iv. labour market institutions and wage...

Fortin – Econ 561 Lecture 4B

IV. Labour Market Institutions and Wage Inequality

15. Decomposition Methodologies

1. Measuring the extent of inequality

2. Links to the Classic Analysis of Variance (ANOVA)


1. Measurement

The theoretical literature on income inequality has developed sophisticated measures

(e.g. Gini coefficient) on inequality according to some desirable properties such as

decomposability into income groups.

o In a diagram with the cumulative proportion of

total income on the vertical axis and the

cumulative share of the population on the

horizontal axis, the Gini coefficient is defined

as the ratio of the area between the 45 degree

line and the Lorenz curve to the total area

under the 45 degree line,

o where the Lorenz curve tracks the cumulative

total of y divided by total population size

against the cumulative distribution function

and the generalized Lorenz ordinate can be

interpreted as the proportion of earnings going

to the 100p% lowest earners.


o A Gini coefficient of 0 (Lorenz curve is 45 degree line) expresses perfect equality

where all values are the same . A Gini coefficient of 1 expresses maximal

inequality among values (for example where only one person has all the income).

More formally, the Gini coefficient is defined as

𝐺 = 1 − 2

𝜇∫ 𝐺𝐿(𝑝, 𝐹𝑌)𝑑𝑝

1

0

with p(y) = FY (y) and where 𝐺𝐿(𝑝, 𝐹𝑌) the Generalized Lorenz ordinate of FY is

given by 𝐺𝐿(𝑝, 𝐹𝑌) = ∫ 𝑧𝑑𝐹𝑌(𝑧)𝐹−1(𝑝)

−∞ .

While these measures are very useful for the purpose of cross-country comparisons,

they are less compelling when trying to assess the relative importance of competing

explanations.

o Essentially, the Gini coefficient shows relatively little change over time

for countries such as the United States and Canada.

Canadian Inequality: Recent Developments and Policy Options 123

Canadian PubliC PoliCy – analyse de Politiques, vol. xxxviii, no. 2 2012

income earners received 45 percent of total income earnings, while by 2007 they received 52 percent.3

Second, inequality rises sharply during reces-sions because it is low-income earners who bear the brunt of bad economic times. This happened in both the busts of 1981–83 and the early 1990s. We might expect inequality to then decline as we come out of recessions, but this did not happen in either the 1980s or ’90s. Instead, the level of inequality ratcheted upward over time. In the strong labour market before the downturn in 2008, inequality did decline somewhat but has been rising again in the current slowdown. If recent history is any guide, we could be witnessing another ratcheting up in inequality, though the wage patterns discussed later in this section suggest that the pattern may not repeat itself this time.

The lower line in the figure shows the Gini for after-tax and transfer family income, or disposable income. It is noteworthy how much lower this line is: taxes and transfers really can reduce inequality. In 2009, inequality in disposable income was 28 percent lower than market income inequality. To put the numbers underlying this line in perspective, in international comparisons, Canada stands roughly in the upper middle of the pack of developed coun-tries. Our Gini for disposable income is about 25 percent higher than Sweden’s but about 14 percent lower than the US value. Our growth in inequal-ity over the past three decades, though, has been substantial, albeit somewhat less than that in the United States. Between 1980 and 2009, Canada’s disposable income inequality grew by 13 percent, while south of the border inequality grew by 17.5 percent (US Census Bureau 2011).

Figure 1Canadian Inequality Trends

Source: Statistics Canada, CANSIM Table 202-0709.

0.5

0.45

0.4

0.35

0.3

0.25

Gin

i Coe

�ci

ent

1976 1981 1986 1991 1996 2001 2006 2011

Year

Market Income Disposable Income

CPPVol38No2.indb 123 31/05/12 4:13 PM

Source: Fortin, Green, Lemieux, Milligan and Riddell (2012)


Source: Bee (2012)


0.2

.4.6

.81

Cu

m.

Dis

trib

uti

on/L

og

Wag

e0 .2 .4 .6 .8 1

Cum. Pop. Prop.

