Advanced International Economics
Advanced International Economics ECON 872
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1216 REVIEW OF ECONOMIC STUDIES TABLE 1 Crosscountry correlations ofstock market and exchange rate returns during tranquil times and crises Stock market returns Exchange rate returns Tmnquil times Crises Tranquil times Crises Emerging economies 270 465 56 204 Developed economies 374 429 549 579 Source Authors calculations2 full extent of international comovement in the data Direct measures of the terms of trade channel fail to explain contagion during crises3 For example only 02 of Brazil s exporm go to Russia and yet Brazil was one of the worstaffected countries during the 1998 Russian crisis with its equity prices plummeting more than 20 Testing for the common discount factor channel is much harder as such tests require an asset pricing model Yet the extensive evidence from reducedform speci cations suggests that a single common factor cannot explain contagion4 Moreover neither theory can explain t 39 439 in r quotntry 39 of sovereign bond yields following credit rating changesievents that do not reveal any economic news For instance when Mexican debt was upgraded from noninvestment grade to investment grade in March 2000 its correlation with bonds of other Latin American countries dropped by 30 percentage poinm The shortfalls of the standard theories inspired a search for alteniative channels of propa gation For example Calvo 2002 argues that Wall Street was the carrier of the Russian virus in the fall of 1998 when bindin margin constraints forced leveraged investors most notably LongTerm Capital Management hedge fund to curtail their exposure to all emerging markets Kaminsky and Reinhart 2000 document that Thailand s 1997 currency crisis led to capital losses for Japanese banks forcing them to curb their lending to other Asian countries This newer literature highlights the prominent role that nancial frictions and in particular institutionally or govenimentimposed portfolio constraints play in the propagation of turmoil via nancial centres In this paper we analyse formally the effects of portfolio constraints on asset prices within a uni ed framework that encompasses both the terms of trade and the common discount fac tor channels Our generalequilibrium model allows us to disentangle the effects of portfolio constraints from those of the traditional channels and de ne precisely the notion of contagion as the comovement in excess of that in the unconstrained economy Within our model we can understand why portfolio constraints amplify stock price uctuations and how changes in their tightness contribute to excess comovement From a methodological viewpoint this paper offers a tractable model solved in closed form that can be used to study the impact of a broad class of constraints on the terms of trade and stock prices 2 endix B describes our sample and the details of the estimation The correlations are adjusted for heteroskedasticity as in Forbes and Rigobon 2002 The magnitudes of the crisisrinduced increases in the correlations in our sample are consistent with the recent literature eg Pan and Singleton 2008 Fostel 2005 ee Kaminsky Reinhart and Vegh 2003 for a survey 4 For instance principal component estimates show that the weight on the rst principal component changes around crises and type of country eg Calvo and Reinhart 1996 Kaminsky and Reinhart 2007 For evidence based on reducedrform linear latent factor models see for example Corsetti Pericoli and Sbracia 2005 Dungey Fry Gonzalezr Hermosillo and Martin 2005 and Rigobon 2003 and for that based on the copula approach see for example Hartmann Stmetmans and de Vries 2004 and Rodriguez 2007 5 See Kaminsky and Schmukler 2002 and Rigobon 2002 2008 The Review of Economic Studies Limited PAVLOVA amp RlGOBON ROLE OF PORTFOLIO CONSTRAINTS 1217 Our starting point is a canonical ArrowiDebreu economy which we generalize to include portfolio constrains6 There are three countriesione Centre and two Periphery countriesieach populated by a representative consumeriinvestor We think of the Centre as a large developed economy and of the Periphery countries as small emerging markem Each country is endowed with a Lucas tree producing a countryspeci c good Each tree s output is driven by its own supply shock and stocks are claims to the Lucas trees There is also a riskless bond in zero net supply Each agent has loglinear preferences over all three goods with a preference bias for the home good The goods markem are frictionless but nancial markets are imperfect in that the Centre s agent faces constraints limiting his portfolio choice We assume a general form for the constraints that nests among others portfolio concentration constraints VaR constraints margin requirements and collateral constraints Absent portfolio constraints all comovement in our model is due to the common discount factor and the terms of trade channels The former works through changes in the investors aver sion to risk Consider for instance a negative shock to the Centre s stock7 Because of their log linear preferences all investors hold identical meanivariance ef cient portfolios which include positive amounts of the Centre s stock Therefore the negative shock implies a negative return on their portfolios As the investors become poorer they become more averse to risk and lower their demand for the stocks However they cannot all sell Hence prices of all three stocks must fall Since all investors hold the same portfolios shocks leave wealth distribution unchanged This mechanism illustrates the propagation of shocks across stock markets due to the investors desire to diversify internationally An alternative intuition for the same phenomenon makes use of the terms of trade channel The terms of trade improve for the country experiencing a negative shock and therefore deteriorate for all others which in turn reduces their stock market prices Section 2 To understand the additional effects that constraints introduce let us now impose portfolio constraints on the Centre s agent While the two Periphery countries continue to hold identical portfolios the constraints force the Centre s agent to hold a different one As a result the returns on portfolios of the Centre and the Periphery countries implied by a common shock now differ Suppose for example that the constrained portfolio is such that the Periphery countries lose more This implies a change in wealth distribution which we call a wealth transfer from the Periphery to the Centre Wealth transfers absent in the frictionless economy work much like income transfers in the classic Transfer Problem also known as the Keynes effect8 That is as the Periphery countries wealth drops so does their demand for all three goods but because of their preference biases towards their own goods the demand for the Periphery goods suffers more Hence both Periphery countries see their goods prices fall relative to that of the Centre s good deteriorating their terms of trade Consequently the value of the dividends on the Periphery countries trees falls depressing the prices of their stocks The Centre s terms of trade improve and its stock market rises Hence the Keynes effect i increases the comovement of the stock markets and of the terms of trade across the Periphery and ii decreases the comovement between 6 See Cass Siconol and quot 39 f mquot anal i fa general a 39 quot39L 39 model with portfolio constraints They show existence and nite local uniqueness of equilibrium For more applied analyses see also Boyer Kumagai and Yuan 2006 Geanakoplos 2003 Gromb and Vayanos 2002 Mendoza and Smith 2002 and Yuan 2005 This recent litemture is rapidly growing and has already produced many important insights however to deal with the technical dif culties involved in solving dynamic models with multiple risky assets it has resorted to a static or a partial equilibrium fmmework relied on behaviouml assumptions or abstmcted away from international trade and other crossrcountry linkages intrinsic to the phenomenon of international nancial contagion 7 nsidering a shock directly to an endogenously determined stock price mther than a supply shock here and below is a shortcut allowing us to bypass some technical steps of our formal analysis while conveying correct intuitions e Tmnsfer Problem stems from the argument made originally by Keynes that in a world with a home bias in consumption an income tmnsfer from one country to another will deteriorate the terms of trade of the sender of the tmnsfer 2008 The Review of Economic Studies Limited 1218 REVIEW OF ECONOMIC STUDIES those of the Periphery and the Centre relative to that in the unconstrained model We note that for these results to hold portfolio constraints in the Centre must be binding which we associate with periods of nancial crises The Keynes effect may also give rise to important phenomena such as ampli cation and ighttoquality effects9 Consider a constraint that limits the fraction of wealth the Centre may invest in the stock markets of the two Periphery countries Relative to the frictionless case the constraint forces the Centre to decrease its holdings of the Periphery stocks and to increase its holdings of the Centre s stock First consider a negative shock to the Centre s stock In that case all agents experience a negative return as in the frictionless model However the Centre s port folio has a lower retuni as it is now overweighted in the Centre s stock relative to the Periphery countries This implies a wealth transfer from the Centre to the Periphery which in tuni causes a deterioration of the Centre s terms of trade and a further decline of its stock market Hence the portfolio constraint leads to an ampli cation of negative shocks to the Centre s stock marketi consistent with the literature arguing that portfolio constraints exacerbate market volatility Now consider a negative shock to one of the Periphery countries stocks The Periphery countries are overweighted in their stocks and lose more than the Centre which generates a wealth transfer away from the Periphery This further depresses both Periphery stocks but has an incrementally positive effect on the Centre s stockia phenomenon often referred to as a ight to quality To illustrate both the model s workings and its t in the context of recent crises we con sider two examples of constrains a concentration constraint and a market share constraint The former puts a ceiling on the fraction of the Periphery stocks in the Centre s portfolio in absolute terms while the latter speci es it in proportion to the market share of the Periphery in the world Both constraints produce similar dynamic behaviour of prices but they have different implica tions for capital ows The impact on ows is driven by how the tightness of a constraint changes following a given shock For instance in the case of the market share constraint a negative shock in the Periphery reduces the market capitalization of the Periphery tightening the constraint10 This tightening generates capital withdrawals and large price drops across the Periphery through two mechanisms First the Periphery agents became poorer and hence want to downsize their stock market positions Second the Centre s agent who would have chosen to invest more in the Periphery has to curb his position to respect his portfolio constraint Therefore the change in the tightening of the constraint generates an additional source of price comovement across the Periphery To quantify these effects we parameterize the model and nd that the correlation be tween