# How skilled immigration may improve economic equality

- Martin Kahanec
^{1}and - Klaus F Zimmermann
^{2}Email author

**3**:2

**DOI: **10.1186/2193-9039-3-2

© Kahanec and Zimmermann; licensee Springer. 2014

**Received: **16 September 2013

**Accepted: **14 January 2014

**Published: **18 February 2014

## Abstract

Mobile workers involve flows of labor and human capital and contribute to a moreefficient allocation of resources. However, migration also changes relative wages,alters the distribution of skills and affects equality in the receiving society. Thepaper suggests that skilled immigration promotes economic equality in advancedeconomies under standard conditions. This is discussed and theoretically derived in acore model, and empirically supported using unique data from the WIID database andOECD.

### JEL codes

D33; E25; F22; J15; J61; O15

### Keywords

Inequality Income distribution Human capital Skill allocation Migration Ethnicity Minority Gini coefficient## 1. Introduction

Economic migration involves flows of labor, human capital, and other production factors.At least in theory, it contributes to a more efficient allocation of resources and alarger welfare of nations. However, the distributional effects of migration may changethe skill composition of labor in the receiving and sending countries. This is the caseif, for example, a country experiences a steady inflow of workers whose skill level ison average higher than the skill level of the average native worker. The induced changesin the labor force have the direct effects on inequality through changing the shares of“poor” and “rich” people in the economy, as long as skills arecorrelated with wealth. They affect the wages of high and low skilled labor in theeconomy. Individuals may react to such changes in labor force quality by changing theirinvestment decisions, including those regarding their investment into human capitalacquisition. As another example, low skill immigration may increase the overall qualityof the labor force, if it brings about a larger increase in the quality of the nativelabor force. We measure the quality of the labor force by the incidence of skilledworkers in it. We define skilled and unskilled workers by their highest attained levelsof education, albeit we understand that skill is a broader category than education.

The economic consequences of migration have been one of the central topics of laboreconomics since Chiswick (19781980) andBorjas (19831985). While variousdistributional effects have been considered in the ensuing literature summarized inKahanec and Zimmermann (2009), there is little empirical evidenceon the relationship between migration and inequality, although the distributionaleffects of migration drive public attitudes towards immigration and the related policydiscourse (Constant, Kahanec, and Zimmermann, 2009; Epstein, 2013).

We consider the relationships between economic inequality, the quality of the laborforce, and international migration from the perspective of developed countries receivinginflows of migrants. We first start from Kahanec and Zimmermann (2009) to link inequality to the share of skilled workers in the labor force.In the Appendix, we prove in a theoretical framework that skilled immigration promotesincome equality. We econometrically investigate the relationship between (i) labor forcequality and immigration and (ii) inequality and labor force quality using countrystatistics from the OECD Statistical Compendium and a unique compilation of inequalitydata provided by the WIDER institute at the United Nations University. We find evidencesupporting the hypothesis that skilled immigration supports equality.

## 2. A mathematical analysis

We consider an economy with a labor force of size one with *L* low-skilledworkers earning wages *w*_{
l
} and *S* = 1– *L* high-skilled workers earning wages *w*_{
h
}.We define*θ* = *w*_{
l
}/*w*_{
h
}.L denotes also the share of low-skilled workers. For a constant elasticity ofsubstitution (CES) production function $C={\left({L}^{1-\rho}+{\left(\mathit{\alpha S}\right)}^{1-\rho}\right)}^{\frac{1}{1-\rho}}$, where *ρ* = 1/*ϵ*and *ϵ* > 0 is the (finite) elasticity of substitution of high-and low-skilled labor in a competitive industry and *α* > 1 isthe efficiency shift factor of skilled relative to unskilled labor,*θ* = (*L*/(*α*(1 - *L*)))^{- ρ}and the earnings of an unskilled relative to a skilled worker are*θ*/*α*. When high-skilled workers earn more than low-skilledworkers, *θ*/*α* < 1.

