Foreign Trade and Equilibrium Indeterminacy

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1 WORKING PAPER SERIES Foregn Trade and Equlbrum Indeermnacy Luís Aguar-Conrara and Y Wen Workng Paper A hp://research.slousfed.org/wp/2005/ pdf June 2005 FEDERAL RESERVE BANK OF ST. LOUIS Research Dvson 411 Locus Sree S. Lous, MO The vews expressed are hose of he ndvdual auhors and do no necessarly reflec offcal posons of he Federal Reserve Bank of S. Lous, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Lous Workng Papers are prelmnary maerals crculaed o smulae dscusson and crcal commen. References n publcaons o Federal Reserve Bank of S. Lous Workng Papers (oher han an acknowledgmen ha he wrer has had access o unpublshed maeral) should be cleared wh he auhor or auhors. Phoo couresy of The Gaeway Arch, S. Lous, MO.

2 Foregn Trade and Equlbrum Indeermnacy Luís Aguar-Conrara Deparmen of Economcs Cornell Unversy and NIPE, Unversdade do Mnho, 4704 Braga Y Wen Deparmen of Economcs Cornell Unversy (Frs Verson: May 2004) Absrac We show ha dependence of producon on foregn npus (or non-producble naural resources) can sgnfcanly ncrease he lkelhood of ndeermnacy. Paymen of mpored foregn facors of producon may ac as a sem-fxed cos, amplfyng producon exernales and reurns o scale, makng selffulfllng expecaons drven busyness cycles easer o arse. Ths s demonsraed usng a sandard neoclasscal growh model. Calbraon exercse shows ha he requred ncreasng reurns o scale can be reduced by as much as 64% based on esmaed share of foregn npus n producon for OECD counres. Keywords: Indeermnacy, Facor Impors, Naural Resources, Capacy Ulzaon, Exernaly, Reurns o Scale, Open Economy, Sunspos, Self- Fulfllng Expecaons. JEL Classfcaon: E13, E20, E30. We hank Karl Shell for commens. Correspondence: Y Wen, Deparmen of Economcs, CornellUnversy,Ihaca,NY14853,USA.Emal:yw57@cornell.edu.

3 Based on npu-oupu ables of OECD counres, mpors of nermedae goods and raw maerals accoun for a sgnfcan fracon of oal npus n domesc producon. Table 1 repors he average share of he value of mpors n domesc producon across 35 producon secors for each of he en OECD counres consdered. These cos shares of foregn npus range from 5.4% (Japan) o 21.1% (Neherlands). The average cos share of foregn npus among he en counres s abou 13%. Hence, mpored producon facors play a non-rval role n an economy s producon process. 1 Table 1. Cos Share of Foregn Inpus n Domesc Producon Ausrala Canada Denmark France Germany Ialy Japan Neherlands U.K. U.S. 10% 16% 20% 13% 14% 12% 5% 21% 16% 6% Wha does he dependence of producon on foregn npus mply for economc flucuaons? Paymen o foregn npus acs as a ax for producon, hence has no only a wealh effec bu also a subsuon effec. Under he wealh effec, a hgher facor prce for mpored maerals reduces domesc ncome, leadng o lower consumpon and hgher labor supply. Under he subsuon effec, he hgher facor prce leads o lower demand for foregn maerals, depressng producvy of capal and labor, resulng n lower employmen. The subsuon effec can be dramacally amplfed f here exs producon exernales n an economy. Ths paper shows ha dependence of producon on foregn npus s an mporan channel leadng o ndeermnacy n a sandard neoclasscal growh model wh mperfec compeon or producon exernales. In parcular, we prove ha ndeermnacy s easer o oban he larger he share of mpored facors s n domesc producon. The key s ha paymens for foregn mpors can ac as asem-fxedcoshaamplfes reurns o scale, renderng he balanced growh pah of a neoclasscal economy more lkely o be ndeermnae. In such a model, a fear or speculaon of an ncrease n he mpored facor prce, say due o polcal nsably n he foregn counry, can rgger pessmsms, makng economc recessons self-fulfllng. The mpored foregn facor of producon n our model can also be nerpreed as non-reproducble naural resources exraced domescally. Hence he mplcaon of our model s no lmed o open economes wh rade. We call he non-reproducble producon facor foregn npu n hs paper because we have beer daa o calbrae s share n GDP. 2 Equlbrum ndeermnacy n a sandard neoclasscal growh model wh exernales or ncreasng reurns o scale has receved sgnfcan amoun of aenon n he recen busness cycle leraure due o he poneerng work of Benhabb 1 Daa are based on npu-oupu ables from OECD (1995) Repors. Each npu-oupu able has 35 secors and conans value of mpored npus used by each secor. The fgures shown n he ex were calculaed by dvdng he oal value of mpored npus by he oal value of producon of all secors. 2 We hank Karl Shell for suggesnghsnerpreaonous. 2

