Portfolio Choice and Home Bias in Equities in a Monetary Open- Economy DSGE Model

Transcription

1 Porfolio Choice and Home Bias in Equiies in a Moneary Open- Economy DSGE Model Charles Engel Universiy of isconsin and NBE Akio Masumoo Inernaional Moneary Fund May (Firs version Sepember ) Absrac This paper develops a wo-counry moneary DSGE model in which households choose a porfolio of home and foreign equiies and a forward posiion in foreign exchange. I presens a poenial soluion o he home bias puzzle in he conex of a model in which some goods prices are se wihou full informaion of he sae. In response o echnology shocks sicky prices generae a negaive correlaion beween labor income and he profis of domesic firms leading o home bias in equiy holdings. In conras under flexible prices labor income and he profis of he domesic firms are posiively correlaed. eurns on human capial and equiies may be posiively correlaed under sicky prices when he source of shocks is moneary bu his risk is hedged hrough nominal asses raher han hrough equiies. JEL classificaion: F3 F4 G. Keywords: home bias; inernaional porfolio choice e have benefied from discussions wih Olivier Jeanne Karen Lewis Alessandro ebucci Cedric Tille ober Vigfusson and Frank arnock. e also hank paricipans a numerous seminars. Engel acknowledges he suppor of he Naional Science Foundaion hrough a gran o he Universiy of isconsin. Some of he work on his projec was compleed while Engel was a visiing scholar of he Inernaional Moneary Fund and Masumoo was a graduae inern a he Board of Governors of he Federal eserve Sysem. The views expressed in his paper are hose of he auhors and do no necessarily represen hose of he Federal eserve Sysem he IMF or IMF policy. cengel@ssc.wisc.edu amasumoo@imf.org Phone:

2 . Inroducion In an open macroeconomy in which asse rade is possible he porfolio choice of households may play an imporan role in undersanding macro flucuaions. In conras o a closed economy model in which a represenaive agen simply holds he marke porfolio agens in each counry may hold differen porfolios depending on he counry-specific risks and reurns ha hey encouner. Porfolio choice migh maer for a number of quesions: Does he inernaional ransmission mechanism depend on who owns firms? Do changes in valuaions of inernaionally raded asses play a role in he macroeconomic adjusmen o shocks? Is here an ineracion beween he sock marke and exchange raes? e believe a successful model mus capure wo feaures. The firs is ha equiy prices respond o moneary policy shocks. The empirical evidence makes abundanly clear ha equiy prices depend on he marke s percepion of moneary policy. 2 e model he channel hrough which moneary policy affecs real reurns as sicky nominal prices: some firms mus se nominal prices wihou full informaion abou he sae. The second key feaure ha a model mus explain is he home bias puzzle. The home bias puzzle is one of he major puzzles in inernaional finance. Empirical sudies have found ha foreign equiies comprise a small proporion of invesors' porfolios. 3 This finding is puzzling because i appears ha invesors are forgoing imporan opporuniies for diversificaion of risk. 4 hile here have been many suggesed resoluions o he puzzle none seem able o explain enirely he exen of home bias. Our model offers a new explanaion ha may conribue o an undersanding of home bias. In a framework in which some nominal prices are sicky i may be naural for households o bias heir porfolios srongly oward home equiies as a hedge agains shocks o heir labor income. e do no build he model specifically o explain he home-bias puzzle bu we find ha in he framework we examine opimal porfolios exhibi home bias. The inuiion is sraighforward: If all nominal prices are sicky in he shor run he level of oupu is demand deermined. Produciviy shocks have no effec on shor-run oupu if he firm adjuss oupu only in response o changes in demand. For example if home firms experience a posiive produciviy shock heir demand for labor will decline. Employmen On he firs quesion see for example Baxer and Crucini (995) Kehoe and Perri (2002). Gourinchas and ey (2005) and Tille (2004) have recenly addressed he second quesion. Pavlova and igobon (2003) and Andersen Bollerslev Diebold and Vega (2004) are recen papers ha have ried o answer he hird quesion. 2 See for example Bernanke and Kuner (2004) Andersen Bollerslev Diebold and Vega (2004) Bordo and heelock (2004). 3 French and Poerba (99) Tesar and erner (995) and arnock (2002) for example. 4 Lewis ( ) surveys he lieraure on his puzzle and discusses he losses from non-diversificaion. 2

3 and wages will fall bu profis o he firm will increase. An effecive hedge agains employmen and wage risk is ownership of he firm. If oupu is demand deermined he shor-run reurns o labor and firm owners are negaively correlaed in conras o he usual presumpion in neoclassical models. The fac ha produciviy shocks creae a negaive correlaion beween reurns o workers and hose o firm owners is a key implicaion of he model. Gali (999) builds a closed economy model under sicky prices and shows ha i can generae a fall in labor hours in response o he posiive echnology shock which rarely arises in a flexible price model. 5 His empirical work demonsraes ha labor hours decline in response o posiive echnology shocks in mos G7 counries. The relaed empirical work by Boazzi Peseni and van incoop (996) and Julliard (2002) find ha reurns o human capial and equiies are negaively correlaed in mos OECD counries. 6 Our explanaion can be considered par of one hread of he lieraure ha has aemped o explain home bias as a hedge agains non-radable risks. 7 Our non-radable risk is flucuaions in labor income. The lieraure has no ye reached a consensus on how much of home bias can be explained by he need o hedge agains non-radable risk. There is a lieraure ha specifically examines wheher home asses can be a hedge for labor risk. Bu in neoclassical models because labor income is correlaed more wih domesic firms' profis han wih hose of foreign firms' he opimal porfolio will be more foreign-weighed han he classical endowmen model predics as shown in Baxer and Jermann (997). Hence pas aemps o explain home bias by using labor income have been largely unsuccessful. 8 In our model he uncondiional correlaion beween reurns o equiies and human capial need no be negaive. Moneary shocks lead o consumpion risk bu we show ha hose shocks can be hedged effecively wih bond porfolios (or by aking a forward posiion in foreign exchange.) Unexpeced changes in he relaive supplies of money (a home and abroad) creae nominal exchange rae changes ha in urn aler he value of reurns on home and foreign bonds. So he real risk creaed by moneary shocks can be diversified hrough he 5 For example of flexible price models which generae a negaive correlaion see Francis and amey (2003) and Dosey (999). 6 The US is one of he excepions. 7 For example Eldor Pines and Schwarz (988) Sockman and Dellas (989) Tesar (993) Baxer Jermann and King (998) Serra (200) and Peseni and van incoop (2002). A relaed analysis by Obsfeld and ogoff (200) argues ha ransacions coss o rade in inernaional goods can help accoun for home bias in equiies. 8 See also Jermann (2002). However Palacios-Huera (200) claims ha a subsanial fracion of home bias can be explained when he differenial human capial of sockholders and non-sockholders is aken ino accoun along wih human capial fricions. Julliard (2002) claims ha Baxer and Jermann did no properly accoun for reurn correlaions across counries. 3

