Improvemen in Informaion, Income Inequaliy, and Growh by Bernhard Eckwer Deparmen of Economics Universiy of Bielefeld Germany Izhak Zilcha The Eian Berglas School of Economics Tel Aviv Universiy Israel Absrac We analyze he imporance of informaion abou individual skills for undersanding economic growh and income inequaliy. The paper uses he framework of an OLG economy wih endogenous invesmen in human capial. Agens in each generaion differ by random individual abiliy, or alen, which realizes in he second period of life. The human capial of an agen depends on boh his alen and his invesmen in educaion. The invesmen decision is based on a public signal es oucome) which screens all agens for heir alens. We analyze how a beer informaion sysem, which allows more efficien screening, affecs he co-movemens of indicaors for income inequaliy and human capial accumulaion. Keywords: Informaion, human capial, inequaliy. JEL classificaion numbers: D80, J24, J30. Mailing address: Prof. Izhak Zilcha The Eian Berglas School of Economics Tel Aviv Universiy Israel
1 Inroducion In recen decades we have winessed a growing body of research on he role of informaion in economic analysis. In paricular, he welfare implicaions of informaion have been sudied exensively. These sudies have revealed he ambiguous naure of informaion wih regard o economic welfare when markes for risk sharing are operaive Hirshleifer 1971), Green 1981), Campbell 2001), Schlee 2001), Eckwer and Zilcha 2003)). Surprisingly, he quesion how beer informaion affecs economic growh and income inequaliy has received hardly any aenion, even hough his quesion is no unrelaed o he welfare problem. Typically, he effecs of informaion in general equilibrium models depend on he scope of risk sharing opporuniies Hirshleifer 1971,1975), Orosel 1996), Schlee 2001), Eckwer and Zilcha 2001)). In his paper we consider an economy where no explici risk sharing arrangemens are operaive. Neverheless, due o imperfec informaion, in equilibrium some individual risks will be shared across agens: he marke reas all agens, who are equal on he basis of he available informaion, idenically since heir rue characerisics are currenly unknown. Thus, even wih risk sharing markes absen, beer informaion goes hand in hand wih less risk sharing. Therefore, informaion has imporan effecs on he allocaion of risks and, hence, on invesmen in human capial formaion. In addiion, since beer informaion provides more reliable idenificaion of individual characerisics, i also affecs in a very naural way he inequaliy of income disribuion. This se up provides a heoreical plaform where he co-movemens of economic growh and income inequaliy can be analyzed. Our analyical framework is an OLG economy in which privae invesmen in educaion say, non-compulsory schooling), while young, affecs an agen s human capial and, hence, his lifeime earnings. Individual s human capial depends also on innae abiliy, or alen, which is sill unknown when he agen decides how much effor o inves in his/her educaion and raining. The invesmen decision is made afer observing a signal es oucome) which screens agens for heir abiliies. Each signal conains imperfec public informaion abou an agen s random alen. Since individual abiliies are no ye known agens differ only by he signals hey have 1
received. As a consequence, in our framework, all agens wih he same signal are grouped ogeher, and hey are paid a wage equal o he mean marginal produc of human capial in his group. The idea ha growh and income inequaliy are sysemaically relaed hrough equilibrium marke mechanisms has simulaed a considerable amoun of empirical and heoreical research. So far, hese sudies have produced mixed resuls. Based on empirical evidence, Persson and Tabellini 1994) concluded ha higher growh induces less income inequaliy. Oher papers e.g., Forbes 2000), Quah 2002)) found a posiive correlaion beween growh and inequaliy. In par, he inconclusiveness of he evidence is due o a lack of consensus wih regard o he main facors by which inequaliy and growh are deermined. Various sudies focus on differen facors and hereby produce conflicing resuls. Our paper also follows his roue by singling ou informaion abou individual skills as an explanaory facor: we idenify he effecs of such informaion on indicaors for economic growh