Productive Government Spending and its Consequences for the Growth-Inequality Tradeoff*

Transcription

1 Producve Governmen Spendng and s Consequences for he Growh-Inequaly Tradeoff* Yoseph Y. Geachew Unversy of Preora, Preora, Souh Afrca Sephen J. Turnovsky** Unversy of Washngon, Seale WA, 9895 Absrac Ths paper nvesgaes he effecs of producve governmen spendng on he relaonshp beween growh and nequaly n an economy subjec o dosyncrac producon shocks and heerogeneous endowmens. Assumng lognormal dsrbuons, we derve racable closed form soluons descrbng he equlbrum dynamcs. We show how he effec of governmen nvesmen on he equlbrum dynamcs of boh nequaly and growh depends crucally upon he elascy of subsuon beween publc and prvae capal n producon. Ths has mporan consequences for he growh- and welfaremaxmzng raes of governmen nvesmen. Fnally, we supplemen our heorecal analyss wh numercal smulaons, calbraed o approxmae he producve characerscs of a real world economy. Wh he emprcal evdence srongly supporng he complemenary beween publc and prvae capal, our smulaons sugges ha conclusons based on he commonly employed Cobb-Douglas producon funcon may be serously msleadng. Key words: Governmen nvesmen; Idosyncrac shocks; Growh; Inequaly JEL Classfcaon: D3, O4 Revsed verson June 05 *The paper has benefed from commens receved on earler versons from semnar parcpans a he Unversy of Durham and he Unversy of Preora. We also hank Sananu Chaerjee for useful suggesons. Research for hs paper was carred ou whle Geachew was vsng he Unversy of Washngon n he Fall of 03. Turnovsky s research was suppored n par by he Van Voorhs endowmen a he Unversy of Washngon. **Correspondng auhor: surn@u.washngon.edu

2 . Inroducon The role of publc nvesmen n nfrasrucure as a source of economc growh connues o be wdely debaed n boh developng and advanced economes, wh dfferen economes adopng dfferen polcy opons. Several emergng-marke counres ncludng Inda, Chna, and Brazl have underaken exensve publc nvesmen, o whch her hgh growh raes of recen years may a leas n par be arbued. In conras, several European counres have reduced publc spendng, as hey have pursued ausery measures nended o deal wh concerns relaed o her rsng deb levels. Conemporaneously wh hese dverse polces oward publc nvesmen, we have wnessed ncreasng ncome nequaly, boh n emergng markes and n mos OECD counres. Ths rases he mporan queson ha we address n hs paper, namely he relaonshp beween publc nvesmen dreced oward growh enhancemen and s consequences for ncome nequaly. Ineres n he relaonshp beween publc nvesmen, oupu, and growh has a long hsory, dang back o Arrow and Kurz (970), who examned n he conex of a neoclasscal economy. Begnnng wh Barro (990), an exensve leraure has evolved addressng he ssue n an endogenous growh framework, wh a general consensus ha governmen spendng on nfrasrucure can yeld sgnfcan producvy gans and hus enhance growh. See e.g. Fuagam e al. (993), Glomm and Ravkumar (994), Turnovsky (997), and more recenly Agénor (0), who provdes a dealed survey of he heorecal leraure on hs opc. The emprcal leraure s even more exensve. Mos of focuses on esmang he producve elascy of governmen expendure n producng oupu. The overwhelmng consensus s ha nfrasrucure conrbues posvely and sgnfcanly o oupu, hough he producve elascy s consderably smaller han Aschauer s (989) orgnal esmae of Bom and Lghar (04) provde an exhausve revew of he leraure and place he elascy somewhere beween 0.0 and 0.0, a range also shared by mos oher sudes. By mpacng facor producvy, and hence relave facor reurns, publc nvesmen also plays a crcal role n he evoluon of wealh and ncome dsrbuons as he economy grows over me. Indeed, nfrasrucure by vrue of s dverse naure s lkely o have sgnfcan redsrbuve There s a much brefer leraure analyzng he effec of nfrasrucure on he growh rae. Whle much of hs s nconclusve, Sachez-Robles (998) and Calderón and Servén (004, 00) oban a subsanal posve growh effec.

3 consequences, snce dependng on s ype wll nevably confer dfferenal benefs across agens n he economy. Publc nvesmen n publc ransporaon, dreced oward he needs of he less affluen agens, s lkely o reduce nequaly, whle publc expendure on enhanced communcaon, by favorng he wealher owners of capal, wll end o exacerbae nequaly. As our analyss hghlghs, he dsrbuonal consequences depend crcally upon he subsuably-complemenary relaonshp beween prvae and publc capal n producon. In conras o he publc nvesmen-growh relaonshp, emprcal evdence on he relaonshp beween nfrasrucure nvesmen and nequaly s less conclusve and more anecdoal. For example, Calderón and Chong (004), Calderón and Servén (004), Fan and Zhang (004), Ferran e al. (004), and Lopez (004) fnd ha publc nvesmen has boh promoed growh and helped mgae nequaly. On he oher hand, Brakman e al. (00) fnd ha governmen spendng on nfrasrucure has ncreased regonal dspares whn Europe, whle Arad and Sala--Marn (003) sugges excessve publc nvesmen has conrbued o rsng ncome nequaly n Afrca. In he case of Inda, Banerjee and Somanahan (007) repor ha access o crcal nfrasrucure servces and publc goods s n general posvely correlaed wh socal saus, whle a World Bank (006) sudy also fnds ha he qualy and performance of sae-provded nfrasrucure servces end o be he wors n Inda's poores saes. The dversy of hese emprcal fndngs emphaszes he need for a well-specfed analycal framework, whn whch he neracon beween nfrasrucure spendng, economc growh, and nequaly can be sysemacally addressed. Ths paper examnes he dsrbuonal and growh effecs of publc nvesmen whn such a unfed framework, where boh growh and nequaly are endogenously deermned. In dong so, we address a key lmaon of he exsng leraure. Mos of he leraure, boh heorecal and emprcal, assessng he producvy of publc capal, employs a Cobb-Douglas producon funcon. Ths s a serous shorcomng n lgh of he emprcal evdence suggesng ha he elascy of subsuon beween publc and prvae capal n aggregae oupu s mos lkely sgnfcanly less han uny, conrary o he Cobb-Douglas specfcaon. Thus, snce he elascy of subsuon urns ou For example, usng daa for low and hgh ncome counres, beween he perod 960 and 997, Calderón and Chong (004) fnd a negave and sgnfcan relaonshp beween nfrasrucure measures (such as roads, ralways, elecommuncaons, and energy) and ncome nequaly.

