Jean-Baptiste Gossé
email: gosse@univ-paris13.fr

The Real and Financial Implications of the Global Saving Glut: A Three-Country Model

Keywords: F21, F32, F41, F47 International Macroeconomics, Global Imbalances, Balance of Payments, International Finance , Simulation and Forecast

The model presented in this paper has two objectives. First, it models global imbalances in a simple way while conserving real andnancial approaches. This double approach is necessary because Global Imbalances are due to the conjunction of nancial and real phenomena: the increase in the price of commodities, the accumulation of foreign reserves by the Asian central banks, the limited absorption capacity of the OPEC countries, the insucient development of the Asian nancial system and the perception of better returns in the US.

The second objective is to model the global saving glut hypothesis and to show its implications. We start with a model which consists of three identical countries and then we replicate the current pattern of global imbalances in introducing three asymmetries: a xed exchange rate between Asia and the United States, a limited absorption capacity in Asia and endogenous propensity to spend in the United States. In order to avoid the recession linked to the increase of their propensity to import, the United States increase their propensity to spend. This adjustment has a cost: (i) the Global Imbalances grow quickly with an increase of current account imbalances and net foreign assets in both the US and Asia ; (ii) the euro area supports an appreciation of its exchange rate which put it in a long depression.

Introduction

During the last years, the Global Imbalances have been increasing. The weight of both current account imbalances and net foreign assets in the 2006 world economy is twice what it was in 1996 (gure 1). If we choose an approach by the real sector, the two big reasons for this growth of Global Imbalances are the increase of oil prices (gure 2) and the under evaluation of the Renminbi (gure 3). The three major players of the GI are the US, for decits, and the OPEC and the Asian countries, for surpluses. The US suered the brunt of rising price of oil and the under evaluation of Renminbi and, since the beginning of the millennium, their propensity to spend has been increasing. This paper presents the alternatives of the US to respond to this shock and the consequences of their reactions, rst, on the countries with exible exchange rates and, second, on the countries with xed exchange rates.

The model presented in this paper has two objectives. First, it models global imbalances in a simple way while conserving real and nancial approaches (the Three-Country model presented here is comprised of 50 equations). This double approach is necessary because Global Imbalances are due to the conjunction of nancial and real phenomena: the increase in the price of oil, the accumulation of foreign reserves by the Asian central banks (gure 5), the limited absorption capacity of the OPEC countries, the insucient development of the Asian nancial system and the perception of better returns in the US. The second objective is to model the global saving glut hypothesis and to show its implications. We want to demonstrate that in the present context, the increase in the US propensity to spend would oset negative impact on their propensity to import, but with an increase in their net foreign debt. The United States can temporarily avoid a recession by accumulating debts on the rest of the world as long as xed exchange rate countries accept to accumulate dollar assets. In contrast, the exchange rate of the euro area is the only one to adjust and the impact on European GNP is negative.

The second part briey describes the evolution in the economic literature on global imbalances. The third part presents the model with three identical exible exchange rate countries. We introduce the Global Saving Glut Hypothesis in the fourth part.

A brief review of the literature on Global Imbalances

The article by Dooley, Folkerts-Landau and Garber (2003) marks the return at the forefront of the issue of current imbalances in economic literature. They argue that the Bretton Woods system has returned so their work has been named Bretton Woods II. The authors consider that the global imbalances of the 2000s result from an export-led growth strategy of Asian countries that is similar to the one adopted by Europe and Japan after the Second World War. In both cases, the surplus countries accumulate reserves so that their exchange rate remains undervalued relative to the dollar. They can maintain their competitiveness and benet from growth by exports. However, as [START_REF] Eichengreen | Global Imbalances and the Lessons of Bretton Woods[END_REF] pointed out, the resemblance to Bretton Woods is limited. The current international monetary system is a oating exchange rates one and the consistency between the Asian countries in the years 2000 is lower than that of Europe during the 1960s. In addition, the thesis of Bretton Woods II considers the global imbalances resulting from the real sector, i.e. the top of the balance of payments, and the nancial sector -i.e. the bottom of the balance of payments -adjust to it.

This vision of global imbalances from the top of the balance of payments is not adopted in the following papers. The Global Saving Glut Hypothesis, supported by [START_REF] Bernanke | The Global Saving Glut and the U.S. Current Account Decit[END_REF], assumes that the global imbalances result from both the nancial and real sectors. The current account surpluses of the emerging countries have two origins. On the one hand, following the nancial crisis of 1997, the Asian emerging countries are accumulating reserves to cover against a possible sudden exit in foreign capital. At the same time, they maintain the under evaluation of their exchange rates, and the exports led their growth. On the other hand, the emergence of new great industrial countries weighs on oil prices. OPEC countries also emit a surplus of savings that they want to invest abroad, since their absorption capacity is limited due to their small population. The saving surplus of the emerging countries leads to the United States, the economy that presents the best features. Bernanke lists the special features that make the U.S. nancial sector the most attractive. This is the growth of productivity linked to the development of new technologies, the low political risk, the strong property rights protection or still favorable institutional environment. These inows of investments lead an appreciation of the dollar, an increase in asset prices and, after 2000, lower interest rates and an increase in household wealth. The combination of these factors motivates households to reduce their savings and increase consumption. Thus, the global savings glut is absorbed by the increase in spending of U.S. households that avoids a recession due to an excess of supply.

