The Overlapping Generations growth model. of Blanchard and Weil

Transcription

1 1 / 35 Th Ovrlapping Gnraions growh modl of Blanchard and Wil Novmbr 15, 2015 Alcos Papadopoulos PhD Candida Dparmn of Economics Ahns Univrsiy of Economics and Businss papadopalx@aub.gr I prsn a daild soluion of h Ovrlapping Gnraions modl of Blanchard and Wil, somhing ha I hav no found in any xbook. As an xnsion, I calcula h disribuion of individual incom and walh along h balancd growh pah, as wll as h corrsponding Gini inqualiy cofficins. Th xnsion allows a dpr comparison of his modl wih h Rprsnaiv Houshold on. Exnding h modl o includ h Govrnmn is lf as an xrcis for h radr. 1. Daild soluion 2. Long-run quilibrium and is propris 3. Avrag and individual magniuds in long-run quilibrium: h disribuion, and h inqualiy, of consumpion and walh 4. Comparison wih h Rprsnaiv Houshold modl 5. Tchnical Appndix: Th Gini cofficin for h Paro and Lomax disribuions

2 2 / 35 1.Daild soluion W considr a closd drminisic conomy ha has sard in h infini pas, and which w follow in coninuous im and in ral rms. Th conomy producs a singl oupu ha can b consumd or ransformd ino capial wihou cos. Each insan, nw housholds nr h conomy, oghr comprising a gnraion. Th housholds of a gnraion ar idnical, and hy hav infini horizon (no possibiliy of dah). Populaion L () grows a a fixd and xognous ra qual o n 0, whil w sandardiz h iniial lvl of populaion o uniy, lim L ( ) 1. Populaion growh coms - xclusivly from h nranc of h nw housholds, whil h siz of ach houshold rmains fixd. Each gnraion is indxd by h lr j which is h im of is nranc ino h conomy. Wih w dno h conomy's im Firms Firms solv a saic profi maximizaion problm undr prfc compiion, and spcially pric-aking bhavior in boh h goods and inpu facor marks. Thy hav a noclassical producion funcion (consan rurns o scal, diminishing marginal produc of capial, and saisfacion of Inada condiions), Y F K ( ), A( ) L ( ) [1] i i i Labor fficincy A () is common for all firms, and xognous. Using h propris of h producion funcion w can r-wri i in fficincy unis, Yi yi F Ki( ), 1 f i( ) [2] A( ) L ( ) i whr () i Ki () A( ) L ( ) i Profis pr fficincy uni ar

3 3 / 35 f ( ) r( ) ( ) w( ) i i i whr w hav assumd ha h dprciaion ra is zro (alrnaivly, h producion funcion can b sn as a "n" on, afr subracing capial dprciaion). r () is h inrs ra whil w () is h wag pr labor fficincy uni. Th profi maximizaion goal in condiions of prfc compiion lads o f ( ) r( ) [3] i as wll as o zro profis, so w( ) f ( ) ( ) f ( ) [4] i i i Thrfor, givn h idnical producion funcion and h pric-aking bhavior, all firms will choos h sam raio of capial-fficin labor, and so w will hav, in ach insan of im ( ) ( ) και w( ) f ( ) ( ) f ( ) f r [5] K () No ha h raio () is an avrag for h whol conomy, and no h raio A( ) L( ) ha holds for ach houshold sparaly, as w xplain blow Inrmporal uiliy maximizaion of h houshold Each houshold sars is lif wih zro capial, and offrs in h labor mark a uni of homognous labor inlasically (h labor supply curv is indpndn of h wag). Labor fficincy () A incrass xognously a h consan growh ra g 0, and i is common o all housholds irrspciv of h gnraion hy blong o. In ohr words, hr dos no xis

4 4 / 35 individual -and hrfor possibly diffrn- accumulaion of human capial. W sandardiz h iniial lvl of fficincy o uniy, lim A ( ) 1 -. Th insananous uiliy funcion of h housholds is logarihmic. Th ra of pur im prfrnc is common for all gnraions, fixd and qual o 0. A ach insan of h conomy ach houshold of ach gnraion j has incom from his labor, and from h capial ha i owns, wih h xcpion of h housholds born in j which hav incom only from hir labor. Du o our assumpions so far, i follows ha in ach insan, labor incom is h sam for all housholds, irrspciv of gnraion. This mans ha nw housholds nr producion wih h currn fficincy lvl characrizing h conomy. Bu incom from capial diffrs, sinc whil h ra of rurn o capial is common for all h conomy a a givn insan, h housholds of ach gnraion hav diffrn lvls of capial undr hir ownrship, du o hir diffrn ag. Each houshold of gnraion j allocas is incom o posiiv consumpion and (possibly ngaiv) savings, solving h following inrmporal uiliy maximizaion problm:, max U z j ln c ( j, z ) d z j k( j, z) r( z) k( j, z) w( z) A( z) c( j, z), k( j, j) 0 [6] whr h variabl z dnos any poin in im in h ingraion inrval,, and so i is h variabl ha rprsns h "individual" im, sn and masurd from h houshold's prspciv. No ha h drivaiv of capial is akn wih rspc o z, as will b h drivaivs o follow, o h dgr ha hy rla o h rprsnaiv houshold of a gnraion, sinc h symbol rprsns h "objciv" im of h conomy. Th variabls rz () and wz () ar drmind as shown prviously. Th currn-valu Hamilonian for his problm is H ln c( j, z) ( j, z) k( j, z) [7] whr is h shadow valu of capial in currn rms. Th firs-ordr maximizaion condiions ar

5 5 / 35 H c 1 0 ( jz, ) c( j, z) [8] and so also (, ) ( j, z) ( j, z) c j z [9] c( j, z) Morovr, w mus hav H k ( j, z) ( j, z) ( j, z) r( z) ( j, z) ( j, z) [10] whil h ransvrsaliy condiion sanding a im (again in currn-valu rms) imposs lim z ( j, z ) k ( j, z ) 0 [11] z From [8], [9] and [10] w g h rul for opimal voluion of consumpion (h Eulr quaion for consumpion) as rgards h rprsnaiv houshold of gnraion j, c( j, z) r( z) c( j, z) [12] a diffrnial quaion ha has as soluion (givn ha rz () is im-varying) z z r( v) d v r( v)dv z c( j, z) c( j, ) c( j, ) [13] Insring [13] ino [8] and h rsul ino h ransvrsaliy condiion, w rquir z z k( j, z) 1 r( v)dv lim z 0 lim k( j, z) 0 z r( v)dv (, ) z z c j c( j, ) [14]

