Educational Systems, Growth and Income Distribution: A Quantitative Study

Transcription

1 Educaional Sysems, Growh and Income Disribuion: A Quaniaive Sudy Hung- ju Chen * Deparmen of Economics, Naional Taiwan Universiy, Hsu-Chou Road, Taipei 000, Taiwan Absrac This paper sudies how human capial connecs economic growh wih income disribuion under imperfec credi markes. We consruc a model where households can choose beween privae and public schools. The governmen provides public educaion and public suppor for privae educaion hrough voucher programs. We find ha in a mixed educaional sysem, he governmen can conrol economic performance by changing he srucure of he educaional sysem. If he governmen wishes o reduce income inequaliy, a policy o increase he enrollmen rae in public schools should be adoped. However, having a high enrollmen rae in privae schools wih he liberalizaion of credi markes can keep he growh rae high. JEL Classificaion: E7, I8, O, O5, H3. Keywords: Income Disribuion; Imperfec Credi Markes; Mixed Educaional Sysem. * Tel: (0) ex Fax: (0) hjc@ccms.nu.edu.w. This paper is based on he hird chaper of my Ph. D. hesis a Universiy of California, Los Angeles. I am indebed o my advisor, Cosas Azariadis, for his guidance and advice. I am also hankful o Andrew Akeson, Harold Cole, Gary Hansen, Arleen Leibowiz, Lee Ohanian and several seminar paricipans for heir commens and suggesions.

2 . Inroducion The relaionship beween economic developmen and income inequaliy has always been an imporan issue in macroeconomics. One of he mos famous hypoheses relaed o his issue is Kuznes hypohesis: income inequaliy firs increases and hen decreases wih he developmen of he economy. While Kuznes hypohesis emphasizes he effec of growh on income disribuion, he opposie causal link beween growh and inequaliy is also possible because income disribuion can affec he allocaion of capial and herefore has an impac on growh. Alhough mos of he growh lieraure focuses on he imporance of echnological change o explain he linkages beween growh and inequaliy, we argue ha human capial is also a key facor and sudy how human capial connecs growh wih income disribuion. In paricular, we analyze how he educaional sysem affecs he accumulaion of human capial and how his in urn affecs growh and income disribuion in he long run. Boh causal effecs work in our model. The impac of growh on income disribuion will depend on who will benefi from economic developmen. If rich agens benefi from economic growh and inves more in heir human capial, income inequaliy will increase. On he oher hand, income disribuion will also have an impac on growh hrough he role of human capial and financial developmen. Due o he imperfecions of credi markes, invesmens in human capial are subjec o borrowing consrains. Hence, some poor agens will underinves in educaion if hey are crediraioned. Empirical suppor for his invered U-shape beween he level of income and inequaliy for he Unied Saes and Grea Briain can be found in Williamson and Linder (980) and Williamson (985). For surveys of he cross-secional lieraure on Kuznes hypohesis, see Fields (980), Bigsen (987) and Kanbur (999).

3 Wha disinguishes his paper from he lieraure ha uses human capial o sudy growh and inequaliy is ha we allow for he coexisence of privae and public educaion regimes. Public educaion is provided by he governmen while he amoun of invesmen in privae educaion is decided by households. Table presens daa on privae school enrollmens as a percenage of secondary school enrollmens for boh developed and developing counries for 985. None of he counries in he daa has only privae or public schools. Insead, all counries have boh ypes of schools. In order o mimic he educaional sysem in he real world, we allow households o choose beween public and privae schools. A modified endogenous growh model is used and wo main feaures are inroduced ino our model. Firs, financial developmen is included in he model o deermine he ighness of credi consrains. This disors agens decisions regarding invesmens in educaion. Second, some resources from he public secor can be ranspored o he privae secor hrough voucher programs and his allows he governmen o conrol he size of he privae/public secor 3. We find ha income inequaliy is lower under a public educaion regime han under a privae educaion regime. This is because, under a public educaion regime, everyone has he same invesmen in educaion. For an economy wih a mixed educaional sysem, income inequaliy decreases as he size of he public secor increases and vice versa. In his paper, we are able o quanify he effecs of financial developmen on income inequaliy hrough he role of human capial. Our second finding is ha income inequaliy will increase during he process of financial developmen. The reason for his is ha rich We refer o privae schools as he privae secor and o public schools as he public secor. 3 Wha we mean by he size of he privae secor is he enrollmen in privae schools as a percenage of school enrollmens. Also, he size of he public secor refers o he enrollmen in public schools as a percenage of school enrollmens. Because he summaion of hese wo is one, he size of he privae secor increases while he size of he public secor decreases. 3

4 or able households will benefi from he liberalizaion of credi markes and hey will inves more in educaion. Our model also allows us o quanify he impacs of differen governmen policies regarding ax raes, financial developmens and he scales of voucher programs. A hird finding is ha he governmen is able o conrol economic performance (high economic growh or low income inequaliy) by using combinaions of policies. If he governmen wishes o reduce income inequaliy, i should adop policies ha increase he size of he public secor. However, fas growh akes place if he enrollmen rae in privae schools is high and credi markes are liberalized. Glomm and Ravikumar (99), Glomm (997) as well as Zhang (996) analyze longrun economic performance wih a privae educaion regime and a public educaion regime. Because analyical soluions only exis for a pure public or privae educaion regime, wihou a quaniaive sudy, hey are no able o examine he impac of a mixed educaional sysem on he evoluion of growh and inequaliy. Papers in he lieraure ha sudy school choices wih peer effecs include Fernandez and Rogerson (996), Caucu (000), Epple and Romano (998) and Snipes (998). Fernandez and Rogerson (996) only se up saic models while inergeneraional seings are used in Epple and Romano (998) and Snipes (998). Our work differs from hose papers in erms of he way in which we presen a dynamic model wih a mixed educaional sysem o explore he relaionship beween growh and income disribuion. Mckinnon (973), Shaw (973) and ohers (Kapur (976), Galbis (977), Fry (978, 995), Mahieson (980)) sudy he imporance of financial developmen in he process of economic growh. While he economy grows, he deregulaion of he financial secor 4

