The Interaction between Human and Physical Capital Accumulation and the Growth-Inequality Trade-off

Transcription

1 The Ineracon beween Human and Physcal Capal Accumulaon and he Growh-Inequaly Trade-off Ad Mra, Unversy of Washngon, Seale WA Sephen J. Turnovsky* Unversy of Washngon, Seale WA November 2011 Absrac Ths paper analyzes he effecs of echnologcal change on growh and nequaly n a wo-secor endogenous growh model. We fnd ha he effecs on nequaly depend upon: () wheher he underlyng source of nequaly sems from dfferenal nal endowmens of human capal or physcal capal, () he me horzon over whch he producvy ncrease occurs. Our resuls sugges ha an ncrease n he growh rae resulng from producvy enhancemen n he human capal secor wll ncrease nequaly. Producvy enhancemen n he fnal oupu secor, alhough no havng permanen growh effecs, wll reduce nequaly. In eher case he responses of nequaly ncrease, he more gradually he producvy ncrease akes place. The model can generae a posve or negave relaonshp beween nequaly and growh, dependng upon he relave sze of hese effecs, conssen wh he dverse emprcal evdence. *Mra s research was suppored n par by he Corkery Fellowshp and Turnovsky s research by he Casor endowmen a he Unversy of Washngon.

2 I. Inroducon The early leraure on economc growh denfed hs process wh he accumulaon of physcal capal. However, he sze of he Solow resdual esmaed n growh accounng exercses hghlghed how n fac only a relavely small fracon of economc growh could be explaned by he accumulaon of capal. Oher facors clearly were playng an mporan, and lkely domnan, role n deermnng a counry s growh performance. As a consequence, smulaed by he poneerng work of Becker (1964) and Schulz (1963), economss began o devoe aenon o he role of educaon, knowledge, and more generally human capal, as a crcal source of economc growh. Indeed, research conduced by Goldn and Kaz (1999, 2001), Abramovz and Davd (2000) has suggesed ha durng he 20 h cenury n he Uned Saes he conrbuon of human capal accumulaon o he overall growh rae almos doubled, whle he conrbuon of physcal capal declned correspondngly. Moreover, hs process ended o accelerae durng he las wo decades of he cenury; see Jorgensen, Ho, and Sroh (2005). Along wh hese developmens, mos advanced economes have experenced ncreasng ncome nequaly snce he 1980s; see Aknson (1999), Goldn and Kaz (2008). Evdence of hs can also be seen n he rsng skll premum n he form of an ncrease n reurns o sklled versus unsklled labor ha has been emergng; see e.g. Mchell (2005). As a resul, he role of human capal has receved ncreasng aenon, boh as a source of economc growh and of he observed rsng ncome nequaly. 1 The key underlyng explanaon of hs nvolves he acceleraon of echnologcal progress, requrng more educaon and sklls; see e.g. Galor and Wel (2000), Galor and Moav (2004), and Goldn and Kaz (1998). Several specfc channels relang he accumulaon of human capal and nequaly have been denfed and exensvely dscussed. These nclude: dspares n educaonal opporunes and aanmen; (e.g. Becker and Tomes, 1979, Benabou, 1996, Durlauf, 1996, Fernandez and Rogerson, 1998, Galor and Moav, 2002, and Vaene and Zlcha, 2009); cred marke consrans (e.g. Galor 1 The changng mporance of physcal and human capal wh respec o he overall growh process s suded n deal n an elegan model developed by Galor and Moav (2004). Subsequen work by Galor (2011) has progressed furher oward he developmen of a unfed growh heory. 1

3 and Zera, 1993, Banerjee and Newman, 1993, Levne, 2005); healh and demographc facors (e.g. Becker and Barro, 1998, de la Crox and Doepke, 2003, Ehrlch and Lu, 1991, Ehrlch and Km, 2007); polcal consderaons and educaon (e.g. San-Paul and Verder, 1991, Glomm and Ravkumar, 1992, Ecksen and Zlcha, 1994). 2 Mos of hs leraure employs some form of overlappng generaons model n whch he ransfer of human capal across generaons s a key elemen of he growh process. These models ypcally have he characersc ha he economy evenually converges o a saonary equlbrum, alhough mporan excepons o hs characerzaon ceranly exs. 3 Ths paper akes a somewha dfferen approach. We adap he semnal Lucas (1988) wosecor producon model of human capal and growh, o nclude boh sklled and unsklled (raw) labor, as well as physcal capal, as producve facors. Each agen s equally endowed wh raw labor bu has dfferen nal endowmens of boh physcal and human capal. We assume ha aggregae human capal generaes an exernaly, enablng he economy o susan ongong growh, characersc of he Lucas model and endogenous growh models n general. 4 We also assume ha physcal capal s specfc o he producon of fnal oupu, n whch case he human capal producon secor becomes he fundamenal engne of growh, conssen wh much of he emprcal evdence. Ths model focuses on he echnologcal srucure raher han he socal aspecs noed above, hghlghng he role of he secoral producon characerscs whch need no be unform across he economy as poenally mporan deermnans of long run growh and assocaed nequaly. As Lucas argued, hs exenson of he Uzawa (1965) model and he nroducon of nersecoral facor mobly provdes mporan nsghs and yelds a subsanal mprovemen over he sandard one-secor neoclasscal model n explanng he process of economc developmen. Our focus s on deermnng he dynamc responses of he economy o producvy ncreases, and on conrasng he mpacs on he growh-nequaly relaonshp beween a producvy ncrease n he fnal oupu secor, on he one hand, and and a producvy ncrease n he human capal 2 A recen paper by Caselló-Clmen (2010) provdes an emprcal nvesgaon of he varous channels whereby nequaly n human capal nfluences he growh rae. 3 Excepons o hs characerzaon nclude Ehrlch and Lu (1991), Glomm and Ravkumar (1992), Ehrlch and Km (2007), who embed he overlappng model no an endogenous growh framework. 4 See also Rebelo (1991). 2