6 6 / 35 Obsrv ha h ransvrsaliy condiion rquirs, a ach insan in im, ha h plannd voluion of capial is such ha as individual im nds o infiniy, h valu of capial nds o zro in prsn-valu rms. Rmmbr ha h soluion o h inrmporal maximizaion problm in a drminisic nvironmn, dos no jus giv us h insananous opimal rul, bu h whol squnc of consumpion-capial as i is drmind by h rul (in a sochasic nvironmn, hings ar mor complicad, bu h ssnc of h ransvrsaliy condiion rmains). Chcking h rsul "a h nd" of h im horizon, in rms of h prsn, xcluds mulipl inrmporal pahs ha could b "corrcd" a som lar da, and imposs on h houshold o ac a ach insan in a way consisn wih h rmo fuur. To h dgr ha h modl has a uniqu long-run quilibrium poin (namly, a uniqu fixd poin ha a h sam im saisfis h opimal condiions, and so also h ransvrsaliy condiion), hn h currn acs of h houshold mus b such ha h houshold movs owards his poin (ohrwis i would viola h ransvrsaliy condiion). In pracic, his maks h houshold o follow a uniqu pah (h "saddl" pah, as w will s in a whil, ha lads o h long-run quilibrium), and, if som xognous aspc of h modl changs ha also changs h quilibrium poin, o mov immdialy o h nw saddl pah. Morovr, w s ha, hrough opimal bhavior, h ransvrsaliy condiion covrs also h solvncy condiion, sinc h righ-hand sid of [14] conains h inrs ra and no h ra of pur im prfrnc: h solvncy condiion says ha if capial is ngaiv (h houshold is a dbor), h db canno incras a a ra qual or highr han h inrs ra. Th solvncy condiion rquirs ha [14] is qual or highr han uniy. Bu if i was highr, h ransvrsaliy condiion would b violad. In pracic, h condiion says ha i suffics ha h houshold pays a par of h inrs charg (vrify i for consan inrs ra and consan growh ra of h db, i.. for h long-run quilibrium cas) Th inrmporal budg consrain of h houshold W can us h abov o driv h inrmporal budg consrain of h houshold, in ordr o vnually arriv a an xprssion for consumpion as a funcion of walh. To simplify noaion, w dfin R, z z r( v) dv facor o solv h diffrnial quaion rlad o capial. No ha, which can b usd as an ingraing

7 7 / 35 z z d d r( v)d v r( v)dv d z R, z r( v)d v R, zr( z) dz dz dz [15] hav R-arranging h law of moion of capial, and muliplying by h ingraing facor, w R z k j z r z k j z R z w z A z c j z, (, ) ( ) (, ), ( ) ( ) (, ) d dz R, zk( j, z) R, zw( z) A( z) c( j, z) Ingraing from o infiniy: d dz R, z k( j, z) d z R, z w( z) A( z) c( j, z) dz lim R, z k( j, z) R, k( j, ) R, z w( z) A( z)d z R, z c( j, z)dz z From h ransvrsaliy condiion [14] w hav lim R, z k( j, z) 0. Also, i is asy o z dduc ha R, 1. Using hs and r-arranging w nd up wih R, z c( j, z)d z k( j, ) R, z w( z) A( z)dz [16] Sanding a im poin, h prsn valu of currn and fuur consumpion of h houshold ha blongs o gnraion j quals h valu of is currn capial k( j, ) plus h prsn valu of currn and fuur incom from labor. No ha h discouning is don hrough h us of h inrs ra, and no of h ra of pur im prfrnc ρ. This happns bcaus hr w rcord h budg consrain, which is xprssd in rms of valu, no of uiliy. Thrfor, h prsn valu of consumpion is discound by is opporuniy cos, in rms of capial accumulaion forgon (by consuming w do no sav, do no accumula capial, losing h fuur rurns of h capial w did no accumula).

8 8 / 35 No also ha h prsn valu of fuur incom from labor, is, i oo, idnical for all houshold in h conomy irrspciv of gnraion (h indx j is no includd in h ingral). This is du o h assumpions prsnd in h bginning, bu also o h fac ha all houshold, irrspciv of gnraion, hav infini im horizon. So no mar how much ach has livd up o now (and hy hav livd diffrn lnghs of im sinc hy wr born a diffrn poins in h conomy's im), hy sill hav o liv xacly h sam amoun of im (infiniy). So w will wri R, z w( z) A( z)d z w( ) [17] 1.4. Currn consumpion as a funcion of walh W wan o xprss currn consumpion as a funcion of wid-sns walh (capial plus prsn valu of fuur labor incom). W can calcula h prsn valu of fuur consumpion in rms of currn consumpion. W r-arrang h Eulr quaion [12] and muliply by h ingraing facor: R, z c( j, z) r( z) c( j, z) R, zc( j, z) d R, zc( j, z) R, zc( j, z) dz Ingraing from o infiniy d dz R, z c( j, z) d z R, z c( j, z)dz lim R, z c( j, z) R, c( j, ) R, z c( j, z)dz z To calcula h limi w us h rlaion [13] ha drmins consumpion and w hav

