Exogenous Growth Under International Capital and Labor Mobility

Transcription

1 Exogenous Growh Under Inernaional Capial and Labor Mobiliy The world economy is characerized by diverse levels of income and paerns of growh across counries and across ime. To undersand he evoluion of such diversiy and wheher here are endencies oward cross-counry equaliy, we have o sudy he mechanics of he growh process in each counry individually and in he world equilibrium a large. In order o provide a sysemaic analysis of such process, we devoe his chaper o a simplified world where he engine of growh is exogenously deermined. As a naural exension, his growh engine will be endogenized in he nex chaper. Evidenly, facor mobiliy could poenially influence he convergence/ divergence endencies across counries wihin a growing world economy. We herefore focus our analysis on he roles ha capial mobiliy and labor mobiliy can play in such a process. In addiion, we examine he poenial welfare gains ha capial and labor mobiliy can generae. Also sudied are he effecs of facor mobiliy on he speed of adjusmen o long run equilibria and on he rae of convergence (if any) across counries The Closed Economy In his secion, we provide an exposiion of he sandard growh model ha emphasizes he role of saving and physical capial accumulaion in driving shor run oupu growh and ha of (exogenous) human capial accumulaion in driving long run growh. 1 Consider a closed economy wih a Cobb-Douglas echnology ha ransforms physical - 1 -

2 capial (K ) and human capial (H ) ino a single composie good (Y ): Y ' A K 1&" H ". (12.1) The produciviy level A is, in general, ime-varying and can be a source of growh and flucuaions. To make hings simple a his sage, we assume ha A = A, a deerminisic consan. In Secion 12.4 below, we will assume i o follow a sochasic process in order o 2 analyze growh under uncerainy. Human capial H is a produc of he size of he labor force N (reaed synonymously wih he size of populaion, assuming consan labor force paricipaion and inelasic work hours) and heir skill level (h ). An imporan feaure of his model is consan 3 reurns o scale echnology. While increasing reurns o scale will give rise o unbounded growh (which is hus inconsisen wih he exisence of a sable long run equilibrium), decreasing reurns o scale canno generae susainable long run growh. [See Appendix A.] respecively: Physical and human capial are accumulaed according o he following laws of moion K %1 ' I % (1&* k )K. (12.2) where I is gross invesmen in, and * k he rae of depreciaion of, physical capial. H %1 ' (1%g H )H. (12.3) where g is he exogenous rae of growh of human capial. Since H = N h and boh N and h H are assumed o grow a exogenous consan raes g and g, we can decompose (13.3) ino N N h +1 = (1+g )N and h = (1+g )h, wih (1+g )(1+g ) = 1+g. N +1 h N h H - 2 -

3 The represenaive consumer is he owner of boh physical and human capial. He/she accumulaes hese wo forms of capial over ime, and supplies hem as facor inpus o he represenaive firm for producion a he compeiive wage (w ) and renal (r k) raes, period by d d period. To derive is demand for physical capial (K ) and human capial (H ), he firm in urn solves a saic profi maximizaion problem 4 : Max {K d,h d } A(K d )1&" (H d )" & w H d & r k K d This yields he sandard marginal produciviy-facor price relaions: w ' "Ax 1&" and r k ' (1&")Ax &", (12.4) where x is he raio of physical capial o human capial (K /H ). Preferences of he dynasic head of he family are given by an isoelasic uiliy funcion: U ' j 4 '0$ N c 1&F &1 1&F, (12.5) where 0 < $ < 1 is he subjecive discoun facor, and F > 0 he reciprocal of he ineremporal elasiciy of subsiuion in consumpion (c). In he limiing case where F = 1, he uiliy funcion is logarihmic. The household derives wage income from he supply of human capial (wh ) and renal income from he supply of physical capial (r K). He/she can eiher consume (N c ) or save k hese incomes in he form of physical capial (K! (1!* )K ). He/she can also borrow B a +1 k +1 he rae of ineres r in any period, and repay (1+r )B for his/her borrowing B from he +1 5 previous period!1. The household's ne saving is herefore given by [K +1! (1!* k)k ]! [B+1! (1+r )B ]. His/her period- budge consrain can hus be wrien as - 3 -

4 [K %1 & (1&* k )K ] ' w H % r k K % [B %1 & (1%r (12.6) The consumer chooses {c,k,b } o maximize (12.5) subjec o (12.2), (12.3), and (12.6). The firs order condiions for his problem imply: R B%1 ' 1 $ c %1 c F ' R k%1, (12.7) where R = 1+r and R = 1+r!*. Recall ha, as in Chaper 5, in making his B+1 +1 k+1 k+1 k borrowing decision, he consumer equaes he gross rae of ineres (R ) o his/her ineremporal marginal rae of subsiuion (IMRS). In addiion, as an owner of physical capial, he/she also equaes he reurn on physical capial (R k+1 B+1 ) o he same IMRS. Arbirage hrough capial marke forces will drive equaliy beween he ineres rae and renal rae on capial (ne of depreciaion, i.e., r = r!*. +1 k+1 k The equilibrium values of he wage rae, renal rae, and ineres rae are deermined by d d he following marke clearing condiions: H = H in he labor marke, K = K in he physical 6 capial marke, and B +1 = 0 in he financial capial marke. By he Walras law, he consumer budge consrain (12.6), he profi-maximizing condiions (12.4), and hese marke clearing condiions ogeher imply he economy-wide resource consrain: N c % K %1 & (1&* k )K ' F(K,H ). (12.8) Tha is, oupu is spli beween consumpion and saving/invesmen

