Nominal Exchange Rates and Net Foreign Assets Dynamics: the Stabilization Role of Valuation Effects

Transcription

1 MPRA Munic Personal RePEc Arcive Nominal Excange Rates and Net Foreign Assets Dynamics: te Stabilization Role of Valuation Effects Sara Eugeni Duram University Business Scool April 2015 Online at ttps://mpra.ub.uni-muencen.de/63549/ MPRA Paper No , posted 13 April :53 UTC

2 Nominal Excange Rates and Net Foreign Assets Dynamics: te Stabilization Role of Valuation Effects Sara Eugeni Duram University Business Scool, Mill Hill Lane, DH1 3LB, Duram, UK First draft: May 2013 April 12, 2015 Abstract Recent empirical studies ave igligted tat valuation effects associated wit fluctuations of nominal excange rates are one of te key components tat drive te beavior of te net foreign assets position of a country. In tis paper, we propose a two-country overlapping-generations model of nominal excange rate determination wit endogenous portfolio coice in line wit tis evidence. We sow tat a country runs a current account deficit wen its sare of world GDP decreases. As te domestic currency depreciates in equilibrium, a positive wealt effect partially offsets te current deficit and terefore as a stabilizing impact on te net external position of te country. Te model rationalizes te deterioration of te US external position over te past 20 years as a consequence of te rise of emerging market countries in te world economy, wile being consistent wit te fact te US ave experienced positive valuation effects. Numerical results indicate tat valuation effects are quantitatively relevant as tey account for more tan alf of te cumulated US current account deficits, consistently wit te data. address: sara.eugeni@duram.ac.uk. An earlier version of tis paper was circulated under te title Portfolio coice and nominal excange rate determination in a stocastic OLG model. Tis paper is a development of te tird capter of my PD tesis. I tank my supervisor Subir Cattopadyay for is advice and Herakles Polemarcakis, Steve Spear, Mark Guzman, Neil Rankin, Gabriel Talmain, seminar participants at Duram, Warwick, Reading and te Nort American Summer Meeting of te Economic Society 2014 for teir comments and insigts. 1

3 1 Introduction Cross-border oldings of assets and liabilities ave substantially increased since te early 1990s, for bot developed and emerging countries (Lane and Milesi- Ferretti, 2001, 2007). It is well known tat one of te consequences of te iger degree of financial integration across countries is te increasing importance of te so called valuation cannel in te dynamics of net foreign assets (e.g. Gourincas and Rey, 2007, 2015). Traditionally, te net foreign assets position of a country was simply computed by cumulating current account balances over time. Wile tis measure reflects canges in te stocks of foreign assets and liabilities, it is imperfect as it ignores canges in te value of foreign assets and liabilities wic can arise due to fluctuations of nominal excange rates and asset prices. For instance, Figure 1 sows te divergence between te cumulated current accounts and te net foreign assets position of te United States. According to te former measure, te net foreign assets position of te United States amounted to almost 60% of GDP in However, direct estimates of net foreign assets and liabilities suggest tat te net external position was muc lower and equal to around 20% of GDP. Tis sows te significance of te valuation cannel in te dynamics of te net foreign assets of te US. In particular, te US ave experienced a substantial wealt transfer from te rest of te world over te past 20 years as te value of teir foreign assets as risen relatively to te value of teir foreign liabilities. Te importance of tis cannel is not specific to te US: it is interesting to observe tat emerging countries in East Asia ave experienced te opposite situation (Figure 2). Wile teir net external positions ave considerably improved over te past decades because of current account surpluses, tey ave experienced negative valuation effects 1. For all te above countries, valuation effects seem to ave a stabilizing effect on te net foreign assets position. One of te callenges in international macroeconomics is to come up wit a new generation of portfolio balance models microfounded and embedded in a general equilibrium set up so as to explain tis and oter facts in international financial markets (Gourincas and Rey, 2015). In tis paper, we propose a 1 Gourincas and Rey (2015) make similar observations for oter emerging countries. 2

4 two-country overlapping-generations model wit endogenous portfolio coice in wic te nominal excange rate is endogenously determined. Te main novelty of our work is tat it seds ligt on te role of nominal excange rate in countries portfolio coices and its impact on te dynamics of net external positions troug valuation effects 2. Te nominal excange rate is an important factor as it operates troug two different, but related cannels. Firstly, it as an impact on te decision of an agent to allocate is savings across a menu of currencies (or assets denominated in different currencies). It is rational for agents to buy assets denominated in currencies tat depreciate, as tey are relatively ceaper, but also to buy tose assets denominated in currencies wic are expected to appreciate, as tey ave a iger purcasing power in te future. Terefore, te nominal excange rate matters for quantity decisions. Secondly, fluctuations of nominal excange rates ave an impact on te net foreign assets position of a country, generating positive or negative valuation effects. It is known tat te effect of e.g. a currency depreciation will depend on te currency composition of a country s balance seet. Lane and Sambaug (2010) ave recently sown tat te balance seet of emerging countries is increasingly similar to te balance seet of te US and oter developed countries, i.e. foreign assets are mainly denominated in foreign currencies wile foreign liabilities are mainly denominated in te domestic currency. Figure 3 sows te depreciation of te dollar against te currencies of emerging market economies, especially since As a consequence, a dollar depreciation does imply positive valuation effects for te US and negative valuation effects for emerging economies, as observed in te data. Tis paper provides a teoretical framework suitable to analyze te joint beavior of nominal excange rates and portfolio coices in a general equilibrium setting, and it is also able to rationalize te above stylized facts. As 2 In oter open economy papers wit endogenous portfolio coice, money does not play any role and valuation effects are instead driven by capital gains and losses. For instance, see Pavlova and Rigobon (2007), Heatcote and Perri (2013), Devereux and Suterland (2010), Tille and Van Wincoop (2010). Tille (2008) makes a first step towards analyzing te wealt effects of excange rate fluctuations, but portfolios are exogenous in is analysis. 3 Wile Cina and Malaysia do not ave a fully flexible excange rate regime, controls on foreign excange markets are easing over time (see e.g. IMF, 2014) leading to considerable currency appreciations. 3

