The Pond Dilemma with Heterogeneous Relative Concerns

1. Problem Statement

The paper investigates team formation in an economy where workers differ not only in skills ($x_1$) but also in the strength of their relative concerns ($x_2$). Relative concerns refer to the utility workers derive from the difference between their own wage and that of their co-worker.

The core economic problem is how these heterogeneous preferences distort the sorting of workers into teams. In standard assignment models, sorting is driven solely by output maximization (complementarity or substitutability of skills). However, Gola argues that when workers care about status, a match that maximizes output (e.g., pairing a high-skill worker with a low-skill worker) may be unstable or inefficient if the low-skill worker suffers a significant status loss. This creates a trade-off between output maximization and welfare maximization regarding social comparisons.

The paper specifically addresses:

• How heterogeneous relative concerns affect equilibrium sorting patterns.
• The impact on wage inequality and worker welfare.
• The role of these concerns in determining firm boundaries (outsourcing vs. in-house production).
• How Skill-Biased Technological Change (SBTC) interacts with these dynamics to explain trends in outsourcing and inequality.

2. Methodology and Model Setup

The Environment

• Agents: A continuum of workers with types $x = (x_1, x_2)$, where $x_1$ is skill and $x_2$ is the strength of relative concerns.
• Teams: Workers form teams of size two (one-sided matching).
• Production: A symmetric, increasing, twice-differentiable production function $F(x_1^k, x_1^j)$ generates output.
• Utility: Workers maximize a utility function combining their wage ($w_k$) and the wage gap relative to their co-worker ($w_j$):
  $$U(w_k, w_j; x_2) = w_k + x_2(w_k - w_j)$$
  Here, $x_2$ represents the weight on relative concerns. The assumption $x_2 > -0.5$ ensures utility increases with one's own wage.

Key Theoretical Challenge: Imperfectly Transferable Utility (ITU)

Because workers have different $x_2$, utility is imperfectly transferable. An extra dollar of output increases the utility of a worker with high $x_2$ differently than it decreases the utility of a co-worker with low $x_2$.

• Solution: Gola demonstrates that despite ITU, the model admits a Transferable Utility (TU) representation. By rescaling individual utilities by a factor dependent on $x_2$ (specifically $1+2x_2$), the problem can be transformed into a standard TU assignment problem where the surplus is perfectly transferable.
• Surplus Function: The rescaled surplus $\Pi(x^j, x^k)$ is defined as:
  $$\Pi(x^j, x^k) = F(x_1^k, x_1^j) \left[ \frac{1}{1+2x_2^k} + \frac{1}{1+2x_2^j} \right]$$
  Equilibrium sorting maximizes the aggregate surplus of this transformed economy.

Assumptions for Characterization

To derive closed-form solutions and general results, the paper imposes:

1. Assumption 1 (Common Rankings): The surplus function is supermodular in the worker's skill and a "skill-preference index" $v_1(x)$, which aggregates skill and relative concerns.
2. Assumption 2 (Symmetric Copula): The joint distribution of skill and the skill-preference index is symmetric (exchangeable).
3. Assumption 3 (Sufficient Conditions): The model holds for three specific classes:
   • Additive production with exchangeable distributions.
   • Binary skills.
   • Multiplicative production with log-elliptical distributions.

3. Key Results

A. Equilibrium Sorting Patterns

The equilibrium sorting is positive and assortative between a worker's skill ($x_1$) and their co-worker's skill-preference index ($v_1$).

• The Trade-off: High-skill workers are matched with co-workers who have high values of $v_1$. Since $v_1$ is decreasing in $x_2$ (relative concerns), this implies high-skill workers are matched with co-workers who have weak relative concerns.
• Deviation from Standard Models:
  • Additive Production: Output is independent of sorting. The equilibrium is purely driven by minimizing status costs, resulting in negative assortative matching in skills (Frank, 1984): High-skill workers with strong relative concerns match with low-skill workers with weak relative concerns.
  • Supermodular Production: Standard theory predicts positive assortative matching (PAM). However, if low-skill workers have very strong relative concerns, high-skill workers may still match with them (to avoid status costs) or self-match, potentially leading to negative assortative matching even with supermodular production.
  • Proposition 2: The correlation between co-workers' skills can be manipulated to any value ($\rho \in [-1, 1]$) simply by altering the distribution of relative concerns, even if the production function remains fixed. This explains why empirical data shows strong positive skill correlation even if production is submodular.

