PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS * Jorge Durán and Alexandra Rillaers **

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1 PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS * Jorge Durán nd Alexndr Rillers ** WP-AD Corresponding uthor: Jorge Durán, Deprtmento de Fundmentos del Análisis Económico, Universidd de Alicnte, Cmpus de Sn Vicente, Alicnte, Spin, e-mil: durn@merlin.fe.u.es Editor: Instituto Vlencino de Investigciones Económics, S.A. Primer Edición Octubre 2002 Depósito Legl: V IVIE working ppers offer in dvnce the results of economic reserch under wy in order to encourge discussion process before sending them to scientific journls for their finl publiction. * We thnk Philippe Askenzy, Antoine d'autume, Fbrice Collrd, Dvid de l Croix, Pierre Dehez, V\UNICODE{0xed}ctor R\UNICODE{0xed}os-Rull, Henri Sneessens, nd Clude Wmpch, for comments nd suggestions. This pper ws prtly written t the IRES nd the CEPREMAP. Jorge Dur\UNICODE{0xe1}n cknowledges finncil support from PAI P4/01 project from the Belgin government t IRES nd from Europen Mrie Curie fellowship (Grnt HPMF-CT ) t CEPREMAP. Alexndr Rillers cknowledges finncil support from FDS nd from PAI P4/01 project from the Belgin government t IRES. ** J. Durán nd A. Rillers: Deprtment of. Fundmentos de Análisis Económico, University of Alicnte.

2 PHYSICAL AND HUMAN CAPITAL INVESTMENT: RELATIVE SUBSTITUTES IN THE ENDOGENOUS GROWTH PROCESS Jorge Durán nd Alexndr Rillers A B S T R A C T This pper ims t studying the interction between growth of rel output nd humn cpitl ccumultion when eduction requires investment of physicl resources. To this end we investigte the ggregte implictions of individul specific uncertinty bout returns to investment in eduction in the bsence of insurnce mrkets. We do so in generl equilibrium OLG model in which physicl resources must be devoted to eduction in order to ccumulte humn cpitl. We find tht uncertinty with incomplete finncil mrkets my strongly ffect individul behvior but not the ggregte of the economy: different degrees of uncertinty will induce different intensities of humn to physicl cpitl but will not hve significnt impct on the long run growth rte of the economy. This frmework llows us to conclude tht investing less in eduction in reltive terms does not necessrily led to less growth: the ccumultion of physicl nd humn cpitl disply some degree of substitutbility s n engine for long run growth. Keywords: Overlpping genertions, Investment in eduction, Uninsured shocks, Humn cpitl, Sustined growth. JEL\ clssifiction numbers: E13, O41, I29. 2

3 1 Introduction The objective of this pper is to better understnd the interction between growth of rel output nd humn cpitl ccumultion when eduction requires investment of physicl resources. To this end we explore the role of uncertinty in individul eduction decisions, nd ultimtely in long run growth ptterns. The deprture point will be model economy in which gents invest rel resources in eduction, thus leving the door open to the possibility tht output growth cuses the ccumultion of humn cpitl by creting resources fterwrds llocted to eduction. In this frmework we will exmine whether different degrees of uncertinty my generte differences in investment in eduction nd in cpitl intensities while leving the output growth rte unchnged. Ever since the seminl contributions of Schultz (1960, 1961) nd Becker (1962), investment nd ccumultion of humn cpitl hs been widely regrded s wy to increse individul lbor productivity nd therefore lbor ernings, on one hnd, nd s n engine for sustined ggregte growth on the other hnd. If individul investment in eduction is positively ffected by the environment (the current stock of humn cpitl), the individul ct of investment becomes n ccumultion process t the ggregte. In the minstrem view, eduction increses lbor productivity which in turn cuses the economy to grow. A recent U.S. Congress report reflects the view tht forml eduction is n importnt determinnt of individul ernings s well s economic growth. Institutions like the World Bnk or the United Ntions stress the importnce of investment in eduction for growth nd foster policies directed towrds the increse of forml eduction. There is, however, evidence tht the cusl reltionship between eduction, ernings, nd growth is fr from cler nd tht it my be more subtle. At the individul level there re serious qulifictions s for the role of forml eduction in productivity. Recent contributions regrd eduction s consequence rther thn cuse: eduction would result from chrcteristics lredy determined in erlier stges of the individul s life (Stokey (1998) or Heckmn (2000)). At the ggregte level, we observe shrp differences both in schooling nd in expenses in eduction between developed nd underdeveloped countries. U.N. estimtes for 3

