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

Transcription

1 Nominal Exchange Rates and Net Foreign Assets Dynamics: the Stabilization Role of Valuation Effects Sara Eugeni Durham University Business School First draft: May 2013 April 6, 2016 Abstract Recent empirical studies have highlighted that wealth effects associated with fluctuations of nominal exchange rates (valuation effects) are one of the key components that drive the net foreign assets position of a country. We propose a theory of nominal exchange rate determination and endogenous portfolio choice in line with this evidence. Our overlapping-generations model is able to rationalize the dynamics of the US external position over the past 20 years, which has been characterized by persistent current account deficits as well as stabilizing valuation effects, as a consequence of the increase in emerging market countries share of world GDP. Keywords: nominal exchange rate determination, valuation effects, endogenous portfolio choice, incomplete markets, overlapping-generations economies JEL classification: F31, F32, F36, F41, G11, G15, D52 1 Introduction One of the most relevant developments that characterize the global economy of the recent decades is the rising importance of the so called valuation channel in accounting for the dynamics of net foreign assets of many countries 1. address: sara.eugeni@durham.ac.uk. An earlier version of this paper was circulated under the title Portfolio choice and nominal exchange rate determination in a stochastic OLG model. This paper is a development of the third chapter of my PhD thesis. I thank my supervisor Subir Chattopadhyay for his advice, Michael Reiter for his very useful discussion and Mauro Bambi, Luis Catão, Andrea Ferrero, Mark Guzman, Roberto Pancrazi, Herakles Polemarchakis, Katrin Rabitsch, Neil Rankin, Steve Spear, Gabriel Talmain, Harald Uhlig, seminar participants at Bologna, Durham, Reading, Warwick, the Royal Economic Society conference 2016, the CEPR conference Macro-Financial Linkages and Current Account Imbalances held in Vienna in 2015 and the North American Summer Meeting of the Econometric Society 2014 for their comments and insights. 1 See Gourinchas and Rey (2015) for a recent survey of the literature. 1

2 Since the early 1990s, cross-border holdings of assets and liabilities have substantially increased and the traditional method of computing the net foreign assets position of a country, which relied on the cumulation of current account balances over time, has proved to be inaccurate 2. In fact, the balance of payments data do not record changes in the value of foreign assets and liabilities which can arise due to fluctuations of nominal exchange rates and asset prices. For instance, Figure 1 shows the discrepancy between the cumulated current account balances and the net foreign assets position of the United States. According to the former measure, the net foreign assets position of the United States amounted to almost 60% of GDP in However, direct estimates of net foreign assets and liabilities suggest that the net external position was much lower and equal to around 20% of GDP. This shows the significance of the valuation channel for the US: the US have experienced a substantial wealth transfer from the rest of the world over the past 20 years as the value of their foreign assets has risen relatively to the value of their foreign liabilities. The importance of this channel is not specific to the US: it is interesting to observe that emerging countries in East Asia have faced exactly the opposite situation (Figure 2). While their net external positions have considerably improved over the past decades because of current account surpluses, they have experienced negative valuation effects 3. For all the above countries, valuation effects seem to have had a stabilizing effect on the net foreign assets position. The hypothesis of this paper, which is supported by the empirical literature, is that part of these valuation effects can be attributed to fluctuations of nominal exchange rates. Lane and Shambaugh (2010) have recently documented that the wealth effects associated with nominal exchange rates fluctuations are substantial as they account for a significant fraction of the overall valuation effects. Moreover, Gourinchas and Rey (2007) have shown that a significant part of the US cyclical external imbalances are eliminated via predictable movements in nominal exchange rates. Figure 3 shows the depreciation of the dollar against the currencies of East Asian, emerging economies, especially since Lane and Shambaugh (2010) have found that the currency composition of emerging countries balance sheet is increasingly similar to the US and other developed countries, in the sense that foreign assets are mainly 2 Lane and Milesi-Ferretti (2001, 2007) constructed estimates of foreign assets and liabilities for 145 countries for the period and were among the first to notice the discrepancy between the stock measures and the balance of payments data. 3 Gourinchas and Rey (2015) make similar observations for other emerging countries. 4 While China and Malaysia do not have a fully flexible exchange rate regime, controls on foreign exchange markets have eased over time (see e.g. IMF, 2014) leading to considerable currency appreciations. 2

3 denominated in foreign currencies while foreign liabilities are mainly denominated in the domestic currency. As a consequence, a dollar depreciation does imply positive valuation effects for the US and negative valuation effects for emerging economies, as observed in the data. Figure 1: Net foreign assets position and cumulated current accounts of the United States as a percentage of GDP, Year Cumulated current account Net foreign assets Source: Lane and Milesi-Ferretti's database (2007). This paper proposes a theory of nominal exchange rate determination which sheds some light on the role of nominal exchange rate in countries portfolio choices as well as its impact on the dynamics of net external positions through valuation effects. To the best of our knowledge, this is the first paper that models nominal exchange rate-driven valuation effects in a general equilibrium framework 5. Setup and main results In our two-country model, overlapping generations of agents receive an endowment of the country-specific good when born and they gain utility from consuming both goods in the two periods of life. Markets are incomplete in the sense that the young cannot insure against the realization of output that they receive, as well as they lack of a complete set of assets to ensure against the risk that they face when old. The only way in 5 See Gourinchas and Rey (2015) for a recent account of the theoretical challenges in this field. In other open economy papers with endogenous portfolio choice, money does not play any role and valuation effects are instead driven by capital gains and losses. For instance, see Pavlova and Rigobon (2007, 2010), Heathcote and Perri (2013), Nguyen (2011), Devereux and Sutherland (2010), Tille and Van Wincoop (2010). Tille (2008) makes a first step towards analyzing the wealth effects of exchange rate fluctuations, but portfolios are exogenous in his analysis. As the focus of this paper is modeling valuation effects due to nominal exchange rate fluctuations, we abstract from other sources of valuation effects. 3