4 and Farmer (1994). I s now wdely vewed as a promsng vehcle for sudyng endogenous busness cycles and sunspos drven flucuaons. 3 Alhough hs frs-generaon ndeermnae RBC model requres mplausbly large degrees of exernales o generae ndeermnacy (hereby casng doub on her emprcal relevance, see e.g., Schm-Grohe 1997), subsequen work by Benhabb and Farmer (1996), Benhabb and Nshmura (1997), Benhabb, Nshmura and Meng (2000), Benne and Farmer (2000), Perl (1998), Weder (1998 and 2001) and Wen (1998), among many ohers, show ha addng oher sandard feaures of real economes no he model of Benhabb and Farmer (1994) can reduce he degree of exernales requred for nducng local ndeermnacy. 4 Ths lne of research dscovers ha feaures such as addonal secors of producon, durable consumpon goods, non-separable uly funcons, small open economy, or varable capacy ulzaon can reduce he requred exernales for local ndeermnacy o a degree ha s whn emprcally admssble range. In hs paper we add o hs fas growng leraure anoher mechansm for ndeermnacy: he dependence of producon on foregn npus. Based our calbraons, we show ha when he share of foregn npus n domesc producon ncreases from zero percen o fve percen, he requred degree of ncreasng reurns o scale for ndeermnacy s reduced by as much as 17%; and f he share ncreases o 20%, hen he correspondng reducon can be as large as 64%. 5 The res of he paper s organzed as follows. To llusrae he basc mechansm of ndeermnacy due o dependence on foregn mpors as producon facors, we frs nvesgae a benchmark model (Benhabb and Farmer, 1994) by assumng ha producon of nermedae goods requres no only capal and labor, bu also a hrd facor, say ol, mpored from ousde he economy. For smplcy, we assume ha hs hrd facor s perfecly elascally suppled. Laer on, we wll nroduce a foregn monopoly power ha supples he hrd facor wh arbrary elascy of supply, so as o sudy he robusness of our resuls. Fnally, we wll calbrae a more realsc model wh varable capacy ulzaon (Wen, 1998) and show ha dependence of foregn mpored facors of producon can sgnfcanly decrease he degree of exernaly requred for ndeermnacy so ha he socal reurns o scale are essenally consan for mulple equlbra o emerge. 3 For he broader leraure on sunspos, please see Shell (1977, 1987), Cass and Shell (1983), Shell and Smh (1992), Azarads (1981), Azarads and Guesnere (1986), and Woodford (1986a, 1986b, 1991). 4 See Wen (2001) for a recen analyss of hs class of models regardng mechansms gvng rse o local ndeermnacy from he vewpon of he permanen-ncome hypohess. For open economy models wh ndeermnacy, see Weder (2001), Meng (2003), and Meng and Velasco (2003). 5 Our model dffers from ha of Weder (2001). In Weder (2001), ndeermnacy s easer o oban for a small open economy due o perfec or nearly perfec world capal markes ha keep neres rae more or less consan. In hs model, he ably o use nernaonal cred markes o dsconnec savngs and nvesmen s an mporan mechansm for ndeermnacy. Weder s model also requres some form of negave exernales o ensure sably of he seady sae. In our model, he economy does no need o be small and he foregn facor markes do no need o be perfecly compeve. The mechansm for ndeermnacy n our model s hrough producon coss due o paymens for foregn facors, such as ol, whch are no producble a home. 3