9 9 / 35 z z r( v)d v ( )d z r v v z lim R, zc( j, z) lim c( j, ) c( j, ) lim 0. z z z Also, as bfor, R, 1. Givn hs, w r-arrang and obain 1 (, ) c j R, zc ( j, z )d z [18] Insring in h budg consrain [16] and using h xprssion for labor incom [17] w finally g c( j, ) k( j, ) w( ) [19] Currn consumpion of ach houshold is a consan proporion and qual o of widsns walh. No ha whil h lvl of consumpion dpnds on h inrs ra (hrough h prsnc of i in h discouning facor R,zwhich is includd in w () and discouns h fuur incom from labor), h marginal propnsiy o consum dos no dpnd on h inrs ra. This is a consqunc of h assumpion ha h uiliy funcion is logarihmic, and so ha h inrmporal lasiciy of subsiuion in consumpion is qual o uniy Aggrgaion Givn ha populaion grows a a consan ra n (sricly du o h nranc of nw housholds), oal populaion a im j is uniy). Th populaion of h gnraion born a im j is hn h populaion of ach gnraion rmains fixd. jn (sinc w hav normalizd h iniial populaion o d dj jn n jn. W rmind ha So oal consumpion and oal capial pr gnraion a im ar givn by h rlaions C( j, ) n jn c( j, ) and K( j, ) n jn k( j, ) corrspondingly, whil aggrga consumpion and aggrga capial a im ar

10 10 / 35 jn C( ) n c( j, )dj [20] jn K( ) n k( j, )dj [21] whr in ordr o aggrga w ingra wih rspc o h gnraion indx j, and no wih rspc o im (alhough sinc a ach poin in im a nw and immoral gnraion is born, in pracic h inrval of ingraion coincids wih h whol pas of h conomy. Nvrhlss, aggrgaion is of cours cross-scional, no mporal). Toal incom from labor a im is w( ) A( ) n sinc i is h sam for all housholds, irrspciv of gnraion. Thrfor h prsn valu of fuur labor incom of all housholds currnly aliv is ( ), ( ) ( ) n n W R z w z A z d z w( ) [22] Insring [19] in [20] w hav jn C( ) n k( j, ) w( ) dj jn jn n k( j, ) dj n w( )dj Using also [21] w hav jn 1 C( ) K( ) nw( ) d j K( ) nw( ) n jn C( ) K( ) w( ) n w( ) lim j - jn Th las gos o zro sinc w () is fini (ohrwis from [19] w would g infini n consumpion). From quaion [22] w hav w( ) W( ), so finally C( ) K( ) W( ) [23]

11 11 / 35 namly, a macroconomic rlaion "analogous" o h quaion [19] ha holds for ach houshold individually (vn hough housholds ar diffrniad from gnraion o gnraion wih rspc o h capial ha hy own). W ar inrsd in h ra of chang of aggrga consumpion: C( ) K( ) W ( ) [24] whr hr drivaivs ar akn wih rspc o h conomy's im. To calcula h im-drivaiv of aggrga capial and of aggrga prsn valu of fuur incom from labor, w apply Libniz's formula: L bs ( ) H ( x, s ) h ( x, s )d x. Thn h drivaiv of Η wih rspc o s quals as ( ) d H( x, s) d b( s) d a( s) bs ( ) h( x, s) h( b( s), s) h( a( s), s) dx ds ds ds as ( ) s In our cas, w hav, for aggrga capial, diffrniaing [21]: d K ( ) d jn n jn n k( j, )d j n k(, ) n k( j, )dj d d Th firs rm is zro by assumpion, sinc nw housholds sar lif wihou capial. For h scond rm, w us h law of moion of individual capial (now wrin in rms of h variabl ) and w hav: d K ( ) jn n r( ) k( j, ) w( ) A( ) c( j, ) dj d and using [21] and [20] w g n K( ) r( ) K( ) w( ) A( ) C( ) [25]

12 12 / 35 Equaion [25] jus says ha h chang in aggrga capial quals oal savings, sinc h firs wo rms qual h oal oupu of h conomy bcaus of closd conomy => oupu is xhausd in consumpion and savings, consan rurns o scal in producion and prfcly compiiv marks => producion inpus g paid hir marginal produc xhausing oal oupu (zro profis) inlasic supply of labor and capial => h mployd inpus qual o oal availabl sock (full mploymn of labor and capial) and givn also ha w hav assumd ha h dprciaion ra is zro (or subsumd in r). W could hrfor driv [25] dircly. Bu by obaining i hrough h ingraion procss, h final ncssary condiion was rvald, namly ha nw housholds bring no capial in h conomy (which would rprsn an "xrnal" infusion of capial). For h prsn valu of incom from labor w hav: d d d ( ), ( ) ( ) d, ( ) ( )d d W n n d R z w z A z z d R z w z A z z d n n n w( ) R, zw( z) A( z)dz d n n R, z n w( ) R, w( ) A( ) w( z) A( z) dz No ha h drivaiv of h discoun facor, dos no qual [15] hr w R z diffrnia wih rspc o h lowr bound of h inrval of ingraion ha is includd in R, z. W hav z z r( v)d v r( v)dv z R, z r( v)d v R, zr( ) Subsiuing and rmmbring ha R, 1 w g

13 13 / 35 d n n n W( ) n w( ) w( ) A( ) w( z) A( z) R, zr( )dz d d n n n W( ) n w( ) w( ) A( ) r( ) w( ) d and so n W( ) r( ) n W ( ) w( ) A( ) [26] Insring quaions [25] and [26] in h xprssion for h voluion of aggrga consumpion [24] w g C( ) r( ) K( ) w( ) A( ) n C( ) r( ) n W ( ) w( ) A( ) C( ) r ( ) K( ) W( ) C( ) nw ( ) n R-arranging [23] in ordr o rplac W () and subsiuing w hav r( ) C( ) K( ) C( ) C( ) C( ) n and simplifying, C( ) r( ) n C( ) nk( ) [27] which oghr wih h law of moion of aggrga capial [25] characrizs h macroconomic sysm. No ha w wr abl o limina from h xprssion for aggrga consumpion and capial h prsnc of discound fuur valu of labor incom.

