WP/09/38 Inernaional Risk Sharing: Through Equiy Diversificaion or Exchange Rae Hedging? Charles Engel and Akio Masumoo
2009 Inernaional Moneary Fund WP/09/38 IMF Working Paper Research Deparmen Inernaional Risk Sharing: Through Equiy Diversificaion or Exchange Rae Hedging? Prepared by Charles Engel and Akio Masumoo Auhorized for disribuion by Sijn Claessens July 2009 Absrac This Working Paper should no be repored as represening he views of he IMF. The views expressed in his Working Paper are hose of he auhor(s) and do no necessarily represen hose of he IMF or IMF policy. Working Papers describe research in progress by he auhor(s) and are published o elici commens and o furher debae. Well-known empirical puzzles in inernaional macroeconomics concern he large divergence of equilibrium oucomes for consumpion across counries from he predicions of models wih full risk sharing. I is commonly believed ha hese risk-sharing puzzles are relaed o anoher empirical puzzle he home-bias in equiy puzzle. However we show in a series of dynamic models ha he full risk sharing equilibrium may no require much diversificaion of equiy porfolios when here is price sickiness of he degree ypically calibraed in macroeconomic models. This conclusion holds under a range of assumpions abou home bias in preferences price seing as PCP or LCP and wih or wihou nominal wage sickiness as long as here is some price rigidiy. JEL Classificaion Numbers: F30; F4 Keywords: Inernaional porfolio choice inernaional risk sharing inernaional diversificaion Auhor s E-mail address: cengel@ssc.wisc.edu; amasumoo@imf.org We are graeful for he commens from Gianluca Benigno Nicolas Couerdacier and paricipans a he CEPR Workshop on Inernaional Risk Sharing and Porfolio Diversificaion and he CEPR Conference on Inernaional Macroeconomics and Finance. Engel acknowledges he suppor of he Naional Science Foundaion hrough a gran o he Universiy of Wisconsin. Some of he work on his projec was compleed while Engel was a visiing scholar a he Inernaional Moneary Fund and a Senior Fellow a he Globalizaion and Moneary Policy Insiue of he Federal Reserve Bank of Dallas.
2 Conens Page I. Inroducion...3 II. A General Resul in a Saic Framework...8 III. A Dynamic Sicky-Price Model wih Local-Currency Pricing...6 A. Household Problem...7 B. Firms...2 C. Equilibrium Porfolios under LCP and Flexible Wages...23 D. Equilibrium Porfolios under LCP and Sicky Wages...30 E. A Dynamic Sicky-Price Model wih Producer-Currency Pricing...32 IV. Conclusion...34 Tables. Opimal Porfolios under LCP Flexible Wages...35 2. Opimal Porfolios under LCP Sicky Wages...37 3. Opimal Porfolios under PCP Flexible Wages...39 4. Opimal Porfolios under PCP Sicky Wages...4 References...43
3 I. INTRODUCTION The puzzle of home-bias in equiy holdings ha households hold a disproporionae share of heir equiy porfolios in heir own counry s equiies is frequenly linked wih puzzles of inernaional risk sharing. I is naural o make such a link because diversificaion in equiy holdings seems o be a naural way o diversify income risk. Bu here we argue ha he risk sharing puzzles may be more closely linked o exchange rae hedging by households. In a world in which nominal goods prices are sicky we make he case ha much of he burden of risk sharing is borne by exchange rae hedging behavior hrough he porfolio of bonds denominaed in differen currencies currency forward posiion or oher financial insrumens affeced by exchange raes. The resul is surprising as nominal sickiness and shor-erm foreign exchange hedges play a key role in inernaional risk sharing insead of permanenlylived equiies. We show ha when exchange rae hedging is possible price sickiness plays a dominan role in deermining he equiy porfolio relaive o oher effecs such as wage seing currency of pricing and home bias in preferences. Foreign exchange risk can be hedged hrough forward conracs swaps or he porfolio of nominal bonds in differen currencies so ha equiy porfolios hedge risks no associaed wih foreign exchange rae movemens. Bu when prices are sicky much of he differenial real income risk of residens in differen counries is associaed wih nominal exchange rae movemens. To he exen ha exchange rae changes are passed hrough o impor prices hey influence he relaive inernaional prices of commodiies. If pass-hrough is incomplee exchange rae movemens influence he profis per uni sold in he exporer s currency. Moreover relaive oupu across counries is influenced by demand shocks under sicky prices bu hese shocks in urn influence exchange raes. The value of asses is deermined by he enire fuure expeced pah of payoffs. Equiy prices for example are deermined in mos asse-pricing models as a presen value of expeced discouned dividends. Under ypical calibraions of macroeconomic models prices adjus quickly compared o he rae a which markes discoun fuure dividends. So why should goods price sickiness have much of an influence on asse prices and porfolios? Indeed we consider a model in which were goods prices flexible he equiy porfolio of households could be used o share risk efficienly. In our 2-counry model he real allocaions ha could be achieved under complee markes will be replicaed in a model in which only equiies are raded. There is no role for nominal bonds o hedge risk when goods prices are flexible. Indeed a simple Lucas ree model predics ha full risk sharing can be achieved hrough equiy porfolio diversificaion. (Lucas 982)
4 When prices are flexible much of he burden of risk sharing falls no on he porfolio choices of individuals bu insead on changes in relaive goods prices (as Cole and Obsfeld 99 have noed.) When a firm experiences a produciviy increase i will increase he supply of is produc. The consequen drop in he price of is oupu can parly offse he income gains from increased produciviy. Under sandard calibraions of macroeconomic models Cole and Obsfeld show ha he gains from diversificaion of he equiy porfolio migh no be very large. However his erms-of-rade effec is no operaive when prices are sicky. Under sicky prices he porfolio of asses socks and bonds are herefore an imporan way o diversify risk. Indeed even hough goods prices may be sicky only emporarily he asse demand decision migh be dominaed by he risk-sharing consideraions under sicky prices. 2 We demonsrae ha only a small amoun of price sickiness will have large effecs on he porfolio choices of individuals. We build a wo-counry dynamic sochasic general equilibrium model wih rade in equiies and nominal bonds. We consider price sickiness of wo ypes ha have been examined in he inernaional macroeconomics lieraure: producer currency pricing (PCP) in which prices are se in he producers currencies and local-currency pricing (LCP) in which prices are se in he currency of consumers. We examine he opimal equiy and bond porfolios under nominal wage flexibiliy and nominal wage sickiness. We also consider he role of home-bias in preferences. Our main conclusion is ha in a log-linearized version of he model rade in only equiies and bonds will replicae he allocaions achieved under rade in a complee se of nominallydenominaed sae-coningen bonds. For many reasonable parameerizaions he porfolios ha replicae he complee-markes oucome do no require much diversificaion of he equiy porfolio. Tha is households opimally should hold porfolios ha are heavily weighed oward heir own counry s equiies. Bu he efficien allocaion does require ha agens ake he correc open posiion in foreign exchange. In paricular under all plausible parameerizaions agens should be long in heir own currency and shor in foreign currency. An unexpeced currency depreciaion should cause a negaive wealh shock o he porfolio. This shock serves o balance he posiive income effecs from a depreciaion under sicky nominal prices. We begin our analysis in Secion II wih a model ha illusraes he economic forces a work. This secion exends he example presened in Engel and Masumoo (forhcoming). I examines he linearized version of a wo-counry model in a one-period economy. Oupu is produced using labor by monopolisic firms. Households are endowed wih claims o profis 2 We have demonsraed his poin in deail in a dynamic model of sicky prices ha is a special case of he general one considered in his paper. See Engel and Masumoo (forhcoming).
