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Fearing the worst: the importance of uncertainty for inequality

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Abstract

We present an overlapping generations model in which aspirational agents face uncertainty about the returns to human capital investment. This uncertainty implies the prospect that aspirations will not be fulfilled, the probability of which is greater the lower is the human capital endowment of an agent. We show that agents with sufficiently low human capital endowments may experience such a strong influence of loss aversion that they abstain from human capital investment. We further show how this behaviour may be transmitted through successive generations to cause initial inequalities to persist. These results do not rely on any credit market imperfections, though they may appear as if they do.

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Notes

  1. For the purposes of this paper, we use the terms risk and uncertainty interchangeably, and do not distinguish between the two.

  2. This research developed alongside other work on income distribution that signalled a general revival of interest in the subject. Amongst this work are models of redistribution based on political motives (e.g. Alesina and Rodrik 1994; Persson and Tabellini 1994; Perotti 1993) and models of inequality based on neighbourhood effects (e.g. Benabou 1992; Durlauf 1993; Fernandez and Rogerson 1996).

  3. This has become the dominant descriptive theory of decision-making under uncertainty. It is part of the broader literature on non-expected utility, a comprehensive review of which can be found in Starmer (2000).

  4. A selection of alternative functional forms can be found in Maggi (2004).

  5. Two recent macroeconomic applications of prospect theory are presented by Foellmi et al. (2011) and Rosenblatt-Wisch (2008). The focus of each of these is on the aggregate implications of loss aversion in stochastic growth models.

  6. For a broad comparison of aspiration level models and prospect theory, see Lopes and Oden (1999). In a wider context, we note that aspiration-based preferences bear some similarity to, but are distinct from, maxi–min-type preferences. Unlike the former, the latter do not generate discontinuities and do not involve reference dependence. Compared to loss aversion, maxi–min preferences may similarly give rise to kinks, but these kinks would be observed in the indifference curves, rather than utility. The extent to which our analysis (based on aspiration theory) is transferable to other behavioural models of uncertainty is an interesting issue for further research.

  7. Note that this argument does not necessarily mean that poor agents save more in total than rich agents. The argument refers specifically to precautionary savings, the motive for which intensifies as wealth declines because of the greater inability to buffer one’s consumption against bad shocks (e.g. Carroll 2001). This is not inconsistent with total savings being an increasing function of wealth.

  8. This is an investment of physical resources, rather than time or effort. The latter may be treated as being already subsumed into the behaviour of agents, who may be thought of as devoting a fixed amount of time or effort to human capital production when they are young.

  9. If the concept of human capital is confined to schooling and education, then the target outcome, \(x^{*}\), may be thought of as reflecting an underlying target for academic achievement to which individuals aspire. For example, \(x_{t+1}\ge x^{*}\) may correspond to the case in which a student graduates successfully, whilst \(x_{t+1}<x^{*}\) may represent the case in which a student barely passes (or perhaps drops out of college altogether).

  10. Having said this, we note that our approach is well motivated by our earlier discussion about the various risks associated with human capital investment. The formulation in (2) can be likened to the stochastic human capital production technologies used by Grossman (2008) and Krebs (2003), to whom we referred in our discussion. The main focus of these authors is on the role of human capital risk in influencing growth, though the former establishes this role with reference to initial wealth inequalities. Our own focus is centred primarily on distribution in an environment with initial human capital inequalities.

  11. This can be ensured by assuming that \(A[\beta h_{0}^{L}+b(1-c)]>x^{*}\), where \(h_{0}^{L}\) denotes the lowest initial human capital endowment amongst agents.

  12. Note that the effect of \(c\) is positive because \(\widehat{\gamma }_{t+1}<0\) in (8). This follows from the fact that \(A(\beta h_{t}+B)-(1+r)k-x^{*}>0\) must hold if human capital investment is ever to be chosen (otherwise, the expected income from this investment would be less than target income).

  13. For example, the restriction rules out the possibility that all agents automatically end up either investing or not investing in human capital, and also means that any lineage that chooses to invest at some point will never alter its choice subsequently.

  14. To be sure, observe from (13) that the first integral term on the right-hand side of (15) is equal to \(\int \nolimits _{-c}^{\overline{\gamma }_{t+1}}(1+R_{t+1})kf(\gamma _{t+1})\mathrm{d}\gamma _{t+1}\) which measures the expected amount of non-repayment when defaulting occurs. Conversely, the second integral term on the right-hand side of (15) gives the expected amount of income that is seized from a defaulter, net of enforcement costs.

