CHAPTER II THE BASICS OF INTERTEMPORAL GENERAL EQUILIBRIUM

Transcription

1 file:chp2-v3.word6, 10/13/97 CHAPTER II THE BASICS OF INTERTEMPORAL GENERAL EQUILIBRIUM II.1 Inroducion The purpose of his chaper is o provide he concepual fundamenals of iner emporal general equilibrium which are necessary for he developmen and use of he empirical policy based models presened in he following chapers. These models ypically conain far more economywide and secoral specific deail han can be considered in an analyical model. Moreover, hey aemp o accoun for he effecs of policy inervenions embodied in he vary daa used o obain many of he model parameers. I will be seen ha while he dynamic analyical model developed in his chaper amouns o a skech of he more deailed empirical models, an undersanding of he fundamenals his skech provides is criical in guiding he specificaion, fiing o daa and he inerpreing of he empirical model resuls. We focus on fundamenals of consumpion and capial accumulaion in a neoclassical ineremporal general equilibrium model ha was iniially developed by Frank Ramsey and exended by Cass (1965) and Koopmans (1965), and has now become a key par of he lieraure leading o oher ypes of dynamic general equilibrium models as in Grossman and Helpman (1992) and Borro and Sala-I-Marin (1995). In he nex secion, we begin wih a very simple one secor model of a decenralized closed economy and derive he iner emporal condiions of he equilibrium pah. We specify he model in discree ime. This choice is based on wo reasons: ypically, policy based models are fi o annual daa which makes he ransiion from a coninuous ime analyical model o he specificaion of a discree ime empirical model raher awkward. Second, he compuaion mehod used o solve he empirical model is beer undersood in erms of a discree ime model. The secion concludes wih a discussion of he effecs on invesmen and savings when he economy is small and open. This secion lays he ground work for he nex and key secion which exends he single secor model o a muli-secor closed economy, and hen o a muli-secor small open economy. Wih a number of simple ricks i will be seen ha his exension can be made in a naural and simple way which preserves virually all of he insighs and srucure of he single secor model. The chaper concludes wih a discussion of a number of implicaions of he heory o he developmen of policy models. II.2 The Dynamic General Equilibrium Model of a Closed Economy Frank Ramsey posed he quesion of how much a naion should save and addressed i by solving a model ha is now a prooype for sudying he opimal iner emporal allocaion of resources. For our purposes, we do no presen he sandard Ramsey social planner model. Insead, we use he model o sudy a decenralized (marke ) economy. We begin wih a simple one secor model of a closed economy wih wo facors: labor and 1

2 capial services. The economy is assumed o be perfecly compeiive in he sense ha here exis many households and producers, each of whom make decisions on he assumpion ha prices in each ime period are given and beyond he individual s conrol. There is freedom of enry and exi for firms a each ime period, and profi-seeking aciviies ensure ha profis are eliminaed in he long-run equilibrium. Individuals are assumed o have an infinie horizon over which hey allocae spending o maximize an iner emporal uiliy funcion subjec o an iner emporal budge consrain. Differen form he saic model, households can borrow from and lend o each oher, i.e., hey can allocae heir income over ime. In making hese plans, individual households can be viewed as aking ino accoun he welfare and resources of heir curren and prospecive descendans. The infinie horizon assumpion urns ou o have srong implicaions: ogeher wih he assumpion of compeiive markes, consan reurns o scale in producion, and homogenous agens, he allocaion of resources achieved by a decenralized economy will be he same as ha chosen by a cenral planner who maximizes he uiliy of he represenaive economic agen. Nex, we discuss firs he behavior of he households and firms, and hen derive he condiions for equilibrium and define he seady sae for he economy. II.2.1 The Households II Seup of he Model There are many idenical households. Alhough each household conains one or more aduls wih a finie life span, hey ake accoun of he welfare and resources of heir acual and prospecive descendans by providing forward ransfers o heir children. This exended and immoral household corresponds o finie-lived individuals who are conneced via a paern of iner generaional ransfers. Hence, as a household, i faces an infinie horizon. The size of he exended family is viewed o grow indefiniely a he exogenous and consan rae n for simplificaion. If we normalize he number of aduls a ime 0 o uniy, hen he household size a ime -- which corresponds o he adul populaion -- is L = (1+ n ). If C is oal consumpion a ime, hen c = C /L is consumpion per person. A household s overall (iner emporal) uiliy, U, is given by U 0 = ' ( =0! 1+ n $ # & u(c ). " 1+! % This formulaion assumes ha he household s uiliy a ime 0 is a weighed sum of all fuure flows of uiliy, u(c ). The funcion u(c ) -- ofen called he feliciy funcion -- relaes he flows of uiliy per person o he quaniy of consumpion per person, c. The funcion u(c ) is nonnegaive, coninuous, concave and increasing. The concaviy assumpion generaes an eagerness o smooh consumpion over ime: households prefer a relaively uniform paern o one in which c is very low in some periods and very high in ohers. We also assume ha u(c ) saisfies Inada condiions: u (c ) as c 0, and u (c ) 0 as c. 2