5 te focus of tis paper is modeling valuation effects due to nominal excange rate fluctuations, we abstract from oter sources of valuation effects. It is important to stress tat excange-rate driven valuation effects are known to be empirically important. Lane and Sambaug (2010) ave recently documented tat te wealt effects associated wit nominal excange rates fluctuations are substantial as tey account for a significant fraction of te overall valuation effects. Moreover, Gourincas and Rey (2007) ave sown tat a substantial part of te US cyclical external imbalances are eliminated via predictable movements in nominal excange rates. Our framework as two important ingredients: incomplete markets and imperfect substitutability of assets. Models wit complete markets generate very strong predictions, as it is always optimal not to adjust portfolios following a new realization of uncertainty 4. Te fact tat portfolio rebalancing is instead observed in te data is evidence tat tere is some degree of market incompleteness in te real world. In our model, markets are incomplete in te sense tat te young cannot insure against te realization of output tat tey receive wen tey are born. Moreover, te young lack of a complete set of assets to ensure against risk wen old. Wen markets are incomplete and assets are nominal, te equilibrium allocation can be indeterminate (Balasko and Cass, 1989; Geanakoplos and Mas-Colell, 1989; Polemarcakis, 1988) and tis poses particular callenges for applied work. However, if an asset in positive net supply suc as money is introduced, te price level can be pinned down and te indeterminacy problem can be avoided (Magill and Quinzii, 1992; Gottardi, 1996; Neumeyer, 1998). Our setting does not suffer from te indeterminacy problem as agents transfer wealt across periods using te two national currencies 5. As te asset structure is simple, we are able to obtain some analytical results and terefore to gain a very good understanding of agents portfolio coices as well as te beaviour of te nominal excange rate. Anoter advantage of our framework is 4 See Lucas (1982) and Judd et al. (2003) for a more general version of te Lucas asset pricing model. 5 If we introduced nominal bonds in zero net supply as well as te currencies, currencies and bonds would be perfect substitutes. As a consequence, te exact allocation of savings between money and bonds would not be determined. For instance, see Gottardi (1994). Terefore, we do not introduce nominal bonds as it would not add too muc to our analysis. 4

6 tat we can compute te global solution of te model, wic is known to be more accurate 6. In particular, we solve for te stationary equilibrium of te model, wic is defined as a time-invariant distribution (across state of nature) of nominal prices, excange rates, consumption and portfolio allocations. Te second important feature of our model is tat currencies are imperfect substitutes, as old agents can only buy te country-specific good wit te local currency. Since te seminal paper of Kareken and Wallace (1981), it is known tat te equilibrium excange rate and portfolios are not determinate in te absence of some form of legal restrictions in currency trading 7. Tis restriction, along wit te timing structure, guarantees tat te nominal excange rate is determinate. Te timing is structured as follows. Wen young, agents receive a statedependent endowment of te domestic good. Tey spend part of te domestic output for consumption of bot goods in te current period and te rest of teir income to buy a portfolio of currencies in view of consuming wen old. Wen old, agents are not allowed to readjust teir portfolio after uncertainty realizes so tat tey are restricted to use te domestic (foreign) currency accumulated in te previous period to buy te domestic (foreign) good. In our model, agents face genuine excange rate uncertainty, as no portfolio adjustments are possible in te old age. Te restriction tat agents must buy eac good wit te local currency is also a feature in Lucas (1982). However, money is not used to transfer wealt across periods in te cas-in-advance literature but to carry out excange witin a given period. As a consequence, te nominal excange rate is simply a function of current state and does not affect agents intertemporal decisions 8. In tis paper, te nominal excange rate is a forwardlooking variable wic depends on te expected purcasing power of te two currencies weigted by te old s marginal utilities. In sections 2 and 3, we present te model and define net foreign assets 6 In te open economy literature, recent work as proposed local solution metods to analyze incomplete markets model (e.g. Devereux et al. (2010) and Tille et al. (2010)). Wile tese metods can deal wit any state space, Rabitsc et al. (2014) sowed tat te global solution does not always coincide wit te local one. See also Coeurdacier and Rey (2012) for a critical assessment of local solution metods. 7 Sargent (1987) sowed tat te indeterminacy result olds more generally and is not due to te OLG structure in Kareken and Wallace (1981). 8 See also Svensson (1985) and Alvarez et al. (2009). 5

7 as well as valuation effects in te context of our framework. In section 4, we derive our main analytical result. We sow tat te country tat runs a current account surplus in equilibrium is te country wose sare of world GDP as increased over time. As te country is wealtier wit respect to te past, te young accumulate more foreign assets and old less foreign liabilities: at country level, tere is a positive cange in net foreign assets. We also point out tat te surplus country can be poorer tan te oter country in equilibrium. However, as te country s output grows relatively more tan te oter country s, its sare of world GDP increases. Terefore, our model rationalizes te deterioration of te US external position as te result of te rise of emerging market countries in te world economy. In section 5, we parametrize te model to illustrate te impact of te nominal excange rate on te net external position of te US and Cina. Our finding is tat te nominal excange rate stabilizes te net foreign assets position of eac country. Te intuition is very simple and can be explained as follows. Because tere is persistence in te stocastic process for output, te young expect tat prices will stay relatively low in te surplus country (Cina). As te currency of te surplus country as a iger purcasing power in expectation, te demand for te Cinese currency increases. To restore equilibrium, te currency as to appreciate. Terefore, te surplus (deficit) country experiences negative (positive) valuation effects, consistently wit te stylized facts presented above. Our result is also quite robust as it requires mild assumptions suc as persistence in te stocastic process for output and te elasticity of substitution between traded goods to be bigger tan one so tat we avoid episodes of immiserizing growt. Anoter important result is te quantitative relevance of valuation effects. Wile te model can explain more tan a tird of te US-Cina trade imbalances, valuation effects reduce te impact of te US current account deficit on te net foreign assets position by more tan a alf, consistently wit te data. 6