5 The Mechanics of Economic Growh The dynamics of our example economy are driven by he wo fundamenal laws of moion (12.2) and (12.3), which can be combined o yield a single difference equaion in he physical capial-human capial raio (or he capials raio, X ): X %1 ' A 1%g H s X 1&" % 1&* k 1%g H X, (12.9) where s is he saving rae a ime, defined as he raio beween saving S (gross and ne) and oupu Y (= I /Y since S = I in he closed economy equilibrium). In general, he equilibrium saving rae is deermined by solving he consumer's ineremporal opimizaion problem and imposing he marke clearing condiions. We spell ou he dynamics of he sysem in Appendix B, and illusrae how he saving rae is derived in an example economy in Appendix C. 1!" We can express per capia oupu as y (= Y /N ) = A(X ) h, where h = h 0 (1+g h ). Evidenly, he dynamic pah of y depends on ha of X and h. In paricular, he growh rae of y (g y) can be approximaed (using ln(1+g) = g) by ln y %1 ' (1&")ln X %1 % g y X h A ' (1&")ln s 1%g X &"% 1&* k % g H 1%g h H ' (1&")6ln[As X &" %(1&* k )]&g N > % "g h. (12.10) Observe ha g depends posiively on he saving rae (s ) and he skill growh rae (g ), and y h negaively on he capials raio (X ) and he populaion growh rae N(g ). If, however, he economy converges o a long run ime-invarian equilibrium where X = X so ha ln(x /X ) = 0, hen g y will converge o g h

6 Seady Sae Growh and Transiional Dynamics: Policy Implicaions While equaion (12.9) describes he enire growh pah of he capials raio X, i can convenienly be divided ino wo componens: he ransiion (o seady sae) pah and he seady sae growh pah. Seady sae growh is defined as he paricular paern of growh where growh raes of all variables are consan over ime, bu possibly differen from one anoher. (I is someimes called `balanced growh' as well.) The consan reurns o scale assumpion, combined wih isoelasic preferences (which implies equaliy beween he average and marginal propensiies ^ o consume), are necessary for he exisence of such long run equilibrium. Wih X = X +1 = X ^ and s = s +1 = s in he seady sae, he capials raio and he level of per capia oupu can be expressed as (posiive) funcions of he saving rae as follows: ˆX ' sa g H %* k 1/" and ŷ ' s g H %* k 1&" " A 1 " h 0 (1%g h ). ^ ^ ^ ^ ^ The capial-(raw) labor raio k (= K /N ) is equal o Xh = Xh 0 (1+g h ). Observe ha y and k boh grow a he same rae g along he seady sae growh pah, implying ha he capial-oupu h ^ ^ ^ raio mus be consan. Since he rae of reurn on capial (r k) is given by (1!")y /k, i mus also ^ ^ ^ be consan. Indeed, hese seady sae properies of y, k, and r k are all consisen wih a se of empirical regulariies abou paerns of growh esablished by Kaldor (1963). Anoher componen of he growh pah exhibis ransiional dynamics described by equaion ^ (12.9) for X /= X +1 /= X and porrayed by Figure For any iniial socks of physical and human capial (hence X ), he capials raio in he nex period (X ) can be read off he concave 0 1 curve for he corresponding equilibrium value of he saving rae (s, say s'). Given X, he

7 capials raio in he period afer (X ) can be obained from a poin on he concave curve 2 correspending o he equilibrium saving rae in period 1 (s, say, s"). Evenually, he capials raio 1 0 will converge o he seady sae a he inersecion beween he 45 line and he concave curve ^ wih consan saving rae s (poin A). As shown in he example in Appendix C, he Solow (1956) model is a special case of our model (when F = * = 1) where he saving rae is ime-invarian k ^ ^ (say, s). In his case, for X < X, X > X (as indicaed by he arrows poining o he righ); +1 ^ and for X > X, X +1 < X (as indicaed by he lefward poining arrows). [inser Figure 12.1 here] In Appendix B, we derive wo measures of he speed of convergence o he long run equilibrium. The firs one measures he half-life of he dynamical sysem, i.e., he ime i akes o reach half of he disance beween he iniial posiion and he seady sae. The second one, analogous o Barro and Sala-i-Marin (1992), measures he raio beween he deviaions of oupu from is seady sae value in wo consecuive periods. [Explain dependence of (1+g )8 on underlying parameers.] h From he argumens jus presened, i should now be eviden ha, in he exogenous growh paradigm, policies which arge saving raes (such as axaion of capial income) can influence growh only along he ransiion pah, bu no in he long run seady sae equilibrium. They can noneheless affec he long run level of oupu per capia. In oher words, hese policies only have level effecs, bu no growh effecs The Open Economy: Capial Mobiliy From a global perspecive, here is an issue as o wheher counries wih differen levels - 7 -