14 14 / 35 EXERCISES-QUESTIONS 1.1 Th xprssion for consumpion as a funcion of walh appars o b dircly analogous bwn houshold lvl (q.[19]) and aggrga lvl (q. [23]). Is i indd? Discuss. In any cas, h law of moion appars o b srucurally diffrn (quaions [12] and [27] rspcivly). Discuss how and why his diffrnc mrgs. (Hin: which is h basic characrisic of nw housholds?) 1.2 Th modl prdics ha currn consumpion a houshold/individual lvl is a funcion also of fuur incom from labor, incom ha has no marializd y. Would ha b logically consisn in a modl whr borrowing was no allowd? 1.3 As in many ohr macroconomic modls, h ovrlapping-gnraions modl of Blanchard and Wil assums fully inlasic labor supply. Is his an accpabl simplificaion of raliy whn w ar inrsd in h long-rm voluion and quilibrium of an conomy? 1.4 Th modl assums "immoral" housholds. Is his assumpion rlad o h individuals ha compris h housholds, or i could rla o h houshold as an insiuion, a "dynasy"? Undr which addiional assumpions could w adop his las inrpraion, wihou affcing a all h modl's srucur and rsuls? 1.5 Why dos i suffic o pay only par of h inrs charg on our db, no vn all of i, in ordr o rspc h solvncy condiion, whn h im horizon is infini? Would our crdiors b conn?

15 15 / Long-run Equilibrium and is propris 2.1. Long-run quilibrium in rms of avrag consumpion and capial Givn ha h populaion and h fficincy of labor incras consanly and xognously, and also du o h infini horizon, his conomy dos no possss a long-run macroconomic quilibrium in rms of lvls of macroconomic variabls, or vn in "pr capia" rms, which also grow consanly as on can dduc from quaions [25] and [27]. Bu w can xprss h variabls as avrags in fficincy unis. W hav C () c( ) C( ) c( ) A( ) L( ) K () ( ) K( ) ( ) A( ) L( ) ng ng Diffrniaing wih rspc o im, ( ) ( ) n g ng C c n g c( ) [28] ( ) ( ) n g ng K n g ( ) [29] Equaing [28] wih [27] w hav ng ng c( ) n g c( ) r( ) n C( ) nk( ) c( ) f ( ) g c( ) n( ) [30] and quaing [29] wih [25] w g ng ng n ( ) n g ( ) r( ) K( ) w( ) A( ) C( ) ( ) f ( ) n g ( ) c( ) [31]

16 16 / 35 whr w hav also usd h rlaions in [5] for h xhausion of oupu. Rlaions [30] and [31] form h basic sysm of diffrnial quaions of h modl. Th zro-chang loci of avrag consumpion and avrag capial ar givn by h rlaions c( ) 0 c and n f g [32] ( ) 0 c f n g [33] whr w hav usd inqualiy o indica also h dircion of movmn off h zrochang loci. In h c phas diagram, h locus c ( ) 0 sars from h bginning of h axs and incrass xponnially as avrag capial incrass, wih an asympo vrical o h axis a h poin whr f g. Th locus ( ) 0 sars from h bginning of h axs, has a maximum a h poin whr f n g and crosss h axis a h poin whr f n g Sabiliy of h long-run quilibrium To xamin h sabiliy of h long-run quilibrium of his non-linar sysm of diffrnial quaions, and whhr i is characrizd by "saddl-pah" sabiliy or no, w xamin h ignvalus of h Jacobian marix of firs drivaivs of h sysm valuad a h fixd poin: J c c c, c, f c g f c n 1 f n g c, c, c

17 17 / 35 2 Th corrsponding characrisic quaion is ignvalus ar r J d J 0 and h 1,2 2 J J J r r 4d 2 W know ha in a 2 X 2 sysm of non-linar diffrnial quaions, ncssary and sufficin condiion for saddl-pah sabiliy is ha h drminan of h Jacobian is ngaiv, so ha boh ignvalus ar ral, and on is ngaiv and h ohr is posiiv (h sign of h rac of h Jacobian hn plays no rol). W hav J d f g f n g f c n Using h zro-chang rlaions (as qualiis) w hav from [32] f g n c whil from [33] w can g f c f c n g f n g f Insring hs ino h drminan d J w hav d J n c f f f c n c n d J f f n f c n c n d J w f c 0 c

18 18 / 35 whr w hav usd [4] for h drminaion of h ral wag, bu also h concaviy of h producion funcion f characrizd by saddl-pah sabiliy. 0. Th drminan is ngaiv, and so h sysm is 2.3. Dynamic infficincy of h long-run quilibrium An conomy is characrizd by "dynamic infficincy" whn a rducion in is long-run quilibrium lvl of capial lads o an incras in long-run quilibrium consumpion. In h bnchmark rprsnaiv houshold modl, such a possibiliy is xcludd by h assumpion n, ha is ndd hr in ordr no o viola h ransvrsaliy condiion, sinc in h rprsnaiv houshold modl h paramr n rprsns h growh ra of h siz of ach houshold (and so i nrs h inrmporal uiliy funcion and vnually h ransvrsaliy condiion). In h ovrlapping-gnraions modl n rlas o h incras in h numbr of housholds, and i dosn' affc h opimizaion problm of ach houshold. Thrfor h rlaion bwn and n dos no affc h ransvrsaliy condiion. Morovr, as on can vrify, h assumpion n dos no affc h saddl-pah sabiliy propry of h modl, and so i rmains consisn wih h basic assumpions of h modl and i can b allowd. In cas n holds, hn h long-run quilibrium may b characrizd by dynamic infficincy. W prsn wo phas diagrams, wih h firs rflcing h usual assumpion n. c n c 0 c OGM 0 OGM :fg :f n g

19 19 / 35 If n, h following siuaion is possibl : c n c OGM 0 c 0 OGM :f g :fng Th conomy is dynamically infficin, i ovr-savs: i could incras is long-run quilibrium consumpion by rducing h long-run quilibrium capial. Nvrhlss, usually w assum n so ha h modl is comparabl o h rprsnaiv houshold modl. EXERCISES-QUESTIONS 2.1 A dynamic sysm ha is characrizd by "saddl-pah sabiliy", in raliy is considrd unsabl from a mahmaical poin of viw. Why, in conomic modls do w "prfr" a saddl-pah sabl sysm, from on ha is proprly sabl? Wha would i imply for h ral-world conomy and h aciviy of h humans in i, if i was characrizd by sabiliy propr? 2.2 Labor and incom from labor do no affc dircly h long-run quilibrium of h conomy. Discuss. 2.3 Prov ha in h long-run quilibrium h inqualiy n r g holds. Thn, combin i wih h usual assumpion n h usual arihmic valus hs paramrs ak?. Ar hs inqualiis validad by