4 school life expectncy in the nineties re roughly 10 yers for developing countries nd 16 yers for industrilized countries; expenses in eduction s percentge of GDP were round 3.9% for developing countries nd round 5.4% for industrilized countries. Nevertheless, recent reserch suggests tht there is no bsolute poverty trp: the poorest countries in hve been growing on verge t similr rtes s the richest countries (Mddison (1995) nd Prente nd Prescott (1995)). Further, there is some theoreticl controversy s for the cusl reltionship between eduction nd growth. Even if eduction is ccepted s one of the determinnts of the stedy growth of lbor productivity, growing economy genertes resources tht cn be llocted ex post to eduction (Bénbou (1996)). In this pper we focus on the ggregte implictions of eduction for growth nd vicevers. The evidence outlined bove suggests both () tht there re wys to ccumulte humn cpitl other thn forml trining nd (b) tht time nd resources devoted to eduction reltive to GDP cnnot fully ccount for long run growth in ctul economies s suggested by schooling theories. The first observtion motivtes the serch for mechnisms of humn cpitl ccumultion complementry to tht of forml eduction. For instnce, Lucs (1993) stressed the potentil role of lerning-by-doing in development processes. The second observtion motivtes closer look t the reltionship between rel (resources) investment in eduction nd growth. In prticulr, to the possibility tht growing economies my generte the rel resources necessry to invest in eduction nd to ccumulte humn cpitl which in turn will cuse the economy to grow. The rel nture of investment in humn cpitl hs lredy been emphsized in erly contributions to this literture. Schultz (1960) nd Uzw (1963) hve brod notion of investment in humn cpitl of which eduction is only one spect: they include helth cre nd ny other rel expenditure ffecting the qulity of lbor. 2 From the individul point of view eduction is lso regrded s long run project chrcterized by uncertinty. Different innte bilities to 2 Schultz (1961) trces the ide bck to John Sturt Mill. This view is shred by modern pproches to the question of eduction t the individul level. Stokey (1998) emphsizes tht expenditures in school fees re just prt of the prent s investment in their children. 4

5 tke dvntge of formtion, life length, the impct of fmily environment, or simply unpredictble events, more likely to occur in such long period, re some of the identified sources of uncertinty ffecting eductionl choices (see Schultz (1961) nd Becker (1962) or more recently Kodde (1986)). Mny uthors hve nlyzed the impct of uncertinty on individul decisions of eduction: Levhri nd Weiss (1974) crried out the first forml nlysis nd were followed by mny others like Willims (1978, 1979) or more recently Snow nd Wrren (1990). Under vrious finncil frmeworks, they bsiclly confirm the intuition tht uncertinty negtively ffects the level of investment in eduction. These re, however, prtil equilibrium nlyses. Thispperisintendedsfirst step in the development of generl equilibrium model in which similr rtes of growth re comptible with different frctions of output devoted to eduction. Following Michel (1993) we nlyze n OLG model in which gents hve to invest resources in their eduction to produce humn cpitl. The ccumultion of humn cpitl is the source of sustined growth. Furthermore, individuls mke their choice under uncertinty bout the returns to eduction nd with no insurnce (n idiosyncrtic shock ffects the individul production of humn cpitl). We will discuss tht different degrees of uncertinty cn generte differences in expenses in eduction (s frction of output) while leving the growth rte reltively unchnged: The negtive impct of uncertinty on eduction t the individul level is not trnslted to the ggregte becuse the ccumultion of physicl nd humn cpitl ct s reltive substitutessnengineforgrowth. 2 Individul investment in humn cpitl The economy is represented by n overlpping genertions model, stochstic version of tht described in Michel (1993). The demnd-side of the economy is represented by sequence of genertions, ech composed of lrge number of ex nte identicl gents. The supply side of the economy is represented by single competitive firm endowed with technology of constnts returns to scle in physicl nd humn cpitl. 5

6 2.1 Households nd humn cpitl Agents live for three periods. Consider the typicl gent born in period t 1. In the first period she decides eductionl investment E t 0. To finnce her eduction she borrows in the finncil mrket: she issues the single sset of the economy (there is no insurnce). The resulting level of individul humn cpitl will be conditionl on productivity shock z t U[, b] with >0, ndonthe ggregte level of humn cpitl H t 1, nd it is given by h t (z t )=z t H 1 β t 1 E β t (1) where β (0, 1). Returns to scle re ssumed to be constnt to ensure fesibility of sustined growth. 3 All gents re identicl ex nte nd their productivity shock is independent from the others. 4 Then E t is the sme for ll gents of genertion t 1. Further, there is lrge number of gents with mss normlized to one. Hence, ex nte expecttions re ex post verges nd ggregtes. For exmple, ggregte humn cpitl is H t = µh 1 β t 1 E β t where µ is the expecttion of the shock. Herefter we will often work with vribles per unit of stock of humn cpitl writing e t = E t /H t 1. Preferences over consumption will be homogeneous so tht this trnsformtion pplies to both consumptions s described below. There is no ggregte uncertinty nd prices re perfectly forecsted by the 3 This function meets the condition in Levhri nd Weiss (1977, pge 956) for uncertinty to hve negtive effect on eduction. The ssumption is quite ppeling: it would be difficult to justify positive effect of z t on returns nd negtive effect on mrginl returns becuse it would suggest tht higher skills (humn cpitl) would be ssocited with lower mrginl bility to increse skills (further ccumultion of humn cpitl through schooling). 4 As discussed in the introduction, this individul specific bility to ccumulte humn cpitl cn be interpreted to be prtilly innte or cn be ssumed to represent externl fctors: Schultz (1961) cites ccess to helth services, Crd nd Krueger (1992) nd Kodde (1986) point out the qulity of schooling while Altonji nd Dunn (1996) stress the fmily bckground (see gin Stokey (1998) for relted discussion). These interprettions seem to fit well the ssumption of independence. 6

