Edouard Challe
email: edouard.challe@polytechnique.edu

Benoit Mojon
email: benoit.mojon@banque-france.fr.

Xavier Ragot
email: xavier.ragot@pse.ens.fr.

Andrew Atkeson

Regis Breton

Hans Gersbach

Michel Guillard

Robert Kollmann

François Larmande

Etienne Lehmann

Jose-Luis Peydro

Jean-Charles Rochet

Equilibrium Risk Shifting and Interest Rate in an Opaque Financial System

Keywords: JEL codes: E44, G01, G20 Risk shifting, Portfolio correlation, Systemic risk, Financial opacity

We analyse the risk-taking behaviour of heterogenous intermediaries that are protected by limited liability and choose both their amount of leverage and the risk exposure of their portfolio. Due to the opacity of the …nancial sector, outside providers of funds cannot distinguish "prudent" intermediaries from those "imprudent" ones that voluntarily hold high-risk portfolios and expose themselves to the risk of bankrupcy.

We show how the number of imprudent intermediaries is determined in equilibrium jointly with the interest rate, and how both ultimately depend on the cross-sectional distribution of intermediaries'capital. One implication of our analysis is that an exogenous increase in the supply of funds to the intermediary sector (following, e.g., capital in ‡ows) lowers interest rates and raises the number of imprudent intermediaries (the

Introduction

The 2007-2010 …nancial crisis has rejuvenated the interest in systemic risk in the …nancial system, its dramatic spill over to the real economy and whether and how it should be addressed by public policies. We contribute to this debate with an analysis of the risk taking behaviour of …nancial intermediaries that have limited liabilities and may deliberately choose a level of risk in excess of the social optimum. We show how the level of economywide risk taking depends on the distribution of equity among intermediaries and the level of interest rate in the economy.

Our key assumption is that outside providers of funds cannot tell apart "prudent" and well diversi…ed banks from "imprudent" ones overly exposed to one particular asset, because the balance sheets of individual intermediaries is imperfectly observable, or opaque.

This assumption is consistent with the view of several commentators of the crisis including [START_REF] Brunnermeier | Deciphering the liquidity and credit crunch 2007-08[END_REF], [START_REF] Acharya | Causes of the …nancial crisis[END_REF], and [START_REF] Dubecq | Fuzzy capital requirement, risk shifting, and the risk-taking channel of monetary policy[END_REF] 1 . In the decade prior to the crisis, risk transfer instruments, which have reached a very large scale in the U.S., have increased the opacity of banks'leverage and risk-taking incentives [START_REF] Acharya | How banks played the leverage game, in Restoring Financial Stability: How to Repair a Failed System[END_REF]. First, regulatory loopholes allowed banks evade capital requirements by securitising assets and providing (unregulated) liquidity support to "shadow" (i.e., o¤balance-sheet) entities [START_REF] Acharya | Causes of the …nancial crisis[END_REF]. Second, the …nancial sector as a whole e¤ectively repurchased much of the senior tranches of structured products, whose payo¤ distributions was particularly di¢ cult to assess (see, e.g., [START_REF] Coval | Economic catastrophe bonds[END_REF]. Third, some banks actively relied on "window dressing" to manipulate leverage …gures -by selling asset before the books releases to repurchase them at a later date (see, e.g., The Financial Crisis Inquiry Commission, 2011). Last but not least, this opacity may have taken the form 1 And the references therein.

2

of shadow subsidiaries that were used to absorb poorly performing assets, as was revealed by the investigation on Lehman's bankruptcy. 2 While the opacity of the …nancial sector may have reached unprecedented level during the run-up to the crisis and the crisis itself, it has long been recognised as a key issue in that industry and one of the fundamental reasons for why it should be regulated. For example, [START_REF] Morgan | Rating banks: Risk and uncertainty in an opaque industry[END_REF] shows that bond raters disagree signi…cantly more about U.S. …nancial intermediaries than they do over other …rms, and interprets this result as evidence that banks are intrinsically more opaque -essentially because their assets are di¢ cult to observe and change at a fast pace. 3 This feature of the industry severely limits the ability of outsiders (investors and rating agencies alike) to assess changes in bank's capital structure in real time. To illustrate this point, Figure 1 reports the standard errors of an AR(2) regression of the idiosyncratic component of the capital-asset ratio of 90 French banks over the period 1993Q2-2009Q1. These standard deviations are sorted from the smallest to the largest. It is striking that for more than a quarter of these banks the one-quarter-ahead standard deviation of the forecast error in the capital-asset ratio is higher than 2%. To summarise, the intrinsic nature of the banking industry combined with the recent trends in …nancial innovations have made it especially di¢ cult for outside providers of funds to accurately assess the true net worth level of individual banks.

