Supply, Demand, Institutions, and Firms: A Theory of Labor Market Sorting and the Wage Distribution

Layer 1 — Overview

Research question. How do workforce composition (labor supply), labor demand, and minimum wage policy jointly determine the wage distribution in imperfectly competitive labor markets, and what were the quantitative contributions of each force to the dramatic decline in Brazilian wage inequality between 1998 and 2012?

Motivation. Brazil’s formal-sector wage inequality fell sharply over this period. Three candidate shocks are well-documented: (1) a large increase in educational attainment — the share of adults completing at least secondary school rose by 20 percentage points (a 68 percent increase) between 1998 and 2012; (2) labor demand shocks, primarily the commodities boom of the 2000s; and (3) a 93.7 percent (66.1 log point) real increase in the federal minimum wage. Existing frameworks analyze these shocks separately — competitive supply/demand models on one side and imperfectly competitive minimum wage models on the other — and therefore cannot detect interactions or jointly explain all observed patterns, including the novel finding that assortative matching between high-wage workers and high-wage establishments rose in 104 out of 151 microregions, a fact inconsistent with the predictions of leading minimum wage models.

Data. The paper uses the RAIS (Relação Anual de Informações Sociais), a confidential linked employer-employee dataset covering the Brazilian formal sector, together with Brazilian Census data for 1991, 2000, and 2010. Statistics are computed for 151 microregions (analogous to US commuting zones) with at least 15,000 workers in RAIS in both base years and at least 1,000 formal workers per educational group. The final sample covers 73 percent of the adult population. Firm wage premiums and assortative matching are measured via AKM two-way fixed effects regressions using the bias-corrected KSS (Kline, Saggio, Sølvsten 2018) estimator, run separately for each microregion and period on three-year panels.

Theoretical framework. The paper develops a unified general-equilibrium model featuring: (i) a task-based production function with distance-dependent complementarity between worker types; (ii) monopsony power arising from idiosyncratic worker preferences for firms, generating constant firm-level labor supply elasticity β (calibrated at 4, implying markdowns of 20 percent); (iii) heterogeneous firms differentiated by their production “blueprints” (the complexity of tasks they require), with blueprint shape parameterized as a Gamma distribution; and (iv) free firm entry, endogenous participation, and goods market general equilibrium with CES consumer preferences (elasticity σ). A key result is that firms with different blueprints exhibit different within-firm substitution patterns: worker types that are substitutes at low-skill, low-wage firms may be complements at high-skill, high-wage firms.

Estimation. A parsimonious parameterization is estimated by simultaneous-equation nonlinear least squares, targeting 26 endogenous outcomes per region (13 per period) including between- and within-group wage inequality, variance of establishment effects, covariance of worker and establishment effects, formal employment rates by education, and minimum wage bindingness. The model requires solving for equilibrium more than 15,000 times per optimization step (151 regions × 2 periods × 53 Jacobian columns). The elasticity of substitution between goods is estimated at σ = 8.36 (significantly above 1), and the aggregate labor supply parameter λ implies formal-sector elasticities of approximately 0.6–0.7 for college workers and around 1.1 for less-than-secondary workers. The model fits the data well, with R² above 0.5 for most targeted moments and perfect fit for the six moments used in the inversion procedure.

Main findings.

1. Demand shocks and the minimum wage are the primary drivers of falling inequality. In counterfactual simulations, the minimum wage alone (a 66.1 log point increase) reduces the variance of log wages by 0.13. Demand shocks reduce it by a further 0.18. Supply shocks (rising education) increase the variance by 0.04, leaving their net inequality-reducing contribution negligible.
2. Supply shocks increase assortative matching despite compressing within-firm skill premiums. Within-firm task reassignment would reduce the variance of log wages by 0.221 and the correlation between worker and establishment effects by 0.165, holding production levels and firm entry fixed. However, scale, entry, and price adjustments — driven by the large estimated σ = 8.36 > β + 1 = 5 — reallocate skilled labor toward high-wage, skill-intensive firms, counteracting within-firm compression and raising assortative matching by 0.189. These two channels largely offset each other.
3. Concurrent supply and demand changes attenuate minimum wage impacts by roughly half. When the minimum wage is the only shock, it would have reduced the variance of log wages by 0.13; in the presence of supply and demand changes, its incremental contribution is approximately 0.07. Minimum wage effects on sorting (which would reduce assortative matching when acting alone) disappear when accompanied by supply and demand transformations.
4. Minimum wage effects are concentrated in the bottom two productivity deciles. Wage effects for workers in productivity deciles three through ten from the minimum wage are approximately 1 percent or less once all channels are considered. Strong wage gains are concentrated at the bottom, primarily through the monopsony channel. The wage-posting channel (within-firm returns to skill) reduces wages for low- and middle-skill workers and raises them at the top two deciles due to the reallocation of low-skilled workers toward high-wage firms, which reduces those workers’ marginal products there.
5. Cross-firm differences in substitution patterns generate non-standard minimum wage spillovers. Conditional on the task demands of the firm employing them, a pair of worker types may be substitutes in low-skill firms and complements in high-skill firms. This firm-heterogeneity channel causes minimum wage impacts to be non-monotone across the productivity distribution, contrasting with the smooth inequality-reducing effects predicted by both competitive task-based models and frictional minimum wage models.

