The Pond Dilemma with Heterogeneous Relative Concerns

This paper explores team formation when workers differ in skills and their desire to out-earn co-workers. I cast this question as a two-dimensional assignment problem with imperfectly transferable utility and show that equilibrium sorting optimally trades off output maximisation with the need to match high-skill workers to co-workers with weak relative concerns. This can lead to positive (negative) assortative matching in skill even with submodular (supermodular) production functions. Under supermodular production, this heterogeneity in preferences benefits all workers and reduces wage inequality. With submodular production, the distributional consequences are ambiguous, and some workers become worse off. The model reveals that skill-biased technological change (SBTC) incentivises domestic outsourcing, as firms seek to avoid detrimental social comparisons between high- and low-skill workers, thus providing a compelling explanation for the long-term increase in outsourcing. Finally, the benefits of SBTC can trickle down to low-skill workers-but only those whose relative concerns are weak.

💡 Research Summary

The paper studies how teams are formed when workers differ not only in skill but also in the intensity of their relative‑earnings concerns. The author models the problem as a one‑sided, two‑dimensional assignment with imperfectly transferable utility (ITU). Each worker’s utility is a weighted sum of his wage and the wage gap with his teammate; the weight (the “relative‑concern parameter”) varies across individuals. Because these weights differ, a wage increase for one worker does not translate one‑for‑one into total utility, creating ITU. Nevertheless, the model admits a transferable‑utility representation: by rescaling each individual’s utility, a surplus function can be defined whose sum across the two partners is perfectly transferable. Equilibrium matching maximises aggregate surplus, not necessarily output.

Two key assumptions drive the analysis. (1) The surplus is super‑modular in a worker’s skill and a “skill‑preference index” that combines the partner’s skill and relative‑concern intensity. This implies a planner would like to pair high‑skill workers with partners who have high values of the index (i.e., weak relative concerns). (2) The joint distribution of skill and the index is symmetric (the copula is symmetric). This symmetry guarantees that the optimal matching is feasible and that equilibrium sorting exists.

Under these conditions the paper derives a full characterisation of equilibrium sorting, wages and payoffs for a broad class of production functions (additive, multiplicative, binary‑skill, log‑elliptical distributions). The equilibrium balances two forces: (i) output maximisation, which in standard matching models leads to positive assortative matching when high‑skill workers are complements (sub‑modular production) and negative assortative matching when they are substitutes (super‑modular production); and (ii) welfare from social comparisons, which pushes high‑skill workers toward partners with weak relative concerns to minimise the cost they impose on teammates. When production is additive, output does not depend on sorting, so the second force dominates and sorting between skill and partner’s relative‑concern intensity is negative and assortative, confirming Frank’s (1984) conjecture. More generally, the strength and direction of sorting in the skill dimension depend jointly on the production technology and the joint distribution of traits, making it possible for high‑skill workers to be complements yet still be matched negatively (or vice‑versa).

The model yields several novel implications for wage inequality. Because a high‑skill worker matched with a low‑skill partner must compensate the partner for lower status, low‑skill workers can earn more than in a self‑matching benchmark. The magnitude of this “status compensation” rises with the productivity gap between skill types, implying a potential trickle‑down effect: productivity gains for high‑skill workers can raise low‑skill wages when they are paired. Consequently, relative‑concern heterogeneity can reduce within‑firm wage inequality (since compensation smooths gaps) but may increase between‑firm inequality, potentially raising overall inequality if high‑skill workers have weak concerns while low‑skill workers have strong concerns. This mirrors results from inequity‑aversion models but arises here from heterogeneous status concerns.

A further contribution is a theory of firm boundaries based on social comparison costs. High‑skill workers with strong relative concerns prefer to keep low‑skill teammates in‑house (to avoid costly outsourcing), whereas high‑skill workers with weak concerns find it optimal to outsource low‑skill partners, thereby eliminating the comparison externality. The paper shows that skill‑biased technological change (SBTC), by expanding the wage gap between skill groups, makes outsourcing more attractive precisely when the high‑skill worker’s relative concerns are weak. This mechanism offers a fresh explanation for the long‑run rise in domestic outsourcing and predicts that SBTC‑driven productivity gains can “trickle down” only to low‑skill workers whose relative concerns are weak.

The author situates the work within three strands of literature: (i) sorting with multidimensional traits under transferable utility, (ii) models of inequity aversion or status concerns, and (iii) the theory of the firm based on transaction‑cost and property‑right arguments. The paper extends each by allowing a social‑preference dimension, providing closed‑form solutions for non‑Gaussian trait distributions, and analysing the impact of production technology on sorting, inequality and outsourcing.

In sum, the paper demonstrates that heterogeneous relative‑concern intensities fundamentally reshape team formation, wage dispersion, and firm‑boundary decisions. It highlights that policies addressing skill‑biased technological change should also consider workers’ social comparison preferences, as ignoring them may mis‑estimate the distributional consequences of productivity growth and the incentives for outsourcing.