and for income inequaliy; and we analyze he co-movemens of boh indicaors due o changes in informaion. We find ha income inequaliy always increases wih beer informaion. The effecs on economic growh depend on he properies of he individual invesmen decisions which, in urn, are deermined by he degree of ineremporal subsiuion in consumpion: if individual preferences exhibi high elasiciy of ineremporal subsiuion, agens wih more favorable signals will choose higher invesmen levels. Under his consellaion more efficien screening resuls in higher aggregae socks of human capial, hus enhances growh; consequenly, higher growh goes hand in hand wih more income inequaliy. By conras, growh and inequaliy are inversely relaed when he elasiciy of ineremporal subsiuion is small. The paper is organized as follows. In secion 2 we describe he OLG economy and define our concep of informaiveness. In secion 3 we sudy he informaioninduced link beween inequaliy and growh. All proofs are relegaed o a separae Appendix. 2
2 The Model Consider an overlapping generaions economy wih a single commodiy and a coninuum of individuals in each generaion bu no populaion growh). The commodiy can be eiher consumed or used as an inpu physical capial) in a producion process. Individuals live for hree periods: youh where hey obain educaion while sill suppored by parens), middle-age where hey work and consume, and reiremen where hey only consume. We denoe generaion by G, = 0, 1,.G consiss of all individuals born a dae 1. One of he main feaures of our economy is he heerogeneiy of individuals wih regard o heir human capial generaed by a random innae abiliy. When individual i is born his abiliy is ye unknown. The uncerainy abou he agen s abiliy is described by some random variable Ãi which realizes a he beginning of he nex period and akes values in some inerval A R +. We assume ha he random variables Ãi, i G, = 0, 1,, are i.i.d; hus, in paricular, he ex ane disribuion of abiliy is he same for all agens and does no depend on ime or on he hisory of he economy. Human capial of individual i G depends on abiliy Ãi which is random), effor e i R + invesed in educaion by his individual, and he environmen, represened here by he average human capial of agens in he previous generaion who are currenly acive economically). Thus we wrie, h i = ϕãi )gh 1, e i ) 1) where i belongs o generaion, and H 1 is he average human capial of G 1, see he role of H 1 in generaing human capial of G, for example, in Lucas 1988), Azariadis and Drazen 1990)). Assumpion 1 The funcion gh, e) is sricly increasing and g 12 0, g 22 < 0. ϕ : A R + is increasing and differeniable. A priori he disribuion of random abiliy Ãi is he same for all agens i boh wihin he same generaion and across generaions. 1 However, before choosing opimal 1 In he sequel we will herefore suppress he index i and wrie à insead of Ãi. Noe, however, 3
effor in he youh period each individual receives a signal es oucome) which conains public informaion abou his own abiliy. We model he informaional srucure of he economy as follows: le ỹ be a real-valued random variable which akes values in Y R and is correlaed o Ã. Each agen i G, = 0, 1,, wih abiliy A observes an individual signal y i which is drawn randomly from he disribuion of he random variable ỹ A). 2 By consrucion, his individual signal is correlaed o i s abiliy. Therefore, when agen i makes his decision abou how much effor e i o inves in educaion, he relevan c.d.f. for random abiliy is he poserior disribuion of à given he individual signal yi. For convenience we normalize he measure of agens in each generaion o 1: νa) da = 1, A where νa) is he Lebesgue)-densiiy of agens wih abiliy A. Denoe by f A) he densiy of he random variable ỹ A), and by ν y ) he densiy of he random variable à y). Using his noaion, he disribuion of signals received by agens in he same generaion has he densiy 3 µy) = fy A)νA) da. 2) A And average abiliy of all agens who have received he signal y is ϕy) := E[ϕÃ) y] = ϕa)ν y A) da. 3) The agens are expeced uiliy maximizers wih von-neumann Morgensern lifeime uiliy funcion A Ue, c 1, c 2 ) = ve) + u 1 c 1 ) + u 2 c 2 ). 4) Individuals derive negaive uiliy from effor while hey are young and posiive uiliy from consumpion in he working period, c 1, and from consumpion in he reiremen period, c 2. ha in general he random variables Ãi and Ãj differ for i j; only heir disribuions are he same. 2 Throughou he paper we shall refer o he realizaions of ỹ as signals, and o he realizaions of he ỹ i s as individual signals. 3 Noe ha, by he law of large numbers, µ does no depend on. 4