B. Wage Inequality

• Supermodular Production: Relative concerns reduce wage inequality compared to a benchmark where workers self-match. High-skill workers must compensate low-skill partners for status loss, raising the low-skill wage.
• Submodular Production: The effect is ambiguous. If low-skill workers have stronger relative concerns than high-skill workers, they may self-match to avoid status loss. This can lead to higher overall wage inequality because the "between-firm" inequality (between self-matching teams and mixed teams) increases, outweighing the reduction in "within-firm" inequality.
• Trickle-Down Effect: Increases in high-skill productivity can trickle down to low-skill workers, but only if those low-skill workers have weak relative concerns. Those with strong relative concerns may self-match and miss out on these gains.

C. Welfare

• Supermodular Production: All workers are better off with social comparisons than without (relative to self-matching), as they can always choose to self-match if the comparison is too costly.
• Submodular Production: Some workers (specifically high-skill with weak concerns and low-skill with strong concerns) may be worse off. They choose to self-match to avoid detrimental comparisons, but this yields lower payoffs than the mixed-team benchmark.

4. Theory of the Firm and Outsourcing

The paper extends the model to explain firm boundaries.

• Mechanism: Social comparisons are salient within a firm but negligible across firm boundaries (outsourcing). Outsourcing incurs a cost $c$ but eliminates the utility loss from wage gaps.
• Decision Rule: Teams consisting of a high-skill worker (weak relative concerns) and a low-skill worker (strong relative concerns) will choose to outsource if the cost of outsourcing is lower than the surplus loss from self-matching ($c < s_F$).
• Impact of SBTC: Skill-Biased Technological Change increases the productivity gap between high and low skills. This widens the potential wage gap within a team, increasing the "pain" of social comparisons. Consequently, SBTC increases the incentive to outsource, leading to a rise in domestic outsourcing.
• Sorting Implications: SBTC increases the number of single-worker firms (outsourced teams). Since these firms consist of single types, the observed sorting of workers into firms becomes more positive and assortative, even if the underlying production teams remain mixed.

5. Significance and Contributions

1. Theoretical Innovation: It is the first paper to provide analytical expressions for sorting and wages in a setting with heterogeneous relative concerns and multidimensional traits (skill + preference). It successfully bridges the gap between assignment models and theories of social comparison.
2. Reconciling Empirical Puzzles:
   • Sorting: Explains how positive skill correlation can coexist with submodular production functions (a common empirical finding) by introducing preference heterogeneity.
   • Outsourcing: Provides a novel micro-foundation for the long-term rise in domestic outsourcing, linking it directly to SBTC and social comparison costs, rather than just transaction costs or trade liberalization.
   • Inequality: Offers a mechanism where within-firm inequality reduction (via compensation) can paradoxically increase overall inequality due to increased between-firm segregation.
3. Policy Implications: The model suggests that technological progress (SBTC) does not automatically benefit all workers. Low-skill workers with strong relative concerns may be left behind or forced into self-employment/outsourcing, exacerbating inequality.
4. Methodological Contribution: The derivation of a TU-representation for a specific class of ITU problems (relative concerns) allows the application of powerful optimal transport and sorting tools to a broader class of economic problems involving social preferences.

In summary, Gola's paper demonstrates that preferences matter as much as skills in determining economic outcomes. The "Pond Dilemma"—the tension between wanting to be the biggest fish in a small pond versus being a small fish in a big pond—fundamentally reshapes team formation, wage structures, and the very boundaries of the firm.