Mouhamadou Sy
email: mouhamadou.sy@strategie.gouv.fr

Daniel Cohen

Salifou Issoufou

Mathias Lé

Richard Portes

Romain Rancière

Exchange Rate Regimes, Capital Controls and the Pattern of Speculative Capital Flows

Keywords: Short-term Capital Flows, Exchange Rate Regimes, Financial Openness JEL classification: F31, F32, F41 Short-Term Capital Flows, Exchange Rate Regimes, Financial Openness

This paper proposes theoretical and empirical analysis of the effect of capital controls and alternative exchange rate regimes on the patterns of speculative capital. I argue that the exchange rate regime and its interaction with the monetary regime can explain the patterns of speculative capital around the world. I show that speculative capitals are more likely to flow into countries in which there is a contradiction between the monetary and the exchange regimes, e.g. more likely in countries with managed exchange rates. I model exchange-rate as a jump process in a stochastic dynamic portfolio optimization. Through this approach, the influence of the frequency and the size of "jumps" in the exchange rate on the allocation of speculative capital can be determined. It will also allow inflows to be endogenous. By linking the jumps to the frequency of exchange rate movements, this paper determines the effectiveness of different exchange rate regimes in fending off "hot money" for a given monetary regime. On the empirical side, I use a newly constructed data set to verify the theoretical predictions of the determinants and the patterns of speculative capital. Capital controls do not affect speculative capital.

Introduction

"Since the adoption of this regime, the nature of capital flows to Mexico has shifted from mainly short-term to long-term in nature. To give you an idea, before the crisis of 1990 to 1994, foreigners had over $34 billion in short-term investments in Mexico. As of today, they have less than $2 billion. That reduction can be attributed to the floating exchange-rate regime." Agustin G. Carstens (1998) a a Governor of Central Bank of Mexico

The above quote highlights the fact that between 1994 and 1998, the amount of short-term capital inflows to Mexico was divided by seventeen because of the exchange rate flexibility.

This paper tries to rationalize this theoretically, empirically and look beyond Mexico.

The 1990s is still generally called the lost decade for some emerging countries, particularly Asian countries. The financial turmoils in Mexico (1995), Korea, Indonesia, and Thailand (1997), Russia (1998), andBrazil (1999) were mainly due to foreign capital flight. As noted by [START_REF] Rodrik | Short-term capital flows[END_REF], all these countries had one thing in common: a large level of short-term foreign debt. [START_REF] Komulainen | What drives financial crises in emerging markets?[END_REF] and others found that foreign direct investment reduces the likelihood of financial crises, whereas short-term capital flows increase it. In retrospect the ''key question is therefore why do some economies at certain time receive much more short-term flows?" [START_REF] Hoggarth | Capital Flows: Causes, Consequences and Policy Responses[END_REF]. This paper attempts to provide the answers to this question by examining the role of capital controls and exchange rate regimes from a short term perspective on the distribution of speculative capital worldwide both theoretically and empirically. In this paper, I adopt the definition of speculative capital given by [START_REF] Kant | Foreign direct investment and capital flight[END_REF]: "Money that responds quickly to political or financial crisis, to expectations of tighter capital controls or the devaluation of the domestic currency, or to changes in after-tax real return". With this definition of speculative capital flows, I can approximate it by short-term capital flows.

The main characteristic of short-term capital inflows is their dependence on short-term interest rates, thus it also depends on monetary policy when it is used to fight inflation.

In other words, short-term capital inflows must be a result of speculators' positions taking advantage of heterogeneity in monetary regimes across different countries. But, nowadays, many countries conduct ''active monetary policies" which consist of fighting inflation through changes in their nominal interest rates, e.g. when inflation increases, countries choose to fight inflation by raising their nominal interest rates. This paper argues that even if monetary regimes become more and more homogeneous across countries, huge differences between their exchange rate regimes can explain the pattern of speculative capital inflows. I build a portfolio balance model which takes into account these differences in the exchange rate regimes. In a stochastic dynamic optimization framework, inflation is a diffusion process and exchange rate adjustment a jump process. This framework focuses, in particular, on the uncovered interest rate parity condition (UIP). Since a violation of UIP allows for arbitrage profits, it is traditionally assumed to hold at all times. Empirical researches such as [START_REF] Chinn | The (partial) rehabilitation of interest rate parity in the floating rate era: Longer horizons, alternative expectations, and emerging markets[END_REF] show, however, that deviations do occur. For reasons such as exchange rates forecasting errors [START_REF] Campbell-Pownall | Irving fisher and the uip puzzle: Meeting the expectations a century later[END_REF]), UIP does not always hold in the short term. The model allows for temporary deviations from UIP and thus identifies, among other factors, the role of exchange rate regime and the frequency of exchange rate adjustments on speculative capital1 .

The influence of monetary policy on exchange rates is an important pillar of this mechanism. This literature starts with [START_REF] Dornbusch | Expectations and exchange rate dynamics[END_REF]. [START_REF] Obstfeld | Exchange rate dynamics with sluggish prices under alternative price-adjustment rules[END_REF] finds that the Mussa and Barro-Grossman rules yield structurally equivalent exchange rate models. Unexpected currency depreciations are explained as a result of a surprise inflation policy by the central bank in [START_REF] Penati | Optimal portfolio choice and the collapse of a fixedexchange rate regime[END_REF]. Finally, there are more technical strands on portfolio management. The first consists of technical treatments for the solution of the stochastic dynamic optimization. The seminal contribution of this sort is [START_REF] Merton | Continuous-time finance[END_REF]. The second concerns the importance of skewness -unexpected jumps -in returns on asset prices such as [START_REF] Harvey | Conditional skewness in asset pricing tests[END_REF] and [START_REF] Martinez | Optimal portfolio and consumption decisions under exchange rate and interest rate risks[END_REF]. I model the link between the monetary regime and the exchange rate regime in a simple way: by making the unexpected size of the currency depreciations endogenous to the inflation rate and to the frenquency of devaluations.

The existing empirical literature, [START_REF] Montiel | Do capital controls and macroeconomic policies influence the volume and composition of capital flows? evidence from the 1990s[END_REF], [START_REF] Ariyoshi | Capital controls: Country experiences with their use and liberalization[END_REF], [START_REF] Kamil | Are capital controls effective in the 21st century? the recent experience of colombia[END_REF], and others focused mainly on the role of capital controls. Most of the studies reached the conclusion that capital controls do not play an important role in fending off hot money and do not give another channel through which speculative capital inflows can be avoided. The empirical part of this paper is in line with this literature, it shows that capital controls do not affect short-term capital inflows. It also shows that managed exchange rates are more likely to attract these types of capitals compared to flexible exchange rates.

The central message of this paper is that if the primary objective of policymakers is to fend off ''hot money", moving from a pegged exchange rate to a flexible exchange rate is more efficient that imposing controls on inflows. This paper is also linked to [START_REF] Levy-Yeyati | To float or to fix: Evidence on the impact of exchange rate regimes on growth[END_REF]. They studied the rela-tionship between exchange rate regimes and economic growth and showed that less flexible exchange rate regimes are associated with slower growth. But [START_REF] Levy-Yeyati | To float or to fix: Evidence on the impact of exchange rate regimes on growth[END_REF] did not give the insights -the channels -through which countries with a more flexible exchange rate regimes grow faster than countries with less flexible exchange rate regimes1 . This paper shows that short-term capital are more likely to flow in countries with pegged exchange rate regimes which implicitly means that countries with flexible exchange rate regimes attract long-term capital inflows which can affect economic growth positively (see [START_REF] Eichengreen | Capital account liberalization, financial development and industry growth: A synthetic view[END_REF]). In other words, this paper may give the channel between exchange rate regimes, long-term capital inflows and economic growth.

To my limited knowledge, this is the first paper which investigates the geography of speculative capital flows. [START_REF] Rodrik | Short-term capital flows[END_REF] studied the determinants of short-term capital flows but focused on one form of capital: short-term external debt. Moreover, they did not distinguish the difference between gross and net capital flows and they did not consider the role of capital controls or the nature of the exchange rate regimes in their analysis. They identified the share of money supply over gross domestic product as the main determinant of short-term external debt. In this paper, I focus on gross rather than net capital inflows. The reason is that recent literature (Gourinchas and Rey (2007a) and [START_REF] Forbes | Capital flow waves: Surges, stops, flight, and retrenchment[END_REF] highlights the importance of gross capital inflows that measure positions. This may help us to better understand the abrupt movements in capital inflows. Recently [START_REF] Gourinchas | Global imbalances and global liquidity[END_REF] highlights that to understand vulnerabilities it is necessary to focus on gross capital inflows.

He also highlights that the measure of short-term liabilities should be more broader than the standard measure of short-term external debt as in [START_REF] Rodrik | Short-term capital flows[END_REF] and should include bank lending and money funds.

Empirically, I use an innovative data set on speculative capital to take into account these issues and to determine the influence of the theoretically identified factors on speculative capital.

The remainder of this paper is organized as follows. Section 2.2 describes the data and gives descriptive statistics on the pattern of short-term capital inflows. The third section shows a simple model which describes the mechanism, while section 4 examines the determinants and the pattern of short-term capital using a new dataset. The last section 5 concludes.

Data and Stylized Facts

Description of the Data

To document worldwide patterns of speculative capital flows, I have gathered a comprehensive data set on capital flows by distinguishing short-term from long-term inflows, and gross inflows from net inflows. The sample covers annual observations of 70 countries from 1980 to 2009. The sample is mainly dictated by the availability of the variable of interest: shortterm capital inflows. Except for this variable, the remaining variables mainly stem from the World Development Indicators (WDI, World Bank). They are commonly used variables and are well described in macroeconomic literature. Therefore, I devote more time describing the short-term capital inflows.

Capital inflows:

There is no data on international gross capital flows by maturities. In order to document the pattern of short term capital flows, several choices were made while gathering data. To distinguish between short-term capital flows and long-term capital flows, I use the Balance of Payments Statistics (BOPS) of the International Monetary Fund (IMF). The BOPS is divided into the three categories: Foreign direct investment (FDI), Portfolio investment, and the Other investments. Each category is divided between assets and liabilities. I consider FDI as a long term investment since it is ''an investment involving a long-term relationship and reflecting a lasting interest and control of a resident entity in one economy" according to the United Nations World Investment Report (UNCAD, 1999). In the BOPS, FDI assets are called ''foreign direct investment abroad" and liabilities are called ''foreign direct investment in reporting economy". Even if Portfolio investment could be considered as short-term investment, some Portfolio investments can have long term motive. Therefore, for this study, I did not consider Portfolio investments1 . The third category which is Other investments, is mainly composed by trade credit, loans and, currency and deposits. The BOPS provides information on their maturities. Short-term investments are characterized by maturities less than one year while the others are long-term investment. In this paper, long-term capital flows are Foreign direct investment plus long-term flows from the category ''Other investment" in the BOPS while short-term capital flows are short-term flows in the category ''Other investment". In the BOPS, all these categories, including assets and liabilities, are not aggregated. Consequently, all these categories result from an aggregation of sub-categories.

