Ensayos sobr POÍTICA ECONÓMICA PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET HERNANDO VARGAS YANNETH ROCÍO BETANCOURT ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 PP. 22-55 os drchos d rproduccón d st documnto son propdad d la rvsta Ensayos Sobr Polítca Económca (ESPE). El documnto pud sr rproducdo lbrmnt para uso académco, smpr y cuando no s obtnga lucro por st concpto y admás, cada copa ncluya la rrnca bblográca d ESPE. El(los) autor(s) dl documnto pud(n) admás ponr n su propo wbst una vrsón lctrónca dl msmo, pro ncluyndo la rrnca bblográca d ESPE. a rproduccón d sta rvsta para cualqur otro n, o su colocacón n cualqur otro wbst, rqurrá autorzacón prva d su Edtor d ESPE.
AS ADMINISTRADORAS DE FONDOS DE PENSIONES Y A ESTRUCTURA DE MERCADO CAMBIARIO HERNANDO VARGAS YANNETH ROCÍO BETANCOURT as opnons xprsadas corrspondn a los autors y no rprsntan los puntos d vsta dl Banco d la Rpúblca n d su Junta Drctva. Agradcmos las valosas sugrncas y comntaros d los valuadors anónmos. os autors son, n su ordn, Grnt Técnco, y Asstnt d la Grnca Técnca dl Banco d la Rpúblca. Corros lctróncos: hvargah@banrp.gov. co y ybtanga@banrp. gov.co Documnto rcbdo l 3 d abrl d 2007; vrsón nal acptada l 24 d agosto d 2007. El comportamnto d las admnstradoras d ondos d pnsons (AFP) pud tnr ctos mportants n l mrcado cambaro dbdo al tamaño d su portaolo y su posbl podr d mrcado. Algunos autors sñalan qu las accons d grands nvrsonstas, como las AFP, n l mrcado cambaro pudn nlur n las dcsons d otros agnts ncrmntando l mpacto sobr la tasa d cambo; sn mbargo, cuando las AFP tnn podr d mrcado modran su volumn transado al tnr n cunta su nlunca sobr la tasa d cambo. Est artículo dmustra la xstnca d st cto modrador sobr drnts structuras dl mrcado cambaro. Clascacón JE: G23; F3; D43. Palabras clav: ondos d pnsons, mrcado cambaro, podr d mrcado.
PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET HERNANDO VARGAS YANNETH ROCÍO BETANCOURT Th cts o th Pnson Fund Managrs (PFM) bhavor on th Forgn Exchang (FX) markt may b mportant, gvn th sz o thr portolo and thr possbl markt powr. Som authors argu that whn bg nvstors lk PFM trad larg volums n th FX markt, thy may nlunc othr agnts dcsons, ncrasng th mpact on th xchang rat. Howvr, whn PFM hav markt powr, thy wll tak nto account thr nlunc on th xchang rat and wll modrat thr tradng volum. Ths papr sks to dmonstrat th xstnc o ths mtgatng ct undr drnt thortcal FX markt structurs. JE Classcaton: G23; F3; D43. Kywords: pnson unds, orgn xchang markt, markt powr. Th opnons xprssd ar thos o th authors and do not rprsnt th vws o th Banco d la Rpúblca or o ts Board o Drctors. W ar gratul to anonymous rrs or suggstons and commnts. In ordr, th authors ar, Dputy Govrnor, and Assstant to th Dputy Govrnor, Banco d la Rpúblca. E-mals: hvargah@ banrp.gov.co, and ybtanga@banrp.gov.co. Documnt rcvd 3 Aprl 2007; nal vrson accptd 24 August 2007.
24 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 I. INTRODUCTION Th ntrvnton o local nsttutonal nvstors, lk Pnson Fund Managrs (PFM), n th orgn xchang markt could hav cts on th xchang rat volatlty, gvn th sz and th markt powr o ths agnts. As notd by J. P. Morgan (2002) th ct o ths nsttutonal nvstors on th orgn xchang markt can b mportant, gvn th sz o thr portolos. For th Chlan cas, Zahlr (2005) stats that th mpact o PFM on th orgn xchang markt can b strong du to th concntraton o managrs and th rgulaton on th rat o rturn o pnson unds. Addtonally, ths author argus that th prssurs on th xchang rat could b xacrbatd bcaus th PFM s nvstmnt dcsons ar domnatd by th short run xtd rturns, dspt thr long run am. Accordng to Zahlr (2005), larg shts o th orgn currncy holdngs n th PFM s portolos ovr a short span may dstablz th orgn xchang markt and, hnc, may justy lmts on th sz o th PFM orgn nvstmnts or on th spd o such shts. Th act that ths larg nvstors trad bg volums n th orgn xchang markt may also nlunc othr agnts dcsons, ncrasng th mpact on th xchang rat. Howvr, ths ct can b mtgatd bcaus agnts wth markt powr wll consdr th mpact that thr actons hav on th xchang rat, on th valu o thr orgn currncy transactons and on othr agnts actons. Thror, PFM wth markt powr wll tak nto account ths cts, rducng th volum o Ths rgulaton can gnrat a hrd ct.