Mnmum Invesmen Requremen, Fnancal Inegraon and Economc (In)sably: A Refnemen o Masuyama (2004) Hapng Zhang Dec 203 Paper No. 09 203 ANY OPINIONS EXPRESSED ARE THOSE OF THE AUTHOR(S) AND NOT NECESSARILY THOSE OF THE SCHOOL OF ECONOMICS, SMU
Mnmum Invesmen Requremen, Fnancal Inegraon and Economc (In)sably: A Refnemen o Masuyama (2004) Hapng Zhang November 203 Absrac Ths noe proposes a smple, more precse, necessary condon for symmery breakng n Masuyama (Fnancal Marke Globalzaon, Symmery-Breakng, and Endogenous Inequaly of Naons, Economerca, 2004 ),.e., he posve neres rae response o ncome changes, whch essenally arses from he assumpons of fnancal frcons and mnmum nvesmen sze requremen of ndvdual projecs. Ths condon also holds under he more general sengs. Thus, hs noe offers an emprcally esable hypohess,.e., Masuyama s symmery breakng s more lkely, f he neres rae response o ncome changes s posve and suffcenly large. Keywords: fnancal frcons, fnancal marke globalzaon, mnmum nvesmen sze requremen, symmery breakng JEL Classfcaon: E44, F4 School of Economcs, Sngapore Managemen Unversy. 90 Samford Road, Sngapore 78903. E- mal: hpzhang@smu.edu.sg
Inroducon Masuyama (2004) shows ha, under ceran condons, counres wh dencal fundamenals converge, ndependen of nal ncome levels, o he same, unque, and sable seady sae under nernaonal fnancal auarky (IFA, hereafer), whle fnancal marke globalzaon (FMG, hereafer) may desablze hs symmerc seady sae n he sense ha counres wh relavely hgh (low) nal ncome levels may converge o a new seady sae wh he ncome level hgher (lower) han ha under IFA. Masuyama (2004) summarzes hree general condons for symmery breakng n secon 7:. For a fxed domesc neres rae, he domesc nvesmen s an ncreasng funcon of he wealh held by he domesc enrepreneurs n he lower range. 2. Domesc nvesmen ncreases he wealh held by domesc enrepreneurs (more han ha of foregn enrepreneurs). 3. The domesc neres rae adjuss o balance domesc supply and domesc demand for cred n he absence of he nernaonal fnancal marke, whle s lnked o he foregn neres rae n he presence of he nernaonal fnancal marke. In hs noe, I frs se up a model sasfyng he hree general condons menoned above and show ha FMG does no lead o symmery breakng here. Then, I denfy a smple, necessary condon for Masuyama s symmery breakng from he cred marke perspecve,.e., he posve neres rae response o ncome changes, whch essenally arses from he assumpons of fnancal frcons and mnmum nvesmen sze requremen of ndvdual projecs. As my frs conrbuon, hs posve relaonshp should be augmened as a more precse condon no secon 7 of Masuyama (2004). The nuon s as follows. Suppose ha he world economy consss of a connuum of counres wh dencal fundamenals excep for he nal ncome level. In each counry, some agens have boh he echnology and he funds o run he nvesmen projecs and are called enrepreneurs; whou eher he echnology or he funds, oher agens lend her ne wealh o he cred marke and are called households. If he neres rae s below he margnal rae of reurn o nvesmen, enrepreneurs prefer o fnance her nvesmen usng exernal funds. However, due o lmed commmen, hey are subjec o borrowng consrans and mus also pu her ne wealh n he projec. The hgher he enrepreneural ne wealh, he more hey can borrow and nves. In hs model, capal accumulaon and he resulng changes n aggregae ncome affec he neres rae hrough wo channels. Frs, gven ha he neoclasscal producon funcon has he decreasng margnal produc of capal, capal accumulaon reduces he margnal rae of reurn o nvesmen and he neres rae ends o fall. Ths s called he neoclasscal effec. Second, capal accumulaon rases he ndvduals ncome and ne wealh, rggerng he cred marke adjusmen. 2
In he absence of mnmum nvesmen sze requremen, capal accumulaon affecs he cred marke on he nensve margn. The hgher ndvdual s ne wealh allows enrepreneurs (households) o borrow (lend) more. The neoclasscal effec reduces he pledgeable value per un of enrepreneural nvesmen and dampens he expanson of her deb capacy, mplyng ha he rse n he aggregae cred demand s domnaed by ha n he aggregae cred supply. Thus, capal accumulaon reduces he neres rae and he neres rae s lower n he rch han n he poor counry under IFA; under FMG, fnancal capal flows are from he rch o he poor counry, narrowng he nal cross-counry ncome gap. Evenually, counres wh dencal fundamenals excep for he nal ncome converge o he same seady sae as under IFA. Thus, symmery breakng does no arse, alhough he hree condons menoned above are sasfed. In he presence of fxed or mnmum nvesmen sze requremen, capal accumulaon affecs he cred marke no only on he nensve margn and bu also on he exensve margn. Take he case of fxed nvesmen sze requremen as an example. The hgher ndvdual s ne wealh reduces he cred demand of ndvdual enrepreneur as well as allows more agens o become enrepreneurs wh leveraged nvesmen. The nensvemargn (exensve-margn) effec ends o reduce (rase) he aggregae cred demand. In he ne erm, he sze of he expanson n aggregae cred demand s dencal as n he absence of mnmum nvesmen sze requremen. Meanwhle, he hgher ndvdual s ne wealh rases he lendng of ndvdual household and reduces he mass of lenders (households). The nensve-margn (exensve-margn) effec ends o rase (reduce) he aggregae cred supply. The nensve-margn effec s dencal as n he absence of mnmum nvesmen sze requremen, whle he exensve-margn effec s new here. If he negave exensve-margn effec on he cred supply sde domnaes he negave neoclasscal effec on he cred demand sde, capal accumulaon rases he neres rae and he neres rae s hgher n he rch han n he poor counry under IFA; under FMG, fnancal capal flows are from he poor o he rch counry, wdenng he crosscounry ncome gap. Evenually, counres wh dencal fundamenals excep for he nal ncome level may converge o he seady saes wh dfferen ncome levels,.e., FMG leads o symmery breakng. Here, fxed or mnmum nvesmen sze requremen gves rse o he dsnc exensve-margn effec on he cred supply sde, whch s key o he posve relaonshp beween he neres rae and ncome changes. Kkuch (2008), Kkuch and Sachursk (2009), Masuyama (2005, 2007, 2008, 202, 203) apply he mechansm of symmery breakng o he opcs on endogenous flucuaons, nequaly, cred raps, cred cycles, and oher aggregae mplcaons of cred marke mperfecons. However, s no clear how o emprcally es he heorecal condons supporng hs mechansm n hese papers. As my second conrbuon, a sraghforward, emprcally esable hypohess s proposed,.e., Masuyama s symmery breakng s more lkely o arse f he real neres rae response o ncome changes s posve and suffcenly large. A comprehensve emprcal nvesgaon on he neres rae response o ncome changes s beyond 3
