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Is Bilingual Education Desirable in Multilingual countries?

  • Kazuhiro Yuki ORCID logo EMAIL logo
Published/Copyright: October 12, 2022
The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy Volume 22 Issue 4

Abstract

In many countries, people use their mother tongue in local business, but use the language of the former colonizer in national business. How much weight should be placed on teaching one’s mother tongue and the lingua franca is a critical issue in these countries. This paper develops a model to examine these issues theoretically. It is shown that balanced education of the two languages is critical for skill development of those with limited wealth. It is also found that balanced bilingual education yields higher earnings net of educational expenditure than lingua-franca-only education only when a country has favorable educational and technological conditions (productivity is reasonably high and education is reasonably effective) and only for those with adequate wealth. Policy implications of the results are also discussed.

Keywords: language policy; bilingual education; human capital; economic development
JEL Classification: I25; J24; O15; Z13
Corresponding author: Kazuhiro Yuki, Faculty of Economics, Kyoto University, Yoshida-hommachi, Sakyo-ku, Kyoto 606-8501, Japan, E-mail:

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 10197395

Acknowledgment

Invaluable comments from an anonymous referee and helpful comments from participants at the 2017 Asian Meeting of the Econometric Society are gratefully acknowledged.

  1. Research funding: Financial support from JSPS through Grants-in-Aid for Scientific Research 10197395 is acknowledged. Data sharing is not applicable to this paper as no new data were created or analyzed in this study.

  2. Conflict of interest statement: The author declares no conflicts of interest regarding this article.

  3. Author contributions: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

Appendix A: Analysis of the Effect of sl on Earnings When e+<ē

This appendix examines in detail the effect of sl on earnings when e+<ē.

A.1 When the Return to Education for Local Jobs is Constrained to be Positive

First, as in Section 4.2, the effect is examined under the assumption that the return to educational investment for local jobs is positive for any sl. The next proposition summarizes analytical results, based onProposition A1 in Appendix B, which presents a fully detailed statement. This proposition is the counterpart of Proposition 3 when e+=ē. In the proposition, symbol e+(sl) is used to signify the dependence of e+ on sl (e+′(sl) > 0 from Lemma 1).

Proposition 5

Suppose that the return to education for local jobs is positive and e+<ē holds.

    1. If the proportion of those with limited wealth is high enough that e+(0)≤α1−αh̲lδl holds, d(wnhn)dsl<0 for any sl.

    2. Otherwise, d(wnhn)dsl>(<)0 for small (large) sl. sl maximizing wnhn is smaller than sl⋆⋆, the critical sl when e+=ē.

    1. If ē≤1+α1−αh̲lδl, d(wlhl)dsl>(<)0 for small (large) s l when a(= e) is large, while d(wlhl)dsl<0 for any sl when a(=e) is small. In the former case, sl maximizing wlhl is greater than sl⋆(e), the critical sl when e+=ē.

    2. Otherwise, d(wlhl)dsl is negative for small sl, positive for middle sl, and negative for large sl when a(=e) is intermediate, while when it is small and large, results are similar to (a). In the former case, sl maximizing (minimizing) wlhl is greater (smaller) than sl⋆(e)(sl°(e)), the critical sl when e+=ē.

    3. The maximum a(=e) such that d(wlhl)dsl<0 holds for any sl is lower than when e+=ē.

As shown in Proposition A1 in Appendix B, unlike the case e+=ē, analytical results cannot be obtained for some ranges of sl and as for wlhl, a(=e). However, the above proposition and numerical simulations suggest that results for workers with a local job are qualitatively the same as the case e+=ē: when ē≤1+α1−αh̲lδl, wlhl increases (decreases) with sl for small (large) sl when a(=e) is large, and wlhl decreases with sl for any sl when it is small; when ē>1+α1−αh̲lδl, wlhl decreases with sl for small sl, increases with sl for middle sl, and decreases with sl for large sl when a(=e) is intermediate, while the results when it is large and small are similar to the previous case. As for the relationship between s l maximizing wlhl and a(=e), unlike the case e+=ē, an analytical result is not obtained but numerical simulations suggest that the relationship is positive as before.

There exist minor differences from the case e+=ē. First, sl maximizing earnings of workers with a local job and a above the threshold is greater than sl⋆(e), the critical sl when e+=ē. Second, the maximum level of a(=e) such that the earnings decrease with sl for any sl (and thus sl = 0 maximizes the earnings) is lower than when e+=ē. From Proposition 1, for given sl, e+<(=)ē holds when F(ē) is relatively large (small). Hence, these results suggest that individuals choosing a local job are more likely to benefit from the education of the skill useful in their future jobs when the share of those who face the wealth constraint on educational investment, including themselves, is high, i.e. e+<ē, than when the share of such individuals is low, i.e. e+=ē.

Results for workers with a national job are also similar to the case e+=ē, but some differences exist. First, if the proportion of those with limited wealth is high enough that e+(0)≤α1−αh̲lδl, wnhn decreases with sl for any sl. Numerical simulations suggest that wlhl of those who actually choose a local job (i.e. those with a ≤ e+(sl)) also decreases with sl in this case. These results indicate that earnings of all workers decrease with sl when a large proportion of people have limited wealth for education. Second, when e+(0)>α1−αh̲lδl, sl maximizing wnhn is smaller than sl⋆⋆ when e+=ē. Hence, in contrast to workers with a local job, workers who have abundant wealth to choose a national job are less likely to benefit from the education of the skill for local jobs when e+<ē than when e+=ē.

A.2 When the Return to Education for Local Jobs is Endogenous

Now, as in Section 5, the effects of sl on earnings net of educational spending is examined by taking into account the fact that whether the return to education for local jobs is positive or not is endogenously determined, drawing on Propositions 1 and 2 as well as Proposition 5. The following proposition is the counterpart of Proposition 4 when e+=ē. Note that unlike when e+=ē, the proposition does not cover intermediate ranges of A, δn, δl, and a and as for (iii) of the proposition, some ranges of sl. In what follows, sl̲(f) (sl̄(f)) denotes the lower (higher) sl such that the return to education for local jobs is 0 when e+<ē. (“f” is to indicate the dependence of their values on the distribution of wealth.) Appendix B provides a fully detailed statement of the proposition.

Proposition 6

Suppose that e+<ē holds.

  1. When A, δn, or δl is small, net earnings of all workers decrease with sl except at sl=sl̲(f), sl=sl̄(f), or both, where they increase discontinuously.[35] The net earnings are maximized at sl = 0 if A, δn, or δl is sufficiently small.

  2. Those with wealth below a certain level choose a local job for any sl. Their net earnings decrease with sl except at sl=sl̄(f), where they increase discontinuously, and are maximized at sl = 0.

  3. When A, δn, and δl are not small, net earnings of those with wealth greater than a certain level decrease with sl for small sl, increase with sl for middle sl, and decrease with sl for large sl. When δ l is sufficiently large, their net earnings are maximized at sl∈(sl̲(f),sl̄(f)).[36]

The first part of the proposition shows that, if A, δn, or δl is small, net earnings of everyone decrease with sl except at either sl=sl̲(f), sl=sl̄(f), or both, where, differently from the case e+=ē, they increase discontinuously. But, as before, net earnings are highest at sl = 0 if A, δn, or δl is sufficiently small. The second part shows that irrespective of values of A, δn, and δl, net earnings of those with little wealth decrease with sl except at sl=sl̄(f), where, unlike when e+=ē, the earnings increase discontinuously, but as before, are highest at sl = 0. The last part shows that, when A and δn are not small and δl is sufficiently large, net earnings of those with sufficient wealth are highest at sl∈(sl̲(f),sl̄(f)).

Appendix B: More Precise Statements for Several Propositions and a Claim

This appendix presents more precise statements for Proposition 1 in Section 3.3, Proposition 4 in Section 5, and Propositions 5 and 6 in Appendix A. The appendix also presents Claim 1 that is used for drawing Figure 3 in Section 2.

Proposition 1

Suppose that A is not extremely low so that wlδlsl−1=(1−α)×Aαδn(1−sl)e+(1−α)h̲l+δlsle+αδlsl−1>0 at sl satisfying h̲l(1−sl−αsl)h̲lF(e+)+1−sl−αδlsl∫0e+ef(e)de+1α(1−sl−αsl)h̲l+(1−sl−α)slδle+hl(e+,sl)×e+f(e+)=0 when e+<ē.[37] Then,

    1. There exist two critical values of sl ∈ (0, 1) at which the return to educational investment for local jobs equals 0, and for sl smaller (greater) than the lower (higher) critical value, the return is negative and individuals with wealth a < e+ do not spend on education, while the return is positive and they spend e = a on education for s l between the critical values.