*z*(

*λ*), depicting the share ofeconomy’s income accruing to the

*λ*poorest individuals, divided bythe area between the line of perfect equality and the line of perfect inequality. Theline of perfect inequality attains zero for any

*λ*∈ [0, 1) and

*z*(1) = 1. Inthe Appendix, we show that the Gini coefficient is

*L*

_{1},

*L*

_{2}], where values

*L*

_{1}and

*L*

_{2}satisfy0 ≤

*L*

_{1}≤

*L*

_{2}≤ 1,on which

*G*(

*L*) is increasing in

*L*. Whenever

*ϵ*∈ (0, 1],

*dG*(

*L*)/

*dL*> 0 for any

*L*∈ (0, 1). For

*ϵ*> 1,

*G*(

*L*) is increasing within and decreasing outside of[

*L*

_{1},

*L*

_{2}]. The range [

*L*

_{1},

*L*

_{2}] is large. For example, if the substitutability of skilled andunskilled labor is about 2.5, as estimated by Chiswick (1978b),and high skilled labor is twice as productive as its low skilled counterpart, thecorresponding values

*L*

_{1}= 0.07 and

*L*

_{2}= 0.83. This is further corroborated byTable 1, which provides the values of

*L*

_{1}and

*L*

_{2}for a range of values of

*ϵ*. Parametric values determine which

*L*∈ (0, 1) are admissible with respect to the condition

*θ*/

*α*< 1 and which are not. We denote

*L*

^{*}the value of

*L,*where

*θ*/

*α*= 1.

**The range of**
L
**for which**
G
**(**
L
**) is increasing**

ϵ | ϵ ≤ 1 | 1.1 | 2.5 | 5 | 10 | 100 | ϵ → ∞ |
---|---|---|---|---|---|---|---|

| 0 | 0.02 | 0.07 | 0.02 | 0 | 0 | 0 |

| 1 | 0.98 | 0.83 | 0.73 | 0.66 | 0.59 | 0.59 |

In the Appendix, we show that*L*^{*} = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}),*L*_{1} < *L*^{*} < *L*_{2},and *θ*/*α* < 1 for any*L* ∈ (*L*^{*}, 1) and*θ*/*α* > 1 for any*L* ∈ (0, *L*^{*}). If*ϵ* > 1 (*ϵ* ∈ (0, 1)), itmust be that *L* < 0.5 (*L* > 0.5) for*θ*/*α* < 1 to hold. *L*^{*} =0.26 if *ϵ* = 2.5 and *α* = 2 as in the example above. For thevalues of *L* ∈ (0, *L*^{*}), the Ginicoefficient equals –*G*(*L*): for OECD economies with a large shareof skilled labor, the relevant segment of *G*(*L*) is decreasing in theshare 1-*L* of skilled labor, for the most part and may become decreasing in1-*L* for *L* ∈ (0, *L*^{*}),where, counterfactually, the low-skilled earn more than the high-skilled.

This enables us to consider the effects of changes in *L* that occur whenimmigrants of different skill composition from that of natives enter or leave thecountry under the conditions of flexible wages. For example, for*L* ∈ (*L*^{*}, *L*_{2}),an inflow of immigrants who are on average more skilled than the natives decreasesinequality, in case of *ϵ* > 1. We then predict that inequalityis decreasing with skilled immigration, or more generally immigration that increases thequality of the labor force, for moderate to high values and may be increasing for veryhigh values of the share 1-*L* of skilled labor. In OECD countries where skilledlabor is abundant and earns more than unskilled labor, skilled immigration shoulddecrease inequality.

Skills develop with age, and age and migration are related through the migrationdecision. Therefore, skilled immigrants may first not directly compete with natives,since they are typically male, young and often over-skilled for the job they do. Theirinteraction with natives also depends on their willingness of investing incountry-specific knowledge and human capital. Natives may also react with educationaldecisions. Hence, skilled immigrants can increase the share of skilled workers in thecountry right upon arrival, but also after they or the natives adjust.

In a similar way, even mixed or less-skilled immigration may increase the average skilllevel in the receiving labor market through immigrants’ or natives’adjustment. Natives may react not only ex post by adjusting their educational ortraining decisions, but also before actual immigration takes place in expectation ofincreased labor market competition.

## 3. Empirical specification and data

G stands for inequality measured as the Gini coefficient, *S* is the share ofskilled labor force as in our theoretical model, and *F* is the share offoreigners in the labor force measuring migration. X and Z are vectors of contextualvariables, and μ_{G} and μ_{S} are error terms. Equation (2)captures the derived trade-off between inequality and educational attainment, whileEquation (3) measures the optimal relationship between the share of skilled workers inan economy and the share of foreign labor of total employment resulting from thestandard firm optimization principle.