8 2 Te Model We consider a two-country pure excange overlapping-generations economy 9. In eac period, an agent wit a two-period lifetime is born in eac country. Terefore, two young and two old populate te world economy at eac t. Te young are born wit an endowment of te country-specific good l, wic is also te total output of te country. Output is denoted as y l (s) as it depends on te state of nature realized, were s = {1,..., S}. We will use te superscript l to indicate goods and currencies, wile we will refer to agents wit te subscript. We assume tat output follows a first-order stationary Markov process, were ρ(ss ) indicates te probability of transiting from state s to s. Agents gain utility from te consumption of bot goods altoug tey are only endowed wit te country-specific good, as in Lucas (1982). At time 0, te two governments issue fiat money and distribute it to te initial old. M l is te stock of money issued in country l. As te old ave no endowment, money is valued in equilibrium as agents would not be able to consume in teir second period of life oterwise. For simplicity, we assume tat monetary autorities are inactive after te first period. As we study te stationary equilibria of te model, prices will not depend on te istory of te socks but only on te current state of nature. Te timing is organized as follows. In te first period of life, young agents consume part of teir endowment of te domestic good and sell te rest to buy te foreign good and te two currencies for saving purposes. Terefore, tere is bot intra-generational and inter-generational trade in tis economy. Te two young engage in trade in order to consume te foreign good in te present period. Moreover, tey sell part of teir endowment to te current old in excange for money to finance future consumption. We now state te key assumptions of our model. Assumption 1 Te old can buy good l only wit currency M l. Assumption 2 Te old cannot adjust teir portfolio after te realisation of uncertainty. 9 Tis is wit no loss of generality. Te model can easily be extended to L countries. 7

9 Te first restriction tat we impose is tat agents need te local currency to buy te local good. However, Assumption 1 alone is not useful as agents could old all teir savings in te domestic currency and ten buy te foreign currency tat tey need to buy te foreign good wen old in te following period, after uncertainty is realized. In tis case, tere would be no actual portfolio coice to be made wen young and terefore tis scenario is not interesting for our purposes. Te addition of Assumption 2 guarantees tat te young old a portfolio of two currencies at te end of te period. Moreover, Assumption 2 is important as it introduces an element of excange rate risk in te agents decision problems, as uncertainty is realized after te currencies are cosen. Tese Assumptions are a crucial aspect of te model, as tey allow to pin down te equilibrium excange rate and countries portfolios. Currencies are not perfect substitutes in te sense tat eac of tem as a specific role, tat is to allow agents to consume a particular good. On te contrary, in a world of no legal restrictions in wic portfolios and excange rates are indeterminate, only total money oldings matter and not te currency composition (see Kareken and Wallace, 1981). Moreover, tese assumptions are also important for te existence of a stationary equilibrium in itself (see Eugeni, 2013). We assume te following functional form for te utility function: U (s) = l c l 1 1 (s) β ρ(ss ) s l c l 2 (ss ) > 0, 1 (1) Taking as given te vector of transition probabilities and te goods and currencies prices, agent born in state s cooses te consumption vectors and te portfolio of currencies tat maximise te above utility function subject to te following constraints: p 1 (s)c 1 1(s) + p 2 (s)e(s)c 2 1(s) w (s) = m 1 (s) e(s)m 2 (s) (2) p 1 (s )c 1 2(ss ) = m 1 (s) s (3) p 2 (s )c 2 2(ss ) = m 2 (s) s (4) Te budget constraint of te young is expressed in units of currency 1, wic is our numéraire. p l (s) is te nominal price in country l expressed in units of te domestic currency. e(s) is te price of currency 2 in units of currency 1 or te 8

10 nominal excange rate. Terefore, we say tat if e(s) rises ten currency 2 (1) appreciates (depreciates). w (s) is te wealt of agent in units of currency 1, wic is equal to te value of te domestic output: w 1 (s) = p 1 (s)y 1 (s) and w 2 (s) = p 2 (s)e(s)y 2 (s). Notice tat, wen agents are old, tey face two constraints in eac state of nature as tey use te currencies tat tey bougt in te previous period to purcase eac good in te local market wit te appropriate currency. Let λ (s) be te multiplier associated to te young s budget constraint, λ l (ss ) te multiplier of te constraint of te old related to good l in state s. Te necessary and sufficient conditions for a maximum are te following first-order conditions: c 1 1(s) : c 1 1(s) 1 = λ (s)p 1 (s) (5) c 2 1(s) : c 2 1(s) 1 = λ (s)p 2 (s)e(s) (6) c l 2(ss ) : β ρ(ss )c l 2(ss ) 1 = λ l (ss )p l (s ) l, s (7) m 1 (s) : λ (s) + s λ 1 (ss ) = 0 (8) m 2 (s) : λ (s)e(s) + s λ 2 (ss ) = 0 (9) λ (s) : p 1 (s)c 1 1(s) + p 2 (s)e(s)c 2 1(s) w (s) + + m 1 (s) + e(s)m 2 (s) = 0 (10) λ 1 (ss ) : p 1 (s )c 1 2(ss ) m 1 (s) = 0 s (11) λ 2 (ss ) : p 2 (s )c 2 2(ss ) m 2 (s) = 0 s (12) In te Appendix B, we sow ow to find te following closed-form solutions for te agents portfolios: m 1 (s) = m 2 (s) = β [ ] σ ρ(ss )p 1 (s ) 1 s A (s) [ β e(s)1 ρ(ss )p 2 (s ) 1 s A (s) w (s) (13) ] σ w (s) e(s) (14) 9