5 of firms in he counry where hey live and work bu purchase and/or sell a porfolio of equiies and a forward posiion in foreign exchange before he realizaion of shocks. The model assumes preferences are homoheic in each counry bu Home and Foreign residens preferences may be differen. The general resul of his model is ha when households can rade asses ha hedge relaive price risk real exchange rae risk and risk from he changes of relaive prices of Home o Foreign goods in each counry hen a full risk-sharing equilibrium can be achieved wih no rade in equiies. One applicaion of his resul is o he case in which goods prices are flexible and he law of one price holds for all goods. The complee markes allocaion can be achieved in his case wih only rade in wo non-sae-coningen bonds one denominaed in unis of he Home good and one in unis of he Foreign good. 3 Our focus is primarily on exchange rae hedging in sicky-price models. We show in he saic seing ha he complee markes equilibrium can be suppored wih only rade in a forward posiion in foreign exchange or nominal bonds denominaed in differen currency. Tha is full risk sharing does no require any diversificaion of he equiy porfolio in his special model. Price seing of goods can be LCP PCP or here can be indexaion of he price o he exchange rae. Bu a crucial assumpion in his example model is ha all goods prices mus be se in advance of he realizaion of shocks. The model pus no resricions a all on he labor marke or in general on how firms and workers share revenues. Secion III hen ses up a dynamic wo-counry model. Oupu is produced using only labor and firms are monopolisic. In his secion we firs assume ha some firms se prices in advance wih local-currency pricing while oher firms are flexible-price firms. We consider wo differen specificaions of he labor marke. In he firs wages are flexible and he labor marke is compeiive. In he second all wages are fixed in advance and households supply labor as demanded. In boh cases rade in equiies and a forward posiion in foreign exchange (or equivalenly a bond porfolio of zero ne value ex ane) will lead o allocaions ha are idenical o hose achieved under rade wih a complee se of nominal coningen claims. We show ha under plausible parameerizaions he opimal equiy porfolio does no reflec much inernaional diversificaion. Tha is here is a high degree of home bias in equiies. 4 Bu his efficien allocaion requires agens ake sufficienly large posiions in foreign exchange o diversify risk fully. We also repea he analysis under he assumpion ha hose firms ha have sicky prices se hem one period in advance according o producer-currency pricing. We show ha he equiy 3 Our resuls in his secion exend resuls presened in our earlier paper o allow for differen asymmeric preferences beween households in each counry. 4 Moreover home bias in preferences does no generae a disconinuiy in he equiy porfolio wih respec o he degree of home bias in preferences. See he discussion of Obsfeld (2007).
6 posiion under PCP is idenical o ha under LCP bu he opimal posiion in foreign exchange is differen. Inuiively he PCP and LCP models are differen because of he way exchange raes are passed hrough o prices. Demand for producs from home and foreign firms and hence profis will differ under LCP and PCP pricing. Bu he effecs of hese exchange rae changes on wealh can be hedged hrough he forward marke in foreign exchange. The opimal equiy porfolio is no influenced by LCP vs. PCP because he difference is all in how exchange rae changes affec he economy. We emphasize ha he degree of equiy diversificaion needed o achieve full risk sharing does depend on he degree of price and wage sickiness. We know from he work of Baxer and Jermann (997) ha when goods prices and wages are perfecly flexible he opimal equiy porfolio should be heavily weighed oward he equiies of he oher counry even o he exen ha households may wish o go shor in heir own counry s equiies. Our general equilibrium model encompasses he Baxer-Jermann model as a special case bu more generally allows us o examine how he degree of price and wage sickiness influence he composiion of he porfolios ha opimally share risk. Finally even hough we find ha he opimal risk-sharing equiy porfolios under some plausible parameerizaions reflec a high degree of home bias in equiies we are no advancing our model as an explanaion for he degree of home bias observed in he daa. In he conex of our model home bias in equiies is only opimal if agens also hold he correc forward posiion in foreign exchange. In fac counry bond porfolios do no look much like he opimal porfolios we consruc. Our poin is ha he paern of bond holdings may be as puzzling as he paern of equiy holdings. However we are also cauious o sae ha his is indeed a puzzle as wha really maers is currency exposure in our model bu we do no have good daa on he acual currency exposure including derivaives. 5 Noneheless we do claim ha exchange rae hedging and nominal rigidiy play an imporan role in (he lack of) inernaional risk sharing. In he pas few years here have been many new dynamic models of porfolio choice in general equilibrium. Perhaps hese models provide a basis for a more posiive analysis of porfolio allocaion. Paricularly relevan are he sudies ha use approximaion mehods no unlike hose used in his paper o analyze he dynamics of porfolios. Devereux and Suherland (2006) derive an approximaion mehod for an economy wih incomplee markes bu consan porfolio shares and apply i o a 2-counry general equilibrium model wih producion and rade in equiies and o a 2-counry endowmen model wih rade in real 5 Recenly Lane and Shambaugh (forhcoming) consruc a daabase of inernaional currency exposures for a large panel of counries over 990-2004 and Lane and Shambaugh (2008) provide sylized facs concerning he cross-counry and ime-series variaion in aggregae foreign currency exposure. They find ne currency exposure for advanced counries is posiive implying hey are long in foreign currency. However hey also find ha ne exposure from deb porfolio is negaive in line wih our predicion a les in erms of sign. While heir daa are informaive and useful he acual exposure is ambiguous wihou derivaives daa.