For the reasons highlighted in the introduction, I am interested in gross capital inflows by foreigners in this paper. But the balance of payments statistics report assets and liabilities in net terms so they can be negative or positive, which is a challenge when trying to give gross capital inflows a value. For example, an increase on the liabilities side vis-à-vis non-residents is entered as positive value while a decrease is a negative value. Therefore, the negative values of liabilities can be considered as capital outflows while negative values of assets can be considered as capital inflows. Thus, I compute gross capital inflows -short and longusing the following methodology:

Gross Capital In f lows = max(Liabilities, 0)min(Assets, 0) I compile datasets from 1980 to 2009 for 160 countries using this methodology. I follow [START_REF] Broner | Gross capital flows: dynamics and crises[END_REF] by dropping very small or very poor countries. Small countries are deleted because "they might display an artificially high volume of financial transactions due to their role as offshore financial centers or tax havens" and poor countries are deleted because they "generally depend heavily on official aid flows that behave differently than private capital flows". Small countries had a gross national income (GNI) in 2005 that was less than four billion U.S. dollars PPP adjusted, while poor countries had a GNI per capita smaller than 2,000 U.S dollars PPP adjusted. I ended up with 90 countries after applying this elimination method.

The are some countries with large amount of missing data in the BOPS. All countries that do not have at least 20 consecutive observations for capital inflows are dropped. This eliminates another 20 countries and we are finally left with a sample of 70 countries that have useful annual data. See table 7 for the list of countries included in the final sample.

Capital controls:

The measure of capital controls is based on [START_REF] Chinn | Capital account liberalization, institutions and financial development: Cross country evidence[END_REF]. The index depends on four variables: restrictions on the current account, restrictions on the financial account, if the country has multiple exchange rates or not, and if there is a requirement to surrender export receipts. The index is the principal component analysis factor of these three variables.

Exchange rate regimes:

The exchange rate regime classification is based on [START_REF] Levy-Yeyati | To float or to fix: Evidence on the impact of exchange rate regimes on growth[END_REF]. They use a de facto classification of exchange rate regimes based on cluster analysis techniques. Countries are sorted according to three variables: (i) Exchange rate volatility, (ii) Volatility of exchange rate changes, and (iii) Volatility of reserves. They are classified into three categories: 1 = float; 2 = intermediate and 3 = fixed.

Quality of institutions:

The Political Rights Index of the NGO Freedom House is used as a proxy for the quality of institutions. A country with the highest score adheres to political rights that are close to certain ideals (free and fair elections, competitive parties, minorities have reasonable self government, etc.). I transform this index via a logistic transformation to the interval between zero and one, where one is the best possible score for quality of institutions.

Other variables: The remaining variables are taken from the World Development Indicators (WDI) of the World Bank for September 2010. The inflation rate is in the form of annual log differences of the consumer price index. Money and quasi money is the total money supply.

Real GDP growth and population size are also taken from the WDI. The nominal interest, defined as the average of the deposit interest rate and the lending interest rate come from also to the WDI.

Stylized Facts

Prior to a formal model or a formal econometric analysis, let us compute some basic descriptive statistics. Table 1 shows, in billions of dollars, the average amount of short-term capital inflows across exchange rate regimes during the last three decades. For each decade, I compute the average amount of short-term capital inflows conditional on the nature of the exchange rate regimes, e.g. peg, intermediate, or flexible. Note that for calculating the average speculative capital, each country is weighted by the country's GDP, PPP adjusted. A simple comparison of the values in the table across line anticipates the main results of the paper.

These figures clearly show that there is a lot of heterogeneity in short-term capital across exchange rate regimes and the message is very clear: the more the exchange rate is flexible, the lower the amount of short-term capital inflows. How can these differences across exchange rate regimes be explained? As I highlighted in the introduction, the main characteristic of short-term capital inflows is their dependence on nominal short-term interest rates set by monetary policies. However, diverging nominal short-term interest rates across countries reflects only diverging inflation rates across countries since countries with constantly higher long-term inflation rates have steadily higher nominal interest rates, e.g. inflation differentials explain most of the nominal interest rate differentials, see figure 1. The correlation between the two variables is equal to 83% in the sample. In other words, short-term capital inflows are driven mainly by inflation rates. Why then, are these differences between exchange rate regimes as shown in table 1? The answer is that the higher level of inflation attracts short-term capital inflows only in the absence of exchange rate risk, e.g. only if the exchange rate is pegged. In this case, the value of the exchange rate is stable and foreign investors lower the risk of potential loses because there are rare adjustments. Indeed, this is what I found in the data, see figure 2. In panel 2a, the relationship between short-term capital inflows and inflation conditional on the fact that the exchange rate is pegged is drawn. As we can see, the link between the two variables is strongly positive. However, once the risk is introduced by allowing the exchange to be flexible, the relationship disappears, see panel 2b. While this analysis is intentionally simple, it intuitively demonstrates a clear relationship between short-term capital inflows, inflation rates, and the nature of the exchange rate regimes. Simple descriptive statistics already confirm the main hypothesis of this paper. Section 3 builds a simple model which explains the mechanism. Section 4 checks to see if the result holds ceterus paribus under statistical inferences.

A Simple Stylized Portfolio Balance Model

I built a simple model in order to show the link between exchange rate regimes, capital controls and speculative capital inflows. In the model, the monetary regime of the country is given: the country adjusts its nominal interest rate for any increase in its inflation rate. This hypothesis comes from the fact that countries with consistently higher long-term inflation rates have steadily higher nominal interest rates, see figure 1. Under this assumption, if the exchange rate is flexible there is no exchange rate policy. I call this situation the pure monetary regime. If the government decides to peg its currency, the government has to manage the exchange and the monetary regimes. I call this situation the hybrid monetary regime. By taking as given the monetary policy1 , I model the fact that speculative capital inflows is possible only if the government moves from a pure to a hybrid monetary regime, which can help to explain table 1 and figure 2.

The model that takes into account capital controls, the monetary regime, and the frequency of the exchange rate adjustment is a little technical because it uses a stochastic calculus. Before I present it, I will show a simple two period model which provides insights for the general and more realistic model.

A Model in Two Periods

Before considering a fully dynamic model in section 3.2, I first illustrate the main mechanism in a very simple two period model. The main research question remains the same: why do some countries receive much more speculative capital inflows than others? The objective is to determine the international distribution of speculative capital. I therefore consider an investor with two different investment alternatives, one of them being a speculative investment. Even in this simple framework, we can see how policy choices such as the exchange rate regime, capital controls and macroeconomic characteristics such as inflation, influence a country's attractiveness to speculators. In this case, I use a portfolio balance approach to model speculative capital flows since they are easily reversible compared to foreign direct investment. I use long-term capital flows as a given in this model because I'm only focusing on short-term capital inflows.

Wealth and Utility

I model an international investor with initial wealth W 0 who maximizes expected utility from wealth in period 1. He has two investment projects to distribute his wealth. A long-term project (L) and a short-term project (S).

W 0 = S + L (1)
The share invested in the two projects are noted by

ω = L W 0 and (1 -ω) = S W 0 (2)
The long term project is not risky and pays a return r r f . The return on the speculative project is noted r spec and will be detailed in section 3.1.2. Wealth in the next period will therefore be

W 1 = r r f L + r spec S = r r f ωW 0 + r spec (1 -ω)W 0 = (ωr r f + (1 -ω)r spec )W 0 . (3)
The investor thus maximizes

max ω E(U(W 1 )) (4)
in which utility is assumed to be logarithmic.

Return on Speculation in Two Periods

I start with a short illustration using two periods and later generalize in the continuous case. A speculator living in home country H and investing in the foreign country F gets the following return r spec from investing a dollar abroad:

r spec = R F + g s -π H . ( 5
)
R F is the foreign nominal interest rate. g s is the change in the exchange rate that the investor has to take into account since he has to convert his own currency into the foreign currency in period 0 and back in period 1. Due to home inflation, the home investor looses π H .

For a quick experiment, I plug g s in two extreme scenarios: fully fixed and fully flexible exchange rates in the foreign country (the home country is assumed to have flexible exchange rates in both cases). I then return to the general scenario, which uses the extreme scenarios as limiting cases.

If the currency and the good markets could function as a perfect market, the change in the exchange rate would equals the difference between home inflation π H and foreign inflation π F . This is the relative purchasing power parity (RPPP):

g f lex s = π H -π F . ( 6
)
in which I denote the variables under a perfectly flexible exchange rate with the index f lex .

Substituting this into (5) we see that the speculator investing abroad earns exactly the real interest rate of the foreign country:

r f lex spec = R F + (π H -π F ) -π H (7) = R F -π F = r F
This means that foreign investors simply earn the real interest rate of the foreign country, because the perfectly flexible exchange rate takes away all exchange rate risk. This case is illustrated in figure 2b: under a pure flexible exchange rate regime, equation ( 7) says that there is no relationship between short-term capital inflows (r f lex spec ) and foreign inflation (π F ), e.g. correlation(r f lex spec , π F ) = 0. Indeed in the data, see figure 2b, this correlation is equal to zero. Now, let us assume that the foreign country fully pegs its currency to that of the home country, a scenario we denote with peg . Then the exchange rate remains constant

g peg s = 0. ( 8
)
Substituting this into (5) we obtain

r peg spec = R F -π H (9) = r F + (π F -π H )
In this case the investor earns a nominal interest rate from abroad minus home inflation and therefore benefits from a high inflation abroad. In other words, under a pegged exchange rate, any positive inflation differential with respect to the anchor country is beneficial to investors.

This case is illustrated by the figure 2a: under a hard peg exchange rate regime, equation ( 9)

says that there is a positive relationship between short-term capital inflows (r peg spec ) and foreign inflation (π F ), e.g. correlation(r peg spec , π F ) > 0. Indeed in the data, see figure 2a, this correlation is equal to 44.35%. 1 .

Let's assume that the foreign country in principle wants to keep the exchange rate pegged.

But once in a while it is forced to adjust the exchange rate by the difference in inflation rates.

The speculator knows that eventually the exchange rate will adjust, but cannot anticipate exactly when the government chooses to do so. He therefore attaches a probability ρ to the event of an adjustment occurring. This feature of occasional exchange rate adjustments is based on the empirical literature concerning the uncovered interest rate parity (cf. Alexius 1 But if the peg is credible then π F = π H and so r f lex spec = r peg spec = r F , see [START_REF] Barro | A positive theory of monetary policy in a natural rate model[END_REF].