the stock retunis of the two Periphery countries goes up by 15720 while the correlation between the Centre and the Periphery decreases by 578 The literature closest to our work is the twogood twocountry assetpricing models of Helpman and Razin 1978 Cole and Obstfeld 1991 Pavlova and Rigobon 2007 and Zapatero 1995 which feature both the terms of trade and the common discount factor chari nels of inteniational propagation but no portfolio constraints Also related is the literature on portfolio constraints in asset pricing Basak and Croitoru 2000 Basak and Cuoco 1998 Detemple and Murthy 1997 Detemple and Serrat 2003 Gallmeyer and Holli eld 2008 and Shapiro 2002 among others all consider the effects of portfolio constraints on asset prices in dynamic economies11 Their solution methodology is similar to ours however they consider singlegood closed economies and hence have no implications for the terms of trade We employ 9 See for example Bernanke Gertler and Gilchrist 1996 Eichengreen Hale and Mody 2001 and Vayanos 2004 The de nitions of these effects differ somewhat depending on the application 10 The concentration constmint works in exactly the opposite direction After a negative shock to the Periphery the constmint becomes looser The market share constmint provides a better t for the stylized facts that have emerged after recent nancial crises most notably the Russian default in 1998 For earlier litemture examining the effects of constraints on crossrcountry holdings in a static setting see for example Errunza and Losq 1985 1989 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1219 techniques developed in Cvitanic and Karatzas 1992 to solve the partial equilibrium dynamic optimization problem of an investor who is facing portfolio constrains Our paper illustrates the usefulness of these techniques in solving dynamic equilibrium models with portfolio constraints The rest of the paper is organized as follows Section 2 describes the model and characterizes its equilibrium in the benchmark unconstrained case Section 3 investigates the equilibrium in the economy with portfolio constraints Section 4 argues that our results hold even if the Periphery countries do not trade with each other Section 5 specializes the model to examine the effect of two particular portfolio constraints Section 6 discusses caveats and possible extensions and the appendices contain all proofs and other supplementary material 2 THE MODEL We now present the formal model and before turning to portfolio constraints analyse its solution in the unconstrained case 21 1718 economic setting We consider a continuoustime pureexchange world economy with a nite horizon 0 T in the spirit of Lucas 1982 The main advantage of using continuous time is tractabilityian anal ogous discretetime model does not admit a closedform solution Uncertainty is represented by a ltered probability space Q F E P on which is de ned a standard threedimensional Brownian motion 100 1000 1010 w2tT t E 0 T All stochastic processes are assumed to be adapted to f5 1 E 0 T the augmented ltration generated by 10 All stated inequalities involving random variables hold P almost surely In what follows given our focus we assume all processes introduced to be well de ned without explicitly stating regularity conditions ensuring this There are three countries in the world economy indexed by j E 0 1 2 Country 0 rep resents a large Centre country eg an industrialized economy and countries 1 and 2 represent smaller Periphery countries 8 g emerging economies Each country j produces its own perish able good via a strictly positive output process modelled as a Lucas tree rm 2 M 0 Wm 01 ltrY1ltrdwfltr jero12t 1 where 1 and 01 J gt 0 are arbitrary adapted processes12 The price of the good produced by country j is denoted by 1 Since prices are not pinned down in a real model such as ours we need to adopt a numeraire We x a basket containing 8 e 0 1 units of the good produced in country 0 and 1 7 2 unis of each of the remaining two goods and normalize the price of this basket to be equal to unity We think of 8 as the size of the large Centre country relative to the world economy It is convenient to de ne the terms of trade from the viewpoint of the Centre country country 0 q1 E p1 j0 and q2 E 117210 are the terms of trade of the Periphery countries 1 and 2 respectively with the Centre country Each country is endowed with a stock S j a claim to its output stream All stocks are in unit supply Additionally we assume that there is a riskless bond so that markets are complete The 12 For genemlity we allow the parameters of the output processes to depend on all information genemted up to time t including that generated abroad This may be useful for a potential extension of our model in which unlike here these parameters are unobserved and agents estimate them using all available information see Detemple and Murthy 1994 for a detailed analysis of such an economy For the purposes of the current model however we may instead assume that the parameters are either constant or depend only on the history genemted by the domestic output processes 2008 The Review of Economic Studies Limited 1220 REVIEW OF ECONOMIC STUDIES bond market and the stock markets of the three countries follow dBt Btrtdt B0 l 2 dsf r p1ltrYfltrdr s1ltrgtltmltrgtdr 0139 WW j 0 1 2 3 respectively where the interest rate r the stocks expected returns 1 E 0 1 2T and the volatility matrix of stock retunis a E 01 j 0 l 2 are to be determined in equilibrium A representative consumeriinvestor of each country is endowed at time 0 with a total supply of the stock market of his country the initial wealth of agent i is denoted by Wi0 Each consumeri chooses nonnegative consumption of each good C190 C110 Ci2 1 i E 0 l 2 and a portfolio of the available risky securities xi 1 E xiSO If x510 x52 1 T where X denotes a fraction of wealth Wi invested in security j so as to maximize his timeadditive utility E juxcf a cm 03mm with 0 uo08ltr 030 030 aologch 1 010gcgltr 1 010gcgo u1Cltrgt cm cm 1 1 logci ltrgt a1l 10gCil 1 1 logcltrgt 0620 1 logC t 2 l 7 l 0 logcioa a2ltrgtlogc ltrgt u2C Cga 030 subject to the budget constraint dWiQ 7 Wm quot5 0 dSOtp0tYotdt 1 dSltp1tY1tdt 3 is I 310 2 dSZtp2tY2tdt 5 y t lt17x5 lt07x51ltrgt7x52ltrgtgt dBt 1 0 o 1 1 BO mqy 0C 01 DQO pzmc r dr 4 with WiT 2 0 where the subscript i i 0 l 2 indexes the consumer in countries 0 l and 2 respectively Two features of the preferences speci cation are noteworthy First we assume that each at is between 13 and l or in other words that there is a home bias in preferences13 Second xi 139 l 2 are stochastic that is we allow for demand shifts modelled along the lines of Dornbusch Fischer and Samuelson 1977 This assumption is useful because in the absence of demand uncertainty free trade in goods may imply excessively high correlation of stock market 13 This assumption is common in open economy macroreconomics It may be replaced by explicitly accounting for the demand of nonrtmdables and assuming that the nonrtmdables are produced using domestically produced inputs or by explicitly modelling tmnsport costs The implications of these richer models for the properties of the terms of tmde and other pertinent quantities are known to be very similar see eg Samuelson 1954 Obstfeld and Rogoff 2000 and we hence adopted the more parsimonious speci cation Furthermore note that the purpose of the assumption is to generate a home bias in consumption and not in portfolios 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1221 returns and irrelevaricy of a nancial market structure14 Moreover empirical evidence indicates that demand uncertainty is of the same order of magnitude as supply uncertainty Pavlova and Rigobon 2007 Formally we assume that each at is a martingale ie Eai S 1ft at t s gt t and hence can be represented as da10a1wat da20a2Td1D where am t and a t are such that our restriction that al and 12 take values between 13 and 1 is satis ed15 Since our primary focus is on the Periphery countries for expositional clarity we keep the preference parameter of the Centre country 10 xed The loglinear speci cation of the preferences is adopted for tractability While investment policies of the Periphery countries are unconstrained the Centre faces a portfolio constraint that we allow to take the most general form suggested by Cvitanic and Karatzas 1992 Namely portfolio values x0 are constrained to lie in a closed convex non empty subset K e R3 Moreover the deterministic subset K may be replaced by a family of stochastic constraints so that X00810 EIKtw I w E 0 T X91 Making the constraint set stochastic and in particular dependent on exogenous variables in the Centre s optimization problem eg 5 12 Y i 0 1 2 allows for more exibility in spec ifying constrains which we exploit in Section 5 16 Examples of portfolio constraints belonging to this class include prohibitions to trade certain stocks or some less severe provisions such as lim its on the fraction of the portfolio that could be invested in the emerging markets S1 and 52 This speci cation can also capture constraints on borrowing VaR constrains margin requirements collateral constrains and so forth A body of empirical literature has argued that regulation or risk management practices adopted in nancial centres were largely responsible for nancial contagion in emerging markets in the past two decades17 This motivates our choice of imposing the constraint on the Centre In this paper we do not provide a model supporting the economic rationale behind imposing portfolio constrains Typically such constraints are either goveniment imposed or arise in response to an agency problem in institutional money management18 22 Countries optimization The solution method and main results In this section we sketch the method for solving the partial equilibrium optimization problems of the Centre and the Periphery countries All technical details are provided in Appendix A The consumptioniportfolio problem of the Periphery countries is standard because both countries face complete markem see eg Duf e 2001 The only difference here is that we 14 As established by Helpman and Razin 1978 Cole and Obstfeld 1991 and Zapatero 1995 For other recent attempts to break the nancial market structure inelevancy result see Engel and Matsumoto 2006 Ghironi Lee and Rebucci 2007 Pavlova and Rigobon 2007 Sermt 2001 and Soumare and Wang 2007 ur speci cation the demand shifts are driven by the same Brownian motions as output shocks We do so to reduce the dimensionality of uncertainty in the model and hence to require fewer assets for market completeness remain unchanged the explicit expressions in Proposition 3 require some modi cation This extension may be useful for future applications that would focus more closely on demand shifts Finally we comment on the bounds we im ose on at i l 2 An example of a martingale process that does not exit the interval 13 1 is ort EorT 1 with or T E 13 1 We thank Mark Loewenstein for this example 16 See Cvitanic and Kamtzas for minor regularity conditions imposed on the constmint set 17 See Calvo 2002 Kaminsky and Reinhart 2000 and Van Rijckeghem and Weder 2003 18 Forthe latter see for example recent papers by Basak Pavlova and Shapiro 2008 and Dybvig Farnsworth and Carpenter 2006 Such constraints are panicularly prevalent in developed countries where risk management pmctices are more sophisticated 2008 The Review of Economic Studies Limited 1222 REVIEW OF ECONOMIC STUDIES work in a multigood framework as opposed to a singlegood framework typically adopted in the nance