Agnès Bénassy-Quéré

Yeganeh Forouheshfar

The impact of yuan internationalization on the euro-dollar exchange rate

Keywords: China, yuan, exchange-rate regime, euro, dollar. JEL classification: F31, F33. Résumé Chine, yuan, régime de change, euro, dollar. Classification JEL : F31, F33

We study the implication of a multipolarization of the international monetary system on crosscurrency volatility. More specifically, we analyze whether the internationalization of the yuan could modify the impact of asset supply and trade shocks on the euro-dollar exchange rate, within a threecountry, three-currency portfolio model. Our static model shows that the internationalization of the yuan (defined as a rise in the yuan in international portfolios) would be either neutral or stabilizing for the euro-dollar rate, whatever the exchange-rate regime of China. Moving to a dynamic, stockflow framework, we show that the internationalization of the yuan would make exchange-rate variations more efficient to stabilize net foreign asset positions after a trade shock.

Introduction

Since World War II, the US dollar has remained the key currency of the international monetary system (IMS, hereafter). The yen never succeeded in challenging the US currency as an international means-of-payment, unit-of-account or store-of-value. As for the euro, it has emerged mostly as a diversification and as a regional currency (see ECB, 2010).

Such resilience of the dollar was benign as long as the US economy was clearly dominant in terms of GDP, trade and financial markets. Monetary pegs and reserve accumulation did not weigh much in global trade and capital flows. This is no longer the case today, and the mismatch between a unipolar IMS and a multipolar real economy has sometimes been pointed as one key ingredient of the macroeconomic environment that led to the 2007-09 financial crisis.

Against this background, China has taken significant steps towards the progressive internationalization of the yuan: it has allowed domestic exporters to invoice their cross-border sales in yuan; it has timidly opened the gate to foreign capital inflows (as a fist step, only on the off-shore market and through foreign direct investment in mainland China); it has developed bilateral swap agreements in yuan with central banks in partner countries; it has liberalized the use of deposits in yuan in Hong Kong. Although the path towards internationalization will still be long (see, e.g., [START_REF] Dobson | Will the yuan become a world currency?[END_REF][START_REF] Eichengreen | Exorbitant Privilege[END_REF][START_REF] Prasad | Will the yuan rule?[END_REF][START_REF] Yu | Revisiting the internationalization of the yuan[END_REF], it is significant that the first steps have been carried out while the currency remained carefully managed with respect to the US dollar. According to [START_REF] Vallée | The path Internationalisation path of the RMB[END_REF], China could go a long way in the direction of internationalization without switching to a free floating exchange-rate regime.

The implications of such developments for the IMS have generally been studied from the angle of global imbalances and the Triffin dilemma (Triffin, 1961). The debate started with [START_REF] Dooley | The revived Bretton Woods system[END_REF] suggesting that the rise of China with a fixed exchange rate regime on the dollar was beneficial both to China (which could develop through exports, thanks to an undervalued currency) and to the United States (which could cheaply finance its current-account deficit through capital inflows from China). Whether this large imbalance played a role in the 2007-08 financial crisis is still debated (see [START_REF] Angeloni | Global currencies for tomorrow: a European perspective[END_REF]. As a matter of fact, the crisis triggered a renewal of the debate on the IMS. The dangers of a large mismatch between the growing role of Asia in the global economy and the persistent supremacy of the US dollar in the IMS were increasingly recognized [START_REF] Bénassy-Quéré | What international monetary system for a fastchanging world economy?[END_REF]. In particular, Farhi et al. (2011) argued that the IMS would face a new version of the Triffin dilemma: the increasing demand for international liquidity (due to the growth of emerging economies and their appetite for reserve accumulation) would keep interest rates low for the US economy, hence encourage continuous current-account deficits in this country and, more specifically, excess government indebtedness. At some stage, international investors would lose confidence in US solvency (or equivalently, would fear massive monetization of US bonds), which would trigger a crash of the dollar. To avoid such an outcome, it would be necessary to develop alternative sources of international liquidity through the internationalization of other currencies or the development of the special drawing right [START_REF] Mateos Y Lago | The debate on the international monetary system[END_REF]. However the implications of such move towards a multipolar monetary system have little been studied. On the one hand, more substitutability across key currencies would mean more frequent portfolio reallocations, hence higher exchange-rate volatility. On the other one, a given shock to a country could be adjusted Documents de Travail du Centre d'Economie de la Sorbonne -2013. [START_REF] Kindleberger | Dominance and leadership in the international economy[END_REF] through lower exchange-rate variation, and multipolarity could act as a discipline device if investors were offered a choice between several currencies with equivalent liquidity features (Bénassy-Quéré and Pisani-Ferry, 2011). 1 In this paper, we study the implication of a multipolarization of the IMS on cross-currency volatility. More specifically, we analyze whether the internationalization of the yuan could modify the impact of wealth and balance-of-payment shocks on the euro-dollar exchange rate. We rely on the model proposed by [START_REF] Blanchard | International Investors, the U.S. Current Account, and the Dollar[END_REF] -a portfolio equilibrium model based on fixed portfolio allocations. Then, to the extent that domestic investors display a home bias (i.e. a preference for home-currency assets), a wealth transfer from country A to country B involves an appreciation of currency B against currency A, because the wealth transfer involves a rise in the global demand for currency A. Blanchard et al. study the implications of China pegging its currency to the dollar through foreign exchange interventions and capital controls. In this case, a wealth transfer from the United States to China (due to a bilateral trade imbalance) no longer triggers an appreciation of the yuan against the dollar because the People's Bank of China buys all the dollar released by impoverished US residents, hence the global demand for both dollar and yuandenominated assets stays constant, and the dollar-yuan does not move. However, the global demand for euro-denominated assets falls because US residents sell euros while euro area residents do not buy them since their wealth is not directly affected by the shock: the euro depreciates against both the dollar and the yuan. Now if China switches to a free floating regime with full capital mobility, these effects are blurred: following a wealth transfer from the United States to China, the global demand for dollars falls, that for the yuan increases and that for the euro may either increase or decrease, depending on the relative preferences of the United-States and China vis-à-vis the European currency; the euro may appreciate against the dollar while depreciating against the yuan. Blanchard et al. conclude that the Chinese peg on the dollar tends to maintain the euro relatively weak against the dollar in a context of cumulated current-account deficits in the United States. However they do not consider the possibility that the Chinese currency grows international: in their setting, foreign investors are not allowed to hold Chinese assets.

Although there is considerable debate on the path of yuan internationalization (see, e.g., [START_REF] Dobson | Will the yuan become a world currency?[END_REF][START_REF] Yu | Revisiting the internationalization of the yuan[END_REF][START_REF] Eichengreen | Exorbitant Privilege[END_REF], it is difficult to envisage a fully multipolar global economy without a considerable reshaping of the IMS in the long run [START_REF] Angeloni | Global currencies for tomorrow: a European perspective[END_REF]. In the next decades, we consider it likely that the yuan will reap at least some international role, something like the Swiss franc or the yen today. We are interested in the implications of such move for the stability of the IMS. If European residents hold yuans in their portfolios, then a wealth transfer from the United States to China will make Europeans react to the shock. In the case where the Chinese currency floats, the Euro area investor will react to the appreciation of the yuan against the dollar by 1 Another strand of the literature focuses on the incentive of the hegemonic country to to assume responsibility for stabilising the system. [START_REF] Kindleberger | Dominance and leadership in the international economy[END_REF] argues that a hegemonic system may be more stable for this reason, whereas [START_REF] Eichengreen | Hegemonic stability theories of the international monetary system[END_REF] suggests that the hegemon may rather exploit its position. According to [START_REF] Cohen | Dollar dominance, euro aspirations: recipe for discord?[END_REF], monetary power fragmentation involves economic risks (e.g. increasingly antagonistic relationships between currency blocs, possibly leading to de-globalisation) and geopolitical ones (e.g. a breakdown of fragile equilibria, such as oil and support for the US dollar in return for military protection in the Middle East). Documents de Travail du Centre d'Economie de la Sorbonne -2013.16 selling yuans against dollars, which will stabilize the euro-dollar exchange rate. Hence our intuition is that, in a multipolar IMS, the Chinese currency regime may be more benign for third countries than in a dollar-centered IMS, because portfolio behaviors in third countries compensate for the distorted portfolio choices of China, following a wealth transfer.

To investigate this question, we extend Blanchard et al.'s model to allow for the internationalization of the yuan. We work on a model with three countries (US, Euro area, China) and three currencies (dollar, euro, yuan). We successively study two polar exchange-rate regimes: a fixed peg on the dollar, and a free floating exchange rate. 2 In each case, we study the impact of shocks to asset supplies and to net foreign asset positions on the euro-dollar exchange rate, depending on the international status of the Chinese currency.

We find that a simultaneous rise in the yuan's weight in US and Euro area's portfolios would make the euro-dollar exchange rate less vulnerable to bilateral imbalances between China and either the US or the Euro area. Furthermore, it would attenuate the distortions arising from China pegging its currency to the dollar, while not fundamentally modifying the reaction of the euro-dollar exchange rate to asset-supply shocks.