(2001), [START_REF] Chinn | The (partial) rehabilitation of interest rate parity in the floating rate era: Longer horizons, alternative expectations, and emerging markets[END_REF][START_REF] Campbell-Pownall | Irving fisher and the uip puzzle: Meeting the expectations a century later[END_REF] ), which says that there are deviations from UIP in the short run, but that it holds in the long run.

In this scenario, the expected return to speculation becomes a weighted average of the two limiting cases in ( 7) and ( 9):

r spec = (1 -ρ)r peg spec + ρr f lex spec = R F -π H + ρ(π H -π F )
ρ can also be interpreted as the exchange rate regime of the foreign government. A flexible exchange rate is indicated by ρ = 1. As adjustments become every more rare, ρ decreases and finally reaches a fixed exchange rate with ρ = 0. The link between ρ and the expected return to speculation can be summarized as follow:

{ρ, r spec }          ρ = 1 if fully flexible exchange rate ⇒ r spec = r F ρ = 0 if fully pegged exchange rate ⇒ r spec = r F + (π F -π H ) 0 < ρ < 1 if adjustable peg exchange rate ⇒ r spec = r F + (π F -π H )(1 -ρ)
Wealth in period 1 is with probability 1ρ equal to

W peg 1 = ωr r f + (1 -ω)(R F -π H )
and with probability ρ:

W f lex 1 = ωr r f + (1 -ω)(R F -π F ) .
Normalizing W 0 = 1 and substituting ( 7) and ( 9) into (3) we obtain for expected utility:

E[U(W 1 )] = (1 -ρ) log(W peg 1 ) + ρ log(W f lex 1 ) = (1 -ρ) log(ωr r f + (1 -ω)(R F -π H )) + ρ log(ωr r f + (1 -ω)(R F -π F )) (10)

Results in Two Periods

By maximizing the expected utility, it is possible to determine how the parameters affect the allocation of international investors between short-term and long-term capital flows:

Proposition 1 (Discrete time) If inflation in the foreign country is higher than inflation in the home country and that the real foreign interest rate is lower than the safe asset then:

The share of wealth invested in short-term capital (1ω) will depend (i) positively on foreign inflation π F (ii) negatively on the degree on the exchange rate flexiblity ρ Proof See appendix 1'

A higher foreign inflation increases the profits of the speculator in fear of no devaluation. This increases the overall attractiveness for the speculator. Since a devaluation decreases the speculator's profits, an increase in the probability of this event reduces speculative investments.

A very flexible exchange rate is characterized by a large probability for a devaluation, so ρ is close to 1. In this case, arbitrage profits are minimal. However, a peg that is certain of not experiencing a devaluation, that is, ρ = 0, allows speculators to earn arbitrage profits from high foreign inflation since it drives up the nominal interest rate abroad and the profits made abroad can be converted back using the same exchange rate used when the investment was made.

The Continuous Time Model

A continuous time framework is necessary since in practice all these variables are dynamic.

The continuous time allows also to introduce more "ingredients" in the previous model. This section extends the analysis from section 3.1 and includes it in a dynamic framework. Based on the work from the two periods already studied, we can analyze to what degree it is affected by the dynamic structure.

In order to theoretically find the determinants for the distribution of speculative capital across countries, I use the point of view of an individual international investor. In the most basic form considered here, the investor can choose between a speculative investment abroad and a long-term investment project, which I call long-term capital flows (LCF) . The speculative investment is subject to exchange rate changes since it must be moved twice across currency borders between countries with potentially very different inflation rates. The LCF project has a long-term horizon and its return is therefore less affected by inflation and exchange rates risks.

The profitability of speculation crucially depends on the interplay between the exchange rate and relative purchasing power parity (RPPP). If RPPP holds at all times (the exchange rate is perfectly flexible or the peg is perfectly credible1 ) then speculators could make no arbitrage profits between different currency zones and could ignore inflation differentials because they

will not be able to exploit exchange rate regime heterogeneity across different countries. Motivated by empirical results, I modeled skewed deviations of the exchange rate by introducing a jump process for the exchange rate in the target country of the investor. This also allows us to model the importance of skewness on the returns of portfolio choices.

The model is solved using stochastic dynamic optimization based on [START_REF] Merton | Lifetime portfolio selection under uncertainty: The continuous-time case[END_REF] and similar treatments of portfolio theory.

Wealth and Utility

I model an international investor with initial wealth W 0 and utility function

U(W t ) = log(W t ) (11)
The objective function is thus

V 0 = E ∞ 0 log(W t )e -βt |I 0 ( 12
)
where I 0 is all available information at t = 0 and β is the discount rate. He maximizes utility by allocating optimally between a speculative investment S t and long term investment, L t :

W t = S t + L t (13)
The investment shares into the two projects are noted by

ω t = L t W t (14)
(1

-ω t ) = S t W t (15)
The return of the LCF project is noted α:

d(lc f t ) lc f t = αdt (16)
In this basic form, the LCF project is equivalent to a risk-free rate 1 . It is modeled to be independent of inflation and exchange rates, so we can think of it as an investment in the home country of the investor, another member country of a currency union or another country 1 This paper focuses on the determinant of short-term capital inflows and abstract from the determinant of long-term flows so the process of the last capital is simply modeled. Another rationale of this hypothesis is that foreign direct investment is the main component of long-term capital inflows in the data presented earlier and the main determinants of FDI are the levels of financial development, the quality of institutions, property rights, productivity, etc. See [START_REF] Antras | Multinational firms, fdi flows, and imperfect capital markets[END_REF]. These determinants make long-term capital more independent to shortterm monetary policy. The long-term capital can include a broader type of assets. For this, the reader can see [START_REF] Kraay | Country portfolios[END_REF], [START_REF] Alfaro | Capital flows in a globalized world: The role of policies and institutions[END_REF], Gourinchas and Rey (2007b) and [START_REF] Lane | The external wealth of nations mark ii: Revised and extended estimates of foreign assets and liabilities, 1970-2004[END_REF] which focused on long-term capital flows with similarly low inflation.

In order to determine the return to the speculative investment, I calculate its profitability depending on the frequency of exchange rate adjustments:

Return to Speculation in Continuous Time

I generalize the previous illustration for a continuous time and I also introduce taxes on speculation. Without loss of generality, let us assume that the inflation rate in the home country is zero, π H = 0. Second, assume that the foreign country uses the following exchange rate policy: it pegs the exchange rate for long times, but at random points devaluates the currency very quickly. Then, speculators will benefit from the foreign inflation as long as the exchange rate stays constant and also experience a sudden loss at times when the currency devaluates.

Given these two assumptions, the continuous time can be written the same as in equation ( 5) and individual terms specified as:

dR t R t = dr Ft r Ft + π Ft + g st (17
)

dr Ft r Ft = (1 -τ)r 0 dt + σ 0 dx t (18) π Ft = εdt + σdz t (19)
g st = νdq t . ( 20
)
The foreign real interest rate r F is modeled as a deterministic interest rate r 0 plus a random part. σ 0 is the instantaneous volatility of the expected rate of return and x t is a standard Wiener process. The deterministic rate can be reduced by the foreign government through a tax τ ∈ [0, 1] on speculation in order to maintain an autonomy on monetary policy. The tax τ on speculative investments represents the possibility of the foreign government to introduce capital controls. For τ = 0 speculative capital is unrestricted, while for τ = 1 it is effectively banned. This allows the foreign government to keep a fixed exchange rate, an autonomous monetary policy and yet restrict speculative capital, thus respecting the "impossible trinity" introduced by [START_REF] Mundell | The appropriate use of monetary and fiscal policy for internal and external stability[END_REF] and [START_REF] Mundell | Capital mobility and stabilization policy under fixed and flexible exchange rates[END_REF].

Similarly, inflation is given as a deterministic trend ε and zero-mean noise, where σ is volatility and z t is a Wiener process. Without loss of generality, let normalize inflation in the home country to zero. In this case, ε is equivalent to the inflation differential in the two periods model (π Fπ H ). Taylor rule in which the central bank focused only on inflation and not on output1 .

Finally, the random devaluations by the foreign government are modeled as a Poisson process q t with intensity λ and amplitude ν. This type of modeling allows for the exchange rate to be endogenous and for investment strategies in speculative capital to be risky. The endogenous characteristics and the uncertainty of the exchange rate are a result of the random devaluations. The number of devaluations per unit of time is

Pr{one unit jump during dt} = Pr{dq t } = λdt + o(dt)
Combining equations ( 17) to ( 20), I obtain the following profit from speculation

dR t R t = (1 -τ)r 0 dt + σ 0 dx t + εdt + σdz t + νdq t (21)
The interest rate process and the exchange rate process will be linked through the uncovered interest parity condition. Concerning the size of the exchange rate adjustments, I can take guidance in the empirical literature on the uncovered interest rate parity (cf. [START_REF] Alexius | Uncovered interest parity revisited[END_REF], [START_REF] Chinn | The (partial) rehabilitation of interest rate parity in the floating rate era: Longer horizons, alternative expectations, and emerging markets[END_REF], and Campbell-Pownall et al. ( 2007) ), which says that there are deviations from the UIP in the short run, but that it holds in the long run. More interestingly, [START_REF] Campbell-Pownall | Irving fisher and the uip puzzle: Meeting the expectations a century later[END_REF] provide the reasons for why the UIP condition does not hold in the short run but is validated in the long run. First, they show that deviations from the uncovered interest parity condition are mainly explained by exchange rates forecasting errors and not by the risk premium factor. They also found that in the long run, exchange rate forecasting errors decrease progressively. In other words, in the short run, deviations from the UIP condition are mainly driven by the exchange rate forecasting errors2 . I formulate the long run condition as:

E dR t R t = E dr F r F (22)
The expected return on a speculative investment equals the expected real interest rate in the target country. Using the property of the Poisson process with intensity λ, E(dq t ) = λ and the fact that the relative purchasing power parity holds when the change in the exchange rate equals the difference between home inflation (normalized to zero) and foreign inflation ( ), I can solve ( 22) for ν:

ν = ε λ (23)
I define the size of the exchange rate adjustments as |ν|, e.g. the absolute amount of the jump.