literature The problem faced by the Centre country on the other hand is nonstandard because of the presence of portfolio constraints To solve it we rely on Cvitanic and Karatzas 1992 who show that the optimization problem of an investor subject to portfolio constraints is formally equivalent to an auxiliary problem with no constraints but the investor facing a modi ed investment opportunity set dBO Btrt5vtdh 301 detPjlYjl dlSjtjtvjl5vtdl01tdwl l j012 where the function 5 and an endogenously determined stochastic process 12 E 120 121 122T are de ned in Appendix A There are two key differences between this investment opportunity set and the one faced by the unconstrained countries speci ed in equations 2 and 3 i the effective expected retunis on the stocks and ii the interest rate on the bond are tilted away from the values faced by the unconstrained investors One may think of the process 12 as the Lagrange multiplier associated with the set of portfolio constrains For example if a portfolio constraint imposes an upper bound on investment in stock S 1 the corresponding 121 is going to be negative making the expected retuni on this stock less favourable from the viewpoint of the constrained investor and hence convincing him to invest less in this stock so that the constraint is satis ed This is the fundamental idea behind Lagrange multipliers In Appendix A we specify the minimization problem that this process 12 has to solve and then derive its solution 12 for each example of the portfolio constraints we consider in this paper Section 5 Assuming that the process 12 has been determined optimal consumption and portfolios of a constrained investor have the same characterizations as those of an unconstrained except that in each instance the interest rate r is replaced by the effective interest rate r t 502 and the expected retunis on the stocks by in vector notation 2 12 612 T where T l l l In particular we can claim the following result and its corollary Lemma 1 The optimal consumption allocations and wealth are linked as follows C80 1 aOWOOT c520 W Wla C90 p WZO Clo 1 SFWW Cl W a1tW1l 2 cm p WN C30 1 12 1W00 l 2 7 l C20 pzwrir 1 2 QW1 020 a2ltrW2ltr As is to be expected in a model with loglinear preferences the consumption expenditure on each good is proportional to wealth However in our economy the marginal propensity to consume out of wealth is stochastic due to possible demand shifts Lemma 1 allows us to easily generalize the standard implication of the singlegood models that logarithmic agents follow myopic trading strategies holding only the Merton l97l meanivariance ef cient portfolio 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1223 Corollary 1 The countries portfolios of risky assets are given by x00 unsafrlwvltr7rltrT mt ltoltoaltrfr1ltyltrgt4m ie 1 2t Note that the portfolio of the investor in the Centre generally differs from that chosen by the investors in the Periphery because his investment opportunity set is augmented by the portfolio constraint through the multiplier 12 Only when the constraint is absent or at times when it is not binding 12 t 0 all investors in the world economy hold the same portfolio19 23 Benchmark unconstrained equilibrium To establish a benchmark we solve for an equilibrium in an economy with no portfolio con straints In equilibrium each agent maximizes his utility subject to the budget constraint and stock bond and goods markets clear The derivation of equilibrium consumption allocations and the terms of trade in this economy is standard It is usually convenient to consider the planner s problem The planner s utility is T UC C1C2 1112E uC0tC1tC2tAlAzdt 0 with uC0 C1 C2 1112 max u0C8 Cg C Alu1C C11 C12 2 2050 all 10 12u2C3Ci 0 where ii gt 0 i l 2 are constant weights on consumers l and 2 respectively re ecting the value of their endowments The planner is endowed with the aggregate supply of all assets and consumes the aggregate output that is C1 t Y1 t all j In frictionless pureexchange models the problem of solving for optimal allocations in this dynamic economy reduces to a static maximization problem see eg Backus Kehoe and Kydland 1994 The sharing rules for aggregate endowment are given by C0t 0 0 M 0 7 mu C10 7 a01117a1tl1217a2t Al 21 5 030 2 2 1241 0amp0 Y1 0 t C10 f M110 6 i 142 106109ianl azzgl I it C20 12 2 030 m 13 2 7 17 t C1t 7 HO 111mm M120 11 7 egg 2 2 Amer 19 This result may appear surprising because the investors in our model are heterogenous and in panicular have a home bias in consumption However it follows from Lemma 1 that their total consumption expenditures constitute the same fraction of wealth Thus 39 39 39 quot quot r dquot L39 4 requires the same portfolios and then 39 r 4quot goods 2008 The Review of Economic Studies Limited 1224 REVIEW OF ECONOMIC STUDIES These consumption allocations are similar to familiar sharing rules arising in equilibrium models with logarithmic preferences In the benchmark economy with perfect risk sharing the correla tion between consumption of a particular good and its aggregate output would have been perfect if not for the demand shifts Since consuming the aggregate output must be optimal for the representative agent the terms of trade are given by the pertinent marginal rates of substitution processes 107UCIY0IY1IY2I Ablz713a011a112l M 8 q iuC0Y0tY1tY2tlllz7 011AZ Y1t 20 7 C2Y0Y117 Y2t1112 7 13 11Q12a2t Y0t 9 q ucoltY ltrY1ltrWynn owlwmz W Since in our model the terms of trade would play a central role in linking together the countries stock markets we structure our benchmark economy so as to be able to capture some of their most important properties highlighted in international economics First the terms of trade of the Periphery countries with the Centre decrease in their domestic output and increase in the Centre s output This is a standard feature of Ricardian models of international trade Terms of trade move against countries experiencing an increase in productivity or output as their goods become relatively less scarce20 Second we attempt to capture the dependent economy effects highlighted in open economy macroeconomics The terms of trade improve for a country i that has experienced a positive demand shift an increase in at The intuition for this result is that a higher demand for domestic goods increases the price of domestic goods relative to foreign goods improving the terms of trade The key to the tractability of our model is that the stock prices can be computed in closed form We report the resulting expressions in the following lemma Lemma 2 The prices of the stocks of the Centre and the Periphery countries are given by so Y0t Tit 10 lt r w m ltlt ltgt 1 51 q mYlt Tit 11 lt r w m ltlt ltgt 2 520 q 0 Y2tT7t 12 qIO H 1120 Equations 5712 summarize the prices and allocations that would prevail in the com petitive equilibrium in our economy The expressions for all these quantities are explicit but involve endogenous weights Al and 12 It tunis out that in our model these weights can also be computed in closed form we report them in Appendix A At this point it is important to note that the expressions for neither the prices nor the allo cations feature the wealth distribution in the economy as a state variable This is because wealth distribution is constant determined by the weights in the planner s problem W10 7 W20 7 WW11 and WW 12 13 20 TI r u cthe m 4 39L AoL ro39 shares 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1225 The equalities in equation 13 follow from for example equation 5 combined with Lemma 1 This is a convenient feature of our benchmark equilibrium allowing us to easily disentangle the effects of the timevarying wealth distribution in the economy with portfolio constraints presented in the next section To facilitate the comparison with the economy with portfolio constraints we need the following proposition Proposition 1 i The joint dynamics of the terms of trade and the three stock markets in the benchmark unconstrained economy are given y 17111117532 at bt 1 71 0 171 at 30 1 0 1 1 Iltrgtdr 49w Axum MO 1 EMltrq1ltrgt1 Mltrgtq2ltrgt 5032 altrgtixmlto faka MO 1 EMltrgtq1ltrgt 1 Mqum 031 ammo banana MO Mmla Mmza 2810 W110 da2t gtlt 0Yodw0t 0Y1tdw1t layzwwzm The drift term I and quantities Xal Xaz M a a b and hare de ned in Appendix A Proposition 1 decomposes stock and commodity markets retunis into responses to ve underlying factors demand shifts in Periphery countries 1 and 2 and output supply shocks in all three countries These responses are captured in matrix OM henceforth referred to as the ancon strained dynamics The exact form of the elements of OM need not conceni us at this pointiwe are primarily interested in their signs Understanding the responses of the terms of trade to the shocks is key to understanding the transmission of the shocks to the remaining quantities The directions of transmission of supply shocks are unambiguous and easy to sign On the other hand those of the demand shifts depend on the relative sizes of the countries involved Our leading interpretation of the economy involves a large Centre country a developed economy and two small and relatively similar Periphery countries emerging markets Such interpretation allows us to get sharper predictions for the signs of the responses of the terms of trade and therefore the stock market prices to the demand shocks It justi es considering the following conditions21 21 In the sequel we always specify whether a sign is unambiguous or occurs under a speci c condition Condir tion Cl is necessary and suf cient and Condition C2 is suf cient Condition C2 is imposed on exogenous quanti es while Condition Cl involves the distribution of wealth endogenously determined within the model Further discussion of these conditions is presented in Appendix A The conditions affect none of the derivations they are used only for presenting the directions of responses of the stocks and the terms of trade to the underlying shocks 2008 The Review of Economic Studies Limited 1226 REVIEW OF ECONOMIC STUDIES TABLE 2 Terms of trade and stock returns in the benchmark unconstrained economy Variableeffectsof da1t da2t moo dw1t W20 l Zest 1 t g TC 2 0 T 5 7 7 C1 7C2 m 7C2 C1 4 3 Notes Where a sign is ambiguous we specify a suf cient or a necessary and suf cient condition for the sign to obtain C1 stands for the small country condition Cl and 4 stands for the similar country condition C2 Condition C1 The Periphery countries are small relative to the Centre 3060 7 l lt 3a2t 7 l 3060 7 l lt 3m t 7 l 2 1 Condition C2 The Periphery countries are similar 3a07l Y2t 3a0l 3a0l lt Y1t lt 30071 Let us now discuss the details of the transmission mechanisms in our model and relate them to the literature Table 2 summarizes the patterns of responses of the terms of trade and stock prices to the underlying shocks22 One immediate implication of Table 2 is that supply shocks create comovement among stock market prices worldwide The comovement is generated by two channels of international transmission the terms of trade and the common worldwide discount factors for cash ows common state prices To illustrate the workings of the former channel consider a positive supply shock in country j Such a shock has a direct positive effect on country j s stock market Additionally it has an indirect also positive effect on the remaining stock markets through the terms of trade Indeed as discussed earlier a supply shock in country j creates an excess supply of good j and hence causes a drop in its price relative to the rest of the goods This implies that the prices of all the other goods increase relative