4 Figure 2: Net foreign assets position and cumulated current accounts of selected East Asian countries as a percentage of GDP China Malaysia Philippines Thailand Source: Lane and Milesi-Ferretti database (2007). Figure 3: The depreciation of the US dollar against the currencies of selected East Asian countries China Thailand Philippines Malaysia Source: Lane and Milesi-Ferretti database (2007). Updated to

5 which agents can transfer wealth across periods is to buy a portfolio of currencies. Since the two currencies are stores of value, the nominal exchange rate (and portfolios) would be indeterminate in the absence of some form of legal restrictions in currency trading as pointed out by Kareken and Wallace (1981) 6. In this paper, the nominal exchange rate is determined as a result of two key ingredients: firstly, the currencies are not substitutable as each currency can only buy the country-specific good; secondly, the old cannot adjust their portfolio of currencies after uncertainty realizes but they directly use the currencies accumulated in the previous period to buy the two goods in the respective markets. The first ingredient is a feature that we have in common with the cash-in-advance literature started by Lucas (1982) 7. Differently from the cash-in-advance literature, money has the double role of medium of exchange and store of value in this paper and the presence of a friction in the currency markets is necessary to guarantee determinacy of the nominal exchange rate. In relation to the dynamics of net foreign assets, the nominal exchange rate operates through two different, but related channels. Firstly, it has an impact on the portfolio decisions of the agents. In fact, while it is rational for agents to buy currencies that depreciate, as they are relatively cheaper, they also have an incentive to buy those currencies which are expected to appreciate, as they have a higher purchasing power in the future. The inability of the old to adjust their portfolio of currencies after uncertainty realizes introduces a further element of risk in the portfolio decision that agents make when young. Secondly, fluctuations of the nominal exchange rate have an impact on the net foreign assets position of a country, in generating positive or negative valuation effects. This paper provides an interpretation of the stylized facts presented in Figures 1-3, as it explains both the deterioration of the US net external position against emerging market economies and the positive valuation effects that they experienced over the past twenty years as the consequence of emerging countries increase in their share of world GDP. In fact, we show that the country that runs a current account surplus in equilibrium is the country whose share of world GDP has increased over time. In the numerical exercises, we will focus on China as it is the country whose share of world GDP has increased the most since the early 1990s, because of its rapid output growth. As the current generation in China is wealthier with respect to the previous generation, the 6 Manuelli and Peck (1990) extended Kareken et al. s result to a stochastic framework. Sargent (1987) showed that the indeterminacy result holds more generally and is not due to the OLG structure. 7 See also Svensson (1985) and Alvarez et al. (2009). 5

6 young Chinese accumulate more domestic as well as foreign assets than in the past and this causes an improvement of the net foreign assets position of the country, consistently with Figure 2. The overlapping generations structure is crucial to obtain this result, as the trade balance is driven by the consumption of the old as well as the current generation. In a framework with infinitely-lived agents, Engel and Rogers (2006) emphasize the role of the expected share of world output as opposed to the past share of world GDP. In particular, they show that a country borrows from the rest of the world if the country is expected to grow in the future. Their mechanism is different because the trade balance is entirely driven by the desire of consumption smoothing of the representative agent. In a two-period overlapping generations economy, the young are always net savers whenever their income when old is low as compared to their income when young. However, at aggregate level, the country could be either borrowing or lending from abroad as part of the aggregate consumption also comes from the old people in the economy. In this sense, it is the comparison between the current and the past distribution of wealth that matters when it comes to determining the net position of the country. As Engel et al. (2006) recognize, their model is able to capture the US position vis-à-vis other industrialized countries but it falls short of explaining the deterioration of the US external position against East Asian economies. In fact, rapidly growing countries are net borrowers in their framework, which is at odds with the East Asian experience. In our model, while the United States net foreign assets position against China deteriorates, the dollar depreciates and this generates positive valuation effects for the US. The mechanism is very intuitive and can be explained as follows. Suppose that the young living both in China and the US expect that the world economy will remain in the current state of nature with high probability. As the Chinese goods are cheaper than in the past, because of a supply-side effect, the demand for the Chinese currency increases because it is expected that it will have a higher purchasing power in the future. As the currency appreciates in equilibrium, the value of the foreign currency held by US residents increases relatively to the value of the US currency held abroad. Therefore, the surplus (deficit) country experiences negative (positive) valuation effects, consistently with the stylized facts presented above for the US and East Asian economies. The result that valuation effects are stabilizing is obtained under the mild condition that the probability that next period s output is the same as the current period is no less than half, which means 6