Masuyama (2004) clams n subsecon 7. and 7.2 ha symmery breakng may also exs n he presence of wealh nequaly and mnmum (nsead of fxed) nvesmen sze requremen, whle a complee characerzaon of mulple seady saes s hopelessly complcaed. As my hrd conrbuon, I ncorporae wealh nequaly and mnmum nvesmen sze requremen n a generalzed seng and provde a complee, analycal characerzaon of symmery breakng, formally provng Masuyama s conjecure. As a sde noe, here s a echncal error n one of he boundary condons for symmery breakng n fgure 5 of Masuyama (2004). The res of he paper s srucured as follows. Secon 2 ses up he basc model and compares he mpacs of FMG on he model dynamcs n he absence and n he presence of fxed nvesmen sze requremen, respecvely. Secon 3 checks he robusness n a generalzed seng. The appendx collecs some exensons and echncal proofs. 2 The Basc Model Consder a wo-perod overlappng generaons model. 2 The world economy consss of a connuum of dencal counres, ndexed by [0, ]. Agens lve for wo perods, young and old. There s no populaon growh and he populaon sze of each generaon s normalzed a one n each counry. When young, each agen s endowed wh one un of labor whch s suppled nelascally o aggregae producon. A fnal good s nernaonally radable and chosen as he numerare. I can be consumed or ransformed no capal goods. Capal goods are non-radable and used ogeher wh labor o produce fnal goods conemporaneously. Capal fully deprecaes afer producon. The markes for fnal goods, capal goods, and labor are perfecly compeve. Y denoes aggregae oupu of fnal goods, L = and K denoe he aggregae npus of labor and capal goods, ω and v denoe he wage rae and he prce of capal n counry [0, ]. ( ) K Y α ( ) α = L, where α (0, ), () α α v K = αy and ω L = ( α)y, (2) Agens only consume when old and hey save her enre labor ncome when young. In order o show he crcal role of fxed nvesmen sze requremen n deermnng Masuyama s symmery breakng, 3 I compare wo alernave model sengs as follows. In he frs seng, a fracon η (0, ) of agens n each generaon are endowed wh he lnear projec o ransform fnal goods n perod no capal goods n perod + a he rae of R, and are called enrepreneurs. The mass of enrepreneurs η s exogenously fxed 4, whle he nvesmen sze of ndvdual projec m s endogenous. Wh he scope of hs noe and s lef for fuure research. 2 The model seng closely resembles ha of Masuyama (2004). 3 Secon 3 shows n a generalzed model ha mnmum (nsead of fxed) nvesmen sze requremen s an crcal assumpon for symmery breakng. 4 Appendx A. endogenzes he mass of enrepreneurs and he resuls n hs seng sll hold here. 4
no producve projecs, oher agens lend ou her labor ncome and are called households. Gven he fxed mass of enrepreneurs and he lneary of ndvdual projecs, aggregae nvesmen akes place on he nensve margn, K+ = Rm η. Wh no fxed nvesmen sze requremen for ndvdual projecs, s called seng N. In he second seng, each agen s endowed wh an ndvsble projec 5 o ransform m uns of fnal goods n perod no Rm uns of capal goods n perod +. 6 If ω < m, an agen mus borrow m ω o sar s projec and he aggregae resource s no suffcen o allow all agens o sar he projecs. Accordng o Masuyama (2004), random cred raonng allows a fracon η (0, ) of agens o sar he projecs wh he loans, m ω, and hey are called enrepreneurs, whle oher agens only lend ou he labor ncome and are called households. Dfferen from seng N, he projec sze m s exogenously fxed, whle he mass of enrepreneurs η s endogenous. Alhough he fxed nvesmen sze resuls n he non-convexy of he ndvdual producon se, Masuyama (2007, 2008) argues ha assumng a connuum of homogeneous agens convexfes he producon se and aggregae nvesmen akes place on he exensve margn, K+ = Rmη. Wh fxed nvesmen sze requremen for ndvdual projecs, s called seng F. Seng N Seng F Seng M k k Rm k Rm k Rm k Rm k Rm O m m O m O m m Fgure : Projecs n Varous Sengs Fgure shows he producve funcon of ndvdual projec n varous sengs. The projec oupu n seng N s lnear n he npu, k + = Rm. Wh fxed nvesmen sze requremen, he projec oupu s zero for he npu m < m; s consan a Rm for he npu m m n seng F. In secon 3, I analyze a seng wh mnmum nvesmen sze requremen (seng M),.e., he projec oupu s zero for m < m; s lnear n he npu k + = Rm for m m. I use η and m o denoe he mass of enrepreneurs and he projec sze n he model descrpon. Seng N s characerzed by he fxed mass of enrepreneurs, η = η, whle seng F s characerzed by he fxed projec sze, m = m. In each seng, I analyze he dynamc properes of he equlbrum allocaons under wo scenaros,.e., IFA where agens can only borrow or lend domescally, nernaonal capal flows are forbdden, and he gross neres rae r clears he cred marke a he 5 Secon 3 relaxes he assumpon of projec ndvsbly and he resuls n hs seng sll hold. 6 Masuyama (2004) mplcly normalzes he ndvdual projec sze a m =, whle I allow o be a free parameer and consder how may affec he possbly of symmery breakng. 5
counry level; FMG where agens can borrow or lend domescally and aboard, here are no barrers o nernaonal borrowng/lendng, and he gross neres rae r = r equalzes across counres. 7 Le Φ denoe he fnancal capal ouflows from counry, wh negave values ndcang he fnancal capal nflows. The neres rae canno exceed he margnal rae of reurn o nvesmen, r v+r; oherwse, enrepreneurs would lend ou her funds raher han sar he projec. Masuyama (2004) calls he enrepreneurs profably consrans. Consder he agens of he generaon born n perod. If r < v+r, enrepreneurs prefer o fund her projecs wh loans,.e., d = m ω. Due o lmed commmen, hey are subjec o he borrowng consran, r d = r (m ω ) λv +Rm. (3) λ (0, ) denoes he level of fnancal developmen, whch s dencal for all counres. 8 Le ψ ω denoe he equy-nvesmen rao. The equy rae s defned as, m Γ = v +Rm r d ω ( ) The erm (v+r r ) ψ ( = v+r + (v+r r) ψ ). (4) capures he excess reurn due o he leveraged nvesmen. If r < v+r, he leveraged nvesmen leads o Γ > v+r > r so ha enrepreneurs borrow up o he lm o maxmze he leverage rao ( ); f r ψ = v+r, he zero excess reurn leads o Γ = r so ha enrepreneurs do no borrow o he lm. In he followng, he prvae raes of reurn refer o he equy rae and he neres rae, whle he socal rae of reurn refers o he margnal rae of reurn o nvesmen. Households save he labor ncome when young and consume he fnancal reurn when old; enrepreneurs fnance he projec usng he loans and her labor ncome when young, and hen, consume he projec revenue ne of deb repaymen when old, c,h + = r ω, and c,e + = v +Rm r d = Γ ω. (5) The markes for capal goods, cred, and fnal goods clear smulaneously, ( η )c,h K + = Rm η, (6) η (m ω ) = ( η )ω, (7) + η c,e + η m = Y. (8) Defnon. A marke equlbrum under IFA s a se of allocaons of households, {c,h }, enrepreneurs, {m, c,e }, and aggregae varables, {Y, K, ω, v, Γ, η, r}, sasfyng equaons ()-(7). η = η s exogenously fxed n seng N, whle m = m s exogenously fxed n seng F. 7 Followng Masuyama (2004), I exclude FDI flows by assumpon. von Hagen and Zhang (20, 204) analyze he jon deermnaon of fnancal capal flows and FDI flows n seng N. 8 See Masuyama (2007, 2008) for dealed dscusson on formulang he borrowng consran n such a way. von Hagen and Zhang (20, 204) analyze he case where counres dffer n λ. 6
