Bas Jacobs 1 Sweder J.G. van Wijnbergen 2

Transcription

1 TI /3 Tnbergen Inttute Dcuon Paper Captal Market Falure, Advere Selecton and Equty Fnancng of Hgher Educaton Ba Jacob 1 Sweder J.G. van Wjnbergen 2 1 European Unverty Inttute, Unverty of Amterdam, and Tnbergen Inttute, 2 Unverty of Amterdam, Tnbergen Inttute, and CEPR.

2 Tnbergen Inttute The Tnbergen Inttute the nttute for economc reearch of the Eramu Unvertet Rotterdam, Unvertet van Amterdam, and Vrje Unvertet Amterdam. Tnbergen Inttute Amterdam Roetertraat WB Amterdam The Netherland Tel.: +31(0) Fax: +31(0) Tnbergen Inttute Rotterdam Burg. Oudlaan PA Rotterdam The Netherland Tel.: +31(0) Fax: +31(0) Pleae end queton and/or remark of noncentfc nature to Mot TI dcuon paper can be downloaded at

3 CAPITAL MARKET FAILURE, ADVERSE SELECTION AND EQUITY FINANCING OF HIGHER EDUCATION * Ba Jacob European Unverty Inttute, Unverty of Amterdam, and Tnbergen Inttute (b.jacob@uva.nl) Sweder J.G. van Wjnbergen Unverty of Amterdam, Tnbergen Inttute and CEPR (vw.hea@planet.nl) March, 2005 Abtract: We apply theore of captal market falure to analyze optmal fnancng of rky hgher educaton. In the market oluton, tudent can only fnance ther educaton through debt. There undernvetment n human captal, becaue ome tudent wth ocally proftable nvetment n human captal wll not nvet n educaton due to advere electon problem n debt market and becaue nurance market for human captal related rk are abent. Legal lmtaton on the ue of human captal n fnancal contract caue th undernvetment; wthout them prvate market would optmally fnance thee rky nvetment through equty rather than debt and upply ncome nurance. The government, however, can crcumvent th problem and mplement equty and nurance contract through the tax ytem ung a graduate tax. Th paper how that publc equty fnancng of educaton coupled to provon of ome ncome nurance the optmal way to fnance educaton when prvate market fal due to advere electon. We how that educaton ubde to retore market neffcence are ub-optmal. Key word: human captal, captal market mperfecton, credt ratonng, fnancng rky nvetment, optmal educaton fnance, graduate taxe, educaton ubde. JEL-code: H21, H24, H52, H81, I22, I28, J24 * Comment by Lan Bovenberg, Amber Dav, Caper van Ewjk, Joer Gorter, Rck van der Ploeg and emnar partcpant of the Unverty of Amterdam, the CPB Netherland Bureau for Economc Polcy Analy and the CEPR Economc of Educaton Conference, May , Bergen, Norway are gratefully acknowledged. Ba Jacob thank the NWO Prorty Program Scholar fnanced by the Dutch Organzaton for Scence for fnancal upport. Correpondng author: Department of Economc, Unverty of Amterdam, Roetertraat 11, 1018 WB Amterdam, the Netherland. Phone: , Fax:

4 Invetment n profeonal tranng wll not necearly be puhed to the margn becaue earnng power eldom explctly treated a an aet to be captalzed and old to other by the uance of tock. [ ] If ndvdual old tock n themelve,.e., oblgated themelve to pay a fxed proporton of future earnng, nvetor could dverfy ther holdng and balance captal apprecaton agant captal loe. Fredman and Kuznet (1945, p.90) 1 Introducton In mot countre of the world, hgher educaton heavly ubdzed by the government. Apart from mert motve and the preumed preence of externalte of educaton 1, the man argument n favor of thee ubde that the government hould guarantee acceblty of hgher educaton. Captal market can fal to delver a uffcent upply of fund to graduate to fnance ther educaton. And, tudent are generally avere to nvet n rky educaton whle ncurrng debt. 2 Falure of captal and nurance hamper acce epecally for tudent from lower oco-economc background. Educaton ubde ndeed avod fnancal market falure by reducng the need for tudent to borrow and thereby reducng the rk of debt fnanced nvetment n educaton. The queton that reman to be anwered however whether educaton ubde are the mot effcent mean to warrant acce to hgher educaton. Economt have often advocated more effcent form of educaton fnance uch a ncome contngent loan or graduate taxe. The dea that both captal and nurance market falure wll be drectly addreed by provdng the fund to tudy and by (partally) nurng tudent ncome rk wthout ung educaton ubde, ee Nerlove (1972, 1975), Barr (1991, 1993), Chapman (1997), van Wjnbergen (1997), Ooterbeek (1998), and García-Peñaloa and Wälde (2000). Fredman and Kuznet (1945) and Fredman (1962) argue that graduate hould be allowed to ue equty to fnance ther nvetment n human captal. Except for García-Peñaloa and Wälde (2000), none of thee tude ha yet appled formal analy to the problem of optmal fnancng of educaton and to the oluton propoed. And García-Peñaloa and Wälde (2000) do not pay attenton to the underlyng mcro-economc caue of 1 Potve externalte may ndeed jutfy at leat ome educaton ubde. Although, Morett (2004) fnd emprcal evdence favorng external effect of educaton, other are more keptcal, ee for example Heckman and Klenow (1998), Acemoglu and Angrt (2000), Krueger and Lndahl (2002), amongt other. 2 The mportance of captal market falure tll a matter of emprcal controvery. Some argue that captal market are hghly mperfect baed on the gnfcant and potve aocaton between ocoeconomc tatu and enrollment n (hgher) educaton, ee e.g. Haveman and Wolfe (1995). On the other hand, th relaton may be due to unoberved charactertc uch a parental educaton and ablte. After ntrumentng for th, Shea (2000) fnd weak evdence for the unmportance of credt contrant. Cameron and Taber (2000), Cameron and Heckman (2001) and Carnero and Heckman (2002) argue that credt contrant are not really mportant emprcally. Plug and Vjverberg (2005), on the other hand, fnd trong evdence for the mportance of captal market falure whle correctng for unoberved charactertc ung adopted chldren a a natural experment. 2