    2. The lower [higher] critical value of sl decreases [increases] with A, δn and δl.

  2. When sl is small or large enough that the return to educational investment for local jobs is negative, e+<(=)ē holds iff F(ē) is large (small) enough that F(ē)−(1−α)F(ē)α1−F(ē)1−αAδn(1−sl)αh̲lē1−α>(≤)1. When the return is positive, e+<(=)ē holds iff F(ē)>(≤)1−αh̲l+δlsl∫0ēef(e)deh̲l+δlsl(1−α)ē. When F(ē)≤1−α, e+=ē always holds.

Proposition 4

Suppose that F(ē)≤1−α and thus e+=ē hold.

  1. If A, δn, or δl is small enough that sl̲>sl⋆⋆, net earnings of all workers decrease with sl.[38]

  2. Otherwise,

    1. Net earnings of workers with a national job decrease with sl for sl<sl̲, increase with sl for sl∈(sl̲,sl⋆⋆), and decrease with sl for sl>sl⋆⋆. The net earnings are maximized at sl=sl⋆⋆ when A, δn, and δl are large enough that (1−α)2α21−αα1+h̲lδlēδl1−α>F(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]ē1−α.[39]

    2. Net earnings of workers with a local job and wealth above a certain level decrease with sl for sl<max{sl̲,sl°(e)}, increase with sl for sl∈(max{sl̲,sl°(e)},sl⋆(e)), and decrease with sl for sl>sl⋆(e), while net earnings of workers with wealth below the threshold decrease with sl for any sl. The threshold equals e satisfying sl⋆(e)=sl̲ if ē≤1+α1−αh̲lδl and if ē>1+α1−αh̲lδl and sl⋆Λ(ē)≤sl̲, where Λ(ē)≡αh̲lδl+ē1+δl4h̲ē(1−α)ē−(1+α)h̲lδl2, otherwise, it equals Λ(ē). Net earnings of the former workers with wealth a = e are maximized at sl=sl⋆(e), when A, δn, and δl are large enough that Aδnēα(1−α)α(1−sl⋆(e))1−ααh̲l+δlsl⋆(e)ē1−α−αF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]1−α+ē−e>0.

    3. sl̲ in (a), max{sl̲,sl°(e)} and the threshold wealth in (b) decrease with A, δn, and δl.

Proposition 6

Suppose that e+<ē holds.

  1. When A, δn, or δl is small enough that either e+(0)≤α1−αh̲lδl or sl̲(f)≥sl♯(sl♯ satisfies sl♯=(1−α)−αh̲lδle+(sl♯)) and sl̲(f)≥sl,h▽(e) hold,[40] net earnings of all workers decrease with sl except at sl=sl̲(f), sl=sl̄(f), or both, where they increase discontinuously.[41] The net earnings are maximized at sl = 0 if A, δn, or δl is small enough that 1−ααh̲lδne+(0)1−α>(1−α)2Aδl1−ααα.

  2. Those with wealth below a certain level choose a local job for any sl. The threshold wealth equals mine+(0),αh̲lδl+E(e|e<e+(0)) when E(e|e<e+(0))≤1+α1−αh̲lδl and equals mine+(0),Ω(e+(0)) when E(e|e<e+(0))>1+α1−αh̲lδl, where Ω(e+(0))≡αh̲lδl+E(e|e<e+(0))1+δl4h̲lE(e|e<e+(0))(1−α)E(e|e<e+(0))−(1+α)h̲lδl2. Their net earnings decrease with sl except at sl=sl̄(f), where they increase discontinuously, and are maximized at sl = 0.

  3. When A, δn, and δl are large enough that e+(0)>α1−αh̲lδl, E(e|e<e+(0))>αmax11−αh̲lδl,e+(0)1+α, and min{sl♭,sl,h△(e)}>sl̲(f)(sl♭ satisfies sl♭=(1−α)−αh̲lδlE(e|e<e+(sl♭))), net earnings of those with wealth greater than maxαh̲lδl+e+(0),Λ(ē) decrease with sl for small sl, increase with sl for middle sl, and decrease with sl for large sl.[42] When δl is large enough that h̲le+(0)1−α<h̲l+δlsl♭ēē1−α(1−sl♭)α for those with a national job and (1−α)Aδnα1−αα{(αe)α[h̲l+δl(1−α)e]1−α −e+(0)α(h̲l)1−α}−e>0 for those with a local job and wealth a = e, their net earnings are maximized at sl∈(sl̲(f),sl̄(f)).[43],[44]

The following proposition presents more precise results for Proposition 5. Note that except (i)(a) of the proposition, conditions are sufficient but not necessary. Proofs of the proposition and the claim below are contained in Appendix D posted on the author’s webpage (http://www.econ.kyoto-u.ac.jp/yuki/english.html).

Proposition A1

Suppose that the return to education for local jobs is positive and e+<ē holds.

    1. If e+(0)≤α1−αh̲lδl, which is true when the proportion of individuals with limited wealth is high, d(wnhn)dsl<0 for any sl.

    2. Otherwise, d(wnhn)dsl<0 for sl≥sl♯, where sl♯∈(0,sl⋆⋆) satisfies sl♯=(1−α)−αh̲lδle+(sl♯), and when E(e|e<e+(0))≡∫0e+(0)ef(e)deF(e+(0))>maxα1−αh̲lδl,αe+(0)1+α, d(wnhn)dsl>0 for sl≤sl♭, where sl♭∈(0,sl♯) satisfies sl♭=(1−α)−αh̲lδlE(e|e<e+(sl♭)).

    1. If ē≤1+α1−αh̲lδl, d(wlhl)dsl>0 for sl≤sl,h△(e)∈(sl⋆(e),sl,h▽(e)) when e>maxαh̲lδl+e+(0),Λ(ē), d(wlhl)dsl<0 for sl≥sl,h▽(e) when e>maxαh̲lδl+E(e|e<e+(0)),Ω(ē)(Ω(ē)<Λ(ē)), and d(wlhl)dsl<0 for any sl when e≤αh̲lδl+E(e|e<e+(0)), where sl,h△(e) is the greater solution of L(sl)≡−ee+(sl)sl2+(1−α)e+(sl)−(1+α)h̲lδlesl+−αh̲lδl+e+(sl)+eh̲lδl=0 and Λ(ē)≡αh̲lδl+ē1+δl4h̲lē(1−α)ē−(1+α)h̲lδl2, while sl,h▽(e) is the greater solution of M(sl) = 0, where M(sl) equals L(s l ) with e+(sl) replaced with E(e|e < e+(sl)), and Ω(ē) equals Λ(ē) with ē replaced with E(e|e<ē).

    2. Otherwise, d(wlhl)dsl>0 for sl∈max{0,sl,l△(e)},sl,h△(e) when e>maxαh̲lδl+e+(0),Λ(ē),[45] d(wlhl)dsl<0 for sl≤max{0,sl,l▽(e)} and sl≥sl,h▽(e) when e≥maxαh̲lδl+E(e|e<e+(0)),Ω(ē),[46] and d(wlhl)dsl<0 for any sl when e ≤ Ω(e+(0)) and when E(e|e<e+(0))≤1+α1−αh̲lδl and e≤αh̲lδl+E(e|e<e+(0)), where sl,l△(e)(sl,l▽(e)) is the smaller solution of L(sl) = 0 (M(sl) = 0).

Claim 1

Suppose ∫0ēef(e)de∈(0,(1−α)ē]. As illustrated in Figure 3, on the (sl,F(ē)) plane, the dividing line between e+<ē and e+=ē when the return to educational investment for local jobs is positive is located below the dividing line when the return is negative on the loci for zero return.

Appendix C: Proofs of Lemmas and Propositions

Proof of Proposition 1

    1. Suppose that education is worthwhile for local jobs. Consider case e+<ē first. From (2)–(4) and (7), the marginal return to education for local jobs when e+<ē equals

(14)wlδlsl−1=(1−α)Aαhne+,sl(1−α)hle+,slαδlsl−1=(1−α)Aαδn(1−sl)e+(1−α)h̲l+δlsle+αδlsl−1.

In the above equation,

(15)dδn(1−sl)e+h̲l+δlsle+dsl=h̲l+δlsle+−δne++δn(1−sl)de+dsl−δn(1−sl)e+δle++δlslde+dslh̲l+δlsle+2=−h̲l+δle+δne++h̲lδn(1−sl)de+dslh̲l+δlsle+2,

where the numerator equals, from (37) in the Proof of Lemma 1 below,

(16)−h̲l+δle+δne++h̲lδn(1−sl)de+dsl=−h̲l+δle+δne++h̲lδn(1−sl)δle+(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=δne+−h̲l+δle+(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)+δl(1−sl)h̲l(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=−δne+hle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+h̲l+δle+hle+,sle+f(e+)+δl(1−sl)h̲lα∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)<0.