When intermediaries'balance sheet is opaque, those with relatively low levels of capital may be tempted to hold high-risk asset portfolios, or even to gamble for resurrection in the 2 On April 13, 2010, the New York Time reported that "Lehman Brothers operated a side business that allowed the defunct brokerage to transfer risky investments o¤ its books in the years leading up to its collapse, according to a report published Tuesday 13 April 2010. The …rm, called Hudson Castle, appeared to be an independent company, but played an important "behind-the-scenes role" at Lehman, .... Hudson is part of a "vast …nancial system" that operates largely beyond the reach of banking regulators. But banks can use such entities to raise cash by trading investments and, at times, make their …nances look arti…cially strong.

The report said Lehman conducted several transactions greater than $1 billion with Hudson vehicles, but added that it is unclear how much money was involved since 2001. Critics charge that this type of creative …nancing allowed Lehman and other major banks to temporarily transfer risky investments in subprime mortgages and commercial real estate, the report said. While most of the deals done through operations such as Hudson are legal, the report points out that bank examiners have recently raised questions about other dubious accounting practices at Lehman." 3 See also [START_REF] Iannotta | Testing for opaqueness in the european banking industry: evidence from bond credit ratings[END_REF] for similar evidence about European banks, and [START_REF] Flannery | The 2007-09 …nancial crisis and bank opaqueness[END_REF] on the increase in the opacity of U.S. banks during the crisis. (principal components). Second, the residual of this regression (which approximate the idiosyncratic movements of this ratio), is modelled as an AR(2) process, from which the standard deviation of the forecast error is computed. Data source: Banque de France, Commission Bancaire (see [START_REF] Jimborean | Banks'…nancial conditions and the transmission of monetary policy: A FAVAR approach[END_REF], for further desciption of the data).

face of worsening economic conditions. In our model, intermediaries'limited liability creates an incentives to correlate asset portfolios and raise leverage, thereby allowing intermediaries to raise their return on equity in case of success while transferring much of their losses to their creditors in case of failure. This tendency, however, is alleviated by intermediaries' shareholders'initial equity stake, which disciplines risk-taking, thereby limiting leverage and favouring diversi…cation. We show that this trade-o¤ gives rise to an endogenous sorting of intermediaries along the equity dimension, with well capitalised intermediaries holding diversi…ed portfolios and keeping a limited level of leverage (that is, behaving prudently), and poorly capitalised ones heavily resorting to leverage to invest in correlated assets (i.e., behaving imprudently). Opacity implies that the former are not readily distinguishable from the latter, so that risk-prone behaviour may prosper without being immediately sanctioned by higher borrowing rates.

One property of our model is that the proportion of imprudent intermediaries and, therefore, the level of systemic risk in the …nancial system, crucially depend on both the crosssectional distribution of capital and the prevailing interest rate. The endogenous determination of the number of imprudent intermediaries jointly with the (equilibrium) interest rate is our key contribution. Equipped with this joint equilibrium outcome, we analyse the impact on the interest rate and the number of imprudent intermediaries of two exogenous shocks: a lending boom that shifts the loan supply curve rightwards; and an equity squeeze that shifts the distribution of banks' capital leftwards. As we show, the downward pressure on the equilibrium interest rate that follows the lending boom raises the number of imprudent intermediaries and hence the level of economywide risk shifting (the risk-taking channel of low interest rates). An "equity squeeze", that is, a reduction in the equity value of intermediaries' shareholders after a negative aggregate shock, has the same e¤ect provided that the supply of funds is su¢ ciently elastic (the gambling-for-resurrection channel of falling equity). The evidence strongly suggests that both shocks occurred in the run-up to the current crisis. In the …rst half of the 2000 decade, both capital in ‡ows from China and oil-exporting countries into the U.S. and the accommodative monetary policy of the Fed contributed to keep the yield curve very low; according to our model, this would have favoured imprudent behaviour by an increasing number of banks -those at the lower end of the capital distribution-thereby raising their risk exposure and the amount of systemic risk in the economy. 4 Second, the tightening of U.S. monetary policy in 2004 and the rise in delinquency rates on subprime mortgages from 2006 onwards may have deteriorated the equity position of exposed intermediaries, and hence favored gambling-for-resurection strategies. [START_REF] Landier | Going for broke: New Century Financial Corporation[END_REF] provide direct evidence of this behaviour for New Century Financial Corporation, a major subprime originator prior to its bankruptcy in 2007. 5Related literature. In our model, systemic risk in the …nancial sector arises from the interaction between i. intermediaries'limited liability and option to default (the risk shifting problem); ii. their incentive to correlate their risk exposure (the endogenous correlation problem); and iii. the di¢ culty for outside lenders to discriminate individual institutions on the basis of their true net worth level (the opacity problem). While our model is the …rst to explicitly connect these three dimensions, we build on many contributions that have studied each of them separately.