Under FMG, he equlbrum condons are dencal as under IFA excep for he cred marke clearng condons a he counry and a he world level, η (m ω ) = ( η )ω Φ, (9) 0 Φ d = 0. (0) Defnon 2. A marke equlbrum under FMG s a se of allocaons of households, {c,h }, enrepreneurs, {m, c,e }, and aggregae varables, {Y, K, ω, v, Γ, η, Φ }, sasfyng equaons ()-(6) and (9), and he world neres rae r s deermned by he world cred marke equlbrum condon (0). η = η s exogenously fxed n seng N, whle m = m s exogenously fxed n seng F. 2. Equlbrum Allocaon under IFA In seng N, accordng o equaon (7) and (6), he equy-nvesmen rao s consan a ψ = η and domesc nvesmen s fully fnanced by domesc savng K+ = Rω. The dynamc equaon of wages 9 s ω + = ( α) L Y + = ( ) Rω α, where α α. () In seng F, for ω < m, aggregae savng s no enough o allow all agens o run her projecs. Accordng o equaon (7) and (6), he mass of enrepreneurs and he equynvesmen rao are endogenous, η = ψ = ω <, and domesc nvesmen s fully m fnanced by domesc savng. Thus, he phase dagram of wages s he same as n seng N. For ω m, all agens can self-fnance her projecs, η = ψ =. Gven he fxed projec sze, aggregae oupu of capal goods s consan a K+ = Rm and he phase ( dagram of wages s fla a ω+ = Assumpon. m > ω IF A. Rm ) α. Proposon. In boh sengs, counres wh dencal fundamenals excep for he nal ncome levels converge o he same seady sae under IFA, whch s unque and sable; for λ (0, ψ ), he borrowng consrans are bndng and here s a wedge beween he socal and prvae raes of reurn, Γ > Rv + > r ; for λ ( ψ, ), he borrowng consrans are slack and Γ = r = Rv +. In he followng analyss, I focus on he case of he bndng borrowng consrans. Use he bndng borrowng consrans o rewre he neres rae as r = λ v ψ +R < v+r. (2) 9 The model dynamcs can also be characerzed by he dynamc equaon of capal, K+ = Rω = ( ) R α L Y K α. = R For α (0, ), he phase dagram of capal s globally concave, mplyng he exsence of a unque and sable seady sae under IFA wh K IF A = Rω IF A. For noaonal smplcy, I use he phase dagram of wages o analyze he exsence, unqueness, and sably of he seady sae. 7
The neres rae depends on hree facors,.e., he socal rae of reurn, v+r, he level of fnancal developmen λ, and he equy-nvesmen rao ψ. As long as assumpon s sasfed, fxed nvesmen sze requremen does no maer for he dynamc properes under IFA. However, does maer under FMG and he key s how he neres rae responds o ncome changes. Fgure 2 shows how a hgher labor ncome affecs he cred marke equlbrum. For smplcy, I suppress he counry ndex. The cred marke equlbrum s nally a pon E. A hgher labor ncome ω > ω rases he aggregae nvesmen, leadng o a lower socal rae of reurn, Rṽ + < Rv +, due o he concavy of he neoclasscal aggregae producon funcon wh respec o capal. I s called he neoclasscal effec. r ~ N r λrv w r D N S Seng N λrv~ r E D ~ w ~ ~ N S E ~ N ~ F r r λrv w r D F S Seng F λrv~ r E D ~ w ~ ~ F S F E ~ ( η)w ( η)w ~ ( η )w ( ~ η ) w ~ Fgure 2: Income Changes and he Cred Marke Adjusmen In seng N, he masses of households and enrepreneurs are fxed a η and η, respecvely. Thus, a hgher labor ncome affecs he cred marke only on he nensve margn. The aggregae cred supply, S = ( η)ω, rses proporonally n ω, whle he aggregae cred demand, D = ηd = λrv +ω r, rses less-han-proporonally n ω, due o he negave neoclasscal effec. Thus, he rghward shf of he cred demand curve s domnaed by ha of he cred supply curve and hence, he equlbrum moves from pon E o pon ẼN wh a lower neres rae, r N < r. See he lef panel of fgure 2. Le ln X ln X ln X. Rewre he cred marke equlbrum condon D = S, ln λ + ln Rv + ln r + ln ω = ln ω + ln( η) (3) ln Rv }{{ + ln r } + he neoclasscal effec ln ω }{{} he wealh effec = ln ω }{{} he wealh effec ln r = ln Rv }{{ + } he neoclasscal effec. (5) Wh he wealh effecs cancelng ou on boh sdes, he neres rae responds o ncome changes only hrough he neoclasscal effec n seng N. 8 (4)
Lemma. In seng N, he neres rae srcly decreases n ω under IFA. In seng F, gven he fxed nvesmen sze a he ndvdual level, a hgher ω affecs he cred marke on he nensve and he exensve margns. Frs, a rse n ω reduces he ndvdual enrepreneur s cred demand, d = m ω, as well as allows more agens o become enrepreneurs, η = ω. The nensve-margn (exensve-margn) effec ends m o reduce (rase) he aggregae cred demand. Overall, he aggregae cred demand curve shfs o he rgh a he same magnude as n seng N, D = η d = λrv +ω r. Second, a rse n ω rases he ndvdual household s savng, whle also reduces he mass of households, η. The nensve-margn (exensve-margn) effec ends o rase (reduce) he aggregae cred supply, S = ( η )ω. The posve nensve-margn effec s dencal as n seng N, whle he negave exensve-margn effec s new n seng F. Same as n seng N, he negave neoclasscal effec dampens he expanson of he cred demand; dfferen from seng N, he negave exensve-margn effec dampens he expanson of he cred supply. If he negave exensve-margn effec domnae he negave neoclasscal effec, he cred marke equlbrum moves from pon E o pon Ẽ F wh a hgher neres rae, r F > r. See he rgh panel of fgure 2. Rewre he cred marke equlbrum condon D = S as ln λ + ln Rv + ln r + ln ω = ln ω + ln( η ) (6) ln Rv }{{ + ln r } + he neoclasscal effec ln ω }{{} he wealh effec = ln ω }{{} + ln( η ) }{{}, (7) he wealh effec he exensve-margn effec ln r = ln Rv }{{ + } ln( η ) }{{}. (8) he neoclasscal effec he exensve-margn effec Wh he wealh effecs cancelng ou on boh sdes, he neres rae responds o ncome changes hrough he neoclasscal effec and he exensve-margn effec n seng F. Lemma 2. In seng F, he equy-nvesmen rao ψ = ω rses n m ω under IFA; gven λ (0, ψ F ), he neres rae rses n ω f ψ ( ψ F, λ), where ψ F α Fgure 3 shows he parameer confguraon n seng F. Accordng o proposon, for (λ, ψ ) n regon SD, he borrowng consrans are slack and he neres rae, concdng wh he socal rae of reurn, declnes n ω, due o he neoclasscal effec. Accordng o lemma 2, for (λ, ψ ) n regon BI, he borrowng consrans are bndng and he neres rae ncreases n ω, as he exensve-margn effec domnaes he neoclasscal effec; for (λ, ψ ) n regon BD, he borrowng consrans are bndng and he neres rae declnes n ω, as he neoclasscal effec domnaes he exensve-margn effec. Fgure 4 llusraes proposon and lemmas -2 graphcally. The lef panel shows ha he phase dagram of wage sars from zero and s concave crosses he 45 lne once from he lef n he wo sengs. Thus, he fxed nvesmen sze requremen does no maer for he exsence, unqueness, and sably of he seady sae under IFA. Gven λ < η, he mddle panel shows ha he neres rae n seng N, proporonal o he 9 2 α.