The remainder of the paper is organized as follows. The static model is presented in Section 2. In Section 3, it is solved under the two exchange-regimes, successively, and the impact of several shocks on the euro-dollar exchange rate is analysed. Section 4 discusses the impact of internationalizing the yuan on the results obtained in the previous section. Section 5 presents the dynamic, stock-flow model and some simulations. Section 6 concludes. [START_REF] Blanchard | International Investors, the U.S. Current Account, and the Dollar[END_REF] first propose a portfolio-balance model with two countries (the United States and the Euro area) to study the impact of trade shocks and shocks to portfolio preferences on the euro-dollar exchange rate in the short run and in the long run. In their model, the residents of both areas allocate their wealth in fixed shares between domestic-and foreign-currency denominated assets. On average, they have a preference for assets denominated in their home currency (home bias). Hence, a transfer of wealth from the United States to the Euro area (deriving from US bilateral trade deficits) tends to reduce global demand for dollar-denominated assets, triggering a progressive depreciation of the dollar against the euro. In turn, a rise in the share of the dollar in investors' portfolio leads the dollar to appreciate in the short run but depreciate in the long run (in line with the fall in the net foreign asset position). In the last section of their paper, Blanchard et al. extend their model to four currencies (US dollar, euro, yen and yuan) to study the impact of US current-account and preference shocks on exchange rates, depending on China's exchange-rate regime. However they assume that only three of the four currencies are traded internationally, the yuan remaining purely national due to strict capital controls. Here we use the same framework which we limit to three countries (United States, Euro area, China) to specifically analyze the implications of the internationalization of the yuan.

The model

The accounting framework is summarized in Table 1. Denoting by i the country of residence (i=U,E,C) and by j the currency of investment (j=$,€,¥), W i represents the wealth of asset holders in country i, D i their holdings in domestic currency and F j i their holdings in foreign currency j. Finally, the asset supply in currency j is denoted A j . All these demands and supplies are expressed in the currency of denomination, whereas wealths are expressed in the home currencies of their holders. Using the yuan as the numeraire, S $ denotes the nominal exchange rate between the dollar and the yuan and S € the nominal exchange rate between the euro and the yuan. A rise in S $ (resp. S € ) denotes and appreciation of the dollar (resp. of the euro) against the yuan.

United States (in USD) D U F € U S € /S $ F ¥ U /S $ W U Europe (in EUR) S $ /S € F $ E D E F ¥ E /S € W E China (in RMB) S $ F $ C S € F € C D C W C
Total (asset supply in each currency)

A $ A € A ¥ -
Each line of Table 1 represents the budget constraint of a country, in its own currency. As for the columns, they represent the three market-clearing conditions once the demands are converted into the same currency. Specifically, the three market-clearing conditions write:

C E U F F D A $ $ $ + + = (1a) C U E F F D A € € € + + = (1b) U E C F F D A ¥ ¥ ¥ + + = (1c)
We denote by f j i the share of currency j in the portfolio of country i's residents, and f i the total share of foreign currency assets in the portfolio of country i. 3 For instance, we have, for the United States:

U U U W S F S f $ € € € / = U U U Y W S F f $ ¥ / = and U U U f f f ¥ € + =
Note that, to the extent that currencies are floating, these shares are assumed exogenous: the residents of each country are assumed to keep constant the domestic currency counterpart of their foreign-currency holdings, as a share of total wealth. This means for instance that US residents will buy more euro-denominated assets whenever the euro depreciates, in order to keep the dollar-counterpart of their euro holdings constant as a share of their wealth. 4 Consistently, the three market-clearing conditions can be re-written as follows:

( )

$ $ $ $ € $ 1 S W f S W f S W f A C C E E U U + + - = (2a) ( ) € € € € $ € 1 S W f S W f S W f A C C U U E E + + - = (2b) ( ) E E U U C C W f S W f S W f A ¥ € ¥ $ ¥ 1 + + - = (2c)
Due to the Walras law, having two markets in equilibrium ensures that the third one will be in equilibrium. Denoting by NFA i the net foreign asset position of country i expressed in its own currency, we have

NFA U = W U -A $ , NFA E = W E -A € and NFA C = W C -A ¥ = -(S $ NFA U + S € NFA E ).
Equations ( 2a) to (2c) then re-write:

( ) $ $ $ $ € $ 1 S W f S W f S NFA f A f C C E E U U U + + - = (3a) ( ) € € € € $ € 1 S W f S W f S NFA f A f C C U U E E E + + - = (3b) ( )(
)

E E U U E U C C W f S W f S NFA S NFA S f A f ¥ € ¥ $ € $ ¥ 1 + + + - - = (3c)
where again the last equation is redundant due to the Walras law. As a first step, NFA U and NFA E are considered exogenous (we relax this assumption in Section 5). Under a flexible exchange-rate regime, Equations (3a) and (3b) jointly determine S € /S $ (the euro-dollar rate) and S $ (the dollar-yuan rate), for given portfolio choices. Under a fixed dollar peg in China, S $ is exogenous but the portfolio allocation of China (f $ C , f € C ) becomes endogenous. We now consider each regime, successively.

Resolution under each regime

A free floating regime

The most standard way of solving a portfolio-balance model is under flexible exchange rates. Consistently, we first consider all portfolio shares f as exogenous and solve Equations ( 2a) and ( 2b) for the dollar-yuan exchange rate (S $ ) and the euro-dollar exchange rate (S € /S $ ). We get:

( ) ( )( ) C C U U E E E E E U U U E E C E E E W f W f W f NFA W f NFA W f W f NFA W f S $ € $ $ $ - - - + - = ϕ (4a) ( ) (
)

E E E C E U C U U C U NFA W f f NFA W f f S S - + - + = $ € $ € ϕ ϕ ϕ (4b) 7 With C C C f f $ € / = ϕ .
In the following, it will be assumed that NFA U , NFA E <0, hence NFA C >0, consistent with the stylized facts (see Figure 1).5 Under this assumption, all terms under parenthesis in Equation (4a) are positive. Since f U >f € U and f E >f $ E , the denominator of Equation ( 4a) is also positive. This means that a balanced rise in China's wealth (increase in W C with constant NFAs) unambiguously makes the dollar appreciate relative to the yuan (except if simultaneously f $ C falls). Strikingly, though, China's wealth has no impact on the euro-dollar exchange rate (Equation 4b): a rise in China's wealth makes both the dollar and the euro appreciate in equal proportions against the RMB, because the rise in Chinese demand for dollar and euro assets is not matched by any increase in their supply. (  )

E E E C E U U U C U NFA A f f NFA A f f S S Φ - + Φ - + = € $ $ € $ € ϕ ϕ (5) With ( ) U C U U f f € 1 - - = Φ ϕ and ( ) E C E E f f $ 1 ϕ - - = Φ .
Φ U and Φ E represent the impact of a transfer of wealth to the United States (resp. to the Euro area)

on the global demand for dollars (resp. euros). In the standard case, we assume that the home bias (captured by 1-f i with i=U,E) is large enough so that Φ U , Φ E > 0. Then, given that both NFA U and NFA E of the world" as a fourth country, which would make the analysis much more complicated. In our framework, we will rather consider China a shortcut for large surplus emerging economies. All in all, since the internationalization of the yuan and opening up of China's financial account will be only gradual, we can consider that when our simplification applies, China's NFA will have increased considerably.

are negative, the numerator and the denominator of Equation ( 5) are both positive. Holding all NFAs constant, a rise in asset supply in dollars (resp. in euros) triggers an appreciation (resp. depreciation) of the euro against the dollar, whereas the supply of yuan-denominated assets has no impact on the euro/dollar exchange rate. In turn, a fall in the US NFA position, which means a transfer of wealth from the United States to the rest of the world, translates into a depreciation of the dollar against both the yuan and the euro. Similarly, a fall in the Euro area NFA position makes the euro depreciate against both the dollar and the yuan.

In the case of a large bias of China in favor of dollar assets, however, ϕ C is close to zero, so that Φ U < 0. In this case, a fall in the US NFA position, holding the Euro area's NFA constant, means a rise in the global demand for dollar relative to the euro, hence an appreciation of the dollar against the euro. This is because China's NFA increases while the Chinese asset holder has a strong preference for the dollar.7

Table 2 depicts the sign of the various partial derivatives.

A $ + NFA U -if Φ U >0 A € - NFA E + if Φ E >0
A ¥ 0

Source: Authors' calculations.

A peg on the dollar

We now assume that the yuan is pegged to the dollar, i.e. S $ is exogenous. The model is solved for the exchange rate of the euro against the dollar (S € /S $ ) and for the share of the dollar in China's portfolio, f $ C . Starting again from Equations (2a) and (2b), we get:

( )

E E E U U U C C NFA W f NFA W f S W f S S - - - = ¥ ¥ $ $ € (6)
A balanced increase in China's wealth (rise in W C with constant NFAs) now makes the euro appreciate against the dollar. The reason is given by the equation for f $ C in Appendix A: when W C increases, China reduces the share of the dollar in its portfolio so as to maintain its peg (otherwise the yuan would depreciate against the dollar, since the rise in W C here is triggered by a rise in the supply of yuan-denominated assets). For a fixed share of foreign assets in the portfolio (f C ), this means raising the share of the euro.