11 were A (s) p 1 (s) 1 + [p 2 (s)e(s)] 1 + β + β e(s)1 [ s ρ(ss )p 2 (s ) 1 [ ] σ s ρ(ss )p 1 (s ) 1 ] σ + Agent s demand functions can be derived using (13), (14) and te budget constraints (calculations of te demand functions wen young are provided in te Appendix): c 1 1(s) = p1 (s) A (s) w (s) l (15) c 2 1(s) = [p2 (s)e(s)] w (s) l (16) A (s) [ ] σ ρ(ss )p 1 (s ) 1 s c 1 2(ss ) = c 2 2(ss ) = β β e(s) w (s) p 1 (s ) A (s) [ ρ(ss )p 2 (s ) 1 s A (s) ] σ s (17) w (s) p 2 (s ) s (18) As preferences are omotetic, te demand for eac good is a linear function of wealt as we would expect. Wealt is premultiplied by a complicated nonlinear function of current and future prices as well as te current nominal excange rate. 2.1 Te role of te excange rate: partial equilibrium Using equations (8, 9) and (7), we can obtain te following expression for te nominal excange rate: e(s) = s ρ(ss ) c2 2 (ss ) 1 p 2 (s ) s ρ(ss ) c1 2 (ss ) 1 p 1 (s ) s = 1,..., S (19) In our model, te nominal excange rate is a forward-looking variable, as it depends on te expected marginal utilities derived from te consumption of te two goods as well as from te expected purcasing power of te two currencies. In fact, 1 p l (s ) gives ow many units of good l we can afford in state s per unit of currency l eld. In oter words, te nominal excange rate is 10

12 te ratio of te expected purcasing power of currency 2 over te expected purcasing power of currency 1, weigted by agent s marginal utilities. Te more a currency can buy tomorrow relatively to te oter currency, te iger will be its price today. In oter words, te nominal excange rate follows some sort of asset pricing equation, given tat te currencies are used to transfer wealt across periods. In te cas-in-advance literature, te spot excange rate simply depends on te current realization of te stocastic variables and not on expectations of future variables (see e.g. Lucas (1982)). Tis is due to te transaction role tat it is attributed to money, wic is only used to carry out excange in a given period. In te cas-in-advance literature, money is a veil and te excange rate does not ultimately affect te real allocation, wic is te same as in te barter economy. Let us now consider te role of te nominal excange rate in te portfolio decision of an agent. We combine te demand for te two currencies (13) and (14) to get: m 1 (s) m 2 (s) = e(s) [ ] σ ρ(ss )p 1 (s ) 1 s [ s ρ(ss )p 2 (s ) 1 ] σ (20) Te above equation sows tat te iger is te (relative) price of currency 2 (i.e. te nominal excange rate) te iger is te (relative) demand for currency 1. In a sense, te two currencies are substitutes, altoug not perfectly. Moreover, te iger is te expected purcasing power of currency 1, te iger is te relative demand for currency 1 as long as te degree of substitutability between te two goods is ig enoug ( > 1). Obviously, our arguments about te role of te nominal excange rate in te portfolio coice of te agents are of a partial equilibrium nature as we assume tat te nominal excange rate is fixed. Below, we will sow te importance of general equilibrium analysis as te nominal excange rate does act as a sock absorber in tis model. 11

13 3 Equilibrium Definition 1 A stationary equilibrium is a system of prices (p, e) R 2S ++ R S ++, consumption allocations and portfolios (c 1 (s), c 2 (ss ), m (s)) R 2 ++ R 2S ++ R 2 ++ for every = 1,..., H and s = 1,..., S suc tat: (i) agent maximizes is utility function subject to te budget constraints in every s; (ii) c l 1(s) + c l 2(s s) = y l (s) s, s and l (iii) ml (s) = M l s, l were c l 1(s) cl 1 (s) and cl 2(s s) cl 2 (s s). Notice tat we ave 3S endogenous variables, i.e. 2S nominal price levels as well as S nominal excange rates. On te oter and, we ave 2S 2 + 2S equations. Goods markets ave to clear for any pair of s and s, as te consumption of te old does depend on te previous state as well as on te current state. Moreover, 2S monetary equations ave to clear. First, te system can be reduced by applying Walras Law. In particular, S 2 equations can be made redundant. If we sum across agents te budget constraints of te young and te old and combine tem, we get: p 1 (s)[c 1 1(s) + c 1 2(s s) y 1 (s)] + p 2 (s)e(s)[c 2 1(s) + c 2 2(s s) y 2 (s)] = 0 s, s Terefore, if for every pair of (s, s) te market for good 1 clears, te market for good 2 clears automatically. However, we still ave S 2 S equations more tan te number of endogenous variables. Tis is te issue raised by Spear (1985), wo proved tat a steady state equilibrium does not generically exist in a stocastic OLG economy wit money and multiple goods. Heuristically speaking, te non existence result is due to te fact tat tere are too many equations wit respect to te number of unknowns It is important to stress tat is generic result does not rule out te possibility tat a stationary equilibrium may exist under some restrictions. For example, e sowed tat economies wit additively time-separable utility functions and one type of agent per generation do ave a stationary equilibrium. In an open economy setting, we ave eterogenous agents terefore existence of equilibrium is not guaranteed. 12