Our modelling of the risk shifting problem closely follows [START_REF] Allen | Bubbles and crises[END_REF] and [START_REF] Acharya | A theory of systemic risk and design of prudential regulation[END_REF], who show that limited liability leads …nancial institutions to overweight risky assets in their portfolio, relative to the …rst best. 6 There are two main di¤erences between earlier models of risk shifting and ours. In Allen and Gale, market segmentation and limited-liability debt contracts twist intermediaries'risk-taking incentives and leads to an overvaluation (or "bubble") in the price of the risky asset (a feature that is also in Challe and Ragot and Dubecq et al.). In Acharya, risky assets are in ‡exible supply so that their quantity (rather than price) adjust to clear the market. All these models share the property that intermediaries' excessive risk-taking is ubiquitous: risky assets always have excessive space in intermediaries portfolios, leading all of them to be exposed to bankruptcy risk.

We see this property as somewhat extreme, which leads us to emphasise the disciplining role of initial shareholders' equity stake and to endogenise each intermediary's (discrete) choice of adopting or not a bankruptcy-prone behaviour based on the expected costs and bene…ts of doing so. The second di¤erence with earlier contributions concerns the way we model excessive risk taking: while earlier models rely on intermediaries'overexposure to a risky asset relative to a safe one, in ours excessive risk taking exclusively takes the form of insu¢ cient portfolio diversi…cation in equilibrium.

This asset correlation problem has been the focus on several recent contributions, both empirical and theoretical. [START_REF] Acharya | Causes of the …nancial crisis[END_REF] notably document the overexposure of the U.S. banking sector to securitised mortgages prior to the current crisis, with the risk associated with those securities being e¤ectively kept within the sector (via the use of unregulated liquidity enhancements or the repurchase of CDO tranches) rather than rising interest rates lower asset values, which in turn deplete bankowners'net equity positions ex post and may trigger the gambling-for-resurection logic. Our model is consistent both views. 6 see also [START_REF] Rochet | Capital requirements and the behaviour of commercial banks[END_REF] for an early analysis of bank risk taking under limited liability.

transferred to other investors and disseminated throughout the economy. Any unit of investment in a 1 pays R 1 = R h 1 > 0 if s = s 1 and 0 otherwise, while any unit of investment in a 2 pays R 2 = R h 2 > 0 if s = s 2 and R l 2 > 0 otherwise. 7 State s 1 (s 2 ) occurs with probability p (s 1 ) p = 0:5 , > 0 (p (s 2 ) = 1 p.) Finally the two assets are assumed to have identical expected payo¤s, i.e.,

pR h 1 = pR l 2 + (1 p) R h 2 : (1)
Our assumed joint payo¤ distribution has the following properties: when considered in isolation, a 1 is more risky (in the sense of mean-preserving spread) than a 2 ; however, the strict negative correlation between the two assets implies that one of them may be used as a hedge against the portfolio risk generated by the other. In particular, a suitably diversi…ed portfolio pays the certain gross return pR h 1 -thereby entirely eliminating bankruptcy risk for a leveraged investor. This simple payo¤ structure allows us to focus on the joint choice of leverage and portfolio correlation as the ultimate source of endogenous aggregate risk in the economy.

