The Mundell-Fleming Model: Stochastic Dynamics

Transcription

1 Th Mundll-Flming Modl: Sochasic Dynamics Th Mundll-Flming modl, which is sill h workhors modl of inrnaional macroconomics, can now b cas in a sochasic framwork. Such a framwork assums a s of xognous sochasic procsss (.g., mony supply) which drivs h dynamics of h quilibrium sysm. Sinc conomic agns ar forward looking, ach shor rm quilibrium is basd on xpcaions abou fuur shocks and h rsuling fuur shor rm quilibria. 4.1 Th Sochasic Framwork L us bgin wih a dscripion of h sochasic vrsion of h Mundll-Flming modl. For simpliciy, w xprss all variabls in logarihmic forms (xcp for h inrs ras) and assum all bhavioral rlaions ar linar in hs log variabls. This linar sysm (similar o h ons in Clarida and Gali (1994)) can b viwd as an approximaion from an original nonlinar sysm. d Aggrga dmand in priod, y, spcifid as a funcion of an xognous dmand componn, d, h ral xchang ra, q, and h domsic ral ra of inrs, r, is givn by y d ' d % 0q & Fr. (4.1) whr 0 and F ar posiiv lasiciis. This quaion is an analogu of quaion (3.5) of h prvious chapr. As is usual, h ral variabls ar drivd from h following nominal * variabls: s, h spo xchang ra (h domsic valu of forign currncy); p, h forign - 1 -

2 pric lvl; p, h domsic pric lvl; and i, h domsic nominal ra of inrs. Mor * spcifically, q = s + p! p and r = i! E (p +1 * pric lvl, p, o b consan ovr im.!p ). For simpliciy, w assum h forign Aggrga dmand is posiivly rlad o h xognous dmand shock, capuring xrnal, fiscal xpansion and ohr inrnal shocks. Th ral xchang ra affcs posiivly aggrga dmand by simulaing h radd scor (xporabls and domsic producion of imporabls). Th ral inrs ra affcs ngaivly aggrga dmand by discouraging invsmn and consumpion. Mony mark quilibrium is spcifid as: m s & p ' y & i, (4.2) s whr m is h mony supply a im, and (> 0) h inrs smi-lasiciy of h dmand for mony. This quaion is an analogu of quaion (3.7) of h prvious chapr. As usual, h domsic nominal ra of inrs (i ) has a ngaiv ffc on h dmand for mony, whil domsic oupu (y ) has a posiiv ffc. To simplify mars, h oupu dmand lasiciy is assumd o b uniy. Pric sing is basd on a mix of aucion marks and long rm conrac marks. Th mark claring pric in h aucion mark is p. Th pric in h long rm conrac mark is s on priod in advanc according o xpcaions of h fuur mark claring pric in ha mark, E!1p. Accordingly, h gnral pric lvl in h domsic conomy, p, is givn by a wighd avrag of hs wo prics: p ' (1&2)E &1 p %2p, (4.3) - 2 -

3 whr 0 < 2 < 1 is h shar of h aucion mark in domsic oupu. Th long rm conrac lmn is akin o Taylor (191) and Fischr (191). This inroducs an lmn of pric rigidiy ino h sysm. Du o fr capial mobiliy, inrs pariy prvails. Assuming risk nuraliy, uncovrd inrs pariy should hold. Tha is, i ' i ( % E (s %1 &s ), (4.4) * whr i is h world ra of inrs, assumd for simpliciy o b consan ovr im. Through coslss arbirag, h rurn on invsing on uni of domsic currncy in domsic scuriy, i, is mad qual o h xpcd valu of h domsic currncy rurn on invsing h sam * amoun in forign scuriy, which yilds a forign currncy rurn, i, plus an xpcd dprciaion of domsic currncy, E (s!s ). +1 Th quilibrium sysm consiss of h four quaions (4.1)!(4.4) a ach poin in im. Obsrv ha domsic oupu is dmand-drmind, as in all modls wih pric rigidiy. sysm ar: Th shock (or forcing sochasic) procsss ha driv h dynamics of h quilibrium y s ' g y % y s &1 %, y, (4.5a) d ' g y % d &1 %, d, (4.5b) m s ' g m % m s &1 %, m, (4.5c) - 3 -