4 nominal asse porfolio. Moneary shocks lead o posiively correlaed changes in labor paymens and profis bu ha risk is no hedged wih he equiy porfolio. Of course nominal prices do no remain fixed forever when produciviy or moneary shocks occur. Evenually an adjusmen is made and neoclassical resuls obain in he long run. Indeed our model has real labor income posiively correlaed wih produciviy shocks in he long run. So he abiliy of our model o explain home bias depends on he persisence of price sickiness he persisence of produciviy shocks and he weigh ha households assign o fuure consumpion. e show ha home bias is greaer when prices adjus more slowly when produciviy shocks (in one counry relaive o he oher) are less persisen and when he fuure is discouned more heavily. e do no claim ha our model offers he only explanaion for home bias. The lieraure has aken many differen approaches o explaining he phenomenon. In addiion o he papers cied above ha consider diversificaion agains non-radable risk several oher avenues have been explored. One group of sudies has argued ha he gains from inernaional diversificaion are in fac small so ha small ransacions coss of diversificaion will lead o heavily concenraed porfolios. 9 Ohers have claimed ha acquisiion of informaion abou foreign firms is more cosly han for informaion on home firms. 0 Anoher se of sudies shows ha home bias can be explained in he conex of generalized preferences or prior beliefs. Some claim ha home bias is parly due o empirical mismeasuremen. 2 All of hese facors may help explain home bias. In he following secions we presen wo kinds of models. The firs is saic and gives us he inuiion for home bias. In he saic model here is complee home bias in equilibrium. The second model is a more realisic dynamic one in which we focus on persisen echnology shocks and differenial price sickiness. The secion on he dynamic model analyzes he condiions under which home bias occurs and he degree of bias. 9 For example Cole and Obsfeld (99) Tesar (995) Buler and Joaquin (2002) and many ohers. However van incoop ( ) for example finds large unexploied gains from inernaional risk sharing. 0 For example Kang and Sulz (997) and Hasan and Simaan (2000). A relaed recen sudy is van Nieuwerburgh and Veldkamp (2005). For example van incoop (994) Aizenman (999) as examples of he former and Pasor (2000) for he laer. 2 For example owland and Tesar (2004) find ha mulinaionals may have provide diversificaion opporuniies for some counries. 4

5 2. The Simple Saic Model e build a general-equilibrium wo-counry model wih sicky prices. In his secion we consider a saic model ha provides he inuiion behind our explanaion for home bias. The nex secion hen builds a dynamic model o invesigae more realisically wha deermines wheher and how much home bias is generaed. In our model here are wo counries which we call Home and Foreign. The world populaion is normalized o uniy; half he populaion lives in Home and half in Foreign. Their preferences are idenical. Households provide labor and own firms hrough equiy. Firms use labor as he only inpu o produce a good monopolisically and prese heir prices in he consumers' currency. Markes are segmened so ha only firms can expor goods. All goods are radable and perishable. e adop local currency pricing here. e observe in he daa a leas for developed counries ha consumer prices are sicky in he consumers' currencies raher han in he producers' currencies. However he pricing assumpion is no paricularly imporan in deermining he equiy porfolio. In fac we would have exacly he same equiy porfolio when prices are prese in producers' currencies even hough he equilibrium number of forward conracs differs. 3 In our model we consider wo kinds of shocks: moneary and echnology shocks. The disribuion of shocks is idenical beween Home and Foreign. Finally we assume ha before he realizaion of shocks only forward conracs in he foreign exchange and equiies are raded. 2. Households Households in boh counries have idenical preferences over he consumpion baske he real money of he domesic counry and leisure. There are wo sages o he household decision problem. In he firs sage households choose a porfolio posiion: shares of home equiies ( γ h ); shares of foreign equiies ( γ ) and a forward posiion in foreign exchange (δ ). f These are chosen before he resoluion of uncerainy. Afer shocks are realized households choose consumpion labor supply and money balances o maximize: M M η U C L = C L ρ ψ χ ln P ρ P ψ (2.) 3 See Masumoo (2004). 5