8 of iniial income will converge (wih income gaps narrowing) over ime. We have jus shown ha income convergence can occur even among isolaed economies if he long run (capial accoun auarky) seady sae posiions hey converge o are similar. This enails similar echnologies (A, ", and * k) and preferences ($ and F), and mos imporanly similar exogenous growh raes (g H, or g N and g h). When he economies are open, i is more likely ha such converegence will ake place because openness can poenially enhance echnology ransmission. Moreover, when capial markes are inegraed ino he world capial marke, he process of convergence (is speed as well as he paricular ransiion pah) will likely be alered. Inernaional borrowing and lending can faciliae consumpion smoohing and creae addiional invesmen opporuniies, which will improve he efficiency in he allocaion of consumpion over ime and he allocaion of capial across counries. For hese reasons, he capial marke inegraion may generae welfare gains as well. In his secion, herefore, we inroduce capial mobiliy in his secion o sudy is role for he growh process and for inernaional convergence of income levels. Perfec Capial Mobiliy Consider inegraion of our example economy ino he world capial marke. Assume free rade in commodiies. Since we have a single composie good here, alhough a counry eiher expors or impors (bu will no do boh) wihin any given period, i can definiely engage in ineremporal rade in his single commodiy (exporing in some periods and imporing in ohers). This implies mobiliy in financial capial (in he form of various financial securiies). Naurally, rade in goods can occur in he form of physical capial as well. In oher words, physical capial - 8 -

9 also becomes inernaionally mobile. Accordingly, we modify he consumer budge consrain (12.6) as follows: H % r( k K H( N c % (K H H( %K %1 % (B H %1 ) & (1&* )(K H H( %K ) %1 k H( %B%1 &B(H ) & (1%r )(B H %1 %B(H ) & (12.11) H H K and B denoe he domesic consumer's claims on physical capial residing in he home counry H* H* and domesic deb respecively. Likewise, K and B sand for he domesic consumer's claims on physical capial residing in he foreign counry and foreign deb. The domesic rae of ineres * * and renal rae are denoed by r and r, and heir foreign counerpars by r and r respecively. modified. k Similarly, he resource consrain of he domesic economy (12.8) will have o be k N c % (K H H( %K %1 H( %K ) ) & (1&* )(K H %1 k ) % r ( K H( & r k k K (H % (B H( %1 &B(H ) & %1 [(1%r( )B H( &(1 (12.12) H *H where K = K + K, he laer being he sock of foreign direc invesmen in he domesic economy. We can rearrange erms in (12.12) in order o derive he counry's balance of paymens accouns CA + KA = 0 as follows: [F(K,H )%(r B (H &r ( B H( )%(r( K H( k ' &[K H( %1 &(1&* k CA ' GNP & N C & I H &r k K(H KA ' &(FDI % FPI ) H( )K ] % [(B(H )] & N C & [K H &(1&* )K H ], an %1 k %1 &B(H )&(B H( %1 &B H( )]. Here, CA sands for he curren accoun balance, KA he capial accoun balance, FDI foreign direc invesmen, and FPI foreign porfolio invesmen

10 In his deerminisic seing, capial flows will be unidirecional, he direcion being deermined by he relaive magniudes of he domesic and foreign raes of ineres. Furhermore, * * since r = r k! * k and r = r k! * k in equilibrium, physical capial and financial capial are perfec subsiues. Assume wihou loss of generaliy ha r > r, so r > r. Then, B = 0 and K = 0. * * *H H* k k To highligh he role of capial mobiliy for inernaional convergence of per capia oupu levels, we assume ha he res of he world is iniially in a long run seady sae growh equilibrium of he sor sudied in he previous secion, whereas he home economy sars iniially * ^ * wih a lower physical capial-human capial raio (X 0 < X 0 = X ). Assume furher ha he res of he world is large so ha he home counry is a price-aker * ^ *!" in he world marke. Capial marke inegraion hen ensures ha r = r (= (1!")A(1X )! * from (13.4)). As a resul, he capial inflow will generae an immediae convergence of he ^ * ^ * capials raio (X 1 = 1X ) and hence convergence of per capia GDPs (y = y ). From (13.7), one can verify ha he consumpion growh rae will converge a he same ime o is long run value, g. Per capia GNPs and consumpion will no converge, however, because he capial inflow h effecively equalizes labor income beween he home counry and he res of he world, while he former sars wih a lower K. Evidenly, he home economy reaps all he benefis from he 0 ineremporal rade!!since prices in he res of he world remain unchanged while, for he domesic economy, he iniial marginal produciviy of capial ne of depreciaion exceeds he * world rae of ineres. Therefore, he consumpion raio c /c will iniially rise, and hen say consan hereafer (given equaliy beween heir consumpion growh raes). 7 Free capial flows mus generae welfare gains from ineremporal rade, akin o he k