SD ψ BI ~ ψ F BD O λ ψ~ F Fgure 3: Parameer Confguraon for he Ineres Rae Paerns n Seng F Phase Dagram of Wages N F S w + ψ ψ ψ O w w IFA Fgure 4: Paerns of Wage Rae and Ineres Raes under IFA socal rae of reurn, declnes n ω, due o he neoclasscal effec. Gven λ < ψ F, he rgh panel shows ha he neres rae n seng F s a non-monoonc funcon of ω, due o he neracons of he neoclasscal effec and he exensve-margn effec. Le us focus on he neres rae response o ω around he seady sae. The equynvesmen rao n he seady sae s ψ IF A = ω IF A m = R m. If ψ IF A = ψ h ( λ, ) or ψ IF A = ψ l (0, ψ F ),.e., n regon SD or BD of fgure 3, he neres rae declnes n ω around he seady sae; f ψ IF A = ψ m ( ψ F, λ),.e., n regon BI of fgure 3, he neres rae rses n ω around he seady sae. 0 In seng F, he non-monoonc neres rae response o ncome changes under IFA wll lead o he non-monoonc paerns of fnancal capal flows, whch s he key mechansm behnd Masuyama s symmery breakng, as shown n subsecon 2.2. 0 Noe ha he assumpon of he Cobb-Douglas producon funcon s no essenal here. The wo key effecs n seng F,.e., he neoclasscal effec and he exensve margn effec, exs as long as he aggregae producon funcon s neoclasscal,.e., f (k) > 0 and f (k) < 0, where k K L. 0
2.2 Equlbrum Allocaon under FMG In hs subsecon, I analyze wheher and under wha condons FMG may desablze he seady sae under IFA. For ha purpose, I assume ha all counres are n her respecve seady sae under IFA before agens are allowed o borrow or lend globally n perod = 0. Agens ake he world neres rae as gven, r = r IF A = λ <, where ψ IF A = η (ψ IF A = ω IF A ) n seng N (F). ψ IF A m Phase Dagram of Wages (Seng N) 0.4 Capal Flows (Seng N) 0.2 w FMG + S 0 S w IFA + 0.2 0.4 w IFA w IFA O w w Fgure 5: Phase Dagram of Wages and Capal Flows under FMG n Seng N The sold and he dash-doed curves n he lef panel of fgure 5 show ha he phase dagrams of wage n seng N are concave and pon S s he unque and sable seady sae under IFA and under FMG. For a margnal rse (declne) n ω around pon S, he neres rae ends o reduce (rase), due o he neoclasscal effec, and hence, gven he world neres rae consan a r = r IF A under FMG, fnancal capal flows ou of (no) counry, dampenng he rse (declne) n domesc nvesmen and ω+. See he rgh panel. Essenally, FMG makes he phase dagram of wage flaer so ha counres converge o he same seady sae as under IFA bu faser. Alhough seng N sasfes he hree condons menoned n secon 7 of Masuyama (2004), symmery breakng does no arse and he key reason s he negave neres rae responses o ncome changes. Proposon 2. In seng N, FMG manans he unqueness and sably of he seady sae under IFA. The lef panel of fgure 6 shows he parameer confguraon for fve cases n seng F n he (λ, ψ IF A ) space. By rescalng he vercal axs from ψ IF A no R = (mψ IF A ), he rgh panel of fgure 6 replcaes he same resul n he (λ, R) space, correspondng o fgure 5 n Masuyama (2004). For parameers n regon C (A), he borrowng consrans are
BC R + BC C C B ψ IFA B R c AB ^ ψ F ~ ψ F AB A A O λ ψ ^ F O λ λ c Fgure 6: Parameer Confguraon for Symmery Breakng n Seng F slack (bndng), he seady sae under FMG s dencal as under IFA, whch s unque and sable. In he followng, I focus on regon B, AB, and BC where FMG leads o mulple seady saes. The rgh panel of fgure 6 s dencal as fgure 5 of Masuyama (2004) excep for he boundary beween regon AB and A. By defnon, he mass of enrepreneurs canno exceed he oal mass of populaon n each generaon, η. Takng ha no accoun, he boundary beween regon AB and A s characerzed by a pecewse funcon wh wo subfuncons. 2 Ths resul s confrmed n he generalzed seng n secon 3. Thus, here s a echncal error n fgure 5 of Masuyama (2004). Case B Case AB Case BC H M w + H w + w + S S S M L L O w L w w IFA w H O w IFA w w M w H O w L w IFA w M w Fgure 7: Phase Dagrams of Wage under FMG n Seng F The sold and he dashed curves n fgure 7 show he phase dagrams of wage under IFA versus under FMG n he hree cases. 3 As shown n fgure 3, for (λ, ψ ) n regon BI, The analyss for regon A and C s n he proof of proposon 3 n he appendx. 2 See he proof of Proposon 3 n he appendx for he explc characerzaon of he wo subfuncons. 3 The hree cases dffer only n erms of he fxed nvesmen ( ) sze,.e., m BC < m B < m AB, and have. R he same seady sae (pon S) under IFA wh ω IF A = 2
he neres rae rses n ω under IFA; under FMG, a margnal rse (declne) n ω leads o fnancal capal nflows (ouflows), amplfyng he change n domesc nvesmen and ω+. Thus, he phase dagram of wage s convex for ω below a hreshold value. Le ˆψ F α > ψ F. Consder regon B of fgure 6. The amplfcaon effec of FMG s so srong ha ω + > a pon S. As shown n he lef panel of fgure 7, FMG ω desablzes he nal seady sae and creaes wo new sable seady saes,.e., pon H wh a hgher ncome and pon L wh a lower ncome, respecvely. Consder regon AB of fgure 6. Accordng o fgure 3, f ψ IF A (0, ψ F ), he neres rae declnes n ω around pon S under IFA; under FMG, fnancal capal flows end o dampen he change n domesc nvesmen, makng he phase dagram of wage flaer around pon S so ha he nal seady sae (pon S) s more sable; f ψ IF A ( ψ F, ˆψ F ), alhough he neres rae rses n ω around pon S, he amplfcaon effec of FMG s no srong enough,.e., ω + < around pon S, so ha he nal seady sae (pon S) ω s sll sable. However, as shown n he mddle panel of fgure 7, for ω ω IF A, ψ eners n regon BI of fgure 3 and FMG generaes he suffcenly large amplfcaon effec so ha here exss wo oher seady saes,.e., pon M (unsable) and pon H (sable). Consder regon BC of fgure 6. The borrowng consrans are slack n he seady sae under IFA; FMG makes he phase dagram of wage fla around pon S so ha he nal seady sae (pon S) s sll sable. However, as shown n he rgh panel of fgure 7, for ω ω IF A, ω crosses regon BI of fgure 3 along he convergence pah where he amplfcaon effec makes he phase dagram of wage convex and here exss wo oher seady saes,.e., pon M (unsable) and pon L (sable). Proposon 3. In seng F, FMG may lead o mulple seady saes. To sum up, alhough he nal ncome level does no maer for he seady sae under IFA, does maer under FMG. In case B, sarng wh he ncome level slghly hgher (lower) han he seady-sae one under IFA, a small open economy converges o a new, sable seady sae wh he ncome much hgher (lower) han n he seady sae under IFA; n case AB (BC), sarng wh an ncome suffcenly hgher (lower) han he seady-sae one under IFA, a small open economy converges o a new, sable seady sae wh he ncome much hgher (lower) han n he seady sae under IFA. Ths way, FMG amplfes he cross-counry oupu gap. Techncally, Masuyama (2004) s symmery breakng arses from he convexy of he phase dagram of wage under FMG, whch s a resul of he posve and suffcenly large neres rae responses o ncome changes under IFA. Thus, one may es he emprcal relevance of Masuyama s symmery breakng by esmang he sgn and he sze of neres rae responses o ncome changes. 