Equation ( 23) makes the depreciations of the foreign currency endogenous since it depends (positively) on the inflation rate and (negatively) to the frequency of devaluations, which is consistent intuitively. This is a simple way to model the interaction between the exchange rate policy and the monetary policy. With this we can write the return to speculation as

dR t R t = ((1 -τ)r 0 + ε)dt + σ 0 dx t + σdz t + ε λ dq t ( 24
)
This equation shows that as long as the government keeps its exchange rate fixed, speculators will make arbitrage profits and "earn the inflation rate" as an additional profit. But whenever the exchange rate adjusts to restore long-run equilibrium, speculators will experience sudden losses. See figures 2 and 3. 16) and ( 24), the evolution of wealth can be written as:

dW t = ω t dlc f t f t W t + (1 -ω t ) dR t R t W t ⇔ dW t W t = ω t αdt + (1 -ω t )((ε + (1 -τ)r 0 )dt + σdz t + σ 0 dx t + ε λ dq t ) (25) = (ω t α + (1 -ω t )(ε + (1 -τ)r 0 ))dt + (1 -ω t )(σdz t + σ 0 dx t ) + (1 -ω t ) ε λ dq t

Results from Continuous Time

The first results concern the determinants of the investment shares. To obtain them, I solve the model with a Hamilton-Jacobi-Bellman equation. Since this is standard procedure and rather technical, the solution for this can be found as the proof of proposition 1' in the appendix.

Proposition 1' (Continuous time)

The share of capital invested in short-term capital (1ω) depends:

(i) positively on the average level of inflation ε in the target country (ii) negatively on the frequency of exchange rate adjustments λ Proof See appendix 1'

Part (i) means that since countries with steady higher long-term inflation rates have steadily higher nominal interest rates, a high level of inflation attracts speculator's capital even though the exchange rate is adjusted occasionally. In this model, this is the case because inflation pressures require a tighter monetary policy which for a given exchange rate, requires higher interest rates that in turn attract more short-term capital inflows.

Part (ii) links the number of exchange rate adjustments to inflows of speculative capital. Recall from equation ( 23) that the size of devaluating jumps ν is endogenously determined. But the frequency of these jumps, λ is a free choice parameter belonging to the foreign government (cf. equation ( 20)). This policy instrument allows the government to control the amount of speculative capital without requiring restrictions on inflows. In the continuous time, λ can be interpreted as the exchange rate regime of the foreign government, so λ is equivalent to ρ in the discrete time. In the presence of short-term capital inflows, an increase in λ means a more flexible exchange rate which increases the risk for investors and therefore eliminates the ''onesided bet" that a less flexible exchange rate would provide, see figures 2a and 2b for empirical evidences.

From the perspective of the exchange rate regime, proposition 1' (ii) can be expressed as:

Lemma 1 The amount of short-term capital inflows is minimized by

argmin λ (1 -ω) = ∞ maximized by argmax λ (1 -ω) = 0 .
In economic terms, a fully flexible exchange rate regime minimizes the inflow of speculative capital. A fully pegged exchange rate triggers large inflows of speculative capital. In between the amount of speculative capital inflows increases as the exchange rate regime becomes less flexible. A very flexible exchange rate is characterized by a large number of exchange rate adjustments (λ big), each adjustment being small (ν small). In this case, arbitrage profits are minimal. In the limiting case (λ → ∞), there is a permanent adjustment and no arbitrage profits. However, peg that has rare and large exchange rate corrections (λ small, ν big), allows speculators to earn arbitrage profits from high foreign inflation since it drives the foreign nominal interest rate up. This theoretical reasoning is empirically illustrated in figure 4. Australia and Malaysia's exchange rates with respect to the U.S. dollar are illustrated. Australia is characterized by continuous devaluations and therefore, the frequency of its exchange rate adjustments is high and as figure 4a shows, the size of each jump is small. For Malaysia, which has a pegged exchange rate, exchange rate adjustments are rare but when they do occur, they are large in size compared to continuous devaluations, see figure 4b. This is consistent with the peso problem hypothesis. Large nominal interest rate differentials are possible even though there is an absence of exchange rate changes. However, when these changes occur, they tend to be large because of the uncovered parity condition, equation 22.

Since it is a given that the exchange rate regime is a parameter choice for the government, why would the government want to choose anything other than a fully flexible exchange rate regime? The reason for this, is that governments' profit from fixed exchange rate because it could increase exports. This can potentially lead to a positive net profit since tax revenues from exporters exceed losses to foreign speculators. The details of this trade-off are however beyond the scope of the current model, so government policy will remain exogenous.

The exchange rate regime is not the only option for controling speculative capital inflows, at least theoretically. Established by the theory of the "impossible trinity", a second option is of course capital controls. The following proposition highlights another government policy instrument. The tighter capital controls are, the lower the share invested.

Proposition 2 The share of capital invested in speculative investments (1ω) depends negatively on the tax on speculative capital (τ)

Proof See apprendix 2

Speculative capital inflows imply a loss for the foreign government since it is the source for the speculator's arbitrage profits. In other words, taxing capital inflows allows the foreign government to reduce the share of speculative capital flows, e.g. to increase the share of longterm capital flows. In other words, capital inflows can affect the composition of aggregate capital. This is really not a new idea. It was first proposed by [START_REF] Tobin | A proposal for international monetary reform[END_REF] in order to ''throw sand in the wheels" of speculative capital. [START_REF] Magud | Capital controls: myth and reality, a portfolio balance approach to capital controls[END_REF] find similar theoretical results.

Similarly, we can identify the determinants of long-term investment, LCF:

Lemma 2 The share of capital invested in a long-term project (ω) depends positively on the return in the LCF target country (α)

Next, we consider the difference between a fully flexible exchange rate and a completely pegged exchange rate. The difference is characterized by λ, the frequency of jumps occurring in the Poisson process dq t . If λ → 0, the jump occurs more and more rarely and the size of the jump becomes bigger and bigger (recall that the size of the jump is λ ). In the case of λ = 0, jumps never occur and the UIP never holds. If λ → ∞, the jumps occur very frequently, but each jump is very small. For λ = ∞, the UIP will always hold. In all the intermediate cases -0 < λ < ∞ -the UIP will only hold at the moment of a jump but never in between. [START_REF] Flood | Uncovered interest parity in crisis[END_REF] empirically found that the UIP functions poorly in countries with a fixed rate regime. This model rationalizes Flood and Rose's finding. The link between λ, ν and the UIP condition can be summarized as follow:

{λ, ν}

         λ = 0 ⇒ ν = ∞ UIP never holds 0 < λ < ∞ ⇒ ν = λ
UIP holds only at the moment of a jump λ = ∞ ⇒ ν = 0 UIP will always hold Proposition 3 A country that keeps its exchange rate perfectly pegged (λ → 0) gives a speculative investor a premium equal to the inflation differential (ε), compared to a country with a fully flexible exchange rate (λ → ∞).

Proof See appendix 3

By formulating proposition 3 less technically, we can conclude that a country with a pegged exchange rate pays an extra interest rate to the speculators' capital that is equal to its inflation rate. If the peg is fully credible, the inflation differential is zero and there is no premium for foreign investors. Proposition 3 highlights the difference between pure monetary regimes and hybrid monetary regimes. Under a pure monetary regime, speculators only earn the real interest rate. Under a hybrid monetary regime, speculators benefit from a real interest rate and from inflation differentials since the exchange rate is not flexible enough to offset this difference. In other words, since both regimes suppose that there is a one-to-one relationship between the nominal interest rates and the inflation rates of countries, ceteris paribus, crosscountry differences in the exchange rate regimes can explain the differences in the pattern of speculative capital inflows. Proposition 3 therefore rationalizes the findings in section 2.2: allows me to take into account liquidity. [START_REF] Rodrik | Short-term capital flows[END_REF] and [START_REF] Komulainen | What drives financial crises in emerging markets?[END_REF] have shown that this variable is among the main determinants of short-term debt.

The above equation is estimated using different regression techniques to address the various short-comings of standard OLS. For example, to control for the potential endogeneity of inflation, this variable is instrumented by using its lagged values.

Since the goal of this paper is first and foremost to explain the geography of speculative capital inflows, I include the year fixed effect instead of the country fixed effects: the identification strategy relies on differences across countries within each year. The model predicts that

β 1 > 0, β 2 > 0,

Estimation Results

Tables 2 and3 are my baseline results and are presented in the following paragraph. Additional control variables and robustness checks follow in the following paragraph.

Main Results

Table 2 is the benchmark regression in which pegged and flexible exchange rate regimes are compared. The first column is an ordinary least square with the year fixed effect, while the third column presents general method of moment to deal with endogeneity. The regressions are performed for the full sample, and for both developed and developing countries. All variables have expected signs. Money supply which is the proxy of liquidity is positively linked to short term capital inflows. We now arrive at the variables of interest for our proposed model on the pattern of speculative capital inflows. As predicted by the model, higher inflation is associated to higher short-term capital inflows. Crucial for our theory, the regression for the full sample shows that countries with pegged exchange rate regimes received significantly higher short-term capital inflows than countries with flexible exchange rate regimes. A deeper analysis shows that this conclusion is valid for both developing and developed countries.

The former literature focused mainly on the role of capital controls on speculative capital.

Of course, having a direct measure of capital controls is difficult. But we may get an idea of the countries' financial account restrictions by using the Chinn-Ito's index. We expected countries with more capital controls to receive less speculative capital inflows. By including this index (see table 3), it is found that it is indeed the case, but that the coefficient is not significant, which means that capital controls seem to play an insignificant role in fending off ''hot money". The former literature reached the same conclusion1 . Our novelty is to show that the nature of the exchange rate regimes is more effective. This conclusion does not depend on the choice of the sample. By comparing table 2 and table 3 we see that these results are valid for both developing and developed countries. The ineffectiveness of capital controls is theoretically documented by [START_REF] Gros | The effectiveness of capital controls: Implications for monetary autonomy in the presence of incomplete market separation[END_REF]. Capital controls are innefective because financial operators always find a way to get round controls in order to make the same transactions.

Robustness Check and Extension

Robustness Check: The results discussed above are broadly consistent with the main thesis of this paper, the fact that the nature of the exchange rate regimes is more important than capital controls on the allocation of speculative capital inflows. This section seeks to check how robust this conclusion is and its level of sensitivity by adding more controls. Extension: Through the theoretical and the empirical parts of this paper, I demonstrated that it is possible to explain the pattern of speculative capital inflows across the world. The main conclusion is that: speculative capital inflows are more likely to flow into countries with managed exchange rates, higher inflation rates, higher ratio of money supply over gross domestic product, poor quality of institutions, low gross domestic product growth, and low population size through its link with the nature of exchange rate regimes. In line with former literature, I also showed that capital controls do not affect speculative capital. To explain the geography of speculative capital, I did not exploit the time variation of the series. It is possible to move beyond cross country comparison by investigating within-country variation: is a country more likely to receive less speculative capital inflows when its exchange rate regime becomes relatively less pegged? This can be achieved only by introducing country fixed effect and year fixed effect. I began by creating a new classification of the exchange rate regime that combines the pegged dummy, the intermediate dummy, and the flexible dummy in one index.

This new index takes a value of zero if for a given year in a given country, the exchange rate is pegged. It takes a value of one if for a given year in a given country, the exchange rate is an intermediated pegged. And it takes a value of two if for a given year in a given country, the exchange rate is fully flexible. In other words, an increase in this index means that the exchange rate is becoming more flexible. The index is called "Exchange regime".

