Minimum Wages, Efficiency, and Welfare

Overview

Research question. Can minimum wages improve welfare through efficiency — by correcting monopsony-driven under-employment — and, if so, by how much? What is the optimal minimum wage, and how much of the welfare gain from a higher minimum wage comes from efficiency versus redistribution?

Model and methodology. The paper develops a tractable general equilibrium oligopsony model with heterogeneous workers (four types: non-high-school, high-school, college workers, and capital owners) and heterogeneous firms (varying in total factor productivity), embedded in a continuum of local labor markets where firms compete strategically in Cournot fashion. Firms face downward-sloping labor supply curves; their market power generates wages below the marginal revenue product of labor (markdowns). The model is calibrated to US data using the Census Longitudinal Business Database (LBD, 2014), the Bureau of Labor Statistics Current Population Survey (CPS, 2019), and the Survey of Consumer Finances (SCF). Key calibration targets include: average firm size of 22.83 workers (LBD), 29 percent of workers earning below $15/hr (CPS), labor and capital income shares, and household-level earnings and capital income ratios. The model is validated by quantitatively replicating four strands of empirical evidence: (i) reallocation effects of the German minimum wage introduction (Dustmann et al., 2021); (ii) employer spillover responses to Amazon’s voluntary $15 minimum wage (Derenoncourt et al., 2021); (iii) wage distribution compression evidence from Brazil (Engbom and Moser, 2021); and (iv) heterogeneous employment effects by market concentration (Azar et al., 2019).

Three channels for efficiency gains. The model identifies three mechanisms through which a minimum wage can improve efficiency under oligopsony: (1) a direct effect in which constrained firms with monopsony markdowns increase wages and expand employment toward the competitive level (Region II firms); (2) a spillover effect in which unconstrained competitor firms narrow their own markdowns in response to constrained firms’ increased wages and market shares; (3) a reallocation effect in which employment is shifted away from low-productivity firms (which enter Region III — constrained on labor demand) toward more productive firms.

Main findings on efficiency versus redistribution. Under the $15.12/hr minimum wage that maximizes social welfare under utilitarian weights (population-share weights), less than 5 percent of the welfare gains come from improved efficiency, while more than 95 percent come from redistribution. When the government is additionally given access to budget-neutral lump-sum transfers that fully address redistribution goals, the efficiency-maximizing minimum wage narrows to a range of approximately $7.50–$10.00 per hour, which is robust across social welfare weight specifications. The welfare gains attributable to efficiency alone are approximately 0.16–0.20 percent in consumption-equivalent terms, representing only about 1–2 percent of the welfare gains achievable in an economy with no labor market power at all (which would be 15.26 percent in consumption-equivalent terms under the same conditions with optimal transfers).

Why efficiency gains are small. Three structural reasons limit efficiency gains: (i) low-productivity firms — which are the firms most affected by a binding minimum wage in Region II — have endogenously narrow markdowns even absent a minimum wage, because they face more elastic labor supply and command small market shares; (ii) the calibrated production function has relatively flat marginal revenue product of labor schedules (decreasing returns parameter α = 0.940), so once firms enter Region III, employment rationing occurs rapidly; (iii) the large, high-productivity firms with the widest markdowns are not materially affected by the minimum wages of their small, low-wage competitors because those competitors have small market shares — making spillovers quantitatively negligible even though the model matches empirical cross-employer wage elasticities.

Optimal minimum wages under alternative frameworks. Without transfers and under utilitarian weights, the optimal minimum wage is $15.12. Without transfers but under Negishi weights (which rationalize the observed competitive equilibrium and load approximately 62 percent of weight on college workers and owners versus their 35 percent population share), the optimal is $6.97. Under a 97 percent weight on high-school graduates, the optimal rises to $18.32. With optimal lump-sum transfers, the optimal collapses to $7.76–$10.11 regardless of social welfare weights — a range robust across Frisch elasticity variants (ϕ ∈ {0.30, 0.62, 0.86}), regional decompositions (low, medium, and high income US states), short-run capital-fixed scenarios (where the optimum declines by approximately $1 under utilitarian weights), and the removal of household heterogeneity entirely (which yields an optimum of $7.74).

Distributional proxies versus welfare. Wage inequality (college–non-college log wage premium, cross-sectional variance of log wages) and the labor income share are monotonically improving as the minimum wage rises, even as welfare is hump-shaped and eventually declining. A rise in the minimum wage from $7.50 to $15 reduces the college–non-college log wage premium from 0.53 to 0.43 (roughly one-fifth), reduces the cross-sectional variance of log wages by nearly half, and raises the aggregate labor income share by approximately 3 percentage points — all while welfare (under utilitarian weights with no transfers) reaches its maximum at $15.12 and then declines. These standard proxies therefore do not reliably indicate welfare.