7 that the stochastic process is either i.i.d. or there is some degree of output persistence. The stabilizing nature of valuation effects is also in line with the empirical findings of Lane et al. (2002) and Devereux et al. (2010). The numerical results show that valuation effects are quantitatively relevant in our model. While we can explain almost a third of the US-China trade imbalances, valuation effects reduce the impact of the US current account deficit on the net foreign assets position by more than a half, consistently with the data. This result is also not particularly sensitive to our chosen parametrization of the model. Another main novelty is that the model can generate predictable valuation effects. Gourinchas and Rey (2007) have shown that the adjustment of the net external position of a country can occur via future changes in asset returns (i.e. future valuation effects) as well as future changes in the trade balance positions. Therefore, they stressed that it is important to come up with a model that is able to generate expected valuation effects in order to be able to understand the relative importance of the different channels of adjustment (Gourinchas and Rey, 2015). In the context of our numerical analysis, we are able to show that part of the valuation effects realized over the past 20 years were indeed predictable. In particular, these have contributed to the adjustment of the US net foreign assets position as from 1990 for almost a half. In Devereux et al. (2010) and Tille et al. (2010), asset price-driven expected valuation effects are quantitatively small, as they only arise at higher orders of approximation. This paper does not suffer from this particular issue as we compute the stochastic steady state of the model. Related literature There is a growing literature on open economy models with endogenous portfolio choice and incomplete markets. In fact, complete markets models are not suitable to explain the dynamics of gross foreign assets and liabilities: as portfolios are constant across states of nature in equilibrium, their predictions are at odds with the data 8. Therefore, incomplete markets has become an essential ingredient of open economy models that aim at explaining net and gross foreign assets dynamics. Unfortunately, incomplete markets models have typically the disadvantage that they are not analytically tractable. However, our simple asset structure allows us to obtain some analytical results and to compute the global solution of the model numerically when this is not possible. To be more specific, we solve for the stationary equilibrium of the model, which is defined as a time-invariant distribution (across 8 See Lucas (1982) for an open economy version of the Lucas asset-pricing model and Judd et al. (2003) for a proof of the same result under a more general version of the model. 7

8 state of nature) of nominal prices, exchange rates, consumption and portfolio allocations 9. Pavlova and Rigobon (2007, 2010) are a notable exception in the literature, as their model is fully tractable given the logarithmic utility function. Our model is also analytically solvable with logarithmic utility, but log utility implies that portfolios are constant across states of nature if the only source of uncertainty is output uncertainty (see also Cass and Pavlova, 2004). Pavlova and Rigobon (2007, 2010) then study the role of demand shocks in generating time-varying portfolios. In this paper, we fully explore the role of output shocks by relaxing the log utility assumption. Another main difference is that their model has real assets and is not a model of nominal exchange rate determination. Finally, this paper is related to the general equilibrium literature with incomplete markets. When markets are incomplete and assets are nominal, the equilibrium allocation can be indeterminate and this poses particular challenges for applied work 10. However, this result hinges on the fact that the currency denominated assets are in zero net supply. If an asset in positive net supply such as money is introduced, the price level in each state of nature can be pinned down and the indeterminacy problem can be avoided 11. As agents transfer wealth across periods using the two national currencies, the stationary equilibrium is determinate in this paper 12. In sections 2, we present the model and define net foreign assets as well as valuation effects in the context of our framework. In section 3, we derive our main analytical result on the relationship between the trade balance and the share of world GDP. In section 4, we parametrize the model so as to show that it can rationalize the above stylized facts on the dynamics of the net foreign assets of the US and East Asian countries, while in section 5 we conduct a sensitivity analysis which demonstrates that the results are quite robust to alternative parameter specifications. Section 6 concludes the paper. 9 Most of the open economy literature use local solution methods to analyze incomplete markets model (e.g. Devereux et al. (2010) and Tille et al. (2010)). While these methods can deal with any state space, Rabitsch et al. (2015) showed that the global solution does not always coincide with the local one. See also Coeurdacier and Rey (2012) for a critical assessment of local solution methods. 10 See Balasko et al. (1989) and Geanakoplos et al. (1989). Polemarchakis (1998) derives an indeterminacy result in the presence of multiple units of account or currencies. 11 See e.g. Magill and Quinzii (1992) for a two-period incomplete markets economy with money and Neumeyer (1998) for an open economy extension. Cass et al. (1992) and Gottardi (1996) proved the determinacy of equilibrium in a stochastic OLG economy with finitely lived assets when there is outside money. 12 If we introduced bonds in addition to the two currencies, currencies and bonds would be perfect substitutes. As a consequence, the exact allocation of savings between money and bonds would not be determined. For instance, see Gottardi (1994). Therefore, we do not introduce nominal bonds as it would not add too much to our analysis. 8

9 2 The Model We consider the following two-country overlapping generations economy 13. In each period, an agent h with a two-period lifetime is born in each country. Therefore, two young and two old populate the world economy at each date. The young are born with an endowment of the country-specific good l, which is also the total output of the country. Output is denoted as y l (s) as it depends on the state of nature realized, where s = {1,..., S}. We will use the superscript l to indicate goods and currencies, while we will refer to agents with the subscript h. We assume that output follows a Markov chain, where transition probabilities are time-invariant and ρ(ss ) indicates the probability of transiting from state s to s. Agents gain utility from the consumption of both goods although they are only endowed with the country-specific good, as in Lucas (1982). At time 0, the two governments issue fiat money and distribute it to the initial old. M l is the stock of money issued in country l. As the old have no endowment, money is valued in equilibrium as agents would not be able to consume in their second period of life otherwise. For simplicity, we assume that monetary authorities are inactive after the first period. Our objective is to characterize the stationary equilibrium of the model, therefore prices will not depend on the history of the shocks but only on the current state of nature. Moreover, agents born in the same state of nature although at different dates have the same consumption allocation. Agent h born in state s has the following utility function: U h (s) = c 1h(s) 1 γ 1 γ + β s ρ(ss ) c 2h(ss ) 1 γ 1 γ where γ is the coefficient of relative risk aversion and c 1h (s) and c 2h (ss ) are the constant elasticity of substitution (CES) aggregators: [ c 1h (s) a 1 1 h c 1 1h(s) 1 + a 2 1 h c 2 1h(s) 1 [ c 2h (ss ) a 1 1 h c 1 2h(ss ) 1 + a 2 1 h c 2 2h(ss ) 1 ] 1 ] 1 where a 1 h + a2 h = 1, > 0 and 1. Before presenting the budget constraints that each agent faces, it is useful to present the problem in terms of the main underlying assumptions. Assumption 1 Agents can buy good l only with currency M l. 13 This is with no loss of generality. The model can easily be extended to L countries. (1) (2) (3) 9

