3 Models of earnings inequality

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A general mechanism of examining earnings inequality and earnings instability was introduced by Gottschalk and Moffitt (1994) in a study of the growth of earnings instability in the United States. It was further developed in Baker (1997), Haider (2001), Moffitt and Gottschalk (2002), and Baker and Solon (2003), who added considerably more flexibility into the earlier models.

The basic idea of the approach is that individual earnings (or rather, log-earnings) in period t can be thought of as a sum of two orthogonal components, permanent and transitory, that evolve independently over time. A simple life-cycle model that incorporates dynamic changes in both components can be written as

where yit represents the (log) earnings of an individual i in period t, αi and vit are permanent and transitory components, and pt and λt are period-specific factor loading on each of these components.

Note that cov(vit, vis) = 0 in (1) implies that, unlike var(yit), cov(vit, vis) does not depend on λt, so the source of cross-sectional inequality can be identified in a dynamic context from changes in autocovariances (Baker and Solon 2003). Put otherwise, an increase in pt leads to an increase in earnings inequality, both current and long term; an increase in λt, on the other hand, does not imply a long-term effect. Such an increase can be thought of as an increase in a person's earnings instability. Assuming that the permanent component measures the life-time earnings potential or skill, pt can be interpreted as the price of skill, which changes with changes in demand and supply for skill, due to technological transformation or other types of economic restructuring (Moffitt and Gottschalk 2002). In the context of immigrants' earnings, changes in pt may reflect the general 'quality' of immigrants' human capital, affecting their ability to adjust to technological changes in the host country, as well as the diversity of immigrants' skills determined by immigration policies.

The model above can incorporate several additional features of earnings growth. For instance, the first term in (1) can incorporate heterogeneity in individual growth rates (Haider 2001), or a random walk component that would allow for permanent changes (Moffitt and Gottschalk 2002), or both (Baker and Solon 2003), so (1) may take the following form

where xit is a set of variables determining growth rates, uit = ui, t−1 + rit and εit represents the transitory component.

The last term in (2) can also take a more flexible specification. Baker and Solon (2003) allow for serial correlation in the transitory component

and model the variance of vit as a quartic function of age. Haider (2001) and Moffitt and Gottschalk (2002), on the other hand, assume an ARMA2 (1,1) specification.

By specifying the functional form of yit, we also specify the functional form of the variance– covariance matrix of individual earnings, Ω, so that each element in Ω is expressed as ωi = f(xi;θ), where θ is a set of parameters that includes pt and λt . Crucially, unlike the model in (2), θ does not include individual specific parameters αi and βi ; instead, it includes σ2α2β as well as σαβ .

The parameters of the resulting model are usually estimated using the generalized method of moments (GMM) based on minimizing the distance between the observed sample moments (elements of Ω) and f( xi;θ). The parameter estimates θ are used to construct the profiles of earnings inequality and earnings instability.

 

2 The acronym ARMA stands for autoregressive moving average. ARMA models describe changes in a variable in terms only of its past value.