Assumpion 2 The uiliy funcions v and u j, j = 1, 2, have he following properies: i) v : R + R is decreasing and sricly concave, ii) u j : R + R is increasing and sricly concave, j = 1, 2. In each period, producion in our economy, is carried ou by compeiive firms who use wo producion facors: physical capial K and human capial H. The process is described by an aggregae producion funcion F K, H), which exhibis consan reurns o scale. If individual i supplies l i unis of labor in his working period, his supply of human capial equals l i h i. We assume inelasic labor supply, i.e., l i is a consan and i is equal o 1 for all i. Assumpion 3 F K, H) is concave, homogeneous of degree 1, and saisfies F K > 0, F H > 0, F KK < 0, F HH < 0. We assume hroughou his paper full inernaional capial mobiliy, while human capial is assumed o be immobile. Thus he ineres rae r is exogenously given a each dae. This implies ha marginal produciviy of aggregae physical capial K mus be equal o 1 + r assuming full depreciaion of capial in each period). On he oher hand, given he aggregae sock of human capial a dae, H, he sock K mus adjus such ha 1 + r = F K K, H ) = 1, 2, 3, 5) holds. Bu his implies, by Assumpion 3, ha K H is deermined by he inernaional rae of ineres r. Hence he wage rae w price of one uni of human capial), given in equilibrium by he marginal produc of aggregae human capial, is also deermined once r is given. Thus we may wrie K ) w = F L, 1 =: ζ r ) = 1, 2, 3,. 6) H Labor conracs are concluded afer agens have learned heir signals bu before heir abiliies become known. 5
Obviously, he wage income specified in a labor conrac canno be made coningen on individual human capial because individual abiliy is ye unknown. Therefore agens are unable o appropriae he full marginal produc of heir human capial. Insead, individuals are grouped according o he signals hey have received. And, in he absence of any furher informaion, he marke reas all agens in he same group idenically. Under hese circumsances each individual will receive a wage equal o he mean marginal produc of human capial of hose wih whom he is grouped. Wihin he group of all agens who have received he signal y, average abiliy is given by ϕy) in equaion 3). Therefore, he wage income of agen i G wih signal y is w hi, where h i = ϕy)gh 1, e i ). 7) In equilibrium, all agens wih he same signal y choose he same effor level. As a consequence, aggregae wage income and aggregae human capial in his group are given by µy)w hi ) and µy) h i, respecively. The firm herefore pays he compeiive wage in 6), µy)w hi )/µy) h i = w, for each uni of aggregae human capial supplied by agens wih signal y. Now le us consider he opimizaion problem ha each i G faces, given r, w, and H 1. A dae 1, when young, his individual chooses an opimal level of savings, s i, and an opimal level of effor employed in obaining educaion. These decisions are made under random abiliy Ã, bu afer he individual signal y i has been observed. For given levels of H 1, w and r, he opimal saving and effor decisions of individual i G are deermined by max E[ve i ) + u 1 c i 1) + u 2 c i 2) y i ] 8) s i,e i s.. c i 1 = w hi s i c i 2 = 1 + r )s i. Since income is deermined by average abiliy, given he signal y i, saving s i based on average human capial h i and no on h i ); as a consequence, period 2 consumpion c i 2 is non-random when e i is chosen. 6 is
