They have taken different forms: tightened restrictions on access to international capital markets, accumulation of large stocks of financial liabilities, debt restructuring, bailouts packages provided by the international financial organizations, debt rescheduling, delays in paying debt service, strong increases in bond spreads, arrears of principal or interest on external obligations. The question regarding the policies designed to tackle sovereign debt crises has given rise to a voluminous literature. The issue is not only the ways out of debt distress episodes (cleaning up after the crises occur), but more importantly how to lean against the wind (understand the chain of events leading to external debt crises and try to act in a preventive way). In the empirical literature, two approaches are usually explored.

A first branch of the literature on sovereign debt distress in the emerging economies attempts to identify those factors that are triggers of debt crises and define threshold values on the driving variables that permit early alert of debt stress outcomes. This literature builds upon a signal extraction approach by looking at the behavior of different macroeconomic variables that have been historically associated with debt crises: inflation, trade imbalances, fiscal and monetary policies, financial stress, extreme episodes of capital flows, etc. Debt crises have also often been associated with currency and banking crises. This literature therefore explores the triggers of sovereign debt crises and debt stress episodes using early warning models. The literature is massive and we refer the reader to some seminal papers by [START_REF] Berg | Predicting currency crises: the indicators approach and an alternative[END_REF], [START_REF] Herrera | A user guide to an early warning system for macroeconomic vulnerability in Latin American countries[END_REF], [START_REF] Kamin | The contribution of domestic and external factors to Latin American devaluation crises: an early warning approach[END_REF], [START_REF] Kaminsky | The twin crises: the causes of banking and balance of payment problems[END_REF], [START_REF] Ciarlone | Designing an early warning system for debt crises[END_REF] and to recent contributions by [START_REF] Arellano | Default risk and income fluctuations in emerging economies[END_REF], [START_REF] Boonman | Currency crises in Mexico, 1990-2009: an early warning system approach[END_REF], [START_REF] Manasse | Rules of thumb for sovereign debt crises[END_REF], [START_REF] Savona | Fitting and forecasting sovereign defaults using multiple risk signals[END_REF].

A second branch of the literature examines over-indebtedness through indicators of debt sustainability. The frequency of situations of debt distress in the emerging countries has been explained by the snowball effect (high interest rates above the economic growth rate), the original sin (countries borrow in foreign currency), currency mismatch (the currency denomination of the countries assets differ from that of their liabilities), the fiscal sin (fiscal policy pro-cyclical bias), the initial stock of debt. Debt sustainability analysis (DSA) has been developed as a way of getting insight into the probability that external debt becomes unsustainable in the face of plausible shocks. Probabilistic frameworks are therefore used to provide estimation of upper and lower intervals of the future evolution of debt. The idea is to assess, prospectively, which changes in policies should be undertaken today to prevent the future occurrence of debt crises if adverse shocks were to happen.

Here also the literature is massive. For recent contributions, we refer the reader to the papers by [START_REF] Adler | External conditions and debt sustainability in Latin America[END_REF], [START_REF] Aguiar | Measuring, explaining, and controlling tax evasion: lessons from theory, experiments, and field studies[END_REF], [START_REF] Tanner | Probabilistic sustainability of public debt: a vector autoregression approach for Brazil, Mexico and Turkey[END_REF].

This paper proposes a new approach of debt distress or debt crises in the emerging economies, based on the study of the distribution of external debts. We investigate debt distress as changes in the ratio of external debt over fiscal revenue that are extreme or large (relative to normal situations), but not rare. A situation of debt distress or debt crisis is defined by the probability that the debt ratio jumps above a given threshold. We consider distributions in the family of the so-called extreme values distribution (GEV) of [START_REF] Jenkinson | The frequency distribution of the annual maximum (or minimum) values of meteorological elements[END_REF], with a specific attention to distributions with a decay like a power function rather than those with an exponential rate of decay. The reasons are twofold. First, debt crises in the emerging countries have been found to happen fairly often (they are not rare). Secondly, changes from normal to abnormal debt situations do not necessarily occur abruptly. Indeed, unlike in the developed countries, extreme debt events in the emerging countries are not events occurring only under extraordinary economic events (shocks hitting the economies or the financial systems implying a huge variance in the data that disappear when the shocks are switched off). Conversely, the sources of uncertainty causing changes in the ratio of external debts are endogenous to the economic system, since the factors determining the dynamics of debt in normal time are the same factors triggering debt stress episodes. For both reasons, we need distributions whose tails decay more slowly than those of Weibull or Gumpel distributions. We therefore consider distributions in the basin of attraction of Fréchet distributions. The candidate distributions being potentially large (Pareto, Cauchy, Student-t etc.) we focus on a class of distributions that we motivate theoretically: power-law functions. See [START_REF] Gabaix | Power laws in economics and finance[END_REF] for a detailed characterization of power-laws and a survey of empirical power-laws.

After providing empirical evidence that the distributions of their external debt-to-fiscal revenues ratio is represented by a power-law function, we propose a theoretical explanation of the occurrence of such a power-law. More specifically, we develop a continuous time stochastic optimal growth model where the dynamics of debt is described by an Itô diffusion process. The sources of uncertainty are intrinsic to the economic system and come from both tax evasion (households only report a fraction of their income to the fiscal administration) and corruption (a fraction of fiscal revenues is stolen by corrupted bureaucrats, see [START_REF] Célimène | Tax evasion, tax corruption and stochastic growth[END_REF] for similar assumptions in a closed economy framework). Fraud activity is risky because the households can be caught with a given probability and fined, if this happens. Similarly, the fiscal administration faces a situation of uncertainty because the corrupted bureaucrats are not known with certainty. It only has a suspicion that some of its civil servants are corrupted. We examine the asymptotic behavior of our variables using the concept of steady state in a stochastic sense. The steady state is defined in terms of a probability density function which is shown to be a power-law.

The remainder of the paper is organized as follows. Section 1 establishes empirical evidence that the external debt-to-fiscal revenue ratio has a power-law distribution in the emerging countries. We go beyond the visual inspection of a log-log graph of the distribution of the series, by estimating the scale coefficient and by testing formerly the hypothesis of a power-law distribution. Section 2 contains our theoretical model. Finally, Section 3 concludes.

Empirical power-laws in foreign debt-to-public revenue ratio

Let us look at the data in Figures 1 to 3. We plot the ratio of external debt over public revenues for some major emerging countries. The data are quarterly from 1980 (1990 for some countries) to 2014 and are taken from Oxford Economics and national sources (central banks and statistical institutes). Foreign debt data do not only comprise the general government's external debt but also include external debt issued by the private sector. Public revenues data correspond to central and local government revenues in local currency, including different types of taxes, interest receipts and dividends from state owned enterprises. To compare them with the ratio of external debt on GDP, these public revenues are then divided by nominal GDP (at market prices, in local currency).

We show the non-parametric distribution of the ratio using both a Gaussian and Epachenikov Kernel. The countries are shared over three regional areas : Latin America, Asia and Europe. We see that many of the distributions have fat tails in their right side and do not seem to fit with a Normal law. The probability of finding extreme values for the external debt ratio is higher than for a Gaussian distribution. These extreme values reflect situations of external debt vulnerability, which may cause financial tensions (rising interest rates, currency depreciation, capital outflows). To prevent such situations, policymakers may seek to estimate the probability that the debt ratio exceeds a certain value which is believed to correspond to episodes of debt stress or crises. This is usually done thanks to early warning indicators. A more accurate way to proceed is to estimate a parametric law that approximates the observed distributions. In what follows, we show that the extreme values observed in Figures 1 to 3 follow power-law distributions, more specifically Pareto type 1 distributions.

Formally, a continuous variable x obeys a power-law with parameter α if its probability density function is written as

p(x) = Cx -α , x ≥ x min 0, x < x min , (1)
with the normalizing constant C = α-1 (x min ) -α+1 , x min being the lower bound of the power-law behavior and α the scaling parameter.

The m-th order moment is given by

x m = ∞ x min x m p(x)dx = α -1 α -1 -m
x m min , which is well defined for m < α-1.

(2) For finite size samples, the central moments like the mean and the variance are not finite. Typical values for Pareto laws in economics are found for α in the interval (2, 3) for which the mean is well defined and exists but the variance and higher-order moments are infinite.

Estimates of power-law based on bootstrap methods

We first estimate the lower bound x min from which the data can be considered to follow a power-law. We follow [START_REF] Clauset | Power-law distributions in empirical data[END_REF]. For a given x min (assumed to be known), we compute the maximum likelihood estimator of α : whose standard error is equal to α-1 √ n + O(1/n), with n being the number of observed values being higher than x min . α corresponds to the shape parameter of the best fitting power-law, for this given value of x min . Then, we compute the weighted Kolmogorov-Smirnov statistic, defined as the distance between the CDFs of the empirical data and the best fitting power-law model:

α = 1 + n n i=1 ln( x i x min ) -1 , (3)
D * (x min ) = max x≥x min |S(x) -P (x)| P (x)(1 -P (x)) , (4)
where S(x) is the CDF of the data for the values higher than x min , and P (x) is the CDF of the power-law model that best fits the data for x ≥ x min . Using a scale of possible values for x min , the "optimal" x min is finally computed so as to make the distributions of the empirical data and of the best fitting power-law as close as possible:

xmin = argmin x min D * (x min ). (5)
It minimizes D * on a range of possible values for x min . The mean and standard errors of the estimated parameters xmin and α(x min ) are then obtained by a bootstrap procedure. This consists in drawing N draws (N=1000) uniformly at random from the original dataset, estimating α and x min again for each of these series.

Regression-based approaches

The previous method is based on the assumption that a power-law can fit the data well, which is not necessarily true. To evaluate whether a Pareto law really fits the data, we proceed by using a rank-regression approach proposed by [START_REF] Gabaix | Rank-1/2: a simple way to improve OLS estimation of tail exponents[END_REF].

By OLS, we run the log-log rank-size regression (for γ = 0.5):

log(t -γ) = a -b log(Z(t)) ⇔ log(Rank -γ) = a -b log(Size), ( 6
)
where t is the rank of the observations Z(t) (ranked in decreasing order). The standard error of the shape coefficient b id 2 n b, with n the "total" rank, i.e. the number of observations higher than x min (i.e. following a power-law).

Another way to compute this standard error is to estimate the opposite regression:

log(Z(t)) = a 2 -b 2 log(t -γ), (7)
where the shape coefficient is obtained by 1 b2 whose standard error is 2 n 1 b2

. One can plot log(t -0.5) against log(Z(t)), as well as the straight line whose equation we have estimated, such that we see if the data fits well with the power-law on this log-log plot. [START_REF] Gabaix | Rank-1/2: a simple way to improve OLS estimation of tail exponents[END_REF] show that the bias is optimally reduced for γ = 0.5.

Evidence of Pareto laws in foreign debt data

Tables 1 and2 show the estimates for our sample of countries. Graphs of log-log plots of the estimates are not reported to save place but are available upon request to authors. The estimates suggest that many of the distributions of the ratio of the external debt over public revenues can be approximated by power-laws. The scale exponent lies between 2 and 4 which are typical values for which power-law behaviors are usually observed. This has several implications for the analysis of the emerging countries' external debt.