3 The muliplicaion of u(c ) in Eq. (2.1) by family size L = (1 + n), represens he adding up of uile for all family members alive a ime. The oher muliplier, (1 +ρ) -, involves he rae of ime preference, and ρ > 0. A posiive ρ means ha uiles are valued less he laer hey are received. We assume ρ > n, which implies ha U in Eq. (2.1) is bounded if c is consan over ime. One reason for ρ o be posiive is ha uiles far in he fuure correspond o consumpion of laer generaions. Suppose ha, saring from a ime period a which he levels of consumpion per person in each generaion are he same, parens prefer a uni of heir own consumpion o a uni of heir children s consumpion. This parenal selfishness corresponds o ρ > 0 in Eq. (2.1). In a fuller specificaion, we would also disinguish he rae a which individuals discoun heir own flow of uiliy a differen ime periods (for which ρ = 0 migh apply) from he rae ha applies over generaions. Eq. (2.1) assumes, only for reasons of racabiliy, ha he discoun rae wihin a person s lifeime is he same as ha across generaions. Households own labor and capial and earn income by rening he services of hese resources o firms. A any ime period, hey decide he proporion of income o save and consume. They can save by eiher accumulaing capial or lending o oher households. Because he wo forms of asses, capial and loans, are assumed o be perfec subsiues as sores of value, households are indifferen o he composiion of heir wealh. Thus, he ineres rae on deb mus be equal o he renal rae on capial. Households are presumed o have perfec foresigh and hence know boh curren and fuure values of prices, and ake hem as given. Since here is only one aggregae good in he economy, he curren price for his good is normalized o one. More formally, le {w, r }, = 1, 2,, be he sequence of wages and renal raes. Then, given his sequence, each household maximizes a any ime period (2.1') U 0 = ' ( =s! 1+ n $ # & " 1+! % )s u(c ); subjec o he budge consrain: (2.2) *a = w +r a ) c ) na +1, for all, k 0 given, where a = k - b p. Nonhuman wealh, which is given by a, is equal o he holding of capial, k, minus family deb, b p, and hence, he change in a, i.e., Δa = a +1 - a, is household savings. a is measured in real erms, ha is, in uni of consumables. As each household is assumed o have 1 uni of labor, he income from labor services is w. The equaion saes ha asses per person rise wih per capia income, w + ra, fall wih per capia consumpion, c, and fall because of expansion of he populaion in accordance wih na. (We can derive Eq. (2.2) by imposing he budge consrain in erms of he change in he level of oal asses, A +1 - A, and hen noing ha for given A, a +1 = A +1 /L +1 income falls a rae n due o 3

4 populaion growh.) A any ime, he household supplies boh capial and labor services inelasically. The curren sock of capial is he resul of his/her previous decisions and is given a ime. Thus, as menioned, he decision he household has o make a each ime period is how much o consume and save. However, a consrain needs o be placed on he household s borrowing decisions. If each household can borrow an unlimied amoun a he going ineres rae, r, hen i has an incenive o pursue a form of chain leer or Ponzi game. I is opimal for households o borrow sufficienly o mainain a level of consumpion such ha he marginal uiliy of consumpion equals zero (or an infinie level of consumpion if marginal uiliy is always posiive) and o le he dynamic budge consrain deermine he dynamic behavior of a, wealh. From he budge consrain i follows ha his pah of consumpion will lead o higher and higher levels of borrowing (negaive a), wih borrowing being used o mee ineres paymens on he exising deb. For example, household can borrow, say $1, o finance curren consumpion and hen use fuure borrowing o roll over he principal and pay all ineres charges. In his case, he household s deb grows forever a he rae of ineres, r. Since he principal never ges repaid, oday s added consumpion of $1 is effecively free. Thus, a household ha can borrow in his manner would be able o finance an arbirary high level of consumpion in perpeuiy. To rule ou chain-leer possibiliies, we need herefore an addiional condiion ha prevens household from choosing such a pah, wih an exploding deb relaive o he size of he household. A he same ime we do no wan o impose a condiion ha rules ou emporary indebedness. A naural condiion is o assume ha he deb marke imposes a consrain on he amoun of borrowing. The appropriae resricion urns ou o be ha he presen value of asses mus be asympoically nonnegaive, ha is, (2.3) lim =0 (1+ n) a " 0.! ( 1+ r s ) s=0 This condiion means ha, in he long run, a household s deb per person (negaive value of a ) canno grow as fas as (1+r )/(1+n) - 1, so ha he level of deb canno grow as fas as r. This resricion rules ou he ype of chain-leer finance ha we described above. We show laer how he credi-marke consrain expressed in Eq. (2.3) emerges naurally from he marke equilibrium. II Firs-Order Condiions The condiion characerizing he maximizaion of Eq. (2.1) subjec o Eq. (2.2) and Eq. (2.3), can be derived from he Lagrangean 4

5 (2.4) L = ' ( =0! 1+ n $ ' # & u( c ) + (" [ w + (1+ r )a ) c ) (1+ n)a +1 ]; " 1+! % =0 where λ is he shadow price of income. Differeniaing Eq. (2.4) wih respec o c and c -1 obains (2.5)!L! c = 0 " ( 1+n 1+! ) # u ( c ) = " ; (2.6)!L = 0 "! c $1 $1 ( 1+n 1+! ) u #( c $1) = " $1. Differeniaing Eq. (2.4) wih respec o a yields (2.7)!L! a = 0 "! = -1! 1+n 1+ r. Rearranging Eqs.(2.5)-(2.7) obains he following necessary and sufficien condiions, he famous Euler Equaion: (2.8) u!( c ) u!( c -1 ) = (1+!) (1+ r ). The Euler equaion describes a necessary condiion ha has o be saisfied along any equilibrium pah. I is he sandard efficiency condiion ha he marginal rae of subsiuion -- he raio of marginal uiliy of consumpion a ime and -1, -- is equal o he raio of he ime discoun rae over he rae of reurn o is savings a hese wo adjacen ime periods, i.e., (1+ρ)/(1+r ). Households prefer o consume oday raher han omorrow for wo reasons. Firs, he erm (1+ρ) appears because households discoun fuure uiliy a his rae. Second, suppose c > c -1, i.e., consumpion is high oday relaive o yeserday. Since agens like o smooh consumpion over ime -- because u (c) < 0 -- hey would like o even ou he flow by bring some fuure consumpion forward o he presen. If agens are opimizing, hen Eq.(2.8) says ha hey have equae he wo raes of reurn and are herefore indifferen a he margin beween consuming and saving. If r = ρ, households would selec a fla consumpion profile. Households are willing o depar from his fla paern and sacrifice some consumpion oday for more consumpion omorrow only if hey are compensaed by an ineres rae, r, ha sufficienly exceeds ρ. The elasiciy of marginal uiliy, which someime is called he reciprocal of he elasiciy of ineremporal subsiuion, gives he required amoun of compensaion o forgo a uni of oday s consumpion for a uni of fuure consumpion. In oher words, i deermines he amoun by 5