Men 1988 Men 1979

45o line

The empirical literature on wage inequality has favored the use of measures that are

easy to interpret such as the 90-10, 90-50, 50-10 log wage differential and the

variance or standard deviation of log wages.

0.2

.4.6

.8

.69 1.61 2.3 3.22Log(Wage)

Men 1988 Men 1979

Minimum Wages

1979 1988


.1.2

.3.4

.5.6

.7.8

.9

Cu

mula

tive

Dis

trib

uti

on

-

Men

19

88

.69 1.61 2.3 3.22lwage

Let )( ttt wF be the percentile number of the ranking (a non-parametric measure)

of log wage tw in the cumulative wage distribution tF , then since a cumulative

distribution is monotonic, it can be inverted and )(1

ttt Fw . So that

)10()90( 111090 FFd

is the 90-10 log wage differential between the 10th and the 90th centile.

Similarly

)50()90( 115090 FFd ,

)10()50( 111050 FFd

designate the 90-50 and 50-10

log wage differential and are

meant to describe upper end

and lower end wage inequality,

respectively.


0.2

.4.6

.8

Den

sity

-

M

en 1

988

.69 1.61 2.3 3.22Log(Wage)

Men 1988 Normal Density

Figure 5: Overall U.S. Wage Inequality, 1940-98

90-1

0 W

age R

atio

Males Females

1940 1950 1960 1970 1980 1990 1998

2.7

3

3.5

4

4.5

4.8

Source: Estimates are for the weekly wages of full-time, full-year workers not employed in agriculture and earning at least half of the federal minimum wage. The estimates for 1940 to 1990 are from Katz and Autor (1999, Table 8), and the estimated changes from 1990 to 1998 are from Bernstein and Mishel (1999). The 90-10 wage ratio is the ratio of the earnings of the worker in the 90th percentile of the earnings distribution to the earnings of the worker in the 10th percentile.

Source: Katz (1999)

Figure 2. Three Measures of Wage Inequality: College/High School Premium,Male 90/10 Overall Inequality and Male 90/10 Residual Inequality

A. March CPS Full-Time Weekly Earnings, 1963 - 2002

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

Log E

arnin

gs

Rat

io

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Log C

olle

ge/

HS W

age

GapOverall 90/10

Residual 90/10

1973 1979 1992

College/HS Gap

B. MORG CPS Hourly Earnings, 1973 - 2003

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003

Log E

arnin

gs

Rat

io

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Log C

olle

ge/

HS W

age

Gap

Overall 90/10

Residual 90/10

1979 1992

College/HS Gap

Source: Autor, Katz and Kearney (2005)


To describe changes over time, plots of the percentiles of the wage distribution on

the horizontal axis and the change in the log wage on the vertical axis are used

(AKK, fig1.)

Alternatively, indexes of some chosen percentiles of the log wage distribution are

plotted (JMP, fig 1.)

DFL have used kernel density estimate of the log wage distribution and presented a

succession of plots by years (DFL, fig1a and 1b).

The standard deviation of log wage is another popular measure of wage inequality,

which is decomposable into a between and within group components.

Figure 1. Change in Log Real Weekly Wage by Percentile, Full Time Workers, 1963 - 2003 (March CPS)

0.00

0.15

0.30

0.45

0.60

0.75

0.90

0 10 20 30 40 50 60 70 80 90 100

Percentile

Male Female

Source: Autor, Katz and Kearney (2005)

Source: Juhn, Murphy and Pierce (1993)


Figure 1b. Kernel Density Estimates of Women's Real Log Wages 1973-92

1973

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1974

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1975

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1976

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1977

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25

1978

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1979

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1980

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1981

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1982

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25

1983

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1984

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1985

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1986

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1987

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25

1988

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1989

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1990

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1991

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25 1992

ln(2) ln(5) ln(10) ln(25)

0

.25

.5

.75

1

1.25


Figure 2b. Density of Women's Real Wages in 1979 and 1988

Log Wage ($1979)

ln(2) ln(5) ln(10) ln(15) ln(25)