5 1. The Benchmark Model Ths s a slghly modfed verson of he Benhabb-Farmer (1994) model. There are wo producon secors n he economy, he fnal goods secor and he nermedae goods secor. The fnal goods secor s compeve and uses a connuum of nermedae goods o produce fnal oupu accordng o he producon echnology, µz 1 Y = y λ d =0 where λ (0, 1) measures he degree of facor subsuon among nermedae goods. Le p be he relave prce of he h nermedae goods n erms of he fnal good, he profs of fnal good producer are gven by Z 1 Π = Y p y d. =0 Frs order condons for prof maxmzaon lead o he followng nverse demand funcons for nermedae goods: 1 λ p = Y 1 λ y λ 1. The echnology for producng nermedae goods s gven by y = k a k n a n o a o, where he hrd facor n producon, o, s mpored, and (a k + a n + a o ) 1 measures reurns o scale a he frm level. Assumng ha frms are prce akers n he facor markes, he profs of he h nermedae good producer are gven by π = p y (r + δ)k wn p o o, where (r + δ) denoes he user cos of renng capal, w denoes real wage, and p o denoes he real prce of ol (he mpored good). The nermedae goods producers are monopolss facng downward slopng demand curves for nermedae goods, hence he prof funcons can be rewren as π = Y 1 λ y λ (r + δ)k wn p o o, whch s concave as long as λ(a k + a n + a o ) 1. Prof maxmzaon by each nermedae goods producng frm leads o he followng frs order condons: r + δ = λa k p y k w = λa n p y n p o = λa o p y o. 4

6 In a symmerc equlbrum, we have n = n, k = k, o = o, y = y = Y,π = π,p =1, and µz 1 Π = Y y d 1 λ λ =0 =0 π =(1 λ(a k + a n + a o )) Y. In words, perfec compeon n he fnal goods secor leads o zero prof and mperfec compeon n he nermedae goods secor leads o posve prof f λ(a k + a n + a o ) < 1. A represenave consumer n he economy maxmzes uly, Ã! X β log c b n1+γ 1+γ subjec o =0 c + s +1 =(1+r )s + w n + π, where s s aggregae savng. Snce he aggregae facor paymen, p o o, goes o he foregners, s no ncluded n he consumer s ncome. The frs order condons for uly maxmzaon wh respec o labor supply and savngs are gven respecvely by bn γ = 1 w, c 1 = β 1 (1 + r +1 ). c c +1 In equlbrum, s = k, and facor prces equal margnal producs, he frs order condons and he budge consran hen become bn 1+γ = 1 λa n y c (1) 1 = β 1 µ y +1 1 δ + λa k c c +1 k +1 (2) c + k +1 =(1 δ)k +(1 λa o )y (3) y = k a k 2. Condons for Indeermnacy n an o ao. (4) Assumng ha he foregn npu s perfecly elascally suppled, hen he facor prce, p o, s ndependen of he facor demand for o. 6 Hence we can subsue ou o n he producon funcon usng o = λa o y p o, 6 Ths assumpon wll be relaxed n secon 3. 5

7 o oban he followng reduced-form producon funcon: y = Ak a k 1 ao an 1 ao n, (4 0 ) ³ ao where A = λao 1 ao p acs as he echnology coeffcen n a neoclasscal growh o model, whch s nversely relaed o he foregn facor prce. In hs reduced-form producon funcon, he effecve reurns o scale s measured by a k + a n, 1 a o whch exceeds he rue reurns o scale, (a k + a n + a o ), provded ha (a k + a n + a o ) > 1. Hence, he relance on foregn facors amplfes he rue reurns o scale. I can be easly shown ha a unque seady sae exss n hs economy. To sudy ndeermnacy, we subsue y by ulzng equaon (4 0 ) and log lnearze equaons (1)-(3) around he seady sae. Ths gves µ 1+γ a n ˆn = a k 1 a o µµ ak ĉ = ĉ +1 +(1 β(1 δ)) (1 s)ĉ + s δ ˆk +1 = ˆk ĉ 1 a o 1 ˆk a o ˆk + µ ak 1 a o + s 1 δ δ a n ˆn +1 1 a o a n 1 a o ˆn where s s he adjused seady-sae savng rae (nvesmen-o-naonal ncome rao) gven by δk s = (1 λa o )y = δβλa k (1 λa o )(1 β(1 δ). The above sysem of lnear equaons can be reduced o k+1 k M 1 = M λ 2 +1 λ where M 1 = M 2 = 1 β(1 δ) ³a k + a o 1+ a na k (1+γ)( ) a n 1+ (1 β(1 δ))an (1+γ)( ) a n s 1 δ ³ a k a 1+ n (1+γ)() a n + s 1 δ δ (1+γ)() (1+γ)() a n s Denoe B = M 1 1 M 2, a necessary and suffcen condon for ndeermnacy s ha boh egenvalues of B are less han one n modulus. Ths s rue f and only f he deermnae and he race of B sasfy 1 < de(b) < 1 (1 + de(b)) < r(b) < 1 + de(b) 6