6 ρ > χ > 0 ψ > 0 and η > 0. subjec o he consrain: PC M = γ Π γ S Π L δ( S F ) Tr (2.2) h f C denoes he consumpion baske for Home; M denoes Home money; he price index; and L he labor supply. C is a consumpion baske of a represenaive Home household defined as /( ω ) ( )/ ( )/ ( ) ( ) ( ω ω ω ω ω )/ C C C (2.3) 2 h f where ω > 0 is he elasiciy of subsiuion beween Home produced goods and Foreign produced goods. C h Foreign produced goods. is he consumpion baske of Home produced goods and λ/( λ ) ω P λ/( λ ) C f is ha of /2 / λ ( λ )/ λ Ch 2 Ch ( i) di / λ ( λ )/ λ C 2 ( ) (2.4) 0 f Cf /2 i di where λ denoes he elasiciy of subsiuion among varieies wih λ >. Then we can wrie he CPI as follows: where /( ω ) ( ) ( ) /( ω ω ω ) P P P (2.5) 2 h f where /2 λ Ph 2 P 0 h ( i) di /( λ) λ Pf 2 P /2 f ( i) di /( λ ) (2.6) Ph () i is he price of Home goods i sold in Home in erms of he Home currency and P i is he price of Foreign goods i f () sold in Home in erms of he Home currency. Home households receive he following: wages ( L where 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 Π γ Π γ S Π h f is he profi (dividend) of Home firms and he Foreign currency. 4 S Π is ha of Foreign firms in erms of is he Home currency price of Foreign currency. Home and Foreign households rade forward conracs in he foreign exchange. The forward rae F is 4 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. 6

7 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. 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. Prior o he realizaion of shocks he households choose he porfolio posiion o M maximize expeced uiliy ( E U C L P ) 5 subjec o he consrain: γ γ = (2.7) h Noe ha here is no consrain on he forward posiion δ. e assume ha he ex ane disribuion of shocks are idenical beween Home and Foreign. This assumpion ogeher wih he assumpions of idenical size and idenical preferences gives us an equilibrium in which he equiy prices of Home and Foreign firms are he same prior o he realizaion of shocks. 6 In our normalizaion he represenaive household of each counry is endowed wih an ownership share of of heir own firms bu hey may rade some of heir shares wih households in he oher counry which implies consrain (2.7). Given he symmery in he model here is home bias when γ f <. 2 Given prices and he oal consumpion baske allocaions are f C he opimal consumpion C h P h = 2 P ω C C f Pf = 2 P ω C (2.8) λ C P () i = () i C h h 2 h P h The remaining firs order condiions are M P λ Pf () i Cf () i = Cf 2 P. (2.9) f = χc (2.0) ρ η = (2.) χ M L ψ 5 e use he noaion ha expecaions are aken a ime - in his secion even hough he model is saic for noaional convenience so ha we can refer o some of he same equaions ha arise in he dynamic model. 6 If prices are differen hen one counry is richer han he oher ex ane a siuaion ha conradics symmery. 7

8 ρ ρ C C E S = FE (2.2) P P E C E S ρ Π = Π P ρ C. (2.3) P 2.2 Firms Firms engage in monopolisic compeiion as in Blanchard and Kiyoaki (987). A firm in his economy monopolisically produces a specific good indexed by i using a linear echnology: 7 Y() i = AL () i (2.4) 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. Labor is assumed o be homogeneous and o be supplied elasically. Home and Foreign markes are segmened and only he producer can disribue is produc. 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]. Given he CES uiliy sub-funcion he demand for Home good i from he Home marke denoed by Yh () i is λ P () i P = ω h h Yh () i C 2 P h P while he demand for Home good i from he Foreign marke is λ Ph () i Ph h () 2 P h P ω (2.5) Y i = C. (2.6) Firm i s profi maximizaion problem is max E D ( i) Ph ( iy ) h ( i) Ph ( iy ) h ( i) ( Yh ( i) Yh ( i)) A Ph () i Ph () i 7 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. 8

9 where D () i is he sochasic discoun facor for he firm i. For example if firms are owned by Home residens i will be C P ρ owned we use a more general noaion.. However because firms are no always domesically The opimal price of Home goods for he Home marke 8 is P h E DC λ A =. (2.7) λ E ( DC ) Similarly he opimal price of Home goods for he Foreign marke is E DC λ A h λ E P = ( DC S ). (2.8) Because firms are all alike hey will se he idenical prices for each marke. The marke clearing condiion can be obained by equaing he oupu wih he sum of he demands for Home goods: ω ω Ph P h = 2 P 2 P AL C C. (2.9) Given hese prices we can calculae profis. Using he opimal consumpion allocaions we can wrie he profis for he firms in each counry in erms of he Home currency as: ω ω Ph P h Ph C SPh C 2 P 2 P Π = L (2.20) ω ω P f Pf Π = f f 2 P 2 P S S P C P C S L. (2.2) Firms will pay ou all of heir profis as dividends. e assume ha A ( ( )) ( ) and ( ) 2 a A are drawn from idenical lognormal disribuions wih var ln A = var ln A = σ cov ( ln A ln A ) = σ. e also assume ha M and aa M are drawn from idenical lognormal disribuions wih 8 e will omi index i since Home firms are idenical. 9

10 ( ) ( ( )) ( ) var ln M var ln M σ cov ln M ln M σ and we assume ha he = = ( ) 2 m = mm money shocks are independen of he echnology shocks. The labor marke is compeiive and he wage moves freely o equae demand and supply of labor afer he shocks. The oupu of each good is deermined by demand. Firms adjus oupu afer he shocks o saisfy demand holding prices consan. The money marke is assumed o equilibrae so money demand equals money supply. 2.3 Soluion of he Saic Model An equilibrium in he saic model saisfies equaions (2.2) and (2.5)-(2.2) and heir foreign counerpars. These 39 equaions (one is redundan by alras' Law) solve for C f Ch () i Cf () i L P P h P f Ph () i Ph () i Y ( i ) Yh () i Y () i Π C h γ and heir foreign counerpars and δ F and S. 9 f e will no in fac solve for his equilibrium bu will insead solve he equilibrium for a se of equaions ha approximae hese 39. e ake firs-order approximaions o he budge consrain (2.2) he definiions of he consumpion and price indexes (2.3)-(2.6) he equilibrium condiion (2.9) and he definiion of profis (2.20)-(2.2). Under our assumpion ha he driving variables are lognormally disribued and wih he loglinearizaion of hese equaions we can solve equaions (2.7)-(2.8) exacly. Our focus is on he equilibrium porfolio choice of equiy shares and forward foreign exchange posiion. e proceed in his secion o consruc he equilibrium soluions for hese variables in an inuiive manner o exposi he economic facors ha lead o home bias. e will firs derive he porfolio demands for households aking prices as given. ih hese in hand we will use equilibrium condiions in goods labor and asse markes o derive he equilibrium porfolio posiions. e rely on ex ane symmery in he derivaions below. Lower case leers refer o logs of heir upper case counerpars. e use var o denoe variance and cov covariance. 20 e use he noaion x = Ex ( ). In he linearized equaions below we suppress he inercep erms for convenience. h C γ h 9 e have also implicily assumed ha here is a money marke equilibrium condiion ha holds bu we have no inroduced separae noaion for money demand and money supply and ha here is a forward marke clearing condiion which can be guaraneed here by seing δ = F δ. As he Appendix demonsraes equilibrium acually requires bu by symmery in he saic model F equals one. 20 e drop he subscrip on expecaions for he res of his secion. 0