to those of good j boosting the value of the dividends and therefore the stock mar kets in the rest of the world This explanation of the transmission of shocks across countries appears to be solely based on goods markets clearing where the terms of trade act as a propa gation channel This channel however is not unrelated to the second transmission vehicle the wellfunctioning nancial markets creating the common discount factor for all nancial assets eed in our model clearing in good markets implies clearing in stock and bond markets as 22 n our speci cation demand and supply shocks are correlated We nonetheless nd it useful to report their effects sepamtely in Table 2 We do so because the implications of the supply shocks for the stock market comovement are ofthe opposite nature as those ofthe demand shocks and 39 39 f p f 39 for understanding the mechanism behind international propagation 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1227 well and hence the abovementioned intuition could be restated in terms of equilibrium re sponses of the stock market prices Such intuition for nancial contagion was highlighted by Kyle and Xiong 2001 who see contagion as a wealth effect see also Cochrane et al 2004 An output shock in one of the countries always increases its stock market price and hence each agent s wealth because all agents have positive positions in each stock market At a partial equilibrium level a wealth increase triggers portfolio rebalancing In particular it is easy to show that for diversi cation reasons our agents demand more of all stocks At an equilibrium level of course no rebalancing takes place because the agents have identical portfolios and they must jointly hold the entire supply of each market Therefore prices of all stocks move upwards to counteract the incentive to rebalance So the two transmission channelsithe terms of trade and the common discount factoriinteract and may potentially be substitutes for each other Note that none of these arguments makes any assumption about the correlation of output shocks across countriesiin fact in our model they are independent The existing literature would identify the phenomenon we described here as contagion the comovement in stock markets in excess of the comovement in fundamentals In our personal views this comovement is not contagioniwe view it as nothing else but a simple consequence of market clearing and hence a natural propagation that is to be expected in any international general equilibrium model Our de nition of contagion is the comovement in excess of the natural propagation described above While supply shocks induce comovement among the countries stock markem demand shocks potentially introduce divergence Consider for example a positive demand shift occurring in country 1 Country 1 now demands more of the domestically produced good and less of the foreign goods which unambiguously increases the price of the domestic good The direction of the response of the other Periphery country s terms of trade depends on its wealth relative to the Centre 12 If the country is small Condition Cl it suffers disproportionately more due to a drop in demand for its good and its terms of trade with the Centre deteriorate The impact on the stock markets however requires a more detailed discussion We can repre sent the stock market prices of the countries in the following form SO I p00 Y0t T 7 t 510 q1tp0t Y1tT 7 r and 520 q2tp0t Y2tT 7 r A demand shift in coun try 1 improves its relative price q1 and deteriorates the other Periphery countries relative price qz pushing S1 up and S 2 downithis is the direct effect However there is also an indirect effect due to a fall in the price level in the Centre country The conditions of similar and small Periph ery countries ensure that the impacts of these demand shocks on the Centre price p0 are small forcing the terms of trade effect to dominate However small there is a drop in the price of the Centre s good p0 and hence the stock price of the Centre falls 3 EQUILIBRIUM IN THE ECONOMY WITH PORTFOLIO CONSTRAINTS Having established a reference point by examining a frictionless economy we are now ready to explore the role of portfolio constraints in propagation and ampli cation of shocks We stress the importance of addressing this question within a general equilibrium framework which highlights the critical role of wealth redistribution in the transmission mechanism 31 The solution method We now brie y sketch our solution method both to provide a roadmap for our formal analysis and to highlight its generality The benchmark unconstrained model is solved by rst consider ing the planner s problem with weights ii i l 2 and deriving consumption sharing rules as 2008 The Review of Economic Studies Limited 1228 REVIEW OF ECONOMIC STUDIES functions of these weights 577 The terms of trade prevailing in the unconstrained equilib rium are then identi ed from the consumers marginal rates of substitution 8 and 9 and the weights in the planner s problem from their budget constraints This solution method is standard The main new element that we bring over much of the inteniational economics literature is our ability to solve in closed form for the countries portfolios and for stock prices see also Pavlova and Rigobon 2007 This has been possible because of three modelling features i loglinear preferences over multiple goods ii endowments given by shares in Lucas trees and iii demand shocks Solving for equilibrium in the economy with portfolio constraints is a more formidable task because we are dealing with a general equilibrium model with investor heterogeneity market frictions and multiple risky assets But again the three key elements i7iii highlighted above render the model tractable To characterize equilibrium with portfolio constraints we rst solve for the investors portfolios at a partial equilibrium level Section 22 The resulting portfolios of constrained investors incorporate the multipliers on the portfolio constraints which we can pin down when we specialize to a particular constraint Then building on recent literature in asset pricing theory we again consider a planner s problem but with the planner s utility now featuring stochastic weights ii i l 2 These weights are again endogenous We then proceed as in the unconstrained benchmark and derive the consumption sharing rules and the terms of trade as functions of exogenous variables and endogenous weights ii We also show that stock prices are given by the same expressions as in the unconstrained benchmark Lemma 2 but with the weights 1 now stochastic We can thus isolate the effects of the portfolio constraints on the stock prices by examining the incremental contribution brought about by a change in is The economics behind a positive change in A is that it is a wealth transfer from the Centre to the Periphery country 139 since A t Wit W0 1 Finally we use our characterization of optimal policies and prices as a function of is to pin down the processes 11 and 12 This is done in Section 5 for two particular portfolio constraints This solution technique applies more generally than just to the economic phenomena we highlight in this paper 8 g excess comovement In particular it does not depend on the number of countries the speci c form of the portfolio constraints the probability distribution of output or market completeness We comment on possible generalizations of our model throughout the paper and especially in Section 6 32 The common factor due to constraints In the economy with nancial markets imperfections the equilibrium allocation would not be Pareto optimal and hence the usual construction of a representative agent s utility as a weighted sum with constant weights of individual utility functions is not possible Instead we employ a representative agent with stochastic weights introduced in an important contribution by Cuoco and He 1994 with these stochastic weights capturing the effects of market frictions23 This representative agent has utility function T UC0 C1 C2 1112E uC0l C10 C20 110 120dl 0 23 The construction of a representative agent with stochastic weights has been employed extensively in dynamic asset pricing See for example Basak and Croitoru 2000 Basak and Cuoco 1998 and Shapiro 2002 A related approach is the extmrstatervariable methodology of Kehoe and Perri 2002 For the original solution method using weights in the representative agent see Negishi 1960 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1229 ultC C1C2 1112 2 max uoltC8 c3 C 11u1ltc Cl C 2050allj 10 12u2C Cl Cl where 1 gt 0 i l 2 are yet to be determined weighting processes which may be stochas tic The advantage of employing this approach is that a bulk of the analysis of the previous section can be directly imported to this section In particular the only required modi cation to equations 579 is that the constant weights 11 and 12 are now replaced by their stochastic counterparts The expressions for stock market prices 107l2 also continue to hold in the con strained economy see the proof of Lemma 2 in Appendix A Furthermore as a consequence of the consumption sharing rules and Lemma 1 we again conclude that 11 t W1 t W0 t and 12t W2 t W0 t So in the constrained economy the wealth distribution captured by the quantities 11 and 12 becomes a new state variable Finally in the constrained economy we also have an analogue of Proposition 1 except now the weighting processes 11 and 12 enter as additional factors These factors capture the effects of the portfolio constraint imposed on the Centre s consumer Proposition 2 i In an equilibrium with the portfolio constraint the weighting processes 11 and 12 are the same up to a multiplicative constant ii When such equilibrium exists the joint dynamics of the terms of trade and three stock markets in the economy with the portfolio constraint are given by q 0 AG w da1t 1120 At d I 0 X 0133003 ICWH XN M r on 0 0 10 11031 Amixm Y 1 m Myron J 1090 t L 520 J i0y2tdw2t where 1t E 11 t C and X2 are reported in Appendix A and where the unconstrained dynamics matrix OM t is as de ned in Proposition 124 Proposition 2 reveals that the same transmission channels underlying the benchmark econ omy are present in the economy with portfolio constraints Ceteris paribus the sensitivities of the terms of trade and stock prices to the demand and supply shocks are exactly the same as in Proposition 1 The only difference from the benchmark economy comes in the rst d1 1 term This term summarizes the dynamics of the two stochastic weighting processes 11 and 12 which end up being proportional in equilibrium and hence represent a single common factor we 24 Existence of equilibrium can be shown for the case in which the portfolio constraint does not bind the unconr strained benchmark and for the case of speci c constmints considered in our examples in Section 5 but would be ve dif cult to show for the general speci cation of the constmint considered in this section Still we feel that our analysis i this section is important as it chamcterizes properties of equilibrium that obtain for any constraint imposed on the Centre co untry 2008 The Review of Economic Studies Limited 1230 REVIEW OF ECONOMIC STUDIES labelled A25 Thus the process A should be viewed as an additional factor in stock prices and the terms of trade dynamics arising as a consequence of the portfolio constraints One can already note the crossmarket effect of portfolio constrains The constraint affects not only the Centre s stock market but also the Periphery stocks as well as the terms of trade This nding is of course to be