8 The deermnae and he race of B are gven by (see Appendx 1): de(b) = 1 (1 λ) (1 + γ)(1 β(1 δ)) λ(1 a 1+ o ) β 1+γ β(1 δ) a (5) n ³ (1 β(1 δ))(1 + γ) 1 ao a k δ 1 s s r(b) = 1 + de(b)+ 1+γ β(1 δ) a (6) n Noce ha when λ = 1, hen de(b) = 1/β > 1, ndcang saddle-pah-sably as n a sandard RBC model. Hence, wha s crucal for ndeermnacy s no ncreasng reurns o scale per se, bu also he degree of marke power or mperfec compeon. The common denomnaor n he second erm n expresson (5) and he hrd erm n (6) suggess ha when he labor s elascy of oupu n he reducedform producon funcon, n a, ncreases, he model may go hrough a pon of dsconnuy a whch 1 + γ β(1 δ) an =0andde(B) andr(b) boh change sgn from + o, f he condon 1 a o a k > 0sllholds. Clearly, when hese erms are negave nfny, he condons for de(b) < 1andr(B) < 1 + de(b) are rvally sasfed. Bu o reach he dsconnuy pon such ha he second erm n (5) and he hrd erm n (6) are negave, we need a n β(1 δ) > 1+γ. (7) 1 a o (7) s an mporan necessary condon for ndeermnacy. Clearly, he larger s a o, he easer hs condon can be sasfed. To faclae nerpreng hs condon, we map he monopolsc compeon model no a one-secor compeve model wh producon exernales (see Benhabb and Farmer, 1994), n whch he aggregae producon funcon s replaced by y = k α k(1+η) n α n(1+η) o α o(1+η), and he reduced-form producon funcon s replaced by y = Ak α k (1+η) 1 αo(1+η) n αn(1+η) 1 αo(1+η) where (α k + α n + α o ) = 1 and he parameer η measure he degree of producon exernales. Ths model s dencal o he monopolsc compeon model f λa k = α k, λa n = α n, λa o = α o, and (a k + a n + a o )=1+η. Thsgvesλ(a k +a n + a o )=λ(1 + η) =1, mplyng ha n he correspondng monopolsc compeon model he nermedae goods producng frms earn zero profs. In he exernaly verson of he model, aggregae reurns o scale are measured by 1 + η. Whhs change n framework, equaons (5) and (6) become de(b) = 1 1+ (1 + γ)(1 β(1 δ)) η 1 α o(1+η) β 1+γ β(1 δ) αn(1+η) (5 0 ) 1 α o (1+η) 7

9 (1 β(1 δ))(1 + γ) r(b) =1+de(B)+ ³ 1 (αo+α k )(1+η) 1 α o(1+η) δ 1 s s 1+γ β(1 δ) α n(1+η) 1 α o(1+η) Clearly, ndeermnacy s no possble f η = 0, whch mples de(b) =1/β > 1. Ths shows ha monopoly power n he prevous verson of he model perans o exernaly n he curren verson of he model. Condon (7) hus becomes whch can also be expressed as α n (1 + η) β(1 δ) 1 α o (1 + η) 1 > γ, (6 0 ) η > (1 + γ)(1 α o) β(1 δ)α n β(1 δ)α n +(1+γ)α o. (7 0 ) Condon (7 0 ) s analogous o ha derved by Benhabb and Farmer (1994) n a connuous me model when δ 0andβ 1. In a connuous me verson of he model, hs condon smplfes o η > 1 (α o + α n )+γ(1 α o ) (α o + α n )+γα o. If α o = 0 (.e., producon does no requre he mpored facor), hen hs condon for ndeermnacy s dencal o ha n Benhabb and Farmer (1994). Snce he rgh hand sde s a decreasng funcon of α o, hs necessary condon for ndeermnacy s easer o sasfy han ha n he Benhabb-Farmer model. To furher pn down he full se of condons for ndeermnacy, noe ha as long as (α o + α k )(1 + η) < 1, he second erm n he deermnae of B and he hrd erm n he race of B mus pass hrough for large enough η and moves o a fne negave number as η keeps ncreasng. Snce we are neresed only n he smalles value of η ha gves rse o ndeermnacy, we can herefore lm our aenon o he followng smpler one-sded condons as necessary and suffcen condons for ndeermnacy: de(b) > 1 andr(b) > (1 + de(b)), assumng he necessary condon (7 0 )ssasfed. The condon de(b) > 1 sequvaleno η > (1 + γ)(1 α o ) β(1 δ)α n β(1 δ)α n +(1+γ)α o 1+γ 1+β (1 β(1 δ)). Noe ha f hs condon s sasfed, hen condon (7 0 )salsosasfed snce hey dffer only by a posve erm, 1+γ 1+β (1 β(1 δ)). In a connuous me verson of hs model (δ 0, β 1), hs condon smplfes o η > 1 (α o + α n )+γ(1 α o ) (α o + α n )+γα o, 8

