¶ Wage inequality and segregation by skill

NBER link

NBER pdf link

¶ BibTeX

@misc{kremer1996wage,
  title={Wage inequality and segregation by skill},
  author={Kremer, Michael and Maskin, Eric},
  year={1996},
  publisher={National bureau of economic research Cambridge, Mass., USA}
}

¶ Abstract

Evidence from the US, Britain, and France suggests that recent growth in wage inequality has been accompanied by greater segregation of high- and low-skill workers into separate firms. A model in which workers of different skill-levels are imperfect substitutes can simultaneously account for these increases in segregation and inequality either through technological change, or, more parsimoniously, through observed changes in the skill-distribution

¶ My Notes

• growing wage gap based on education
• less common for low and high skill workers to work in same firm
• model low and high as imperfect substitutes
• When mean and variance of skill increase, firms have to specialize in one over the other.

¶ Simple model of worker pairs

Consider firm which needs two tasks done, and hires workers with skills $q$ (assistant) and $q’$ (manager) to do each.

(f:This is the most important piece of the model. Simple idea that does a lot of work. Problem with CES literature is that changes are made at top level without looking at micro foundations.)

Better to assign high skill employee to manager position.

• one-good economy
• exogenous distribution of workers of different skills
• indefinite supply of firms, same production funciton
• dispersion of skill distribution decreases dispersion of skill between manager and assisstant.
  • Example: two people with skill L, two with H
  • Should the be matched by skill or should low-skill be paired with high
  • cross-matching makes sense if \(L^3 + H^3 < 2LH^2\), which happens if \(H < (1+\sqrt{5})L/2\)
• Two competing effects:
  • Assymetry between tasks promotes cross-matching (high skilled all managers),
  • but complementarity promotes assortive matching

¶ Example with wages

• three skill levels: $L < M < H$
• population $x$ with skill $L$, $x$ with skill $H$, more than $2x$ with skill $M$.
• $H < (1+\sqrt{5})L/2$, so no self-match for $H$ or $L$ types
• Both H and L types should be matched with M types because $LH^2 + M^3 < LM^2 + MH^2$
• There will be some M types left over.
• Assume workers absorb all revenue. Then:
  • M-types in M,M firms earn $w(M) = M^3 / 2$
  • Other M types earn the same
  • L types in L,M firms earn $w(L)=LM^2-w(M)=LM^2-M^3/2$
  • H types in H,M firms earn $w(L)=MH^2-w(M)=MH^2-M^3/2$

• what happens when M skill increases?
  \(\frac{d}{dM}w(L) = 2LM - \frac{3}{2}M^2\)
  \(\frac{d}{dM}w(H) = H^2 - \frac{3}{2}M^2\)

• If low dispersion $L\approx H$, then a marginal increase in M causes $w(L)$ to rise and $w(H)$ to fall. Wages converge.
• If high dispersion where \(L < \frac{3}{4}M < \frac{\sqrt 6}{4}H\), opposite effect. wages diverge.
• Intutition:
  • If L close to M,
    • Increase in M’s share decreases more slowly than total product.
    • “trickle-down” effect, where improving productivity of high-paid workers accrues benefits to low-paid
    • I think this is the only time I’ve actually seen that term used by an economist.
  • If L and M far apart
    • product of LM firms rises more slowly than MM firms, so wages of M increase, and wages of L fall.
    • At the same time, boosts wage of H for similar reasons.

¶ segregation index

• $J$ firms indexed by $j$
• $z_j$ workers in firm $j$
  • $Z_j$ denotes the set of these
• Denote mean skill level $\bar q$
• Define index of correlation or segregation $\rho$:

• $\rho=0$ indicates all firms have same skill mix.
• $\rho=1$ indicates complete segregation, in which workers within each firm have no variation.
• Equivalent to
  • one minus the variance within firms divided by overall variance
  • regressing $q$ on firm dummies and looking at $R^2$
• With finite firms, $\rho > 0$ expected due to random chance.
  • Even with random employee assignment, some firms will just get lucky.
  • If all firms had $K$ workers, randomly chosen from pool, expectation is that $\rho = 1/K$
• index adjusted for firm size

¶ measuring segregation

• Want to measure at plant level
  • obfuscated by outsourcing
  • French data more similar to firm level
• France Wage Structure Surveys (ESS)
  • correlation of low wages and experience rose within business units
  • segregation by worker category rose.
• Similar evidence from US and BRitish worker surveys
• segregation indices by US state
  • Kremer Trosker 1995
  • Worker-Establishment Characteristics Database (WECD)
  • states with more education variance have more segregation by education
  • Prediction doesn’t work for varaince/segregation by age
  • when using log wage or predicted wage as skill indicator, prediction correct but not statistically significant.

¶ Generalized model

• two distributions - pre and post shift - on worker skill level $n\in\mathbb Z$
  • support bounded by $\underline n, \bar n$
• parameter $\theta=0$ representing pre-shift in distributions, $\theta=1$ post-shift
• $\mu(\theta)$ for means
  • assume equal to $n^*(\theta)$ modes
• Recall $A=2/(1+\sqrt 5)$ to be cutoff ratio that makes assortive better than cross-matching
• $$
• Some additional assumptions to ensure some modal workers will self-match
  • makes computation of equilibrium wages easier.
• Some additional assumptions to ensure there is a range of workers who will match with modal workers.
• shift in distributions reflects stylized facts
  • mean skill rises
    • n^(1)=n^(0)+1
  • dispersion increases
    • tails get symetrically fatter but retain relative shapes
    • $p(n;1)/p(n;0)=\alpha$ for all n outside some central range.

¶ Propositions

¶ Prop1 (trickle down)

With tight bounds on the distribution of skill, a shift of the kind described decreases wage dispersion. If $\underline n > \sqrt{2/3} \bar n$, then

• $w(n;0) < w(n;1)$ for all $n < n^*(0)$
• $w(n;0) > w(n;1)$ for all $n > n^*(1)$

¶ Prop2

Given large enough bounds in the story above, the wage in the low tail decreases, and wage in high tail increases

¶ segregation

Increase in skill dispersion increases segragation index, provided shift in mean is small relative to shift in dispersion

Tangential related thoughts

To what extent does a private school paying teachers lower the quality of public schools.

High skill dispersion makes it difficult to sustain unions. (TODO)

TODO: read Shirwen Rosen’s papers. Simple models are good at capturing ideas.

• KS stereotypes
• interesting that political polarization is maybe just an instance of general societal polarization
• Kremer likes his two-type dispersion models
• In Kremer model, seems between-firm segregation happens because you hire really high quality janitors.
• prefential mixing in Newman model