Interet rate party and no arbtrage a equvalent equlbrum condton n the nternatonal fnancal aet and good market Stefano Bo, Patrce Fontane, Cuong Le Van To cte th veron: Stefano Bo, Patrce Fontane, Cuong Le Van. Interet rate party and no arbtrage a equvalent equlbrum condton n the nternatonal fnancal aet and good market. Document de traval du Centre d Econome de la Sorbonne 016.63 - ISSN : 1955-611X. 016. <halh- 01391013> HAL Id: halh-01391013 http://halh.archve-ouverte.fr/halh-01391013 Submtted on Nov 016 HAL a mult-dcplnary open acce archve for the depot and demnaton of centfc reearch document, whether they are publhed or not. The document may come from teachng and reearch nttuton n France or abroad, or from publc or prvate reearch center. L archve ouverte plurdcplnare HAL, et detnée au dépôt et à la dffuon de document centfque de nveau recherche, publé ou non, émanant de établement d enegnement et de recherche frança ou étranger, de laboratore publc ou prvé.
Document de Traval du Centre d Econome de la Sorbonne Interet rate party and no arbtrage a equvalent equlbrum condton n the nternatonal fnancal aet and good market Stefano BOSI, Patrce FONTAINE, Cuong LE VAN 016.63 Maon de Scence Économque, 106-11 boulevard de L'Hôptal, 75647 Par Cedex 13 http://centredeconomeorbonne.unv-par1.fr/ ISSN : 1955-611X
Interet rate party and no arbtrage a equvalent equlbrum condton n the nternatonal fnancal aet and good market Stefano Bo EPEE, Unverty of Evry Patrce Fontane Eurofda, CNRS, Leonard de Vnc Reearch Center Cuong Le Van IPAG Bune School, CNRS, PSE, VCREME Aprl 1, 016 Abtract In th paper, we conder a two-perod conumpton model wth many fnancal aet. In the prt of Hart [5], conumer purchae fnancal aet n perod 0 and conume n perod 1. We dffer from Hart by conderng that each agent a country. We provde condton for the extence of an equlbrum n both nternatonal fnancal aet and good market. Frt, we ntroduce a weaker noton of Uncovered Interet rate) Party UIP) called Weak Uncovered Interet rate) Party WUIP), and we how t equvalence to the no-arbtrage condton n the nternatonal fnancal market. Second, we ntroduce the concept of common no arbtrage and we how t equvalence to UIP. Thee reult brdge concept of no arbtrage n general equlbrum theory and fnancal mcroeconomc, and of nteret party n nternatonal fnancal macroeconomc. In a mult-country model wth many currence and only one good, we ntroduce a country-pecfc converon rate whch tranform the return on aet valued n local currency nto unt of phycal good. We the defne alo the exchange rate between currence of dfferent countre. The UIP condton requred for the extence of an equlbrum n both nternatonal fnancal aet and good market and for the extence of the Law of One Prce. We are grateful to the two Referee for ther thoughtful remark and queton. We are alo grateful to Saqb Jafarey for valuable comment and to the partcpant of the NBER General Equlbrum Conference held n Bloomngton on September 01. Th reearch ha been conducted a part of the project LABEX MME-DII ANR11-LBX-003-01). Correpondng author: levan@unv-par1.fr. 1 Document de traval du Centre d'econome de la Sorbonne - 016.63
Keyword: general equlbrum, no arbtrage, return on fnancal aet, exchange rate, Law of One Prce, Uncovered Interet rate Party. JEL Clafcaton: D53, F31, G11, G15. Document de traval du Centre d'econome de la Sorbonne - 016.63
1 Introducton In th paper, we revt a two-perod fnancal model by Hart [5] to ntroduce the nternatonal trade. The frt perod wll be called perod 0, the econd, perod 1. In perod 0, agent buy or ell fnancal aet to dverfy ther portfolo and maxmze an expected utlty functon. They know the tate of nature of perod 1 and have belef on the occurrence of thee tate. In perod 1, they buy or ell conumpton good wth ther endowment and the gan from fnancal nvetment. Thee gan are the return on the fnancal aet the agent receve n each tate of the nature. In our paper, we have many aet but only one good for conumpton purpoe. There are many countre whch have acce to the ame fnancal market. In one-country model, t equvalent to expre the return n term of conumpton good or currency money), whle n mult-country model a our, the prce may dffer from one country to another becaue of the exchange rate. Therefore, we requre an ndependent numerare. The return on fnancal aet are ntally valued n local currency. However, each agent form expectaton on the converaton rate between the currency of her country and the conumpton good. We ntroduce alo the exchange rate between the currency of her country and a reference country, ay country 1. We aume that agent ue ther expectaton on the gan of perod 1 valued n term of conumpton good to purchae fnancal aet n perod 0. Each agent maxmze a utlty functon n perod 0 dependng on her conumpton n any tate nature. We defne the condton for the extence of a general equlbrum of both the trade balance and the nternatonal fnancal market. Frt, we addre the ue of equlbrum extence n the nternatonal fnancal market. We apply the uual no-arbtrage condton of general equlbrum fnancal lterature, whch refer to the extence of approprate fnancal aet prce that prevent agent from makng gan wthout cot. In the prt of Allouch, Le Van and Page [1], Dana, Le Van and Magnen [3], Page and Wooder [6], Werner [8] among other, we how that, under the trct concavty of utlty functon, no-arbtrage condton are neceary and uffcent to enure the extence of an equlbrum n the nternatonal fnancal market. In a econd tage, we ntroduce two condton. The frt one the Uncovered Interet rate Party UIP) whch potulate the extence of exchange rate under whch the return on the fnancal aet of all the countre become equal n each tate of nature f they are expreed n term of conumpton good. A weaker noton of UIP, called Weak Uncovered Interet rate Party WUIP), potulate that ) there ext a ytem of exchange rate and ) all the countre hare the ame probablty dtrbuton over the tate of nature uch that the expected value of the return on fnancal aet n term of conumpton 3 Document de traval du Centre d'econome de la Sorbonne - 016.63