The necessary and sufficien firs order condiions are where h i is given by equaion 7). u 1w hi s i ) + 1 + r )u 21 + r )s i ) = 0 9) v e i ) + w g 2 H 1, e i ) ϕy)u 1w hi s i ) = 0, 10) Observe ha he signal y eners he firs order condiions only via he erm ϕy). Thus we may express he opimal decisions as funcions of ϕy) raher han as funcions of he signal iself, i.e., s i = s ϕy) ), e i = e ϕy) ). Similarly, hi = h ϕy) ). as Using 2) and 3) he aggregae sock of human capial a dae can be expressed H = E y [ h ϕy) ) ] = h ϕy) ) µy)dy, 11) where h ϕy) ) := ϕy)g H 1, e ϕy) )) Y 12) is he average human capial of agens in G who have received he signal y. Definiion 1 Given he inernaional ineres raes r ) and he iniial sock of human capial H 0, a compeiive equilibrium consiss of a sequence {e i, s i ) i G } =1, and a sequence of wages w ) =1, such ha: i) A each dae, given r, H 1, and w, he opimum for each i G in problems 9) and 7) is given by e i, s i ). ii) The aggregae socks of human capial, H, = 1, 2,, saisfy 11). iii) Wage raes w, = 1, 2,, are deermined by 6). 2.1 Informaion Sysems The abiliy of each individual i is a random variable Ãi. We assume ha he random variables Ãi are i.i.d. across individuals in G, = 0, 1, 2..., and ha hey all have he same disribuion as Ã. We shall refer o he realizaions of à as he saes 7
of naure. Before a young agen wih abiliy A chooses an opimal effor level he observes an individual signal which is drawn randomly from he disribuion of he random variable ỹ Ãi = A) = ỹ à = A) =: ỹ A). Thus, ex ane he condiional disribuions of he individual signals are idenical. For convenience, we shall refer o he realizaions of ỹ simply as signals. An informaion sysem, which will be represened by f : Y A IR + hroughou he paper, specifies for each sae of naure A a condiional probabiliy funcion over he se of signals. The posiive real number fy A) defines he condiional probabiliy densiy) ha if he sae of naure is A, hen he signal y will be sen. F y A) denoes he c.d.f. for he densiy fy A). We assume hroughou he paper ha he densiies {f A), A A} have he sric monoone likelihood raio propery MLRP): y > y implies ha for any given nondegenerae) prior disribuion for Ã, he poserior disribuion condiional on y dominaes he poserior disribuion condiional on y in he firs-order sochasic dominance. This implies ha higher signal is good news see Milgrom 1981)). As a consequence, A ϑa)ν y A) da > A ϑa)ν ya) da holds for any sricly increasing funcion ϑ. By he law of large numbers, he prior disribuion over A coincides wih he ex pos disribuion of abiliy across agens. Also he prior disribuion over Y coincides wih he ex pos disribuion of individual signals across agens and, hence, is given by equaion 2). Finally, he densiy funcion for he updaed poserior disribuion over A is ν y A) = fy A)νA)/µy). 13) Nex we define our crierion of informaiveness. Le GA y) be he c.d.f. for he condiional densiy νa y). Remark 1: GA y) is a decreasing funcion of y. This follows from MLRP. Choose  A arbirarily bu fixed and define { UA) = 0 ; A  1 ; A > Â. Since EUà y) = 1 G y) is increasing in y by virue of MLRP, GA y) is decreasing in y for all A A. 8
Consider he ransformaion π := F ỹ, where F is he c.d.f. for he probabiliy densiy µ defined in 2) under informaion sysem f, µy) = fy A)νA) da. A Thus for any y Y, he ransformed signal π = F y) represens he probabiliy ha under he informaion sysem f an agen receives a signal less han y. Obviously, π is uniformly disribued over [0, 1], i.e., he disribuion of he ransformed signal across agens does no depend on he informaion sysem f. We will exploi his fac laer when we define our concep of income inequaliy. An informaion sysem will be regarded as more informaive if he observable signal realizaions have a uniformly sronger impac on he poserior disribuion of saes: Definiion 2 informaiveness) Le f and ˆf be wo informaion sysems wih corresponding c.d.f s ḠA y), ĜA y) for he densiies νa y), ˆνA y). f is more informaive han ˆf expressed by f inf ˆf), if Ḡ π A F 1 π) ) Ĝπ A ˆF 1 π) ) 14) holds for all A A and π 0, 1). According o Remark 1, G A F 1 π) ) = prob à A F 1 π) ) is decreasing in he ransformed) signal π. Inequaliy 14) says ha under a more informaive sysem he poserior disribuion over saes is more sensiive wih respec o changes in he signal. In he economics lieraure various conceps of informaiveness have been used, daing back o he seminal work by Blackwell 1951,1953) where he ordering of informaion has been linked o a saisical sufficiency crierion for signals. More recenly, conceps have been developed which represen informaiveness as a sochasic dominance order over poserior disribuions Kim 1995), Ahey and Levin 1988), Demougin and Flue 2001)). Some of hese parial orderings conain he Blackwell ordering as a subse. 4 Our concep of informaion in 14) also imposes a resricion 4 E.g. Kim s crierion can be shown o be sricly weaker han Blackwell s crierion. 9
