Jornal of Economic Dynamic & Conrol 51 (215) 42 444 Conen li available a ScienceDirec Jornal of Economic Dynamic & Conrol jornal omepage:.elevier.com/locae/jedc Tax mooing in a bine cycle model i capial-ill complemenariy Konanino Angelopolo a, Syliano Aimaopolo b,n, Jame Malley a,c a Univeriy of Glago, Unied Kingdom b Univeriy of Noingam, Scool of Economic, Sir Clive Granger Bilding, Room B15, Univeriy Par, Noingam, NG7 2RD, Unied Kingdom c CESifo, Germany aricle info Aricle iory: Received 11 Jly 214 Received in revied form 25 Sepember 214 Acceped 4 November 214 Available online 12 November 214 JEL claificaion: E62 E32 J31 Keyord: Sill premim Tax mooing Opimal fical policy abrac Ti paper nderae a normaive inveigaion of e qaniaive properie of opimal ax mooing in a bine cycle model i ae coningen deb, capial-ill complemenariy and endogeno ill acqiiion nder ecnology and pblic expendire oc. We find a illed and nilled labor ax mooing mainain qaniaively nder exernaliie and exogeno oc in ill acqiiion, a ell a en e relaive ill pply i exogenoly deermined. We frer find a e governmen find i opimal o redce bo e ize of e edge beeen e marginal rae of biion and ranformaion in ill aainmen in e long-rn and e andard deviaion of i edge over e bine cycle. Ti i acieved by bidiing ill creaion and axing bo ype of labor income. & 214 Te Aor. Pblied by Elevier B.V. Ti i an open acce aricle nder e CC BY licene (p://creaivecommon.org/licene/by/3./). 1. Inrodcion Te celebraed ax mooing rel of Barro (1979) in a parial eqilibrim eing a led o a nmber of imporan die on opimal fical policy over e bine cycle in repreenaive agen general eqilibrim model. For example, Lca and Soey (1983) formalied labor ax mooing iin a complee mare neoclaical ep io capial en e governmen ad acce o ae-coningen deb. Cari e al. (1994) generalied i rel in a model i capial and labor axaion and oed a Ramey policy dicaed a e labor income ax flcaed very lile in repone o aggregae oc and e ex ane capial income ax flcaed arond zero. Te lierare a alo examined e implicaion of policy fricion and incomplee ae mare for opimal ax and deb policy rog a variey of rericion o e policy inrmen e, governmen deb and capial income axaion (ee e.g. Socman, 21; Aiyagari e al., 22; Angeleo, 22; Beraa and Nicolini, 24; Fari, 21). In conra, aming complee ae mare and a complee inrmen e, Arenea and Cg (212) conidered labor mare fricion aociaed i a diviion of e labor force ino employed and nemployed orer. Teir model, i ae-coningen deb b no capial, ggeed a opimal labor ax volailiy depended on eer age ere e efficienly. n Correponding aor. Tel.: þ44 79485471. E-mail addre: yliano.aimaopolo@noingam.ac. (S. Aimaopolo). p://dx.doi.org/1.116/j.jedc.214.11.2 165-1889/& 214 Te Aor. Pblied by Elevier B.V. Ti i an open acce aricle nder e CC BY licene (p://creaivecommon.org/licene/by/3./).
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 421 Anoer imporan diviion of e labor force i i repec o e ype of labor ervice orer provide and, in pariclar, o ee complemen capial in e prodcion proce. Ti i epecially perinen given e empirical relevance of e age premim accring o illed labor and e role aribed o capial-ill complemenariy, e relaive pply of illed labor and capial agmening ecnical progre (ee e.g. Kaz and Mrpy, 1992; Krell e al., 2; Hornein e al., 25). In an imporan conribion, ic alo conider non-omogeno labor, Werning (27) eablied e condiion nder ic opimal labor ax mooing eld in a model i rediribion nder complee ae mare en orer differed i repec o eir prodciviy. Hoever, ince e diribion of prodciviy differenial i aen a given, i approac doe no accon for e endogeno deerminaion of employmen ype (ee e.g. Mayama, 26, o alo revie e lierare on pard profeional mobiliy). In i paper e aim o conribe o e ax mooing lierare by focing on e difference in e complemenariy beeen capial and illed/nilled labor a ell a e endogeno deerminaion of e relaive ill pply for Ramey ax policy. Compared o Werning (27), e foc on aggregae ocome and abrac from rediribion incenive, by folloing e lierare a examine a diviion of e labor force ino o ype of orer. To i end, e or i a repreenaive oeold ic garanee i member e ame level of conmpion (ee e.g. Arenea and Cg, 212). We ay a cloe a poible o e repreenaive agen Ramey analyi of Cari e al. (1994) and exend eir model o allo for capial-ill complemenariy and endogeno ill formaion. Or goal i o nderae a normaive inveigaion of e qaniaive properie of opimal axaion of capial and labor income, a ell a ill-acqiiion expendire, in e preence of aggregae oc o oal facor prodciviy (TFP), invemen-pecific ecnological cange and governmen pending. We ame complee ae mare, b conider e implicaion of differen ampion regarding e mecanim driving ill acqiiion ic can lead o labor mare imperfecion. Moivaed by e analyi and dicion in Goldin and Kaz (28), e foc on ree main alernaive cae. Fir, e evalae e effec of exernaliie in ill creaion given e relevance of peer effec relaed o neigborood and ocial cla. Second, e conider e imporance of oc o exogeno non-fical policy inervenion and ocio-poliical facor affecing ill formaion. Finally, e examine e implicaion of rericing e raio of illed o oal orer o remain conan. In or ep, e governmen can borro, ax (or bidie) ill acqiiion expendire, capial, illed and nilled labor income eparaely, o finance exogeno pblic pending. All policy inrmen are alloed o be ae-coningen. In i environmen, e opimal axe on labor income and ill acqiiion expendire are niqely deermined. Hoever, a i ell non, en e governmen a acce o bo ae coningen deb and ae coningen capial axaion, e econd-be Ramey allocaion do no niqely pin don opimal deb and capial axe (ee Cari e al., 1994). Hence, folloing e lierare, e dic e properie of e ex ane capial ax rae. Moreover, e alo examine e cae ere deb i rericed o be ae nconingen, ic allo o calclae e ex po capial ax or, if e alo allo for ae-coningen axaion of income from bond, e privae ae ax. 1 We fir find a labor income axe remain moo nder capial-ill complemenariy and e differen ampion on ill acqiiion conidered. Te ax on illed labor income i more volaile en relaive ill pply i endogeno, erea e nilled labor income ax become more volaile en relaive ill pply i fixed. Hoever, given e mall magnide of e andard deviaion, ee cange are no qaniaively ignifican for e volailiy of labor axe. Alog opimal labor axe do no opimally cange mc over e bine cycle, e frer find a capial-ill complemenariy lead o differen correlaion of e axe on illed and nilled labor income i e exogeno prodciviy oc. In pariclar, o moo labor or for bo ype of ill, e governmen a o mae e policy edge in e illed and nilled labor mare move in e ame direcion a e oc for illed labor. In conra, e edge need o move in e oppoie direcion afer e very or-rn for nilled labor. Te difference arie becae ecnology oc increae e prodciviy of illed labor more an a of nilled, ic creae an incenive o bie illed labor or for nilled. In rn, i implie a e ax on illed labor income i poiively correlaed i exogeno ecnological oc, ile e ax on nilled labor income i effecively ncorrelaed. We nex find a e ill-acqiiion bidy i e lea moo of e policy inrmen a apply o aic margin of coice and a i correlaion properie follo oe of e ax on illed labor income. Ti bidy i ed o affec e edge beeen e marginal rae of biion and ranformaion in relaive ill pply. Te governmen find i opimal o redce i edge in e eady-ae and alo o redce i andard deviaion over e bine cycle. Or rel frer o a e ill eerogeneiy conidered, irrepecive of e ampion regarding relaive illed pply a e examine, doe no affec e rel obained in e lierare regarding e cyclical beavior of ae axe. In pariclar, e ex ane ax rae on capial flcae arond zero and e ae coningen privae ae and ex po capial axe are near zero and are e mo volaile of e ax inrmen. Finally, e find a irrepecive of e model varian examined, all of e policy inrmen excep for e ex po capial ax and e privae ae ax ineri e perience properie of e oc. 1 A on by Z (1992) and Cari e al. (1994), ae-coningen capial income axe allo e governmen o implemen e complee ae mare ocome, depie e lac of acce o ae-coningen deb.
