Notes on Heterogeneity, Aggregation, and Market Wage Functions: An Empirical Model of Self-Selection in the Labor Market

Transcription

1 Notes on Heterogeneity, Aggregation, and Market Wage Functions: An Empirical Model of Self-Selection in the Labor Market Heckman and Sedlacek, JPE 1985, 93(6), James Heckman University of Chicago Econ 345 January / 31

2 The Model There are two sectors, i = 1, 2. Each agent is endowed with skills s R J +. The population distribution of s is g (s Θ) where Θ is a vector of parameters. Agents cannot invest in order to change skills s. t i (s) is a function that expresses the amount of sector i specific tasks a worker with endowment of skills s can perform. 2 / 31

3 The Model T i is total amount of tasks employed in sector i. A i is the vector of non-labor inputs in sector i. F (i) (T i, A i ) is the aggregate output of sector i F (i) is twice continuously differentiable, increasing and concave. Furthermore, F (i) (0, A i ) = F (i) (T i, 0) = F (i) (0, 0) = 0. P i is the price of sector i output. π i is the price of one unit of sector i specific task. 3 / 31

4 The Model Optimization implies: An agent with endowment s works in sector i if: π i = P i F (i) T i (1) π i t i (s) π j t j (s), i, j = 1, 2 and i j. (2) Let L i denote the set of agents working in sector i : L i = {s : π i t i (s) π j t j (s), i j}. 4 / 31

5 The Model The log wage in sector i of an individual with endowment s is: ln w i (s) = ln π i + ln t i (s) (3) The proportion of the population working in sector i is: pr (i) = g (s Θ) ds, i = 1, 2 L i Roy assumes that g (s Θ) and t i (s) are such that: [ ] [( ) ] ln t1 (s) µ1 N, Σ ln t 2 (s) µ 2 5 / 31

6 The Model In the Roy model agents choose between two possible wages: or ln w 1 = ln π 1 + µ 1 + u 1 ln w 2 = ln π 2 + µ 2 + u 2 Workers enter sector 1 if ln w 1 ln w 2. Otherwise they enter sector 2. 6 / 31

7 The Model Let σ = var (u 1 u 2 ) ( ) π ln i π j + µ i µ j c i =, i, j = 1, 2 and i j. σ ( ) λ (c i ) = ci ( 1 2π 1 2π e 1 2 c2 i ) e 1 2 x2 dx Note that λ (c i ) is a convex monotone decreasing function of c i with λ (c i ) 0, lim λ (c i) =, lim λ (c i) = 0. c i c i 7 / 31

8 The Model The mean of log wages observed in sector i is: ( σii σ ij E (ln w i ln w 1 ln w 2 ) = ln π i + µ i + σ ) λ (c i ) (4) The variance of log wages observed in sector i is: var (ln w i ln w 1 ln w 2 ) = σ ii [ ρ 2 i (1 c i λ (c i ) λ 2 (c i )) + (1 ρ 2 i ) ] (5) The linear projection of ln t 2 conditional on ln t 1 is: ln t 2 = µ 2 + σ 12 σ 11 (ln t 1 µ 1 ) + ε 2 (6) ] where E (ε 2 ) = 0, var (ε 2 ) = σ 22 [1 σ2 12 σ 11 σ / 31

9 The Model 9 / 31

10 The Model 10 / 31

11 Estimating the Model The goal is to estimate: a) parameters of the distribution of tasks g and functions t i. b) parameters of the sectoral demand functions. 11 / 31

12 Estimating the Model The data available is: (i) time-series data on the aggregate amount of compensation paid to workers in each sector. (ii) microeconomic repeated cross-section data on the wages of workers by sector and their associated demographic and productivity characteristics (iii) time-series data on sectoral determinants of the demand for tasks. 12 / 31

13 Estimating the Model The task function is: The log real wages are: ln t i = β i X + u i, i = 1, 2 (7) ln w i = ln π i + β i X + u i, i = 1, 2 (8) Unless σ ii σ ij = 0, OLS estimators are inconsistent because of selection bias. 13 / 31

14 Estimating the Model The intercept of equation (8) combines two parameters: the log of the real price of task i, ln π i, and the intercept of the task function, denoted β 0i. To obtain the quantities of log task employed in each sector in each period, subtract the estimated intercept from the log real wage bill in each sector i, ln WB i. This produces an estimated of labor aggregate ln T i up to a known additive constant β 0i. 14 / 31