on he sensiiviies of he poserior sae disribuions. I has an advanage in erms of racabiliy over he above menioned crieria as i involves only signal derivaives of he poseriors raher han more complex measures of sochasic dominance. 3 Income Inequaliy and Growh: The Role of Informaion Our analysis of income inequaliy focuses on he disribuion of labor income wihin a given generaion G. Labor income depends boh on he informaion sysem and on he ransformed) signal received by an agen, I f π) = w ϕ f π)g H 1, e ϕ f π) )), 15) where ϕ f π) := E f[ ϕã) F 1 π) ]. 16) To sudy he impac of informaion on income inequaliy we use a concep which is based on he following comparison of disribuions: Definiion 3 Le Y and X be real-valued random variables wih zero-mean normalizaions ˇY = Y EY and ˇX = X EX. The disribuion of Y is more unequal han he disribuion of X if ˇY differs from ˇX by a MPS. This definiion of inequaliy differs from he requiremen ha one Lorenz curve is sricly above he oher one, which is equivalen o second degree sochasic dominance, SDSD, see, Akinson 1970)). Insead, our definiion is based on a locaion-free concep of dispersion. The induced ordering is implied by he Bickel- Lehman sochasic ordering see, Landsberger and Meilijson 1994)) which is a concep commonly used in saisics. 5 5 Denoe by F and G he c.d.f s of of X and Y, respecively. The disribuion F is less dispersed han G in he Bickel-Lehmann sense, if for any 0 < α < β < 1, F 1 β) F 1 α) G 1 β) G 1 α). Namely, he inerval beween he α-quanile and he β-quanile of F is less han or equal o ha for G. This implies ha G 1 θ) F 1 θ) is a non-decreasing funcion on 0, 1). I is easy o verify ha for each consan k, F θ k) and Gθ) cross a mos once, and if hey cross hen 10
The following lemma faciliaes he applicaion of our inequaliy concep in Definiion 3: Lemma 1 Le π be a random variable which is disribued over [0, 1]. Le y : [0, 1] R, x : [0, 1] R be coninuous increasing funcions such ha i) ỹ := y π differs from x := x π by a MPS, ii) yπ) xπ) is monoone in π. For any coninuous sricly increasing funcion ϑ : R R he disribuion of ϑ ỹ is more unequal han he disribuion of ϑ x. Our concep of income inequaliy is based on he dominance crierion for normalized disribuions in Definiion 3. Definiion 4 Le f and ˆf be wo informaion sysems. Income inequaliy under f is higher han under ˆf, if he disribuion of I f π) is more unequal in he sense of Definiion 3) han he disribuion of I ˆf π) for all 1. Under a beer informaion sysem individual abiliy can be assessed more accuraely a he ime when labor conracs are concluded. We may conjecure, herefore, ha firsly he income disribuion will be more discriminaing wih respec o differences in abiliies, and secondly ha i will be beer in line wih he disribuion of human capial across agens. The following proposiion confirms our firs conjecure, i.e., he informaional mechanism resuls in higher income inequaliy. Proposiion 1 Le f and ˆf be wo informaion sysems such ha f inf inequaliy is higher under f han under ˆf. ˆf. Income Nex we look ino he effecs of beer informaion on he aggregae sock of human capial and, hence, on economic growh. Aggregae human capial of generaion is H f = 1 h ϕ f π) ) dπ, 17) 0 F θ k) lies below Gθ) o he lef of he crossing poin see, Lansberger and Meilijson 1994)). If < x dgx) x df x) < holds in addiion o he above inequaliy) hen F dominaes G in he sense of SDSD. 11
where h ϕ f π) ) := ϕ f π)g H 1, e ϕ f π) )). 18) Since h 0) = 0, he funcion h ) is convex concave) in ϕ f decreasing) in ϕ f. if e ) is increasing Depending on he well-known ineracion beween an income effec and a subsiuion effec, effor e ) may be increasing or decreasing in ϕ f. If he elasiciy of ineremporal subsiuion is sufficienly small, he income effec will be dominan and, hence, a beer signal resuls in lower effor. By conras, he subsiuion effec will be dominan, if preferences exhibi sufficienly high elasiciy of ineremporal subsiuion. In ha case agens sep up heir effors when hey receive more favorable signals. In Secion 3.1 we will characerize he monooniciy properies of he effor decision for he special case of consan elasiciy of subsiuion. The expeced marginal produc of invesmen in educaion, ϕ f π)g 2 H 1, e ), is higher for agens wih beer signals. Thus, from he viewpoin of economic growh an increasing effor funcion would be more efficien i.e., more conducive o he growh of he human capial sock) han a decreasing effor funcion. We shall herefore call individual behavior efficiency-inducing if good news higher signal) induces higher invesmen in educaion; and individual invesmen behavior will be called inefficiency-inducing if good news resuls in lower effor. 