422 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 Te remainder of e paper i organied a follo. Secion 2 and 3 preen e eoreical model and e Ramey problem repecively. Secion 4 decribe e qaniaive implemenaion and Secion 5 analye e rel. Secion 6 dra e conclion. 2. Model We develop a model a exend e complee mare neoclaical ep in Z (1992) and Cari e al. (1994) by alloing for a diviion of e labor force ino illed and nilled orer, an endogeno ill pply and exernaliie in ill-acqiiion on e oeold ide, and capial-ill complemenariy on e good prodcion ide. Ti ep implie a age premim for illed labor, e relaive pply of ic can be increaed by a co o e oeold in e form of earmared raining expendire. 2 A in Cari e al. (1994) oeold ave in e form of pyical capial and aeconingen governmen bond. Te oeold i modelled a an infiniely-lived repreenaive dynay. Te ead of e oeold mae all coice on bealf of i member by maximiing e aggregae elfare of e family, enring a eac oeold member experience e ame level of conmpion irrepecive of individal labor mare a. Ti i a commonly employed ampion ince Merz (1995), given a i allo for racabiliy en dying aggregae flcaion nder eerogeneiie in e labor mare (ee e.g. Arenea and Cg, 212 for an example i opimal ax policy). Firm e capial, illed and nilled labor o prodce a omogeneo prodc. Folloing Kaz and Mrpy (1992), Krell e al. (2) and Hornein e al. (25), illed labor i amed o be more complemenary o capial an nilled labor. Hence, capial accmlaion a ell a ecnological developmen and governmen policie a are capial agmening, end o increae e illed age premim. In conra, increae in e relaive pply of illed labor ac o redce e ill premim. Finally, e governmen can borro and ax capial, illed and nilled labor income eparaely, o finance bidie on ill-acqiiion expendire and exogeno pblic pending. 2.1. Noaion Te noaion employed rogo follo Ljngqvi and Sargen (212). In pariclar, e ame a in every period Z, ere i a realizaion of oc (ocaic even) AS. Terefore, a eac period ere i a iory of even ¼ ½ ; 1 ; 2 ; ; Š ic i non. Te ncondiional probabiliy of oberving a pecific iory of even i defined a π.for4τ, e condiional probabiliy of aving eqence of even given e realizaion of τ i defined a: π τ. 2.2. Hoeold A repreenaive oeold i compoed of o ype of member o provide illed and nilled labor ervice. 3 Te iliy fncion of e oeold i given by U c n; ; l n; ; l ere Uð: i increaing, ricly concave and ree ime coninoly differeniable i repec o i argmen; c i average conmpion of all oeold member a ime given e iory of even ; 4 l n; ¼ ψ ð l n; and l ¼ 1ψ ð l, denoe, repecively, average oeold leire ime from illed and nilled member; l and l are leire ime per illed and nilled member repecively; and ψ ð i e are of illed o oal oeold member or e relaive ill pply. Te ime conrain facing eac ype of member are given by þl ¼ 1 ð1 þl ¼ 1 ð2 ere, and denoe, repecively, illed and nilled labor or per member. Te oeold can deermine i relaive ill pply by incrring an average (over all i member) ill-acqiiion expendire, e, according o e folloing relaion: ψ ð ¼qe ; e e ; ξ ð3 ere qð: i increaing, ricly concave and ree ime coninoly differeniable i repec o e and e e ; e e denoe aggregae, economy-ide ill-acqiiion expendire a e repreenaive oeold ae a given; and oξr1 i a parameer a capre e exen of exernaliie in ill creaion, i iger vale of ξ denoing le exernaliie. Exernaliie aociaed i ill acqiiion expendire capre for example peer effec relaed o neigborood and ocial cla. Te cae ere all expendire on ill-acqiiion i inernalied, i.e. en e ¼ e e can be obained by eing ξ ¼ 1. 2 Ti i conien i e lierare on pard profeional mobiliy, ere ere i a co aociaed i acieving a iger profeional a (ee e.g. Mayama, 26 for a revie of everal model). 3 Noe a e ni ma of oeold member i eqal o e m of i illed and nilled member. 4 Since conmpion i e ame for all member of e oeold, average and per member conmpion are eqal.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 423 We can re-arrange (3) for e o obain ill-acqiiion expendire a a fncion of relaive ill pply, i.e. e gð ¼ g ψ ð ; e e ; ξ, ere gð= ψ gψ 4, and gð= e e ge e o. 5 We ill denoe e olion for e in (3) in e cae ere e e ¼ e by gð ¼ g ψ ð. Finally noe e fncion gð a e propery a gð ¼ g½ψ ð ; e e Š en ξ ¼ 1. Te oeold can inve in capial and in ae-coningen eqenially raded governmen bond a mare flly iin a period. Te evolion of capial i given by þ 1 ð ¼ð1δ ð 1 þa n; ð i ð ð4 ere 1 i e per member oc of capial a ime given e iory of even 1 ; i ð i invemen in capial a ime given e iory of even ; and oδo1 i e capial depreciaion rae. Te capial evolion eqaion allo for an exogeno proce, A n; ð, capring an invemen-pecific ecnological cange, ic a been on o conribe o op flcaion (ee e.g. Greenood e al., 2), a ell a e cange in e ill premim (ee e.g. Krell e al., 2; Lindqi, 24; Porporide, 211). By ing Eq. (4) and defining a A ð 1=A n; ð, e ge e oeold' eqence of bdge conrain: c þa ð þ 1 þ p þ 1 b þ 1 þ 1 þ 1τ a gð ¼ 1τ ψ ð þ 1 þ 1τ 1ψ ð þ 1δ A ð 1 þ 1τ i r 1 þb 1 8 ð5 ere p þ 1 i e pricing ernel for governmen bond in erm of good and b þ 1 þ 1 i e ae þ 1 coningen payo vale of bond bog per member a period ; 6 ; τ, τ, τ are e ax rae on illed and nilled labor, capial income repecively; and τ a i a bidy on ill-acqiiion expendire; and are e age rae of illed and nilled labor repecively; and r i e rern o capial. 2.3. Fir order condiion for oeold Te objecive of e repreenaive oeold i o maximie: 1 β π c ; ; ; ψ ð ð6 ¼ ere oβ o1 i e oeold' bjecive dicon facor, bjec o e eqence of conrain in (5), by cooing fc ð ; ð ; ð ; ψ ð ; þ 1 ð 8 g 1 ¼ and fb þ 1ð þ 1 ; ; 8 g 1 ¼, given iniial vale for b,. In eac ime period and given iory, fb þ 1 ð þ 1 ; g 1 ¼ i a vecor of governmen bond i one elemen of e vecor for eac poible realiaion of þ 1.In(6), ð: i obained by biing e ime acconing ideniie for l n; n; and l and e ime conrain (1) (2) ino Uð:. Ti maximiaion problem yield ix fir-order condiion ic are repored in Appendix A. Combining e fir-order condiion for conmpion, illed and nilled labor pply give e folloing aemporal eqilibrim condiion, ic eqae e marginal rae of biion beeen conmpion and eac ype of labor i e average rern o illed and nilled labor ne of axe: ð c ð ¼ ψ 1τ ð ð7 ð c ð ¼ 1ψ ð 1τ ð ð8 Sbiing e fir-order condiion for conmpion ino e fir-order condiion for ψ ð give e folloing aemporal condiion for relaive ill pply: ψ ð c ð ¼ 1τ 1τ 1τ a gψ ð9 Condiion (9) ae a e marginal rae of biion beeen conmpion and e relaive ill pply i eqal o e ne marginal benefi of increaing e oeold' are of illed orer. Te laer inclde e ne increae in labor income, given by e difference beeen e po-ax labor income from an addiional illed member, 1τ, and e po-ax labor income from one le nilled member, 1τ. From i, e oeold need o dedc e po-bidy co for an addiional illed member, 1τ a gψ. Sbiing e fir-order condiion for conmpion and i one-period lead ino e fir-order condiion for e o ae give e folloing ineremporal condiion eqaing e crren co of inveing in bond and capial o e fre ae-coningen and expeced benefi repecively: c ð p þ 1 ¼ βπ þ 1 þ 1 c ð þ 1 ð1 5 Noe a e follo Ljngqvi and Sargen (212) in ing e noaion Xð= x Xx for fncion X and variable x in ime for iory. 6 Given e period ae 1 (or ele e iory ), e income ide of e oeold bdge inclde revene from bond daed b 1.