10 whch s dencal o (7 0 ). Hence, an ncrease n α o, eher holdng α n consan or holdng (α o + α n ) consan, wll decrease he rgh hand sde, makng ndeermnacy easer o arse. The condon r(b) > (1 + de(b)) leads o 1+η 1 α o (1 + η) > 2(1 + γ)(2 δ)+(1+γ)δ 1 s s (1 β(1 δ)) 2(1 + β)(1 δ)α n (1 + γ) δα k s (2 (1 s)(1 β(1 δ))). Clearly, he presence of α o on he lef-hand sde makes he nequaly easer o sasfy he larger he value of α o s. Alernavely, we can consder a connuous me verson of he model (δ 0, β 1), hen he above condon smplfes o 1+η 1 α o (1 + η) > 1+γ α n, whch mples η > 1 (α o + α n )+γ(1 α o ). (α o + α n )+γα o Ths s dencal o he condon mpled by de(b) > 1. Hence, he necessary and suffcen condons for ndeermnacy are all easer o be sasfed f α o > Robusness The necessary and suffcen condons show ha ndeermnacy s easer o occur he larger he share of he mpored producon facor n aggregae oupu s. However, he above resuls are obaned under he assumpon ha he supply of foregn facor s perfecly elasc. Ths secon examnes he robusness of our resul when mperfecly elasc supply of he mpored facor o s allowed. To ncorporae a less elasc supply of foregn facors, we assume ha ol s suppled by a monopols foregn counry whose objecve funcon s o maxmze prof: Π f = p o o b 1+ζ o1+ζ, where he cos funcon of ol producon s concave (ζ 0). Gven he nverse aggregae demand funcon of ol from home counry, p o = αoy o, prof maxmzaon of he foregn counry leads o he followng frs order condon: y α o o = boζ. Ths mples ha he supply curve for ol s gven by p o = b α o o ζ, where 1/ζ measures he elascy of supply. 9

11 I s easy o show ha when supply mees demand, he home counry s reduced form producon funcon becomes: y = Ak α k (1+η) 1 α o(1+η) 1+ζ αn(1+η) 1 α o(1+η) 1+ζ n, (8) where A s a producvy parameer ha depends negavely on he cos parameer b. Noehafζ = 0, hen he model s reduced o he prevous one wh perfecly elasc supply of he foregn facor. Clearly, as long as he supply elascy of he foregn facor, 1 ζ, s no oo small (or ζ no oo large), he mplcaon for ndeermnacy s he same: namely, he dependence of producon facors on foregn mpors (α o > 0) makes ndeermnacy easer o occur snce he share parameer of foregn facor n producon connues o magnfy he aggregae reurns o scale of he home counry f ζ < η: α k (1 + η) 1 αo(1+η) 1+ζ + α n(1 + η) 1 αo(1+η) 1+ζ > (1 + η). Hence, as long as ζ < η, ndeermnacy s easer he larger α o s. Noe ha excep for he producon funcon, none of he frs order condons of he prevous model (equaons 1-3) s affeced by he fac ζ > 0. For example, he resource consran of he home counry remans he same as before: c + k +1 (1 δ)k = y p o o =(1 α o )y, n spe of a less elasc supply curve of o. In oher words, he reduced form producon funcon (8) s a suffcen ndcaor for he effec of facor supply elascy on ndeermnacy. 4. Calbraon wh Capacy Ulzaon The above analyss based on a smple benchmark model provdes he essenal undersandng on he mechansm as o how he dependence of producon on foregn mpored facors can ncrease he lkelhood of ndeermnacy under exernales or mperfec compeon. Now we calbrae a more realsc neoclasscal growh model wh varable capal ulzaon, so as o show ha ndeermnacy can easly occur under essenally consan aggregae reurns o scale. Ths s he represenave-agen verson of he model of Wen (1998) 7 n whch a represenave agen chooses sequences of consumpon (c), hours (n), capacy ulzaon (e), and capal accumulaon (k) osolve Ã! X max β log c b n1+γ 1+γ =0 7 For a smple proof for he equvalence beween a represenave-agen verson wh exernaly and a monopolsc-compeon verson of he model wh ncreasng reurns o scale, see Benhabb and Wen (2004). Ths equvalence connues o hold when ceran facors of producon are mpored from foregn counres, as proved n he prevous secons. 10