Firstly, the standard analysis of debt sustainability, that consists of examining how some economic shocks can make the debt ratio deviate from their "normal" levels, can be misleading. Indeed, with power-laws, each new event changes the central and higher moments of the distribution, so that debt does not necessarily stabilize around an equilibrium value. It seems more appropriate to focus on finding critical levels around which the debt oscillates and from which debt crises can occur.

Secondly, in power-law distributions, extreme events are not rare. This accords with the finding that some emerging countries have been found to be "serial defaulters" on their foreign debt : if they have defaulted in the past, they are very likely to default again in the future (the literature on serial sovereign default is abundant). For a recent literature review, see [START_REF] Asonuma | Serial sovereign defaults and debt restructurings[END_REF].

Tax evasion and corruption as causes of power-laws in foreign debt ratios: a theoretical model

What causes high external debt in the emerging countries? The literature suggests that episodes of foreign debt distress have been systematically as- (2005), [START_REF] Reinhart | This time is different: a panoramic view of eight centuries of financial crises[END_REF], [START_REF] Reinhart | From financial crash to debt crisis[END_REF], [START_REF] Reinhart | Banking crises: an equal opportunity menace[END_REF].

x
What is distinctive in our paper relative to previous works is that we explain theoretically how outcomes of high external debt ratio depicted by a power-law distribution can arise from loose civic capital or governance, captured by tax evasion and bureaucratic corruption in the fiscal administration. Both these factors weaken a country's fiscal position by contracting government's revenues. Public spending is therefore financed through a forcible foreign borrowing. Since tax revenues account for the greatest part of public revenues and given that interest receipts and dividends from state owned enterprises are exogenous (in the sense that they do not directly depend upon tax evasion and corruption behaviors), we focus on tax revenues as the denominator of the external debt ratio. We propose a simple continuous time stochastic growth model from which the exact asymptotic distribution of foreign debt-to-fiscal revenue ratio is obtained using results from the theory of stochastic differential equations.

Two types of uncertainties are introduced into the model. First, we consider lotteries on tax evasion and corruption behaviors which give us the expected return to cheating for both households and bureaucrats, given the risks of being a fraudulent and a corrupt government. Secondly, we consider diffusion processes as follows.

Consider a random variable X(t). Its change between time t and t + dt is defined by the Itô stochastic differential equation:

dX(t) = f (X(t))dt + g(X(t))dZ. (8)
f and g are two continuous functions and Z is a Brownian motion : dZ ≈ N (0, dt) and dt is of infinitesimal order. Moreover, we assume that Z is a Markovian process in the sense that dZ depends neither on t nor on its past values. This assumption will allow us to use the general properties of diffusion processes to derive a probability distribution for our main variable of interest.

Production

We consider an open economy specialized in the production of a single good. This good is the numeraire and its price is normalized to 1. The country comprises N inhabitants (consumer/taxpayers and workers). All the variables in the model are expressed in per-capita terms. The individuals differ in their behaviors as taxpaypers (some are fraudulent, others are not), but their consumption and production decisions are the same.

The single domestic good of the economy is produced with capital, labor (fixed amount) and public spending

y(t) = k(t) β g(t) 1-β , (9)
where y(t) is per-capita income, k(t) is private capital-labor ratio, g(t) is percapita productive public spending (infrastructure, public goods and services).

We assume constant returns to scale (0 < β < 1).

Domestic capital yields a stochastic return. The flow of private rate of return on capital over the period (t, t + dt) is subject to a stochastic disturbance:

dR k (t) = r k dt + du k , (10)
where the drift component r k = ∂F ∂k is a constant return to private capital and the diffusion component du k is a Brownian motion.

Tax evasion

A consumer is faced with the decision of whether or not to hide a share of his actual income y(t) to evade taxes. The income tax rate τ is constant. He hides a fraction e(t) of the income (0 < e(t) < 1). Therefore [1 -e(t)]y(t) is the amount of reported income. The probability of being detected as a fraudulent by the administration is p (0 < p < 1) and, if caught, the taxpayer pays the due tax plus a penalty. b is the penalty rate on the amount of evaded tax bτ e(t)y(t). Therefore, the consumer pays τ [1 -e(t)]y(t) if he is not detected and τ y(t) + be(t)y(t) if he is caught. The return of each unit of evaded tax is described by the following lottery:

x = 1, w.p. 1 -p -b, w.p. p . ( 11
)
The expected gain and risk of the lottery are defined by

x = (1 -p) -pb, σ 2 = E(x 2 ) -E(x) 2 = E(x 2 ) -x2 . ( 12
)
x and σ 2 are function of p and b.

Corruption and government's choice

The tax administration consists of bureaucrats whose number is the same as the consumer3 . A fraction p 1 of them are corrupted and a fraction 1 -p 1 who are non-corrupted (0 < p 1 < 1). Bureaucratic corruption arises when civil servants charged with the responsibility for collecting taxes choose to extort a fraction θ(t) of tax revenues (it happens with a probability p 1 ). They steal a fraction of tax resources which normally serves to finance public goods that enter as inputs in the production activity.

A fraction p 1 θ(t) of tax revenues is stolen and converted to unproductive public spending (or public consumption). Per-capita public spending is therefore described by

g(t) = (1 -p 1 θ(t)) τ y(t) + be(t)y(t) , w.p. p (1 -p 1 θ(t))τ (1 -e(t))y(t), w.p. (1 -p) , (13)
and the dynamics of productive public spending is given by the following stochastic differential equation (SDE):

dg(t) = ḡ(t)dt + g(t)dW g , (14)
where W g is a Brownian motion, ḡ(t) is the expected productive public expenditure from the above lottery and g2 (t

) = V(g(t)) = E(g(t) 2 ) -E(g(t)) 2
is the variance of the lottery. Unproductive public spending (the share of diverted tax revenues which we consider to be public consumption) is given by

C G (t) =    0, w.p. (1 -p 1 ) θ(t) τ y(t) + be(t)y(t) , w.p. pp 1 θ(t)τ (1 -e(t))y(t), w.p. p 1 (1 -p) . ( 15
)
The dynamics of public consumption is described by the following SDE

dC G (t) = CG (t)dt + CG (t)dW C G , (16)
where W C G (t) is a Brownian motion and

CG (t) = E(C G (t)) = p 1 θ(t)y(t) pbe(t) + pτ e(t) + τ (1 -e(t)) , ( 17
) CG 2 = V(C G (t)) = E(C G (t) 2 ) -E(C G (t)) 2 = E(C G (t) 2 ) -CG (t) 2 . (18)
In this stochastic framework, the balanced budget constraint describes how the flow of tax revenues is assigned to the flows of productive and unproductive public spending:

dT (t) = ḡ(t) + CG (t) dt + gdW g (t) + CG dW C G , (19)
where

CG (t) = p 1 θ(t)y(t) pbe(t) + pτ e(t) + τ (1 -e(t)) (20)
p 1 θ(t) can be interpreted as a corruption index ranging from 0 (low corruption) to 1 (high corruption). It depends upon the proportion of corrupted bureaucrats and the degree of prevarication in the use of public resources. We assume that the bureaucrats' punishment for being corrupt is nil. This is a distinguishing feature of countries with poor governance (lack of transparency and accountability, inefficient court systems). Moreover, it is assumed that θ(t) > 0 in the sense that corruption is a rent seeking activity that is privately beneficial, and that θ(t) < 1, because corruption inhibits growth (by reducing the productivity of private capital) and therefore reduces the amount of the rent. Therefore, the bureaucrats' aim is to define an optimal degree of prevarication (how much to steal) that maximizes public consumption.

The government maximizes its intertemporal expected utility function defined on public consumption subject to the constraint (19):

max n G ,n C E +∞ 0 1 γ (T (t)n C (t)) γ e -ρt dt, ρ > 0, -∞ < γ < 1, (21)
subject to dT (t)

T (t) = ψ T (t)dt + σ w T (t)dw T , T ∈ (0, ∞), (22)
and

n g (t) + n C (t) = 1, ( 23
)
where

n g (t) = ḡ(t) T (t) , n C (t) = CG (t) T (t) , ψ T (t) = ḡ(t) T (t) + CG (t) T (t) , (24)
σ 2 w T (t) = ñg (t) 2 σ 2 W g (t) + ñ2 C (t)σ 2 W C G (t), ( 25
) ñg (t) = g(t) T (t) , ñC (t) = CG (t) T (t) . ( 26
)
The solution to the maximization problem of the government gives an optimal path of the share of public consumption out of total fiscal revenues defined by (we omit the index t for purpose of simplification)

4 nC = σ 2 W g + σ 4 W g -4(σ 2 W C G + σ 2 W g )( ρ-γ 1-γ + 1 2 σ 2 w T γ) 2(σ 2 W C G + σ 2 W g ) (27)
This solution illustrates the role of the relative aversion risk coefficient (1γ). When corruption takes place, there is a chance that lean times could occur in the future. A bureaucrat could suffer from loss in his future consumption because the outcome of diverting tax revenues is lower productive public spending, lower production and thus lower tax base.

The open economy and external debt

Insofar as tax evasion and corruption do not allow the domestic country to produce the amount of the single domestic good that households would like to consume5 , the country needs to buy a foreign good from abroad. We adopt the convention that imports of goods is equivalent to external borrowing, so that external debt is defined as the domestic country's net international current account position (i.e. C(t) M = D(t)). We make the following assumptions:

• P M is the price of imports and therefore 1/P M is a proxy of the terms of trade. Import price changes are described by the following SDE:

dP M (t) P M (t) = πdt + dP, (28)
where π defines changes in the terms of trade and P is a Brownian motion.

• Debt is denominated in terms of the foreign output and its price is expressed in terms of the price of the numeraire P M D. The interest rate paid on foreign debt is stochastic and described by

dR f (t) = r f dt + du f , r f = i * + π and du f = dP. ( 29
)
where r f is the real world interest rate assumed to be constant.

Consumer's choice

The domestic country comprises agents who consume the domestic and the imported goods. The agent bears domestic assets (he owns the firm) and foreign liabilities (he pays back the foreign debt). His wealth constraint is given by

W (t) = K(t) -P M (t)D(t), W ∈ (0, ∞). ( 30
)
This implies

K(t) W (t) - P M (t)D(t) W (t) = n K (t) -n f (t) = 1, (31)
where n K (t) and n f (t) are the shares of capital and foreign debt in total wealth with 0 < n K (t) < 1 and n f (t) > 0 (domestic country assumed to be indebted vis-à-vis the rest of the world).

The stochastic wealth accumulation equation is given by

dW (t) = 1 -τ + xτ e(t) k(t) β g(t) 1-β dt -(C P (t) + P M (t)C M (t))dt +σ τ e(t)k(t) β g(t) 1-β dW W +k(t)dR k (t) -P M (t)D(t)dR f (t) . ( 32
)
This equation has several drift and diffusion components:

• Line 1 describes the drift component of the disposable income. The household can inflate his disposable thanks to tax evasion activity. Given (9), we have (1

-τ )k(t) β g(t) 1-β = (1 -τ )y(t)
. This is the disposable income when there is no tax evasion. e(t)τ (t)y(t) is the amount of tax evaded. Since tax evasion is a risky activity, we need to consider the expected return of tax evasion that is xe(t)τ y(t) where x is defined by (12).

