How skilled immigration may improve economic equality

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

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 defineθ = 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/ϵand 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)))^- ρand the earnings of an unskilled relative to a skilled worker areθ/α. When high-skilled workers earn more than low-skilledworkers, θ/α < 1.

The Gini coefficient is the area between the line of perfect equality, the 45 degreeline, and the Lorenz curve 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 anyλ ∈ [0, 1) and z(1) = 1. Inthe Appendix, we show that the Gini coefficient is

$$G\left(L\right)=\frac{L\left(1-L\right)\left(\alpha -{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho }\right)}{\alpha -\mathit{\alpha L}+{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho -1}}$$

(1)

and that there is a nondegenerate range [L₁,L₂], where values L₁ and L₂satisfy0 ≤ L₁ ≤ L₂ ≤ 1,on which G(L) is increasing in L. Wheneverϵ ∈ (0, 1],dG(L)/dL > 0 for anyL ∈ (0, 1). For ϵ > 1,G(L) is increasing within and decreasing outside of[L₁, L₂]. The range [L₁,L₂] 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 L₁ = 0.07 andL₂ = 0.83. This is further corroborated byTable 1, which provides the values ofL₁ and L₂ for a range of values ofϵ. Parametric values determine whichL ∈ (0, 1) are admissible with respect to the conditionθ/α < 1 and which are not. We denoteL^* the value of L, whereθ/α = 1.

Table 1 The range of L for which G ( L ) is increasing

In the Appendix, we show thatL^* = α^1 - 1/ρ/(1 + α^1 - 1/ρ),L₁ < L^* < L₂,and θ/α < 1 for anyL ∈ (L^*, 1) andθ/α > 1 for anyL ∈ (0, L^*). Ifϵ > 1 (ϵ ∈ (0, 1)), itmust be that L < 0.5 (L > 0.5) forθ/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, forL ∈ (L^*, L₂),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

The relationship between inequality, the quality of the labor force, and migration ismodeled using a recursive econometric specification of the following type:

$$G=f_1\left(S,X\right)+{\mathrm{\mu }}_{\mathrm{G}}$$

(2)

$$S=f_2\left(F,Z\right)+{\mathrm{\mu }}_{\mathrm{S}}$$

(3)

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.

The WIID 2007 dataset reports Gini coefficients for many countries covering many yearsof collection and estimation of this inequality indicator. In those cases where WIID2007 reports multiple Gini coefficients per year and country, we prefer those of thehighest quality if based on gross rather than net takings and earnings rather thanbroader measures of income to quantify those components of economic inequality that stemfrom the labor market as precisely as possible. Whether earnings inequality is measuredat the individual or household level is a non-trivial issue in the context of measuringthe relationship between inequality and immigration. In particular, immigrants oftenhave larger households and different family structures than natives. Measures ofinequality based on individual and household earnings may give different pictures ofinequality. We control for individual against household level at which the Ginicoefficient was measured. The combined dataset covers 29 OECD member states and provides109 observations with non-missing information on the Gini coefficient and the shares ofthe labor force with at least upper secondary or post-secondary education.Table 2 reports descriptive statistics of the mainvariables. The mean Gini coefficient is 32%, the mean share of workers with uppersecondary or higher education is 73%, the corresponding figure for post-secondary orhigher education is 51%, and the mean share of foreigners in the labor force is about7%.

Table 2 Descriptive statistics of key dependent and independent variables

4. Labor force quality and migration

Figures 1 and 2 showing line plots ofnonparametric locally weighted regressions reveal that inequality is mostly a negativefunction of labor force quality for both quality measures that we apply. Indeed, thisrelationship is negative for about 80% of the observations in case of post- secondary orhigher education. The corresponding percentage for upper secondary or higher educationis about 60%. The relationships are not too different from simple quadratic fits.

Figure 1

Scatter plot of the Gini coefficient as a function of the share of labor forcewith upper secondary or higher education. OECD members except for Iceland.Data on Gini coefficients are from the WIID 2007 database. Data on the shares oflabor force with given education are from the OECD Compendium. 1992–2003.Line plot of the nonparametric locally weighted regression of the Gini coefficientas a function of the share of labor force with upper secondary or highereducation.

Figure 2

Scatter plot of the Gini coefficient as a function of the share of labor forcewith post-secondary or higher education. OECD members except for Icelandand Mexico. Data sources see Figure 1. 1992–2003.Line plot of the nonparametric locally weighted regression of the Gini coefficientas a function of the share of labor force with post-secondary or highereducation.

Before we scrutinize the relationship between labor force quality and inequality moredeeply, we first investigate how labor force quality relates to migration.Figures 3 and 4 show that acrossOECD countries the share of labor force with upper secondary or higher educationalattainment is a predominantly positive function of the share of foreign labor force inthe economy, while the same relationship is monotonously increasing in case ofpost-secondary or higher education.