11 The household firs-order condiion (2.2) can be wrien as: ρ cov( c s) var( s) = 0 (2.22) 2 where we have used ex ane symmery o give us f = 0 and Es ( ) = 0. e can use similar seps and recognize ha symmery implies ha var( π ) var( π ) = and cov( s π ) cov( s π ) = o derive from equaion (2.3): π = π ρcov( c π ( s π )) cov( s π ( s π )) = 0. (2.23) 2 e approximae he budge consrain (2.2) using condiion (2.7) o arrive a: e where ζ π e e p c = γ ζ π γ ζ s π ζ w l δs (2.24) ( )( ) ( )( ) ( ) w l δ w l π w l e δ e consrain around a poin where x and γ γ f. Here we have approximaed he budge = x for x = s c π π w l. e use equaion (2.24) o subsiue ou for c in equaions (2.22) and (2.23) and recognize ha p is predeermined. Then we solve ou for γ and δ : γ = cov( π π ( s π )) var( π ( s π )) cov( w l π ( s π )) cov( s π ( s π )) ( δ ) s s )) ζ ζ var( π ( π )) ζ 2ρ var( π ( π ( ζ)cov( π s) ζ cov( w l s) ( ζ)cov( π ( s π ) s) δ = γ var( s ) var( s ) var( s ) 2ρ = ( ζ) βπ s ζβw l s γ( ζ) β π s π s 2ρ cov( x s) where we have used he noaion β xs. var( s ) (2.25) e can hen use hese o equaions o solve ou for γ using he properies of orhogonal projecions o ge: cov( π βπ ss π π ) ζ cov( w l βw l ss π π ) var( π π β s ) var( ) s ζ π π β s π π π π s γ =. (2.26) Consider expression (2.26). From he poin of view of he household he equiy posiion is deermined by he covariances and variances of shocks o profis and labor income ha are orhogonal o exchange raes. Any variance in he porfolio ha is aribuable o

12 exchange rae changes is hedged hrough he forward posiion so he equiy posiion is deermined only by hose risks ha are uncorrelaed wih exchange rae risk. If he componen of labor income ha is orhogonal o exchange raes were uncorrelaed wih relaive profis of home and foreign firms (or if labor s share were zero) he second erm in equaion (2.26) would drop ou. Then he share γ of equiies held in foreign firms would increase as home profis (orhogonal o he exchange rae) have a higher covariance wih relaive home and foreign profis. Under our symmery assumpion his erm will equal /2 so he porfolio would be balanced beween home and foreign equiies if only he firs erm maered. I is he second erm of equaion (2.26) ha will deermine home bias. Tha erm ells us ha he share of foreign equiies will be larger he greaer he covariance beween wage income and home profis relaive o foreign profis. If his covariance is posiive here will be ani-home bias ( γ > ) as in Baxer and Jermann (997). 2 In ha case reurns o home equiies (compared o reurns on foreign equiies) are posiively correlaed wih labor income so he variance of oal income (reurns o equiies and human capial) is reduced by holding a relaively large share of foreign equiies. There is home bias when ha covariance is negaive. In ha case home equiies serve as a hedge agains labor income shocks. So far o arrive a equaion (2.26) we have only used he households' firs-order condiions and budge consrains along wih he symmery assumpion and he assumpion ha nominal prices are fixed. Now we can bring in one more equaion from he res of he economy he linearizaion of he profi equaion for home firms. e have from (2.20): ( ζ) π ζ( w l) = p c s 2 (2.27) where c ( c ) = c. 2 2 Taking covariances on boh sides of equaion (2.27) we ge ζ w l s cov( π ( ) π π ) cov( π π ) ζ = 2( ζ) (2.28) where we have used symmery o infer ha cov( ) 0 c π π =. Also 2 In deriving (2.27) we use symmery o ge c = c and which we have used o derive (2.27). p =. The Appendix shows ha p p = 0 p h 2