424 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 ( " β c ð þ 1 c ð A ð A þ 1 ð þ 1 1δ þ 1τ þ 1 ð þ 1 r þ 1 þ 1 i) ¼ 1 E ð11 ere π þ 1 þ 1 =π ¼ π þ 1 þ 1 ; E i e expecaion condiional on informaion a ime (i.e. iory ), E x þ 1 ð þ 1 ¼ π þ 1 þ 1 =π x þ 1 ð þ 1 ; and e mmaion over þ 1 denoe e m over all poible iorie þ 1 ~ þ 1 c a ~ ¼. By combining e ineremporal condiion (1) and (11) e obain A ð ¼ p þ 1 f 1τ þ 1 þ 1 i r þ 1 þ 1 þ 1δ A þ 1 ð þ 1g ð12 þ 1 ic enre no-arbirage beeen e invemen opporniie in bond and capial. 2.4. Firm Firm ren capial a ell a illed and nilled labor from oeold o maximize eir profi ing a prodcion ecnology, Fð, a exibi conan rern o cale in i ree inp: i Π ¼ F ð ;f ð ; ;f ð ; f ð ; ð ;f ð ð ;f ð r ð f ð ð13 ere Fð incorporae a ocaic oc,, and an f percrip denoe firm qaniie. Ti yield e andard firorder condiion: ð ¼F ;f ð ð ¼F ;f ð r ð ¼F f ð : ð14 ð15 ð16 2.5. Governmen bdge and eqilibrim condiion Given a iory, e governmen finance an exogeno ream of expene, g e ð, bidie o ill-acqiiion a expendire, τ, and i deb obligaion. b 1, by axing capial and labor income and by iing ae-coningen deb. Hence, e iin-period governmen bdge conrain i given by g e ð ¼τ ð ð ψ ð ð þτ ð ð 1ψ ð ð þτ ð r ð ð 1 τ a gð þ þ 1 b þ 1 þ 1 b 1 : ð17 þ 1 p Te aggregae coniency condiion and mare clearing condiion for illed labor, nilled labor and capial are given repecively by Fð ¼ c ð þg e ð þgðþa ð þ 1 ð 1δ ð 1 ð18 ψ ð ;f ¼ ð ð19 1ψ ð ;f ¼ ð ð2 ð 1 ¼ f ð : ð21 Noe a in eqilibrim a e aggregae level e e e. Te eqilibrim condiion for e e can be obained by eing e e ¼ e in (3) and re-arranging for e, ic i eqivalen o e e g ψ ð : ð22 Hence, o obain e eqilibrim condiion for e decenralied economy given economic policy, e bie e e ing (22) ino e fir-order condiion of e privae economy. Ti implie a in (5) e replace gð by gð and (9) become ψ ð c ð ¼ 1τ 1τ 1τ a ~gψ ð23 ere ~g ψ i obained by biing (22) ino gψ. 7 7 Noe a e aggregae condiion (17) and (18) ave already been rien in erm of gð.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 425 Finally, o obain e eqilibrim condiion in erm of oeold qaniie e bie e mare clearing condiion (19) (21) ino (14) (16) and (18). In pariclar, e bie ;f ð, ;f ð, f ð ino Fð, F ;f ð, F ;f ð and F f ð and denoe e reling qaniie repecively by Yð, F ~ ; F ~ ; F ~. Terefore, (14) (16) and (18) become ð ¼ F ~ ð24 ð ¼ F ~ r ð ¼ F ~ Yð ¼ c ð þg e ð þgðþa ð þ 1 ð 1δ ð 1 ð25 ð26 ð27 2.6. Model edge In Appendix B e preen e ocial planner' olion for i model economy and expre economic efficiency in erm of marginal rae of biion and marginal rae of ranformaion a explicily rea ill acqiiion a a primiive. Ti frer allo, in Appendix C, o obain e relevan edge a preven e privae economy from acieving e efficien ocome, by comparing e fir-order condiion a caracerie eqilibrim ocome in e privae economy o e repecive efficien condiion. Ti analyi demonrae a, for e eqilibrim margin caraceriing illed and nilled labor pply and capial accmlaion, e edge are idenical i e relevan labor and capial income axe. Hoever, i i no e cae for e coice of e relaive ill pply. In i inance, e edge, ψ, i compoed of mare, ψ m, and policy, ψ p, componen ψ ¼ ψ m þψ p ð28 ere ψ m ¼ ~g ψ gψ ð29 ψ p ¼ τ F ;f ð τ F ;f ð τ a ~gψ : ð3 Hence, a fficien condiion for eliminaing ψ i τ ¼ τ ¼ τ a ¼ and ξ ¼ 1. In pariclar, noe a given e Pigovian nare of axaion in e model i exernaliie, if ξa1, eing e axe o zero ill no eliminae e edge beeen e mare and efficien ocome, ince eing axe o zero ill eliminae ψ p b no ψ m. Hoever, if ξa1, e governmen can maniplae e axe o affec e oal edge. Aming a e governmen ad acce o a lmp-m inrmen o finance pblic pending, all edge cold be eliminaed from e mare economy by eing τ ¼ τ ¼ τ ¼ and τ a g ψ þ ~gψ ¼ ~g ψ ð : ð31 Te analyi in Appendix C alo demonrae a e ax yem i ic e governmen i endoed i complee, ince for eac margin of adjmen in e mare economy ere i a niqe policy edge. In oer ord, e available policy inrmen deermine eac edge niqely. 3. Te Ramey problem Te governmen cooe labor and capial income axe, e bidy on ill-acqiiion expendire and nex period ae-coningen deb o maximie e oeold' elfare bjec o e eqilibrim fir-order condiion a mmarie e reacion of e privae economy. 8 Noe a e Ramey planner inernalie e exernaliie aociaed i illacqiiion expendire. To olve e Ramey problem e fir derive e preen diconed vale (PDV) of e oeold' lifeime bdge conrain ing e Arro Debre price of e bond and e ranveraliy condiion for bond and capial. Second, e derive e implemenabiliy conrain by biing o price and ax rae from e oeold' preen vale bdge conrain ing e fir-order condiion for e oeold and firm. Finally, e derive e opimal Ramey allocaion by maximiing e planner' objecive fncion bjec o e implemenabiliy conrain and e eqence of aggregae reorce conrain. 8 In Angelopolo e al. (213), e develop a model ere fricion in bo labor and capial mare lead o income ineqaliy beeen differen ype of oeold. We en dy e problem of a governmen ic cooe axe on oal income for differen oeold and non-coningen deb. Hoever, folloing mo of e lierare on opimal axaion, a ep doe no allo for mobiliy beeen e differen agen.