The equilibrium model of Caballero, Fahri and Gourinchas (2006) explains growth of GI by the inability of nancial systems of certain areas to achieve sucient investment to use all the savings available. The current pattern of GI and the low global interest rates result from the dierences on the nancial institutions development in the world and from the greatest potential growth of the United States compared to other nancially developed areas. This thesis provides a new element to explain the accumulation of foreign assets by Asian countries beyond the level of reserves to provide assurance against a ight of foreign capital. Furthermore, Caballero et al. ( 2006), all kinds of models have been developed to represent the evolution of global imbalances. In Obstfeld and Rogo (2005) the adjustment occurs by changing the preferences between tradable and non tradable goods and between domestic and foreign goods. The model of Blanchard, Giavazzi and Sa (2005) stressed the role of the exchange rate in the allocation of international portfolio. These three models have a major disadvantage that should be overcome: they assume that the GDP are not aected by the adjustments of current imbalances. In the continuous time model with two countries Asada, Chiarella, Flaschel and Franke (2003) and Proaño (2008), the GDP reacts to the adjustment of current imbalances, but they only model the real sector. Finally, models of [START_REF] Lavoie | A Three-Country Stock-Flow Consistent model: A Study of the Diversication of China's Foreign Reserve[END_REF], [START_REF] Godley | A simple model of three economies with two currencies: Euroland and the USA[END_REF] and [START_REF] Zhao | A Three-Country Model with Fixed Exchange Rates under a Stock-Flow Coherent Approach[END_REF] have the advantage of being stock-ow consistent and of representing both the top and the bottom of the balance of payments. However, they should be simplied [START_REF] Lavoie | Les apports des approches stocks-ux à l'analyse postkeynésienne[END_REF]): these three-country models comprise 91, 79 and 89 equations, respectively. In his review of the state of macroeconomics, Blanchard (2008) also argues for smaller models to capture a specic mechanism. He cites Solow including: " My general preference is for small, transparent, tailored models, often partial equilibrium, usually aimed at understanding some little piece of the (macro-) economic mechanism. " [START_REF] Solow R | The State of Macroeconomics[END_REF] In order to replicate the global imbalances of the 2000s, we use an absorption model to describe the real sector and a portfolio model to describe the nancial sector. The interest of such a model is to take into consideration simultaneously factors linked to several interpretations of global imbalances:

• The trade balance approach: the increase of the propensity to import Asian products in the United States and in Europe.

• the absorption approach: the limited absorption capacity in Asian and OPEC countries.

• The saving-investment approach: the consumption of the global saving glut by the United States.

• The portfolio balance approach: the accumulation of reserves in Asian countries to cover against a risk of sudden exit of capitals and the inability of Asian nancial systems to absorb the whole domestic saving.

We start with a model which consists of three identical countries and then we replicate the current pattern of global imbalances by introducing three asymmetries: a xed exchange rate between Asia and the United States, a limited absorption capacity in Asia and an endogenous propensity to spend in the United States.

A model of three identical countries with exible exchange rates

The model is composed of three identical countries. These economies exchange goods and services and hold foreign assets that provide incomes. We assume the three countries are at equilibrium before shocks: trade equilibrium, current account equilibrium, zero net foreign debt and stable exchange rates. The model is divided into two sectors. The real sector (3.1) describes the equations of national income, domestic demand, imports, exports and income balance. The nancial sector (3.2) presents the evolution of supply and demand for foreign assets and allows determining the net external position and the exchange rate. Simulations (3.3) illustrate the eect of an increase in the propensity to spend and the impact of a competitiveness shock.