• Line 2 defines the drift component of consumption, that is the household's consumption of domestic and foreign goods that are assumed to proceed at non-stochastic rates over the interval (t, t + dt).

• Line 3 corresponds to the diffusion component of disposable income or the risk associated to fiscal evasion. The standard error of the Brownian motion W W is στ e(t)y(t) where σ is defined by ( 12).

• Line 4 (first expression) k(t)dR k (t) describes both the deterministic and the stochastic shocks to the productivity of private capital.

• Line 4 (second expression) P M (t)D(t)dR f (t) describes the influence of the international financial markets and of the terms of trade shocks which impact the interest rate paid on external debt.

The consumer's objective is to choose his domestic and foreign consumption, the amount of income to hide and his portfolio of net assets (assets minus liabilities) in order to maximize the expected value of discounted utility subject to the constraint given by (C.2): max

C P ,C M ,e,n K ,n f = E +∞ 0 1 µ C P (t) η C M (t) 1-η µ e -ρt dt, ρ > 0, -< µ < 1, (33) subject to dW (t) W (t) = ψ W (t)dt + σ z W (t)dz W , n K (t) -n f (t) = 1, (34)
and W (0) = w 0 , where

ψ W (t) = 1 -τ + xτ e(t) n β k (t)n 1-β g (t)( T (t) W (t) ) 1-β -CP I(t)C(t) W + n k (t)r k -(i * + π)n f (t), (35)
σ z W dz W = στ e(t)n β k (t)n 1-β g (t)( T (t) W (t) ) 1-β dW W + n k (t)du k -n f (t)du f . (36) n k (t) = k(t) W (t) and n f (t) = P (t) M D(t) W (t)
are respectively the share of capital and foreign debt in total wealth 6 . The optimal path is described by the following system of recursive equations 7 :

                     Ĉ(t) = (δµ) 1 µ-1 (P M (t)) 1-η µ-1 W (t) ê(t) = xτ nk (t) β ng(t) 1-β ( T (t) W (t) ) 1-β (1-µ) στ nk (t) β ng(t) 1-β ( T (t) W (t) ) 1-β σ W W (t) 2 nk (t) = r k -(i * +π)+(1-µ)σ W f (t) 2 + √ 2(1-µ)(σ W k (t) 2 +σ W f (t) 2 ) nf (t) = nk (t) -1 (37) with = r k -i * -π + (1 -µ)σ 2 W f 2 -4(1 -µ)(σ 2 W k + σ 2 W f ) (1 -µ)βσ 2 τ 2 ê2 σ 2 W W n 2 y -β 1 -τ + xτ ê n y .
From the first equation, we see that total private consumption (private domestic goods plus imports) increases with wealth. Moreover, an increase in import prices (deterioration of the terms of trade) raises the nominal debt and accordingly reduces the nominal net wealth (since W (t) = K(t) -P M (t)D(t)). In this case the households reduces his consumption (balance sheet effect).

From the second equation of the system, it is seen that the incentive for tax avoidance increases when the return on each unit of hidden income increases relative to the risk of being caught (ê(t) is an increasing function of x/σ 2 ). Consumers are also prone to cheat more if their relative risk aversion 1 -µ decreases. Moreover the taxpayer is more likely to cheat when production over total wealth n y = n 1-β g n β k ( T W ) 1-β decreases. This happens, either: i) when private capital decreases (the households hold less assets in their portfolio); ii) when corruption by bureaucrats increases (in this case n g diminishes or, equivalently, n c increases); iii) when T /W decreases (as it measures the implicit corruption and fraud tax on the household's net wealth). Indeed, these illegal activities imply that the consumer must borrow from abroad (in the form of imports) and therefore bears the costs of the interest rate charged on external debt. This cost is internalized by the consumer and has an inhibiting effect on tax fraud (we see that ê is a decreasing function of the ratio T /W ).

7 See Appendices B and C.

From the third equation, we see that the capital share in total wealth depends upon the following factors:

• r k -(i * + π) is the expected net rate of return of capital (the capital owned as share of the domestic firm minus the capital borrowed from abroad). It is inversely proportional to the risks of both types of capital

(σ W k (t) 2 + σ W f (t) 2
) and to risk aversion.

• the ratio σ W f (t) 2 /(σ W k (t) 2 + σ W f (t) 2
) captures the household's dehedging behavior. When the risk of debt holding increases, the households want to own more domestic capital (which is possible by reducing tax evasion).

• We have an additional term ∆ 1/2 that depends on ê, W/T , nc .

Power-law as steady state distribution for the external debt ratio

From Equation ( 30), the ratio of nominal external debt over nominal fiscal revenues can be defined by

P M (t)D(t) T (t) = K(t) T (t) - W (t) T (t) . ( 38
)
For a given value of the capital/tax income, changes in the external debt/tax revenues ratio are negatively correlated to those in W (t)/T (t). Given the invariance properties of power-law distributions, for a given value of the ratio K(t)/T (t), the existence of a Pareto law for W (t)/T (t) implies that the external debt ratio will also follow a Pareto law. Thus, we define the macroeconomic equilibrium in terms of the ratio of net wealth-to-tax revenues.

A power-law probability density function is derived by proceeding in several steps. First, we specify the dynamics of the ratio W (t)/T (t) by using the Itô's lemma. We thus obtain a Itô stochastic differential equation formulation. Then, a theorem of the existence of a stochastic steady state is proved by using some properties of Markov chain diffusion processes. After that, we show that the stochastic steady state can be defined by a power-law with shape parameter depending upon our key variables of interest in the model related to tax evasion and corruption. Finally, we propose some simulations of the model.

Macroeconomic equilibrium as a diffusion process

Definition 4.1. Stochastic macroeconomic equilibrium A stochastic macroeconomic equilibrium is defined as a path of wealthto-fiscal revenues ratio {λ(t)}, where

λ(t) = W (t) T (t) ∞ t=0 , (39)
along which the agents' choice (bureaucrats and households) are optimal and the current account is balanced (imports are financed by foreign borrowing).

Theorem 4.1. The stochastic dynamics for the wealth-fiscal revenue ratio is a diffusion process defined by the stochastic differential equation ( SDE)

dλ(t) = Ω 1 (λ)dt + Ω 1 2 2 (λ)dB, λ(t) = W (t) T (t) ∈ 0, ∞ , (40)
where

Ω 1 (λ) = λ ñ2 g (λ) + ñ2 C (λ) -ng (λ) -nC (λ)+ 1 -τ + xτ ê(λ) ny (λ) -CP I Ĉ W + nk (λ)r k -(i * + π) nf (λ) , Ω 2 (λ) = λ 2 ñg (λ) 2 + ñC (λ) 2 + σ 2 τ 2 ê(λ) 2 ny (λ) 2 + nk (λ) 2 + nf (λ) 2 .
Proof. We first write the bureaucrats' and consumers' constraints (Equations ( 22) and( 34) which give us the separate dynamics of dT (t) and dW (t)):

dW(t) = W(t) 1-τ + xτ ê(λ) ny (λ) -CP I Ĉ W + nk (λ)r k -(i * + π) nf (λ) dt + W(t) στ ê(λ) ny (λ)dW W + nk (λ)du k -nf (λ)du f , dT (t) = T (t) ng (λ) + nC (λ) dt + T (t) ñg (λ)dW g (t) + ñC (λ)dW C G .
To obtain the dynamics of the ratio λ(t) = W (t)/T (t), we use the Itô's lemma.

Itô's lemma. Let X(t) in R 2 be a diffusion process and F(X) a C 2 map from R 2 to R, then

dF (X) = F x dX + 1 2 dX F xx dX, (41)
with F x and F xx representing, respectively, the matrix of partial derivatives of F and the Hessian matrix.

We define X = (T, W ) , dX = (dT, dW ) ,

F (X) = W T , F x = ∂F ∂T ∂F ∂W = -W T 2 1 T , F xx = ∂ 2 F ∂T 2 ∂ 2 F ∂T ∂W ∂ 2 F ∂W ∂T ∂ 2 F ∂W 2 = 2W T 3 -1 T 2 -1 T 2 0 .
From ( 41) we obtain

d( W T ) = -W T 2 dT + 1 T dW + W T 3 dT 2 - 1 T 2 dT dW. ( 42
)
We get the final form of Equation ( 40) by using the Levy characterization of diffusion processes and by considering the following properties of Wiener processes. Consider two Wiener processes w i and w j . We have:

(dw i ) 2 = dt, < dt.dw i >= 0 ∀i = j, dt 2 = 0.
For purpose of simplicity, we assume the following correlation structure of two Wiener processes: d < w i , w j >= 0, where < w i , w j > is the quadratic variation process for the components of the Wiener processes.

Existence of a steady state distribution

The concept of steady state for diffusion processes is defined in a stochastic sense. Instead of a point, λ(t) converges to a set of values in a basin of attraction. This means that, once λ has reached its long-term attractor λ * , all the variables in the model which depend upon λ will also reach their own basin of attraction. For Itô diffusion processes, the properties of the values in the basin of attraction of the variable of interest can be studied by constructing their distribution called the steady state distribution of the diffusion process. The issue here is therefore to study the convergence in distribution of the variable λ(t). To do this, we use the mathematical tools of the theory of Markov chains to prove the existence and derive stationary probability measures. The techniques are similar to those used in a few papers dealing with continuous time stochastic growth models (see Bourguignon (1974), [START_REF] Merton | An asymptotic theory of growth under uncertainty[END_REF], [START_REF] Chang | Asymptotic growth under uncertainty. Existence and uniqueness[END_REF], [START_REF] Jensen | Stochastic One-Sector and Two-Sector Growth Models in Continuous Time[END_REF]).

Asymptotic stochastic solutions of Itô diffusion processes

We first recall some mathematical properties of steady state distributions of Itô diffusion processes (see [START_REF] Feller | The parabolic differential equation and the associated semigroup of transformations[END_REF][START_REF] Feller | Diffusion processes in one dimension[END_REF], [START_REF] Ito | Diffusion processes and their sample paths[END_REF]).

Let us consider the following SDE:

dx = a(x)dt + b 1/2 (x)dz, dz ≈ N (0, dt), x ∈ 0, ∞ , (43)
with a(•) and b(•) being continuous and differentiable functions of x.