5 market falure. But any dcuon of optmalty ncomplete f the underlyng caue of the market falure that gve re to nterventon to begn wth, are not explctly ncorporated n the analy. Some mght argue that moral hazard effect explan the abence of properly workng captal market for fnancng educaton and the abence of nurance for human captal rk. 3 Student and graduate may not exert enough effort to tudy and work after graduaton. A bank and nurance compane are not able to montor tudent and graduate effort, they are afrad that tudent wll ht and run when they apply for a loan, or tudent mght become lethargc durng and after graduaton becaue they are nured agant falure outcome. Although moral hazard certanly an ue to be taken nto conderaton n the degn of publc nterventon, we alo thnk that moral hazard not the man problem explanng the captal market falure blockng effcent fnancng mechanm for prvate educaton, for two reaon. Frt, and mot mportantly, moral hazard effect would reult n overnvetment n human captal, nce the hgh return nvetment cro-ubdze the low return nvetment n thoe crcumtance. 4 Aggregate overnvetment n human captal however not a good decrpton of realty, becaue the return to n partcular hgher educaton are very hgh and for example ealy approach the return to equty. 5 We therefore thnk that undernvetment n educaton a more lkely poblty, certanly n the abence of government nterventon. Conequently, moral hazard not the domnant nformaton problem n the market. Second, moral hazard n fnancal marker wll encourage the poorer tudent to overnvet n educaton at the expene of the rcher one, nce the latter wll looe more n the cae of default. Hence, moral hazard cannot explan the well documented overrepreentaton of the well-to-do n hgher educaton. In th paper we argue that advere electon n educatonal debt market n combnaton wth the mpoblty for prvate parte to wrte equty and nurance contract coverng return on human captal explan why and how government hould ntervene n the fnancng of hgher educaton. Advere electon n debt market lead to under-nvetment becaue ome tudent wth ocally proftable nvetment wll not nvet or wll not get credt (credt ratonng). Bank may not ncreae the nteret rate to meet exce demand for credt becaue th reult n large hft n the overall rkne of tudent applyng for a loan becaue the low-rk tudent drop out of the credt market, ee e.g. Stgltz and We (1981), Mankw (1986), and Hellman and Stgltz (2000). 6 Rk avere ndvdual further requre a rk premum on ther nvetment. Hence, f thee ncome rk cannot be nured under-nvetment exacerbated, cf. Eaton and Roen (1980). We how that an equty 3 See for example De Meza (2000) for th lne of reaonng n the context of mall frm and Judd (2000) n the context of nurng human captal rk. 4 Bernanke and Gertler (1990) and Hoff and Lyon (1995) modfy De Meza and Webb (1987, 1990) by addng addtonal cot for nvetor to verfy ther type. Thee may reult n under-nvetment of ome nvetor whch are ocally derable, but thee model all dplay overnvetment at the aggregate level. 5 See for example Card (1999) and Harmon et al. (2003). 6 Aymmetrc nformaton may alo play a role n the nurance market. Indvdual earnng capacte and ablte are generally well known before ncome nurance contract can be wrtten o that advere electon occur and the good rk eparate themelve from the bad rk and the market for nurance contract may break down, cf. Rothheld and Stgltz (1976), and Snn (1995). 3

6 partcpaton model a propoed by Fredman and Kuznetz (1945), Fredman (1962) and García-Peñaloa and Wälde (2000) ndeed the optmal way of fnancng hgher educaton n the preence of advere electon n captal market and the abence of nurance market to nure human captal rk. In practce, th oluton bol down to a graduate tax for the fnancng of hgher educaton. Our analy buld on the credt ratonng lterature poneered by Stgltz and We (1981). We alo contrbute to the lterature on advere electon n credt market and credt ratonng by allowng for rk avere tudent. Stgltz and We (1981), Mankw (1986) and De Meza and Webb (1987), and other have generally analyzed rk neutral nvetor. However, rk averon of tudent a real lfe phenomenon and we how that the ntroducton of rk averon ha non-trval conequence. Credt ratonng le lkely to occur, and may even dappear when tudent are uffcently rk avere. The ntuton that hgh-rk tudent alo requre a large rk-premum on ther nvetment. When bank ncreae nteret rate, potve electon effect may become domnant over advere electon effect f hgh-rk tudent drop out of the credt market frt becaue they requre a larger rk premum on ther nvetment. 7 Furthermore, we how that debt fnancng of hgher educaton not optmal and that tudent would prefer equty fnancng where t avalable. However, legal problem prevent the executon of both equty and nurance contract by the prvate ector n the cae of educaton fnancng. The reaon for th that the ue of human captal a collateral or clamng t proceed a dvdend mpoble becaue lavery and ndentured labor are outlawed n all cvl ocete. Th effectvely preclude fnancal contract that are contngent upon the return of human captal nvetment. Therefore, only debt fnance provded n market, credt ratoned, and under-nvetment preval alo due to rk averon. For the cae of rk neutral nvetor Cho (1986) and De Meza and Webb (1987) have hown that equty contract are ndeed optmal n the Stgltz and We (1981) model. Looely peakng, a bank offerng a debt contract only attract the hgh-rk tudent, whle an equty contract attract only the low-rk tudent (.e. nvetor wth low-rk project). Therefore, only equty contract are offered. However, wth rk avere tudent, th le obvou. If the potve electon effect of hgher nteret rate alway domnate the advere electon effect due to lmted lablty, one mght expect that debt contract and not equty contract are the equlbrum contract, becaue debt contract then attract the low-rk tudent. We how that th doe not happen and an equty contract alway preferred to a debt contract no matter how rky the tudent are. The reaon that equty contract offer more ncome nurance than debt contract and avod dtortonary redtrbuton of ncome from low to hgh-rk tudent. A a conequence, the undernvetment problem mtgated. Student wth low-rk nvetment wll now nvet n hgher educaton whle they would under-nvet wth debt fnancng. And, more tudent wth rky educaton enroll becaue they are better protected agant falure outcome. Government nterventon n fnancng educaton warranted, becaue only the government can mplement equty contract. The crucal dtncton between prvate parte and the government that the government can montor and enforce clam on all return from human captal through the tax ytem, a t already doe through the regular ncome tax. Equty partcpaton by the government come down 7 De Meza (2000, p.f21) alo peculate that th may occur but doe not refer to work howng th formally. 4