11 sandard gains from rade argumen. Obviously, he magniude of he gains is direcly relaed o he difference in he iniial capials raios beween he home counry and he res of he world. (Recall ha his is assumed o be he only source of heerogeneiy in our analysis.) In Appendix D, we compue such gains and relae hem o he iniial cross-counry difference. These gains are measured in uiliy erms. An alernaive welfare concep T, measured in erms of compensaing variaions in consumpion, is defined implicily by j 4 '0$ U(c auarky (1%T)) ' j 4 '0$ U(c kmobiliy ). ^ [calculae numerical values of T as a funcion of Y 0/Y 0] Consrained Capial Mobiliy Free capial mobiliy is an exreme siuaion based on ideal marke srucure wih full informaion and absence of risk, defaul, and ime inconsisen behavior (boh for privae agens and governmens). When such elemens are aken ino accoun, borrowing and lending may be subjec o significan consrains and regulaions. To accoun for hese possibiliies in our aggregaive model, we now inroduce an upper bound on financial capial flows. No such bound is imposed, however, on foreign direc invesmen. We assume ha households (in heir capaciy as owners of he firms) can only borrow up o a cerain limi. The limi is he collaeral based on he ownership by he domesic households H 8 of he domesic firm's ne worh, which is equal o K. Recall, however, ha an increase in domesic consumpion can come from hree sources: inernaional borrowing, domesic oupu ne of reurns o FDI, and reducion in invesmen by domesic residens in domesic capial

12 Therefore, if borrowing from abroad is consrained, domesic consumers can sill offse he effec of his consrain by reducing heir invesmen in domesic capial. Only when a lower bound on ha form of invesmen is imposed will he consumpion boom following he open-up of he inernaional capial marke become consrained. We herefore make he realisic assumpion ha gross invesmen of his kind be non-negaive. (Effecively, his consrain will eliminae he possibiliy of using he domesic firm's capial sock, which is parly owned by foreigners, as an addiional collaeral for borrowing from abroad o finance domesic consumpion.) If binding, hese wo consrains ogeher wih he resource consrain (12.12) will imply ' F(K,H ) & r k K(H & r ( k K H ' w H % (r k &r( k )K (12.12) A he same ime, he domesic rae of ineres will no be se equal o he world rae of ineres because of he borrowing consrain. From he ineremporal condiion (12.7), we have c * 1/F /c > [$(1+r )]. In comparison o he free capial mobiliy case, herefore, while he jump +1 in consumpion immediaely following he open-up of inernaional capial markes is smaller, he consumpion growh rae is higher as long as he consrains remain binding. Thus, while he borrowing consrain lengehens he ransiional dynamics, i generaes a more acceleraed rae of growh in consumpion and income during ha phase compared o he long run growh rae. Evenually, he consrains will become slack, and he growh rae will converge o he same long run value, g. h This is a sylized example of real world consrains ha end o slow down he adjusmen process. The welfare gains from he opening up of credi markes will evidenly be lower han

13 hose under free capial mobiliy Open Economy: Labor Mobiliy Since boh labor and physical capial are primary inpus in producion, labor mobiliy is expeced o play a similar role in he growh process as capial mobiliy. The skill levels are ypically embodied in labor. Consequenly, we have inernaional ransfer of skills and human capial as an inegral par of labor mobiliy. We now open up he home counry o free labor flows. For simpliciy, we assume ha capial (financial as well as physical) is immobile inernaionally. As before, he res of he world is assumed o be in he long run seady sae iniially, whereas he home economy sars wih a * ^ * * ^ * lower physical capial-human capial raio (X 0 < X 0 = 1X ). This implies ha w 0 < w 0 = w. We reain he assumpion ha he res of he world is large so ha he home counry is a price- ^ * ^ * 1!" aker in he world marke. Labor marke inegraion hen ensures ha w = w (= "A(1X ) from (12.4)). As a resul of he labor ouflow, we obain an immediae convergence of he capials raio ^ * ^ * (X 1 = 1X ) and hence convergence of per capia GDPs (y 1 = y ). Again, he consumpion growh rae will converge a he same ime o is long run value, g because wage rae equalizaion implies h ineres rae equalizaion. Per capia GNPs and consumpion will no converge, however, because he home counry sars ou wih a lower K, hence a lower iniial wealh. Evidenly, he home 0 economy reap all he benefis from he ineremporal rade since prices in he res of he world remain unchanged while, for he domesic economy, he iniial marginal produciviy of labor falls * shor of he world wage rae. Therefore, he consumpion raio c /c will iniially rise, and hen say consan hereafer (given equaliy beween heir consumpion growh raes). Wha we have