The real sector

The real sector is represented synthetically in order to focus only on the adjustment mechanisms vis-à-vis the rest of the world. We use a model of absorption à la [START_REF] Gossé | Eects of a Devaluation on a Trade Balance[END_REF]. The interest of this approach is to focus on macroeconomic, based around expenditure and production in the economy as a whole, in the domestic and international perspectives. When absorption is lower than national income, the country has a current account surplus and vice versa. In the propensity to spend (c), it does not distinguish the propensity to spend of households (1 -s), rms (i) and government (g):

c = 1 -s + i + g
The subject of the paper is not to explain the government's way of intervening but to dene the level of domestic spending that it should attempt to achieve through scal, monetary and exchange rate policies. The scal policy acts on the budget decit (g). The monetary policy changes saving and investment levels, in particular, playing on interest rates. Finally, changes in central bank reserves allow to compensate for current account imbalances without modifying the exchange rate. In the model with exible exchange rates, the level of domestic expenditure (D) is determined like [START_REF] Samuelson | Interactions between the Multiplier Analysis and the Principe of Acceleration[END_REF] and [START_REF] Hicks | A contribution to the Theory of the Trade cycle[END_REF], the propensity to spend of the country (c) and its GNP in the previous period (Y t-1 ). We assume that before the shock the propensity to spend is equal to one in the three countries 1, 2 and 3:

c 1 = c 2 = c 3 = 1. D i t = c i × Y i t-1 with i = 1, 2, 3 (1)
Imports are dened as standard by levels of expenditure (D) and relative prices (e), approached by the nominal exchange rate (that implies P 1 = P 2 = P 3 = 1). Country i's currency is the CU i and the exchange rate between country i and country j is:

1CU i = e ij CU j .
For instance, we use the exchange rate e 12 to convert CU 2 to CU 1 .

IM ij t = m0 ij (D i t-1 ) m1 ij (e ij t-1 ) m2 ij with i = 1, 2, 3 ; j = 1, 2, 3 and i = j (2)
We determine country i's propensity to import from country j (µ) by the ratio of imports on expenditures.

µ ij t = IM ij t D i t ( 3
)
The balance of investment incomes between country i and country j is calculated as the dierence between incomes received and incomes paid taking into account exchange rate variations:

IN C ij t = (ω ij t-1 W i t-1 r j t-1 e ij t-1 e ij t ) -(ω ji t-1 W j t-1 e ij t-1 r i t-1 ) (4)
Incomes received from country j are dened by the share (ω ij ) of country i's wealth (W i ) invested in country j at the previous period, times the rate of return in country j (r j ), times the exchange rate variation (

e ij t-1 e ij t
). Incomes paid by country i to country j are dened by the share (ω ji ) of country j's wealth (W j ) invested in country i at the previous period, times the rate of return in country i (r i ).

Then, we determine the GNP by the sum of the absorption (or expenditure Overall residents) A, the trade balance (the dierence between exports and imports) X -IM and the income balance INC.

Y = A + (X -IM ) + IN C with D = A + IM
The GNP corresponds to domestic demand (the rst member), plus exports (the second and third members), plus net income investment (fourth and fth members).

Y i t = (1 -µ ij t -µ ik t )D i t + µ ji t (D j t )e ij t + µ ki (D k t )e ki t + IN C ij t + IN C ik t ( 5
) with i = 1, 2, 3 ; j = 1, 2, 3 ; k = 1, 2, 3 and i = j = k
The real sector allows for determining the trade balance. The trade decit (TD) is the dierence between imports and exports expressed in domestic currency:

T D ij t = IM ij t -e ji t × IM ji t (6)

The nancial sector

A portfolio model à la [START_REF] Kouri | Balance of Payments and the Foreign Exchange Market: A Dynamic Partial Equilibrium Model[END_REF] which incorporates the mechanical of Blanchard et al. model (2005) represents the nancial sector. It is used to determine exchange rates. The propensity to hold foreign assets is determined in accordance with the horizontal constraint of [START_REF] Godley | Money, Finance and National Income Determination: An Integrated Appoach[END_REF], i.e., for each equation, the sum of all rates of return coecients (λ) is equal to zero. For the sake of simplicity, we assume that the rates of return are the same in the three countries (r 1 = r 2 = r 3 ).

ω ij t = λ0 ij -λ1 ij (r i t ) + λ2 ij (r j t ) -λ3 ij (r k t ) (7)
Net foreign debt is equal to the value of assets held by foreign investors in the country, minus the value of assets held by domestic investors abroad, plus the trade decit vis-à-vis the foreign country.

N F D ij t = ω ji t-1 W j t-1 e ij t-1 (1 + r i t-1 ) -ω ij t-1 W i t-1 (1 + r j t-1 ) e ij t-1 e ij t + T D ij t ( 8
)
The quantity of the assets held by investors from country j in country i is equal to the share (ω ji ) of country j's wealth (W j t-1 ) held in country i during the previous period, times the exchange rate between country i and country j (1CU j = e ji CU i ), times the rate of return of country i's assets in CU i (1 + r i t-1 ). The value of assets held by investors from country i in country j is equal to the share of country i's wealth of the previous period (W i t-1 ), times the propensity to hold assets of country j (ω ij t-1 ), times the rate of return on assets of country j in CU i ((1 + r j t-1 ) ×

e ij t-1 e ij t
).