The sign of the derivative of wlδlsl − 1 with respect to sl is same as the sign of the following derivative, which, by using the above equations, can be expressed as

(17)dδn(1−sl)e+h̲l+δlsle+(sl)1αdsl=dδn(1−sl)e+h̲l+δlsle+dsl(sl)1α+δn(1−sl)e+h̲l+δlsle+1α(sl)1αsl=(sl)1αδne+h̲l+δlsle+−hle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+h̲l+δle+e+f(e+)+δl(1−sl)h̲lα∫0e+ef(e)de(h̲l+δlsle+)(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)+1α1−slsl=−(sl)1αhle+,sl2δne+hle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+h̲l+δle+e+f(e+)+δl(1−sl)h̲lα∫0e+ef(e)de−1α1−slslhle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=(sl)1αδne+hle+,sl2hle+,sl1−sl−αslαsl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+1−sl−αslαslh̲l+1−sl−ααδle+e+f(e+)−δl(1−sl)h̲lα∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)

Since (1−α)∫e+ēef(e)de+1−F(ē)ēhl(e+,sl)=α∫0e+(h̲l+δlsle)×f(e)dee+ from (7), (5), (3), and (4), the numerator of the above equation becomes (sl)1αδne+ times

(18)1−sl−αslslh̲l∫0e+(h̲l+δlsle)f(e)de+1−sl−αslαslh̲l+1−sl−ααδle+hle+,sle+ ×f(e+)−δl(1−sl)h̲lα∫0e+ef(e)de=1slh̲l(1−sl−αsl)h̲lF(e+)+1−sl−αδlsl∫0e+ef(e)de +1α(1−sl−αsl)h̲l+(1−sl−α)slδle+hle+,sle+f(e+).

The following lemma presents the critical result on (18).

Lemma A1

There exists an sl∈(1−α,11+α) such that (18) equals zero and the equation is positive (negative) for lower (higher) sl.

Proof of Lemma A1

Clearly, (18) is positive for sl ≤ 1 − α and negative for sl≥11+α. (18) is positive for sl greater than 1 − α and weakly lower than the unique sl∈(1−α,11+α) satisfying (1−sl−αsl)h̲l+(1−sl−α)slδle+=0 too, because, for such sl, (1−sl−αsl)h̲l+(1−sl−α)slδle+≥0 and (1−sl−αsl)h̲lF(e+)+1−sl−αδlsl∫0e+ef(e)de≥−1−sl−αδlsle+F(e+)−∫0e+ef(e)de>0, where the former statement is true from

(19)d(1−sl−αsl)h̲l+(1−sl−α)slδle+dsl=−(1+α)h̲l+(1−2sl−α)δle++(1−sl−α)δlslde+dsl<−(1+α)h̲l+(1−2sl−α)δle+<0  for  sl>1−αsince de+dsl>0.

Thus, the lemma is proved if the derivative of the expression inside the curly bracket of (18) with respect to sl is negative for sl greater than the critical value and lower than 11+α, which equals

(20)h̲ld(1−sl−αsl)h̲lF(e+)+1−sl−αδlsl∫0e+ef(e)dedsl+1αd(1−sl−αsl)h̲l+(1−sl−α)slδle+dslhle+,sle+f(e+) +1α(1−sl−αsl)h̲l+(1−sl−α)slδle+dhle+,sle+f(e+)dsl,
(21)where  d(1−sl−αsl)h̲lF(e+)+1−sl−αδlsl∫0e+ef(e)dedsl=−(1+α)h̲lF(e+)+(1−2sl−α)δl∫0e+ef(e)de +(1−sl−αsl)h̲l+(1−sl−α)slδle+f(e+)de+dsl<−(1+α)h̲lF(e+)+(1−2sl−α)δl∫0e+ef(e)de<0 for sl greater than the critical value,

where the first inequality sign is from (19). Hence, the derivative of (18) is negative if the last term of (20) is negative, which holds unless f′(e+) is negative and f′(e+) is very large, since (1−sl−αsl)h̲l+(1−sl−α)slδle+<0 for such sl. □

From the lemma, there exists an sl∈(1−α,11+α) such that the derivative of the marginal return wlδlsl − 1 with respect to sl equals zero, and the marginal return increases (decreases) with sl for sl smaller (greater) than the critical value. Because the marginal return equals −1 at sl = 0, 1 from (14), if A is high enough that wlδlsl − 1 > 0 holds at sl such that (18) equals zero, there exist two critical values of sl satisfying wlδlsl − 1 = 0 and the marginal return is negative for sl smaller than the lower critical value and greater than the higher one and positive for sl between them. Next, consider case e+=ē. In this case, from (8)–(10),

(22)Hl(ē,πn,sl)=(1−α)1−F(ē)hl(ē,sl)+∫0ēhl(e,sl)f(e)de,Hn(ē,πn,sl)=α1−F(ē)hl(ē,sl)+∫0ēhl(e,sl)f(e)dehn(ē,sl)hl(ē,sl).

Thus, from (2)–(4), the marginal return when e+=ē equals

(23)wlδlsl−1=(1−α)Aαhn(ē,sl)(1−α)hl(ē,sl)αδlsl−1=(1−α)Aαδn(1−sl)ē(sl)1α(1−α)h̲l+δlslēαδl−1.

In the above equation,

(24)dδn(1−sl)ēh̲l+δlslē(sl)1αdsl=dδn(1−sl)ēh̲l+δlslēdsl(sl)1α+δn(1−sl)ēh̲l+δlslē(sl)1ααsl=(sl)1αδnēh̲l+δlslē−h̲l+δlē+h̲l+δlslē1−slαslh̲l+δlslē=(sl)1αδnē1αslh̲l+δlslē(1−sl−αsl)h̲l+δlēsl(1−sl−α)h̲l+δlslē,

which is positive (negative) for sl smaller (greater) than the critical value satisfying (1−sl−αsl)h̲l+δlēsl(1−sl−α)=0. Hence, the statement is true as in the case of e+<ē. (b) The result when e+=ē is straightforward from (23). When e+<ē, the marginal return depends on e+ from (14), thus how these exogenous variables affect the return through e+ must be examined. From (7), (5), (3), and (4), e+ is a solution to

(25)α∫0e+h̲l+δlslef(e)dee+=(1−α)∫e+ēef(e)de+1−F(ē)ēh̲l+δlsle+.

Thus, e+ does not depend on A and δn and the result on these variables is straightforward from (14). e+ depends positively on δl from (35) in the Proof of Lemma 1 and the derivative of the RHS − LHS of the above equation with respect to δl, which equals

(26)(1−α)∫e+ēef(e)de+1−F(ē)ēsle+−α∫0e+slef(e)dee+=αsle+∫0e+h̲l+δlslef(e)dee+h̲l+δlsle+−∫0e+ef(e)de>0.

The result on δl is clear from (14) and de+dδl>0. (ii) When the return to education for local jobs is positive, e+=ē iff (10) satisfies πn ≤ 1, i.e.

(27)α∫0ēhl(e,sl)f(e)de≤(1−α)[1−F(ē)]hl(ē,sl)⇔α∫0ēh̲l+δlslef(e)de≤(1−α)[1−F(ē)]h̲l+δlslē⇔h̲lF(ē)+δlslα∫0ēef(e)de+(1−α)ēF(ē)≤(1−α)h̲l+δlslē⇔F(ē)≤(1−α)h̲l+δlslē−δlslα∫0ēef(e)deh̲l+δlsl(1−α)ē=1−αh̲l+δlsl∫0ēef(e)deh̲l+δlsl(1−α)ē.

When ∫0ēef(e)de>(1−α)ē, which implies F(ē)>1−α, e+=ē cannot hold, because the RHS of (27) decreases with sl and is smaller than 1 − α. When ∫0ēef(e)de≤(1−α)ē, the RHS weakly increases with sl, hence e+=ē could hold and e+=ē always when F(ē)≤1−α. The rest of the statement is straightforward from (27) (note that F(ē) raises ∫0ēef(e)de and lowers the RHS). When the return is negative, from (12), (4), and (13), wnhn(ē,sl)−ē=wlh̲l is expressed as

(28)αAF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]hn(ē,sl)1−αδn(1−sl)−1ē=(1−α)Aπn[1−F(ē)]hn(ē,sl)F(ē)+(1−πn)1−F(ē)h̲lαh̲l.
πn ∈ (0, 1] satisfying the above equation exists, that is, e+=ē holds iff the LHS of the above equation is weakly smaller than the RHS at πn = 1:
(29)αAF(ē)h̲l[1−F(ē)]hn(ē,sl)1−αδn(1−sl)−1ē≤(1−α)A[1−F(ē)]hn(ē,sl)F(ē)h̲lαh̲l⇔αAF(ē)h̲l[1−F(ē)]ē1−α−δn(1−sl)−αē≤(1−α)A[1−F(ē)]ēF(ē)h̲lαh̲l⇔δn(1−sl)−α≥A[1−F(ē)]ēF(ē)h̲lαh̲l[1−F(ē)]ēF(ē)−(1−α)⇔F(ē)−(1−α)F(ē)α1−F(ē)1−αAδn(1−sl)αh̲lē1−α≤1.

