Sorting along Business Cycles

We develop an analytically tractable model featuring heterogeneous workers and firms, where labor markets clear through a one-to-many sorting mechanism. Firms determine both the number and composition of their employees, shaping (1) the income distribution among workers and (2) the productivity distribution across firms. We study business cycles driven by market efficiency shocks that disproportionately benefit more productive firms. The model’s implications are consistent with empirical regularities on the cyclical behavior of wage and productivity distributions.

💡 Research Summary

This paper, “Sorting along Business Cycles,” presents a novel theoretical model that endogenizes firm productivity through the sorting of heterogeneous workers and firms, offering an explanation for the observed cyclical patterns of wage and productivity inequality.

The core innovation lies in challenging the standard assumption of exogenous firm-level productivity. Instead, the model posits that a firm’s productivity is an equilibrium outcome determined by whom it hires. The framework features a continuum of firms and workers, each with exponentially distributed types (θ for firms, x for workers). Production combines capital and labor with a Cobb-Douglas technology, where the idiosyncratic productivity component is a function of both the firm’s type and the type of its matched workers: q(x, θ) = exp{x^ψ θ^(1-ψ)}. The labor market clears through a one-to-many matching process where firms choose both the quantity and the skill composition of their workforce.

The primary driver of business cycles in the model is a reduced-form “market efficiency shock” (z_t). This shock manifests as time-varying, firm-specific labor wedges that are correlated with firm type (τ_1 = exp(z_t θ + ε_1)). A positive shock (improved efficiency) disproportionately lowers the effective labor cost for high-type firms, mimicking empirical evidence that more productive firms expand more aggressively during booms.

The model’s key analytical result reveals a paradoxical mechanism. During an expansion (z_t increases), high-type firms expand and create more jobs. Due to the one-to-many sorting structure, this forces them to hire workers of relatively lower skill types on average. Consequently, the realized productivity of these expanding high-type firms decreases toward the mean. Conversely, low-type firms contract and shed jobs, retaining a higher average skill level among their remaining employees, which increases their realized productivity. This convergence dampens the cross-sectional dispersion of measured firm productivity, making it countercyclical.

Simultaneously, the aggressive expansion of high-type firms increases the relative demand for high-skill workers, bidding up their wages more than those of low-skill workers. This widens the wage distribution, making wage inequality procyclical. Thus, the same aggregate shock generates opposing cyclical movements in wage dispersion (procyclical) and productivity dispersion (countercyclical), aligning with documented empirical regularities.

The paper further discusses how alternative shocks—such as aggregate productivity shocks or shocks to the variance of firm types—could generate different co-movement patterns between wage and productivity distributions, providing a broader framework for understanding varied business cycle episodes.

In summary, by re-framing productivity as an endogenous result of talent allocation, the model provides a unified explanation for how business cycle fluctuations, mediated through shifts in labor market sorting, can simultaneously drive procyclical wage inequality and countercyclical productivity dispersion.