426 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 3.1. Preen vale of bdge conrain We ar i e oeold' flo bdge conrain in eqilibrim. Beginning from period and by repeaedly biing forard one-period bdge conrain for e oeold, e obain e PDV of e oeold' lifeime bdge conrain: 1 ¼ 1! p i i þ 1 i c ð ¼ 1 1! p i i þ 1 i f ð1τ ψ ð þ ð1τ i ¼ ¼ i ¼ 1ψ ð 1τ a n i gðgþb þ ð1τ ð o r ð þ 1δ A ð ð32 ereeaveimpoedeerieofno-arbiragecondiionfrom(12) 8 and e folloing ranveraliy condiion for any 1 : lim 1 p i -1 i ¼ lim -1 þ 1 i þ 1 i 1 p i i ¼! þ 1 ¼! i þ 1 i p þ 1 b þ 1 þ 1 j ¼ ð34 ic pecify a for any poible fre iory e oeold doe no old poiive or negaive valed eal a infiniy. Defining i ¼ 1 p i i þ 1 i q ð, 8 Z1, i q ð 1, ere q ð i e Arro Debre price, e can re-rie (32) a 1 q ð c ð ¼ 1 q ð ð1τ ψ ð þ ð1τ 1ψ ð 1τ a gð ¼ ¼ n i þb þ ð1τ ð o r ð þ 1δ A ð ; ð35 Noe a e Arro Debre price aifie e recrion: q þ 1 ð þ 1 ¼p þ 1 q ð : ð36 Uing e fir-order condiion from e eqenial eqilibrim for pricing coningen claim (1) and noing a π ¼ 1, ince, a period e ae i non, e above recrion can be rien a ¼ β þ 1 π þ 1 þ 1 c ð þ 1 c ð : q þ 1 þ 1 ð33 ð37 3.2. Implemenabiliy conrain Fir, noe a (37) implie q ¼ β π c ð c ð : ð38 Sbiing (38) for q ð ; e fir-order condiion of e firm, (24), (25) and (26) for, and r, repecively; and e fir-order condiion of e oeold, (7), (8), (23) for τ, τ and τ a repecively ino e preen vale bdge conrain (35), e obain e implemenabiliy conrain 9 : 1 β π ½c c þ þ þc gðω ŠA ¼ ð39 ¼ ere A Aðc ; ; ; ψ ; b ; ; τ ¼ c fb þ½ð1τ F ~ ð þ 1δ A ð Š g and Ω ¼ ð c ð ψ ð þ ð c ð ð1ψ ð ψ ð þ c ð = ~g ψ : 3.3. Pedo vale fncion Te governmen maximie (6) bjec o e implemenabiliy conrain (39) and e eqence of aggregae reorce conrain in (27) 8 by cooing c ; ; ; ψ ð ; þ 1 8 1 ¼, given b ; ; τ. 1 To acieve i, e 9 Noe a e ineremporal fir-order condiion (12) a been ed already in deriving (35), ile e governmen bdge conrain i redndan, ince i i a linear combinaion of e oeold' bdge conrain and e aggregae reorce conrain. Terefore, (27) and (39) mmarie all e conrain a e governmen need o repec. 1 Noe a folloing e lierare e do no examine e problem of iniial capial axaion and do no allo e governmen o cooe τ.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 427 follo Ljngqvi and Sargen (212) and fir pecify e folloing iin-period pedo vale fncion: Vc ; ; ; ψ ð ; Φ ¼ c ; 1 ; 1 ; ψ ð þφ½c c þ þ þc gðω Š ere Φ i e Lagrange mliplier i repec o e implemenabiliy conrain. 11 Te Lagrangian of e Ramey planner i defined a J ¼ 1 β π fv c ; ; ; ψ ð ; Φ þθ ½Yðc g e ð gðð þ 1 ð þð1δ ð 1 A ð ŠgΦA ¼ ð41 ere fθ ð ; 8 g 1 ¼ i a eqence of Lagrange mliplier aaced o e aggregae reorce conrain. For a given level of b ; ; τ, J i maximized i repec o fc ; ; ; ψ ð ; þ 1 ; 8 g 1 ¼ 1 and c,,, ψ ð, 1 yielding e folloing fir-order condiion repecively: V c ¼ θ ; Z1 ð42 V ¼θ Y ; Z1 ð43 V ¼θ Y ; Z1 ð44 V ψ ¼ θ gψ Yψ i ; Z1 ð45 A ð ¼βE θ þ 1 þ 1 Y þ 1 i þ 1δ A þ 1 ð þ 1 ; Z ð46 θ ð4 V c ¼ θ þφac ð47 V ¼θ Y þφa ð48 V ¼θ Y þφa ð49 V ψ ¼ θ gψ Yψ i þφa ψ : ð5 Te fir-order condiion derived in (42) (5) imply a e yem of eqaion o be olved ill be differen for ¼ and for 4. Tee condiion in a non-ocaic environmen are preened in Appendix D. 3.4. Capial and ae axe A i ell non (ee e.g. Z, 1992; Cari e al., 1994; Ljngqvi and Sargen, 212), e Ramey problem i aeconingen deb canno niqely pin don e capial ax rae. Hence, e follo e lierare and calclae e opimal exane capial income ax rae (ee Appendix E for deail): τ ;a βe c ð þ 1 ~ i F ð þ 1 þ 1δ A þ 1 ð þ 1 c ð A ð þ 1 ¼ βe c ð þ 1 F ~ : ð51 ð þ 1 Alernaively, by aming a governmen deb i no ae-coningen, e can calclae e ex po ae coningen capial ax (ee Appendix F for e derivaion): τ ;p 1 ¼ fg e r ð ð 1 ð þτ a b þ 1 gðð R ð þb 1 τ ð ð ψ ð τ ð ð 1ψ ð g ð52 ere R ð i e ae nconingen or e ri free rern o olding governmen deb. Alernaively, aming a e governmen employ a ae-coningen ax on income from governmen bond, e can calclae e privae ae ax, ξ þ 1 j a applie o axing joinly e income from ae a (ee Appendix F for e derivaion): ξ þ 1 j 1 ¼ fg e F ð þ 1 þ 1 ð þb þ 1 ð þ 1 ð þ 1þb þ 1 b þ 2 þ 1 R þ 1 þ 1 τ þ 1 ð þ 1 þ 1 ð þ 1 ψ þ 1 þ 1 þ 1 ð þ 1 τ þ 1 ð þ 1 þ 1 ð þ 1 1ψ þ 1 þ 1 þ 1 ð þ 1 þτ a þ 1 þ 1 gðð þ1g: ð53 11 Noe a e mliplier Φ i non-negaive and meare e diiliy of fre ax diorion.