Scope conditions. All results are long-run steady-state comparisons unless otherwise noted. Results assume no price passthrough and a unit elasticity of substitution between capital and labor. The paper abstracts from capital–labor substitution responses and occupational choice. The redistribution channel quantified here is specific to the utilitarian welfare criterion and to the existing distribution of capital and profit income, in which owners (6 percent of households) earn 92 percent of dividends.

Q&A

Q1: What are the three regions of firm behavior in response to a binding minimum wage, and what are their efficiency implications?

A: A firm can be in one of three regions. In Region I the minimum wage is not binding: the firm pays its optimal monopsony wage and employment is inelastically below the competitive level. In Region II the minimum wage binds and exceeds the firm’s optimal monopsony wage, but labor supply at the minimum wage still falls short of labor demand: employment and efficiency improve as the shadow markdown narrows. In Region III the minimum wage exceeds the competitive wage, so unconstrained labor supply would exceed demand: the firm rations employment and the rationing constraint binds, reducing efficiency. At the boundary of Region II and Region III, the shadow markdown equals one and the firm is at its efficient employment level. Only a firm-specific minimum wage targeting each firm’s competitive wage could deliver economy-wide efficiency.

Q2: How does the paper define and use “shadow wages” to characterize equilibrium?

A: The shadow wage for a firm is the effective wage that rationalizes equilibrium employment given rationing constraints. Formally, when a firm rations employment (Region III), households act as if facing a shadow wage equal to the actual minimum wage multiplied by a rationing factor p < 1 (the Lagrange multiplier on the rationing constraint, normalized as a fraction). Shadow wages aggregate across firms into market- and type-level shadow wages via CES aggregation. The key insight is that shadow wages, not observed wages, are allocative: aggregate labor supply for each worker type is determined by the type-level shadow wage, not by the minimum wage that firms actually pay. This allows the paper to express aggregate efficiency via two wedges — the aggregate shadow markdown (capturing average market power) and a misallocation term — without tracking all firm-specific constraints individually.

Q3: What are the two aggregate efficiency wedges and how do they behave as the minimum wage rises?

A: The two wedges are: (i) the aggregate shadow markdown µ̃, which is a productivity-weighted average of firm-level shadow markdowns and measures the extent to which aggregate wages fall short of marginal revenue products; and (ii) the misallocation term ω, which measures whether employment is allocated toward more productive firms and equals one when all shadow markdowns are identical. As the minimum wage rises from zero, µ̃ initially narrows (improving efficiency) because firms in Region II expand toward their competitive employment level and constrained firms’ market shares rise, tightening the residual labor supply of unconstrained competitors and narrowing their markdowns. But as the minimum wage rises further, Region III rationing causes shadow markdowns to widen rapidly — first for low-productivity firms and then progressively for more productive ones — so µ̃ turns back downward. The misallocation term ω first improves as low-productivity firms are pushed out, but then worsens because rationing at intermediate-productivity firms redirects employment from high- to medium-productivity firms.

Q4: What does the model validation exercise on the German minimum wage (DLSUB 2021) show?

A: The paper calibrates the model to the German context by setting a minimum wage of $8.95/hr equivalent to 48 percent of the pre-reform median wage — matching Germany’s 8.50 euro introduction in 2015, where 15 percent of workers earned below the threshold. The model produces employment effects that are slightly positive (consistent with empirical findings of no disemployment), average wage increases consistent with both constrained and unconstrained firms raising wages, a negative elasticity of the number of operating firms with respect to minimum wage exposure (correctly signed, moderately smaller than data), and a positive elasticity of average firm size with respect to exposure (slightly larger than the data). The reallocation direction — small unproductive firms shrinking and workers moving to larger, more productive firms — matches the data qualitatively and within the range of data estimates across specifications.

Q5: What does the Amazon spillover replication (DNWT 2021) show, and what does it imply about the minimum wage spillover channel?

A: Derenoncourt et al. (2021) estimate a cross-employer wage elasticity of 0.26: when Amazon raised wages by approximately 18.1 percent, competitors raised wages by 4.7 percent on average. The model replicates this by treating Amazon as the largest (or second-largest) firm in each market, exogenously narrowing its markdown by a fraction ζ calibrated to deliver an 18.1 percent wage increase. Competitors in the model raise wages through the strategic interaction mechanism: Amazon’s higher wage and market share tightens competitors’ residual supply curves, inducing them to narrow their own markdowns. The model matches the 0.26 cross-employer elasticity when Amazon is the largest firm in markets with at least 36 competitors, or the second-largest in markets with at least 12. Critically, the authors note that this empirical evidence concerns responses to a large firm raising wages; for minimum wages the question is whether large firms respond to their small wage competitors, which the model shows they do not substantially, because small firms have negligible market shares.

