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Poutvaara, Panu

Helsingin yliopiston digitaalinen arkisto

öMmföäflsäafaäsflassflassflasffffffffffffffffffffffffffffffffffffDiscussion PapersOn Human Capital Formation with Exit Options:Comment and New ResultsPanu PoutvaaraUniversity of Helsinki and HECERDiscussion Paper No. 89December 2005ISSN 1795-0562HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,E-mail info-hecer@helsinki.fi, Internet www.hecer.fiHECERDiscussion Paper No. 89On Human Capital Formation with Exit Options:Comment and New ResultsAbstractKatz and Rapoport (2005) conclude that with linear production technology and thepossibility of unilateral migration, region-specific shocks may increase the average level ofeducation. Previously, Poutvaara (2000) derived a corresponding result with Cobb-Douglas technology and migration which may go in both directions. This paper shows thatthe exit option may reduce human capital formation with a quadratic productiontechnology.JEL Classification: F22, J24, I21Keywords: Human capital formation, migration, economic volatility.Panu PoutvaaraDepartment of EconomicsUniversity of HelsinkiP.O. Box 17 (Arkadiankatu 7)FI-00014 University of HelsinkiFINLANDe-mail: panu.poutvaara@helsinki.fi1 IntroductionIn their recent contribution, Katz and Rapoport (2005) explore the rela-tionship between economic volatility and human capital formation in a two-country framework. In one country, the rate of return to human capital iscertain. In the other country, it is uncertain but of the same expected value.Katz and Rapoport …nd that increased variability in the unstable country,which they call undeveloped, increases investment in education there. Thereason for this is that the exit option provides an insurance for those whohave su¢ciently low migration costs.Previously, also Poutvaara (2000) studied the e¤ect of region-speci…cshocks on human capital formation when the regions have the same ex-pected rate of return to human capital. Poutvaara (2000) assumes thattwo regions face symmetric and opposite shocks, and both have ex ante anidentical chance of a positive and a negative shock. Also Poutvaara (2000)…nds that the individual investment in human capital is increasing in themagnitude of shocks when migration is allowed. Unlike Katz and Rapoport(2005), Poutvaara (2000) assumes that migration can go in either direction.Another di¤erence is in production technology: Katz and Rapoport (2005)assume that production is linear in human capital, while Poutvaara (2000)assumes a Cobb-Douglas production technology which combines human cap-ital and a region-speci…c …xed factor. Katz and Rapoport assume ex anteheterogeneous and risk-neutral individuals, while Poutvaara (2000) assumesthat those who became educated are ex ante identical and that they may berisk-averse. The third di¤erence is that Poutvaara (2000) allows everyone toemigrate, while Katz and Rapoport (2005) assume that the costs of adjust-ment and preferences for living in the home country restrict emigration.This comment extends the …nding by Poutvaara (2000) and Katz andRapoport (2005) by showing that the results that they derive with linearand Cobb-Douglas production technology may be reversed with other pro-duction technologies. The comment follows Katz and Rapoport (2005) byfocusing on risk-neutral individuals in the absence of taxation. As Poutvaara(2000), the comment derives the results when individuals are ex ante iden-tical. The results could be easily generalized to the case of ex ante di¤erentproductivities, in line with the appendix A in Köthenbürger and Poutvaara(forthcoming).12 The Model2.1 A Cobb-Douglas TechnologyThere are two countries, A and B. In both countries, production combinesa …xed factor and human capital. Denoting human capital in country i,i 2 fA; Bg by Hi, the total production is H®i , where 0 < ® < 1. As inWildasin (1995) and Poutvaara (2000), both regions face uncertainty aboutthe price of the exported goods. This uncertainty may take two values: pricesare high when they are 1+v and low when they are 1¡v, where the volatilityterm v satis…es 0 · v < 1. There are no taxes.In both countries, the total population is normalized to unity. Invest-ments in education are made before region-speci…c shocks are revealed. How-ever, the educated migrate costlessly. They take into account that migrationequalizes the marginal productivity of human capital in the two countries.Denoting the country which faces a positive (negative) shock by P (N), wecan write the migration equilibrium condition as(1¡ v)®H®¡1N = (1 + v)®H®¡1P : (1)In both countries, individuals invest in education to maximize their ex-pected income. Investment in education is denoted by e. The resultingindividual human capital is denoted by h(e). The marginal productivity ofinvestment in education is positive and non-increasing, so that h0 > 0 andh00 · 0. Individuals decide privately on their own investment in education,taking the market rate of