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 25 thr transactons and th prssur on th xchang rat. For th sam rason, thy wll rspond lss to shocks on xchang rat xtatons and on th ntrst rat drntal than undr prct comptton. Morovr, th qulbrum xchang rat undr non-compttv structurs wll rspond to xognous shocks n th orgn currncy nt supply (. g. xports changs, trms o trad, tc ), n contrast to th compttv cas. Th da o ths papr s to show th xstnc o ths mtgatng ct undr drnt thortcal orgn xchang markt structurs. Th rsults o th non compttv structurs wll b contrastd wth th bnchmark cas o prct comptton. Th rmandr o ths papr s organzd as ollows. Scton II rvws som mcroconomc ltratur about th orgn xchang markt. Scton III analyss th bhavor o PFM and ts mpact on th xchang rat undr drnt markt structurs. Scton IV concluds and summarzs th rsults o ths papr. II. MICROECONOMIC ITERATURE ON THE FOREIGN EXCHANGE MARKET In contrast to th tradtonal macroconomc ltratur, th mcro ltratur analyzs th bhavor and ntracton o ndvdual agnts n th orgn xchang markt, takng nto account th cts o asymmtrc normaton and th agnts htrognty. Ths ltratur rrs thn to mcro asts o th orgn xchang markt, such as th transmsson o normaton among markt agnts, thr bhavor and th mplcatons o th agnts xtatons htrognty on tradng volum and xchang rat volatlty. Bcaus ths ltratur trs to undrstand th mchansms that gnrat shts n th xchang rat that ar unxpland by movmnts n macroconomc undamntals, t ocuss on th mchancs o orgn xchang tradng. Much o th arly ltratur o th orgn xchang markt mcrostructur s concrnd wth th analyss o th procss by whch agnts orm xtatons about th xchang rat. By contrast wth th smplyng assumpton o ratonal xtatons that undrls th tradtonal ltratur, th mcrostructur ltratur uss drct masurs o xtatons, lk survys o markt partcpants. Ths orgn xchang survy data studs suggst a strong htrognty and an ncrasng dsprson o xtatons sally at longr orcast horzons. Thy also nd a rvrson n xtatons, so that th drcton o th long run xtatons drs
26 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 rom th drcton o th short run xtatons. 2 Addtonally, Froot and Ito (989) nd an ovrracton o th short trm xtatons wth rst to th long trm xtatons whn th xchang rat changs. Ths ltratur suggsts that th dscrpancy btwn xtatons at drnt horzons may b xpland by th us o drnt orcastng tchnqus n th short and n th long run. Th chartst or tchncal analyss, whch dnts broad rangs wthn whch xchang rats ar xtd to mov by usng trnds, graphc pattrns and dscrptv statstcs, s th prdomnant tchnqu usd or short run orcastng. In contrast, undamntal analyss and convntonal xchang rat modls ar usd or long-run xchang rat orcastng. 3 Th act that standard xchang rat modls hav poor prormanc and that chartst tchnqus ar usd by many orgn xchang practtonrs to prdct th xchang rat at short run horzons suggsts that non undamntals actors could domnat th short run movmnts o th xchang rat. Insoar as ths short run bhavor s gnralzd, th orgn xchang markt can b pron to sulatv movmnts ollowng unxtd dvatons o th xchang rat wth rst to th conomc undamntals. 4 Many studs n th mcrostructur ltratur hav ound a strong contmporanous corrlaton btwn tradng volum and th xchang rat volatlty (s Sarno and Taylor [200] or a survy o ths ltratur). In th contxt o portolo choc modls, yons (99) xplans th tradng volum manly by drncs n valuatons, whch can b du to drnt xtatons or 2 Ths mans that xtatons o addtonal dprcatons n th short run ollow an obsrvd dprcaton whl t s ollowd by xtatons o a modrat apprcaton n th long run. 3 Taylor and Alln (992) suggst a broad consnsus o th partcpants n th ondon orgn xchang markt wth rst to th mportanc gvn to th chartst analyss. Thy nd that almost 90% o th rspondnts us som chartst tchnqu whn thy orm thr short run xchang rat xtatons (ntraday to on wk) and 60% o th rspondnts rgard charts at last as mportant as undamntals. Morovr, th mportanc gvn to conomc undamntals ncrass or longr horzons (rom on month to on yar). Chung and Wong (2000) nd that th majorty o rspondnts n th orgn xchang markts o Hong Kong, Tokyo and Sngapor rcognzs th xstnc o sgncant cts o non undamntals actors on short run xchang rats xtatons. u and Mol (998) rport dalrs n Hong Kong hav a skw toward tchncal analyss at short horzons and toward undamntal analyss at longr horzons. 4 A numbr o rsarchrs hav constructd drnt modls, n th vn o Frankl and Froot (987), to show th rol o non undamntalst tradrs n th gnraton o bubbls n th orgn xchang markt.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 27 bcaus th volum o transactons thmslvs may nlunc valuatons ovr tm. In ths contxt, larg volums o orgn currncy ar tradd du to non undamntal rasons (sulatv bhavor), th movmnts o th xchang rat can b vry mportant. Ths s sally th cas whn th transactons o agnts wth markt powr can nlunc th xtatons and th valuatons o othr agnts and thr tradng volum, gnratng a strongr mpact on th xchang rat. Although th markt powr o som agnts n th orgn xchang markt can b vry mportant to xplan changs n th tradng volum and n th xchang rat, th mcro ltratur has not ocusd on non compttv practcs n th ntrnatonal orgn xchang markts. 