Consider X(t) the solution of the SDE and define the transition probability as P (x, t; x 0 , t 0 ) = P r X(t) ≤ x X(t 0 ) = x 0 .( 44)

The probability density π(x, t, x 0 ) satisfies the Kolmogorov-Fokker-Planck equation:

1 2 ∂ 2 ∂x 2 b(x)π(x, t, x 0 ) - ∂ ∂x a(x)π(x, t, x 0 ) = ∂π(x, t, x 0 ) ∂t . ( 45
)
The steady state density function, obtained by integrating ( 45), must satisfy

p(x) = c 1 m(x) + c 2 S(x), p(x) = lim t→∞ π(x, t, x 0 ), ( 46
)
where

M (x) ≡ x x 0 m(u)du, S(x) ≡ x x 0 s(u)du, (47)
with

m(x) ≡ exp 2J(x) b(x) , s(x) ≡ exp -2J(x) , J(x) ≡ x x 0 a(u) b(u) du. ( 48
)
c 1 and c 2 are constants of integration ensuring that p(x) is a true probability density. s(x), S(x) and m(x) are called, respectively, the scale density function, the scale function and the speed density function of the stochastic process X(t). (49) The existence of steady state distribution implies that the boundaries of the process are inaccessible. A corollary is that, if the boundaries are inaccessible, then π(x, t, x 0 ) converges towards a probability density function defined by ( 46) with c 2 = 0 (for rigorous proofs, see [START_REF] Feller | The parabolic differential equation and the associated semigroup of transformations[END_REF][START_REF] Feller | Diffusion processes in one dimension[END_REF], [START_REF] Ito | Diffusion processes and their sample paths[END_REF]).

Existence of steady state

A theorem of the existence of a steady state distribution for the wealthto-tax revenues ratio λ(t)

Now, we must prove that the boundaries 0 and +∞ are inaccessible for the wealth-to-tax revenues ratio λ(t). To do that, several preliminary remarks are in order.

First, λ(t) is the ratio of two variables W (t) and T (t). We assume that T ∈ (0, +∞) and W ∈ (0, +∞). By assumption 0 is thus an inaccessible boundary for W and T . This means that we do not consider the extreme situation in which corruption and tax evasion are so important that this yields a depletion of tax revenues (T → 0) and therefore entirely annihilates net wealth (W → 0, because of a high level of foreign debt).

Second, the fact that by assumption, T and W do not reach the zero boundary, does not mean that 0 and +∞ are inaccessible for λ. Indeed, for a finite T, lim W →+∞ λ = +∞, and for a finite W, lim

T →∞ λ = 0. ( 50
)
Therefore to prove that 0 and +∞ are inaccessible boundaries for λ, a sufficient condition consists in proving that +∞ is an inaccessible boundary for both W and T .

Theorem 4.2. Let us consider the SDE of T and W in a compact form using the Levy characterization

dT (t) = a 1 (T, λ * )dt + b 1 2 1 ((T, λ * ))dw T , where a 1 (T, λ * ) = T (t) n g (λ * ) + n C (λ * ) = T (t)a(λ * ), b 1 (T, λ * ) = T 2 (t) ñ2 g (λ * ) + ñ2 C (λ * ) = T 2 (t)b(λ * ), dW (t) = d 1 (W, λ * )dt + h 1 2 1 ((W, λ * ))dz W ,
where

d 1 (W, λ * ) = W (t) 1 -τ + xτ ê(λ * ) ny (λ * ) -CP I Ĉ W + nk (λ * )r k -(i * + π)n f (λ * ) = W (t)d(λ * ), h 1 (W, λ * ) = W 2 (t) σ 2 τ 2 ê(λ * ) 2 n2 y (λ * ) + n2 k (λ * ) + n2 f (λ * ) = W 2 (t)h(λ * ).
Sufficient conditions for the existence of a steady state distribution for the ratio of wealth-to-fiscal revenue are:

a) 2a(λ * ) -b(λ * ) < 0, and b) 2d(λ * ) -h(λ * ) < 0 (51) Proof. i)
We first prove that lim

T →+∞ S(T, λ * ) = +∞
Using the Levy representation of the SDE of T , as given in the theorem, we compute the scale density function of T (t) as

s(T, λ * ) = exp -2 T T 0 a 1 (u, λ * ) b 1 (u, λ * ) du , T 0 = T (0), (52)
or

s(T, λ * ) = exp -2 a(λ * ) b(λ * ) T T 0 1 u du = T T 0 -2 a(λ * ) b(λ * ) , (53)
Then, we calculate the scale function

S(T, λ * ) = T T 0 s(u)du = b(λ * )(T 0 ) 2a(λ * ) b(λ * ) -2a(λ * ) + b(λ * ) [T ] -2a(λ * )+b(λ * ) b(λ * ) -[T 0 ] -2a(λ * )+b(λ * ) b(λ * ) , (54)
We see that lim

T →+∞ S(T, λ * ) = +∞, if 2a(λ * ) -b(λ * ) < 0. ( 55
)
ii)

Using a similar approach by considering the SDE of W(t), we get lim 51) means that to avoid an infinite increase of wealth, the risk-adjusted return of net wealth must be capped, which implies that the risk-adjusted return of tax evasion and the marginal productivity of capital should not exceed a threshold value, and that the cost of borrowing abroad cannot be too low. Condition a) implies that there is a minimal risk associated to fiscal evasion and corruption. Indeed, would they be no risk for corrupted bureaucrats and frauding taxpayers, no tax revenues would be left to finance public spending. It would lead to the extreme case described in section 4.2.2.

W →+∞ S(W, λ * ) = +∞, if 2d(λ * ) -h(λ * ) < 0 (56) Remark 1. Condition b) in (

Invariant density function for the external debt-to-fiscal revenues ratio

Theorem 4.3. Let f (λ) be the invariant steady-state density function of λ(t).

Assume that T > 0, W > 0 and that the conditions a) and b) of Theorem 4.2 hold. Then, f (λ) for the SDE of the wealth-to-fiscal revenues ratio

dλ(t) = Ω 1 (λ)dt + Ω 1 2 2 (λ)dB(t), Ω 1 = K * 2 λ = λ ñ2 g (λ) + ñ2 C (λ) -ng (λ) -nC (λ) + 1 -τ + xτ ê(λ) ny (λ) - CP I Ĉ W + nk (λ)r k -(i * + π)n f (λ) , Ω 2 = K * 3 λ 2 = λ 2 ñ2 g (λ) + ñ2 C (λ) + σ 2 τ 2 ê2 (λ)n 2 y (λ) + n2 k (λ) + n2 f (λ) is f(λ) = C 0 m(λ) = 1-2 K * 2 K * 3 λ ( λ λ ) -2(1- K * 2 K * 3 ) = α-1 λ ( λ λ ) -α , α = 2(1 -K * 2 /K * 3
), λ is an arbitrary λ defined as the lower bound from which the power-law holds.

Proof.

We compute the speed density function as

m(λ) = exp(2J(λ)) Ω 2 (λ) , J(λ) = λ λ Ω 1 (u) Ω 2 (u) du, (57)
We therefore get

C 0 m(λ) = C 0 1 K * 3 λ 2 exp 2 λ λ K * 2 u K * 3 u 2 du = C 0 1 K * 3 λ 2 exp 2 λ λ K * 2 K * 3 u du = C 0 1 K * 3 λ 2 exp 2 K * 2 K * 3 ln( λ λ ) = C 0 1 K * 3 λ 2 λ λ 2 K * 2 K * 3 = C 0 λ-2 K * 3 ( λ λ ) -2(1- K * 2 K * 3 ) (58)
This corresponds to a power-law with α = 2(1 -K * 2 /K * 3 ).

We then calculate the normalizing constant C 0 , such that

+∞ λ f (u)du = 1. This yields C 0 = -λ(2K * 2 -K * 3 )
. By replacing in the expression above, we get f (λ).

The µ th -order moment is given by

+∞ λ u µ f (u)du = α -1 α -1 -µ λµ . (59)
All moments, except the mean, diverge when 2 < α < 3. This means that they do not converge as more observations are added to the sample and that the distribution has time-varying tails.

In the expression of α, K * 2 and K * 3 refer respectively to the mean and volatility of net wealth. The effect of a change in a parameter of the model on the shape parameter thus depends upon a trade-off between the mean level of wealth and its variability. One cannot exclude to have nonlinear effects depending upon the respective response of the mean and variance component of wealth to changes in the parameters. Rather than perform analytical exercises, we do some simulations of the model.

Some simulations of the model

We now simulate the model to examine the impact of tax evasion and corruption on the scale parameter α. Since the shape parameter is positive (by definition of a power-law function), the ratio (K * 2 /K * 3 ) varies in the interval (-∞, 1). When (K * 2 /K * 3 ) varies from -∞ to 1, α decreases, thereby implying that the pdf of λ will tend to Pareto laws as the ratio diminishes. This ratio can be interpreted as the combined net returns of illegal activities (fraud and corruption) relative to their risk. Since, we have seen that net wealth and debt are negatively correlated, one can understand that, an increase in the returns to illegal activity relative to their risk implies heavier tails in the pdf of the external debt ratio and therefore a higher likelihood of the occurrence of extreme debt events.

We first calibrate the model to have a benchmark situation (baseline scenario). The macroeconomic variables are chosen in such a way to simulate an economy with some plausible macroeconomic characteristics of emerging economies. The aggregate income is assumed to be 1000 billion dollars with aggregate tax revenues of 400 billions. The share of tax revenues over total wealth is 50%. The marginal productivity of capital is 5% and the interest rate on external debt is 7%. Corrupt bureaucrats steal 30% of tax revenues, tax evaders hide 10% of their income. The probability to be caught as a tax fraudulent is one-half, the probability of being a corrupt bureaucrat is 0.5. If they are caught, the consumer/tax evader must pay a penalty of 20% of the hidden income. The values of the other parameters (variances, tax rate, and the parameters computed from the calibrated variables) are given below (see Table 3).

Impact of fighting fiscal evasion on the shape parameter Fighting against fiscal evasion reduces the frequency of extreme events of external debt (see Figure 4). This happens for a higher probability p of catching those consumers convinced of fraud, or a higher penalty rate b. A corollary result is that tax avoidance have more negative repercussions on foreign debt (with more heavy tails indicating more frequent extreme events), if the evaders take advantage of favorable conditions: deficiencies of administrative surveillance, opaque fiscal environment.

Meanwhile, as the figures suggest, a highly punitive system can create a harmful situation in terms of debt. Indeed, the inverted bell curves (especially for b) suggest that, above a given threshold, there may be a fiscal cost for the government: the higher the penalty rate, the more consumers will try to figure out ways to stop paying their income tax. Such a nonlinear effect is in line with the literature suggesting that there is no compelling evidence of how tax compliance is affected by punishment through more efficient tax audits, or tax penalties (see Alm (2012), [START_REF] Murphy | Enforcing tax compliance: to punish or persuade? Economic Analysis and[END_REF], [START_REF] Slemrod | Cheating ourselves: the economics of tax evasion[END_REF]).