Agents and market structure

There are two types of agents in the economy: "lenders" and "intermediaries", both riskneutral and in mass one. Our market structure (and implied decisions) is similar to that in [START_REF] Allen | Bubbles and crises[END_REF] and [START_REF] Acharya | A theory of systemic risk and design of prudential regulation[END_REF]. In particular, markets are segmented, in the sense that intermediaries have exclusive access to the menu of assets a (due, for example, to 7 Assuming that asset a 1 has no liquidation value in state s 2 greatly simplify the analysis of Section 4, where we need to sum up the repayments of heterogenous intermediaries to the lenders across states.

Alternatively, one may assume that the asset has some liquidation value but that the latter cannot be recovered in case of default (due, e.g., to bankrupcy costs). asymmetric information, di¤erence in asset management abilities, regulation etc.). Intermediaries may borrow from the lenders to achieve their desired level of asset investment, and are protected by limited liability debt contracts. Once lending has taken place, the portfolio chosen by the intermediaries is out of the control of the lenders.

We modify this basic framework in two directions. First, we assume that an intermediary's funding partly comes out of inside equity, which will serve both to bu¤er the intermediary's balance-sheets against adverse shocks and to discipline its shareholders'risk-taking attitude. 8 Second, we study the equilibrium of an economy populated by a large number of intermediaries with heterogenous equity levels that are imperfectly observed by outside providers of funds.

Intermediaries. Intermediaries' shareholders maximise value, given their (exogenously given) initial equity stake e > 0. Denoting by (x i ) i=1;2 0 the portfolio of an intermediary, its balance sheet constraint may be written as:

P x i e + b; (2)
with i = 1; 2 and where b is the intermediary's debt. Intermediaries face a convex, nonpecuniary investment cost c ( P x i ), which satis…es c 0 (:) > 0, c 00 (:) < 0 and c (0) = 0. For the sake of tractability, our analysis in the body of the paper is carried out under the assumption that c (:) is quadratic, but we show in Appendix B that all our results carry over to the more general isoelastic case. More speci…cally, c (:) takes the form:

c ( P x i ) = (2 ) 1 ( P x i ) 2 ; > 0: (3)
Given its initial equity stake e and a contracted gross interest rate r on borrowed funds, an intermediary chooses (x i ) and, by implication, b -i.e., it chooses both the size and the structure of the balance sheet. Limited liability implies that an intermediary's payo¤ net of debt repayment is bounded below by zero, so the ex post net payo¤ generated by the portfolio (x i ) is:

max [ P x i R i rb; 0]
Substituting (2) (with equality) into the latter expression, we …nd the date 1 value (including the nonpecuniary cost) of an intermediary with initial equity e to be:

V (e) = max

x i 0 P s p (s) (max [re + P x i (R i r) ; 0]) c ( P x i ) ; (4)
with s = 1; 2.

In solving (4), intermediaries di¤er in the amount of the inside equity stake of its shareholders, e. The cross-sectional equity distribution is assumed to be characterised by a continuous density function f (e; ) with support [0; e max ] and c.d.f. F (e; ) = R e 0 f (i; ) di. Since the number of intermediaries is normalised to one we have F (e max ; ) = 1, while

E R emax 0 ef (e;
) di is the total capital of the intermediary sector. The parameter indexes the location of the density function, with an increase in being associated with a rightward shifts in the distribution of equity level (so that F (e; ) < 0.)

Lenders. Funds are supplied by households (the "lenders"), who lend their funds to the intermediary sector at date 1 in order to collect repayments at date 2. Each lender enjoys labour income w > 0 at date 1 and maximises u (c 1 ) + c 2s , where c 1 is date 1 consumption, c 2s consumption at date 2 in state s, and u (:) a twice continuously di¤erentiable, strictly increasing and strictly concave function. Let s denote lenders' ex post date 2 return in state s from lending to the intermediary sector, and P s p (s) s the corresponding ex ante return (Note that both in general di¤er from the face lending rate r due to the possibility of intermediaries' default.) Lenders choose their loan supply B s , where B s = arg max u (c 1 ) + P s p (s) c 2s , subject to c 1 = w B and c 2s = B s . The implied loan supply curve is:

B s ( ; w) = w u 0 1 ( ) ; (5)
which is continuous and strictly increasing in both arguments. In short, risk neutrality implies that lenders value the expected return on loans, , with the implied loan return curve being shifted by date 1 income, w. We impose speci…c parameter restrictions later on ensuring that B s ( ; w) > 0 in equilibrium.

