Chapter 5 Trends in between- and within-group
5.3.3 The effect of supply changes
In order to incorporate changes in the relative supply of different skill groups, a slightly more complicated production technology must be specified. A simple assumption is a CES- production function with M groups and Nht workers in each group, all providing labor input Xit. Further, assume that workers are perfect substitutes within same sex-education-experience cell but imperfectly substitutable across groups. This is a special case of the two level CES- production function (Hamermesh 1993, p. 39, eq. 2.41).
( )
Yt iht Xiht
i N
h
M ht
=
=
= ∑
∑ exp
/
θ
ρ ρ
1 1
1
(9)
Assuming that the relative labor supply is exogenous7, the wage of an individual i is determined by the marginal productivity at full employment:
( )
( )
( ) ( )
( ) ( )
w Y
X Y
iht t iht
X iht
i N
h M
iht i
N
iht
t i iht
N iht
iht
ht ht
ht
= =
×
=
= = =
−
=
−
=
−
∑
∑ ∑
∑
∂
∂ ρ θ ρ θ θ
θ θ
ρ ρ ρ ρ
ρ 1
1 1
1
1
1
1
1
1 exp exp exp
exp
exp
/
.
(10) Taking logarithms and using approximation i exp( )θih hθh
N=h =N
∑ 1 yields
( ) ( ) [ ( ) ( ) ]
log wiht = −1 ρ log Yt −θht −log Nht +θiht. (11)
Mean log wages in cell h at period t are then
( ) ( ) [ ( ) ( ) ]
log wht = −1 ρ logYt −θht −log Nht +θht, (12)
and log wages of qth quantile of cell h
( ) ( ) [ ( ) ( ) ] ( )
log whtq = −1 ρ logYt −θht −log Nht +θht +zq ⋅s aiht +εiht . (13) s() denotes the standard deviation of the within-cell wage distribution. Let us suppose again that the relative productivity is related to the relative productivity at the base period by a linear transformation θt = αt + βt θ0. Using the expressions above, cell means and quantiles of wages can then be written as a function of the base period wages and supply changes
( ) ( ) ( ) [ ]
log wht =const+βtlog wh0 − −1 ρ logNht −logNh0 (14)
( ) ( ) ( ) [ ]
log whtq =const+βt log whq0 − −1 ρ logNht −βt logNh0 + ⋅sh zq ⋅δh. (15)
7This is not very far from reality in Finland were the supply of educated labour mainly depends on the number of slots in the government-run education system.
Both of the above equations are nonlinear in parameters, but with β close to one, the terms in square brackets can be reasonably approximated with changes in log supplies.
The equations (14) and (15) are the equations to be estimated. Extending the single-skill model to incorporate supply effects simply adds in the equations an additional cell-specific term that reflects changes in the relative supplies. If the productive value of the skills increases (βt > 1) and relative supplies stay constant, both the between-group wage differences and the within-group wage dispersion increase. However, a change in relative supply only affects between-group differences with no effects on the within-group differences. Note also that the supply change identifies the intergroup elasticity of substitution, σ (Hamermesh 1993):
( )
−1/σ = − −1 ρ . (16)
5.4 Empirical results
To estimate the augmented single-skill model, I first divided the data from each survey into 48 cells based on sex, six education categories8 and ten-year age-intervals9. The self- employed and those who had worked for less than six months in the survey year were excluded from the data. For this restricted sample, a measure of monthly wage was constructed based on the annual taxable wage and salary income and information on the number of months spent at paid employment, counting every month in part-time work as half a month. The individuals whose calculated monthly earnings were less than 3000 marks in the 1995 prices were dropped. This restricted sample, weighted by the sampling weights, was used to calculate means and quantiles of the cell wages.
In order to calculate changes in supply, fever restrictions were applied. Following Katz and Murphy (1992), all members of the labor force were included, no matter whether they were employees, self-employed or unemployed. A weighted sum of months in the labor force over all individuals in the cell was used as a measure of labor supply. Normalizing the labor
8Those with postgraduate education are merged with those with Master's level education.
9Groups are 25-34, 35-44, 45-54 and 55-64. The youngest (below 25) and the oldest workers (over 64) are excluded.
supplies in the cells by dividing each by the total supply in each year resulted to a cell employment share, a log of which was included in the estimated equations. Changes in the supply measure are, therefore, driven by the changes in the age structure, the changes in the labor force participation rate and the changes in the education composition of the successive cohorts. On the other hand, changes in the demand conditions only affect this supply measure if the changes in employment opportunities induce changes in participation.
There are some obvious problems in explaining mean cell wages with mean the cell wages in the previous survey and change in employment shares. Both mean log wages and employment shares were estimated from sample data. Both estimates contain sampling errors that may not be negligible since the cohort size is relatively small. The problem is potentially more serious in the employment shares, since they enter into the equation as differences. Errors in the explanatory variables cause bias in the estimates. However, error variances for the estimators can be estimated from the data, and these estimates can then be used to derive consistent estimators using errors in variables techniques (Deaton 1985).
Following this strategy, I first estimated the sampling variances of the estimators using standard formulas for variances of means in complex surveys.10 The across-cell average of the error variance of means provided an estimate for measurement error in cell means.
V
M N s
t
ht ht h
= M
∑=
1 1 2
1
where sht2 is the within-cell variance of log wages in cell h A sampling variance of wage quantiles was similarly calculated using a normal approximation.
V
M N k s
t
ht q
ht q
h
= ∑M ∑
=
1 3
1 2
1
where kq =q(1−q) / (φ zq)2 and zq is the qth quantile of the standard normal distribution and φ(.) the density of the standard normal.
10The survey mean estimator is programmed in STATA and takes in the account of stratification, clustering and sampling weights.
To estimate the sampling variance in the change of supply, first the sampling variance of the sum of months in the labor force were calculated. The delta method was then used to derive the sampling error in the change of log supplies.
Var N N Var N
N
Var N
ht h N
ht ht
h h
(log log ) ( ) ( )
− 0 = 2 + 0
0 2
Finally, the estimates of the sampling errors and the across-cell variances were used to calculate reliabilities of the explanatory variables and the regressions were run with a standard measurement error correction with known (estimated) reliability ratios (Fuller, 1987). The method is computationally somewhat simpler than the one used by Card and Lemieux (1996), where an estimate of a matrix of measurement errors is deducted from an observed moment matrix. The only difference is that relying on reliabilities ignores covariances between the sampling errors in different variables, which should have little practical importance.