5 usually akes place. Financial developmen can help an economy grow by allocaing financial resources among he bes uses 4. Alhough he posiive correlaion beween financial developmen and growh is well known, he relaionship beween financial developmen and income inequaliy is no clear. The quaniaive sudy performed in his paper shows ha he correlaion beween financial developmen and inequaliy is also posiive. Under a privae educaion regime, inequaliy (which is measured by he Gini coefficien) will increase by.7% when an economy experiences a financial reform by ransiioning from an imperfec credi marke wih exogenous consrains o a perfec marke. We consruc an overlapping generaions model where agens differ in heir innae abiliies and in erms of parenal human capial. Boh innae abiliies and parenal human capial affec households choices of invesmens in educaion and each of hem will affec he accumulaion of human capial. Our work in relaion o a privae educaion regime wih marke imperfecions is closely relaed o ha of Galor and Zeira (993). Their heoreical work shows ha, in he presence of credi marke imperfecions and indivisibilies in human capial invesmen, he iniial disribuion of wealh affecs aggregae oupu and invesmen boh in he shor run and in he long run. In his paper, we firs sudy an economy wih perfec credi markes. This means ha households can borrow as much as hey wan o inves in educaion and we also assume ha no one will defaul. Anoher exreme case is ha none of he agens can borrow. This migh happen when here is no enforcemen of punishmen for defauling. Wihou any enforcemen of punishmen, i is opimal for households o defaul. Because here is no punishmen o 4 Empirical sudies conduced by Demeriades and Hussein (996) and Luinel and Khan (999) find ha here is a bi-direcional causaliy beween financial developmen and economic growh. 5

6 preven borrowers from defauling, no one is willing o lend. We refer o hese hings as exogenous borrowing consrains. Unlike Galor and Zeira (993), borrowing consrains do no only raion poor agens, bu also hose able individuals. Agens who are crediraioned will underinves in educaion. Our sudy indicaes ha, wih exogenous credi consrains, income inequaliy will be ransmied from one generaion o anoher. However, exogenous credi consrains are misleading because people wih higher fuure income are more likely o repay heir loans and i should also be easier for hem o borrow. Following Lochner and Monge (00), we assume ha agens will lose a fracion of heir income if hey defaul 5. Given he punishmen for defauling, an endogenous credi consrain is derived as a fracion of an agen s fuure income 6. Far fewer agens are credi raioned wih endogenous credi consrains han wih exogenous credi consrains. By allowing an economy o be ransformed from one wih exogenous credi consrains o one wih perfec markes hrough a sage of endogenous credi consrains, we find ha loosening credi consrains improves economic growh bu also increases income inequaliy. By swiching from a privae educaion regime o a public educaion regime, i is possible o avoid he problem of underinvesmen in educaion when financial markes are imperfec. Under a public educaion regime, ax revenue is colleced from wage income ax and is used for public educaion. Wih he coexisence of public and privae schools, we allow he governmen o provide public suppor for privae educaion hrough 5 Lochner and Monge (000) sudy he life-cycle behavior of consumpion, labor supply and human capial accumulaion in an economy where credi consrains arise endogenously. 6 Kehoe and Levine (993) sudy he case where households are indifferen beween repaying loans and defauling. 6

7 voucher programs 7. The role of voucher programs is like ha of a subsidy from he governmen o households who choose privae schools. We find ha he srucure of he educaional sysem is imporan for deermining he growh and income inequaliy boh in he shor run and he long run. The model is calibraed by using 980 U.S. daa. The iniial income disribuion is calibraed o mach he median households income and he Gini coefficien in 980. The ax rae, borrowing consrains and he scale of voucher programs (hese are insrumens conrolled by he governmen) are se o mach he raio of public spending on educaion o gross naional produc, he raio of consumer credi o ne naional produc and he public school enrollmen rae, respecively. Changing he values of hese hree insrumens allows us o analyze he impac of fiscal policies. Secion presens he model of a privae educaion regime. We firs analyze a privae educaion regime wih perfec credi markes. We hen impose an exogenous borrowing consrain: every young agen can borrow up o a cerain limi regardless of his learning abiliy and parenal human capial. Endogenous borrowing consrains are inroduced ino he model a he end of his secion. The definiion of general equilibrium is also given in secion. The model of a public educaion regime is presened in secion 3. A mixed educaional sysem is analyzed in secion 4. In order o sudy he impac of he srucure of he educaional sysem on economic growh and income disribuion, we simulae equilibrium in secion 5. The calibraion of he parameers and he resuls of he simulaion are given in secion 5. Secion 6 sudies he impac of governmen policies. The conclusion is given in secion 7. 7 An alernaive way of providing public suppor for a privae educaion regime is o subsidize privae schools. 7

8 . The Model of a Privae Educaion Regime We consider an infinie-horizon, discree-ime overlapping generaions model where agens live for hree periods. Each period is approximaely 0 years, corresponding o youh, middle age (adulhood) and old age. Every adul gives birh o a young person. We refer o young agens as children, o aduls as parens, and o old people as reirees. There is no populaion growh and we normalize populaion size o one. Agens have he same uiliy funcion over heir lifecycle. The uiliy funcion is ln( c ) + β ln( c+ ) + β ln( c+ ), () where β ( 0, ) is he discoun facor, and c, c, c represen he consumpion of he + + young, he middle aged and he old for he cohor born a ime > 0. Each individual is endowed wih one uni of ime. Young agens devoe all of heir ime o sudying, middle-aged agens use all of heir ime for work, and old agens enjoy leisure. Aduls use a fixed amoun of ime o work a home and he res of heir ime o work for earnings. We assume ha he oupu produced a home equals each adul s human capial ( h ). Hence, he home producion funcion is hom e Y = h. Aduls give heir home producs o heir children as endowmens. Each young agen, afer he/she is born, receives parenal endowmen and his/her innae abiliy is revealed. Boh h and he innae/learning abiliy ( z ) of each young agen are public knowledge. Given hese endowmens and innae abiliies, he young individuals in cohor choose how much hey wan o spend on consumpion ( c ) and educaion ( q ), and how much 8