Shape parameter α = 2(1 -K 2 /K 3 ) 2,80 K 2 -9,
The fact that a severe punishment may increase the way people behave in not fulfilling their tax obligations (thereby implying here a higher foreign debt level) has been motivated by different arguments. First, people do not necessarily report their tax liabilities on the basis of legal obligations, but in good faith according to what they believe to be correct. High penalty rate can be though of as reflecting abusive transactions between the tax administration and tax payers, especially if government officials are known to be corrupted (they respond to tax fraud by ratcheting up penalties). A second explanation is more in line with our model. The penalty regime does affect the taxpayer's behavior. Here the tax evaders pays twice: a fixed penalty of the evaded tax (provided they are caught), plus an interest rate on debt (they internalize the effect of fraud on debt, because this reduces their net wealth). The fact that they bear this additional interest rate cost should persuade the consumers to limit their tax noncompliance below a certain level (if the amount of debt service paid from foreign borrowing increases, their compliance rate is likely to rise). In this context, attempts by the government to coerce and threaten taxpayers into compliance through high penalty rates could undermine the legitimacy of the tax administration's authority. Impact of corruption on the shape parameter Figure 5 suggests that more corrupt bureaucrats (increase in p 1 ) and a higher tax diversion (increase in θ) imply a thiner tail on the pdf of λ (α increases) and therefore more frequent extreme events for the external debt ratio.

The negative harmful effects are, however, reversed when the share of resources stolen becomes higher above a certain threshold. The humped curve representing a nonlinear relationship between α and θ suggests that tax diversion of tax resources can have de-stabilizing or stabilizing effects on foreign indebtedness (stabilizing meaning a lower likelihood of adverse high debt scenarios in the right-hand side of the pdf of the external debt ratio, that is a decrease in α). An increase in θ biases the composition of spending by raising the share of unproductive spending relative to productive spending. The country needs to borrow more from abroad and the mean net wealth (expressed as share of total taxes) diminishes (portfolio-adjustment effect). But, as θ takes higher values, the expected lifetime path of returns from corruption becomes more decreasing (the bureaucrats exhaust rapidly their corruption opportunity). This contributes to stabilize foreign borrowing.

Conclusion

Our intention in this paper has been to suggest that the emerging countries' external debt ratio can well be described by power-law distributions, and to close the gap between the empirical evidence and the theoretical framework within which such power-law functions can be obtained analytically. We consider the role of corruption and tax evasion about which there is still much to learn concerning their implications on the emerging countries' debt crises or debt stress episodes.

Our claim that foreign debt ratio can be the outcome of Pareto laws is obtained rigorously, by using econometric tests rather than simply plotting log-log graphs of the presumed distributions of the data. This is important, since a non-negligible amount of empirical papers estimates shape parameters by assuming that the distribution is a Pareto law, while this assumption needs to be tested formally.

Our purpose was then to provide a simple illustration of how a corrupt economy with tax evasion might explain this empirical observation. Our model incorporates the essential features of emerging economies: foreign aid and borrowing is often a substitute to domestic taxation, bureaucrats officially appointed to make tax audits are sometimes inclined to seek private ways of using the collected tax rather than leaving them available for productive investment, the detection of illegal taxation activities by the governments is not necessarily efficient. Though they are present in the model, for sake of completeness, our analysis is not meant to consider the role of macroeconomic shocks such as terms of trade, supply shocks, interest rates shock on foreign debt. We prefer to focus on the role of endogenous uncertainty, meaning that the tails of the Pareto law do not only depend upon the stochastic components of the diffusion processes, but also on the probability of detecting fraudulent people and their risk-taking behavior, on the probability of facing a corrupt bureaucrat and on the risk-taking behavior of bureaucrats.

This paper could be extended by investigating other distributions that are likely to characterize foreign debt, notably the family of upper incomplete Gamma distributions. Enriching the class of distributions for external debt would help in the detection of early warning stress debt episodes in the emerging countries.

Appendix A. Government's optimal choice

This appendix and the next ones present how the government's and household's optimal choices are calculated. The methodology is based on techniques used in continuous time optimization models (see for instance [START_REF] Chang | Stochastic optimization in continuous time[END_REF], [START_REF] Turnovsky | On the Role of Government in a Stochastically Growing Open Economy[END_REF], [START_REF] Turnovsky | Methods of macroeconomic dynamics[END_REF]).

The government's choice is as follows:

max

CG (t) = E +∞ 0 1 γ CG (t) γ e -ρt dt (A.1) subject to dT (t) = ḡ(t) T (t) + CG (t) T (t) T (t)dt + gdW g (t) T (t) + CG dW C G (t) T (t) T (t). (A.2)
This constraint can be expressed as:

dT (t) T (t) = ψ T dt + σ w T dw T , (A.3) with ψ T = ḡ(t) T (t) + CG (t) T (t) , σ w T dw T = gdW g (t) T (t) + CG dW C G (t) T (t)
.

(A.4) and To solve it, we substitute the optimized value of n C and n g :

T γ n γ-1 C = T 2 X T T (σ 2 W C G + σ 2 W g )n C -σ 2 W g . (
1 γ T γ nC γ -ρX(T ) + T X T + 1 2 σ 2 w T T 2 X T T = 0. (A.17)
To solve the resulting equation in X(T), we postulate X(T) of the form: (A.18) with δ to be determined. This yields to T X T = γX(T ), T 2 X T T = γ(γ -1)X(T ).

X(T ) = δT γ ,
( A.19) Using this, the Bellman equation becomes: 1 γ T γ nC γ -ρX(T ) + γX(T ) + 1 2 σ 2 w T γ(γ -1)X(T ) = 0. (A.20)

According to (A.15), (T nC ) γ-1 is given by:

T γ n γ-1 C = T 2 X T T (σ 2 W C G + σ 2 W g )n C -σ 2 W g = γ(γ -1) (σ 2 W C G + σ 2 W g )n C -σ 2 W g X(T ).
(A.21)

We can substitute it in (A.20) and divide by X(T ): (A.22) which leads to the second-order differential equation with:

(σ 2 W C G + σ 2 W g ) nC 2 -σ 2 W g nC + ρ -γ 1 -γ + 1 2 σ 2 w T γ = 0,
= σ 4 W g -4(σ 2 W C G + σ 2 W g )( ρ -γ 1 -γ + 1 2 σ 2 w T γ). (A.23) Solutions (if is positive) are of the form nC = σ 2 W g ± √ 2(σ 2 W C G + σ 2 W g ) . (A.24)
With nC positive we have nC =

σ 2 W g + σ 4 W g -4(σ 2 W C G + σ 2 W g )( ρ-γ 1-γ + 1 2 σ 2 w T γ) 2(σ 2 W C G + σ 2 W g ) . (A.25) AR(W ) = (1 -µ) constant.
The third equation leads to

β 1 -τ + xτ ê ( T W ) 1-β W V (W )n 1-β g n β k +βσ 2 τ 2 ê2 ( T W ) 2(1-β) σ 2 W W W 2 V (W )n 2(1-β) g n 2β k + n 2 k (σ 2 W k + σ 2 W f ) -n k σ 2 W f W 2 V (W ) + r k -(i * + π) W V (W )n k = 0.
(B.17) Define n y as the share of production over total wealth, that is n y = Y W . We then have n 1-β g n β k ( T W ) 1-β = n y , which leads to rewrite the condition as follows

β 1 -τ + xτ ê W V (W )n y +βσ 2 τ 2 ê2 σ 2 W W W 2 V (W )n 2 y +n 2 k (σ 2 W k + σ 2 W f )W 2 V (W ) +n k r k -(i * + π) W V (W ) -σ 2 W f W 2 V (W ) = 0, (B.18) or (1 -µ)(σ 2 W k + σ 2 W f )n 2 k + -r k -(i * + π) -(1 -µ)σ 2 W f n k -β 1 -τ + xτ ê n y + (1 -µ)βσ 2 τ 2 ê2 σ 2 W W n 2 y = 0. (B.19)
This is a second-order differential equation in n k . The discriminant is

= r k -i * -π + (1 -µ)σ 2 W f 2 -4(1 -µ)(σ 2 W k + σ 2 W f ) (1 -µ)βσ 2 τ 2 ê2 σ 2 W W n 2 y -β 1 -τ + xτ ê n y .
(B.20) Considering the solutions for > 0, we obtain:

n 1,2 k = r k -(i * + π) + (1 -µ)σ 2 W f ± √ 2(1 -µ)(σ 2 W k + σ 2 W f ) . (B.21)
Among both solutions, the following one satisfies the condition n k > 0:

n k = r k -(i * + π) + (1 -µ)σ 2 W f + √ 2(1 -µ)(σ 2 W k + σ 2 W f ) . (B.22)
The household's optimal choice is therefore described by the following system :

             Ĉ = (δµ) 1 µ-1 P 1-η µ-1 W ê = xτ nk β n 1-β g ( T W ) 1-β (1-µ)(στ nk β n 1-β g ( T W ) 1-β σ W W ) 2 n k = r k -(i * +π)+(1-µ)σ 2 W f + √ 2(1-µ)(σ 2 W k +σ 2 W f ) n f = n k -1 . (B.23)

Appendix C. Transversality conditions

The agents' choices must also satisfy the transversality condition. We consider this condition for the consumer (the proof is similar for the government).

For the constant elasticity utility function, the transversality condition is given by: lim ψ W (t) and σ z W (t) dz W (t) (defined by Equations ( 35) and ( 36)) converge to constant terms when when t → ∞, so we omit t. We first compute the solution of C.2 for W (0) = w 0 (initial condition of wealth), given.

We This is a geometric brownian motion. Assuming z W (t) is independent of w 0 , one of the properties of such a motion is that, E W (t) µ exp(-ρt) = E w µ 0 exp (µψ W -ρ)t .

(C.13)

In the end, the transversality condition can be rewritten as This puts a lower bound on the rate of impatience ρ, since the above condition is satisfied for ρ > µψ W .

For the government, the transversality condition is obtained in a similar way and implies that ρ > γψ T with ψ T is defined by Equation( 24).

Figure 3 :

 3 Figure 3: Distribution of External debt as share of public revenues (%), Europe, Russia and South Africa

  distribution. A time-invariant distribution function P (x) exists if and only if lim x → 0 x → ∞ S(x) = ∓∞ and M(x) is finite at the boundaries 0 and +∞, i.e. lim x → 0 x → ∞ S(x) = ∓∞ and |M (b)| = b x 0 m(u)du < ∞ for b = 0 and +∞.

Figure 4 :

 4 Figure 4: Simulations: impact of changes in p, b on α

Figure 5 :

 5 Figure 5: Simulations: impact of changes in θ, p 1 on α

  (T n C ) γ + L e -ρt X(T ) = 0. (A.16)

t→∞E

  W (t) µ e -ρt = 0. (C.1)The stochastic differential equation in W isdW (t) = ψ W W (t)dt + σ z W W (t)dz W (t).(C.2)

Table 1 :

 1 min s.e. (bootstrap) Estimated lower bound x min and associated α (bootstrap)

	α	s.e. (bootstrap)

Table 2 :

 2 Estimated α using rank-1/2 regression sociated with other macroeconomic and financial imbalances, among which hyper-inflation, banking crises, asset price bubbles, credit booms, capital flows, the cyclical nature of fiscal and monetary policies. SeeKaminsky et al.