Like in the flexible regime case, we now solve the model in terms of asset supplies and NFA positions:

( ) ( ) ( ) ( ) E C E E E U U U C U C NFA f f A f f A f f NFA f f S A f S S - - - - - + - - + = ¥ € $ $ € ¥ $ ¥ $ € 1 1 (7)
Due to limited internationalization of the yuan, we can assume that f ¥ U and f ¥ E are small enough so

that 0 1 ¥ > - - C U f f and 0 1 ¥ > - - C E f f
. With NFA E < 0, the denominator of the expression in Equation ( 7) is positive. Hence, the euro appreciates against the dollar whenever the supply of yuandenominated assets increases (rise in A ¥ ) or when the Chinese portfolio opens up (rise in f C ). The reason is the same as above: to maintain the peg on the dollar, Chinese authorities reduce the share of the dollar in their growing portfolio, which means that they increase the share of the euro, triggering a euro appreciation against both the dollar and the yuan. Now, a fall in the US NFA position (holding the European NFA constant) triggers a depreciation of the euro against the dollar. This is because there is a wealth transfer from the United States to China, which means a fall in the global demand for dollar to the benefit of the yuan (except if f C is very large). To avoid an appreciation of the yuan against the dollar, Chinese authorities buy more dollardenominated assets, hence they reduce the share of the euro in their portfolio, which in turn weakens the euro against the dollar.8 A fall in the NFA of the Euro area also leads to a depreciation of the euro through the more traditional channel (transfer of wealth from the Euro area to China, fall in the global demand for euro-denominated assets).

In a similar way, a rise in the supply of dollar-denominated assets (A $ ) now triggers a depreciation of the euro against the dollar. The reason is the increase in the US demand for yuan-denominated assets, which makes Chinese authorities increase the share of the dollar in their portfolio to oppose an appreciation of the yuan.9

Table 3 summarizes the impact of the various variables on the euro/dollar exchange rate.

A $ - NFA U + A € - NFA E + A ¥ +
Source: Authors' calculations.

The impact of yuan internationalization on the euro-dollar exchange rate

We now study the impact of the internationalization of the yuan on the reaction of the euro/dollar exchange rate to the various variables mentioned in the previous tables. To do so we calculate how the derivatives presented in Tables 2 and3 ). In most cases, the impact of yuan internationalization derives from the reduced forms obtained in the previous section. In a few cases, however, we have to simulate the model to get clear-cut result. We first present the calibration of the model before discussing the results obtained for the impact of yuan internationalization.

Model calibration

In four cases, we have to simulate the model to know whether yuan internationalization will increase or decrease the reaction of the euro-dollar exchange rate to shocks. To get robust results, we successively simulate the model around a symmetric and an asymmetric equilibrium.

In the symmetric equilibrium, the three countries have equal sizes: asset supplies A j are normalized to 100. Additionally, the extent of financial openness is the same in the three countries (f i =0.2), and symmetric across the different currencies (f i j =0.1). Bilateral exchange rates are normalized to unity and NFAs are all equal to zero (see Appendix B, Table B.1). This first, symmetric equilibrium allows us to highlight possible asymmetries stemming from the Chinese asymmetric exchange-rate regime (in the peg case) while eliminating any asymmetry arising from the initial equilibrium. It is however far from the real world where the United States and the Euro area display comparable sizes, but China's size is about half the US size, at current exchange rates. 10 Additionally, as illustrated in Figure 1, both the US and the Euro area's NFA positions are negative, and the former is more negative than the latter, whereas the Chinese NFA is positive. Consistently, we simulate the model around a second, asymmetric equilibrium where asset supplies are 110 in the United States, 100 in the Euro area, but only 50 in China, and the US and Euro area's NFAs are negative. To close the model, we assume asymmetric portfolio choices where the yuan represents a small share (only 2.5%) of international portfolios, whereas the dollar represents as much as 40% of the Chinese portfolio (see Appendix B, Table B.2). These two sets of simulations are used to determine the impact of a variation in the share of the yuan in the Euro area's portfolio on the sensitivity of the euro-dollar exchange rate to some of the shocks. The results are depicted in Table 4 for both the floating and the peg regimes.

The impact of yuan internationalization under a floating regime

Shocks to asset supplies

In a floating regime, the dollar depreciates against both the euro and the yuan whenever there is an increase in the supply of dollar-denominated assets (A $ ). Following the dollar depreciation, US investors cut their demand for foreign assets in order to keep constant shares of each type of assets (in value) in their portfolios. When the share of the yuan is higher in US portfolios, the selling of Chinese assets by US investors is greater, which makes the dollar depreciate relatively less against the yuan than against the euro. In turn, Chinese investors see the yuan appreciate less against the dollar than against the euro. Hence they buy relatively more euros. On the whole, the euro appreciates more against the dollar than in the case where the yuan plays a minor role in the US portfolio.

Conversely, when there are more yuans in the European portfolio, then an increase in the dollar supply has less impact on the euro-dollar exchange rate. In this case European investors buy more yuan assets following the euro appreciation, which puts downward pressure on the euro.

The results are symmetrical for a shock on the supply of euro-denominated assets (A € ).

Finally, it can be observed from Table 4 that, whatever the share of yuan-denominated assets in US and European portfolios, the euro-dollar exchange rate is insulated to shocks on the supply of Chinese assets.

On the whole, then, a higher share of the yuan simultaneously in US and in European portfolios has benign effect on how the euro-dollar exchange rate reacts to shocks on asset supplies.

Shocks to NFAs

The results are somewhat different for shocks on NFAs. In the base case, a rise in NFA U (holding NFA E constant) makes the dollar appreciate against both the yuan and the euro. This is because, due to the home-bias, a transfer of wealth from China (whose NFA falls) to the United States (where it increases) raises the global demand for dollar-denominated assets. Suppose now that the share of the yuan increases in the US portfolio. Then, following the same transfer of wealth, US investors will increase by more their demand for yuan-denominated assets, which stabilizes the dollar-yuan exchange rate. If the yuan depreciates less, then Euro area's investors buy less of it, which also stabilizes the euro.

A higher share of the yuan in Euro area's portfolio also attenuates the impact of the shock on the euro-dollar exchange rate: since the euro depreciates, Euro area residents cut their demand for both dollar and yuan-denominated assets. If the yuan weighs more in their portfolio, they sell more yuan, which stabilizes the euro. Hence, the internationalization of the yuan unambiguously reduces the vulnerability of the euro-dollar exchange rate to a US NFA shock. Symmetrically, it makes the eurodollar rate also less reactive to a Euro area NFA shock.

Summing up

It can be concluded from this sub-section that, in a free-floating regime, a simultaneous rise of the yuan's weight in both US and Euro area's portfolio would make the euro-dollar exchange rate less vulnerable to bilateral imbalances between either the US or the Euro area and China; conversely, RMB internationalization would not fundamentally change the reaction of the euro-dollar exchange rate to shocks on the supply of either dollar or euro-denominated assets, while keeping the eurodollar rate insulated to shocks to yuan-denominated asset supply.

∂(∂(S € /S $ )/∂x)/ ∂f Y U ∂(∂(S € /S $ )/∂x)/ ∂f Y E Float A $ + + - A € - - + (a) A Y 0 0 0 NFA U - + + NFA E + - -(a) Peg on USD A $ - - + A € - + + (a) A Y + 0 - NFA U + - - NFA E + - -(a)
Reading: in the model, a rise in the supply of USD-denominated assets (rise in A $ ) makes the euro appreciate against the USD (rise in S € /S $, first column). This effect is magnified if the US residents hold more yuandenominated assets in their portfolios (higher f Y U , second column) but attenuated if Euro area residents hold more yuan-denominated assets in their portfolios (higher f Y E , third column). Source: Authors' calculations.

Notes: we assume NFA U <0, NFA E <0, 1 -f Y U -f C > 0 and 1 -f Y E -f C > 0.

Yuan pegged on the dollar

Shocks to asset supplies

We know from Section 3 that when the yuan is pegged on the dollar, a rise in the supply of dollardenominated assets (A $ ) leads the dollar to appreciate rather than depreciate against the euro. The reason is that Chinese authorities buy dollars to prevent the yuan from appreciating against the dollar. This counter-intuitive effect is magnified when there is more yuan in US portfolios, 11 because the rise in US wealth puts more pressure on the yuan (US investors want more Chinese assets), hence Chinese authorities buy more dollars to keep the bilateral exchange rate stable. Conversely, if there is more yuan in the European portfolio, then the counter-intuitive effect of A $ on the euro-dollar exchange rate is attenuated. In this case, the depreciation of the euro makes Euro area residents cut relatively more their demand for yuan-denominated assets, which alleviates the pressure on the 11 Here we limit ourselves to the case that the f Y U and f Y E stay small enough so that 1 -

f Y U -f C >0 and 1 -f Y E -f C >0.
Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

yuan and reduces the needs for Chinese authorities to buy dollars. On the whole, the simultaneous rise in yuan-denominated assets in US and Euro area portfolios has an ambiguous and probably limited effect on the reaction of the euro-dollar rate to shocks on the supply of dollar-denominated assets.

As for shocks on the supply of euro-denominated assets (A € ), they have the usual impact on the eurodollar exchange rate: a rise in A € leads to a depreciation of the euro. The internationalization of the yuan in either US or Euro area's portfolios tends to mitigate this impact: both types of investors cut by a larger amount their demand for yuan-denominated assets, which reduces the necessity for the PBoC to buy more dollars.