12 subjec o c +[k +1 (1 δ )k ]+p o = y(e k,n,o ), where he home counry pays he amoun p o o n erms of oupu o foregners o receve he amoun o as facor npus, 8 and where he producon echnology s gven by y(e k,n,o )=Φ (e k ) α k n α n o α o, α k + α n + α o =1; n whch e [0, 1] denoes capal ulzaon rae, and Φ s a measure of producon exernales and s defned as a funcon of average aggregae oupu whch ndvdual frms ake as paramerc: Φ =[(e k ) α k n α n o α o ] η, η 0. The rae of capal deprecaon, δ, s me varable and s endogenously deermned n he model. In parcular, s assumed ha capal deprecaes faser f s used more nensvely: δ = 1 θ eθ, θ > 1; whch mposes a convex cos srucure on capal ulzaon. 9 Proposon 4.1. The necessary and suffcen condons for ndeermnacy under varable capacy ulzaon are gven by η > θ [(1 + γ)(1 α o) βα n ] (1 + γ)α k θβα n +(1+γ)(α k + α o θ) θ(1 + γ) 1 β, (9) 1+β (2(1 + γ)+(1 β)φ) θ 1+η > α n (1 + β)θ α k (1 + β)φ(θ 1) + (2(1 + γ)+(1 β)φ)(α o θ + α k ) ; (10) where φ δ 2 ³(1 (1 + γ) α o ) θ α k 1. Proof. See Appendx We calbrae he model s srucural parameers followng Benhabb and Wen (2004) and Wen (1998). Namely, we se he me perod n he model o a quarer, he me dscounng facor β = 0.99, he seady-sae rae of capal deprecaon δ =0.025 (whch mples θ =1.404), he nverse labor supply elascy γ =0, and he labor elascy of oupu α n =0.7. We also assume ha he supply of he foregn facor of producon s perfecly elasc (.e., ζ =0). Gvenhese parameer values, he followng able shows ha as he share of foregn facor n 8 Noe ha rade s balanced n every perod snce he cos of nermedae goods energy mpors are pad for wh expors of oupu. Hence naonal ncome s gven by y po, whch equals domesc consumpon and capal nvesmen. 9 See Benhabb and Wen (2004) and Wen (1998). 10 Noe ha condons (9) and (10) are dencal n he lm as β 1 (whch mples θ 1 also). 11

13 domesc producon ncreases, he hreshold value of he producon exernaly for nducng ndeermnacy (η ) decreases dramacally. For example, when we ncrease he share parameer of foregn npu α o from zero percen o 10 percen, he reducon n he exernaly s 33%. And f we ncrease he share parameer o 20 percen, hen he reducon n he exernaly s 64%. Table 2. Effec of α o on Indeermnacy Facor Share (α o ) Exernaly (η ) % Reducon of η The above able s based on he assumpon ha he foregn mpored facor s manly a subsue for capal, hence when α o ncreases, α n remans consan bu α k decreases such ha α k + α o remans consan (assumng consan reurns o scale a he frm level). If we assume ha he mpored foregn facor s manly a subsue for labor nsead (.e., α n + α o s fxed), hen a larger α o also mples asmallerη alhough he reducon of exernaly s less dramac as α o ncreases. If we fx heraoofα k /α n whle ncrease α o, hen he reducon n η s beween he frs case and he second case. In he real world, however, a hgher value of α o presumably affecs α k more han affecs α n, snce mpored producon facors or maerals are ofen reaed by he leraure as beer subsues for capal goods han for labor. The model, however, can also be appled o he case of foregn labor. 5. Concluson In hs paper we showed ha dependence of domesc producon on foregn facors can sgnfcanly reduce he requred degree of reurns o scale for ndeermnacy when he supply of foregn facors s suffcenly elasc. As a resul, mulple equlbra and self-fulfllng expecaons drven flucuaons can arse much more easly under very mld exernales or wh essenally consan reurns o scale. 12