14 jus shown is ha labor mobiliy is a perfec subsiue for capial mobiliy, wih he same welfare implicaions. The assumpion of free labor flows absracs, however, from he significan coss of adjusmen associaed wih such mobiliy (e.g., culural, ehnic, family, and oher differences). When such coss are accouned for, labor mobiliy becomes a less efficien convergence mechanism in comparison wih capial mobiliy (even afer incorporaing he above-menioned borrowing consrains) Sochasic Growh So far, we have been assuming ha growh is deerminisic. In order o highligh he effecs of random shocks on he growh dynamics, we inroduce in his secion sochasic elemens. To simplify he analysis, we rever o he closed economy seing developed in Secion The complexiy in characerizing he sochasic dynamics excludes he possibiliy of an analyic inspecion of he growh mechanism excep for relaively low dimensional cases, one of which is eleganly analyzed by Campbell (1994) and is presened in his secion. A he same ime, however, we develop here ools of analysis ha will be useful also for analyzing higher dimensional problems such as hose in an open economy. These echniques will be furher developed in Chaper 13. To incorporae random disurbances ino he model of Secion 12.1, we assume ha he produciviy level a (= ln(a )) follows a firs order auoregressive process: a %1 ' Da %, %1, (12.13)

15 where, is an i.i.d. shock wih zero mean and consan variance, and 0 # D # 1 measures he +1 persisence of he shock. In he presence of such shocks, he expecaion operaor will have o be insered ino equaions (12.5) and (12.7). Defining X = K/H and Z = c/h, we can rewrie he wo dynamic equaions (12.7) and (12.8) in K and c as (B.1) and (B.2) in Appendix B. Togeher wih equaion (12.13), hese wo laws of moion form a sysem of nonlinear expecaional difference equaions in (Z,X,A ). To ge a handle on he soluion, we ake a loglinear approximaion of his nonlinear sysem around is deerminisic seady sae (wih, = 0) o obain a linear sysem of expecaion difference equaions in x (=ln(x )) and z = (ln(z )). Deails of he soluion procedure are laid ou in Appendix E, from which we derive he approximae dynamic behavior of he economy as a funcion of he shock as follows: a ' 1 (1&DL),. (12.14) x %1 ' 0 xa (1&0 xx L)(1&DL),. (12.15) z ' 0 %(0 0 &0 0 )L za zx xa za xx, (1&0 xx L)(1&DL). (12.16) ln y ' (1&")x h %a ' 1%[(1&")0 &0 ]L xa xx (1&0 xx L)(1&DL),. (12.17) Above, `L' denoes he lag operaor, and 0, 0, 0, and 0 represen he parial elasiciies of zx za xx xa

16 z wih respec o x, z wih respec o a, x wih respec o x, and x wih respec o a respecively. The explici expressions for hese elasiciies in erms of he fundamenal parameers are spelled ou in Appendix E. Observe from equaions (12.15)!(12.17) ha he logarihm of he raio beween physical capial and human capial (x ) follows an AR(2) process, while he logarihm of he raio beween concumpion and human capial (z ) as well as he logarim of he raio beween oupu and human capial (ln(y /h )) follow an ARMA(2,1) process. Noe also ha hey have idenical auogressive roos 0 and D. xx We now flesh ou some properies of he parial elasiciies: (1) 0, which measures he effec on curren consumpion of an increase in he capial sock for zx a fixed level of produciviy, naurally does no depend on he persisence of he produciviy shock D; (2) 0, which measures he effec on curren consumpion of a produciviy increase wih a fixed za sock of physical capial, is increasing in D for low values of F bu decreasing for high values of F. The iniuiion behind his is as follows. When subsiuion effecs are weak (small F), he consumer responds mosly o income effecs, which are sronger he more persisen he shock. When subsiuion effecs are srong (high F), a persisen shock which raises he furure rae of ineres will simulae saving a he expense of curren consumpion, making he response in consumpion small; (3) 0, which measures he effec on he fuure level of capial sock of an increase in he curren xx level of capial sock wih a fixed level of produciviy, does no depend on D bu declines wih F. In fac, in our model of Secion (12.1) where echnology is ime-invarian, 1!0 reflecs he xx speed of convergence o he seady sae, similar o our second measure of convergence on p

17 We now urn o he general equilibrium implicaions of he persisence parameer D. Observe ha if produciviy follows a random walk (D = 1), we have 0 +0 = 1 and 0 +0 zx za xx xa = 1. Consequenly, capial, consumpion, and oupu will follow coinegraed random walk processes, so ha he difference beween any wo of hem is saionary

18 Appendix A: Reurns o Scale and Long Run Growh Consider a Cobb-Douglas producion funcion ha allows for decreasing, increasing, and consan reurns o scale. Y ' AK $ H". (A.1) where $+" can be < 1, > 1, or = 1. Following Solow (1956), assume for simpliciy a consan saving rae s, so ha invesmen (I) is equal o sy. The physical capial-human capial raio (X) can be shown o evolve according o: X %1 ' s 1%g H AX $ H"%$&1 % 1&* K 1%g H X. If "+$ > 1 (increasing reurns), we will have unbounded growh in he long run. If "+$ < 1 (decreasing reurns), he economy will vanish over ime