Clearly, the condition is satisfied when F(ē)≤1−α. It holds when F(ē) is low because the derivative of the first part of the LHS of (29) with respect to F(ē) equals

(30)F(ē)α1−F(ē)1−αf(ē)1−F(ē)−(1−α)αF(ē)−1−α1−F(ē)F(ē)α1−F(ē)1−α2=F(ē)α1−F(ē)1−αf(ē)1−F(ē)−(1−α)α−F(ē)F(ē)1−F(ē)F(ē)α1−F(ē)1−α2=F(ē)α1−F(ē)1−αf(ē)α(1−α)F(ē)α1−F(ē)1−α3>0.

□

Proof of Lemma 1

[When the return to educational investment for local jobs is positive]. When e+<ē, by totally differentiating (7), one obtains

(31)αHle+,sl∂hne+,sl∂e+−(1−α)Hne+,sl∂hle+,sl∂e+ +α∂Hle+,sl∂e+hne+,sl−(1−α)∂Hne+,sl∂e+hle+,slde++αHle+,sl∂hne+,sl∂sl−(1−α)Hne+,sl∂hle+,sl∂sl +α∂Hle+,sl∂slhne+,sl−(1−α)∂Hne+,sl∂slhle+,sldsl=0,
(32)where ∂Hle+,sl∂e+=hle+,slf(e+)>0,∂Hne+,sl∂e+=−hne+,slf(e+)<0.
(33)∂Hle+,sl∂sl=∫0e+∂hl(e,sl)∂slf(e)de=δl∫0e+ef(e)de>0.
(34)∂Hne+,sl∂sl=∫e+ē∂hn(e,sl)∂slf(e)de+[1−F(ē)]∂hn(ē,sl)∂sl=−δn∫e+ēef(e)de+[1−F(ē)]ē<0.

In (31), the term of de+ equals

(35)αHle+,sl∂hne+,sl∂e+−(1−α)Hne+,sl∂hle+,sl∂e+ +hne+,slhle+,slf(e+)=1e+αHle+,slhne+,sl−(1−α)Hne+,slhle+,sl−h̲l+hne+,slhle+,slf(e+)=1e+(1−α)Hne+,slh̲l+hne+,slhle+,slf(e+)(from(7))=δn(1−sl)(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+e+hle+,slf(e+)>0,

The terms of dsl equals (δ≡δnδl)

(36)αHle+,sl∂hne+,sl∂sl−(1−α)Hne+,sl∂hle+,sl∂sl+αhne+,sl∫0e+∂hl(e,sl)∂sl×f(e)de−(1−α)∫e+ē∂hn(e,sl)∂slf(e)de+[1−F(ē)]∂hn(ē,sl)∂slhle+,sl=δl−e+αHle+,slδ+(1−α)Hne+,sl+αhne+,sl∫0e+ef(e)de +(1−α)δ∫e+ēef(e)de+[1−F(ē)]ēhle+,sl=δl−e+(1−α)Hne+,slhne+,slδhle+,sl+hne+,sl+αhse+,sl∫0e+ef(e)de+(1−α)δ∫e+ēef(e)de+[1−F(ē)]ēhle+,sl(from(7))=δl−(1−α)∫e+ēef(e)de+[1−F(ē)]ēδhle+,sl+hne+,sl+αhse+,sl∫0e+ef(e)de+(1−α)δ∫e+ēef(e)de+[1−F(ē)]ēhle+,sl(from(5)and(4))=−δlhne+,sl(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de<0,

where the last inequality holds because

αh̲l∫0e+f(e)de+δlsl∫0e+ef(e)dee+=(1−α)∫e+ēef(e)de+[1−F(ē)]ē×h̲l+slδle+
⇔αF(e+)e+−(1−α)∫e+ēef(e)de+[1−F(ē)]ēh̲l=slδle+(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de from(7)and thus sign(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de=signαF(e+)e+−(1−α)∫e+ēef(e)de+[1−F(ē)]ē.

Hence,

(37)de+dsl=δlhne+,sl(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de1e+(1−α)Hne+,slh̲l+hne+,slhle+,slf(e+)=δle+(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)>0.

When e+=ē, from (10),

(38)dπndsl=αhl(ē,sl)∫0ē∂hl(e,sl)∂slf(e)de−∂hl(ē,sl)∂sl∫0ēhl(e,sl)f(e)de[1−F(ē)]hl(ē,sl)2=αδlh̲l∫0ēef(e)de−ē∫0ēf(e)de[1−F(ē)]hl(ē,sl)2<0.

[When the return is negative] First consider case e+<ē. From (12), (4), and (11), wnhne+,sl−e+=wlh̲l can be expressed as

(39)αAF(e+)h̲l∫e+ēef(e)de+1−F(ē)ēδn(1−sl)1−αδn(1−sl)−1e+=(1−α)A∫e+ēef(e)de+1−F(ē)ēδn(1−sl)F(e+)h̲lαh̲l⇔αAF(e+)h̲l∫e+ēef(e)de+1−F(ē)ē1−α−δn(1−sl)−αe+=(1−α)A∫e+ēef(e)de+1−F(ē)ēF(e+)h̲lαh̲l

Since the LHS increases with e+, the RHS decreases with e+, and the LHS decreases with sl, e+ satisfying the above equation increases with s l . When e+=ē, from (12), (4), and (13), wnhn(ē,sl)−ē=wlh̲l can be expressed as

(40)αAF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]ē1−αδn(1−sl)−1ē=(1−α)Aπn[1−F(ē)]ēF(ē)+(1−πn)1−F(ē)h̲lαh̲l⇔αAF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]ē1−α−δn(1−sl)−αē=(1−α)Aπn[1−F(ē)]ēF(ē)+(1−πn)1−F(ē)h̲lαh̲l.

Since the LHS decreases with πn, the RHS increases with πn, and the LHS decreases with sl, πn satisfying the above equation decreases with sl. □

Proof of Lemma 2

[When the return to educational investment for local jobs is positive]. When e+<ē, from (2) and (32) in the Proof of Lemma 1,

(41)∂wn∂e+=(1−α)wn1Hle+,sl∂Hle+,sl∂e+−1Hne+,sl∂Hne+,sl∂e+=(1−α)wnf(e+)hle+,slHle+,sl+hne+,slHne+,sl=wnf(e+)hne+,slHne+,sl (from (7))=wne+∫e+ēef(e)de+[1−F(ē)]ēf(e+)>0.

From (2), (33), and (34) in the Proof of Lemma 1,

(42)∂wn∂sl=(1−α)wn1Hle+,sl∂Hle+,sl∂sl−1Hne+,sl∂Hne+,sl∂sl=(1−α)δlwn1Hle+,sl∫0e+ef(e)de+δHne+,sl∫e+ēef(e)de+[1−F(ē)]ē=(1−α)δlwn∫0e+ef(e)deh̲l∫0e+f(e)de+δlsl∫0e+ef(e)de+1δl(1−sl)=(1−α)δlwnh̲lδl∫0e+f(e)de+∫0e+ef(e)de(1−sl)h̲l∫0e+f(e)de+δlsl∫0e+ef(e)de=wne+∫e+ēef(e)de+[1−F(ē)]ēαh̲l∫0e+f(e)de+δl∫0e+ef(e)de(1−sl)hle+,sl>0.

The last equality is because αh̲l∫0e+f(e)de+δlsl∫0e+ef(e)dee+=(1−α)∫e+ēef(e)de+[1−F(ē)]ēhle+,sl from (7). From (41), (42), and (37) in the Proof of Lemma 1,

(43)dwndsl=∂wn∂sl+∂wn∂e+de+dsl=wne+∫e+ēef(e)de+[1−F(ē)]ēαh̲lF(e+)+δl∫0e+ef(e)de(1−sl)hle+,sl+δlf(e+)e+(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=wne+∫e+ēef(e)de+[1−F(ē)]ē×αh̲lF(e+)+δl∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)+(1−sl)hle+,slδlf(e+)e+(1−α)∫e+ēef(e)de+[1−F(ē)]ē−α∫0e+ef(e)de(1−sl)hle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=wne+∫e+ēef(e)de+[1−F(ē)]ē×αh̲lF(e+)+δl∫0e+ef(e)de(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)δl(1−sl)(1−α)∫e+ēef(e)de+[1−F(ē)]ē+αh̲lF(e+)+δlsl∫0e+ef(e)de(1−sl)hle+,sl(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)=wnhle+,sle+11−sl(1−α)αh̲le+h̲lF(e+)+δl∫0e+ef(e)de+h̲le++δlhle+,sle+f(e+)(1−α)h̲le+∫e+ēef(e)de+[1−F(ē)]ē+hle+,sle+f(e+)>0,

where the last equality is again from (7). When e+=ē, from (8)–(10),

(44)Hl(πn,sl)=∫0ēhl(e,sl)f(e)de+1−α1+∫0ēhl(e,sl)f(e)de[1−F(ē)]hl(ē,sl)1−F(ē)hl(ē,sl)=(1−α)∫0ēhl(e,sl)f(e)de+1−F(ē)hl(ē,sl) ,
(45)Hn(πn,sl)=αhn(ē,sl)hl(ē,sl)∫0ēhl(e,sl)f(e)de+1−F(ē)hl(ē,sl) .