428 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 4. Qaniaive implemenaion In i ecion e qaniaively olve bo e non-ocaic and ocaic opimal policy model. Or olion approac follo Arenea and Cg (212). In pariclar, e fir calibrae e non-ocaic model i exogeno policy. Nex, e olve e deerminiic Ramey problem, aring from e exogeno policy eady-ae, ing nonlinear meod. Since e are inereed in ax mooing over e bine cycle, e en approximae arond e eady-ae of e deerminiic Ramey problem o olve e ocaic problem and obain near eady-ae dynamic. 4.1. Fncional form Folloing Cari e al. (1994) and Socman (21), e e a CRRA iliy fncion: n c 1 σ1 σ 2 ψ ð l σ1 1ψ ð l σ2 o σ3 ð¼ σ 3 ere σ 1 and σ 2 are e eig o leire in e iliy fncion and σ 3 i e relaive ri averion parameer. Te prodcion ide i given by a CES prodcion fncion a allo for capial-ill complemenariy, ince e laer a been on o mac e dynamic of e ill premim in e daa (ee e.g. Krell e al., 2; Lindqi, 24; Porporide, 211): ( Fð ¼ A ð μ ;f α ð þ 1μ ρ f ν ð þ 1ρ ;f ν i α=νg 1=α ð ere A ð i oal facor prodciviy; αo1, and νo1 are e parameer deermining e facor elaiciie, i.e. 1=ð1α i e elaiciy of biion beeen capial and nilled labor and beeen illed and nilled labor, erea 1=ð1ν i e elaiciy of biion beeen eqipmen capial and illed labor; and oμ; ρo1 are e facor are parameer. In i pecificaion, capial-ill complemenariy i obained if 1=ð1α41=ð1ν ic alo implie a e ill premim i decreaing in ψ ð and increaing in ð 1. Te fncional form for e relaive ill pply i q½š ¼ Ψ e γξ e e γð1 ξ ð56 ere Ψ 4 i e prodciviy of ill-acqiiion; rγ o1 meare e prodciviy of ill-acqiiion expendire; and oξr1 i e parameer a capre e exen of exernaliie in creaing ill. ð54 ð55 4.2. Exogeno policy and calibraion We nex preen e calibraion and eady-ae for e exogeno policy model. In pariclar, e obain e eadyae of e decenralied compeiive eqilibrim (DCE). Given iniial level of, and b, and e five policy inrmen fτ ; τ ; τ τ a ; ge g, e non-ocaic DCE yem i caracerized by a eqence of allocaion fc ; ; ; ψ ; þ 1g 1 ¼,pricef ; ; r ; R g 1 ¼, and e reidal policy inrmen fb þ 1g 1 ¼ c a (i) oeold maximie eir elfare and firm maximie eir profi, aing policy, price and aggregae ocome a given; (ii) e governmen bdge conrain i aified in eac ime period and (iii) all mare clear. T, e non-ocaic DCE i compoed of e non-ocaic form of e fir-order condiion of e oeold (7), (8), (1), (11)and(23), e governmen bdge conrain (17), e ree fir-order condiion of e firm (24) (26), and e aggregae reorce conrain (27). 4.2.1. Calibraion Te non-ocaic model i exogeno policy i calibraed o a i eady-ae i conien i e annal US daa. Uiliy: Table 1 belo repor e model' qaniaive parameer along i an indicaion of eir orce. Saring i e are of leire for eac ill ype in iliy, σ 1 and σ 2, e calibrae ee o.35 eac o a, in e eady-ae, e oeold devoe abo one ird of i ime o oring. Te relaive ri averion parameer, σ 3 ¼2 i commonly employed in bine cycle model. Prodcion: Te elaiciie of biion beeen illed labor and capial and beeen nilled labor and capial (or illed labor) ave been eimaed by Krell e al. (2). Folloing e lierare (ee e.g. Lindqi, 24; Porporide, 211), e alo e ee eimae o e a¼.41 and ν ¼:495. Te remaining parameer in e prodcion fncion are calibraed o enre e eady-ae predicion of e model in ae and labor mare are conien i e daa. More pecifically, e labor eig in e compoie inp are μ ¼ :268 i calibraed o obain a labor are of income of approximaely eqal o 69% and e capial eig in e compoie inp are, ρ ¼ :52, i calibraed o obain a ill premim of 1.64. Bo of ee arge are conien i e U.S. daa for e period 197 211. Te arge vale for e ill premim i from U.S. Cen daa and e are of labor income in GDP i
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 429 Table 1 Model parameer. Parameer Vale Definiion Sorce oσ 1 o1.35 Weig o illed leire in iliy Calibraion oσ 2 o1.35 Weig o nilled leire in iliy Calibraion σ 3 o 2. Coefficien of relaive ri averion Ampion 1 1α 4 1.669 Cap. eqip. o nilled labor elaiciy Ampion o 1 1ν o 1.669 Cap. eqip. o illed labor elaiciy Ampion 1α o1μo1.732 Sare of compoie inp o op Calibraion oρo1.52 Sare of cap. eqip. o compoie inp Calibraion A4 1. TFP Ampion A n; 4 1. Invemen-pecific prodciviy Ampion rδr1.7 Depreciaion rae of capial Calibraion oβo1.96 Time dicon facor Calibraion rγ o1.172 Relaive ill pply elaiciy Calibraion oξr1.7 Te are of exernaliy Calibraion Ψ 4 1. Prodciviy of ill-acqiiion Ampion τ.31 Capial income ax rae Daa τ.2 Unilled labor ax rae Daa τ.25 Silled labor ax rae Daa τ n.22 Effecive labor ax rae Daa τ a. Sill-acqiiion expendire ax rae Ampion g e 4.46 Governmen pending Calibraion from e BEA daa on peronal income. 12 We alo normalize e eady-ae vale of TFP and invemen on eqipmen o niy (i.e. A ¼ A n; ¼ 1). Depreciaion and ime preference: Te depreciaion rae of capial δ ¼ :7 i calibraed o obain an annal capial o op raio of abo 1.96, ic i conien i e annal daa repored by e BEA on capial oc. 13 Te ime dicon facor, β ¼ :96, i e o obain a po-ax po-depreciaion annal real rae of rern on capial of rogly 4.17%, ic coere i e 4.19% obained in e daa from e World Ban. 14 Relaive ill pply: We normalie ill-acqiiion prodciviy, Ψ, o niy. To mac e are of illed orer in oal poplaion, ψ, of rogly 44% in e daa, e e e prodciviy of ill-acqiiion expendire, γ, eqal o.172. Ti are i conien i e daa from e 21 U.S. Cen ic indicae a 43% of e poplaion a a college degree. 15 I alo correpond i a relaed daa e by Acemogl and Aor (211) ic implie a e average are of e labor force i a college degree i approximaely 45%. We e ξ eqal o.7, ic prodce a eady-ae ill expendire o op raio of abo 4.3%, ic i conien i e daa on eriary edcaion expendire for e la for decade from e U.S. Naional Cener for Edcaion Saiic, Dige of Edcaion Saiic. 16 Tax rae and governmen pending: Finally, e e e ECFIN effecive capial and labor ax rae from Marinez-Mongay (2) o obain an average ax rae for capial and labor. 17 Terefore, e e e ax rae for capial income τ ¼ :31 and e o labor income ax rae τ ¼ :2 and τ ¼ :25. 18 Given a i i difficl o obain daa ic mac ell i e ill-acqiiion expendire ax (bidy) rae, τ a, e e i o zero for e exogeno policy model. We finally e e eady-ae vale g e ¼ :46, o obain a eady-ae deb-o-op raio, b=y ¼ 53%, ic i eqal o e average deb o GDP raio obained in e daa. 19 Seady-ae: Te eady-ae of e DCE defined and calibraed above i preened in Table 2. Te rel indicae a e model' predicion for e grea raio mac oe implied by e daa qie ell. For example, in e daa for 197 211: =y ¼ 1:895, c=y ¼ :64, i=y ¼ :146, g e =y ¼ :23 and b=y ¼ :53. 2 Moreover, e are of ill-acqiiion expendire in GDP, e=y, coere i US oal expendire for college and niveriie a a are of op of abo 12 Te daa orce i e Crren Poplaion Srvey, 211 Annal Social and Economic Spplemen from e U.S. Cen Brea. 13 Specifically, e BEA Table 1.1 on fixed-ae a been ed o obain e ime erie for capial oc for 197 211. 14 Te daa refer o e annal real inere rae from World Ban Indicaor daabae for e period 197 211 (i.e. FR.INR.RINR). 15 Ti informaion i obained from Table 4 of e Cen Brea, Srvey of Income and Program Paricipaion. 16 For inance, in e verion of e model io e exernaliy (ξ ¼ 1) a ill be diced belo, ill expendire i abo 5.3% of op. 17 In pariclar, e e e LITR and KITN rae for effecive average labor and capial axe repecively for 197 211, a ey rea elf-employed income a capial income in e calclaion. 18 Noe a e calclaion of e effecive labor income ax rae i eqal o.22. B ince e ame a e illed and nilled labor income i axed differenly e decompoe e labor income ax ino illed and nilled ax o a e eiged average of e o ax rae eqal.22. 19 Te orce of a ime erie i: FRED Economic Daa on Gro Federal Deb a a percenage of GDP, 197 211. 2 Noe a if model predicion for e co of becoming illed, e=y ¼ :433, i added o e c=y raio from e model, e m i very cloe o e c=y raio in e daa.