10 Assumption 2 The old cannot adjust their portfolio after the realization of uncertainty. The first restriction implies that agents face cash-in-advance constraints à la Lucas (1982). Assumption 2 implies that timing is structured as follows: first, the agents decide their portfolio of currencies; then, uncertainty realizes; and finally the currencies are spent in the respective goods markets. This involves a friction in the currency markets which introduces an element of exchange rate risk in the agents decision problems, as uncertainty is realized after the portfolio decision. This assumption is very important because if old agents were able to adjust their portfolio after uncertainty is realized, this would be equivalent to a model in which there are no restrictions in currency trading, which implies nominal exchange rate indeterminacy and portfolio indeterminacy as pointed out by Manuelli et al. (1990) 14. Therefore, Assumption 2 guarantees that the portfolio decisions and the nominal exchange rates can be pinned down in equilibrium. Taking as given the vector of transition probabilities, the goods prices and the nominal exchange rate, agent h born in state s then chooses the consumption vectors and the portfolio of currencies that maximize (1) subject to (2), (3) and the following constraints: m 1 h(s) + e(s) m 2 h(s) = w h (s) (4) m 1 h(s) + p 1 (s)c 1 1h(s) = m 1 h(s) (5) m 2 h(s) + p 2 (s)c 2 1h(s) = m 2 h(s) (6) p 1 (s )c 1 2h(ss ) = m 1 h(s) s (7) p 2 (s )c 2 2h(ss ) = m 2 h(s) s (8) Agent h is born with an initial wealth w h (s), which is equal to the value of the domestic output expressed in units of currency 1 (the numéraire): w 1 (s) p 1 (s)y 1 (s) and w 2 (s) p 2 (s)e(s)y 2 (s). With his wealth, he buys the two currencies for the purpose of financing the consumption of the domestic and foreign good as well as for saving purposes. m h (s) is the portfolio of currencies held at the beginning of the period and e(s) is the price of currency 2 in units of currency 1 or the nominal exchange rate. Therefore, we say that if e(s) rises then currency 2 (1) appreciates (depreciates). The second and 14 See the Online Appendix for further considerations on the role of Assumption 2. Manuelli et al. (1990) find the same indeterminacy result as Kareken et al. (1981) in a stochastic OLG model. The main difference between a stochastic and a deterministic world is that the nominal exchange rate could fluctuate in equilibrium in the former as opposed of being simply constant. However, the equilibrium path for the nominal exchange rate would still be indeterminate. 10

11 the third constraints are the cash-in-advance constraints faced by the young, where m h (s) is the end-of-period portfolio and p l (s) is the nominal price level in country l expressed in units of the domestic currency. Assumption 2 comes into play in the last two lines. Because they cannot readjust their portfolio, the old directly use the currencies accumulated in the previous period to buy the goods in the respective markets. Therefore, they face as many constraints as the number of goods instead of a single budget constraint for each state of nature. We can consolidate the constraints of the young (4), (5) and (6) into a single budget constraint and rewrite the problem as follows: p 1 (s)c 1 1h(s) + p 2 (s)e(s)c 2 1h(s) w h (s) = m 1 h(s) e(s)m 2 h(s) (9) p 1 (s )c 1 2h(ss ) = m 1 h(s) s (10) p 2 (s )c 2 2h(ss ) = m 2 h(s) s (11) For analytical convenience, we will assume that the intertemporal elasticity of substitution, which is the inverse of the coefficient of relative risk aversion γ, is equal to the elasticity of substitution between the traded goods 15 : Assumption 3 1 =. γ Let λ h (s) be the multiplier associated to the young s budget constraint, λ l h (ss ) the multiplier of the constraint of the old related to good l in state s. The necessary and sufficient conditions for a maximum are the following: 1 c 1 1h(s) : a 1 h c 1 1h(s) 1 = λ h (s)p 1 (s) (12) 1 c 2 1h(s) : a 2 h c 2 1h(s) 1 = λ h (s)p 2 (s)e(s) (13) 1 c l 2h(ss ) : βa l h ρ(ss )c l 2h(ss ) 1 = λ l h(ss )p l (s ) l, s (14) m 1 h(s) : λ h (s) + λ 1 h(ss ) = 0 (15) s m 2 h(s) : λ h (s)e(s) + λ 2 h(ss ) = 0 (16) s λ h (s) : p 1 (s)c 1 1h(s) + p 2 (s)e(s)c 2 1h(s) w h (s) + + m 1 h(s) + e(s)m 2 h(s) = 0 (17) λ 1 h(ss ) : p 1 (s )c 1 2h(ss ) m 1 h(s) = 0 s (18) λ 2 h(ss ) : p 2 (s )c 2 2h(ss ) m 2 h(s) = 0 s (19) 15 This parameter restriction allows us to derive the demand functions for the two currencies analytically, which is very helpful to gain intuition of the main mechanisms. In the sensitivity analysis, we will show that the qualitative and quantitative results obtained under this restriction are robust for commonly assumed values of γ and. 11