4 producng secor, on he oher. We emphasze wo crcal, and dsnc, sources of dynamcs ha we can erm exernal and nernal. The frs peran o he me pah over whch he exogenous producvy change akes place. We compare he convenonal assumpon ha all occurs a once, wh he alernave ha he same overall ncrease accumulaes gradually. Ths dsncon s relevan when one consders ha echnologes are always changng, and ha economes approach he boundares of known echnologes gradually raher han nsananeously. Moreover, he ransonal me pah also urns ou o have mporan permanen consequences for wealh and ncome nequaly. As a resul, hs source of dynamcs perms srucurally smlar economes o have subsanally dfferen levels of nequaly, dependng upon he me pah over whch her (common) producvy levels may have evolved. Ths characersc s conssen wh he experences of counres n Eas Asa and Lan Amerca, whch have smlar levels of per capa ncome, bu very dfferen ncome dsrbuons. The nernal dynamcs relae o he neracon beween he wo capal goods n he accumulaon process. Whle we can provde a general analycal characerzaon of he long-run equlbrum, he model s suffcenly complex o requre ha he dynamcs be suded numercally. Our smulaons brng ou he sark conrass beween he effecs of producvy ncreases n he wo secors, as well as her sensvy o he me horzons over whch hey are mplemened. Overall, our conclusons are robus and our fndngs wh respec o aggregaes and dsrbuon can be summarzed as follows: () A producvy ncrease n he fnal oupu secor has only a ransonal effec on he growh rae. I has no long-run effec on he relave usage of sklled o unsklled labor, whch s a key deermnan of he long-run growh rae. Insead, rases he long-run raos of physcal capal and oupu o human capal, leadng o a correspondng ncrease n consumpon. () In conras, a producvy ncrease n he human capal secor wll lead o a change n he relave use of sklled and unsklled workers and a posve permanen effec on he growh rae. () Whle he long-run responses of he aggregae varables are ndependen of he me pah followed by he producvy ncrease (n eher secor), here are sharp conrass n he 3

5 ransonal pahs beween he cases where he ncreases occur fully nsananeously (he convenonal case) and where hey are mplemened gradually over me. For producvy ncreases n eher secor, he adjusmens nally proceed n oppose drecons. Ths s because he allocaon of labor, whch drves he adjusmens, responds o boh he relave prce of human o physcal capal and o he shocks hemselves. Wh full nsananeous ncreases, boh effecs operae mmedaely. Bu where he producvy ncrease proceeds gradually, he former domnaes n he early sages. Ths s because he producvy enhancemen akes me o buld up and only begns o become effecve laer n he ranson. We consder hree measures of nequaly: wealh nequaly, ncome nequaly, and welfare nequaly. In conras o he aggregaes, a producvy ncrease n he fnal oupu secor does have permanen dsrbuonal consequences, and oban he followng conclusons: (v) In general, he responses of all hree nequaly measures depend upon he underlyng source of nal nequaly n he economy,.e. wheher s due o unequal endowmens of physcal capal, of human capal nequaly, or some combnaon. (v) The dynamcs of wealh nequaly consss of an nal jump followng by a gradual ranson. The drecon of he nal jump sems from he nal change n he relave prce of human capal and depends upon (a) he source of he nal nequaly, and (b) he secoral locaon of he producvy ncrease. The drecon of ransonal pah depends upon wheher consumpon adjuss faser or slower han do wages. For a producvy ncrease occurrng n he fnal oupu secor, consumpon does ndeed adjus faser and wealh nequaly declnes, whle he oppose s rue for a producvy ncrease n he human capal secor. Producvy ncreases ha are spread ou over me exacerbae hese effecs. (v) Long-run wealh nequaly may eher rse or fall, wh eher form of producvy ncrease, dependng upon he nal underlyng source of he nequaly and he speed wh whch he producvy ncrease occurs. (v) Income nequaly reflecs he evoluon of boh wealh nequaly and he share of ncome orgnang from wealh. In he long run, our smulaons sugges ha he former effec 4

6 domnaes so ha he long-run response of ncome nequaly reflecs ha of wealh nequaly. Ths conrass wh resuls obaned usng one-secor models where he oppose ends o be he case. 5 The dfference s largely accouned for by he nal jump n wealh nequaly arsng from he prce response, denfed n (v). (v) Long-run wealh nequaly exceeds long-run ncome nequaly, whch n urn exceeds long-run welfare nequaly. The remander of he paper s srucured as follows. Secon 2 ses ou he analycal framework, whle he followng secon characerzes he aggregae equlbrum. Secon 4 derves he long-run effecs of he producvy ncreases on he equlbrum quanes of he aggregaes, ncludng he growh rae. Secon 5 ses ou he evoluon of he hree nequaly measures we consder. Secons 6 and 7 descrbe he numercal smulaons, whle Secon 8 concludes. Addonal echncal deals, ncludng he mplcaons for he long-run relave posons of physcal and human capal are se ou n he Appendx. 2. The analycal framework The analycal framework we employ s an adapaon of he wo-secor producon economy poneered by Lucas (1988), augmened o nclude raw (unsklled) labor as well as human capal Households The economy consss of a connuum of nfnely-lved consumers. Agens are ndexed by and are dencal n all respecs excep for her nal endowmens of human capal, H,0, and physcal capal, K,0. Snce we consder a growng economy, we are neresed n he shares of ndvdual n he accumulang oal socks of human capal, h() H () H(), and physcal capal, k () K () K(), where H (), K() denoe he correspondng economy-wde average 5 Ths s rue boh for he one-secor Romer model and he one-secor Ramsey model; See e.g. García-Peñalosa and Turnovsky (2006, 2011). 6 The model also exends Turnovsky (2011) whch adops a smlar framework excep ha absracs from physcal capal. Wh only one capal good (human capal), here are no nernal dynamcs and n he absence of exernal dynamcs, he economy s always on s balanced growh pah. All dsrbuon measures reman unchanged over me. 5

7 quanes. The nal relave endowmens across agens have mean 1, sandard devaons h,0, k,0, and covarance kh,0 (possbly zero). 7 We make wo mporan assumpons wh respec o he underlyng source and naure of he heerogeney. Frs, here are clearly many sources of heerogeney, of whch nal endowmens, n hs case of boh human and physcal capal, are arguably he mos sgnfcan. Emprcal evdence supporng hs, n he form of nherance, s provded by Pkey (2011). 8 Second, he dsrbuons of nal endowmens can be arbrary, and herefore conssen wh any requred nonnegavy consrans. As wll become apparen n due course, hese nal dsrbuons wll be refleced n he evolvng dsrbuons of wealh and ncome. Each ndvdual s endowed wh a un of raw labor ha can be allocaed eher o employmen n he fnal oupu secor, L X,, or o acqurng more human capal, L,, hus, L X, L, 1 (1a) A any pon of me he agen has accumulaed a sock of human capal, allocaed eher o he fnal oupu secor or o he furher accumulaon of human capal H, ha smlarly can be H H H (1b) X,, Physcal capal, however, s employed only n he producon of fnal oupu and herefore does no nvolve a secoral allocaon decson. 9 The wage rae earned by raw (unsklled) labor s W, whle he reurns o human capal and physcal capal are r H, r K, respecvely, all hree beng expressed n uns of fnal oupu. The agen s budge consran, descrbng hs marke acves, assers ha he ncome earned from hs 7 Beng he sandard devaon of he relave capal sock, h, k are n fac coeffcens of varaon. 8 Oher papers o generae heerogeney from agens nal endowmens of wealh nclude Sorger (2000), Malar and Malar (2001), Casell and Venura (2000) and Turnovsky and García-Peñalosa (2008). Wh respec o human capal endowmens, Becker and Tomes (1986) assume ha parenal nvesmen n human capal s he domnan source of heerogeney among famles, whle Han and Mullgan (2000), n sudyng nergeneraonal mobly consder heerogeney n ably. An alernave source of heerogeney n he earler leraure was he rae of me preference, whch led o he concluson ha he mos paen ndvdual ends up holdng all he capal; see Becker (1980). 9 We can also nerpre he producon funcon for human capal as ncludng a fxed amoun of physcal capal [emboded n he erm B ()], whch s no accumulaed furher n he human capal secor. 6