7 to allowng tudent to fnance educaton n exchange for a clam on part of the tudent future ncome through a tax on the return of the nvetment,.e. a graduate tax. Alo, we how that ntroducng a graduate tax n general not uffcent to attan the optmal level of nvetment n human captal, nce rk averon of graduate mple that they tll under-nvet. Although both equty and debt fnancng feature ncome nurance, not all ncome rk are elmnated, o that ome undernvetment due to rk averon reman. Therefore, addtonal ncome nurance welfare mprovng. The government may retore ocal effcency by reducng ncome rk through a hgher graduate tax. We how that educaton ubde are at mot econd-bet ntrument to retore ocal effcency n nvetment n human captal. We fnd that effcency n nvetment n human captal can only be retored by gvng very large educaton ubde (on educatonal cot or nteret cot), becaue educaton ubde do not nure ncome rk. An equty partcpaton cheme wll be far more effectve to gve the hgh rk tudent ncentve to nvet n human captal, becaue of the aocated ncome nurance. An unfortunate by-product of ubdzed hgher educaton alo that t mple revere redtrbuton. The ncdence of ubde fall on the average tax payer, wherea the beneft accrue to the mot talented, and hence generally better pad. Addtonally, a dproportonal number of graduate belong to the mot wealthy famle. Hence, equty fnancng of educaton avod th pervere redtrbuton of ncome, nce n prncple no ubde are needed. Fnally, we preent ome calculaton on the lkely conequence of ntroducng a graduate tax n the Netherland. We how that n a graduate tax ytem payment uncertante are gnfcantly reduced compared to a loan ytem and ubtantal avng on government outlay could be acheved wth only modet tax rate. The etup of the paper a follow. In ecton 2 we preent the model and analyze the role of captal market mperfecton and rk on decon to nvet n learnng. Optmal fnance of educaton analyzed n ecton 3. In ecton 4 we dcu ub-optmal way of fnancng educaton. Secton 5 preent ome calculaton of a graduate tax ytem ung Dutch data. In ecton 6 we dcu the conequence of moral hazard for our reult and Secton 7 conclude. 2 Invetment n human captal wth captal and nurance market mperfecton 2.1 Student The benchmark model the mplet poble model wth captal and nurance market mperfecton. We extend Stgltz and We (1981) by allowng for rk avere nvetor. Conder a ma of graduate wth ndex, of unt meaure. Each graduate decde whether to enroll n hgher educaton whch requre an nvetment of K. K can be thought of a tuton cot and foregone earnng. The return to the nvetment n human captal random. We only conder two-outcome project and denote the return under the ucceful outcome R,, and R f f the nvetment n human f captal fal. We aume wthout lo of generalty that R = R f for all. Expected return are the ame for all graduate: f (1) R R = p R + ( 1 p ) R = cont., 5

8 where p n [0,1] the probablty of a ucce for graduate. We ay that graduate rker than graduate j f p < p j. 8 9 All graduate have dentcal ntal wealth W = W whch aumed nuffcent to cover all cot of educaton: W < K. Therefore, addtonal fnance requred. We make the followng mportant non-lavery aumpton. Prvate fnancal contract between tudent and fnancal nttuton cannot be made contngent upon the return R of the nvetment n human captal. Only debt fnance therefore allowed, nce a debt contract (r, B), that pecfe the prncpal B and nteret rate r, ndependent of the return of the nvetment. Furthermore, ncome nurance mpoble nce th would alo requre contract dependent on the return to human captal. If graduate decde to nvet n educaton they borrow B = K - W at nteret rate r. If the nvetment n educaton fal, bank receve the falure return R f. If educaton ucceful, bank receve prncpal plu nteret. We aume that R > (1 + r)b > R f alway hold. Graduate have lmted lablty, therefore the return for graduate gven by: (2) π max[ (1 r) B, 0]. R + Graduate are rk avere, wth a tandard expected utlty functon EU( ) wth U(0) = 0, U' > 0, U'' < 0, U''' 0. We alo mpoe Inada type condton on U: lm 0 U'( ) =, lm U'( ) = 0. Graduate are wllng to nvet n rky educaton fnanced wth debt a long a: (3) EU ( π ) = p U ( R (1 + r) B) U ((1 + ρ) W ) ), where the afe real return on non-human nvetment (avng). Expected utlty ether monotoncally ncreang n p or non-monotonc; frt ncreang, then reachng a maxmum and fnally decreang n p. To ee th, dfferentate (3) whle ubttutng (1): deu ( π ) f (4) = U ( R (1 + r) B) U (' R (1 + r) B)( R R ) < dp U ( R (1 + r) B) U (' R (1 + r) B)( R (1 r) B). + The lat lne equal zero for rk neutral nvetor and potve for rk avere nvetor. The gn of (4) therefore trctly negatve for rk neutral nvetor. The gn of (4) however cannot be determned n general. We know that the econd lne alway potve for any concave utlty functon. Therefore, the frt lne may be ether 8 Generally peakng one cannot ay that graduate ha hgher rk than graduate j f p < p j becaue the varance of return frt ncreae and then decreae wth p becaue the return are b-modally dtrbuted. However, wth mean return retrcted to be equal acro all, t ealy hown that the varance decreae wth p. 9 There no ytematc macroeconomc rk and all rk are doyncratc. In the emprcal applcaton below we argue why th a reaonable benchmark n the cae of hgher educaton. 6