14 Next, we sow tat Assumptions 1 and 2 imply tat S 2 S equations can be made redundant. As we end up wit a system aving te same number of equations and unknowns (3LS), we can get around te non-existence problem. Proposition 1 Under Assumptions 1 and 2, furter S 2 S equations are redundant. Proof. Given Walras Law, suppose tat te independent equations in te goods markets are tose for good 1. Sum across agents te budget constraints of te old for good 1 in state s: p 1 (s)c 1 2(s s) = M 1 It is easy to see tat te aggregate consumption of te old does not depend on te previous state (te state realized wen born) as aggregate real money balances only depend on te current state: c 1 2(s s) = M 1 p 1 (s) c 1 2(s s) = c 1 2(s) Suppose tat te S equations for wic s = s clear: c 1 1(s) + c 1 2(ss) = y 1 (s) Given tat te aggregate consumption of te old does not depend on te past, te oter S 2 S clear automatically. 3.1 Definitions Before we discuss te solution metod, we introduce some key definitions and make a couple of useful remarks Portfolio rebalancing and trade imbalances: a unified view To start wit, let us define te balance of trade of country 1 in state s 11 : tb 1 (s s) p 1 (s)[y 1 (s) c 1 11(s) + c 1 21(s s)] p 2 (s)e(s)[c 2 11(s) + c 2 21(s s)] Notice tat te sign of te balance of trade does depend on te coices tat te young make in te current period, but also on te coices made by te 11 Obviously, by Walras Law we ave tat tb 2 (s s) = tb 1 (s s). 13

15 current old in te previous period. Substituting te budget constraints into te trade balance equation, it sould be immediate tat te above definition can be rewritten as: tb 1 (s s) = m 1 1(s) m 1 1(s ) + e(s)[m 2 1(s) m 2 1(s )] (21) Tis leads us to te following two remarks: Remark 1 If portfolios are constant across states, ten trade is always balanced. Remark 2 If today s realized state is te same as yesterday s, ten trade is balanced. Equation (21) sows tat tere is a close relationsip between agents beaviour in te assets markets and te goods markets. If, for some reason, tere is no portfolio rebalancing in equilibrium, ten te balance of trade is always in equilibrium. Our framework is very different from te cas-in-advance literature wit complete markets. In Lucas (1982), trade imbalances arise and yet portfolio rebalancing is never a possibility wit te implication tat te cange in te net foreign assets position of a country is always zero. Te second remark is related to Polemarcakis and Salto s result for deterministic OLG economies (2002). In a one-currency economy, tey sowed tat te balance of trade is in equilibrium at te monetary steady state. In tis paper, te monetary steady state is stocastic and trade imbalances are possible wenever s s. It is reasonable to expect tat te set of parameters of te economy under wic portfolios are state invariant as a very small measure. Constant portfolios implies tat te consumption of an old person does not depend on te state in wic e is born 12. If output is a random variable, agents born in different states of nature are likely to ave different wealt and terefore different demands for te goods. For te consumption of te old to be independent from te state wen born, te demand function must be very special. In Appendix C, we sow tat tis beaviour occurs wen utility functions are 12 In te previous section, we sowed tat te aggregate consumption of te old does not depend on te past, but tis does not imply tat te individual consumption is independent of te past as well. 14

16 logaritmic. Under logaritmic utility, te demand functions are extremely simple and te model is fully tractable. However, tis comes at te cost tat agents beaviour is too simplistic and terefore uninteresting to our purposes. On te oter and, under isoelastic utility functions, constant portfolios will only occur for degenerate values of te endowments but te solution of te model requires a numerical approac. Our findings for te log case are related to Cass and Pavlova (2004), wo ave sown tat logaritmic utility yields peculiar results wen markets are incomplete. In a two-period economy wit N Lucas trees, te matrix of portfolio returns is degenerate and tat te equilibrium allocation is Pareto optimal despite te incompleteness of te markets. Pavlova and Rigobon (2007) extended te model to te infinite-orizon but output socks cannot generate time-varying portfolios. On te oter and, tere is portfolio rebalancing wit demand socks. In our logaritmic version of te model, we could acieve a similar result if we allowed for state-dependent discount factors. Te innovation of tis paper is tat we are able to explain portfolio dynamics as a consequence of output innovations rater tan demand socks, wic is easier to verify in te data. We now define our main variables of interest, i.e. net foreign assets and valuation effects Net foreign assets and valuation effects In tis section, we explore te relationsip between net foreign assets, te balance of trade and valuation effects. Consider te balance of trade of country 1 in state s s, as defined in te previous section (equation (21)): tb 1 (s s) = m 1 1(s) m 1 1(s ) + e(s)[m 2 1(s) m 2 1(s )] (22) Using te fact tat m 1 1(s) + m 1 2(s) = M 1 for every s, we can rewrite te first two terms on te rigt and side as follows: tb 1 (s s) = m 1 2(s ) }{{} current value F L 1 (s ) m 1 }{{ 2(s) + e(s)m 2 }}{{ 1(s) } F L 1 (s) F A 1 (s) e(s)m 2 1(s ) }{{} current value F A 1 (s ) (23) F A(s) are oldings of foreign assets in state s and F L(s) are foreign oldings of te domestic currency, i.e. foreign liabilities. Now, define net foreign assets 15

17 as NF A(s) F A(s) F L(s) and rewrite te above as follows: NF A 1 (s) = current value NF A 1 (s ) + tb 1 (s s) (24) Equation (24) states tat te end-of-period net foreign assets in country 1 is equal to te current value of te net foreign assets accumulated in te previous period and te balance of trade 13. Te next step is to rewrite equation (23) in order to igligt valuation effects. In te rigt and side, sum and subtract te foreign assets of country 1 in te previous state (e(s )m 2 1(s )) and use te definition of net foreign assets to obtain: tb 1 (s s) = NF A 1 (s) NF A 1 (s ) + [e(s ) e(s)]m 2 1(s ) (25) Tis equation can be rewritten as: were NF A 1 (s s) = tb 1 (s s) + r(s s)e(s )m 2 1(s ) }{{} valuation effects r(s s) = R(s s) 1 e(s) e(s ) 1 (26) Terefore, te cange in te net foreign assets position of country 1 will be determined by te beaviour of te balance of trade and te valuation effects, were r(s s) is te return on te foreign assets accumulated in te previous period. In tis model, valuation effects are entirely determined by excange rate movements 14. If foreign currencies ave appreciated wit respect to te past (i.e. e(s) > e(s )), ten te return on te foreign assets accumulated in te previous period is positive and terefore we say tat te country experiences positive valuation effects 15. Conversely, a country experiences negative valuation effects if foreign currencies ave depreciated. In tis framework, currencies are te only assets available and terefore our setting can capture a scenario in wic te majority of domestic assets 13 Tis equation is equivalent to equation (1) in Gourincas and Rey (2007, footnote 2). 14 Moreover, tere is no net income from abroad and terefore te trade balance position is equivalent to te current account position. 15 As te price of te foreign asset is defined in units of te domestic asset, i.e. te excange rate, te above rate of return as to be interpreted as te return of foreign assets relatively to te return on foreign liabilities. 16