First best

The key contractual friction in this economy is that an intermediary maximises the expected terminal payo¤ to its (risk-neutral) shareholders who are protected by limited liability -and hence transfer losses to the debtors in case of default. Before further analysing the implications of this distortion, we compute the …rst-best outcome, where this distortion is removed.

Planner' s problem. Since a 1 and a 2 have identical expected payo¤ and are perfectly negatively correlated, a fully diversi…ed portfolio that entirely eliminates the payo¤ risk is always e¢ cient, at least weakly. A portfolio (x i ) i=1;2 pays the certain payo¤ ( P x i ) pR h 1 if it pays identical payo¤s across states, that is, if it satis…es

x 1 R h 1 + x 2 R l 2 = x 2 R h 2 ; (6)
where the left and right hand sides are the portfolio payo¤s in states s 1 and s 2 , respectively.

Equation ( 6) implies that the riskless portfolio must be composed of the following asset shares:

x1 P xi = p R h 2 R l 2 R h 2 ; x2 P xi = p R h 1 R h 2 : (7)
Equation ( 7) characterises the structure of the portfolio but not its optimal size. Let C denote the total consumption of intermediaries ('shareholders), and C 1 and C 2 the total consumption of lenders at date 1 and 2, respectively. The planner solves max u (C 1 )+C 2 + C, subject to the following …rst-and second-period resource constraints:

( P xi ) + C 1 = w + E; C 2 + C = ( P xi ) pR h 1 c ( P xi ) :
Hence, the optimal level of investment P xi satis…es:

P xi = pR h 1 u 0 (w + E P xi ) : (8)
Decentralisation. Since the limited-liability constraint is the only friction a¤ecting intermediaries portfolio choice, the planner's problem can be decentralised by removing this constraint -or equivalently, by punishing default su¢ ciently severely. When the option to default is not operative, the value of an intermediary in (4) becomes:

V (e) = max

x i 0 re + ( P x i ) pR h 1 r c ( P x i ) ; (9)
where we have used the fact that P s p (s)

P x i (R i r) = ( P x i ) pR h 1
r under full diver-si…cation (and given the payo¤ distribution (1)). From ( 9), the e¢ cient portfolio satis…es 1) and ( 6), we …nd it to be:

P xi = pR h 1 r . Combined with (
(x 1 ; x2 ) = pR h 1 r p R h 2 R l 2 R h 2 ; pR h 1 r p R h 1 R h 2 : (10)
Portfolio ( 10) is only weakly dominant because, due to agents'risk neutrality, a continuum of portfolios in fact achieve the same welfare outcome as [START_REF] Bernanke | What explains the stock market's reaction to Federal Reserve policy?[END_REF]. Indeed, for lenders to enjoy the face return r in both states, it is enough that intermediaries be solvent in both states, which is possible (to some extent) with an imperfectly diversi…ed portfolio thanks to the bu¤ering role of intermediaries'capital. The solvency conditions impose that a given portfolio (x i ) i=1;2 never generates a negative net payo¤ ex post, i.e.,

re + P x i (R i r) 0; s = s 1 ; s 2 ; (11)
where

(R 1 ; R 2 ) = R h 1 ; R l 2 if s = s 1 and 0; R h 2 if s = s 2 .
Combining [START_REF] Brunnermeier | Deciphering the liquidity and credit crunch 2007-08[END_REF] with the optimal balance-sheet size

P xi = pR h 1 r
, we …nd that a solvent portfolio must be such that

x 1 2 [x 1 ; x 1 ] ; 0 < x 1 < x 1 < 1,
where x 1 and x 1 are:

x 1 = pR h 1 r r R l 2 re R h 1 R l 2 ; x 1 = pR h 1 r R h 2 r + re R h 2 ;
and where x 1 < pR h 1 r whenever the intermediary is leveraged. In short, given the optimal balance-sheet size pR h r ; x 1 cannot be too low, otherwise the intermediary would default in state s 1 ; it cannot be too high either (and hence x 2 too low), otherwise default would occur in state s 2 . Intermediaries that deviate from the riskless portfolio (x i )

while still satisfying

P xi = pR h 1 r and x 1 2 [x 1 ;
x 1 ] will bear some asset risk but are indi¤erent to it thanks to risk neutrality. These portfolios, which we refer to as "prudent", lie along a closed subinterval of the x 2 = pR h 1 r x 1 line and include the riskless portfolio