20 20 / Avrag and individual magniuds in long-run quilibrium: h disribuion, and h inqualiy, of consumpion and walh An imporan conribuion of h ovrlapping-gnraions modl of Blanchard and Wil is ha i allows us o disinguish bwn individual variabls and hir avrags. In basic modls of rprsnaiv houshold, such a disincion is los sinc all housholds ar assumd idnical. W will sudy h issu a h long-run quilibrium. Morovr, w will driv h disribuions of consumpion, walh and capial, as wll as h corrsponding Gini indics of inqualiy Individual consumpion and capial in fficincy unis. In ordr o conras individual variabls wih hir avrags, w dfin a nw variabl, individual consumpion pr fficincy uni: g c ( j, ) c( j, ) [34] W ar inrsd in how consumpion is disribud among housholds in h sam insan of im. Sinc our modl is drminisic, in raliy w will driv h (coninuous) "rlaiv frquncy" funcion, rahr han h "probabiliy dnsiy" funcion. Sill w can us h rminology and h concps ha prain o random variabls, rmmbring ha hr hy hav dscripiv rahr han infrnial foundaions. Firs, w vrify ha h variabl c ( j, ) has as is avrag h variabl c () (avrag consumpion pr fficincy uni) a ach poin in im. W mus adjus, for ach gnraion j, h variabl c ( j, ) by h rlaiv wigh ha his gnraion has in h populaion. Th (fixd) siz of gnraion j is jn n, whil oal populaion a im is n. So h wighing facor is n n jn n n( j). Thrfor, ( ) ( ) n j n jn g n g jn E c ( j, ) n c ( j, )d j n c( j, ) d j n c( j, )dj

21 21 / 35 ( ) n g E c ( j, ) C( ) c( ) [35] So indd h variabl "individual consumpion pr fficincy uni" c ( j, ) has as is xpcd valu h variabl "avrag consumpion pr fficincy uni" c (). Thn, w wan o driv h disribuion of c ( j, ), spcifically is rlaiv frquncy funcion, say hc c (, ) j, a ach poin in im (afr birh). Sinc h modl is in coninuous im, w can us h mhods for obaining a probabiliy dnsiy funcion. Obsrv ha h funcion h ( j) n j n j, which was usd in h calculaion of h xpcd valu of c ( j, ), saisfis h propris o b a probabiliy disribuion funcion for h variabl j wih suppor,, sinc i is vrywhr non-ngaiv, and w n j. H( j) n dj 1 If w could xprss c ( j, ) as a coninuous, diffrniabl, monoonic (and dircly invribl) funcion of j for vry, c ( j, ) ( j),, hn w could apply h "changof-variabls" formula, according o which, 1 1 c (, ) j dc d c h c j h c [36] and w would g h probabiliy dnsiy funcion of c ( j, ), which is wha w ar afr. R-arranging h Eulr quaion for individual consumpion [13] w hav c( j, ) c( j, z) z r( v)d z v Sing z j w hav ( )d r v v j j and insring his ino [34] w obain c( j, ) c( j, j) r ( v )d v j c ( j, ) c( j, j) j g [37]

22 22 / 35 Nx, from quaion [19] ha xprsss consumpion as a funcion of wid-sns walh, c( j, ) k( j, ) w( ), sing j w hav, givn also ha k( j, j) 0, h rlaion c( j, j) w( j) and so r ( v )d v j j c ( j, ) w( j) g whr w( j) is h prsn valu of fuur labor incom for ach houshold, irrspciv of gnraion, whn h conomy's im aks h valu j, z r ( v ) dv j w ( j ) R j, z w ( z ) A ( z )d z w ( z ) A ( z )d z j j So z r( v)d v r( v)dv c ( j, ) w( z) A( z)dz j j j g [38] j This rlaion canno b invrd o giv us j as a dirc funcion of c ( j, ). To mov forward, w will consrain ourslvs o h xaminaion of only hos gnraions ha ar born on and afr h sing of long-run quilibrium Th disribuion of individual consumpion in long-run quilibrium In long-run quilibrium, ral wag and h ral inrs ra ar consan: r( ) r, w( ) w. W xamin only housholds born on or afr long-run quilibrium Th prsn valu of fuur labor incom whn j is ( ) 1 r z j gz jr ( r g) z w jg w( j) w dz w [39] j r g j r g No ha hr j dos no rflc h gnraion, bu i is h arihmic valu ha h variabl aks, sinc ssnially w ar calculaing h valu of c( j, j ).

23 23 / 35 Morovr, r d v j r j. Insring his and [39] ino [38], w g w jg w c ( j, ) r g r g j r j g rg j [40] whr w hav usd h sar o rflc ha w xamin only gnraions of h long-run quilibrium. Wih rspc o j, quaion [40] is a coninuous, diffrniabl and monoonic funcion. I can also b invrd dircly. Spcifically, w w c( j, ) ln c( j, ) ln r g j r g r g r g j 1 1 c j c j r g r g r g w ln ln (, ) 1 Also, 1 d c 1 dc r g c ( j, ) Insring hs wo rsuls in [36] w hav n 1 w ln c ( j, ) hc c( j, ) xp n ln r gc ( j, ) r g r g r g n n n w r g (, ) (, ) r g c 1 h c j c j r g r g [41] No ha quaion [40], ha dfins individual consumpion pr fficincy uni for h gnraions of long-run quilibrium, is sricly incrasing in h conomy's im, and h smalls valu ha can ak for a houshold of gnraion j is j, for which w obain h

24 24 / 35 minimum consumpion ha h houshold of his gnraion will hav. This quals min c ( j, ) w r g. Sing for compacnss r g n, w can wri h c ( j, ) c min c ( j, ) [42] 1 c ( j, ) Th funcion c (, ) h c j is now rcognizabl as h probabiliy dnsiy funcion of a Paro disribuion wih shap paramr, (a disribuion ha is ofn usd in mpirical modling of incom and walh), wih gnral form our cas n r g. Th xpcd valu of his disribuion xiss if pdf ( X ) x x min 1 and shap paramr in 1 n r g 1, somhing ha holds in our cas (s xrcis 2.3) Thn xmin EX ( ) 1 and so n w E c ( j, ) c n r g r g [43] Tha his xprssion quals c can b vrifid if ons manipulas algbraically h fundamnal quaions of h modl c ( ) 0 και ( ) 0. If morovr w hav n r g 2 hn h varianc xiss also, Var( X ) ax 2 min 2 a1 a2, and in our cas, Var( c ( j, )) 2 2 n r g n 2r g n r g w r g 2 [44]