5 It s dcult to thnk o non compttv bhavor whn thr ar a larg numbr o partcpants n th markt and th transmsson o normaton s ast. Howvr, non compttv bhavor can xst n small domstc orgn xchang markts, 6 snc th agnts portolo sz may act th markt through th tradng volum and ts mpact on th xtatons o othr agnts. Ths papr gos n ths drcton and trs to xplan th consquncs o non compttv markt structurs on th xchang rat. Scally w wll dmonstrat th xstnc o an ct o markt powr that modrats th mpact o agnts dcsons on th xchang rat, whch w call th mtgatng ct. I agnts wth markt powr consdr th cts that thr actons hav on othrs valuatons and on th xchang rat, thn thy wll modrat thr tradng volum, mtgatng th prssur on th xchang rat and makng th orgn xchang markt lss pron to sulatv movmnts. In partcular, gvn th ncrasng sz o assts managd by prvat pnson unds 7 w consdr th cas whr thy ar wthn th most mportant nvstors n th orgn xchang markt, such that thy may hav som markt powr and may nlunc th othr agnts dcsons. 5 Ths ar th markts or currncs wth ntrnatonal dmand lk dollars or uros. 6 Ths ar markts or currncs lk th Colomban pso, whch ar not ntrnatonally dmandd. 7 Ths has bn wdly documntd (. g. Davs and Stl, 200)
28 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 III. PENSION FUND MANAGERS AND DIFFERENT FOREIGN EXCHANGE MARKET STRUCTURES In ths scton, drnt structurs o th orgn xchang markt, n a statc ramwork, 8 wll b modld to show th consquncs on th xchang rat o th PFM markt powr and th xstnc o th mtgatng ct. Frst, w ar gong to prsnt th bnchmark cas whr prct comptton s assumd. Thn, a smpl modl o monopsony wll b ntroducd to s th ct o a bg PFM on th xchang rat. Atr that, an olgopsony à la Cournot wll b analyzd n ordr to dnty th cts o N PFM wth markt powr. Fnally, an olgopsony à la Stacklbrg wth on PFM as a ladr and N - PFM as ollowrs wll b analyzd. Th rsults o ach on o ths structurs wll b compard wth rst to th bnchmark cas, n whch thr ar not markt powr cts. A. GENERA ASSUMPTIONS W assum that thr ar two typs o agnts n th orgn xchang markt: PFM who dmand orgn currncy and othr agnts lk xportrs who supply orgn currncy. Although thr can b othr agnts dmandng orgn currncy, w ar ntrstd n th ky rol playd by th PFM whch ar among th bg playrs n th FOREX markt. 9 For ths rason, th bhavor o othr agnts drnt rom PFM, lk mportrs, s takn nto account n th orgn currncy nt supply uncton Q(,, whr Q s th nt amount o orgn currncy suppld, s th xchang rat dnd as domstc currncy/orgn currncy, and z corrsponds to xognous varabls that act th nt supply o orgn currncy (.g. xports changs, trms o trad, tc ). Ths uncton s assumd to b ncrasng n th xchang rat Q (, z ) > 0 and postvly rlatd to th xognous varabls, Q (, z ) > 0. W can also wrt ths z uncton n th nvrs orm, (Q,, wth Qz (, ) < 0. z 8 Undr ths statc ramwork th trms o nvstmnt ar not rlvant. 9 It mans that gvn th portolo sz o PFM thy can dmand mor than th suppld amount by othr agnts n th orgn xchang markt.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 29 In ordr to analyz th dcsons o th PFM n th FOREX markt, w assum that thr ar two knds o assts n whch thy can allocat thr unds: assts n domstc currncy (l) and orgn currncy assts (x). W also assum that ach PFM has a xd amount (m) to nvst n ths two assts, whch mans that th portolo sz o th PFM dos not chang. 0 Addtonally an ntal composton o thr portolo btwn th two assts (l 0 and x 0 ) s assumd, thus whn a PFM dmands an amount x o orgn currncy hs nt purchas o ths asst s x - x 0, th sam or th othr asst. B. PERFECT COMPETITION In ths sttng all th agnts shar th sam xchang rat xtatons and PFM tak prcs as gvn. In ths cas, th sz o agnts portolos dos not mattr and all ract n th sam way wth rst to changs n th ntrst rats and othr varabls. Th optmal dcson o a rsk nutral PFM s to choos hs portolo composton btwn assts n domstc currncy l and orgn currncy assts x, n ordr to maxmz hs portolo xtd rturn (n local currncy ) gvn a portolo ntal composton (l 0 and x 0 ) and a xd amount to nvst m. Thus, th managr rallocats hs portolo gvn th obsrvd xchang rat,, th domstc and orgn ntrst rats ( and, rstvly) and th xchang rat xtaton,, whch s th sam or vrybody. 2 Th maxmzaton problm o a tyal PFM s thn: Max[( l + ) + x ( + )] lx, st.. l= m x m= l + x 0 0 Th rducd orm o ths problm s: Max l0+ x0 x ( )+ x + x + 0 It also mpls that ach PFM can nvst a maxmum amount m n ach asst. Ths s mportant bcaus th pnsons ar gvn n local currncy. 2 Snc our objctv s to dtrmn th cts on th xchang rat o PFM s actons n th FOREX markt, w only analyz th partal qulbrum or ths markt. Thus, w do not analyz othr markts lk th mony markt and, or ths rason; th domstc ntrst rat s xognous.