9 potve or negatve, nce R R f > R (1 + r)b. Whether (4) wll be potve or not depend on the ze of (1 + r)b R f and margnal utlty of ncome U' (whch lowet for low-rk nvetor). If borrowng cot are large compared to the return (mall R (1 + r)b), return n the bad outcome relatvely low (hgh R R f ), then (4) may be negatve, and vce vera. Therefore, (4) wll be typcally negatve for low-rk nvetor wth relatvely mall rk averon due to relatvely afe nvetment and who have a large margnal utlty of ncome nce they have relatvely hgh borrowng cot relatve to the return. We can ketch the graph of EU(p ); cf. Fgure 1 and 2 below. We know that EU(0) = 0, and EU(1) = U(R (1 + r)b) > 0. The graph of EU(p ) ether alway ncreae monotoncally, or frt ncreae and then decreae to reach EU(1). The ntuton for the hape of EU(p ) can be undertood mot ealy by alo plottng U(E ), whch denote utlty from the certanty equvalent and correpond to the cae where there perfect ncome nurance. Th lne alo correpond to the Stgltz and We (1981) cae wth rk neutral nvetor. A we move along the horzontal ax from p = 1 to p = 0 (from rght to left), we know that nvetment become more rky. If graduate could elmnate ncome rk o a to obtan the certanty equvalent of ncome, the utlty (of expected ncome) would ncreae for graduate wth lower p. Equaton (4) alway negatve for rk neutral graduate (U lnear), nce only the lmted lablty effect allow the hgh rk graduate to hft the downde rk to bank. However, f graduate are rk avere, expected utlty lower than the utlty of expected ncome. Expected utlty may ntally ncreae f p lowered due to the potve effect of havng lmted lablty. Th lmted lablty effect more mportant when rk averon mall, ncome n the bad tate of nature are lower (R f lower) or f nteret rate are hgher o that debt cot are hgher ((1 + r)b larger), nce then the welfare gan of beng able to hft default cot to bank ncreae. Eventually, however, expected utlty mut become decreang f p decreae, becaue rk averon become domnant n lowerng expected utlty. Th becaue the 'utlty cot of beng rk avere ncreae quadratcally wth lower p, wherea the utlty beneft of havng lmted lablty only ncreae lnearly wth decreang p. For example, f utlty feature contant relatve rk averon (CRRA), (EU( ) = p (R (1 + r)b ) 1 - /(1 )), then (4) may be alway potve (low nteret rate, hgh return n bad outcome) for rk avere graduate,.e. when 0 < < 1, ee Fgure 1. Stronger rk averon (hgher ) decreae the lope of the EU lne. We plotted the cae n whch the nteret rate hgher (r = 1.5) n Fgure 2. Hence, for hgh p the potve effect of lmted lablty domnate the negatve rk averon effect on rk takng, o that EU( ) frt ncreang and then decreang a p fall. 7

10 Fgure 1 - Invetment decon wth debt fnancng and wth hgh and low rk averon (R f =.5, R = 3, W =.6, B = 1, = 0, r =.5) EU(P) CRRA =.5 U(EP) CRRA =.5 EU(P) CRRA =.75 U((1+rho)W) probablty of ucce Rk averon may have mportant conequence for the equlbrum of the model. For the margnal graduate,.e. the graduate who ndfferent between nvetng n educaton or puttng money n the bank, (3) hold wth equalty. The ucce probablty at whch a tudent margnally ndfferent between nvetng and tayng out, p m, may declne or ncreae f bank ncreae nteret rate dependng on whether (4) > 0. Th follow from totally dfferentatng p m U((R - R f )/p m (1 + r)b - R f )) = U((1 + )W): dpm pmu '(.) B (5) =. f dr U (.) U '(.)( R R ) m Conequently, dp m /dr > 0 when (5) > 0 and vce vera. In the lmtng cae where graduate are rk neutral, p m = (R R f (1 + )W)/((1 + r)b R f ), and therefore, dp m /dr < 0 for rk neutral graduate. 8

11 Fgure 2 - Invetment decon wth debt fnancng and wth low rk averon and hgh nteret rate (R f =.5, R = 3, W =.6, B = 1, = 0, r = 1.5) EU(P) U(EP) U((1+rho)W) probablty of ucce If dp /dr < 0 for all, hgh-rk graduate (wth low p ) are wllng to pay hgher nteret rate on loan. Th the ource of advere electon n Stgltz and We (1981), nce bank cannot oberve p. If bank ncreae nteret rate charged to tudent, the average rk n the pool of loan applcant ncreae. However, f dp /dr > 0 for all, the hgh-rk graduate drop out of the credt market frt, and ncreang the nteret rate create a potve electon effect on the loan applcant. In the ntermedate cae, dp /dr wtche gn f rk averon mall, nteret rate are not uffcently low ((1 + r)b hgh) or ncome n the bad tate of nature are not uffcently hgh (R f low). I.e. when dp /dr > 0 for for low p and dp /dr > 0 for hgh p. Then, there are n fact two margnal graduate p and p*, becaue the EU( ) lne cut the U((1 + )W) lne twce on the nterval [0,1], ee Fgure 2. p correpond to the margnal graduate wth the lowet probablty of ucce who wllng to nvet, where dp/dr > 0. p* correpond to the margnal graduate wth the hghet probablty of ucce who wllng to nvet, wth dp*/dr < 0. Graduate wth p < p or p > p*, do not nvet. 2.2 Bank Identcal and rk neutral bank offer debt ervce to graduate o a to maxmze expected proft. The credt market compettve n the ene that there free entry and ext. Crucal for our expoton that bank uffer from an nformaton problem. A n Stgltz and We (1981), graduate know the probablty of ucce p wherea bank cannot oberve p. Whle bank do not oberve p, they know the common expected return R to nvetment n human captal and the dtrbuton of ucce probablte denoted f(p ). Alternatvely, one may ay that the bank have clafed all graduate n group wth dfferent mean return to ther educaton. In practce, th reult n bank chargng dfferent nteret rate to tudent n dfferent dcplne wth dfferent mean return to ther educaton. Bank obtan fund at afe real (depot) rate. For mplcty we aume that the upply of fund to bank totally elatc at. Expected (average) proft for the bank are then gven by: 9

12 * p f (6) Π = (1 + r) B p f ( p ) dp + R (1 p ) f ( p ) dp (1 + ρ) B. p * p p In equlbrum, perfect competton between bank drve expected proft down to zero. 2.3 Equlbrum In equlbrum bank offer debt contract uch that no equlbrum contract make negatve expected proft and there no other debt contract uch that th can make non-negatve expected proft. From Stgltz and We (1981), we know that credt may be ratoned n equlbrum. The mportant aumpton drvng thee reult that dp m /dr < 0, whch alway hold true f graduate are rk neutral. The ntuton that even though there may be an exce demand for credt, bank are not wllng to ncreae the nteret rate to choke off exce demand for credt, nce the good rk graduate (hgh p ) drop out of the market frt (advere electon). The average default rk on loan may ncreae o much that th caue loe that are larger than the ncreaed revenue from hgher nteret rate. The ntuton for the credt ratonng reult can alo be derved from dfferentaton of (5) wth repect to the nteret rate note that n th cae p = 0, and p m = p*, nce all hgh-rk graduate apply for credt: p m dπ f dpm (7) = B p f ( p ) dp + [(1 + r) Bpm + R (1 pm )] f ( pm ). dr dr 0 The frt term meaure the ncreae n repayment of thoe graduate who do n fact repay ther debt. The econd term yeld the advere electon effect of ncreae n the nteret rate on the average rk of borrower. The term n quare bracket potve. Hgher nteret rate caue the probablty of the margnal graduate who repay debt n full to declne f dp m /dr < 0. Maxmum proft occur at the nteret rate at whch (6) equal to zero. Propoton 1 (Stgltz and We) A credt ratonng equlbrum ext f dp m /dr < 0 for all, whch alway for hold rk neutral nvetor. Proof See De Meza and Webb (1987). However, we have jut een that dp m /dr < 0 doe not hold n general. The reaon that debt contract allow the graduate to take more rk becaue bank aborb the downde rk. Th potve nurance effect on rk-takng may actually be uch that probablty of ucce of the margnal graduate ncreae when bank ncreae nteret rate. Th create a potve electon effect and credt ratonng may ceae to be an equlbrum. In cae rk averon large, nteret rate reman uffcently low, or R f uffcently hgh, dp /dr > 0, and therefore alo for p = p m. 10