The results are displayed in table 6. As we can see, all variables carry expected signs. When the within-country variation is exploited, all the variables continue to behave as predicted. The variable of interest, that is, the exchange rate regime index is negative and significant across all the specifications. This means that when a country moves relatively from a pegged to an intermediate regime, or from an intermediate to a flexible regime, this country will receive less speculative capital inflows.

Conclusion and Policy Implication

A large body of literature studied the causes and the consequences of the Asian financial crisis.

As noted by [START_REF] Rodrik | Short-term capital flows[END_REF] all these countries had on thing in common: large level of short-term foreign debt. This paper studied a broader type of short-term capital inflows and focused on gross capital inflows rather than net capital inflows. I show that it is possible to explain the geography of short-term capital inflows. Although the previous literature had focused on capital controls, I demonstrated how important the nature of exchange rate regimes is and the link between the monetary regime and the exchange rate regime. The main policy implication of this paper is the following: if the only goal is to fend off "hot money", as in Mexico in 1994, moving from a pegged exchange rate to a flexible exchange rate is more efficient than imposing controls on inflows. But adopting a flexible exchange rate can induce other costs and trade-offs between a pegged and a flexible exchange rate when fending off "hot money" is not the primary objective of the policy makers is open to future research.

A Appendix

A.1 Proofs of the Different Propositions

Proof of Proposition 1'

The first order condition of (4) using ( 12) is

0 = (1 -ρ) r r f -(R F -π H ) (ωr r f + (1 -ω)(R F -π H ) + ρ r r f -(R F -π F ) (ωr r f + (1 -ω)(R F -π F ) This is equivalent to M = 0 with M = (1 -ρ)[r r f -(R F -π H )][ωr r f + (1 -ω)(R F -π F )] + ρ[r r f -(R F -π F )][ωr r f + (1 -ω)(R F -π H )]
From this I get

∂M ∂ω = (1 -ρ)[r r f -(R F -π H )][r r f -(R F -π F )] + ρ[r r f -(R F -π F )][r r f -(R F -π H )] < 0 ∂M ∂ρ = -[r r f -(R F -π H )][ωr r f + (1 -ω)(R F -π F )] + [r r f -(R F -π F )][(ωr r f + (1 -ω)(R F -π H )] < ∂M ∂π F = (1 -ρ)[r r f -(R F -π H )][-(1 -ω)] + ρ[(ωr r f + (1 -ω)(R F -π H )] > 0
we can now write the FOC as

0 = H ω t = e -ρt [δ 1 (α -ε -(1 -τ)r 0 ) + δ 1 (1 -ω t )X -δ 1 λ λ + (1 -ω t )
]

In order to compute the effect of ε and λ on ω t we define

M = (α -ε -(1 -τ)r 0 ) + (1 -ω t )X -λ λ + (1 -ω t ) (33)
and use the implicit function theorem:

∂M ∂ω = -X - λε 2 (λ + (1 -ω t )) 2 < 0; ∂M ∂ε = -1 - λ 2 (λ + (1 -ω t )) 2 < 0; ∂M ∂λ = ε 2 (1 -ω) (λ + (1 -ω t )) 2 > 0.
which implies: ∂ω ∂ε = -∂M/∂ε ∂M/∂ω < 0 and ∂ω ∂λ = -∂M/∂λ ∂M/∂ω > 0

Proof of Proposition 2

Using the proof for proposition 1' and ∂M ∂τ = r 0 > 0 we obtain ∂ω ∂τ = -∂M/∂τ ∂M/∂ω > 0 .

Proof of Proposition 2

Using the proof for proposition 1' and ∂M ∂α = 1 we obtain ∂ω ∂α = -∂M/∂α ∂M/∂ω > 0

Figure 1 :

 1 Figure 1: Nominal Interest Rate vs Inflation Rate Notes: The y-axis is the average between the lending and the deposit rates between 1980 and 2009. The x-axis is inflation rate between 1980 and 2009. The variables are defined in section 2.1. Sources: World Development Indicators (WDI) and author's calculation.

Figure 2 :

 2 Figure 2: Sort-Term Capital Inflows and Inflation Conditional on Exchange Rate Regimes Notes: The y-axis is the share of short-term capital inflows over total capital inflows. The x-axis is inflation in log-differences of CPI. The variables are defined in section 2.1. For each year, I separated the sample between peg, intermediate, and flexible exchange rate regimes. I excluded intermediate exchange rate regimes in order to compare hard peg and pure flexible exchange rate regimes. Conditional on the exchange rate regime -peg and flexible -for each country, I computed the average share of short-term capital inflows over total inflows and the average inflation between 1980 and 2009. This leads to these two graphs. Sources: World Development Indicators (WDI, WB), Balance Of Payments Statistics (BOPS, IMF), and author's calculation.

Figure 3 :

 3 Figure 3: Illustration of the parameters describing speculative profits

Table 1 :

 1 Short-Term Capital Inflows across Exchange Rate Regimes

	Exchange Rate Regimes

Sources: Balance Of Payments Statistics (BOPS) and author

's calculation. 1980-2009

  The linear relationship between the trend in the inflation process (ε) and the nominal return (dR t /R t ) is clear thanks to figure 1. Furthemore, the relathionship can be rationalized by using the standard linear

dx t and dz t are temporally independent normally distributed random variables with E[dx t ] = E[dz t ] = 0 and Var[dx t ] = Var[dz t ] = dt.

table 1

 1 and figure 2. The empirical part seeks to test if the results hold under statistical inferences.

	4

Estimation and Methodology 4.1 Predictions derived from the theoretical model

  Through the different propositions and lemmas, this paper has shown that the main determinants of short-term capital inflows are: controls on inflows, nominal interest rate differentials between countries because of inflation differentials, and the nature of the exchange rate regimes. In return I have also shown that profits from inflation differentials depend on the exchange rate regimes. In other words, the model predicts that ceteris paribus, capital controls, inflation rates and exchange rate regimes should be the main determinant of short-term capital inflows. The dependent variable is defined as in the theoretical part: (1w t ) = S t /W t i.e. the share of short-term capital inflows over total capital inflows.Total Capital Inflowsi,t = β 0 + β 1 Inflation i,t + β 2 ERR i,t + β 3 Capital Controls i,t + β X i,t + µ t + U i,twhere i = 1, • • • , 70 indexes for countries and t = 1, • • • , 28 indexes the years (from 1980 to 2009). The dependent variable, the share of short term capital over total capital inflows is regressed on the inflation rate, capital account openness, and exchange rate dummies for hard peg and intermediate peg. The flexible regime is thus the reference. I intend to compare pegged and flexible exchange rate regimes. Another control variable is money supply, which

	Short-Term Capital i,t

  and β 3 < 0. In other words, speculative capital are more likely to flow into countries with high inflation, managed exchange rate regimes, and open financial account.

Table 4

 4 and table 5 add more macroeconomic controls. The first variable is the quality of institutions.[START_REF] Alfaro | Capital flows in a globalized world: The role of policies and institutions[END_REF] documented that the main determinant of long term capital flows is the quality of institutions. The regressions confirm this result. Speculative capital inflows and the quality of institutions are negatively correlated. The GDP growth rate is also added to take into account economic perspectives. Higher growth is associated with good/long-term perspectives that attract more long-term capital. As expected, speculative capital inflows and GDP growth are negatively correlated. Population is also added as a control for size. Countries with fixed exchange rate regimes are mainly small countries and hence attract more speculative capital. Indeed, the correlation between speculative capital inflows and population is negative.If we compare table 4 and table 5 with table 2 and table 3, we see that even though the new variables carry the signs expected and have significance, they do not strongly affect the results of the benchmark regression. Quantitatively, they remain significant and carry the signs expected, and qualitatively the main conclusion holds: the nature of the exchange rate regime plays a role on the allocation of speculative capital inflows. Capital controls do not affect this allocation.

  I start out with the general solution of the model. In order to maximize (12) subject to (26) I need to maximize the Hamilton-Jacobi-Bellman equation H with respect to ω t (cf. MertonThe first-order condition for an interior solution is: H ω t = 0. I postulate I(W t , t) in a timeseparable form asI(W t , t) = e -ρt [δ 1 log(W t ) + δ 0 ] (30)where δ 0 and δ 1 are to be determined from (29). Substituting (30) into (29) and simplifying yieldsH = e -ρt [log(W t ) + δ 1 (ω t α + (1ω t )(ε + (1τ)r 0 ))ρ(δ 1 log(W t ) + δ 0 ) ω t ) 2 (σ 2 + σ 2 0 + Cov(dx t , dz t ))] + λ(δ 1 log( ε λ (1ω t ) + 1)δ 0 ))] (31)Abbreviating the variance term as X = (σ 2 + σ 2 0 + Cov(dx t , dz t ))

	Applying the implicit function theorem we obtain ∂ω ∂ρ = -∂M ∂ω ∂M ∂ρ > 0 and Proof of Proposition 1' (1992), equation (4.17a) for example): max ω t H + 1 2 E t ∞ t I ω t -1 2 δ 1 [(1	∂ω ∂π F log(W t )e -ρs ds. = -∂M ∂ω ∂M ∂π F	< 0	(27) (29)	(28)

where

H = log(W t )e -ρt + I W (W t , t)W t (ω t α + (1ω t )(ε + (1τ)r 0 )) + I t (W t , t) WW (W t , t)W 2 t [(1ω t ) 2 (σ 2 + σ 2 0 + Cov(dx t , dz t ))] + λ[I(W t ( ε λ (1ω t ) + 1), t) -I(W t , t)]

where I(W t , t) = max

Throughout the paper, I will not consider sterilization operation in order to manage capital inflows since it requires widening the interest differential between the domestic and the international interest rates. This is more likely to attract more short-term capital inflows than to prevent them

As[START_REF] Levy-Yeyati | To float or to fix: Evidence on the impact of exchange rate regimes on growth[END_REF] acknowledge, their ''paper opens more questions than it answers."

As a supplement of this study, the reader can see[START_REF] Lane | International investment patterns[END_REF] "International Investment Patterns" which focused exclusively on portfolio equity holdings across countries

Contrary to a currency board or a dollarization, there is room for monetary policy under an adjustable peg. For example, China raised its interest rates for the fifth time by

basis points basis on July 7, 2011 in order to fight inflation. While having a pegged exchange rate vis-à-vis United States.

Or prices are not sticky[START_REF] Dornbusch | Expectations and exchange rate dynamics[END_REF]) but I abstract here from this last consideration

In the new Keynesian framework, this allow also to stabilize output. In other words, central banks can only focus on inflation management. This property is called divine coincidence by[START_REF] Blanchard | Real wage rigidities and the new keynesian model[END_REF]. But this is beyond the scope of this paper.