(x i ) -see
are repaid r with certainty. Hence, we may rewrite (5) as:

Bs (r; w) = w u 0 1 (r) : (13)
Bd (r; ) is continuous and linearly decreasing in r, while Bs (r; w) is continuous and strictly increasing in r (since u 00 (:) < 0). Hence the two curves cross at most once and, if they do, give a unique equilibrium interest rate r. In the remainder of the paper, we focus on equilibria in which all intermediaries are active and leveraged. Lemma 1 provides a su¢ cient condition for the existence of a …rst-best equilibrium with this property. All proofs are in the Appendix. Essentially, a unique equilibrium with all intermediaries being leveraged exists if both expected asset payo¤s (i.e., pR h 1 ) and lenders'income (i.e., w) are su¢ ciently large. This equilibrium is depicted in Figure 2(b) below.

3 Loanable funds equilibrium under risk-shifting

Intermediaries'behaviour

The presence of the limited-liability debt contracts a¤ect investment choices by altering intermediaries' shareholders payo¤s relative to the …rst best. Namely, value maximisation under limited liability may lead an intermediary to choose a high risk/high expected payo¤ strategy, thereby maximising its own payo¤ in case of success while transferring losses to the lenders in case default.

We work the problem of an intermediary (i.e., equation ( 4)) backwards. Let us refer to as "prudent" an intermediary whose asset portfolio satis…es both solvency constraints in [START_REF] Brunnermeier | Deciphering the liquidity and credit crunch 2007-08[END_REF], and denote its value as V (e). Similarly, let us call "imprudent" an intermediary whose portfolio violates one of the two inequalities in (11) -thereby triggering default in one of the two states-, and denote its value by V (e). The intermediary chooses the best option, giving a value to the initial equity holders of V (e) = max [V (e) ; V (e)].

Prudent intermediaries. Trivially, the absence of the option to default implies that the portfolio choice of a prudent intermediary is the same as in the …rst best:

P x i = pR h 1 r ; x 1 < x 1 < x 1 ; (14) b (r; e) = pR h 1 r e: (15)
Substituting ( 14)-( 15) into (9), we …nd the value of a prudent intermediary to be:

V (e) = re + ( =2) pR h 1 r 2 (= V (e)): (16)
Imprudent intermediaries. Imprudent intermediaries, unlike prudent ones, correlate their portfolio and consequently default in one of the two states. Consider …rst the optimal portfolio choice of an intermediary having chosen to overweight asset a 1 in its portfolio, and thus defaults at date 2 if state s 2 occurs. Ex ante, this intermediary earns zero with probability 1 p, so the objective (4) becomes:

max x i 0 V (e) = p re + x 1 R h 1 r + x 2 R l 2 r c ( P x i ) (17)
Since x 1 and x 2 enter symmetrically in the cost function while R h 1 > R l 2 , the intermediary must entirely disregard a 2 , leading to the optimal portfolio:

(x 1 ; x 2 ) = p R h 1 r ; 0 ; (18) b (r; e) = p R h 1 r e: (19)
An alternative investment strategy for an imprudent intermediary would be to overweight a 2 ; and hence to default if s 1 occurs ex post. However, it is straightforward to show that it is never optimal to do so under our distributional assumptions. Indeed, imprudent behaviour implies that the intermediary earns zero if the wrong state occurs, and accordingly only values the state corresponding to the asset being invested in. Since the univariate distribution of a 1 is a mean-preserving spread of that of a 2 , a 1 has more value to the imprudent intermediary than a 2 . 9 Substituting ( 18) into ( 17), we …nd the optimised value of an imprudent intermediary to be:

V (e) = pre + ( =2) p R h 1 r 2 :
(20) 9 An intermediary choosing the default in state s 1 does not value payo¤s in that state and hence max-

imises (1 p) x 1 (0 r) + x 2 R h 2 r + re c (x 1 + x 2 )
, leading to the optimal portfolio (x 1 ; x 2 ) = 0; (1 p) R h 2 r : Computing and comparing the ex ante utility levels associated with (x 1 ; x 2 ) and (x 1 ; x 2 ) leads the former to be prefered, provided that is not too large.