13 Propoton 2 (Abence of credt ratonng wth uffcent rk averon) A credt ratonng equlbrum cannot ext f dp /dr > 0 for all. Th may happen f there uffcently hgh rk averon. Proof Suppoe that the equlbrum nteret rate r', uch that ome graduate wth ocally proftable nvetment do not get credt,.e. for nvetor where p U((R R f )/p (1 + r')b R f )) > U((1 + )W). If th the cae, the bank can ncreae t ntal volume of outtandng loan f t make a mall ncreae n the nteret rate charged. If r' ncreaed, the probablty of ucce of the margnal graduate ncreae becaue dp /dr > 0; conequently we have from (6) that d/dr > 0. Th contradct the aumpton that r' the equlbrum rate becaue th aumpton mple that d/dr = 0. Therefore a credt ratonng equlbrum cannot ext f dp /dr > 0. The lat propoton contrat wth Stgltz and We (1981). The credt ratonng equlbrum apparently not robut to the ntroducton of rk avere nvetor. In the ntermedate cae, f rk averon mall, nteret rate are not uffcently low ((1 + r)b hgh) or ncome n the bad tate of nature are not uffcently hgh (R f low), and dp /dr wtche gn on the nterval [0,1], a credt ratonng equlbrum may tll occur. Graduate wth p < p or p > p * do not nvet. See Fgure 3. Clearly, ncreang the nteret rate ha both a potve and a negatve (advere) electon effect. Potve becaue the probablty of the graduate wth the lowet probablty of ucce p ncreae; and an advere electon effect, becaue the probablty of ucce of the graduate wth the hghet probablty of ucce p* decreae. Th can alo be een by dfferentatng the bank proft functon: (8) dπ = B dr p * [(1 + r) B p + R p p f ( p ) dp + [(1 + r) Bp f m d p (1 p)] f ( p). dr + R f * dp (1 pm)] f ( pm) dr The frt term the tandard term denotng ncreaed revenue on all outtandng loan, the econd term meaure the advere electon effect occurrng becaue the good rk drop out of the credt market (dp * /dr < 0), wherea the thrd term meaure the potve electon electon effect becaue ome bad rk graduate alo drop out of the market (dp/dr > 0). There can only be a credt ratonng equlbrum f the econd effect domnate the frt two effect. Th depend on the number of good rk graduate droppng out of the market relatve to the number of bad rk droppng out of the market. 11

14 Fgure 3 - Invetment decon wth debt fnancng. Effect of ncreang nteret rate (R f = 3, W =.6, B = 1, = 0) =.5, R EU(P) r = 1.5 EU((1+rho)W) EU(P) r = probablty of ucce Propoton 3 (Extence of a credt ratonng equlbrum wth uffcently low rk averon) A credt ratonng equlbrum ext f there ext a p below whch dp /dr > 0 for all p < p, and above whch dp /dr < 0 for all p > p Proof Suppoe that the equlbrum nteret rate r', uch that ome graduate wth ocally proftable nvetment do not get credt,.e. for nvetor where p U((R - R f )/p -((1 + r')b R f )) > U((1 + )W). A neceary condton for r' to be an equlbrum nteret rate that d/dr = 0. If bank charge hgher nteret rate n order to meet exce demand for credt they make loe. Therefore, f r ncreaed above the equlbrum rate, bank get more revenue from ucceful graduate, and ome low-rk graduate may drop out of the market nce dp/dr > 0. At the ame tme, there are alo relatvely fewer low-rk graduate that apply for credt nce p * fall. Therefore, d/dr < = > 0 depend on whether the revenue and potve electon effect are larger or maller than the advere electon effect. If at r' the advere electon effect equal the revenue and potve electon effect, d/dr = 0, and credt ratoned n equlbrum. 2.4 Effcency of the market equlbrum We can how that nvetment n human captal below t ocally optmal level, whether credt ratoned or not. Invetment n human captal ocally effcent when graduate wth expected gro return on nvetment n human captal larger than the cot of that nvetment ndeed nvet n human captal,.e.: f (9) p R + ( 1 p ) R (1 + ρ ) K,. Snce all project have the ame mean gro return, (8) hould hold wth equalty for all project. 12