9 hey wan o save (s ). When s is negaive, i refers o deb. The budge consrain for a young agen is c + q + s = h. () Human capial is accumulaed according o he consan-reurns-o-scale learning echnology: γ δ γ δ h + = z q ) ( h ) ( H ), γ, δ (0,). (3) ( where H is he average human capial for he cohor. We assume ha he disribuion of learning abiliy (g ) is log-normal wih mean µ and variance. We also assume ha z he iniial disribuion of human capial (g σ h. z σ z ) is log-normal wih mean µ and variance h h z ~ LN( µ, σ ), z z h ~ LN( µ, σ ). h h We assume ha he accumulaion of human capial does no affec he realizaion of innae abiliy. The human capial accumulaion funcion in eq (3) depends on learning abiliy, educaional invesmen, parenal human capial and average human capial. One exernaliy arises, because communiy qualiy, H, is a funcion of he human capial of all of he residens in his economy. Assume ha w and R are he real wage rae per uni of human capial and real ineres rae in period, respecively. Middle-aged agens can earn. Given he real ineres rae and he real wage rae, an adul decides how much he wans o consume and save for old age. The budge consrain for an adul is hus w + h + 9

10 c+ s+ = w + h + + R+ s +. (4) Old agens consume wha hey save. Hence, he budge consrain for an old person is c R s (5) + = Perfec Credi Markes We firs consider he case of perfec credi markes. Under a privae educaion regime, young agens decide how much hey wan o inves in educaion. If we assume ha households can commi o repay all debs (and hence here is no upper limi on borrowing), heir opimal choice of invesmen in educaion is w + γz δ γ δ γ ( h H ) R + q =. (6) Eq (6) ells us ha invesmen increases as learning abiliy/parenal human capial increases and decreases as he real ineres rae increases. Hence, people who are smarer or have bigger endowmens will inves more in educaion. The opimal choice of consumpion in middle and old age saisfies he sandard Euler equaions: βr = + + i c + c+ + i, i=0,. (7) The human capial accumulaion funcion becomes = w + γ γ δ γ δ γ + [ z ( ) h H ] R+ h. (8) Since young agens wih high abiliy or high parenal human capial inves more in educaion if hey can borrow freely, heir adul human capial will be higher. Eq (8) also 0

11 shows ha, if he real ineres rae is high, young individuals will borrow less o inves in educaion and he adul human capial will be low. This is because a high real ineres rae means a high opporuniy cos of borrowing. The ineremporal budge consrain is c c c = h q. (9) R+ R+ R + R + w h This implies ha he opimal saving and consumpion for a young agen are: w + γz δ γ δ γ + γβ ( + β ) s = [ β ( + β ) h ( ) ] h H (0) + β + β R γ + w + z γ γ δ γ δ γ [ h + ( ) q h H R+ c = ]. () + β + β Le us now define g as he growh rae of average human capial from period o + period + (g + H + = ), z as he average of learning abiliy and b = H h H as relaive human capial. Proposiion. If he ineres raes are consan over ime ( R = R, 8 ), hen under a privae educaion regime wih perfec markes, here is a balanced growh pah ( =, z = z ) wih a growh rae ( Proof: See Appendix. * g ) equal o [ wγ γ γ z ( ) ]. R b.. Exogenous Credi Raioning 8 This implies ha he real wage rae is also consan, = w,. We will explain his in secion.4. w

12 Suppose ha agens are subjec o borrowing consrains. Wihou he commimen of paying back loans, i is opimal for borrowers o defaul. Therefore, savings for young agens are resriced o being non-negaive ( s 0) 9 because no one is willing o lend. We refer o he zero borrowing as he exogenous credi consrain. From eq (0), for a given z, saving is non-negaive under a perfec marke if h γ + γβ ( + β ) R + γ δ + γβ ( + β ) w γz [( ) ] H. () Eq () means ha for a cerain learning abiliy, poor agens will wan o borrow. Hence, he exogenous credi consrain is binding on poor young agens, i.e. hose wih a low sock of inheried parenal human capial. Given h, saving is non-negaive under he perfec marke if γβ ( + β ) γ R + γ δ ( ) ( h z )( ). (3) + γβ ( + β ) w + γ H Eq (3) says ha for a cerain level of parenal human capial, agens wih high innae abiliy will wan o borrow. The consrain is binding on he able young agens. To summarize, young agens migh face problems wih credi limis if hey have low parenal human capial or high learning abiliies or boh. Everyone in middle age chooses o save o provide for consumpion in old age. For young individuals who are credi-raioned, heir savings are zero. Their invesmens in educaion are βγ ( + β ) q = h. (4) + βγ ( + β ) 9 We can also assume ha young individuals can borrow up o a cerain fixed level. However, he resul will be similar o wha we ge when we se ha level o zero.