25 25 / 35 A qusion ha can b askd hr is h following: From [35] w know ha E c (, ) ( ) j c, whil w also obaind quilibrium implis E( c ( j, )) c. Th pah owards long-run c c E c j E c j ( ) (, ) (, ) Bu h variabl c ( j, ) rprsns all housholds in h conomy, housholds ha, having infini im horizon, nvr sop paricipaing in i. On h ohr hand, h variabl c ( j, ) rprsns only housholds born on or afr long-run quilibrium, a sric subs of h formr. How is i possibl ha h xpcd valu of c ( j, ) is h limi of h xpcd valu of c (, ) j ; Th answr lis wih h consanly and xponnially growing populaion, somhing ha maks oldr gnraions asympoically ngligibl for h disribuion, vn hough hy ar hos ha njoy h highs consumpion. Bu hir siz rlaiv o h populaion gos o zro fasr han hir consumpion incrass (s xrcis 3.1). Morovr, in ordr o calcula h xpcd valu of c ( j, ), w ingrad ovr min c ( j, ),. Bu in ordr for h consumpion of h gnraions born ino long-run quilibrium o go o infiniy, h lngh of im for which h conomy is in long-run quilibrium mus also nd o infiniy. In such a cas h conribuion of h housholds bfor long-run quilibrium vanishs asympoically. A scond qusion is: w know from [40] ha h variabl c ( j, ) incrass for all houshold as h conomy's im passs (his mans ha in his modl, vn individual magniuds pr fficincy uni incras consanly and don' convrg). How is i possibl ha h disribuion mainains a consan xpcd valu and a consan varianc, so ha w ar abl o alk abou "long-run quilibrium" in his "avrag" sns? Th answr lis again in h coninuing nranc of nw housholds: in ach insan, h consumpion pr fficincy uni of pr-xising housholds incrass bu in h sam insan nw housholds nr which consum in hir firs priod of lif h minimum lvl w r g and hy ar h mos populous gnraion of all. I is hrough his "counrbalancing" ha h avrag and h varianc of h disribuion rmain consan. W hav a siuaion whr h

26 26 / 35 consumpion lvl of all incrass, bu h corrsponding avrag (bing a wighd avrag wih rspc o h disribuion of populaion among gnraions) rmains fixd! W no ha from h consrain r g 0 1 w g ha r g min (, ) c j w. So all nw housholds (ha do no posss capial) do no consum all hir incom during h firs priod (incom ha coms only from work), bu hy sar immdialy o sav. W no finally, ha h mdian of h Paro disribuion is mdian( X) 2 1/ xmin and in our cas w r g rg mdian( c ( j, )) 2 n [45] Th Paro disribuion is non-symmric, and is mdian lis o h lf of is xpcd valu. This mans ha h prcnag of housholds consuming lss han avrag is grar han 50%. Finally h Gini cofficin, (which is an incrasing masur of inqualiy wih minimum valu zro and maximum valu uniy), quals for h Paro disribuion Tchnical Appndix). For our cas, 1 G 2 1 (s Gc ( ) r g 2n r g [46] 3.3. Th disribuion of capial in long-run quilibrium W urn now o sudy h disribuion of h individual/houshold capial, pr fficincy uni. As bfor, w look only a h housholds ha ar born in long-run quilibrium. Analogously wih consumpion w dfin h variabl k ( j, ) k( j, ) g as wll as h variabl k ( j, ) ha rlas o hs housholds. A individual lvl, h xprssion of consumpion as a funcion of wid-sns walh [19] bcoms now

27 27 / 35 w c( j, ) k( j, ) r g [47] Dfining h wid-sns walh for housholds born in long-run quilibrium as w V ( j, ) k( j, ) rg, w can invr and xprss i as a linar funcion of c ( j, ), 1 V ( j, ) c( j, ) [48] of Thrfor, applying again h "chang-of-variabl" formula, h probabiliy dnsiy funcion V ( j, ) is w hv V j hc V j V j r g (, ) (, ) (, ) 1 [49] which is again a Paro disribuion sinc V j rg min (, ) w, and wih h sam shap paramr ha characrizs h disribuion of consumpion. Th xpcd valu and varianc can b obaind dircly from [48] as 1 1 E V ( j, ) E c ( j, ), Var V ( j, ) Var 2 c ( j, ) [50] Turning o capial k ( j, ), i quals k ( j, ) V ( j, ) w V ( j, ) min V ( j, ) r g [51] Thrfor, capial follows a Lomax disribuion (or Paro yp ΙΙ), wih probabiliy dnsiy funcion h k j w k j r g w (, ) (, ) 1 k r g [52] Is xpcd valu is

28 28 / 35 w n w w E k( j, ) E V ( j, ) r g n r g r g r g w r g E k ( j, ) r g n r g [53] From h zro-chang loci of h phas diagram, on can vrify ha (, ) As rgards h varianc of capial, w hav E k j. 2 2 n r g n 2r g n r g w Var k( j, ) Var V ( j, ) r g 2 k j c j Var (, ) Var (, ) [54] 2 as i should b xpcd, sinc h capial variabl is an affin funcion of consumpion. Th Gini cofficin for h Lomax disribuion is (s Tchnical Appndix) Gk ( ) n 2 1 2n r g [55] Givn ha 1 h modl prdics a rlaivly grar inqualiy in h disribuion of capial han in consumpion, in long-run quilibrium. Th diffrnc may b larg for xampl, for usual valus of h paramrs, h Gini cofficin for capial may b doubl h on for consumpion. Th ovrlapping-gnraions modl of Blanchard and Wil lads o a Paro disribuion for consumpion and a Lomax disribuion for capial in long-run quilibrium. Th srngh of h rlaiv inqualiy as rflcd in h Gini cofficin dpnds on h various modl paramrs, bu also on h inrs ra. Inqualiy is always highr as rgards capial, whr h sourc of inqualiy originas: h diffrn lvls of capial ownd by diffrn gnraions du o hir diffrn ag, and h absnc of any inrgnraional alruisic/bqus moiv. Inqualiy xiss as a propry of h long-run quilibrium. I is consan as rgards is rlaiv siz, bu i is no saic: vry priod all housholds improv hir posiion.