  15. Details of the derivations can be found in an “Appendix”.

  16. Verification of these results is again contained in the Appendix. Note that one does not need to assume that financial intermediaries are able to observe human capital directly. As shown in (17) and (18), \(\overline{\gamma }_{t+1}\) and \(R_{t+1}\) are functions of \(Ah_{t}\), which is the output produced by parents. One needs only to assume that this is observable (in which case, of course, \(h_{t}\) can be trivially inferred anyway).

  17. For example, there is a large body of research that shows how uncertainty (or volatility) can influence long-term growth (either positively or negatively) through various factors (e.g. Aghion and Saint-Paul 1998; Blackburn and Varvarigos 2008; de Hek 1999; Jones et al. 2005; Martin and Rogers 2000). Whilst we do not consider long-term growth, our analysis identifies another factor that can create a link between uncertainty and macroeconomic performance—the impact of uncertainty on distribution outcomes.

  18. From (29), the first integral term on the right-hand side of (31) is equal to \(\int \nolimits _{-c}^{\widehat{\gamma }_{t+1}}(1+R_{t+1})kf(\gamma _{t+1})\mathrm{d}\gamma _{t+1}\), which is the expected amount of non-repayment when bankruptcy is declared. Conversely, the second integral term on the right-hand side of (31) is the expected amount of income that is claimed in the case of bankruptcy.

  19. The solution can be obtained directly from (17) and (18) by setting \(\delta =1\) and \(e=0\).

  20. These results, and the intuition for them, are the same as those obtained in our model of strategic defaulting.

  21. See Krasa and Villamil (2000) for conditions under which debt is optimal in a costly state verification model.

References

  • Aghion, P., Bolton, P.: A theory of trickle-down growth and development. Rev. Econ. Stud. 64, 151–172 (1997)

    Article  Google Scholar 

  • Aghion, P., Saint-Paul, G.: On the virtue of bad times: an analysis of the interaction between economic fluctuations and productivity growth. Macroecon. Dyn. 2, 322–344 (1998)

    Google Scholar 

  • Aiyagar, S.R.: Uninsured idiosyncratic risk and aggregate savings. Q. J. Econ. 109, 73–92 (1994)

    Google Scholar 

  • Alesina, A., Rodrik, D.: Distributive politics and economic growth. Q. J. Econ. 109, 465–490 (1994)

    Article  Google Scholar 

  • Banerjee, A.V., Newman, A.F.: Occupational choice and the process of development. J. Polit. Econ. 101, 274–298 (1993)

    Article  Google Scholar 

  • Benabou, R.: Heterogeneity, Stratification and Growth. MIT, Mimeo (1992)

    Google Scholar 

  • Benartzi, S., Thaler, R.H.: Myopic loss aversion and the equity premium puzzle. Q. J. Econ. 110, 73–92 (1995)

    Article  Google Scholar 

  • Blackburn, K., Bose, N.: A model of trickle-down through learning. J. Econ. Dyn. Control 27, 445–466 (2003)

    Article  Google Scholar 

  • Blackburn, K., Varvarigos, D.: Human capital accumulation and output growth in a stochastic environment. Econ. Theor. 36, 435–452 (2008)

    Article  Google Scholar 

  • Bleichrodt, H., Schmidt, U., Zank, H.: Loss aversion in multi-attribute utility theory. Manag. Sci. 55, 863–873 (2009)

    Article  Google Scholar 

  • Carroll, C.D., Kimball, M.S.: On the concavity of the consumption function. Econometrica 64, 981–992 (1996)

    Article  Google Scholar 

  • Carroll, C.D.: A theory of the consumption function, with and without liquidity constraints. J. Econ. Perspect. 15, 23–45 (2001)

    Article  Google Scholar 

  • de Hek, P.A.: On endogenous growth under uncertainty. Int. Econ. Rev. 40, 727–744 (1999)

    Article  Google Scholar 

  • Diamond, D.W.: Financial intermediation as delegated monitoring. Rev. Econ. Stud. 51, 393–414 (1984)

    Article  Google Scholar 

  • Diecidue, E., Van de Ven, J.: Aspiration level, probability of success and failure, and expected utility. Int. Econ. Rev. 49, 683–700 (2008)

    Article  Google Scholar 

  • Durlauf, S.: Neighbourhood Feedbacks, Endogenous Stratification and Income Inequality. University of Wisconsin, Mimeo (1993)

  • Edwards, W.: The theory of decision making. Psychol. Bull. 51, 380–417 (1954)

    Article  Google Scholar 

  • Fernandez, R., Rogerson, R.: Income distribution, communities and the quality of public education. Q. J. Econ. 111, 135–164 (1996)

    Article  Google Scholar 

  • Foellmi, R., Rosenblatt-Wisch, R., Schenke-Hoppe, K.R.: Consumption paths under prospect utility in an optimal growth model. J. Econ. Dyn. Control 35, 273–281 (2011)