30 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Th ntror soluton o ths problm ( x> 0, l> 0) s gvn by th ntrst rat party condton: = + ( + ) It corrsponds to th qulbrum xchang rat undr prct comptton ( ), at whch th tyal PFM dmands an amount x o orgn currncy and hs nt purchas o ths asst s x - x 0. 3 For all PFM th nt dmand o orgn currncy corrsponds to th sum o th ndvdual nt purchass X - X 0. In qulbrum, th total amount o orgn currncy suppld by othr agnts n th conomy, Q, s qual to th total dmand by PFM, X - X 0. It mpls that th qulbrum condtons n th orgn xchang markt ar: = ( Q, z ) (2) Q = X X0 To smply and bcaus X 0 s gvn, th orgn currncy nt supply can b wrttn as (X,, at ach prc th suppld amount s qual to th dmandd amount. By smplcty w also assum that ( X, z ) = a, wth a a postv constant (Graph ). 4 Rsult : In th prct comptton qulbrum, th xchang rat s dnd by th ntrst rat party condton () and th amount o orgn currncy tradd s gvn by quaton (2). Rsult 2: Th qulbrum xchang rat undr prct comptton,, s dtrmnd only by th ntrst rat drntal, d = + ( ), and th xchang rat xtatons, (quaton []). Ths mpls that any chang n ths varabls acts ( + ) proportonally th qulbrum xchang rat. () 3 Th soluton undr prct comptton s smlar whn agnts ar rsk-avrs and hav ( + ) CARA utlty unctons xcpt or th xstnc o a rsk prmum: E = [ ] x ( + ) ρ. In ths cas, th ntuton about th mtgatng ct s rnorcd by rsk avrson. Intutvly, a larg PFM wth markt powr wll also hav a orgn xchang xposur that s gratr than that o a smallr, compttv PFM. Hnc th rspons o th ormr to a shock n or E[ ] wll b mor mutd du to both th markt powr ct and th rsk avrson ct (s Appndx ). 4 Takng th mor gnral assumpton ( X, z ) > 0 th rsults rman th sam, thy do not dpnd on th lnarty o th supply uncton.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 3 Graph Qz (, ) X Q = X X 0 Sourc: Authors calculatons. Rsult 3: Th qulbrum xchang rat undr prct comptton dos not ract to an xognous shock n th orgn currncy nt supply, d = 0. Thror any xognous chang n th supply s totally absorbd by th dmandd amount o orgn dz currncy; thr s not an ct on prc. Ths rsult mpls that whn thr s an xognous shock on th orgn currncy nt supply (. g. xports changs, trms o trad, tc ), PFM absorb compltly th addtonal supply untl th ntrst rat party condton s rstablshd (Graph 2). It mans that ( X X ) = (, ) 0 Q z. z z C. MONOPSONY Undr ths structur thr s only on PFM who acs th total nt supply o orgn currncy; thn th total nt dmand o ths asst (X - X 0 ) s xclusvly hs dcson. As a rsult, ths agnt knows that any chang n hs tradng volum has an mpact on th xchang rat and hnc on th valuaton o hs nt purchass. Th PFM wth monopsony powr taks nto account th ct o hs transactons on th xchang rat, thus hs optmzaton problm s:
32 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Graph 2 Qz Qz, 0, Δz Sourc: Authors calculatons. Q Q X Max 0+ X ( 0 X, X( X, ( + )+ X + X rom th rst ordr condton, th qulbrum xchang rat undr monopsony s: m= + a X0 X (3) Rsult 4: Snc (). = a > 0, th qulbrum xchang rat undr monopsony s lowr than n prct comptton whn th monopsonst has a postv nt dmand o orgn currncy ( X X)<. 0 0 Intutvly, whn th optmal dcson o th monopsonst s to buy orgn currncy, h knows that th addtonal purchas o a unt o orgn currncy wll ncras th xchang rat and hnc th cost o hs purchass. For ths rason, th monopsonst wll dmand lss than th total dmandd amount undr prct comptton, thus th qulbrum xchang rat wll b lss dprcatd than n th prct comptton cas (ths s th mtgatng ct ). By th sam tokn, whn th monopsonst nds t optmal to sll orgn currncy (( X0 X)> 0), h knows that sllng an addtonal unt o ths asst rducs th xchang rat and thus th valu o hs total sals. To avod ths advrs ct on hs
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 33 sals, th monopsonst rducs hs orgn currncy supply and th xchang rat wll b mor dprcatd than n prct comptton. Rsult 5: Th rspons o th monopsony qulbrum xchang rat, m, to shocks to th ntrst rat drntal or to th xchang rat xtatons s lss (hal) than th rspons o th prct comptton xchang rat,, to th sam shocks: dm = (s Appndx or a proo). (4) d 2 Ths ollows bcaus th monopsonst rsponds lss to ths shocks than n prct comptton gvn that h taks nto account th mpact o hs transactons on th xchang rat and on th valu o hs total purchass or sals (th mtgatng ct ). Rsult 6: An xognous shock to th orgn currncy nt supply has a ngatv mpact on th monopsony qulbrum xchang rat: d X z m = (, ) dz 2 z (s Appndx or a proo). (5) In contrast to th prct comptton cas whr d = 0, undr monopsony th dz xchang rat racts to xognous shocks to th nt supply o orgn currncy. Intutvly, whn thr s an xognous shock, th monopsonst dos not absorb all th xcss supply bcaus h consdrs th ct o hs purchass on th xchang rat. Thus, th shock s not compltly absorbd by th tradng volum and th qulbrum xchang rat wll adjust to a nw qulbrum. For xampl, thr s an xognous ncras n xports (or trms o trad) th orgn currncy supply wll rs by mor than th dmand o th monopsonst, and thr wll b an xcss o supply that apprcats th xchang rat. Th nw qulbrum xchang rat wll b lowr than n prct comptton, whr th addtonal supply s absorbd totally by PFM. D. COURNOT OIGOPSONY In ths cas th markt structur s charactrzd by N pnson und managrs who act smultanously and tak th valu o th othrs nt purchass as gvn. Th optmzaton problm or a PFM j s as ollows:
34 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Max l0 + x ( 0 X, x ( X, ( + )+ x + x j st.. X= j j j j j N h= x h Th rst ordr condton s now: = + a x x c j 0 j j (6) Whr th xchang rat xtatons among PFM ar allowd to dr, so that: j = j + +. Rsult 7: As n th monopsony cas, th qulbrum xchang rat undr a Cournot olgopsony structur s lowr than n prct comptton whn PFM hav a postv nt dmand or orgn currncy, ( x x )<. 0 j j 0 As bor, ths rsult s xpland by th xstnc o markt powr. Agnts wth markt powr consdr th ct o thr transactons on th xchang rat and thus on th valu o thr purchass (or sals), nducng thm to rduc thr tradng volum and modratng th mpact on th xchang rat. Ths s agan th mtgatng ct o markt powr. Th cts o gnralzd or sc changs n xtatons can b analyzd undr drnt scatons o th modl, dpndng on whthr thr s convrgnc or dvrgnc o xtatons. Rsult 8: I xtatons convrg j = h = h j, th mpact o a gnralzd shock to xtatons or to th ntrst rat drntal on th qulbrum xchang rat s smallr n olgopsony than n prct comptton: dc N = (7) d N + In addton, gvn that d c N = > or N >, th ct o a gnralzd shock d N + 2 to xtatons on th olgopsony qulbrum xchang rat s gratr than n monopsony (s Appndx or a proo).
