Notes on Differential Rents and the Distribution of Earnings

Transcription

1 Notes on Differential Rents and the Distribution of Earnings from Sattinger, Oxford Economic Papers 1979, 31(1) James Heckman University of Chicago AEA Continuing Education Program ASSA Course: Microeconomics of Life Course Inequality San Francisco, CA, January 5-7, 2016 from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 1 / 61

2 This is a version of an hedonic model. It features 1-1 matches. Assume that we can rank workers and firms by a skill scale: l is amount of labor skill, c is amount of capital owned by firm. F (l, c) is output. Assume a common production technology. One worker - one firm match F l > 0, F c > 0, F ll < 0, F cc < 0, no need to make scale restrictions. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 2 / 61

3 Can be increasing returns to scale technologies. Homogeneous output of firms, identical technologies. Let G(l) be cdf of l in population. Let K(c) be cdf of c in population. Assume both monotone strictly increasing, density has positive support no mass points. Let W (l) be wage for worker of type l. Let π(c) denote profit for a firm of type c. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 3 / 61

4 Assume 2 F > 0 (opposite sign produces negative sorting). l c Assume wage function exists. This is something to be proved. Firm indexed by c. Profit maximization requires that max(f (l, c) W (l)) l FOC: F l = W (l) SOC: 2 F l 2 W (l) < 0 Defines demand for worker of type l for firm type c. rom Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 4 / 61

5 Differentiate FOC totally with respect to l: W (l) 2 F (l, c) l 2 2 F dc l c dl = 0 ( ) ( ) W (l) 2 F (l, c) 2 F dc = l }{{ 2 l c dl }}{{} >0, from SOC + (1) dc dl > 0 ( best firms match with best workers ) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 5 / 61

6 Opposite true if we have 2 F < 0 (dc/dl < 0). l c Retain 2 F > 0 for specificity. l c Profits residually determined: π(c) = F (l(c), c) W (l(c)). Observe that the roles of l and c can be reversed (labor hires capital) and labor incomes could be residually determined. rom Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 6 / 61

7 The continuum hypothesis for skills = local returns to scale df = F l dl + F c dc we get product exhaustion locally. Residual claimant gets marginal product, no matter who is claimant. Now suppose number of workers (N l ). Number of capitalists (N c ). rom Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 7 / 61

8 Let W R be the reserve price of workers (what they could get not working in the sector being studied). Let π R be reserve price of capitalist. Let l be the least productive worker (employed). We need W (l ) W R. If all capital employed, and c [c, c], l works with assuming that π(c) π R. c, least productive capitalist from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 8 / 61

9 How to establish that decentralized wage setting is optimal and a wage function exists? Solve Social Planner s Problem. 2 F (l, c) > 0 l c maximize total output by matching the best with the best. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 9 / 61

10 Proof: trivial based on proof by contradiction Take a discrete example two workers l 1 < l 2 two firms c 1 < c 2 From complementarity (or supermodularity) F (l 2, c 2 ) + F (l 1, c 1 ) > F (l 2, c 1 ) + F (l 1, c 2 ) because F (l 2, c 2 ) F (l 1, c 2 ) > F (l 2, c 1 ) F (l 1, c 1 ) due to 2 F (l, c) l c > 0. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 10 / 61

11 Using the fact that the best matches with the best, sort top-down. Assume densities continuous (absolutely continuous). G 1 [1 ( Nc N l g(l) dl = N c k(c) dc l(c) N l (1 G(l(c))) = N c (1 K(c)) ( ) Nc (1 G(l(c))) = (1 K(c)) N ) ] l (1 K(c)) = l(c) N l This defines the optimal sorting function. c from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 11 / 61

12 Use survivor function: S(x) = Pr [X x] S G (l) = 1 G(l) S K (c) = 1 K(c) S G (l(c)) = ( Nc N l l(c) = S 1 G ) S K (c) ( Nc N l S K (c) ) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 12 / 61