The supply of domestic assets BS is given by a ratio κ of GNP Y :

BS i t = κ i (Y i t ) (9)
For the sake of clarity, we set: κ 1 = κ 2 = κ 3 = 1 (assuming that the supply of assets is the same size as GNP). The domestic wealth (W ) is equal to the supply of domestic assets (BS ) minus the net foreign debt NFD expressed in home currency.

W i t = BS i t -N F D ij t -N F D ik t ( 10
)
The exchange rate e ij (to convert country j's currency into country i's currency) is dened so as to equalize the liabilities of country i and the assets of country j:

e ij t [N F D ij t + ω ij t (W i t )] = ω ji t (W j t )
We replace W i and W j by their expressions:

W i t = BS i t -N F D ij t -N F D ik t W j t = BS j t + (N F D ij t )e ij t -N F D jk t
We get the following expression:

e ij t [N F D ij t +ω ij t (BS i t -N F D ij t -N F D ik t )] = ω ji t [BS j t +(N F D ij t )e ij t -N F D jk t ]
This equation determines the exchange rate between country i and country j:

e ij t = ω ji t (BS j t ) -N F D jk t ω ij t (BS i t -N F D ik t ) + (1 -ω ji t -ω ij t )N F D ij t ( 11
)
We remark that as in the Blanchard et al. model, the higher the assets supply, the lower the exchange rate variation resulting from current account imbalances. Furthermore, as country j's net foreign debt vis-à-vis country k increases, the country i's currency is weakened compared to country j's currency, because the country j's assets supply available in country i decreases.

Similarly, when country i's net foreign debt vis-à-vis country k increases, its exchange rate apprises because country i's assets supply available in country j decreases.

3.3

The simulations in a world without asymmetries 3.3.1 Scénario 1: a shock on country 1's propensity to spend Country 1's propensity to spend increases from 1 to 1.005 (gure 3). This rise provokes an increase of country 1's GNP and a growth of assets supply. As a result, country 1's currency is depreciated compared to two other countries currencies. The competitiveness of country 1 increases so it releases a trade surplus, a current account surplus and its net foreign position improves. Its trade surplus is shrinking gradually with the growth of its GNP but its current account surplus continues to increase because trade surplus reduction is oset by an elevation of the net income related to the depreciation of its currency.

Scenario 2: a shock on the propensity to import of countries 1 and 2

In countries 1 and 2, the propensity to import goods made in country 3 passes from 0.05 to 0.055 (gure 4). The GNP of countries 1 and 2 decrease while that of country 3 increases. The currencies of countries 1 and 2 depreciate relative to the country 3 to return to current account equilibrium. As a rst step, the country 3's trade balance surplus allows it to accumulate assets in the rest of the world. As a second step, the country 3 recorded a trade decit which is oset by the receipt of net income. A new equilibrium is established in which the country 3 consumes more goods than it produces because the balance of investment incomes procures him a rent. Thus, in a "perfect world" without asymmetries, productivity shocks are adjusted by exchange rate changes and do not generate global imbalances. Results of simulations under exible exchange rates are presented in table 2.

Modeling the global saving glut hypothesis

We start from the previous three-country model to describe the relationships between the three major areas that currently interact. The country 1 is the United States who hold the currency on which some countries are pegged. The country 2 is named the euro area and it comprises exible exchange rates countries. The country 3 is named Asia and it includes xed exchange rates countries. However, this model does not describe the current global imbalances since it ignores several key features of the global economy. We introduce three asymmetries in the previous model in order to replicate the global imbalances of the 2000s and to show their implications on growth.

First, some countries have xed exchange rates so the Asian propensity to hold foreign securities must be determined to leave unchanged its exchange rate vis-à-vis the dollar (4.1). Second, the limited absorption capacity of OPEC and Asia should be take into account (4.2). Third, the Bernanke's global saving glut hypothesis is described by a model endogenizing the American propensity to spend to maintain the income of the United States unchanged (4.3). Simulations show the eects of expansionary politics, of under-evaluation of the Renminbi and of the absorption by the United States of the global saving glut (4.4).

First asymmetry: Asia pegs its currency on the dollar

In this case, the adjustment is made by modifying the Asian home bias as long as it agrees to acquire securities issued to oset the current account imbalance. The home bias compatible with a xed exchange rate allows equalizing supply and demand for U.S. assets without modifying exchange rates:

[N F D 13 t + ω 13 t (W 1 t )]e 13 t = ω 31 t (W 3 t )
We replace W 1 and W 3 by their expressions and we get the following expression:

[N F D This equation gives the level of home bias that adjusts current account imbalances while maintaining the exchange rate unchanged:

ω 31 t = ω 13 t (BS 1 t -N F D 12 t ) + (1 -ω 13 t )N F D 13 t BS 3 t -N F D 32 t e 13 t + N F D 13 t ( 12
)
Since Asia is in xed exchange rates vis-à-vis the United States, this equation replaces the equation of the exchange rate between the United States and Asia in the previous model.