4 o be a key mechansm whereby publc nvesmen nfluences he dsrbuon of ncome and growh, becomes even more mporan ha he producon funcon be generalzed beyond he resrcve Cobb- Douglas form. We develop an overlappng generaons model n whch ndvduals are subjec o wo sources of heerogeney: () her nal endowmens of prvae capal, and () dosyncrac producvy shocks. To examne he poenal mpac of he subsuably-complemenary relaonshp beween publc and prvae capal on he growh-nequaly relaonshp, we assume ha hese wo facors conrbue o fnal oupu va a consan elascy of subsuon (CES) producon funcon. By assumng ha he echnology has consan reurns o scale n publc and prvae capal, he macroeconomc equlbrum we generae s one of endogenous growh. We assume ha he wo sources of heerogeney can be represened by lognormal dsrbuons. Ths assumpon faclaes aggregaon, enablng us o derve he jon dsrbuonal and aggregae dynamcs of he CES economy n a very racable closed form, hereby provdng subsanal nsgh no s evoluon. 3 The equlbrum dynamcs has a smple recursve srucure. The dynamcs of nequaly drves he growh of he aggregae varables prvae capal, publc capal, and oupu bu no vce versa. There are no cred or nsurance markes, he absence of whch s a key elemen generang nequaly, as n Loury (98), and Bénabou (000, 00). When ndvduals canno fully nsure hemselves from fuure ncome uncerany and are unable o borrow or lend unlmed amouns, nequaly assocaed wh he dosyncrac producvy shocks persss. 4 Ths s because dmnshng reurns o nvesmen mply ha he poorly endowed can oban a hgher margnal reurn o her capal han do he wealhy. Bu, snce hey canno borrow and nves effcenly due o cred consrans, Pareo effcency canno be acheved, as ofen s mplcly assumed n represenaveagen models wh complee markes. Consequenly, nequaly persss, leadng o less effcen resource allocaon and slower growh. However, f redsrbuve publc nvesmen polces can mgae he equy-effcency rade- 3 Some emprcal suppor for he lognormal s ha for several economes descrbes que well he dsrbuon of ncome below he op 3%; see Clemen and Gallega (005). 4 In conras, he nequaly orgnang wh he nal endowmens gradually dsappears. 3

5 off hey could poenally yeld addonal effcency gans. To he exen ha hs s so depends upon he degree of subsuably beween publc and prvae capal. Ths s ye anoher reason why s so mporan o generalze he producon echnology o he CES funcon, raher han resrc o he Cobb-Douglas funcon as so much of he relevan growh leraure does. We show ha an ncrease n publc nvesmen wll decrease or ncrease nequaly, boh n he shor run and over me, accordng o wheher he elascy of subsuon beween publc and prvae capal s greaer han, or less han, uny. The underlyng nuon s sraghforward. Publc capal ha s hghly subsuable for prvae capal provdes poor households wh he opporuny o replace her resrced capacy o nvesmen, due o her nably o borrow, by engagng n facor subsuon. Ths leads o a more equable dsrbuon of wealh and a subsequen posve mpac on producve effcency. In conras, publc nvesmen hghly complemenary o prvae capal wll exacerbae nequaly and have an adverse effec on producve effcency. Ths s because publc nvesmen wll now dsproporonaely favor he rch who, because hey own much of he prvae capal n he economy, oban lower margnal reurns on her nvesmens. The ne effcency gans obaned from enhancng he margnal producvy of aggregae prvae capal, whle aggravang nequaly n hs laer case, neverheless sll mos lkely reman posve. Thus, nvesmen n nfrasrucure wll mpac he growh rae hrough wo channels. The frs s he famlar drec producvy effec, emphaszed by Barro (990). Bu, n addon, wll have an ndrec mpac hrough s effec on nequaly. The presence of he second channel nfluences he choce of opmal (boh growh-maxmzng and welfare-maxmzng) ax, whch now also depends on he degree of wealh heerogeney as well as he elascy of subsuon beween he facors. As a resul, he opmal ax rae s hgher n socees wh more nequaly f he elascy of subsuon exceeds uny. Ths s because n hese crcumsances, n addon o enhancng he producvy of prvae capal, publc nvesmen also serves o offse he neffcency arsng from nequaly. Ths paper s relaed o several bodes of leraure dealng wh ssues peranng o publc capal, economc growh, and ncome dsrbuon. Frs, s he leraure on publc capal and endogenous growh, some of whch s referred o earler. Second, are papers sudyng dsrbuon and growh wh mperfec cred markes, (e.g., Loury, 98, Galor and Zera, 993, Aghon and Bolon, 4

6 997, Aghon, e al., 999, Bénabou, 996, 000, 00, and Seshadr and Yuk, 004). The paper s also relaed o he recen, bu sparser, leraure on he relaonshp beween publc expendure and he dsrbuons of wealh and ncome. In parcular, Chaerjee and Turnovsky (0) employ a CES producon funcon o examne he mpac of producvy-enhancng fscal polces on nequaly. Ths analyss, followng Casell and Venura (000), assumes he exsence of perfec facor markes and unlmed borrowng, he effec of whch s o fundamenally reverse he srucural dynamcs from ha developed here. Specfcally, under hose assumpons, perfec aggregaon across agens as n Gorman (953) s sraghforward, and he macroeconomc (aggregae) equlbrum s deermned ndependenly of any dsrbuon characerscs. Insead, he facor reurns generaed by he aggregae equlbrum now deermne he evoluon of wealh and ncome nequaly. In addon, hese models rea he nal capal endowmens as he prmary source of nequaly, and show furher ha, n conjuncon wh he evoluon of facor reurns, hey reman a key deermnan of long-run nequaly. These wo aspecs he dynamc srucure and he long-run relevance of he heerogeneous endowmens are fundamenally dfferen n hese wo classes of models. The conrasng conclusons derved from hese alernave approaches underscore he fac ha he growhnequaly relaonshp s a complex one, and confrms he need o sudy from alernave vewpons. By focusng on unnsurable dosyncrac shocks as he key underlyng source of perssen nequaly, he paper s also relaed o he approach pursued by Krusell and Smh (998) and ohers. 5 However, he complexy of her framework precludes analycal soluons, and hence he analyss s enrely based on numercal compuaons. In addon, he specfc naure of he sochasc srucure and he focus of he analyses are very dfferen, especally snce ha leraure does no address ssues peranng o publc polcy. In hs las respec he paper s relaed o Geachew (00, 0) who consders publc nvesmen and nequaly n endogenous growh models and mperfec cred markes, alhough ha work s hghly resrcve. Frs, does no consder dosyncrac unnsurable rsk, he key elemen generang he non-degenerae wealh dsrbuon. Insead, he only source of heerogeney nroduces s nal endowmens, he effecs of whch on nequaly are only ransory. Also, by assumng Cobb-Douglas producon funcons, does no address he role of he 5 See, for example, Casañeda, Díaz-Gménez, and Ros-Rull (998), Díaz, Pjoan-Mas, and Ros-Rull (003). 5