4 whr g and g ar h drminisic growh ras of oupu and mony, and,,,,, ar y m y d m indpndnly and idnically disribud (i.i.d.) supply, dmand, and mony shocks wih zro 1 mans and consan variancs. Accordingly, our spcificaion assums ha h sysm is bombardd by prmann shocks (in a random walk fashion) Flx-Pric Equilibrium Sinc our sochasic framwork is boh forward and backward looking, a sysmaic procdur is rquird o obain a soluion. W hus apply a wo-sag procdur for solving h quilibrium sysm (4.1)!(4.5). In h firs sag, w solv for a flxibl pric quilibrium ha corrsponds o his sysm. In h scond sag, w us h flx-pric quilibrium o arriv a a full-fldgd soluion for h mixd fix-flx-pric sysm. Using suprscrip `' o dno flx-pric quilibrium valus, w can xprss h firs sag soluion in h following form. y ' y s. (4.6) q ' 1 0 (y s &d %Fi( ). (4.7) p ' m s & y s % (i( %g m &g y ). (4.) B (/ E P %1 & p ) ' g m & g y. (4.9) r ' i(. (4.10) - 4 -

5 i ' i( % g m & g y. (4.11) s ' m s % 1 0 &1 y s & 1 0 d % F 0 % i( & p ( % (g m &g y ). (4.12) Th flx-pric quilibrium is conomically inuiiv. Whn prics ar flxibl and h supply of oupu is xognous, oupu mus b supply-drmind, hnc (4.6). Wih consan mony dmand lasiciis, h xpcd ra of inflaion (which urns ou also o b h acual inflaion ra) mus b qual o h diffrnc bwn mony growh and oupu growh, hnc (4.9). Sinc world prics ar consan in h forign counry (hnc, zro world inflaion), h * world ral and nominal ras of inrs mus b qual o i. Undr h assumpion of fr * capial mobiliy, h domsic ral ra of inrs mus b qual o i as wll, hnc (4.10). From (4.9) and h Fishr quaion linking h nominal ra of inrs o h ral ra and h xpcd ra of inflaion, w can obain h corrsponding domsic nominal inrs ra as (4.11). Using h domsic ral inrs ra xprssion in (4.10), h ral xchang ra ha quas oupu dmand o h xognous supply of oupu can b solvd from h aggrga dmand quaion (4.1) o yild (4.7). Givn h domsic nominal ra of inrs (4.11) and oupu (4.6), h domsic pric lvl which is consisn wih mony mark quilibrium (4.2) can b xprssd as in (4.). Finally, w can driv h nominal xchang ra from (4.7) and (4.) oghr wih h dfiniion of ral xchang ra in rms h nominal xchang ra and domsic and forign pric lvls, hnc (4.12)

6 As an applicaion, considr an xpansionary fiscal policy indicad by a posiiv )(d ), whr )(.) is a diffrnc opraor. From (4.7) and (4.12), on can vrify ha h ral and nominal xchang ras will apprcia, wihou any ffcs on oupu, prics, and inrs ras. This should b familiar o h radr from h rsul sablishd in h prvious chapr ha, undr a flxibl xchang ra sysm wih prfc capial mobiliy, fiscal policis ar nural. s Considr nx an xpansionary monary policy indicad by a posiiv )(m ). From (4.), h domsic pric lvl will go up. From (4.12), h domsic nominal xchang ra will dprcia. Oupu, inrs ras, and h domsic ral xchang ra will no b affcd. This is obviously consisn wih h classical dichoomy bwn ral and nominal magniuds associad wih monary policy undr flxibl prics, in addiion o h familiar Mundll- Flming ffcs of monary policy on h nominal xchang ra discussd in h prvious chapr. 4.3 Full-fldgd Equilibrium Following our wo-sag soluion procdur, w can now us h flx-pric quilibrium valus obaind in h firs sag o solv for h full-fldgd quilibrium in his scond sag. Th quilibrium, drivd in Appndix A, is as follows. y ' y % F%0 %F%0 (1%)(1&2)(, m &, y ). (4.13) q ' q % 1 %F%0 (1%)(1&2)(, m &, y ). (4.14) - 6 -