8 hree producve facors can be spen eher on consumpon or accumulang physcal capal K rkrh WL C (1c) K H X, X, In addon, he agen accumulaes human capal n accordance wh he Cobb-Douglas producon funcon (common o all agens) 1,, H B() L H H (1d) where H denoes he average (aggregae) sock of human capal, and B () denoes he level of echnology n he producon of new human capal. Ths funcon s analogous o Lucas (1988), n ha s homogeneous of degree one n he accumulang asse (human capal), wh he economywde average sock of human capal provdng an exernaly ha enhances he producvy of each ndvdual s raw labor. The only way an agen can accumulae human capal s by devong hs own physcal resources o hs acvy; here s no marke where human capal can be purchased. The represenave agen chooses consumpon, C, raw labor allocaons, LX,, L,, human capal allocaons, HX,, H,, and he raes of human capal accumulaon, H and physcal capal accumulaon, K, o maxmze he so-elasc uly funcon: 1 Ce 0 d 1 (2) subjec o he consrans (1a) (1d). The opmaly condons are respecvely: 1 C (3a) L H H, 1,, W B(1 ) H (3b) r H L B H 2,, H, 1 (3c) 2, (3d) 7

9 r K (3e) ogeher wh he ransversaly condons lm He 0, lm Ke 0 (3f) where s he agen s shadow value of physcal capal, s hs shadow value of an exra un of nvesmen n human capal, 1, s hs shadow value of raw labor, 2, s hs shadow value of human capal. Equaons (3a)-(3c) are sandard sac effcen allocaon condons, whle (3d) and (3f) are famlar arbrage condons, equang raes of reurn across he alernave nvesmen opons. 2.2 Frms There s a sngle represenave frm producng fnal oupu,, usng he aggregae producon funcon: A K H L H 1 (4a) () X X where A(), s he level of echnology n he fnal oupu secor, L X s raw labor, and H X s human capal, boh employed n he fnal oupu secor. The producon funcon, (4a), s analogous o he ndvdual producon funcons for human capal, wh economy-wde human capal provdng a producvy-enhancng exernaly for raw labor. The frm chooses raw labor, physcal capal, and human capal o maxmze profs, () X X X K H H A K H L H WL r K r H (4b) yeldng he opmaly condons W K L X A () H HX HX H 1 (5a) r K 1 K L X A() HX HX H (5b) 8

10 r H K LX A() HX HX H (5c) Thus, he equlbrum real wage s proporonal o he economy-wde sock of human capal, a posve funcon of he physcal capal-human capal rao, and a decreasng funcon of he rao of raw labor o human capal employed n he fnal oupu secor, whle he raes of reurn on human and physcal capal are boh ncreasng funcons of ha same rao. In conras, he rae of reurn on physcal capal (human capal) s a decreasng (ncreasng) funcon of he rao of physcal capal o human capal employed n he fnal oupu secor. From (1d) and (4) we see ha producon n boh secors exhbs consan reurns o scale n he wo forms of capal, so ha he economy s capable of susanng ongong (endogenous) growh. The focus of our analyss s o consder he effec of mprovemens n echnology n he wo producve secors on he pahs of growh and dsrbuon n he economy. A cenral nsgh of he model s ha he me pahs followed by he ncreases n producvy have permanen dsrbuonal effecs. To demonsrae hs, we shall consder ncreases n he producvy level from nal gven levels A0, B 0 o hgher long-run levels, AB,, specfed by he (known) deermnsc growh pahs 0 A A () A A A e 0 (6a) 0 A B B () B B B e 0 (6b) The parameers, hus defne he me pahs followed by he ncreases n producvy, and he A B speeds wh whch hey occur. The convenonal approach o specfyng producvy ncreases s o assume ha hey are compleed mmedaely, as an nsananeous dscree change n level. Ths s obaned as a specal case by leng, n (6). 10 Bu he more general specfcaon A B nroduced n (6) s mporan. Ths s because, as we wll demonsrae subsequenly, here s a sharp conras beween how, affec he dynamcs of aggregae quanes and of dsrbuons across A B B 10 The assumpon ha he ncrease n producvy occurs a a consan proporonae rae, and s compleed only asympocally, s made purely for analycal convenence. I s sraghforward o generalze (6) o he case where he new level of producvy s reached n fne me, T. The analyss could also be modfed o allow for he echnology ncrease o follow a more general me pah, and he same general qualave conclusons would emerge. 9

11 agens. As one would expec, hey affec he ransonal pah of he aggregae economy, bu no he aggregae seady sae. In conras, hey nfluence boh he me pahs and he seady-sae levels of boh wealh and ncome nequaly Aggregaon We have now lad ou he basc componens, and he nex ask s o aggregae hem o derve he economy-wde equlbrum. To do hs we frs noe he followng aggregaon relaonshps and marke clearng condons, descrbng he raw labor and human capal markes, respecvely L L L L (7a) X,, X 1; H H H H H (7b) X,, X ; For noaonal convenence, le L X X HX H ;, H L,, ; H denoe he rao of raw labor o human capal employed n he fnal oupu secor, and he rao allocaed by ndvdual agen n he producon of human capal, respecvely. Usng hs noaon enables us o express he ndvdual opmaly condons n he form 1 C (3a) 1, K W A H B H 1 X (1 ), H X r A K B 1 2, H X, H X (3b ) (3c ) 11 The reason for hs s he homogeney of he uly funcon (2) whch causes ndvduals o manan fxed relave consumpon over me; see (10 ). Ths nroduces a zero roo no he dynamcs of he dsrbuonal measures, as a resul of whch her equlbrum values become pah dependen; see Aola, Chaerjee, and Turnovsky (2011) where hs ssue s dscussed n deal n he conex of a Ramsey model. 10