Second asymmetry: the limited absorption capacity of xed exchange rates countries

The absorption capacity of country 3 is limited for two reasons. On the one hand, we assume that Asia reach its maximum absorption capacity because its nancial system is not able to use domestic saving. On the other hand, the absorption capacity of OPEC countries is limited due to their small population. We model this dual limit by assuming that the level of expenditure of the country 3 is xed ∆D 3 = 0. Then the propensity to spend adjusts changes in income:

c 3 t = D 3 Y 3 t-1 (13)
This equation replaces equation 3 of the previous model.

4.3 Third asymmetry: the U.S. propensity to spend is endogenized to maintain constant U.S. GNP

We determine the level of the U.S. propensity to spend that can avoid a recession in the United States resulting from country 3's current account surplus. The level of propensity to spend which can absorb the shock c 1 * is determined in [START_REF] Brender | Les déséquilibres nanciers internationaux, La découverte[END_REF]. We highlight the equilibrium values (pre-shock). As a rst step, we equalize pre-shock income Y1 with post-shock income Y1 :

Y 1 = Ȳ 1 With D 3 = D3 , D 1 = c 1 (Y 1 t-1
) and c 1 = 1, so it comes:

(1 -µ 12 t -µ 13 t )c 1 * (Y Then we determine the level of c1* which maintain the American GNP constant after a competitiveness shock:

c 1 * = μ21 ( D2 )ē 21 -µ 21 t D 2 t e 21 t +D 3 [μ 3 1(ē 31 )-µ 31 t (e 31 t )-IN C 12 t -IN C 13 t ] (1-µ 12 t -µ 13 t )Y 1 t-1 + 1 -μ12 -μ13 1 -µ 12 t -µ 13 t ( 14
)
To avoid a global recession, the U.S. must increase their propensity to spend c 1 in order to compensate for the reduction in Asia c 3 .

The simulations

The rst two simulations are under asymmetries 1 and 2. In this case, the productivity shock implies a growth of global imbalances and a recession in the United States and in Europe. When we introduce the third asymmetry, the US GNP is stabilized, but the global imbalances are bigger and the negative eect on the European GNP is stronger. The results of these simulations are summarized in table 3 The U.S. propensity to spend rises from 1 to 1.005. The supply of U.S. securities and the American GNP increase (gure 5). The result is a depreciation of the dollar vis-à-vis the euro and an increase in Asian reserves to avoid an appreciation of &. The increase in the GNP generates a trade decit vis-à-vis Asia that is partly oset by a surplus vis-à-vis Europe. The euro area has a trade decit with the United States and Asia as the euro appreciates against currencies of both countries. The net external debt of the United States and Europe increase and their net income is negative. Finally, Asia has a current account surplus and the United States and Europe a decit. The European GNP decreases and that of Asia expands.

Scenario 2: a shock on the propensity to import of countries 1 and 2 without global saving glut hypothesis

The propensity to import Asian products to the United States and Europe increased from 0.05 to 0.055 (gure 6). The trade balances of both economies decline while Asia has a surplus. The trade balances tend to return to equilibrium with the depreciation of the euro vis-à-vis both Asian and American currencies and with the decline of European and U.S. GNP. Asia accumulates U.S. assets in order to avoid an appreciation of its currency. This accumulation leads to an increase of Asian net incomes. The U.S. GNP diminishes gradually to adjust the shock. The European GNP is reduced and then stabilized after the euro has depreciated. Finally, the increase of U.S. and European propensity to import causes a global recession and growing global imbalances.

Scenario 3: a shock on the propensity to import from countries 1 and 2 with global saving glut hypothesis

In this case, the propensity to spend of the United States adjusts itself in order to maintain their GNP unchanged following the increase in their propensity to import Asian products (gure 7). The dollar depreciates against the euro since the eect of the propensity to spend is stronger than the one of the propensity to import. Asia accumulates U.S. assets to maintain the level of its exchange rate with the dollar. The trade and current account imbalances persist. Thus, global imbalances grow rapidly: the current account decit and the net foreign debt of the United States continue to rise as the current account surplus and the net foreign assets of Asia. External imbalances in the Euro area are less important. The adjustment takes place through the gradual reduction of European GNP that is linked to the deteriorating competitiveness.

The results of simulations under the global saving glut hypothesis are very close to the evolution of the pattern of global imbalances in the 2000s. The trend in the current account imbalances are similar to a surplus in Asia and a decit in the United States (gure 11). However, according to the simulation, the euro area should be in decit but the observations show a dierent trend around the equilibrium. The trends in net foreign debts are very akin observations: a growing net debt in the United States, a soaring net stock of assets in Asia and a smoothly increasing net debt in the euro area (gure 12).