7 p ' p & (1&2)(, m &, y ). (4.15) B ' B % (1&2)(, m &, y ). (4.16) r ' r & 1 %F%0 (1%)(1&2)(, m &, y ). (4.17) i ' i % F%0&1 %F%0 (1&2)(, m &, y ). (4.1) s ' s & F%0&1 %F%0 (1&2)(, m &, y ). (4.19) Th full-fldgd quilibrium valus in quaions (4.13)-(4.19) rval inrsing faurs: (1) Pric Rigidiy and h Classical Dichoomy Pric rigidiy is rflcd in (4.15) sinc a posiiv xcss mony shock gnras a pric incras which falls shor of h mark claring pric. Wih pr-s prics, h classical dichoomy no longr holds. Accordingly, in (4.13), on can obsrv ha oupu rsponds o h innovaion in h mony supply in xcss of h innovaion in domsic oupu supply. Th ral xchang ra is posiivly affcd, and h domsic ral ra of inrs, ngaivly affcd by h diffrnc in innovaions. Th magniuds of hs ffcs dpnd on h dgr of pric flxibiliy, indicad by 2. Indd, in h xrm cas of compl pric flxibiliy (2 = 1), hs ral ffcs of monary policy will vanish (as shown also in h prvious chapr). (2) Th Phillips Curv - 7 -

8 Dfin xcss oupu capaciy (which is dircly rlad o h ra of unmploymn) by u as y!y. Thn w can obain an xpcaions-augmnd Phillips curv rlaion bwn inflaion (B ) and xcss capaciy (u ) as follows: B ' B & (1%)(F%0) % 1 1% u. (4.20) Th flar lin in Figur 4.1 porrays h opn-conomy Phillips curv undr fr capial mobiliy. Equaion (4.20) shows ha h Phillips curv is flar whn h aggrga dmand lasiciis 0 (wih rspc o h ral xchang ra) and F (wih rspc o h domsic ral ra of inrs) ar largr. Th ffc of h inrs smi-lasiciy of mony dmand () on h slop of h Phillips curv is, howvr, ambiguous, dpnding on whhr F+0 xcds or falls shor of uniy. Th sourc of his ambiguiy is drivd from h mor fundamnal ambiguous ffcs of xcss innovaions on h domsic nominal inrs ra (4.1) and spo xchang ras (4.19). (3) Ral Exchang Ra and Ral Ra of Inrs Subsiuing (4.17) ino (4.14) yilds a conmporanous ngaiv rlaion bwn h ral xchang ra and h domsic ral inrs ra as follows: q ' q & (r &r ). (4.21) This unambigous prdicion has bn subjc o a larg body of mpirical sudis (s Campbll and Clarida (197), Ms and Rogoff (19), and Edison and Pauls (1993)), wih mixd - -

9 rsuls. I hus sms ha h Mundll-Flming modl should accoun for his inconsisncy wih daa bfor i will b usd wih gra confidnc for policy advic. (4) Expcd Long Run Valus Applying h xpcaion opraor as of priod o h sysm of quaions (4.13)!(4.19) in priods +1 and on rvals ha h xpcd long run quilibrium valus ar qual o h flxpric soluion. Equaion (4.19) hn shows ha an xcss mony innovaion will lad o xchang ra ovrshooing a la Dornbusch (1976) if h sum of dmand lasiciis (F+0) falls shor of uniy. 4.4 Capial Conrols Th hallmark of h Mundll-Flming modl is h disinc rol playd by inrnaional capial movmns on h ffcivnss of policis. Thus, rsricing capial flows should hav a significan ffc on h working of h inrnaional macro sysm. Considr h xrm cas whr capial flows ar complly rsricd. In his cas, h inrs pariy (4.4) will no longr hold, and rad balanc will b quilibrad fully by h mark claring xchang ra. Th final form of h aggrga dmand quaion (4.1), drivd from h srucural quaion, will hav o b modifid. W can wri h original srucural quaion as: y d ' ( d A % A y d y % A r ) % (d X r % X y d y % X q ). q - 9 -