6 Wih his erminology, he funcion h ) in 12) is convex under efficiencyinducing invesmen behavior, and concave under inefficiency-inducing invesmen behavior. Proposiion 2 Le f and ˆf be wo informaion sysems such ha f inf ˆf. Consider he compeiive equilibrium for a given iniial H 0. i) Under efficiency-inducing behavior beer informaion weakly) enhances growh, i.e., H f H ˆf for all 1. 6 As an example, consider he case where he uiliy funcions belong o he CRRA family wih parameer γ. Then, for γ < 1 we have efficiency enhancing behavior while for γ > 1 we have inefficiency enhancing behavior. 12
ii) Under inefficiency-inducing behavior beer informaion weakly) reduces growh, i.e., H f H ˆf for all 1. From proposiions 1 and 2 we obain, as a corollary, a characerizaion of he informaion-induced link beween growh in income inequaliy: Corollary 1 As he resul of an improvemen of he economy s informaion sysem, i) under efficiency-inducing invesmen behavior, higher growh goes hand in hand wih more income inequaliy; ii) under inefficiency-inducing invesmen behavior, higher growh goes hand in hand wih less income inequaliy. The characerizaion in Proposiion 2 can be inerpreed in erms of a simple economic mechanism. Consider par i), i.e., assume ha invesmen behavior is growh-efficien. The implemenaion of a beer informaion sysem enhances he reliabiliy of he individual signals. As a consequence, high signals become even beer news and induce higher invesmen in educaion. Similarly, under a beer informaion sysem he bad news conveyed by a low signal becomes even worse because now he news is more reliable). As a resul, invesmen in educaion declines. Thus, under growh-efficien invesmen behavior, beer informaion ends o increase he effors of agens wih high signals and decrease he effors of agens wih low signals. Since he expeced marginal produc of effor in erms of human capial) is higher for agens wih higher signals, aggregae human capial increases when he informaion sysem becomes more informaive. If invesmen behavior is growh-inefficien, he same mechanism resuls in lower aggregae human capial under a more informaive sysem. The relaionship beween economic growh and income inequaliy has been widely debaed in he lieraure in he las decade. Based on empirical evidence, Persson and Tabellini 1994) show ha higher growh resuls in less income inequaliy a finding ha was challenged by oher auhors, e.g., Forbes 2000) and Quah 2002). Our sudy conribues o his conroversy from a narrow, informaion-based perspecive: we idenify he effecs of informaion on indicaors for economic growh 13
and income inequaliy; and we analyze he co-movemens of boh indicaors due o changes in informaion. In his sense, growh and income inequaliy are posiively relaed if agens respond o beer signals wih higher invesmen in educaion. Ye, he model is also consisen wih an inverse relaionship beween growh and income inequaliy. Such a paern arises when beer signals induce agens o reduce invesmen in educaion. 3.1 An Example: CEIS Preferences To illusrae he criical role of he elasiciy of ineremporal subsiuion for he informaion-induced link beween income inequaliy and growh we resric he uiliy funcions u 1 ), u 2 ), and v ) o be in he family of CEIS Consan Elasiciy of Ineremporal Subsiuion) : u 1 c 1 ) = c1 γu 1 ; u 2 c 2 ) = β c1 γu 2 ; ve) = eγv+1 1 γ u 1 γ u γ v + 1. 19) γ u and γ v are sricly posiive consans. 1/γ u paramerizes he elasiciy of ineremporal subsiuion in consumpion. We also assume ha he funcion g in 1) has he form where ĝ is sricly increasing in H, and α 0, 1). gh, e) = ĝh)e α, 20) Using he funcional forms of u j, j = 1, 2, in 19), i follows from equaion 9) ha, given r and w, he saving s i is proporional o he human capial level h i. In oher words, for each here is a consan m such ha for all i G we have: s i = m h i, 0 < m < w, = 1, 2, 21) Seing s = F y), he specificaions in 19), 20) and 21) allow us o solve equaion 10) for he opimal effor level as a funcion of average abiliy ϕ f s): e ϕ f s) ) = δ ϕ f s) ) ρ1 γ u)/α 22) where [ ] αw ĝh 1 )) 1 γu ρ/α δ := ; ρ = w m ) γu α γ v + αγ u 1) + 1. 14