expected in a general equilibrium model The effects of constraints in nancial markets get transmitted to all other stock bond and commodity markets via pertinent market clearing equations Our contribution is to fully characterize these spillover effects and identify their direction The signs of responses to the supply and demand shocks are of course the same as in the benchmark unconstrained equilibrium Additionally we can sign the responses of all markets to innovations in the new factor some signs are unambiguous and some obtain under the following condition Condition C3 The re ect of the portfolio constraint on p0 is small26 1 7 2 TL IAlAl lt Al 14 l 7 1 5 TL IAlAl lt AO 1 Table 3 reveals the contribution of portfolio constraints to international comovement The rst striking implication is that the terms of trade faced by both Periphery countries move in the same direction in response to an innovation in the A factor A movement in A should be viewed in our model as a tightening or a loosening of the portfolio constraint Given the de nition of A such innovation re ecm a wealth redistribution in the world economy to or away from the Periphery countries Parallels may be drawn to the literature studying the effects of wealth transfers on the terms of trade It is well known from the classic quotTransfer Problem of the inteniational economics literature that an income wealth transfer from one country to another improves the terms of trade of the recipient As wealth of the recipient of the transfer goes up his total demand increases but because of the preference bias for his own good the demand for the domestic good increases disproportionately more Hence the price of the home good rises relative to that of the foreign goods improving the terms of trade of the recipient This effect is also known as the Keynes effect27 In our model a decrease in the factor A is interpreted as a wealth transfer to the Centre country Just like in the Transfer Problem it results in an improvement of its terms of trade against the world and hence a deterioration of the terms of trade 25 TL39 2 quot r 39r 39 39 39 mHere they are both unconstmined If 39 39 in geneml one would r L 39 weighting processes to be proportional and hence both A1 and A2 would enter as relevant factors Furthermore in our speci cation the new factorA is not independent from the existing factorsifor example an innovation to any underlying Brownian motion affects the distribution of wealth and hence A However for the purposes of separating the incremental effect of the portfolio constmint relative to the dynamics occurring in the unconstrained benchmark we nd it useful to treat A as an additional factor The condition is necessary and suf cient It is likely to be satis ed under the leading interpretation of the Centre country being big In Appendix A we investigate this condition further representing it as a combination of two effects i the impact of a change in A the implied wealth transfer on the demand forgood 0 and ii the crossrcountry demand reallocation in the Periphery countries The condition affects none of the derivations it is used only for presenting the directions of responses of the Periphery countries stocks to innovations in 27 The original Transfer Problem was the outcome of a debate between Benjl Ohlin and John Maynard Keynes if all countries have the exact same demands in our model this requires an assumption that or 13 i 0 l 2 See Krugman and Obstfeld 2003 for an elabomtion and references 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1231 TABLE 3 Terms of trade and stock returns in the economy with portfolio constraints Variableeffectsof 1 da1t da2t moo dw1t W20 1 7C1 0 Mg 7C1 0 I I also I c2 c2 7 7 7 501 5123 C3 C1 7C2 Z azs2 I Q 7C2 C1 520 Notes Where a sign is ambiguous we specify a suf cient or a necessary and suf7 cient condition for the sign to obtain C1 stands for the small country condition C1 C2 for the similar country condition C2 and C3 for the small effect on p0 condition C3 of both Periphery countriesithe reverse for an increase in A The main difference between our work and the Transfer Problem literature is that the latter considers exogenous wealth transfers while wealth transfers are generated endogenously in our model as a result of a tightening of the portfolio constraint The direction of such a transfer to or from the Periphery countries in response to a tightening or a loosening of the constraint depends on the form of a constraint We determine whether a particular constraint tightens or loosens in response to a shock in Section 5 in which we consider two speci c examples of portfolio constrains The intuition behind the occurrence of the wealth transfers in our model is simple Assume for a moment that there is no constraint Then each country holds the same portfolio When a binding constraint is imposed on the investors in the Centre their portfolio has to deviate from the benchmark and now the portfolios of the Centre and Periphery investors differ This means that stock market price movements will have differential effects on the investors wealth The movements of wealth obviously depend on the type of the constraint For any constraint that binds however one can say that the distribution of wealth will uctuate becoming an additional transmission vehicle Moreover since the Periphery countries hold identical portfolios their wealth shares move in tandem That is the resolution of uncertainty always affects the Periphery countries in the same way They both either become poorer or become richer relative to the Centre The portfolio constraint also generally induces the comovement between the stock markets of the Periphery countries This comovement may be partially confounded by the Centre good price effect which is of the same nature as the one encountered in the case of the demand shifts in the benchmark model Section 2 Consider for example a response to a positive shock in A While the improving terms of trade effect boosts the Periphery stock markets the associated downward move in p0 may potentially offset this However under our Condition C3 the terms of trade effect dominates If we were to quote stock market prices of the Periphery in terms of the production basket of Centre rather than the world consumption basket the two Periphery markets would always comove in response to a tightening or a loosening of the portfolio constraint On the other hand the response of the stock market of the Centre is unambiguous and goes in the opposite direction of A re ecting the effects of an implicit wealth transfer to or from the Centre So in summary the implicit wealth transfers due to the portfolio constraint create an additional comovement among the terms of trade of the Periphery countries as well as their stock market prices while reducing the comovement between the Centre and the Periphery stock markem This 2008 The Review of Economic Studies Limited 1232 REVIEW OF ECONOMIC STUDIES implication resembles the phenomenon known as contagionithe comovement in excess of that occurring in the unconstrained economy 4 CONTAGION WITHOUT TRADE In the previous section we have considered a model in which each Periphery country allocates equal expenditure shares to the two goods it imports This may appear unrealistic in the context of our leading interpretation where the Centre country represents a large developed economy and the Periphery countries two emerging markets because emerging economies trade with iri dustrialized economies much more than among themselves Moreover recent empirical studies of emerging markets have cast doubt on the ability of trade relationships to generate inteniational comovement of observed magnitudes and have documented that contagion exists even among countries with insigni cant trade relationships Since the movements in the terms of trade are an essential ingredient of the contagion mechanism in our model it is natural to ask whether our results still hold under alteniative assumptions regarding the extent of trade in goods between the Periphery countries In this section we take our setting to the limit and show that even when Periphery countries do not trade with each other at all their stock markets comove as described in the baseline analysis To examine this scenario we modify the countries preferences as follows u008ltr 050 030 logcgm u1Cltr cm 01 lt17 a1t10gCta1t10gClt u203ltr cm Cm 17 062010gC30a2l10gC l That is we assume that the goods produced by the Periphery countries are nontraded and the only trade occurring in the model is that between each Periphery country and the Centre We continue to assume that there is a home bias in consumption by restricting at to be a martingale lying between 12 and 1 As before the Centre country s portfolios are constrained to lie in a closed convex nonempty subset Ktw t w E 0 T X Q Under this speci cation the terms of the trade of each Periphery country with the Centre are Mr 1lt W Yoa 111t17 10 A20 10lt2l where the relative weights Al and 12 are possibly stochastic It is straightforward to show that the expressions for the stock prices remain the same given by equations lO7l2 In the interest of space we do not provide the dynamics of the terms of trade and stock prices in this economy we just present a table Table 4 that mimics Table 3 of Section 3 In contrast to Table 3 only two signs in Table 4 are ambiguous related again to demand shifts the remaining implications do not require any further conditions The effects of the demand shocks on the terms of trade are now clearcut because a demand shift in a Periphery country 1 not only increases the world demand for good 1 relative to all other goods as before but also decreases the demand for good 0 while leaving the demand for good 2 unchanged Therefore the price of good 0 drops relative to that of both goods 1 and 2 Another set of signs that becomes unambiguous is that for the effects of the innovation in the wealth shares of the Periphery countries captured by A on the stock prices in the Periphery JE LZL 16 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1233 TABLE 4 Terms oftrade and stock returns in the economy withportfoiio constraints and no trade between the Periphery countries Variableeffects of M dal t da2t moo dw1t W20 1 H1351 7 0 q 21 111 0 7 115 z 5011 7 7 115 Z 10 A 2 A 520 Note A stands for ambiguous Within this economy it is easy to derive the real exchange rates faced by the Periphery countries Remark 2 real exchange rates The price indexes in each country derived from the countries preferences are given by 1 0110 01t P00071200 P1ltiamp 1210 1 i 0610 0610 1 P20 p00 20 p1tazt 1 0620 0620 i The real exchange rates expressed as functions of the terms of trade are then Pl 5 7 7 I M 7 P00 7 lt17ajltma1lt0 1am 1 q t 1 J e 1 2 Our primary conceni is the incremental effect of a change in A on the real exchange rates as in the rst column of Table 4 Since the utility weights on are positive the real exchange rates respond to a change in A in the same direction the terms of trade do This implies that the excess comovement in the terms of trade due to the portfolio constraint translates into the excess comovement of the real exchange rates of the Periphery countries 5 EXAMPLES OF PORTFOLIO CONSTRAINTS The purpose of this section is to illustrate the applicability of our general framework to studying speci c portfolio constraints Under a speci c constraint we can fully characterize the countries portfolios and hence identify the direction of the constraintnecessitated wealth transfers This allows us to address questions of the following