10 whr h firs parnhical xprssion rfrs o domsic absorpion (A), and h scond o n ~A X rad balanc (X), d dnos h auonomous componn of absorpion, A y > 0, A r < 0, d dnos h auonomous componn of rad balanc, X y < 0, and X q > 0. To arriv a h d final form (4.1), w simply solv for y as a funcion of r and q. Dfin h sum of marginal ~A X propnsiis o sav and impor, 1!A y!x y, as ". Noic ha d = (d +d )/", 0 = X q/" > 0, and F =!A /" > 0. r In h prsnc of full capial conrols, h n rad balanc (X) is zro. Hnc, d + X d Xyy + Xqq = 0, which can b rwrin as d X & µy d % "0q ' 0, (4.1)' whr µ =!X y and "0 = X q. Subsiuing his ino h srucural quaion for aggrga dmand, w can modify h final form as y d ' d A & r, (4.1)" A ~A whr d = d /(1!A y) and ( = (1!A y!x y)/(1!a y) < 1. As an analogu o (4.5b), w spcify A h sochasic procss for d as d A ' g y % d A &1 %,A d, (4.5b)' A whr, is assumd o hav similar propris as,. d W lay ou h soluions for h flx-pric and full-fldgd quilibria in Appndix B. Hr, w focus on h ffc of capial conrols on oupu y, h ral xchang ra q, and h d

11 inflaion ra B. Th full-fldgd quilibrium oupu, ral xchang ra, and inflaion ra ar givn rspcivly by y ' y % % (1%)(1&2), m & %, y %,A d, (4.13)' s whr y = y. ' q % µ "0 % (1%)(1&2), m & %, y %,A d (4.14)' s X whr q = (µy!d )/"0. B ' B % (1&2), m & %, y %,A d, (4.16)' whr B = g m!g y. Comparing hs quilibrium valus wih hos undr fr capial flows (4.13) and (4.14), w can highligh hr main diffrncs. 1. Dmand Shocks A Undr capial conrols, h absorpion shock, d has a posiiv ffc on y!y, q!q, and B!B, hrough is ngaiv ffc on h domsic ral ra of inrs r. In conras, i has no ffc in h cas of fr capial flows sinc h ral ra of inrs hr is naild down by h world ra of inrs. 2. Monary Shocks

12 Sinc /(+) < (F+0)/(+F+0), h monary shock, has a smallr ffc on m y!y (hrough a srongr ngaiv ffc on r ) undr capial conrols. Th rlaiv snsiiviy of q!q o h shock undr h wo capial mobiliy rgims is ambiguous in gnral, dpnding on h rlaiv magniuds of µ()/"0(+) and 1/(+F+0). Finally, h shock has h sam ffc on B!B as in h fr capial mobiliy cas. 3. Supply Shocks Sinc (F+0)/(+F+0) < 1, h produciviy shock, has a biggr ngaiv ffc on y aggrga dmand y!y (hrough a srongr posiiv ffc on r ) undr capial conrols. Th rlaiv snsiiviy of q!q o h shock undr h wo capial mobiliy rgims is again ambiguous. Th shock will nonhlss produc a mor pronouncd ffc on B!B undr capial conrols. 4. Th Phillips Curv Subsiuing quaion (4.13)' ino quaion (4.16)' and dfining u =!(y!y ) as bfor, w can xprss h Phillips curv undr capial conrols as follows: B ' B & (1%) % 1 1% u. (4.20)' Th spr lin in Figur 4.1 porrays h opn-conomy Phillips curv undr capial conrols. In ohr words, flucuaions in inflaion ras will b associad wih smallr variaions in unmploymn

13 Th inuiion bhind h spr slop has o do wih h impac ffc of capial conrols on aggrga dmand. Comparing h aggrga dmand funcions undr capial conrols (4.1)" and undr fr capial mobiliy (4.1), w obsrv ha in h formr cas h inrs smilasiciy bcoms smallr ( < F sinc ( = 1!X y/(1!a y) < 1) and h ral xchang ra ffc disappars (0 < 0) from h rducd form quaion for aggrga dmand du o h zro n rad balanc rsricion undr capial conrols. On h ohr hand, capial conrols do no alr h mchanisms undrlying h drminaion of prics (i.., h pric sing quaion (4.3) and h mony mark quaion (4.2)). Combind wih h srucural chang in aggrga dmand, rsricions on capial flows will gnra lss variaions in unmploymn ras (xcss oupu capaciy) a h xpns of mor variaions in inflaion ras. Indd, a comparison bwn quaions (4.20) and (4.20)' rvals ha h diffrnc in h slops of h Phillips curv undr fr capial mobiliy and capial conrols dpnds solly on h aggrga dmand paramrs F+0 vrsus, and no on h mony mark paramr and h pric sing paramr 2. Naurally, h naural ra of unmploymn (= 0 in our cas) and h xpcd ra of inflaion (B ) ar unaffcd by srucural changs such as capial conrols. This is rflcd by h inrscion of h wo Phillips curvs a h poin (0,B ). Whil h various shocks will mov conomy away from his poin along h rspciv Phillips curv (dpnding on h capial mobiliy rgim), changs in h xpcd ra of inflaion du o prmann changs in h rlaiv mony-oupu growh ras (g m!g y) will shif h Phillips curv around