Q6: How does the paper separate efficiency from redistribution, and what is the key methodological innovation?

A: The paper gives the government access to budget-neutral, unrestricted lump-sum transfers across households in addition to the minimum wage. With transfers available, the government can use them to meet any redistributive objective encoded in arbitrary social welfare weights. Whatever is left for the minimum wage to do must be purely efficiency-improving. The paper shows (via aggregation theorems) that optimal lump-sum transfers can be computed in closed form for any social welfare weights, and that the social welfare maximizing allocation subject to transfers can be decentralized by transfers that sum to zero across households. Under this framework, the efficiency-maximizing minimum wage lies between $7.50 and $10.00 per hour regardless of whether utilitarian, Negishi, or 97 percent high-school-weighted social welfare functions are used — collapsing the original $0–$31 range to a tight interval.

Q7: How are Negishi weights computed, and why are they important for interpreting the results?

A: The Negishi weights are the social welfare weights under which a planner would choose the observed competitive equilibrium with zero lump-sum transfers. They are computed by inverting the planner’s first-order conditions: for the competitive equilibrium to be optimal under some set of weights, the implied consumption ratios must match observed data. The calibrated Negishi weights assign a combined weight of approximately 62 percent to college workers and owners, who constitute only 35 percent of the population. This means the competitive equilibrium is disproportionately aligned with higher-income households. A utilitarian planner, which weights households by population shares, therefore sees large scope for redistribution toward non-college workers — which is exactly why the utilitarian-optimal minimum wage is $15.12 and why 94 percent of its welfare gains come from redistribution rather than efficiency.

Q8: What are the quantitative welfare gains from the efficiency-maximizing minimum wage, and how small are they relative to the potential gains from eliminating monopsony?

A: With optimal lump-sum transfers, the welfare gains from the efficiency-maximizing minimum wage are approximately 0.16–0.20 percent in consumption-equivalent terms, robust across social welfare weight specifications, Frisch elasticity variations, and regional decompositions. The welfare gains associated with an economy in which all firms’ markdowns are set to one (no labor market power at all), also evaluated with optimal transfers, are 15.26 percent in consumption-equivalent terms. The efficiency-maximizing minimum wage therefore recovers approximately 1–2 percent of the potential welfare gains from eliminating monopsony. Equivalently, the efficiency gains correspond to roughly a 0.1 percent increase in TFP. These gains are small despite the model matching all empirical evidence on the channels through which efficiency gains could occur.

Q9: How do employment effects of minimum wages vary by market concentration, and why?

A: In concentrated markets (upper tercile of HHI), firms have larger monopsony markdowns, so a binding minimum wage pushes them into Region II — where employment expands — over a wider range of minimum wage values before entering Region III. This produces large, positive employment effects in concentrated markets. In less concentrated markets, firms already have narrow markdowns (they are closer to competitive), so even small minimum wage increases push them into Region III, where employment contracts. The model replicates the statistically significant positive effects in high-concentration markets and negative effects in low-concentration markets documented by Azar et al. (2019), for initial minimum wages below approximately $8/hr. At higher initial minimum wages, however, even high-concentration markets exhibit negative employment effects as more firms enter Region III.

Q10: What does the robustness exercise for Mississippi reveal?

A: Mississippi has the lowest per capita income in the US, and a $15 minimum wage would bind for 41.3 percent of its workers (versus 29.4 percent nationally). Despite this, the model finds that Mississippi would benefit from a $15 federal minimum wage under utilitarian weights, and the Mississippi-specific optimal minimum wage is $14.89 — nearly identical to the national optimum. The reason is an offsetting compositional effect: while Mississippi has lower average wages (pushing toward a lower optimal), it has a larger share of high-school graduates (63 percent versus 52.8 percent nationally) who prefer higher minimum wages (around $17 in the model). These two forces wash out, producing a stable optimal close to the national figure.

Q11: What happens to common empirical proxies for inequality and worker power as the minimum wage rises?

A: The college–non-college log wage premium declines from 0.53 to 0.43 (a fall of roughly one-fifth) as the minimum wage rises from $7.50 to $15. The cross-sectional variance of log wages falls by nearly half over this range, driven equally by declining within- and between-type inequality. The aggregate labor income share rises by approximately 3 percentage points, and the share of output created in non-high-school jobs paid to non-high-school workers rises by 7 percentage points. All of these proxies are monotonically improving in the minimum wage throughout, even as aggregate welfare under the model’s social welfare function is hump-shaped and declining past the optimum. The paper concludes that observations of declining inequality or a rising labor share are consistent with falling welfare, so these proxies cannot serve as reliable welfare indicators.