12 2, r K (3d) (3e) Dvdng (3c ) by (3b ) mples rh W X 2,, 1, (1 ) (8) from whch we mmedaely see ha, and 2, 1, are common across all ndvduals. Ths s a consequence of he assumpon ha all agens face a common raw wage rae, W, and reurn on human capal, r H. Thus we have, L (9) H H where denoes he common economy-wde rao of raw labor o human capal employed n he producon of human capal. 12 Combnng he lef hand equaly of (3c ) wh (3d) and (9) we oban B( ) 1 (3d ) Thus, a furher consequence of he common facor nenses across agens n human capal producon,, s ha he growh rae of he shadow value,, s also common o all. We hen mmedaely deduce from (3b ) and (3d) ha 1,, 2, and hence and herefore are also common across agens. Dfferenang (3a) C C ( 1) ( 1) C C (10) so ha by choosng a common growh rae for he margnal uly of ncome, agens choose a common growh rae of consumpon, whch herefore concdes wh he aggregae (average) economy-wde consumpon growh rae. Equaon (10) furher mples ha each agen manans a 12 We shall refer o he erm,, as descrbng he facor nensy n he human capal secor 11

13 consan rao of hs consumpon o average consumpon C C for each and 1 (10 ) where he consan s o be deermned by hs relave seady-sae wealh [see (26b) below]. Le q ( ) denoe he common relave shadow prce of human capal o ncome. Wh hs noaon, we may use (3b ) and (3c ) o express he economy-wde equlbrum allocaon of raw labor and human capal: W H K A X HX 1 qb(1 ) (11a) K r 1 H A X qb H X (11b) In addon, subsung for rk, rh, W, X, no (1c) and (1d), he ndvdual raes of accumulaon of physcal and human capal accumulaon are respecvely 1 K HX, LX, X HX HX LX K A K K K C (12a) 1, H B H (12b) whch n each case s lnear n he agen s facor supply devoed o producng ha oupu, wh he proporonaly facor beng dependen upon he facor allocaon nensy and common across agens. Thus, summng over all agens yelds he correspondng aggregae relaonshps 1 K K A X K C (12a ) H X 1 H B H (12b ) The fac ha he sysem aggregaes exacly s a consequence of wo assumpons: () he 12

14 homogeney of he uly funcon, and () he assumpon ha all agens face common reurns Dervaon of aggregae equlbrum The aggregae economy possesses many of he characerscs of a sandard wo-secor endogenous growh model, as poneered by Lucas (1988), and exended by Bond, Wang, and p (1996). The macro equlbrum ncludes he followng four condons: () labor marke clearance: L L 1 (13a) X () human capal marke clearance: HX H H (13b) () goods marke clearance: (v) human capal accumulaon: 1 K K A X K C (13c) H X H B H (13d) In addon, shor-run and long-run effcency condons mus hold and he echnology levels evolve n accordance wh (6a) and (6b). Our procedure s o follow Bond, Wang, and p (1996) and o express he macroeconomc dynamc equlbrum n erms of he rao of physcal capal o human capal, k K H, he rao of consumpon o human capal, c C H dynamcs of A, B. To do so, s convenen o le, and he rao of he shadow value, q, augmened by he u HX H denoe he allocaon of human capal o he producon of fnal oupu. Wh hs noaon, we may express he secoral allocaon condons for raw labor and human capal, (11a) and (11b), ogeher wh he clearance of boh he raw labor and human capal markes, (13a) and (13b), n he form W H k A X u 1 qb(1 ) (14a) k r 1 H A X qb u (14b) 13 These assumpons also accoun for he fac ha he aggregae equlbrum s ndependen of dsrbuon measures, whch led Casell and Venura (2000) o characerze hs ype of model as a represenave agen heory of dsrbuon. 13

15 u X (1 u) 1 (14c) These hree equaons can be solved for he secoral allocaons of raw and sklled labor, X,, u, n erms of kqab,,,, whch for convenence we wre n he form 14 ( kqab,,, ), ( kqab,,, ), u ukqab (,,, ) (15) X X Dvdng (11b) by (11a) we oban X (1 ) (11 ) yeldng he sandard resul ha for gven producve elasces,,,, he relave nenses of raw labor o human capal wll move proporonaely n he wo secors. The quany measures he relave nenses of sklled o unsklled labor n he wo secors. If 1, human capal s relavely more nensve han unsklled labor n he human capal secor, and vce versa. Equaons (15) are crcal n deermnng he mpac effecs of producvy changes. Takng dfferenals of (14) one can show ha: dx d dq dk da db X u( 1) q k A B 1 (16a) du u 1 d 1 u( 1) (16b) These relaonshps hghlgh how he nal responses of boh sklled and unsklled labor allocaons depend upon her relave secoral nenses and ha whaever ha response, ncreases n A and k operae n one drecon, whle ncreases n B and q have he oppose effecs. Equaon (16a), n parcular, s mporan n undersandng he conrasng shor-run effecs of gradual versus dscree producvy changes ha we shall denfy n Secon 7, below. Takng he me dervave of k K H and subsung from (13c) and (13d) yelds he followng equaon for k 14 Throughou, we shall assume ha all producon elasces reman consan and hus om hem from he soluons. 14

16 1 k k 1 c A ( X) B(1 u)( ) k u k (17a) Nex, akng he me dervave of c C H, and usng (3e), (10), and (13d), we oban 1 c 1 k A X B u c 1 u (1 ) 1 (17b) Fnally, akng he me dervave of q, combnng wh (3c), (3d), (3e) and (11d) mples 1 q rh () () k 1 k rk A X A X q q u q u (17c) Subsung he soluons for,, from (15), equaons (17a)-(17c), descrbe he evoluon of X u he macroeconomc equlbrum, condonal on he gven me pah for he echnology parameers, A(), B(), as specfed by (6a) and (6b). The macrodynamc equlbrum s a modfcaon of ha analyzed by Bond e al. (1996), he dfferences beng: () he dsncon beween sklled and unsklled labor, and () he assumpon ha physcal capal s specfc o he fnal oupu secor, and () he gradual evoluon of A(), B(). Bu he key observaon s ha he evoluon of he aggregae economy s ndependen of any dsrbuonal measures. As k, c, and q evolve, hs wll generae adjusmen pahs for he raes of reurn and he secoral allocaon of resources. One varable of parcular neres s he skll premum, whch we defne as s r ( W / H). The reason for hs s because he supply of raw labor s fxed, whereas human capal grows ndefnely over me, as a resul of whch n he long run he reurn o human capal s consan, whle he reurn o raw labor grows a he equlbrum rae. We herefore defne skll premum as he rao of ncome earned by human capal o he ncome earned by raw labor. Dvdng (14b) by (14a), we see ha he crucal deermnan s he equlbrum rao of raw o sklled labor employed n he human capal secor,. H 3.1 Characerzaon of macrodynamc equlbrum In secon 7 below we shall analyze he local dynamcs followng a producvy ncrease by 15