6 which r mus exceed ρ. If we follow he common pracice of assuming he funcional form (2.9) u(c) = c 1-! - 1, for! > 0,! 1, 1 -! = lnc, for! = 1. Then, he opimal condiion from Eq.(2.8) simplifies o (2.10) c c = -1 1/!! 1+ r $ # &. " 1+ " % The elasiciy of subsiuion for his uiliy funcion is he consan σ = 1/θ. Hence, his form is called he consan ineremporal elasiciy of subsiuion (CIES) uiliy funcion. The relaion beween r and ρ deermines wheher households choose a paern of per capia consumpion ha rises over ime, says consan, or falls over ime, while a lower willingness o subsiue ineremporally (a higher value of θ and hence a lower value of σ) implies a smaller responsiveness of change in c o he gap of r and ρ. II The Transversaliy Condiion From Lagragian equaion (2.4), he ransversaliy condiion is (2.11) lim!"! a = 0. Eq.(2.11) saes ha he value of he household s asses -- which equals he quaniy a imes he shadow price of income λ -- mus approach 0 as ime approaches infiniy. If we hink of infiniy loosely as he end of he planning horizon, hen he inuiion is ha opimizing agens do no wan o have any valuable asses lef unconsumed a he las period. The shadow price of income evolves overime in accordance wih! = (1+ n)! 0.!(1+ r s ) s=0 If we subsiue he resul for λ ino Eq. (2.11) and ignore he erm λ 0, which is equal o u (c 0 ) and is posiive, hen he ransversaliy condiion becomes (2.12) lim!" ( 1+ n) a = 0. # ( 1+ r s ) s=0 6

7 This equaion implies ha he quaniy of asses per person, a, does no grow asympoically a rae as high as r - n or, equivalenly, ha he level of asses does no grow a a rae as high as r. I would be subopimal for households o accumulae posiive asses forever a he rae r or higher, because uiliy would increase if hese asses were insead consumed in finie ime. In he case of borrowing, where a is negaive, infinie-lived households would like o violae Eq.(2.12) by borrowing and never making paymens for principal or ineres. Tha is why we needed he consrain in Eq.(2.3) o rule ou chain-leer finance, ha is, schemes in which a household s deb grows forever a he rae r or higher. In order o borrow on a perpeual basis, households would have o find willing lenders; ha is, oher households ha are willing o hold posiive asses ha grow a he rae r or higher. Bu we already know from he ransversaliy condiion ha hese oher households are unwilling o absorb asses asympoically a his high a rae. Therefore, in equilibrium, each household will be unable o borrow in a chain-leer fashion. In oher words, he inequaliy resricion shown in Eq.(2.3) is no arbirary and would, in fac, be imposed in equilibrium by he credi marke. Faced by his consrain, he bes hing ha opimizing households can do is o saisfy he condiion shown in Eq.(2.12). Tha is, his equaliy holds wheher a is posiive or negaive. II The Role of Expecaions and Consumpion Funcion The Euler equaion given in Eq.(2.8) or (2.10) gives he rae of change of consumpion as a funcion of variables known a he curren momen. I could be inerpreed as suggesing ha households need no form expecaions of fuure variables in making heir consumpion/saving decisions and ha he assumpion of perfec foresigh is no necessary. However, if we derive an ineremporal budge consrain from he household s budge consrain for each ime period, hen we can see ha he household canno plan wihou knowing he enire pah of boh he wage and he ineres rae. To derive household ineremporal budge consrain, we firs re-wrie household curren budge consrain in he following form: (2.2') a = ( 1+n ) a +1! ( 1+ r ) 1 ( 1+ r ) ( w! c ). We ierae Eq.(2.2') forward ino ime. By successive subsiuion ino he las erm on he righhand side, and sar his process from ime period 0, we deduce ha: 7

8 a 0 = ( 1+ n) 1+ r 0 ( ) a 1! 1 ( ( 1+ r 0 ) w 0! c 0 ), ( ) ( ) a! 1 2 ( 1+ r 1 ) w! c 1 1 a 1 = 1+ n 1+ r 1 ( ), ( 1+ n) 2 a 0 = ( 1+ r 0 ) 1+ r 1 " " " ( 1+ n) T (2.13) a 0 = a T!1 T! # 1+ rs s=0 ( ) ( ) a 2! 1 ( 1+ r 0 ) w 0! c 0 ( ) T $ 1+ n =0 s=0 ( )! ( w! c ). # ( 1+ r s ) ( 1+ n) ( 1+ r 0 ) 1+ r 1 ( ), ( ) w 1! c 1 Taking he limi as T, he firs erm on he righ-hand side becomes zero by he ransversaliy condiion. Thus we obain: (2.13') ( 1+ n) " # =0! ( 1+ r s ) s=0 c = ( 1+ n) " # =0! ( 1+ r s ) s=0 w + a 0 Expecaions hus are crucial o he allocaion of resources in he decenralized economy. In erms of he Euler equaion, i only deermines he rae of change no he level of consumpion. Alhough i is difficul in general o solve explicily for he level of consumpion, his can be done easily when he uiliy funcion is of he form defined in Eq.(2.9). For a given value of iniial consumpion c 0, we can deduce he similar process from Eq.(2.9) as we did for Eq.(2.2') o obain " (2.14) c =! % $ ( 1+ r s ) s=0 ' $ ( 1+!) ' # $ &' 1/! c 0. Replacing c in he ineremporal budge consrain equaion (2.13') gives he value of c 0 consisen wih Euler equaion and he budge consrain: 8