15 Propoton 4 (Under-nvetment) Under-nvetment n human captal reult whether credt ratoned or not. The market equlbrum therefore alway ocally neffcent. Proof For every graduate we have U((1 + )W) EU( ) < U(E ),.e. p (R (1 + r)b) > (1 + )W. Summng over all graduate we have: p * p * (10) (1 + ρ ) W f ( p ) dp < pr f ( p ) dp (1 + r) B p f ( p ) dp. p p When we add p p* (1 p )R f f(p )dp to both de and ubttute the zero proft condton for bank, we arrve at: p R + (1 p )R f > (1 + )K and undernvetment n human captal reult. If graduate are rk neutral, the ame dervaton hold, except for the fact that we ntegrate over 0 pm rather than p p* nce p m = p * and p = 0 for rk neutral nvetor. The nformaton problem reult n drop out of the good rk graduate, for whom nvetment n human captal are ocally effcent. Therefore, there under-nvetment n equlbrum a a reult of aymmetrc nformaton and due to rk averon of graduate. 3 Optmal educaton polcy Debt contract are obvouly unlkely to be the optmal type of contract n the model preented above. For example Cho (1986) and De Meza and Webb (1987) have hown that wth rk neutral nvetor equty fnancng optmal n the Stgltz and We model. We how below that equty fnancng of nvetment alo optmal when graduate are rk avere. The reaon not trval and mportantly dfferent from Cho (1986) and De Meza and Webb (1987). Prvate equty and nurance contract that effectvely requre collateralzaton of the return of human captal nvetment are mpoble to mplement for legal reaon. Although fnancer may oberve return to nvetment n human captal through, for example,, pay check, they cannot nclude the return of the nvetment n human captal n a fnancal contract, nce bank lack the poblty to enforce the contract n a non-lave tate. Therefore, only debt contract can be wrtten by prvate parte nce thee are ndependent from the return to human captal nvetment. Th lkely to be an mportant contrant for the fnancng of nvetment n human captal. The pont of th paper that the government can buy hare n graduate earnng propect becaue the government can ecure clam on human captal. The reaon that the government ha the power of compulon through the ncome tax, ee alo Stgltz (1989; 1994). By beng able to tax ncome, the government already ha an effectve clam on (part of the return on) all human captal tock n ocety. Therefore, t actually able to crcumvent the barrer to wrtng of equty and nurance contract that prvate parte face. 10 p * p 10 Further, the tax authorte can do o at relatvely low tranacton cot; admntratve cot were a low a 1% of the total loan repad n the ncome contngent loan ytem n Autrala, ee Chapman (1997, p. 747). 13

16 3.1 Equty fnancng of educaton Suppoe that the government buy hare n graduate human captal. It provde equty E = K - W to all graduate obtanng an educaton n exchange for a clam on the return from human captal. We aume that th clam executed through a proportonal tax t on the return of the nvetment n human captal,.e. a graduate tax. A graduate who fnance h/her educaton by ung an equty take to the government ha expected utlty: f (11) EU ( π ) = pu ((1 t) R ) + (1 p ) U ((1 t) R ). We aume for the moment that f the government fnance educaton, graduate do not ue debt at the ame tme. We etablh later that he/he wll ndeed never chooe to do o, makng th a vald aumpton. We frt etablh the optmalty of equty fnancng when graduate are rk neutral, cf. De Meza and Webb (1987). The ntuton that when bank offer equty contract, they attract the low-rk nvetor, becaue debt contract would attract the hgh-rk nvetor only. Propoton 5 (Optmal fnancng rk neutral nvetor) For rk neutral nvetor the optmal contract to fnance educaton an equty contract. Proof Aume that the government act a f t were a proft maxmzng bank offerng both debt and equty contract. Aume that the margnal graduate ndfferent between debt and equty fnancng of educaton. Let h/her probablty of ucce be denoted p o. We derve the expected return for the ndfferent graduate wth p = p o : f (12) ( 1 t)( p R + (1 p ) R = p ( R (1 r) B). o o o o o + Ung (1) yeld p o (tr R f )/((1 + r)b R f ), o that p o /t > 0, p o /r > 0. Due to the contancy of expected return, the left-hand-de of (11) ndependent from p. However, the rght-hand-de decreang n p. The government know therefore that graduate obtanng debt fnance have lower probablte of ucce than the graduate who ndfferent,.e. p < p o. Smlarly, graduate obtanng equty fnance have p > p o. Expected proft from equty contract are alway hgher than the proft on the ndfferent nvetor wherea expected proft from debt contract are alway lower than on the debt contract offered to the ndfferent nvetor. Therefore, an equlbrum wth both debt and equty contract mpoble. For the ndfferent graduate, expected proft on equty contract are equal to expected proft on debt contract: (13) o f f o + (1 po) R = po(1 + r) B + (1 po R, t ( p R ) where the left-hand-de gve expected proft from equty and the rght-handde gve expected proft from debt contract. Although expected proft on equty fnanced nvetment n human captal for the ndfferent margnal graduate are equal to expected proft on debt fnanced nvetment, all debt fnanced nvetment n human captal taken together are wore than the equty 14

17 fnanced nvetment, at leat from the government pont of vew. Th o becaue the frt type of contract only attract the wort rk and wll conequently make loe. Therefore, n the optmal oluton all nvetment wll be equty fnanced. Wth rk avere graduate and dp o /dr > 0 one mght expect exactly the oppote outcome. Ung mlar argument a thoe above, one could argue that the equlbrum type of contract a debt contract due to the potve electon effect of offerng debt on the margn to the ndfferent graduate. Th however not the cae. The reaon that there no graduate ndfferent between debt and equty fnancng; all graduate prefer equty. In order to llutrate whether graduate would prefer debt or equty to fnance ther nvetment n human captal, we plotted expected utlty of educaton fnanced wth both equty and debt n Fgure 4. We ubttuted the government budget contrant for the tax rate and et the nteret rate charged to graduate equal to the afe rate. (If no default rk exted, graduate would obtan fund at rate.) We ee that equty fnancng of educaton alway yeld hgher expected utlty than debt fnancng. Th can be etablhed formally by notng that default rk ncreae the nteret rate above the rk free rate,.e. r >. Expected utlty for debt fnancng then alway le below expected utlty wth equty fnancng. The ntuton that debt fnancng mple an nurance premum that not actuarally far, nce debt fnancng reult n redtrbuton from low-rk to hgh-rk graduate. Equty fnancng of educaton doe not have th property nce every graduate ha the ame expected return and no dtortonary redtrbuton occur. Fgure 4 - Invetment decon wth debt or equty fnancng (R f =.5, R = 3, W =.6, = 0) EU(P) - debt r > rho =0 EU(P) - equty t = (1+rho)E/W EU(P) - equty t = 1 EU(P) - debt r = rho = probablty of ucce Moreover, the varance of ncome wth debt fnanced contract ( d (p (1 p )) 1/2 (R (1 + r)b) decreae fater wth the probablty of ucce compared to the varance of equty fnanced project ( e (p (1 p )) 1/2 (1 t)(r R f )): 15