7 subsuably-complemenary relaonshp beween publc and prvae capal, whch s a cenral elemen of he presen analyss. 6 Fnally, one of he key conrbuons of he presen paper s ha o our knowledge s he frs sudy o derve a closed form soluon for he equlbrum growh and dsrbuonal dynamcs n a suaon where heerogeney s generaed by dosyncrac shocks and a more general producon funcon, such as he CES, s employed. Bu despe beng able o conduc mos of our analyss formally, we also carry ou numercal smulaons, n par o subsanae s emprcal relevance. To do so requres relable esmaes of he elascy of subsuon beween prvae and publc capal, on whch emprcal evdence s sparse. Thus, because hs s such a crucal parameer we underake a dealed calbraon, by relang o he wdely avalable emprcal esmaes of he producve elascy of publc capal. We supplemen hs by also obanng our own emprcal esmaes, based on a cross secon of Afrcan counres. The upsho of hs s ha, a he aggregae level, he evdence suggess ha publc and prvae capal are complemens, whch we herefore ake as he emprcally more relevan case. As a consequence, our smulaons exendng over a wde range of elasces of subsuon sugges ha compuaons based on he Cobb-Douglas producon funcon may serously undersae he growh and welfare consequences of governmen nvesmen polcy. The res of he paper s organzed as follows. Secon develops he analycal framework. Secon 3 examnes he dynamcs and equlbrum of he model. Secon 4 analyzes he effecs of governmen nvesmen on he shor-run and long-run evoluon of growh and nequaly. Secon 5 descrbes he calbraon employed o esmae he elascy of subsuon beween publc and prvae capal as well as a bref summary of he new emprcal esmaes obaned. Secon 6 complemens he heorecal analyss by carryng ou exensve numercal smulaons. In conducng hese, we pay parcular aenon o relang hem o he relevan emprcal evdence, and fnd ha despe s parsmony he model performs well n hs dmenson. Secon 7 concludes, whle echncal deals are provded n he Appendx. 6 There s also a vas leraure ha sudes he relaonshp beween publc educaon and ncome dsrbuon, o whch hs paper may also be relaed o by vrue of he common neres of examnng he dsrbuonal effec of a producve publc good n growh models, (e.g., Glomm and Ravkumar, 99, 003, San-Paul and Verder, 993, and Ecksen and Zlcha, 994). Fnally, s also relaed o sudes of he mpac of he elascy of subsuon (beween capal and labor) on growh; see e.g. Klump and de la Grandvlle (000), Myagwa and Papageorgou (003). 6

8 . The analycal framework The analycal framework we employ s ha of an overlappng generaons economy populaed by a connuum of heerogeneous households.. Preferences and echnology A any pon n me, each household, ndexed by [0,], consss of a young he offsprng and an adul he paren. Populaon sze hus remans consan over me. Each paren of he nal generaon (a =0) s endowed wh prvae capal, k 0, a un of me, and has access o publc nfrasrucure, g 0, whch s a pure publc good equally avalable o all. The dsrbuon of wealh assumes a known (gven) nal probably dsrbuon bu evolves endogenously over me. Indvduals lve for wo perods. In he frs perod, when agens are young, hey acqure capal from her parens, whle her consumpon s ncluded n ha of her parens. They do no ake any economc decsons. Raher, all such decsons are made durng he second perod, when chldren have become aduls. Parens earn ncome by supplyng capal and labor o prvaely-held frms, whch produce oupu, by combnng hese facors wh publcly provded capal. 7 The governmen axes ncome a a fxed fla rae n order o fnance he publc nvesmen good ha s used n fnal goods producon. Agens allocae her afer-ax ncome beween consumpon and savng, wh he accumulaed capal a he end of he second perod beng endowed o her offsprng. where The h household s uly funcon s specfed by he logarhmc preference funcon, W ln c ln l ln k () c denoes s consumpon durng perod, l denoes s labor supply, and k s he amoun of capal leaves o s offsprng for producve use n perod (+). The coeffcen can be vewed as reflecng wo elemens: () he radeoff n uly beween s own consumpon vs. bequess, reflecng he joy of gvng and () some neremporal dscounng due o he fac ha he bequess 7 Ths ype of ndvdual enrepreneurshp s common n models wh ncomplee markes (see, for e.g., Loury, 98, Bénabou, 000, 00 and Angeleos and Calve, 005, 006). 7

9 are o be enjoyed by he nex generaon. 8 Households choose c, s, and l o maxmze he uly funcon, (), subjec o her budge consran and capal accumulaon relaonshp 9 c s ( ) y (a) y a k g l k 0, a 0 (b) k s (c) where s denoes he ndvdual household s savng, and y s s ncome, whch s axed a he fla rae,. Oupu s produced by he wo-level producon funcon specfed n equaon (b). A he frs level, he ndvdual s prvae capal and publc capal, g, are combned n accordance wh he CES producon funcon, wh elascy of subsuon beng ( ). We assume ha he wo capal goods are cooperave n producon, meanng ha y kg 0, whch mples (d) and mposes an upper bound on he degree of subsuably,. 0 Ths nermedae oupu s hen combned wh labor n accordance wh a Cobb-Douglas echnology o produce fnal oupu. Labor producvy s augmened by he aggregae capal sock, k, n accordance wh he Romer (986) echnology, hereby renderng (b) consan reurns o scale n k, k, g and generang an equlbrum of ongong growh. The exponen reflecs he producvy of labor n fnal oupu, whle reflecs he mporance of publc capal n he nermedae oupu good. Producon by he h agen (frm) s subjec o an dosyncrac producvy shock, assumed o be..d. and lognormal wh mean one, so ha dsrbuon of capal s also assumed o be lognormal, ln N(, ) 0 0 0,. The nal ln k N(, ). Fnally, for analycal 8 As dscussed a lengh by Galor and Zera (993), he general resuls and nsghs are robus wh respec o varaons n he specfcaon of hs class of uly funcon. 9 The assumpon of a unary ner-emporal elascy of subsuon uly funcon of alrusc agens wh a "joy of gvng" move s wdely used n he leraure of ncome dsrbuon dynamcs (see, for nsance, Glomm and Ravkumar, 99, Galor and Zera, 993, San-Paul and Verder, 993, and Bénabou, 000). 0 Seng 3mples.5, whch he emprcal evdence dscussed n Secon 5 suggess s no a severe resrcon. 8