Finally, we note here that more yuan in Euro area's portfolio tends to attenuate the asymmetric impact of the supply of yuan-denominated assets on the euro-dollar exchange rate: when A Y rises, Chinese authorities sell dollar-denominated assets to prevent the yuan from depreciating against the USD. In doing so, they put downward pressure on the dollar against the euro. Hence the euro appreciates equally against the yuan and against the dollar. The European investor reacts by buying more foreign assets. If the weight of the yuan in the European portfolio is higher, there is relatively more demand for Chinese assets, which reduces the downward pressure on the yuan, hence also reduces the sale of dollars by China's central bank: the dollar depreciates less against the euro. Conversely, more yuan-denominated assets in US portfolios does not affect the asymmetric effect of a rise in A Y . This is because any additional purchase of these assets by US investors (following a dollar appreciation) is matched by an additional sale by Chinese authorities.

On the whole, it can be concluded that a simultaneous rise in the share of yuan-denominated assets in US and European portfolios has either a neutral or a mitigating impact on the impact of assetsupply shocks on the euro-dollar exchange rate.

Shocks to NFAs

Table 4 shows that the counter-intuitive effect of a shock on NFA U (a rise in US NFA triggers a dollar depreciation against the euro) is unambiguously attenuated by the internationalization of the yuan, be it in US or European portfolios. This is because the PBoC buys less dollars (to avoid a depreciation of the yuan) since both US and Euro area investors tend to buy more yuan in reaction to the increase in their wealth (US investors) or to the appreciation of their currency (Euro area's investors).

Summing up

One striking feature of the internationalization of the yuan is the attenuation of the counter-intuitive effects of US shocks on the euro-dollar exchange rate stemming from the asymmetric exchange-rate regime of China. Combining these results with those obtained under a free floating regime, it can be concluded that the internationalization of the yuan is able to (i) attenuate the reaction of the eurodollar exchange rate following bilateral, external shocks between the US (or Euro area) and China; (ii) attenuate the distortions generated by a fixed yuan-dollar peg; and (ii) be relatively neutral vis-à-vis shocks to asset supplies. It can then be concluded that yuan internationalization would help stabilize the euro-dollar rate, whatever the exchange-rate regime of China.

Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

A dynamic model

So far, we have considered net foreign asset positions (NFAs) as exogenous. Still, exchange-rate variations are deemed to impact on NFAs through trade imbalance accumulation. In this section, we consider NFA accumulation with endogenous trade balance. Like [START_REF] Blanchard | International Investors, the U.S. Current Account, and the Dollar[END_REF], we successively describe portfolio balance, NFA accumulation and trade balances.

The portfolio balance

Indexing by t NFAs and exchange rates at the end of period t, the dollar and euro market equilibria write (see Equations ( 2a) and ( 2b) where wealths are expressed as their functions of asset supplies and NFAs)12 :

( )( ) ( )         - - + + + + - = E t t t U t t Y C t E t E t U t U NFA S S NFA S A f S NFA A f S NFA A f A $, €, $, $ $, € $ €, $ $ 1 (8a) ( )( ) ( )         - - + + + + - = U t t t E t t Y C t U t U t U t E NFA S S NFA S A f S NFA A f S NFA A f A €, $, €, € €, $ € $, € € 1 (8b)
As usual, the equilibrium of the yuan market is ensured by the Walras law.

NFA accumulation

To see how NFAs accumulate over time, it is useful to start from the following expressions:

t C t C t E t E t U t U U t S W f S W f S W f NFA $, $ $, $ , € = - - (9a) t C t C t U t U t E t E E t S W f S W f S W f NFA , $ , $ $, = - - (9b)
NFA accumulation then depends on valuation effects (due to exchange-rate variations), on interest rates (r i , i=U,E,C) and on the trade balance of each country (TB i , i=U,E). For the United States, we have:

U t t t U E t E U C t t C t t C U t U Y t t t t E U t U U t TB S S r W f r W S f S S r W f S S S S r W f NFA + + - + - + + + - - - - - - - - - - 1 $, 1 , € 1 $ 1 1 $, $ $, 1 $, 1 1 $, 1 €, $, €, 1 € ) (1 ) (1 ) (1 / / ) (1 = (10)
By using Equation (9a) at time t-1, we can simplify the accumulation of the US NFA as:

U t U t U U C t t U Y U E t t t t U U t U t TB NFA r r r S S f r r S S S S f W NFA + + +                 + - + +         + - + - - - - - 1 $, 1 $, 1 $, 1 €, $, €, € 1 ) (1 ) (1 ) (1 ) (1 ) (1 / / = (11a)
Similarly, for the Euro area:

E t E t E E t t C E Y E t t t t U E E t E t TB NFA r r S S r f r S S S S r f W NFA + + +                 + - + +         + - + - - - - - 1 , € 1 , € 1 $, 1 €, $, €, $ 1 ) (1 ) (1 ) (1 ) (1 / / ) (1 = (11b)

Trade balance

Finally, we assume that the trade balance of each country depends on its bilateral exchange rate visà-vis each partner (price competitiveness effect) and on the comparative wealth of each partner (income effect13 ). Assuming price and income elasticities are homogenous across the three countries, and starting from a symmetric and balanced equilibrium, we have:

U E t t t t C t U t t t t U t z W S S S W W S S S TB +                 + + - +         - = $, €, $, $, $, €, 2 1 γ θ (12a) E U t t t t C t E t t t t E t z W S S S W W S S S TB +                 + + - +         - = €, $, €, €, €, $, 2 1 γ θ (12b)
Where θ,γ > 0, and z U , z E represent exogenous shocks to the trade balances. Replacing wealths by their functions of (exogenous) asset supplies and (endogenous) NFAs, we get: . In the pegged regime, S $ is exogenous but f $ C becomes endogenous. To further simplify the model, we assume all interest rates to be equal (r U =r E =r C =r=0.02). We then linearize the model around the symmetric and asymmetric equilibria described in Appendix B (Tables B.1 and B.2), successively, with θ=1 and γ=0.1. This allows us to study the impact of a permanent, asymmetric trade shock: a negative shock on the US trade balance (z U = -2) and a positive, albeit smaller shock on the Euro area's trade balance (z E =+1). 14 This means a positive shock of +1 on China's trade balance. Hence the shock is symmetric between the Euro area and China but asymmetric between these two and the United States. Here we present the results of the simulations around the symmetric equilibrium. Those obtained around the asymmetric equilibrium are qualitatively similar (see the graphs in Appendix B).

U U t t t t Y t t t U t z NFA S A S S A A S S S TB +         - + + - +         - = 2 3 2 2 $, € €, $, $ $, $, €, γ θ (13a) E E t t t t Y t t t E t z NFA S A S S A A S S S TB +         - + + - +         - = 2 3 2 2 €, $ $, €, € €, €, $,

a. Floating regime

The results under a floating exchange-rate regime are depicted in Figure 2. As expected, the dollar progressively depreciates against both the yuan and the euro (the euro stays stable against the yuan). The decline of the US' NFA, combined with a weaker dollar, progressively pushes the trade balance back to balance, and the NFA stabilizes. In the Euro area, the NFA rises but the appreciation of the euro against the dollar, combined with the income effect, pushes the trade account back to balance, so the NFA stabilizes.

b. Yuan pegged to the dollar

The results for a pegged regime are reported in Figure 3. In this case, the euro appreciates against the yuan despite the fact that the shock is symmetric between the Euro area and China. What happens is that the share of dollar-denominated assets increases in China's (official) portfolio in order to oppose any depreciation of the dollar against the yuan. Then, the euro can no longer appreciate against the dollar without appreciating against the yuan. In this regime, the US NFA falls much more than in the floating regime because the dollar only depreciates against the euro.

c. Internationalization of the yuan

To study the impact of the internationalization of the yuan on the dynamics of the model, we simulate the same shock as above under a floating and a pegged regime, successively, while setting the share of the yuan in both Euro area and US portfolio at zero (scenarios with no internationalization of the yuan, see Appendix B, Table B.3). The results are reported on Figures 4 and5. In both regimes, the internationalization of the yuan magnifies the impact of the trade shock on the euro-dollar exchange rate, but it reduces considerabily the impact of the shock on Euro area's net foreign asset positions. Hence the exchange rate plays better its role as a stabilizing device when the yuan is internationalized. The reason for this is the fact that, when the yuan is internationalized, its depreciation against the euro impacts negatively on the Euro area's NFA position, which compensates for the positive trade shock. The US NFA is also stabilized by the internationalization of the yuan, but only in the floating regime (where the dollar depreciates against the yuan, with positive valuation effects on the US NFA).

Conclusion

Based on a three-country, three-currency portfolio model, we have studied the impact of an internationalization of the yuan on the sensitivity of the euro-dollar exchange rate to various shocks, in two different exchange-rate regimes. We have shown that the counter-intuitive effects arising due to a fixed exchange rate in China (e.g. a transfer of wealth from China to the United States makes the dollar depreciate against the euro) are more limited when the yuan internationalizes, and that yuan internationalization also stabilizes the euro-dollar exchange rate in a floating regime. The dynamic version of the model further shows that yuan internationalization helps euro-dollar variations to stabilize net foreign asset positions following a trade shock, through enhanced valuation effects.