13 Eq (4) shows ha he expendiu re on educaion of a consrained agen is a fracion of he parenal human capial. Such an invesmen is independen of learning abiliy, he real ineres rae and average human capial. The law of moion of human capial for consrained agens is βγ ( + β ) γ γ + δ γ δ h + = z[ ] h H. (5) + βγ ( + β ).3. Endogenous Credi Consrains Alhough he assumpion of exogenous credi raioning makes i easier o analyze human capial accumulaion wih imperfec credi markes, i does no seem very reasonable o make his assumpion. Jappelli and Pagano (994), for example, sudy households savings in an overlapping generaions model in which liquidiy consrains are a fracion of discouned lifeime income. Wihou inroducing human capial ino heir model, heir finding ha he relaionship beween credi consrains and growh is posiive is quie differen from much of he lieraure ha saes capial marke imperfecions end o deer growh. In his secion, we analyze he siuaion where borrowing consrains differ across individuals. There are wo ypes of young agen: hose who are consrained and hose who are no. If young agens choose o borrow, hey can choose o repay heir loans or defaul in heir middle age. All aduls and old agens are unconsrained because aduls need o save for heir old age and old people only consume heir savings. In order o make credi consrains arise endogenously, we follow he assumpions made by Lochner and Monge-Naranjo (000) who sudy he effecs of endogenous credi consrains on he accumulaion of human capial. We assume ha, if agens defaul in 3

14 heir middle age, hey will lose a fracion φ (0, ) of heir income 0. There are wo main differences beween heir work and ours: firs, we sudy he impacs of educaional sysem and allow for he coexisence of privae and public schools while hey only consider a privae educaion regime. Second, he relaionship beween growh and inequaliy is no he concern of heir paper while i is he objec of his paper. Given he income and saving/deb decision made in his/her youh, an adul chooses o save and consume β ( w + h + + R+ s ) s + = (6) + β w + h + + R+ s c + =. (7) + β Recall ha aduls are no credi raioned because hey need o save for old age. Using eq (7) and eq (7), he value funcion for an adul who does no defaul is V adul ( h +, s ) = ( + β )[log( w + h + + R+ s ) log( + β )] + β log( R+ β ). (8) If an adul needs o repay a huge deb, he migh choose o defaul. If he defauls ( s =0), a fracion of his income will be gone. The value funcion for an adul who defauls is d V adul ( +, s ) = ( + β )[log( φ) w + h + log( + β )] + β log( R+ β ) h. (9) Eq (9) is independen of. The size of he credi limi is deermined by leing R + borrowers be indifferen beween repaying and defauling. So he value funcion of paying back deb is a leas as big as he value funcion of defauling. Crediors se he 0 Using a hree period OLG model, an alernaive punishmen for defauling ha we can use is ha for aduls who defaul, hey can only obain he lower rae of reurn of heir savings. However, he resuls will no be differen from hose in his paper. Lochner and Monge (00) use boh punishmens for defauling. 4

15 upper boundary of deb up o his level so ha no defaul will ake place. By seing d V ( h, s ) V (, ), he credi limi is adul + adul h + s - ρ where s w + h + φ ρ =. (0) R + Eq (0) gives he maximal deb ha a young individual can carry on o he nex period. I is a fracion of discouned fuure income. Noe ha ρ is an increasing funcion of φ. When φ increases, he cos of defauling also increases and people are less willing o defaul. Hence, lenders are willing o lend more. For unconsrained young agens, he opimal choices of expendiures on educaion, saving and consumpion are he same as hose under perfec markes. The value funcion for an unconsrained young agen is V youh ) = (h, z [ + β log β R+ + β log( β R + R+ )] + ( + β + β ) log c. Poor or able young agens are consrained. The maximal deb hey are allowed o have is ρ wh +. Thus, a young agen s problem becomes max.log( h + ρw + h + q, s + q + β log(( R ρ) w h s ) β log( R s ). ) There are no analyical soluions for and. The firs-order condiions are q s + s β + β + = ( R+ ρ) w + h + (a) γρ w + h + q = βγ ( + β )( q h ρw + h + ). (b) 5

16 .4. Equilibrium For a closed economy, he average human capial in his economy is h + H + = g h, zdhdz. () Aggregae capial is deermined by marke clearing condiions in he capial marke: K s + s+ ) g h, z = ( dhdz. (3) Here s + i (i =, +) represens saving in ime + while he agen was born a ime i and g h,z is he join disribuion of human capial and learning abiliy. Using aggregae physical capial ( K ) and average human capial ( H ) as inpus, aggregae domesic oupu is produced by a sandard neoclassical producion funcion: Q = AF( K, H ) where A is he oal facor produciviy Assuming ha he producion funcion has consan reurns o scale, hen Q = A* H * F( K / H ) = A* H * f ( k ), where k = K / H. The marke clearing prices in every period saisfy R w F = A - depreciaion rae = A f '( k ) depreciaion rae, K F = A = A( f ( k ) k f '( k )). H Definiion. Equilibrium under a Privae Educaion Regime in a Closed Economy Given he iniial disribuion of human capial g, he disribuion of learning abiliy g, preferences, a home producion funcion, human capial accumulaion echnology, producion echnology and financial regulaions, an equilibrium consiss of aggregae h z 6

17 capial socks { H, K }, sequences of prices { w, R }, he disribuion of human capial g and individual decisions { q, s, c, c, c } such ha : h +. Given { w, R }, he { q, s, c, c+, c+ } solves he households problems (maximizing he uiliy funcion subjec o eqs (), (3), (4) and (5)).. { w, R } clear markes Eqs () and (3) hold. 4. Given g and g, he disribu ion of human capial a +,, is deermined by h z g h+ h γ δ + = z q ) ( h ) ( ( H ) γ δ. For a small open economy, if we assume ha here is no depreciaion of physical capial and he world ineres rae is consan ( R = R ), he equilibrium value k * of physical capial per uni of human capial is deermined by he facor-price equaion of R and is consan. From he facor-price equaion of w, he wage rae is also consan over ime ( w = w ). Definiion. Equilibrium under a Privae Educaion Regime in a Small Open Economy Given he iniial dis ribuion of human capial, he disribuion of learning g h abiliy g z, consan ineres rae R and consan real wage rae per uni of human capial w, preferences, a home producion funcion, human capial accumulaion echnology, producion echnology and financial regulaions, an equilibrium consiss of average capial socks { H, { q, s, c, c +, c + } such ha : K }, he disribuion of human capial g h and individual decisions 7