10 convenence equaon (c) assers ha prvae capal fully deprecaes whn he perod, so ha he capal sock accumulaed by household concdes wh s savngs. There are no markes for nsurance or cred. Though hese are exreme forms of marke ncompleeness, adoped o maxmze racably and ransparency, wha maers s he exsence of some forms of ncompleeness; see, e.g., Aghon e al. (999), Bénabou (000 and 00). In hs respec we vew our analyss as beng parcularly relevan for developng economes. The governmen provdes npus for goods producon, and collecs proporonal axes on markeed oupu. Is budge s always balanced. Publc capal also compleely deprecaes whn he perod, so ha s accumulaon akes he form, g y d y (3) 0 where y s aggregae ncome. In reang publc capal as a flow, we are followng he orgnal Barro (990) approach, raher han some of he subsequen leraure whch reas as a sock. Summng (a) and (c) over all he households, aggregae consumpon and he prvae capal flow are gven by, respecvely c ( ) y s (4) and where c c d, s 0 0 k s d, k k d s (5). 0. Indvdual opmal capal accumulaon Performng he opmzaon se ou n () and () yelds household s frs order condons s ( ) y (6a) There s mounng evdence ha cred markes are mperfec n mos of he developng counres (see, for e.g., Mel e al., 008, McKenze and Woodruff, 006 and Banerjee and Duflo, 004). Bu even n he developed world, he cos of admnsrang cred s subsanal suggesng he presence of cred marke frcons n hese counres as well. See e.g. Fuagam e al; (993), Turnovsky (997). The model can be exended o nroduce publc capal as a gradually accumulang sock, bu lle nsgh s los by adopng he smpler flow specfcaon. 9

11 l l ( ) (6b) where ( ). Eq. (6a) descrbes he h household opmal savng allocaed o capal accumulaon, whch for he logarhmc preference funcon smplfes o savng beng a consan fracon of he agen s afer ax ncome, and s ndependen of he rae of reurn o nvesmen. 3 Equaon (6b) deermnes he household s allocaon of me o labor, whch s smlarly consan across agens and over me. 4 Subsung for s, y, and l from (b), (c), and (6b) yelds he followng relaonshp descrbng he dynamcs of capal accumulaon for he h ndvdual k ( ) a k g k (7) where a a Thus, capal accumulaon of he h offsprng a me s a funcon of he paren s capal ( k ), he level of publc capal n he prevous perod ( g ), he sock of aggregae prvae capal n he prevous perod ( k ), and he dosyncrac producvy shock ( ). Wh ha and k are uncorrelaed. 3. Infrasrucure, nequaly and aggregae capal dynamcs beng..d., (7) mples In hs secon we characerze he aggregae dynamcs and seady-sae equlbrum. Of parcular neres are he effecs of nfrasrucure nvesmen on he ransonal dynamcs of nequaly, as measured by he varance of ln( k ) and denoed by, and he growh rae, denoed by. 5 An appealng propery of he lognormal dsrbuon s ha faclaes aggregaon, deals of whch are provded n he Appendx. There we show ha he macroeconomc equlbrum s 3 As s well known, when he ner-emporal elascy of subsuon s uny, he ncome effec exacly offses he subsuon effec. 4 Ths conrass sharply wh he opmaly condons obaned n he heerogeneous agen models based on he nfnelylved represenave agen, where he dfferenal allocaon of me across agens s he crucal deermnan of he equlbrum dsrbuon of ncome. 5 There are many measures of nequaly havng varyng qualave arbues; see Aknson (970). Our choce of s essenally he squared coeffcen of varaon, and s convenen, gven he assumed underlyng lognormal dsrbuon. Alhough places more wegh on exreme observaons han does he coeffcen of varaon,, s qualave mplcaons are he same and can be easly convered o he laer f one so chooses. 0

12 summarzed by he followng relaonshps, where suppresson of he ndex, denfes aggregaes: 6 () Dynamcs of nequaly: z ln e z (8a) where ( ) z e (8b) k g (8c) () Dynamcs of aggregae prvae capal accumulaon: / ln k ln( a ) ( ) ln( ) ln ln g ln( z) z ln e z (9) () Dynamcs of publc capal accumulaon: / ln g ln( a ) ln( ) ( ) ln ln g ln( z) z ln e z (0) (v) Dynamcs of aggregae oupu: / ln y ln( a ) ln( ) ln ln g ln( z) z ln e z () 6 We should noe ha by vrue of (6a), measures of wealh nequaly and ncome nequaly concde (hough he laer lags by one perod) and hence we jus refer o nequaly.

13 3. Transonal dynamcs A number of observaons follow from hese equaons. Frs, he consancy of he rao of prvae o publc capal, assered n (8c), follows drecly from equaons (9) and (0). In addon, equaons (0) and () reflec he consancy of he rao of publc capal o lagged oupu, as specfed n (3). Thus (0) mples ha durng he perod (, +) prvae and publc capal grow a he same rae k / ln g ln g ln( a ) ln( ) ( ) ln ln( z) z ln e z (a) whle durng he same perod oupu and consumpon boh grow a he common rae 7 y / ln y ln y ln( a ) ln( ) ( ) ln ln( z ) z ln e z (b) k y From (a) and (b) we see ;.e. he growh of oupu leads ha of capal by one perod.8 From equaons (8)-() we see ha he dynamcs of he economy are drven enrely by nequaly, whch operaes hrough wo channels. The frs s hrough s mpac on he average rao of prvae o publc capal employed n producon, refleced n z, whle he second s he drec sysemac sochasc componen of oupu as deermned by var(ln ) n he aggregaon of he producon funcon, (b). Sarng from an nal nequaly endowmen dynamc me pahs for and z These soluons for and 0, (8a)-(8c) generae z hen feed no he remanng equaons o generae he me pahs for k, g, y and her growh raes, wh k beng subjec o an nal condon, k 0, obaned by aggregang over k 0. Comparng equaons (8) and (), we see ha here s a conemporaneous relaonshp 7 To see ha consumpon and oupu grow a he same rae subsue (6a) no (a) and aggregae, from whch mmedaely follows ha c y ( ) ( ) mplyng a common growh rae. 8 Ths s a consequence of he lag nroduced n eq. (3); s no parcularly consequenal.

14 beween cross-seconal nequaly and oupu growh, whle nequaly mpacs he growh of capal smlarly, bu wh a one perod lag. The nfluence of nequaly on growh reflecs wo facors cred marke mperfecons and dmnshng reurns o ndvdual nvesmen. The nably o borrow mples ha producve nvesmen opporunes may be foregone. Wh dmnshng reurns o nvesmen, he poor have a hgher margnal produc han do he rch. Therefore, greaer nequaly s assocaed wh a loss of producve effcency, leadng o lower growh. Ths causaly conrass sharply wh ha obaned n he class of nequaly-growh models developed by Casell and Venura (000), and Turnovsky and García-Peñalosa (008), for example, where her underlyng assumpons perm exac aggregaon as poneered by Gorman (953). 9 Under hose assumpons, he macroeconomc equlbrum s deermned ndependenly of he dsrbuon across agens, whle he dsrbuon s hen deermned by reurns o capal and labor generaed by he aggregaes, causng Casell and Venura o characerze hs as a represenave agen heory of dsrbuon. 3. Seady sae Assumng ha he sysem s sable, he seady-sae equlbrum wealh nequaly, growh rae,, are deermned jonly (ogeher wh z ) by, and z z e ( ) e ln( a ) ln( ) ( )ln ln( z) / (3a) (3b) ( ) z e (3c) From hese condons nequaly,, s seen o mpac he long-run growh rae hrough wo channels. The frs channel s drec, reflecng he basc propery of he lognormal dsrbuon ha expeced value s an ncreasng funcon of s varance; see (3b). The second channel occurs va he 9 The key assumpons are homogeney of he underlyng uly funcons and perfec facor markes, n whch all agens earn he same raes of reurn. 3