43 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 4.3% ince e mid-evenie. 21 A poined o above, e remaining eady-ae variable in e exogeno model ave been calibraed o mac eir vale in e daa. 4.3. Deerminiic Ramey Te deerminiic verion of e Ramey problem in (42) (5) i mmaried in Appendix D, (ee Eq. (D.1) (D.16)) and i olved ieraively, condiional on e calibraion decribed in e previo ecion. In pariclar, e fir ge a vale for Φ and olve Eq. (D.1) (D.15) for an allocaion fc ; ; ; ψ ; þ 1g T ¼. Ten e e eer Eq. (D.16) i binding and increae or decreae e vale of Φ if e bdge i in defici or rpl repecively. Te iniial condiion for e model' ae variable are given by e non-ocaic exogeno eady-ae (ee Table 2). For e erminal vale of e forard looing variable, e ame a afer T year e dynamic yem a converged o i Ramey eady-ae. Ti implie a e appropriae erminal condiion are obained by eing e vale for ee variable eqal o oe of e preceding period. Te final yem i given by ½ð5 Tþ1Š eqaion, ic i olved non-linearly ing andard nmeric meod (ee, e.g. Adjemian e al., 211). Ti give e dynamic raniion pa from e exogeno o e opimal eady-ae. We e T¼25 o enre a convergence i acieved. Or rel o a i occr for all endogeno variable iin 15 year. 22 Afer e find e opimal allocaion for fc ; ; ; ψ ; þ 1g T ¼ e obain ¼ F ~ ð, ¼ F ~ ð and r ¼ F ~ ð. a Addiionally, e olve for τ, τ, τ and τ ing e non-ocaic form of (7), (8), (11) and (23) repecively. Te Ramey eady-ae i repored in Table 3. Te rel are conien i e meage from e lierare iniiaed by Camley (1986) on dynamic Ramey axaion in a deerminiic environmen (ee e.g. Ljngqvi and Sargen, 212, c. 16 for a revie of i lierare). A expeced, alloing e governmen a complee inrmen e rel in a zero capial ax rae in e long-rn. Compared i e eady-ae of exogeno policy, a Ramey governmen old increae capial accmlaion in e eady-ae, by eliminaing e ineremporal edge. Moreover, given e exernaliie in ill creaion, e governmen find i opimal o bidie expendire on ill-acqiiion by abo 4% in i ep, ic lead o an increae in e relaive ill pply. Since i increae i encoraged by e ill bidy, ic imlae overall effecive illed labor or, e governmen can ax a a ligly iger rae e iger labor income orce, i.e. illed labor income, ic allo for more ax revene o be generaed. Noice a e progreiviy of opimal labor income axaion i very mall qaniaively. Te fall in e ill premim nder Ramey policy gge a e increae in e relaive ill pply a a relaively ronger qaniaive impac an e increae in e capial oc. Finally, e governmen i able o finance par of e reqired pblic pending in e long-rn from accmlaed ae. We nex dy e raniion dynamic aociaed i Ramey policy. Fig. 1 illrae e dynamic pa implied by opimal policy for e capial ax, e o labor axe, e ill-acqiiion expendire ax and deb o op a e economy evolve from e exogeno eady-ae o e Ramey eady-ae. Te fir panel of Fig. 1 o a in period 1 illed and nilled labor are bidied a rae of 24.1% and 21.3% repecively; and ill-acqiiion expendire i axed a a rae of 3.6%. In period 2, illed and nilled labor axe are 29.2% and 27.5% repecively and evenally converge o eir eady-ae vale repored in Table 3. Alo in period 2, illacqiiion i bidied a a rae of 37.7% and converge o abo 4% in e eady-ae. Te econd panel of Fig. 1 o a in period 1, ince capial i already in place, capial income i axed a a conficaory rae (approximaely 36%). In period 2, e capial income ax i.9% and en converge loly o zero. Te ig capial axaion in e fir period allo e governmen o creae a oc of ae by lending o e oeold. Governmen ae increae in fre period and eir income i ed o bidie ill-acqiiion expendire and o compenae for e loe from foregone capial income axaion, io e need o reor o ig labor income axe. Tee raniion pa are conien i previo reearc. 4.4. Socaic procee To move o e analyi of e ocaic Ramey problem, e need o define e ocaic procee a drive economic flcaion. In a follo e deignae a ocaic ae a ime a deermine exogeno oc o oal facor prodciviy, A ð, invemen-pecific ecnological progre, A n; ð, and governmen expendire, g e ð. Folloing e lierare, A ð, A n; ð and g e ð are amed o follo ocaic ARð1 procee: log A þ 1 ð þ 1 ¼ 1ρ A log AþρA log A ð þε A þ 1 ð57 log A n; ð þ 1 þ 1 ¼ 1ρ A log A n; þρ A n; log A n; ð þε An; þ 1 ð58 21 Teriary edcaion expendire in e USA a iger in e early 197, b declined coninoly ince 197 o converge o le an 4% over e la o decade. 22 See Fig. 1 belo for an illraion of convergence ing e policy inrmen.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 431 Table 2 Seady-ae of exogeno policy. c y y i y e y b y g e y r ne ψ.5823 1.9629.1374.433.53.2371 1.64.417.44 Table 3 Seady-ae of opimal policy. c y y e y b y g e y.5565 2.6859.65.8348.195 1.4285 τ τ τ τ a r ne ψ.2654.2577..45.417.4819 percen percen 5 4 3 2 1 1 2 3 5 1 15 2 25 3 35 4 45 5 ime 4 35 3 25 2 15 1 5 5 1 15 2 25 3 35 4 45 5 ime Fig. 1. Traniion pa of e policy inrmen. 6 4 2 2 4 6 8 1 log g e þ 1ð þ 1¼ 1ρ g e log g e þρ g e log g e ð þε ge þ 1 ð59 A ere ε, ε A n; and ε ge are independenly and idenically diribed Gaian random variable i zero mean and andard deviaion given repecively by σ A, σ A n; and σ g e. Te vale for e ARð1 coefficien and e andard deviaion for e governmen expendire are daa-baed and are eimaed o be: ρ g e ¼ :65 and σ g ¼ :17. 23 Te AR(1) coefficien and e andard deviaion for e invemen pecific ecnological progre are calibraed o ρ A n; ¼ :7, σ A n; ¼ :12 o a e correlaion of invemen i op and e relaive andard deviaion of invemen o op are approximaely eqal o.85 and 4.3 repecively. 24 Te aocorrelaion parameer of TFP i e eqal o.8 folloing Cari e al. (1994) and Socman (21), ile σ A i calibraed o mac e volailiy of op oberved in e BEA daa. 25 More pecifically, e andard deviaion for TFP i e σ A ¼ :8% o obain a volailiy for op from 197 o 211 of abo 1%. 23 Te governmen pending erie refer o governmen conmpion expendire and gro invemen from NIPA Table 1.1.5 (197 211). To calclae e aiical properie of e cyclical componen of e erie, e ae log and apply e HP-filer i mooing parameer eqal o 1. 24 Te invemen erie refer o Privae Fixed Invemen and i i obained from NIPA Table 5.3.5 (197 211). Cyclical invemen i calclaed ing e HP-filer a above. 25 Te ime erie for GDP from 197 o 211 i obained from NIPA Table 1.1.5. Cyclical op i calclaed ing e HP-filer a above.