Conclusion

The model includes a real and nancial approach of global imbalances. The three-country model only has 50 equations or about half of the models of [START_REF] Zhao | A Three-Country Model with Fixed Exchange Rates under a Stock-Flow Coherent Approach[END_REF] A rst series of simulations is conducted in a model with three identical countries under exible exchange rates. The increase in the propensity to import country 3's products in countries 1 and 2 causes a reduction of the GNP of both countries and provokes small external imbalances that stabilize after the depreciation of exchange rates. Then, we introduce three asymmetries in order to model the global saving glut hypothesis. First, the Asia-OPEC area is pegged on the dollar. Second, we assume a limited absorption capacity in xed exchange rates countries tied to the small population in the OPEC countries and to the limited nancial development in Asia. Third, under the Bernanke's global saving glut hypothesis, it is assumed that the United States raise their propensity to spend in order to maintain their GNP stable following an increase in their propensity to import.

Under the rst two constraints, the productivity shock implies the stagnation of Asian GNP and a strong negative eect on US GNP and that of the euro area. By adding the third asymmetry, it appears that the recession can be avoided in the United States if they increase their propensity to spend. However, the consequences for the global economy would be disastrous: global imbalances grow very fast and the euro area experiences a deep recession. Finally, we nd that the trend of simulations are very close to the observed trends of the pattern of global imbalances both in stocks and in ows.

• in the exible exchange rates model there are 50 endogenous equations for 50 equations,

• the model with a xed exchange rate between the U.S. and Asia has 49 equations for 49 endogenous variables,

• when we introduce the global saving glut hypothesis, there are 50 equations for 50 unknowns. ex/e exchange rate (1 £ = ex/e &) implicit 1 1/ex exchange rate (1 & = 1/ex $) implicit 1 m120 constant in the equation of imports of European products exogenous 0.05 m210 constant in the equation of imports of American products exogenous 0.05 m310 constant in the equation of imports of American products exogenous 0.05 m121 income elasticity in the equation of imports of European products exogenous 1 m211 income elasticity in the equation of imports of American products exogenous 1 m311 income elasticity in the equation of imports of American products exogenous hold European assets equation exogenous 0.2 λ211 coefficient on r1 in propensity to hold American assets equation exogenous 0.4 λ311 coefficient on r1 in propensity to hold American assets equation exogenous 0.4 λ122 coefficient on r2 in propensity to hold European assets equation exogenous 0.4 λ212 coefficient on r2 in propensity to hold American assets equation exogenous 0.2 λ312 coefficient on r2 in propensity to hold American assets equation exogenous 0.2 λ123 coefficient on r3 in propensity to hold European assets equation exogenous 0.2 λ213 coefficient on r3 in propensity to hold American assets equation exogenous 0.2 λ313 coefficient on r3 in propensity to hold American assets equation exogenous 0.2 exogenous 100 IM12 imports of European products endogenous 5 IM21 imports of American products endogenous 5 IM31 imports of American products endogenous 5 IM13 imports of Asian products endogenous 5 IM23 imports of Asian products endogenous 5 IM32 imports of European products endogenous 5 µ12 propensity to import European products endogenous 0.05 µ21 propensity to import American products endogenous 0.05 µ31 propensity to import American

United States Euro area Asia

  .

4. 4 . 1

 41 Scenario 1: a shock on the U.S. propensity to spend without global saving glut hypothesis

  , Godley and Lavoie (2007) and Lavoie and Zhao (2008) although, unlike the models of Obstfeld and Rogo (2005), Blanchard et al (2005) and Caballero et al (2006), it does not involve a constant GNP. Thus, we can observe both the real and nancial implications of the shocks.

1 m122

 1 price elasticity in the equation of imports of European products exogenous 1 m212 price elasticity in the equation of imports of American products exogenous 1 m312 price elasticity in the equation of imports of American products exogenous 1 m130 constant in the equation of imports of Asian products exogenous 0.05 m230 constant in the equation of imports of Asian products exogenous 0.05 m320 constant in the equation of imports of European products exogenous 0.05 m131 income elasticity in the equation of imports of Asian products exogenous 1 m231 income elasticity in the equation of imports of Asian products exogenous 1 m321 income elasticity in the equation of imports of European products exogenous 1 m132 price elasticity in the equation of imports of Asian products exogenous 1 m232 price elasticity in the equation of imports of Asian products exogenous 1 m322 price elasticity in the equation of imports of European products exogenous λ120 constant in the propensity to hold European assets equation exogenous 0.1 λ210 constant in the propensity to hold American assets equation exogenous 0.1 λ310 constant in the propensity to hold American assets equation exogenous 0.1 λ121 coefficient on r1 in propensity to

  coefficient on r3 in propensity to hold European assets equation exogenous 0.2 ω12 propensity to hold European assets endogenous 0.2 ω21 propensity to hold American assets endogenous 0.2 ω31 propensity to hold American assets endogenous* 0.2 ω13 propensity to hold Asian assets endogenous 0.2 ω23 propensity to hold Asian assets endogenous 0.2 ω33 propensity to hold