15 adjusmen n he publc o prvae capal rao as refleced n z, and s drecon depends upon wheher he elascy of subsuon. If 0, so ha here are no dosyncrac producvy shocks, (3a) mples 0, so ha seady-sae nequaly s zero. In he absence of such shocks, he long-run mpac of nal wealh nequaly dsappears, and he long-run dsrbuons of wealh and ncome degenerae, as relavely resource poor ndvduals rapdly accumulae wealh due o her relavely hgh margnal producvy, whch n urn s due o he presence of dmnshng reurns o nvesmen. In conras, f 0, so ha he economy s subjec o dosyncrac producvy shocks of consan varance, hen he long run equlbrum s characerzed by non-degenerae wealh and ncome nequaly ha exceeds. In he presence of such unnsurable dosyncrac rsks, ndvduals may be unable o overcome boh her dfferences n luck as well as some of her nal dfferences n endowmen. Ths n urn wll mpac he equlbrum growh rae. The fac ha s mmedaely seen by wrng (3a) n he form z ln e z (3a ) To deermne he consequences of he dosyncrac producvy shocks for he long-run nequaly-growh relaonshp we dfferenae equaons, (3a) - (3c), wh respec o o oban: d d D d z ( )( ) ( ) D d ( D) z (4a) (4b) where z ( ) D e e 0 z z z e ( z) (4c) I s sraghforward o show D 0, n whch case, as we wll see n Secon 3.3, D s a necessary and suffcen condon for he aggregae economy o be locally sable. In fac, for any plausble ~ d parameerzaon we oban 0D, n whch case / d. Ths wll be assocaed wh a negave effec on growh f and only f hs condon s srenghened o D z ( z) ( ), a weak 4

16 condon ha we shall henceforh mpose. Moreover, as shown n (5 ) below, D s closely approxmaed by approxmaon leads o: D( ) z ( z), clearly meeng hs consran, and nvokng hs d ( z) [ ( ) ][ ] d z z (4a ) d ( ) z d z 0 d ( z) d (4b ) Thus we may summarze hese responses n he followng: Proposon : () An ncrease n he varance of he dosyncrac producvy shocks wll ncrease nequaly and wll be assocaed wh a lower growh rae, boh n he shor run and n seady sae. () An ncrease n he nequaly n nal endowmens wll ncrease nequaly and reduce he growh rae emporarly, declnng over me, and vanshng n he long run. I s nsrucve o relae he naure of he long-run dsrbuon of wealh obaned here wh ha obaned usng oher approaches. Frs, L and Sare (004) consder an economy n whch agens have heerogeneous nal endowmens of capal. They hen show ha f agens are subjec o progressve axes bu have a common rae of me dscoun, he long-run wealh dsrbuon wll degenerae, precsely as (3a) mples n he absence of producvy shocks. Bu hey also show ha he long-run wealh dsrbuon wll no degenerae f agens have dfferen raes of me preference. 0 Tha oo urns ou o be he case here, as one can easly show for he Cobb-Douglas echnology, assumed by L and Sare. In effec, he assumpon of ncomplee fnancal markes beng adoped here s playng he role of he progressve ax srucure n her analyss. Anoher srand of leraure, nally developed by Becker and Tomes (979), Krusell and Smh (998) and ohers, shows f agens are nally dencal bu are subjec o dosyncrac shocks, hen hs wll lead o a long-run non- 0 Followng hs general approach, Carroll and Young (009) show ha he assumpons of: () complee markes, () progressve axaon, and () heerogenous labor producvy can also produce a nondegernerae wealh dsrbuon. 5

17 degenerae wealh dsrbuon, jus as we oban here. A hrd body of leraure, whch assumes ha he only source of heerogeney s due o he nal endowmens of capal, obans non-degenerae long-run dsrbuons of wealh and ncome ha are drecly ed o he nal dsrbuon. Ths hyseress arses because of he assumpon of complee fnancal markes ha hese models also assume, makng he dsrbuonal dynamcs be pah-dependen; see e.g. Casell and Venura (000), and Aola, Chaerjee, and Turnovsky (0). These dfferences hghlgh how he assumpons peranng o he presence or absence of fnancal markes play a crucal role n assessng he causaly and naure of he growh-nequaly radeoff. 3.3 Local sably To deermne he local sably of he dynamcs we evaluae d d n he neghborhood of he seady sae equlbrum, usng (8a) and (8b), o oban: d D (5) d where D s defned n (4c) and for local sably D. In he absence of producon rsk, when 0, D ( ) z ( z) and (5) smplfes o d d ( ) z z (5 ) and s clearly locally sable. Moreover, n he case ha: () he producon funcon s suffcenly close o Cobb-Douglas, and () he dosyncrac producon rsk s suffcenly small so ha e, hen (8a) can be approxmaed by ( ) z z (6) and s also clearly sable. Equaon (6) reflecs he conrary effecs of he wo underlyng sources of In dervng (6) we are also usng he fac ha f e, hen ln( ) for 0. For he 6 Cobb-Douglas funcon z, n whch case (6) converges o ( )(.

18 heerogeney nal endowmens and dosyncrac producvy shocks. Whle he effecs of he former declne over me, vanshng n he seady sae, he effecs of he laer grow over me. As a praccal maer we vew hs approxmaon as beng hghly relevan, snce n he mos plausble nsances s lkely o be small. Bu whle our numercal smulaons ndcae ha for all plausble parameers D, we canno rule ou D, and for (6) o dverge; hs may occur f he varance of he dosyncrac producon shocks s mplausbly large and he elascy of subsuon,, s suffcenly small. 4. Effecs of publc nvesmen on growh and nequaly We now employ he equlbrum relaonshps (8)-(3) o deermne boh he shor-run and seady-sae effecs of publc nvesmen on nequaly and growh. 4. Shor-run effecs We shall assume ha he economy s nally n seady sae and n perod governmen nvesmen ncreases by d. From (8a) we see ha wh he one perod lag, nequaly remans unchanged unl perod, when usng (8a) and (8b) we derve e z z e ( ) ( z) 3 ( ) ( z) ( ) (7a) from whch we nfer ha he shor-run effec of an ncrease n nvesmen n publc goods on nequaly depends upon he elascy of subsuon, ; governmen nvesmen wll ncrease nequaly f and decrease f. The underlyng nuon s sraghforward. The presence of dmnshng reurns o nvesmen and cred marke mperfecons make dfferen households face dfferen problems wh respec o her producvy and nvesmen opporunes n he economy. The poor face nvesmen consrans due o he lack of a cred marke, whle he rch oban relavely lower margnal reurns due o dmnshng reurns o nvesmen. Whch group benefs more from he For our prefered calbraon and 0.33, n whch case 7 e.08, s closely approxmaed by