By substituting the above equations into (2),

(46)wn=αA1−ααhl(ē,sl)hn(ē,sl)1−α.

Thus,

(47)dwndsl=(1−α)wn1hl(ē,sl)δlē+1hn(ē,sl)δnē=(1−α)wn11−slhl(ē,sl)+δlδnhn(ē,sl)hl(ē,sl)>0.

Since wl=(1−α)AHnHlα=(1−α)AαAwnα1−α from (2), dwldsl=−α1−αwlwndwndsl<0. [When the return is negative] Straightforward from Lemma 1 and the first equation of (39) when e+<ē and of (40) when e+=ē. □

Proof of Proposition 2

When e+<ē, wlh̲l=wnhne+,sl−e+=wnhn(1,sl)−1e+ decreases with sl from Lemma 2. Then, wnhn(e,sl)−e=wnhn(1,sl)−1e for e > e+ also decreases with sl, because wnhn(1, sl) − 1 decreases with sl from the above equation and de+dsl>0 (Lemma 1). When e+=ē, wlh̲l=wnhn(ē,sl)−ē decreases with sl from Lemma 2. □

Proof of Proposition 3

  1. From (47) in the Proof of Lemma 2,

(48)d[wnhn(ē,sl)]dsl=dwndslhn(ē,sl)+wndhn(ē,sl)dsl=wnhn(ē,sl)1−sl(1−α)hl(ē,sl)+δlδnhn(ē,sl)hl(ē,sl)−1=wnhn(ē,sl)1−sl(1−α)δlδnhn(ē,sl)−αhl(ē,sl)hl(ē,sl).

Thus,

(49)d(wnhn)dsl⋛0⇔(1−α)δl(1−sl)ē−αh̲l+δlslē⋛0⇔sl⪋(1−α)−αh̲lδlē.

  1. From (47) in the Proof of Lemma 2 and (2),

(50)d[wlhl(e,sl)]dsl=dwldslhl(e,sl)+wldhl(e,sl)dsl=−α1−αwlwndwndslhl(e,sl)+wldhl(e,sl)dsl=wl1−sl1hl(ē,sl)−αhl(ē,sl)+δlδnhn(ē,sl)hl(e,sl) +δle(1−sl)hl(ē,sl).

Thus,

(51)d(wlhl)dsl⋛0⇔−αh̲l+δlēh̲l+δlsle+δle(1−sl)h̲l+δlslē⋛0⇔−eē(sl)2+(1−α)ē−(1+α)h̲lδlesl+−αh̲lδl+ē+eh̲lδl⋛0.

  1. Suppose that the LHS of (51) is positive at sl = 0, i.e. e>αh̲lδl+ē. Because the derivative of the LHS at sl = 0 is non-positive, i.e. (1−α)ē−(1+α)h̲lδl≤0 and the LHS at sl = 1 equals −eē+(1−α)ē−(1+α)h̲lδle+−αh̲lδl+ē+eh̲lδl=−αh̲lδl+ēh̲lδl+e<0, there exists an sl⋆(e)∈(0,1) such that d(wlhl)dsl⋛0⇔sl⪋sl⋆(e), where sl⋆′(e)>0, since, from (51),

(52)sl⋆(e)=(1−α)ē−(1+α)h̲lδle+(1−α)ē−(1+α)h̲lδl2e2+4eē−αh̲lδl+ē+eh̲lδl2eē=(1−α)ē−(1+α)h̲lδl+(1−α)ē−(1+α)h̲lδl2+4ē−αh̲lδl+ēe−1+1h̲lδl2ē.
sl⋆(e)<sl⋆⋆ for e<ē, since wlhl(ē,sl)=wnhn(ē,sl) (thus sl⋆(ē)=sl⋆⋆) and sl⋆′(e)>0. Suppose instead that the LHS of (51) is non-positive at sl = 0, i.e. e≤αh̲lδl+ē. Because the derivative of the LHS at sl = 0 is non-positive and the LHS at sl = 1 is negative, d(wlhl)dsl<0 for any sl > 0 (and d(wlhl)dsl<(=)0 at sl = 0 when e<(=)αh̲lδl+ē). (b) The case in which the LHS of (51) at sl = 0 is positive, i.e. e>αh̲lδl+ē, can be proven as in (a). Suppose that the LHS at sl = 0 is zero, i.e. e=αh̲lδl+ē. Since the derivative of the LHS at sl = 0 is positive, i.e. (1−α)δlē−(1+α)h̲l>0, there exists an sl⋆(e) in (0, 1) such that the LHS of (51) equals 0, and the LHS is zero at sl = 0, positive for sl∈0,sl⋆(e), and negative for sl>sl⋆(e). Thus, d(wlhl)dsl⋛0 for positive sl⪋sl⋆(e) and d(wlhl)dsl=0 at sl = 0. Instead, suppose that the LHS of (51) at sl = 0 is negative, e<αh̲lδl+ē. Since the derivative of the LHS at sl = 0 is positive, i.e. (1−α)δlē−(1+α)h̲l>0, the LHS is positive (negative) when e>(<)αh̲lδl+ēh̲lδl−ēsl+(1−α)ē−(1+α)h̲lδlsl+h̲lδl, where αh̲lδl+ēh̲lδl−ēsl+(1−α)ē−(1+α)h̲lδlsl+h̲lδl<αh̲lδl+ē when sl > 0 and −ēsl+(1−α)ē−(1+α)h̲lδl>0⇔sl∈(0,(1−α)−(1+α)h̲lδlē). So the LHS of (51) is negative for any e<αh̲lδl+ē when sl≥(1−α)−(1+α)h̲lδlē. For sl<(1−α)−(1+α)h̲lδlē, the LHS of (51) is positive (negative) when e>(<)αh̲lδl+ēh̲lδl−ēsl+(1−α)ē−(1+α)h̲lδlsl+h̲lδl, where the RHS is lowest when −ēsl+(1−α)ē−(1+α)h̲lδl−ēsl=0⇔sl=(1−α)ē−(1+α)h̲lδl2ē. Hence, the LHS of (51) is negative, i.e. d(wlhl)dsl<0, for any sl when e≤αh̲lδl+ēh̲lδl−ēsl+(1−α)ē−(1+α)h̲lδlsl+h̲lδl at sl=(1−α)ē−(1+α)h̲lδl2ē=αh̲lδl+ēh̲lδlh̲lδl+(1−α)ē−(1+α)h̲lδl24ē=αh̲lδl+ē1+δl4h̲ē(1−α)ē−(1+α)h̲lδl2, except at sl=(1−α)ē−(1+α)h̲lδl2ē and e=αh̲lδl+ē1+δl4h̲ē(1−α)ē−(1+α)h̲lδl2, in which the derivative is zero. When e∈αh̲lδl+ē1+δl4h̲ē(1−α)ē−(1+α)h̲lδl2,αh̲lδl+ē, there exist sl°(e) and sl⋆(e), where 0<sl°(e)<sl⋆(e)<(1−α)−(1+α)h̲lδlē<sl⋆⋆, such that the LHS of (51) equals 0, and the LHS is negative for sl < sl°(e) and sl>sl⋆(e) and positive for sl∈sl°(e),sl⋆(e). sl⋆′(e)>0>sl◦′(e) from (52). □

Proof of Proposition 4

e+=ē always holds when F(ē)≤1−α from Proposition 1 (ii). The following lemma is used in the proof of the proposition.

Lemma A2

When e+=ē, net earnings change continuously when the return to educational investment for local jobs turns from negative to positive with a change in sl.

Proof of Lemma A2

When e+=ē and the return is negative, wnhn(ē,sl)−ē=wlhl(0,sl)⇔αAHlHn1−αδn(1−sl)−1ē=(1−α)AHnHlαh̲l, where Hl=F(ē)+(1−πn)1−F(ē)h̲l and Hn=πn[1−F(ē)]hn(ē,sl), from (13) and (12). When e+=ē and the return is positive, wnhn(ē,sl)−ē=wlhl(ē,sl)−ē⇔αAHlHn1−αδn(1−sl)−1ē=(1−α)AHnHlαh̲l+δlslē−ē, where Hl=∫0ēhl(e,sl)f(e)de+(1−πn)1−F(ē)hl(ē,sl) and Hn=πn[1−F(ē)]hn(ē,sl), from (2), (8), and (9). When the return is zero, i.e. wlδlsl − 1 = 0, this equation becomes αAHlHn1−αδn(1−sl)−1ē=(1−α)AHnHlαh̲l, the same equation as the case of the negative return, and net earnings of all workers are the same. Since HnHl satisfying this equation is uniquely determined for given sl, net earnings when the return is zero are same as net earnings when the return is negative and sl→sl̲, sl̄. □

(i) Earnings decrease with sl when the return to education for local jobs is negative from Proposition 2, while when it is positive and e+=ē, earnings of all decrease with sl for sl>sl⋆⋆≡(1−α)−αh̲lδlē from Proposition 3. From Proposition 1 (i)(b), the lower critical value for the negative return, sl̲, decreases with A, δl, and δn. Hence, from Lemma A2, net earnings of all decrease with sl when A, δl, and δn are small enough that sl̲≥sl⋆⋆=sl⋆(e+)=sl⋆(ē)≡(1−α)−αhlδlē. Note that sl⋆⋆ is smaller than the higher critical value sl̄, since sl⋆⋆<1−α<sl̄ from Lemma A1 in the Proof of Proposition 1.