This is why this paper abstracts from risk premia considerations since deviations from UIP are almost explained by forecasting errors of the exchange rates rather than the risk premia. In this model, I do not model these errors of forecasting but in the model the only sources of uncertainty are the ''jump" and the ''frequency" of the exchange rates adjustment. So we may think that the forecasting errors of the exchange rate come from these two sources of uncertainty.

The exception is[START_REF] Magud | Capital controls: myth and reality, a portfolio balance approach to capital controls[END_REF]. They showed theoretically conditions under which capital controls can work or not, their effectiveness depending mainly on country-specific characteristics

Proof of Proposition 3

I determine the difference between the two limiting cases for λ in the function M from equation (33):

Consequently: lim λ→0 M(λ)lim λ→∞ M(λ) = ε


Marc Dambrine
email: marc.dambrine@univ-pau.fr

Djalil

D Kateb

Persistency of wellposedness of Ventcel's boundary value problem under shape deformations

come

Persistency of wellposedness of Ventcel's boundary value problem under shape deformations

Introduction and statement of the results

Let Ω be a smooth bounded domain of R d with d ≥ 2. Let α and β denote two real numbers and fix f ∈ L 2 (Ω). The Ventcel boundary value problem for the Laplace operator reads as follows

-∆u = f in Ω, ∂ n u + αu + β∆ τ u = 0 on ∂Ω, (1)
where ∆ τ stands for the Laplace-Beltrami operator on ∂Ω . The boundary condition appears in asymptotic models for coated structures: the second order term β∆ τ u represents surface diffusion on the boundary which models the tangential effects of the diffusion in the coating layer. Surface and volume diffusion both induce similar effects, therefore the coefficient β is naturally signed. In this case, β is nonpositive and a variational approach is available. Define the bilinear form A and the linear form B by

A(u, v) = Ω ∇u(x) • ∇v(x)dx + ∂Ω α u(x)v(x) -β ∇ τ u(x) • ∇ τ v(x)dσ(x), B(v) = Ω f (x)v(x)dx,
on the variational space

H(Ω) = u ∈ H 1 (Ω), u |∂Ω ∈ H 1 (∂Ω) .
Endowed with the norm

u 2 H(Ω) = u 2 H 1 (Ω) + u 2 H 1 (∂Ω) ,
the space H(Ω) is a Hilbert space. Then, the weak formulation of problem [START_REF] Arendt | The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions[END_REF] is the following: Find u ∈ H(Ω) such that for all v ∈ H(Ω), A(u, v) = B(v).

When β < 0 and α > 0, the bilinear form A is coercive. The existence and uniqueness of a solution to (1) is a consequence of the Lax-Milgram theorem. A large literature has been devoted to that case of great importance: the condition β < 0 is generally satisfied in the applications since the pioneering works of Feller and Ventcel, [START_REF] Feller | The parabolic differential equations and the associated semi-groups of transformations[END_REF][START_REF]Generalized second order differential operators and their lateral conditions[END_REF][START_REF]On boundary conditions for multi-dimensional diffusion processes[END_REF][START_REF] Ventcel | Semigroups of operators that correspond to a generalized differential operator of second order[END_REF]. For the specific case of the Laplace operator, we refer to [START_REF] Arendt | The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions[END_REF] and [START_REF] Engel | The Laplacian on C(Ω) with generalized Wentzell boundary conditions[END_REF][START_REF] Goldstein | Derivation and physical interpretation of general boundary conditions[END_REF][START_REF] Kennedy | A Faber-Krahn inequality for the Laplacian with generalised Wentzell boundary conditions[END_REF].

In the case β > 0, the quadratic form u → A(u, u) is neither positive, nor negative. Lax-Milgram's theorem does not apply. To the best of our knowledge, the condition β > 0 appear ed for the first time in a recent work of D. Bresch and V. Milisic [START_REF] Bresch | High order multi-scale wall-laws, Part I: the periodic case[END_REF] on wall laws in fluid mechanics. Marigo and Pideri also found such a boundary condition when studying equivalent boundary conditions for a elastic body damaged on surface [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF].

In the recent work [START_REF] Bonnaillie-Noël | On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain[END_REF], a first study of the case β > 0 has been performed. The main idea is to study the boundary value problem (1) as a nonlocal equation on the boundary. Take as new unknown w the trace of the previous unknown u on ∂Ω. After a lifting, ( 1) is rewritten as a nonlocal equation on the boundary:

β∆ τ w + Λw + αw = ϕ, on ∂Ω, (2)
where Λ denotes the Dirichlet-to-Neumann map. This equation has a sense in the space H 1 (∂Ω). The original unknown u is then recovered by solving a usual Dirichlet boundary value problem. Applying Fredholm alternative to this pseudodifferential equation, Bonnaillie-Noël and her coauthors obtained the following result:

Theorem 1.1 The operator P α,β = -β∆ τ -Λ -αId is an elliptic self-adjoint semibounded from below pseudodifferential operator of order 2. Besides, for fixed β > 0, there exists a sequence (α n (Ω)) n∈N growing to infinity such that for any φ ∈ H s (∂Ω) with s ∈ R, we have 1. If α ∈ {α n (Ω)}, then equation -P α,β w = φ admits a unique solution in S ′ (∂Ω) which, in addition, belongs to H s+2 (∂Ω);

2. If α ∈ {α n (Ω)}, then there is either no solution or a complete affine finite dimensional space of H s+2 (∂Ω) solutions.

Let us give an illustration of this result: consider the case of the unit ball in dimension three. As proved in the annex, the Ventcell boundary value problem is uniquely solvable if and only if

α / ∈ βn 2 + (β -1)n, n ∈ N . (3)
In this work, we consider α / ∈ {α n (Ω)}, i.e., α is chosen in such a way that the boundary value problem (1) has a unique solution. Deform Ω into another domain close to Ω. We address the following question: does the boundary value Problem (1) also have a unique solution on the perturbed domain?

Assume that Ω is a C 2 domain of R d , we prove that for small deformations of Ω the Ventcel boundary value problem remains uniquely solvable. Precisely, our result is the following.

Theorem 1.2 Let Ω be a C 3 domain of R d and consider α / ∈ {α n (Ω)}. Consider a C 2 vector field h and the application T h : R d → R d defined by T h = I R d + h. Then, there exists ε 0 > 0 such that the Ventcel boundary value problem

-∆u = f in Ω h = T h (Ω), ∂ n u + αu + β∆ τ u = 0 on ∂Ω h . ( 4
)
is uniquely solvable for all vector fields h satisfying h C 2 ≤ ε 0 .

This result is useful for numerical simulation when computations are made on a close but distinct domain: it ensures that the boundary value problem remains solvable on the approximating domain. It is also required to prove that the solution of ( 1) is differentiable with respect to the shape. Such a differentiability result is interesting for optimization purposes. In the particular case of the sphere, this theorem implies the existence of a neighborhood of the sphere in which the Ventcel boundary value problem is uniquely solvable.

Our strategy to prove Theorem 1.2 is the following. First, we transport the boundary value problem (4) defined on Ω h onto the fix domain Ω. We obtain a new boundary value problem written now on ∂Ω, this modified problem is not of the type of (1). Differential operators are modified by the change of variable. However, the key is the following: if the geometric deformation is sufficiently small, then the new operators are perturbations of the original one and we use a perturbation argument around the configuration on Ω.

The main difficulties are directly connected with the transport of Ω. First, the transport on ∂Ω of the Laplace-Beltrami operator on ∂Ω h has to be derived and linked to the Laplace-Beltrami operator on ∂Ω. Second, orthogonality is not preserved in the transport and once transported the Dirichlet-to-Neumann map on ∂Ω h is not a usual Dirichlet-to-Neumann map on ∂Ω. The paper is organized as follows. After introducing some definitions and notations, we present in section 3 preliminary results on the transport for the Laplace-Beltrami operator.After, we consider the transport of the Dirichlet-to-Neumann map. The last section is devoted to the proof of Theorem 1.2.

Let us introduce some notations. We denote by H s the usual Lebesgue and Sobolev spaces. The transpose and the determinant of a matrix A is denoted respectively by A * and det(A). The space of real-valued square matrices of size N is denoted by M N ×N . We denote in bold the vectorial quantities: for example, n and n h stand respectively for the unit normal vector field on ∂Ω and ∂Ω h . The notation a • b stands for the Euclidian scalar product of the vectors a and b. The tangential differential operators will be denoted by the subscript τ : for functions ϕ and V defined in a neighborhood of ∂Ω. We recall that

• ∇ τ ϕ := ∇ϕ -∂ n ϕ n is the tangential gradient of the scalar function ϕ. As usual, we have set ∂ n ϕ = ∇ϕ • n, • div τ V := div V -(DV n) n is the tangential divergence of the vector field V , • ∆ τ ϕ := div τ (∇ τ ϕ) is the Laplace-Beltrami operator on ∂Ω.
These quantities are only defined on the boundary ∂Ω. As the deformed domain Ω h depends on a parameter h, the operators related to ∂Ω h also depend on h, and are denoted by ∇ τ,h , div τ,h and ∆ τ,h . Through the paper, we will use the notation DT h for the Jacobian matrix of the transformation T h . Here,

DT h = I + Dh since T h = I R d + h.

Boundary perturbations and transported problem

In this section, h is fixed so that T h is a diffeomorphism from R d into itself. We aim to transport Equation (2) set on the perturbed domain's boundary ∂Ω h into a new equation set on the boundary ∂Ω of the reference domain. Therefore we have to compute the transport of both the Laplace-Beltrami operator and the Dirichlet-to-Neuman map under the deformation T h . Let fix some notations: we set

ω h (x) = det(DT h )(x) (DT h (x) * ) -1 n(x) .
The density ω h is the surface jacobian. It can be easily checked that ω h is a smooth function of h satisfying

ω |h=0 (x) = 1 and (Dω h ) |h=0 .ξ = div τ ξ. (5)
We also set

A h (x) = (DT h (x)) -1 (DT h (x) * ) -1 and C h (x) = ω h (x)A h (x).
We first prove a useful technical lemma.

Lemma 2.1

Let Ω be a C 3 smooth bounded domain of R d and let Ψ be a function in

C 2 (∂Ω), then there is a extension ψ of Ψ in C 2 (R d ) such that n • ∇ψ = 0 on ∂Ω.

