Peng (2021) explains both the Great Gatsby curve (the cross-country positive association between income inequality and intergenerational income persistence) and the Easterlin paradox (average happiness failing to rise with average income over time) through a single mechanism: positional competition, in which productive opportunities are allocated by relative performance and whose intensity rises endogenously as opportunities become more dispersed. In that model, however, the additive form of the utility function makes each generation's bequest a fixed fraction of its income, so wealth is proportional to income by construction and the model cannot speak to the wealth distribution, wealth-to-income ratios, or wealth mobility. The author explicitly leaves open the task of separating wealth from income. The open problem is to construct and solve an extension of the positional-competition model in which wealth is an endogenous state variable distinct from income, to characterize the stationary joint distribution of wealth and income, and to determine how the dispersion of productive opportunities shapes wealth inequality, the wealth-income correlation, the wealth-to-income ratio, and the comparison between income mobility and wealth mobility.

Background

Two empirical regularities motivate the source model. The Great Gatsby curve is the observed positive cross-sectional relationship between income inequality and intergenerational persistence of income (children's incomes echoing parents' incomes) across countries and regions; the Easterlin paradox is the observation that, although richer people report higher well-being at a point in time, average well-being in a population does not trend upward as average income grows over time. Peng (2021) proposes a single mechanism for both. The economy consists of a continuum of one-period-lived family dynasties. Each agent draws talent zz from a Pareto distribution (a power-law distribution often used for right-skewed quantities such as income, parameterized by a scale and a tail exponent) and receives a bequest bb from the previous generation. Productive opportunities have quality AA, also Pareto distributed, with dispersion parameter θ=1/βA\theta = 1/\beta_A: larger θ\theta means a heavier tail, i.e. more 'superstar' opportunities concentrated at the top. The source restricts θ\theta to 0<θ<10 < \theta < 1 so that the expected value of AA is finite, and its Assumption 1 requires θ<(βz−1)/βz\theta < (\beta_z - 1)/\beta_z (equivalently βAβz−βA−βz>0\beta_A\beta_z - \beta_A - \beta_z > 0) so that total output remains bounded even when the allocation of talent is perfect; its short-run result further divides the admissible parameter space into three cases according to whether the competition intensity γ\gamma and the income exponent βy\beta_y are determined by βz\beta_z or by the bequest exponent βb\beta_b. Opportunities are allocated by an economy-wide contest: performance is p=k zp = k\,z, where kk is effort (contest expenditure) chosen optimally, and agents are matched to opportunities by performance rank, so an agent's income is y=A(ρP) zy = A(\rho_P)\,z with ρP\rho_P the agent's performance rank. The disutility of effort is χ(k,b)=k/b\chi(k,b) = k/b, following Loury (1981): a larger inherited bequest lowers the cost of effort, which is the channel of intergenerational transmission of advantage. Preferences over consumption cc and a warm-glow bequest to the next generation (a bequest valued for its own sake by the giver rather than derived from the recipient's utility) are U=cm+bt+1m−k/bU = c^m + b_{t+1}^m - k/b with 0<m≤10 < m \le 1. Because utility is additively separable, the consumption-savings stage has the closed-form solution c=bt+1=y/2c = b_{t+1} = y/2: bequests are exactly half of income. The effort stage is solved as a Nash equilibrium in which each agent optimizes against the (endogenous) distribution of performances; the equilibrium intensity of positional competition is captured by a parameter γ\gamma that rises with θ\theta. In the long-run (stationary) equilibrium — a time-invariant distribution of bequests that the dynastic dynamics reproduce period after period — the distribution of bequests is endogenously determined, income follows a power law with exponent βy\beta_y decreasing in θ\theta, and intergenerational mobility — measured by the copula, i.e. the joint distribution of parent and child income ranks with the marginals stripped away — falls as θ\theta rises (the Great Gatsby curve, Theorem 1). When the dispersion θ\theta grows along with productivity, the intensifying competition drives a wedge between the cross-sectional and time-series slopes of utility on income, reproducing the Easterlin paradox (Theorem 2). The structural limitation is that wealth never becomes an independent object: bequests are proportional to income each period, so the model identifies the wealth distribution with the income distribution and can say nothing about wealth-to-income ratios, the wealth-income correlation, or how income mobility differs from wealth mobility. A separate literature derives endogenous wealth distributions from bequest motives and idiosyncratic returns on savings (e.g. Benhabib, Bisin and Zhu 2011), but it contains no contest-based allocation of opportunities and no endogenous competition intensity. The author states in both a footnote and the conclusion that separating wealth from income — endogenizing the distribution of wealth alongside income, which would be required to model bequests as inter vivos transfers (transfers made during the parent's lifetime rather than at death) that relax the child's constraints — is beyond the scope of the paper and is left for future research.

Problem Statement

Extend Peng's (2021) positional-competition framework to an economy in which wealth is an endogenous state variable distinct from income, while retaining the model's defining features: allocation of productive opportunities by relative performance in an economy-wide contest, effort cost falling in inherited resources, and the endogenously determined intensity of positional competition. In any such extension the wealth state must be coupled to the intergenerational transmission channel rather than evolving independently of the contest: inherited wealth, and not only the proportional bequest bt+1=y/2b_{t+1} = y/2, must enter the child's effort cost χ(k,⋅)\chi(k,\cdot) and/or the child's income determination, so that appending a wealth stock that appears in no other equation of the model does not constitute the requested extension. Within such an extended model, derive the stationary joint distribution of wealth and income (its marginal distributions and tails, the wealth-to-income ratio, and the rank dependence between wealth and income across households), and establish how the dispersion of productive opportunities θ\theta — over the admissible range inherited from the source, 0<θ<10 < \theta < 1 together with Assumption 1's bound θ<(βz−1)/βz\theta < (\beta_z - 1)/\beta_z, plus any further admissibility conditions imposed by the wealth technology — and the induced intensity of positional competition affect (i) wealth inequality, (ii) the wealth-income correlation and the wealth-to-income ratio, and (iii) the comparison between income mobility and wealth mobility. The effects on (i), (ii), and (iii) are to be derived from the extended model rather than assumed. As a headline restatement of targets (i), (ii), and (iii): does the positional-competition mechanism that generates the Great Gatsby curve and the Easterlin paradox for income extend to wealth — that is, do more dispersed opportunities raise or lower wealth inequality, raise or depress wealth mobility, and is there a systematic wedge between the wealth and income distributions?

References

  1. Baochun Peng (2021), 'Positional competition: A theory of the Great Gatsby curve and the Easterlin paradox', Journal of Economic Behavior & Organization 186, 562-575. DOI: 10.1016/j.jebo.2021.04.003, https://doi.org/10.1016/j.jebo.2021.04.003
  2. Steven N. Durlauf, Andros Kourtellos, Chih Ming Tan (2022), 'The Great Gatsby Curve', Annual Review of Economics 14, 571-605. DOI: 10.1146/annurev-economics-082321-122703, https://doi.org/10.1146/annurev-economics-082321-122703
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