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 35 As PFM wth markt powr know that any chang n thr dmand or orgn currncy has an advrs ct on th valu o thr purchass (or sals), thy rspond lss to ths shocks than whn ths ct s not ralzd (.g. undr prct comptton). Howvr, th mpact on th xchang rat s gratr than n monopsony bcaus mor, smallr ndvduals ar ractng smultanously to a gnralzd shock to xtatons (or to th ntrst rat drntal), takng th actons o othrs as gvn. Hnc, th sum o th changs n th quantts tradd by all th Cournot compttors s largr than th chang n th quantty tradd by th monopsonst n th ac o th sam shock. Rsult 9: I xtatons dr among PFM ( j h h j), th mpact o a shock to th xtatons o only on PFM on th qulbrum xchang rat s lowr than th mpact o a gnralzd shock to xtatons: d d c j = N + dc Snc = < N >, th ct o a chang n th xtatons o on dj N + 2 agnt on th qulbrum xchang rat undr Cournot comptton s lss than th ct n monopsony (s Appndx ). Ths rsult mpls that whn thr s a shock to th xtatons o only on ndvdual, th mpact on th Cournot qulbrum xchang rat wll b lss than n th cas whr all PFM ar ht smultanously by th sam shock. At th sam tm, th qulbrum xchang rat wll rspond mor to changs n th monopsonst s xtatons than to movmnts o th xtatons o only on agnt n th olgopoly structur. 5 Rsult 0: As n monopsony, an xognous shock on th orgn currncy nt supply has a ngatv mpact on th Cournot qulbrum xchang rat: (8) d ( X, c = dz ( N + ) z (9) 5 Although th dmand o th agnt wth changng xtatons wll vary mor than th monopsonst s dmand, th othr agnts wll ract nvrsly to th chang o th dmand o ths agnt. Thus th racton o th monopsonst s dmand s gratr than th chang n th total dmand o all th Cournot compttors.
36 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Howvr, snc N >, mor ndvduals can absorb th xcss supply, so that th qulbrum xchang rat has to adjust lss than n th monopsony cas. E. STACKEBERG OIGOPSONY In ths structur, thr ar N pnson und managrs wth markt powr, but now on o thm acts as a ladr and th othrs N - act as ollowrs. Ths s a squntal gam n whch th ladr movs rst and thn th ollowrs obsrv th ladr s acton and tak t nto account to mak thr dcsons. At th sam tm, th ladr consdrs th racton o th ollowrs n hs optmal dcson. In ths sttng, t s assumd that all ollowrs rspond n th sam way to th ladr s transacton and that thr ar two cts o ths acton: a drct ct that acts only th tradng volum o th ollowrs and an ndrct ct that nluncs th ollowrs xtatons. In th rst cas, th ollowrs wll ract nvrsly to a chang n th ladrs tradng volum, whl n th scond cas th chang n th ollowrs nt dmand s n th sam drcton as th ladr s movmnt. Th analyss o ths squntal gam bgns by th ollowrs dcson, whch s th sam as th dcson o a Cournot compttor. Th maxmzaton problm o a tyal ollowr s thn: Max l0 + x ( 0 X, x ( X, ( + )+ x ( x ) + x st.. X = x + x whr th ollowrs xtaton may dpnd on th ladr s transacton = ( x ), thr xst th abov mntond ndrct ct. I not, s xognous. Th rst ordr condton mpls that: S = ( x)+ a x0 x (0) + In gnral: ( x )= ( x ). ( + ) Now th ladr s maxmzaton problm s gvn by:
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 37 Max l0 + x ( 0 X, x ( X, ( + )+ x + x st.. X= x x x + and hs optmal dmand s: ( X, x = x0 + a + ( N ) wth = +. ( + ) Th qulbrum xchang rat n th Stacklbrg olgopsony ollows rom quatons (0) and (). Rsult : I only th drct ct o th ladrs transacton s takn nto account ( s xognous) th mpact o a gnralzd shock to xtatons xtatons convrg j = h = = h j on th Stacklbrg qulbrum xchang rat s lowr than n prct comptton: ds N = 2 (2) d 2N Ths conrms th xstnc o a mtgatng ct o markt powr on th qulbrum xchang rat (s Appndx or a proo). Howvr, ths mpact s hghr than n th monopsony and Cournot qulbra. Although th ladr s racton to a gnralzd shock to xtatons s smlar to th monopsonst s, th convrgnc o xtatons mpls that th ollowrs wll ract n th sam way as th ladr (dmand mor orgn currncy n th ac o ncrasd xtatons o dprcaton). Ths hghr dmand mpls that th xchang rat wll chang mor than n monopsony. In contrast to th Cournot playr, who taks th othrs dmand as gvn, th Stacklbrg ladr knows that any ncras n hs dmand wll rduc th dmand o th ollowrs (drct ct). Howvr, h also knows that ths dcrasng n th ollowrs dmand s partally ost by th gnralzd xtatons o dprcaton. Hnc, th ladr wll dmand mor than th Cournot compttor, ollowrs wll dmand lss, but th ct o th ormr wll b strongr. Th total dmand wll b hghr than undr Cournot and thus th qulbrum xchang rat. ()