We believe these results to be encouraging for a multipolar international monetary system: even if it does not come immediately with a free floating regime, the internationalization of the yuan yields some stabilizing properties.

( ) ( ) ( ) (
)

Y E U U E E U U E E U E E C C E E C E C A NFA NFA NFA A NFA A A A f f f f NFA f f f f A f f f f S Γ + Γ - Γ - - - - + + = $ $ $ $ $ $ $ $ 1 With ( ) ( ) C E U C E U U f f f f f f $ $ € € 1 - + - - = Γ Γ E = f E 1 -f U -f $ C ( ) + f $ E f € U -f € C ( ) ( )( ) ( )( ) C E C U C E C U f f f f f f f f $ $ € € € $ 1 1 - - - - - - - = Γ
In normal circumstances, the Γs are positive, so the denominator of the expression for S $ is also positive (given the negative signs of NFA U and NFA E ). However, if China's portfolio choices differ greatly from those of the US or the Euro area, the Γs can turn negative.

A.2. Fixed peg of the yuan against the dollar

Resolution for the share of the dollar in China's portfolio:

(

)

E E Y C E E E E Y U U Y E E U U U U Y C C W f A f W f W f A W f A W f W f W f A W S f ) 1 ( ) 1 ( ) 1 ( ) 1 ( € $ € $ $ € $ $ $ - - -         - - - - + + - - =
Replacing W C by its function of NFA U and NFA E , we get:

( ) ( ) ( ) E U Y E U E Y E Y E Y Y E Y U C U E Y E C U U E E U E Y E C E U E U E C NFA A f NFA NFA A f S NFA A f A A f NFA A f f f S A A f f NFA A f f NFA NFA NFA S A f f NFA A A A f f f f f $ € $ € € $ $ € $ € 2 $ $ $ 1 € $ $ $ 1 ) ( 1 + + - - - + - - - Ψ + - Ψ - - = with (
)

U E C E U f f f f f $ 1 1 + - - = Ψ and ( )( ) ( ) U C E C E U f f f f f f $ 2 1 1 - + - - - =
f j i f € U = f Y U = 0.1 f $ E = f Y E = 0.1 f $ C = f € C =
f j i f € U =0.2 , f Y U = 0 f $ E =0.2 , f Y E = 0 f $ C = f € C =

Figure 1 :

 1 Figure 1: Net foreign asset positions, 1999-2007, in USD bn

  (a) Obtained by simulation of the linearized model around two alternative initial points -see Appendix B.

  ,b) (portfolio balance), (11a,b) (NFA accumulation) and (13a,b) (trade balance) jointly determine our six endogenous variables under a floating regime:

Figure 2 :

 2 Figure 2: impact of the trade shock, floating regime (Deviations from initial, symmetric equilibrium)

Figure 3 :

 3 Figure 3: impact of the trade shock, pegged regime (Deviations from initial, symmetric equilibrium)

Figure 4 :Figure 5 :

 45 Figure 4: Impact of the trade balance shock, floating regime (Deviations from symmetric equilibrium)

Ψ

  Documents de Travail du Centre d'Economie de la Sorbonne -2013

  du Centre d'Economie de laSorbonne -2013.16

Figure

  Figure B.1: Impact of the trade balance shock, floating regime (Deviations from initial, asymmetric equilibrium -Table B.2)

Table 1 : Accounting framework

 1

	USD	EUR	RMB	Total (wealth)

Table 2 : Impact of asset supplies and NFAs on the euro/dollar exchange rate, flexible regime

 2

x ∂(S € /S $ )/∂x x ∂(S e /S $ )/∂x

Table 3 : impact of asset supplies and NFAs on exchange rates, dollar peg X ∂(S € /S $ )/∂x x ∂(S € /S $ )/∂x

 3

  react to changes in the share of the yuan in US and Euro area portfolios, holding the share the euro (resp. the dollar) in the US (resp.

		European) portfolios
	constant (hence, constant f € U and f $	E

Table 4 : impact of RMB internationalization on euro/dollar behavior

 4

	x	∂(S € /S $ )/∂x

Table B .1: Symmetric equilibrium

 B

		United States	Euro area	China
	Exogenous variables
	A j	100	100	100
	f i	0.2	0.2	0.2

Table B .2: Asymmetric equilibrium

 B

			United States	Euro area	China
	Exogenous variables
	A j	110	100	50
	f i		0.2	0.2	0.5
	f j	i	f € U = 0.175 , f Y U =0.025	f $ E =0.175 , f Y E = 0.025	f $ C = 0.4 , f € C = 0.1
	Endogenous variables: baseline values
	NFA i	-24.6307	-7.614	-
	S j	1.2756	1.0738	-

Table B .3: Symmetric equilibrium with no internationalization of the yuan United States Euro area China

 B

	Exogenous variables
	A j	100	100	100
	f i	0.2	0.2	0.2

We also performed an analysis under a basket peg, but the non-linearity arising in this regime did not allow us to derive clear-cut results. Documents de Travail du Centre d'Economie de laSorbonne -2013.16

In Blanchard et al., we have f ¥ U =f ¥ E =0: non-Chinese investors do not hold yuan-denominated assets. Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

Hence we do not account for possible bandwagon effects due to extrapolative expectations. Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

In Figure1, the three NFAs do not sum to zero. To get a balanced framework, we would need to add the "rest

Here, we concentrate on the impact of China on the euro/dollar exchange rate. Results for the dollar/yuan exchange rate are reported in Appendix A. Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

This result is in line with Blanchard et al. (2005) ("extreme dollar preference" case). Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

In a more sophisticated model, Mazier and Tiou-Tagba Aliti also find the dollar to appreciate vis-à-vis the euro whenever there is a negative shock on the US trade balance, if the yuan is pegged on the dollar.

Note that this effect disappears if f U =f $ U (no internationalization of the yuan). Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

In 2011, GDP figures are USD bn 15,000, USD bn 13,100 and USD bn 7,300 for the United States, the Euro area and China, respectively (sources: World Bank and Eurostat). Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

Asset supplies are kept exogenous and independent of time. Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

Strictly speaking this is not an income effect since it depends on wealth rather than income. However, since interest rates are assumed exogenous in our framework, there is no difference between income and wealth effect. Note that[START_REF] Blanchard | International Investors, the U.S. Current Account, and the Dollar[END_REF] only consider the exchange-rate effect. Documents de Travail du Centre d'Economie de laSorbonne -2013.16

These shocks represent -2 and +1 percent of US and Euro area's wealth, respectively. Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

Documents de Travail du Centre d'Economie de la Sorbonne -2013.16

We are grateful to Jean Pisani-Ferry for his contribution at the early stage of this research, and to the participants in the CESIfo-CEPII-GEP conference on China

Appendix A

Analytical resolution of the static model

A.1. Flexible regime

Resolution for the dollar-yuan exchange rate:


Carmen Camacho

Spatial migration

Keywords: Migration, Spatial dynamics, Economic Growth, Parabolic PDE, Optimal Control, C61, R11, R12, R13

We develop a model economy adapting Hotelling's migration law to make individuals react to the gradient of their indirect utility. In a first version, individuals respond uniquely to utility differences. In a second phase, we insert our migration law as a dynamic constraint in a spatial model of economic growth in which a policy maker maximizes overall welfare. In both cases we prove the existence of a unique solution under certain assumptions and for each initial distribution of human capital. We illustrate some extremely interesting properties of the economy and the associated population dynamics through numerical simulations. In the decentralized case in which a region enjoys a temporal technological advantage, an agglomeration in human capital emerges in the central area, which does not coincide with the technologically advanced area. In the complete model, initial differences in human capital can trigger everlasting inequalities in physical capital.

Introduction

In 1929 Harold Hotelling proposed a provocative vision of economics on space. He postulated a diffusion nature for human migration. Under this vision, individuals react to changes in salaries and move like a flow along space. Our objective in this paper is to frame this behavior in a standard economic environment. Agents obtain utility from their lifetime consumption and they know the mechanisms behind migration and production.

In 1951, Skellam rediscovered and applied the very same migration law in Biology.

He proposed different diffusion laws for the time-space dynamics of forests and various species of small animals, that is non-human populations. The difference with Hotelling is that population diffusion depends on the population concentration, the fertility and mortality rates which are location specific. Noteworthy, the resulting dynamics coincide with the trajectories of a model in which individuals move randomly, or more precisely, in which individuals move following a brownian motion.

There was no application to economics until 1985 and the publication of two works: [START_REF] Puu | A simplified model of spatiotemporal population dynamics[END_REF] and [START_REF] Beckmann | Spatial economics: density, potential, and flow[END_REF]. Retaking Hotelling's law, they believe that population diffusion is due to spatial productivity differentials and not just to fertility and mortality rates. [START_REF] Puu | On growth and dispersal of populations[END_REF] puts Hotelling's law in perspective, integrating both Skellam and Hotelling's law. Indeed, Skellam's law is better suited to describe migration in an economy where there is no growth. Hence Puu analyzes migration systematically starting by Skellam's no-growth case and then adding production to the framework.