Figure 3

Scatter plot of the share of labor force with upper secondary or highereducation as a function of the share of foreigners in the labor force. OECDmembers. Data on the shares of labor force with given education and foreigners arefrom the OECD Compendium. 1992–2003. Line plot of the nonparametric locallyweighted regression of the Gini coefficient on the share of labor force with uppersecondary or higher education.

Figure 4

Scatter plot of the share of labor force with post-secondary or highereducation as a function of the share of foreigners in the labor force. OECDmembers. Data sources see Figure 3, 1992–2003.Line plot of the nonparametric locally weighted regression of the Gini coefficienton the share of labor force with post-secondary or higher education.

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.

Table 3 contains our findings. The sample included also thoseobservations for which the information on the Gini coefficient was missing. Luxembourgwas dropped from the analysis due to its unusually high share of foreigners. The resultsare robust with respect to inclusion of Luxembourg. The analysis strongly confirms thatthe quality of the labor force increases with the share of foreigners in the laborforce. This finding is valid for all econometric models and for any measure of education(post-secondary or higher and upper-secondary or higher) that we have considered. It isalso robust with respect to the fixed effects model specification as well as for therestricted sample of observations for which the Gini coefficient is available.

Table 3 Share higher education as a function of share foreign labor force

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.

Our regression analysis reported in Table 4 confirms that theobserved decreasing and convex relationship is robust for both considered measures ofeducation and across a number of model specifications, including the standard OLS model,the weighted least squares model with quality weights for the Gini coefficient from theWIID database, and the model with random country effects. This result remains robust inalternative models with weighting by country size, clustering, and fixed effects. Thecoefficients on post-secondary or higher education measure of labor force quality retainthe correct signs, but become insignificant in the fixed effect model. The share ofeducated labor force is negatively and its square positively associated with inequalityin all specifications. The estimated coefficients yield the minimum of the U-shapedrelationship between the share of skilled labor and the Gini coefficient to lie at about80% of the labor force with upper secondary or higher education and 66% of the laborforce with post-secondary or higher education. In our sample these numbers imply adownward sloping relationship between the share of skilled labor and inequality for 67%and 84% of the observations for the two applied measures of skilled labor.

Table 4 Gini coefficient as a function of labor force quality

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 wagesw _l and S = 1 –L high-skilled workers earning wages w _h . Wedenoteθ = 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 thatθ = (L/(α(1 - L)))^- ρwhere ρ > 0.

Proposition

ForL ∈ [α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1)the Gini coefficient equals

$$G\left(L\right)=\frac{L\left(1-L\right)\left(\alpha -{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho }\right)}{\alpha -\mathit{\alpha L}+{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho -1}}.$$

(A1)

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

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

For 0 < ρ < 1 andL ∈ (0, 1), there existL₁ ∈ (0, α^1 - 1/ρ/(1 + α^1 - 1/ρ))andL₂ ∈ (α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1)such that dG(L)/dL > 0 forL ∈ (L₁, L₂),dG(L)/dL < 0 forL ∈ (0, 1) - [L₁, L₂],and dG(L)/dL = 0 forL ∈ {L₁, L₂}.Also,L₁ < L^* < L₂,whereL^* = α^1 - 1/ρ/(1 + α^1 - 1/ρ).

Proof

Givenθ = (L/(α(1 - L)))^- ρ,L ∈ (α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1)impliesθ/α = w _l /αw _h  < 1,that is, high-skilled workers earn more than low-skilled ones. Then the Lorenz curveis defined by

$$z\left(\lambda \right)=\frac{\mathit{\theta \lambda }}{\mathit{\theta L}+\alpha \left(1-L\right)}\mathrm{for}\lambda \in \left[0,L\right]\mathrm{and}$$

(A2)

$$z\left(\lambda \right)=\frac{\mathit{\theta L}+\alpha \left(\lambda -L\right)}{\mathit{\theta L}+\alpha \left(1-L\right)}\mathrm{for}\lambda \in \left[L,1\right]$$

(A3)

We integrate the Lorenz curve over λ ∈ [0, 1]and substitute for θ to obtain

$$G\left(L\right)=\frac{L\left(1-L\right)\left(\alpha -{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho }\right)}{\alpha -\mathit{\alpha L}+{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho -1}}$$

(A4)

to depict the Gini coefficient in this case and

$$\frac{\mathit{dG}\left(L\right)}{\mathit{dL}}=\frac{{\alpha }^2{\left(1-L\right)}^2{L}^{2\rho }+L^2{\alpha }^{2\rho }{\left(1-L\right)}^{2\rho }-{\alpha }^{\rho +1}{L}^{\rho }{\left(1-L\right)}^{\rho }\left(1-\rho -2L\left(1-L\right)\right)}{{\left({\alpha }^{\rho }{\left(1-L\right)}^{\rho }L+\alpha \left(1-L\right)L^{\rho }\right)}^2}.$$