What is the empirical relationship between inequality and educational attainment levelsin the labor force? To address this question, we combine data on education, labor forcecharacteristics and other national indicators from the OECD Statistical Compendium 2007with the Gini measures reported in the World Income Inequality Database (WIID 2007) version 2.0b compiled by the WIDER institute at the UnitedNations University and published in May 2007. The OECD Statistical Compendium providesstatistics on labor force characteristics, national accounts, and education, mainly fordeveloped country members of OECD.

**Descriptive statistics of key dependent and independent variables**

Mean | Standard deviation | Number of observations | |
---|---|---|---|

Gini coefficient | 31.95 | 6.14 | 109 |

Share of upper secondary or higher education | 72.84 | 17.17 | 109 |

Share of post-secondary or higher education | 50.64 | 20.26 | 109 |

Share of foreign labor force | 5.11 | 3.85 | 110 |

Inflation rate | 2.63 | 2.50 | 109 |

Share of population 15–64 years of age | 66.86 | 1.56 | 109 |

Unemployment rate | 7.49 | 3.53 | 109 |

Women’s unemployment rate | 8.37 | 4.61 | 109 |

Participation rate | 73.01 | 6.24 | 109 |

Women’s participation rate | 65.00 | 8.44 | 109 |

Share of labor force in agriculture | 5.68 | 4.00 | 109 |

Government size | 20.25 | 3.38 | 109 |

GDP per capita, 1000s USD | 19.95 | 12.15 | 109 |

## 4. Labor force quality and migration

To consider this relationship (Equation 3) as a causal phenomenon requiresaccounting for the endogeneity of the decision of migration, the effects of migration onthe educational attainment of the native labor force, and the skill level of theimmigrants relative to native workers. While such causal evaluation would require a muchmore detailed dataset than we have, we evaluate the association between the share offoreign labor force and its quality controlling for a number of potential covariatessuch as the size of the government and the age composition of the labor force.

**Share higher education as a function of share foreign labor force**

(1) | (2) | (3) | (4) | (5) | (6) | |
---|---|---|---|---|---|---|

Upper secondary and higher | Post-secondary and higher | |||||

OLS | OLS | Random effects | OLS | OLS | Random effects | |

Share of foreign labor force | 0.91** | 1.14** | 0.65** | 2.62** | 2.88** | 1.43** |

(0.29) | (0.30) | (0.23) | (0.33) | (0.43) | (0.42) | |

Share of population 15-64 years of age | 1.99 | -0.16 | 1.85 | -0.16 | ||

(1.57) | (0.46) | (1.77) | (0.86) | |||

Government size | 1.79** | -0.57** | 1.38* | -1.58** | ||

(0.56) | (0.27) | (0.76) | (0.50) | |||

GDP per capita, 1000 s USD | 0.68** | -0.00 | 0.47** | -0.13** | ||

(0.14) | (0.03) | (0.18) | (0.06) | |||

Year dummies | Yes | Yes | Yes | Yes | ||

Constant | 66.28** | -112.35 | 85.60** | 37.51** | -110.04 | 87.21 |

(2.75) | (115.00) | (34.39) | (2.71) | (128.11) | (63.90) | |

Observations | 110 | 110 | 109 | 110 | 110 | 109 |

R-squared | 0.04 | 0.27 | 0.73 | 0.22 | 0.30 | 0.52 |

Government size as well as GDP per head have positive effects on the quality of laborforce in the OLS models in columns 2 and 5 of Table 3, butthese effects have a different sign in the random effects models. This reversal isconsistent with the hypothesis that the association of these variables is positivebetween but negative within countries.

## 5. Inequality and the quality of the labor force

The question that remains to be addressed is whether inequality indeed tends to be anegative function of labor force quality as suggested by our theoretical argument aswell as Figures 1 and 2. We thereforenow estimate Equation 2, accounting for a number of potential confounding factors.Besides the distribution of educational levels in the labor force, Katz and Murphy(1992) report that increased demand for skilled workers andwomen as well as changes in the allocation of labor between industries contributed toincreasing inequality in the US in recent years. Gustafsson and Johansson (1999) provide evidence that the share of industry in employment, perhead gross domestic product, international trade, the relative size of the publicexpenditures, as well as the demographic structure of the population affect inequalitymeasured by the Gini coefficient across countries and years. Topel (1994) finds that technological and economic development determines economicinequality.