nature Does a positive shock in the Centre entail a wealth transfer to the Centre How does the origin of a shock affect stock retunis worldwide and Does the constraint amplify the shocks 51 Pare wealth transfers A portfolio concentration constraint Here we retuni to our workhorse model presented in Section 2 and specialize the constraint set K to represent a portfolio concentration constraint That is the resident of the Centre country 2008 The Review of Economic Studies Limited 1234 REVIEW OF ECONOMIC STUDIES now faces a constraint permitting him to invest no more than a certain fraction of his wealth y into the stock markets of Periphery countries 1 and 2 x 1ltoxgzlto n n ER 17 While this constraint is prevalent in practice we do not intend to argue that such a constraint is necessarily behind the pattenis of correlations observed in reality Our goal is to merely illustrate the workings of our model We feel that equation 17 is particularly well suited for this purpose since its impact on the portfolio composition and hence the entailed wealth transfers is easy to understand28 For the concentration constraint we can fully characterize the process A and hence the remaining equilibrium quantities Note that the consumption allocations terms and trade and stock prices all depend on the primitives of the model and the unknown stochastic weighs There fore once the process A and the constants A10 and A20 are determined we would be able to pin down all these equilibrium quantities This step is inevitably somewhat technical A reader interested primarily in economic mechanisms and intuitions may skip directly to Section 511 The set of equations required to fully close the model in the economy with portfolio constraints is presented in the following two propositions Proposition 3 When equilibrium exists the wealth distribution A follows dAO Atrt VOO mtTmot mO Dldt MUWOO mO DwaO L 18 where r0 and mo are interpreted as the interest rate and the market price of risk faced by the Centre and m is the market price of risk faced by the Periphery countries29 The quantities m0 and m are related as follows When n i2TltaltrT 1mltr n mog mg W 0 constraint not binding 19 otherwise moo mt70t 1i1i2wt 20 V i1i2T0tT 1mI W ili2TltoltroltrT 1lti1i2 gt 0 constraint binding 28 We concede that other constmints especially government imposed may be more economically relevant but in this section we consider only two possible constmints Another set of restrictions absent from the model is those on the Periphery countries We believe that the model possesses suf cient exibility to accommodate these alternative conr r m n 1 c ll39Lo39 cc 139 rr 29 39 of r39 fquot quot me 39K see the beginning of Appendix A Proving existence consists of showing existence of a solution to algebmic equations l9721 given our state variables and then showing that this solution implies existence and uniqueness of a solution to 39h 39 ha ti 439 quot quot l The c r J 39 h 39 39 quot quot39 of matrix 6There is a possibility that this matrix may not be invertible in our model which happens when there are no demand shifts However existence for that case has been established in the previous literature Zapatero 1995 Cass and Pavlova 2004 To highlight and characterize the behaviour of asset prices in our model in the following subsections we compute the solution to our model for speci c parameter values In all the examples the matrix 6 was always invertible The second step amounts to verifying that Lipschitz and growth conditions see eg kaendal 2003 theorem 521 are satis ed for the drift and diffusion terms in equation 18 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1235 where i0 2 1 0 of i1 2 0 1 of i2 2 0 0 1T and the volatility matrix 0 is provided in Appendix A Furthermore it timzl l im ma 7 mom 7 we r 7 Mo 0110013907 a0l 042170 1t 12l170 2t 7 Xtlttltmltt 7moltt Xa1ltoaellto amount Mltoltq1lto q2lttoolttio 7 Mlttgtq1lttgte lttgtit 7 mttqzmemie molttgt 21gt Equations 19 and 20 are the complementary slackness conditions coming from the con strained portfolio optimization of the resident of the Centre At times when the constraint is not binding the market price of risk faced by the Centre coincides with that faced by the Periphery Therefore the portfolio of the Centre is given by the same equation as the unconstrained portfo lios When the constraint is binding however there is a wedge between the market prices of risk faced by the Centre and the Periphery 20 The quantity 1 is to be interpreted as the multiplier on the portfolio constraint speci ed in equation 17 Equation 21 is the direct consequence of market clearing in the consumption goodsiit is a consumptionCAPMtype equation Together equations 19721 allow us to pin down the equilibrium market prices of risk of Centre and Periphery and hence the responses of all three stock markets to innovations in the underlying Brownian motions 100 ml and 102 as functions of the state variables in the economy Once the market prices of risk processes m0 and m are determined it is straightforward to compute the effective interest rate differential faced by the Centre country Proposition 4 which completes our description of the dynamics of the process A equation 18 This together with the countries portfolio holdings reported in Corollary 1 concludes the full characterization of the economy Proposition 4 When equilibrium exists the di erential between the interest rates faced by countries 1 and 2 and that effectivelyfuced by country 0 is given by V i ii i2T0l T71ml 040m iti2TltelttalttT 1ltiti2 22 From equations 19 and 20 and equation 22 one can easily show that the interest rate differential is always nonpositive That is the interest rate effectively faced by the constrained country is higher than the world s unconstrained interest rate This accounts for the effects of the portfolio constraints Recall from Section 22 that the optimization problem of the Centre subject to a portfolio constraint is formally equivalent to an auxiliary problem with no constraints but the Centre facing a ctitious investment opportunity set in which the bond and the Centre s stock the unrestricted investments are made more attractive relative to the original market and the stocks of the Periphery countries the restricted investments are made relatively less attractive In this ctitious market the Centre optimally invests more in the bond and in the Centre s stock relative to the original market and less in the Periphery countries stocks 511 Analysis of equilibrium with the portfolio concentration constraint The dy namics of our model implied by Proposition 3 are best illustrated by means of plots The para meters used in the analysis are summarized in Table 5 All timedependent variables in Table 5 are the state variables in our model In the interest of space in our gures we x all of them but the wealth shares of the Periphery countries 11 t and 12t These stochastic wealth shares are 2008 The Review of Economic Studies Limited 1236 REVIEW OF ECONOMIC STUDIES TABLES Parameter choices 09 y 035 10 075 11 10 6100 03 1 075 110 03 6110 01 120 03 6120 03 mt 6 005035 120 6 005035 6x1 t 0 02 0 and 0 0 02 behind the additional common factor driving the stock prices and terms of trade that we identify in our model and it is of interest to highlight the dependence of the prices and portfolios in our model on these wealth shares Hence the horizontal axes in all the gures measure 11 and 12 The reasoning behind the choice of our parameterization is the following In our leading interpretation the Periphery countries are small so for the choice of the numeraire consumption basket we decided that they represented 5 of the world We have chosen 75 as the share of expenditures on the domestic good which is a conservative estimate given the share of the service sector in GDP In terms of output the Periphery countries are onetenth of the Centre and three times as volatile We assume that the wealth ratios relative to the Centre for both Periphery countries may range from 005 to 035 Finally we need to specify the parameters of the demand shocks Recall that in our model there are only three primitive sources of uncertaintyi the Brownian motions in w and w iand so the supply and demand shocks are necessarily correlated In Pavlova and Rigobon 2007 we nd that in the data demand shocks are positively correlated with domestic supply innovations Therefore we assume that a demand shift in country j has a positive loading on 10 and zero loadings on the remaining Brownian motions To develop initial insight into the solution we examine the region where the constraint is binding The tightness of the constraint is measured by the multiplier 11 from equations 19 and 20 As is evident from Figure l for small wealth shares of the Periphery countries the portfolio constraint is not binding and the multiplier is 0 As their wealth shares increase the constraint tightens The multiplier is increasing in both is In the unconstrained economy larger is imply that Periphery countries constitute a larger fraction of world market capitalization and hence they command a larger share of the investors portfolios Therefore given the same upper bound constraint on the investment in the Periphery countries the larger these countries are the tighter the constraint Let us now concentrate on how the portfolio constraint affects portfolio decisions by the Centre s investor In our parameterization the Periphery countries are symmetric and therefore we only show gures for one of the Periphery countries Figure 2 depicts the changes in portfo lio weights relative to the unconstrained economy The excess weight in the Centre country s stock is shown in panel a and the excess weight in the Periphery country l s stock is shown in panel b For the range of is where the constraint is not binding the portfolio holdings are identical to those in the unconstrained equilibrium For the range where it becomes binding the investor in the Centre is forced to decrease his holdings of the Periphery markets The freedup assets get invested in the stock market of the Centre country and the bond making the Centre country overweighted in the Centre s stock market relative to its desired unconstrained position 30 L H mmn unaltered 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1239 The incremental effect on the stock prices brought about by the portfolio constraint mimics the effects on the terms of trade A country experiencing an improvement of its terms of trade enjoys an increase in its stock market and that experiencing a deterioration sees its stock drop Now we can fully address the issue of the comovement among the stock markets that the portfolio constraint induces These results are presented in Figure 4 Panel a demonstrates the impact that a negative shock to the Centre has on the Centre s stock market beyond the already negative effect that takes place in the unconstrained economy In the region where the constraint is not binding the effect is zero but it is negative over the remainder of the state space That is the negative effect of a shock to the Centre is ampli ed in the presence of the constraint Furthermore the magnitude of the effect is increasing with A which is to be expected because the higher the wealth shares of the Periphery countries are the tighter the constraint The exact same intuition applies to the effects of the shocks in the Periphery on domestic stock prices panel d The transmission of shocks across countries is depicted in panels b c and e The impact of a productivity shock in the Centre on the Periphery stock prices is shown