return as given. This follows as there is a con-tinuum of individuals. An individual i chooses investment in education tomaximize:¡ei + h(ei)·12(1¡ v)®H®¡1N +12(1¡ v)®H®¡1N¸: (2)By inserting (1), this yields the …rst-order condition¡1 + h0(ei)(1¡ v)®H®¡1N = 0: (3)As all individuals face the identical optimization problem, they all choosean identical education in both countries. From now on, denote this by be. Thecondition that pre-migration and post-migration stocks of human capital areequal is2HN +HP = 2h(be): (4)Note that the total value of production in the two regions is (1+ v)H®P +(1 ¡ v)H®N . Solving HN and HP from (4) and (1), we obtain as the totalvalue of production in the two countriesY W =h(1 + v)11¡® + (1¡ v) 11¡®i1¡®(2h(be))®:Note that@@vh(1 + v)11¡® + (1¡ v) 11¡®i1¡®=h(1 + v)11¡® + (1¡ v) 11¡®i¡® h(1 + v)®1¡® ¡ (1¡ v) ®1¡®i> 0:Therefore, an increased volatility increases the total value of productionin the two countries with any given investment in education. As a constantfraction ® of this production accrues to the educated, this implies that therate of return to any given stock of human capital increases. As investment ineducation equalizes the marginal cost and the return, this implies an increasein the investment in education. We can summarize the result asProposition 1 With a Cobb-Douglas production technology, the investmentin education is increasing in the magnitude of symmetric and opposite region-speci…c shocks.Note that as ® approaches unity, the production technology approachesthe linear case. In the linear case, the marginal productivity of all humancapital would equal that of the country experiencing a positive shock. There-fore, an increase in the positive shock would increase investment in humancapital, in line with the …ndings by Katz and Rapoport (2005).2.2 A Quadratic Production FunctionAssume next that the production technology is quadratic. The total pro-duction in country i is aHi ¡ bH2i . The region-speci…c shocks are the sameas in the case of a Cobb-Douglas technology. Denoting again the country3which faces a positive (negative) shock by P (N), we can write the migrationequilibrium condition as(1¡ v)(a ¡ 2bHN) = (1 + v)(a ¡ 2bHP ): (5)An individual i chooses investment in education to maximize:¡ei + h(ei)·12(1¡ v)(a ¡ 2bHN) + 12(1 + v)(a ¡ 2bHP )¸:By inserting (5), this yields the …rst-order condition¡1 + h0(ei)(1¡ v)(a ¡ 2bHN) = 0:As the maximization problem is the same in both ex ante identical coun-tries, the solutions are identical. Solving HP from (4) and inserting it into(5) yieldsHN = h(be)¡ v2b(a ¡ 2bh(be)):The rate of return to human capital is then(1¡ v)(a ¡ 2bHN ) = (1¡ v2)(a ¡ 2bh(be)):Note that this is both the expected and the realized rate of return: asthe two countries face opposite shocks, there is no uncertainty about thepost-migration productivity of human capital. Di¤erentiation yields@@v£(1¡ v2)(a ¡ 2bh(be))¤ = ¡2v(a ¡ 2bh(be)): (6)Note that a ¡ 2bh(ei) has to be positive, as otherwise the marginal pro-ductivity of human capital would be negative. The right-hand side of (6)is thus negative. This implies that with any given investment in education,an increase in the region-speci…c shocks reduces the expected rate of returnto human capital investment. Thus, it would reduce investment in humancapital. We can summarize the result asProposition 2 With a quadratic production technology, the investment ineducation is decreasing in the magnitude of symmetric and opposite region-speci…c shocks.4Contrary to the …nding with linear and Cobb-Douglas production tech-nologies, region-speci…c shocks would reduce investment in human capitalwith a quadratic production technology.3 ConclusionPrevious literature has concluded that the possibility of migration boostshuman capital formation with region-speci…c shocks (see Poutvaara (2000)and Katz and Rapoport (2005) for analysis of countries with same expectedreturns to education, and references therein on contributions where the ex-pected rates of return di¤er). This paper shows that this result is sensitiveto the assumptions about the production technology. The results that Pout-vaara (2000) derives with a Cobb-Douglas technology and Katz and Rapoport(2005) with a linear production technology may be reversed with quadraticproduction technology.References[1] Katz, E, Rapoport, H (2005) On Human Capital Formation with ExitOptions. Journal of Population Economics 18(4):267-274[2] Köthenbürger, M, Poutvaara, P (forthcoming) Social Security Reformand Investment in Education: Is there Scope for a Pareto-Improvement?Forthcoming in Economica[3] Poutvaara, P (2000) Education, Mobility of Labour, and Tax Competi-tion. International Tax and Public Finance 7(6):699-719[4] Wildasin, DE (1995) Factor Mobility, Risk and Redistribution in theWelfare State. Scandinavian Journal of Economics 97(4):527-5465

On Human Capital Formation with Exit Options : Comment and New Results

Panu Poutvaara

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