432 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 4.5. Socaic Ramey We nex approximae e dynamic eqilibrim pa de o ree exogeno oc ing a fir-order approximaion of e deciion rle of e eqilibrim condiion nder opimal policy in (42) (46), arond e opimal deerminiic eadyae of ee condiion decribed above. 26 A i common in e lierare en caracerizing policy dynamic, e alo mae e axiliary ampion a e iniial ae of e economy a ¼ i e eady-ae nder opimal policy. To calclae e bine cycle aiic of e relevan qaniie of e model nder opimal policy, e condc imlaion by ocing all of e exogeno procee, obain e reqired momen for eac imlaion and en calclae eir mean vale acro e imlaion. We nderae 1 imlaion, eac 4 period long, o enre a e ave enog period o approximae lifeime qaniie and a e model generaed daa i aionary. 5. Cyclical properie We nex preen e rel regarding e ey econd momen of e ocaic opimal policy problem. We condc i analyi for e bencmar model developed above and e alo examine e robne of e main rel o cange in e ill creaion ecnology. 5.1. Capial and labor axe We ar i e cyclical properie of Ramey axaion in e bencmar model. Table 5 preen rel on e mean, andard deviaion relaive o op and correlaion for e opimal allocaion, policy and edge. Te rel regarding opimal capial and labor axe are largely conien i e lierare and exend previo finding o a ep i capial-ill complemenariy and endogeno ill pply. In pariclar, e ex ane ax rae on capial i effecively zero and a e ige volailiy of all ax inrmen en deb i ae-coningen. Moreover, en deb i no ae-coningen, e ae coningen privae ae and ex po capial axe are near zero, ave lo correlaion i op and are e mo volaile of e ax inrmen. Alo conien i e labor ax-mooing lierare, bo labor axe ave very lo andard deviaion relaive o op, a e governmen find i opimal o minimie e diorion inrodced by labor axe by eeping em relaively moo over e bine cycle and by leing e remaining ae-coningen policy inrmen repond o exogeno oc. Ti finding alo coere i e rel regarding e volailiy of labor axe in Werning (27), albei in a differen ep, ince e allo for orer, a oppoed o oeold, eerogeneiy. 27 Depie e imilariy regarding eir andard deviaion, e labor income axe exibi differen correlaion i op in or ep. In pariclar, e ax rae on illed labor income i rongly pro-cyclical, erea e ax rae on nilled labor income a a lo and negaive correlaion i op. Ti difference mainain for bo ype of ecnology oc, ince e correlaion of τ i A and A n; are poiive and ig, erea e correlaion of τ i A and A n; are cloe o zero. Noe a e difference in e correlaion of e o labor income axe i e ecnology oc are mirrored in e difference in e correlaion of illed and nilled labor or i e ecnology oc. In pariclar, illed labor or are rongly correlaed i bo ecnology oc, ere nilled labor or exibi very lo correlaion. 28 Finally, e correlaion for all ax inrmen i g e are very lo, a e governmen e aeconingen governmen ae o accommodae emporary pre pblic finance oc. 5.2. Sill-acqiiion bidy We nex find a e ill-acqiiion bidy i abo for ime more volaile an e remaining axe a affec aic margin (i.e. τ and τ ), b no a volaile a e ax inrmen a affec e ineremporal margin of adjmen (i.e. e capial and ae axe). Qaniaively, i andard deviaion implie very lile flcaion in τ a over e bine cycle, a, 95% of e ime i varie beeen 39.8% and 4.3%, given a σ Y ¼ :47. Moreover, e find a τ a i pro-cyclical and i poiively correlaed i bo ecnology procee. 29 I old be noed ere a e volailiy of τ a i de o e exience of e policy diorion in e labor mare and erefore i a caraceriic of e econd-be nare of Ramey opimal policy. Recall from e analyi in Secion 2.6 a if e governmen cold finance pblic pending i lmp m inrmen, o a τ ¼ τ ¼ τ ¼, e opimal ill-acqiiion bidy ic old eliminae e edge inrodced by e exernaliy in ill creaion, τ a, i given by (31). 26 We e e perrbaion meod in Scmi-Groe and Uribe (23) o olve e dynamic model. 27 Werning (27) o a e opimal volailiy of labor axe for oeold of differen abiliy i zero en e iliy fncion incorporae a conan Fric-labor pply elaiciy. In or model, i can alo be on a e andard deviaion of e labor axe are opimally zero for e conan Fric-labor pply elaiciy cae. 28 Te dynamic beavior of e ax rae and eir relaionip i e remaining endogeno variable are frer diced belo en e preen repone o emporary oc. 29 Te repone of τ a o exogeno ecnology oc ill be diced in more deail belo in e imple repone analyi.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 433 Table 4 Parameer for ocaic procee. Parameer Vale Definiion σ A.8 Sd. dev. of TFP ρ A.8 AR (1) coef. of TFP σ A n;.12 Sd. dev. of capial eqipmen ρ A n;.7 AR(1) coef. of capial eqipmen σ g e.17 Sd. dev. of pblic pending ρ g e.65 AR (1) coef. of pblic pending Table 5 Socaic rel bae model. x i x i σ xi σ Y ρðx i ; y ρðx i ; A ρ x i ; A n; ρðx i ; g e Allocaion c.1315.5386.8655.7787.656.993.6347 3.4917.644.39.2194.495.37.3963.7114.638.7424.2452.1715.468.689.616.3771.2313 ψ.4819.2796.9252.788.1826.22.6878 2.6241.9283.8346.444.571.4815 2.725.9275.8471.131.64 1.4285.7358.8752.8978.2194.1126 Policy and edge τ.2654.776.7996.6379.7352.2121 τ.2577.595.25.88.1834.1817 τ ;a 6.7e5 2.7963.937 1..37.2 τ ;p.47 26.2437.3532.4371.859.983 ξ 7.51e4 13.7352.3526.4349.835.984 τ a.45.3144.8791.6621.5663.192 ψ.215.3125.9242.7629.2442.36 ψ m.74.269.9252.788.1826.22 ψ p.525.17.911.7185.364.321 I can be on analyically a for a ill-acqiiion ecnology of e form of (56), i qaniy i conan and eqal o e ize of e exernaliy, i.e. τ a ¼ 1ξ. 3 Terefore, ince e do no ave a lmp m ax inrmen in e model, e volailiy in e opimal τ a repored in Table 5 i de o e preence of non-zero labor income axe. 5.3. Sill acqiiion edge Or rel o far gge a en e baeline neoclaical model i exended o allo for capial-ill complemenariy and endogeno ill pply, e governmen ill find i opimal o eep e labor mare edge moo, ile e capial ax edge i effecively eliminaed. Hoever, e correlaion of e labor edge i e exogeno ecnology procee differ for illed and nilled labor. A Eq. (28) mae clear, e edge in relaive ill pply, ψ, depend on e bidy o ill acqiiion a ell a on e labor axe ic define ψ p. Moreover, becae of e exernaliy in ill creaion, ψ alo depend on ψ m. To frer inveigae opimal policy in relaion o e edge in relaive ill pply, e preen in Table 6 e mean and andard deviaion of ψ and i componen for e model i exogeno policy and nder e Ramey planner. Te rel in Table 6 o a opimal policy redce e oal edge in e eady ae, by rning e policy componen ino a bidy, and alo redce e andard deviaion of e oal componen, by eing e policy inrmen over e bine cycle o moo e policy componen of e edge. 5.4. Imple repone To frer analye e effec of opimal policy over e bine cycle and examine e opimal repone of axaion o cange in exogeno prodciviy, e plo e imple repone of ey endogeno variable, a percen deviaion from 3 Noe a (56) implie a g ψ ¼ 1=γΨ pi ð1=γ1 and ~g ψ ¼ð1=γξΨ ð1=γ1. ψ