9 $ & (2.15) c 0 =! 0& %& " # =0! s=0 ' ( 1+ n) ) w + a 0 ), ( 1+ r s ) () ( ) ( ) 1/# " 1+ n where!!1 0 = # % $ 1+ r s s=0 1+ " & ' =0 ( ) 1/#! 1 ( ) *. Combining Eqs. (2.14) and (2.15), i is clear ha consumpion a each ime period is a linear funcion of human (he firs erm in he bracke of Eq. (2.15)) and nonhuman wealh, a 0. The parameer, β 0, is he propensiy o consume ou of wealh and depends for he mos par on he pah of ineres raes. An increase in he ineres rae, given wealh, has wo effecs. The firs is o make laer consumpion more aracive: his is he subsiuion effec. The second is o allow for higher consumpion now han laer: his is he income effec. In general, he ne effec on he marginal propensiy o consume is ambiguous. For he logarihmic uiliy funcion, however 1/θ = 1, he wo effecs cancel; he propensiy o consume is simplified o (ρ - n)/(1 + ρ) and is independen of he pah of ineres rae, i.e., -1 0! = ' ( =0! 1+ n $ # & " 1+ " % = 1+" " ) n. In general, he pah of ineres raes affec boh he marginal propensiy o consumer ou of wealh and he value of wealh iself, hrough he erm of human wealh. Expecaions of wages also affec c 0 hrough his erm. Given hese expecaions, households decide how much o consume and save. This in urn deermines capial accumulaion and he sequence of facor prices. II.2.2 The Firms There are many idenical firms, each wih he idenical consan reurns o scale echnology. For simpliciy, we assume no physical depreciaion of capial. We inroduce a posiive capial depreciaion rae laer. Each firm rens he services of capial and labor from households o produce oupu. The oupu is a homogeneous good ha can be consumed or invesed o creae new unis of physical capial. The consan reurns o scale assumpion means ha he number of firms is of no consequence, provided he firms behave compeiively, aking prices (he real wage and renal rae on capial) facing hem as given. Firms choose quaniies of labor and capial as inpus o maximize heir profis of each ime period 1. Each ime period s profi is defined as he difference beween he revenue firms receive 1 Tha here are many alernaive ways o describe he firms behavior and hey have he same oucome in 9

10 and coss hey occur. Technology is characerized by where Y is he flow of oupu, K is capial inpu (in unis of commodiies), and, chronological ime, represens he effecs of exogenous echnological progress. Y exhibis consan reurns o scale in L and K, and each inpu exhibis posiive and diminishing marginal produc. The exogenous echnological progress is presumed o ake he labor-augmening form, i.e., if we define ˆL! L " A, as he effecive amoun of labor inpu, where A is he level of he echnology and grows a he consan rae g 0. Hence, A = (1 + g). Then, he producion funcion can be wrien as In per capia erms: where ŷ! Y / ˆL, and ˆk! K / ˆL. Y = F( ˆL, K ). ŷ = f ( ˆk ), Thus, he firm s problem can be defined as follows: Max < L, K > ˆL [f ( ˆk )! w! r ˆk ] ; = 1, 2,.... From he firs order condiion for he firm s problem, i can be readily verified ha he marginal producs of he facors are given by: erms of resource allocaion under he perfec foresigh assumpion. For example, firms can own he capial and finance invesmen by eiher borrowing or issuing equiy. Or, insead of operaing wih spo facor markes, he economy may operae in he Arrow-Debreu complee marke framework in which markes for curren and all fuure commodiies are open a he beginning of ime; all conracs are made hen, and he res of hisory merely execues here conracs. In he following chapers, we will presen some differen model seups for firms behavior dending on he purpose of he policy analysis. 10

11 (2.16) f!( ˆk ) = r, (2.17) [f ˆk ( ) " ˆk f! ( ˆk )] 1+ g ( ) = w. II.2.3 The Equilibrium By combining he behavior of households and firms, we can analyze he srucure of a compeiive marke equilibrium. We assume in his secion of he chaper ha he economy is closed; here is no inernaional commodiy rade and households canno borrow from or lend o foreigners. Thus in equilibrium (in each ime period), aggregae privae deb b p mus always be equal o zero. Even hough each household presumes i can freely borrow and lend, in equilibrium no household has an incenive o do so. Thus, a = k, and Δ a = Δk,, i.e., he only asses (nonhuman wealh) owned by households as a group is he sock of capial. The amoun saved by households equals he amoun invesed on capial creaion. Also, assuming for he momen ha he rae of exogenous echnological progress, g, is zero, using Eq. (2.2) for w and r and subsiuing in Eqs.(2.8), (2.16) and (2.17) gives (2.18) f ( k ) = c +! k + nk +1, (2.19) ( ) ( ) u" c +1 u" c = 1+! 1+ f " k +1 ( ). Eqs.(2.18) and (2.19) form a sysem of wo difference equaions in c and k. This sysem, ogeher wih he iniial condiion, k 0, and he ransversaliy condiion, deermines he ime pahs of c and k, i.e., he dynamic behavior of he economy. We can wrie he ransversaliy condiion in erms of k by subsiuing a = k ino Eq.(2.12): (2.12') lim!" s=0 ( 1+ n) # ( 1+ f $ ( k s )) k = 0. II.2.4 The Seady Sae In he seady sae, boh he per capia capial sock and he level of consumpion per capia are consan. This implies by Eq.(2.19) ha he marginal produc of capial (ineres rae) in he seady sae is equal o he rae of ime preference, i.e., r = ρ. Corresponding o he seady sae capial sock, k*, is he seady sae level of per capia consumpion implied by Eq (2.18) 11