12 In the Appendix A, we show how to find the following closed-form solutions of the agents demand functions for the two currencies: where m 1 h(s) = m 2 h(s) = [ a 1 h β s ρ(ss )p 1 (s ) 1 A h (s) ] [ a 2 h β e(s) 1 s ρ(ss )p 2 (s ) 1 A h (s) w h (s) (20) ] w h (s) e(s) [ A h (s) a 1 hp 1 (s) 1 + a 2 h[p 2 (s)e(s)] 1 + a 1 hβ ρ(ss )p 1 (s ) 1 s [ + a 2 hβ e(s) 1 ρ(ss )p 2 (s ) 1 s ] ] + (21) Agent h s demand functions for the goods can be derived using (20), (21) and the budget constraints 16 : c 1 1h(s) = a1 h p1 (s) w h (s) l (22) A h (s) c 2 1h(s) = a2 h [p2 (s)e(s)] w h (s) l (23) A h (s) c 1 2h(ss ) = c 2 2h(ss ) = [ a 1 h β s ρ(ss )p 1 (s ) 1 ] w h (s) A [ h (s) p 1 (s ) a 2 h β e(s) ] s ρ(ss )p 2 (s ) 1 A h (s) s (24) w h (s) p 2 (s ) s (25) As preferences are homothetic, the demand for each good is a linear function of wealth where the multiplicative term is a complicated non-linear function of current and future prices as well as the nominal exchange rate. 2.1 The role of the nominal exchange rate: partial equilibrium Using equations (14), (15) and (16), we can obtain the following expression for the nominal exchange rate: e(s) = a2 h a 1 h s ρ(ss ) c2 2h (ss ) 1 p 2 (s ) s ρ(ss ) c1 2h (ss ) 1 p 1 (s ) s = 1,..., S (26) In our model, the nominal exchange rate is a forward-looking variable, as it depends on the expected marginal utilities derived from the consumption of the two goods as well as from the expected purchasing power of the two 16 The procedure to calculate the demand functions when young is provided in Appendix A. 12

13 1 currencies. In fact, gives how many units of good l we can afford in state p l (s ) s per unit of currency l held. In other words, the nominal exchange rate is the ratio of the expected purchasing power of currency 2 over the expected purchasing power of currency 1, weighted by agent h s marginal utilities. The more a currency can buy tomorrow relatively to the other currency, the higher will be its price today. This means that the nominal exchange rate follows some sort of asset pricing equation, given that the currencies are used to transfer wealth across periods. In the cash-in-advance literature, the spot exchange rate simply depends on the current realization of the stochastic variables and not on expectations of future variables (see e.g. Lucas (1982)). This is due to the transaction role that it is attributed to money, which is only used to carry out exchange in a given period. In Lucas (1982), money is a veil and the exchange rate does not ultimately affect the real allocation, which is the same as in the barter economy. Let us now look at the portfolio decision of an agent in more detail. We combine the demand for the two currencies (20) and (21) to get: [ ] s ρ(ss )p 1 (s ) 1 m 1 h (s) m 2 h (s) = e(s) a1 h a 2 h [ s ρ(ss )p 2 (s ) 1 ] (27) The portfolio decision depends on three sets of variables: the nominal exchange rate, the expected nominal price levels in the two countries and the weight of the two goods in the utility function. Firstly, as currency 2 becomes more expensive (e(s) increases) the (relative) demand for currency 1 falls. In this model, note that the two currencies are substitutes albeit not perfectly. Moreover, the higher is the expected purchasing power of currency 1 relatively to currency 2, the higher is the relative demand for currency 1 as long as the degree of substitutability between the two goods is high enough ( > 1). Finally, it is intuitive that the relative demand for currency 1 rises with the weight given to good 1. Obviously, these arguments about the role of the nominal exchange rate in the portfolio choice of the agents are of a partial equilibrium nature as we assume that the nominal exchange rate, as well as the prices, are given. Below, we will show the importance of general equilibrium analysis as the nominal exchange rate acts so to offset the expected price differentials across countries in equilibrium. 13

14 2.2 Stationary equilibrium Definition 1 A stationary equilibrium is a system of prices and nominal exchange rates (p, e) R 2S ++ R S ++, consumption allocations and portfolios (c 1h (s), c 2h (ss ), m h (s)) R 2 ++ R 2S ++ R 2 ++ for every h = 1,..., H and s = 1,..., S such that: (i) agent h maximizes his utility function (1) subject to the budget constraints (9), (10) and (11) in every s; (ii) c l 1(s) + c l 2(s s) = y l (s) s, s and l (iii) h ml h (s) = M l s, l where c l 1(s) h cl 1h (s) and cl 2(s s) h cl 2h (s s). In the previous section, we have shown how to compute analytically the demand functions for the goods and the currencies. Therefore, the number of endogenous variables that we need to compute reduces to 3S, i.e. the nominal price levels and the exchange rate in each state of nature. The number of equations is instead 2S + 2S 2, where 2S refer to the two money markets that have to clear in each state and 2S 2 are the two goods markets that have to clear for any pair of s and s, as the consumption of the old depends on the state when born as well as the current state. In Appendix B, we show that the goods markets equations that are apparently in excess are actually redundant, so that solving the model actually reduces to handling a non-linear system of 3S equations and unknowns. Before solving the model, next we introduce some key definitions and make some useful remarks. 2.3 Portfolio rebalancing and trade imbalances: a unified view To start with, let us define the balance of trade of country 1 in state s 17 : T B 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 that the sign of the balance of trade depends on the choices that the young make in the current period, but also on the choices made by the current old in the previous period. Substituting the budget constraints into the trade balance equation, it should be immediate that the above definition can be rewritten as: T B 1 (s s) = m 1 1(s) m 1 1(s ) + e(s)[m 2 1(s) m 2 1(s )] (28) 17 Obviously, by Walras Law we have that T B 2 (s s) = T B 1 (s s). 14

