Assortative Matching with Large Firms

Transcription

1 Assortative Matching with Large Firms Jan Eeckhout 1 and Philipp Kircher 2 1 UCL and UPF 2 EUI LACEA, Medelĺın November 10, 2016

2 Motivation Two cornerstones of analyzing firms in Macro, Labor, IO, Trade, Firm size: productive firms are larger and produce more 2. Sorting of workers: firms compete for skilled workers These two aspects are usually treated independently 1. Firm Size (Lucas 1978, Hopenhayn 1992) intensive margin 2. Matching: one-to-one (e.g. Becker 1973) extensive margin Needed: Trade-off better workers vs. more workers

3 Motivation Two cornerstones of analyzing firms in Macro, Labor, IO, Trade, Firm size: productive firms are larger and produce more 2. Sorting of workers: firms compete for skilled workers These two aspects are usually treated independently 1. Firm Size (Lucas 1978, Hopenhayn 1992) intensive margin 2. Matching: one-to-one (e.g. Becker 1973) extensive margin Needed: Trade-off better workers vs. more workers Apply theory to technological change: SBTC vs. QBTC

4 The Model Intensive and Extensive Margin Population Workers of type x X = [x, x], distribution H w (x) Firms of types y Y = [y, y], distribution H f (y) Production of firm y F (x, y, l x, r x ) l x workers of type x,r x fraction of firm s resources F increasing in all, concave in last two arguments F constant returns to scale in last two arguments Denote: f (x, y, θ) = rf (x, y, lr ), 1, where θ = l r

5 The Model Intensive and Extensive Margin Population Workers of type x X = [x, x], distribution H w (x) Firms of types y Y = [y, y], distribution H f (y) Production of firm y F (x, y, l x, r x ) l x workers of type x,r x fraction of firm s resources F increasing in all, concave in last two arguments F constant returns to scale in last two arguments Denote: f (x, y, θ) = rf (x, y, lr ), 1, where θ = l r Key assumption: no peer effects satisfies GS Total output: F (x, y, l x, r x )dx

6 The Model Intensive and Extensive Margin Population Workers of type x X = [x, x], distribution H w (x) Firms of types y Y = [y, y], distribution H f (y) Production of firm y F (x, y, l x, r x ) l x workers of type x,r x fraction of firm s resources F increasing in all, concave in last two arguments F constant returns to scale in last two arguments Denote: f (x, y, θ) = rf (x, y, lr ), 1, where θ = l r Key assumption: no peer effects satisfies GS Total output: F (x, y, l x, r x )dx Preferences transferable utility (additive in output goods and numeraire)

7 The Model Equilibrium Hedonic wage schedule w(x) taken as given. Optimization: Firms maximize: ( max ) lx,r x [F (x, y, lx, r x ) w(x)l x ]dx r x > 0 only if x, lx r x = arg max f (x, y, θ) θw(x) ( ) Feasible Resource Allocation (market clearing) under PAM: x x h w (s)ds = y µ(x) θ(s)h f (s)ds Competitive Equilibrium: optimality + market clearing

8 Assortative Matching Proposition (Condition for PAM) A necessary condition to have equilibria with PAM is that F xy F lr F yl F xr holds along the equilibrium path. The reverse inequality entails NAM.

9 Assortative Matching F xy F lr F yl F xr Interpretation (F lr > 0 by assumption): 1. F xy > 0: bet. manag. produce more w/ bet. workers (Becker) 2. F yl > 0: bet. manag., larger span of control (as in Lucas) 3. F xr > 0: bet. workers produce more w/ manag. time

10 Assortative Matching F xy F lr F yl F xr Interpretation (F lr > 0 by assumption): 1. F xy > 0: bet. manag. produce more w/ bet. workers (Becker) 2. F yl > 0: bet. manag., larger span of control (as in Lucas) 3. F xr > 0: bet. workers produce more w/ manag. time Quantity-quality trade-off by firm y with resources r: 1. F xy : better manager manages quality workers better vs. 2. F yl : better managers can manage more people Marginal increase of better marginal impact of more workers

11 Sketch of Proof of PAM-Condition Assume PAM allocation with resources on (x, µ(x), θ(x)). Must be optimal, i.e., maximizes: max f (x, µ(x), θ) θw(x). x,θ First order conditions: f θ (x, µ(x), θ(x)) w(x) = 0 f x(x, µ (x), θ(x)) θ(x)w (x) = 0 The Hessian is ( f Hess = θθ f xθ w (x) Second order condition requires Hess 0: f xθ w (x) f xx θw (x) ). f θθ [f xx θw (x)] (f xθ w (x)) 2 0 Differentiate FOC s with respect to x, substitute: µ (x)[f θθ f xy f yθ f xθ + f yθ f x/θ] 0 Positive sorting means µ (x) > 0, requiring [ ] < 0 and after rearranging: F xy F lr F yl F xr

12 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr

13 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr Multiplicative Separability F (x, y, l, r) = A(x, y)b(l, r) sorting if AAxy BB lr A x A y B l B r 1 AA If B is CES with elast. of substitution ɛ: xy A x A y ɛ (root-sm)

14 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr Multiplicative Separability F (x, y, l, r) = A(x, y)b(l, r) sorting if AAxy BB lr A x A y B l B r 1 AA If B is CES with elast. of substitution ɛ: xy A x A y ɛ (root-sm) Becker s one-on-one matching F (x, y, min{l, r}, min{r, l}) = F (x, y, 1, 1) min{l, r}, Like inelastic CES (ɛ 0), so sorting if F 12 0

15 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr Multiplicative Separability F (x, y, l, r) = A(x, y)b(l, r) sorting if AAxy BB lr A x A y B l B r 1 AA If B is CES with elast. of substitution ɛ: xy A x A y ɛ (root-sm) Becker s one-on-one matching F (x, y, min{l, r}, min{r, l}) = F (x, y, 1, 1) min{l, r}, Like inelastic CES (ɛ 0), so sorting if F 12 0 Sattinger s span of control model { r F (x, y, l, r) = min };, l write as CES between both arguments t(x,y) Our condition converges for inelastic case to log-supermod. in qualities

16 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr Multiplicative Separability F (x, y, l, r) = A(x, y)b(l, r) sorting if AAxy BB lr A x A y B l B r 1 AA If B is CES with elast. of substitution ɛ: xy A x A y ɛ (root-sm) Becker s one-on-one matching F (x, y, min{l, r}, min{r, l}) = F (x, y, 1, 1) min{l, r}, Like inelastic CES (ɛ 0), so sorting if F 12 0 Sattinger s span of control model { r F (x, y, l, r) = min };, l write as CES between both arguments t(x,y) Our condition converges for inelastic case to log-supermod. in qualities Extension of Lucas span of control model F (x, y, l, r) = yg(x, l/r)r, sorting only if good types work less well together ( g 1g 22 g 2g 12).

17 Special Cases Efficiency Units of Labor Skill = Quantity: F (x, y, l, r) = F (y, xl, r) F xy F lr = F yl F xr Multiplicative Separability F (x, y, l, r) = A(x, y)b(l, r) sorting if AAxy BB lr A x A y B l B r 1 AA If B is CES with elast. of substitution ɛ: xy A x A y ɛ (root-sm) Becker s one-on-one matching F (x, y, min{l, r}, min{r, l}) = F (x, y, 1, 1) min{l, r}, Like inelastic CES (ɛ 0), so sorting if F 12 0 Sattinger s span of control model { r F (x, y, l, r) = min };, l write as CES between both arguments t(x,y) Our condition converges for inelastic case to log-supermod. in qualities Extension of Lucas span of control model F (x, y, l, r) = yg(x, l/r)r, sorting only if good types work less well together ( g 1g 22 g 2g 12). Spacial sorting in mono-centric city: F (x, y, l, r) = l(xg(y) + v(r/l)) higher earners in center.