12 * c = f ( k * ) - n k * The ransiional dynamics of his model are ypically shown using a phase diagram drawn in (k, c) space. Since we uilize a discree raher han a coninuous ime horizon model, we skip he discussion of ransiional dynamics. This ask insead consiues a major focus of he following chapers. II.3 Invesmen and Saving in an Open Economy II.3.1 Seup of he Model In his secion we exend he closed economy model o he open economy. The world now conains many counries. For convenience, we hink of one of hese counries, counry i, as he home counry, and he res as foreign. Households and firms in all counries have idenical preferences and echnologies which are of he same form as above. Domesic and foreign claims on capial are assumed o be perfec subsiues as sores of values; hence, each mus pay he same rae of reurn, r. Since loans and claims on capial in any counry are sill assumed o be perfec subsiue as sores of value, he variable r is he single world ineres rae. For he same reasons as in he closed economy model, borrowing and lending among each counry s households, i.e., he family deb, b p, is zero. If he home counry s capial per person, k i, exceeds per capia asses a i, hen he difference, k i - a i, mus correspond o ne claims by foreigners on domesic economy. Conversely, if a i exceeds k i, hen he difference, a i - k i, mus correspond o ne claims by domesic residens on foreign economies. We define b i o be he home counry s ne deb o foreigners (Foreign claims on he home counry ne of domesic claims on foreign counries), hen (2.20) b i = k i! a i. Equivalenly, domesic asses equal domesic capial less he foreign deb: a i = k i - b i. The model sill conains only a single good, bu foreigners can buy domesic oupu, and home counry residens can buy foreign oupu. The only funcion of inernaional rade in his model is o allow domesic producion o diverge from domesic expendiure on consumpion and invesmen. In oher words, we consider he ineremporal aspecs of inernaional rade bu neglec he implicaions for paerns of specializaion in producion of differen goods. We coninue o assume ha labor is immobile; ha is, residens of he home counry canno emigrae nor can foreigners immigrae o home counry (or immigrae). 12

13 The budge consrain for he represenaive household in counry i is he same as ha given in Eq. (2.2): (2.2")! a = w + ra " c " na +1, for all, k 0, b 0 given. For noaional convenience, we drop counry subscrip i for all variables in Eq. (2.2") and in he following equaions. The only new elemen is ha r is he world ineres rae. We assume ha he form of households preferences are as in Eq. (2.9). Therefore, he firsorder condiion for consumpion is idenical o he one shown in Eq. (2.10): (2.10') c c = -1 1! ( 1+r 1+" ). The ransversaliy condiion again requires a o grow asympoically a a rae less han r - n, as in Eq. (2.12). The opimizaion condiion for firms, once again, enail equaliy beween he marginal producs and he facor prices (Eqs. [2.16] and [2.17]): (2.16') f!( k ) = r, (2.17) f k ( ) " k f!( k ) = w. If we subsiue for w from Eq. (2.17) ino Eq. (2.2") and use Eq. (2.16'), hen he change in asses per capia is derived as (2.21)! a = f ( k ) " r( k - a ) " c " na +1, for all, b 0 and k 0 given. Noe from Eq. (2.20) ha (k - a ) = b, and foreign deb equals zero for a closed economy. For he purpose of policy applicaions discussed in he following chapers, i is useful o derive a simply relaion among he curren accoun defici, savings and invesmen from household s curren budge consrain in Eq. (2.21). Consequenly, a brief refresher in naional income accouning ideniies and definiions may be useful a his poin. The curren accoun defici is equal o he change in foreign deb, which in urn equal o ineres paymens minus ne expors of goods, he rade surplus (X):! B = rb " X. 13

14 A counry s ne expor equals is oupu minus consumpion and invesmen, i.e., X = F ( L, K )! C! " K. GDP is equal o is oupu, F(L, K ), while GNP equals GDP minus ineres paymens on foreign deb, i.e., F(L, K ) - rb. Savings is equal o GNP minus consumpions: S = F ( L, K )! rb! so ha! B = curren accoun defici =! K " S. C = " K + X! rb = " K! " B, If L grows a he rae n, hen he per capia curren accoun defici equals Δb + nb. I is easy o see ha per capia saving is exacly Δa defined in Eq. (2.21), i.e., s = Δa. Then, i is again a Ponzi-like soluion. Households should borrow unil he marginal uiliy of consumpion is equal o zero, and hen borrow furher o mee ineres paymens on heir foreign deb. I is unlikely ha he lenders are willing o coninue lending if he home counry s households only means of paying off heir deb were o borrow more. Tha is why he ransversaliy condiion in Eq. (2.12) is needed o resric foreign deb, b so ha i does no grow asympoically a a rae higher han r - n. The ransversaliy condiion can be defined as a resricion on foreign deb: (2.22) lim!" ( 1+ n) b = 0. # ( 1+ r s ) s=0 II.3.2 Behavior of a Small Economy s Capial Sock and Consumpion For a small open economy, we can rea he pah of r as exogenous. Given his pah, Eqs. (2.16') and (2.17) deermine he pahs of k and w, wihou regard o he choice of consumpion and saving by he home counry households. Given he ime pah for w, Eqs. (2.2") and (2.10') and he ransversaliy condiion deermine he pahs of c and a. Finally, he pahs of k and a prescribe he behavior of he ne foreign deb, b, from Eq. (2.20). For simpliciy, we assume ha he world ineres rae equals a consan r. In effec, he world economy is in a seady sae of he naure derived for a single closed economy. If r is consan, hen Eq (2.16') implies ha k equals a consan, which saisfied f (k) = r. In oher words, he speed of convergence from any iniial value, k 0, o k is infinie or insananeous. An excess of k over k 0 causes capial o flow in from he res of he world so ha he gap disappears insananeously. Similarly, an excess of k 0 over k induces a massive ouflow of capial. This counerfacual predicion of an infinie speed of convergence for k is one of he problemaic implicaions of he open-economy version of he Ramsey model. Laer, we discuss several ways of overcome his shor coming. 14