15 This leads us to the following remarks: Remark 1 The balance of trade is zero if: (i) portfolios are constant across states of nature; (ii) this period s realized state is the same as last period s. Equation (28) shows that there is a close relationship between currency markets and the goods markets. If, for some reason, there is no portfolio rebalancing in equilibrium, then the balance of trade is always balanced. In the Online Appendix, we show that this behaviour occurs e.g. when utility functions are logarithmic. The demand functions become extremely simple as they do not depend on future prices, and the model is fully tractable. Constant portfolios imply that the consumption of an old person does not depend on the state in which he is born but only on the state realized when old 18. As we explained in the introduction, this is a prediction at odds with the reality of international financial markets. When the elasticity of substitution is different than one, we will show that agents born in different states of nature have different demands for the goods which is then reflected in their demand for the two currencies 19. The second part of the remark is related to Polemarchakis and Salto s result for deterministic OLG economies (2002). In a monetary union, they showed that the balance of trade is always in equilibrium at the monetary steady state. In this paper, the monetary steady state is stochastic and trade imbalances are possible whenever s s. Next, we decompose the trade balance equation to highlight valuation effects and the change in net foreign assets. 2.4 Net foreign assets and valuation effects In this section, we explore the relationship between net foreign assets, the balance of trade and valuation effects. Consider the balance of trade of country 1 in state s s, as defined in the previous section (equation (28)). Using the fact that m 1 1(s) + m 1 2(s) = M 1 for every s, we can rewrite it as follows: T B 1 (s s) = m 1 2(s ) m 1 }{{} 2(s) + e(s)m 2 }{{} 1(s) e(s)m 2 }{{} 1(s ) }{{} F L 1 (s ) F L 1 (s) F A 1 (s) current value F A 1 (s ) (29) 18 In the previous section, we showed that the aggregate consumption of the old does not depend on the past, but individual consumption can vary across states and change with the distribution of money holdings. 19 Our finding for the log case is related to Cass and Pavlova (2004), who showed that the matrix of portfolio returns is degenerate in a two-period economy with N Lucas trees. A similar result holds in the infinite-horizon setting of Pavlova and Rigobon (2007, 2010), which then introduce demand shocks to generate time-varying portfolios. 15

16 where F A(s) and F L(s) stand respectively for foreign assets and foreign liabilities, which in this context is a country s holdings of foreign currency and the foreign country s holdings of the domestic currency. Next, define net foreign assets as NF A(s) F A(s) F L(s) and rewrite the above as follows: NF A 1 (s) = current value NF A 1 (s ) + T B 1 (s s) (30) Equation (30) states that the end-of-period net foreign assets in country 1 is equal to the current value of the net foreign assets accumulated in the previous period and the balance of trade 20. The next step is to rewrite equation (29) in order to highlight the valuation effects in this model. We can do that by summing and subtracting the foreign assets of country 1 in the previous state (e(s )m 2 1(s )) in the right hand side and using the definition of net foreign assets: T B 1 (s s) = NF A 1 (s) NF A 1 (s ) + [e(s ) e(s)]m 2 1(s ) (31) This equation can be rewritten as: where NF A 1 (s s) = T B 1 (s s) + r(s s)f A 1 (s ) }{{} valuation effects (32) r(s s) = R(s s) 1 e(s) e(s ) 1 Therefore, the change in the net foreign assets position of country 1 will be determined by the behaviour of the balance of trade and the valuation effects, where r(s s) is the return on the foreign assets accumulated in the previous period 21. In this model, valuation effects are entirely determined by exchange rate movements. If foreign currencies have appreciated with respect to the past (i.e. e(s) > e(s )), then the return on the foreign assets accumulated in the previous period is positive and therefore we say that the country experiences positive valuation effects 22. Conversely, a country experiences negative valuation effects if foreign currencies have depreciated. In this framework, currencies are the only assets available and therefore our setting can capture a scenario in which the majority of domestic assets are denominated in the foreign currency while domestic liabilities are denominated in the domestic currency. As from the findings of Lane and Shambaugh (2010), 20 This equation is equivalent to equation (1) in Gourinchas and Rey (2007, footnote 2). 21 Notice that there is no net income from abroad and therefore the trade balance position is equivalent to the current account position. 22 As the price of the foreign asset is defined in units of the domestic asset, i.e. the exchange rate, the above rate of return has to be interpreted as the return of foreign assets relatively to the return on foreign liabilities. 16