434 K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 Table 6 Sill acqiiion edge. Mean Sandard deviaion (%) Exog. policy Ramey Exog. policy Ramey ψ.842.215.17.15 ψ m.477.74.8.1 ψ p.365.525.1.5 Ramey: A oc Exogeno: A oc % dev. from. 1.5 1 2 3 4 5.2.1 ψ 1 2 3 4 5.2.2 1 2 3 4 5.5.5 1 2 3 4 5 % dev. from..2.1 τ a 1 2 3 4 5 1.5 ψ edge 1 2 3 4 5.6.4.2 τ 1 2 3 4 5.2.2.4 1 2 3 4 5 τ Ramey: A *, oc Exogeno: A *, oc ψ % dev. from. 1.5 1 2 3 4 5.4.2 1 2 3 4 5.2.2 1 2 3 4 5.5.5 1 2 3 4 5 % dev. from..1.5 τ a 1 2 3 4 5.5 ψ edge.5 1 2 3 4 5.6.4.2 τ 1 2 3 4 5.2.2 1 2 3 4 5 τ Fig. 2. Imple repone o 1% emporary prodciviy oc. A eir repecive eady ae, afer a emporary 1% oc o e exogeno diribion in ε, ε A n;. Tee plo are on in Fig. 2 above. 31 To conexalie ee effec e alo plo e ame imple repone for e model i exogeno policy diced in Secion 4. Te conino line o e repone nder Ramey policy, ile e daed line e repone nder exogeno policy. Te fir poin o noe in Fig. 2 i a e governmen opimally moo e reacion of e inp in e prodcion proce, in repone o exogeno ecnology oc, relaive o e cae i exogeno policy. To acieve i increaed moone in e labor mare, e governmen need o mae e edge, ic affec e oeold coice for, and ψ, move in e ame direcion a ee qaniie. In addiion o e direc impac of e invemen-pecific oc, bo ecnology oc increae e prodciviy of illed labor relaive o a of nilled indirecly, via e increae in e capial oc. Tee prodciviy effec creae more incenive for e oeold o increae e relaive qaniy of illed member. Tee effec alo increae e labor or of e more prodcive ype of labor and decreae e or ime of e lea prodcive (afer an iniial increae), ince bo ype of orer provide eqally valed leire ime. To mediae e above reacion, e governmen need o increae e edge in illed labor or, τ, and relaive ill pply, ψ, ile concrrenly decreaing e edge in nilled labor or, τ. Noe a depie e increae in τ a, e relaive ill pply edge a increaed becae ψ alo depend on τ and τ poiively and negaively, repecively, o e cange in e labor axe end o raie ψ. Tee movemen give rie o e correlaion for axe and edge mmaried in Table 5. 31 To ave on pace, e do no preen e imple repone o ε ge, ince e ax inrmen repond very lile qaniaively o ee oc, a e governmen e ae-coningen governmen ae o accommodae emporary pre pblic finance oc.
K. Angelopolo e al. / Jornal of Economic Dynamic & Conrol 51 (215) 42 444 435 5.5. Aocorrelaion Te rel in Table 7 gge a e labor income axe and e ex ane capial income ax in i model ineri e properie of e exogeno procee. For example, e aocorrelaion of ee inrmen follo e aocorrelaion of e exogeno procee. T, en e oc are aocorrelaed a in Table 4, o are e ax rae. Hoever, if e ame a e oc follo iid procee, e aocorrelaion of e ax rae generally become very mall. In conra, e aocorrelaion of e ex po capial ax and e privae ae ax do no follo e aocorrelaion of e exogeno procee. Ti i again imilar o previo finding in e lierare. Te rel in Table 7 finally o a e ill-acqiiion bidy alo ineri e properie of e exogeno procee, oring in a imilar faion o e remaining axe a affec e aic margin. 5.6. Alernaive ampion regarding ill pply To frer evalae e imporance of ill creaion and endogeno relaive ill pply for opimal axaion in e bencmar model analyed above, e obain e mean and e andard deviaion relaive o op for axe and edge for ree alernaive ampion regarding relaive ill pply. Tee are preened in Table 8. 32 5.6.1. No exernaliie Te fir cae conidered i a model io exernaliie in ill creaion, obained by eing ξ ¼ 1 in qð. In i cae, ψ ¼ ψ p, ic implie a e ill-acqiiion bidy reflec only policy diorion and old be ep o a minimal level a e eady-ae. Moreover, labor income axaion become mildly regreive in e eady-ae. Depie ee long-rn adjmen, e relaive andard deviaion of axe and bidie do no cange ignificanly, erea e edge in relaive ill pply a loer volailiy, ince i i only driven by e volailiy of e policy componen of e edge. 5.6.2. Prodciviy oc in ill creaion We nex conider a cae ere e ill creaion ecnology inclde a prodciviy erie, inended o capre exogeno facor a affec ill creaion and ocial mobiliy. For inance, i i generally acceped (ee e.g. Goldin and Kaz, 28) a ocial mobiliy and enrolmen in eriary edcaion, ic in rn deermine relaive ill pply, can depend on policy inervenion. Tee may be e proviion of acce o ill-relaed edcaion o member of deprived commniie and rcral cange in e proviion of primary and econdary edcaion (e.g. e crriclm of die and e definiion of cool cacmen area). To broadly capre ee poenial inervenion, e allo for Ψ o be deermined by an exogeno AR(1) proce: log Ψ þ 1 ð þ 1 ¼ 1ρ Ψ log Ψ þρψ log Ψ ð þε Ψ þ 1 ð6 ere ε Ψ i an independenly and idenically diribed Gaian random variable i zero mean and a andard deviaion given by σ Ψ. We e ρ Ψ ¼ :95 and σ Ψ ¼ :9, o a e model generaed erie for relaive ill pply i conien i e perience and andard deviaion (relaive o op) of e daa, for e model i exogeno policy calibraed a in Secion 4. We e annal daa for e are of college edcaed o oal oring poplaion meared in efficiency ni from e 197 from Acemogl and Aor (211) and GDP daa from e US NIPA accon o find a e aocorrelaion of e cyclical componen of relaive ill pply i.47 and i andard deviaion relaive o op i.27. Te rel in Table 8 gge a opimal labor ax mooing i no affeced in i cae and ere i only a mall increae in e volailiy of e ill-acqiiion bidy. Te andard deviaion of e edge in relaive ill pply and of bo of i componen ave increaed ignificanly. Hoever, given a e increaed volailiy in ill creaion affec efficien a ell a mare ocome, e governmen doe no find i opimal o cange e cyclical properie of e ax inrmen ignificanly. 5.6.3. Fixed relaive ill pply Te final cae e conider i en e relaive ill pply i fixed, ψ ¼ ψ f ¼ :44 and exogeno o e oeold. Ti ampion implie a ill creaion expendire alo remain fixed a e level a i reqired o ppor ψ ¼ :44. Hoever, ince ere i no margin of adjmen for e oeold regarding e coice of ψ in i cae, e alo normalie τ a ¼, ince i doe no ave a role o play in affecing a edge in e model. 33 In pariclar, aming a i rericion i impoed on e mare economy, il e primiive of e model allo e ocial planner o cooe relaive ill pply, e can define e edge in relaive ill pply a e difference beeen e efficien and e 32 In all cae conidered belo, e do no re-calibrae e model, excep for e pecific cange e dic in e pecificaion for ψ ð in (3), o iolae e effec of e cange in ill pply. In eac cae, e re-calclae e deerminiic Ramey eady-ae and obain e reqired momen a e economy flcae arond i eady-ae. We do no preen rel for correlaion and aocorrelaion a ee are imilar i oe obained nder e bencmar model. 33 Noe a if τ a a lef a explici coice of e governmen, i old be eqivalen o a lmp-m ax, ic old violae e econd-be nare of e problem a e an o dy.