15 If r = ρ, hen Eq. (2.10') implies consumpion, c, is consan. Oherwise, c asympoically approaches 0 if r < ρ. The home counry households borrow o enjoy a high level of consumpion early on because hey are impaien in he sense ha ρ > r. Bu hey have o pay he price laer in he form of lowering heir consumpion. (If ρ < r, he domesic economy would evenually accumulae enough asses o violae he small-counry assumpion ha we made). The resul is ha c ends o zero, which is anoher problemaic feaure of he open-economy Ramsey model. If r = ρ, hen a is consan by Eq. (2.2"). Oherwise, a evenually falls o 0, so ha b = k, if a 0 > 0. Subsequenly, a becomes negaive; ha is, he home counry becomes a debor no only in he sense of no owning is capial sock bu also of borrowing agains he presen value of is wage income as collaeral. In oher words, an impaien counry asympoically morgages all is capial and all of is labor income. This counerfacual behavior of asses is ye anoher difficuly wih he model. The specificaions of capial adjusmen cos or a consrain on inernaional borrowing has been used o address his problem We also discuss hese issues in laer chapers. II.4 From he Single Secor Model o he Muli Secor Model An aggregae single secor dynamic general equilibrium model focuses on resource or wealh allocaion over ime wih respec o changes in he price/ineres rae over ime. In he real economy, resources are also allocaed cross secors in response o a policy which may cause a change in he relaive prices of differen commodiies. Hence, for many purposes, a single secor model is inadequae for policy analysis. For example, a highly aggregaed single secor model does no allow us o invesigae rade policy effecs on a counry s rade paern. Also, i is impossible o use a single secor model o analyze he differen effecs of some domesic policies, such as fiscal and indusrial policy on differen indusries, or differen groups of households. In his sub-secion, we exend he sandard single secor dynamic model ino a wo secor model and derive he major linkages beween inra- and iner-emporal equilibrium. The wo secor model can be easily exended o a muli-secor model provided i is appropriae o assume ha here exiss more han wo facors of producion in he economy. II.4.1 The households As in secion II.3.1.1, we assume ha here are many idenical households, he number of which grow a he consan rae, n, and he iniial period s number is normalized o uniy. The wo goods produced correspond o he wo secors of he economy. While good 1 can be eiher consumed by household or used for invesmen o form new capial, good 2 can only be consumed as a consumpion good. The ineremporal uiliy funcion is given by 15

16 ' U 0 = ( =0! 1+ n $ # & " 1+! % u( c 1,,c 2, ), where c i, represens consumpion of good i a ime. The feliciy funcion u(.) has he same properies as defined in he above wo secions, ha is, i is increasing in c i and concave. As in he previous secions, households own labor and capial and make decisions on he consumpion of wo final goods and savings o maximize heir ineremporal uiliy. The economy is assumed o be a closed and borrowing and lending among households are ignored. Thus, he curren budge consrain for each household a ime is:! k = w + r k " 1, c " pc 2, " nk +1, for all, k 0 given. Differen from he budge consrain defined in Eq. (2.2) for he single secor model, commodiy price, p = p 2, /p 1,, expressed as he relaive price of good 2, appears in he equaion. From he corresponding Lagragian problem, as in Secion II.3.1, we obain (2.23)!L! c = 0 " # 1+ n & % ( 1, $ 1+ p '!u! c 1, =! ; (2.24)!L = 0 "! c 2, # 1+ n & % ( $ 1+ p '!u! c 2, =! p, from he derivaives wih respec o c 1, and c 2,. From he raio of Eq. (2.24) o Eq.(2.23), we obain he relaionship beween marginal rae of subsiuion and relaive prices of he wo final goods:!u/! c 2,!u/! c 1, = p. This equaion is idenical o ha of a saic wo good model. I implies ha he marginal rae of subsiuion beween wo consumpion goods wihin each ime period is equal o he relaive price of he respecive goods in ha period. This imporan inra-emporal propery of he household s problem allows us o separae he inra-emporal problem from he iner-emporal problem by consrucing an aggregaed consumpion good, c, as a composie of he wo final goods. The consumpion/saving decision based on he aggregae good is achieved hrough ineremporal opimizaion behavior, as in he single good model, while he choices of he quaniies of he wo final commodiies are made wihin each period, given he level of he aggregae good 2. 2 When discussing he se-up of he equaions o solve an empirical model in he following 16

17 By consrucing he aggregae good, c, he represenaive household, in fac, faces a wo-level uiliy maximizaion problem. In he firs level, he household maximizes an ineremporal uiliy by choosing c and savings, while in he second level (wihin period), i choose c 1, and c 2, for given c. Formally, he household problem becomes: (2.25) Max U 0 = ' ( =0! 1+ n $ # & " 1+! % u( c ) (2.26) s.. )k = w + r k * pc c * nk +1 ; (2.27) Min c 1, + p c 2, (2.28) s.. c = h( c 1,,c 2, ); = 0, 1, 2,.... where pc is he shadow price for c, he funcion h(.) is a consruced echnology mapping he wo final goods ino a composie. h(.) is assumed o have consan reurns o scale and increasing in c i. The homogeneiy propery of funcion h(.) deermines ha pc c = c 1, + p c 2,, and pc depends only on p, i.e., (2.29) pc = pc( p ). The Euler equaion, obained from he firs-order condiion of he household problem becomes: (2.30) u!(c ) u!(c -1 ) = " 1+! % pc $ ' # 1+ r & pc. -1 The only difference beween he Euler equaion in (2.30) and ha in Eq.(2.8) of Secion II.2.1 is ha pc, he shadow price of c, appears in he righ hand side of he equaion. In Eq.(2.30), pc is endogenously deermined by he relaive price of he wo final goods, while in Eq.(2.8) he price for he single commodiy is normalized o uniy as he subsiuion of consumpion occurs only over ime. The shadow price, pc, links he wo levels of he maximizaion problem ogeher. Thus, he opimal level of demand for final goods, c 1, and c 2, canno be independenly deermined by he second level (inra emporal) maximizaion problem wihou opimizing household s ineremporal uiliy. In oher words, hese wo levels of maximizaion problem have o be solved simulaneously. Thus, he demand for he wo commodiies can be derived as funcions of pc and c by using Shephard s lemma: chapers, i is usefull o place equaions ino blocks as his srucure implies. 17