Layer 2 — Q&A

Q1: What is the novel empirical fact that motivates the unified framework? A: Using KSS bias-corrected AKM decompositions performed separately for each of 151 microregions, the paper documents that assortative matching — measured as the correlation between worker and establishment fixed effects — rises in 104 out of 151 regions between 1998 and 2012. The covariance term accounts for less than 7 percent of the average decline in the variance of log wages. This finding is inconsistent with the leading imperfectly competitive minimum wage model (Engbom and Moser 2022), in which minimum wages reduce assortative matching. It is also inconsistent with purely competitive supply/demand models, which have no role for firm wage premiums or sorting. The divergence from prior national-level studies (which do not find rising sorting) is explained by the fact that national-level sorting conflates geographical sorting with supply-demand dynamics.

Q2: What is the key mechanism through which the task-based production function generates cross-firm differences in substitution patterns? A: In the task-based production function, each firm assigns workers to tasks assortatively — lower types handle lower-complexity tasks, higher types handle higher-complexity tasks, with cutoff thresholds determined by the firm’s blueprint. When a firm has a blueprint concentrated in complex tasks (a high-skill, high-wage firm), adjacent worker types are more differentiated in the tasks they perform, making them complements. When a firm has a blueprint concentrated in simple tasks (a low-skill, low-wage firm), adjacent worker types are assigned to a narrow, similar range of tasks and are therefore closer substitutes. The elasticity of complementarity between any pair of worker types is thus endogenous, depending on which tasks the firm uses and, in the monopsony case, on the firm’s skill intensity — a prediction validated empirically using nonroutine cognitive task content data for Brazilian occupations.

Q3: Under what conditions can a positive supply shock (rising educational attainment) widen the aggregate skill wage premium rather than compress it? A: The paper’s Proposition 4 and Corollary 2 show that a supply shock that increases the relative supply of skilled workers can widen the aggregate skill wage premium when the elasticity of substitution between goods (σ) exceeds the firm-level elasticity of labor supply plus one (β + 1). Intuitively, when σ is large, the reduction in prices for skill-intensive goods generated by the supply shock shifts consumption toward those goods, causing net entry of skill-intensive firms. If the gains in firm wage premiums earned by skilled workers reallocated to those firms outweigh the compression in within-firm productivity differentials, the aggregate skill premium can rise. This mechanism does not require non-convexities from endogenous innovation; it operates through imperfect competition and firm entry alone. In the estimated Brazilian model, σ = 8.36 substantially exceeds β + 1 = 5, so this condition holds, explaining why rising education increases rather than compresses assortative matching in the data.

Q4: How does the model generate positive employment effects from minimum wages, and how do these interact with reallocation? A: In the monopsonistic baseline without a minimum wage, firms post wages below workers’ marginal revenue products, causing some workers to choose non-employment. A minimum wage increase raises posted wages at constrained firms, shifting some workers from non-employment (or home production) to formal employment, generating positive employment effects at the margin where the minimum wage binds. Simultaneously, minimum wages price out the least productive workers at low-wage firms (disemployment), while workers in the intermediate productivity range reallocate from low- to high-wage firms, because high-wage firms have higher revenue productivity and can profitably hire workers that low-wage firms can no longer afford. The net employment elasticity for the lowest productivity decile with respect to the log minimum wage is −0.61 (Table 7), while the mean wage for that decile rises substantially through the monopsony channel.