Proof of Lemma 2.1:

By assumption, ∂Ω is a compact C 2 manifold, then its cut locus ρ is non negative. Any x ∈ R d such that the distance d(x, ∂Ω) from x to the boundary ∂Ω is strictly less than ρ has a unique orthogonal projection on ∂Ω denoted by p ∂Ω (x). Let d ∂Ω be the signed distance function to ∂Ω and fix χ : R → R + a C ∞ cutoff function such that:

χ(t) = 1 if |t| < ρ/3 and χ(t) = 0 if |t| > ρ/2. The function ψ(x) = χ • d ∂Ω (x)Ψ • p ∂Ω (x)
satisfies the requirements since the functions x → d ∂Ω (x) and x → p ∂Ω (x) are respectively C 3 and C 2 in the tubular neighborhood of ∂Ω of radius ρ. This is a consequence of the implicit functions theorem, see for example Theorems 4-2 & 4-3 in Chapter 5 of [START_REF] Delfour | Shapes and geometries[END_REF] for precise statements and detailled proofs.

To describe the transport of the Laplace-Beltrami operator ∆ τ , we first tackle the transport on the manifold ∂Ω of the tangential gradient of a function defined on ∂Ω h .

Transport of the tangential gradient. Let y be a point on ∂Ω h and x := T -1 h (y) be the corresponding point on ∂Ω. Given ϕ ∈ C 2 (R d ), we give an explicit formula of (∇ τ,h ϕ) • T -1 h , that is the back transport on ∂Ω of the tangential gradient of a quantity defined on ∂Ω h . By definition, the tangential gradient on ∂Ω h is

∇ τ,h ϕ(y) = ∇ϕ(y) -n h (y) • ∇ϕ(y) n h (y)
for all y ∈ ∂Ω h , that is to say for all x ∈ ∂Ω

(∇ τ,h ϕ) • T h (x) = (∇ϕ) • T h (x) -n h • T h (x) • (∇ϕ) • T h (x) n h • T h (x).
Let ϕ b = ϕ • T h be the back transport of ϕ on ∂Ω. By the chain rule, one has

(∇ϕ) • T h (x) = (DT h (x) * ) -1 (∇(ϕ • T h )(x)) that is to say ∇ϕ(y) = (DT h (x) * ) -1 ∇ϕ b (x).
Let us point out that the normal fields are linked with each other through the relation

n h (y) = n h • T h (x) = (DT * h ) -1 (x)n(x) (DT * h ) -1 (x)n(x)
.

Plugging the expression of the transported fields in the equation defining the tangential gradient, we obtain the expression of the transported tangential gradient at a point of ∂Ω as

(∇ τ,h ϕ) • T h (x) = (DT * h ) -1 (x)∇ϕ b (x) - A h (x)n(x) • ∇ϕ b (x) (DT * h ) -1 (x)n(x) 2 (DT * h ) -1 (x)n(x). ( 6
)
Note that if ∇ϕ.n h = 0, then this expression simplifies since

(DT * h ) -1 (x)n(x) • (DT * h ) -1 (x)∇ϕ b (x) = (DT * h ) -1 (x)n(x) n h (y) • ∇ϕ(y).
To summarize, we have proved the following lemma.

Lemma 2.2 For a function φ ∈ C 2 (∂Ω h ), we consider any extension ϕ ∈ C 2 (R d ) such that ∇ϕ.n h = 0. Then we have

(∇ τ,h φ) • T h (x) = (DT * h ) -1 (x)∇ϕ b (x).
The transport of the Laplace-Beltrami operator. Let us also recall from [START_REF] Henrot | Variation et optimisation de formes[END_REF] how to integrate by parts on ∂Ω the boundary of a domain Ω. Let ϕ be a function in H 2 (R d ) and V be a vector field supposed to be sufficiently regular, then

∂Ω (∇ϕ(x) • V (x) + ϕ(x) div τ V (x))dσ(x) = ∂Ω ∂ϕ ∂n (x) + H(x)ϕ(x) V n (x)dσ(x). ( 7
)
where H denotes the mean curvature of ∂Ω. We now are in position to derive the expression of the transported Laplace-Beltrami operator for a function defined only on ∂Ω h .

Lemma 2.3 For all functions φ ∈ H 2 (∂Ω h ), it holds

∆ τ,h φ(y) = 1 ω(h)(x) div τ C h (x)∇ τ ϕ b (x) on ∂Ω. (8)
Proof of Lemma 2.3: Fix φ in H 2 (∂Ω h ) and an extension ϕ such that ∇ϕ • n h = 0. We seek to compute the quantity ℵ such that for any ψ ∈ H 2 (∂Ω h ), it holds

∂Ω h ∆ τ,h ϕ(y) ψ(y)dσ(y) = ∂Ω ℵ(x)ψ b (x)ω h (x)dσ(x).
Here, the exponent b denotes the back transport on ∂Ω. To compute ℵ, we use the variational characterization of the Laplace-Beltrami operator. We fix a test function ψ in H 2 (∂Ω h ) and an extension Ψ such that ∇Ψ • n h = 0. By the variational definition of the Laplace-Beltrami operator, we get

∂Ω h ∆ τ,h ϕ(y) ψ ( y)dσ(y) = - ∂Ω h ∇ τ,h ϕ(y) • ∇ τ,h ψ h (y)dσ(y) = - ∂Ω (DT h (x) * ) -1 ∇ϕ b (x) • (DT h (x) * ) -1 ∇Ψ b (x) ω h (x)dσ(x) (by Lemma 2.2) = - ∂Ω C h (x)∇ϕ b (x) • ∇Ψ b (x)dσ(x).
Integrating [START_REF]Generalized second order differential operators and their lateral conditions[END_REF] by parts, we obtain that

∂Ω C h (x)∇ϕ b (x).∇Ψ b (x)dσ(x) = ∂Ω ∂ n Ψ b (x) + H(x)Ψ b (x) C h (x)∇ϕ b (x)•n(x)dσ(x) - ∂Ω Ψ b div τ C h (x)∇ τ ϕ b (x) dσ(x).
Let us simplify the formula by eliminating the first term of the right hand side: the symmetry of C h enables to get

C h (x)∇ϕ b (x)•n(x) = ∇ϕ b (x)•C h (x)n(x) = 0
Thus, we have obtained the announced expression for the transported Laplace-Beltrami operator.

In the application we are interested in, the function φ is defined as the trace of a Sobolev function ϕ defined in Ω h . The choice of a specific extension (such as in the proof of Lemma 2.3) cannot be done since ϕ is already given. Thus, Lemma 2.3 cannot be applied in our case and we need to adapt it to a function defined on Ω h . Before we state our result, we need to introduce the operator L(h) defined by

L(h) [ϕ • T h ] (x) (9) = 1 ω h (x) div τ C h (x)∇ τ [ϕ • T h ] (x) - C h (x)∇ [ϕ • T h ] (x).n(x) A h (x)n(x).n(x) A h (x)n(x)
for ϕ ∈ H 5/2 (Ω h ). We have Lemma 2. [START_REF] Delfour | Shapes and geometries[END_REF] The identity

[∆ τ,h φ] • T h = L(h) [ϕ • T h ] ( 10
)
holds for all functions ϕ belonging to H 5/2 (Ω h ).

Proof of Lemma 2.4:

Let ϕ ∈ H 5/2 (Ω h ), its trace also denoted by ϕ belongs to H 2 (∂Ω h ). We follow the proof of Lemma 2.3: we fix a test function ψ ∈ H 2 (∂Ω h ) and an extension Ψ such that ∇Ψ.n h = 0. By the variational definition of the Laplace-Beltrami operator, we get

∂Ω h ∆ τ,h ϕ(y) ψ h (y)dσ(y) = - ∂Ω h ∇ τ,h ϕ(y).∇ τ,h ψ h (y)dσ(y).
By the change of variables y = T h (x), we get integrals defined on the fixed boundary ∂Ω.

Thanks to the chain rule (see formula ( 6)), we obtain

∂Ω ((∆ τ,h ϕ) • T h ) (x)Ψ b (x) ω h (x)dσ(x) = = - ∂Ω (DT h (x) * ) -1 ∇ϕ b (x) • (DT h (x) * ) -1 ∇Ψ b (x) ω h (x)dσ(x) + ∂Ω A h (x)n(x).∇ϕ b (x) (DT * h ) -1 (x)n(x) 2 (DT * h ) -1 (x)n(x) • (DT h (x) * ) -1 ∇Ψ b (x) ω h (x)dσ(x)
Since ∇Ψ(y) • n h (y) = 0 : we have

(DT * h ) -1 (x)n(x) (DT * h ) -1 (x)n(x) • (DT h (x) * ) -1 ∇Ψ b (x) = ∇Ψ(y) • n h (y) = 0 hence ∂Ω ((∆ τ,h ϕ) • T h ) (x)Ψ b (x) ω h (x)dσ(x) = - ∂Ω C h (x)∇ϕ b (x).∇Ψ b (x)dσ(x).
We focus on the right side of the equation : we have to make some computations in order to get a formula depending only on Ψ b . Integrating ( 7) by parts, we obtain

∂Ω C h (x)∇ϕ b (x)•∇Ψ b (x)dσ(x) = ∂Ω ∂ n Ψ b (x) + H(x)Ψ b (x) C h (x)∇ϕ b (x)•n(x)dσ(x) - ∂Ω Ψ b div τ C h (x)∇ τ ϕ b (x) dσ(x). ( 11
)
It remains to deal with the term involving ∂ n Ψ b (x). From the property ∇Ψ • n h = 0 satisfied by the extended test function, we back transport it on ∂Ω h and decompose ∇Ψ b :

0 = A h (x)∇Ψ b (x) • n(x) = A h (x) ∇ τ Ψ b (x) + ∂ n Ψ b (x)n(x) • n(x).
Hence, it comes that

∂ n Ψ b = - A h (x)n(x) • ∇ τ Ψ b (x) A h (x)n(x) • n(x) .
Then we inject this expression in the right hand side of [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] and integrate by parts:

∂Ω ∂ n Ψ b (x)C h (x)∇ϕ b (x) • n(x)dσ(x) = ∂Ω - C h (x)∇ϕ b (x) • n(x) A h (x)n(x) • n(x) A h (x)n(x) • ∇ τ Ψ b (x)dσ(x) = ∂Ω div τ C h (x)∇ϕ b (x) • n(x) A h (x)n(x) • n(x) A h (x)n(x) Ψ b (x) -H(x)C h (x)∇ϕ b (x) • n(x) dσ(x).
Gathering the terms, we obtain

∂Ω ((∆ τ,h ϕ) • T h ) (x) ω h (x) Ψ b (x)dσ(x) = ∂Ω div τ C h (x)∇ τ ϕ b (x) - C h (x)∇ϕ b (x) • n(x) A h (x)n(x) • n(x) A h (x)n(x) Ψ b (x)dσ(x).
This achieves the proof.