13 Defines a relationship: l = ϕ(c) (most productive match with each order) This function has an inverse from strictly decreasing survivor function assumption (density has no mass points or holes). from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 13 / 61

14 Feasibility requires, using ϕ 1 (l) = c, that the lowest quality capitalist cover his/her reserve income outside the sector π(c) = F (l(c), c) W (l ) π R. If not satisfied we have unemployed capital. Jack up c > c until constraint satisfied. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 14 / 61

15 From the allocation derived from the social planner s problem, we can derive the hedonic equation (instead of assuming it). The slope of the wage function is given by FOC (using ϕ) W (l) = F l (l, ϕ 1 (l)) (the right-hand side determined by the equilibrium sorting). This defines the slope of hedonic line with a continuum of labor. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 15 / 61

16 Note that if we totally differentiate the right-hand side, W (l) = F ll <0 dc + F lc + dl + SOC satisfied, because W (l) F ll 0 as required. The marginal wage at minimum quality l satisfies W (l ) = F l (l, ϕ 1 (l )). from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 16 / 61

17 Competitive labor market forces W (l ) = W R. You cannot pay any less than reserve wage. If you pay more, all workers from the reserve will want to work in the sector being studied and hence it forces wages down. W (l) = l F l x (x, ϕ 1 (x))dx + W R. hedonic function Similarly π(c) = c c df dz (ϕ(z), z)dz + π R. (Reserve value of capital is nonnegative; π R 0.) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 17 / 61

18 Under our assumptions (more workers than firms and unemployed worker, N c > N l ), rents are assigned to firms. Density of earnings is obtained from inverting wage function w(l) = η(l) η 1 (w) = l (exists under our assumptions) Density of earnings is g(η 1 (w)) dη 1 (w) dw Density of profits obtained in a similar way. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 18 / 61

19 Cobb Douglas Example F (l, c) = l α c β, α > 0, β > 0. Assume Pareto distribution of endowments: g(l) = jl γ γ > 2, l 1 k(c) = hc σ σ > 2, c 1. This ensures finite variances. Obviously F lc > 0. The higher γ, the more equal is the distribution of l. The higher σ, the more equal is the distribution of c. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 19 / 61

20 Equilibrium: N c hx σ dx = N l jη γ dη c(l) c(l) = [ ] 1 Nl j (σ 1) 1 σ 1 γ (l) 1 σ. N c h (γ 1) l from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 20 / 61

21 FOC (for wages) αl α 1 c β = W (l). Substitute for c(l) to reach [ ] β W Nl j(σ 1) 1 σ (l) = α l P N c h(γ 1) P = (α 1)(1 σ) + β(1 γ) 1 σ [ Nl j(σ 1) α(1 σ) W (l) = N c h(γ 1) α(1 σ) + β(1 γ) ] β 1 σ } {{ } g 1 (l) 0 ( ) α(1 σ)+β(1 γ) 1 σ and where k 1 is a constant of integration, determined by W R : W (l ) W R. + k 1, from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 21 / 61

22 Obviously W (l) as l. Convexity or concavity in labor quality hinges on whether P 0 P = (α 1) + β (1 γ) 1 σ. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 22 / 61

23 If α + β = 1 (CRS) [ P = β γ ] 1 σ [ ] [ ] σ γ γ σ = β = β 1 σ σ 1 If γ > σ, W (l) is convex in l. (More firms out in tail than workers workers get scarcity payment). Firms less equally dispersed (more productive firms out in tail). If β (from CRS) reinforces effect (Renders capital relatively more productive). rom Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 23 / 61

24 If γ = σ and β + α > 1 (β big enough), P > 0 and hence produces convexity. Increasing returns to scale gives rise to convexity (scale of productivity of resources effect). from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 24 / 61

25 Profits can be written as π(c) = l α c β w(l) From the equilibrium matching condition we obtain l = g 0 (c) 1 σ 1 γ g 0 = [ ] 1 Nc h(γ 1) 1 γ N l j(σ 1) π(c) = [g 0 (c) (1 σ) (1 γ) ] α c β ( g 1 g 0 (c) (1 σ) (1 γ) ) α(1 σ)+β(1 γ) 1 σ k 1 α(1 σ) 1 γ + β = α(1 σ) + β(1 γ) 1 γ from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 25 / 61

26 π(c) = ] [g α0 g 1 (g 0 ) α(1 σ)+β(1 γ) 1 σ c α(1 σ)+β(1 γ) 1 γ k 1 For positive marginal productivity of capital, this requires that α + β(γ 1) σ 1 > [ ] γ(β 1) Nc h(γ 1) (σ 1)(γ 1) N l j(σ 1) Otherwise, coefficient on c α(1 σ)+β(1 γ) 1 γ is negative. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 26 / 61

27 π(c) = ac α(1 σ)+β(1 γ) 1 γ k 2 a = (g 0 ) α g 1 (g 0 ) α(1 σ)+β(1 γ) 1 σ > 0 (True if N c N l, for example.) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 27 / 61

28 convexity of π(c) is determined by sign of α(1 σ) + β(1 γ) 1 1 γ α(1 σ) + (β 1)(1 γ) 1 + γ = 1 γ (γ 1)(β 1) + (σ 1)α = γ 1 ( ) σ 1 = (β 1) + α. γ 1 Observe if α + β > 1 then both π(c) and W (l) can be convex in their arguments. With CRS one must be concave, the other convex. Linearity arises when we have γ = σ and α + β = 1. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 28 / 61

29 γ big relative to σ (scarcity of labor at top firms (high c firms)). α, β big scale effects we get convexity at top of distribution. Suppose we invoke full employment conditions for capital: N l > N c π(1) π R from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 29 / 61

30 We need to determine the constants for the wage equation. Minimum quality labor earns its opportunity cost outside of the sector. Rents accrue to other workers. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 30 / 61

31 At lowest level of employment, we have (from matching function c(l)) [ ] 1 Nl j(σ 1) 1 σ 1 = (l ) 1 γ 1 σ N c h(γ 1) [ ] 1 l Nl j(σ 1) γ 1 = N c h(γ 1) k 1 = W R W (l ) = W R [ ] β α(1 σ) Nl j(σ 1) 1 σ (l ) α(1 σ)+β(1 γ) 1 σ. α(1 σ) + β(1 γ) N c h(γ 1) π(c) defined residually. (Need to check π(1) > π R ). from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 31 / 61

32 Pigou s Problem: Why doesn t the distribution of earnings resemble the distribution of ability? Distribution of earnings: (generated from distribution of endowments by the pricing function). Look at distribution of translated earnings (translated around the constant k 1 ). (γ 1)(σ 1) [1+ (W (l) k 1 ) (W k 1 ) α(σ 1)+β(γ 1)] Distribution of raw skills l γ. Higher γ is associated with more equality in the distribution of labor skills. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 32 / 61

33 One way to measure the market-induced change in inequality is the change in the wage distribution from γ. Example: 1 + (γ 1)(σ 1) α(σ 1) + β(γ 1) < γ (wage inequality > inequality in l) For this to happen, 1 (γ 1) α + β (σ 1) < 1 The higher α + β, the more unequal the distribution of wages. Higher γ > σ (capital more unequally distributed) the greater the wage inequality. rom Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 33 / 61

34 If γ = σ, α + β = 1, no induced change in inequality. If γ = σ, α + β > 1, more inequality in wages than skills. If σ γ, then more inequality in wages than skills (Demand for top talent). It is not superstars but superfirms. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 34 / 61

35 The wage equation is an hedonic function. Hedonic Functions (Tinbergen, 1951, 1956; Rosen, 1974). What can you estimate when you regress W on l? Obviously we can estimate k 1, and slope coefficient (g 1 ). α(σ 1) + β(γ 1) (σ 1) Do not recover any single parameter of interest. We get lowest l in market and from distribution of l and c, we can get γ, σ, h (if c fully employed). If we assume α + β = 1 (CRS) and we observe distributions of the factors, we get σ, γ and hence α, β. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 35 / 61