18 . Given { w, R }, he { q, s, c, c +, c + } solves he households problems (maximizing he uiliy funcion subjec o eqs (), (3), (4) and (5)).. Facor price eq uaions hold. 3. Eq () holds and K = (k*) H. 4. Given g h and g z, he disribuion of human capial a +, g h+, is deermined by h γ δ + = z ( q ) ( h ) γ δ ( H ). In he remainder of his paper, we focus our research on a small open economy and assume ha he world ineres rae is consan. 3. A Public Educaion Regime Our discussion so far has only focused on a privae educaion regime. Under a privae educaion regime, young people decide how much o spend on educaion. However, because of borrowing consrains, some of hem are credi-raioned and underinves in educaion. By inroducing a public educaion regime, we can overcome he problem of underinvesmen in educaion due o credi consrains. Under a public educaion regime, aduls need o pay ax on wage income. We assume ha he ax rae is consan ( τ = τ ) over ime and is deermined by he governmen. Young agens do no need o decide how much hey wan o spend on educaion since qualiy schooling is provided by he governmen. We rule ou he case of governmen deb by assuming ha he governmen always mainains a balanced budge. Hence he ax revenue equals he school expendiure. So he school expendiure is 8

19 q = τw H. (4) u The human capial accumulaion under a public educaion regime becomes h γ δ δ + = z ( τw) h H. (5) Definiion 3. Equilibrium under a Public Educaion Regime in a Small Open Economy Given he iniial disribuion of human capial, he disribuion of learning abiliy, consan ineres rae R and consan real wage rae per uni of human capial w, preferences, a home producion funcion, a consan ax rae, human capial accumulaion echnology, producion echnology and financial regulaions, an equilibrium is average capial socks { H, K }, a di sribuion of human capial h g and individual decisions { s, c, c +, c + } such ha :. Given { w, R }, { s, c, c +, c + } solves he households problem.. q = τw H. u 3. The facor price equaions hold. 4. eq () holds and K = ( k*). H 5. Given g and g z, he disribuion of human capial a +,, is deermined h γ δ γ δ by h + = z ) ( h ) ( H ). ( q u g h+ Proposiion. In a small open economy wih a public educaion regime, a balanced growh pah ( b =, * z = z ) exiss wih a growh rae ( g ) equal o γ z ( τ w). Proof. See Appendix. 9

20 The ax on he fuure income makes young agens save and consume more han is he case wihou any income ax. However, aduls will save less due o he loss of ax on wage income. Proposiion ells us ha, under a public educaion regime, he governmen plays an imporan role in economic developmen. The governmen can conrol economic g rowh by using differen ax policies. Because we assum e ha all ax revenue is used for public schools, a high ax rae will increase human capial accumulaion and his will in urn increase he economic growh rae. Alhough a high ax rae can generae a high growh rae, i may no be preferred by households. Along he balanced growh pah, he growh rae is higher under a public educaion regime han under a privae educaion regime wih perfec credi markes if he ax rae is greaer han r γw γ z [ R ]. Because of he ax rae, he endogenous credi consrain becomes - S ' w + h + ρ where ρ ' = ( τ ) ρ. (6) Since aduls need o pay a fracion of heir income in he form of ax, he amoun ha hey can commi o repay decreases as he ax rae increases. No many households are credi-raioned under a public educaion regime because endogenous credi consrains are no very igh consrains and young agens do no need o pay for heir educaion under a public educaion sysem. However, if we include he voing sysem in our model, he preferred ax rae is chosen by solving γ δ δ he maximizaion problem: maxτ log( h s ) + β log{( τ ) w[ z ( τ w) h H ] + Rs s+ }. The firs-order condiion gives rise o a consan opimal ax rae, γ τ = τ = γ. 0

21 4. A Mixed Educaional Sysem As we show in Table, mos counries have boh privae and public schools. In his secion, a mixed educaional sysem of privae and public schools will be analyzed. Besides providing public educaion, he governmen can also subsidize privae educaion by giving uiion reimbursemens (vouchers) o he households who choose o aend privae schools. Epple and Rmano (996) analyze he compeiion beween privae and public schools in erms of a voucher program and peer effecs. They inroduce he voucher ino he model as a fixed lump-sum subsidy. Their resuls show ha uiion vouchers increase he relaive size of he privae secor and he exen of suden soring, and benefi high-abiliy sudens relaive o low-abiliy sudens. Wes (997) claims ha mos voucher programs in he world are fracions of school expendiure in public schools. Hence, we assume ha he uiion voucher is a fracion ( v ) of he expendiure on public educaion. However, he way we rea he voucher is as if i were a lump-sum income subsidy as in Epple and Rmano (996). If he amoun of he voucher is larger han he uiion for he privae school, hen households can use vouchers for consumpion or 3 saving. The governmen runs a balanced budge and in period, he governmen budge consrain is: τ w H = p q + ( p ) vq, u u u u where p u is he percenage of young agens going o public schools. Wihou voucher programs, only households whose ideal educaion invesmens are higher han hose in public educaion will choose privae schools. Hence, privae schools will provide higher school qualiies han public schools. Bu wih voucher programs, in order o qualify for vouchers, some households whose desired educaion invesmens are lower han public educaion expendiure will choose privae schools. 3 We make his assumpion o simplify he model.