19 Appendix B: Local Dynamics of Our Example Economy The dynamics of our example economy is governed by he sysem of difference equaions ~ (12.2), (12.3), and (12.7) in c, K, H, and A. For he sake of he analysis in Secion 12.4, he produciviy level A is allowed o be ime-varying. Defining X as K /H as before and Z as c /h, we can combine (12.2) and (12.3) as one single difference equaion in X. (1%g H )X %1 ' A X 1&" %(1&* k )X & Z. (B.1) Dividing (12.7) hroughou by h yields a difference equaion in Z. Z &F ' $E [(1%g h )Z %1 ] &F [(1&")A %1 X &" %1 %(1&* k )] (B.2) ^ ^ In he seady sae, A = A = A, X = X = X and Z = Z = Z. These seady sae values are given by ˆX ' (1&")A (1%g h ) F /$&(1&* k ) 1/" and Ẑ ' A ˆX 1&" &(g H %* k ) ˆX. ^ ^ Log-linearizing (B.1) and (B.2) around he seady sae (Z,X) yields %s x (1%g H )E dx %1 X %1 &s x (1&* k ) dx X ' (1&") dx X ' ˆX " A ' 1&" r%* k and s z ' Ẑ Ŷ ' 1&(g H %* k )s x ' 1& (B.3)

20 E dz %1 Z %1 & dz Z ' r%* k 1%r E da %1 A %1 &" dx %1 X %1 (B.4) We can group (B.3) and (B.4) ino a marix difference equaion as follows: dx %1 dx AE X %1 dz %1 ' B X dz % CE da %1 A %1 % D da A, (B.5) Z %1 Z where he marices A, B, C, and D are given by ) 0 ) F, B ' (1&")%s x (1&* k ) &s z 0 F, C ' 0 r%* k 1%r, Equaion (B.5) can be rewrien as: dx %1 dx E X %1 dz %1 ' W X dz % RE da %1 A %1 % Q da A, (B.6) Z %1 Z!1!1!1 where W = A B, R = A C, and Q = A D. To sudy he deerminisic dynamics of his sysem, we rever o he case where A is imeinvarian (hence he expecaion operaors and he erms da /A and da /A can be dropped from (B.6)). We can hen compue he eigenvalues (call hem 8 and 8 ) by solving he equaion

21 de(w!8i) = 0, which is a quadraic equaion in 8. The values of he wo roos 8 and 8 are 1 2 complicaed funcions of he parameers ",F,$,A,*,r, and g. One can show ha 8 and 8 k H 1 2 saisfy he following condiions ' 1%r 1%g H, and 8 1 %8 2 ' 1% 1 1%g H (1%r)% " F r%* k 1&" &(g H %* k ) r%* k 1%r. Of he wo eigenvalues of he fundamenal marix W, one exceeds uniy (unsable roo) and he oher is less han uniy (sable roo, call i 8). Imposing he ransversaliy condiion: 1!F $(1+g ) = (1+g )/(1+r) < 1, we can eliminae he unsable roo from he soluion o obain H he following. H x & ˆx ' (x 0 & ˆx)8, and (B.7) z &ẑ ' (z 0 &ẑ)8. (B.8) where x = ln(x) and z = ln(z). diagram'. 9 The dynamics in he viciniy of he seady sae can be porrayed graphically by a `phase [inser `phase diagram'] The saionary schedules corresponding o x = x and z = z are given by he verical and he ^ ^ invered-u curves respecively. The seady sae (z,x) is aained a he inersecion of hese schedules. The unsable rajecories diverge from he viciniy of his seady sae in he norhwes and souheas direcions, while he sable rajecories approach he seady from he norheas and

22 souhwes direcions. The equilibrium ransiional dynamics correspond o movemens along he sable pahs, described by equaions (B.7) and (B.8). As a measure of he speed of adjusmen o he seady sae, we use he concep of half life * ( ), defined as x (& ˆx x 0 & ˆx ' 1 2 ' z (&ẑ z 0 &ẑ. * * Subsiuing from (A2.3) and/or (A2.4) and solving for, we ge =!ln(2)/ln(8). ^ o y!y. ^ Alernaively, we can measure he speed of convergence from he implied raio of y +1!y+1 y %1 &ŷ %1 ' (1%g y &ŷ h ) x 1&" & %1 ˆx1&" x 1&" & ˆx 1&". (1%g h ) [1%(1&")x ]&[1%(1&")ˆx] %1 ' (1%g [1%(1&")x ]&[1%(1&")ˆx] h )8. 1!" 1!" where we have used y = A(x ) h 0(1+g h) in he firs equaliy and he approximaion x = 1+(1!")x in he second equaliy. This is he discree ime analogue of he convergence rae ($) in Barro and Sala-i-Marin (1992)'s coninuous ime seup, where ln y ŷ ' e &$ ln y 0 ŷ