17 this is entirely consistent with the currency denomination of the foreign assets and liabilities of the US while it would be less realistic when applied to developing countries. Lane and Shambaugh (2010) also find that emerging market countries are becoming more similar to advanced economies as they issue less foreign-currency denominated debt than developing countries and are accumulating foreign-currency denominated assets in the form of foreign exchange reserves. Therefore, our setting is appropriate to interpret the stylized facts on the dynamics of the net foreign assets of the US and East Asian emerging countries (Figures 1-3). Before discussing the behavior as well as the quantitative importance of valuation effects, in section 3 we show that portfolio rebalancing, and therefore trade imbalances, occurs in equilibrium whenever there are changes in countries share of world GDP. 3 The distribution of world GDP, portfolio rebalancing and trade imbalances In this section only, we assume that there is no home bias (a 1 h = a2 h ) as this case is very helpful to develop some key intuitions about the model. The following Proposition establishes that there is a strong relationship between the distribution of world GDP across countries, portfolio holdings and trade imbalances when preferences are identical across countries. Proposition 1 Assume that a 1 h = a2 h. (i) A country s portfolio holdings at the end of a period are linearly related to its current share of world GDP. (ii) If a country has a higher (lower) share of world GDP with respect to the past, it runs a trade surplus (deficit). Proof. (i) The demand of agent h for the two currencies is linear in wealth (see equations (20) and (21)): m l h(s) = k l (s)w h (s) where k l (s) is identical across agents if a 1 h = a2 h. Summing across h and assuming that the money markets clear, we get the following equation: M l = k l (s) h w h (s) 17

18 Dividing the first equation by the second equation and rearranging, we obtain the desired result: where w(s) = h w h(s) 23. m l h(s) = w h(s) w(s) M l l = 1, 2 (ii) Suppose that today s realized state is s and yesterday s state was s and assume that w h(s). Our previous result implies that: > w h(s ) w(s) w(s ) m l h(s) > m l h(s ) l = 1, 2 As the young born in the current period hold a higher share of both currencies than the previous generation, then country h runs a trade surplus by equation (28). The other case can be worked out in a similar way. The end-of-the-period wealth of the young is equal to their total money holdings, as the two currencies are the means by which they save and therefore finance future consumption. Therefore, Proposition 1 suggests that the distribution of world GDP is the same as the distribution of world wealth at the end of a period 24. If the distribution of world GDP changes across states of nature, then portfolio rebalancing occurs over time and the distribution of wealth will change too. Proposition 1 sheds further light on the behavior of the trade balance. If a country is in surplus, it is because the young are relatively wealthier with respect to the past although this does not rule out the possibility that a country is poorer than the other country in all states of nature 25. Therefore, our model offers a novel explanation of the fact that emerging countries run trade surpluses against the United States: global imbalances simply reflect the rise of emerging countries in the world economy. If a country is classified as emerging, then its share of world GDP should have increased over time. Using a sample of 146 countries, we find that the share of world GDP of all emerging countries except Argentina has increased 23 World GDP is defined as the sum of countries nominal GDP expressed in units of the numéraire currency. 24 Because nominal interest rates are zero, then domestic GDP is equal to domestic income. Therefore, the distribution of world GDP is also equal to the distribution of world income. 25 On the other hand, the poor country is always in trade deficit in cash-in-advance models under isoelastic utility (see Eugeni (2013) for a derivation). The reason is that the sign of the trade balance only depend on the current shock, and not on the past. 18

19 over the past 20 years 26. As expected, China is the emerging country whose share of world GDP has increased the most, as it has gained 7.81% points over the past 20 years. On the other hand, the share of world GDP of the US has fallen by 3.54%. Without calculating the equilibrium prices, we are not able to say how much of the increase in wealth is due to output growth and how much to changes in prices. Our numerical results will show that output changes dominate the terms of trade effect as long as the elasticity of substitution between traded goods is greater than 1, which is supported by empirical evidence as we explain in the next section. Therefore, the reason behind the Chinese surplus is that China s real GDP has grown more than in the US, which has implied that the distribution of world GDP has changed in favour of China over the past 20 years. Finally, the allocation of savings across currencies deserves some comment. Agents do not have very sophisticated portfolio strategies according to Proposition 1. In fact, agents hold the same share of both money stocks 27. As we explain next, this is due to the shock absorbing role that the nominal exchange rate plays in this model. 3.1 Nominal exchange rate determination and the role of money We plug the demand for the two currencies into the equilibrium conditions for the money market and get the following two equations: M 1 = 1 2 e(s)m 2 = 1 2 [ β s ρ(ss )p 1 (s ) 1 A(s) ] [ β e(s) 1 s ρ(ss )p 2 (s ) 1 A(s) w h (s) (33) h ] w h (s) (34) Combining equations (33) and (34), we obtain the following expression for the exchange rate: h ( M 1 e(s) = M 2 ) 1 s ρ(ss )p 2 (s ) 1 s ρ(ss )p 1 (s ) 1 s = 1,..., S (35) Although the above expression is not a closed-form solution, we can gain some intuition about the role of the nominal exchange rate and the importance 26 We calculate the change in the share of world GDP as follows: share h = GDP h,1990 h GDP. h,1990 GDP h,2010 h GDP h,2010 GDP is taken from the IMF World Economic Outlook database and it is measured at current national prices converted in US dollars. We use the IMF classification of emerging countries. 27 Notice that we have assumed away home bias in the preferences, therefore agents hold the two currencies in the same share as they like the two goods equally. 19