17 lnearzng he above sysem abou s seady sae. The formal srucure of hs sysem s se ou n he Appendx. A necessary condon for he local dynamcs o be a saddlepon and o generae a unque sable adjusmen pah s D (1 )(1 ) (1 )(1 ) (1 ) 0 (18) where lde denoes he seady sae. The sable adjusmen pah s hen as characerzed n he Appendx. 15 Durng he ranson, physcal capal, human capal, oupu, and consumpon grow a dfferen raes, alhough f he sysem s sable hey converge o a long-run common growh rae. 3.2 Balanced growh equlbrum The seady-sae balanced growh equlbrum s reached when k cq A B 0 and s summarzed by he followng condons 1 Secoral allocaon of raw labor: A qb (1 ) W H k u Secoral allocaon of human capal: 1 k A X qb u X (19a) (19b) Full employmen: u (1 u) 1 (19c) X k u Equlbrum growh: A B (1 u ) 1 X c k k A X B u 1 u (1 ) Equlbrum raes of reurn: 1 1 k 1 k 1 r A A r u q u q K X X H (19d) (19e) (19f) 15 The use of lnearzaon rases he queson of accuracy. Aola, Chaerjee, and Turnovsky (2010) examne hs ssue n deal for growh models of hs dmensonaly, and conclude ha for small exogenous shocks, such as hose nroduced here, lnearzaon nvolves accepably small errors, relave o more compuaonally nensve shoong algorhms. 16

18 Gven he long-run levels of echnology, AB,, hese equaons deermne he equlbrum values of u,,, k, c, q, whch hen mply he correspondng equlbrum facor raes of reurn, W H, r, r, X skll premum, s, and equlbrum growh rae,. In Table 1, whch we shall dscuss n Secon 4, we summarze he long-run effecs of ncreases n echnology on hese equlbrum values. Two crcal condons consran he equlbrum value of (and ransversaly condon, (3f), ha each agen mus sasfy. Ths wll be me f and only f X K H ). The frs s he H H lm 0 lm 0 H H (20) whch n seady sae s equvalen o r K. 16 Usng (19b), (19d) and (19f) hs n urn s equvalen o u 1. The oher condon s ha wh no deprecaon o human capal, he equlbrum growh rae s always posve, whch n urn mples ha 1 u. Combnng hese wo condons, ogeher wh he full employmen condon (19c) and (16), yelds he followng bounds on for a feasble soluon o exs: If 1: If 1 : 1 1 (1 ) (21a) 1 1 (1 ) (21b) 4. Long-run effecs of producvy ncreases on aggregae equlbrum As Table 1 clearly llusraes, he producvy ncreases n he wo secors have dramacally dfferen long-run effecs on he aggregae economy. These shall be brefly summarzed n urn. 4.1 Producvy ncrease n fnal oupu secor, A These resuls may be summarzed n 17 Proposon 1: A producvy ncrease n he fnal oupu secor: 16 Ths condon also apples o he ransversaly condon applcable o physcal capal. 17 Ths proposon s analogous o ha obaned by Turnovsky (2011), he dfference beng ha he proporonaly ncreases are by a facor 1 (1 ), raher han 1, reflecng he producvy of physcal capal n fnal oupu. 17

19 I. Leads o equ-proporonae ncreases n () he rao of physcal o human capal (k), () he prce of human capal (q), () he rae of reurn o human capal ( r H ), (v) he reurn o raw labor (W/H), and (v) he consumpon o human capal rao (c). Ths proporonae ncrease exceeds uny by an amoun ha reflecs he producvy of physcal capal n fnal oupu. II. Has no effec on () he secoral raos of sklled o unsklled labor (, ), () allocaon of human capal across secors (u), () he reurn o physcal capal ( r K ), (v) he equlbrum growh rae ( ), or (v) he skll premum (s). X The nuon underlyng hese responses s sraghforward. An ncrease n he producvy, A, of he fnal oupu secor aracs resources o ha secor. Ths rases he producvy of raw labor and human capal proporonaely n ha secor, ncreasng her respecve raes of reurn, W H, r H equally, so ha he skll premum, s, remans unchanged. There s herefore no ncenve o subsue beween he wo classes of labor, so ha he secoral labor nenses,, reman unchanged. The ncrease n producvy n he fnal oupu secor leads o a long-run ncrease n capal and oupu, whch s fully consumed. Ths ncrease n capal exacly offses he effec of he ncrease n producvy on he reurn o capal, r K, whch remans unchanged, leavng he long-run growh rae unchanged as well. X 4.2 Producvy ncrease n he human capal secor, B In hs case he effecs of he producvy ncrease are more complex and nvolve wo crcal parameers, he secoral nensy of labor, 1, and he producvy of physcal capal. They may be summarzed n Proposon 2: A producvy ncrease n he human capal secor has he followng long-run effecs: () () I leads o an unambguous ncrease n he equlbrum growh rae. If he human capal secor s relavely more nensve n sklled labor han s 18

20 he fnal oupu secor ( 1), he wage rae on raw labor wll decrease, rasng he skll premum, and causng a subsuon oward raw labor n boh secors. The fracon of human capal employed n he fnal oupu secor, u, declnes and he rae of reurn on physcal capal ncreases. () If ( 1) he responses n,,, are all reversed. Oher varables X us experence offseng effecs due o he producvy of physcal capal n he fnal oupu secor The followng nuon apples. An ncrease n he producvy of human capal B wll arac resources from he fnal oupu secor o he human capal secor, causng he oupu of ha secor o rse a he expense of fnal oupu. If 1, so ha he human capal secor s relavely more nensve n human capal, human capal ncreases n scarcy, relave o raw labor, causng he rae of reurn on human capal o rse and ha of raw labor o fall, resulng n an ncrease n he skll premum and nducng a subsuon away from human capal o raw labor n boh secors. There s, however, anoher effec n operaon. The producvy of raw labor and human capal are enhanced by her neracon wh physcal capal n he fnal oupu secor. As resources move away from hs secor hs effec s reduced, as represened by he erm (1 ) for he correspondng expressons n Table 1. Ths renforces he declne n he wage of raw labor and offses he ncrease n he reurn o human capal. The mos srkng conras beween he producvy ncreases n he wo secors s n he mpac on he long-run growh rae. To see he reason behnd hs s consrucve o subsue (19b), (19c), and (19f) no (19e), rewrng as 1 1 B ( 1) B (19e ) 1 1 From he lef-hand sde equaly, we may solve for long-run rao of raw labor o human capal n he form ( ) and subsung no he rgh-hand sde equaly, we may wre B ( B, ( B)), hghlghng how, boh drecly and hrough, he producvy of he human capal secor s he crucal long-run deermnan of growh, whch by he same oken s ndependen 19