    Article  Google Scholar 

  • Gale, D., Hellwig, M.: Incentive compatible debt contracts: the one period problem. Rev. Econ. Stud. 52, 647–663 (1985)

    Article  Google Scholar 

  • Galor, O., Zeira, J.: Income distribution and macroeconomics. Rev. Econ. Stud. 60, 35–52 (1993)

    Article  Google Scholar 

  • Genicot, G., Ray, D.: Aspirations and inequality. NBER Working Paper 19976 (2014)

  • Grossman, V.: Risky human capital investment, income distribution and macroeconomic dynamics. J. Macroecon. 30, 19–42 (2008)

    Article  Google Scholar 

  • Guiso, L., Jappelli, T., Terlizzese, D.: Income risk, borrowing constraints and portfolio choice. Am. Econ. Rev. 86, 158–172 (1996)

    Google Scholar 

  • Guiso, L., Paiella, M.: Risk aversion, wealth and background risk. J. Eur. Econ. Assoc. 6, 1109–1150 (2008)

    Article  Google Scholar 

  • Holthausen, D.M.: A risk-return model with risk and return measured as deviations from a target return. Am. Econ. Rev. 71, 182–188 (1981)

    Google Scholar 

  • Jones, L.E., Manuelli, R.E., Siu, H.E., Stacchetti, E.: Fluctuations in convex models of endogenous growth I: growth effects. Rev. Econ. Dyn. 8, 780–804 (2005)

    Article  Google Scholar 

  • Kahnemen, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979)

    Article  Google Scholar 

  • Krasa, S., Villamil, A.: Optimal contracts when enforcement is a decision variable. Econometrica 68, 119–134 (2000)

    Article  Google Scholar 

  • Krebs, T.: Human capital risk and economic growth. Q. J. Econ. 118, 709–744 (2003)

    Article  Google Scholar 

  • Langer, T., Weber, M.: Prospect theory, mental accounting and differences in aggregated and segregated evaluation of lottery portfolios. Manag. Sci. 47, 716–733 (2001)

    Article  Google Scholar 

  • Lopes, L.L., Oden, C.: The role of aspiration level in risky choice: a comparison of cumulative prospect theory and SP/A theory. J. Math. Psychol. 43, 286–313 (1999)

    Article  Google Scholar 

  • Maggi, M.A.: A Characterisation of S-Shaped Utility Functions Displaying Loss Aversion. Università degli Studi di Pavia, Mimeo (2004)

    Google Scholar 

  • Martin, P., Rogers, C.A.: Long-term growth and short-term economic instability. Eur. Econ. Rev. 44, 359–381 (2000)

    Article  Google Scholar 

  • Mezias, S.J.: Aspiration level effects: an empirical investigation. J. Econ. Behav. Organ. 10, 389–400 (1988)

    Article  Google Scholar 

  • Mezias, S.J., Chen, Y.R., Murphy, P.R.: Aspiration level adaptation in an American Financial Services Organisation: a field study. Manag. Sci. 48, 1285–1300 (2002)

    Article  Google Scholar 

  • Payne, J.W.: It is whether you win or lose: the importance of the overall probabilities of winning or losing in risky choice. J. Risk Uncertain. 30, 5–19 (2005)

    Article  Google Scholar 

  • Perotti, R.: Political equilibrium, income distribution and growth. Rev. Econ. Stud. 60, 755–776 (1993)

    Article  Google Scholar 

  • Persson, T., Tabellini, G.: Is inequality harmful for growth? Am. Econ. Rev. 84, 600–621 (1994)

    Google Scholar 

  • Piketty, T.: The dynamics of wealth distribution and the interest rate with credit rationing. Rev. Econ. Stud. 64, 173–189 (1997)

    Article  Google Scholar 

  • Rabin, M.: Risk aversion and expected utility theory: a calibration theorem. Econometrica 68, 1281–1292 (2000)

    Article  Google Scholar 

  • Rabin, M., Thaler, R.H.: Anomolies: risk aversion. J. Econ. Perspect. 15, 219–232 (2001)

    Article  Google Scholar 

  • Rosenblatt-Wisch, R.: Loss aversion in aggregate macroeconomic time series. Eur. Econ. Rev. 52, 1140–1159 (2008)

    Article  Google Scholar 

  • Samuelson, W.F., Zeckhauser, R.J.: Status quo bias in decision making. J. Risk Uncertain. 1, 7–59 (1988)

    Article  Google Scholar 

  • Schmidt, U., Zank, H.: What is loss aversion? J. Risk Uncertain. 30, 157–167 (2005)

    Article  Google Scholar 

  • Starmer, C.: Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. J. Econ. Lit. 38, 332–382 (2000)

    Article  Google Scholar 

  • Thaler, R.H.: Toward a positive theory of consumer choice. J. Econ. Behav. Organ. 1, 39–60 (1980)

    Article  Google Scholar 

  • Townsend, R.: Optimal contracts and competitive markets with costly state verification. J. Econ. Theory 21, 265–293 (1979)

    Article  Google Scholar 

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Authors and Affiliations

  1. Exeter College, University of Oxford, Oxford, OX1 3DP, UK

    David Chivers

  2. Department of Economics, Centre for Growth and Business Cycles Research, University of Manchester, Manchester, M13 9PL, UK

    Keith Blackburn

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  1. Keith Blackburn
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  2. David Chivers
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Correspondence to David Chivers.