23 Appendix C: An Example Economy wih Full-Fledged Soluion In order o highligh he mechanics of growh in his model, we specialize for simpliciy by seing F = * = 1. From equaions (12.4) and (12.7), we ge k c %1 $c ' (1&")AX &" %1 ' (1&") Y %1 K %1. (C.1) We can wrie: c = (1!s )Y and K = s Y. Subsiuing hese ino (12.7)' yields +1 1&s %1 $(1&s ) ' 1&" s. This can be resaed as a firs order difference equaion in s, wih an unsable roo. Thus, he unique (economically plausible) soluion is a consan saving rae s = $(1!"). Subsiuing his saving rae ino (12.9) gives he following fundamenal difference equaion in X. X %1 ' $(1&")A 1%g H X 1&". (C.2) We display his dynamic equaion graphically in Figure The concave curve, depicing he o righ hand side of (B.2), inersecs he 45 line a wo poins o produce wo seady saes, a he origin (unsable) and E (sable). For any given iniial value X > 0, he economy mus converge 0 o he long run equilibrium poin E along he rajecories as indicaed by he arrows. A E, he ^ seady sae value of he raio beween he wo capial socks is given by 1X = 1/" {[$(1!")A]/(1+g H)}. Given his closed form soluion, i is sraighforward o compue he consumer's uiliy along he enire growh pah. Recursive subsiuion of (B.2) implies

24 x ' 1&(1&") " Subsiuing his ino c = (1!s)Y = [1!$(1!")]AX ino he uiliy funcion, we ge ln [1&$(1&")]A 1%g H % (1&") x 0. 1!" 0 h h (1+g ) and hen he resuling expression % 1 1&$ 1% U ' j 4 '0 ln(c ) ' 1&" 1&$(1&") x 0 1&" 1&$(1&") ln{[1&$(1&")]a}%ln(h )%$g & 1&" 0 h 1&$(1&") g. H This welfare measure will be used in he capial mobiliy secion o evaluae he gains from ineremporal rade

25 Appendix D: Welfare Gains from Free Capial Flows To evaluae he welfare gains from capial flows, we firs compue he equilibrium consumpion pah under such regime. From he ineremporal condiion (12.7), we have c = * c 0[$(1+r )], where c 0 can be solved from he consumer's presen value budge consrain, obainable by consolidaing he flow budge consrain in (12.11). j 4 '0 N c (1%r ( ) ' j 4 '0 Y (1%r ( ) % K 0. As we make clear in he ex, he capials raio will jump o is seady sae in one period. Consequenly, oupu per capia is given by Y ' AX 1&" h ' AX 1&" 0 h 0 ' 0 A( ˆX) 1&" h 0 (1%g h ) > 0 * Subsiuing his and c = c 0[$(1+r )] ino he presen value budge consrain, we have c 0 ' [1&$(1%g N )] X 0 h 0 % AX 1&" 0 h 0 % A ˆX 1&" h 0 1%r ( r ( &g h. Subsiuing ino he uiliy funcion yields U ' j 4 '0 ln(c ) 1 1&$ ln(1&$) % ln X 0 h 0 % AX 1&" 0 h 0 % A ˆX 1&" h 0 1%r ( r ( &g h % $ 1&$ ln[$(1%r( )] where r ( ' (1&")A [1&$(1&")]A 1%g H 1/" &

26 Appendix E: Sochasic Dynamics We follow Campbell (1994) by solving he equilibrium dynamics of he sysem of equaions (B.3), (B.4), and (12.13) by he mehod of undeermined coefficiens. We guess a soluion for z (= ln(z )) of he form: z ' 0 zx x %0 za a, (E.1) where 0 and 0 are unknown fixed coefficiens, and a = ln(a ). To verify his soluion, zx za subsiue (E.1) ino (B.3) and (B.4) and rearrange o ge he following: x %1 ' 0 xx x % 0 xa a, where 0 xx ' 1%r 1%g H % (g H %* k )& r%* k 1&" 0 zx (E.2) and 0 xa ' r%* k (1&")(1%g H ) % (g H %* k )& r%* k 1&" 0 za. E (x %1 &x )%0 za E (a %1 &a ) ' F r%* k 1%r E (a %1 &x %1 (E.3) Subsiuing (E.2) ino (E.3) and using he fac ha Ea +1 = Da from (12.13), we arrive a one equaion in x and a : )0 zx x %[0 zx 0 xa %0 za (D&1)]a ' F r%* k 1%r [(D&0 xa )a (E.4) Recall from (E.2) ha 0 is a linear funcion of 0 and 0 is a linear funcion of 0. xx zx xa za Comparing he coefficiens on x on he wo sides of equaion (E.4), we can obain a quadraic