3 The Generalzed Model The ndvdual projec sze s exogenous n seng F. I exend seng F n wo ways o allow for he endogenous ndvdual projec sze and wealh heerogeney. Frs, he 3
ndvdual projec has a mnmum (nsead of fxed) nvesmen sze requremen m 0, as shown n he rgh panel of fgure ; second, he labor endowmen for agen j [0, ] s ndvdual specfc, l j = θ+ θ ɛ j, where ɛ j (, ) follows he Pareo dsrbuon wh he cumulave dsrbuon funcon F (ɛ j ) = ɛ θ j and θ >. The aggregae labor npu s L = l j df (ɛ j ) =. Wh mnmum nvesmen requremen, s called seng M. 4 Gven agen j s labor ncome n j, = ωl j = ω θ+ and he borrowng consrans as θ specfed by equaon (3), s maxmum nvesmen sze s m j, = ɛ j n j, λ Rv + r = ω λ Rv + r For a hgh m and/or a low ω, he labor ncome of agens wh ɛ j > ɛ s so low ha hey canno mee he mnmum nvesmen sze requremen, m j, < m, and hence, hey become households and lend ou he labor ncome; agens wh ɛ j (, ɛ ] can mee he mnmum nvesmen sze requremen, m j, m, and hey become enrepreneurs. The mass of enrepreneurs s η = (ɛ ) θ. In equlbrum, f he cuoff value ɛ s suffcenly low, he mass of households η = (ɛ ) θ s large and he relavely hgh aggregae cred supply depresses he neres rae, r < Rv +. In hs case, enrepreneurs borrow o he lm and he equy-nvesmen rao s dencal among hem, θ+ θ ɛ j. ψ j, = n j, m j, = λ Rv + r = ψ = ω m ɛ θ +. (9) θ The condon for he bndng borrowng consrans s he same as n seng F. See proposon. In he followng, I focus on he case of he bndng borrowng consrans. Under IFA, domesc nvesmen s fnanced by domesc savng, ɛ K+ = R m j,df (ɛ j ) = Rω (ɛ ) (+θ) = λ Rv +, (20) r and he oupu dynamcs s characerzed by equaon (). The seady-sae properes are ndependen of λ and m. See proposon. Lemma 3. The equy-nvesmen rao ψ and he cuoff value ɛ rse monooncally n he wage rae ω under IFA. A rse n he wage rae affecs aggregae nvesmen on he exensve and he nensve margns. Frs, allows more agens o become enrepreneurs and sar he projecs wh leveraged nvesmen and hence, he cuoff value ɛ s hgher; second, he hgher aggregae nvesmen reduces he margnal rae of reurn and he declne n he mass of households ends o reduce he cred supply, whch ghens he borrowng consrans for enrepreneurs and rases he equy-nvesmen rao. Gven he mass of enrepreneurs η = (ɛ ) θ, equaon (20) can be rewren as, 4 Seng F s a specal case here,.e., for θ, he dsrbuon of labor endowmen degeneraes no a un mass a ɛ j = and hence, l j =. 4
r = λrv +(ɛ ) +θ = λrv +( η ) +θ θ (2) ln r = ln λ + ln Rv + ( θ + ) ln( η ). (22) ln r = ln Rv+ ( }{{} θ + ) ln( η ) }{{} he neoclasscal effec he exensve-margn effec (23) Same as equaon (8) n seng F, ncome changes affec he neres rae hrough he neoclasscal effec and he exensve-margn effec. In parcular, for θ, equaon (23) concdes wh (8). Le ψ M α > ψ 2 α α F. +θ Lemma 4. Gven λ (0, ψ M ), he neres rae rses n ω f ψ ( ψ M, λ). Fgure 8 shows he regon where he neres rae responds posvely or negavely o ncome changes n he (λ, ψ ) space, qualavely dencal as fgure 3 for seng F. For θ, lm θ ψm = ψ F and lemma 4 s dencal as lemma 2. SD ψ BI ~ ψ M BD O λ ψ ~ M Fgure 8: Parameer Confguraon for he Ineres Rae Paerns n Seng M Agens save he labor ncome when young and consume he fnancal ncome when old, c,h j,+ = r ω l j, and c,e j,+ = Γ ω l j. (24) Defnon 3. A marke equlbrum under IFA s a se of allocaons of households, {c,h j, }, enrepreneurs, {m j,, c,e j, }, and aggregae varables, {Y, K, ω, v, Γ, r, ɛ, ψ}, sasfyng equaons ()-(4), (9)-(20), and (24). Under FMG, he equlbrum condons are dencal as under IFA excep for he cred marke clearng condon a he counry and a he world level, 5
0 ɛ K+ = R m j,df (ɛ j ) = R(ω Φ ), (25) Φ d = 0. (26) Defnon 4. A marke equlbrum under FMG s a se of allocaons of households, {c,h j, }, enrepreneurs, {m j,, c,e j, }, and aggregae varables, {Y, K, ω, v, Γ, r, ɛ, ψ, Φ }, sasfyng equaons ()-(4), (9), (24), and (25). The world neres rae r s deermned by he world cred marke equlbrum condon (26). Followng he analyss n subsecon 2.2, I assume ha he world economy s nally n he seady sae under IFA wh he neres rae consan a r λ = r IF A = ψ IF A, before agens are allowed o borrow and lend globally n perod = 0. Proposon 4. In seng M, FCM may lead o mulple seady saes. Fgure 9 shows he parameer confguraon for fve cases n he (λ, ψif A ) space. As shown analycally n he proof of proposon 4 n he appendx, for θ, he respecve boundares of he fve regons converge o hose n he lef panel of fgure 6. ^ ψ + M BC C ψ IFA B ^ ψ M ~ ψ M AB A O λ ψ ^ M Fgure 9: Parameer Confguraon for Symmery Breakng n Seng M Fgure 0 shows he phase dagrams of wage under IFA versus under FMG n he hree cases of symmery breakng, qualavely he same as fgure 7 n seng F. As shown n he proof of proposon 3, he phase dagram of wage n seng F consss of wo subfuncons,.e., a convex par due o he posve neres rae response o ncome changes, and a fla par due o he profably consrans Rv r+ r or he upper lm for he mass of enrepreneurs η. As a resul, Masuyama (2004) shows ha n he case of symmery breakng, he borrowng consrans are srcly bndng (slack) n he poor 6