We examine the effects of the aggregate and women’s labor force participationrates, aggregate and women’s unemployment rates, share of the population between15 and 64 years of age, labor force in the agricultural sector, share of the governmentin the economy, defined as the expenditures of the central government divided by theaggregate GDP, gross domestic product, and inflation rate. We control for the year,country, and the method of computing the Gini coefficient, distinguishing various incomemeasures, net and gross figures and the unit of analysis used to calculate the Ginicoefficient.

**Gini coefficient as a function of labor force quality**

(1) | (2) | (3) | (4) | (5) | (6) | |
---|---|---|---|---|---|---|

Upper secondary and higher | Post-secondary and higher | |||||

OLS | Quality weighted | Random effects | OLS | Quality weighted | Random effects | |

Share of highly educated in the labor force | -0.83** | -0.75** | -0.81** | -0.31** | -0.32** | -0.29** |

(0.16) | (0.17) | (0.17) | (0.13) | (0.12) | (0.13) | |

Share of highly educated in the labor force, sq/100 | 0.56** | 0.49** | 0.54** | 0.24** | 0.24** | 0.22* |

(0.15) | (0.14) | (0.14) | (0.11) | (0.11) | (0.12) | |

Inflation rate | 0.21 | 0.18 | 0.18 | 0.11 | 0.09 | 0.08 |

(0.26) | (0.24) | (0.25) | (0.24) | (0.26) | (0.28) | |

Share of population | -0.52 | -0.44 | -0.60 | -0.68 | -0.44 | -0.78 |

15-64 years of age | (0.48) | (0.48) | (0.49) | (0.57) | (0.51) | (0.53) |

Unemployment rate | 2.95** | 2.92** | 2.95** | 2.09** | 2.19** | 2.11** |

(0.83) | (0.52) | (0.54) | (0.71) | (0.56) | (0.59) | |

Women’s unemployment rate | -1.87** | -1.86** | -1.88** | -1.34** | -1.42** | -1.36** |

(0.59) | (0.39) | (0.40) | (0.53) | (0.42) | (0.44) | |

Participation rate | 0.11 | 0.32 | 0.16 | 0.47 | 0.67* | 0.54 |

(0.40) | (0.39) | (0.39) | (0.34) | (0.38) | (0.41) | |

Women’s participation rate | -0.31 | -0.44 | -0.35 | -0.47* | -0.59** | -0.52* |

(0.32) | (0.30) | (0.31) | (0.24) | (0.28) | (0.30) | |

Share of labor force in agriculture | -0.34 | -0.29 | -0.32* | -0.20 | -0.18 | -0.16 |

(0.26) | (0.18) | (0.18) | (0.21) | (0.18) | (0.19) | |

Government size | -0.41 | -0.36* | -0.40** | -0.43* | -0.36* | -0.41** |

(0.25) | (0.19) | (0.19) | (0.23) | (0.19) | (0.20) | |

GDP per capita, 1000 s USD | 0.06 | 0.05 | 0.05 | -0.08 | -0.08 | -0.10 |

(0.06) | (0.07) | (0.08) | (0.07) | (0.07) | (0.08) | |

Gini definition controls | Yes | Yes | Yes | Yes | Yes | Yes |

Year dummies | Yes | Yes | Yes | Yes | Yes | Yes |

Constant | 115.40** | 99.31** | 118.89** | 93.42** | 68.78* | 95.98** |

(34.24) | (34.20) | (34.02) | (40.97) | (35.82) | (37.06) | |

Observations | 109 | 109 | 108 | 109 | 109 | 108 |

R-squared | 0.70 | 0.71 | 0.70 | 0.62 | 0.66 | 0.62 |

The aggregate unemployment rate is positively associated with inequality, butwomen’s unemployment rate affects inequality negatively. The same should hold foraggregate and women’s participation rates, but we do not find this. One reasoncould be the effect of women’s selection into the labor force, whereby highwomen’s unemployment and participation rates indicate that women with lessfavorable earnings opportunities are joining the labor force, increasing the dispersionof earnings. The size of the government, government spending as a percentage of GDP, isnegatively associated with inequality, which is consistent with the hypothesis thatredistribution decreases inequality.