14 4.5 Som Policy Implicaions Thr xis hr yps of gains from rad: rad of goods/srvics for goods/srvics; rad of goods/srvics for asss (inrmporal rad); and rad of asss for asss (for divrsificaion of risk). Evidnly, capial conrols limi h ponial gains from h las wo yps. Howvr, hr ar scond-bs and ohr subopimal (du o, say, non-mark-claring) siuaions whr capial conrols can improv fficincy. Whn, for xampl, axaion of forign-sourc incom from capial is no nforcabl, i provs fficin o `rap' capial wihin naional bordrs so as o broadn h ax bas and o allvia ohr ax disorions (s Razin and Sadka (1991)). This chapr inroducs anohr argumn in favour of consrains on capial mobiliy. W show ha capial conrols alr h inflaion-unmploymn radoffs. In paricular, oupu/ mploymn variaions ar rducd a h xpns of biggr variaions in inflaion ras. Whn h policy makr pus havir wigh on sabl mploymn han on sabl inflaion, his/hr objciv can b aaind mor asily undr capial conrols

15 Appndix A: Drivaion of h Full-fldgd Equilibrium Soluion This appndix drivs h soluion for h full-fldgd quilibrium (4.13)!(4.19), aking as givn h flx-pric quilibrium. 1. Drivaion of p (4.15) To g h soluion for h domsic pric lvl (4.15), w simply subsiu (4.5c) and (4.) ino (4.3). 2. Drivaion of q (4.14) Subsiuing (4.4) and (4.7) ino (4.1), using h dfiniions of ral xchang ra q = s * + p! p and ral inrs ra r = i! E (p +1!p ), and subracing (+F)E q +1 and adding (+F)q, w g q ) ' (%F)[E (q %1 &q %1 )&(q &q )] % (1%)(1&2)(, m & (A.1) Obsrv, from (4.5a), (4.5b), and (4.7) and h propris of, and,, ha E q = q. W guss a soluion of h form q = q + 6(, m!, y) and apply i o (A2.1). y+1 d (, m &, y ) ' (%F)E 6(, m%1 &, y%1 )&6(, m &, y ) % (1%)(1&2)(, m &, y ). Sinc E(,!, ) = 0, w can obain 6 = (1+)(1!2)/(+F+0) from h abov quaion. m+1 y+1 This valu of 6 in our guss soluion yilds h soluion for h ral xchang ra (4.14)

16 3. Drivaion of y (4.13) and i (4.1) Subsiuing h soluions for p and q from (4.15) and (4.14) jus drivd abov ino h aggrga dmand quaion (4.1) whil using h inrs pariy (4.4) yilds h soluions for y (4.13) and i (4.1). 4. Drivaion of B (4.16) Applying h dfiniion of h xpcd ra of inflaion, B = E (p!p ), o (4.15) +1 drivd in sp 1 and (4.9) yilds h soluion for h inflaion ra (4.16). 5. Drivaion of r (4.17) Using i drivd in sp 3 and B in sp 4 and h Fishr quaion yilds h soluion for h domsic ral ra of inrs r. 6. Drivaion of s (4.19) Using h soluions for p drivd in sp 1 and q in sp 2 and applying hm o h * dfiniion of h ral xchang ra q = s + p! p yilds a soluion for h nominal xchang ra s