(rch) counry so ha, under FMG, he equy premum s always zero, Γ = r = Rv +, n he rch counry. As shown n he proof of proposon 4, he phase dagram of wage n seng M may conss of hree subfuncons,.e., a convex par, a concave par and a fla par. Thus, n he case of symmery breakng, he borrowng consrans can be srcly bndng n he poor and n he rch counry so ha, under FMG, he equy premum can sll be posve Γ > Rv + > r, n he rch counry bu smaller han n he poor counry. Thus, one can es emprcally he ghness of he borrowng consrans by esmang he spread beween he equy rae and he neres rae across counres. Case B Case AB Case BC H H w + w + M w + S S S M L L O w L w IFA w w H O w IFA w M w w H O w L w M w IFA w Fgure 0: Phase Dagrams of Wage under FMG n Seng M To sum up, fnancal frcons and mnmum (nsead of fxed) nvesmen sze requremen are key o he posve neres rae response o ncome changes whch essenally underpns Masuyama s symmery breakng. Furhermore, I prove formally ha he symmery breakng resuls sll hold n he presence of wealh nequaly and endogenous projec sze, confrmng Masuyama s conjecures n subsecon 7. and 7.2. References Kkuch, T. (2008): Inernaonal asse marke, nonconvergence, and endogenous flucuaons, Journal of Economc Theory, 39(), 30 334. Kkuch, T., and J. Sachursk (2009): Endogenous nequaly and flucuaons n a wo-counry model, Journal of Economc Theory, 44(4), 560 57. Masuyama, K. (2004): Fnancal Marke Globalzaon, Symmery-Breakng, and Endogenous Inequaly of Naons, Economerca, 72(3), 853 884. (2005): Cred Marke Imperfecons and Paerns of Inernaonal Trade and Capal Flows, Journal of he European Economc Assocaon, 3(2-3), 74 723. (2007): Cred Traps and Cred Cycles, Amercan Economc Revew, 97(), 503 56. (2008): Aggregae Implcaons of Cred Marke Imperfecons, n NBER Macroeconomcs Annual 2007, ed. by D. Acemoglu, K. Rogoff, and M. Woodford, vol. 22. MIT Press. 7
(202): Insuon-Induced Producvy Dfferences and Paerns of Inernaonal Capal Flows, Journal of he European Economc Assocaon, forhcomng. (203): The good, he bad, and he ugly: An nqury no he causes and naure of cred cycles, Theorecal Economcs, 8(3), 623 65. von Hagen, J., and H. Zhang (20): Inernaonal Capal Flows wh Lmed Commmen and Incomplee Markes, CEPR Dscusson Paper No. 8750. (204): Fnancal Developmen, Inernaonal Capal Flows, and Aggregae Oupu, Journal of Developmen Economcs, 06, 66 77. A Appendx A. Endogenze he Mass of Enrepreneurs n Seng N The mass of enrepreneurs s exogenous n seng N. Here, I endogenze by assumng ha all agens can produce capal goods usng fnal goods and he producvy for agen j [0, ] s ndvdual specfc, kj,+ = R j m j,, and R j = θ + R (27) θ ɛ j where ɛ j (, ) follows he Pareo dsrbuon wh he cumulave dsrbuon funcon F (ɛ j ) = ɛ θ j and θ >. R j (0, R) has he mean E(R) = R j df (ɛ j ) = R and he upperbound R +θ R. Wh producvy heerogeney, s called seng P.5 θ Rv+ r Le ɛ θ+. The profably consran, R θ j v+ r, mples ha he agens wh ɛ j (, ɛ ] choose o become enrepreneurs and fnance her projecs wh loans, whle he agens wh ɛ j > ɛ choose o become households and lend ou he labor ncome. Ths way, he mass of enrepreneurs η = (ɛ ) θ s endogenzed. Suppose ha he borrowng consrans are bndng for all enrepreneurs. The projec nvesmen sze rses n he ndvdual-specfc producvy, m j, = Under IFA, aggregae nvesmen s fnanced purely by domesc savng, ω λ R j v + r = ω. λ ɛ ɛ j ɛ m j,df (ɛ j ) = ω ɛ λ ɛ ɛ j df (ɛ j ) = ω ɛ λ ɛ ɛ j df (ɛ j ) =, (28) mplyng ha he cuoff value s me nvaran ɛ = ɛ IF A and depends only on he level of fnancal developmen λ and he dsrbuon funcon F (ɛ j ). Aggregae oupu of capal goods s K + = ɛ R j m j,df (ɛ j ) = ω R IF A, where R IF A ɛ θ+ θ ɛ j λ ɛ ɛ j df (ɛ j ). (29) 5 Seng N s a specal case here,.e., for θ, he dsrbuon of ɛ j degeneraes no a un mass a ɛ j = and hence, R j =. 8
R IF A (R, R) measures he aggregae producvy under IFA and s me nvaran. The dynamcs of aggregae oupu can be characerzed by ( ) ω+ ( α) = L Y + RIF A ω α =, (30) ( ) R, and here exss a unque and sable seady sae wh ω IF A = IF A smlar as equaon () n seng N. Agen j can ge he loan λ R jv+ = λ ɛ r IF A ɛ j per un of s projec nvesmen. As long as λɛ IF A <, even he mos producve agens ɛ j = canno fnance her enre projec nvesmen by loans and hence, he borrowng consrans are bndng for all enrepreneurs. In equlbrum, he neres rae s deermned by he rae of reurn of he margnal enrepreneurs wh ɛ j = ɛ IF A, r = Rv+ = θ + Rv+ = θ + θ ɛ IF A θ R ɛ IF A ( R IF A ω ) α. (3) Lemma 5. For λ (0, ɛ IF A ), he borrowng consrans are bndng. The neres rae, proporonal o he prce of capal goods, decreases n ω. Inuvely, alhough he mass of enrepreneurs η = (ɛ ) θ s endogenously deermned, he me-nvaran cuoff value ɛ = ɛ IF A mples he me-nvaran mass of enrepreneurs n equlbrum under IFA. Thus, ncome changes only affec he cred marke on he nensve margn, same as n seng N. Due o he absence of exensve-margn effec, he neoclasscal effec leads o he negave neres rae response o ncome changes under IFA; FMG makes he phase dagram of wage flaer and he nal seady sae under IFA s sll he unque and sable seady sae under FMG. To sum up, wh no fxed or mnmum nvesmen sze requremens, FMG does no lead o Masuyama s symmery breakng n seng P as well as n seng N. A.2 Proofs Proof of Proposon Proof. Accordng o equaon (), he phase dagram of wage n seng N sars from zero and s srcly concave, gven α (0, ). As shown n he lef panel of fgure( 4, ) crosses he 45 R. lne once and only once from he lef wh he wage a ω IF A = Gven assumpon, he phase dagram of wage n seng F s dencal as n seng N, excep for a knk a ω = m. Thus, here exss a unque and sable seady sae under IFA n boh sengs. Rewre he bndng borrowng consrans (2) as r v +R = λ. (32) ψ 9