Transport of the Dirichlet-to-Neuman map. Let us consider the Dirichlet-to-Neumann operator defined on its natural space Λ h : H 1/2 (∂Ω) → H -1/2 (∂Ω). It maps a function φ in H 1/2 (∂Ω h ) onto the normal derivative of its harmonic expansion in Ω h , i.e., Λ h (φ) = ∂ n h u, where u solves the boundary value problem:

-∆u = 0 in Ω h , u = φ on ∂Ω h . (12)
To compute the quantity ℵ such that ℵ(φ b ) = [Λ h (φ)] • T h , we back transport the boundary value problem [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF] on the domain Ω. Setting u h = u • T h , we check from the variational formulation, that the function v h is the unique solution of the transported boundary value problem:

-div ( Ãh ∇u h ) = 0 in Ω, u h = φ b on ∂Ω, (13)
where Ãh = det(T h )A h . Hence, we get formally

Λ h (φ)(y) = ∇u(y) • n h (y) = (DT h (x) * ) -1 ∇u h (x) • (DT * h ) -1 (x)n(x) (DT * h ) -1 (x)n(x) = 1 (DT * h ) -1 (x)n(x) A h (x)n(x) • ∇u h (x).
Here again, we can give a sense to the conormal derivative A h n.∇v thanks to the boundary value problem [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF]. This quantity is defined in a weak sense as the previous Dirichlet-to-Neumann operator Λ h . To be more precise, we have the following result Lemma 2.5 For φ ∈ H 1/2 (∂Ω), we define D h φ as the element of

H -1/2 (∂Ω) such that f ∈ H 1/2 (∂Ω) → D h φ, f H -1/2 (∂Ω)×H 1/2 (∂Ω) := Ω Ãh (x)∇u h (x) • ∇E(f )(x)dx,
where E is a continuous extension operator from H -1/2 (∂Ω) to H 1 (Ω). Then, for all functions ϕ ∈ H 1/2 (Ω h ), it holds

Λ h ϕ = D h [ϕ • T h ] . (14)
Now, we consider the restriction of the operator D h to H 1 (∂Ω). It will still be denoted by Λ h and is now to be considered as a linear continuous operator in L(H 1 (∂Ω), H -1 (∂Ω)).

We can now state the main result of this section.

Proposition 2.6 Let w be a function on ∂Ω h . One has

β∆ τ w + Λw + αw = ϕ on ∂Ω h (15)
if and only if w h = w • T h its back transport on ∂Ω satisfies

βL h w h + D h w h + αw h = ϕ • T h on ∂Ω. ( 16
)
3 The transported problem seen as a perturbation.

Let us introduce the operator L h defined on H 1 (∂Ω) with values in H -1 (∂Ω) :

L h w = βL h w + D h w + αw.
Our goal is to prove that L h is a perturbation of L 0 . To that end, we now want to express that the operator L h (resp. D h ) is a perturbation of the Laplace-Beltrami operator ∆ τ (resp. of the Dirichlet-to-Neumann map Λ). The persistency of the existence and uniqueness result under shape deformation is deduced from these two results.

Study of L h

Lemma 3.1 There exists a constant C > 0 such that

L h -∆ τ L(H 1 (∂Ω),H -1 (∂Ω)) ≤ C h C 2 holds for all h ∈ C 2,∞ (R d ; R d ) with h C 2 sufficiently small.
Proof of Lemma 3.1: Let φ ∈ H 1 (∂Ω), we write

L h ϕ = L h,1 ϕ + L h,2 ϕ + L h,3 ϕ + L h,4 ϕ, where L h,1 ϕ := 1 -ω h ω h div τ {C h (x)∇ τ ϕ} , (17)
L h,2 ϕ := div τ {(C h (x) -I)∇ τ ϕ} , (18)
L h,3 ϕ := div τ {∇ τ ϕ} = ∆ τ ϕ, (19)
and

L h,4 ϕ := - 1 ω h (x) div τ C h (x)∇ τ ϕ(x) • n(x) A h (x)n(x) • n(x) A h (x)n(x) . (20) Hence
(L h -∆ τ )ϕ = L h,1 ϕ + L h,2 ϕ + L h,4 ϕ.
We point out that if T h is a small perturbation of the identity in the norm of W 1,∞ , then DT h and DT -1 h belong to L ∞ (∂Ω, M N ×N ). It follows that C h belongs also to L ∞ (∂Ω, M N ×N ). Let ψ ∈ H 1 (∂Ω). After integration by parts, we get

∂Ω L h,1 ϕ ψ dσ = - ∂Ω C h ∇ τ ϕ • ∇ τ 1 -ω h ω h ψ dσ;
hence there exists a strictly positive constant C > 0 such that when h is small:

∂Ω L h,1 ϕ ψ dσ ≤ 1 ω h -1 W 1,∞ C h L ∞ (∂Ω,M N ×N ) ϕ H 1 (∂Ω) ψ H 1 (∂Ω) ≤ C div τ h W 1,∞ (∂Ω) ϕ H 1 (∂Ω) ψ H 1 (∂Ω) ≤ C h 2 ϕ H 1 (∂Ω) ψ H 1 (∂Ω) .
Here, we used the fact that the derivative of the surface jacobian is the tangential divergence as stated in [START_REF] Engel | The Laplacian on C(Ω) with generalized Wentzell boundary conditions[END_REF].

We focus now on L h,2 . A straightforward calculation shows that h

∈ W 1,∞ → (DT h ) -1 ∈ L ∞ (∂Ω, M N ×N ) is C ∞
when h is sufficiently small (see [START_REF] Henrot | Variation et optimisation de formes[END_REF], p 184). Hence, there exists a strictly positive constant C > 0 such

∂Ω L h,2 ϕ ψ dσ ≤ C h -I L ∞ (∂Ω,M N ×N ) ∇ τ ϕ L 2 (∂Ω) ∇ τ ψ L 2 (∂Ω) ≤ C |ω h -1| L ∞ (∂Ω) ∇ τ ϕ L 2 (∂Ω) ∇ τ ψ L 2 (∂Ω) ≤ C div τ h W 1,∞ (∂Ω) ϕ H 1 (∂Ω) ψ H 1 (∂Ω) ≤ C h 2 ϕ H 1 (∂Ω) ψ H 1 (∂Ω) .
Let us sketch the proof of that classic claim by reductio ad absurdum. We define

ω(δ) = sup h ≤δ u h -u 0 H 1 (Ω) .
We assume by contradiction that lim δ→0 ω(δ) = 0. Then, there exists a sequence h n such that h n ≤ 1/n and u hn -u 0 H 1 (Ω) ≥ α > 0 for all n ∈ N. Using the classical elliptic estimates, we check that the sequence u hn is bounded in H 3/2 (Ω) . Then, by the compact imbedding of H 3/2 (Ω) into H 1 (Ω), this sequence has to converge, up to an extraction, to a limit u ∞ in H 1 (Ω). Noting that div (A hn ∇u hn ) -∆u ∞ = div ([A hn -I]∇u hn ) -∆[u hn -u ∞ ], and passing to the limit n → +∞, we check that u ∞ satisfies -∆u = 0 in Ω, u = φ on ∂Ω.

By uniqueness of the solution of this boundary value problem, u 0 = u ∞ that contredicts u hn -u 0 H 1 (Ω) ≥ α > 0 for all n ∈ N.

Conclusion Gathering Lemmas 3.1 and 3.2, we have shown that there is a modulus of continuity ω such that L h -L 0 L(H 1 (∂Ω),H -1 (∂Ω)) ≤ ω( h ).

We are now in position to prove our main result Theorem 1.2.

Proof of Theorem 1.2.:

We place ourselves under the assumptions and notations of Theorem 1.2. For a given deformation field h, we have shown in Proposition 2.6 that the boundary value problem (4) is equivalent to the nonlocal equation (16). To prove that this equation has a unique solution, it suffices to prove that L h is invertible. Now, we remark that L 0 being invertible by assumption, we can write

L h = L 0 [I + L -1 0 (L h -L 0 )].
Then, for h small enough, the operator L h is invertible and its inverse can be written in terms of the Neumann's series

L -1 h = ∞ n=0 (I -L -1 0 L h ) n L -1 0 ,
expression that provides a uniform bound for the norm of the inverse.

A Case of the ball: justification of (3).

We denote by r the radius which is the distance from the point to the origin, by θ and ϕ the Euler angles. Since u is harmonic, we write it as a sum of spherical harmonics

  u(r, ϕ, θ) = +∞ l=0 r l m m=-l u m l Y m l (ϕ, θ),

Concerning L h,4 ϕ, we use the fact that n.∇ τ φ = 0 and rewrite L h,4 ϕ as

We then obtain

ψ dσ, and following the same arguments as before, we get

Study of D h Our aim is to prove the following result stating that the perturbed D i richlet-to-Neumann map depends continuously on the perturbation on the natural space for the Laplace-Beltrami operator .

Lemma 3.2 There is a modulus of continuity ω 2 such that for all h with h C 2 < 1, one has

Proof of Lemma 3.2: Take φ and f two functions in H 1 (∂Ω). Let us notice (see [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]) that H 1 (∂Ω) is the trace space of H 3/2 (Ω) and that the normal derivative of an element of H 3/2 (Ω) is not defined in general except under regularity properties of its Laplacian. Hence, the Dirichlet-to-Neumann map is still defined by the weak formulation and we will prove the estimation first in the norm of L(H 1/2 (∂Ω), H -1/2 (∂Ω)). The passage to L(H 1 (∂Ω), H -1 (∂Ω)) is then a consequence of the hierarchy of the norms. By definitions of the operators D h and Λ, one has

where u h and u are the respective solutions of the boundary value problem

It remains to estimate the variations

To that end, the key arguments are the two following continuity results. First, the application h → A h is continuous from W 2,∞ (R N ) in L ∞ (R N ). Second, we claim that there exists a modulus of continuity ω such that

where the spherical harmonics (Y m l ) are defined as

where P m l (cos θ) are the sequence of the associated Legendre functions. We recall [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF] that both the Laplace-Beltrami operator ∆ τ and the Dirichlet-to-Neuman operator Λ are diagonal on the basis of spherical harmonics, i.e., one has

To solve the Ventcel boundary value problem, we begin first to expand φ ∈ L 2 (S) into a series of spherical harmonic functions

In the next step, we have to solve the decoupled projected equations for the Fourier coefficient u m l of u : it comes obviously that

Since each projected equation should have a unique solution in order to obtain a unique solution to the original problem, we obtain that -βl(l + 1) + l + α = 0, ∀l ∈ N or equivalently [START_REF] Bresch | High order multi-scale wall-laws, Part I: the periodic case[END_REF]. Concerning the regularity of the found solution, we recall the caracterisation of Sobolev spaces on the sphere in terms of Fourier's coefficients: for an arbitrary f ∈ D ′ (S) we have when

Hence, for the admissible values of α, Fourier coefficients u m l of the solution u satisfy

We obtain for φ ∈ H s (S):