(ii) (a) When sl̲<sl⋆⋆=sl⋆(e+)=sl⋆(ē), from Propositions 2 and 3 and Lemma A2, net earnings of workers with a national job decrease with sl for sl<sl̲, increase with sl for sl∈(sl̲,sl⋆⋆), and decrease with sl for sl>sl⋆⋆. Thus, their net earnings are maximized at either sl = 0 or sl=sl⋆⋆. From the Proof of Lemma A2, net earnings of such workers at sl = 0 equal

(53)αAF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]δnē1−αδn−1ē,

where πn is a solution to (28) in the Proof of Proposition 1. From (2) and (22) in the Proof of Proposition 1, net earnings of such workers at sl=sl⋆⋆ equal

(54)αA(1−α)h̲l+δlsl⋆⋆ēαδn1−sl⋆⋆ē1−αδn1−sl⋆⋆−1ē=αA(1−α)2α2δlδn1−αδnα1+h̲lδlē−1ē.

From these equations, the net earnings are maximized at sl=sl⋆⋆ if

(55)(1−α)2α21−αα1+h̲lδlēδl1−α>F(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]ē1−α,

which holds when A, δn, and δl are large. This is because πn increases with A and δn from (28), 1+h̲lδlēδl1−α increases with δl from (1−α)1δl−1+h̲lδlē−11δl2h̲lē=(1−α)−αh̲lδlēδl1+h̲lδlē=sl⋆⋆δl1+h̲lδlē>0, and the condition does hold when πn = 1, i.e. (1−α)2α21−αα1+h̲lδlē>F(ē)h̲l[1−F(ē)]δlē1−α. Since F(ē)≤1−α at sl = 0 from (27) in the Proof of Proposition 1, the last statement is proved if (1−α)2α21−αα1+h̲lδlē>(1−α)h̲lαδlē1−α⇔(1−α)1−ααα1+h̲lδlē−1h̲lδlēα>1 holds, which is true, because sl⋆⋆≡(1−α)−αh̲lδlē>0⇔h̲lδlē<1−αα and the LHS decreases with h̲lδlē:

(56)αh̲lδlē−h̲lδlē−21+h̲lδlē−1=α−(1−α)h̲lδlē−1h̲lδlē1+h̲lδlē−1<0 from h̲lδlē<1−αα.

(b) Consider case ē≤1+α1−αh̲lδl first. Since sl⋆(e) increases with e, sl⋆(α(h̲lδl+ē))=0, and sl⋆(ē)=sl⋆⋆ from (52) in the Proof of Proposition 3, there exists an e†∈(α(h̲lδl+ē),ē), such that sl⋆(e†)=sl̲. Then, the relationship between sl and net earnings of workers with a local job and e > e† is similar to that of workers with a national job: their net earnings decrease with sl for sl<sl̲, increase with sl for sl∈(sl̲,sl⋆(e)), and decrease with sl for sl>sl⋆(e) from Propositions 2 and 3 and Lemma A2. As for workers with e ≤ e†, net earnings decrease with sl. Now consider case ē>1+α1−αh̲lδl. If A, δl, and δn are small enough that sl⋆(α(h̲lδl+ē))=1−α−(1+α)h̲lδlē<sl̲, the result is same as the case of ē≤1+α1−αh̲lδl. Let Λ(ē)≡αh̲lδl+ē1+δl4h̲ē(1−α)ē−(1+α)h̲lδl2. If sl⋆(α(h̲lδl+ē))>sl̲>sl⋆Λ(ē)=(1−α)−(1+α)h̲lδlē2, there exists an e‡∈Λ(ē),αh̲lδl+ē such that sl⋆(e‡)=sl̲. The results for those with e>αh̲lδl+ē and those with e≤Λ(ē) are same as the corresponding cases (e > e† and e ≤ e† respectively) of ē≤1+α1−αh̲lδl. As for those with e∈Λ(ē),αh̲lδl+ē, sl°(e)≤(1−α)−(1+α)h̲lδlē2=sl⋆Λ(ē)<sl⋆(e‡)=sl̲ holds for any e from (51) and (52) in the Proof of Proposition 3. Hence, the result of those with e > e‡ is similar to that of those with e>αh̲lδl+ē, and the result of those with e ≤ e‡ is same as that of those with e≤Λ(ē). In sum, the result is similar to when ē≤1+α1−αh̲lδl except that the critical wealth level is e‡, not e†. Finally, if sl⋆Λ(ē)>sl̲, since sl°(e) decreases with e, sl°(α(h̲lδl+ē))=0, and sl°Λ(ē)=sl⋆Λ(ē)=(1−α)−(1+α)h̲lδlē2 from (51) and (52), there exists an e⋇∈Λ(ē),αh̲lδl+ē such that sl◦(e⋇)=sl̲. Hence, as for workers with e∈Λ(ē),αh̲lδl+ē, net earnings of those with e < e⋇ decrease with sl for sl < sl°(e), increase with sl for sl<(sl◦(e),sl⋆(e)), and decrease with sl for sl>sl⋆(e), while net earnings of those with e ≥ e⋇ decrease with sl for sl<sl̲, increase with sl for sl∈(sl̲,sl⋆(e)), and decrease with sl for greater sl. The results for those with e>αh̲lδl+ē and those with e≤Λ(ē) are same as the previous case. In sum, net earnings of workers with wealth greater than a certain level decrease with sl for sl<max{sl̲,sl°(e)}, increase with sl for sl∈(maxsl̲,sl◦(e),sl⋆(e)), and decrease with sl for sl>sl⋆(e), while net earnings of workers with a = e smaller than the threshold decrease with sl. Thus, net earnings of workers with wealth greater than a threshold are maximized at either sl = 0 or sl=sl⋆(e). From (28) in the Proof of Proposition 1, net earnings of such workers at sl = 0 is same as net earnings of workers with a national job, which equals (53). From (23) in the Proof of Proposition 1, net earnings of such workers with e at sl=sl⋆(e) equal

(57)(1−α)Aαδn(1−sl⋆(e))ē(1−α)h̲l+δlsl⋆(e)ēαh̲l+δlsl⋆(e)ē−e.

Thus, the net earnings are maximized at sl=sl⋆(e) if

(58)Aδnēα(1−α)α(1−sl⋆(e))1−ααh̲l+δlsl⋆(e)ē1−α −αF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]1−α+ē−e>0,

which holds when A, δn, and δl are large, because πn increases with A and δn, the derivative of the first term of the expression inside the large curcly bracket with resepct to sl is 0 at sl=sl⋆(e) and negative for sl>sl⋆(e), and thus the expression is positive when A and δl are large from (55):

(59)(1−α)α(1−sl⋆(e))1−ααh̲l+δlsl⋆(e)ē1−α−αF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]1−α>(1−α)α1−sl⋆⋆1−ααh̲l+δlsl⋆⋆ē1−α−αF(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]1−α(from Proposition 3 (ii)(a))=αδlē1−α(1−α)2α21−αα1+h̲lδlē−F(ē)+(1−πn)1−F(ē)h̲lπn[1−F(ē)]δlē1−α>0.

(c) From the proof of (b), the threshold wealth when ē≤1+α1−αh̲lδl is e†∈(α(h̲lδl+ē),ē) such that sl⋆(e†)=sl̲. The threshold when ē>1+α1−αh̲lδl is e†∈(α(h̲lδl+ē),ē) if sl⋆(α(h̲lδl+ē))<sl̲, e‡∈Λ(ē),αh̲lδl+ē such that sl⋆(e‡)=sl̲ if sl⋆(α(h̲lδl+ē))>sl̲>sl⋆Λ(ē), and Λ(ē) if sl⋆Λ(ē)>sl̲. When ē>1+α1−αh̲lδl, case sl⋆(α(h̲lδl+ē))<sl̲ is realized when A, δl, and δn are small, and case sl⋆Λ(ē)>sl̲ is realized when they are large, because sl̲ decreases with A, δl, and δn from Proposition 1, and sl⋆(α(h̲lδl+ē))=(1−α)−(1+α)h̲lδlē and sl⋆Λ(ē)=12(1−α)−(1+α)h̲lδlē increase with δl. Further, e† and e‡ decrease with A, δl, and δn, because sl⋆(e) increases with δl at e = e†, e‡ from Lemma A3 below and increases with e, and sl̲ decreases with A, δl, and δn. Hence, the threshold of wealth decreases with A and δn (except when the threshold is αh̲lδl+ē and Λ(ē)) and δl. □

Lemma A3

∂sl⋆(e†)∂δl>0 and ∂sl⋆(e‡)∂δl>0.