  commercial deficit vis-à-vis the United States endogenous 0 DC31 commercial deficit vis-à-vis the United States endogenous 0 DC13 commercial deficit vis-à-vis Asia endogenous 0 DC23 commercial deficit vis-à-vis Asia endogenous 0 DC32 commercial deficit vis-à-vis the euro area endogenous 0 NFD12 net foreign debt vis-à-vis the euro area endogenous 0 NFD21 net foreign debt vis-à-vis the United States endogenous 0 NFD31 net foreign debt vis-à-vis the United States endogenous 0 NFD13 net foreign debt vis-à-vis Asia endogenous 0 NFD23 net foreign debt vis-à-vis Asia endogenous 0 NFD32 net foreign debt vis-à-vis the exchange rate (1 & = e/ex £) implicit 1 * the first term indicates the status of the variable in the first model in flexible exchange rates. the second term indicates the status of the variable when the country is in 3 fixed exchange and the assumption of surplus global savings is made.

Figure 2 :Figure 3 :

 23 Figure 2: Evolution of the crude oil price (dollars a barrel)Source: IMF, IFS

Figure 4 :

 4 Figure 4: Evolution of current account imbalances (in billions of dollars) Source: CEPII-Chelem

Figure 5 :

 5 Figure 5: Evolution of reserves minus gold (in billions of dollars) Source: Lane and Milesi-Ferretti (2007)

Figure 9 :

 9 Figure 9: The propensities to import Asian goods to the United States and to the euro area passe from 0.05 to 0.055, with xed exchange rates and limited absorption capacity

Figure 11 :

 11 Figure 11: Evolution of current account imbalances (% of GDP) Source: CEPII-Chelem

Figure 12 :

 12 Figure 12: Evolution of Net Foreign Assets (% of GDP) Source: Lane and Milesi-Ferretti (2007)

Table 1 :

 1 Values of model parameters


Kenta Ogawa

Thomas Schmugge

Frederic Jacob

Andrew French

Estimation of Broadband Emissivity from Satellite Multi-Channel Thermal Infrared Data Using Spectral Libraries

HAL is

I. INTRODUCTION

Surface broadband emissivity is an important parameter for estimating the longwave surface energy.

Mostly, the broadband emissivity may vary significantly, because the spectral emissivity variation ranges from 0.7 to 1.0 for bare soils and rocks (see Fig. 1) in the 8-12 µm range, there is smaller variation in the 5-8 µm and 12-14 µm ranges.

A constant emissivity is often used for land surface in energy balance study and general circulation models (GCMs). But, the spatial variation of broadband emissivity may cause a feedback and affect surface temperature in GCMs.

Recent spaceborne thermal infrared multispectral sensors, such as ASTER, allow the estimation of spectral emissivities from local to global scales. We are helped in this effort by the fact that the peak in the thermal emission is at ~10 µm, i.e. the region we have measurements.

In this study, we propose to express the broadband emissivity as a linear combination of channel emissivities of ASTER/TIR. Firstly, we produced a dataset of the broadband emissivity (3-14 µm), and channel emissivities for ASTER/TIR, using spectral libraries. Then we calibrate and validate a regression using the dataset. Finally, we computed broadband emissivities over the Jornada Experimental Range [START_REF] Schmugge | Temperature and emissivity separation from multispectral thermal infrared observations[END_REF], New Mexico with ASTER/TIR using the calibrated regression.

II. METHOD & DATA

Broadband emissivity is defined in (1):

∫ ∫ = = = = - ≡ 2 1 2 1 2 1 ) , B( ) , B( ) ( λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ε ε d T d T (1).
Where ε is surface thermal infrared emissivity, λ is the wavelength, B is the Planck function, and T is surface temperature. In this study, we selected a wavelength range, λ 1 =3.3 µm to λ 2 =14.0 µm, since spectral libraries covered this wavelength range. We assume T = 300 K in this analysis. We computed the difference of broadband emissivity caused by variation of target temperature. It was smaller than 0.005 over the range from 270 K to 330 K. We note that the emission at wavelengths longer than 14 µm is about 40 % to 50 % of the total blackbody radiation and that we have no spectral emissivity data of these longer wavelengths.

AS5TER channel emissivity ε ch is defined in (2) as:

∫ ∫ = = = = ≡ 2 1 2 1 ) , B( ) ( ) , B( ) ( ) ( λ λ λ λ λ λ λ λ λ λ λ λ λ λ ε λ ε d T f d T f ch ch ch (2).
Where f ch is the spectral response function of ASTER/TIR channel.

We assume that broadband emissivity can be expressed in (3) as:

c a ch ch ch + = ∑ = - 14 10 .0 14 3 . 3 ε ε (3).
The center wavelengths of ASTER channel 10 to 14 are 8.3, 8.65, 9.1, 10.6, 11.3 µm, respectively. Then we calibrate and validate the coefficients of (3) using two spectral libraries.