Accordng o equaon (4), enrepreneurs prefer o fnance her nvesmen usng loans ff r < v+r, or equvalen, λ < ψ. In seng N, he cred marke clearng equaon (7) mples ha ψ = ω = η. Thus, for λ (0, η), he prvae raes of reurn m are proporonal o he socal rae of reurn, r = λ η Rv + < Rv+ < Γ = λ η Rv +; for λ = η, he prvae raes of reurn concde wh he socal rae of reurn, r = Γ = Rv+, and he borrowng consrans are weakly bndng; for λ ( η, ), he borrowng consrans are slack and enrepreneurs do no borrow o he lm, because r = Γ = Rv+. Smlar analyss apples o seng F. Proof of Lemma Proof. Accordng o equaon ()-(2), v + = ( ) K α + = ( ) Rω α = R ( ωif A ω ) α. (33) Accordng o he proof of proposon, for λ (0, η), he borrowng consrans are ( ) bndng and r = λ η Rv + = λ ω α; IF A η ω for λ ( η, ), he borrowng consrans ( ) α are slack and r = Rv+ = under IFA. Due o he neoclasscal effec, he ω IF A ω socal rae of reurn Rv + declnes n ω and so does he neres rae, r ω Proof of Lemma 2 Proof. Accordng o proposon, for λ (0, ψ ), he borrowng consrans are bndng under IFA. Combne he borrowng consrans (3) and equaon (33) o ge r = λ Rv ψ + = λ ψ = ln r ψ r ψ ω = ( ψ r ω < 0. ( ) α ωif A (34) ψm α ) r ψ m > 0, ff ψ ( ψ F, λ). (35) Proof of Proposon 2 Proof. Combnng equaons ()-(2) wh he bndng borrowng consrans (3), I derve he phase dagram of wage n seng N and s properes are as follows, [ ] r (m ω) = λv+rm r R (ω +) α ηω = λω+, (36) [ ] ω+ = η (ω+) ω αr λ = ηv +R > 0, (37) r + ω m ( ) 2 ω+ ω 3 (ω) = + (ω+) < 0. (38) 2 ηαr ω 20
Gven he world neres rae r, equaon (36) mples ha, for ( ω ) 0, he phase dagram. of wage has a posve nercep on he vercal axs a ω+ = Defne a hreshold value ω N λ ( R ). For ω η r (0, ω N ), he borrowng consrans are bndng and he phase dagram of wage s ncreasng and concave, accordng o equaons (37)-(38). For ω > ω N, aggregae savng and nvesmen are so hgh ha he socal rae of reurn s equal o he world neres rae, Rv+ = r and he borrowng consrans are slack. The = r phase dagram s fla a ω+ = ω + N = ( R ). Gven r r < and ( λ) > η, ω + N < ω N so ha he knk pon on he phase dagram s below he 45 lne. Graphcally, he phase dagram of wage crosses he 45 lne once and only once from he lef, and he nersecon s n s concave par. ( Gven ) r = r IF A, he seady sae concdes wh he one under IFA R. a ω F MG = ω IF A = See he lef panel of fgure 5. Use equaon () and (37) o evaluae he slope of he phase dagram a he seady sae under IFA and under FMG, ω + ω F MG = α + ( α)( η) η Rλ r < α = ω + ω IF A. Thus, FMG makes he phase dagram of wage flaer han under IFA, whch speeds up he convergence o he same seady sae as under IFA. Proof of Proposon 3 Proof. I frs prove he shape of he phase dagram of wage and hen descrbe he condons for symmery breakng. For Rv+ > r or equvalenly ψ < λ, he borrowng consrans are bndng. Use equaons ()-(2) o rewre he bndng borrowng consrans (3) as ω m = λv +R r ω + ω = λ r = ω+ ω ψ ( ωif A ω + ), (39) > 0, and 2 ω + (ω ) 2 = ( ) ω 2 + ω αω +ω > 0. (40) Combne equaon (39) wh ()-(2) and hen compue he mass of enrepreneurs, [ ] ω+ λ = ω IF A η r ( ψ) = K + Rm = (ω [ ] α +) Rm = ψ λ α IF A. (4) r ( ψ) The mass of enrepreneurs canno exceed he populaon sze of each generaon, η. For ψ IF A (0, λ), he borrowng consrans are bndng n he seady sae under IFA and r = λ ψ IF A < ; under FMG, accordng o equaon (4), η mples ha ψ ˇψ F ψ α IF A ( ψ IF A). For ψ IF A > λ, he borrowng consrans are slack n he seady sae under IFA and r = ; under FMG, accordng o equaon (4), η mples ha ψ λψ α IF A. Thus, he phase dagram of wage under FMG s a pecewse funcon wh wo subfuncons and here are wo cases. Case : f ˇψ F > λ, For ψ (0, λ), he borrowng consrans are bndng, some agens become [ enrepreneurs, η <, and he phase dagram of wage s convex, ω+ ψ ] = ω IF A IF A ; 2 ω m
for ψ > λ, he borrowng consrans are slack, some agens become enrepreneurs, η <, and he phase dagram of wage s fla a ω+ = ω ψif A [ ]. IF A λ Case 2: f ˇψ F < λ, For ψ (0, ˇψ F ), he borrowng consrans are bndng, some agens become [ enrepreneurs, η <, and he phase dagram of wage s convex, ω+ ψ ] = ω IF A IF A ; for ψ > ˇψ F, he borrowng consrans are bndng, all agens ( ) become enrepreneurs, α η =, and he phase dagram of wage s fla a ω+ = = ω IF A. Accordng o equaon (4), for ω 0, ψ ( ) 0 and he phase dagram has a posve nercep on he vercal axs a ω+ λ. = ω IF A r The convex par of he phase dagram creaes he possbly of mulple seady saes. Rm ψ α IF A ω m Case C Case A w + w + S S O w IFA w O w IFA w Fgure : Phase Dagrams of Wage under FMG n Seng F Fgure 6 shows he parameer confguraon of fve regons n he {λ, ψ IF A } space. Besdes he hree symmery-breakng cases shown n fgure 7, fgure shows wo cases where he seady sae under IFA s sll he unque, sable seady sae under FMG. In he followng, I derve he boundary condons for he fve regons n fgure 6. Gven r = r IF A, he seady sae under IFA s sll a seady sae under FMG, hough may no be sable or unque. For he parameers n he lower-lef (upper-rgh) rangle of fgure 6, he borrowng consrans are bndng (slack) around he seady sae under IFA. Sar wh he upper-rgh rangle of fgure 6,.e., ψ IF A ( λ, ). Compare he rgh panel of fgure 7 and he lef panel of fgure. Gven r = r IF A =, he phase dagram of wage under FMG s fla a he nal seady sae (pon S); he boundary beween regon BC and C s defned as he case where he convex par of he phase dagram of wage s angen wh he 45 lne,.e., ω = ω+ = ω F < ω IF A. Rewre equaons (39) and (40) a he angen pon, ωf m = λ(ωf ) R ( ) (, ωf ω F m m ω+ = ψf ω ψ = ( F λrm (wf ) ω F α =, m 22 ) = ψ IF A λ (42) ) α λ = ψ IF A. (43)
Combne hem o ge ω F ( α ) m = α and ψ IF A = ( α), (44) λ ω F < ω IF A ωf m < ψ IF A and λ < α. (45) Equaons (44)-(45) jonly defne he boundary beween regon BC and C. Consder he lower-lef rangular of fgure 6,.e., ψ IF A (0, λ). Case B arses f he slope of he phase dagram of wage under FMG s larger han uny a he nal seady sae, ω+ ω F MG = ψ IF A > ψ IF A > ˆψ F α, (46) whch specfes he boundary beween regon B and AB. If ω + ω F MG <, he nal seady sae s locally sable. However, FMG may sll generae a mulple seady-sae equlbrum f he knk pon of he phase dagram of wage s above he 45 lne,.e., ω + F > ω F. There are wo cases. Case : f ˇψ ( F > λ, he knk pon s a ω F = ( λ)m and ω + F = ω ψif A ). IF A λ ω F + > ω F, ( ψ IF A ) ψ IF A > ( λ)λ. (47) Case 2: f ˇψF < λ, he knk pon s a ω F = [ ( ψ IF A )ψ α IF A ]m and ω + F = ω IF A. ψif α A ω F + > ω F, ψ α IF A (2 ψ IF A). (48) Equaons (47) and (48) defne he boundary condons beween AB and A. Proof of Lemma 3 Proof. Combnng equaons (9) and (20), he cuoff value ɛ s he soluon o equaon (49) and he equy-nvesmen rao ψ s an ncreasng funcon of he cuoff value, ɛ (ɛ ) θ = ω m θ +, ln ɛ θ ln ω ψ = (ɛ ) (+θ), ln ψ ln ω = (ɛ ) (+θ) + θ(ɛ ) = + θ (+θ) (0, ), (49) + θ = ln ɛ ln ω ψ ψ = + θ (0, ). (50) + θ Proof of Lemma 4 23