(A5)

IfL ∈ (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

$$z\left(\lambda \right)=\frac{\alpha \left(1-L\right)}{\mathit{\theta L}+\alpha \left(1-L\right)}\mathrm{for}\lambda \in \left[0,L\right]\mathrm{and}$$

(A6)

$$z\left(\lambda \right)=\frac{\alpha \left(1-L\right)+\theta \left(L-\lambda \right)}{\mathit{\theta L}+\alpha \left(1-L\right)}\mathrm{for}\lambda \in \left[L,1\right].$$

(A7)

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 fordG(L)/dL that this derivative is positive for anyL ∈ (0, 1).

For 0 < ρ < 1, firstG(L) and dG(L)/dL are continuousfunctions for L ∈ (0, 1),G(L) → 0 for L → 1 orL → 0 and substitutingL = α^1 - 1/ρ/(1 + α^1 - 1/ρ)into G(L) in equation (A4) yieldsG(α^1 - 1/ρ/(1 + α^1 - 1/ρ)) = 0,because

$$\underset{L\to 0^{+}}{lim}G\left(L\right)=\underset{L\to 0^{+}}{lim}\frac{L^{1-\rho }\left(1-L\right)\left(\alpha L^{\rho }-{\left(\alpha \left(1-L\right)\right)}^{\rho }\right)}{\alpha -\mathit{\alpha L}+{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho -1}}=0,\mathrm{and}$$

(A8)

$$\underset{L\to 1^{-}}{lim}G\left(L\right)=\underset{L\to 1^{-}}{lim}\frac{L{\left(1-L\right)}^{1-\rho }\left(\alpha -{\left(\alpha \left(1-L\right)\right)}^{\rho }/L^{\rho }\right)}{\alpha {\left(1-L\right)}^{1-\rho }+{\alpha }^{\rho }{L}^{1-\rho }}=0,$$

(A9)

where we made use of 0 < ρ < 1.

dG(L)/dL → - ∞ whenL → 1 or L → 0 andsubstitution yields dG(L)/dL > 0 atL = 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 evaluatingdG(L)/dL at L^*, simplifying it, andrealizing thatdG(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 theintervalL ∈ (0, α^1 - 1/ρ/(1 + α^1 - 1/ρ))and a maximum on the intervalL ∈ (α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1),where dG(L)/dL = 0.

To show the uniqueness of each and the maxima of dG(L)/dL,assume for the moment that α = 1; we extend the argument tothe case where α > 1 below. First

$$\frac{d^{2}G\left(L\right)}{d{L}^2}=-\frac{\left(\rho -1\right){\left(L\left(1-L\right)\right)}^{\rho -1}}{{\left(-L{\left(1-L\right)}^{\rho }+L^{\rho }\left(L-1\right)\right)}^3}\left(L^{\rho }\left(1-L\right)\left(2L-\rho \right)+L{\left(1-L\right)}^{\rho }\left(2L+\rho -2\right)\right).$$

(A10)

The ratio $\frac{\left(\rho -1\right){\left(L\left(1-L\right)\right)}^{\rho -1}}{{\left(-L{\left(1-L\right)}^{\rho }+L^{\rho }\left(L-1\right)\right)}^3}$ is positive for0 < ρ < 1 andL ∈ (0, 1), then the second derivative has the signof

$$-\left(L^{\rho }\left(1-L\right)\left(2L-\rho \right)+L{\left(1-L\right)}^{\rho }\left(2L+\rho -2\right)\right).$$

(A11)

For 0 < ρ < 1 andL ∈ (0, 0.5), equation (A11) becomes

$$-L^{\rho }\left(1-L\right)\left(\left(2L-\rho \right)+{\left(\frac{L}{1-L}\right)}^{1-\rho }\left(2L+\rho -2\right)\right).$$

(A12)

As 2L + ρ - 2 < 0 andL/(1 - L) < 1 we write

$$\left(2L-\rho \right)+{\left(\frac{L}{1-L}\right)}^{1-\rho }\left(2L+\rho -2\right)\le \left(2L-\rho \right)+\frac{L}{1-L}\left(2L+\rho -2\right)=\rho \frac{2L-1}{1-L}\le 0,$$

(A13)

which, together with- L^ρ(1 - L) < 0,implies

$$\begin{array}{l}-\left(L^{\rho }\left(1-L\right)\left(2L-\rho \right)+L{\left(1-L\right)}^{\rho }\left(2L+\rho -2\right)\right)>0\mathrm{for}0<\rho \\ <1\mathrm{and}L\in \left(0,0.5\right).\end{array}$$