## 6. Conclusion

First, our theory predicts that inequality is decreasing in labor force quality foradvanced economies under standard conditions. This effect is mainly a consequence of thestandard economic law of diminishing marginal product of production factors: as theshare of skilled workers in the economy increases, its value decreases and thus also thewage differential between high and low skilled labor decreases. In our theoreticalmodel, migration affects inequality in the economy as it changes the quality of thelabor force. In particular, inflows of workers with average skill level above that ofthe receiving country decrease inequality, and the opposite holds for low-skilledimmigration.

Second, we confirm empirically that the relationship between inequality and the qualityof the labor force is predominantly negative. The econometric analysis accounting formany covariates confirms what already appears from the raw data. We show that, in thesample of OECD countries, inequality decreases with a higher labor force quality formost values of educational attainment; and a positive relationship shows up forobservations with the quality of the labor force above a certain high threshold level aspredicted by the theory.

Third, we empirically evaluated the relationship between migration and labor forcequality as observed across OECD countries. We find that the share of foreigners in thelabor force and its quality as measured by educational attainment are throughoutstrongly positively associated. Given our finding that labor force quality andinequality are negatively associated, this result implies that immigration is negativelyassociated with inequality.

## Appendix

### Gini coefficient and immigration

Consider an economy of size 1 with *L* low-skilled earning wages*w*_{
l
} and *S* = 1 –*L* high-skilled workers earning wages *w*_{
h
}. Wedenote*θ* = *w*_{
l
}/*w*_{
h
}and normalize the total income to unity,*w*_{
l
}*L* + *w*_{
h
}(1 - *L*) = 1.Consider the case with endogenous wages such that*θ* = (*L*/(*α*(1 - *L*)))^{- ρ}where *ρ* > 0.

### Proposition

*L*∈ [

*α*

^{1 - 1/ρ}/(1 +

*α*

^{1 - 1/ρ}), 1)the Gini coefficient equals

For*L* ∈ (0, *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ})]the Gini coefficient equals –*G*(*L*).

If *ρ* ≥ 1,*dG*(*L*)/*dL* > 0 for any*L* ∈ (0, 1).

For 0 < *ρ* < 1 and*L* ∈ (0, 1), there exist*L*_{1} ∈ (0, *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}))and*L*_{2} ∈ (*α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}), 1)such that *dG*(*L*)/*dL* > 0 for*L* ∈ (*L*_{1}, *L*_{2}),*dG*(*L*)/*dL* < 0 for*L* ∈ (0, 1) - [*L*_{1}, *L*_{2}],and *dG*(*L*)/*dL* = 0 for*L* ∈ {*L*_{1}, *L*_{2}}.Also,*L*_{1} < *L*^{*} < *L*_{2},where*L*^{*} = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}).

### Proof

*θ*= (

*L*/(

*α*(1 -

*L*)))

^{- ρ},

*L*∈ (

*α*

^{1 - 1/ρ}/(1 +

*α*

^{1 - 1/ρ}), 1)implies

*θ*/

*α*=

*w*

_{ l }/

*αw*

_{ h }< 1,that is, high-skilled workers earn more than low-skilled ones. Then the Lorenz curveis defined by

*λ*∈ [0, 1]and substitute for

*θ*to obtain

*L*∈ (0,

*α*

^{1 - 1/ρ}/(1 +

*α*

^{1 - 1/ρ})),

*θ*/

*α*=

*w*

_{ l }/

*αw*

_{ h }> 1and high-skilled workers earn less than low-skilled ones. The Lorenz curvebecomes

Integrating the Lorenz curve over *λ* ∈ [0, 1] weobtain that the Gini coefficient is –*G*(*L*).*L* = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ})is the case of perfect equality.

For *ρ* ≥ 1 we see from the expression for*dG*(*L*)/*dL* that this derivative is positive for any*L* ∈ (0, 1).

*ρ*< 1, first

*G*(

*L*) and

*dG*(

*L*)/

*dL*are continuousfunctions for

*L*∈ (0, 1),

*G*(

*L*) → 0 for

*L*→ 1 or

*L*→ 0 and substituting

*L*=

*α*

^{1 - 1/ρ}/(1 +

*α*

^{1 - 1/ρ})into

*G*(

*L*) in equation (A4) yields

*G*(

*α*

^{1 - 1/ρ}/(1 +

*α*

^{1 - 1/ρ})) = 0,because

where we made use of 0 < *ρ* < 1.