22 The budge consrains for young agens and aduls are c + q + s = h + vq u (7) c+ s+ = ( ) w + h + + R+ + τ s. (8) For unconsrained people, because he ax on fuure income decreases he reurn on invesmen in educaion, he opimal choice of school spending decreases as he ax rae increases. The uiion voucher is simply a lump-sum subsidy and so i does no affec an agen s choice of school spending. However, he saving of young agens increase and fewer young people are consrained because of vouchers. For a consrained agen, he upper limi on wha he/she can borrow is he amoun ha he/she commis o repay. Eq (6) says ha his amoun is equal o ρ' wh +. Therefore, he bud ge consrains for young agens and aduls, respecively, are c, (9) + q = h + vqu + ρ' w +h+ c. (30) + + s+ = [( τ ) ρ' R+ ] w + h + 5. Simulaion Having consruced he model, our inenion o sudy is long-run implicaions for growh and inequaliy. One way o do his is o simulae he equilibrium. By simulaing he model, we can quanify he impacs of credi consrains on growh and income inequaliy hrough he accumulaion of human capial. Before performing a simulaion, we need o calibrae he parameer values. 5.. Calibraion

23 We sar by calibraing he human capial accumulaion funcion. The parameers γ and δ are income elasiciies wih respec o expendiure on educaion and parenal income, respecively. The resul of an empirical sudy by Johnson and Safford (973) gives income elasiciy of s chool expendiure ( γ ) of The number used by Fernandez and Rogerson (997) based on Card and Kreuger s esimaes (99) is 0.. These wo numbers are close and so we se γ =0.. We hen follow Caucu (000) and pick δ =0.4. This corresponds o an inergeneraional correlaion of income of roughly he same number and is consisen wih he resuls of Solon (99) and Zimmerman (99). 0 Because one period consiss of 0 years, he discoun facor is β = (0.98). To make hings easier, we consider a small open econom y. As we discussed before, he real wage rae per uni of human capial is consan if he world real ineres rae is consan over ime. We assume ha he real ineres rae is 3.5% per year. Thus, R = (.035) 0. The 4 wage rae is se as 3.0. We need o calibrae wo disribuions: a log-normal disribuion of innae abiliy and a log-normal disribuion of iniial human capial. We use he model under a privae educaion regime wih perfec credi markes as our baseline model and calibrae hese wo disribuions under he environmen of he baseline model. The average growh rae of oupu per capia for he U.S. from 960 o 998 is abou %. Choosing a mean of innae abiliy equal o.733 approximaes he growh rae of income as being equal o 0 (.0) along he balanced growh pah in he baseline model. We calibrae he variance 4 We assume ha he capial share in he final good secor equals /3 and ha here is no depreciaion of capial, and we also assign oal facor produciviy as being equal o 5 as he value used by Lochner and Monge N. (000). This implies ha he equilibrium physical o human capial raio ( k * ) is According o he facor-price equaion, he wage rae is

24 of log(z) so ha he variance of log human capial is roughly consan under perfec markes. We calibrae µ h and σ h o mach he median of U.S. household incomes ($7,70) 5 and he Gini coefficien (35.) in 980 under he baseline model. Due o he srucure of he model, he households income is ( + w)h. This gives he median human capial as being equal o $ Because we assume ha he iniial human capial sock is log-normally disribued, he median of human capial is exp( µ ). 6 µ h h Accordingly, is se o and σ is se o We also use wo differen h numb ers for he variance of log h (0.645 and ) o sudy he impac of iniial inequaliy on economic growh and income inequaliy wihin an economy. Jappelli and Pagano (994) repor ha in 980, consumer credi was 6.% of ne naional produc in he U.S. Hence, we choose φ o be 8.5% o mach his daa o allow agens o borrow up o 6.% of heir fuure income. Using he parameer values we have jus calibraed, he annual growh rae along he balanced growh pah under a public educaion regime wih average innae abiliy will be less han % if he ax rae is less han 4.7%. In 980, he raio of public spending on educaion o gross naional produc was 6.7% 7. Accordingly, τ is se as 8.86%. This implies ha he growh rae equals.5% along he balanced pah under a public educaion regime. 5 Daa source: U.S. Census Bureau. 6 If he iniial human capial is log-normally disribued, he average human capial H = exp( = Source: UNESCO, Unied Naions. σ h µ h + ) 4

25 The las parameer we need o calibrae is he scale of he voucher. In his survey, Wes (997) shows ha he scales of mos voucher programs are beween 30% and 00%. We choose a number ha maches he enrollmen rae in public schools in he U.S of 88%. This makes v equal o 63%. Laer, we will ry differen scales of voucher programs o sudy he impac of voucher programs on economic performance. Table gives a summary of he calibraion of parameers. 5.. Resuls The simulaion resuls are given in Table 3. Figure shows ha he ransiions in he annual growh rae and he Gini coefficien for he youh under perfec markes over 0 periods wih homogeneous innae abiliy. The annual growh rae converges o he growh rae along he balanced pah (%) and he Gini coefficien decreases o zero. Using eq (8), he growh of human capial is h = [ z wγ ( ) R H ( + γ h h ) γ δ γ ]. Wih homogeneous innae abiliy, income inequaliy decreases over ime because he growh of human capial is a decreasing funcion of human capial (his means ha agens wih high human capial will accumulae heir human capial slowly and vice versa). Figure shows regions for agens being consrained or unconsrained in he case of a privae educaion regime wih exogenous credi consrains and heerogeneous innae abiliy. Using he parameer values calibraed in he previous secion, abou 93% of young agens are credi-consrained in he firs period. We find ha he paern of he 5