Proof of Lemma A3

From (52) in the Proof of Proposition 2, the derivative of sl⋆(e) with respect to δ l equals a constant times

(60)(1+α)h̲l−12−(1+α)h̲l2(1−α)ē−(1+α)h̲lδl +4ē−αh̲lδl+ēe−1+1h̲l−4ēαh̲lδle−1h̲l ×(1−α)ē−(1+α)h̲lδl2+4ēδl−αh̲lδl+ēe−1+1h̲l−1/2.

Thus,

(61)∂sl⋆(e)∂δl⋛0⇔−(1+α)h̲l2(1−α)ē−(1+α)h̲lδl+4ē−αh̲lδl+ēe−1+1h̲l−4ēαh̲lδle−1h̲l⋚2(1+α)h̲l×(1−α)ē−(1+α)h̲lδl2+4ēδl−αh̲lδl+ēe−1+1h̲l1/2.

The lemma is proved if it is shown that ∂sl⋆(e)∂δl≤0 cannot hold at e = e†, e‡. From the above equation, ∂sl⋆(e)∂δl≤0 is possible only when the LHS of the equation is positive, which is true only when (1−α)ē−(1+α)h̲lδl<0⇔ē<1+α1−αh̲lδl or −αh̲lδl+ēe−1+1−αh̲lδle−1>0⇔e>α2h̲lδl+ē. When the LHS is positive, the above equation can be expressed as

(62)∂sl⋆(e)∂δl⋛0⇔−(1+α)h̲l2(1−α)ē−(1+α)h̲lδl +4ē−αh̲lδl+ēe−1+1h̲l−4ēαh̲lδle−1h̲l2⋚2(1+α)h̲l2×(1−α)ē−(1+α)h̲lδl2+4ēδl−αh̲lδl+ēe−1+1h̲l
(63)⇔−(1+α)(1−α)ē−(1+α)h̲lδl−α2h̲lδl+ēe−1+1+−α2h̲lδl+ēe−1+12ē⋚(1+α)21δl−αh̲lδl+ēe−1+1h̲l⇔−α2h̲lδl+ēe−1+1−2h̲lδl+ēe−1+αē−(1+α)21δlh̲lδle−1h̲l⋚0.

The expression is clearly negative when e≥α2h̲lδl+ē. Hence, ∂sl⋆(e)∂δl≤0 is possible only when ē<1+α1−αh̲lδl and e<α2h̲lδl+ē. In this case, the LHS of (61) is weakly smaller than (1+α)h̲l2(1+α)h̲lδl−(1−α)ē, while, since e†>αh̲lδl+ē (e‡ does not exist) when ē<1+α1−αh̲lδl, the RHS of (61) is greater than (1+α)h̲l2(1+α)h̲lδl−(1−α)ē. Hence, ∂sl⋆(e†)∂δl>0 holds in this case. The fact that e‡ does not exist when ē<1+α1−αh̲lδl proves that ∂sl⋆(e‡)∂δl≤0 cannot happen. □

Proof of Corollary 1

Since F(ē)=0<1−α, the Proof of Proposition 4 can be applied with a>ē for anyone. The result is straightforward from the proof, since e=ē=e+ and thus sl⋆(e)=sl⋆(ē)=sl⋆⋆ for those choosing a local job, and ē>αh̲lδl+ē by assumption (see footnote 20). □

Proof of Proposition 6

In order to prove the proposition, the following lemma is used.

Lemma A4

When e+<ē, net earnings of workers with given wealth and a national job increase discontinuously and those of workers with a local job decrease discontinuously, when the return to educational investment for local jobs turns from negative to positive with a change in sl.

Proof of Lemma A4

When e+<ē and the return is negative, wnhne+,sl−e+=wlh̲l⇔αAHlHn1−αδn(1−sl)−1e+=(1−α)AHnHlαh̲l, where Hl=F(e+)h̲l and Hn=∫e+ēhn(e,sl)f(e)de+1−F(ē)hn(ē,sl), holds from (11) and (12). When e+<ē and the return is positive, wnhn(e+,sl)−e+=wlhl(e+,sl)−e+⇔αAHlHn1−αδn(1−sl)−1e+=(1−α)AHnHlαh̲l+δlsle+−e+, where Hl=∫0e+hl(e,sl)f(e)de and Hn=∫e+ēhn(e,sl)f(e)de+1−F(ē)hn(ē,sl), holds from (2) and (5). When the return is zero, i.e. wlδlsl − 1 = 0, this equation becomes αAHlHn1−αδn(1−sl)−1e+=(1−α)AHnHlαh̲l, the same equation as the case of the negative return. Because HnHl under the negative return is greater than under the positive return for given e+ and HnHl decreases with e+, e+ and HnHl satisfying the above equation are greater when the return is negative. Hence, net earnings of those with a local job are greater and of those with given a(>e+) and a national job are smaller when the return is negative and sl approaches a value at which the return is zero than when the return is zero. That is, net earnings of those with a national (local) job increase (decrease) discontinuously when the return turns from negative to positive with a change in sl. □

(i) Let the lower (higher) sl such that the return to education for local jobs is 0 when e+<ē be sl̲(f) (sl̄(f)), whose existence is shown in the Proof of Proposition 1. As for those who have abundant wealth and choose a national job for any sl, i.e. a≥min{e+(1),ē}, net earnings decrease with sl except at sl=sl̲(f), where they increase discontinuously from Lemma A4, if e+(0)≤α1−αh̲lδl or sl̲(f)≥sl♯ satisfying sl♯=(1−α)−αh̲lδle+(sl♯) from Propositions A1 (i) and 2. The former condition holds when δl is small, because e+(0) does not depend on A, δn, and δl from (D3) in the Proof of Proposition A1 in Appendix D. The latter condition holds when A, δn, and δl are small, since sl̲(f) decreases with A, δn, and δl from Proposition 1 and is greater than 1 − α when these variables are small from the part of the Proof of Proposition 1 (i)(a) just after Lemma A1, while sl♯<1−α. As for those who choose a local job for any sl, i.e. a = e < e+(0), net earnings decrease with sl except at sl=sl̄(f), where net earnings increase discontinuously from Lemma A4, when E(e|e<e+(0))≤1+α1−αh̲lδl and e≤αh̲lδl+E(e|e<e+(0)), when E(e|e<e+(0))>1+α1−αh̲lδl and e ≤ Ω(e+(0)), and when sl̲(f)≥sl,h▽(e) for greater e, where sl,h▽(e) is the greater solution of M(sl) = 0 (M(sl) equals L(sl) with e+(sl) replaced with E(e|e<e+(sl))≡∫0e+(sl)ef(e)deF(e+(sl))), from Propositions A1 (ii) and 2. Thus, irrespective of a = e, net earnings decrease with sl except at sl=sl̄(f), if sl̲(f)≥sl,h▽(e). The condition holds when A, δn, and δl are small, because as shown above, sl̲(f)>1−α when they are small, while 1−α≥sl,h▽(e) holds when δl is small. The latter statement is true because M(1−α)=−αh̲lδl+E(e|e<e+(1−α))−αeh̲lδl<−αh̲lδl−αeh̲lδl≤0 when δl is sufficiently small. Finally, as for those who choose a national (local) job at small (large) sl, if the above conditions hold, net earnings decrease with sl except at sl=sl̲(f), sl=sl̄(f), or both depending on at which s l the switch to a local job occurs, where net earnings increase discontinuously from Lemma A4. Under the above condition, net earnings of workers with a < e+(0) are maximized at sl = 0, because their earnings when the return is negative decrease with sl from Proposition 2 and thus the earnings when sl→sl̄(f) from above are lower than the earnings at sl = 0. Net earnings of workers with a≥min{e+(1),ē} are maximized at either sl = 0 or sl=sl̲(f), since the earnings increase discontinuously at sl=sl̲(f). From the Proof of Lemma A4, net earnings of such workers with educational spending e at sl = 0 equal

(64)αAHle+,slHne+,sl1−αδn−1e=αA1−ααh̲lδne+(0)1−αδn−1e,

where the equality sign is from (7). From (14) in the proof of Proposition 1 and (2), their net earnings at sl=sl̲(f) equal