Additional information

The authors are grateful to Paul Madden, Horst Zank and an anonymous referee for helpful comments on an earlier version of the paper.

Appendix

Appendix

The results in (17) and (18) are derived as follows. Combining (13) and (16) yields the quadratic equation

$$\begin{aligned} 0= & {} \delta AB\overline{\gamma }_{t+1}^{2}-2(\delta ABc-e)\overline{\gamma } _{t+1} \nonumber \\&\quad -\,\left\{ 4c[\delta A(\beta h_{t}+B)-(1+r)k]-\delta ABc^{2}-2ec\right\} . \end{aligned}$$
(38)

Hence

$$\begin{aligned} \overline{\gamma }_{t+1}=c-\frac{e}{\delta AB}\pm \frac{\sqrt{4\delta ABc[\delta A(\beta h_{t}+B)-(1+r)k-e]+e^{2}}}{\delta AB}. \end{aligned}$$
(39)

A sufficient condition for ruling out complex roots is \(A(\beta h_{t}+B)\ge (1+r)k+e\). Given this, together with the fact that \(\overline{ \gamma }_{t+1}\le c\), the only possible solution to (39) is when \(\sqrt{ \cdot }\) enters negatively, as shown in (17). The restriction \(\delta A(\beta h_{t}+B)\le (1+r)k+\delta ABc\) ensures that \(\overline{\gamma }_{t+1}\ge -c\) as well. Having obtained (17), the result in (18) is obtained by appropriate substitution in (16).

The properties of the functions \(\gamma (c,e,h_{t})\) and \(R(c,e,h_{t})\) are established as follows. From (17) and (18), one finds that

$$\begin{aligned} \gamma _{c}(\cdot )= & {} 1-\frac{2[\delta A(\beta h_{t}+B)-(1+r)k-e]}{\sqrt{4\delta ABc[\delta A(\beta h_{t}+B)-(1+r)k-e]+e^{2}}}, \end{aligned}$$
(40)
$$\begin{aligned} \gamma _{e}(\cdot )= & {} -\frac{1}{\delta AB}\left[ 1-\frac{(2\delta ABc-e)}{\sqrt{4\delta ABc[\delta A(\beta h_{t}+B)-(1+r)k-e]+e^{2}}}\right] \end{aligned}$$
(41)
$$\begin{aligned} \gamma _{h}(\cdot )= & {} -\frac{2\delta A\beta c}{\sqrt{4\delta ABc[\delta A(\beta h_{t}+B)-(1+r)k-e]+e^{2}}}, \end{aligned}$$
(42)
$$\begin{aligned} R_{c}(\cdot )= & {} \frac{(\gamma (\cdot )+c)}{c^{2}}\left\{ \frac{\delta AB[(c\gamma _{c}(\cdot )+c)+(c\gamma _{c}(\cdot )-c)]}{4k}+\frac{e(c\gamma _{c}(\cdot )-c)}{2}\right\} , \end{aligned}$$
(43)
$$\begin{aligned} R_{e}(\cdot )= & {} \frac{\delta AB(\gamma (\cdot )+c)\gamma _{e}(\cdot )}{2ck}+ \frac{e\gamma _{e}(\cdot )+\gamma (\cdot )+c}{2c} \end{aligned}$$
(44)
$$\begin{aligned} R_{h}(\cdot )= & {} \frac{\delta AB[\gamma (\cdot )+c]\gamma _{h}(\cdot )}{2ck}+ \frac{e\gamma _{h}(\cdot )}{2c}. \end{aligned}$$
(45)

Under the above parameter restrictions, it is deduced that \(\gamma _{c}(\cdot )>0, \gamma _{e}(\cdot )>0, \gamma _{h}(\cdot )<0, R_{c}(\cdot )>0, R_{e}(\cdot )>0\) and \(R_{h}(\cdot )<0\).

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Blackburn, K., Chivers, D. Fearing the worst: the importance of uncertainty for inequality. Econ Theory 60, 345–370 (2015). https://doi.org/10.1007/s00199-015-0876-9

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