18 are denominated in te foreign currency wile domestic liabilities are denominated in te domestic currency. As from te findings of Lane and Sambaug (2010), tis is entirely consistent wit te currency denomination of te foreign assets and liabilities of te US wile it would be less realistic wen applied to developing countries. Lane and Sambaug (2010) also find tat emerging market countries are becoming more similar to advanced economies as tey issue less foreign-currency denominated debt tan developing countries and are accumulating foreign-currency denominated assets in te form of foreign excange reserves. Te consensus in te empirical literature is tat valuation effects are very important in explaining te dynamics of net foreign assets of te US and oter countries (see e.g. Gourincas and Rey (2015)). Lane and Sambaug (2010) ave also sown tat te valuation effects stemming from nominal excange rate canges are an important driver of te overall valuation effects 16. Wile most of te teoretical literature as focused on oter sources of valuation effects (e.g. Devereux and Suterland (2010)), te novelty of tis paper is tat it provides a teoretical framework in wic excange rates related-valuation effects can arise wile countries adjust teir portfolios over time because of te market incompleteness. In section 5, we will discuss te interaction between excange rates and net foreign assets, as well as te quantitative importance of valuation effects. 4 Portfolio oldings, te distribution of world GDP and te role of te nominal excange rate From now onwards, we focus on te case in wic preferences are identical across countries: = σ and β = β. Plugging te demand functions for te goods and te currencies into te equilibrium conditions, we get te following 16 Tey also observed tat valuation effects associated to excange rate fluctuations tend to move in te same direction as valuation effects associated to capital gains and losses. 17

19 system of 3S equations, wic will be solved numerically: p 2 (s)e(s) p 1 (s) M 1 = e(s)m 2 = = ω1 (s) ω 2 (s) [ [p 2 (s)e(s)] 1 σ + β σ e(s) 1 σ s ρ(ss )p 2 (s ) 1 σ σ [ β σ s ρ(ss )p 1 (s ) 1 σ σ [ p 1 (s) 1 σ + β σ s ρ(ss )p 1 (s ) 1 σ σ A(s) ] σ [ β σ e(s) 1 σ s ρ(ss )p 2 (s ) 1 σ σ A(s) ] σ ] σ (27) w (s) (28) ] σ w (s) (29) Te following proposition establises tat tere is a strong relationsip between te distribution of world GDP across countries, portfolio oldings and trade imbalances wen preferences are identical across countries. Proposition 2 If = σ and β = β: (i) country s portfolio oldings at te end of te period depends on its current sare of world GDP; (ii) if country as a iger (lower) sare of world GDP wit respect to te past, it runs a trade surplus (deficit). Proof. (i) Wen = σ and β = β, te demand of agent for te two currencies as te following form (see equations (13) and (14)): m l (s) = k l (s)w (s) were k l (s) is identical across agents. Summing across, we get te following equation: M l = k l (s) w (s) Dividing te first equation by te second equation, we obtain te desired result: were w(s) = w (s) 17. m l (s) M l = w (s) w(s) l = 1, 2 (ii) Suppose tat today s realized state is s and yesterday s state was s. By ypotesis, w (s) > w (s ) w(s) w(s ). Te first part of te proof implies tat: m l (s) M l > ml (s ) M l l = 1, 2 17 World GDP is defined as te sum of countries nominal GDP expressed in units of te numéraire currency. 18

20 Finally, equation (21) implies country as a trade surplus in state s. Te oter case can be worked out in a similar way. Te end-of-te-period wealt of te young is equal to teir total money oldings, as te two currencies are te means by wic tey can save and terefore finance future consumption. Terefore, Proposition 2 suggests tat te distribution of world GDP is te same as te distribution of world wealt at te end of te period 18. If te distribution of world GDP canges across states of nature, ten te distribution of wealt will cange as well and portfolio rebalancing occurs over time. As we discussed above, tis is a likely outcome of te model as markets are incomplete. As a matter of fact, portfolios are constant across states of nature in complete markets models precisely because wealt is identical across agents and terefore agents do not adjust teir portfolio oldings following new socks. Proposition 2 seds furter ligt on te beavior of te trade balance. If a country is in surplus, it is because it is relatively wealtier wit respect to te past. Tis does not rule out te possibility tat suc country is poorer tan te oter country in all states of nature 19. Terefore, our model offers a novel explanation of te fact tat emerging countries run trade surpluses against te United States: global imbalances simply reflect te rise of emerging countries in te world economy. If a country is classified as emerging, ten its sare of world GDP sould ave increased over time. Using a sample of 146 countries, we find tat te sare of world GDP of all emerging countries except Argentina as increased over te past 20 years 20. As expected, Cina is te emerging country wose sare of world GDP as increased te most, as it as gained 7.81% points over te past 20 years. On te oter and, te sare of world GDP of te US as fallen by 3.54%. At tis stage, te model does not say weter an increase in 18 As domestic GDP is equal to domestic income, te distribution of world GDP is equal to te distribution of world income. 19 On te oter and, te poor country is always in trade deficit in cas-in-advance models under isoelastic utility (see Eugeni (2013) for a derivation). Te reason is tat te sign of te trade balance does only depend on te current sock, and not on te past. 20 We calculate te cange in te sare of world GDP as follows: sare = GDP,1990 GDP.,1990 GDP,2010 GDP,2010 GDP is taken from te IMF World Economic Outlook database and it is measured at current national prices converted in US dollars. We use te IMF classification of emerging countries. 19

