Wages, Unemployment and Inequality with Heterogeneous Firms and Workers

Wages, Unemployment and Inequality with Heterogeneous Firms and Workers Elhanan Helpman Oleg Itskhoki Stephen Redding Harvard and CIFAR Harvard LSE and CEP Penn State June 12, 2008 1 / 28

Introduction Three prominent features of product and labor markets are: 1 substantial differences in size, performance and workforce composition across firms; 2 variation in wages and other labor market outcomes for workers with the same observed characteristics; 3 positive unemployment We develop a general equilibrium model that captures these features We then use the model to study: allocation of workers across firms size and productivity distribution across firms distribution of wages and income within and between sectors sectoral and economy-wide unemployment rates labor market outcomes for workers with similar ability 2 / 28

Ingredients of the model 1 Firm productivity heterogeneity 2 Unobserved worker ability heterogeneity general worker ability match-specific productivity 3 Random search and matching 4 Costly screening 5 Production technology with complimentarities Rosen (1982), Kremer (1993), Garicano (2000) 3 / 28

Main Findings Labor market frictions: Search friction increases sectoral unemployment, while screening friction reduces it Labor market frictions have no affect on sectoral wage inequality, but they do affect economy-wide inequality Both labor market frictions reduce welfare Interdependence between product and labor market: Size distribution of firms is more dispersed in sectors with more productivity and ability dispersion Wage inequality increases in the dispersion of firm productivity, but may increase or decrease in the dispersion of worker ability Labor market outcomes for workers with similar ability: Workers with higher ability are less likely to end up unemployed On average they receive higher wages and have higher dispersion of wages (but not necessarily of income) 4 / 28

Utility function: Preferences and Demand I U = q 0 + i=1 1 ζ i Q ζ i i, ζ i > 0 Preferences for differentiated products in sector i: [ ] 1 Q i = q i (ω) β βi i dω, ζ i < β i < 1 ω Ω i Demand functions: Indirect utility function: q i (ω) = Q β i ζ i 1 β i i p i (ω) 1 1 β i V = E + I i=1 1 ζ i Q ζ i ζ i, i where Q (1 ζ i ) i = P i 5 / 28

Market Structure All goods are produced with labor The homogeneous product requires one unit of labor per unit output and the market for this product is competitive: p 0 = w 0 = 1 6 / 28

Market Structure All goods are produced with labor The homogeneous product requires one unit of labor per unit output and the market for this product is competitive: p 0 = w 0 = 1 The market for brands of the differentiated product is monopolistically competitive: Fixed entry cost f e in terms of the homogenous good The firm then learns its productivity θ, drawn from a Pareto distribution: G θ (θ) = 1 ( θ min /θ ) z, z > 2 Fixed production cost f d in terms of the homogeneous good Revenue of the firm with output y r = Q (β ζ) y β If revenue is insufficient to cover all costs, the firm exits 6 / 28

Production Technology Output of a firm with productivity θ and h employees with average ability ā: ( ) 1 1 γ h y = θh γ ā = θ h a i di, 0 < γ < 1 0 Interpretation: human capital externalities or fixed managerial time at the level of the firm Ability of workers a is unobservable and distributed Pareto: G a (a) = 1 ( a min /a ) k, k > 2 By paying bn a firm can match randomly with n workers By paying ca δ c /δ a firm can screen workers with ability below a c 7 / 28

Firm s Problem Wage bargaining as in Stole and Zwiebel (1996): 1/(1 + βγ) is the firm s share of revenues Firm s problem: π(θ) max n 0, a c a min 1 1 + βγ Q (β ζ) [( ) ] γk β ka min θn γ ac 1 γk bn c k 1 δ aδ c f d Firms that sample more workers are also more selective: (1 γk)bn(θ) = γca c (θ) δ Assuming δ > k, these firms also hire more workers: (1 γk)bh(θ) = γcamin k a c(θ) δ k 8 / 28

Wages, Productivity and Profits As a result of bargaining, the wage rate is: w(θ) = Size-wage premium in the model: βγ r(θ) 1 + βγ h(θ) = b [ ac (θ) a min ] k d ln w(θ)/d ln h(θ) = k/(δ k) > 0 9 / 28