7 bonds. Devereux and Suherland (forhcoming) apply his model o a sicky-price moneary model ha allows for porfolios of bonds and equiy rade. Devereux and Suherland (2007) exend he approximaion mehod o allow for ime-varying porfolios and apply he mehod o a wo-counry endowmen model wih rade in real bonds. Devereux and Suherland (2008) examine a similar model wih a focus on he role of changes in valuaion for he inernaional disribuion of wealh. Tille and van Wincoop (2008) use a similar approximaion o solve a wo-counry general equilibrium model wih capial and producion and rade in equiies. Tille and van Wincoop (2009) use hese mehods o examine he response of he curren accoun and ne foreign asses o changes in saving. Also Evans and Hnakovska (2008) examine a similar model wih a relaed soluion mehodology. Heahcoe and Perri (2008) and Pavlova and Rigobon (2007 2008) presen neoclassical models in closed form using special assumpions on funcional forms (such as log preferences) o examine equilibrium porfolios. Like he model of his paper Heahcoe and Perri allow for home bias in preferences. In heir model firms have value because hey own capial. Pavlova and Rigobon consider a coninous-ime version of he wo-counry endowmen model. Kollmann (2006) considers an endowmen model wih home bias in preferences and rade in equiies. Similarly Coeurdacier (2009) reconsiders he issue of home bias in consumpion in he conex of an approximaed general equilibrium model wih endowmens. Obsfeld (2007) argues ha Coeurdacier s resul may no be robus in he presense of non-raded goods. Indeed Collard Dellas Diba and Sockman (2007) exend an endowmen model wih raded and nonraded goods of Baxer Jermann and King (998) o incorporae home bias in preferences and numerically solve i o address he home bias in equiy puzzle. Masumoo (2008) analyically solves a producion economy model wih raded and nonraded goods o sudy he role of human capial and nonseparable uiliy in inernaional porfolio allocaions. Tha paper emphasizes he role of human capial wealh. All of hese models assume flexible prices and no exchange rae hedging. As a resul he equilibrium porfolio is ofen highly sensiive o preference parameers and someimes super home bias or ani-home bias may arise depending on he values of parameers. In closely relaed work Coeurdacier and Gourinchas (2009) focus on rade in real bonds denominaed in unis of oupu. 6 They consider one-period endowmen models wih home bias in preferences under flexible goods prices. They sudy he bond and equiy porfolio when here is a erms-of-rade shock and oher shocks. In conras o he earlier sudies hey 6 Tha paper and ours were wrien simulaneously and independenly. Alhough some resuls are very similar he emphasis of he papers is differen. Our focus is primarily on exchange rae hedging under sicky prices. Coeurdacier and Gourinchas (2009) derive some general resuls abou porfolios of equiies and bonds in a flexible price endowmen model wih home bias in preferences when rade in real bonds is permied.
8 find ha inroducing real bonds may make he equilibrium equiy porfolios less sensiive o changes in parameers. While boh heir paper and his paper emphasize he role of he exchange rae he mechanism of flucuaion of exchange raes is differen. Coeurdacier and Gourinchas (2009) adop home bias in consumpion o generae real exchange rae flucuaions. In conras we focus on nominal rigidiy and nominal exchange rae flucuaions. Indeed we show ha in he presence of nominal rigidiy home bias in preferences does no have much impac on equiy porfolios. van Wincoop and Warnock (2008) empirically sudy wheher home bias in preferences can explain home bias in equiy porfolios. In heir model (which absens human wealh) he magniude of he effec of preferences on home bias in equiies is small especially when a real exchange rae hedge is available. We inroduce home bias in preference in our model. As we have shown in our previous paper (Engel and Masumoo (forhcoming)) when exchange rae hedging is possible home bias in preferences does no maer a all in a saic model. In his paper we exend our analysis in fully dynamic general equilibrium and we find ha i does change he opimal equiy porfolio bu no much if here is price rigidiy and human capial. Coeurdacier Kollmann and Marin (2009) presen a 2-counry 2-good model wih a number of ypes of shocks and rade in equiies and bonds. They focus on redisribuive shocks shocks ha redisribue income beween firm owners and workers as a source of home bias in equiy holdings. Couerdacier Kollmann and Marin (2008) consider a similar model bu wih invesmen specific echnological change which hey argue can explain he home bias in equiy holdings. II. A GENERAL RESULT IN A STATIC FRAMEWORK We build a general-equilibrium model wih sicky prices. There are wo counries Home and Foreign each wih populaion ½. The model of his secion is a one-period model in which porfolio choices are made prior o he resoluion of uncerainy. We also assume ha goods prices mus be se prior o he realizaion of he sae. In Secion III we fully specify dynamic general equilibrium models. In he example model of his secion we use only some of he feaures of he model demand funcions for goods marke-clearing condiions and a general assumpion abou nominal price sickiness o demonsrae ha () rade in equiies and a foreign exchange hedge will allow households o achieve he complee markes equilibrium allocaions; (2) his oucome may always be achieved wih a porfolio allocaion such ha households have complee ownership of heir own counry s firms and hold no foreign equiies; and (3) his oucome wih complee home bias in equiy holdings is he only porfolio ha will replicae he complee markes allocaion excep in he special case in which labor income is perfecly correlaed wih he nominal exchange rae. This example exends he one presened in Engel and Masumoo (forhcoming).