(A14)

Similarly, rewriting equation (A11) as

$$-L{\left(1-L\right)}^{\rho }\left({\left(\frac{1-L}{L}\right)}^{1-\rho }\left(2L-\rho \right)+\left(2L+\rho -2\right)\right),$$

(A15)

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

Thatd²G(L)/dL² > 0for any L ∈ (0, 0.5) (G(L) isstrictly convex) andd²G(L)/dL² < 0for any L ∈ (0.5, 1) (G(L) isstrictly concave), dG(L)/dL < 0 forL → 1 or L → 0 anddG(L)/dL > 0 forL = α^1 - 1/ρ/(1 + α^1 - 1/ρ) = 0.5,and the continuity of dG(L)/dL forL ∈ (0, 1) imply the uniqueness of the extrema andthe properties of dG(L)/dL forα = 1.

To extend the argument to the case where α > 1, fordG(L)/dL = 0 to have at most two solutionswithin L ∈ (0, 1), it suffices to show thatd²G(L)/dL² = 0has at most one solution.

$$\frac{d^{2}G}{d{L}^2}=\frac{L^{\rho -1}{\left(1-L\right)}^{\rho }{\alpha }^{\rho +1}\left(\rho -1\right)}{\left(L-1\right){\left(\left(L-1\right)L^{\rho }\alpha -L{\left(1-L\right)}^{\rho }{\alpha }^{\rho }\right)}^3}\left(\left(L-1\right)L^{\rho }\alpha \left(-2L+\rho \right)+L{\left(1-L\right)}^{\rho }{\alpha }^{\rho }\left(-2+2L+\rho \right)\right)$$

(A16)

and

$$\begin{array}{c}\left(L-1\right)L^{\rho }\alpha \left(-2L+\rho \right)+L{\left(1-L\right)}^{\rho }{\alpha }^{\rho }\left(-2+2L+\rho \right)\hfill \\ =\alpha L^{\rho }\left(1-L\right)\left(2L-\rho +{\left(\frac{L}{\left(1-L\right)\alpha }\right)}^{1-\rho }\left(2L+\rho -2\right)\right).\hfill \end{array}$$

(A17)

We need to show that

$$H\left(L\right)=2L-\rho +{\left(\frac{L}{\left(1-L\right)\alpha }\right)}^{1-\rho }\left(2L+\rho -2\right)=0$$

has at most one solution within L ∈ (0, 1) forα > 1 and0 < ρ < 1. For this to be true itsuffices that H (L) is monotonous forL ∈ (0, 1), that is, forL′ > L it must be thatH(L′) > H(L). ConsiderL′ > L. Then

$$2L^{\prime }-\rho +{\left(\frac{L^{\prime }}{\left(1-L^{\prime }\right)\alpha }\right)}^{1-\rho }\left(2L^{\prime }+\rho -2\right)>2L-\rho +{\left(\frac{L}{\left(1-L\right)\alpha }\right)}^{1-\rho }\left(2L+\rho -2\right),$$

(A18)

which is:

$$2\left(L^{\prime }-L\right)+{\alpha }^{\rho -1}\left({\left(\frac{L^{\prime }}{\left(1-L^{\prime }\right)}\right)}^{1-\rho }\left(2L^{\prime }+\rho -2\right)-{\left(\frac{L}{\left(1-L\right)}\right)}^{1-\rho }\left(2L+\rho -2\right)\right)>0.$$

(A19)

Equation (A19) holds whenever

$$\left({\left(\frac{L_1}{\left(1-L_1\right)}\right)}^{1-\rho }\left(2L_1+\rho -2\right)-{\left(\frac{L_2}{\left(1-L_2\right)}\right)}^{1-\rho }\left(2L_2+\rho -2\right)\right)$$

(A20)

is non-negative. If the term in equation (A20) is negative, we already know that theinequality in equation (A19) holds for α = 1. Asα^ρ–1 is decreasing forα ∈ (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,d²G(L)/dL² = 0has at most one and dG(L)/dL = 0 at most twosolutions and thus G(L) has at most two interior extrema withinL ∈ (0, 1). We already know that there exists atleast one minimum of G(L) onL ∈ (0, α^1 - 1/ρ/(1 + α^1 - 1/ρ))and at least one maximum onL ∈ (α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1).Therefore, these extrema are unique and we denoteL₁ ∈ (0, α^1 - 1/ρ/(1 + α^1 - 1/ρ))the minimum andL₂ ∈ (α^1 - 1/ρ/(1 + α^1 - 1/ρ), 1)the maximum. It also follows thatL₁ < L^* < L₂,whereL^* = α^1 - 1/ρ/(1 + α^1 - 1/ρ).