21 of he producvy n he fnal oupu secor. 18 The reason ha B wll ncrease he growh rae s because of he lmed subsuon possbles n he producon funcon for human capal. Wh no possbly for subsung oward physcal capal, ncreasng B drecly mpacs he rao of raw labor o human capal, and hereby ncreases he equlbrum growh rae. 5. Evoluon of wealh nequaly, ncome nequaly, and welfare nequaly We urn now o he dsrbuonal mplcaons. Snce everyhng s drven by he dynamcs of relave wealh, hs s he naural place o sar. 5.1 Relave wealh dynamcs A any nsan of me, he oal wealh of agen s defned by V() K () q() H () (22) Dfferenang (22) wh respec o me, he agen s rae of wealh accumulaon s V() K () qh () () qh () () where K ( ), H ( ), and q () are gven by (1c) (12b) and (17c), respecvely. We may decompose (12b) and express n he form 1 1 (1 ) H B H B H B HL,,, where we are usng he equlbrum condon L, H H, Usng he equlbrum prcng relaonshps (14a) and (14b), we may express agen s rae of accumulaon of human capal as rh () W () H H, L, q () q () (23) Combnng (23) wh (1c) and (12b), whle recallng (1a) and (1b), we may express agen s rae of wealh accumulaon as 18 Through, he producve elasces n he fnal oupu secor play a role n deermnng he equlbrum growh rae. 20

22 V () r () V() W() C() (24) K Boh hese relaonshps ndcae ha n he absence of any marke mpedmens, he shadow value of human capal, q (), behaves lke a prce n a compeve marke. Summng (24) over all agens, aggregae wealh evolves n accordance wh V () r() V () W () C () (24 ) K Now defne agen s relave wealh as v() V() V(). Takng he me dervave of v and combnng (24) and (24 ), ogeher wh he fac ha C C [see (10 )], we fnd 1 C ( ) v () C() W() v() 1 (1 ) V () V () (25) In seady sae, when all aggregae quanes grow a he same rae V C r K W C r K V C 1 V V so ha 19 C W r K 0 V V 1 (26a) In addon, seng v ( ) 0, (25) mples ha agen s relave consumpon, whch s consan hroughou he ranson, s r K V V 1 1 C (26b) Equaon (26b) mples ha f agen has above-average long-run wealh, he has above-average long-run consumpon. As we wll see shorly, he dynamcs of relave wealh are drven by he aggregae consumpon-wage rao. For noaonal convenence we shall denoe C () W () by z (). Ths 19 The rgh hand sde of (26a) s posve by he ransversaly condon whch requres r K 21.

23 enables us o wre z 1 1 v 1 z where z 1 and express (25) n he form W () 1z v() z() 1 v() 1 z() v 1 V () z (25 ) whch lnearzed around he seady sae can be approxmaed by () () W v z 1 v() v v 1 z z V z (27) The key pon o observe abou (27) s ha he coeffcen of v ( ) 0. In order for he agen s relave wealh o reman bounded, he soluon for v ( ) s gven by he forward-lookng soluon W z( ) z WV z 1 v() 1v 11 e d V z (28a) Seng 0 we oban he relaonshp 20 W z( ) z WV z 1 v(0) 1 v 11 e d V 0 z (28b) where K0 q(0) H0 v(0) k,0 h,0 V(0) V(0) (28c) Equaon (28b) deermnes v, and aken n conjuncon wh (28a) deermnes he enre me pah for v ( ), gven v (0). Ths laer erm s a weghed average of he agen s nal relave endowmens of he wo forms of capal, and s subjec o an nal jump, hrough q (0), n response o any srucural change; 20 We should pon ou ha he effec of he me pah followed by he ncrease n echnology on he wealh dsrbuon, s embedded n he me pah of z( ). 22

24 see Appendx (A.4c). Suppose ha he economy s nally n seady sae, and a me =0 experences a srucural change. One mmedae effec of hs s o generae a jump n he relave prce q (0) as par of he adjusmen o ensure ha he economy les on s new sable saddle pah. From (28c) hs causes a jump n agen s nal relave wealh, KH 0 0 dv(0) h,0 k,0 dq(0) (29) V (0) The drecon of he nal jump depends upon: () he dfference n he agen s nal relave endowmens of human vs. physcal capal ( h,0 k,0 ), and () he drecon of he change n he relave prce [sgn ( dq (0) ]. Followng hs jump, v ( ) evolves n accordance wh (28a) and (28b). In vew of he lneary of (28a) and (28b) n v, we can sum over he agens and ransform hese expressons no sandard devaons across agens, as convenen measures of nequaly W z( ) z WVz1 W z( ) z WVz1 v() 1 e d 1 e d v (0) V z V 0 z 1 (30a) W z( ) z WV z 1 v 1 e d v(0) V 0 z 1 (30b) where (0) 0 (0) 0 0 (0) (,0) K (,0) q H 2 q K H v k h kh,0 2 V(0) V(0) V(0) (30c) The soluons (28) and (29) hghlgh how agen s relave wealh a each pon of me,, and herefore he enre dsrbuon of wealh, s drven by he (expeced) fuure me pah of he consumpon o wage rao from me forward, as hese respond o he underlyng srucural change, n hs case he ncrease n he level of echnology. As a resul, he pah followed by z () wll have a permanen effec on he relave sock of wealh and herefore on s dsrbuon across agens. If z () z nsananeously, hen (0) and he dsrbuon of wealh wll reman unchanged v v durng he ranson. For he Romer echnology hs wll occur f here s only one form of capal 23

25 and he srucural change occurs fully on mpac, n whch case here are no ransonal dynamcs. 21 Boh he CH rao, he W H rao, and herefore he CW rao jump mmedaely o her seadysae and he dsrbuon of wealh remans unchanged. In conras, n he presen case, followng s nal jump, he evoluon of wealh nequaly durng he ranson wll depend upon he relave speeds of adjusmen of consumpon and wage, whch n urn depend upon he naure of he adjusmen pah assumed by he ncrease n echnology, o whch hey are respondng. As we shall dscuss n conjuncon wh he more specfc shocks below, hese comprse a combnaon of he acual mplemenaon of he srucural change and s ancpaon, where occurs gradually. A hs pon we can sae he followng proposon: Proposon 3: If consumpon adjuss more (less) rapdly han do wages along he ransonal pah, so ha z approaches z from below (above), hen followng an nal jump, wealh nequaly wll declne (ncrease) durng he ranson. The nuon for hs resul s sraghforward. If consumpon grows faser han do wages on raw labor, savngs grow a a slower rae. Snce wealher people end o save more, her relave rae of wealh accumulaon declnes and wealh nequaly declnes as well. I s also possble o derve he componens, k, h of an agen s relave wealh, v, he seady-sae values of whch are provded n Appendx B. These enable us o address ssues peranng o ndvduals radeoffs beween human capal and physcal capal, hough space lmaons preclude furher dscusson of ha here. 5.2 Income nequaly We nex urn o personal ncome nequaly, where agen s personal ncome as measured by ncome from wealh plus ncome from raw labor s gven by () r () V() W() (31) K and summng over all agens gves us he average economy wde personal ncome () r() V () W () (31 ) K 21 See e.g. García-Peñalosa and Turnovsky (2006). Ths s also he case n Turnovsky (2011). 24