18 (2.31) c 2, = p c!( p )c ; (2.32) c 1, = pc( p )c " p c!(p )c. II.4.2 The Firms The wo final good producion secors are characerized by many price aking compeiive firms. Firms wihin each secor produce a single oupu wih he same consan reurns o scale echnology (while he echnologies are differen beween secors). Each firm rens he services of labor and capial which are perfecly mobile cross secors. For simpliciy, we assume no physical depreciaion of capial and he rae of echnological progress is zero. While he oupu of secor 1 can be consumed or invesed o creae new unis of physical capial, oupu of secor 2 is only consumed by households. Firms make similar decision as hey did in Secion II.2.2, i.e., hey maximize profis a each ime period. Formally, heir echnologies are i Y i, = F ( L i,, K i,); i = 1,2. Le l i, = L i, /L be he share of oal labor employed by secor I, i.e., l 1, + l 2, = 1, and y i, denoe per capia oupu of secor i, i.e., y i, = Y i, /L. Then in each secor, firm s echnology is characerized by: i (2.33) y i, = l i, f ( k i, ); i = 1,2, where k i, = K i, /L i,. From he firms cos minimizaion wihin each ime period, we have: (2.34) i i g ( w, r ) y! Min " l, k # i, l ( w + r k i,) s.. l i, f i ( k i, ) = y i ; i = 1, 2; = 1,2,.... II.4.3 The Equilibrium II The inra-emporal equilibrium Since firms only face an inra-emporal (wihin ime period) problem, he inra-emporal equilibrium, for a close economy, can be discussed independenly given he household s ineremporal choices on aggregae consumpion and savings. Much of he analysis in he following discussion dealing wih firm s problem in he inraemporal equilibrium involves he use of dualiy heory. This approach is convenien as i ofen 18

19 yields resuls in a simple way by pushing unwaned variables behind he scene. Thus, we define he miminum cos for firms given he echnology as a funcion of he oupu and facor prices, raher han dealing wih he condiions of profi maximizaion expressed in erms of producion funcion. The equilibrium condiions for he producion secors wih posiive oupu require: (2.35) 1 g ( w, r ) = 1, (2.36) 2 g ( w, r ) = p. Solving Eqs. (2.35) and (2.36) for w and r as funcions of p yields (2.37) w = w( p ), (2.37) r = r( p ). The facor marke clearing condiions, (2.39)! 1 g ( w, r ) y 1, +! g 2 ( w, r ) y 2, = 1,! w! w (2.40)! 1 g ( w, r ) y 1, +! g 2 ( w, r ) y 2, = k,! r! r mus hold for he posiive w and r. By hese wo equaions, ogeher wih Eqs.(2.37) - (2.38) which define facor renal raes as funcions of oupu prices, y i, can be expressed as a funcion of oupu price, p and he level of oal capial sock per capia, k, i.e., i (2.41) y i, = y ( p, k ); i = 1,2. From Eqs.(2.35) - (2.36) and (2.41), we can solve he inpu demand for labor and capial in per capia erms, l i, and k i,, as funcions of p and k using Shephards lemma, i.e., (2.42) l i, =! g i y =! w i, l i ( p, k ); (2.43) k i, =! g i! r y i, l i, = l i ( p, k ); i = 1, 2. 19

20 As good 2 is a pure consumpion good, he supply of good 2 equals o he household s consumpion demand for i. Thus, from Eqs.(2.31) and (2.41), he marke clearing condiion for good 2 requires, 2 y ( p, k ) = p c!( p )c. From his equaion, we obain p as a funcion of k and c, i.e., (2.44) p = p( k,c ). II The iner-emporal equilibrium Equaion (2.44) implies ha he inra-emporal equilibrium is condiional on he wo variables, k and c. In a saic general equilibrium model, as he sock of capial is fixed, Eqs. (2.37)-(2.44) are sufficien for deermining an equilibrium. However, in he dynamic model, he sock of capial accumulaes over ime and is deermined by he household s iner-emporal behavior. Hence, wihou deriving his iner-emporal equilibrium condiion, an inra-emporal equilibrium canno be verified. Recalling in Secion II.2.3, he dynamic equilibrium depends on a sysem of wo difference equaions in he consumpion and capial sock. In he muli secor model, we can derive a similar sysem of wo difference equaions in he aggregae consumpion and he capial sock. Since good 2 is a pure consumpion good, an increase in he sock of capial is equivalen o he difference beween he oupu of good 1 and he amoun consumed by he households, i.e., 1 1 (1+n) k +1! k = y! c. From Eq.(2.32), we know ha c 1 = pc(p )c - pc (p )c. Using his relaionship, ogeher wih Eq. (2.41) for y 1 and Eq. (2.44) for he oupu price, p, we derive he firs difference equaion as follows: (2.45) (1+n) k +1 = k + 1 y [p( k, c ), k ]![pc[p( k, c )] c - p c "[p( k, c )] c ]. Subsiuing Eq.(2.38) for he ineres rae (he capial renal rae), r, and Eq. (2.29) for he shadow price of he aggregae consumpion, pc, ino Eq. (2.30) yields he second differen equaion: (2.46) u!( c +1) u! c ( ) = 1 +! 1 + r[p( k, c ), pc[p( k +1, c +1)]. k ] pc[p( k, c )] Eqs.(2.45) and (2.46) form a sysem of wo difference equaions in c and k. This sysem, 20

21 ogeher wih he iniial condiion, k 0, and he ransversaliy condiion, deermines he ime pahs of c and k, i.e., he dynamic behavior of he economy. Once he dynamic pahs of c and k are defined, he equilibrium soluions for he oher variables derived in Eqs.(2.29), (2.31) - (2.32), (2.37) - (2.38) and (2.41) - (2.44) in each ime period are obained simulaneously. Corresponding o he seady sae per capia capial sock, k *, is he seady sae level of per capia aggregae consumpion, and he seady sae levels of oher variables in per capia erms, such as secoral oupus, secoral inpu demand for labor and capial, he demand for each final good, he wage and capial renal raes (ineres rae), and he relaive price implied by Eqs.(2.29), (2.31) - (2.32), (2.37) - (2.38), and (2.41) - (2.45). II.5 The Small Open Economy wih Two Producion Secors In he previous secion, inernaional rade in commodiies was no permied. In he single commodiy model of Secion II.3.2, inernaional rade only occurred in erms of borrowing and lending. These flows served o smooh domesic consumpion by allowing consumpion o diverge from domesic producion. We now reurn o his srucure for he case of a wo commodiy small-open economy o show he consequen implicaions o ou-of-seady sae adjusmen and he resuling rade paerns. As in Secion II.3, we define a as he per person s asses in he home counry, and b = k - a as foreign deb, and we assume ha labor is immobile inernaionally. The budge consrain for he represenaive household in he home counry becomes: and! a = w + ra " pcc " na +1, pcc = c 1, + pc 2,, where r is now he world ineres rae. A common r presumes ha claims on he foreign asses and domesic capial are perfec subsiues as sores of values. p is he world relaive price for good 2. Given an exogenous p, he shadow price for overall consumpion, pc, which is dependen only on p, is also exogenously deermined. From he household s uiliy maximizaion problem, an Euler equaion resul similar o Eq (2.10') is obained: (2.47) c c -1 1! = ( 1+r 1+" ), where pc no longer appears since i is ime independen. Consumpion is deermined by: (2.48) c 2, = p c!(p)c ; (2.49) c 1, = pc(p)c " p c!(p)c. 21