In practical terms this means that the diffusion term is made of two factors: the first directs population towards less populated areas and the second towards areas offering higher wages. After describing the different laws, [START_REF] Puu | On growth and dispersal of populations[END_REF] focuses on the stationary solutions and their stability.

Several authors have produced spatial models in the same direction, analyzing the economic motives that urge people to migrate. [START_REF] Beckmann | On spatial population dynamics[END_REF] enunciates and solves the spatial problem presented by Hotelling assuming a convex-concave production function. Although he can prove that the spatially homogeneous distribution of individuals is a stable equilibrium solution, he cannot prove the unicity of the solution. The work closest to ours in the migration literature is [START_REF] Alvarez | Estimation of a Continuous Spatio-Temporal Population Model[END_REF], who propose a migration law which captures three decisive demographic factors: population growth, diffusion and migration drift towards preferred regions; and two other relevant factors 1 Documents de Travail du Centre d'Economie de la Sorbonne -2013.17 in migration: climate and economic outcomes. They test their theoretical intuition with US data, which confirm their hypothesis. There are however remarkable differences between this and the present work. First, their setting is built over a discrete space-time structure, for which the existence and uniqueness of the solution is not a problem. Second, the decisive factors in migration are exogenous, that is, population movements do not affect the economic environment: production, productivity, technology or environmental quality.

All afore mentioned literature assumes that individuals follow demographic flows, or at best a combination of the gradients corresponding to the climate and the economy.

In the present work we shall perform a similar exercise first. There is however an important difference since we endogenize locations' economic performance. Indeed, since production requires labor, the final outcome at each location depends on the size of its population. Afterwards, we consider an economy led by a policy maker who decides on the optimal trajectory for consumption per capita in an open and bounded one dimensional region, during a time span that stretches from 0 to T . Solving this problem we can obtain the optimal distribution of population in space and time. The optimal solution represents the best economic outcome since the policy maker has perfect foresight and can take into account all simultaneous movements, economic decisions and their consequences on current and future outcomes.

We prove the existence and uniqueness of the solution to both problems, that is the centralized and the decentralized versions, but unfortunately, we cannot provide detailed descriptions of the behavior of the dynamic trajectories. Using an algorithm developed in [START_REF] Camacho | On the dynamics of capital accumulation across space[END_REF], we illustrate the dynamical properties of our models in section (5). In the decentralized case we have focused on two issues: the relevance of an heterogenous initial distribution and of technological differences. As expected, we show that initial disparities in human capital disappear with time, if this is the unique difference across space. On the other hand, we show that a population agglomeration is possible if a region enjoys, even if temporarily, a technological advantage. Most striking: the agglomeration does not coincide with the technologically advanced region. When we simulate the solution to the centralized economy, we find another interesting result.

Starting from an heterogenous distribution of population and a spatially homogenous distribution of physical capital, human capital reaches an homogenous distribution on space whereas we observe ever-lasting differences in physical capital. This article is structured as follows. We present our adapted version of Hotelling's migration law in section 2. Section 3 studies the decentralized problem while section 4 tackles the centralized economy. Numerical exercises are reproduced in section 5.

Finally, we conclude in section 6.

2 A version of Hotelling's migration law [START_REF] Hotelling | Stability in competition[END_REF] migration law postulates that individuals move according to the gradient of salaries. In this logic, individuals move towards the locations with the highest salaries. As mentioned in the introduction, this is also the modeling strategy adopted by [START_REF] Puu | On growth and dispersal of populations[END_REF], Puu and Beckmann (1989), [START_REF] Beckmann | On spatial population dynamics[END_REF], etc. In contrast with this assumption, we assume that individuals move following the gradient of their utility, that is, individuals move towards locations which provide higher utility, that we call by u (we provide more details on u in section 3). Denoting by h(x, t) the amount of individuals at location x and time t, the dynamics of human capital are described by the following parabolic partial differential equation

h t (x, t) -h xx (x, t) = ∇ x u(c(x, t)), (1)
where x is the gradient in x. We can rewrite (1) as

h t (x, t) -h xx (x, t) = u (c(x, t))c x (x, t). (2)
In this article, we do not study human capital formation so that we are implicitly assuming that one individual possesses one unit of human capital.

A simple model for spatial migration

Let us start with a simple model to describe individuals migration in an open region R ⊂ R of space as a response to changes in their utility. 1 We assume that the initial distribution of individuals is known and on the production side, that the production of the unique consumption good only requires labour. Following the classic literature, labour transforms into the consumption good according to a neoclassical production function. Denoting output by Y , we have that

Y (x, t) = A(x, t)F (h(x, t)),
where A(x, t) is the technology available at location x at time t. F is a function that satisfies the following assumption:

(H1) F is a positive, increasing, concave function:

F (•) ≥ 0, F (•) ≥ 0 and F (•) ≤ 0.
To simplify our framework, we assume there is no trade in the final good so that individuals at any location can only consume their production, that is

c(x, t) = A(x, t)F (h(x, t)).
Individuals measure their welfare by means of a utility function u, which verifies (H2) u is a positive, increasing and concave function of individual's consumption.

u(•) ≥ 0, u (•) ≥ 0 and u (•) ≤ 0.
With all these elements in hand, we can write the migration law in (2) as:

h t -h xx = u (AF (h)) [A x F (h) + AF (h)h x ] . (3)
Gathering the partial derivatives of h on the left hand side, we can rewritte (3) as

h t -h xx -u (AF (h))AF (h)h x = u (AF (h))A x F (h). ( 4
)
We can assume without loss of generality that points in space are ordered such that

A x (x, t) ≥ 0, ∀t.
The assumption A x ≥ 0 ensures that the right hand side of (4) is positive.2 In addition to (4), we need boundary conditions to describe the behavior of the state variable along δR, R's border. Following [START_REF] Boucekkine | Bridging the gap between growth theory and the new economic geography: The spatial Ramsey model[END_REF], we adopt the following boundaries of the Neuman type:

lim x→δR h x (x, t) = 0, for all t ∈ [0, T ]. ( 5
)
We believe this condition on the boundary is less restrictive the any other alternative since we are only forbidding migration at the borders. Indeed, the alternative would be to provide the values of human capital at the borders for every t ∈]0, T ]. This second option is far more constraining since it will condition the solution very heavily.

We know that under certain assumptions, the problem (4)-( 5) has a unique solution:

Theorem 1 Let assumptions (H1) and (H2) hold and assume that

u (AF (0))A x F (0) ≥ 0, h(x, 0) ≥ 0,
hold but not all the two are identically zero. Then, problem (4)-( 5) has a unique positive solution.

Before providing the theorem's proof, note that in a standard economic framework, u (AF (0))A x F (0) = 0, because it is assumed that F (0) = 0. We do not need this assumption here. On the contrary, it would be meaningful to assume F (0) = 0. Indeed, if F (0) = F > 0, then the economy could produce a minimum amount F without human capital. A strictly positive F translates the existence of productive factors other than human capital, which are not modeled but which are necessary to produce like physical capital (introduced in the following section), infrastructures, social capital, etc.

Proof of theorem 1. Our proof relies on theorem 2.5.2 in [START_REF] Pao | Nonlinear Parabolic and Elliptic Equations[END_REF]. In order to apply Pao's result we only need to ensure the existence of a positive upper solution to (4). By definition, ĥ is an upper solution of (4) if ĥtĥxx -u (AF ( ĥ))AF ( ĥ) ĥx ≥ u (AF ( ĥ))A x F ( ĥ).

Next, we need to name the borders of R. We can assume without loss of generality that R is connected so that δR is made of two real numbers: R 1 and R 2 .3 Let us define a positive upper solution ĥ on R:

ĥ(x) = (R 2 -R 1 )β -β(x -R 1 ),
where β = sup {x,t}∈R×[0,T ] A(x, t). This ends the proof.

Optimal population dynamics

We present in this section a more complete version of the migration model: there exists a policy maker in our economy that maximizes aggregated welfare over a finite time horizon. The policy maker has to choose the optimal trajectory for consumption, taking into account the consequences that her choices will have on the dynamic evolution of human and physical capital, denoted by k. To measure the region's welfare the policy maker aggregates discounted utility. The policy maker maximizes:

max c R T 0 u(c(x, t))e -ρt dxdt + R Ψ(h(x, T ))e -ρT dx + R Φ(k(x, T ))e -ρT dx.
The term e -ρt in the objective function serves to discount utility with time. It is assumed that the further in time consumption takes place, the less it matters. Time discount helps the policy maker evaluate consumption trajectories, providing its present value.

The functions Ψ and Φ value the final states of human and physical capital, respectively. We assume that both Ψ and Φ are continuously differentiable, Ψ, Φ ∈ C 1 .

As in the previous model, there exists a unique good which is produced at each location endowed with positive amounts of physical and human capital. However, contrary to the first model, the final good can be either consumed or invested in the generation of future capital. We assume a neoclassical production function, f (k, h), which satisfies (H3) f (•, •) is continuous and continuously differentiable in each of its arguments.

In this first attempt to model migration in continuous space, physical capital does not move in space and consequently, individuals can only invest in their own location.

Therefore, we can describe the dynamics of physical capital as

k t (x, t) = f (k(x, t), h(x, t)) -δk(x, t) -c(x, t), (6)
in which we take into account that physical capital depreciates over time at a rate δ.