0

.25

.5

.75

1

1.25

Min. wage 1988 Min. wage 1979

1988

1979


0.2

.4.6

.8

Den

sity

0 1 2 3 4lwage

0.2

.4.6

.8

Den

sity

0 1 2 3 4lwage

0.5

11.5

Den

sity

0 1 2 3 4lwage

𝑓(𝑥) =1

𝑛ℎ(no. 𝑜𝑓 𝑋𝑖 in the same bin as 𝑥)

(bin=10, start=.00935174, width=.44223485)

(bin=50, start=.00935174, width=.08844697)

(bin=100, start=.00935174, width=.04422348)

Source: Silverman (1986)


2. Links to the Classic Analysis of Variance

• Katz and Author (1999) refer to the following common approach to evaluate different explanation starts with a simple wage equation of the form

ittitit uBXW += , where itW is the log wage of individual i in year t , itX is a vector of observed

individual characteristics (e.g. education and experience), tB is the vector of estimated (OLS) returns to observable characteristics in t , and itu is the log wage residual.

• Then the variance of log wages can be decomposed into two components:

o a component measuring the contribution of observable prices (or returns to characteristics) and quantities (or characteristics) and

o a component measuring the effect of unobservables.


• That is, we can write )()()( ittitit uVarBXVarWVar += ,

between within given that by construction the errors are orthogonal to the predicted values (

0),( =itit uXCov ).

• Between two periods, the change in the variance of log wages can be decomposed

into the change in the variance or the predicted values (change in between-group inequality) and the change in the residual variance (change in within-group inequality) (e.g. Table 5, KA, 1999).

As the classic analysis of variance, where when treatments were applied to different groups, a relative high ratio (F-test) of between-group variance to within-group implied that the treatments had a significant effect.

• Another famous paper Juhn, Murply and Pierce (1993) set out to summarize the rising

dispersion of earnings in the U.S. during the 1970s and 1980s.

Source: Katz and Autor (1999)

Nicole

Rectangle

Nicole

Rectangle


• They wanted a tool for describing the components of wage density changes that could be attributed to measured prices, measured quantities and residuals (which they referred to as unmeasured prices and quantities).

• The results of Juhn et al. (1993) suggest that the contribution of changes in inequality

due to unobservables in more important at the bottom than at the top of the wage distribution.

• This interpretation was challenged by Lemieux (2006) who argues that the

assumption in JMP’s analysis of homoskedastic earnings residual is contributing to the larger role attributed to residuals vs. labour force composition. 1. Because within wage dispersion is substantially larger for older and more educated

workers than for younger and less educated works, he shows that a large fraction of the increase in residual wage inequality is a spurious consequence of the fact that the work force has grown older and more educated since the early 1980s.

2. JMP's procedure consists of replacing each period t residual by a period s residual at the same position in the residual wage distribution, thus by assumption, it imposes that the growth in the residual variance is solely due to changes in skill prices.


• Consider a simple model with only schooling,

where is individual I schooling level, there a rise in the return to ability (that is a rise in ), will raise between-group inequality but will not not raise residual inequality.

• Now, assume that is not observed. Instead we proxy for using a measure of

human capital such as schooling, .

• Assume that , where is an iid error term. If we estimate the model

• the variance of this expression will be

between within • Thus, if ability is imperfectly measured, a rise in the returns to ability will cause

both between and within-group inequality to rise.

• Under this single index assumption, these two error terms ought to move together. Is this what happened?

VOL. 96 NO. 3 LEMIEUX: INCREASING RESIDUAL WAGE INEQUALITY 485

TABLE 3--ESTIMATES OF MEASUREMENT ERROR IN THE MAY/ORG AND MARCH CPS

Men Women

May/ORG March May/ORG March

1. Average measurement error variance (1976-2003)* a. Paid by the hour 0.017 0.087 0.024 0.077

[8.4] [33.2] [14.3] [35.0] b. Not paid by the hour 0.052 0.065 0.045 0.054

[16.8] [20.4] [19.4] [23.2] 2. 1976-2003 change in measurement error variance

a. Paid by the hour 0.006 0.020 0.011 0.016 b. Not paid by the hour 0.000 0.017 0.015 0.011

3. Spurious change in variance due to a. Growth in fraction of hourly workers** -0.004 0.002 -0.003 0.003 b. Growth in measurement error variance*** 0.004 0.018 0.013 0.014 c. Total (3a + 3b) 0.000 0.020 0.010 0.017

4. 1976-2003 change in residual variance 0.046 0.079 0.057 0.074 5. Change adjusted for measurement error (4 - 3c) 0.046 0.059 0.047 0.057

Note: Measurement error estimated using the matched March-May/ORG sample. See text for detail. * Numbers in square brackets represents the percentage of the overall variance of wages due to measurement error. ** Based on Figure 8, it is assumed that the growth in the fraction of workers paid by the hour is 10 percent for men and

15 percent for women. These proportions are then multiplied by the difference in the estimated measurement error variances for hourly (row la) and nonhourly (row lb) workers.

*** Change in the weighted average of the measurement error variances for hourly and nonhourly workers.

Appendix Figure 3A shows the estimated measurement error variances for men paid by the hour and not paid by the hour. Appendix Figure 3B reports the same estimates for women. As expected from Figure 5, the measurement error variances for nonhourly workers are comparable in the March and May/ORG CPS. Table 3 shows that measurement error accounts for about 20 percent of the variance of wages for these workers. Also, as expected, the measurement error variance for hourly workers is much larger in the March than in the May/ ORG CPS. Measurement error represents about a third of total variance of wages in the March CPS compared to only about 10 percent of the total variance of wages in the May/ORG CPS. Interestingly, the measurement error variance for nonhourly workers lies more or less in between the measurement error variance for hourly workers in the May/ORG and March CPS. This is inconsistent with the suggestion of Autor et al. (2005) that the variance of measurement error is the same for hourly and nonhourly workers in the March CPS.

The estimates suggest that the growth in the fraction of workers paid by the hour both biases upward the growth in inequality in the March CPS, and biases downward the growth in inequality in the May/ORG CPS. The magnitude

of these biases is shown in row 3A of Table 3 under the assumption that the fraction of hourly workers increased by 10 percent for men and 15 percent for women (see Figure 8).

Interestingly, Appendix Figure 3 and Table 3 (rows 2A and 2B) also indicate that the measurement error variance generally has been growing over time. This suggests that part of the increase in residual wage inequality is simply a consequence of the fact that wages are increas- ingly badly measured in both the March and the May/ORG CPS. Row 3B of Table 3 shows that, for men, the variance of measurement error increased by 0.018 in the March CPS compared to 0.004 in the May/ORG CPS. For women, the corresponding measurement error variances grew by 0.014 (March CPS) and 0.013 (May/ ORG CPS). Since the two sources of measurement error go in opposite directions for men in the May/ORG CPS, the adjusted change in the residual variance (row 5) is the same as the change unadjusted for measurement error (row 4). By contrast, the adjusted change is system- atically smaller than the unadjusted change in the March CPS, as both sources of measurement error tend to inflate the growth in the residual variance.

I conclude from this detailed examination of the measurement of hourly wages in the CPS

Nicole

Rectangle

Nicole

Rectangle


• The regressions in Table 3 of Lemieux (2006) test the single index hypothesis for the consistency of within and between-group trends using both May/MORG and March CPS and find that the May/MORG accepts this hypothesis while the March data reject it.

• The often-discussed trends in residual inequality are less robust across data sources

than trends in between-group inequality. • Hence, hypotheses for the growth of inequality that hinge critically on the timing of

residual versus between-group inequality are also somewhat fragile.

• A further issue concerning the decomposition of changes in wage inequality into observables and unobservable components is the extent to which changes in between-group wage inequality reflects changes in the returns to observed skills (wage structure effects) as opposed to changes in the distribution of worker characteristics (composition effects).

o i.e. separating the effects of the Δ from the ΔX.

iv. labour market institutions and wage...

Documents