20 of general equilibrium analysis as opposed to partial equilibrium analysis. Recall the equation that linked the portfolio choice of the agents to the nominal exchange rate (equation (27)). In a partial equilibrium setting, an increase in the nominal price levels in country 1 means that the purchasing power of currency 1 is lower and therefore the relative demand for currency 1 falls (provided that the elasticity of substitution is bigger than 1). In equilibrium, the nominal exchange rate will behave in such a way to counteract expectations on price movements. In particular, currency 1 depreciates to compensate the lower demand for good 1 as equation (35) shows. In the context of the numerical exercises, we will clarify how the nominal exchange rate behaves in response to output shocks. We conclude this section with a discussion of the role of money in our economy. Since the old face separate budget constraints for each good, then the aggregate consumption of the old of good l is equal to the real money balances of currency l. As a consequence, we can write the following expressions for the nominal price levels in the two countries using the goods markets clearing conditions: p 1 (s) = p 2 (s) = M 1 y 1 (s) c 1 1(s) M 2 y 2 (s) c 2 1(s) It can be observed that the level of the money stocks does not matter for the real allocation in this economy. Suppose that the money stock of country 1 doubles. Given the first of the above two equations, the nominal price level will be doubled as well as it is homogenous of degree 1 with respect to the domestic money supply. The nominal exchange rate doubles as well (see equation (35)) as currency 1 becomes cheaper. Therefore, the wealth of both agents is twice as before: agent 1 s wealth increases because the nominal price of good 1 is higher; agent 2 s wealth increases because the nominal exchange rate appreciates. It can be checked from the demand functions that this change in prices does not affect the consumption of both agents. Although the level of the money stocks does not matter for the real allocations, money is not neutral in our model. If we removed money from the economy, the equilibrium allocation would be very different. While the young would still be able to engage in barter (intragenerational trade), the old would not be able to consume anything. Even if the old had an endowment, such as a pension, he would not be able to trade it because of the multiple budget constraints, so he would limit his consumption to the domestic good. Money 20

21 is important in our framework as in standard Samuelsonian OLG economies, but in an even stronger sense because of the multiple budget constraints which prevent barter among the old. 4 The deterioration of the United States external position against China: the role of valuation effects The aim of this section is to show that our two-country model is able to explain the dynamics of the net external position of the United States and China over the past twenty years as the consequence of China s increase in the share of world GDP. The intention is not to provide a fully-fledged calibration exercise, as the model is stylized in many aspects. The purpose is rather to demonstrate that this model can offer an interpretation of the stylized facts presented in the introduction as well as a very clean mechanism that indicates why and when exchange rate-driven valuation effects tend to stabilize the net external position of a country. The reasons why we choose the United States and China for our numerical exercise are several. Firstly, China is widely known as one of the main creditors of the US and the US deficit against China account for a significant fraction of the overall US current account deficit (e.g. Eugeni, 2015). Second, the US and China are the countries whose share of world GDP has changed the most (see section 3). Moreover, the US-China imbalances are persistent and our two-period OLG model is especially suitable to capture low-frequency trends in international financial markets. The currency composition of the US and China s balance sheet can also be captured by our model, as their foreign assets are mainly denominated in foreign currencies while foreign liabilities are mainly denominated in the domestic currency. In fact, foreign assets is a country s foreign currency holdings in our setting, while foreign liabilities is the domestic currency held abroad. According to the Benetrix, Lane and Shambaugh (2015) database, 64% of US foreign assets were denominated in foreign currencies while 88% of US foreign liabilities were denominated in dollars in As far as China is concerned, 100% of the Chinese foreign assets are denominated in foreign currency, 61% of which are dollar denominated. On the other hand, 76% of Chinese liabilities were issued in renmimbi in This reflects a general trend which sees emerging market economies increasingly able to borrow in their domestic currency (Lane and Shambaugh, 28 The first figure reflects the fact that many developing economies still borrow in US dollars as they are unable to issue debt in domestic currency-denominated assets. 21

22 2010) 29. Given the currency denomination of foreign assets and liabilities for the US and China, a depreciation of the dollar implies a positive wealth effect for the US and a negative wealth effect for China. Although the Chinese currency has considerably appreciated over the past 10 years (Figure 3), the Chinese exchange rate is not freely floating therefore it is reasonable to expect that the model will tend to over predict valuation effects. It is also important to stress that our model can only capture low-frequency movements of the exchange rate and the balance of trade and does not aim at explaining high-frequency movements (or lack of) in foreign exchange markets. Since agents live for two periods in our OLG economy, we assume that a period is 20-years long. As we wish to explain the deterioration of the US external position against emerging economies over the past 20 years, we adopt the following strategy. We consider an economy with two states of nature, where state 1 corresponds to the state of the world economy in 1990 while state 2 is the state of the world economy in Therefore, we will focus on what happens in the world economy in the transition from state 1 to state We report the parameter values that we choose for the numerical exercise in Table 1. Table 1: Parameter Values. Baseline model. US real GDP in 1990 y 1 (1) = 31, 432 US real GDP in 2010 y 1 (2) = 41, 627 Chinese real GDP in 1990 y 2 (1) = 2, 005 Chinese real GDP in 2010 y 2 (2) = 7, 693 Money supply M 1 = M 2 = 1 Elasticity of substitution 1 = 2 = 4 Share of home goods a 1 1 = a 2 2 = 0.72 Discount factor β 1 = β 2 = 1 Output persistence ρ(ss) = 0.9 We take the real GDP per capita of the United States and China in 1990 and 2010 to parametrize output in the two states 31. Notice that while US 29 Another signal of the increased ability of emerging countries to borrow in their own currency is that a third of the foreign currency-denominated US foreign assets are denominated in currencies other than the Euro, the Yen, the Pound and the Swiss Franc. Therefore, these are assets held in emerging economies and denominated in local currencies. 30 This is not to argue that the world economy can only be in a state that matches the situation of the world economy either of the 1990 or the However, a two-states example is enough to illustrate our arguments while adding more states of nature would not provide neither more information nor intuition. 31 We take the output-side real GDP at chained PPPs and the population from the Penn World Tables 22