13 ζ cov( π ( w l) s) = var( s) (2.29) ζ 2( ζ) using symmery o infer ha cov( c s ) = 0. Dividing (2.29) hrough by v ar( ) we can wrie β π s w l s ζ β =. (2.30) ζ 2( ζ) Subsiue (2.28) and (2.30) ono he righ side of (2.26) and we derive γ = 0. To ge he equilibrium value of δ subsiue γ = 0 ino equaion (2.25) and use equaion (2.29): ( ζ ) cov( π s) ζ cov( w l s) δ = =. (2.3) var( s) var( s) 2ρ 2 2ρ e find complee home bias in equiy holdings γ = 0. Equaion (2.26) indicaes ha he share of equiies held in he foreign firm is deermined by he covariance of he componen of home firm revenues ( ζ ) π ζ ( w ) ha is orhogonal o he exchange rae wih he relaive profis of home o foreign firms. If ha covariance is zero hen no foreign equiies are held. In ha case reurns o home equiies are a perfec hedge for labor income. In fac he residual from projecing ( ζ ) π ζ ( w ) on is orhogonal o l l s s π π. Tha is because equaion (2.27) ells us ha he revenue of he home firm ( ζ ) π ζ ( w ) is deermined by world consumpion and he exchange rae: c l s. 2 Oupu is demand deermined. Demand depends on he overall level of consumpion a home and abroad. Addiionally he home-currency revenue of he home firm increases when he currency depreciaes because he depreciaion increases he home-currency value of foreign sales. The projecion residual is simply world consumpion relaive profis by symmery. c and ha is uncorrelaed wih Noe ha if we subsiue he soluions for γ and δ back ino he budge consrain (2.24) we obain (using (2.27)): p c = ( ζ) π ζ( w l) ( ) s p c s 2ρ 2 = 2ρ. (2.32) Using he definiion of world consumpion his expression can be wrien as: ρc s c = ρ. (2.33) 3

14 This condiion indicaes ha he linearized model replicaes he equilibrium in which a full se of nominal coningen bonds is raded. As is well known in his case (and assuming symmery) he marginal uiliy of a uni of home (or foreign) currency is equalized beween home and foreign residens: C P ρ ρ C =. SP Equaion (2.33) akes he log of his condiion using symmery o infer. The rading of home and foreign equiies and forward conracs for foreign exchange are enough o deliver he same allocaion as rading a full se of nominal coningen claims in he linearized economy. c p = p e can noe here ha he facor ha firms use o discoun expeced profis when seing prices does no depend on which households own he firm because he discoun facors of home and foreign households are equal in equilibrium. e have derived he complee home bias resul using only he nominal price sickiness assumpion he definiion of home profis he budge consrain of home households and he wo firs-order condiions (2.2 and 2.3) ha perain o asse choice. (The derivaions in his subsecion all arise from equaions (2.22) (2.23) (2.24) and (2.27) which are he approximaed versions of he wo firs-order condiions for asse choice he household budge consrain and he definiion of firm profis. In performing he approximaions we have used he fac ha prices are prese.) e have no relied on oher feaures of he model so our home bias resul is robus o alernaive assumpions. For example he resul does no depend on money demand arising from real balances in he uiliy funcion. Oher specificaions ha mainain equaions (2.2) and (2.3) will deliver he same resul. As long as symmery is mainained he resul does no depend on he assumpions abou moneary policy (ha money supplies are deermined exogenously wih shocks ha are independen of equiy shocks.) The resul also does no depend on our specificaion of he labor marke as compeiive wih flexible wages. For example a sicky-wage model in which employmen was demand-deermined would no aler he condiions ha we used in he derivaion of he home-bias resul. Furher insighs can be obained from making use of some of he oher equaions of he model. Specifically he firs-order condiion for holdings of money balances (and again using he fac ha nominal prices are prese) is wrien as: m = ρ. (2.34) 4

15 Using his equaion along wih is foreign counerpar and equaion (2.33) we derive: s m m Exchange raes are deermined by relaive money supplies. =. (2.35) The fac ha equiy demand depends only on he covariances afer projecing on he exchange rae means ha he equiy porfolio is used only o hedge produciviy shocks. Produciviy shocks do no influence he amoun of produc he firm sells which is demand deermined in a sicky-price model. Nor do produciviy shocks affec he exchange rae which influences firm revenue as well. So firm revenue depends only on moneary shocks. A posiive produciviy shock for example allows he firm o produce he quaniy demanded wih less labor. Boh wages and employmen fall in equilibrium. Profis increase by he exac amoun of he drop in labor income. Bu he effec of hose shocks on household income is fully hedged when home households hold 00 percen of home firms. Moneary shocks have real consequences in his model. Indeed equaion (2.34) shows ha in equilibrium consumpion is deermined only by money supplies. As we have noed produciviy shocks only affec he disribuion of revenues beween labor income and profis bu in equilibrium he effecs of ha redisribuion is nullified by he complee home bias in equiy holdings. The real effecs of moneary shocks are hedged hrough he forward posiion in foreign exchange. Suppose for example ha here is a negaive home money shock. In equilibrium income of home households falls because boh labor and profi income fall. Bu he drop in he home money supply also causes a home currency appreciaion ( declines.) The equilibrium value of δ is negaive given our assumpion of ρ >. In his case a decline in s s leads o a posiive pay-off from he forward posiion. Tha is when δ is negaive he home residen is shor in foreign currency and long in home currency. So an appreciaion yields a posiive payoff which hedges he effecs of moneary shocks on labor and profi income. I is ineresing ha our model implies ha invesors ake a long posiion in home currency. Anoher well-esablished empirical fac is ha home residens hold a disproporionae share of home-currency denominaed bonds in heir porfolios. In he saic seing of course here are no ne bond holdings. Subjec o he consrain of zero ne bond holdings home bias in nominal asses implies being long in home asses and shor in foreign asses. This is he configuraion implied by a negaive value of δ. 5

16 Noice ha he forward posiion does no compleely eliminae he effecs of moneary shocks on income. From equaion (2.27) we have ha ( ζ ) π ζ ( w ) falls by imes he decrease in (because falls by m c l 2ρ 2 2ρ and s by.) Including reurns from 2 2 he forward posiion solved from equaion (2.3) δ = we find ha income sill falls 2ρ 2 by imes he drop in ρ m. hy? In his model he Home and Foreign consumpion markes are compleely segmened. A change in he exchange rae causes a change in he relaive prices paid by Home and Foreign households for idenical goods because nominal prices are se in advance in consumers currencies and do no respond o shocks. So home prices rise relaive o foreign prices (expressed in a common currency) when falls. Bu households canno rade goods o arbirage he difference in goods prices. As is well known when consumer producs are no radable he efficien configuraion of consumpion (achievable when a full se of coningen nominal bonds is raded) has consumpion levels lower in he Home counry (relaive o he Foreign counry) in hose saes of he world in which is goods prices are higher han hose in he Foreign counry. Tha is why he equilibrium condiion (2.33) does no achieve perfec consumpion correlaion. So wih a negaive Home moneary shock ceeris paribus Home income falls and Home consumpion declines. s 3. Dynamic Model 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. Mos of he assumpions are he same as in he saic model. The price-seing rule is modified 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 bundled ogeher. 6