26 growh rae under perfec markes wih heerogeneous innae abiliy is quie differen from he one wih homogeneous innae abiliy. Wih homogeneous innae abiliy, he annual growh rae converges o %. However, wih heerogeneous innae abiliy, he average annual growh rae over 0 periods is.9645%. This is close o he average U.S. growh rae, bu he variaion in innae abiliy lowers he average growh rae. The explanaion for his is given in Appendix 4. Wih exogenous credi consrains, he average annual growh rae decreases dramaically by 53.6% o 0.889%. This is no surprising because, wih exogenous credi consrains, a grea number of young agens underinves in educaion as we can see from Table 3. The average raio of school expendiure o he income per capia under perfec markes is.59 imes he raio wih exogenous borrowing consrains. The Gini coefficien is lower wih exogenous borrowing consrains han wih perfec markes. Wih exogenous borrowing consrains, he Gini coefficien is lower because more middle income or able young individuals are consrained and canno ge heir desired school invesmens. The average saving rae for youh is wih perfec markes and 0.0 wih exogenous borrowing consrains. Saving raes for he youh wih exogenous borrowing consrains are posiive because exogenous borrowing consrains rule ou negaive savings. In he case where here are endogenous borrowing consrains, only 64.% of young individuals are credi consrained in he firs period. Figure 3 presens he hisogram of educaion invesmens for he firs period 8. The simulaion resuls are also given in Table 3. Because endogenous credi consrains are no as sric as exogenous credi consrains, he average saving rae for he youh, which equals , is much lower han is he case wih exogenous credi consrains, bu i is higher han is he case wih perfec credi 8 We assume ha here are 000 agens for each cohor. 6

27 markes. The raio of school expendiure o per capia GDP is The average raio of school expendiure o per capia GDP is lower han is he case wih perfec markes, and so some young agens underinves heir educaion. The average growh rae (.688%) is less han is he case wih perfec marke. The average of he Gini coefficien is %. Figure 4 shows he ime pahs of he annual growh rae and he Gini coefficien for he youh under a public educaion regime wih homogeneous innae abiliy. Based on our calibraion of parameer values, he annual growh rae wih homogeneous innae abiliy converges o.5% and he Gini coefficien goes o zero. Wih heerogeneous innae abiliy, he simulaion resuls show ha none of he individuals in he economy is crediconsrained. The explanaion of differen paerns of he growh rae wih homogeneous/heerogeneous innae abiliy under a public educaion regime is given in Appendix 4. The annual growh rae under a public educaion regime is lower han ha wih perfec markes because of he low ax rae. Alhough poor agens can benefi from public educaion, some agens who prefer o inves more in educaion now are forced o accep he public educaion. For example, an agen wih high abiliy and a huge parenal endowmen will wan o inves much in educaion and accumulae his human capial more rapidly, bu now he can only enjoy he same invesmen in educaion as oher agens under a public educaion regime. Noice ha, under a public educaion regime, he average Gini coefficien drops o 7.39%. Hence, a public educaion regime can be used o decrease he income inequaliy beween households. The Gini coefficien is smaller under a public educaion regime han under a privae educaion regime because, under a public educaion regime, everyone is forced o have he same invesmen in educaion. 7

28 This is consisen wih he heoreical resuls derived by Glomm and Ravikumar (99). However, he reasons are differen. Because hey do no inroduce imperfecions in credi markes ino heir model, he higher income inequaliy in a privae educaion sysem is generaed because he ime devoed o human capial accumulaion in a privae educaion economy is higher han in a public educaion economy (p.87). Glomm (997) also obains he same resuls by using a wo-period overlapping generaions model where parens make schooling decisions for heir children. When boh public and privae schools exis in he economy, abou 88% of agens choose public schools. Agens migh be endogenously credi-consrained if hey choose privae schools. Figure 5 shows which ype of agens will choose privae/public schools. As we can see, mos rich or able agens will choose privae schools, while mos poor or unable agens will choose public schools. Because Figure 5 does no cover all ypes of agens due o he limied ranges of parenal human capial and innae abiliy shown in he figure, Figure 6 presens he hisogram of educaion invesmens for young agens wih differen parenal human capial in he firs period. I shows ha some people wih low parenal human capial or innae abiliy will choose privae schools in order o qualify for vouchers. Their invesmens in educaion are lower han he expendiure on public educaion. The average annual growh rae (.6067%) lies beween he average growh rae under a privae educaion regime wih endogenous credi consrains and he average growh rae under a public educaion regime. The Gini coefficien is 3.49% and lies beween ha under a privae educaion regime wih endogenous credi consrains and he one under a public educaion regime. We also compare he long-run economic impacs for wo differen iniial disribuions of human capial. Two differen sandard deviaions 8

29 (0.645 and ) of log h are considered. Under a mixed educaional sysem, he iniial income inequaliy will be ransmied from one generaion o anoher. 6. Policies This secion analyzes he impacs of fiscal policies. Our main concern is how hese policies aler he ransiions of growh and income inequaliy. Three insrumens ha he governmen can use o affec he economic performance are he scales of he voucher programs, he regulaion of financial markes and he ax rae. The simulaion resuls for he policies are given in Table 4. Figures 8, 9 and 0 show he ime pah of he average level of human capial and he Gini coefficien over 0 periods. Noe ha he average income is proporional o he average human capial. Each do on he graph indicaes he level of human capial and income inequaliy in each period. Figure 7 shows he dynamic ransiions of he Gini coefficien during he process of developmen for hree differen voucher programs v (0%, 63% and 90%). A high v means large subsidies for agens who choose privae educaion. More rich and able young agens will choose privae schools as he scales of he voucher programs increase. Hence, he Gini coefficien will increase as v goes up. In addiion, as he subsidy becomes larger, he expendiure on public educaion will go down because of he balanced budge. Poor young agens who go o public schools will accumulae heir human capial more slowly while rich or able agens accumulae heir human capial more rapidly. Hence, he large scales of he voucher programs will deer an economy o grow if he former dominaes he laer. 9