We used two spectral libraries; One is those collected by Jack Salisbury of Johns Hopkins University (JHU Library) [START_REF] Salisbury | Emissivity of terrestrial materials in the 8-14 µm atmospheric window[END_REF] and the other is collected by University of California Santa Barbara (UCSB Library) [START_REF] Snyder | Thermal infrared (3-14 µm) bi-directional reflectance measurement of sands and soils[END_REF]. We selected 150 samples spectra from the JHU Library including soils, vegetation, rock (except fine powders), water, and ice emissivity spectra, and selected 107 samples spectra from UCSB library including soils, vegetation, and water.

The JHU library contains directional hemispherical spectral reflectance, ρ λ , therefore we converted to spectral emissivity ε λ using Kirchhoff's law, ε λ =1-ρ λ . The spectra of the fine powdered samples (particle size: <50 µm) were not used in this study, because they may not follow this relation [START_REF] Salisbury | Thermal-infrared remote sensing and Kirchhoff's law, 1, Laboratory measurements[END_REF].

ASTER data were acquired on 12 May 2001 over the Jornada Experimental Range in New Mexico, a desert grassland where the main vegetation components are grass and shrubs. The data were atmospherically corrected using MODTRAN with NCEP atmospheric profiles. Then the MMD method [START_REF] Gillespie | A temperature and emissivity separation algorithm for Advanced Spaceborne Thermal Emission and Reflection radiometer (ASTER) images[END_REF] was applied to separate the surface temperature and spectral emissivity from surface spectral radiance.

III. RESULTS

A. Spectral libraries and broadband emissivity

Fig. 1 displays the average and range of spectral emissivity for data from both libraries. The range is larger both over [3 -5] µm and [8 -10] µm regions than in other wavelength regions. Because the emitted energy is low over 3 to 5 µm at around 300 K, it is expected that the emissivity from 8 to 10 µm (ASTER channel 10, 11, 12) will strongly affect the broadband emissivity.

The calculated broadband emissivities range from 0.885 to 0.994. Mean emissivity of rock, soil, vegetation, and water is 0.938, 0.955, 0.967, and 0.982 respectively.

B. Calibration & Validation

Table I displays the calibrated coefficients of computed from the step-wise regression. The coefficient of 0.0 for channel 13 (a 13 ) means that the variable was dropped, because it was not statistically significant. Table II shows the range of emissivity, and RMSE and maximum error. RMSE is smaller than 0.01. The error was large in some kinds of igneous rocks, especially in ultramafic and mafic rocks samples, such as picrite. Maximum absolute error is 0.023.

Table III shows RMSE, maximum error, and range of emissivity for validation with UCSB library. The agreement between estimated broadband emissivity and calculated ones is consistent. The difference was smaller than 0.01 for 100 of 107 samples (93%).

Figure 2 shows the comparison of the estimated broadband emissivity against the calculated value using spectral libraries. We noticed an underestimation for the samples with higher emissivity (around 0.99), such as vegetation and water. This may results, because of the small number of spectra for these types ain the calibration. We also tried other empirical functions, such as, 1) the linear regression without a constant, and 2) polynomial regression. But we didn't observe significant improvements in either instance.

C. Application to ASTER data

This calibrated regression was finally applied to emissivities computed from ASTER/TIR data acquired over the Jornada Experimental Range in New Mexico. The result is shown in Fig. 3. The histogram of extracted emissivities is shown in Fig. 4. Emissivities range from 0.93 to 0.97 with the average ~0.95.

In Fig. 3, the bright area in the upper center to lower center of the image corresponds to a ridge with vegetation. Another bright area in lower left corresponds to agricultural fields along the Rio Grande River. These areas have higher emissivities range from 0.96 to 0.97.

The darker area at upper left and right edge is bare soil or sparsely vegetated sites. The upper right area is for the gypsum at white sand. These areas have lower emissivities that range from 0.93 to 0.95.

IV. SUMMARY AND CONCLUSIONS

The results of the spectral libraries analysis suggest that the estimation of broadband emissivity of 3-14 µm using the linear regression from emissivities of ASTER/TIR channels is potentially accurate and the expected RMSE is lower than 0.01 for the emissivity range from 0.90 to 0.99.

The estimated broadband emissivity using the regression in Jornada site showed the variation of 0.04, from 0.93 to 0.97.

Fig. 1 .Fig. 2 .

 12 Fig. 1. The mean and range of emissivity in spectral libraries samples.

Fig. 3 Fig. 4

 34 Fig. 3 Broadband emissivity map of Jornada Experimental Range.

ACKNOWLEDGMENTS

This study was supported by the ASTER Project of NASA's EOS-Terra Program.