17 An imporan quesion under he dynamic model is how will persisen shocks affec he opimal porfolio? In a flexible price seing he opimal porfolio is more foreign skewed han i is in he classic endowmen economy case as shown in Baxer and Jermann (997). This effec decreases he degree of home bias in our model as well. In he dynamic model when he elasiciy of subsiuion beween Home and Foreign goods is more han uniy ( ω > ) he opimal Home porfolio should be less home biased han i is in he saic model because households mus ake ino accoun he fuure afer prices have been adjused. 3. Household Problem Home households maximize heir expeced uiliy: M max E0 β U C L = 0 P subjec o he following budge consrain: PC M Qγ S Q γ h f = γ Q Π γ S Q Π S F δ L M Tr h ( ) f ( ) ( ) (3.) where ( ) denoes he price of Home (Foreign) equiies. The uiliy funcion and Q Q consumpion baskes are he same as in he saic model. Households ener ime wih money M equiies ( h γ 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 he ransfer from he governmen as well. Finally he household will choose forward conracs δ and equiy holdings γ h γ f which will deermine he dividends households receive in ime. The firs order condiions for he households are χ C C = (3.2) M P P ρ ρ E β ηl C ρ ψ = P (3.3) ρ ρ C C E S = FE (3.4) P P 7

18 C P C P ρ ρ ρ C Q = E β ( Q Π) (3.5) P ρ C S Q = E β S( Q Π ). (3.6) P Firs le D Le s C C. The no-bubble soluion for equiy prices implies ha ρ ρ s P s P s s = β sπ s SQ = Eβ D ss sπ s (3.7) s= s= Q E D V γ Q γ S h f Q (3.8) s β s sl s s= H E D (3.9) β ( Q Π) (3.0) Q β ( H L ) (3.) H H γ SQ γ Q γ =. (3.2) 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. e can rewrie he budge consrain (3.) for ime : S PC V H = V ( γ ) β V γ β H β δ ( S F ). (3.3) H S e will assume below a process for he money supply in which E M ( ) = M. e noe his now because under his assumpion he firs-order condiion (3.2) can be simplified direcly o ge: I follows from his ha D and (3.6) can be summarized as ρ C C χ = χm = M. (3.4) P β ρ Eβ P M s =. The firs order condiions for equiy holdings (3.5) M s 8

19 M M S E = E =. (3.5) M M S 3.2 Firms Firms use he same linear echnology as in he previous secion. e 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 max P () i Y () i S P () i Y () i ( )[ Y () i Y ()] i. h h h h h h A Because Y () i is no a funcion of Ph () i and Yh () i is no a funcion of P () i he problem h is easy o solve: λ P () i = P h flex h λ A λ P () i = P (3.6) 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. The oher opimal prices are h P P prese h prese h λ ω P h E D C λ A P h P (3.7) λ λ ω Ph E D C P h P λ ω P h E D C λ A P h P (3.8) λ ω λ P h E D C P h P where D is he sochasic discoun facor 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: λ λ λ h [( τ) flexh τ preseh ] P = P P (3.9) 9

20 λ λ λ f [( τ) flex f τ prese f ] P = P P. (3.20) 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: ω ω Ph P h = 2 P 2 P AL C C. (3.2) hile flexible-price firms will have higher profi han prese-price firms in general CES sub-uiliy makes he aggregae profi of each counry he same as before: ω ω Ph P h Ph C SPh C 2 P 2 P Π = L (3.22) ω ω P f Pf Π = f f L 2 P 2 P S S P C P C S e assume ha where ϑ [0) ϑ [0) a m m = m m v m m v = ϑa v a ϑa v. (3.23) = (3.24) = (3.25) are degrees of persisence in world and relaive echnology levels x and where v ( x = mm ) are i.i.d. shocks. e denoe ln( X ) as x and we will also denoe he world variables as x = x x and he relaive variables as x = x x. e 2 2 assume m m 2 = = 2 σ so ha Ev Ev m E ( M ) = M as menioned above. e assume also 2 m m = = σ m and cov( v v ) = σ mm m m var( v ) var( v ) var( ) v σ 2 = var( v) = σ and 2 cov( v v ) = 0. e assume iniial symmery beween Home and Foreign: ha is a 0 = 0 and m = Soluion of he Dynamic Model To solve he model we use approximaions similar o hose in he saic model. The Appendix presens he soluion o he model. There he equilibrium is defined and soluions for all he endogenous variables are given. I shows ha he equilibrium condiions are 22 e assume ha produciviy shocks follow saionary processes. This is for convenience so ha real variables have uncondiional means around which we log-linearize some equaions. I would change nohing in our analysis bu be more noaionally burdensome if we allowed he world produciviy shock o have a uni roo. Then real variables defined in effecive unis would be saionary. 20

