\widehat{A}_{11}\) and \(AK_2\geqslant \widehat{A}_{21}\), then:

(a)
If \(AK_1\geqslant \widehat{C}_{11}\), it is the fulltime KUJ equilibrium (18);

(b)
If \(AK_1<\widehat{C}_{11}\) and \(m_1(AK_1)\leqslant AK_2<m_2(AK_1)\), it is the KUJ equilibrium (17), where
$$\begin{aligned} m_1(AK_1)\equiv \left[ \frac{(1\theta ^2)AK_1(AK_1)^{1/\sigma }}{\theta }\right] ^{\sigma }, \quad m_2(AK_1)\equiv \left[ \frac{(1\tau \theta ^2)AK_1}{\tau \theta }\right] ^{\sigma }; \end{aligned}$$ 
(c)
If \(AK_2<m_1(AK_1)\), it is the KUJ equilibrium (16);

(d)
If \(m_2(AK_1)\leqslant AK_2<2m_2(AK_1)/(1+\theta ^2)\), it is the fear equilibrium (20);

(e)
If \(AK_2\geqslant 2m_2(AK_1)/(1+\theta ^2)\), it is the destructive equilibrium in case 2 of (21).

(a)

3.
If \(AK_1<\widehat{A}_{11}\) and \(\widehat{A}_{22}\leqslant AK_2<\widehat{A}_{21}\), then:

4.
If \(AK_1<\widehat{A}_{11}\) and \(\widehat{A}_{23}<AK_2<\widehat{A}_{22}\), then:

(a)
If \(k\geqslant \tau \theta \), it is the fulltime KUJ equilibrium (18);

(b)
If \(k<\tau \theta \), it is the destructive equilibrium implicitly defined by
$$\begin{aligned} {\left\{ \begin{array}{ll} C_1^{*}=\widetilde{C}_1^d(C_2^{*});\\ C_2^{*}=\min \{C_2^d(C_1^{*}),\widetilde{C}_2^d(C_1^{*})\}. \end{array}\right. } \end{aligned}$$

(a)
Rights and permissions
About this article
Cite this article
Gershman, B. The two sides of envy. J Econ Growth 19, 407–438 (2014). https://doi.org/10.1007/s1088701491068
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1088701491068
Keywords
 Culture
 Economic growth
 Envy
 Inequality
 Institutions
 Redistribution
JEL Classification
 D31
 D62
 D74
 O15
 O43
 Z13