30 Figure 8 illusraes economic performance for hree differen siuaions of financial developmen (φ =0%, 8.5% and 56.7%). Because he upper limi of deb increases while φ increases, fewer young agens are consrained for larger φ (in our model, no one is credi consrained when φ equals 56.7%). If fewer young individuals are consrained, more agens will choose o go o privae schools. Hence, he Gini coefficien increases as φ increases. Since he liberalizaion of financial markes can help he economy grow, a large φ will generae a hig h growh rae. The las insrumen ha we analyze is he ax rae. The ransiions of inequaliy and he level of human capial for hree differen ax raes (8.86%, 5% and 30%) are given in Figure 9. Because higher ax raes imply more expendiure on public educaion and also more subsidies for agens who choose privae schools, he enrollmen raes in public schools will increase if he former dominaes he laer. In our model, enrollmen rae in public schools increases firs when he ax rae increases and decreases afer a urning poin. The average enrollmen rae in public schools is 93.57% when he ax rae equals 5% and only 37.65% when he ax rae equals 30%. Hence, he Gini coefficien firs decreases hen increases. Thus, he growh rae increases as he ax rae increases because of more invesmen in educaion. 7. Conclusion This paper indicaes ha he srucure of he educaional sysem is imporan in explaining he relaionship beween growh and inequaliy. Human capial is an engine for growh, bu no many economiss pay aenion o he composiion of educaional sysems. We develop a dynamic model where households are allowed o make choices 30

31 beween privae and public schools. The exisence of imperfec credi markes prohibis some agens from choosing privae schools. However, hrough voucher programs, he governmen can provide public suppor for privae educaion. By changing he scales of he voucher programs, he governmen is able o conrol he educaion secor. The findings in his paper sugges ways in which he impacs of policies can be considered. Our simulaion resuls show ha low scales of voucher programs can increase he public school enrollmen raes and his in urn will raise he economic growh rae if he ax rae is high enough and financial markes are imperfec. Financial reforms wihou any oher inervenions will increase boh he growh rae and inequaliy. If he governmen inends o use financial reforms o help he poor, a policy o increase he size of he public secor (e.g., decreasing he amouns of vouchers) should also be used. Our resuls provide a cauionary noe o hose developing counries ha are experiencing financial reforms. A high ax rae can conribue o growh, bu i may increase or decrease income inequaliy. This paper also offers anoher approach o hinking abou he educaional sysem. For a developed counry wih low income inequaliy, a mixed educaional sysem wih a higher privae school enrollmen rae may be a beer choice han an educaional sysem wih a lower privae school enrollmen rae because financial markes are already liberalized and a high ax rae is poliically unfeasible. Table 5 gives he average growh raes ( ) and he Gini coefficiens 0 for 6 developed counries. We use 90% of 9 9 This is because in democraic counries, ax raes are deermined by he voing. However, we do no include he voing sysem in his paper. See Peroi (993) for a paper ha considers voing. 0 According o Deininger and Squire (996), here is no significan difference beween Gini coefficiens defined on he basis of ne or gross income and beween household-based or individual-based esimaes. However, he difference is significan beween income-based and expendiure-based esimaes. Following heir empirical resuls, we add 6.6 o he expendiure-based coefficiens. 3

32 he public secondary school enrollmen rae as our crierion. If a counry has a public secondary school enrollmen rae higher han 90% in 985, we include ha counry in a group of counries ha have a larger public secor. On he oher hand, if a counry has a public secondary school enrollmen rae of less han 90% in 985, we include ha counry in a group of counries ha have a larger privae secor. The average growh rae for he counries wih a larger privae secor is 3.355% (3 counries), while he average growh rae for he counries wih a larger public secor is.95% (0 counries). Income inequaliy is higher for hose counries ha correspond o a larger privae secor han for hose counries ha correspond o a larger public secor. For a developing counry, our model predics ha a large public secor is preferred o a large privae secor due o he underdevelopmen of financial markes. Table 6 gives he average growh raes ( ) and he Gini coefficiens for 46 developing counries. The average growh rae for he counries wih a larger public secor is 3.86% (4 counries), while he average growh rae for he counries wih a larger privae secor is 3.58% (3 counries). As wih developed counries, income inequaliy is higher for hose counries which correspond o a larger privae secor han for hose counries which correspond o a larger public secor 3. Alhough he differences in he growh raes and he Gini coefficiens beween he wo groups are no very significan, hese findings provide some suppor for he predicions of our model. If we focus on he daa for low income counries, we find ha mos low income counries do no have high enrollmen raes in public secondary We use 90% of he public school enrollmen rae as our crierion in order o have more equal sizes for each group. The average Gini coefficien for he counries wih a larger privae secor is 34.43% (9 counries), while he average Gini coefficien for he counries wih a larger public secor is 3.57% (7 counries). 3 The average Gini coefficien for he counries wih a larger privae secor is 50.% (0 counries), while he average Gini coefficien for he counries wih a larger public secor is 46.4% (0 counries). 3

33 schools. Table 6 also shows ha hese counries have high income inequaliy and no a very high growh rae as prediced by our model for a developing counry wih a larger privae secor. Given hese resuls, governmens should hink carefully abou he long-run consequences of educaional policies. This paper also provides a heoreical explanaion of Kuznes curve found in ime series and cross-secional regressions. During he early sage of economic developmen, he liberalizaion of credi markes will increase income inequaliy. By increasing he size of he public secor, he governmen can reduce income inequaliy. This will generae an invered U-shape beween levels of income and inequaliies in a ime series regression. In a cross-secion regression, we show ha he high Gini coefficiens of some developing counries migh be a consequence of low public school enrollmen raes and imperfecions in credi markes. 33