9 To be clear his demonsraion is no a proof ha a decenralized marke wih rade in equiies and a forward marke in foreign exchange will achieve he efficien allocaion. We do no in his secion show ha he porfolio ha replicaes he complee marke allocaion is opimal from he household s perspecive. The models of Secion III however do derive household opimaliy condiions and derive an equilibrium in which hose are saisfied. The purpose of his secion is o explore inuiively he economic forces a work ha allow a porfolio wih complee home bias in equiies bu wih an appropriae foreign exchange posiion o replicae he complee-marke oucome. Iniially households are endowed wih ownership of firms in heir own counry and wih human capial. The ex pos budge consrain of a represenaive Home household can be wrien as () PC = REV + X. Under homoheic preferences Home expendiure is he produc of he exac price index P and he consumpion index C. We assume here ha Home agens have full ownership of Home firms and no ownership of Foreign firms here is no rade in equiies. Under ha assumpion Home households receive all of he revenue of Home firms REV in he form of profis capial income and labor income. We pu absoluely no resricion on he mechanism ha divides Home revenues beween equiy owners and workers. In addiion Home households can make conracs ha pay off X in Home currency ex pos bu cos zero ex ane. The adding up consrain implies ha Foreign households mus receive ex pos in Home currency erms or X / S in Foreign currency erms. X Nex we use he fac ha Home revenue is equal o purchases of he Home good by Home and Foreign residens: (2) REV = P C + S P C. h h h h C h is consumpion of he Home good by Home residens and C h is consumpion of he Home good by Foreign residens. P h is he Home currency price of Home goods and P h is he Foreign currency price of Home goods. S is he exchange rae expressed as he Home currency price of Foreign currency. We do no impose ha he law of one price holds. Tha is we do no require SP h = Ph. Moreover C h and C h may be homogeneous-of-degreeone indexes over a se of goods produced in he Home counry in which case he prices should be inerpreed as he corresponding exac price indexes.
0 Using (2) we will log-linearize () around a poin of iniial rade balance. If rade is iniially balanced P C P C = S P C where he 0 subscrip denoes he poin of linearizaion. 0 0 h0 h0 0 h0 h0 Tha is +α = 2 P C h0 h0 P C 0 0 Le + α denoe he share of Home expendiure on Home goods a he poin of linearizaion. 2. 7 Then we can wrie (3) p + c = α( p + c ) + ( α)( s + p + c ) + x h h h h where he lower case leers are he deviaions of he logs of he upper case leers from he poin of approximaion. x = X. We use he = sign in his equaion o mean equal up o PC a firs-order approximaion. 0 0 We now summarize some properies of log-linear approximaions o consumpion funcions and demand funcions ha follow from he assumpion of homoheic preferences. We can wrie he Home aggregae consumpion as + α α (4) c = ch + cf 0 α <. 2 2 ( + α) / 2 is he share of Home goods in Home nominal expendiures a he poin of approximaion. c f is he log of he Home household s consumpion of Foreign goods. The analogous Foreign household s consumpion is given by (5) α + α c c c 2 2 = h + f. In he models of secion III we will have α = α as we assume ha here is home bias in preferences bu i is symmeric beween Home and Foreign households and ha iniial wealh is idenical. We do no even impose ha symmery in his secion. The log of he consumer price indexes for Home and Foreign households are given by respecively + α α (6) p = ph + pf and 2 2 7 α in his secion is very closely relaed o home bias in preference parameer inroduced in secion III and idenical in symmeric case i.e. α = α = α bu α is no exacly same as α in general.
(7) h α + α p p p 2 2 = h + f. p and p are he logs of price indexes in Home currency for Home consumpion of he f aggregaes of Home and Foreign goods respecively. and are expressed in Foreign currency erms. p and h p are defined analogously Nex le ω be he elasiciy of subsiuion beween he aggregaes of Home and Foreign goods in Home households and ω be ha in Foreign households. Then he (log) nominal demand for Home goods by Home households is given by (8) ph ch ( ω)( ph p) p c + = + +. f For he Foreign counry we can wrie (9) p + c = ( ω )( p p ) + p + c. h h h Equaions (8) and (9) follow from he homoheiciy assumpion. Under homoheiciy he demand for Home goods relaive o Foreign goods depends only on heir relaive price and no on he aggregae level of consumpion. Rearranging he demand curves using (4)-(7) gives us (8)-(9). We assume ha he Marshall-Lerner condiion holds so haωω. Subsiue (8) and (9) ino (3) o rewrie Home budge consrain: p + c (0) + α α. = ( ω)( ph p) + p + c + ( ω )( ph p) + s + p + c + x 2 2 We can rearrange his Home budge consrain equaion using he price index equaions (6) and (7) o wrie + α + α 2 () c c = s + p p + ( ω )( pf ph ) + ( ω )( pf ph ) + x 2 2 α The relaive Home o Foreign consumpion given in equaion () depends on he real exchange rae s + p p and he relaive price of Foreign o Home goods in each counry as well. Tha is because he real exchange rae and he relaive prices influence he relaive revenues and profis of Home and Foreign firms. Firs consider he effecs of a home depreciaion which is an increase in s holding prices of goods compensaion o workers and payoffs o heir non-equiy posiions ( x ) consan. Wih goods prices consan he demand for he Home and Foreign goods is consan so he effec of he exchange rae is only on he valuaion of revenues. A depreciaion raises he
2 Home currency value of Home profis because he Home currency value of foreign sales increases. Also he Home currency value of Foreign revenues rises wih an increase in s for he same reason. Because Home households own Home firms Home consumpion increases. Relaive goods prices also have an effec on relaive nominal consumpion. An increase in relaive price of Foreign goods o Home goods in he Home marke p f ph or an increase in he Foreign marke p p will raise demand for Home goods relaive o Foreign f h goods. Since ωω by assumpion his price change also raises Home revenue relaive o Foreign revenue and so Home consumpion rises relaive o Foreign consumpion. Can households in he Home and Foreign counry make conracs ha efficienly allocae risk in his seing? Tha is is i possible o wrie a conrac for an asse ha pays x ex pos and achieves he complee markes allocaion? We will assume he curvaure of he uiliy funcions of Home and Foreign households is he same. Le ρ be he coefficien of relaive risk aversion for households in boh counries evaluaed a he poin of linearizaion (where we are assuming uiliy is separable in aggregae consumpion.) When asse markes are complee ha is when households can ex ane rade a complee se of nominal coningen claims he equilibrium condiion equalizes he marginal uiliy of nominal spending of Home and Foreign. I is well known ha he condiion can be log-linearized as (2) ( c c s + p p ) = 0. ρ I follows from he relaive consumpion equaion () ha his risk-sharing relaionship can be achieved if he non-equiy porfolio payoff x saisfies (3) ρ + α + α 2 ( s + p p) = ( ω )( pf ph ) + ( ω )( pf ph ) + x. ρ 2 2 α Rearranging equaion (3) households can aain he complee markes allocaion if heir non-equiy porfolio has a payou given by (4) x s p p 2 ρ α ρ ( = + ) ( + α) ( + α ) + ( ω )( pf ph ) + ( ω )( pf ph ). 2 2 As in he Lucas (982) model households do no need o rade coningen claims o ge hese payoffs. For example Home and Foreign households could rade asses ex ane whose