19 provson of a specfc form of publc capal depends upon s subsuably or complemenary o prvae capal. Subsuable publc nvesmen dsproporonaely benefs he poor, as provdes he opporuny o crcumven her resource consran by subsung publc capal n producon; nequaly herefore declnes. Complemenary publc nvesmen, however, benefs he rch more, hrough enhancng he producvy of prvae capal, causng nequaly o ncrease. Seng 0 n (b) and combnng wh (8a) he growh rae of oupu n perod s y / ln( a ) ln( ) ( )ln ln( z) (b ) from whch we derve y ( ) ( ) ( z ) (7b) and we see ha publc nvesmen mpacs growh hrough wo channels. The frs s he ne mpac on he producve capacy of he economy, he effec of whch depends upon relave o he producvy of publc capal as refleced n alhough n conras o Barro, for he CES echnology z. Ths s he effec emphaszed by Barro (990), z depends upon he exsng degree of nequaly; see (8b). 3 Bu hs response mus be modfed by s drec effec on nequaly, expressed by he second erm n (7b). If, hs erm s posve. In hs case he ncrease n nequaly caused by ncreasng publc nvesmen ncreases he shor-run growh rae. However, f, publc nvesmen decreases nequaly, and hs causes he shor-run growh rae o declne. Thus, he overall mpac of ncreasng publc nvesmen on he shor-run growh rae wll depend upon he curren magnude of and he degree of subsuably beween he wo producve npus. 4. Seady-sae effecs To derve he long-run effecs of governmen nvesmen on nequaly and growh we reurn o (3a)-(3c), from whch we show: 3 For he Cobb-Douglas producon funcon he erm becomes ( ( ) ) [ ( )], whch reduces furher o ( ) [ ( )] f, as Barro does, one absracs from labor. 8

20 e ( ) ( z) ( D) z ( z) e 3 ( z) ( ) ( ) ( )( ) z ( ) ( z) z (8a) (8b) The parallel beween (8a) and (7a) s clear, wh he qualave mpac of on nequaly sll dependng upon sgn( ), and he scale facor ( ) D reflecng he adjusmen of nequaly over me. Equaon (8b) parallels (7b) n ha ncorporaes he wo channels hrough whch publc nvesmen mpacs growh. Bu now accoun mus be aken of he fac ha n seady sae, he choce of wll also deermne he degree of nequaly refleced n z, reducng he coeffcen of (d) o, and hus mpacng he radeoff beween long-run growh and nequaly. Srenghenng z( ) ( z) ensures ha he seady-sae mpac of nequaly on growh remans posve, alhough wh he modfcaon beng small s only margnally weaker han he shor-run effec. We may summarze hese aggregae effecs n: Proposon : () An ncrease n he rae of publc nvesmen wll ncrease (decrease) boh shor-run and long-run nequaly accordng o wheher he elascy of subsuon s less (greaer) han one. () To he exen ha more publc nvesmen ncreases nequaly, hs wll end o ncrease he growh rae boh n he shor run, and n he long run, alhough n all cases he ne overall effec wll also depend crcally upon wheher he curren rae of expendure ( )( z). 4.3 Growh versus welfare maxmzaon The growh-nequaly radeoff s also manfesed n he choce of long-run growhmaxmzng rae of publc nvesmen. Seng 0, we see ha he growh-maxmzng rae of publc nvesmen, *, and correspondng rao of prvae o publc capal, * z, are relaed by * ( ) * ( )( ) z 0 * * * * ( ) ( z ) z (9) 9

21 For exposonal convenence we shall manan he assumpon z( ) ( z), a condon ha s almos ceranly sasfed by he emprcal daa. 4 Proposon 3: To he exen ha governmen nvesmen n nfrasrucure ncreases (decreases) nequaly, wll se he long-run growh-maxmzng rae of publc nvesmen a a rae rskless economy. * * ( )( z ), he opmaly condon characerzng a The choce of * mpled by (9) hus depends upon he presence and degree of nequaly n hs economy. In comparng (9) o a rskless economy, we need o ake accoun of he fac ha he equlbrum rao of prvae o publc capal wll also be affeced. analogous rskless economy s characerzed by Opmal expendure n he ( )( z) n whch case (9) mples * * ( z)( z ). The fac ha governmen nvesmen ncreases nequaly and he growh rae causes hem o reduce nvesmen, hus ncreasng z, enablng hem o reduce he ax rae. In our numercal smulaons repored n he nex secon, we see ha he growh-maxmzng ax rae s relavely hgher n an economy wh no dosyncrac rsk, and herefore no nequaly, f he elascy of subsuon beween publc and prvae capal s less han uny; he oppose apples f. To see he underlyng nuon suppose ha he governmen has se he approprae growhmaxmzng ax, gven he degree of nequaly. If he degree of nequaly s now ncreased, hs wll decrease he growh rae; see (4b ). In order o offse hs decrease, he governmen should adjus s nvesmen, wh he approprae response dependng upon he elascy of subsuon. If, hs s acheved by ncreasng, whch wll ncrease nequaly, whle smulaneously lowerng z and rasng he long-run growh rae n accordance wh (8b), and correspondngly reducng f. Whle growh and nequaly are mporan, he key ssue n assessng he consequences of srucural changes and polcy responses concerns her mpac on socal welfare. Defnng such a measure as he dscouned sum of he expeced uly of all fuure generaons, W E W ( R ), n Appendx A. we show ha n seady sae he mpled socal welfare funcon s 0 4 Ths condon s clearly me f and he wo capal goods are complemens. For and our benchmark parameerzaon nvolves srenghenng.5 o around.4. 0

22 W ( R)( ) Cln R R (0) where he consan C s defned n equaon (A.0). Thus, seady-sae socal welfare nvolves a radeoff beween he equlbrum growh rae, he degree of ncome nequaly, and he loss n average ncome due o he ax rae used o fnance governmen nvesmen. Maxmzng (0), he welfare-maxmzng rae of governmen nvesmen sasfes R In he absence of dosyncrac producvy shocks, () mples () R( ) 0, so ha maxmzng socal welfare s assocaed wh a rae of governmen nvesmen ha falls shor of he growh maxmzng rae. In oher words, maxmzng he growh rae leads o oo much governmen nvesmen for neremporal uly maxmzaon across generaons. Moreover, governmen nvesmen resulng from growh maxmzaon s even more excessve n he presence of dosyncrac shocks, when for, governmen nvesmen exacerbaes ncome nequaly. 5. Calbraon I s clear from he analyss of he prevous secons ha he magnude of he elascy of subsuon beween publc and prvae capal s crucal n deermnng he effecs of governmen nvesmen n nfrasrucure on he dynamcs of nequaly, and growh. However, he avalably of emprcal esmaes of hs crcal parameer s sparse. Mos emprcal sudes examnng he effec of publc capal smply assume a Cobb-Douglas funcon ( ), whle ohers adop he more general ranslog funcon; see e.g. Bernd and Hansson (99), Lynde and Rchmond (99), Nadr and Mamuneas (994). Alhough he generaly of he laer funcon makes dffcul o map he esmaes o a CES funcon, anecdoal evdence suggess ha he complemenary-subsuably relaonshp beween publc and prvae capal can vary subsanally, dependng upon he specfc ypes of capal. Thus, for example, whle publc and prvae ranspor may be hghly subsuable, roads and cars are complemenary. In conras o he lack of esmaes of, abundan esmaes of he producve elascy of