Table 3 :

 3 Calibration of the parameters for the computation of the shape parameter α

	Note: in blue: coefficients or parameters fixed for the calibration
	in orange: coefficients or parameters computed

  rewrite C.2 as followsdW (t) W (t) = ψ W dt + σ z W dz W (t). W dz W (u) = ψ W t + σ z W z W (t).Since dz W (t) 2 = dt, dt 2 = 0 and dz W (t).dt = 0, we havedW (t) W (t) 2 = ψ W dt + σ z W dz W (t)

	Taking f (t, W (t)) = f (W ) = ln(W ), we obtain
									d ln(W (t)) =	dW (t) W (t)	-	1 2	dW (t) W (t)	2 .	(C.6)
											2 = σ 2 z W dt.	(C.7)
	Thus,
										d ln(W (t)) =	dW (t) W (t)	-	σ 2 z W 2	dt.	(C.8)
	And integrating between 0 and t, we have
								ln(W (t)) -ln(w 0 ) =	0	t	dW (u) W (u)	-	σ 2 z W 2	t.	(C.9)
	Replacing by the expression in C.4, we get
			ln(	W (t) w 0	) +	σ 2 z W 2	t =	0	t	dW (u) W (u)	= ψ W t + σ z W z W (t),	(C.10)
	and thus			W (t) = w 0 exp (ψ W -	σ 2 z W 2	)t + σ z W z W (t) ,	(C.11)
	which yields to
	W (t) µ exp(-ρt) = w µ 0 exp (µψ W -	µσ 2 z	(C.3)
	Integrating this equation between 0 and t gives
	0	t	dW (u) W (u)	=	0	t	ψ W du +	0	t	σ z (C.4)
	We use Itô's formula:
	df (t, W (t)) =	∂f (t, W (t)) ∂t	dt +	∂f (t, W (t)) ∂W	dW (t) +	1 2	∂ 2 f (t, W (t)) ∂W 2	(dW (t)) 2 . (C.5)

W 2 -ρ)t + µσ z W z W (t) .

(C.12)

The assumption that the number of bureaucrats equals those of consumers is made for technical simplicity and does not change our arguments.

See Appendices A and C.

Tax evasion constrains funding of total public spending and corruption erodes its productive share, as corrupted public consumption is unproductive.

Assuming the same rate of time preference for the government and the household is made for simplicity and does not have any consequence on our results.

Let us define n g = ḡ(t) T (t) , n C = CG (t) T (t) , ñg = g T (t) and ñC = CG T (t) . We then have:

, σ w T dw T = ñg dW g (t) + ñC dW C G (t), (A.5) such that

The program becomes

The differential generator of the value function V(T,t) is defined by:

We assume V to be of the following time separable form:

V (T, t) = e -ρt X(T ). (A.10)

And government choses n C and n g maximising the following Lagrangian:

The partial derivative with respect to n C is:

12)

The partial derivative with respect to n g is:

Putting these equations together with 1 = n g + n C leads to:

Appendix B. The consumer's optimal choice

Government and households are assumed to have the same time preference coefficient ρ). The objective function is max

The household maximizes the intertemporal utility function subject to the constraints given by Equation ( 34), and with W (0) = w 0 .

We define the aggregate consumption C = C P (t) η C M (t) 1-η . The consumer price index can be defined as CP I(t) = P P (t) η P (t) 1-η which yields to CP I(t) = P (t) 1-η , as the domestic good is the numeraire We thus have:

3) where n k = k W and n f = P D W (0 < n k < 1 and n f > 0). And we get

The households' decision is as follows:

Define V as the value function

Then, the optimal program satisfies the Hamilton Jacobi Bellman equation

where

Using (B.7), which implies n f = n K -1, we get the following necessary conditions:

(B.14) F has an extremum ( Ĉ, ê, nk ) defined such as to verify the last three equations. From the first two variables we obtain: Ĉ = (CP IV (W )) V (W ) . Assuming V has the form V (W ) = δW µ , where δ is a constant, we get


Jing Yu

Hiba Saada

Neso Sojic
email: neso.sojic@enscbp.fr

Gabriel Loget
email: gabriel.loget@univ-rennes1.fr

Photoinduced electrochemiluminescence at nanostructured hematite electrodes

Keywords: Photoelectrochemistry, electrochemiluminescence, hematite, semiconductors, luminol

In this article, we study the photoinduced electrochemiluminescence (PECL) of a luminol analog (L-012: 8-amino-5-chloro-2,3-dihydro-7-phenyl-pyrido[3,4-d]pyridazine-1,4-dione) on n-type hematite (-Fe 2 O 3 ) nanorod arrays deposited by hydrothermal synthesis on a transparent conducting oxide. We report that this material, when used as a photoanode, triggers PECL at a potential as low as -0.2 V vs Ag/AgCl and allows the amplification of the luminescence signal. The PECL concept should open up new perspectives based on lightaddressable strategies in bioanalysis and imaging.

Introduction

Electrochemiluminescence (ECL) is a light-emitting phenomenon triggered by electrochemical means [START_REF] Liu | Recent advances in electrochemiluminescence[END_REF][START_REF] Sojic | Analytical electrogenerated chemiluminescence: from fundamentals to bioassays[END_REF][START_REF] Hesari | Review-electrogenerated chemiluminescence: light years ahead[END_REF]. In ECL, light generation is induced by the excited state of a luminophore, which is electrochemically-generated by a cascade of redox reactions in the immediate vicinity of the electrode surface. ECL is a well-established analytical method, especially for immunoassays, and is emerging as a powerful imaging technique [START_REF] Zhang | Electrochemiluminescence imaging for bioanalysis[END_REF][START_REF] Ma | Recent progress in electrochemiluminescence sensing and imaging[END_REF][START_REF] Qi | Electrogenerated chemiluminescence biosensing[END_REF][START_REF] Wang | Electrochemiluminescence self-interference spectroscopy with vertical nanoscale resolution[END_REF][START_REF] Ding | Imaging cell-matrix adhesions and collective migration of living cells by electrochemiluminescence microscopy[END_REF][START_REF] Moretto | Epifluorescence imaging of electrochemically switchable langmuir-blodgett films of nafion[END_REF][START_REF] Sentic | Microscopic imaging and tuning of electrogenerated chemiluminescence with boron-doped diamond nanoelectrode arrays[END_REF]. The majority of these applications relies on the use of model ECL luminophores such as the tris(bipyridine)ruthenium(II) ([Ru(bpy) 3 ] 2+ ) or luminol (3-aminophthalhydrazide) and their analogs [START_REF] Dick | Simultaneous detection of single attoliter droplet collisions by electrochemical and electrogenerated chemiluminescent responses[END_REF][START_REF] Glasscott | Visualizing phase boundaries with electrogenerated chemiluminescence[END_REF][START_REF] Swanick | Enhanced electrochemiluminescence from a stoichiometric ruthenium(II)-iridium(III) complex soft salt[END_REF][START_REF] Zanut | Insights into the mechanism of coreactant electrochemiluminescence facilitating enhanced bioanalytical performance[END_REF][START_REF] Zanut | Dye-doped silica nanoparticles for enhanced ecl-based immunoassay analytical performance[END_REF][START_REF] Voci | Surface-confined electrochemiluminescence microscopy of cell membranes[END_REF][START_REF] Shu | Potential-resolved multicolor electrochemiluminescence of n-(4-aminobutyl)-n-ethylisoluminol/tetra(4carboxyphenyl)porphyrin/TiO 2 nanoluminophores[END_REF][START_REF] Liu | Firefly-mimicking intensive and long-lasting chemiluminescence hydrogels[END_REF][START_REF] Cui | Electrogenerated chemiluminescence in submicrometer wells for very high-density biosensing[END_REF][START_REF] Chen | In situ imaging facetinduced spatial heterogeneity of electrocatalytic reaction activity at the subparticle level via electrochemiluminescence microscopy[END_REF][START_REF] De Poulpiquet | Dual enzymatic detection by bulk electrogenerated chemiluminescence[END_REF][START_REF] Wang | Intracellular wireless analysis of single cells by bipolar electrochemiluminescence confined in a nanopipette[END_REF]. For example, L-012 (8-amino-5-chloro-2,3-dihydro-7-phenyl-pyrido [3,4d]pyridazine-1,4-dione), a luminol-based analog, is widely used in vitro and in vivo to measure the activity of oxidase enzymes, to detect reactive oxygen species (ROS), and to measure different analytes at the cell level [START_REF] Marquette | Applications of the luminol chemiluminescent reaction in analytical chemistry[END_REF][START_REF] Kielland | In vivo imaging of reactive oxygen and nitrogen species in inflammation using the luminescent probe L-012[END_REF][START_REF] Zielonka | On the use of L-012, a luminol-based chemiluminescent probe, for detecting superoxide and identifying inhibitors of NADPH oxidase: a reevaluation[END_REF][START_REF] Cui | Attomole antigen detection using self-electrochemiluminous graphene oxide-capped Au@L012 nanocomposite[END_REF][START_REF] Zhang | Electrochemiluminescence-based capacitance microscopy for label-free imaging of antigens on the cellular plasma membrane[END_REF][START_REF] Zhou | Electrochemiluminescence imaging for parallel single-cell analysis of active membrane cholesterol[END_REF]. Compared to luminol, it generates ECL with a higher efficiency [START_REF] Marquette | Applications of the luminol chemiluminescent reaction in analytical chemistry[END_REF][START_REF] Zhang | Electrochemiluminescence-based capacitance microscopy for label-free imaging of antigens on the cellular plasma membrane[END_REF]. These model ECL reactions requires the application of anodic potential, typically > 1.1 V vs Ag/AgCl for the tris(bipyridine)ruthenium(II) ([Ru(bpy) 3 ] 2+ )/tri-n-propylamine system and > 0.3 V for luminol analogs. Depending on the electrode material (e.g., ITO), higher anodic potentials such as +1 V may be required for L-012 to obtain brighter emission [START_REF] Zhang | Electrochemiluminescence-based capacitance microscopy for label-free imaging of antigens on the cellular plasma membrane[END_REF][START_REF] Zhou | Electrochemiluminescence imaging for parallel single-cell analysis of active membrane cholesterol[END_REF]. On the other hand, in semiconductors photoelectrochemistry, a depleted semiconductor absorbs photons of higher energy than its bandgap (E g ) for triggering electrochemical reactions at a solid/liquid interface. In this approach, the high oxidation potential of photogenerated minority carriers allows to considerably decrease the onset potential of the electrochemical reaction [START_REF] Gerischer | Solar photoelectrolysis with semiconductor electrodes[END_REF][START_REF] Tan | Principles and applications of semiconductor photoelectrochemistry[END_REF].