18 d e d σ / σ dp ((1 + r) B R = ((1 t)( R R ) )) < 2 dp ( ) (14) 0 f f dr dr /dp < 0 follow from (1). Snce there no uncertanty at the afet nvetment ( d p=1 = e p=1 = 0), we can mmedately conclude that at p < 1 the varance of debt fnanced nvetment hgher than for equty fnanced nvetment, becaue (13) < 0. Conequently, the cae n whch mean ncome are contant,.e. when there no dtortonary redtrbuton, the requred rk premumon nvetment lower wth equty fnancng compared to debt fnancng becaue the varance of ncome hgher n the latter cae. Snce EU equty EU debt for all, t ealy etablhed that t optmal to fnance all nvetment n human captal wth equty when graduate are rk avere. Propoton 6 (Optmal educaton fnance) In the optmal oluton, all nvetment n human captal are fnanced wth an equty take of the government. Proof For the cae that dp /dr > 0 for all, the proof mple. Equty fnanced nvetment have lower varance compared to debt fnanced nvetment, but the latter feature lmted lablty. However, nce the negatve rk-averon effect alway domnate the potve lmted lablty effect on nvetment f dp /dr > 0 for all, and thereby on expected utlty, equty fnanced project are therefore preferred over debt fnanced project. In cae dp /dr wtche gn, the potve lmted lablty effect domnate the negatve rk averon effect for hgh p and vce vera for low p. Let p' denote the probablty of ucce of the graduate for whom dp /dr = 0. We know from propoton 5 (rk neutral nvetor) that expected utlty for low-rk graduate wth p > p' (and dp /dr < 0) hgher f they chooe equty contract that do not redtrbute ncome to hgh-rk graduate rather than debt fnanced contract. Wth dp /dr < 0 for all equty contract are preferred. The corollary of propoton 5 to rk avere agent traghtforward and all agent wth p > p' prefer equty. From the frt part n th propoton we know that expected utlty for hgh-rk graduate wth p < p' (and dp /dr > 0) hgher f they chooe equty contract becaue the lmted lablty effect domnated by the nurance effect, nce wth dp /dr > 0 for all equty contract are preferred. Conequently, all graduate, both wth p > p' and p < p', prefer equty fnancng over debt fnancng. 3.2 Effcency of equty fnancng Even wth equty fnancng, there reman under-nvetment n human captal nce graduate are rk avere. Although both debt and equty contract alo provde ncome nurance, not all ncome rk are elmnated. Therefore, the margnal graduate requre a rk premum on the nvetment n human captal a a compenaton for rk. Propoton 7 (Under-nvetment wth equty fnancng) Socal effcency n nvetment n human captal not acheved even wth equty fnancng. 16

19 Proof The graduate ha expected utlty from equty fnanced nvetment n human captal at leat equal to utlty from non-human nvetment: EU( ) = p U((1 t)r ) + (1 - p )U((1 t)r f ) U((1 + )W). We can derve that U(p (1 t)r + (1 p )(1 t)r f ) > p U((1 t)r ) + (1 p )U((1 t)r f ) U((1 + )W), from Jenen nequalty. Conequently, p (1 t)r + (1 p )(1 t)r f > (1 + )W. Combne th wth the government budget contrant, o that R > (1 + )K, whch exactly the condton for ocal ub-optmalty of nvetment n human captal,.e. the ocal return on nvetment n human captal are larger than ocal cot. Agan, th contrat wth De Meza and Webb (1987). Wth rk neutral frm, they fnd that equty fnancng yeld the ocally optmal level of nvetment. 3.3 Optmal nurance Gven that there under-nvetment n human captal due to rk averon, ome ncome nurance optmal. Eaton and Roen (1980), Varan (1980), and Snn (1995), amongt other have hown, that a redtrbutve ncome tax optmal f nurance market are abent even f th ncome tax dtort ncentve to nvet n human captal (or to upply labor). In the current et-up the government may provde graduate wth ncome nurance, by ncreang the graduate tax and rebatng the revenue through a hgher level of equty (E). Th enhance the ocal effcency of nvetment n human captal. Not only the captal market mperfecton olved by uffcently hgh level of E, but ncome rk are nured a well. Propoton 8 (Optmal ncome nurance) Optmal government nterventon requre that all ncome uncertanty elmnated and the government take a 100% equty take on the return to human captal,.e. t = 1 and E = R. Proof The government budget contrant can be wrtten down a: tr = (1 + )E. The proportonal ncome tax redtrbute ncome from ucceful tate of nature to unucceful tate. If t = 1 and E = R, there full nurance and no ncome uncertanty. It ealy verfed that n th cae ocal effcency reult,.e. p R + (1 p ) R f = (1 + )K for all. If t < 1 (and E < R) t can be hown that p R + (1 p ) R f > (1 + )K for all, o that only t = 1 yeld ocal optmalty. An equty partcpaton model combned wth ncome nurance olve the undernvetment problem aocated wth mperfect captal and nurance market. Therefore, the author mentoned n the ntroducton were generally correct n ther plea for an ncome contngent loan ytem or a graduate tax. In the current etup, we do not allow for moral hazard effect or other (tax) dtorton, o that full nurance can be acheved. In a more general etup, the preence of moral hazard would reduce the optmal nurance cover. There a tradeoff between ncentve and rk harng, ee alo Eaton and Roen (1980), Varan (1980), and Snn (1995). 17

20 4 Sub-optmal educaton polcy Drect ubde to educaton are far more wdely ued to guarantee acce to hgher educaton than equty partcpaton cheme. Such educaton ubde are often defended by pontng out the extence of captal and nurance market falure that hamper acceblty of educaton epecally for tudent wth poor background. In th ecton we analyze whether ubde can retore ocal effcency n the accumulaton of human captal n the abence of equty and nurance. Let r denote the ubdy on the nteret cot charged by bank, o that graduate only pay (1 + (1 r )r)b. In Fgure 5 we plotted the effect of an nteret rate ubdy on expected utlty for the graduate. Interet ubde hft the expected utlty lne counter-clockwe upward. Th upward hft large for hgh p graduate and mall (approachng zero) for low p graduate. A a conequence of the educaton ubdy, the probablty of ucce correpondng to the margnal graduate who nvet n educaton lowered. The reaon that nteret cot aocated wth nvetment n human captal fall, o that nvetment n human captal become more proftable. Fgure 5 - Invetment decon wth debt fnancng and educaton ubde (R f =.5, R = 3, W =.6, = 0, r =.5) EU(P) - no ubdy U((1+rho)W) EU(P) - nteret ubdy 300% EU(P) - drect ubdy 300% probablty of ucce Educaton ubde hould be ncreaed to nfnty n order to nduce the graduate wth the lowet probablty of ucce to nvet n human captal whoe expected utlty goe to zero whatever the level of ncome n the ucceful tate due to our aumpton of contant mean return to educaton. 11 Therefore, ocal effcency where all graduate nvet n educaton can only be attaned f educaton ubde are nfnte. Although one may regard th reult a pecal cae, the ntuton neverthele traghtforward. Educaton ubde are an mperfect ubttute for 11 Th can be etablhed by takng the lmt of expected utlty under debt fnancng a p 0: lm p0 U(/p )/g(p ), where R R f, (1 + r)b R f, and g(p ) 1/p. Snce the lmt ndetermnate we apply L Hôptal rule to fnd: lm p0 = U'(/p ) = 0, nce lm p0 U'(.) = 0. 18