Q5: What are the three channels through which the minimum wage affects wages and employment in the model, and what does each channel contribute? A: The paper decomposes minimum wage effects into three channels. Channel 1 (monopsony): mechanical wage increases, positive employment effects at firms where the minimum wage binds, disemployment of very low-productivity workers, and reallocation from low- to high-wage firms, holding posted wage schedules, prices, and entry fixed. This channel accounts for nearly all of the strong wage effects at the bottom two productivity deciles. Channel 2 (wage posting): firms reoptimize earnings schedules following changes in worker composition and marginal products induced by Channel 1, holding prices and entry fixed. This channel reduces wages for low- and middle-skill workers (productivity deciles 1–7) by approximately 0.01–0.02 log points and increases wages for top deciles (decile 9: +0.04, decile 10: +0.11), because reallocation of low-skill labor to high-wage firms lowers those workers’ marginal products there. Channel 3 (general equilibrium): firm entry and price responses. The fall in low-wage-firm profits causes entry of high-wage, skill-intensive firms, while the price of low-skill goods falls. General equilibrium effects generate modest positive wage effects for most workers but negative effects for very low-productivity workers due to reduced aggregate demand for low-skill labor.

Q6: Why do the minimum wage’s inequality-reducing effects diminish when accompanied by concurrent supply and demand changes? A: The paper documents that, under concurrent supply and demand transformations, the minimum wage’s reduction of the variance of log wages is approximately 0.07, roughly half the 0.13 reduction it would achieve acting alone. The attenuation occurs through interactions: supply and demand shocks raise the average productivity level of the labor market and shift workers toward high-wage, skill-intensive firms. In this altered equilibrium, the minimum wage binds less tightly (or hits a different part of the distribution), and the reallocation effects of the minimum wage that would normally reduce assortative matching are offset by the sorting-increasing effects of supply and demand changes. The estimated model shows that interactions between the minimum wage and supply/demand changes (columns 6, 7, 8 of Table 5) are economically meaningful, something undetectable without a unified framework.

Q7: How does the model’s prediction regarding minimum wage spillovers differ from Engbom and Moser (2022), and what explains the difference? A: Engbom and Moser (2022) find that the Brazilian minimum wage hike had significant wage effects extending far up the worker productivity distribution, while this paper’s model finds negligible effects (approximately 1 percent) beyond the bottom two productivity deciles. Two structural differences explain this divergence. First, Engbom and Moser (2022) assume perfect substitutability between worker types within firms, so a minimum wage increase at low-wage firms mechanically raises posted wages at all other firms to maintain relative competitiveness. In this paper’s framework, wage-posting responses at high-wage firms can be negative for low-skill workers because the inflow of reallocated low-skill workers reduces their marginal products — a channel absent under perfect substitution. Second, Engbom and Moser (2022) use a national model, allowing displaced low-skill workers to reallocate to top-productivity firms anywhere in the country, dampening disemployment; this paper’s local labor markets approach restricts reallocation to within-region boundaries, consistent with low rates of interregional migration documented for Brazil by Dix-Carneiro and Kovak (2017).

Q8: How are firm wage premiums generated in the model, and why do differences in physical productivity between firms not generate wage differentials? A: Proposition 3 establishes that wage dispersion for similar workers across firms requires either (i) differences in blueprint shapes (firm heterogeneity in skill intensity) or (ii) differences in entry costs. Differences in physical productivity (z_g) or consumer taste parameters alone are insufficient, because with equal entry costs, differences in productivity lead to additional firm entry until the marginal revenue product of labor is equalized across firm types. Wage premiums proportional to entry costs arise because optimal firm creation requires larger-scale operation for higher-entry-cost firms, and hiring more workers forces those firms to post higher wages. Additionally, skill-intensive firms (firms with blueprints tilted toward complex tasks) pay relative wage premiums for the worker types they use most intensively, and if skill intensity and entry costs co-vary, all workers at high-skill firms may receive a wage premium.

Q9: How does the estimation procedure handle unobserved regional heterogeneity in labor demand? A: Demand shocks are not directly observed; they are inferred as a residual from changes in targeted outcomes after accounting for observed supply (education shares from Census) and minimum wage changes. Five region-time-specific demand parameters — TFP (z), blueprint complexities (θ₁, θ₂), relative entry costs (F₂/F₁), and relative consumer preferences (γ₂/γ₁) — are modeled as linear functions of 1998 regional covariates (educational shares, agricultural share, manufacturing share, and initial minimum wage bindingness) with time-specific coefficients. This formulation allows unobserved demand shifters to correlate with initial educational levels, preventing incorrect attribution of demand-supply correlations to causal supply effects. Region-specific parameters (TFP in each period, education-group-specific formal employment shifters) are inverted exactly from six targeted moments within each region, eliminating incidental parameter bias.