23 capal, e dln y dln g are avalable. In Secon 5. below we show how usng hs nformaon, ogeher wh he equlbrum condons repored n Secon 4, can yeld plausble calbraons for he elascy. Bu before dong so, we mus parameerze he remander of he model. Table parameerzes a benchmark economy. The parameers we employ are almos enrely convenonal and reflecve of plausble real economes. Table : Calbraon: Benchmark values Preference and echnology parameers: 0.30,.75, R 0.30 Producon parameers: , e 0. 0., /3 Polcy parameers 0.05 Idosyncrac producvy shocks 0.6 Assumng a perod (generaon) o be of he order of 30 years, a he end of whch parens pass 30 on her bequess, and assumng a psychologcal dscoun facor of 0.96, ; see e.g. de la Crox and Mchel (00, p.55). The wegh.75 assgned o lesure n uly mples an equlbrum allocaon of me o labor of around 0.33, conssen wh he real busness cycle leraure. The dscoun facor ( ) R measures he relave wegh assgned by he socal planner (welfare maxmzer) o he uly of successve generaons and as s well known has no relaonshp o. Seng R 0.3 mples a generaonal dscoun facor of 0.77, whch whle seemngly plausble, s purely llusrave. We should also add ha he only role ha R plays s n assessng he welfare assocaed wh any gven equlbrum. The share of publc capal n he frs level of producon ranges beween 0. and 0.4, whch covers mos of he plausble values parameerzed by Eden and Kraay (04), ncludng her preferred esmae (0.4). As Bom and Lghar (04) documen, emprcal esmaes on he producve elascy of publc capal, e, are far rangng. In her comprehensve sudy hey summarze 578 esmaes and fnd he average producve elascy of publc capal o be around 0.9, whle her mea-regresson analyss yelds an esmae of around 0.0. Oher auhors, usng dfferen daa ses and mehods oban smlar esmaes, so ha our range e brackes he overwhelmng

24 bulk of he emprcal evdence, obaned employng a varey of daa ses and echnques. 5 The producve elascy of labor, /3, s sandard. We se he benchmark governmen spendng rao, g o be 5% of GDP, whch s roughly conssen wh evdence on he rae of publc nfrasrucure spendng for mos OECD counres. 6 The fnal elemen s he specfcaon of he dosyncrac producvy shocks. Daa on hs are sparse, bu Barelsman, Halwanger, and Scarpea (03) provde esmaes of he sandard devaon of whn ndusry log TFP shocks. For he seven counres hey consder hese average 0.4, wh esmaes for he US and UK beng 0.39 and 0.4, respecvely. On he bass of hs we se ). 0.6 ( 0.4 Fnally, a s chosen o mach a long-run growh rae of %. To acheve hs each parameerzaon of he CES producon funcon requres a slgh adjusmen n a Calbrang he elascy of subsuon I s clear ha (or ) s he crcal parameer. Usng he equlbrum condons, n Appendx A.3 we esablsh he followng relaonshp beween he key parameers. ( ) e( ) ln ln ( ) ( e) () Gven assumpons on,,,, and esmaes of he producve elascy e, as refleced n Table, one can nfer a value for and hence an mpled value for ( ). Table repors he correspondng calbraed values for, for he grd of plausble parameer values spannng 0.0,0.30,0.40; e 0.0,0.5, For example, Henderson and Ullah (005) oban an esmae of 0.5. Usng Belgan daa, Everaer (003) fnds e 0.4, whle usng Canadan provncal panel daa, McDonald (008) fnds he elscy o range beween 0.0 and 0.5. Fosu e al. (05), usng daa for Sub Saharan Afrcan counres, esmae e around 0.0. Fnally, usng a sample of daa from developng counres Dessus and Herrer (000) fnd e o range beween 0.. and For developng counres may be slghly hgher. For nsance, based on he World Bank (0), Fosu e al. (05) fnd he average publc nvesmen-gdp percenage for 4 Sub-Saharan Afrcan counres beween 960 and 0 o be around 7.4%. 7 I s worh nong ha n order o arge he same growh rae of %, he producvy parameer, a, mus be reduced as he elascy of subsuon,, ncreases. Ths s a manfesaon of he resul due o Klump and de la Grandvlle (000), and confrmed emprcally by Yuhn (99), ha holdng a consan, he growh rae ncreases wh. 3

25 Table : Calbraed elascy of subsuon e 0. e 0.5 e From Table, he followng can be noed. The calbraed values of are sensve o varaons n boh and e. Over he plausble ranges of boh hese parameers we fnd ha ranges beween 0.45 (publc and prvae capal srong complemens) o.46 (srong subsues). I s also compable wh he Cobb-Douglas producon funcon for 0.3, e 0.. Whle, he case of complemenary s clearly more prevalen, he possbly of subsues canno be ruled ou, and ndeed obans f one combnes he Bom-Lghar (04) mea-regresson esmae of e 0.0 wh he Eden-Kraay (04) preferred calbraon for Provded ( ) ( ), as our parameerzaon clearly mples, () yelds a smple creron for 0 ( ) and for he wo capal goods o be complemens, namely f and only f e ( ) (3) Tha s, publc and prvae capal wll be complemens f and only f he producve elascy of publc capal, e, exceeds s share of oal oupu, ( ). 8 Conversely, subsuably beween prvae and publc capal wll be characerzed by a publc good ha has a low producve elascy bu accouns for a hgh share of oupu. Our fndng ha he elascy of subsuon s lkely o be less han uny s generally characersc of he emprcal esmaes. Eden and Kraay (04), n focusng on he crowdng n of publc nvesmen, draw upon emprcal esmaes o calbrae he CES producon funcon and fnd convncng evdence of complemenary. 8 In he Cobb-Douglas case e ( ) and. 4