Wages, Productivity and Profits As a result of bargaining, the wage rate is: w(θ) = Size-wage premium in the model: βγ r(θ) 1 + βγ h(θ) = b [ ac (θ) a min ] k d ln w(θ)/d ln h(θ) = k/(δ k) > 0 Productivity: t(θ) = r(θ)/h(θ) w(θ) The profit of the firm is π(θ) = Γ 1 + βγ r(θ) f d, Γ 1 βγ β (1 γk) δ and revenue is r(θ) = κ r b βγ Γ c β(1 γk) δγ Q β ζ Γ θ β Γ 9 / 28

Free Entry and Zero Profit Zero-profit productivity cutoff: π(θ d ) = κ π [b βγ c β(1 γk)/δ Q (β ζ) θ β d ] 1/Γ fd = 0, so that π(θ) = f d [ (θ/θd ) β/γ 1 ] Free entry condition: f e = θ d π (θ) dg θ (θ) = f d [ (θ/θd ) ] β/γ 1 dg θ (θ) θ d 10 / 28

Labor Market Tightness of the labor market: x = N/L Hiring cost is increasing in the tightness of the labor market: b = α 0 x α 1, α 0 > 1, α 1 > 0 11 / 28

Labor Market Tightness of the labor market: x = N/L Hiring cost is increasing in the tightness of the labor market: b = α 0 x α 1, α 0 > 1, α 1 > 0 Probability of being sampled is x and the expected wage conditional on being sampled is w(θ)h(θ) = b n(θ) The indifference condition for workers is therefore bx = 1 so that b = α 1/(1+α 1) 0 > 1 and x = b 1 = α 1/(1+α 1) 0 < 1. 11 / 28

Sectoral Equilibrium Productivity cutoff: θ d = [( ) ] β 1/z fd θ zγ β f min e Sectoral Output: Q β ζ = κ Q b βγ c β(1 γk)/δ Number of workers searching for job and number of firms: L = βγ 1 + βγ Qζ and M = 1 L γzf e 12 / 28

Revenue: Variation Across Firms ( ) r(θ) = 1+βγ β/γ Γ f θ d θd Employment: h(θ) = βγ Γ f db [ β(1 γk) Γ f d camin δ ] k/δ ( θ θ d ) β(1 k/δ)/γ Screening cutoff: a c (θ) = [ ] β(1 γk) f 1/δ ( ) β/δγ dc θ Γ θd Wage rate: [ w(θ) = b β(1 γk) Γ f d camin δ ] k/δ ( θ θ d ) βk/δγ 13 / 28

Distribution of Firm Size and Productivity Firm size (in terms of employment and revenue) and measured productivity are distributed Pareto: / ) zγ β(1 k/δ) (h d h F h (h) = 1 for h h d βγ Γ ( / ) zγ/β F r (r) = 1 r d r for r r d 1+βγ Γ f d, ( / ) zδγ/(βk) F t (t) = 1 t d t for t t d b 1+βγ βγ f db [ β(1 γk) Γ [ β(1 γk) Γ f d camin δ f d camin δ ] k/δ, Recall that Γ = 1 βγ β(1 γk)/δ is increasing in k and Γ/k is decreasing in k ] k/δ. Proposition Dispersion of firm size increases in dispersion of productivity (z ) and dispersion of ability (k ). Dispersion of measured productivity decreases in z, but increases in k. 14 / 28

Sectoral unemployment rate: u = L H L Sectoral hiring rate: = 1 H N Sectoral Unemployment N L σ H [ ] [ k N = amin 1 a c (θ d ) 1 + µ = = 1 σx, x = 1/b Γ β (1 γk) ca δ min f d ] k/δ 1 1 + µ, where µ βk/δ (0, 1) zγ β Proposition Unemployment is higher in sectors with higher search cost (b) and lower screening cost (c); unemployment is higher in sectors with more productivity dispersion (z ), but may be higher or lower in sectors with more ability dispersion (k ). 15 / 28

Wage distribution: ( wd ) 1+µ 1 F w (w) = 1 w Sectoral Wage Inequality for w w d b [ β(1 γk) Γ f d ca δ min ] k/δ > 1 16 / 28