17 Appndix B: Soluions for h Capial Conrols Modl This appndix provids h soluion o (4.1)', (4.1)", (4.2) and (4.3), subjc o h sochasic procsss (4.5) and (4.5b)'. Th flx-pric quilibrium condiions ar y ' y s. (B.1) q ' 1 "0 (µy s &d X ). (B.2) p ' 1 (d A &y s )%g m &g y % m s & y s. (B.3) B ' g m & g y. (B.4) r ' 1 (d A &y s ). (B.5) i ' 1 (d A &y s ) % g m & g y. (B.6) ' m s % µ "0 &% y s & 1 "0 d X & d A & p ( % (g m &g y ) (B.7) To solv h full-fldgd quilibrium, w us h flx-pric soluion o obain h quilibrium pric p and inflaion ra B. W hn us h Fishr quaion along wih h aggrga dmand and mony mark quilibrium quaions (4.1)" and (4.2) o g h soluions for h ral inrs ra r and oupu y simulanously. From h rad balanc quaion (4.1)', w can calcula h ral xchang ra q. Th nominal inrs ra i and h nominal xchang

18 ra s ar hn drivd from h Fishr quaion and h dfiniion of h ral xchang ra rspcivly. Blow, w lay ou h soluion for h full-fldgd quilibrium. y ' y % % (1%)(1&2), m & %, y %,A d. (B.) ' q % µ "0 % (1%)(1&2), m & %, y %,A d (B.9) p ' p & (1&2), m & %, y %,A d. (B.10) B ' B % (1&2), m & %, y %,A d. (B.11) r ' r & 1 % (1%)(1&2), m & %, y %,A d. (B.12) i ' i % (1&) % (1&2), m & %, y %,A d. (B.13) s % µ "0 % (1%)&1 (1&2), m & %, y %, (B.14) Comparing h full-fldgd quilibrium undr capial conrols (B.)-(B.14) wih h corrsponding quilibrium undr fr capial flows (4.13)-(4.19), w can assss h significan rol ha capial mobiliy plays in h Mundll-Flming modl

19 Problms 1. Considr h sochasic dynamic vrsion of h Mundll-Flming modl wih prfc capial mobiliy. Inroduc ransiory shocks o h mony supply procss by adding an xra rm!n, (N > 0) o h righ hand sid of (4.5c). Dcompos h varianc of h ral m!1 xchang ra q ino ransiory and prmann componns of h monary shock. 2. Considr h sochasic dynamic vrsion of h Mundll-Flming modl wih prfc s capial mobiliy. Inroduc a corrlaion bwn h mony supply procss m and aggrga dmand procss d by adding Dd o h righ hand sid of (4.5c). On can viw his as a!1!1 fdback rul whrby currn monary policy is condiiond on fiscal impuls in h prvious priod. Wha valu of D will minimiz oupu varianc? Inflaion varianc? 3. Considr h sochasic dynamic vrsion of h Mundll-Flming modl wih and wihou capial conrols. (a) Compar h snsiiviy of h following conomic indicaors o h various shocks bwn h wo capial mobiliy rgims: p, B, r, i, s. (b) (c) Compar h slops of h Phillips curvs undr h wo rgims. Chck whhr h ngaiv rlaion bwn h ral xchang ra and h domsic ral ra of inrs undr prfc capial mobiliy holds also undr capial conrols

20 Rfrncs Campbll, John Y., and Clarida, Richard H Th Trm Srucur of Euromark Inrs Ras: An Empirical Invsigaion. Journal of Monary Economics 19 (January): Clarida, Richard H., and Gali, Jodi Sourcs of Exchnag Ra Flucuaions. NBER Working Ppar No Edison, Hali J. and Pauls, B. Diann A R-assssmn of h Rlaionship bwn Ral Exchang Ras and Ral Inrs Ras: Journal of Monary Economics 31 (April): Ms, Richard A. and Rogoff, Knnh. 19. Was I Ral? Th Exchang Ra-Inrs Diffrnial Rlaion ovr h Modrn Floaing Ra Priod. Journal of Financ 43 (Spmbr): Razin, Assaf, and Efraim Sadka Efficin Invsmn Incnivs In Th Prsnc of Capial Fligh, Journal of Inrnaional Economics 31: Sargn, Thomas J Macroconomic Thory. Scond Ediion. Nw York: Acadmic Prss

21 ENDNOTES 1. To guaran h xisnc of a long run (sady sa) quilibrium for our sysm, h drminisic growh ras of oupu on boh h supply and dmand sids (g y) ar assumd o b idnical. 2. Th problm s a h nd of h chapr considrs also ffcs of ransiory shocks