26 Dvdng (31) by (31 ) gves us he relave ncome of agen y, () rk () V() y() 1 1 v() 1 () r() V () W () K (32) Agan, he lneary of (32) allows us o express he relaonshp beween relave ncome and relave wealh n erms of he correspondng sandard devaons of her respecve dsrbuons, ( ) and (), namely y v rk () V() y() v() () v(); () 1 r () V() W() K (33) Hence a any nsan of me, ncome nequaly can be expressed as he produc of wealh nequaly, and he ncome from wealh as a share of oal ncome, (), mplyng ha ncome s more equally dsrbued han wealh. Seng 0, (0) (0) (0), we see ha nal ncome nequaly may y poenally undergo wo nal jumps, one due o he nal jump n wealh nequaly [see (29)], he oher hrough he adjusmens n he raes of reurn. v The ransonal me pah of ncome nequaly reflecs hose of wealh nequaly and he share of ncome from wealh: y () () v() r K () V() W() W() v() y() () v() rk() V() W() rk() V() W() v() (34) where he laer depends upon he evoluon of V ( ), rk (, ) W(), as hey respond o he shock. 5.3 Welfare Inequaly Recallng (2), agen ' s welfare a me s Z () 1 C. Subsung C C no hs expresson yelds 1 Z() C Z() (35) where Z () s he average welfare level a me. Subsung (35) no (2) yelds an analogous relaonshp for he relave neremporal welfare evaluaed along he equlbrum growh pah. 25

27 U U Z() Z() (36) A each nsan of me, agen ' s relave welfare remans consan, so ha hs neremporal relave welfare, U / U remans consan as well. Usng (18 ) and he fac ha c C H and v V H we can express relave welfare n he form U ( rk ) v Z() Ω 1 ( v 1) z( ) U 1 c Z( ) (37) We can now compue a measure of welfare nequaly. A naural merc for hs s obaned by applyng he followng monoonc ransformaon of relave uly, enablng us o express he relave uly of ndvdual n erms of equvalen uns of relave consumpon flows as 1/ ( ) Ω Ω () 1 r K v u u z ( v 1) 1 c (37 ) from whch we oban (1 W u ) v v C (38) Comparng nequaly measures (23) and (38), he followng rankng among he long-run nequaly measures may be obaned. (39) v y u mplyng n parcular ha ncome nequaly oversaes welfare nequaly. 22 Long-run percenage changes n he hree nequaly measures are relaed by he followng d y 1 dr dvw K d v y rk V W 1 r K V W v (40a) 22 By expressng welfare nequaly n erms of wealh nequaly we are ransformng no a cardnal measure. Whle hs s convenen for purposes of comparson wh ncome nequaly, he ransformaon adoped wll affec relave welfare comparsons, alhough no her rankngs. 26

28 d u u 1 CW d C W 1 CW d v v (40b) The long-run change n ncome nequaly exceeds he change n wealh nequaly f and only f he long-run proporonae changes n he reurn o capal plus he wealh-wage rao are posve. Lkewse he change n welfare nequaly exceeds he change n wealh nequaly f and only f he long-run change n consumpon-wage rao s posve. In he case of a producvy ncrease n he fnal oupu secor hese effecs are all zero, n whch case case hese hree nequaly measures all ncrease by he same proporonae amoun, somehng ha s verfed by our smulaons. 6. Numercal Smulaons Gven he complexy of he model s necessary o employ numercal smulaons o analyze he dynamcs. We begn by calbrang a benchmark economy usng he followng sandard parameer values represenng a ypcal economy. Parameer values Preference parameers 1.5, 0.04 Producon parameers 1/3, 1/3, 1/3, 0.6 Producvy levels A0 0.20, A 0.22 ; B 0.20, B Producvy growh 0.10; 0.10 A B Frs he preference parameers correspondng o a rae of me preference of 4% and an neremporal elascy of subsuon of 0.4 are sandard and nonconroversal. The exponens 1/3 n he producon funcon for fnal oupu approxmae he emprcal esmaes of he generalzed Solow producon funcon obaned by Mankw, Romer, and Wel (1992). 23 Emprcal evdence on he producon funcon for human capal s far more sparse. We feel ha he mos mporan npu n augmenng he sock of human capal s human capal, followed by raw labor, wh physcal capal beng he leas mporan, and whch we have se o zero. Thus we se 0.60, as a plausble benchmark, whch we may noe s very close o he calbraed value of They oban esmaes of 0.43, 0.31,

29 obaned by Manuell and Seshadr (2010). Ths combnaon of producon elasces mples 1.5 1, so ha he human capal secor s relavely nensve n sklled labor versus raw labor, as compared o he fnal oupu secor, whch we vew as beng he more plausble case. The resulng macroeconomc equlbrum s summarzed n Table 2, lne The mpled oupu-physcal capal rao s 0.31, almos 90% of raw labor s employed n he fnal ou secor, whle over 85% of human capal s allocaed o he human capal secor. In addon, he rao of human capal o physcal capal s around 1.2, he skll premum, as measured by he rao of he ncome earned by human capal o he raw wage s 1.05, and he equlbrum growh rae s 2.56%. The equlbrum rae of reurn of boh forms of capal, measured n erms of her respecve own uns s 10.4%. 25 The equlbrum dsrbuon measures (and her evoluon) depend upon he dsrbuons of he nal endowmens, and he hree panels n Table 2B consder hree alernave benchmark cases. In he frs panel, agens nal endowmens of human capal and physcal capal are proporonal, k K K H H h. In hs case nal wealh nequaly, s dsrbued namely,,0,0 0,0 0,0 proporonaely across he wo forms of capal. In he second panel, k,0 1, so ha he nal dsrbuon of physcal capal across agens s unform, mplyng ha nal wealh nequaly s enrely due o dfferenal human capal endowmens. The hrd panel s he reverse case, h,0 1; human capal s nally unformly dsrbued so ha wealh nequaly s enrely due o dfferenal endowmens of physcal capal. Normalzng he nal dsrbuons by h,0 1, k,0 1, we see ha n he nal equlbrum v y w, conssen wh (39). Whle we do no aemp o calbrae o a specfc economy, we vew hese as provdng plausble benchmarks ha wll faclae our undersandng of he dfferen mechansms n operaon as he economy evolves over me n response o producvy ncreases. As we wll see, dfferences n nal dsrbuons wll lead o dfferences n he dsrbuons of all nequaly measures, boh n 24 We have expermened wh oher plausble parameers and fnd our qualave resuls o be robus. In all cases we fnd ha he necessary condon (198) for saddlepon behavor s sasfed. 25 Ths mples ha he rao of earnngs by sklled workers o unsklled workers s approxmaely 2. There s an exensve leraure measurng he skll premum, whch s seen o vary wdely wh he measure adoped, he group under consderaon, and dfferen perods of me. Our equlbrum value of 2 s broadly compable wh he esmaes repored n he comprehensve sudy of Auor, Kaz, and Kearny (2008). In assessng he rae of reurn on physcal capal, we should bear n mnd ha we are absracng from deprecaon. 28