38 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Rsult 2: Whn only th drct ct s takn nto account ( s xognous), th mpact o a sc shock to th ladr s xtatons ( xtatons dvrg h j ) on th qulbrum xchang j h rat s smallr than th mpact o a gnralzd shock to xtatons: ds = d 2 (3) Ths can b vrd by comparng quatons (2) and (3). I only th ladr xprmnts an ncras o hs xtatons o dprcaton, only hs dmand or orgn currncy wll rs. Ths contrasts wth th cas o convrgng xtatons, whr a sht n th lattr s gnralzd and nducs a chang n th dmand or orgn currncy o all th agnts. Rsult 3: Whn both th drct and th ndrct cts ar consdrd ( = ( x ) ), th mpact o a shock to th ladr s xtatons on th qulbrum xchang rat s th sam as n th cas whr only th drct ct s assumd ( s xognous) and xtatons dvrg ( h j ): j h ds = d 2 (4) In ths cas, th ladr consdrs th ct that hs transacton has on th ollowrs xtatons. An ncras n th ladr s xtatons o dprcaton nducs a rs n hs dmand or orgn currncy. Ths, n turn producs hghr ollowrs xtatons o dprcaton and an xpanson o thr dmand or orgn currncy. Ths racton o th ollowrs mpls an addtonal upward prssur on th xchang rat. Thus, n ordr to avod ths advrs ct on th cost o hs purchass, th ladr wll modrat th ncras n hs dmand. Ths mpls that th mtgatng ct o markt powr s strongr snc th ladr not only trs to allvat th drct mpact o hs addtonal dmand on th markt, but also has to ost th ct o hs actons on th ollowrs xtatons. Rsult 4: Th mpact o an xognous shock to th orgn currncy nt supply on th Stacklbrg qulbrum xchang rat s ngatv and lss than th mpact undr monopsony: 6 6 Ths mpact s th sam whn thr both cts or only th drct ct ar consdrd.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 39 d ( X, S = (5) dz 2N z Ths rsult ollows rom comparng quaton (5) wth quaton (5). Intutvly, as n th cass o monopsony and Cournot comptton, an xognous chang n th orgn currncy nt supply s not totally absorbd by PFM bcaus o th mtgatng ct. In sum, th qulbrum rsults undr th drnt markt structurs ar shown n Tabl. Th xstnc o th mtgatng ct o markt powr can b sn by comparng th prct comptton outcoms wth th outcoms undr othr markt structurs. IV. CONCUSIONS Th concrn about th dstablzng ct o PFM on th orgn xchang markt (gvn th ncrasng sz o thr portolos) can b allvatd by th xstnc o a mtgatng ct o markt powr. Ths ct mpls that PFM wth markt powr wll consdr th mpact that thr actons hav on th xchang rat, on th valu o thr orgn currncy transactons and on othr agnts dcsons. For ths rason, PFM wll modrat th volum o thr transactons and, thror, th prssur on th xchang rat. Although th nvstmnt dcsons o PFM can b domnatd by th short run xtd rturns (as suggstd by Zahlr [2005]), and gvn that non-undamntals actors could domnat th short run movmnts o th xchang rat (as th vdnc suggsts), th act that markt powr has a mtgatng ct on th xchang rat can mak th orgn xchang markt lss pron to sulatv movmnts. Thror, th argumnt that larg shts o th orgn currncy holdngs n th PFM portolos ovr a short span may dstablz th orgn xchang markt s waknd whn thr s markt powr. Ths papr shows that, du to th mtgatng ct, PFM wth markt powr wll rspond lss to shocks to xchang rat xtatons (or to th ntrst rat drntal) than undr prct comptton. Furthrmor, n contrast to th compttv cas, th qulbrum xchang rat wll rspond to xognous shocks to th orgn currncy nt supply (.g. xports changs, trms o trad, tc ). In prct comptton
40 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Tabl Summary o Rsults Markt structur Prct comptton Equlbrum xchang rat = + ( + ) Shock on xtatons d d = d dz Exognous shock on nt supply o orgn currncy =0 Monopsony = + a X X m 0 dm = d 2 d X z m = (, ) dz 2 z Cournot = + a x x c j 0 j j Whr: j = j ( + ) ( + ) I xtatons convrg j = h = h j : d N d c c = d N + dz ( X, = I xtatons dvrg ( N + ) z ( j h h j): dc = d N + j I xtatons convrg = = = j h Stacklbrg = ( x )+ a x x S 0 ( X, x= x0 + a + ( N ) Drct ct h j d d S N = 2 2N I xtatons dvrg h j j h ds = d 2 Drct ct: d ( X, S = dz 2N z Indrct ct: ds = d 2 Indrct ct: d ( X, S = dz 2N z X z Nots: a = (, ) s a postv constant. N s th numbr o pnson und managrs n th markt. z corrsponds to xognous varabls that act th nt supply o orgn currncy (.g. xports, trms o trad, tc.). Th drct ct rrs to th mpact o th ladr s transacton on th tradng volum o th ollowrs, so that s xognous.. Th ndrct ct rrs to th nlunc o th ladrs transacton on th ollowrs xtatons = ( x ) Sourc: Authors calculatons.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 4 PFM absorb compltly th addtonal supply untl th ntrst rat party condton s rstablshd. Howvr, whn PFM hav markt powr, th xcss supply gnratd by an xognous shock s not compltly absorbd by th PFM, bcaus thy consdr th ct o thr purchass on th xchang rat.
42 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 APPENDIX A. PERFECT COMPETITION WITH RISK AVERSION I ach PFM s rsk avrs wth rst to th uncrtan utur xchang rat th problm s now: Max Eu l0+ x0 x ( )+ x + x + In th sc cas o a CARA utlty uncton and a utur xchang rat normally dstrbutd N E, Var, w hav: ( ) EW VarW Max Eu W = γ 2 x + whr: W = l + x x ( x 0 0 )+ + problm s quvalnt to th ollowng: Max E W γ Var ( W ) x 2 wth: γ EW =( l+ x x) ( + )+ x + E 0 0 2 Var W = x( + ) Var s also normally dstrbutd and th Th qulbrum xchang rat s obtand rom th rst ordr condton as: = E( + ) x ( + ) ρ γ x + ρ ( x)= ( + ) 2 Var whr γ s th cocnt o Absolut Rsk Avrson, whch s constant. B. THE MONOPSONY CASE Th monopsonst PFM s optmzaton problm s:
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 43 Max 0+ X ( 0 X, X( X, ( + )+ X + X Th rst ordr condton mpls: m = + X z X X ( + ) + (, ) ( 0 ) In th prct comptton qulbrum, t was ound that monopsony qulbrum xchang rat can b wrttn as: = +. Hnc, th ( + ) X X X m= + (). ( 0 ) (A.) whr by smplcty t s assumd that ( X, z ) = a > 0, thus: d a dx m =. dz dz X z Equaton (A.) mpls that: X = X + (, ) 0, and rom hr w hav: a dx X z dx X z = (, ) + (, ) dz a dz z Ths xprsson can b smpld to obtan: dx dz X z = (, ) 2a z Substtutng t n d m, w hav that th mpact o a chang n th xognous varabls on th monopsony qulbrum xchang rat s gvn dz by: d X z m = (, ) dz 2 z From (A.) w hav d s: d dx d = 2a m = a dx, whr dx d d (A.2) X z dx = (, ) whch a d Thus, th ct on th qulbrum xchang rat undr monopsony o a shock to th ntrst rat drntal or to th xchang rat xtatons s: dm = d 2 (A.3)
44 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 C. THE COURNOT OIGOPSONY CASE Th optmzaton problm or th PFM j, who taks as gvn th orgn currncy dmand o othr PFM ( x h h j ), s as ollows: Max l0 + x ( 0 X, x ( X, ( + )+ x + x j st.. X= j j j j j N x h h= th rst ordr condton s: X x x c = j + (). ( j j) (A.4) 0 In ths scaton th xchang rat xtatons among PFM ar allowd to dr, ( + ) so that: j =, whr j ( + ) j corrsponds to th xtatons o th j-pfm. j X z Th orgn currncy dmand o ach PFM s gvn by: x j = (, ) + x0 j. a From hr w hav: j X z dx X z = (, ) + (, ) dz a dz z whr ( X, z ) = a and dx N N j j = = dz dz dz j= j ( X, = dz a( N + ) z. Thus: From (A.4) w hav: d a c j =. dz dz And rom th prvous rsults th mpact o an xognous chang n th orgn currncy nt supply on th qulbrum xchang rat s gvn by: d ( X, c = dz ( N + ) z (A.5) Th cts o gnralzd or sc changs n xtatons can b analyzd undr drnt scatons o th modl, dpndng on whthr thr s convrgnc or dvrgnc o xtatons.
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 45. EXPECTATIONS CONVERGENCE In ths cas thr s not drnc n xtatons, thn: j = h = h j and X z th orgn currncy dmand o ach PFM s gvn by: x j = (, ) + x0 j a. From hr w hav: j X z dx = (, ) d a d whr dx N = and j = j. Thus: d d d d d j = = a( N + ) From (A.4) and takng nto account that j = h = h j, w hav: d d c a j = d (A.6) Substtutng j n th xprsson abov, th mpact o a gnralzd shock to xtatons (or to th ntrst rat drntal) on th qulbrum xchang rat d s: dc N = d N + (A.7) 2. EXPECTATIONS DIVERGENCE Consdr now that xtatons dr among PFM, t mans that h j. j X z From th orgn currncy dmand o ach PFM: x j = (, ) + x th ollowng rsults o comparatv statcs: a j X z dx = (, ) dj a d j whr: N X, z dx j = a, = = + ( N ) and d d d d d j j X z dx = (, ) a d j = j j j j h 0 j, w hav
46 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 dx It mpls that: dj j N = d a( N+ ) j j =. And substtutng ths rsult n N d d From th rst ordr condton (quaton A.4) w obtan: j j j w obtan: (A.8) d d c j a j = d j Thn substtutng (A.8) th mpact o a shock to th xtatons o only on PFM on th qulbrum xchang rat s: d d c j = N + (A.9) D. THE STACKEBERG OIGOPSONY CASE Th maxmzaton problm o a tyal ollowr s gvn by: Max l0 + x ( 0 X, x ( X, ( + )+ x ( x ) + x st.. X = x + x Th rst ordr condton o ths problm s: ( + ) S = ( x)+ a x0 x (A.0) = whr x x whch ndcats that th ollowrs xtatons ( + ) may dpnd on th ladr s transacton x, and ( X, z ) = a. It mpls that th orgn currncy dmand o a tyal ollowr s: x ( x ) ( X, = + x a From ths quaton w hav: 0 d X z dx = (, ) a
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 47 And rom X = x+ x w know that: dx N = +, whr = = and all ollowrs rspond n th sam way to th ladr s acton: =. Thus: dx = + ( N ). Substtutng dx n th quaton abov w obtan th ct o a chang n th ladr s transacton on th dmand o a tyal ollowr: d = (A.) an N Ths ollowrs racton s takn nto account by th ladr n hs maxmzaton problm: Max l0 + x ( 0 X, x ( X, ( + )+ x + x st.. X= x x x + whr th rst ordr condton = + a + ( N ) ( x0 x) mpls that th ladr s optmal dmand s: ( X, x = x0 + a + ( N ) wth = ( + ). ( + ) (A.2) By smplcty assum that th ollowrs xtatons hav th ollowng scaton: ( x)= 0 + α x (A.3) thus = d ( 0 + α x ) ( whr d = + ) s th ntrst rat drntal. ( + )