Q10: What micro-level empirical validations does the paper conduct for the task-based model’s mechanisms? A: The paper tests four micro-level predictions using nonroutine cognitive task content data for Brazilian occupations. First, skill-intensive firms have greater demand for complex tasks (consistent with Figure 1 of the model). Second, within firms, more skilled workers are assigned to more complex tasks (Lemma 1). Third, workers who move to more skill-intensive firms are assigned more complex tasks (Lemma 2, consistent with the monopsony model’s mismatch prediction). Fourth, wage gaps between high- and low-skill firms are larger for skilled workers (Proposition 3). The paper reports finding strong support for all four predictions in the data, lending credibility to the theoretical structure and quantitative results.

Key Concepts

Task-based production function (paper’s definition): A production function in which a firm produces output by assigning workers of different types to tasks indexed by complexity. The assignment is assortatively optimal: lower-type workers handle lower-complexity tasks, with unique threshold complexities separating adjacent worker types. The critical property is distance-dependent complementarity — any pair of worker types that are “close” in skill rank are substitutes, while pairs distant in skill rank are complements. This differs from CES production functions where the elasticity of complementarity is the same for all pairs; in the task-based version, substitutability depends on endogenous assignment and thus on the firm’s blueprint.

Blueprint (paper’s definition): A function b_g(x) that specifies the density of tasks of each complexity level x required to produce one unit of good g. It is the fundamental source of firm heterogeneity in the model: firms producing goods with blueprints tilted toward complex tasks are more skill-intensive, hire workers of higher average type, and pay higher wages. The paper parameterizes blueprints as Gamma distributions with shape parameter θ_g indexing average task complexity; firms with higher θ_g are more skill-intensive.

Firm wage premium (paper’s definition): The component of wages at a given establishment that accrues equally to all workers at that firm regardless of their type, measured as the establishment fixed effect ψ_j in AKM two-way fixed effects regressions. In this model, firm wage premiums arise from heterogeneity in blueprints (skill intensity) and entry costs, not from differences in TFP or consumer tastes. Under monopsony, firms with higher entry costs must operate at larger scale and post higher wages; blueprint heterogeneity generates differential wage premiums by skill type.

Sorting / assortative matching (paper’s definition): The correlation between the worker fixed effect (ν_i,r capturing worker skill) and the establishment fixed effect (ψ_j capturing firm wage premium) in the AKM decomposition, measured as Cov(ν_i,r, ψ_{J(i,r,τ)} | r). In this paper’s framework, sorting arises because firms with blueprints demanding complex tasks (high-wage firms) have a comparative advantage in employing high-skill workers; labor market sorting can therefore change over time due to supply, demand, or minimum wage shocks, even without changes in search frictions.

Monopsony power / markdown (paper’s definition): Arising from idiosyncratic worker preferences for firms (modeled as a nested logit), firms face upward-sloping labor supply curves with constant firm-level elasticity β. Optimal posted wages equal a constant markdown β/(β+1) of the marginal revenue product of labor, set to β = 4 (implying a 20 percent markdown). The macro elasticity of formal sector labor supply is governed by a separate parameter λ, estimated from the data, yielding aggregate formal-sector supply elasticities of approximately 0.6–0.7 for college workers and around 1.1 for less-educated workers.

Wage posting responses (paper’s definition): The second channel of minimum wage effects, in which firms reoptimize their entire earnings schedule following the wage-composition changes induced by the minimum wage’s mechanical and reallocation effects (Channel 1), while keeping goods prices and firm entry fixed. Because task-based production functions are concave, changes in factor proportions (due to reallocation of low-skill workers to high-wage firms) alter marginal products of all worker types within those firms, causing firms to adjust all posted wages — not just those directly constrained by the minimum wage.

Distance-dependent complementarity (paper’s definition): The property, proven as a Corollary to Proposition 1, that for a fixed worker type h, the partial elasticity of complementarity between h and any other type h’ is strictly increasing in h’ for h’ ≥ h (more distant high types are stronger complements) and strictly decreasing in h’ for h’ ≤ h (more distant low types are weaker substitutes / stronger complements). This pattern results from the division of labor: adding a very different worker type allows specialization gains that do not arise when adding similar-type workers competing for the same tasks.