The combination of ECL and semiconductors photoelectrochemistry is referred to as photoinduced-electrochemiluminescence (PECL) [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF]. This concept enables the conversion of an input light signal (photons absorbed by the semiconductor,  exc ) to an output light signal (photons emitted by the ECL process,  em ). Besides, PECL is also promising as it enables the local emission of ECL triggered by a focalized spot of light and can therefore be used for performing multiple assays on a single surface [START_REF] Vogel | Spatiotemporal control of electrochemiluminescence guided by a visible light stimulus[END_REF][START_REF] Terrero Rodríguez | Light-addressable electrochemical sensing with electrodeposited n-silicon/gold nanoparticle schottky junctions[END_REF]. PECL has been initially demonstrated using annihilation ECL in organic solvents by Bard et al. [START_REF] Laser | Semiconductor electrodes. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF][START_REF] Luttmer | Electrogenerated chemiluminescence. 34. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF] and, more recently, in an aqueous electrolyte with the coreactant ECL system [Ru(bpy) 3 ] 2+ /TPrA [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF] and with luminol analogs [START_REF] Vogel | Spatiotemporal control of electrochemiluminescence guided by a visible light stimulus[END_REF][START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF] by our group and others. In most of these works, Si or III-V-based semiconductor surfaces were employed as photoelectrodes [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF][START_REF] Vogel | Spatiotemporal control of electrochemiluminescence guided by a visible light stimulus[END_REF][START_REF] Laser | Semiconductor electrodes. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF][START_REF] Luttmer | Electrogenerated chemiluminescence. 34. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF]. These materials hold some advantages such as the fact that they are commercially available and they have high carrier mobilities. In addition, their electronic structure can be tuned in a very precise way. However, they also exhibit drawbacks such as their price or their instability caused by photocorrosion [START_REF] Loget | Water oxidation with inhomogeneous metal-silicon interfaces[END_REF]. The latter aspect is a particular issue for their long-term use in aqueous electrolytes [START_REF] Bae | Strategies for stable water splitting via protected photoelectrodes[END_REF] and implies the use of protection layers [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF]. Conversely, oxide-based semiconductors usually involve lower processing costs and exhibit better resistance to photocorrosion, which make them particularly interesting for PECL. To the best of our knowledge, so far, only one report on PECL focused on an oxide semiconductor (i.e., BiVO 4 ) [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF]. Among n-type semiconductors that are promising for photoelectrochemical applications, hematite (-Fe 2 O 3 ) attracts attention. It shows a peculiar [START_REF] Formal | Rate Law Analysis of Water Oxidation on a Hematite Surface[END_REF], yet controversial [START_REF] Peter | Interpretation of photocurrent transients at semiconductor electrodes: Effects of band-edge unpinning[END_REF] photoelectrochemical behavior and has been employed in particular in the fields of water splitting [START_REF] Peter | Kinetics of light-driven oxygen evolution at α-Fe 2 O 3 electrodes[END_REF][START_REF] Sivula | Solar water splitting: progress using hematite (α-Fe 2 O 3 ) photoelectrodes[END_REF][START_REF] Tamirat | Using hematite for photoelectrochemical water splitting: a review of current progress and challenges[END_REF][START_REF] Sharma | Key strategies to advance the photoelectrochemical water splitting performance of α-Fe 2 O 3 photoanode[END_REF] and photoelectrochemical sensing [START_REF] Saada | Photoelectrochemical sensing of hydrogen peroxide on hematite[END_REF][START_REF] Li | Gold nanoparticles decorated hematite photoelectrode for sensitive and selective photoelectrochemical aptasensing of lysozyme[END_REF] due to its high theoretical photoconversion efficiency, its low cost, its chemical stability, and its good biocompatibility.

Interestingly, -Fe 2 O 3 also exhibits much lower carrier mobility and higher recombination rates than Si or III-V-based semiconductors. If this is generally considered as a major issue in photoconversion application, it has been recently shown that it can be used as a strong advantage for improving resolution in light-addressable electrochemistry with -Fe 2 O 3 systems [START_REF] Seo | Robust and high spatial resolution light addressable electrochemistry using hematite (α-Fe 2 O 3 ) photoanodes[END_REF], which makes this material particularly relevant for localized PECL.

In this article, we study the PECL of L-012, a luminol analog on nanostructured -Fe 2 O 3 photoanodes. We show that this material allows for luminescence amplification with good temporal stability, which should open up important perspectives for PECL.

Materials and methods

Materials and reagents

Fluorine-doped SnO 2 (FTO)-covered glass plates, urea (≥99%), NaOH (>99.8%), and iron chloride hexahydrate (FeCl 3 .6 H 2 O, ≥98%) were bought from Sigma-Aldrich. Teflon (PTFE) lined hydrothermal synthesis autoclave reactor (25 ml). Acetone (technical grade, VWR) and ethanol (96%, Carlo Erba) were used as received. The resistivity of ultrapure water was 18.2 MΩ cm (Purelab Classic UV, Veolia). 8-amino-5-chloro-2,3-dihydro-7-phenyl-pyrido [3,4d]pyridazine-1,4-dione (L-012) was purchased from FUJIFILM Wako Chemicals Europe GmbH.

Synthesis and characterization of the photoelectrodes

The FTO slides (1×2 cm²) were cleaned with a detergent (110551, Ecolab), rinsed copiously with deionized water, and sonicated for 10 min successively in acetone, ethanol, and ultrapure water. The backside of the FTO substrate (the glass side) was covered with an adhesive tape and vertically placed in the reactor, which was filled with 20 mL of an aqueous solution containing 0.1 M FeCl 3 .6 H 2 O and 0.15 M urea. The reactor was sealed and heated in an oven at 100 °C for 16 h. After the reaction, the films were sonicated in water for 20 min and were annealed in air successively at 500 °C for 3 h and 750 °C for 10 min in a tubular oven. X-ray diffraction (XRD) analysis was performed on a Rigaku Smartlab X-Ray diffractometer (5-axis goniometer, working conditions: 40 kV, 50 mA (2.0 kW), Cu cathode). Scanning electron microscopy (SEM) was performed using a JSM 7100F (JEOL). The colorized SEM pictures were prepared using the Inkscape freeware.

Photoelectrochemical measurements

Cyclic voltammetry (CV) and chronoamperometry (CA) measurements were performed with a μ-Autolab Type III (Metrohm) connected with a 3-electrodes cell. The FTO/-Fe 2 O 3 substrate was the working electrode, an Ag/AgCl (3 M NaCl) electrode was used as the reference electrode and a carbon rod was the counter electrode. The scan rate of the CVs was 20 mV s -1 . The FTO/-Fe 2 O 3 surface was sealed with an O-ring in a home-made open-top cell (the photoelectrochemically active surface area was ~0.3 cm²). A copper tape connected to the FTO was used as an electrical contact. The electrolyte (5 mL), composed of 10 mM L-012 dissolved in 0.1 M NaOH was freshly prepared and disposed in the cell and the other two electrodes were placed on the sides in order to prevent shadowing of the front surface. A 375 nm light-emitting diode (LED, 733-4745, Radiospares) coupled with a low bandpass filter was placed below the photoanode to illuminate the backside of FTO/-Fe 2 O 3 (through the FTO). The LED power was controlled by a power generator. ECL spectra were recorded using a spectrograph Spectra pro 2300i (Princeton Instrument) connected with a collecting optical fiber under different potentials with an exposure time of 1 s. The optical fiber was placed above the electrolyte/air interface and at a distance of about 4 cm from the electrode to fiber. The PECL and fluorescence (FL) spectra were recorded with a constant LED current of 150 mA (corresponding to a power of 15 mW). The FL spectra were acquired when LED was on but without applying a potential to the electrode. In Figures 4 and6, the spectrum obtained in the dark under open-circuit conditions was used as a background and subtracted from the spectra. The PECL intensity (Figure 5) was obtained by subtracting the FL signal (recorded at the open circuit with the LED ON) from the PECL+FL signal (recorded upon the application of a potential). The stability test was done at E = 0 V under intermittent light emission with an exposure time of 10 s. To reduce the noise level and allows for better visualization of the PECL data, the spectra shown in Figure 5 were smoothed by adjacent averaging.

Results and discussion

Preparation of the hematite photoanodes

FTO slides were modified by a previously-reported hydrothermal process in a solution containing FeCl 3 and urea heated at 100 °C for 16 h (see the experimental section for more details) [START_REF] Wang | Direct electron transfer from photosystem II to hematite in a hybrid photoelectrochemical cell[END_REF]. After the modification, the substrates were annealed in the air to crystallize the synthesized layer. Then, the surfaces were analyzed by energy-dispersive X-ray spectroscopy (EDS) and XRD. The EDS spectrum, presented in Figure 1a, shows that the surface comprises O, Fe, Sn, and Si, as expected for a glass-coated FTO slide covered by -Fe 2 O 3 layer [START_REF] Dias | Extremely stable bare hematite photoanode for solar water splitting[END_REF]. In the recorded XRD pattern, shown in the bottom panel of Figure 1b, all peaks of the FTO substrate can be observed (green lines in the top panel) and all other peaks can be attributed to the deposited -Fe 2 O 3 layer (red lines in the top panel), revealing the rhombohedral crystalline nature of our -Fe 2 O 3 coating. The morphological analysis was performed by SEM. Cross-section (Figure 1c) and top-view (Figure 1d) observations revealed that the -Fe 2 O 3 (red color) covered homogeneously the FTO surface. The coating was composed of 550 nm-long nanorods aligned perpendicular to the FTO surface, in good agreement with previous reports [START_REF] Mulmudi | Controlled growth of hematite (α-Fe 2 O 3 ) nanorod array on fluorine doped tin oxide: Synthesis and photoelectrochemical properties[END_REF]. Such a nanorod array morphology is particularly interesting to be used in a photoelectrode design because it is expected to provide a beneficial short pathway (smaller than the rod diameter) for the diffusion of the photogenerated minority carriers, (minimization of bulk recombination rates before reaching the solid/liquid interface) coupled with a transversal path for majority carrier transport to the FTO collector [START_REF] Lewerenz | Photoelectrochemical water splitting[END_REF]. The anodic response of the FTO/-Fe 2 O 3 photoanode was studied in the PECL electrolyte containing L-012 dissolved in an alkaline aqueous solution at a pH that has been shown compatible with -Fe 2 O 3 photoanodes [START_REF] Tamirat | Using hematite for photoelectrochemical water splitting: a review of current progress and challenges[END_REF]. L-012 was chosen as an ECL luminophore for several reasons: i) it allows for ECL generation at a relatively low applied potential at conventional conductive electrodes (e.g., > 0.3 V vs Ag/AgCl at glassy carbon electrode surface), ii) it produces brighter ECL emission than luminol, and, iii) it has been recently demonstrated efficient for triggering PECL at another oxide photoanode (BiVO 4 ) [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF]. Here, we studied "fluorescence-type" PECL, which means that  exc <  em [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF], which differs from "up-conversion-type" PECL, where  exc >  em [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF][START_REF] Vogel | Spatiotemporal control of electrochemiluminescence guided by a visible light stimulus[END_REF][START_REF] Laser | Semiconductor electrodes. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF][START_REF] Luttmer | Electrogenerated chemiluminescence. 34. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF]. In other words, both fluorescence and PECL are photogenerated in the reported configuration. We selected a 10 mM concentration of L-012 in this work because it corresponds to a good compromise between the PECL signal and the fluorescence one. The photoelectrochemical cell was similar to the one reported previously [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF], allowing for a backside illumination (through the FTO) of the photoanode with a 375 nm LED (i.e.,  exc ). Due to the pseudo-transparency of the FTO/-Fe 2 O 3 substrate, as illustrated in the spectra of Figure 2, this configuration enables the excitation of the photoactive -Fe 2 O 3 layer (bandgap = 2 eV, determined by photocurrent spectroscopy) as well as the fluorescence (FL) of L-012 (which absorbs below 465 nm) [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF][START_REF] Zhang | Chemiluminescence chitosan hydrogels based on the luminol analog L-012 for highly sensitive detection of ROS[END_REF][START_REF] Moyon | Luminol fluorescence quenching in biomimicking environments: sequestration of fluorophore in hydrophobic domain[END_REF], to yield emission at ~500 nm. Additional spectra recorded in the absence of L-012, revealed that photoluminescence signals from the semiconductor layer [START_REF] Smandek | Photoluminescence as an in-situ technique to determine solid state and surface properties of semiconductors in an electrochemical cell -application of the "dead layer model[END_REF] were not detected in our experimental conditions. shows that a photocurrent in the µA range appears. In these conditions, the current density starts to increase around -0.2 V and tends to a quasi-plateau at ~0.15 V. This situation strongly differs with that recorded at a bare FTO surface (dashed blue curve). Indeed, voltammetry performed in the same conditions with an FTO electrode revealed that illumination has no effect on the voltammogram curve and an anodic activity that starts at ~0.2 V. This control experiments highlights the importance of the nanostructured -Fe 2 O 3 coating for providing photoelectrochemical activity and promoting anodic behavior at low overpotentials. In this work, we observe the concomitant oxidation of L-012 and the ECL emission (vide infra) at the FTO/-Fe 2 O 3 photoelectrode. Luminol and its analogs such as L-012 may emit ECL under a wide range of experimental conditions [START_REF] Wróblewska | Origin and features of the electrochemiluminescence of luminolexperimental and theoretical investigations[END_REF][START_REF] Miao | Electrogenerated chemiluminescence and its biorelated applications[END_REF]. Indeed, ECL emission may result from several competitive mechanistic pathways with and without additional reagents (e.g., hydrogen peroxide, dioxygen, etc.). In brief, the electrochemical oxidation of L-012 produces a 5-aminophthalazine-1,4-dione form. The addition of OH - associated with the subsequent electrochemical oxidation and N 2 elimination generates the electronically excited state of the monoanions of the 3-aminophthalic acid form [START_REF] Wróblewska | Origin and features of the electrochemiluminescence of luminolexperimental and theoretical investigations[END_REF]. This latter relaxes to the ground state by emitting a photon in the blue region. Emission spectra of L-012 luminophore were recorded from the top side of the cell (i.e., from the electrolyte-air interface side) with a fiber-coupled spectrophotometer. The obtained spectra, corresponding to the typical luminescence spectra from L-012, are shown in Figure 4.