22 The ransversaliy condiion again requires a o grow asympoically a a rae less han r - n, as in Eq.(2.12). The firm s opimizaion condiions in he open economy are idenical o he close economy, i.e., heir opimizaion choices are emporally independen. From Secion II.2.3, we know ha when r = ρ, he speed of convergence from any iniial value of k 0 o he seady sae k * is infinie. In his secion, we will firs check wheher his propery sill holds in a muli secor small open economy. Similar o Secion II.2.3, if r = ρ, hen, Eq.(2.47) implies he aggregae consumpion, c, is consan. Also, from Eqs.(2.48) - (2.49), c 1 and c 2 are consan. To check wheher capial can slowly adjus over ime, we consider wo siuaions for he home counry economy in he following subsecions: (a) he economy produces boh goods, and (b) he economy is specialized in he producion of a single good. II.5.1 The Small Open Economy wihou Producion Specializaion The equilibrium condiions for an economy o produce boh goods are defined in Eqs. (2.35) - (2.36) of he previous secion. These condiions also imply ha w is exogenously given when he economy is a small open economy. To saisfy he firs-order condiions for firm s profi maximizaion, he following equaions are required: (2.50) 1 f! ( k 1 ) = r; (2.51) 2 pf! ( k 2 ) = r; (2.52) 1 f! ( k 1 ) " k 1 f 1! (k 1 ) = w; (2.53) 2 p[ f! ( k 2 ) " k 2 f 2! (k 2 ) = w. Wih consan r and w, k 1 and k 2 have o be consan and, hence, k = k 1 + k 2 is consan. From he facor marke equilibrium condiions, we can derive he following equaions for l 1 and l 2 : 22

23 2 (2.54) l 1= k! k k 2! k ; 1 (2.55) l 2 = k! k 1 k 2! k. 1 Hence, l 1 and l 2 are also consan. Now, i is clear ha we reach a similar resul as in he single secor small open economy model (Secion II.3.2): an excess of k over iniial level of k 0 causes capial o flow in from he res of he world while an excess of k 0 over k causes an insananeous ouflow of capial o close he k 0 - k gap. II.5.2 The Small Open Economy wih Producion Specializaion To consider specializaion, we assume, arbirarily, ha secor 1 is capial inensive and secor 2 is labor inensive, wih no facor inensiy reversal, i.e., k 1, > k 2,, for all. Also, we assume ha, facing a consan world price, he seady sae level of capial sock per capia in he home economy is lower han he capial/labor raio in secor 2, such ha secor 1, he capial inensive secor, is nonprofiable, i.e., he counry is specialized in he producion of good 2. Saisfying his assumpion requires: (2.56) g 1 ( w, r) > 1,! y 1, = 0; (2.57) g 2 ( w, r) = p,! y 2, > 0. Eq. (2.47) implies ha as he uni cos for producing good 1 is greaer han he good 1's world price (which is normalized o uniy), y 1 is no produced in he home counry. Once he economy is specialized in he producion of good 2, household consumpion and invesmen demand for good 1 have o be saisfied hrough inernaional rade. Thus, he seady sae rade paern for his small economy is o expor good 2 and o impor good 1, i.e., (2.58) m 1! c 1 + nk > 0; (2.59) m 2! c 2 " y 2 < 0. However, if an ou-of-seady sae pah exiss, i is possible for he economy o impor boh goods during early periods of ransiion. If his were o occur, he economy would need o borrow from he res of he world, in which case, i would also need o expor more of good 2 in he fuure o cover he ineres paymen on he deb ousanding. As here is now only one secor o produce he final good, he seup of he model for he firm s problem becomes he same as we discussed in Secion II.3.2. The opimizaion condiion for 23

24 firms becomes: (2.60) p f! 2 (k) = r; 2 (2.61) p[ f (k) " kf 2! (k)]= w. For he same reason as discussed in Secion II.3.2.3, k and w are consan o saisfy Eqs.(2.60)- (2.61) wih fixed r and p. Now if we assume ha he iniial level of capial sock in he home counry, k 0, is differen from he seady sae level k, hen, we arrive a he same resul: he gap beween iniial level of capial sock, k 0, and seady sae level k insananeously disappears hrough inernaional borrowing or leanding. Likewise, if we assume insead ha he economy is specialized in he producion of good 1, we will arrive a he same conclusion, namely, he gap of k 0 over k disappears insananeously hrough inernaional borrowing or lending. I is easy o exend he wo secor model o muli-secors and reach similar conclusions, i.e., if he world capial marke is perfec, and he economy is small in he sense i canno affec world marke price and ineres rae, insananeous convergence of k occurs, and hence for he economy as well. To obain ou-of-seady sae dynamics, oher assumpions mus be relaxed. For example, eiher some form of capial adjusmen cos or imperfec capial mobiliy mus be considered (references..). Anoher possibiliy, as we show in he empirical model developed in Chaper 3, ou-of-seady sae dynamics can be he resul when imperfec subsiuabiliy beween he same home and foreign produced goods is assumed. 24