30 ranson as well as n seady sae. The resuls for hese polar cases are abulaed n Table 2B. Regardng he specfcaon of he producvy ncrease, we adop he followng sraegy. Sarng from he nal benchmark, we consder n urn he aggregae and dsrbuonal consequences of 10% ncreases n producvy n he wo secors. In boh nsances, hese ncreases are nroduced n wo alernave ways. The frs s an mmedae one-me unancpaed jump n producvy from 0.2 o We refer o hs as a dscree ncrease and corresponds o he convenonal approach. The second s where he same ncrease akes place gradually, adjusng a he known rae of 10% per year. In he laer case, he hgher accumulaed producvy level s acheved only asympocally, alhough s sraghforward o mpose a fne me horzon. The key pon s ha he momen he producvy sars o ncrease, s subsequen levels along he ransonal pah become fully ancpaed. 7. Increase n producvy n he fnal oupu secor We now consder a 10% ncrease n producvy n he fnal oupu secor, focusng on he wo forms of ncrease n urn. 7.1 Dscree ncrease n he producvy The long-run effecs on he aggregae economy are descrbed n lne 2 of Table 2.A, and as noed are he same wheher he full ncrease occurs nsanly or gradually over me. These responses are very smple. The long-run rao of physcal capal o-human capal, he relave prce of human capal, he consumpon-human capal rao, he rae of reurn on human capal, and he real wage (n erms of uns of human capal) all ncrease by 15.4%, whle he secoral allocaons of human capal, raw labor and he reurn on physcal capal all reman unchanged, conssen wh Table 1. The producvy ncrease n he fnal oupu secor has wo mmedae effecs on facor allocaon. From (16) we see ha gven he relave secoral labor nenses descrbed by 1, facor marke equlbrum wll requre a declne n he relave demand for sklled labor, so ha X, boh fall. A he same me, wh forward-lookng agens, he fac ha he producvy 29

31 ncrease wll ncrease k, and herefore he relave scarcy of human capal, causes q(0) o rse on mpac. On balance, he frs effec domnaes, and X, mmedaely boh declne by around 3%. Wh he shf n resources oward fnal oupu producon, whch s nensve n raw labor, he skll premum also declnes on mpac by 3%. In addon, he producvy ncrease mmedaely rases he real wage. Consumpon also rses, bu by a lesser proporonae amoun, as consumers begn o adjus her consumpon o he hgher permanen ncome, so ha he consumpon o wage rao falls. The producvy ncrease also ncreases he reurns o boh physcal capal, r K, and human capal, r H, hough by mpacng drecly on he oupu secor, he former rses more han does he laer. The nal shf n resources away from he human capal secor oward he fnal oupu secor smulaes he nal growh of physcal capal, whle he growh rae of human capal declnes, so ha he rao of physcal o human capal begns o rse. Turnng o dsrbuon, he mmedae response of wealh nequaly depends crcally upon he nal dsrbuon of he relave endowmens. If hey are proporonal across agens, nal wealh nequaly remans unchanged; f hey are enrely due o dfferences n human capal, he nal ncrease n q (0) wll rase shor-run wealh nequaly, whle f hey are due o physcal capal, wealh nequaly wll mmedaely declne; see (29). Wha happens o ncome nequaly, hus depends upon he sze of he nal posve effec from he share of ncome from capal, relave o he nal response of wealh nequaly. These wo effecs may be eher renforcng or offseng, dependng upon he source of he heerogeney n nal endowmens These nal responses mmedaely rgger he subsequen neremporal adjusmens, whch are llusraed n fgs The fac ha he nal ncrease n r K exceeds ha of r H, requres q (0) 0 n order for he overall raes of reurn o be equaed and hs wll end o cause he nal declne n, o be reversed. Ths s parally offse by he ncreasng physcal o human capal X rao, hough he prce effec domnaes and on balance, rse back oward her orgnal seadysae levels. As resources gradually rever back oward he human capal secor n response o he rsng q, r declnes, whle K r H ncreases. All growh raes gradually converge back o her orgnal seady-sae raes of 2.56%. The fac ha n he long run wages and consumpon mus ncrease by he same proporonae 30 X

32 amoun (15.4%), whle on mpac he wage rae ncreases more han does consumpon, mples ha durng he ranson consumpon mus ncrease a a faser rae han does he wage rae. Indeed, fg 2() llusraes how, afer he nal declne, he C W rao ncreases unformly durng ranson. For reasons dscussed prevously, hs mples ha wealh nequaly declnes durng he ranson. Wheher wealh nequaly ends up beng more or less unequal han nally depends upon he nal jump due o q (0), he effec of whch depends upon he source of he nal wealh nequaly. In he case ha wealh nequaly s due o varaons n he nal endowmen of human capal longrun wealh nequaly, w, wll ncrease by 2.8%, whle f s due o dfferenal endowmens n physcal capal, wll declne by 5.6%; see Table 2.B. The me pah for ncome nequaly generally mrrors ha of wealh nequaly. We may noe ha f he nal dsrbuon of human capal s unform across agens ha ncome nequaly wll ncrease n he shor run bu declne n he long run; see fg. 5 (). Bu wh he reurn on capal unchanged n he long run, long-run ncome nequaly, y wll change by he same proporonae amoun as does wealh nequaly. Moreover, he same s rue for welfare nequaly, as suggesed by (40a), (40b); see Table 2.B Gradual ncrease n he producvy of he oupu secor Whle he long-run responses of he aggregae varables reman as n lne 2 of Table 2.A, he ransonal pahs n hese varables, however, are que dfferen, leadng o subsanal dfferences n he dsrbuons of ncome and wealh, boh durng ranson as well as n he seady sae. From fgs. 1-3 we see ha he me pahs of he aggregae varables under he wo scenaros are characerzed by wo sark dfferences. Frs, for a gradual producvy ncrease hey are generally characerzed by non-monoonc adjusmens; second, he shor-run responses are ypcally n he oppose drecon o hose followed when he producvy ncrease s dscree. The key o undersandng hs dfference s agan provded by equaon (16). In conras o he prevous case, when he full producvy ncrease akes effec a me 0, he level of producvy, A (0), now remans unchanged; nsead, A (0) begns o ncrease. Agens now ancpae an ncrease n he fuure producvy level, as a resul of he accumulaon of he growh 31