26 5. Some new emprcal evdence on he elascy of subsuon In order o oban addonal nformaon on he lkely magnude of he elascy of subsuon beween publc and prvae capal, parcularly for developng counres, we have esmaed an aggregae CES funcon of he form where y, k, and y a k g (4) g are measured n per capa form, and, and a represen aggregae sochasc and non-sochasc TFP, respecvely. 9 We adop he approach orgnally employed by Arrow e al. (96), and regress ln[ y / g ] on he log of he margnal produc of g. Ths s done by akng he frs dervave of (4) wh respec o g, and rewrng he resulng expresson n he form ln y g ln p (5) g where ln a, ln, y y / g, p dy / dg. The esmaon equaon for (5) can hen be easly specfed n panel daa form as follows: ln y ln p (6) g where and denoe unobserved counry-specfc fxed effecs and he error erm, respecvely; refers o ndvdual counres and refers o parcular me perods. We esmae (6) usng he fxed effecs (FE) panel daa esmaon mehod. 30 The panel daa we use cover 4 Afrcan counres, for he perod 960 o 0, from World Bank (0). Eleven Afrcan counres are excluded due o eher lack of daa, nconssences, or ouler effecs. 3 In order o smooh ou any poenal busness cycle effecs we have esmaed he equaon usng 5 year 9 Noe ha wh complee deprecaon of capal k and g are flow varables. 30 Some of he mos wdely employed esmaon mehods for panel daa wh large me are FE, Random Error (RE) or pooled OLS (see Wooldrdge, 00). The advanage of FE over he wo s ha enables one o accoun for counryspecfc me-nvaran unobserved facors. 3 These are Somala, Sao Tome and Prcpe, Ngera, Seychelles, Angola, Algera, Cape Verde, Madagascar, Lbya, Cong, Dem. Rep, and Zamba. Daa are no avalable for Somala, Sao Tome and Prcpe Ngera and Seychelles. Algera, Cape Verde, Madagascar, Lbya, Zamba, and Cong, Dem. Rep have oo hgh average publc or prvae nvesmen or boh over he sample perod. For nsance, prvae nvesmen for Algera, Cape Verde, Madagascar, Zamba and Cong, Dem. Rep are repored o be 09%, 7%, 59%, 79% and 7% of GDP, respecvely. Publc nvesmen for Algera, Cape Verde, Madagascar, Lbya, and Zamba are shown o be 7%, 47%, 7%, 55% and 39% percens of GDP, respecvely. 5

27 averages of he daa. We have esmaed equaon (6) exacly as appears, and also wh a one perod lag o conrol for possble problems of reverse causaly. 3 In he frs case we esmae (sandard error, 0.038), n he laer (sandard error 0.04). These esmaes are n he range characerzng complemenary, summarzed n Table. 6. Publc nvesmen, and he mpac on nequaly and growh 6. Welfare coss of dosyncrac shocks Table 3 repors he effec on he equlbrum growh rae and welfare obaned by elmnang he nequaly arsng from he presence of he dosyncrac producvy shocks. Snce he producvy elascy, e, s ypcally deermned endogenously, raher han specfed as an exogenous parameer, he able s arranged relang he share of publc capal,, o he plausble range of he elascy of subsuon,, based on he calbraon n Secon 5. The mpled producvy elasces are conssen wh he range of emprcal esmaes dscussed by Bom and Lghar (04) and ohers, summarzed n Secon 5. Thus, whle he Cobb-Douglas funcon serves as he convenonal benchmark, we vew as he mos lkely scenaro, and for exposonal purposes wll rea he parameerzaon 0., 0.6 as he base case. Ths poson s conssen wh he emprcal analyss conduced by Lynde and Rchmond (99), who, usng a more general ranslog approach, suggesed ha publc and prvae capal end o be complemens raher han subsues n producon. 33 However, n lgh of our calbraon resuls, we ceranly canno dsmss as a plausble possbly n some nsances. Table 3 ndcaes ha he mpac of elmnang enrely dosyncrac shocks on growh and welfare s raher conssen across hs range of plausble parameers. The long-run growh rae wll ncrease by around.5 percenage pons, whle welfare wll ncrease by around 7%. Boh of hese responses are conssen wh emprcal evdence. The magnude of he growh effec s comparable 3 As robusness checks we also esmaed (6) usng one year averages of he daa. We also employed he approxmaon proposed by Kmena (967); see, Klump e al., 0. In all cases we oban esmaes of he elascy of subsuon, well below uny. 33 See also Baer and Glomm (00). Snce he publc good s an aggregae, comprsng a mx, some of whch are complemens and ohers subsues o prvae capal, we resrc he range of varaon of. 6

28 o he esmae of Alesna and Rodrk (994), who fnd ha a decrease n he Gn coeffcen by 0.6 pons rases he growh rae by around 0.8 percenage pons. 34 In conras o Lucas (987) fndng ha economy-wde supply shocks mpose neglgble welfare coss, Turnovsky and Bancon (005) found ha dosyncrac producvy shocks were many mes more cosly, yeldng coss comparable o hs, dependng upon he assumed degree of rsk averson. In addon, Pallage and Robe (003) provde evdence o sugges ha he welfare coss of economc flucuaons n developng counres may be of he order of up o 5 mes ha of he Uned Saes! 6. Increase n rae of publc nvesmen Alhough he equlbrum descrbed n equaons (8a) generaes ransonal dynamcs (n conras o he Romer-Barro model), he speed of he local dynamcs, summarzed by D, s exceedngly fas. 35 Ths s unsurprsng, snce boh prvae and publc capal deprecae fully each perod and he only source of dynamcs s he small fracon of curren oupu ha s saved and conrbues o nex perod s capal. Table 4 repors he long-run effecs of rasng he rae of publc nvesmen from s benchmark value of 5% by percenage pon o 6%. The seady-sae growh rae s exremely sensve o he producvy parameer, a, and n all cases, we have chosen so as o normalze he nal seady-sae growh rae a %. 36 The frs lne of he able shows he effecs on boh nequaly and growh, correspondng o he calbraed value 0. 6, whle he second lne descrbes he case of zero producvy shocks and herefore no long-run nequaly. Takng he case 0.6, 0. as he benchmark, we see ha he ncrease n governmen nvesmen rases he growh rae by around.4 percenage pons, whch s conssen wh he emprcal evdence ced by Calderón and Servén (04). 37 Lookng across he able we see ha he 34 Reducng he Gn coeffcen o zero would resul n an ncrease n he growh rae of around.6% 35 Ths s easly seen for he Cobb-Douglas case where for he benchmark parameerzaon 0., / 3 when D To oban slower and more plausble ransonal dynamcs, capal mus deprecae a a lower rae. 36 The fac ha we are changng a prevens comparsons of he magnudes of he changes beween he hree cases. 37 Calderón and Servén sugges ha a % ncrease n he sock of nfrasrucure may rase he growh rae by beween - percenage pons. Assumng ha publc capal deprecaes a around 5% per annum, n seady sae he flow of governmen nvesmen wll be approxmaely 5% of he sock. Thus, a % ncrease n he sock s equvalen o around 0% ncrease n he ne flow, whch s comparable o an ncrease n from 5 % o 6% consdered n he presen analyss. 7