Existence and uniqueness

We can gather all the elements and write the policy maker problem as an optimal control problem in which the policy maker has to choose the consumption trajectory

{c(x, t) : x ∈ R, t ∈ [0, T ]} that solves max c T 0 R u(c(x, t))e -ρt dxdt + R Ψ(h(x, T ))e -ρT dx + R Φ(k(x, T ))e -ρT dx (7)
subject to:

                         h t (x, t) -h xx (x, t) = u (c(x, t))c x (x, t), k t (x, t) = f (k(x, t), h(x, t)) -δk(x, t) -c(x, t), h(x, 0) = h 0 (x) ≥ 0, k(x, 0) = k 0 (x) ≥ 0, lim x→δR h x (x, t) = 0, (8)
for all x ∈ R, t ∈]0, T ]. Similarly to theorem 1 our first result ensures the existence of a solution for (8) for given c under certain assumptions:

Theorem 2 Under assumptions H2 and H3 and for given c, the system of PDE described in (8) has a unique positive solution if the initial condition for human capital is not identically zero.

Proof. Notice that the PDE for h does only depend on h and c, so that we can prove the existence of a solution for h given c, independently of k. Then, we shall proceed as in theorem 1 and apply theorem 2.5.2 in [START_REF] Pao | Nonlinear Parabolic and Elliptic Equations[END_REF], that is, we need to ensure the existence of a positive upper solution for

h t (x, t) -h xx (x, t) = u (c(x, t))c x (x, t). (9)
We can build an upper solution ĥ as ĥ(x, t) = u (c)ct, where c = sup x,t c(x, t). By definition, ĥ ≥ 0. Then, we can compute ĥ partial derivatives: ĥt = u (c)c, and ĥxx = 0. We have proven then that ĥ is a positive upper solution, ensuring the existence of a unique solution h to (8) for a given c.

Finally, for the unique solution couple (h, c), we can consider the dynamic equation for k. For every x, (6) is an ordinary differential equation in t with known initial value k 0 (x). Under H3 we can ensure that the solution to ( 6) is unique for every x.

Optimal necessary conditions

We use the Ekeland variational method to obtain necessary conditions in the form of Pontryagin's principles. We follow the method developed by [START_REF] Raymond | Pontryagin's principle for state-constrained control prob-lems governed by parabolic equations with unbounded controls[END_REF][START_REF] Raymond | Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations[END_REF][START_REF] Raymond | Time optimal problems with Boundary controls[END_REF]. We can write the value function V associated to our problem in which we shall penalize the objective function introducing the state constraints:

V (c, h, k, λ, µ) = R T 0 u(c(x, t))e -ρt dtdx + R Ψ(h(x, T ))e -ρT dx + R Φ(k(x, T ))e -ρT dx -R T 0 λ(x, t) (h t (x, t) -h xx (x, t) -u (c(x, t))c x (x, t)) dtdx -R T 0 µ(x, t) [k t (x, t) -f (k(x, t), h(x, t)) + δk(x, t) + c(x, t)] dtdx. ( 10
)
Let us assume that an optimal solution exists {(c(x, t), h(x, t), k(x, t)) : (x, t) ∈ R × [0, T ]}. Then, we can write any other solution to our problem as the addition of the optimal solution plus a deviation from it, measured by a parameter :

           c(x, t) = c * (x, t) + C(x, t), h(x, t) = h * (x, t) + H(x, t), k(x, t) = k * (x, t) + K(x, t). (11)
With this new writing, we can express V as a function of . Then, in order to obtain optimality conditions, we can take the derivative of V with respect to and equate it to zero so that we minimize the distance to the optimal solution. However, before doing so, we need to re-arrange some terms of V using integration by parts:

V ( ) = R T 0 u(c(x, t))e -ρt dtdx + R Ψ(h(x, T ))e -ρT dx + R Φ(k(x, T ))e -ρT dx + R T 0 h(x, t) (λ t (x, t) + h x,x (x, t)) dxdt -R λ(x, t)h(x, t)| T 0 dx + T 0 λ(x, t)h x (x, t)| δR - T 0 h(x, t)λ x (x, t)| δR dt -R T 0 λ x (x, t)u(c(x, t))dtdx - T 0 λ(x, t)u(c(x, t))| δR dt -R µ(x, t)k(x, t)| T 0 dx + R T 0 k(x, t)µ t (x, t)dtdx -R T 0 µ(x, t) [-f (k(x, t), h(x, t)) + δk(x, t) + c(x, t)] dtdx. ( 12
)
Now we can take derivatives with respect to which yields:

V ( ) = R T 0 u (c(x, t))e -ρt C(x, t)dtdx + R Ψ (h(x, T ))H(x, T )e -ρT dx + R Φ (k(x, T ))K(x, T )e -ρT dx + R T 0 H(x, t) (λ t (x, t) + h x,x (x, t)) dxdt -R λ(x, T )H(x, T )dx -R T 0 λ x (x, t)u (c(x, t))C(x, t)dtdx -R µ(x, T )K(x, T )dx + R T 0 K(x, t)µ t (x, t)dtdx -R T 0 µ(x, t) (-f k (k(x, t), h(x, t))K(x, t)dtdx -R T 0 µ(x, t) (-f H (k(x, t), h(x, t))H(x, t) + δK(x, t) + C(x, t)) dtdx. ( 13
)
Note that in parallel to the boundary conditions of the state variable, we need that for all x ∈ δR: h(x, t)λ x (x, t) = 0, λ(x, t)u(c(x, t)) = 0.

(

) 14
One can obtain necessary conditions to our maximization problem grouping together the terms that multiply C, H, K, etc. as follows:

4              C : u (c)e -ρt -λ x u (c) -µ = 0, H : λ t + λ xx + µf h = 0 K : µ t + µ (f k + δ) = 0, (15)
plus an additional terminal condition for λ and µ, for every x ∈ R :

Ψ (h(x, T ))e -ρT -λ(x, T ) H(x, T ) = µ(x, T ) -Φ (k(x, T ))e -ρT K(x, T ).
Since H and K will remain unknown distances, the above condition needs to be split into two:

   λ(x, T ) = Ψ (h(x, T ))e -ρT , µ(x, T ) = Φ (k(x, T ))e -ρT . (16)
We can transform the non-autonomous equation linked to C, re-defining the co-state variables λ as λe ρt and µ as µe ρt . The set of conditions in ( 15) and ( 16) becomes

                         λ t + λ x,x + µf h -ρλ = 0 µ t + µ (f k + δ -ρ) = 0, λ(x, T ) = Ψ (h(x, T )), ∀x ∈ R µ(x, T ) = Φ (k(x, T )), ∀x ∈ R h(x, t)λ x (x, t) = 0, x ∈ δR. ( 17
)
Once we obtain a solution to (17), we compute c using the remaining necessary conditions, that is u (c) (1 -λ x ) = µ plus the boundary condition λ(x, t)u(c(x, t)) = 0 for x ∈ δR .

Theorem 3 Under assumptions H2 and H3, if (h, k) is a solution of (8), then there exists at least a couple (λ, µ), which solves (17).

Proof. We apply here theorem 2.1. in [START_REF] Raymond | Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations[END_REF].

The reverse time parabolic equation

We hit here a cornerstone of economic modeling in a continuous space setting. Contrary to standard growth models, the modeler cannot work with an infinite time horizon. It is the time reversed nature of the parabolic PDE associated to λ which forces the modeler to choose a finite horizon since one can not reverse time in an infinite horizon problem.

We explain the importance of this point next.

The PDE associated to λ in ( 17) does not belong to any of the big families of partial differential equations whose behavior have been analyzed in the literature. Nevertheless, if one reverses time, that is, if we change λ by λ(x, t) = λ(x, T -t), then we obtain a parabolic equation

λt (x, t) -λxx (x, t) -µf h (k(x, T -t), h(x, T -t)) + ρ λ(x, t) = 0. ( 18
)
This change of variable is necessary in order to prove the existence and uniqueness of solution to (17). It is clear now why we cannot study infinite horizon economies: if T = ∞ then we cannot reverse time in the PDE for λ and hence we cannot prove the existence of a solution nor its uniqueness. We say in this case that the problem is ill-posed.

We prepare now the problem for the numerical simulations that follow in section 5.

For the simulation purposes, we are interested in reducing the size of our system. Note that we can substitute µ from the first equation into the second:

λt (x, t) -λxx (x, t) -µf h (k(x, T -t), h(x, T -t)) + ρ λ(x, t) = 0,
and gather the equations linked to C and K to reduce the dimension of the system, eliminating µ. If we do so, we obtain the dynamics of consumption as follows:

u (c) u (c) c t - λx,t 1 -λx = ρ -δ -f k .
We can write our problem as the following system of partial differential equations for 10 Documents de Travail du Centre d'Economie de la Sorbonne -2013.17

which the uniqueness of the solution has been proven for every x ∈ R and t ∈]0, T ]:

                                       h t (x, t) -h xx (x, t) = u (c(x, t))c x (x, t), λt (x, t) -λxx (x, t) -µf h (k(x, T -t), h(x, T -t)) + ρ λ(x, t) = 0, k t (x, t) = f (k(x, t), h(x, t)) -δk(x, t) -c(x, t), u (c(x,t)) u (c(x,t)) c t (x, t) -λx,t(x,t) 1-λx(x,t) = ρ -δ -f k (k(x, t), h(x, t)), h(x, 0) = h 0 (x) ≥ 0 and k(x, 0) = k 0 (x) ≥ 0, λ(x, 0) = Ψ (h(x, 0)), lim x→δR h x (x, t) = 0, lim x→δR λ x (x, t) = 0 and λ(x, t)u(c(x, t)) = 0 if x ∈ δR. (19)
Due to the complexity of the above system, we devote next section to numerical exercises that will help to illustrate its dynamic characteristics.