36 If we know l, we can get j. If we know N l and N c, we can identify γ, σ but α, β are unknown. α + β is known. CRS α, β known. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 36 / 61

37 Idea (Rosen, 1974). Two-stage estimation procedure. Assume perfect data. Assume α 1. No error term in model, no omitted variables. Use FOC for firm, ln α + (α 1) ln l + β ln c = ln W (l) i.e., ln l = ln α α 1 + ln W (l) α 1 β ln c α 1. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 37 / 61

38 Apparently, we can regress ln l on ln W (l). Notice however that from the sorting condition, ( ) σ 1 ln l = ln g 0 + ln c. γ 1 We get no independent variation. ln W (l) is redundant. Alternatively, ln W (l) and ln c are perfectly collinear. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 38 / 61

39 More general principle: FOC: dl = 2 F l dl + 2 F 2 l c dc = dw (l) 1 ( 2 F l 2 )d[w (l)] 2 F l c 2 dc. F l 2 Functional dependence between c and W (l) does not necessarily imply linear dependence. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 39 / 61

40 we might be able to identify the model. Need shifter in regression. Functional dependence linear independence y = α 0 + α 1 X + α 2 X 2. Obviously X and X 2 only dependent but not linearly dependent. We return to this in a bit. from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 40 / 61

41 Pareto Distribution Pareto Distribution k = 1 k = 1.5 k = 3 2 PDF(X) = k. X (-1-k) X X Pareto(k) f X Pareto(k) f X (x) = k x (1+k) X (x)=k x (1+k) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 41 / 61

42 PDF(X) = (k - 1). X (-k) Pareto Distribution Pareto Distribution k = 2 k = 2.5 k = X X Pareto(k) f X Pareto(k) f X (x) = (k 1) x k X (x)=(k 1) x k from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 42 / 61

43 CDF(X) = 1 - X -k Pareto Distribution Pareto Distribution k = k = 1.5 k = (-1-k) X ~ f(x) = k. X X Pareto(k) F X Pareto(k) F X (x) = 1 x k (x)=1 x k from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 43 / 61

44 CDF(X) = 1 - X -k -1 Pareto Distribution Pareto Distribution k = k = 2.5 k = (-k) X ~ f(x) = (k -1). X X Pareto(k) X Pareto(k) F F X X (x) (x)=1 x = 1 x (k+1) (k+1) from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 44 / 61

45 PDF of L (ability) from Sattinger, Oxford Ability Economic Distribution Papers 1979, based 31(1) on [4mm] Sattinger the James parameters (1979) HeckmanUniversity above of Chicago 45 / 61 Ability Distributions Ability Distributions γ =2 γ =3 γ = L (ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

46 10%, 50%, 90% percentile Pareto Percentiles Pareto Percentiles % distribution value 50% distribution value 90% distribution value ( k) k ; X ~ f(x) = (k 1). X Pareto 10%, Pareto 50%, 10%, 50%, 90% Percentiles for k 2 for [2; 4] k [2, 4] from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 46 / 61

47 C(L), matching Capital from Sattinger, Oxford Capital/Ability Economic Papers relation 1979, 31(1) based[4mm] Sattinger onjames the(1979) HeckmanUniversity parameters above of Chicago 47 / 61 Capital/ability relation CapitalandAbilityFunction γ =2 1.1 γ =3 γ = L (ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

48 dw(l)/dl, Marginal Wage from Sattinger, Oxford Economic dw(l) Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 48 / 61 W (L) Wage derivative with Wage respect derivative to ability with respect L to Ability γ =2 γ =3 γ = L (ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

49 W(L), Wage from Sattinger, W(L): Oxford Economic wagefunctionofabilitybasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 49 / 61 Wage as a function of ability Wage function of Ability γ =2 0.5 γ =3 γ = L (ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