7 gent. The rel interest rte r t,r t+1 > 0 nd the rel wge per efficiency unit w t > 0 re tken s given by the gent. Leisure is not vlued: the gent supplies inelsticlly her humn cpitl stock. Dividing by H t 1,netincomenextperiod (lbor ernings minus debt) will be m t (z t ) = z t w t e β t (1 + r t )e t. The gent chooses consumption in the second nd third periods of her life c t (z t ),d t+1 (z t ) 0 s well s svings s t (z t ) contingent to the reliztion of z t so s to verify the set of contingent budget constrints c t (z t )+s t (z t ) m t (z t ) (2) d t+1 (z t ) (1 + r t+1 )s t (z t ). (3) These vribles re expressed in terms of per unit of humn cpitl. Benefits will be zero so tht equilibrium prices of shres will be zero s well. Since there is no insurnce (nd bnkruptcy is not llowed) the non negtivity constrints on consumption nd the budget constrints impose m t (z t ) 0 for ll z t. This is equivlent to impose m t () 0 or e t ē t where ē t is defined implicitly s w t ē β t (1 + r t )ē t =0. Tht is, ffordble choices of eductionl investment should not induce negtive net income in ny stte of nture. We shll refer to this constrint e t ē t s the limit to borrowing: ny higher borrowing would led the gent to bnkruptcy in the worst sttes of nture. This limit to borrowing stems from the no bnkruptcy ssumption underlying ny generl equilibrium model together with the prticulr mrket structure considered The sving rule nd indirect utility The gent hs preferences defined over contingent plns of consumption c t (z t ) nd d t+1 (z t ) for ll z t [, b] represented by the expected utility function U(c t,d t+1 )= b [c t (z t ) ε d t+1 (z t ) 1 ε ] 1 θ 1 1 θ 5 The limit to borrowing is consequence of the bsence of insurnce (incomplete mrkets) nd not exogenously imposed: it should not be mistken with n (exogenously imposed) borrowing constrint (generlly considered cpitl mrkets imperfection). dz t 7

8 where ε (0, 1) nd θ 0. The density of z t is positive constnt nd therefore ignored. When θ =1the integrnd is interpreted to be the logrithm (in the sense tht U converges pointwise to the integrl of the logrithm log(c t (z t ) ε d t+1 (z t ) 1 ε )). We choose these preferences becuse the sving rte is independent of the reltive degree of risk version θ. 6 Since consumption choices cross sttes ofnturereseprble wecnsolve the problem bckwrds. Fixed some z t nd e t, the gent mximizes c t (z t ) ε d t+1 (z t ) 1 ε subject to (2) nd (3). As soon s m t (z t ) > 0 thesolutionisinterior. Further, the budget constrints must be binding t the optimum s the objective function is incresing in both rguments. Hence, the optiml sving rule is s t (z t )=(1 ε)m t (z t ), (4) liner function of income becuse preferences re homothetic. This is lso the optiml rule when m t (z t )=0becuse in tht cse s t (z t )=0. Since the objective function is strictly qusiconcve nd the budget set convex, expression (4) describes the unique solution to the problem given net income. Introducing this optiml rule in the budget constrints nd these in the objective function yields χm t (z t ) for ll m t (z t ) 0 where χ > 0 is constnt from the individul point of view. Use the definition of m t (z t ) in terms of e t nd introduce this expression in the utility function bove to obtin V (e t )= b (z t w t e β t (1 + r t )e t ) 1 θ 1 1 θ the indirect utility function. 7 The function V is twice continuously differentible, strictly concve, nd V (0) =. Moreover, it is continuous nd differentible s function of prmeters, b, ndθ. The proof of these sttements is stndrd nd therefore omitted. 6 If the degree of risk version were relted to the elsticity of intertemporl substitution, it would be difficult to disentngle the effect of risk version from tht of the svings rte in the ggregte. 7 We hve omitted the term χ 1 θ, positive constnt: dd nd substrct χ 1 θ (1 θ) 1 nd operte to conclude tht this is the relevnt indirect utility function in the sense tht it represents the sme preferences. dz t, 8

9 2.3 Optiml choices of eduction Avoid the trivil cse of no uncertinty ssuming herefter tht <bso tht the stndrd devition of the shock is positive: σ > 0. Unless otherwise stted we shll lso ssume tht θ > 0, tht is, we concentrte on the cse of risk version. The optiml rule of investment in eduction hs constnt elsticity with respect to the rtio of prices: Proposition 1 For ll prmeters specifictions nd prices w t /(1 + r t ) > 0 there is unique optiml choice e t > 0. Optiml eduction is given by the rule e wt = ϕ = Q w t 1+r t 1+r t 1 1 β where Q = when the limit to borrowing is binding nd Q = M>0 when the solution is interior; M will then be function of prmeters, b, β, α, ndθ. Proof: Since V is continuous nd [0, ē t ] compct, n optiml solution e t exists. Uniqueness follows from strict concvity of V. This solution cnnot be zero becuse V (0) =. When the solution is interior the first order condition is b (z t w t e β t (5) (1 + r t )e ) θ (z t w t βe β 1 t (1 + r t )) dz t =0. (6) Since r t > 1 nd e t > 0 this condition cn be rewritten s b θ w t z t e β 1 w t t 1 z t βe β 1 t 1 dz t =0. (7) 1+r t 1+r t Implicitly e t = ϕ(w t /(1 + r t )). Since the left hnd side of the eqution is continuously differentible function of the rtio of prices nd of eduction, the implicit function theorem ensures tht + b b θ 1 w t ( θ) z t e β 1 t 1 1+r t e β 1 w t z t + z t (β 1) 1 ϕ wt w t z 1+r t e t βe β 1 t 1 dz t t 1+r t 1+r t θ w t z t e β 1 t 1 βe β 1 w t t z t + z t (β 1) 1 ϕ wt dz 1+r t 1+r t e t =0. t 1+r t 9