21 lackng ncome nurance and the more o for ndvdual wth extremely rky ncome. Conequently, when rk n ncome become unbounded (by aumpton), educaton ubde hould become nfntely large to nduce thee ndvdual to nvet n human captal a well. A mlar tory hold for ubde on the drect cot of educaton. If the government ubdze the cot of educaton K at rate k, o that graduate only face (1 k )K a cot, expected utlty for all graduate ncreae and the probablty of ucce of the margnal graduate fall, ee alo Fgure 5. Clearly, educaton ubde are a very ndrect mean of retorng effcency n nvetment n human captal compared to equty partcpaton cheme. Th not a pecal feature of our model, but a more general property of equty partcpaton cheme. The fundamental reaon that educaton ubde are not very effectve n reducng ncome rk. Th can be hown graphcally a well. An equty partcpaton cheme rotate the expected utlty lne clockwe, wherea educaton ubde rotate expected utlty counter-clockwe, cf. Fgure 4 and 5. An equty partcpaton cheme therefore by defnton more uted to nduce the hgh-rk graduate to nvet than educaton ubde. Generally, very large ubde are cotly. For example, fnancng ubde from tax revenue wll generally caue dead weght loe. Due to the dtortonary cot of revenue rang, the ue of educaton ubde retrcted. Conequently, f very large level of educaton ubde are ruled out for practcal purpoe, t mpoble to acheve the ocally derable level of nvetment n human captal wth educaton ubde f graduate are rk avere. Propoton 9 (Ineffcent educaton polcy) Educaton ubde are at mot a econdbet mean to reolve undernvetment due to mperfect captal and nurance market. The lat propoton contrat wth Mankw (1986) and De Meza and Webb (1987). Thee author argued that ocal effcency n nvetment can n general be retored n mperfect captal market wth debt contract by lettng the government employ fnte (nteret) ubde on educaton loan. We have hown that educaton ubde are a very ndrect mean to retore effcency n nvetment n human captal and hould optmally be non-trvally large, whch we rule out for practcal reaon. 5 Equty partcpaton n practce Suppoe that we ntroduced a graduate tax (GT) for the fnancng of educaton. What would be the level of the tax and the degree of ncome nurance? In th ecton we preent calculaton on the ntroducton of a GT n the Netherland. Computaton are explaned n more detal n a background workng paper (Jacob, 2002). Alo, a complete decrpton of data-ource can be found there. We aume a ubdy fnanced from general government revenue reman. For varou ubdy level we make alternatve calculaton,.e. when = 2,119 euro, = 4,237 euro, and = 6,355 euro. 12 Thee ubdy level correpond to ncreang prvate contrbuton of repectvely 75%, 50%, and 25% of current drect 12 Externalte, publc good, and fcal dtorton n the accumulaton of human captal may alo jutfy at leat ome ubdy element. 19

22 expendture. 13 In the current Dutch ytem, the prvate contrbuton on the part of tudent only 12% of the total outlay on educaton. Under a GT all tudent are treated equally: they all have to pay a contant percentage extra ncome tax durng ther whole workng lve. Acceblty guaranteed becaue all graduate can obtan fund rrepectve of oco-economc charactertc. We bae our computaton on etmated age-earnng profle of graduate wth fxed effect for every educaton type. We do o eparately for men, women, hgher vocatonal, and unverty tranng. Etmated coeffcent are uch that famlar age-earnng profle follow. Under the aumpton that the redual from our etmated wage profle are log-normally dtrbuted we generate ncome clae. We ue 5 quntle wthn each gender educaton type ubject cell, o that each quntle contan 20% of the tudent wthn each cell. We correct the etmated lfe-tme earnng for labor upply effect. We make adjutment for hour worked and (growth n) partcpaton rate. We aume that each graduate enter the labor market drectly after graduaton and reman n the labor market untl 65 year of age. Real wage growth over the lfe cycle (g) aumed to be 2% per annum, n lne wth Dutch htorcal data. There ubtantal heterogenety n earnng between varou educaton type, ubject, men and women, and wthn thee group. Economc, law and techncal educaton have hgher earnng for both men and women. Then follow agrcultural, cence, health and behavoral ubject. Cultural, language and art tude are at the bottom end. Men have teeper earnng profle than women. Men alo work more hour and have hgher partcpaton rate. Profle of worker wth unverty educaton are teeper and le concave than profle for worker wth hgher vocatonal educaton. Educatonal cot are the nttutonal cot excludng the cot of academc hoptal, centfc reearch at unverte, and art at hgher vocatonal chool art academe, conervatore, actng chool, etc. We aume that tudent are enrolled 5 year n hgher educaton rather than the nomnal length of 4 year. Th correpond to realty. All (ncome dependent) grant that are currently gven to tudent wll be replaced by an equty take ued by tudent to the government. The government fnance th through the uance of government bond. We et the graduate tax rate uch that the cheme become elf-fnancng,.e. n expected dcounted value term, tudent wll jut pay out enough dvdend o a to fnance the government debt ncludng nteret. Note that th wll alo ental redtrbuton from hgh ncome earner to low ncome earner. We take nto account the potentally dtortonary effect of GT on lfe-tme labor upply. We aume that the effectve margnal rate wthout the graduate tax repayment equal 50%. A bae-lne value we take for men an uncompenated wage elatcty of lfe-tme labor upply of.1 and for women of.5. We do not conder the conequence of rebatng the avng on government outlay through e.g. lower taxe. Nether do we take nto account ncome effect that may boot labor upply of graduate, nce they wll ee ther ncome reduced becaue educaton ubde are reduced. Our computed revenue loe are therefore a very conervatve upper bound. 13 Our defnton of total expendture on educaton gnore the opportunty cot of educaton. 20