good are equal. Our frt man reult the equvalence of WUIP and the noarbtrage condton n the nternatonal fnancal market and the extence of an equlbrum n th market. We ntroduce the noton of common no-arbtrage fnancal prce whch potulate the extence of S fnancal prce vector q uch that, f the gan generated by the purchae of fnancal aet are potve n tate for any country, then the value of thee aet calculated wth the common no-arbtrage prce q wll be potve. Our econd man reult to prove that UIP equvalent to the extence of a common no-arbtrage condton. Thee reult brdge concept of no arbtrage n general equlbrum theory and fnancal mcroeconomc, and of nteret party n nternatonal fnancal macroeconomc. Our thrd reult the proof that f UIP hold then there ext an equlbrum n both fnancal and good market where the prce of fnancal aet are no-arbtrage prce and the prce of conumpton good n term of numerare atfy the Law of One Prce LOP). Moreover, the trade balance hold n term of nomnal value and n term of phycal good a well. UIP mple WUIP whle the convere not true. 1 Then, one may wonder what apect of equlbrum are preerved under WUIP. Whle UIP ental the equlbrum n both market, under WUIP, the equlbrum of trade balance fal. However, under the probablty dtrbuton of WUIP, the expected value of the trade balance turn out to be zero. Th reult the lat of the paper and hed lght on the equlbrum mplcaton of UIP and WUIP, and ther nterplay. In order to undertand better the role of UIP, we provde three example where th party fal. In the frt one, the no-arbtrage condton hold n the nternatonal fnancal market and an equlbrum ext but t no-trade. In partcular, the LOP not verfed. In a econd example, the no-arbtrage condton hold n the nternatonal fnancal market and an equlbrum ext a well, but the good market mbalanced. In the lat example, the et of noarbtrage prce for the nternatonal fnancal market dffer acro the countre and do not nterect: thu, the no-arbtrage condton fal and there ext no equlbrum. The paper organzed a follow. In Secton, we ntroduce the man aumpton of the model, whle, n Secton 3 we defne the equlbrum. The noarbtrage condton n the nternatonal fnancal market are defned n Secton 4, whle ome noton of party are ntroduced n Secton 5. In Secton 6, we lnk the noton of no-arbtrage and UIP, and we prove two other man 1 Country-pecfc return change over the tate of nature n the cae of equte whle they don t n the cae of bond. Therefore, WUIP mple UIP n the cae of bond. 4 Document de traval du Centre d'econome de la Sorbonne - 016.63
reult of the paper, that the equvalence 1) between the extence of a no-arbtrage fnancal aet prce and WUIP, and ) between the extence of common no-arbtrage condton and UIP. The extence of equlbrum eventually proved n Secton 7 through two equvalent theorem. In Theorem 1, UIP a uffcent condton to the extence of an equlbrum. In Theorem, UIP replaced by the extence of a common no-arbtrage fnancal aet prce. In Secton 8, we compare the equlbrum mplcaton of UIP and WUIP. The mplcaton of UIP falure are recondered n Secton 9 through three example. In Example 1, the only equlbrum we obtan a no-trade equlbrum for both market. In Example, UIP alo fal but the no-arbtrage condton hold for the nternatonal aet market. There ext an equlbrum n the fnancal market but no equlbrum n the good market. In Example 3, the no-arbtrage condton fal and no equlbrum ext n both market. Secton 10 conclude. Model We conder a pure exchange economy where fnancal aet and good are traded n nternatonal market. Before addreng the equlbrum ue, we ntroduce notaton and aumpton, and we decrbe avng dverfcaton and conumpton. Focu on a two-perod exchange economy wth many countre. Fnancal aet are traded n perod 0 and good are conumed n perod 1. The conumpton good uppoed to be perfectly tradable. The repreentatve agent of country {1,..., I} purchae K fnancal aet n perod 0 to mooth conumpton n perod 1 acro S tate of nature. At the begnnng of perod 1, th agent endowed wth an amount of conumpton good. Th amount depend on the tate of nature. Snce we are n preence of many currence, we aume the extence of a numerare uch a gold or one of the currence. The role of th numerare to value the aet purchae and the conumpton good n every country..1 Notaton For the ake of mplcty and readablty, we preent all the varable of the model at the begnnng and currency hould be undertood a currency of country. We ntroduce the fnancal de of the economy through a compact notaton for aet prce and quantte. q q 1,..., q K ) a row of fnancal aet prce where q k denote the prce of aet k n term of numerare n perod 0. x x k) the K I matrx of portfolo where x k denote the amount of 5 Document de traval du Centre d'econome de la Sorbonne - 016.63
fnancal aet k n the portfolo of agent. Column x x 1,..., T K) x the portfolo of the repreentatve agent of country. R Rk) the S K matrx of return where Rk 0 denote the return on fnancal aet k n the tate of nature. R the th row of the matrx. Return Rk correpond to return on aet k n country and tate and are valued n currency. They are expected n perod 1 wth a probablty π. ɛ ɛ ) the I S matrx of converon rate where ɛ denote the converon rate between currency and the phycal good n the tate of nature. Actually, one unt of phycal good, n tate, equal ɛ unt of currency. Therefore, ɛ > 0 for any and any. We denote ɛ ɛ 1,..., S) ɛ the th row of the matrx. Thee converon rate are actually the expected prce of commodty n term of currency. The matrx ɛ ) exogenouly gven. We obtan the expected exchange rate between currency 1 and currency by the expreon τ = ɛ 1 /ɛ. τ τ) the I S matrx of exchange rate where τ denote the exchange rate between currence 1 and currency n the tate of nature. Actually, one unt of currency, n tate, equal τ unt of currency 1. Obvouly, τ > 0 for any and any. We denote τ τ1,..., τ S ) the th row of the matrx. Of coure, the frt row a vector of unt: τ 1 = 1 for any. 3 Gven the ytem of converon rate ɛ, let R denote the return on fnancal aet from the country valued n phycal good. More explctly, R R/ɛ. Thee return are alo expected return n perod 1. We ntroduce now a compact notaton for belef, prce and quantte on the real de of the economy. c c ) the S I matrx of conumpton where c denote the amount of good conumed by agent n the tate of nature n perod 1. c c 1,..., ) T c S the conumpton column of agent. e e ) the S I matrx of endowment where e denote the endowment nature provde to agent n the tate n perod 1. e e 1,..., T S) e the endowment column of agent. π π) the I S matrx of belef where π denote the belef of agent about the occurrence of tate. The ndvdual row of belef π π1,..., ) π S le n the S-unt mplex. p p ) the I S matrx of good prce, n perod 1. The prce p the requred quantty of numerare for the purchae of one unt of good, n country and at tate. Return are alo degned a payout or cah flow. 3 We have τ ɛ = ɛ 1 for any. Th mean that, ex-ante, the commodty prce atfy the LOP f the currency 1 the numerare. 6 Document de traval du Centre d'econome de la Sorbonne - 016.63