Q12: How does the short-run (fixed-capital) analysis differ from the long-run baseline?

A: In the short run, capital at each firm is fixed at the type-specific level chosen under a zero minimum wage. This creates sharper decreasing returns in labor (parameter γα rather than α̃), overhead costs that can make operation unprofitable, and a narrower range of minimum wages over which firms remain in Region II. The result is that firms in the short run enter Region III at lower minimum wages than in the long run, limiting the range of efficiency gains. Quantitatively, the efficiency-maximizing optimal minimum wage declines by approximately $1 under utilitarian weights (from about $10 to about $9 in the short-run exercise) and by only about $0.20 under Negishi weights. The robustness conclusion is that the difference between short- and long-run optimal minimum wages is modest, and the main finding that efficiency gains are small is preserved.

Key Concepts

Shadow wage (w̃ᵢⱼ): The effective wage that rationalizes a firm’s equilibrium employment in the presence of a minimum wage. When labor is rationed at firm ij (Region III), the shadow wage equals the actual minimum wage multiplied by a rationing factor pᵢⱼ < 1, where pᵢⱼ is derived from the Lagrange multiplier on the household’s rationing constraint. The shadow wage is allocative — it determines labor supply decisions — while the observed minimum wage wage is not. When the rationing constraint is slack (Regions I and II), the shadow wage coincides with the observed wage.

Shadow markdown (µ̃ᵢⱼ): The ratio of a firm’s shadow wage to its marginal revenue product of labor. In Region I (unconstrained), this equals the standard monopsony markdown. In Region II (constrained, on the labor supply curve), the shadow markdown narrows as the minimum wage increases, moving the firm toward its efficient employment level. In Region III (constrained, on the labor demand curve), the shadow markdown equals the rationing multiplier pᵢⱼ and widens, reflecting efficiency losses from rationing. An aggregate shadow markdown µ̃ is computed as a productivity-weighted average of firm-level shadow markdowns across all firms in the economy.

Misallocation wedge (ω): A productivity-weighted measure of how well employment is allocated across firms. In an efficient allocation with identical shadow markdowns, ω = 1. When high-productivity firms have wider markdowns than low-productivity firms (the baseline oligopsony outcome), ω < 1 because employment is directed away from productive firms. A minimum wage can improve ω by shrinking low-productivity firms but worsens it when high-productivity firms enter Region III and are over-rationed relative to medium-productivity firms.

Oligopsony with Cournot competition: The specific form of labor market power in this model. In each local labor market (defined as a NAICS 3-digit industry × commuting zone cell), a finite number of firms compete strategically in employment quantities, taking their competitors’ employment levels as given (Cournot assumption). Each firm has an upward-sloping labor supply curve derived from nested CES household preferences, and exercises a markdown on the marginal revenue product of labor. This differs from monopsony (one firm) or perfect competition (infinitely many firms), and generates both direct effects and spillover effects of minimum wages.

Negishi weights: The vector of social welfare weights under which the observed competitive equilibrium allocation would be the solution to a social planner’s problem with zero lump-sum transfers. In this model, the calibrated Negishi weights assign roughly 62 percent combined weight to college workers and owners (who constitute only 35 percent of the population), reflecting the fact that the market equilibrium allocates a disproportionate share of consumption to high-income households. The Negishi weights are used both to identify the gap between market outcomes and utilitarian objectives (motivating redistribution) and as one alternative normative benchmark.

Efficiency-maximizing minimum wage: The minimum wage that maximizes social welfare when the government additionally has access to budget-neutral lump-sum transfers across households. Because transfers can be optimized to handle any redistributive objective encoded in any arbitrary social welfare weights, the minimum wage under this framework serves solely to improve productive efficiency. In the calibrated model, the efficiency-maximizing minimum wage is approximately $7.50–$10.00 per hour, robust to social welfare weight specifications, Frisch elasticity variations (ϕ ∈ {0.30, 0.86}), and regional income differences.

Rationing constraint (n̄ᵢⱼₖ): A firm-specific, type-specific upper bound on the labor a household may supply to a firm in equilibrium. These constraints are taken as given by households and determined in equilibrium by firms’ labor demand decisions. When the minimum wage is above the firm’s competitive wage (Region III), the firm’s labor demand is less than what households would want to supply at that wage, so the rationing constraint binds. The binding rationing constraint generates the shadow wage discount (pᵢⱼ < 1) and is the mechanism by which high minimum wages reduce efficiency in the model.