10 Rerrnging terms nd simplifying it turns out tht ϕ cn be expressed s ϕ wt = e t 1 w 1+r t t 1+r t 1 β. But this implies tht the optiml rule of eductionl investment is of the form wt ϕ = M w t 1+r t 1+r t 1 1 β where M>0 is some constnt. Direct substitution of this expression in (7) yields b zm 1 1 θ zm 1 β 1 dz =0. (8) This eqution hs unique solution becuse optiml eduction is unique nd the rule ϕ is strictly incresing in M. The solution depends on the vlues of, b, β, nd θ, nd indirectly of µ nd σ through nd b. The optiml rule is completed setting, when the limit to borrowing is binding, wt ϕ = w 1 1 β t. (9) 1+r t 1+r t Tht is, the limit to borrowing ē t expressed in terms of its vlue s function of w t, r t, β, nd. An immedite consequence of the proposition bove is tht M<when the solution is interior. The effect of prices on the choice of eduction is obvious. Incresing the wge (or decresing the interest rte) uniformly increses net returns to eduction over ll sttes of nture inducing higher levels of eduction. More interesting is the effect of risk version nd the incomplete structure of finncil mrkets: Risk version reduces (interior choices of) investment in eduction s it will mke the gent cre reltively more bout the bd sttes of nture. Proposition 2 Interior optiml choices of investment in eduction re strictly decresing in risk version s represented by θ. Border solutions re insensitive to infinitesiml chnges of θ. 10

11 Proof: When V (ē t ) > 0 the limit to borrowing is binding. Since V is continuous function of θ, infinitesiml chnges in this prmeter will not ffect this inequlity while the limit to borrowing itself does not depend on θ. Now suppose tht M is the solution to (8) for some θ 0 nd let θ > θ. Let z be the only vlue of the shock for which ( zm 1 1) θ ( zβm 1 1) = 0. Then (8) cn be written s b ( zm 1 1) θ ( zβm 1 1) dz = z z ( zm 1 1) θ ( zβm 1 1) dz. Moreover, since θ θ > 0 we hve 0 < (z M 1 1) (θ θ) < (z M 1 1) (θ θ) for ll z >z nd in prticulr for ll z ( z,b] nd z [, z). Since (zm 1 1) θ (zβm 1 1)(zM 1 1) (θ θ) =(zm 1 1) θ (zβm 1 1) for ll z itmustbethecsetht b z (zm 1 1) θ (zβm 1 1) dz < z (zm 1 1) θ (zβm 1 1) dz. The overll effect is negtive becuse multiplying the integrnd by (zm 1 1) (θ θ) mounts to ssign bigger weight to the negtive prt of the integrl reltive to the positive prt. This inequlity cn be written s b (zm 1 1) θ (zβm 1 1) dz < 0 implying tht M is not optiml for θ.sincev is strictly decresing in e t it must be the cse tht the left hnd side of this expression is strictly decresing in M. Hence, the new optiml M must be such tht M <M. Some nottion: for ny z [, b] we will write e z,t to denote the vlue of investment in eduction tht verifies zw t βe β 1 z,t (1 + r t )=0. Lemm 1 For ll prmeters specifictions it is true tht e,t <e t min{ē t,e µ,t }. In other words, it is true tht β <M min{, βµ}. 11

12 Proof: Supposethte,t >e t,thenz t w t βe β 1 t (1+r t ) w t βe β 1 t (1+r t ) > 0 for ll z t.aninfinitesiml increse of e t would induce mrginl increse of net income in ll sttes of nture thus incresing expected indirect utility nd therefore contrdicting tht e t is optiml. Tht e t ē t follows from the limit to borrowing. Finlly, suppose tht e µ,t < ē t, then follow the proof of proposition 2 for θ =0to show tht e t <e µ,t. Observe tht the proof of proposition 2 could hve been written in terms of the first order condition nd e t with ny other distribution function. First, the constnt elsticity optiml rule of proposition 1 holds for ny distribution F with support infimum >0; second, in the proof of proposition 2, whenever (zm 1 1) θ ppers, it cn be exchnged by mrginl utility in stte z nd (zβm 1 1) by mrginl income in expression (6). In short, the proofs bove cn be reproduced without the uniform distribution ssumption. The uniform distribution ws ssumed for simplicity in the next proof nd in the interprettion of the numericl simultions: the effect of σ is redily interpreted when the shock is uniformly distributed becuse other moments of the distribution re very simple. Proposition 3 Optiml investment in eduction, binding or not, is strictly decresing in σ (the stndrd devition of the shock z t ), µ constnt. Proof: When the solution is e t =ē t,decresing directly decreses the optiml choice s it is cler from expression (9). When the solution is interior, with uniform distribution (8) cn be written in terms of σ s µ+σ 3 µ σ 3 (zm 1 1) θ (zβm 1 1) dz =0. We need the left hnd side of this expression to be decresing in σ for optiml M to be decresing in σ becuse the integrl is decresing in M (see the end of the previous proof). Applying Leibniz rule it is cler tht the integrl is decresing in σ if nd only if (M 1 1) θ (βm 1 1) > (bm 1 1) θ (bβm 1 1). 12