18 Firm Size, Assignment, Wages Proposition Under assortative matching (symmetric distributions of x, y) PAM : θ (x) = F yl F xr F lr ; µ (x) = 1 θ(x) ; w (x) = F x θ(x), NAM : θ (x) = F yl + F xr F lr ; µ (x) = 1 θ(x) ; w (x) = F x θ(x),

19 Firm Size, Assignment, Wages Proposition Under assortative matching (symmetric distributions of x, y) PAM : θ (x) = F yl F xr F lr ; µ (x) = 1 θ(x) ; w (x) = F x θ(x), NAM : θ (x) = F yl + F xr F lr ; µ (x) = 1 θ(x) ; w (x) = F x θ(x), Corollary Under assortative matching, better firms hire more workers if and only if along the equilibrium path F yl > F xr under PAM, and F yl < F xr under NAM.

20 Application: SBTC vs. QBTC How has technology changed: ? Estimate technological parameters that affect size and sorting F (x, y, l, 1) = (ω x x σ 1 σ ) + ω y y σ 1 σ σ 1 σ l ω l. Distribution of types x and y assumed log-normal Estimate parameters ω x, ω y, ω l, σ with parameters of type distributions to match 3 moment conditions: 1. size-wage 2. size-profits 3. size distribution German administrative data for matched employer-employees

21 Results Targeted Moments Firm size, θ 5 4 Firm size, θ 5 4 CDF Wages, w Profits, π Firm size, θ Wages-firm size Profits-firm size Firm size distribution

22 Results Targeted Moments Firm size, θ 5 4 Firm size, θ 5 4 CDF Wages, w Profits, π Firm size, θ Wages-firm size Profits-firm size Firm size distribution

23 Results Estimated Parameters F (x, y, l, 1) = (ω x x σ 1 σ ) + ω y y σ 1 σ σ 1 σ l ω l % change Technology ω x % ω y % ω l % σ % Distributions x LN (2.49, 1.35) LN (2.69, 1.35) y LN (0.08, 1.57) LN (0.03, 1.54)

24 Results Estimated Parameters x - Worker skill y - Firm productivity The Distributions of Worker Types x and Firm Types y.

25 Results Estimated Parameters F (x, y, l, 1) = (ω x x σ 1 σ ) + ω y y σ 1 σ σ 1 σ l ω l % change Technology ω x % ω y % ω l % σ % Distributions x LN (2.49, 1.35) LN (2.69, 1.35) y LN (0.08, 1.57) LN (0.03, 1.54)

26 Results Technology σ < 1 PAM σ 1, technology can be approximated by the Cobb-Douglas F (x, y, l, 1) x ωx y ωy l ω l. but not σ = 1: No sorting!

27 Results Estimated Parameters F (x, y, l, 1) = (ω x x σ 1 σ ) + ω y y σ 1 σ σ 1 σ l ω l % change Technology ω x % ω y % ω l % σ % Distributions x LN (2.49, 1.35) LN (2.69, 1.35) y LN (0.08, 1.57) LN (0.03, 1.54)

28 Results Technological Change ω x 136%: Skill-biased Technological Change (SBTC) ω l 76%: Quantity-biased Technological Change (QBTC) ω y unchanged (1 σ) 14 : Increase in complementarity between x, y

29 Results Complementarities 2.0 1e e F xy 1.0 F lr x & µ(x) x & µ(x) F xy F lr

30 Results Complementarities 1.6 1e e F yl 0.8 F xr x & µ(x) x & µ(x) F yl F xr

31 y Results Firm Size, Allocation, Skill Premium w`(x) log(firm size) x x Size Distribution Allocation Skill Premium w (x)

32 Results Firm Size, Allocation, Skill Premium 1. There is both SBTC and QBTC 2. FOSD in firm size distribution and shift in allocation 3. Skill premium, but polarization (Goos-Manning, Autor-Dorn) 4. SBTC and QBTC interact SBTC increases skill premium QBTC decreases skill premium (concave production) Skill premium increase dampened by QBTC

33 Counterfactuals 1996 economy with one 2010 parameter Median Firm Size % change 1996 Average w (x) % change % % 2010 ω x % % 2010 ω y % % 2010 ω l % % 2010 σ % % 2010 Distributions % %

34 Conclusion Assortative matching with large firms: intensive and extensive margin A simple condition for sorting; nests many known models Equilibrium allocation: system of 3 differential equations Application: Technological Change 1. both SBTC and QBTC 2. effect of QBTC on skill premium: negative 3. effect of SBTC on skill premium would have been 4 times larger