15 Estimating the Model Let l denote a year subscript, assuming that the aggregate derived demand for tasks is loglinear in aggregate tasks and real task prices, write: ( ) ( ) πil PAl ln T il = δ 0i + δ 1i ln + δ 2i ln + e il (9) where: e il is mean zero stationary stochastic process P Al is a vector of real prices for other inputs P il is the real price of output of sector i at time l. P il P il 15 / 31

16 Estimating the Model Set π i1 = 1, 2 defines the units of tasks T il. Using WB il = π il T il, write (9) as: ( ) WBil ln = [δ 0i β 0i (1 + δ 1i )] (10) P il where ln ˆπ il is the estimated ln π i. + (1 + δ 1i ) (ln ˆπ il ln P il ) ( ) PAl +δ 2i ln + e it Because aggregate shocks e il affect P il and π il, OLS is inconsistent in (10). P il 16 / 31

17 Concluding Remarks on the Roy Model When the Roy model is fit on CPS earnings data disaggregated into manufacturing and nonmanufacturing sectors, it is rejected: 1 The proportionality hypothesis is rejected. 2 χ 2 goodness-of-fit strongly rejects distributional assumptions. 17 / 31

18 The Model 1 Assume workers maximize utility. 2 Decompose earnings into hourly wages rates and hours of work. 3 General nonnormal model for (u 1, u 2 ) that nests Roy s model as a special case. 4 Incorporates nonmarket sector as an alternative market. 18 / 31

19 The Model In place of task function (7), consider: t λ i i 1 λ i = β i X + u i (11) Random variable u i is equated to an underlying mean zero normal random variable ui for values of that variable that produce positive values of t i, that is, u i = ui if 1 + λ i (β i X + u i ) 0 (12) When λ i = 0 equation (11) specializes to the Roy s (7). By estimating λ one can determine whether or not the lognormal Roy model fits the data. 19 / 31

20 The Model Let V i denote the utility of participating in sector i, where i = 1, 2, 3, where i = 3 designates the nonmarket sector. An agent chooses to participate in sector i if, and only if: V i > V j, i j, i = 1, 2, 3. (13) Let Z i denote a vector of measured sector-specific consumption attributes and household characteristics variables. 20 / 31

21 The Model Let f = (Z, X, ln π i ). The reduced form linearized index function: ln V i = γ i f + υ i, i = 1, 2, 3 (14) Assume that f is distributed independently of all the υ i and that (υ 1, υ 2, υ 3 ) is a mean zero multivariate normal random variable: (υ 1, υ 2, υ 3 ) N (0, Σ υ ) (15) This specification produces a multivariate probit model. 21 / 31

22 The Model Since only sectoral choices and not the V i are directly measured, it is possible to identify only parameters of the contrasts of utility evaluations among sectors. Without any loss of generality we normalize V 3 = 0 so γ 3 = 0 and υ 3 = 0. Using this convention, sector i is chosen if ln V i ln V j > 0 (γ i γ j ) f + (υ i υ j ) > 0 for all i j (16) If there is at least one nondegenerate regressor in f, it is possible to identify γ 1, γ 2, var (υ 2 ), and cov (υ 1, υ 2 ). 22 / 31

23 Estimates of the Extended Roy Model Estimates of the Model Parameters 23 / 31

24 Estimates of the Extended Roy Model Estimates of the Model Parameters (cont.) 24 / 31

25 Estimates of the Extended Roy Model Estimates of the Model Parameters (cont.) 25 / 31

26 Estimates of the Extended Roy Model Non-manufacturing sector: predicted vs. observed log wage distribution 26 / 31

27 Estimates of the Extended Roy Model Manufacturing sector: predicted vs. observed log wage distribution 27 / 31

28 Estimating the Demand for Aggregate Sector-Specific Tasks Demand functions for aggregate tasks (Eq. [19]) 28 / 31

29 Exploring the Importance of Aggregation Bias in Aggregate Wages Simulation of a 1% increase in the energy price index 29 / 31

30 Assessing the Impact of Self Selection on Inequality in log Wages Assessing the impact of self-selection on the means and variances of log wage rates for white males, / 31

31 Summary 31 / 31