13 The rest of the proof is devoted to show tht this inequlity holds. Let M solve (8) for some given vlue σ > 0 nd let n be the (only) liner function of z tht verifies n() =(M 1 1) θ (βm 1 1) nd n(b) =(bm 1 1) θ (bβm 1 1). where n() < 0 nd n(b) > 0 becuse by lemm 1 interior solutions verify β < M < βµ < βb. Suppose tht n() + n(b) > 0 contrdicting the inequlity bove. Let z be the criticl vlue for which zβm 1 1=0nd note tht M<βµ so tht z <µ. On one hnd, we hve b n(z) dz = 1 2 (n(b)(b z)+n()( z )) > 1 (n(b)(b µ)+n()(µ )) 2 = b (n(b)+n()) > 0. 4 On the other hnd, since the integrnd is strictly concve or decresing it must bethecsetht n(z) < (zm 1 1) θ (zβm 1 1) for ll z (, b). We conclude tht 0 < b n(z) dz < b (zm 1 1) θ (zβm 1 1) dz thus contrdicting tht M issolutionto(8). Observetht(inviewofthelstinequlity intheproof)thenegtiveeffect of σ becomes stronger s the concvity of the integrl, s mesured by θ, increses. In other words, higher degree of risk version will worsen the effect of σ on the optiml choice of investment in eduction. Figure 1 plots the typicl simultion of the individul decision for n rry of vlues of θ nd σ. 8 We set µ =8nd 8 The routines used to simulte individul nd ggregte behvior in this economy re bsed in simple bisection procedure to find solution to (8). The code is written in Mtlb nd is vilble from the uthors upon request. 13

14 Figure 1: The effect of risk version nd the limit to borrowing let = µ n nd b = µ + n. In the third xis we plot different vlues n from zero to 8 so tht n =8represents =0. The degree of risk version rnges from risk neutrl θ =0to θ =3. The strong effect of the lck of insurnce mrkets t the individul level is reflected in this figure: When the gent is risk neutrl she chooses e µ up to the point where is so low tht she hs to choose the corner solution ē. As the gent becomes risk verse, the effect of lowering will operte sooner: her risk version mkes her cre very soon bout the worst sttes of nture. She therefore lowers e in n effort to increse income in those bd sttes of nture. In world with insurnce, lthough one would expect eduction to decrese with σ, theeffect would be less obvious nd not s drmtic when is driven to zero. 9 9 It my be worth stressing tht whether the limit to borrowing is binding or not is irrelevnt. In the plot it is obvious where the constrint is binding: for low vlues of θ nd low vlues of (high vlues of n). It cn proven tht for θ 1 the constrint is never binding. Wht mtters, however, is tht the gent is led to choose zero investment when =0becuse of the contrction of the choice set. 14

15 In the bsence of insurnce, uncertinty strongly discourges investment in eduction. As we will see in the next section, however, t the ggregte this negtive effect is ttenuted nd my even be reversed. As consequence the long run growth rte of the economy will be rther insensitive to the degree of uncertinty. 3 The equilibrium of the economy In this section we introduce the representtive firm nd define competitive equilibrium for this economy. We will prove existence of unique equilibrium nd long run convergence of trnsformed sttionry vribles to unique stedy stte. 3.1 The firm The supply side of the economy is represented by stndrd single competitive firm producing output in t from output in t 1 (physicl cpitl) nd effective lbor in t (humn cpitl). The firm is endowed with technology represented by Cobb-Dougls production function with shre of physicl cpitl α (0, 1), scle fctor A>0, nd full deprecition for simplicity. The firm borrows its stock of cpitl K t 0 inthecreditmrketinperiodt nd returns 1+r t thenextperiod. Sincegentsdonotcreboutleisure,they inelsticlly supply their stock of humn cpitl so tht in equilibrium H t is lso the effective lbor hired by the firm. Define k t = K t /H t, then the firm s first order conditions cn be written s w t = A(1 α)kt α nd 1+r t = Aαkt α Competitive equilibrium ThecreditmrketclerncerequiresK t+1 + E t+1 = S t where S t /H t = (1 ε)(w t (1/µ)(1 + r t )e 1 β t ) is obtined integrting the optiml sving rule (4) over z t (tht is, over individuls) nd dividing it by µe β t mking use of the fct tht H t/ H t 1 = µe β t. Then, in terms of vribles per unit of humn cpitl the credit 15