Proof. Combne equaons () and (2) wh equaon (20) o rewre he neres rae as ( ) ω r = λrv+(ɛ ) +θ α = λ IF A (ɛ ) +θ, (5) ω ln r = ln λ + ( α) ln ω IF A ( α) ln ω + ( + θ) ln ɛ, (52) ln r ln ω = ( α) + ( + θ) ln ɛ ln ω For λ (0, ψ M ), he neres rae rses n ω, ln r ln ω Proof of Proposon 4 = ( α) + ( + θ) + θ( ψ). (53) ψ > 0, f ψ ( ψ M, λ). Proof. The srucure of he proof resembles ha of Proposon 3. I frs prove he shape of he phase dagram of wage and hen descrbe he condons for symmery breakng. For Rv + > r or equvalenly ψ < λ, he borrowng consrans are bndng and he model dynamcs under FMG are feaured by a recursve equaon sysem of {ω, ψ, ɛ }, ψ ɛ ω = + θ θm ω [ (ɛ ) (+θ) ] ψ = ψ IF Aɛ IF A, ω ψ = ω ɛ IF A, (54) ω IF A ψ IF A ɛ [ IF A, ω+ λ = ω IF A ψ = λ Rv + r = K + R, ɛ [ (ɛ ) (θ+) ] ψ IF A ɛ IF A = [ λ ( ψ )r ( ψ )r ] ]. (55) α, (56) Equaon (54) specfes he equy-nvesmen rao, as equaon (9); equaon (55) feaures he bndng borrowng consrans, as equaon (2); equaon (56) shows ha enrepreneurs produce capal goods wh leveraged nvesmen, as equaon (25). Use equaons (54)-(56) o derve he dynamc propery of he phase dagram of wage under FMG, ln ɛ ln ψ ln ω ln ψ ln ω + ln ψ = ψ α ψ = ln ɛ + = ln ψ α = α ψ α ψ (ɛ ) (+θ) > 0, (57) + θ(ɛ (+θ) ) ψ ψ (ɛ ) (+θ) + >, (58) + θ(ɛ (+θ) ) > 0 (59) ω + ω = ω + ω ln ω + ln ψ ln ω ln ψ = ω + ω α (ɛ ) (+θ) +θ(ɛ ) (+θ) + ( α) ψ ψ > 0. (60) 24
Le A (ɛ ) (+θ) +θ(ɛ ) (+θ) and Z ψ A ( A ) ψ θ ( α) A2 [ ] 2 A + θ (ɛ = ) (+θ) ln ɛ > 0 (6) ψ + θ(ɛ ) (+θ) ψ ln ψ [ Z = θa2 ψ α A ψ ( A) + 2θψ A ] < 0, (62) 2 α ( ) 2 ω+ ( ω) = 2 Z A ω 2 ( ) + ω sgn + = ψαω + ω 2 ω sgn(z ). (63) A α + ψ ψ I s rval o prove ha for ψ 0, Z > 0 and he phase dagram of wage s convex. Accordng o equaon (62), Z declnes n ψ and hence, s possble ha Z < 0 and he phase dagram of wage becomes concave. Le ˇψ M defne he hreshold value of ψ such ha Z = 0,.e., he nflecon pon of he phase dagram of wage. There are wo cases. Case : f ˇψ M > λ, he phase dagram of wage s a pecewse funcon wh wo subfuncons: for ψ (0, λ), he borrowng consrans are bndng, he mass of enrepreneurs s sgnfcanly smaller han one, and he phase dagram of wage s convex; for ψ ( λ, ), he borrowng consrans are slack, he mass of enrepreneurs s sgnfcanly smaller han one, and he phase dagram of wage s fla. Case 2: f ˇψ M < λ, he phase dagram of wage s a pecewse funcon wh hree subfuncons: for ψ (0, ˇψ M ), he borrowng consrans are bndng, he mass of enrepreneurs s sgnfcanly smaller han one, and he phase dagram of wage s convex; for ψ ( ˇψ M, λ), he borrowng consrans are bndng, he mass of enrepreneurs s close o one, and he phase dagram of wage s concave; for ψ ( λ, ), he borrowng consrans are slack, he mass of enrepreneurs s close o one, and he phase dagram of wage s fla. Gven he world neres rae r, equaon (56) mples ha, for ω 0 or equvalenly ψ ( 0, ) he phase dagram has a posve nercep on he vercal axs a ω+ λ ; = ω IF A r for ψ = λ, he phase dagram of wage has a knk pon wh ( ) ω+. = ω IF A r The convex/concave par of he phase dagram of wage creaes he possbly of mulple seady saes. 6 6 Alhough he shape of he phase dagram of wage under FMG n seng M may dffer from ha n seng F, hey are fundamenally dencal. In seng M, for a suffcenly low level of ncome, he equy-nvesmen rao s low and so s he cuoff value ɛ, accordng o equaons (57)-(58). Thus, he mass of enrepreneurs η = (ɛ ) θ s very small. Capal accumulaon rases he wage rae and allows more ndvduals o become enrepreneurs. The exensve-margn effec amplfes he rse n domesc nvesmen and ncome, whch makes he phase dagram of wage convex under FMG. For a suffcenly hgh level of ncome, he mass of enrepreneurs s close o one and a margnal rse n he wage rae 25
Fgure 9 shows he parameer confguraon of fve regons n he {λ, ψ IF A } space. Besdes he hree symmery-breakng cases shown n fgure 0, fgure 2 shows wo cases where he seady sae under IFA s sll he unque, sable seady sae under FMG. Case C Case A w + w + S S O w IFA w O w IFA w Fgure 2: Phase Dagrams of Wage under FMG n Seng M In he followng, I derve he boundary condons for he fve regons n fgure 9. Gven r = r IF A, he seady sae under IFA s sll a seady sae under FMG, hough may no be sable and unque. For he parameers n he lower-lef (upper-rgh) rangle of fgure 9, he borrowng consrans are bndng (slack) around he seady sae of IFA. Sar wh he upper-rgh rangle of fgure 9,.e., ψ IF A ( λ, ). Compare he rgh panel of fgure 0 and he lef panel of fgure 2. Gven r = r IF A =, he phase dagram of wage under FMG s fla a he nal seady sae (pon S); he boundary beween regon BC and C s defned as he case where he convex par of he phase dagram of wage s angen wh he 45 lne,.e., ω = ω+ = ω M < ω IF A. Combne equaons (54)-(56) and evaluae equaon (60) a he angen pon wh r = r IF A = { ɛ [ (ɛ ) (+θ) ] ɛ IF A ψ IF A } α = λ ( ) ψ = ɛ α α, (64) ψ ψ IF A ɛ IF A (ɛ ) (+θ) = D λ ψ, and 0 < ψ D < ψ < ψ IF A <, (65) ω+ = ω + α ω ω D + ( α) = (66) λ +θ( D ) D ( αθ + )D2 [ λ + ( θ + )]D + λ ( + ) = 0. (67) θ and he labor ncome canno rase he mass of enrepreneurs very much so ha he neoclasscal effec domnaes he exensve-margn effec and hence, he phase dagram s concave. In seng F, for he suffcenly low level of ncome, he small mass of enrepreneurs allows for he srong exensve-margn effec, explanng he convexy of he phase dagram of wage; for a suffcenly hgh level of ncome, he mass of enrepreneurs reaches one and, due o he fxed nvesmen sze requremen of ndvdual projecs, any furher rse n ncome does no rase domesc nvesmen and fuure ncome so ha he phase dagram of wage becomes fla. In hs sense, he hreshold value ˇψ M n seng M corresponds o ˇψ F n seng F. 26