14 References [1] C. Azarads, 1981, Self-fulfllng propheces, Journal of Economc Theory 25(3), [2] C. Azarads and R. Guesnere, 1986, Sunspos and cycles, Revew of Economc Sudes 53(5), [3] J. Benhabb and R. Farmer, 1994, Indeermnacy and ncreasng reurns, Journal of Economc Theory 63, [4] J. Benhabb and R. Farmer, 1996, Indeermnacy and secor-specfc exernales, Journal of Moneary Economcs 37, [5] J. Benhabb and K. Nshmura, 1997, Indeermnacy and sunspos wh consan reurns, manuscrp, New York Unversy. [6] J. Benhabb, K. Nshmura, and Q. Meng, 2000, Indeermnacy Under Consan Reurns o Scale n Mulsecor Economes, Economerca (November). [7] J. Benhabb and Y. Wen, 2004, Indeermnacy, aggregae demand, and he real busness cycle, Journal of Moneary Economcs 51(3), [8] R. Benne and R. Farmer, 2000, Indeermnacy wh non-separable uly, Journal of Economc Theory 93, [9] D. Cass and K. Shell, 1983, Do sunspos maer? Journal of Polcal Economy 91, [10] R. Farmer, 1999, Macroeconomcs of Self-fullng Propheces, The MIT Press. [11] R. Farmer and J. T. Guo, 1994, Real busness Cycles and he anmal sprs hypohess, Journal of Economc Theory 63, [12] Q. Meng, 2003, Mulple ransonal growh pahs n endogenously growng open economes, Journal of Economc Theory 108 (2). [13] Q. Meng and A. Velasco, 2003, Indeermnacy n a small open economy wh endogenous labor supply, Economc Theory 22 (3). [14] R. Perl, 1998, Indeermnacy, home producon, and he busness cycle: A calbraed analyss, Journal of Moneary Economcs 41. [15] S. Schm-Grohe, 1997, Comparng four models of aggregae flucuaons due o self-fulfllng expecaons, Journal of Economc Theory 72, [16] K. Shell, 1977, Monnae e allocaon neremporelle, Mmeo, Semnare d Economere Roy-Malnvaud, Cenre Naonal de la Recherche Scenfque, Pars. 13

15 [17] K. Shell, 1987, Sunspo equlbrum, n The New Palgrave: A Dconary of Economcs (J. Eawell, M. Mlgae, and P. Newman, eds.), Vol. 4, New York: Macmllan, [18] K. Shell and B. Smh, 1992, Sunspo Equlbrum, n he New Palgrave Dconary of Money and Fnance (J. Eawell, M. Mlgae, and P. Newman, eds.), Vol. 3, London: Macmllan, 1992, [19] M. Weder M, 1998, Fckle consumers, durable goods, and busness cycles, Journal of Economc Theory 81 (1): Jul. [20] M. Weder M, 2001, Indeermnacy n a small open economy Ramsey growh model, Journal of Economc Theory 98, [21] Y. Wen, 1998, Capacy ulzaon under ncreasng reurns o scale, Journal of Economc Theory 81, [22] Y. Wen, 2001, Undersandng self-fulfllng raonal expecaons equlbra n real busness cycle models, Journal of Economc Dynamcs and Conrol 25, [23] M. Woodford, 1986a, Saonary sunspo equlbra: The case of small flucuaons around a deermnsc seady sae. Unversy of Chcago, Sepember, unpublshed manuscrp. [24] M. Woodford, 1986b, Saonary sunspo equlbra n a fnance consraned economy. Journal of Economc Theory 40, [25] M. Woodford, 1991, Self-fulfllng expecaons and flucuaons n aggregae demand. In: N.G. Mankw and D. Romer (eds.), New Keynesan Economcs: Coordnaon Falures and Real Rgdes, Vol. 2. MIT Press, Massachuses, [26] OECD, 1995, The OECD Inpu-Oupu Daabase, OECD Publcaons and Informaon Cener. 14