16 mrket clers when (k t+1 µe β t+1 + e t+1 )=(1 ε) w t 1 µ (1 + r t)e 1 β t. (10) This eqution, the gent s optiml rule ϕ nd the firm s first order conditions, which incorporte the lbor mrket clernce condition, describe competitive equilibri for this economy. Definition 1 Given n initil stock of physicl cpitl k 0 > 0, competitive equilibrium (for the sttionry vribles) is n lloction (k t+1,e t ) t=0 nd sequence of prices (w t,r t ) t=0 such tht e t = ϕ(w t /(1+r t )), w t = A(1 α)k α t, 1+r t = Aαk α 1 t, nd such tht (10) holds for ll t 0. Consumption nd svings cn be recovered from the optiml sving rule (4) nd the gent s budget constrints (2) nd (3). Substituting prices for their vlues s given by the firm s first order conditions yields n equilibrium system of two equtions e t = Q 1 α 1 α k 1 β t (11) k t+1 µe β t+1 + e t+1 = (1 ε)a (1 α)kt α α µ kα 1 t e 1 β t (12) where Q = or M when the solution induced by k t is corner or interior respectively. Equilibrium exists, is unique, nd converges to stedy stte ( k, ẽ). Proposition 4 For ll k 0 > 0 there is unique competitive equilibrium. Competitive equilibrium lloctions converge monotoniclly to single interior stedy stte ( k, ẽ) independently of initil conditions. Proof: Letk t > 0 be ny stock of cpitl. At interior solutions Q = M while M exists nd is unique by proposition 1. At corner solutions Q =. Inbothcses uniquee t of equilibrium is determined by eqution (11). Direct substitution of (11) into (12) yields the ggregte equilibrium trnsition k t+1 = (1 ε)(1 α)a 1 Q 1 β µ µ Q 1 α β 1 β + k α(1 β) Q 1 α 1 t. (13) 1 β α α 16

17 Suppose tht <µ. Then, if the solution is interior Q = M<µbecuse M<βµ by lemm 1, otherwise Q = <µ. Alterntively suppose tht = µ, then the solution is interior nd Q = M = βµ <µ. In both cses Q<µso tht (13) is strictly positive nd well defined. In short, k t uniquely determines k t+1 which in turn, gin through eqution (11), uniquely determines e t+1 : unique competitive equilibrium exists. Write k t+1 = η(k t ) nd observe tht η is differentible, strictly concve, strictly incresing function of k t with η (0) = nd η ( ) =0. Hence, there is unique stedy stte vlue for cpitl k = (1 ε)(1 α)a 1 β 1 Q 1 α(1 β) µ µ Q 1 α β 1 β + (14) Q 1 α 1 1 β α α nd k t k monotoniclly. The stedy stte vlue of eduction ẽ is simply given by eqution (11) evluted t k. An lterntive pproch to the question of existence of competitive equilibrium consists of hving closer look t the left hnd side of eqution (12). Given ny equilibrium lloction (k t,e t ), eqution (12) implicitly describes the combintions of eduction nd cpitl e t+1 = ψ(k t+1,k t,e t ) for the next period tht cler the credit mrket. For ny lloction (k t+1,e t+1 ) to be n equilibrium for the next period it must be true tht e t+1 = ψ(k t+1,k t,e t ) nd tht e t+1 = ϕ(k t+1 ) (the buse of nottion is justified becuse w t+1 /(1 + r t+1 ) is function of k t+1 ). Since ϕ is exponentilly incresing in k nd zero t k =0,itwouldsuffice tht ψ is positive t zero nd decresing to prove tht there is unique equilibrium. If k t+1 µe β t+1 + e t+1 is to remin equl to constnt, when k t+1 =0it must be the cse tht e t+1 > 0 while s k t+1 eduction should converge to zero. Hence, n equilibrium exists. It is unique becuse k t+1 µe β t+1 + e t+1 is incresing in eduction nd cpitl so tht ψ 1 < 0. With this resoning in mind the proof of the following result is strightforwrd: Proposition 5 Fixed n equilibrium (k t,e t ), the equilibrium lloction for cpitl k t+1 (resp. eduction e t+1 ) next period is strictly incresing (resp. decresing) with θ when e t+1 is interior, nd strictly incresing (resp. decresing) with σ. 17

18 Proof: Note tht ψ is independent of θ nd σ while ϕ is strictly decresing with θ when the solution is interior (proposition 2) nd strictly decresing with σ (proposition 3). An increse in σ, nd/orinθ when interior, would cuse ϕ to shift downwrds uniformly: the new equilibrium lloction would result from shifttotherightlongthegrphofψ, decresing function. Fixed ny equilibrium lloction (k t,e t ),nincreseofσ in period t will induce individuls next period to invest less in eduction e t+1, nd more in physicl cpitl k t+1. The increse in k t+1 my stimulte future investment in eduction e t+2. Indeed, t the ggregte the incresed rtio of physicl to humn cpitl will result in higher svings. From eqution (10) we cn write S t+1 =(1 ε)(1 α)kt+1 α 1 Q. H t+1 µ An higher σ increses k t+1 nd decreses Q. As consequence, ggregte svings per unit of humn cpitl rise nd so do the vilble resources to be llocted to k t+2 nd e t+2. This is the resources effect. The subsequent increse in k t+2 will hve in turn consequences s for the individul behvior: higher wge w t+2 nd lower interest rte r t+2 mke investment in humn cpitl e t+2 reltively more ttrctive.thisisthepriceeffect. In short, less investment in eduction tody mens reltively more physicl resources vilble for the future, nd thus potentilly llows more future investment in eduction. As we shll see, this mechnism underlies the mbiguous effect of σ on the long run growth rte. 3.3 The stedy stte growth rte In models of schooling incresing physicl cpitl for tomorrow increses the wge nd therefore the incentive to devote time to schooling tody. Here the price effect is nlogous: it increses the wge nd reduces the interest rte nd therefore stimultes investment in eduction. However, there is n dditionl effect: the higher rtio of physicl to humn cpitl increses svings, nd thus increses the mount of resources vilble for eduction. In contrst with the (fixed) time endowment model, in our economy resources cn be creted (nd subsequently 18