The income of an agen wih signal y = F 1 s) is I f s) = w δ α ĝh 1 ) ϕ f s) ) τ, 23) and aggregae human capial of generaion is H f = δ α ĝh 1 ) 1 0 E f[ ϕa) F 1 s) ]) τ ds, 24) where τ := 1 + ρ1 γ u ) = 1 + γ v γ v + αγ u + 1 α) > 0. 25) Corollary 2 Le f and ˆf be wo informaion sysems such ha f inf ha he specificaions in 19) and 20) are valid. ˆf, and assume i) High EIS: For 1/γ u 1 beer informaion weakly) enhances growh, i.e., H f H ˆf for all 1. ii) Moderae EIS: For 1/γ u 1 beer informaion weakly) reduces growh, i.e., H f H ˆf for all 1. Proof: Since e ϕ f ) in 22) is increasing for 1/γ u 1 and decreasing for 1/γ u 1 he claim is implied by Proposiion 2. From Proposiion 1 and Corollary 2 we obain he following characerizaion of he informaion-induced link beween growh in income inequaliy: Corollary 3 Assume ha he specificaions in 19) and 20) are valid. resul of an improvemen of he economy s informaion sysem, As he i) higher growh goes hand in hand wih more income inequaliy, if he elasiciy of ineremporal subsiuion in consumpion is high, i.e., 1/γ u 1, ii) lower growh goes hand in hand wih more income inequaliy, if he elasiciy of ineremporal subsiuion in consumpion is small, i.e., 1/γ u 1. 15
4 Conclusion The conjecure ha inequaliy in income disribuion is sysemaically relaed o economic growh is an idea which has riggered many debaes and conroversies in economics. Our paper analyzes he inequaliy-growh link from he narrow perspecive of informaion: wha are he join effecs on income inequaliy and economic growh if decisions wih respec o invesmen in educaion are based on more reliable informaion abou he agens abiliies)? I urns ou ha here is no unambiguous answer o his quesion, a fac which reflecs he inconclusiveness of he various empirical sudies in he field Persson and Tabellini 1994), Forbes 2000), Quah 2003)). In our framework he effec of informaion on he inequaliy-growh link depends on a monooniciy propery of individual invesmen in educaion. If consumer preferences exhibi high ineremporal subsiuion in consumpion, agens wih beer es resuls and, hence, beer abiliy prospecs, choose higher invesmens in educaion. Under his consellaion boh income inequaliy and growh increase when he informaion sysem is improved. Similarly, if consumer preferences exhibi low ineremporal subsiuion in consumpion, agens wih more favorable signals inves less and, hence, higher inequaliy goes hand in hand wih lower growh. In he lieraure various formalizaions of wha consiues more informaion in an economic model have been suggesed. Blackwell s prominen sufficiency crierion has been widely used, bu his concep is undersood o be quie demanding and, in fac, sronger han needed for many economic applicaions. In recen years oher conceps based on he sensiiviy of he poserior sae disribuion wih regard o signals have been developed and successfully applied o economic problems e.g., Kim 1995), Ahey and Levin 1998)). Our noion of informaiveness belongs o his class of exensions, i.e., he informaiveness order emerges from a resricion on he disribuion of sae poseriors. The ime srucure of our model implies ha agens receive wage paymens which are condiional on he agens signals raher han on heir rue ex pos) abiliies. Thus individual wage incomes are based on assessmens of each agen s poenial raher han on he human capial he agen acually conribues in he 16
producion process. We believe ha his feaure of our model is reasonably well in line wih various remuneraion schemes ha can be observed in labor markes when individual human capial is no verifiable. Even so, he case where wage conracs can be made coningen on ex pos individual human capial migh be of some ineres as well. In such a seing each agen i is characerized by a pair y i, A i ), bu his economic decisions are based solely on he signal y i while A i is sill random). Under his specificaion, he analysis could focus on he disribuion of income afer individuals alens are observed ex pos income inequaliy). In his case informaion no longer plays a role in some process of risk sharing across agens. Therefore, he impac of informaion on income inequaliy and growh will presumably be weaker han in our