29 29 / 35 EXERCISES - QUESTIONS 3.1 Prov ha h rlaiv wigh in h populaion of h siz of a gnraion born bfor long-run quilibrium, nds o zro fasr han is consumpion (pr fficincy uni) nds o infiniy, afr h conomy is in long-run quilibrium. Hin: Wha is h (fixd) siz of a gnraion? Wha is h (incrasing) siz of populaion? Us q. [38] in long-run quilibrium. 3.2 Wih h conomy in sady-sa, calcula h lngh of h priod j ha a nw houshold of gnraion j nds in ordr o rach a) mdian consumpion and b) man consumpion (i concrns h variabl c ( j, ) ). 3.3 Do h sam for wid-sns walh, as wll as for capial (afr calculaing or finding in h liraur h mdian of h Lomax disribuion). Is hr a diffrnc wih wha you found in 3.2 abov? In any cas, commn on your findings. 3.4 How ar h Gini cofficins for consumpion and capial affcd by an incras in h paramrs ha drmin hm? Suppor h algbraic findings wih conomic argumns. 3.5 Basd on bnchmark valus for h paramrs n,, r, g as you can find hm in h liraur, xamin whhr h inqualiy n r g holds (which prmis us o dfin h xpcd valu), bu also, whhr h inqualiy n 2 r g holds (which prmis us o dfin h varianc) for h Paro and Lomax disribuion. If hy do no hold, which paramr, in your opinion and from your liraur sarch, is rlaivly lss foundd on mpirical daa for is quanificaion, and so on could ak mor libris wih is valu? Rmmbr ha h modl also rquirs r g 0 for i o b wll-dfind and o possss a long-run quilibrium, as wll as infficincy and ovr-accumulaion of capial. n so ha w do no hav dynamic 3.6 Afr you sl on a s of arihmic valus ha allows you o dfin a las h xpcd valu, calcula h rsuling Gini cofficins for consumpion and capial, and compar your findings wih corrsponding simas drmind hrough mpirical daa, and can b found in h liraur. Discuss. 3.7 <<Wha should concrn us is magniuds "pr capia"(pr houshold), no magniuds pr labor fficincy uni.>> Discuss. Can you xamin mars of inqualiy of consumpion and capial in "pr capia" rms?

30 30 / Comparison wih h Rprsnaiv Houshold modl Comparing h ovrlapping gnraions modl (O-G) wih h rprsnaiv houshold modl (R-H) (wih logarihmic prfrncs), i appars hr ar wo imporan diffrncs in hir assumpions: firs, in h O-G modl housholds diffr in h lvl of capial ha hy own. Bu his provs no o b criical. I has bn shown ha h R-H modl can accommoda inqualiy in walh (and vn in produciviy) wihou changing is basic rsuls: in such a varian, hr is of cours inqualiy in walh and consumpion (as is h cas wih h O-G modl), bu h avrags of consumpion and capial ar xprssd wih h sam rlaions ha ar o b found in h bnchmark R-H modl (his implis ha, afr all, in a rprsnaiv houshold sing, w ar no obligd o assum fully idnical housholds, as long as hy shar som criical characrisics). Th scond, and criical diffrnc, is h fac ha in h R-H modl populaion incrass in h sns of an incras in h siz of ach houshold, whos oal numbr rmains fixd ("dynasis"). In h O-G modl, populaion incrass in h sns of an incras in h numbr of housholds, whil h siz of ach houshold rmains fixd (and w can hink abou i as a singl individual). On could obsrv ha as rgards h numbr of naural prsons, populaion is xacly h sam in boh cass a any insan in im (wih sam growh ra and iniial populaion). Bu h populaion growh assumpion in h O-G modl cras an incras in h numbr of disconncd conomic unis in h conomy, and his provs o b h criical fac for is conclusions. (No: mor accuraly, h criical faur is ha w hav an incras in h numbr of disconncd conomic unis ha solv an inrmporal opimizaion problm. Incrasing h numbr of firms for xampl would no mar, sinc hy ar assumd o solv a saic opimizaion problm). A a mahmaical lvl, i is his assumpion ha lads o a diffrn xprssion for avrag consumpion bwn h O-G and R-H modls, wih h apparanc of h rm n() in h diffrnial quaion for avrag consumpion [30], a rm ha dos no xis in h corrsponding quaion in h R-H modl. I is his rm ha maks h lvl of avrag consumpion an incrasing funcion of avrag capial (whil in h R-H modl, i is h ra of chang and no h lvl of avrag consumpion ha dpnds on h lvl of capial, rsuling in h vrical zro-chang locus in h phas diagram).