48 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 Th mpact o a shock to th ladr s transacton on th ollowrs racton s analyzd undr two scnaros dpndng on whch ct o th ladr s transacton w ar consdrng: th drct ct that acts only th tradng volum o th ollowrs or th ndrct ct that nluncs th ollowrs xtatons.. DIRECT EFFECT In ths cas th transacton o th ladr (x ) dos not act th xtatons o th ollowrs, whch mans that α=0 and thus = 0. Hnc, th ladr s transacton acts only th tradng volum o th ollowrs. Th squntal gam mpls that: ( x x a + ) = + X z ( + ) 0 0, (A.4) From ths quaton w drv: X z dx = (, ) a whr: dx = + ( N ) and ( X, z ) = a. Thus th ollowrs racton s gvn by: = (A.5) N Takng nto account ths rsult th ladr s transacton s obtand rom quaton A.2: ( x x N X z (, )) = 0+ (A.6) a From th ollowrs rst ordr condton (quaton A.0) and gvn that hav: S and = + a x x ( + ) + 0 0 d a S = dz dz = 0 w
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 49 Whr dz s obtand rom (A.4) as: dz X z X z = (, ) + ( N ) + (, ) a dz dz z and rom (A.6) w hav: dz N X z X z = (, ) + ( N ) + (, ) a dz dz z Gvn that ( X, z ) = a t can b wrttn as: N( N ) N ( X, = dz ( N + ) dz a( N+ ) z And substtutng ths rsult n th prvous xprsson or dz dz ( X, = 2Na z w obtan: Fnally, th mpact o an xognous chang n th orgn currncy nt supply on th qulbrum xchang rat s gvn by: d ( X, S = (A.7) dz 2N z Th mpact o a shock to xtatons s analyzd undr two drnt assumptons. Frst, whn xtatons convrg (t mans that thy ar th sam or vrybody) ths shock s takn as a gnralzd shock. Scond, whn thr s dvrgnc n xtatons ths shock s takn as a shock only to th ladr s xtatons. a. Extatons convrgnc Th mpact o a gnralzd shock to xtatons (or to th ntrst rat drntal) s analyzd undr th assumpton o convrgnc n xtatons, whch mpls that th ladr and th ollowrs hav th sam xtatons about th xchang rat: ( + ) = = =. 0 ( + ) From (A.4) and (A.6) w hav:
50 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 d X z dx = (, ) a d N X z dx = (, ) d a d whr: dx = + ( N ) and ( X, z ) = a. Thus: d d d = d N a d (A.8) N = ( N ) d ( N + ) a d (A.9) Substtutng A.8 n A.9 w obtan: d = 2a and substtutng ths rsult n A.8: d = 2aN From th ollowrs condton S = + a x0 x w hav: d d S a = d Thn substtutng th mpact o a gnralzd shock to xtatons on th d Stacklbrg qulbrum xchang rat s: d d S N = 2 2N (A.20) b. Extatons dvrgnc In th cas whn xtatons dr btwn ollowrs and th ladr ( ), th ct o a shock to th ladr s xtatons s obtand as ollows. From quaton (A.4) and (A.6) w hav:
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 5 d d whr d d X z dx = (, ) a d N X z dx = (, ) a d dx d = N d = + ( N ) and ( X, z ) = a. Thus: d d (A.2) N = ( N ) ( N + ) a d (A.22) Substtutng (A.2) n (A.22) w obtan: N = d 2 a and substtutng ths rsult n (A.2): (A.23) d = 2a Now, rom th ollowrs rst ordr condton S = + a x0 x w hav: d a x S = d x whr s gvn abov. Thus th ct o a shock to th ladr s xtatons on th Stacklbrg qulbrum xchang rat s gvn by: ds = (A.24) d 2 Comparng ths rsult wth quaton (A.20) w can conclud that th mpact on th xchang rat o a gnralzd shock s gratr than th mpact o a chang on th ladr s xtatons: d d S 2N ds = > = 2N d 2
52 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 2. DIRECT AND INDIRECT EFFECTS Addtonal to th drct ct w consdr an ndrct ct o th ladr s transacton that acts th xtatons o th ollowrs, thn α>0 and = d ( 0 + α x ). Ths ndrct ct lads to an addtonal chang n th ollowrs dmand or orgn currncy n th sam drcton as th ladr movmnt. In ths cas: d = d α From ths rsult and rom quaton (A.) w obtan: an d = α (A.25) N Now substtutng th prvous rsult n th rst ordr condton o th ladr s maxmzaton problm (quaton A.2) w hav: ( X, x = x0 + (A.26) a + ( N ) an d α N From th ollowrs rst ordr condton (quaton A.0) and gvn that ( x )= d ( 0 + α x ) w hav: S = d( 0 + α x)+ a( x0 x ) and d d a S = α dz dz dz whr dz dz s obtand rom x ( x ) ( X, = + x as: 0 a d X z dx X z = (, ) (, ) a dz dz z Gvn that d dz d a = α an dz = dα, ( X, z ) = a X, z z Substtutng (A.28) n (A.27) w hav: and dx dz (A.27) = + ( N ) w obtan: dz dz (A.28)
ENSAYOS SOBRE POÍTICA ECONÓMICA, VO. 25, NÚM. 54, EDICIÓN JUNIO 2007 53 d ( S N ) d + a X z = α + (, ) (A.29) dz N dz N dz From (A.26): X z dx X z = (, ) + (, ) dz a N an d dz z + ( ) α N Gvn ( X, z ) = a, dx = + ( N ) and rom (A.28) t can b wrttn as: dz dz dz ( X, = (A.30) dz 2 ( N ) α d a z + Fnally, substtutng ths rsult n (A.29) th mpact o an xognous chang n th orgn currncy nt supply on th qulbrum xchang rat s gvn by: d ( X, S = (A.3) dz 2N z Bcaus thr s an ndrct ct o th ladr s transacton on th ollowrs xtatons th mpact o a shock to xtatons s analyzd only undr dvrgnc o xtatons ( ), so rom quaton (A.26) w hav: d X z dx = N d X d a + ( ) (, ) α N a ( X, whr = a, (A.25). dx d = + ( N ) and d d s gvn by Thn: d N = 2 a+ α d ( N ) ( x ) ( X, = + Now rom x x w hav: 0 a d X z dx = (, ) d a d d Rplacng th xprssons or ( X, z ) dx,, d (A.32) w obtan: d (A.32) and quatons (A.25) and
54 PENSION FUND MANAGERS AND THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET PP. 22-55 α d a = (A.33) d 2a( a+ α d ( N ) ) Fnally, rom th ollowrs rst ordr condton S = d( 0 + α x)+ a x0 x w hav: d d S d a = α d d whr and ar gvn by quatons (A.25) and (A.32), rstvly. Thus, d th ct o a shock to th ladr s xtatons on th qulbrum xchang rat s: ds = d 2 (A.34)
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