Notce that vector q, R, τ, p, π are row, whle vector x, c, e are column. In the artcle,,, k wll denote unambguouly the explct um I =1, S =1, K k=1.. Portfolo The agent behavor come down to a avng dverfcaton to fnance future conumpton. In perod 1 and tate, agent exchange ther endowment accordng to ther portfolo: c = e + R x 1) where R R/ɛ. We recall that the return R on portfolo are valued n term of phycal good. Preference of agent are ratonalzed by a Von Neumann-Morgentern utlty functon weghted by ubjectve probablte: π u c ), where u the utlty functon and c the quantty of her conumpton good. A a conumpton amount, c requred to be nonnegatve and the utlty functon defned on the nonnegatve orthant. Therefore, the et of portfolo X { x R K : for any, e + R x 0 } ) We ntroduce the et of conumpton allocaton Y + generated by the purchae of a portfolo x of fnancal aet: Y + { c R S : there x R K uch that c = e + R x 0 for any } In perod 0, gven the prce of fnancal aet q), any agent chooe a portfolo x whch maxmze her preference takng nto account the fnancal budget contrant: max πu e x X + R x ) 3) qx 0 The RHS of the budget contrant zero nce we conder agent net purchae..3 Aumpton In order to prove and characterze the extence of a general equlbrum wth fnancal aet, we ntroduce ome mld aumpton. The frt trplet of hypothee pecfe the fnancal fundamental return); the econd trplet pecfe the real fundamental endowment and preference). 7 Document de traval du Centre d'econome de la Sorbonne - 016.63
Aumpton 1 For any country and any tate, k R k > 0. Aumpton For any country and any fnancal aet k, R k > 0. When Aumpton 1 fal, there a country and a tate where any fnancal aet k yeld Rk = 0. In th cae, the repreentatve agent of country wll conume her endowment n the tate. When Aumpton fal, there an aet k yeldng Rk = 0 n any tate of nature n country : the repreentatve agent wll refue to buy th fnancal aet. The followng aumpton tronger and mple Aumpton. Aumpton 3 For any country and any portfolo x 0, the portfolo return nonzero: R x 0. Aumpton 3 mean that there are no nonzero portfolo wth a null return n any tate of nature. In other term, whatever the country we conder, the matrxe R and R are full rank wth rankr = rankr = K. 4 Aumpton 4 Endowment are potve: e > 0 for any agent and any tate. Aumpton 5 Belef are potve: π > 0 for any agent and any tate. Th aumpton mean that any repreentatve agent conder each tate poble. Eventually, preference are requred to atfy regular aumpton. Aumpton 6 For any agent, the utlty functon u concave, contnuou, trctly ncreang from R + to R. 3 Defnton of equlbrum Let u provde a general defnton of equlbrum, then to dtnguh the equlbrum n the aet market and n the good market. Fnally, we ntroduce an mportant and new noton of qua-equlbrum n the ene that trade balanced n expectaton but not tate of nature by tate. Defnton 1 equlbrum) Gven exchange rate, belef and endowment τ, π, e), return and preference R, u ) for any country, an equlbrum a lt of fnancal aet prce and allocaton p, q, c, x), wth p >> 0, q >> 0, and, for any, p = 1 and j q j = 1 uch that: ndvdual plan are optmal pont 1) and ) below) and market clear pont 3) and 4)). 4 Market completene mean that the column of R pan the whole pace R S rankr = S) and mple that a full nurance poble. Redundancy of aet mean that dm ker R > 0, that K > rankr. When market are complete and aet are not redundant, we have K = S = rankr. In th cae, the return matrx quare and nvertble. 8 Document de traval du Centre d'econome de la Sorbonne - 016.63
1) Portfolo are optmzed gven the fnancal aet prce q : n perod 0, any agent chooe a portfolo x whch maxmze her utlty functon under the fnancal budget contrant: max πu e x X + R x ) 4) q x 0 ) In perod 1, for any agent, c = e + R x olve the program max c Y + p c p e πu c ) 5) 3) The nternatonal fnancal aet market clear: x = 0. 4) The trade balance atfed n any tate of nature: c = e for any. We recall that the prce p the quantty of numerare neceary for the purchae of one unt of good, n country and at tate. We wll ee later that, n equlbrum, p R x = q x = 0. Therefore, n perod 1, we can actually replace the Arrow-Debreu contngent contrant p c p e by the contrant nvolvng the ncome obtaned wth the purchae of aet n perod 0: p c p e + p R x An equlbrum n our economy compoed by two equlbra, one n the nternatonal aet market and a econd one n the nternatonal conumpton good market. The defnton of thee equlbra are gven below. The program max x X π u e + R x ) ubject to qx 0 equvalent to the program max x X π u e + R x ) ubject to p e + R x ) p e and e +R x 0. Wth th two-tage approach, we can lnk the aet prce q) to the conumpton good prce p). Alo, th allow u to make a dtncton between the two equlbra, one for the nternatonal aet market, the other for the nternatonal good market. Defnton equlbrum n aet market) A lt of fnancal aet prce and aet portfolo q, x) wth q >> 0 and j q j = 1, an equlbrum for the nternatonal fnancal aet market f: 1) portfolo are optmzed gven the fnancal aet prce q, that any agent chooe her portfolo x whch atfe 15); ) the nternatonal fnancal aet market clear: x = 0. 9 Document de traval du Centre d'econome de la Sorbonne - 016.63
Defnton 3 equlbrum n conumpton good market) A lt of conumpton good prce and conumpton allocaton p, c) wth p >> 0,, p = 1, an equlbrum for the nternatonal conumpton good market f: 1) conumpton utlte are optmzed gven the conumpton good prce p, that any agent chooe her conumpton c whch atfe 16); ) the nternatonal conumpton good market clear: c any. = e Condton ) n the defnton of equlbrum n the good market ret on a trong requrement: the good) market clearng. It eem more reaonable to conder ntead an nternatonal good market whch clear at average : δ [ c e )] = 0, where the weght δ jut the potve probablty of the tate of nature. In other word, the expectaton of the trade balance under the probablty dtrbuton δ null: E δ [ c e )] = 0. Th account for the followng noton of δ-equlbrum. 5 Defnton 4 δ-equlbrum) Let δ >> 0 be a probablty dtrbuton on the et of tate of nature. Gven the exchange rate, belef and endowment τ, π, e), return and preference R, u ) for any country, a δ-equlbrum a lt of fnancal aet prce and allocaton p, q, c, x), wth p, q 0 uch that: ndvdual plan are optmal pont 1) and ) below) and market clear pont 3) and 4)). 1) Portfolo are optmzed gven the fnancal aet prce q : n perod 0, any agent chooe a portfolo x whch maxmze her utlty functon under the fnancal budget contrant: max πu e x X + R x ) q x 0 ) In perod 1, for any agent, c = e + R x olve the program max πu c ) c Y+ p c p e 3) The nternatonal fnancal aet market clear: x = 0. 4) The expectaton under δ of the trade balance atfed: E δ c ) = E δ e). We now provde condton to obtan uccevely an equlbrum n the aet market and an equlbrum n the nternatonal good market. 5 In general, a δ-equlbrum not an equlbrum becaue trade may be mbalanced. When there no uncertanty, δ-equlbrum equvalent to equlbrum. for 10 Document de traval du Centre d'econome de la Sorbonne - 016.63
For aet market, thee condton are uually called no-arbtrage condton. For the nternatonal good market, we requre well-known condton n the theory of nternatonal fnance and trade: the Uncovered Interet rate Party UIP) and the Law of One Prce LOP). Thee concept are well-defned n the cae of no uncertanty. We generalze them to the tochatc cae. 4 No arbtrage n the nternatonal fnancal market We frt defne the ueful portfolo for every agent. The ueful portfolo, called ueful vector n the general equlbrum lterature, were ntroduced by Werner [8]. Defnton 5 ueful portfolo) w a ueful portfolo for agent f, for any µ 0 and any x X, one ha: 1) x + µw X, ) π u e + R x + µw )) π u e + R x ). Let V { v R K : R v 0 for any } = { v R K : R v 0 for any } be the et of portfolo wth nonnegatve return n any tate of nature and W denote the et of ueful portfolo for agent. Propoton 1 Let Aumpton 6 hold. For any agent, V = W. Proof: See Appendx. We ntroduce a No-Arbtrage NA) condton for fnancal aet market. Defnton 6 A vector q a no-arbtrage fnancal aet prce ytem or, more compactly, a NA prce) for agent f qw > 0 for any w W \ {0}. Condton 1 NA) There ext a vector q that a no-arbtrage fnancal aet prce ytem for any agent. Let S denote the cone of no-arbtrage fnancal aet prce for agent. The followng corollary characterze NA. Propoton NA equvalent to S. Proof: See Appendx. A more ueful characterzaton of NA gven n the followng propoton. 11 Document de traval du Centre d'econome de la Sorbonne - 016.63
Propoton 3 Suppoe Aumpton 1, 3, 6 hold. Then q R K NA f and only f ) there ext a I S matrx β β) wth β > 0 for any and uch that q = β R for any, ) or, equvalently, there ext a I S matrx β β ) wth β > 0 for any and uch that q = β R for any. Proof: See Appendx. The coeffcent β can be nterpreted a the prce of a unt of currency, n term of numerare, n country and tate, whle the coeffcent β can be nterpreted a the prce of a unt of good, n term of numerare, n country and tate. It obvou that β = β ɛ. We wll ntroduce no-arbtrage fnancal aet prce relatve to a tate of nature. Defnton 7 ueful portfolo n tate ) w a ueful portfolo for agent n the tate f, for any µ 0 and any x X, one ha: 1) x + µw X, ) u e + R x + µw )) u e + R x ). Propoton 4 w ueful n the tate f and only f R w 0. Proof: See Appendx. Defnton 8 A vector q R K a no-arbtrage fnancal aet prce ytem for an agent n the tate f qw > 0 for any portfolo w whch atfe R w > 0 or equvalently qw > 0 for any ueful portfolo w 0 n tate. Th defnton rele on the no-free-lunch condton. The fnancal aet prce q uch that, f the return yelded by a portfolo w trctly potve, then t value calculated wth q trctly potve. Propoton 5 A vector q a no-arbtrage fnancal aet prce ytem n the tate, f and only f q = µ R wth µ > 0. Proof: See Appendx. Here µ the quantty of good one obtan n country and tate wth one unt of currency. 1 Document de traval du Centre d'econome de la Sorbonne - 016.63
Defnton 9 q R K a common no-arbtrage fnancal aet prce ytem n the tate f t a no-arbtrage ytem for any n th tate,.e., for any, f R w > 0 then q w > 0. We ay that common no-arbtrage hold for the economy f, for any tate, there ext a common no-arbtrage fnancal aet prce ytem q n th tate. Propoton 6 1) If there ext a common no-arbtrage fnancal aet prce ytem n the tate, then there are exchange rate τ uch that τ R = R. 0 ) Converely, f there are exchange rate τ uch that τr = R 0 for any, then there ext a common no-arbtrage fnancal aet prce ytem n tate. Proof: See Appendx. Remark 1 If a common no-arbtrage fnancal aet prce hold then there ext a NA fnancal aet prce. Indeed, let q ) be the lt of S common no-arbtrage fnancal aet prce ytem. Then for any, q = µ R. Let q = q. From Propoton 3, q NA. 5 Uncovered nteret rate parte and the law of one prce We conder two noton of party uually appled n the theory of nternatonal fnance and trade UIP and LOP). If we have two countre a dometc and a foregn one), the UIP hold when the return on dometc fnancal aet are equal to the return on foregn fnancal aet valued n dometc currency. Defnton 10 UIP) The Uncovered Interet rate Party hold f there ext a ytem of exchange rate τ uch that, for any agent and any tate, we have τr = R. 1 In other word, we have R = R 1. UIP mean that, the return on the fnancal aet of all countre valued n conumpton good, are equal for any tate. Th condton may be condered a very trngent nce t mpoe that UIP hold tate by tate. We weaken t wth the noton of Weak Uncovered Interet rate Party WUIP). We ntroduce a convenent notaton. If δ ) S =1 a probablty dtrbuton over the tate of nature and ρ τr ) S the random vector of return, =1 13 Document de traval du Centre d'econome de la Sorbonne - 016.63
then we denote t expected value by E δ ρ ) S =1 δ τ R, that a K- dmenonal row. Before tatng the defnton of WUIP, let u ntroduce a characterzaton. Propoton 7 The followng two tatement are equvalent. 1) There ext a probablty dtrbuton δ >> 0 and a ytem of exchange rate τ wth τ 1 = 1 for any uch that E δ ρ ) = E δ ρ 1 ) for any wth ρ τr ) S =1. ) There ext a probablty dtrbuton δ >> 0 and a ytem of converon rate ɛ uch that E δ R ) = E δ R 1 ) where E δ R ) = δ R and R R/ɛ. Proof: See Appendx. Th propoton allow u to ntroduce two equvalent defnton of Weak UIP. Defnton 11 WUIP) 1) The Weak Uncovered Interet Rate Party hold f there ext a common probablty dtrbuton δ ) S =1 wth δ > 0 for any, and a ytem of exchange rate τ uch that τ 1 = 1 for all, and E δ ρ ) = E δ ρ 1 ) for any wth ρ τr ) S =1. ) Equvalently, the Weak Uncovered Interet Rate Party hold f there ext a common probablty dtrbuton δ ) S =1 wth δ > 0 for any, and a ytem of converon rate ɛ uch that E δ R ) = E δ R 1 ) for any. Propoton 8 UIP WUIP. Proof: Apply the defnton. Remark UIP equvalent to WUIP when there no uncertanty. Defnton 1 LOP) A ytem of conumpton good prce p atfe the Law of One Prce f p = p 1 6) for any and any. The prce of the conumpton good n any country the ame when valued n currency 1. 14 Document de traval du Centre d'econome de la Sorbonne - 016.63
6 No-arbtrage condton and uncovered nteret rate parte: two de of the ame con? The followng propoton lnk the dfferent noton of UIP and the noarbtrage condton. Another man reult of our paper gven jut below. Propoton 9 NA equvalent to WUIP. Proof: See Appendx. Remark 3 Aume any agent ha a lnear utlty: u c ) = c. The equlbrum fnancal aet prce ytem NA nce t atfe q = λ π R wth λ > 0 for any. It nteretng to oberve that we have WUIP wth pecfc probablty dtrbuton: the belef of the agent. Propoton 10 If the return matrce R and the ytem of exchange rate τ atfy UIP, then NA hold. Proof: Snce UIP mple WUIP and, from Propoton 9, WUIP mple NA, the reult mmedate. Propoton 9 tate the equvalence between extence of a NA fnancal aet prce and WUIP. The followng propoton clam that extence of common no-arbtrage fnancal aet prce for the economy equvalent to extence of exchange rate τ uch that the return matrce R and the ytem of exchange rate τ atfy UIP. Th propoton gve uffcent and neceary condton to obtan a ytem of exchange rate for whch UIP hold. In other word t gve uffcent and neceary condton to endogeneze the exchange rate n order to get UIP. It another man reult of our paper. Propoton 11 1) If the common no-arbtrage fnancal aet prce hold for the economy, then there are exchange rate τ uch that the return matrce R and the ytem of exchange rate τ atfy UIP. ) Converely, f there are exchange rate τ uch that the return matrce R and the ytem of exchange rate τ atfy UIP, then common no-arbtrage fnancal aet prce hold for the economy. Proof: The proof follow from Propoton 6. 15 Document de traval du Centre d'econome de la Sorbonne - 016.63
7 Extence of equlbrum In th ecton, we prove the extence of an equlbrum n econome wth nternatonal fnancal aet and good market. 7.1 Equlbrum n the nternatonal aet market We frt decrbe a property of the equlbrum prce q of fnancal aet market. Propoton 1 Let Aumpton 1, 3, 5, 6 hold. Let q, x) be an equlbrum n the fnancal aet market. Then, q atfe the followng equvalent properte. 1) q NA. ) For any, q = β R = β ɛ R wth β > 0 for any and. Proof: See Appendx. We now prove the extence of an equlbrum n the nternatonal fnancal aet market. Propoton 13 Let Aumpton 1, 3, 4, 5, 6, hold. 1) If NA Condton 1) hold, then there ext an exchange rate ytem uch that q, x) an equlbrum n the nternatonal fnancal aet market wth qk > 0 for any k. ) Converely, aume there ext an equlbrum q, x) n the nternatonal fnancal aet market aocated wth an exchange rate ytem. Then NA fnancal aet prce hold. Proof: See Appendx. Remark 4 Oberve that Propoton 13 tate actually the equvalence NA There an equlbrum n the nternatonal fnancal market Remark 5 Combnng Remark 4 and Propoton 9 we have the equvalence WUIP NA There an equlbrum n the nternatonal fnancal market 16 Document de traval du Centre d'econome de la Sorbonne - 016.63
7. Equlbrum n both market We come now to one of our man reult. Theorem 1 Let Aumpton 1, 3, 4, 5, 6, hold. Aume that the return R and the ytem of exchange rate τ atfy UIP. 1) There ext an equlbrum p, q, c, x) Defnton 1) wth q = λp R, λ > 0. ) The conumpton good prce p atfy LOP, that p = p 1 for any country and any tate, and there ext λ > 0 uch that q = λp R for any agent. Proof: See Appendx. From Propoton 11, Theorem 1 equvalent to the followng theorem. Theorem Let Aumpton 1, 3, 4, 5, 6, hold. Aume the common noarbtrage fnancal aet prce hold for the economy. Then, there ext a ytem of exchange rate τ for whch the followng tatement are true. 1) There ext an equlbrum p, q, c, x) Defnton 1) wth q = λp R and λ > 0. ) The conumpton good prce p atfy LOP, that p = p 1 for any country and any tate, and there ext λ > 0 uch that q = λp R for any agent. Remark 6 ) It worthwhle to notce that f the return R and the ytem of exchange rate τ atfy UIP then, n equlbrum, the balance of conumpton good atfed alo n currency 1. Explctly, for any tate, p c = p e and c = e. ) Suppoe the I countre are n the ame monetary zone. They ue the ame currency. In th cae, τ = 1 for any and. UIP mple R = R 1 and R = R 1 for any and. Matrx of return R are gven, but prce q, p) are endogenou. The rgdty of return R R /ɛ depend on that of the converon rate. The equlbrum n the fnancal market enured by the flexblty of the prce q, p), whle that n the good market depend on the converon rate becaue the dfference between the conumpton demand and endowment precely gven by the portfolo return. If the exchange rate atfy UIP, then the equlbrum n the good market enured. 17 Document de traval du Centre d'econome de la Sorbonne - 016.63
8 Trade m)balance under WUIP Accordng to Theorem 1 and Remark 6, we know that, under UIP, there ext an equlbrum and the equlbrum trade balanced,.e., c = e n any tate of nature. WUIP mean that there ext a probablty dtrbuton δ and a ytem of exchange rate τ uch that δ R 1 = δ τr for any. By defnton, WUIP le demandng than UIP. We know that WUIP preerve the equlbrum n the nternatonal aet market Propoton 9 and 13). One may wonder what become the balanced trade under WUIP. Whle UIP mple that the value of trade balance zero, WUIP ental ntead that the expected value of trade balance under δ zero. We ay that the return R and the ytem of exchange rate τ atfy WUIP under a probablty dtrbuton δ f δ R 1 = δ τr for any. Theorem 3 Let Aumpton 1, 3, 4, 5, 6 hold and aume that the return R and the ytem of exchange rate τ atfy WUIP under a probablty dtrbuton δ >> 0. Then, a δ-equlbrum ext. Proof: See Appendx. Theorem 3 brdge the equlbrum mplcaton of UIP and WUIP. Under WUIP, trade may be mbalanced n ome tate, but t expected value zero. A δ-equlbrum a weak noton of dequlbrum, becaue, precely, trade may be mbalanced even f t expected value zero and a fnancal equlbrum hold. The very reaon that, conderng the expected nteret party E δ ρ ) = E δ ρ 1 ) ntead of the nteret party ρ = ρ 1, we obtan the expected trade balance E δ c c ) = E δ e) ntead of the trade balance = e. To undertand why th happen, focu on the proof of Theorem 3: nce the dfference between conumpton and endowment gven by the portfolo return c e = R x ), f the expected) aggregate portfolo return zero becaue of the Weak) UIP, then the expected) aggregate net demand zero, that the expected) trade balance hold. 9 Example wth UIP falure In order to better undertand the role of UIP, we provde three example where UIP fal. In the frt one, the no-arbtrage condton hold n the nternatonal aet market and an equlbrum ext but t no-trade and 18 Document de traval du Centre d'econome de la Sorbonne - 016.63
the conumpton good prce don t atfy LOP. In a econd example, the noarbtrage condton hold n the nternatonal fnancal aet market and an equlbrum ext a well but the good market mbalanced. In a thrd example, the et of no-arbtrage prce for the nternatonal aet market dffer acro the countre and do not nterect: thu, the no-arbtrage condton fal and there ext no equlbrum. Example 1 In th example, where the UIP fal, there ext a unque equlbrum whch no-trade. We conder an exchange economy wth one conumpton good, two countre: = 1,, two aet: k = 1,, and two tate of nature: = 1,. The matrce of return on aet, n term of local currence, are gven: R 1 = [ Rk 1 ) R = [ Rk ) ] [ R11 1 R1 1 R1 1 R 1 = ] [ R11 R1 R1 R = 1 0 1 0 1 1 In th economy, UIP volated becaue τ1 R 1 = R1 1 mple τ 1 0, 1) = 1, 0), a contradcton. Indvdual hare the ame belef and the tate are condered equprobable: [ ] [ ] π1 1 π 1 π π1 π = 1 1 1 9) 1 1 The utlty functon are alo the ame acro the countre: πu c ) 1 = ] ] 7) 8) c 10) but the ntal endowment dffer: [ e 1 1 e 1 e e 1 e ] = [ 1 1 1 ] 11) The et of ueful portfolo W 1 = { x 1 1, x 1 ) : x 1 1 0, x 1 1 + x 1 0 } W = { x 1, x ) : x 0, x 1 + x 0 } determne the cone of no-arbtrage prce: S 1 = nt W 1) 0 { = p 1 1, p 1 ) : p 1 1 > 0, p 1 > 0, p 1 1 p 1 > 0 } S = nt W ) 0 { = p 1, p ) : p 1 > 0, p > 0, p p 1 > 0 } 19 Document de traval du Centre d'econome de la Sorbonne - 016.63
We ee mmedately that 1, 1) S 1 S. From Propoton 13, there ext a par q, x) wth q k > 0 for any k uch that, for any agent, x a oluton of program 3) and x a net trade, that the fnancal aet market clear: =1 x = 0. In Appendx, we prove that there ext an equlbrum whch no-trade and LOP doe not hold. Example In th example, UIP fal and there no equlbrum. A above, we conder an exchange economy wth one conumpton good, two countre: = 1,, two aet: k = 1,, and two tate of nature: = 1,. The matrx of return R 1 reman unchanged and gven by 7), whle R now replaced by a lghtly dfferent matrx: R = [ Rk ) R11 R1 R1 R ] = [ 0 1 ] 1) In th economy, UIP volated becaue τ1 R 1 = R1 1 mple τ 1 0, 1) = 1, 0), a contradcton. The other fundamental reman the ame a n Example 1 belef, preference and endowment are gven by 9), 10) and 11) repectvely). Clearly, the et of ueful portfolo W 1 and the cone of no-arbtrage fnancal aet prce S 1 do not change R 1 the ame), whle the correpondng et for country become now W = { x 1, x ) : x 0, x 1 + x 0 } S = nt W ) 0 { = p 1, p ) : p 1 > 0, p > 0, p p 1 > 0 } We ee mmedately that, 3) S 1 S. From Propoton 13, there ext a par q, x) wth qk > 0 for any k uch that, for any agent, x a oluton of program 3) and x a net trade, that the aet market clear: =1 x = 0. In Appendx we prove that the trade cannot be balanced. Hence equlbrum doe not ext. Example 3 Agan, we conder an economy wth two countre = 1,, two fnancal aet k = 1, and two tate of nature = 1,. The matrce of return on fnancal aet are gven: R 1 = [ ] [ ] Rk 1 ) R 1 11 R1 1 1 0 R1 1 R 1 = 13) 1 R = [ ] [ ] Rk ) = 14) R11 R1 1 R1 R 1 4 1 3 0 Document de traval du Centre d'econome de la Sorbonne - 016.63
The et of ueful portfolo W 1 = { x 1 1, x 1 ) : x 1 1 0, x 1 1 + x 1 0 } W = { x 1, x ) : x 1 + 4x 0, x 1 + 3x 0 } determne the cone of no-arbtrage fnancal aet prce: S 1 = nt W 1) 0 { = p 1 1, p 1 ) : p 1 > 0, p 1 1 p 1 > 0 } S = nt W ) 0 { = p 1, p ) : 4p 1 + p < 0, 3p 1 + p > 0 } Aume there ext p 1, p ) S 1 S. In th cae we have p 1 > 0, p 1 p > 0, 4p 1 + p < 0, 3p 1 + p > 0 Thee nequalte lead to 0 < 3p 1 < p < p 1 whch mpoble. Thu S 1 S = and NA fnancal aet prce doe not ext. In th cae WUIP doe not hold and hence UIP a well. Remark 7 We want to how that when any country want alo to conume n perod 0 then, under NA and UIP, the economy tll ha an equlbrum on both market and LOP hold for any perod, any tate. The reult concernng UIP, WUIP tll hold. Conder the model where any agent olve n perod 0: { max u 0c 0) + πu e c 0,x ) R + X + R x )} ) p 0c 0 e 0) + qx 0 and, n perod 1, we have c = e + R x for any. The functon u 0 and u are trctly concave and ncreang. An equlbrum of th economy a lt p, q, c, x) wth p, q >> 0 uch that ndvdual plan are optmal pont 1) and ) below) and market clear pont 3) and 4)). 1) Portfolo and conumpton n perod 0 are optmzed gven the fnancal aet prce and conumpton prce q and p 0 : n perod 0, any agent chooe a conumpton c 0 and a portfolo x whch maxmze her utlty functon under the budget contrant: { max c 0,x ) R + X ) u 0c 0) + p 0 c 0 e 0) + q x 0 π u e + R x )} 15) 1 Document de traval du Centre d'econome de la Sorbonne - 016.63
) In perod 1, for any agent, c = e + R x olve the program max c Y + p c p e πu c ) 16) 3) The nternatonal fnancal aet market clear: x = 0. 4) The trade balance atfed n perod 0 and n any tate of nature of perod 1: c 0 = c = e 0 e for = 1,..., S Clam 1 Aume UIP hold. 1) There ext an equlbrum p, q, c, x) wth q = p R, for any. ) The conumpton good prce p atfy LOP, that a) n perod 0, p, for any country, 0 = p1 0 b) n perod 1, p = p 1 for any country and any tate, Proof. Defne p 0 = p 0 for any, where p 0 0. At perod 0 agent olve max c 0,x ) R + X ) { u 0c 0) + p 0 c 0 e 0) + qx 0 π u e + R x )} Recall that S denote the et of no-arbtrage aet prce for agent. Then, the et of no-arbtrage prce for perod 0 σ = R ++ S. In addton, S σ. From e.g. Dana, Le Van and Magnen, 1999, there ext p 0, q, c 0, x) ) wth p0, q) >> 0, uch that any agent olve her/h conumpton problem wth prce p 0, q) and c 0 e ) 0 = 0, x 0 = 0. For perod 1, the proof of Theorem 1 apple. We can ealy check that we tll have NA WUIP nce thee reult are relatve only to the econd perod. 10 Concluon Our paper conder a two-perod model à la Hart [5] wth conumpton good and fnancal aet. The conumpton and the aet are nternatonally tradable. Our man reult are the followng. Document de traval du Centre d'econome de la Sorbonne - 016.63
1) We ntroduce a Weak UIP condton and we how that t equvalent to No-Arbtrage Condton fnancal aet prce. Moreover we have the equvalence: NA hold Weak UIP hold An equlbrum ext n the nternatonal fnancal market ) We ntroduce the common no-arbtrage fnancal aet prce and how Extence of a common no-arbtrage fnancal aet prce UIP hold 3) When the UIP hold, we have an equlbrum n both the nternatonal fnancal aet and good market. At th equlbrum, we obtan trade balance n value and LOP. 4) When the UIP fal, we how by mean of three example that ) the no-arbtrage fnancal aet prce condton hold and an equlbrum ext a well but t no-trade and LOP not atfed Example 1); ) the noarbtrage condton prce hold n the nternatonal fnancal aet market and an equlbrum ext a well but there ext no balance n the good market Example ); ) the no-arbtrage condton fnancal aet prce fal and there ext no equlbrum Example 3). Modellng the coextence of an equlbrum n the fnancal aet market jontly wth a dequlbrum n the good market through the falure of parte reman an nteretng queton. Mot of fnancal paper Rogoff [7] among other) conder that the parte are not repected n the hort run. A uggeted by Frenkel and Mua [4] n a monetary model, trade balance dequlbra eem plauble under a regme of pegged rate becaue relatve prce adjutment are acheved through low change n the good market, whle fnancal market are moble and ntegrated. Th pont tackled n the paper through three example. 11 Appendx Proof of Propoton 1 Frt, we want to prove that V W. If v V, R Accordng to ), for any x X and any µ 0, one ha e + R v 0 for any. x + µv ) = e + R x + µr v µr v 0, that Defnton 5, pont 1). From e + R x + µv ) = e + R x + µr v e + R x 17) and the ncreangne of u Aumpton 6), we obtan alo Defnton 5, pont ). 3 Document de traval du Centre d'econome de la Sorbonne - 016.63
Converely, we want to how that W V. Let w W. Then, pont 1) n Defnton 5 requre e + R x + µw ) 0 for any. Dvdng both the de of th nequalty by µ > 0 and lettng µ go to nfnty, we get R w 0 for any, that w V. Fnally, oberve R w 0 R w 0. Proof of Propoton We oberve that S = nt W ) 0 where W ) 0 the polar 6 of W. Under Aumpton 3, the et W do not contan lne and the et S are nonempty ee Dana, Le Van and Magnen [3] among other). Let S be the nterecton of all the cone of no-arbtrage prce. Thu, f NA hold, q belong to S, and, f S nonempty, NA hold. Proof of Propoton 3 1) Let q be NA. Let w W \ {0}. From Aumpton 3, R w 0 for any and R w > 0 for ome. From Dana and Jeanblanc [], there ext a I S matrx β β ) wth β > 0 for any and uch that q = β R for any. ) Converely, aume q = β R for any wth β > 0 for any and. If w W \ {0}, then, from Aumpton 3, R w 0 for any and R w > 0 for ome. Hence, for any, qw > 0 for any w W \ {0} or, equvalently, q NA. Proof of Propoton 4 Suffcency. Rw 0 mple u e + R x + µw )) = u e + R x + µr w ) u e + R x ) becaue u ncreang. Necety. u concave and w ueful n the tate. Then, u e + R x ) µr w u e + R x + µr w ) u e + R x ) 0 Dvdng both the de of th nequalty by µu e + R x ) > 0, we obtan R w 0. It equvalent to R w 0. Proof of Propoton 5 R w 0 mple that w ueful n the tate and, nce q a no-arbtrage aet prce ytem n the tate, qw 0. Accordng to the Farka Lemma, R w > 0 qw > 0 ) mple q = µ R wth µ > 0. The convere obvou. Proof of Propoton 6 1) Snce q = µ R = µ 1 R 1 for any, we can defne τ µ /µ 1 n order to obtan τ R = R 1. 6 The polar cone of a et X R K defned a X 0 { y R K : y T x 0 for any x X }. 4 Document de traval du Centre d'econome de la Sorbonne - 016.63
) Conder a ueful portfolo w 0 whch atfe R w > 0. Defne q R 1 = τ R. Then, q w > 0 for any. Apply Defnton 8 and 9. Proof of Propoton 7 1) ). Aume we have 1). Take ɛ 1 ) >> 0. Defne R R/ɛ, ɛ = ɛ 1 /τ and δ δ ɛ 1 σ δ σɛ 1 σ We get δ R = δ R 1, that ). ) 1). Defne τ = ɛ 1 /ɛ and δ δ /ɛ 1 σ δ σ/ɛ 1 σ Then E δ R ) = E δ R 1 ) become δ τr = δ R, 1 that 1). Proof of Propoton 9 1) If a vector q NA, then from Propoton 3, there ext β > 0 for any and uch that q = β R for any. Defne δ β 1 / σ β1 σ and τ β/β 1. Clearly, the probablty dtrbuton δ well-defned. Conder the random vector ρ 1 R) 1 S =1 and ρ τr ) S =1. Then, E δ ρ 1 ) = S δ R 1 = =1 S =1 β1 R 1 β1 = S =1 β R β1 = S =1 β1 σ β1 σ that WUIP. ) Converely, f E δ ρ ) = E δ ρ 1 ) for any, then the vector NA from Propoton 3. q = E δ ρ 1 ) = E δ ρ ) = S δ τr, for any =1 τ R = E δ ρ ) Proof of Propoton 1 1) Frt oberve that we have q x = 0 for all. Let ε = ε,..., ε) T R K wth ε > 0. Let w W. We then have π u e + R x + w + ε )) > πu e + R x + w )) π u e + R x ) whch mple q w + ε ) > 0. Let ε go to zero. We get q w 0. Now, let w W \ {0}. Then Rw 0 for all. However, from Aumpton 3, there ext wth Rw > 0. Then, from Aumpton 5, we have, for any, π u e + R x + w )) > 5 π u e + R x ) Document de traval du Centre d'econome de la Sorbonne - 016.63