31 31 / 35 In a rprsnaiv houshold modl wih inqualiy in capial ownrship, h avrag consumpion and capial would b a ru avrag, wih individual magniuds disprsd around i. Bu h lvl of avrag consumpion would rmain indpndn of h lvl of avrag capial in h zro-chang locus rlaionship, as happns in h sandard R-H modl. In h O-G modl w hav inqualiy and disprsion, and a srucural chang in how h macroconomic magniuds ar rlad. This dos no com from h opimizing bhavior of individual housholds, bu i mrgs as a propry of h macroconomic variabls. Namly, i is a propry ha ariss only a h macroconomic lvl, bu wihou violaing any of is micro-foundaions. Th mrgnc of nw phnomna a h macroconomic lvl ha ar no jus blown-up vrsions of h microconomic variabls ha ar foundd in opimizing bhavior, is an imporan horical conribuion of h ovrlapping gnraions modl. Don' ry o "xplain inuiivly" his phnomnon, hinking abou h acions of a...rprsnaiv houshold in h ovrlapping-gnraions modl, and hn projc hs individual acions a h macro-conomic lvl. This kind of argumn is usual and prinn in rprsnaiv-houshold modls, bu no hr. In h O-G modl, h macroconomic phnomnon rsuls hrough h inracion and combinaion of h aggrgad individual bhavior wih xognous aspcs of h nvironmn, which in our cas is h apparanc of nw housholds ha ar unrlad o h xising ons. Spcifically, () is h avrag consumpion ha xis du o h xisnc of marial capial (rmmbr ha is h marginal propnsiy o consum), and i is drmind by opimizd individual bhavior. A h sam im, n is h prcnag in h whol populaion, of h siz of h nwborns a ach insan, which by assumpion, hy do no posss capial and hy ar no givn capial by h ldrs. Thrfor, h produc n() is h prcnag of avrag consumpion ha in vry insan, is no ralizd, and xprsss xacly h combind ffc w mniond abov. In long-run quilibrium, and undr h assumpion n, avrag consumpion and avrag capial ar lowr in h O-G modl han in h R-H modl. Th mpaion hr is o ask: sinc in h O-G modl individual masurs ar disribud around is avrags, ar hr housholds in h O-G modl ha consum mor in long-run quilibrium han R-H housholds? Mhodologically, h corrc approach is o compar h variabl "individual consumpion pr fficincy uni" from h O-G modl, wih h variabl "pr capia consumpion in fficincy uni from h R-H modl, and calcula which prcnag of h formr xcds h lar.

32 32 / 35 No h following imporan conclusion ha w arrivd a in h prvious chapr: in h ovrlapping-gnraions modl, individual consumpion pr fficincy uni nvr sops growing (why?). On h conrary, in h R-H modl individual consumpion pr fficincy uni is consan in h sady-sa. Bu hn, shouldn' his man ha, all xising housholds in h O-G modl will consum mor han h housholds in h R-H modl? No. Wha happns in h O-G modl is ha h minimum consumpion pr fficincy uni obsrvd in h conomy rmains consan, and lowr han h sady-sa consumpion of a R- H houshold. Wha incrass is h uppr bound, ha nds o infiniy. Thrfor, in h O-G modl vry houshold/individual will vnually consum mor han h individual/mmbr of h houshold-dynasy of h R-H modl, bu no bfor living hrough a priod ha i will b consuming lss. And hr will always b som individuals/housholds in h O-G modl ha hy will consum lss han h individuals/mmbrs of h houshold-dynasis in h R-H modl. Rurning o our qusion, w know ha h avrag consumpion in h O-G modl follows a Paro disribuion, and so h cumulaiv disribuion funcion is x DF( X ) Pr X x 1 Pr X x x and in our cas min Pr c ( j, ) c 1 n w rg rg c [56] If w s c qual o h sady-sa consumpion in h R-H modl c RH, w can obain h prcnag of housholds in h O-G modl whos consumpion pr fficincy uni will xcd c RH, n n w rg min (, ) r g r g c j RH crh Pr c ( j, ) crh c [57] whr hr oo w inrpr "probabiliy" as "rlaiv frquncy". Bu from [43] w know ha

33 33 / 35 n ( r g) c E c( j, ) min c( j, ) n ( r g) 1 n ( r g) 1 n r g min c ( j, ) c c n ( r g) n Subsiuing in [57] w obain alrnaivly n r g c Pr c ( j, ) crh n crh n rg [58] No ha h wo raios insid h parnhsis ar ach smallr han uniy whil h xponn is grar han uniy, bu also ha h inrs ra rlas o h O-G modl. W srss again ha hr w do no compar housholds, or rahr, ha h maning of "houshold" in h wo modls is ssnially diffrn: To b xac, quaion [58] givs us, in long-run quilibrium, h prcnag of holdrs of on uni of physical labor in h O-G modl, whos consumpion pr fficincy uni is grar han ha of h holdrs of on uni of physical labor in h R-H modl (sinc in all cass h fficincy lvl is idnical bwn individuals and bwn modls). In h O-G modl h "holdr of on uni of physical labor" is idnical wih h concp of "houshold". In h R-H modl, h holdr of on uni of physical labor is a mmbr of a houshold-dynasy. Rvrsly, vry mmbr of a houshold-dynasy in h R-H modl ssnially corrsponds o a houshold in h O-G modl. Bu hy ar comparabl concps basd on h physical labor ha hy offr (again, bcaus fficincy is idnical o all). EXERCISES-QUESTIONS 4.1 Assuming logarihmic prfrncs, driv h sady-sa individual consumpion pr fficincy uni in h rprsnaiv houshold modl (using, if ndd, h sady-sa wag and inrs ra as consans, wihou assuming any paricular form of h producion funcion). Driv h quaion ha xprsss h im ha a nw individual/houshold in h ovrlapping-gnraions modl (wih h xac sam paramrs) will nd in ordr o rach h lvl of sady sa consumpion of h R-H houshold. No ha h sady-sa wag and h inrs ra ar diffrn in h wo modls.

34 34 / Tchnical Appndix: Th Gini cofficin for h Paro and Lomax disribuions For a coninuous, non-ngaiv random variabl X wih xpcd valu and disribuion funcion Fx ( ), h Gini cofficin is dfind as 1 G F( x)[1 F( x)]dx 0 Α. Paro disribuion Th Paro disribuion has disribuion funcion xmin xmin ( ) 1, min 0, 0, for 1 F x x E X x 1 Thrfor 1 x 1 x G x dx min min d x xmin xmin min x x min x 2 1 x 1 x 2 min 1 1 min 1 21 x x x min xmin xmin 1 xmin x x 1 x x min 1 x x min min min min min min x 1 x 2 1 x min x x min min min x min G

35 35 / 35 Β. Lomax disribuion Th Lomax disribuion has disribuion funcion F( x) 1, 0, 0, E X for 1 x 1 W hav F( x)[1 F( x)] 1 x x x x 2 So G 1 dx 1 dx x x Apply h chang of variabl z x, dz d x, x 0 z o g G 1 z dz 1 z dz z z G