3 payoffs were linear in he real exchange rae and relaive prices in Foreign p p f h s + p p relaive prices in Home p f ph. Recall ha x describes he payoff from forward posiions he porfolio of non-equiy asses ha has an ex ane value of zero. By aking he appropriae forward posiion in hese hedges Home households could achieve opimal risk sharing. Many models assume ha he law of one price holds for raded goods. Under ha assumpion condiion (4) reduces o (5) α ρ x = ( α + α ) + ( + α)( ω ) + ( + α )( ω ) ( pf ph ) 4 ρ. The complee markes allocaion can be achieved by rading a hedge on erms of rade movemens. This could be achieved by rading wo bonds one denominaed in he Home good and one denominaed in he Foreign good as Coeurdacier and Gourinchas (2009) have emphasized. Trade in hese bonds consiues a forward posiion as we have defined i as long as he iniial value of he bond porfolio is zero he long posiion in one bond balances he shor posiion in he oher. 8 In realiy markes are no available o hedge erms of rade or even real exchange rae risk explicily. Bu when prices are sicky he log of he real exchange rae and log of he erms of rade in each counry are linear in he log of he nominal exchange rae. So households can hedge hose risks wih only rade in an exchange-rae hedge. Forward markes for foreign exchange do exis or a synheic forward posiion can be obained by rading nominal bonds or wih swaps. We ake an agnosic posiion on he currency of price seing. Prices could all be se in he producer s currency (PCP). Or prices facing consumers migh be se ex ane in he local currency (LCP). Or we migh even have prices indexed o he exchange rae as in Corsei and Peseni (2006) or Engel (2007). We assume p h = 0 and p f = 0. These assumpions mean simply ha he home-currency price of home goods sold in he home currency and he foreign-currency price of foreign goods sold in he foreign counry are consan (independen of shocks) and normalized o one 8 The firs version of his paper June 2008 did no menion how risks could be hedged in he world where he law of one price holds wih real bonds. However he February 2008 version of our earlier paper does show his. Tha model was simpler in ha we assumed preferences were symmeric (bu no idenical) beween Home and Foreign households. We menion his iming because he firs version of Coeurdacier and Gourinchas (2009) came ou i June 2008. We are no rying o claim prioriy for our resuls bu raher declaring ha his is a case of simulaneous discovery. In any even he emphasis of he papers is differen. Our focus is primarily on rade in nominal bonds under sicky prices. Coeurdacier and Gourinchas (2009) derive some general resuls abou porfolios of equiies and bonds when rade in real bonds is permied.
4 (in levels.) We can assume parial pass-hrough for raded prices: p f = bs p h = bs 0 b. b is he degree of indexing of consumer prices of impored goods o exchange raes. LCP corresponds o b = 0 and PCP o b =. Under hese assumpions α + α s + p p = b s and p f ph = pf ph = bs. 2 Subsiue ino equaion (4) o ge (6) x = δ s where α ( )( ) ( )( ) b α + δ α ρ + b α ω + + α ω = +. 2 2 ρ 2 If Home and Foreign agens can ake a forward posiion in foreign exchange hey can achieve he complee-markes allocaion. No rade in equiies is required. This resul holds regardless of degree of home bias in preference iniial wealh wage seup or price seup. A forward conrac coss one uni of he home currency. Wihou loss of generaliy we can normalize he forward price of foreign exchange o be uniy so he log of he forward rae is zero. If home households purchase δ unis of his hedge hey will achieve he payoff given (2.6). To review: We have assumed a one-period horizon and ha all nominal goods prices are fixed ex ane. These wo assumpions are criical o he resul. We will move on o infiniehorizon models wih price adjusmen in Secion III. There rade in equiies is necessary o replicae he complee markes equilibrium and he equilibrium porfolio will no exhibi complee home bias. Bu we argue ha here may be subsanial home bias when a nominal exchange rae hedge is available. Since we have derived (6) from he Home budge consrain alone i is useful o check he derivaion from he sandpoin of he Foreign budge consrain. Recalling he definiion of we have shown x X = δ s. X is he payoff from a forward posiion. In levels we PC 0 0 can rea S 0 as he forward rae i is he ex ane expeced spo rae. The payoff o Home agens is given by X = z( S S0) where z is he quaniy of conracs ha are raded. A log zs0 approximaion gives us x = X s PC PC so P C z = 0 0 δ. S 0 0 0 0 The budge consrain for Foreign household he Foreign equivalen of equaion (3) is 0 (7) p + c = ( α )( p + c s ) + α ( p + c ) + x f f f f
5 where x X PC 0 0. The payoff o he Foreign household in foreign currency erms is X S S = z. Using a log-linear approximaion we can wrie 0 S (8) z PC x s = x. 0 0 PC 0 0 SPC 0 0 0 α Now muliply boh sides of equaion (7) by and add o equaion (3). Afer wriing α ou he expressions for he price indexes we find ha he sum of hese wo equaions reduces o α (9) x + x 0 =. α This condiion is equivalen o equaion (8) given ha we have assumed iniial balanced rade. Hence he foreign hedge is given by (20) x α = δ s where δ = δ. α The resul we have obained holds as we have shown wheher price sickiness is of he PCP LCP or indexing form. We also have made no assumpions abou labor markes. There could be a spo marke in labor wih flexible wages or households could have marke power in labor markes and nominal wages could be sicky. There could be bargaining beween households and firms over revenues. We have no specified a his sage how revenues are spli beween firm owners and workers when here is no rade in equiies. The degree of home bias also does no play any role in deermining he equiy porfolio in his one period model. This is illuminaing given ha Obsfeld(2007) and Coeurdacier (2009) have shown ha home bias in preferences can resul in eiher ani-home bias in equiy or super home bias in equiy ha is aking a shor foreign equiy posiion. We will examine he home bias in preference assumpion also in a dynamic model where prices are fixed a mos one period. We will show ha home bias in preferences does play a minor role in deermining he equiy porfolio bu price sickiness maers more. We also have no specified he sources of shocks o he sysem. There can be real produciviy shocks and nominal moneary shocks. These shocks could influence all of he variables in he sysem: exchange raes labor income profis consumpion ec.