Numerical exercises

In this section, we illustrate numerically some of the characteristics of the economies described in sections 3 and 4. Through the simulations, we try to provide an answer to a crucial question: can initial disparities vanish with time? Although disparities may arise from many fronts, we focus here on disparities on the initial endowment of human capital and on production technology. Our aim is to furnish an answer which takes into account population movements. For example, locations endowed with less initial human capital can offer higher salaries, attracting individuals so that disparities will disappear.

However, if the technology level is not high enough, then these locations cannot offer larger salaries implying further loss of human capital since migration will take place towards more rewarding regions.

The section is divided in two: the first subsection illustrates the decentralized model, in which individuals can freely move in space, whereas the second one concerns the model with a policy maker who computes the optimal distribution of individuals in time and space.

Without loss of generality, space is defined as the interval [0, 1] that we divide in 2 regions of equal size, [0, 0, 5] and [0, 5, 1]. Although the two regions setting may seem oversimplifying , it helps underlining the model's driving forces. The parameters values adopted in the exercises are shown in Table 1.

A

The decentralised economy

The reader familiar with the heat equation knows that initial differences in temperature disappear in time, all else equal. Although the migration law in (4) could look like a heat equation it is not one, due to the term u (c)c x . In effect, the utility gradient involves a term on h x and another on A x once we substitute c for Ah α . We investigate whether initial differences disappear with time. In the first exercise, initial population is unequally distributed:

h(x, 0) =    100, if x ∈ [0, 0, 5], 10, if x ∈ [0, 5, 1]. ( 20
)
As figure 1 shows, initial differences disappear very quickly, as with the heat equation.

Next, we explore the behavior of the model when one of the regions enjoys a technological supremacy. If a region enjoys even if temporarily a sufficiently large advantage, then ever-lasting differences are created. To illustrate this point we assume that technology is defined as

A(x, t) =              10, if x ∈ [0, 0, 5], ∀t, 10 + 10(5 -t), if x ∈ [0, 5, 1], ∀t < 5, 10, if x ∈ [0, 5, 1], ∀t ≥ 5.
An agglomeration of individuals emerges around the central location from the beginning (see figure 2). From start, population strongly agglomerates around the center.

Then, the agglomeration area spreads but the hump shaped distribution of individuals 12 Documents de Travail du Centre d'Economie de la Sorbonne -2013.17 persists in time. One may wonder how this agglomeration of human capital is possible. Indeed, this solution could not be reached nor foreseen in a standard model in economic growth with a finite number of locations. A classical analysis in a setting with a finite of locations would predict a migration flow from the low salary region towards the high salary region until salaries are equalized.

On the one hand, the better technology in [0.5, 1] increases productivity and regional salaries, all else equal. This force attracts human capital into [0.5, 1]. However, on the other hand, once human capital increases their marginal productivity decreases and hence their salaries. In the end, central locations larger than 0.5 enjoy a better technology which will attract individuals. Nevertheless, their agglomeration lower salaries.

As a result, locations close to 0.5 but smaller than 0.5, can benefit from this agglomeration since they may offer higher salaries because human capital marginal productivity is larger.

The policy maker problem

To enrich our understanding of spatial migrations, we would like to illustrate the policy maker problem with a numerical exercise using the algorithm developed in [START_REF] Camacho | On the dynamics of capital accumulation across space[END_REF]. We assume a standard utility function exhibiting constant elasticity of 13 Documents de Travail du Centre d'Economie de la Sorbonne -2013.17 Individuals are heterogenously distributed while physical capital is spatially homogenous. In particular, h(x, 0) is as in (20) and k(x, 0) = 10, for all x ∈ R.

The most salient outcome is that the human capital heterogeneity triggers an unequal accumulation of physical capital (see figures 3 and 4). Indeed, regions with more human capital accumulate more physical capital. Note that since physical capital is not allowed to flow over space, these differences persist in time and this despite the steady homogenization of human capital with time.

Once again, this result is somewhat at odds with the standard literature in growth theory. It is usually predicted that 2 regions endowed with the same production technology, even with unequal levels of physical capital will reach the same level in the long-run, when their marginal productivities are equal. 6 The usual adjustment mechanism that allows convergence is the location's savings: it is expected that a rich region with low marginal productivity of capital will save less and diminish its level of capital.

At the same time, the poorly endowed region, with relative high marginal productivity saves more and increases its stock of capital. In this framework we have an additional adjustment tool, namely human capital migration. A region can adapt its savings but adjustment is faster if human capital flows towards more rewarding regions. We have shown with this simple numerical exercise that spatial convergence is not accomplished even when individuals freely migrate. Nevertheless, we believe that all locations in space will enjoy an equal allocation of human and physical capital if both factors could flow in space without barriers.

Conclussions

Building upon Hotelling's seminal work, we have developed a model economy adapting Hotelling's migration law to make individuals react to the gradient of their indirect utility rather than to the gradient of salaries as in the original law. We have proceeded in two steps, first we analyze the dynamics of human capital as if individuals responded uniquely to utility differences. Under certain assumptions on the utility function, we can prove the existence of a unique solution for each initial distribution of human capital.

In a second phase, we insert our migration law as a dynamic constraint in a standard model of economic growth in which a policy maker maximizes overall welfare. In this case too we can prove the existence of a unique solute under certain assumptions and for each initial distribution of human capital.

Unfortunately, we cannot obtain further analytical results and we need to proceed through numerical simulations. Although these do not constitute formal proves, they shed light on some extremely interesting properties of the economy and the population dynamics. In the decentralized case in which a region enjoys a temporal technological advantage, an agglomeration in human capital emerges in the central area. The most striking part of this result is that this agglomeration surges from the beginning and that the central area is made of locations both from the technologically advanced subregion and from the less-advanced subregion. In the simulation of the complete model, we find that initial differences in human capital can trigger everlasting inequalities in the locations stock of capital.

The framework we have presented pretends to be the first stone of a more sophisticated world description. We leave for further research the comparison of the optimal solution with the decentralized solution, using as measure aggregated welfare over space and time. Also interesting would be to address the issues of social integration, complementarity between different types of labor. One could produce a more sophisticated model introducing the education sector, private or publicly funded, etc.

16 Documents de Travail du Centre d'Economie de la Sorbonne -2013.17

Figure 1 :

 1 Figure 1: Decentralised economy. Human capital dynamics when initial population is heterogeneous.

Figure 2 :

 2 Figure 2: Decentralised economy. Human capital dynamics when technology is heterogeneous

Figure 3 :

 3 Figure 3: Centralised economy. Human capital dynamics when initial population is heterogeneous.

Figure 4 :

 4 Figure 4: Centralised economy. Physical capital dynamics when initial population is heterogeneous.

Table 1 :

 1 Parameter values for section .

	Technology	1
	α Share of human capital 0.75
	δ Depreciation rate	0.05
	ρ Time discount rate	0.015
	Total Population	90

Documents de Travail du Centre d'Economie de la Sorbonne -2013.17

Results can be easily extended to the case R ⊂ R n , n > 1.

Note that we can make this assumption without loss of generality because if A x was non-positive, we could re-order space to obtain A x > 0.

If R was not connected, then we would apply the above proof for each of R's connected components.5Documents de Travail du Centre d'Economie de la Sorbonne -2013.17

For the ease of exhibition, we abstract from spatial-temporal references. We shall write k for k(x, t), h for h(x, t), etc.

Note that with σ = 1.25, the objective function becomes negative. This is by no means a problem since the utility function satisfies all required assumptions.

See for instance the β-convergence concept in[START_REF] Barro | Economic Growth[END_REF].

Appendices

A An algorithm to solve forward-backward systems of parabolic PDE

We start building a grid in discrete time and space that will correspond to the continuous setting [0, R] × [0, T ]. Space will be an interval made of R * Dx, that is, the grid points are separated by a distance Dx, i.e. [0, R] ≈ {Dx, 2 * Dx, 3 * Dx, ..., R * Dx}. Similarly, the time interval will be divided into T * Dt subintervals and we will consider the extreme points of these. If in continuous time variables were evaluated at (x, t), their discrete versions will be evaluated at the grid vertices (n, j) where n = 1, ..., N and j = 1, ..., T * Dt.

Once we have built the grid, we need to discretize all variables and write appropriate definitions for their partial derivatives. We shall adopt the following definitions:

where n is the closest point of the spatial grid to x, which is computed as n = f loor(x * Dx). Then, n is the closest time point in the grid to t, n = f loor(t * Dt).

Then, the discrete version of our original equation in ( 4) is: