More formally, the Gini coefficient is defined as. with p(y) = F Y (y) and where GL(p, F ) the Generalized Lorenz ordinate of F Y is ( )

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1 Fortin Econ 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 p% lowest earners.

2 Fortin Econ 56 o A Gini coefficient of (Lorenz curve is 45 degree line) expresses perfect equality where all values are the same. A Gini coefficient of expresses maximal inequality among values (for example where only one person has all the income). More formally, the Gini coefficient is defined as G = 2 μ GL(p, F )dp with p(y) = F Y (y) and where GL(p, F ) the Generalized Lorenz ordinate of F Y is given by GL(p, F ) = ( ) zdf (z). 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.

3 Source: Osberg (23).45.4 Figure Income Inequality in OECD - 2 Gini Index of Equivalent Disposable Income Iceland Slovenia Norway Denmark Czech Republic Finland Slovak Republic Belgium Austria Sweden Luxembourg Germany Netherlands France Poland Korea Estonia Italy Canada Australia Greece Spain United Kingdom Portugal Israel United States Mexico Source: data extracted on 6 Aug 23 6:4 UTC (GMT) from OECD.Stat

4 Fortin Econ 56 Minimum Wages Cum. Distribution/Log Wage Log(Wage) Men 988 Men Cum. Pop. Prop. Men 988 Men o line The empirical literature on wage inequality has favored the use of measures that are easy to interpret such as the 9-, 9-5, 5- log wage differential and the variance or standard deviation of log wages.

5 Fortin Econ 56 Let t Ft ( wt ) be the percentile number of the ranking (a non-parametric measure) of log wage w t in the cumulative wage distribution F t, then since a cumulative distribution is monotonic, it can be inverted and w ( ). So that t F t 9 d F (9) F () is the 9- log wage differential between the th and the 9 th centile. t Cumulative Distribution - Men Similarly 9 5 d F (9) F (5), 5 d F (5) F () designate the 9-5 and 5- log wage differential and are meant to describe upper end and lower end wage inequality, respectively lwage

6 Fortin Econ 56 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, fig.) Alternatively, indexes of some chosen percentiles of the log wage distribution are plotted (JMP, fig.) DFL have used kernel density estimate of the log wage distribution and presented a succession of plots by years (DFL, figa and b). Density - Men Log(Wage) Men 988 Normal Density

7 Fortin Econ ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) ln(2) ln(5) ln() ln(25) Figure b. Kernel Density Estimates of Women's Real Log Wages

8 Fortin Econ 56 Min. wage 988 Min. wage ln(2) ln(5) ln() ln(5) ln(25) Log Wage ($979) Figure 2b. Density of Women's Real Wages in 979 and 988

9 Weighted Kernel Density Estimation A kernel density estimator can be viewed as a smoothed histogram The kernel density estimate f h of a univariate density f based on a random sample W,..., W n of size n, with weights θ,..., θ n ( i θ i = ), is calculated as follows n ( ) θ f h (w) = i w Wi h K, h i= where h is the bandwidth and K( ) is the kernel function. A critical issue in kernel density estimation is the choice of bandwidth h, which determined the degree of smoothing. The Sheater and Jones (99) plug-in method is a preferred method; it has been shown (Park and Turlach, 992) be among the best data-driven bandwidth selectors for densities with complex structures. The choice of kernel function which give more (or less) weight to observations close (or further) to w i is also important; a common choice is the Gaussian function, but the Epanichov has also given good results. When the data carry sample weights these will be θ i ; alternatively, they can be multiplied by usual hours of work and normalized to add up to one to provide an hours-weighted density.

10 Fortin Econ 56 Density Density lwage (bin=, start=.93574, width= ) lwage f (x) = (no. of X in the same bin as x) Density lwage (bin=5, start=.93574, width= ) (bin=, start=.93574, width= )

11 Fortin Econ 56 The standard deviation (or variance) of log wage is another popular measure of wage inequality, which is decomposable into a between and within group components. Katz and Author (999) refer to the following common approach to evaluate different explanation starts with a simple wage equation of the form Wit X it Bt uit, where Wit is the log wage of individual i in year t, X it is a vector of observed individual characteristics (e.g. education and experience), B t is the vector of estimated (OLS) returns to observable characteristics in t, and uit 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 Var ( Wit it ) Var( X it Bt ) Var( u ), between within

12 Fortin Econ 56 given that by construction the errors are orthogonal to the predicted values ( Cov ( X it, u it ) ). 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, 999). 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. A further issue concerning the decomposition of changes in wage inequality into observables and unobservable components is the extent to which changes in betweengroup 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). Another famous paper Juhn, Murply and Pierce (993) set out to summarize the rising dispersion of earnings in the U.S. during the 97s and 98s.

13 Source: Katz and Autor (999)

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15 Fortin Econ 56 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. (993) 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 (26) 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.. 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 98s. 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.

16 Fortin Econ 56 Consider a simple model with only schooling, W = α + γ S + ε where S is individual I schooling level, there a rise in the return to ability (that is a rise in γ ), will raise between-group inequality but will not raise residual inequality. Now, assume that S is not observed. Instead we proxy for S using a measure of human capital such as schooling, S. Assume that S S = ν, where ν is an iid error term. If we estimate the model W = α + γ S + ε the variance of this expression will be Var( W ) = Var(γ S ) + Var(γ ν ) + Var(ε ) 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?

17 Fortin Econ 56 The regressions in Table 3 of Lemieux (26) 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 betweengroup 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 Δβ from the ΔX.

18 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. Average measurement error variance (976-23)* a. Paid by the hour b. Not paid by the hour [8.4] [33.2] [4.3] [35.] [6.8] [2.4] [9.4] [23.2] change in measurement error variance a. Paid by the hour b. Not paid by the hour Spurious change in variance due to a. Growth in fraction of hourly workers** b. Growth in measurement error variance*** c. Total (3a + 3b) change in residual variance Change adjusted for measurement error (4-3c) 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 percent for men and 5 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.

19 Fortin Econ 56 Basic readings: Katz, L. and D. Autor, Changes in the Wage Structure and Earnings Inequality, in O. Ashenfelter and D. Card, eds., Handbook of Labor Economics, vol. 3A, pp , 999. Piketty, T. and E. Saez, "The Evolution of Top Incomes: A Historical and International Perspective," American Economic Review, Vol. 96 (May 26) 2-25.