6 Simply saed when households in each counry have complee ownership of heir own firms (00% home bias in equiy holdings) relaive consumpion risk is ranslaed hrough relaive prices. When here is full price sickiness he relaive prices adjus only wih changes in he nominal exchange rae. So a forward posiion in foreign exchange can fully hedge risk. The issue we wish o focus on however is he implicaions for he forward posiion in foreign exchange. Recall ha he assumpions of our model require < αα < 0 b ωω > 0. If we make he furher empirically plausible assumpion ha ωω and ρ > hen equaion (6) implies δ < 0. This implies ha a depreciaion in he Home currency has a negaive payoff o Home households. This occurs only if Home households ake a long posiion in home currency and a shor posiion in foreign currency. Tha is hey lend in bonds denominaed in home currency and borrow in bonds denominaed in foreign currency. Inuiively a depreciaion of he Home currency (an increase in s ) as we have discussed above has a posiive effec on he payoffs o Home s equiy porfolio. The opimal hedge hen is o ake a foreign exchange posiion ha offses his exchange rae risk. The inuiion however is a bi more complicaed han his because he opimal porfolio should eliminae purchasing power risk raher han income risk. A depreciaion will also raise he Home price of impored goods if some of he depreciaion is passed hrough o impor prices ( b > 0 ). Bu he sign of δ is negaive for any value of b. We nex fully specify a general equilibrium model in a dynamic seing where we allow prices o adjus. III. A DYNAMIC STICKY-PRICE MODEL WITH LOCAL-CURRENCY PRICING In his secion we build an infinie-horizon model which allows us o examine he effecs of persisen echnology shocks and differen degrees of price sickiness as well as he effecs of home bias in consumpion and differen assumpions abou wage seing. The price-seing rule is defined as follows. A fracion τ of firms in each counry se prices in advance and he res of he firms can adjus heir prices in each period afer he realizaion of shocks. This approach allows us o sudy he porfolio allocaion wih or wihou sicky prices and we can learn how differen degrees of price sickiness affec he porfolio. There are differen ypes of firms in each counry bu we assume he equiies of all firms in each counry are bundled ogeher. In Secions III.A-C we assume ha firms se prices in consumer s currency ha is here is local-currency-pricing (LCP) and wages are flexible. Then we examine he sicky wage case in Secion III.D and he producer-currency-pricing (PCP) case in secion III.E.
7 As in Secion II here are wo symmeric counries Home and Foreign each wih populaion ½. A. Household Problem Home households maximize heir expeced uiliy: subjec o he following budge consrain: M max E0 β U C L = 0 P (2) PC + M + Qγ + S Q γ h + f + = γ ( Q +Π ) + γ S ( Q +Π ) + ( S F) δ + WL + M + Tr h f where Q ( Q ) denoes he price of Home (Foreign) equiies. Households ener ime wih money M equiies ( γ h γ f ) and forward conracs δ. Afer he realizaion of shocks households choose he consumpion level real money balances and labor supply. The dividends from firms are paid a ime and households ge he payoff from he forward conrac. They receive a ransfer from he governmen as well. Finally he households choose forward conracs and equiy holdings γ h + γ f + which deermine he dividends δ + households receive a ime +. Home households receive he following income each period: wages ( WL where W denoes he wage); dividends; ransfers from he governmen ( Tr ) and he gains or losses from forward conracs. Equiy dividends received by a Home household are given by where γ Π + γ S Π h f Π is he profi (dividend) of Home firms and he Foreign currency. 9 Π is ha of Foreign firms in erms of S is he Home currency price of Foreign currency. Home and Foreign households rade forward conracs in foreign exchange. The forward rae F is known a he ime he forward conrac is enered ino prior o he realizaion of shocks. Afer he shocks are realized he Home households receive δ ( S F) unis of Home currency. 9 Theoreically profis can be negaive in he case of a loss bu we have o assume ha he profis of boh Home firms and Foreign firms are posiive o ake logarihms.