*dG*(*L*)/*dL* → - *∞* when*L* → 1 or *L* → 0 andsubstitution yields *dG*(*L*)/*dL* > 0 at*L* = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}).In fact, *dG*(*L*)/*dL* = *ρ*. This lastresult involves tedious algebra; one can show this by evaluating*dG*(*L*)/*dL* at *L*^{*}, simplifying it, andrealizing that*dG*(*L*)/*dL* = 1 + *f*(*α*, *ρ*)(*ρ* - 1)where the term *f*(*α*, *ρ*) = 1.These properties imply that there exists a minimum of *G*(*L*) on theinterval*L* ∈ (0, *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}))and a maximum on the interval*L* ∈ (*α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}), 1),where *dG*(*L*)/*dL* = 0.

*dG*(

*L*)/

*dL*,assume for the moment that

*α*= 1; we extend the argument tothe case where

*α*> 1 below. First

*ρ*< 1 and

*L*∈ (0, 1), then the second derivative has the signof

*ρ*< 1 and

*L*∈ (0, 0.5), equation (A11) becomes

*L*+

*ρ*- 2 < 0 and

*L*/(1 -

*L*) < 1 we write

*L*

^{ ρ }(1 -

*L*) < 0,implies

(*L*^{
ρ
}(1 - *L*)(2*L* - *ρ*) + *L*(1 - *L*)^{
ρ
}(2*L* + *ρ* - 2)) < 0 for 0 < *ρ* < 1 and *L* ∈ (0.5, 1).

That*d*^{2}*G*(*L*)/*dL*^{2} > 0for any *L* ∈ (0, 0.5) (*G*(*L*) isstrictly convex) and*d*^{2}*G*(*L*)/*dL*^{2} < 0for any *L* ∈ (0.5, 1) (*G*(*L*) isstrictly concave), *dG*(*L*)/*dL* < 0 for*L* → 1 or *L* → 0 and*dG*(*L*)/*dL* > 0 for*L* = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}) = 0.5,and the continuity of *dG*(*L*)/*dL* for*L* ∈ (0, 1) imply the uniqueness of the extrema andthe properties of *dG*(*L*)/*dL* for*α* = 1.

*α*> 1, for

*dG*(

*L*)/

*dL*= 0 to have at most two solutionswithin

*L*∈ (0, 1), it suffices to show that

*d*

^{2}

*G*(

*L*)/

*dL*

^{2}= 0has at most one solution.

*L*∈ (0, 1) for

*α*> 1 and0 <

*ρ*< 1. For this to be true itsuffices that

*H*(

*L*) is monotonous for

*L*∈ (0, 1), that is, for

*L*′ >

*L*it must be that

*H*(

*L*′) >

*H*(

*L*). Consider

*L*′ >

*L*. Then

is non-negative. If the term in equation (A20) is negative, we already know that theinequality in equation (A19) holds for *α* = 1. As*α*^{ρ–1} is decreasing for*α* ∈ (1, *∞*), that the term inequation (A20) is negative and the fact that the inequality in equation (A19) holdsfor *α* = 1 imply that the inequality in equation (A19) holdsfor a negative (A20), too.

Given their continuity,*d*^{2}*G*(*L*)/*dL*^{2} = 0has at most one and *dG*(*L*)/*dL* = 0 at most twosolutions and thus *G*(*L*) has at most two interior extrema within*L* ∈ (0, 1). We already know that there exists atleast one minimum of *G*(*L*) on*L* ∈ (0, *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}))and at least one maximum on*L* ∈ (*α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}), 1).Therefore, these extrema are unique and we denote*L*_{1} ∈ (0, *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}))the minimum and*L*_{2} ∈ (*α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}), 1)the maximum. It also follows that*L*_{1} < *L*^{*} < *L*_{2},where*L*^{*} = *α*^{1 - 1/ρ}/(1 + *α*^{1 - 1/ρ}).

## Declarations

### Acknowledgements

This article expands on and complements an earlier chapter that appeared in theOxford Handbook on Economic Inequality (Kahanec and Zimmermann, 2009). Financial support from Volkswagen Foundation for the IZA project on“The Economics and Persistence of Migrant Ethnicity” is gratefullyacknowledged. We thank the anonymous referee and the Editor, Denis Fougère, forhelpful comments on an earlier draft.

Responsible editor: Denis Fougere

## Authors’ Affiliations

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