50 PDF of W (Wage) from Sattinger, Oxford P DF(w) Economic wagepapers distribution 1979, 31(1) based [4mm] Sattinger on James the (1979) HeckmanUniversity parametersof above Chicago 50 / 61 Wage distribution Wage Distribution γ =2 γ =3 γ = w (Wage)

51 PDF of W (Wage) from Sattinger, Oxford P DF(w) Economic wagepapers distribution 1979, 31(1) based [4mm] Sattinger on James the (1979) HeckmanUniversity parameters of Chicago above 51 / 61 Wage distribution Wage Distribution γ =2 γ =3 γ = w (Wage)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

52 PDF of W (Wage) and L (Ability) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 52 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =2 Wage Distribution for γ =3 Wage Distribution for γ =5 Ability Distribution for γ =2 Ability Distribution for γ =3 Ability Distribution for γ = w (Wage) amd l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

53 PDF of W (Wage) and L (Ability) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 53 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =2 Ability Distribution for γ = w (Wage) and l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

54 PDF of W (Wage) and L (Ability) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 54 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =2 Ability Distribution for γ = w (Wage) and l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

55 PDF of W (Wage) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 55 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =3 Ability Distribution for γ = w (Wage) and l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

56 PDF of W (Wage) and L (Ability) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 56 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =5 Ability Distribution for γ = w (Wage) and l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

57 PDF of W (Wage) and L (Ability) from Sattinger, PDF(W), Oxford Economic PDF(L)distributionsbasedontheparametersabove Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 57 / 61 Wage and ability distribution Wage and Ability Distributions Wage Distribution for γ =5 Ability Distribution for γ = w (Wage) and l (Ability)( σ=2, h=1, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

58 (10% of Wage)/median(Wage),(90% of Wage)/median(Wage) 10%of from Sattinger, Oxford Wage Economic, 90%of Papers Wage 1979, ratiosbasedontheparametersabove 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 58 / 61 Wage percentile rations Wage Percentile Ratio (90% of Wage)/median(Wage) (10% of Wage)/median(Wage) σ,(h= σ-1, γ=2, j=1, N =1, N =1, β=0.5, α=0.5 ) c l

59 (10% of Wage)/median(Wage),(90% of Wage)/median(Wage) from Sattinger, 10%of Oxford Wage Economic, 90%of Papers Wage 1979, ratiosbasedontheparametersabove 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 59 / 61 Wage percentile rations Wage Percentile Ratio (90% of Wage)/median(Wage) (10% of Wage)/median(Wage) σ,(h= σ-1, γ=4, j=3, N =1, N =1, β=0.5, α=0.5 ) c l

60 (10% of Wage)/median(Wage),(90% of Wage)/median(Wage) from Sattinger, 10%of Oxford Wage Economic, 90%of Papers Wage 1979, ratiosbasedontheparametersabove 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 60 / 61 Wage percentile rations Wage Percentile Ratio 2.5 (90% of Wage)/median(Wage) (10% of Wage)/median(Wage) σ,(h= σ-1, γ=10, j=9, N =1, N =1, β=0.5, α=0.5 ) c l

61 w(l) = θl ξ +k 1 ξ = P+1= (α 1)(σ 1)+β(γ 1) +1 σ 1 θ = α(σ 1) [ Nl j(σ 1) N ch(γ 1) ] β 1 σ (α 1)(σ 1)+β(γ 1) ( w k1 l= θ )1 ξ butf L (l) = jl γ ( w k1 f W (w)=j θ f W (w) = j ( )1 ξ γ w k1 ξ θξ θ Which is Pareto itself ) γ ξ 1 ξ ( w k1 θ )1 ξ 1 1 θ from Sattinger, Oxford Economic Papers 1979, 31(1) [4mm] Sattinger James (1979) HeckmanUniversity of Chicago 61 / 61