21 wealt is due to eiter output growt or canges in prices and our calculation does not distinguis between te two accordingly. In te next section, we sow tat because Cina s real GDP as grown more tan te US, ten it is wealtier wit respect to te past and terefore it runs a trade surplus in equilibrium 21. Finally, te allocation of savings across currencies deserves some comment. Agents do not ave very sopisticated portfolio strategies according to Proposition 2. In fact, agents old te same sare of bot money stocks 22. Tis is due to te sock absorbing role of te nominal excange rate. If te elasticities of substitution are allowed to differ across countries, te distribution of money oldings is more difficult to caracterize and agents could ave a preference for different currencies in different states. 4.1 Excange rate determination and te role of money Combining equations (28) and (29), we obtain te following expression for te excange rate: ( M 1 e(s) = M 2 ) 1 σ s ρ(ss )p 2 (s ) 1 σ σ s ρ(ss )p 1 (s ) 1 σ σ s = 1,..., S (30) Altoug te above expression is not a closed-form solution, we can gain some intuition about te role of te nominal excange rate and te importance of general equilibrium analysis as opposed to partial equilibrium analysis. Recall te equation tat linked te portfolio coice of te agents to te nominal excange rate (equation (20)). In a partial equilibrium setting, an increase in te nominal price levels in country 2 means tat te purcasing power of currency 2 is lower and terefore te relative demand for currency 2 falls (provided tat te elasticity of substitution is bigger tan 1). As te nominal excange rate is endogenous, it will beave in suc a way to counteract expectations on price movements. In particular, currency 2 appreciates if te price of good 2 increases as equation (30) sows. In fact, if we combine equations (20) and (30), we obtain our previous result tat eac agent olds te money stocks of 21 Tis requires tat te elasticity of substitution between traded goods is greater tan 1, wic is supported by empirical evidence as we explain in te next section. 22 Notice tat we do not impose any ome bias in te preferences, terefore agents old te two currencies in te same sare as tey like te two goods equally. 20

22 bot countries in equal sares. In te numerical section, tese arguments will be furter clarified. It is also interesting to note tat, if te stocastic process is i.i.d., te excange rate is constant as te probabilities tat agents attac to future events are independent from te state in wic tey are born. We conclude tis section wit a discussion of te role of money in our economy. Since te old face separate budget constraints for eac good, ten te aggregate consumption of te old of good l is equal to te real money balances of currency l. As a consequence, we can write te following expressions for te nominal price levels in te two countries using te goods markets clearing conditions: p 1 (s) = p 2 (s) = M 1 ω 1 (s) c 1 1(s) M 2 ω 2 (s) c 2 1(s) Tis also implies tat te level of te money stocks does not matter for te real allocation. Suppose tat te money stock of country 1 doubles. Given te first of te above two equations, te nominal price level will be doubled as well. In oter words, prices are omogenous of degree 1 wit respect to te domestic money supply. Te nominal excange rate doubles as well (see equation (30)) as currency 1 becomes ceaper. Terefore, te wealt of bot agents is doubled. It can be cecked from te demand functions tat tis cange in prices does not affect te consumption of bot agents. Altoug te level of te money stocks does not matter for te real allocations, money is not neutral in our model. If we removed money from te economy, te equilibrium allocation would be very different. Wile te young would still be able to engage in barter (intragenerational trade), te old would not be able to consume anyting. Even if te old ad an endowment, suc as a pension, e would not be able to trade it because of te multiple budget constraints, so e would limit is consumption to te domestic good. Money is important in our framework as in standard Samuelsonian OLG economies, but in an even stronger sense because of te multiple budget constraints wic prevent barter among te old. 21

23 5 Te US external position and valuation effects Te aim of tis section is to gain furter insigts on te beaviour of te nominal excange rate, te balance of trade and te net foreign asset positions. For tis purpose, we parametrize our two-country model. In tis section, we will refer to country 1 as te United States and country 2 as Cina. Te reason wy we coose te United States and Cina for our numerical exercise is tat Cina is one of te main creditors of te US and te US deficit against Cina account for a significant fraction of te overall US current account deficit (e.g. Eugeni, 2015). Moreover, te US-Cina imbalances are persistent and our two-period OLG model is especially suitable to capture low-frequency trends in international financial markets. In our setting, foreign assets is a country s foreign currency oldings, wile foreign liabilities is te domestic currency eld abroad. Terefore, our model is able to capture te currency composition of te US and Cina s balance seet, for wic foreign assets are denominated in foreign currencies wile foreign liabilities are denominated in te domestic currency. According to te Lane and Sambaug (2010) database, 64% of US foreign assets were denominated in foreign currencies wile 93% of US foreign liabilities were denominated in dollars in As far as Cina is concerned, 100% of te Cinese foreign assets are denominated in foreign currency, 70% of wic are dollar denominated. Tis is consistent wit te fact tat Cina is one of te US main lenders. On te oter and, 63% of Cinese liabilities were issued in renmimbi in Tis reflects a general trend wic sees emerging market economies increasingly able to borrow in teir domestic currency (Lane and Sambaug, 2010) 24. Terefore, a depreciation of te dollar in our setting would imply a positive wealt effect for te US and a negative wealt effect for Cina. Altoug te Cinese currency as considerably appreciated over te past 10 years (Figure 23 Te first figure reflects te fact tat many developing economies do still borrow in US dollars as tey are unable to issue debt in domestic currency-denominated assets. 24 Anoter signal of te increased ability of emerging countries to borrow in teir own currency is tat a tird of te foreign currency-denominated US foreign assets are denominated in currencies oter tan te Euro, te Yen, te Pound and te Swiss Franc. Terefore, tese are assets eld in emerging economies and denominated in local currencies. 22