Th mple q w > 0 and q S for any. Equvalently, q NA. ) The econd tatement come from Propoton 3. Proof of Propoton 13 1) Aume that q NA. In th cae, the proof provded by Werner [8], Page and Wooder [6], and Dana, Le Van, Magnen [3] among other. The trct potvty of q reult from the trct ncreangne of u jontly wth Aumpton 1 and 3. ) Refer to Propoton 1. Proof of Theorem 1 1) Snce we have UIP, the prce q = R NA nce t equal R1. It follow from Propoton 13 that we have an equlbrum q, x ) n the nternatonal aet market. From Propoton 1, q = β R for any, wth β > 0 for any and any. Defne for any β We clam that, for any, any, we have β σ k β1 σ Rσk 1. Let λ σ Equvalently, R k = β1 σ σ β k β σ Rσk β R = Rk 1. k β1 σ Rσk 1 = σ β R β R λ k β σ Rσk = β 1 R 1 Hence, for any, σ k β σ R σk β 1 R 1 λ σ β 1 R 1 = Our clam true. Now, defne q β R. We have j q j = 1. Settng p β, for all, all, we fnd. Indeed, for any k and k β1 σ Rσk 1 k β σ R for any. Hence, σk = q = p R = p R 1 = p 1 R 1 18) for any. Let Z { z R S : z R 1 = 0 } and oberve that Z = {0} n the cae of complete aet market rankr 1 = S). From 18), we get, for any, p = p 1 + z wth z Z. Defne ˆp 1 p 1 and ˆp p z = ˆp 1 for any and any. Let c = e + R x for any and any. q x = 0 mple that ˆp c = ˆp e + q x = ˆp e : the budget contrant atfed n any country Defnton 1, pont )). Notcng that, for any and, c over, we obtan for any c = = e +R x = e +R 1 x and ummng e + R 1 6 x = e Document de traval du Centre d'econome de la Sorbonne - 016.63
nce x = 0. We notce that q = p R = ˆp 1 R + z R = ˆp R + z R 1 = ˆp R becaue z Z. We wll how that ˆp, c ) atfe the pont ) of Defnton 1. For that, defne c = e + R x and aume that π u e + R x ) > π u e + R x ) Snce q, x) an equlbrum n the nternatonal aet market, we have q x > q x. Equvalently, ˆp R x > ˆp R x and hence ˆp c = ˆp e + ˆp R x > ˆp e + ˆp R x = ˆp e We have proved that ˆp, c ) atfe pont ) of Defnton 1. Oberve that ˆp >> 0. We now normalze the equlbrum conumpton prce by defnng p = ˆp σ ˆp σ for any and any. Let λ 1/ σ ˆp σ. Then, q = λp R for any. ) Fnally, t clear that the ytem p atfe the LOP. Proof of Theorem 3 Snce the return R and the ytem of exchange rate τ atfy WUIP under a probablty dtrbuton δ >> 0, no-arbtrage condton NA) hold. Hence an equlbrum n the nternatonal aet market q, x) ext. If c denote the aocated conumpton n perod 1, we have c = e + τr x for any and. We can ealy check that they atfy the conumer problem 16). We now check that the expectaton under δ of the trade balance atfed. Indeed, we have δ c = δ e + ) δ τr x = δ e + ) δ R 1 x = ) δ e + δ R 1 x = δ e 7 Document de traval du Centre d'econome de la Sorbonne - 016.63
nce x = 0. Example 1 Trade balance are atfed n any tate of nature f c = e. Let τ [ ] τ ) 1 1 = τ1 1 τ 1 19) be the I S matrx of exchange rate, R 1 = R/ɛ 1 1 and R = R/ɛ and τ = ɛ 1 /ɛ. From equaton 1), we know that c e = R x. Thu, R x = 0 for any. In our example, we get R11 1 ɛ 1 x 1 1 + R1 1 1 ɛ 1 x 1 + R 11 1 ɛ x 1 + R 1 1 ɛ x 1 = 0 or R11x 1 1 1 + R1x 1 1 + τ1 R11x 1 + τ1 R1x = 0 Ung 7) and 8) to replace R11 1, R1 11, R 11 and R 1 by ther numercal value, we fnd x 1 1 + τ 1 x = 0. Snce x1 + x = 0, we obtan alo x1 1 τ 1 x1 = 0, that a contradcton wth q1 x1 1 + q x1 = 0 under prce potvty f x1 1 0. Hence x 1 1 = 0. In th cae, we have x 1 = x = 0. If th allocaton an equlbrum, t olve 3): there are potve multpler µ uch that qk = µ π u c R x ) R k for any k and = 1,. Notcng that c = e + R 1 x 1 + and c 1 1, c1, c 1, c ) = 1, 1, 1, ), we fnd q 1 = µ1 q = µ1 πr 1 1 1 c 1 πr 1 1 c 1 = µ1 and q 1 = µ = µ1 and q = µ π R 1 c π R c = µ = µ 4 τ 0) τ1 + τ ) Th mple τ = τ 1. Thu, we then get a no-trade equlbrum wth, for any, x = 0 and c = e. To compute the prce, we aume, for mplcty, that ɛ 1 1 = ɛ1 = 1. Prce are gven by q = 1, 1) and, ung q = p R, p [ p 1 1 p 1 p 1 p ] = 1 [ 1 1 We oberve that p 1 1 p 1 and p1 p : the LOP not atfed. 1 τ 1 1 τ ] Example Trade balance are atfed n any tate of nature f c = e. Conder the ame generc matrx of exchange rate 19) wth R 1 = R/ɛ 1 1 and R = R/ɛ and τ = ɛ 1 /ɛ. Ung exactly the ame argument than n Example 1, we obtan x 1 1 = x 1 = x 1 = x = 0. 8 Document de traval du Centre d'econome de la Sorbonne - 016.63
If th allocaton an equlbrum, t olve 3): there are potve multpler µ uch that qk = µ π u c ) R k for any k and = 1,. Notcng that c = e + R1 x 1 + R x and c 1 1, c1, c 1, ) c = 1, 1, 1, ), we fnd the ame q1 a n Example 1 ee 0)), whle the q become now q = µ1 πr 1 1 = µ1 c 1 and q = µ πr = µ τ 1 c + τ ) But whch mpoble. q 1 = q µ τ = µ τ 1 + τ ) τ1 = 0 Reference [1] Allouch, N., Le Van, C. and F.H. Page Jr. 00). The geometry of arbtrage and the extence of compettve equlbrum, Journal of Mathematcal Economc 38, 373-39. [] Dana R.A, and M. Jeanblanc 003). Fnancal Market n Contnuou Tme. Sprnger Fnance Textbook. [3] Dana R.A, C. Le Van and F. Magnen 1999). On the dfferent noton of arbtrage and extence of equlbrum. Journal of Economc Theory 86, 169-193. [4] Frenkel J.A. and M.L. Mua 1980). The effcency of foregn exchange market and meaure of turbulence. Amercan Economc Revew 70, 374-381. [5] Hart O. 1974). On the extence of an equlbrum n a ecurte model. Journal of Economc Theory 9, 93-311. [6] Page F.H. and M.H. Wooder 1996). A neceary and uffcent condton for compactne of ndvdually ratonal and feable outcome and extence of an equlbrum. Economc Letter 5, 153-16. [7] Rogoff K. 1996). The purchang power party puzzle. Journal of Economc Lterature 34, 647-668. [8] Werner J. 1987). Arbtrage and the extence of compettve equlbrum. Econometrca 55, 1403-1418. 9 Document de traval du Centre d'econome de la Sorbonne - 016.63