21 saisfied for hose soluions. The derivaion of he soluion is exremely algebra inensive. Here we discuss he salien feaures of he soluion. An imporan feaure of he soluion is ha we are able o replicae he allocaion achieved when a full se of nominal bonds are raded in he linearly approximaed model. e have wo kinds of asses (equiies and forward currency conracs) ha span he space generaed by and. In ha case we have a m ρ( c c ) = s p p. (3.26) This equaion is he familiar condiion ha arises when here is a full se of coningen claims bu in which consumer price levels are no equal (see for example Chari Kehoe and Mcgraan (2002).) Pushing he ime subscrips one period forward and aking expecaions a ime we ge E c ( ) E ( c ) = period and H is he expeced value a ime of reurns o work from onward. So. (3.27) This equaion follows because prices are sicky for a mos one period so purchasing power pariy holds in expecaion. Equaion (3.27) demonsraes a key sor of saionariy ha emerges from our dynamic soluion. Even hough consumpion levels migh differ beween Home and Foreign households a any ime looking forward hey are always expeced o be equal. Tha follows because as we show ζv ( ζ) h s = 0. (3.28) This equaion means ha relaive oal wealh which is he sum of financial wealh and human capial is equalized beween Home and Foreign households. To be clear defined as he value of equiies ha he home household acquires a ime and carries ino equaion (3.28) says ha he wealh levels of Home and Foreign households a he end of period are equal. This equaliy of wealh occurs even hough in equilibrium Home and Foreign households hold differen equiy porfolios. Since he condiionally expeced reurn on equiies depends on he realizaion of shocks v 0 in general. Tha is he condiionally expeced discouned payoffs on he Home and Foreign equiy porfolios differ. In addiion h 0. The value of human capial for Home and Foreign households also depends on he realizaion of shocks and so hey are no in general equal. V is 2

22 hy hen is relaive oal wealh equal? Suppose here is a posiive relaive echnology shock a > 0 bu no change in world produciviy so ha Home produciviy rises and Foreign produciviy falls. Hold moneary shocks equal o zero. In his case we can show ha neiher Home nor Foreign consumpion levels will be changed by he equilibrium which is convenien for his example. a shock in Period wage income of Home workers falls when prices are sufficienly sicky and period wage income of Foreign workers rises as in he saic model. The period profis of Home firms rise and period profis of Foreign firms fall. The curren income of Home relaive o Foreign migh rise or fall. On he one hand Home's relaive labor income falls bu he profis Home households reap may be greaer han ha of Foreign households when here is home bias in equiy holdings. Noneheless under he parameer configuraion ha delivers home bias he overall income of Home falls relaive o Foreign - he relaive loss in wage income mus ouweigh any relaive gain in profi income. Bu in his siuaion in which home bias arises he relaive decline in curren income for Home is precisely offse by he gains Home ges in he value of is human wealh and he gain in he value of he equiies ha i carries ino period. The posiive realizaion of pushes up relaive o Q and H relaive o Q a H. Home's oal wealh - he sum of he income i receives in period from labor and profis plus he value (afer he realizaion of a ) of he equiy posiion i carries ino period plus he value of is human wealh - is unchanged relaive o Foreign. Since consumpion levels are no affeced by a shocks he relaive wealh of Home and Foreign a he end of period is unchanged. As a resul of his saionariy we show ha δ and γ are consan over ime: δ δ = ( ) τ (3.29) 2 ρ where Α γ γ = γ = 2 ζβ ( ζ) Α τ βϑ Α ( ω )[ ] and ω ( τψ ) ωψ βϑ Β τ ω( τψ ). (3.30) 22

23 The share of he equiy porfolio held in foreign asses γ clearly is increasing in Α decreasing in Β. Demand is also decreasing in ζ when Β >Α. In order o have home bias or γ < we generally need Β> Α which implies 23 2 ω( τ) ω βϑ > 0. (3.3) ω( τ) ψ ωψ βϑ Noice ha he condiion (3.3) does no depend on ρ or ζ while ζ deermines he level of home bias. There are inuiive explanaions for how mos of hese parameers affec foreign equiy demand. As labor s share ζ rises hen γ falls when here is home bias ( Β> Α) and rises when here is ani-home-bias ( Β<Α). The inuiion is sraighforward given our discussion above: hen he shor-run effecs ha lead o a negaive covariance of home profis and labor income are sufficienly large ha here is home bias he home bias is amplified he larger is labor s share. The benefis from hedging labor income risk are greaer when labor s share is greaer. Bu when he long-run effecs dominae and reurns o human capial are hedged by having a foreign-equiy bias he effec is again amplified he larger is labor s share. Nex i is helpful o consider wo special cases. βϑ is in a sense a measure of he weigh he fuure receives in he porfolio allocaion decision. βϑ is large when households place a high weigh on he fuure and when he relaive produciviy shocks have a very persisen influence. In he exreme case when all prices are sicky ( τ = ) and he fuure does no maer ( βϑ = 0 ) here is complee home bias ( γ = 0.) This acually is jus he saic model we examined previously ha assumed full price sickiness and placed no weigh on he fuure. On he oher hand if all goods prices were flexible τ = 0 hen he opimal equiy porfolio is γ = >. This oucome is similar o he heoreical resul obained by 2 ζ 2 Baxer and Jermann (997) he inernaional diversificaion puzzle is worse han you hink. Increasing price sickiness implies a larger value of τ -- a greaer fracion of firms se price in advance. A larger τ makes i more likely ha he condiion (3.3) for home bias is 23 e omi he case in which he denominaor in equaion (3.30) is non-posiive: his case can happen only if he price is very flexible and ω. 23