in panel b that of a shock in a Periphery country on the Centre in panel c and nally that of a shock in one Periphery country on the other Periphery country in panel e Again these are incremental effects due to the constraint net of the comovement implied by the unconstrained model The emerging patteni is consistent with the ightto quality and contagion effects observed in the data The ight to quality and contagion refer to a transmission pattern where a negative shock to one of the Periphery countries emerging markets depresses stocks of other countries in the Periphery but boosts the Centre country s stock market an industrialized economy Panels c to e demonstrate that in our model a negative shock to one of the Periphery countries reduces its stock price decreases the stock price of the other Periphery country contagion and increases the stock market price in the Centre ight to quality A similar patteni occurs if the Centre receives a positive shock inally we examine the extent of excess comovement of the Periphery stock markets that our model generates To do so we compute the instantaneous variancexovariance matrix a t0 if from Proposition 3 in the constrained and unconstrained economies and evaluate the instantaneous crosscountry correlations of stock market returns in the constrained economy in excess of those in the unconstrained Figure 5 One can see that the correlations increase in mag nitude as wealth shares of the Periphery countries increase and hence the constraint in the Centre becomes tighter As the theory predicts the comovement between the Centre and the Periphery decreases while the comovement across the Periphery countries goes up This effect is also siz able The decrease in the correlation of the stock retunis of the Centre and the Periphery can be around 477 and the increase in the correlation between the Periphery markets returns around 10715 52 Varying restrictiveness A market share constraint The previous constraint is one of the simplest that can be studied within our framework However it generates some counterfactual implications For instance a negative shock to the Periphery relaxes the constraint instead of tightening it31 We therefore consider a constraint of a different nature a market share constraint which becomes more restrictive when the market share of the Periphery countries in the world drops 1 2 Smlsm y R m 31 S2 x i x i g y F 0 W 9mv 31 It has been argued in the empirical litemture that recent contagious crises in emerging markets may have been caused by the tightenings of constmints in developed countries in response to a crisis in one emerging mar e 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1245 6 DISCUSSION AND FUTURE RESEARCH In this section we discuss the robustness of our main results as well as make suggestions for possible extensions of our framework One important implication of our model is that portfolio constraints give rise to wealth transfers whichithrough the Keynes effectican produce am pli cation of shocks and excess comovement among the Periphery countries stocks One may wonder however whether the same Keynes effect can take place in an environment with no nancial frictions but nonidentical e g home biased portfolios held by the countries Indeed in such a model wealth distribution is no longer constant and hence wealth transfers occur even in the absence of frictions One way to extend our model so that it produces nonidentical portfolios in the unconstrained benchmark is to generalize the expenditure shares in the utility functions of the agents Such a model still admits a closedform solution but in contrast to our model wealth shares of the Periphery countries no longer coincide with the weights in the planner s prob lem Therefore one can disentangle the effect of a timevarying wealth distribution from that of stochastic weights in the planner s problem Interestingly in the world with perfect risk sharing the Keynes effect does not operate even when the portfolios exhibit a home bias This happens because agents can perfectly hedge uctuations in wealth distributioniand hence neither con sumption nor the terms of trade is affected by these uctuations In contrast under imperfect risk sharing the terms of trade are affected by wealth transfers through their effects on the stochas tic weights Al and 12 which in turn cause the response predicted by the Transfer Problem the Keynes effect Throughout the paper we have stressed the importance of portfolio constraints in generat ing an asymmetric pattern of crosscountry correlations depending on whether they are binding or not Portfolio constraints also have implications for the persistence of negative shocks Con sider for example a market share constraint As soon as it starts to bind it forces a capital out ow from the Periphery and reduces the wealth shares of the Periphery Ceten39s paribus these reduced wealth shares make the constraint even more binding in the future thereby increasing the persistence of negative shocks to the Periphery We leave a thorough investigation of this as well as other feedback mechanisms for future research Furthermore in this paper we have maintained the assumption that only one agent is con strained in his portfolio choice Motivated by the contagion literature we have imposed such a constraint on the large Centre country This leaves out an important case in which more than one investor face constrains For example a realistic scenario could be one in which only a small fraction of the population of Periphery country 1 is permitted to hold shares of Periphery coun try 2 and vice versa Our conjecture is that the effects of imposing a constraint on the Centre are going to be qualitatively the same For example a constraint limiting the Centre s investment in the Periphery is still going to force the Centre to hold less of the Periphery and more of the Centreirelative to the new benchmark in which the Periphery countries hold less of each other s stock In response just like in our model the unconstrained residents of the Periphery countries will be persuaded to hold more of both Periphery countries stocks for diversi cation reasons The resulting biases in the countries portfolios relative to the new benchmark are of the same sign as in our model and hence the main implications that we describe in Section 5 should hold Verifying this conjecture is possible but not straightforward and we leave it for future research32 One counterfactual implication of our framework is that the interest rate in the Centre a developed economy is higher than that in the Periphery emerging economies Clearly this is not supported by the data However we have not included important determinants of interest rates such as default or expropriation risk One could model expropriation as a random variable L f quot 39 39 Soumare and Wang 2007 2008 The Review of Economic Studies Limited 1246 REVIEW OF ECONOMIC STUDIES affecting the cash ows of the Periphery counties stocks or bonds This might be an interesting extension to pursue For simplicity our speci cation does not allow for demand shocks to be independent of sup ply shocks This assumption guarantees that nancial markets are complete It is possible how ever to extend the model to the case where demand shocks are driven by independent Brownian motions and hence markets are incomplete33 While the analysis of such an economy becomes more complicated the main results in Sections 24 remain unchanged The examples in Sec tion 5 are still tractable but the formulas we report in Proposition 3 need to be adjusted for the increased dimensionality of uncertainty In this paper we do not pursue this extension and treat the demand shocks primarily as a modelling device that ensures that the stock markets are not perfectly correlated This rules out an investigation of several potentially interesting questions For example one may ask whether demand shocks get ampli ed in the presence of portfolio con straints or whether demand shocks magnify the correlation of the Periphery stock markets when portfolio constraints are binding Finally it would be interesting to take the model to the data and try to address the following issues In recent nancial crises how much of the comovement could be attributed to the common discount factor and how much to the fact that nancial institutions were forced to curb their positions due to the tightening of their constraints Such questions are very dif cult to answer without an underlying theoretical framework Our model offers one such framework APPENDIX A Al Countries optimization Details ofthe analysis Our solution method relies on replacing the dynamic Radnerestyle budget constmint 4 in each country s optimization problem by a static ArrowiDebreuestyle budget constraint This opemtion is routine if investors in a country are uncone strained and face complete markets but more complicated if a country s investment opportunity set is constrained e start with the optimization of the Periphery countries 1 and 2 t at are unconstrained Dynamic market come pleteness implies existence gf a common state price density process 5 consistent with no arbitmge known to admit the follow1ng pammeterization 40 75trtdtmtwat Al where mt E 61ttt 7rtT is the market price of risk process associated with the Brownian motions mo ml and w The quantity 5t w is interpreted as the ArrowiDebreu price per unit probability P of one unit of the numemire delivered in state m E Q at time t We can now convert each country s dynamic optimization problem into a static problem see Kamtzas Lehoczky and Shreve 1987 Cox and Huang 1989 i T E incf t C110 Cftdti A2 I max CCCZ L0 T subjectto E 5tp tctp1tctp2tcftdt 5W0 i12 A3 0 33 A recent work by Pavlova and Rigobon 2008 considers ageneralization of our model in which demand shocks are independent of supply shocks and nancial markets are incomplete 4 trictly speaking we can only claim that markets are potentially complete because it is not guamnteed that in equilibrium all assets are noneredundant To ensure the validity of our solution method this noneredundancy needs to be veri ed in the equilibrium we construct For the representation of the state price density under complete markets see Duf e 2001 ch 6 2008 The Review of Economic Studies Limited PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1247 Its rstrorder conditions are given by 0 l 2 Wy1p1mam i1123 j03132 A4 acf t where y is the scalar Lagmnge multiplier that makes the budget constmint A3 hold with equality We turn to the optimization problem of the Centre country whic faces a portfolio constraint Following Oitanic and Karatzas 1992 we de ne the function 51 in the modi ed investment opportunity set of the Centre to be 5z E supOE 7160 Z Furthermore v is speci ed to be a stochastic process taking values in I z E R3 5z lt 00 A state price density 5quot implied by 39 quot 39 quot Le 39 quot 39 and expected returns on the stocks set is given by dc t 5vtrt 5vtdt mt 6t71vtwatl A5 The quantities r0 E r 61 and mo E m 6 t 71v are interpreted as the interest rate and the market price of risk effectively faced by the Centre country Under the assumption of logarithmic preferences Cvitanic and Kamtzas establish that for a particular vquot E K namely wt mg ruin25v Hmt alttr1vu2x A6 veK the optimization problem of the constrained consumer can also be stated in a static form just as in the unconstmined case but with the personalized state price density 50 E 5quot replacing 5 in equation A3 T 0 l 2 max E uoltcolttgt com comm Cg C3 C3 0 T subject to E 50tp0tC8t p1tct Emcgmm g W00 A7 0 imization of a quadmtic