24 3), te Cinese excange rate is not freely floating terefore it is reasonable to expect tat te model will tend to over predict valuation effects. It is also important to stress tat our model can only capture low-frequency movements of te excange rate and te balance of trade and does not aim at explaining ig-frequency movements (or lack of) in foreign excange markets. Since agents live for two periods in our OLG economy, we assume tat a period is 20-years long. As we wis to explain te deterioration of te US external position against emerging economies over te past 20 years, we adopt te following strategy. We consider an economy wit two states of nature, were state 1 corresponds to te state of te world economy in 1990 wile state 2 is te state of te world economy in Terefore, we will focus on wat appens in te world economy in te transition from state 1 to state We take te real GDP per capita of te United States and Cina in 1990 and 2010 to parametrize output in te two states 26. y 1 (1) = 31, 432 y 1 (2) = 41, 627 y 2 (1) = 2, 005 y 2 (2) = 7, 693 Notice tat wile US output as grown by 32% over te 20-years period, Cina as grown by 384%. Altoug Cina as experienced iger growt over time, te real GDP per capita level is still muc lower tan te US. We coose te rest of te parameter values as follows: M 1 = M 2 = M = 1 σ 1 = σ 2 = σ = 4 β 1 = β 2 = β = 1 ρ(ss) = 0.9 We normalize bot money supplies to 1 since te level of te money supply does not affect te real allocation. Te level of te trade balance does cange in 25 Tis is not to argue tat te world economy can only be in a state tat matces te situation of te world economy eiter of te 1990 or te However, a two-states example is enoug to illustrate our arguments wile adding more states of nature would not provide neiter more information nor intuition. 26 We take te output-side real GDP at cained PPPs and te population from te Penn World Tables

25 te money supplies, as portfolios and te nominal excange rates are affected (equation (21)). However, te size of te money stocks are irrelevant wen we normalize te trade balance as a percentage of domestic GDP. Te same is true for valuation effects as a percentage of GDP. Te elasticity of substitution is assumed to be greater tan 1 as suc parametrization rules out episodes of immiserizing growt. In fact, wen 0 < σ < 1, a country tat experiences a positive sock (everyting else equal) is poorer in value terms since te price of te domestic good falls too muc. In oter words, te terms of trade effect dominates canges in output 27. Empirical work based on low-frequency data found elasticities between 4 and 15, wile estimates at iger frequency suggest tat te elasticity is muc lower and in te range of 0.2 to 3.5 (see Rul (2008)). Our parametrization is more in line wit te low-frequency literature. Below, we sow tat our results are robust to different parameter values for te elasticity of substitution. Te discount factor is set equal to 1 and identical across countries. We ave also assumed tat te Markov process is persistent. In te robustness analysis, we will solve te model for different values of ρ(ss). 5.1 Numerical results We report te equilibrium prices in Appendix D. We can compute relative prices, expressed in te numéraire currency, are follows: p(s) p2 (s)e(s). Tere- p 1 (s) fore: p(1) = p(2) = Country 1 (te US) experiences an improvement of te terms of trade in te transition from state 1 to state 2, as te price of imports fall relatively to te price of exports. Tis is due to a supply effect, as output in country 2 (Cina) as increased relatively more tan output in country 1. At te same time, currency 1 depreciates (see te Appendix). Te intuition beind tis can be explained as follow. Wile bot nominal prices fall in te transition from state 27 A similar issue arises in te simplest possible setting, i.e. a static GE model wit isoelastic utility and corner endowments. See also Cole and Obstfeld (1991) and Lucas (1982). 24

26 1 to state 2, as supply increases in bot countries, te nominal price of country 2 falls at a iger rate. Terefore, currency 2 gives te current old a iger real return. Because te sock is persistent, ten te young expect te current state to realize tomorrow wit a very ig probability. As a consequence, agents would ave an incentive to buy more of currency 2. But since te money supply is fixed, ten currency 2 (yuan) as to appreciate in equilibrium. On te left (rigt) column, we report te money oldings of agents born in country 1 (2) 28 : m 1 1(1) = m 1 2(1) = m 2 1(1) = m 2 2(1) = m 1 1(2) = m 1 2(2) = m 2 1(2) = m 2 2(2) = As Cina is poorer in bot states, ten te country olds a lower sare of bot currencies. However, tis sare increases in te transition from state 1 to state 2. As te country experiences iger growt, its sare of world GDP increases and te young accumulate more assets (see Proposition 2). Using equations (21) and (26), we can compute te balance of trade, te cange in te net foreign assets position and valuation effects. We report results for country 1, terefore we express te above variables as a percentage of te country s GDP, were GDP 1 (s) = p 1 (s)y 1 (s). As we explained above, Table 1: Country 1. Numerical results NF A 1(12) GDP 1(2) % tb 1(12) GDP 1(2) % V AL1(12) GDP 1(2) % 1.87% 6% 4.13% NF A 1(21) GDP 1(1) % tb 1(21) GDP 1(1) % V AL1(21) GDP 1(1) % 2% 5.91% 3.91% we are interested in te transition from state 1 to state 2, terefore we focus on te first two lines. As te US experiences lower growt, it runs a trade deficit of 6% of domestic GDP. Te reason beind tis is tat te country accumulates less foreign assets 28 Since te money supplies are normalized to 1, we can interpret te numbers below as te proportion of te two money stocks eld in te two countries in eac state. 25