24 me. γ is decreasing in τ when ω > -- which can be seen direcly because an increase in τ raises Β and lowers Α. e find hen ha increasing price sickiness leads o greaer home bias in equiy holdings. hen ω > an increase in βϑ leads o an increase in Α which implies a greaer share of foreign equiies in he home household s porfolio. In shor he more he fuure maers he larger he share of foreign equiies. In he limi as βϑ he porfolio approaches he flexible price value γ =. On he oher hand as βϑ 0 he 2 ζ ( ω )( τ ) porfolio approaches γ =. This laer value is precisely he level γ 2 τζ ( ζ )( ω )( τ) would ake in he saic model if a fracion τ of prices were prese. hen ω = he erms of rade adjusmen insures agains he effecs on relaive wealh from produciviy shocks. The share of Home or Foreign goods in consumpion expendiure does no change because of he Cobb-Douglas sub-uiliy funcion. Hence households care only abou he disribuion beween labor and firms as is he case in he saic model. Therefore we ge 00 percen home bias: γ = 0. If ω = and all prices are flexible hen γ is indeerminae. This is similar o he models by Cole and Obsfeld (99) and Obsfeld and ogoff (2002) in which asse rade is no needed because of he Cobb- Douglas specificaion for he consumpion index of Home and Foreign goods. On he oher hand if β is close o one and τ = hen ω he elasiciy of subsiuion beween Home and Foreign goods plays an imporan role. The echnology shock will have a significan impac once prices adjus if Home and Foreign goods are subsiues for one anoher. hen Home receives a negaive echnology shock he demand for Home goods shifs o Foreign goods afer prices are adjused. This fall in demand for Home goods implies ha Home firms will cu heir labor inpus. In order o hedge agains his employmen risk a Home household wans o have Foreign equiies because Foreign firms will generae more profi han will Home firms suffering from he negaive echnology shock. Thus sicky prices lead o home bias as we have seen in saic model while flexible prices lead o foreign bias. If he effec from price sickiness is bigger hen home bias will be opimal. Under flexible prices a posiive echnology shock enables firms o produce goods more cheaply and o sell hem more cheaply so ha nominal sales will increase if ω >. Alhough he demand for labor will decrease from he direc effec of he echnology shock he demand for goods will increase and hus indirecly increase he demand for labor. 24

25 3.4 Properies of he Model e can calibrae he amoun of home bias implied by he model. The share of he Home household s equiy porfolio held in foreign shares γ depends on he price sickiness parameer τ ; labor's share ζ ; he elasiciy of subsiuion beween home and foreign aggregaes ω ; he discoun facor β ; he persisence of relaive produciviy shocks ϑ ; and he elasiciy of labor supply ψ. e se τ = and hen calibrae he lengh of a period by using esimaes of he speed of price adjusmen. ih τ = he half-life of price adjusmen is one-half of a period. In our model he speed of price adjusmen deermines he rae of convergence oward purchasing power pariy. ogoff (996) has noed ha sudies of purchasing power pariy imply a half-life of he real exchange rae of 3-5 years. e will pick a much smaller half-life of year which is far below he lower end of he range cied by ogoff. This implies ha one period is equal o wo years. Following Backus Kehoe and Kydland (992) we se ζ = 23. The esimaes of Backus Kehoe and Kydland (992) give us on quarerly daa ha he auocorrelaion of 8 relaive produciviy shocks is so we se ϑ = (0.855) Likewise he 8 quarerly discoun facor in Backus e al. is 0.99 so we ake β = (0.99) e follow Backus Kehoe and Kydland (994) and Chari Kehoe and Mcgraan (2002) and se ω =.5. e follow Obsfeld and ogoff (2002) and Bergin (2004) and se ψ =. ih his baseline se of parameers we find γ Tha is he model is capable of explaining a subsanial amoun of home bias. The model is perfecly symmeric beween home and foreign counries so an unbiased porfolio would be γ = 0.5. e have been fairly conservaive in picking he degree of price sickiness. A greaer degree of price sickiness would imply even more home bias. In our model negaive condiional correlaion beween labor hours and produciviy condiioning on produciviy shock is he key driving force for home bias. However because households can hedge demand shock hrough forward conracs he uncondiional correlaion can be posiive. I is imporan o disinguish beween condiional and uncondiional correlaion in our model. Gali (999) has addressed precisely his issue. He has noed ha real business cycle models end o imply a posiive correlaion beween hours and produciviy. He shows in a 25

26 simple closed-economy New Keynesian macroeconomic model ha here is a negaive correlaion beween hours and oupu per worker when here is a produciviy shock. The reasoning is much he same as ha in our model. Gali goes on o derive empirical suppor for his implicaion of sicky-price models. He esimaes a srucural bivariae VA on oal labor hours and labor produciviy using U.S. daa. 24 The model was esimaed on quarerly daa from 948:I o 994:IV. There are wo ypes of shocks in he model which Gali classifies as echnology shocks and non-echnology shocks. The non-echnology shocks can be associaed wih aggregae demand shocks. Under his idenificaion scheme only echnology shocks can permanenly increase labor produciviy. Gali finds ha he condiional correlaion beween labor hours and produciviy is negaive for echnology shocks while he uncondiional correlaion is posiive. oemberg (2003) finds similar resuls. If prices were flexible in radiional real business cycle models he correlaion condiional on echnology shocks would be posiive - as i is in our model in he long run. Gali s findings have no gone unchallenged. 25 Chrisiano Eichenbum and Vigfusson (2003) subsiue labor hours per capia for Gali s oal labor hours and reverse Gali s finding on he condiional correlaion. However Francis and amey (2003) use he same measure bu quadraically derended and find he negaive correlaion beween hours per capia and produciviy condiional on echnology shocks. Gali Lopez-Salido and Valles (2003) find a similar resul using firs-differences in hours per capia. Francis and amey (2004) creae a new measure of hours per capia and confirm ha a posiive echnology shock will reduce labor hours in he shor run. hile here is no consensus ye on he sign of he condiional correlaion here is some significan empirical suppor for he conenion ha i is negaive. Home bias does no require ha he uncondiional correlaion of reurns o human capial and reurns o domesic equiy be posiive for wo reasons: Firs as we noe above produciviy shocks may have a low variance relaive o moneary shocks bu i is he covariance holding moneary shocks consan ha maers for home bias. Second i is he correlaion of reurns o human capial wih he relaive home o foreign equiy reurns ha maers for produciviy. If home and foreign produciviy shocks are highly correlaed here may be home bias even when he condiional correlaion of human capial reurns and 24 He also uses employmen insead of labor hours and finds he same resul holds for all G7 counries excep Japan. 25 See Gali and abanal (2005) for deails. 26