19 llocted to eduction) in n unbounded mnner, leving room for sustined resources effect. In the long run this mechnism will cuse the ggregrte impct of uncertinty to be mbiguous despite its strong negtive effect t the individul level. Along blnced growth pth k = K t /H t will be describing whether n economy is more or less physicl cpitl intensive (reltive to humn cpitl) while ẽ will be proxy for the long run growth rte becuse ll ggregte vribles will be growing t the sme rte s the stock of humn cpitl H t+1 /H t = µẽ β. 10 Higher degrees of uncertinty lwys increse k, nd therefore they re ssocited with more physicl cpitl intensive blnced pths. Proposition 6 For ll prmeters specifictions, d k/dσ > 0. Tht is, economies with high σ re more physicl cpitl intensive thn economies with low σ. Proof: Directdifferentition of (14) with respect to Q yields d k Q dq k = 1 β 1+ 1 α Q + 1 Q µ β α α µ µ Q 1 β 1 β α 1 α(1 β) µ < 0. (15) 1 Q µ + 1 α α To see why the negtive sign recll the proof of proposition 4 where it is shown tht Q<µso tht (1 (Q/µ)) > 0. Finlly observe tht k only depends on σ indirectly through Q. Then d k dσ = d k dq dq dσ > 0 becuse dq/dσ < 0: whenq = trivilly, nd when Q = M by proposition 3. In short, n increse of σ, by decresing Q, willincrese k. This is further lso reflected in tht the long run shre of physicl investment in svings ρ k = K S = αµ αµ + Q(1 α), 10 Observe tht unless µ nd A re high enough there is no gurntee tht µẽ β > 1. For the growth rte to be positive the (expected) scle of the humn cpitl production function nd the scle of the physicl good production function should be high enough to generte the resources necessry to sustin growth. 19

20 is incresing with σ (becuse dq/dσ < 0) combinedwiththefctthtggregte svings S/H lso increses with σ. Hence, the resoning of the previous subsection holds in the long run: economies chrcterized by higher σ re more physicl cpitl intensive nd hve more resources to invest. Contrry to the determinte effect of σ on k, itseffect on ẽ, ndthereforeon the growth rte, is mbiguous. Observe tht the increse of ρ k implies decrese of ρ e, the shre of eduction in svings. Nevertheless, ggregte svings S/H re incresing with σ. These two opposite effects explin why the effect of σ on ẽ remins mbiguous. More formlly, differentiting (11) with respect to σ yields dẽ σ dσ ẽ = 1 dq σ 1 β dσ Q + d k σ. (16) dσ k This expression renders explicit the two mechnisms operting in the impct of σ on ẽ. The first term inside the brckets is negtive s dq/dσ < 0, while the second is positive becuse d k/dσ > 0 by the proposition bove. The first term summrizes the effect of the bsence of insurnce t the individul level while the second cptures the price nd resources effects. For resonble prmeters specifictions the stedy stte growth rte ppers to be reltively insensitive with respect to smll chnges in σ. Indeed, using (15) nd fter some cumbersome (nd therefore omitted) lgebr it cn be shown tht either of the two effects cn be dominting. For exmple, in the cse where d k σ dσ k > dq dσ the price nd resources effect dominte the no insurnce effect, nd therefore increse ẽ despite the higher degree of uncertinty. The verifiction of this inequlity implies tht ρ k is growing with σ to smller extent thn ggregte svings. Tht is, both vilble resources nd investment in physicl cpitl (both per unit of humn cpitl) increse with σ, but the former more thn the ltter. When the no insurnce effect domintes the price nd resource effects, the reverse resoning pplies. Figure 2 plots the typicl numericl simultion for the long run growth rte for n rry of vlues for θ nd σ (we considered the sme prmeters vlues s in σ Q, 20

21 Figure 2: The impct of no insurnce in the long run section 2). The mbiguous impct of σ contrsts with the strong negtive impct we observed t the individul level. Of course, s tends to zero the negtive impct will finlly dominte s gents re forced by their budget constrints to invest zero in eduction. 3.4 Eduction expenditures s percentge of GDP: suitble indictor for economic performnce? Investment in eduction is, not surprisingly, n importnt indictor when evluting the economic nd socil performnce of country. From the economic point of view, eduction enlrges the stock of humn cpitl widely regrded s n engine of endogenous growth. From the socil point of view eduction is wy of emncipting people; nd more specificlly, when publicly provided, eduction generlly llows to reduce inequlities. The relevnt figures on eduction in the dt often refer to investment in eduction s percentge of GDP. In this subsection we will hve look t this indictor in the light of our model. More prticulrly, we will exmine the reltion between the proportion of vilble resources devoted to eduction E/Y, nd the long run growth rte. 21