Wage distribution: ( wd ) 1+µ 1 F w (w) = 1 w Lorenz curve and Gini coefficient: Sectoral Wage Inequality for w w d b [ β(1 γk) Γ s w = L w (s h ) 1 (1 s h ) 1/(1+µ), 1 G w = 1 2 L w (s) ds = µ 0 2 + µ f d ca δ min ] k/δ > 1 16 / 28

Sectoral Wage Inequality A sufficient statistic for wage inequality: µ = βk/δ zγ β Proposition The sectoral distribution of wages is more equal in sectors with less productivity dispersion (z ), but it can be more or less equal in sectors with less ability dispersion (k ). 17 / 28

Aggregate Rate of Unemployment u = I I L i i=1 L u L i = i i=1 L (1 σ i x i ). Search and screening costs (b and c) reduce the size of the sector, L i They have opposite effects on the sectoral unemployment rate, u i Proposition An increase in a sector s screening cost (c) reduces aggregate unemployment while an increase in a sector s search cost (b) reduces aggregate unemployment if and only if u i > β ζ 1 u i γβζ. 18 / 28

Aggregate Rate of Unemployment 0.35 Aggregate Unemployment Rate, 0.3 0.25 0.2 0.15 High c Low c 0.1 1 2 3 4 5 6 Search Cost, b 19 / 28

Aggregate Wage and Income Inequality Theil index of aggregate income inequality: T ι = I I L i i=1 L T L ιi = i i=1 L [µ i ln (1 + µ i ) ln (1 u i )] Proposition An increase in screening cost (c i ) reduces aggregate income inequality, while an increase in search cost (b i ) reduces it if and only if T ιi > β ζ γβζ. Contrast this result with the previous result for aggregate unemployment rate Additionally, we show that both c and b can increase or decrease aggregate wage inequality 20 / 28

Aggregate Wage Inequality Theil Index of Aggregate Wage Inequality 0.26 0.24 0.22 0.2 0.18 0.16 0.14 High b Low b 0.12 0.1 0.3 0.5 0.7 0.9 1 Screening cost, c 21 / 28

Unemployment and Wages Distribution by Ability Unemployment rate by ability: u(a) = 1 x σ(a) = 1 1 b [ ( ad ) ] k/µ 1 a for a a d 22 / 28

Unemployment and Wages Distribution by Ability Unemployment rate by ability: u(a) = 1 x σ(a) = 1 1 b [ ( ad ) ] k/µ 1 a for a a d Wage distribution by ability: F w (w a) = 1 (w d /w) 1/µ 1 [w d /w c (a)] 1/µ for w d w w c (a) b ( a Proposition Workers with higher ability have lower expected unemployment, higher average wages and higher wage inequality. Nevertheless, income inequality for workers with higher ability can be higher or lower. a min ) k 22 / 28

Wage and Income Inequality by Ability 1 Thiel Indices, T w (a) and T ι (a) 0.8 0.6 0.4 0.2 T ι (a) T w (a) 0 1 2 3 4 5 Ability Relative to Cutoff, a/a d 23 / 28

Inequality and Unemployment in a Global Economy We extend the analysis to an open two-country economy Exports entail fixed export costs f x and variable costs τ Most productive firms export (θ θ x ), less productive firms serve the domestic market (θ d θ < θ x ) and least productive firms exit (θ < θ d ) In equilibrium with trade, all firm-specific variables jump discontinuously for exporting firms (at θ x ) 24 / 28

Unemployment Rate 0.25 Sectoral Unemployment Rate, u 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 Openness, ρ θ d /θ x 25 / 28

Wage Profile 2 1.8 w x (θ) Wage Rate, w(θ) 1.6 1.4 1.2 w d (θ) w c (θ) 1 θ d θ x 1 1.5 2 2.5 Productivity, θ 26 / 28

Wage Distribution 1 0.8 G w c (w) G w (w) 0.6 0.4 0.2 w (θ ) d x w c d w d w (θ ) x x 0 1 1.5 2 2.5 Wage Rate, w 27 / 28

Inequality of Wages Theil Index of Wage Inequality 0.07 0.06 0.05 0.04 0.03 0.02 0.01 T W T=T W +T B T B 0 0 0.2 0.4 0.6 0.8 1 Openness to Trade Index, θ d /θ x 28 / 28