27 equaion in 0 as follows: zx Q zx % Q 1 0 zx % Q 0 ' 0, where Q 2 ' (g H %* k )& r%* k 1&", Q 1 ' r&g H %F r%* k 1%g H 1%r Q 0 ' (g H %* k )& r%* k 1&", and F(r%* k ) 2 (1&")(1%r). (E.5) Imposing he ransversaliy condiion (which implies ha r > g H), we know ha 0 zx mus be posiive. Taking only he posiive roo, we have 0 zx ' & 1 2Q 2 Q 1 % Q 2 1 &4Q 0 Q 2. (E.6) Given his soluion for 0 zx, we can compare he coefficiens on a on he wo sides of equaion (E.4) o solve for 0 : za a ' & 0 zx (r%* k ) (1&")(1%g H ) % F r%* k 1%r (D&1)% (g H %* k )& r%* k 1&" D& r%* k (1&")(1%g H ) 0 zx %F r%* k 1%r (E.7) The las wo coefficiens 0 xx and 0 xa can be backed ou from (E.2), givem (E.6) and (E.7)

28 Problems 1. Consider he closed economy example in Appendix C (wih F = * = 1). Suppose here k is a ime-invarian ax (J ) levied on capial income (r K) and he ax proceeds are rebaed as lump k k sum ransfers o he households. Show how his ax rae will affec he saving rae (s ), he speed of adjusmen o he long run equilibrium, and he levels and raes of growh of per capia oupu in ha equilibrium. 2. In he previous problem, how will your answers be affeced if he ax proceeds (JkrkK ) are used o finance governmen spending (g ) in each period? 3. Consider he model wih consrained capial mobiliy in Secion Calculae he jump in he iniial level of consumpion resuling from he opening up of he world capial marke, in he presence of he wo consrains described in he secion. Compare his wih he corresponding jump in iniial consumpion in he free capial mobiliy case. 4. Consider he case of labor mobiliy described in Secion (a) Specify he consumer budge consrain and resource consrain, and derive he law of moion for he capials raio (x, redined as he raio beween physical capial and domesically employed labor). (b) Consider a labor migraion quoa se by he receiving counry (he res of he world. Wha is he quoa level (Q in each period ) which makes he consrained labor mobiliy regime equivalen o he consrained capial mobiliy regime?

29 References Barro, Rober J., Mankiw, N. Gregory, and Sala-i-Marin, Xavier Capial Mobiliy in Neoclassical Models of Growh. American Economic Review :???. Cass, David Opimal Growh in an Aggregaive Model of Capial Accumulaion. Review of Economic Sudies 32: Campbell, John Y Inspecing he Mechanism: An Analyical Approach o he Sochasic Growh Model. Journal of Moneary Economics 33: Kaldor, Nicholas Capial Accumulaion and Economic Growh, in Friedrich A. Luz and Douglas C. Hague, eds., Proceedings of a Conference Held by he Inernaional Economics Associaion. London: MacMillan. Koopmans, Tjalling C On he Concep of Opimal Economic Growh, in The Economeric Approach o Developmen Planning. Amserdam: Norh Holland Ramsey, Frank P., "A Mahemaical Theory of Saving," Economic Journal 38 (1928): Solow, Rober M A Conribuion o he Theory of Economic Growh. Quarerly Journal of Economics 70:

30 Endnoes 1. Solow (1956) inerpres his facor as labor-augmening echnological progress. 2. On he oher hand, we can also allow A o follow a deerminisic growh rend, say, A+1 = (1+g A)A (Hicks-neural echnological progress). We choose no o specify i his way because such process has already been subsumed under he accumulaion process of human capial. In 1/" oher words, one can inerpre 1+g H = (1+g A ), i.e., echnological progress is laboraugmening. 3. Indeed, in mos of he discussions below, we need no specialize he producion funcion o he Cobb-Douglas form. We choose his specific form in order o simplify he exposiion. 4. An equivalen specificaion, as in Chaper 7, assumes ha he firm is he owner of physical capial and makes is invesmen and producion decision by solving an ineremporal profi maximizaion problem. 5. Noe ha he ime convenion in our noaions in his chaper is slighly differen from ha used in Chaper 5. In his chaper, we denoe curren period () borrowing by B +1 raher han B and he corresponding rae of ineres by r +1 insead of r. This change is done in order o be consisen wih he ime convenion for facor accumulaion. 6. In he closed economy, equilibrium in he financial capial marke implies ha gross saving (saving in he form of physical capial) amouns o consumer's ne saving. 7. If he res of he world sars from an off-seady-sae posiion iniially, he economies will immmediaely converge and hen grow ogeher a he same raes owards a common long run seady sae in erms of per capia GDP. 8. Barro, Mankiw, and Sala-i-Marin (1995) consider similar inernaional borrowing consrains. Their model focuses on he impossibiliy of financing invesmen in human capial hrough borrowing. 9. Sricly speaking, he phase diagram aparaus applies only o coninuous ime dynmaics described in erms of a sysem of differenial equaions. The discree ime dynamics ha we examine here are jus approximaions o heir coninuous ime counerpars