8 We noe here ha in his secion wages are flexible and deermined in a compeiive labor marke. When we specify firm behavior we will assume ha he labor services of all households are idenical perfecly subsiuable. We will laer examine a version of his model in which each household s labor services are unique and each household is a monopoly supplier of ha ype of labor. There he differen ypes of labor services are no perfecly subsiuable. In ha seing we will also assume nominal wages are se in advance. Uiliy is given by (22) M M η U C L = C + χ ln L ρ + ψ P ρ P + ψ where ρ > 0 χ > 0 ψ 0 and η > 0. C denoes he consumpion baske for Home; M denoes Home money; P he price index; and L he labor supply. C is a consumpion baske of a represenaive Home household defined as (23) C C C 2 2 / ω / ω + α ( ω )/ ω α ( ω )/ ω h + f ω/( ω ) < α < where ω > 0 is he elasiciy of subsiuion beween Home produced goods and Foreign produced goods. C h is he consumpion baske of Home produced goods and C f is ha of Foreign produced goods: (24) λ/( λ ) /2 / λ ( λ )/ λ Ch 2 Ch ( i) di 0 λ/( λ ) / λ ( λ )/ λ Cf 2 Cf ( i) di /2 where λ denoes he elasiciy of subsiuion among varieies wih λ >. There is a represenaive household in each counry and he household preferences in Home and Foreign are symmeric. When α > 0 here is home bias in consumpion. Tha is he Home household pus relaively more weigh on consumpion of Home goods and he Foreign households preferences pu relaively more weigh on Foreign goods. We can wrie he CPI as follows: (25) where /( ω ) + α ω α ω h + f P P P 2 2 (26) /2 λ Ph 2 Ph ( i) di 0 /( λ ) λ Pf 2 Pf ( i) di /2 /( λ )
9 where P () i is he price of Home goods i sold in Home in erms of he Home currency and h Pf () i is he price of Foreign goods i sold in Home in erms of he Home currency. Foreign households have an analogous uiliy funcion for Foreign quaniies and prices which we will denoe by superscrip aserisks. Foreign prices are denominaed in Foreign currency. Our assumpions on consumpion asse acquisiion ec. follow exacly he sandard presenaion of he non-sochasic dynamic model (see for example Obsfeld and Rogoff (996)) wih one excepion: We assume as in he saic model ha households can ake a forward posiion in foreign exchange. Making a conrac o buy foreign exchange forward nex period of course is equivalen o buying a nominal (non-sae-coningen) bond denominaed in he foreign currency and shoring a nominal bond denominaed in he home currency. We could have inroduced nominal bonds denominaed in each currency separaely ino he model raher han forward conracs. However ha would add nohing o our presenaion. We shall see below ha he (linearized) model wih equiies and forward conracs reproduces he allocaion ha would be achieved wih rade in a complee se of nominal sae-coningen bonds. If we inroduced non-sae-coningen nominal bonds insead of forward conracs he posiion held by each household will exacly reproduce heir posiion in he forward marke. Given prices and he oal consumpion baske C he opimal consumpion allocaions are (27) Ch ( ) ( Ph P ) ω = + α C ( α) ( ) ω (28) Ch () i = 2 ( Ph () i Ph ) λ Ch () 2 ( () ) C = P P C f f λ f f f f C i = P i P C. The dynamic firs order condiions for he households are ρ ρ χ C C+ (29) = Eβ M P P + ρ ψ C (30) ηl = W P (3) ρ ρ C C E S = FE P P (32) ρ ρ C C Q = E β ( Q +Π) P P
20 (33) Firs le D + s C P ρ + s ρ C S Q = E β S( Q +Π). P ρ ρ C + s C. The no-bubble soluion for equiy prices implies ha P P s (34) Q = Eβ D + sπ+ s (35) Le s= s = β + s + sπ+ s s= SQ E D S V γ Q + γ SQ h + f + s (36) H β ED + sw+ sl+ s (37) (38) (39) R s= β ( Q + Π) R Q β ( H + WL) H H γ SQ γ Q γ =. + f + h + V V These are respecively financial wealh human capial he rae of reurn on financial wealh and human capial (each muliplied by he uiliy discoun facor for algebraic convenience) and he share of foreign equiy in equiy porfolio. We can rewrie he budge consrain (2) for ime : S H (40) PC + V + H = V ( γ) β R V γβ R H β + + R + δ( S F). S We will assume below a process for he money supply in which E( M+ ) = M. We noe his now because under his assumpion he firs-order condiion (29) can be simplified direcly o ge (4) C P ρ C χ = χm + E = M. ρ + β P+ β I follows from his ha we can wrie sochasic discoun facor as D M + s =. The firs M + s order condiions for equiy holdings (32) and (33) or Euler equaions can be summarized as
2 (42) M M S E R = E R =. M M S B. Firms Firms produce oupu wih labor using a linear echnology. Labor from each household is perfecly subsiuable for labor from any oher household). A firm in his economy monopolisically produces a specific good indexed by i : 0 (43) Y() i = AL () i where Y ( i ) is he producion of firm i A is he counry-specific echnology parameer and L () i is he labor inpu of firm i. Home and Foreign markes are segmened and only he producer can disribue is produc. We have wo ypes of firms in each counry. A fracion τ of firms se he price in advance and he res se he price afer he realizaion of shocks. The profi maximizaion problem of he Home firm wih price flexibiliy is W max Ph ( i) Yh ( i) + SPh ( i) Yh ( i) Yh ( i) + Yh ( i) A. Because Y () i is no a funcion of h is easy o solve: P () i and h Y () i is no a funcion of P () i he problem h h λ W (44) Ph () i = Pflexh λ A λ W P () i = P h flex h λ AS where P flex h is he opimal price for he Home marke of he Home goods produced by he firms ha can adjus prices afer hey observe shocks. P flex h is he opimal price for he Foreign marke. Sicky-price firms se prices one period in advance in he consumers' currencies for each counry. Firms in each counry se prices so as o maximize heir expeced profis aking oher firms' prices as given which is equivalen o aking he price level as given since each firm has measure zero on inerval [0]. The opimal prices are 0 Using a Cobb-Douglas echnology wih oher fixed inpus will no change he resul if he reurns on he oher facors belong o he equiy holders.