form in equation 636 The rest of the chamcterization is routine The rstrorder conditions for the abovermentioned problem are given by Note that the problem of solving for the stochastic process 1quot determining 50 reduces to a simple pointwise mini A auoltC lttgtc1lttgt C20 yaw0500 1 o12 As 8C0 t where yo is the Lagmnge multiplier that makes the budget constraint A7 hold with equality Proof of Lemma 1 It follows from the existing litemture Oitanic and Karatzas I992 Kamtzas and Shreve 1998 that W0t and W t i 12 have representations 72 T 1 W00 m E 5oltsgtltp ltsgtc3ltsgt p1ltsgtcgltsgt p2ltsgt63ltsgtgtds T Wzt1tE5sp sCfsp1sCsp2sC3sds B i12 Z 35 The notation Hz 2 stands for the dot product z z 36 See Kamtzas and Shreve 1998 ch 6 for a detailed discussion of constmined portfolio optimization problems at a partial equilibrium level in geneml and a proof of this result using the duality approach in particular The duality approach involves converting the original constrained consumer s or investor s optimization with respect to consumption into the dual optimization with respect to v such that Efooo Hv2tHdt lt 00 so as to determine the optimal Lagrange multiplier process 1quot re ecting the impact of the portfolio constraints rst and then back out the optimal consumption and portfolio processes See Kamtzas and Shreve for further necessary regularity conditions on 5quot 2008 The Review of Economic Studies Limited 1248 REVIEW OF ECONOMIC STUDIES These expressions combined with equations A4 and A8 yield T 7t 7 T 7 YofoO 7 MW Making use of the rst7order conditions A4 and A8 we arrive at the statement of the lemma H W00 Wt i12 Proof of Corollary 1 This is a standard result for logarithmic preferences over a single good eg Karatzas and Shreve 1998 ch 6 example 42 The modi cation of the standard argument for the case of multiple goods is simple thanks to Lemma 1 In particular we can equivalently represent the objective function of country 0 in the form T W00 1 0 W00 1 0 W00 EOiaol gp lttgtltT7tgtgt 2 1 gp1lttgtltT7tgtgt 2 mgltp2lttgtltT7tgtgtldt T 1 700 1 7 a0 E 10mm 7aologltp lttgtltT 7 7 2 logltp1lttgtltT7tgtgt 7 2 logltp2lttgtltT 70 0 Since the investor of country 0 takes prices in the good markets p1 39 0 12 as given and hence from his viewpoint the last three terms in the integmnd are exogenous this objective function belongs to the family considered by Karatzas and Shreve A similar argument applies to investors I and 2 H A2 Weights in the planner39s problem There are many ways to determine the values ofll and 12 all of which give the same answer Consider for example the expressions for C8 0 and Ci 0 provided in Lemma 1 and combine them with the corresponding ones from the sharing rules 5 and 6 Recalling the assumption that the initial endowments of countries 0 and l are given by W 0 510 i 0 l and substituting the expressions for 300 and 510 from equations 10 and l l we arrive at a system of two equations in two unknowns The system admits an explicit solution 11 l 7a0l 70110 and 12 l 7a0l 7 a2 ProofofLemma 2 It is easy to verify that an equivalent representation of the expressions in Lemma 2 is 5 p lttgtY lttgtltT 7t 51m p1lttgtY1lttgtltT 7t and 5 p2lttgtY2lttgtltT 7 To value the stocks we employ the no7arbitmge conditions of the unconstmined Periphery countries The require u r r holds all f stocks and hence that his wealth is the sum of stock prices implies that each stock s price is equal to its fundamental value the present value of its dividends T SJ t E fsp sYsds F j 0 I 2 A9 I It follows from equations A4 and 5 that 7111quot 7111quot 21lttgtY lttgt Y1p t5t a011t 21 l12t 22 l where A and 12 are constant weights in the unconstmined economy of Section 2 and stochastic in the constmined A10 economy of Section 3 Hence in equilibrium 021lttgt 12lttgt17 3amp Y111tY0t 1 1 170110 170120 120 01 Y1Y t011t 2 l 1 170110 170120 y1gt zo Y1Y t 110 2 2 Y2 2008 The Review of Economic Studies Limited pow0 A11 PAVLOVA amp RIGOBON ROLE OF PORTFOLIO CONSTRAINTS 1249 To derive the last equality we use equations A4 and A8 and Lemma 1 to show that YofoU YofoU A d A A12 1 MEG 3 2 mo Analogous steps can be used to deIive that 1 7 1 7 0 1 170120 p 050 7 y1Y1t 2 1 10 a1t 2 Y2 A13 2t t 1 1700 1 17a1t AIM p gt50 y1y2m 2 1 10 2 M20 gt Making use of the assumption that 11 and a2 axe maningales from equations A9 to A11 we obtain T 0 0 00a0111Li ltiilil2jzllm E U111s i 1716 i 172Szgtds Ft poa m om U 1 1 170110 1 17a2tY1gtT7t a011t 1111quot Azt W Z 11 2 2 Y2 M E Tao 1 dSFz l0T t a011t 1111quot 12t Wa2 I 2 Ms MO 7 0 0 aop t11tY t if 1 i 1 F W WWW 1 W B1 H An analogous aigument can be used to show that hl lplwmwn T 1 1 Sllttp1tY1tltT7t E A d F 77T7t 2011ta1t12t22l Z 1S 10 17a 2 2 T s2lttgtp2lttgtY2lttgtltT7tgtH E dm 77w 20Alt21l12ta2t Z MS 110 Note that the tem EUT T1165 ds 1 7 T1163 T 7 t enteIs the expiession for each stock symmetrically The1efo1e at any time t the prices of all stocks in the economy axe either above or below the value of their dividends augmented by the factor T 7 t 1 113 ds T sflttgt3p1lttgtY1lttgtltT7tgt if E I B 7 ltT7tgt 0 1 7mT7t30 j012 A15 T 1 St 3 ptYtT7t if E ds F 113 1 wheIe we have used the Iesttictions that 0 lt a lt 13 A gt 0 and Y39 gt 0 i 0 1 2 at all times On the other hand from bond market clearing we have that W00 W10 W20 5 510 5 AM 2008 The Review of Economic Studies Limited 1250 REVIEW OF ECONOMIC STUDIES and from lemma 1 and market clearing for goods 0 l and 2 that 1 a0Wot bylaw 532on 7 0 W A17 00 T it 1 1 1Wom a1ltnW1lttgt 143mm 1 p1t Tit T7t Tit Y t A18 171m 170110 1 Twat 2 1W1ltt MW 7 2 mgtiY t A19 Hence by multiplying Al7 A18 and Al9 by p0t p1t and p20 respectively and adding them up We can show that Wot W1lttgt Wm p lttgtY lttgtltT 7tgtp1lttgtY1lttgtltT 7tgtp2lttgtY2lttgtltT7tgt This together With equation A15 yields the required result ProofofPropositions 1 and 2 Since our proofs of the two propositions follow analogous steps We present them together We rst report the quantities At K0 at at bt at Mt X Xal and Xaz omitted in the body of Propositions l and 2 mo 7 551 1111quot A1lttgt m 324120 At E A20 0A1lttgt1 12lttgt1 1A1lttgta1lttgt12lttgt1 3 211 7 WWI 721017 VWHEEQ FMt A21 7 0A1t531912t5 29gt 1 3 A1ltt191 12ltta2lttgt 39 111 2 1 by 1 1 1 A10 1 A22 aoA1lttgt12 AzlttgtJ 1 A1lttgta1lttgt12lttgt1 2 51 2 M 1 7 17 33mm A23 2 on g Q A2lttgt1i 2 lt l 1 310 uvial 1 Human 211 2 m 1 7 who A24 2 on g Hm j 1 1 A1lttgta1lttgt12lttgt gt at E 1 20 1 t l 1 A20 1 z A25 aoA1ltt 2 ZA2ltt 2 l 1 A1ltta1ltt 12lttgt 2 l Mo 2 1 Km 2 MoqunAm 12lttKltt A26 5 qllttgt7 12lttgt 2 X111 2 lg Mwwnlo01120 X112 2 Mmmoqlm 701120 A27 These expressions are the same across Propositions l and 2 except that in Proposition 1 10 are constant Weights To demonstmte that in Proposition 2 1110 and 112 t are the same up to a multiplicative constant We use equation A 12 together With the observation that yl and y2 are constants Taking logs in equation 8 We obtain L f A1ltta1lttzl2lttz gt l 0 l logq t log 7 7 logY t7logY t a02111 J2211 2008 The Review of Economic Studies Limited PAVLOVA amp RlGOBON ROLE OF PORTFOLIO CONSTRAINTS 1251 Applying Ito s lemma to both sides and simplifying We have 11 dq1t lto terms 1 1 dllm 1 121 1110 k2 11ta1tlgt f 1 I 1 I 110 1 I 2 M 170120 d12t 1701t dllt I 1 i 2 220 a021i 2 22i 2 2 2 210 210 17a2lttgtd22lttgt 220 dY lttgt dY1lttgt I i 2 1quot 2 222 2 1102 1112 Substituting 113 26 4lt in the expression above simplifying and making use of equation 1 and the de nitions in equations A 07A2 We anive at the statement in the propositions Of course in Proposition 1 d11td12t 0 and hence the terms involving d11t and d12t drop out The dynamics of 212 are derived analogously To derive the dynamics of SO We restate equations 10712 as logSOt 7log 1110 q2tgt1ugyoo1ugr 7 t A28 7 1 5 1 1 5 2 logSJ t 7 log 21 t 7log 711 t 711 t logYt logT 7t A29 Applying Ito s lemma to both sides of equations A28 and A29 We arrive at ds lttgt 1 1 Mn 2 mm dY lttgt antt dt7 t t n 1 5 111139s ml g quHl g m 1 0 M q 1 11 M 2 1 2 MW Drifttermsdt WW 7 2 1110qu 1 1120 d1 1 d1 1 SW 1110 q1t 1122 2 t q t Yt Substituting the dynamics of q1 and 212 derived above and making use of the de nitions in equations A207A27 We anive at the statement in the propositions Computation of the drift term 15t E 151 152 153 154 155 is stmightforward but tedious so in the interest of space We reportjust the end result 121 t 7 YOU Y1t6yot2 an n2 At6yot01tTi0 7 Amayi on 0 Tu at6Y0t6a1tTi0 7alttgtayiltoau1lttgt7u blttgtayolttgtauzlttgt7io 7blttgtayi twain 152 t yYot71y2t6Yot2 252 02 At6Yot61tTio 7 At6Y2t61tTi2 alttgtayolttgtau1ltt io 712mm mm Mia 5lttgtayo one Nio 75mm 0 Mia 17 17 1 1230tyu 25 down 25 if in T7 17 17 1 Inc yyiow 3 T fm MlttgtGlttgt 7 T Mlttgtq2lttgtGlttgt 7 E 150 7mlttgt 2 q1lttgtMlttgt lttgt 7 lg MomoHo 7 2008 The Review of Economic Studies Limited 1252 REVIEW OF ECONOMIC STUDIES where D0 2 110 Amayo nam o alttgtayolttgtao1lttgt7io home 0 mo ao lt02 130 E 12 t mayo nam o alttgtayolttgtao1lttgt7io 23mm 0 DTio ao 02 Ga 2 110AlttgtaYilttgtailttgtTiialttgtayilttgtao1ltt iiblttgtayilttgtaozlttgt7ii mic 60 2 152m Alttgtayilttgtailttgt7ii alttgtayilttgtao1ltt ii 5lttgtayilttgtao2lttgt i 7610 HO 2 110 Amayz mom alttgtayzlttgtao1lttgt7i2 homo mm 032 762 02 1 E 12 t mayo mom alttgtayzlttgtao1lttgt7i2 23mm mm 032 762 02 and am 2 71tm0tmtio E 1 0 of i1 2 0 1 of and i2 2 0 0 1T In Proposition 1 the weights Al an 2 are constant and hence the drift term I is a special case of I in which 610 0 H Proof of Proposition 3 Recall that due to the portfolio constraint agents in the Centre and the Periphery countries quot r 39 A quot39 and re necti e1 1n particular 39 bentre country s effective interest mte and the market price of risk r0 t and mo t do not coincide with rt and mt faced by the Periphery countries when the constraint is binding It follows from Proposition 2 and A12 that Mt Applying Ito s lemma to this expression and using the de nitions of f and 50 from equations A1 and A5 we obtain equation 18 Substituting equation 18 into the expressions in Proposition 2 we have the following representation for 6 mo m0tT t2 d W 7X10 Exile Exam 2 2 t 0a am Amixm amazon blttgtixozlttgt 13 1 1 M 2 17 17 lttgtayolttgtzo Amixm memo blttgtixozlttgt 13 7 3 7 3 MWIWIOW Y 2 Mlttgtq2lttgta lttgti The 3 x 3 matrix 6 represents the loadings on the three underlying Brownian motions mo ml and w2 of the three stocks 30 captured by the the rst row of 6 S1 the second row and S2 the third row In the benchmark unconstmined economy or at times when the constmint is not binding all countries face the same state price density and hence the market price of risk m0 t coincides with mt and the matrix 6 coincides with its counterpart in the benchmark unconstmined economy Equations 19 and 20 are derived at a par1ial equilibrium level We follow the steps outlined in Section 22 and the beginning of this appendix to solve for the modi ed investment opportunity set of Centre37 We have K x0 6R3 i1i2Txo 7 gt and hence 77 Z ifz3i1i2 forsomeZ S 0 51 00 otherwise 1 z 6R3 z3i1i2 forsomefs 0 This establishes that the process v we are looking for must be of the form vt Wt i1 i2 for some 17t 5 0 To solve for Wt and therefore for the process vquot we make use of equation A6 min 727304 Hmt6t 1Tti1i2H2 A30 vt lt0 Thi optimization is straightforward and results in the expressions reported in equations 19 and 20 where for expositional reasons we have replaced the negative solution 3 t to equation A30 by ut 73 t The quantity 37 For a similar derivation but with different portfolio constraints see Tepla 2000 2008 The Review of Economic Studies Limited