(65)αA(1−α)h̲l+δlsle+αδn(1−sl)e+1−αδn(1−sl)−1e=αA(1−α)Aδlsl1−ααδn(1−sl)−1e
(66)<αA(1−α)2Aδl1−ααδnα−1e (from sl̲(f)<1−α),

where the equality sign is from the fact that the return for local jobs is 0 at sl=sl̲(f). From (64) and (66), the net earnings are maximized at sl = 0 if

(67)1−ααh̲lδne+(0)1−α>(1−α)2Aδl1−ααα,

which holds when A, δn, and δl are small, since e+(0) does not depend on A, δn, and δl. As for those who choose a national job at small sl and a local job at large sl, the proof for those who choose a local (national) job for any sl applies if the switch to a local job occurs at sl≤sl̲(f) (sl≥sl̄(f)). If the switch occurs at sl∈(sl̲(f),sl̄(f)), the proof of those who choose a national job for any sl applies, since earnings when the return is negative decrease with sl from Proposition 2. (ii) From the above results, if a<mine+(0),αh̲lδl+E(e|e<e+(0)) when E(e|e<e+(0))≤1+α1−αh̲lδl or if a<mine+(0),Ω(e+(0)) when E(e|e<e+(0))>1+α1−αh̲lδl, workers choose a local job for any sl, and their net earnings decrease with sl except at sl=sl̄(f), where the earnings increase discontinuously. Net earnings of such workers are maximized at sl = 0, because the earnings when the return is negative decrease with sl from Proposition 2. (iii) As for workers who choose a national job for any sl, i.e. a≥min{e+(1),ē}, their net earnings decrease with sl for sl<sl̲(f) and sl>sl̄(f) from Proposition 2, increase (decrease) discontinuously at sl=sl̲(f) (at sl=sl̄(f)) from Lemma A4, and increase (decrease) with sl for sl∈(sl̲(f),sl♭] (for sl∈[sl♯,sl̄(f))), if e+(0)>α1−αh̲lδl, E(e|e<e+(0))>αmax11−αh̲lδl,e+(0)1+α, and sl♭>sl̲(f), where sl♭∈0,sl♯ satisfies sl♭=(1−α)−αh̲lδlE(e|e<e+sl♭), from Proposition A1 (i). (sl♭,sl♯<1−α<sl̄(f) from Proposition A1 (i) and Lemma A1 in the Proof of Proposition 1.) The condition holds when A, δn, and δl are large, because e+(0) does not depend on these parameters, sl♭ increases with δl (since e+(sl) increases with δl from the Proof of Proposition 1 (i)(b)), and sl̲(f) decreases with A, δn, and δl and approaches 0 as these parameters increase from Proposition 1 and its proof. The net earnings are maximized at sl = 0 or sl satisfying d(wnhn)dsl=0, where the latter satisfies sl∈(sl♭,1−α) and thus sl<1−α<sl̄(f) from Proposition A1 (i) and Lemma A1. From (64) and (65) above, net earnings of such workers with educational spending e at sl = 0 is smaller than their net earnings at sl=sl♭ (and thus the earnings at sl satisfying d(wnhn)dsl=0) iff h̲le+(0)1−α<h̲l+δlsl♭e+(sl♭)e+(sl♭)1−α(1−sl♭)α, which holds if

(68)h̲le+(0)1−α<h̲l+δlsl♭ēē1−α1−sl♭α.

The condition holds when δl is sufficiently large, because e+(0) does not depend on A, δn, and δl, sl♭ increases with δl, and the RHS increases with sl♭:

(69)1−αh̲l+δlsl♭ēδlē−α1−sl♭=δlē1−α−sl♭−αh̲lh̲l+δlsl♭ē1−sl♭=αh̲lēE(e|e<e+sl♭)−1h̲l+δlsl♭ē1−sl♭>0.

As for workers who choose a local job for any sl, i.e. a < e+(0), their net earnings decrease with sl for sl<sl̲(f) and sl>sl̄(f) from Proposition 2, decrease (increase) discontinuously at sl=sl̲(f) (at sl=sl̄(f)) from Lemma A4, and increase (decrease) with sl for sl∈(sl̲(f),min(sl,h△(e),sl̄(f))] (for sl∈[sl,h▽(e),sl̄(f)) when sl,h▽(e)<sl̄(f)), if sl,h△(e)>sl̲(f) and e=a>maxαh̲lδl+e+(0),Λ(ē) from Proposition A1 (ii). The condition holds when A and δn are large, because e+(0), Λ(ē), and sl,h△(e) do not depend on these parameters, and sl̲(f) decreases with A and δn. The condition holds when δl is large, because Λ(ē) decreases with δl, sl,h△(e)∈(0,1) from the Proof of Proposition A1 (ii) in Appendix D, while sl̲(f) approaches 0 as δl increases, which is from sl̲(f) being decreasing in δl, and from (14) and Lemma A1 in the Proof of Proposition 1. Their net earnings are maximized at either sl = 0, or sl∈(sl,h△(e),sl̄(f)) such that d(wlhl)dsl=0. (The earnings at sl=sl̄(f) are smaller than the earnings when sl→sl̄(f) from above and thus cannot be the maximum.) From (2) and (6), net earnings of such workers at sl = 0 equal

(70)(1−α)AHn(e+,0)Hl(e+,0)αh̲l=(1−α)Aδnα1−αe+(0)h̲lαh̲l.

From (2) and (6), net earnings of such workers with e = a at s l satisfying d(wlhl)dsl=0 equal (1−α)Aαδn(1−sl)e+(sl)(1−α)h̲l+δlsle+(sl)αh̲l+δlsle−e. The net earnings at such sl is greater than at sl = 0 iff

(71)(1−α)Aδnα1−αα(1−sl)e+(sl)h̲l+δlsle+(sl)αh̲l+δlsle−e+(0)h̲lαh̲l−e>0,

which is true if

(72)(1−α)Aδnα1−αα(1−sl)eαh̲l+δlsle1−α−e+(0)α(h̲l)1−α−e>0.

When δ l is sufficiently large, 1−α∈(sl̲(f),sl̄(f)) from Lemma A1 in the Proof of Proposition 1. Hence, the above condition holds if

(73)(1−α)Aδnα1−αα(αe)α[h̲l+δl(1−α)e]1−α−e+(0)α(h̲l)1−α−e>0,

which is true when δl is sufficiently large, because the expression inside the curly bracket is large positive (e+(0) does not depend on δl). Finally, as for workers who choose a national (local) job when sl is small (large), i.e. a∈[e+(0),min{e+(1),ē}), the result is clearly similar to workers who choose a national (local) job for any sl, when the shift to a local job occurs at sl>sl̄(f) (sl<sl̲(f)). When the shift occurs at sl∈[sl̲(f),sl̄(f)], their net earnings increase discontinuously at sl=sl̲(f) and sl=sl̄(f), and they are maximized at sl = 0 or sl∈(sl̲(f),sl̄(f)) satisfying either d(wnhn)dsl=0, d(wlhl)dsl=0, or a = e = e+(sl) (sl at which the switch to a local job occurs). (The earnings at sl=sl̄(f) are smaller than the ones when sl→sl̄(f) from above and thus cannot be the maximum.) From the argument above, the net earnings are maximized at the latter sl when δl is sufficiently large. □

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Received: 2022-03-30
Revised: 2022-08-24
Accepted: 2022-09-12
Published Online: 2022-10-12

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Default Behavior and Risk Aversion in Defined Contribution Retirement Systems: Evidence from Chile
  4. Linking Employment and Death: Measuring the Structural Disparity in COVID-19 Deaths for Non-telework Essential Workers
  5. Estimating the Effect of Distance on the Migration of Higher Education Candidates
  6. Do Female Politicians Lead to Better Learning Outcomes?
  7. Understanding Household Consumption Behaviour: What do we Learn from a Developing Country?
  8. Learning with Differing-Ability Peers: Evidence from a Natural Experiment in South Korea
  9. Is Bilingual Education Desirable in Multilingual countries?
  10. Letter
  11. Is the Non-disclosure Policy of Audit Intensity Always Effective? A Theoretical Exploration
https://doi.org/10.1515/bejeap-2022-0123
Keywords for this article
language policy; bilingual education; human capital; economic development

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Default Behavior and Risk Aversion in Defined Contribution Retirement Systems: Evidence from Chile
  4. Linking Employment and Death: Measuring the Structural Disparity in COVID-19 Deaths for Non-telework Essential Workers
  5. Estimating the Effect of Distance on the Migration of Higher Education Candidates
  6. Do Female Politicians Lead to Better Learning Outcomes?
  7. Understanding Household Consumption Behaviour: What do we Learn from a Developing Country?
  8. Learning with Differing-Ability Peers: Evidence from a Natural Experiment in South Korea
  9. Is Bilingual Education Desirable in Multilingual countries?
  10. Letter
  11. Is the Non-disclosure Policy of Audit Intensity Always Effective? A Theoretical Exploration
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