22 (45) (46) P P prese h prese h λ ω W P h E D C λ A P h P λ ω λ Ph E D C P h P λ ω W P h E D C λ A P h P λ ω λ P h E D C P h P where D is he sochasic discoun facor of firm owners and P prese h is he opimal price for he Home marke a ime of he goods produced by he firms ha se prices in advance. Now we can rewrie he price indexes as follows: (47) (48) ( ) λ h = τ λ flex h + τ prese h λ P P P ( ) λ f = τ λ flex f + τ prese f λ P P P. Since we have CES sub-uiliy funcions he marke clearing condiion can be obained by equaing he oupu wih he sum of he demands for Home goods: + AL α Ph P C α = + Ph P ω C. 2 2 Aggregae profis of each counry are (49) ( ) ω ( ) + α α ω Π = Ph Ph P C + SPh Ph P C WL 2 2 ω (50) ( ) ( ) + a α ω SΠ = SPf Pf P C + Pf Pf P C SW L. 2 2 ω (5) ( ) ( ) We assume ha he logs of he money supplies and logs of he echnology shocks evolve according o (where lower-case leers represen he logs of he upper-case counerpars i.e. ln( X ) is denoed as x ) m (52) m+ = m + v+ m + + m = m + v
23 (53) a W W W + = ϑ a + v + a R R R + = ϑ a + v + W R where ϑw [0] ϑr [0) are degrees of persisence in world and relaive echnology levels and where he vecor W 2 2 v ( x = mm W R) is i.i.d. We denoe he world variables as x R m m 2 + + 2 m x = x + x and he relaive variables as x = x x. We assume Ev = Ev = σ so m ha E ( M + ) = M as menioned above. We assume also var( v ) var( v ) σ m m m cov( v v ) σ mm = W var( ) 2 = v σ W R var( ) 2 W R = and cov( ) 0 v σ R R R symmery beween Home and Foreign: ha is a 0 = 0 and m 0 = 0. m 2 = = and v v =. We assume iniial Noe in paricular ha we have no made any assumpions abou he correlaion of moneary shocks and produciviy shocks. As long as here is some independen componen o he money shocks ha is as long as he correlaion beween money and produciviy shocks lies on he inerval (-) our resuls go hrough. In paricular our specificaion allows for W R an inerpreaion in which echnology shocks v + and v + are srucural and moneary shocks respond conemporaneously o echnology shocks: for example m m W w R R + ε+ ξ + ξ + m m W w R R m m v+ ε+ + ξ v+ + ξ v+ and v + v v where ε + and ε + are srucural moneary shocks. C. Equilibrium Porfolios under LCP and Flexible Wages We follow he soluion mehod of Engel and Masumoo (forhcoming) in which we loglinearize he budge consrain resource consrain and he non-porfolio firs-order condiions. We use a second-order approximaion o he asse choice Euler equaions. Noe ha our mehod of soluion is idenical o he one proposed by Devereux and Suherland (2006 forhcoming) for our model. I is imporan o noe ha in his model and in all of he models ha we presen rade in equiies and a forward posiion in foreign exchange is sufficien o replicae he allocaions under complee markes. Tha is because in all cases once we log-linearize he equaions of he dynamics of he model he payoffs from hese asses span he sae space. So as in Engel and Masumoo (forhcoming) and he simple model of secion II risk sharing condiion equaion (2) holds which is repeaed here for convenience: (54) ( c c s + p p ) = 0 ρ
24 The quesion we wish o address in his paper is wha ypes of porfolios will suppor his complee marke allocaion? Our answer is ha under a wide range of plausible assumpions his allocaion can be suppored wih only a small amoun of equiy rade (and hus a lo of home bias in equiies) bu i requires a bond porfolio in which households lend in heir own currency and borrow in foreign currency. The Appendix derives he following expressions for he opimal porfolios of equiies and he forward posiion in foreign exchange. In equilibrium because he log-linearized model replicaes he complee-markes equilibrium he Home/Foreign relaive marginal uiliy of nominal wealh is consan. This impars a saionariy o he economy ha suppors opimal porfolio posiions ha are consan over ime. The opimal share of Foreign equiies in he Home equiy porfolio is given by (55) where ρ τ βϑ R ϖ + α ρ + + ( τ) ψϖ ϖψ + βϑr γ = 2 τ βϑ R τ ( ζ)( ϖ ) + + ζ + ( τψϖ ) ϖψ + βϑr + ( τ ) ψϖ 2 α ϖ + ω( + α)( α). ρ (recalling ha ω is he elasiciy of subsiuion among brands of goods produced wihin each counry ρ is he coefficien of relaive risk aversion and +α is he share of Home 2 goods in Home consumpion if all goods prices were equal so ha α > 0 implies home bias in consumpion.) In his expression also ζ is labor s share of revenue in he non-sochasic seady sae. Recall also ha ψ is he elasiciy of labor supply β is he discoun facor in uiliy ϑ R is he auocorrelaion of relaive produciviy shocks and τ is he fracion of firms ha se prices in advance. To undersand his expression i is useful o consider some special cases. Firs suppose ha all goods prices are flexible so ha τ = 0. Then (55) reduces o (56) ρ ϖ + α ρ γ =. 2( ζ)( ϖ ) Available from he auhors.
25 When goods prices are flexible in general we find ani-home-bias in he opimal equiy porfolio. If here were no home bias in preferences ( α = 0 ) expression (56) simplifies furher o γ = /[2( ζ )] > /2. Tha is Home s holdings of Foreign equiies would consiue more han half he equiy porfolio. This simply reflecs he observaion by Baxer and Jermann (997) ha under flexible prices and wages he reurn o Home human capial is posiively correlaed wih he reurn o Home equiies. So opimally he porfolio should shor Home equiies. In equaion (56) we see ha home bias in preferences does no modify his resul excep under srong home bias in preferences when ϖ <. If uiliy is logarihmic in consumpion (as is assumed in Heahcoe and Perri (2008) or Pavlova and Rigobon (2007)) so ρ = he opimal share of foreign equiies is sill γ = /[2( ζ )] as i was wihou home bias in preferences. More generally under he plausible parameerizaion ρ > we can see ha a high degree of home bias in preferences such ha ϖ < can generae exreme home bias in equiy holdings. 2 Suppose here were complee home bias in preferences so ha α =. Effecively he wo counries are closed economies. Only produciviy shocks maer for real allocaions under flexible prices and wages. Bu if Home agens do no wish o buy any foreign goods hen he Foreign equiy provides no hedge for Home produciviy shocks. To see his noe ha when α = we have ϖ = / ρ. In his case from equaion (56) we find γ = 0. I is imporan o sress ha in his case he porfolio wih no equiy diversificaion achieves he same allocaion as would occur under complee markes bu in boh cases asse markes do nohing o reduce consumpion risk. When households in each counry produce he only good which hey wan o consume i is no possible o hedge produciviy shocks. Now consider he case in which all nominal prices are se one period in advance ( τ = ). Suppose ha eiher agens pu no weigh on he fuure β = 0 or relaive produciviy shocks are serially uncorrelaed ( ϑ R = 0.) Under hese assumpions we effecively replicae he condiions of he model of secion II: all prices are sicky and he fuure does no maer o agens. We see from equaion (56) ha under hese assumpions γ = 0. Tha is he opimal holdings of Foreign equiies by Home households in he porfolio ha replicaes complee marke allocaions is zero. This is he resul illusraed in secion II: equiy rade is no needed (under hese circumsances) o suppor efficien risk sharing. We noe ha in his secion we have derived opimal porfolios aking ino consideraion all of he firs-order condiions of households and firms. Tha is in conras o secion II where we only asked which sors of porfolios would suppor complee marke allocaions wihou asking wheher he equilibrium could be decenralized. 2 A deailed discussion of he disconinuiy around ϖ = can be found in Obsfeld (2007) and Coeurdacier (2009).