Photoelectrochemical characterization

As shown by the black curve, no luminescence was detected when the system was held in the dark. However, when the light from the LED was switched ON, FL emission from L-012 was recorded, as seen in the purple spectrum. Here, the intensity of the emission peak was 138 a.u.

Finally, when a potential was applied to the FTO/-Fe 2 O 3 , the overall luminescence reached a value of 385 a.u., at  em = 498 nm which constitutes a 2.8 times amplification. This amplification is the result of the PECL generation coupled to the FL signal. Next, the PECL emission was investigated as a function of the applied potential. The PECL spectra, which were obtained by subtracting the FL signal from the overall emission are presented in Figure 5a. As it can be seen in this figure, clear PECL emission with  em = 498 nm was detected starting from an applied potential of -0.2 V with an intensity that increased to more than 10 times (from 37 to 388 a.u.) at 0.2 V vs Ag/AgCl. Moreover, Figure 5b demonstrates a clear relationship between PECL intensity and the photocurrent. The latter was calculated by subtracting the current of the voltammogram without LED illumination (black curve of Figure 3) from that of the signal under LED illumination (red curve of Figure 3). This cross-correlation between PECL and the photocurrent demonstrates clearly that PECL originates from the interfacial hole transfer that likely causes L-012 oxidation at the -Fe 2 O 3 /liquid interface, although interfacial pH variation (caused by the concomitant oxygen evolution reaction (OER)) may also play a role in the PECL process.

PECL study

As shown in Figure 6a (recorded at -0.3 V), no PECL was detected at lower potentials, because of the absence of oxidation current at these potentials, even under LED illumination (red curve of Figure 3 corresponding to the voltammetric response). Conversely, Figure 6b (recorded at 0.3 V) shows that ECL was detected above this potential range, caused by the apparition of a dark ECL current, at such bias, as seen in the voltammogram recorded with illumination OFF (black curve of Figure 3). These results show that it is possible to trigger PECL at -Fe 2 O 3 at a potential as low as -0.2 V vs Ag/AgCl. Finally, the PECL stability was assessed on this system by monitoring the luminescence intensity as a function of time under intermittent LED illumination. As reported in Figure 7a, luminescence (FL + PECL) responds instantaneously to illumination and could be observed for 2 h, however, the luminescence intensity faded by 35% over this rather long period. The origin of this decay was investigated by voltammetry after the 2 h-long test. As shown in Figure 7b, the current intensity recorded by LSV after the prolonged stability test also decayed when compared to that recorded before the test, in good agreement with a PECL decrease induced by a degradation of the photoanode in these experimental conditions. Such a trend may originate from photoanodic degradation [START_REF] Bae | Strategies for stable water splitting via protected photoelectrodes[END_REF] of the coating or an increase of surface states and electron-hole recombination as a result of the prolonged electrolysis [START_REF] Peter | Interpretation of photocurrent transients at semiconductor electrodes: Effects of band-edge unpinning[END_REF][START_REF] Peter | Kinetics of light-driven oxygen evolution at α-Fe 2 O 3 electrodes[END_REF]. If these performances are slightly lower than that measured at BiVO 4 photoanodes in similar conditions [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF], such activity is, however, still considerably higher than that obtained at lower band-gap and much more unstable photoelectrodes [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF][START_REF] Vogel | Spatiotemporal control of electrochemiluminescence guided by a visible light stimulus[END_REF][START_REF] Laser | Semiconductor electrodes. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF][START_REF] Luttmer | Electrogenerated chemiluminescence. 34. Photo-induced electrogenerated chemiluminescence and up-conversion at semiconductor electrodes[END_REF].

Conclusion

Photoinduced electrochemiluminescence (PECL) is an original concept that combines ECL and semiconductor photoelectrochemistry. In this article, we have reported the PECL of a luminol analog (L-012) on nanostructured -Fe 2 O 3 photoanodes. Under illumination ( exc = 375 nm) at potentials comprised between -0.2 V to 0.2 V vs Ag/AgCl, photogenerated holes in -Fe 2 O 3 oxidize L-012, and generate the excited state of the monoanions of the 3aminophtalic acid. The excited state relaxes to the ground state emitting a blue photon. We employed an illumination wavelength ( exc = 375 nm) lower than the emission wavelength ( em = 498 nm), resulting in "fluorescence-type" PECL. This is the second example of this type of PECL [START_REF] Yu | Luminescence amplification at BiVO 4 photoanodes by photoinduced electrochemiluminescence[END_REF], which differs from "upconversion-type" PECL [START_REF] Zhao | Photoinduced electrochemiluminescence at silicon electrodes in water[END_REF]. We show that -Fe 2 O 3 allows for PECL at applied potential as low as -0.2 V vs Ag/AgCl and that it can be used for luminescence amplification. Moreover, even if they exhibit signs of degradation, such structured -Fe 2 O 3 photoanodes are active for hours, which should open up an interesting perspective for PECL, in particular for localized electrochemistry.

Figure 1 .

 1 Figure 1. a) EDS spectrum of a FTO/-Fe 2 O 3 photoanode. b) Experimentally-determined XRD pattern of a photoanode (black curve of the bottom panel), theoretical peaks positions and relative intensities for hematite (ICSD #15840, red lines of the top panel) and FTO (ICSD #39178, green lines of the top panel). c) Cross-section and d) top-view colored SEM images of a FTO/-Fe 2 O 3 surface used as a photoanode in this study.

Figure 2 .

 2 Figure 2. Normalized spectra showing the emission of the LED used in this work (orange), L-012 absorption (purple) and the emission of L-012 (black), the two last spectra were recorded in 0.1 M NaOH with a L-012 concentration of 10 mM. The experimentally-determined bandgap of -Fe 2 O 3 is represented by the red dotted line.

Figure 3 .

 3 Figure 3. Cyclic voltammograms recorded on FTO/-Fe 2 O 3 at 20 mV s -1 in the dark (black curve) and under LED illumination at 375 nm (red curve). A linear sweep voltammogram recorded on bare FTO under LED illumination at 375 nm is also reported (dashed blue curve). The measurements were recorded in 0.1 M NaOH with an L-012 concentration of 10 mM.

Figure 4 .

 4 Figure 4. Luminescence spectra recorded at 0 V in the dark (black curve), at open circuit under LED illumination (purple curve), and at 0 V under LED illumination (red curve). The measurements were recorded in 0.1 M NaOH with an L-012 concentration of 10 mM.

Figure 5 .

 5 Figure 5. a) PECL spectra recorded with a constant LED illumination at different applied potentials E. b) Overlay of the intensity of the PECL peak (colored circles) and photocurrent (black curve) as a function of the imposed potential E. The PECL spectra and the intensity of the PECL peak were obtained by subtracting the FL spectra or FL peak values recorded under open-circuit potentials from the luminescence spectra (FL+PECL) or the luminescence peak values recorded under applied potential. All measurements were performed under LED illumination ( exc = 375 nm) in 0.1 M NaOH with an L-012 concentration of 10 mM.

Figure 6 .

 6 Figure 6. a) Luminescence spectra recorded at -0.3 V in the dark (black curve), at open circuit under LED illumination (purple curve), and at -0.3 V under LED illumination (red curve). b) Luminescence spectrum recorded at +0.3 V in the dark. The measurements were recorded in 0.1 M NaOH with an L-012 concentration of 10 mM.

Figure 7 .

 7 Figure 7. a) Stability of the luminescence signal over time. Graph showing the intensity of the luminescence peak as a function of the time at E = 0 V under intermittent LED illumination. The intensity of the FL peak (light ON, potential OFF) is indicated by a purple dashed line. b) Linear sweep voltammograms (LSVs) recorded (blue curve) before and (green curve) after the stability test shown in Figure 7a. All measurements were performed under LED illumination ( exc = 375 nm) in 0.1 M NaOH with an L-012 concentration of 10 mM.

Acknowledgements

This work is funded by ANR (LiCORN -ANR-20-CE29-0006). S. Fryars, F. Gouttefangeas, and L. Joanny (ScanMAT, UMS 2011 University of Rennes 1-CNRS) are acknowledged for their help with the characterization of the samples. XRD and SEM were performed on ScanMAT facilities platforms (UMS 2001, University of Rennes 1-CNRS). J. Yu acknowledges the China Scholarship Council for her Ph.D. fellowship.