model. Ye, if risk sharing markes are inroduced where alen risks can be insured on fair erms, hen he ex pos income disribuion becomes idenical o he inerim disribuion analyzed in his sudy. Individual alen is deermined a birh even hough i remains unknown unil he agens ener heir second period of life. Therefore, since agens differ wih regard o abiliy even in heir firs period, in principle our heoreical framework can be used o analyze ex ane inequaliy i.e., before signals are observed) as well. Some resuls on he role of informaion for ex ane inequaliy can be found in Eckwer and Zilcha 2002). Obviously, any of hese inequaliy conceps has is own economic meaning and normaive implicaions. Appendix In his appendix we prove Lemma 1 and he wo proposiions. Proof of Lemma 1: i) and ii) imply ha ϑ yπ) ϑ xπ) is monoone increasing. Thus here exiss π 0, 1) such ha ϑ yπ) E[ϑỹ)] ) ϑ xπ) E[ϑ x)] for π ) π. This inequaliy implies ha ˇy := ϑ ỹ E[ϑỹ)] differs from ˇx := ϑ x E[ϑ x)] by a MPS and, hence, he disribuion of ϑ ỹ is more unequal han he disribuion of ϑ x. 17
We prove wo preliminary resuls before we proceed wih he proofs of he proposiions. Lemma 2 MPS) Le π be a random variable which is disribued over [0, 1] according o he Lebesgue densiy φ. Le y : [0, 1] [, ] and x : [0, 1] [, ] be differeniable sricly increasing funcions such ha E[y π] = E[x π], i.e., 1 0 yπ)φπ) dπ = 1 0 xπ)φπ) dπ. 26) Assume furher ha yπ) and xπ) have he single crossing propery wih yπ ) = xπ ) =: and yπ) ) xπ) for π ) π. Then Y := y π differs from X = x π by a MPS.... yπ)... xπ). π 1 π Remark 4: If yπ) and xπ) are sricly decreasing and he oher condiions in Lemma 1 are saisfied, hen X = x π differs from Y = y π by a MPS. Proof of Lemma 2: Le G and F be he c.d.f. s for Y and X. Denoe by g and f he Lebesgue) densiies of G and F, and define S := G F. From yπ) ) xπ) 18
for π ) π we conclude S) ) 0 for ) and, hence, 7 ˆ S) d = S) } {{ } = S) d = =0 1 0 [g) f)] d [yπ) xπ)]φπ) dπ = 0. 27) S) d + ˆ S) d 0. 28) The inequaliy in 28) follows from 27) and he fac ha S) ) 0 for ). 27) and 28) ogeher imply ha Y differs from X by a MPS. Lemma 3 Le f and ˆf be wo informaion sysems wih f inf ˆf. For any increasing differeniable funcion ϑ : A R + he random variable θπ) := E f[ϑa) F 1 π)] differs from ˆθπ) := E ˆf[ϑA) ˆF 1 π)] by a MPS. Also, θπ) ˆθπ) is monoone in π. Remark 5: If ϑ is a decreasing funcion, ˆθπ) differs from θπ) by a MPS. Proof of Lemma 3: By he law of ieraed expecaions, 1 0 θπ) dπ = 1 0 ˆθπ) dπ. Therefore, in view of Lemma 1, i suffices o show ha θπ) ˆθπ) is increasing in π. θ π) ˆθ π) = ϑa) [ ν A A π F 1 π) ) ˆν A ˆF 1 π) )] da = ϑ A) [ A ν A A π F 1 π) ) ˆν A ˆF 1 π) ) ] da da A = ϑ A)[Ḡπ A F 1 π) Ĝπ A ˆF 1 π) )] da 0. 7 A The firs equaliy in he second line of 27) follows from The las inequaliy follows from 14), since ϑ 0 has been assumed. g) d = y 1 ) y 1 ) yπ) gyπ))y π) dπ = } {{ } =φπ) 1 0 yπ)φπ) dπ. 19
Proof of Proposiion 1: Incomes under he wo informaion sysems are given by I f π) = w ϕ fπ)g H 1, e ϕ fπ) )), I ˆf π) = w ϕ ˆfπ)g H 1, e ϕ ˆfπ) )), where ϕ fπ) := E f [ ϕã) F 1 π) ], ϕ ˆfπ) := E ˆf [ ϕã) ˆF 1 π) ]. According o Lemma 3, ϕ fπ) and ϕ ˆfπ) differ by a MPS and ϕ fπ) ϕ ˆfπ) is monoone in π. Below we show ha h ϕ) = ϕg H 1, e ϕ) ) is monoone increasing in ϕ. Lemma 1 hen implies ha he income disribuion is more unequal under f han under ˆf. ) ) ) ) Firs observe ha s ϕy), w h ϕy) s ϕy), and h ϕy) are pairwise co-monoone. This observaion is immediae from 9) since u 1 and u 2 are decreasing funcions. Now assume, by conradicion, ha as ϕ increases h ) declines. By comonooniciy, w h ) s ) declines as well. As a consequence, ϕu 1 w h ) s ) ) increases and, according o 10), e ) increases. However, in view of 7), an increase in e ) conradics our assumpion ha h ) declines as ϕ increases. Proof of Proposiion 2: According o Lemma 2, ϕ fπ) differs from ϕ ˆfπ) by a MPS. In addiion, if he invesmen behavior is efficiency-inducing inefficiency-inducing), h ) in 18) is a convex concave) funcion. Therefore, 1 0 h ϕ fπ) ) dπ ) 1 holds see Rohschild/Sigliz, 1970) and, hence, H f H ˆf. 0 h ϕ ˆfπ) ) dπ in 17) is larger smaller) han References 1. Ahey, S. and Levine, J., 1998, The Value of Informaion in Monoone Decision Problems, manuscrip. 20
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