16 Appendx 1. Gven B = M1 1 M2, we have de(b) = de(m2) de(m1). Sraghforward re-arrangemen shows ha de(b) = 1 (1 λ) (1 + γ)(1 β(1 δ)) λ(1 a 1+ o ) β 1+γ β(1 δ) a. n Also, gven b11 b B = 12 b 21 b 22 we have r(b) = b 11 + b 22, where b 11 (1+γ)() a n (1+γ)a k h (1+γ)( ) β(1 δ)a n δ s (1+γ)( ) a n δ gves Appendx 2. r(b) =1+de(B)+ = δ (1+γ)a k h s (1 β(1 δ))(an+(1+γ)(a k +a o 1)) (1+γ)( ) a n +1 δ, and b 22 = (1+γ)( ) β(1 δ)a n ³ (1 β(1 δ))(1 + γ) 1 ao a k δ 1 s s 1+γ β(1 δ) a. n Denoe λ as he Lagrangan mulpler for he budge consran, he frs order condons wh respec o {c, n, e, o, k} and he budge consran are gven respecvely by 1 = λ (A) c an γ = λ α n (e k ) αk(1+η) n αn(1+η) 1 o αo(1+η) y α k = e θ k α o y = p o y +1 λ = βλ +1 α k +1 1 k +1 θ eθ +1. Re-arrangemen (B) (C) (D) (E) c + k +1 (1 1 θ eθ )k =(1 α o )y. (F) To smplfy he analyss, we use equaon (C) o subsue ou e n he producon funcon o ge y = Ak α k(1+η)τ k n αn(1+η)τn o αo(1+η)τn (G) θ 1 where τ k θ α k (1+η), τ θ n θ α k (1+η). Nex, we use equaon (D) o subsue ou o n he producon funcon (G) o ge y = Ãk α k(1+η)τ k 1 αo(1+η)τn αn(1+η)τn 1 αo(1+η)τn n. (H) 15

17 Afer smlar subsuons n all equaons, he above equaon sysem s reduced o c = α n y a n 1+γ (A 0 ) c +1 = βc (1 1 θ )α y +1 k +1 (B 0 ) k +1 c + k +1 k =(1 α o α k θ )y (C 0 ) where he producon funcon s gven by (H). Denoe a α k(1+η)τ k 1 α o(1+η)τ n,b = α n (1+η)τ n 1 α o(1+η)τ n, log-lnearze he above equaons (A 0 -C 0 ) around he seady sae and subsue ou c usng (A 0 ), we have he followng smplfed 2-varable sysem: or (1 + β(a 1))ˆk +1 +(βb (1 + γ))ˆn +1 = a ˆk +(b (1 + γ))ˆn ˆk +1 = ˆk + (1 α o ) θ 1 δ(1 + γ)ˆn α k M 1 ˆk+1 ˆn +1 = M 2 ˆk ˆn where 1+β(a M 1 = 1) βb (1 + γ) 1 0 " a b # (1 + γ) M 2 = h 1 (1 α o ) α θ. k 1 δ(1 + γ) Hence, he Jacoban s gven by h 1 (1 α o ) θ B = M1 1 M α 2 = k 1 δ(1 + γ) h (1 β)(1 a ) 1+γ βb 1+γ b +(1+β(a 1)) (1 α o) θ 1+γ βb α k 1 δ(1+γ) whch mples ha he deermnae and he race of B are gven by (afer smplfcaon and re-arrangemen): " de(b) = 1 1+ η(1 + γ)(1 β) τ n # 1 α o(1+η)τ n β 1+γ βb h (1 β)(1 a ) (1 α o ) α θ k 1 δ(1 + γ) r(b) =1+de(B)+ 1+γ βb. Followng he same dscussons n secon 3, can be shown ha he value of η ha sasfes he condon, de(b) > 1, also sasfes he condon, βb > 1+γ, hence he necessary and suffcen condons for ndeermnacy can be lmed o he value of η ha sasfy: de(b) > 1 andr(b) > (1 + de(b)). These wo condons mply he condons n proposon 4.1, 16