22 In period t the proportion of ntionl product Y t devoted to individul investment in eduction E t+1 is given by E t+1 /Y t = e t+1 /(Ak α t ), orinthelongrunby E/Y = e/(ak α ).Theeffect of n increse in the degree of uncertinty σ on this indictor cn be written s de/y dσ σ E/Y = de σ dσ e αdk σ dσ k As observed before, for resonble prmeters the long run growth rte, determined by e, is reltively insensitive to σ; hence (de/e)/(dσ/σ) 0. Inthtcse n increse in σ hs negtive effect on E/Y given the fct tht dk/dσ > 0. Stted differently, similr growth rtes re comptible with different proportions of ntionl product invested in eduction. This leds us to the following lemm: Lemm 2 Economies tht invest smller proportion of their GDP in eduction do not necessrily disply lower growth rtes. Indeed, when this smller proportion is ccompnied by higher physicl cpitl intensity k, the growth rte does not need to be negtively ffected: humn nd physicl cpitl re reltive substitutes s n engine for growth, mking the link between eductionl investment nd growth less strightforwrd. According to our model the proportion of GDP devoted to eduction is thus less suitble n indictor for economic performnce s fr s economic growth is concerned. Yet this does not in ny wy temper the rtionle for providing public eduction. As mentioned before, eduction does not only serve economic gols; it cn lso constitute powerful tool to fight socil exclusion nd to smooth inequlity. 4 Conclusions Inthispperwehvenlyzedgenerlequilibrium model of investment in eduction when the returns to this investment re uncertin nd gents cnnot insure themselves. Confirming previous prtil equilibrium nlyses, t the individul level the impct of uncertinty is negtive nd very strong. Nevertheless, t the ggregte other mechnisms operte compensting the initil individul incentive 22

23 to reduce investment in eduction when uncertinty is introduced. The model economy describes world in which different degrees of uncertinty cn yield different cpitl intensities but similr long run growth rtes. It is commonly ccepted tht the ccumultion of humn cpitl is t lest one of the chrcteristics tht llow modern mrket economies to grow sustinbly. In this pper it is shown tht the mechnisms by which humn cpitl induces growth my be subtle. When eduction is modeled s n investment in terms of physicl resources, rther thn schooling, eduction cuses growth but growth lso cuses eduction: mking resources vilble tht will eventully be llocted to eduction in the future. In short, the ccumultion of physicl nd humn cpitl disply some degree of substitutbility s n engine for long run growth: two economies identicl except for the vrince of the productivity shock my grow t thesmertelongtwo different pths: one will be more physicl cpitl intensive thn the other, nd will invest smller proportion of its GDP in eduction. In terms of economic policy, these results suggest tht policies of public eduction should be conceived s men to smooth inequlity, rther thn to foster growth. References [1] Aghion, P. nd Bolton, P. (1992) Distribution nd growth in models of imperfect cpitl mrkets, Europen Economic Review, 36, [2] Altonji, J.G. nd T.A. Dunn (1996) The effects of fmily chrcteristics on the return to eduction, Review of Economics nd Sttistics, 78, [3] Becker, G.S. (1962) Investment in humn cpitl: A theoreticl nlysis, Journl of Politicl Economy, 70, [4] Becker, G.S. (1975) Humn cpitl: A theoreticl nd empiricl nlysis with specil reference to eduction. Columbi University Press. [5] Crd, D. nd A.B. Krueger (1992) Does school qulity mtter? Returns to eduction nd the chrcteristics of public schools in the United Sttes, Journl of Politicl Economy, 100,

24 [6] Glor, O. nd Zeir, J. (1993) Income distribution nd mcroeconomics, Review of Economic Studies, 60, [7] Kodde, D.A. (1986) Uncertinty nd the demnd for eduction, Review of Economics nd Sttistics, 68, [8] Kodde, D.A. nd J.M.M. Ritzen (1985) The demnd for eduction under cpitl mrket imperfections, Europen Economic Review, 28, [9] Levhri, D. nd Y. Weiss (1974) The effect of risk on the investment in humn cpitl, Americn Economic Review, 64(6), [10] Lucs, Jr., R.E. (1988) On the mechnics of economic development, Journl of Monetry Economics, 22, [11] Michel, P. (1993) Le modèle à génértions imbriquées, un instrument d nlyse mcroéconomique, Revue d Economie Politique, 103, [12] Schultz, T.W. (1960) Cpitl formtion by eduction, Journl of Politicl Economy, 69, [13] Schultz, T.W. (1961) Investment in humn cpitl, Americn Economic Review, 51(1), [14] Snow, A. nd R.S. Wrren (1990) Humn cpitl investment nd lbor supply under uncertinty, Interntionl Economic Review, [15] Stokey, N. (1998) Shirtsleeves to shirtsleeves: the economics of socil mobility, in Ehud Kli (ed.) Collected Volume of Nncy L. Schwrtz Lectures. Cmbridge University Press. [16] U.S. Congress (2000) Investment in eduction: Privte nd public returns, Joint Economic Committee, Vice Chirmn Jim Sxton (R-NJ), Jnury. [17] Willims, J.T. (1978) Risk, humn cpitl, nd the investor s portfolio, Journl of Business, 51(1), [18] Willims, J.T. (1979) Uncertinty nd the ccumultion of humn cpitl over the life cycle, Journl of Business, 52(4),