To summarise, imprudent intermediaries have two distinguishing characteristics, relative to prudent intermediaries. First, they perfectly correlated their asset portfolio (since x 2 = 0), thereby maximising both their payo¤ in case of success and lenders' losses in case of default. Second, they endogenously choose a larger balance sheet size (since

x 1 > P x i ),
which in turns implies more leverage for any given level of equity e (i.e., b (r; e) > b (r; e)).

This latter property is a direct implication of the fact that imprudent intermediaries avoid repayment with probability 1 p: This e¤ectively lowers the cost of debt ex ante for any given face interest rate r, relative to the cost faced by prudent intermediaries (who repay in both states). In the (x 1 ; x 2 ) plane, the imprudent portfolio lies on the x 1 axis and the left of the x 2 = pR h 1 r x 1 line -see Figure 1(a). Value of an intermediary. Expressions ( 16) and ( 20) re ‡ect the joint roles of equity and the borrowing rate in a¤ecting the intermediary's value and thus incentives to behave prudently or imprudently. For a given level of equity and borrowing rate, imprudent intermediaries buy larger portfolios, consequently earn large payo¤s in case of success, which goes towards raising value (see the second term in the right hand side of both expressions); however, they also risk losing their equity (with probability 1 p), which tends to reduce value for any given initial equity stake (the …rst term). Comparing ( 16) and ( 20) and assuming that indi¤erent intermediaries behave prudently, we …nd that an intermediary engages in imprudent behaviour whenever its equity state is su¢ ciently low, that is, if and only if:

e < ẽ (r) pR h 1 1 + p 2 r (21)
Equation ( 21) implies that a poorly capitalised intermediary, i.e., one with low equity stake and hence relatively little to lose in case of default, will engage in imprudent behaviour, while an intermediary with high shareholders'equity stake, and hence much to lose in case of default, will behave prudently. The implied value of an intermediary as a function of e, i.e., V (e) = max [V (e) ; V (e)] is depicted in Figure 1(b).

A key implication of ( 21) is that lower borrowing rates raise the cut-o¤ equity level below which the intermediary chooses to behave imprudently. To further understand why this is the case, compare the impact of a marginal rise in r on V (e) and V (e) -that is, for each strategy, the loss in the intermediary's value associated with a rise in the face …nancing cost.

Using ( 16) and ( 20), we …nd these falls to be:

V r (e) = b (r; e) ; V r (e) = pb (r; e) :

These expressions follow from the envelop theorem and have a straightforward interpretation. For the prudent intermediary, who never defaults and hence always repays r per unit of debt, the loss in value associated with a marginal rise in r is its total amount debt, b (r; e) : For the imprudent intermediary, who only repays in state 1, the loss in value is the relevant amount of debt, b (r; e), times the probability that it will actually be repaid, p.

For a rise in r to lower the threshold ẽ, it must be the case that V (e) increases more than

V (e) for the marginal intermediary, i.e., that for whom V (ẽ) = V (ẽ) (i.e., that intermediary must turn prudent, rather than imprudent, following a rise in the interest rate). It must be the case that V r (ẽ) > V r (ẽ) or, equivalently by using the two expressions above, b (r; ẽ) < pb (r; ẽ): a switch by the marginal intermediary from the prudent to the imprudent investment strategy must involve a su¢ ciently large increase in leverage. This property can be shown to hold not only in the quadratic case but also for any isoelastic investment cost function (see Appendix B for details).

Aggregate demand for funds

Our key assumption here is that while the distribution of equity levels is perfectly known by outside lenders, …nancial opacity prevents lenders from observing the equity levels any particular intermediary. Hence, lenders cannot condition the loan rate on the speci…c equity level of an intermediary, so that a single borrowing rate r applies to the entire market. 10Then, we may de…ne g (r; )

Z ẽ(r) 0 f (e; ) de = F (ẽ (r) ; ) (22)
as the proportion of imprudent intermediaries in the economy at a given interest rate r.

Note that g r (r; ) = (1 + p) f (ẽ (r) ; ) =2 < 0; that is, a lower face interest rate raises the proportion of imprudent intermediaries in the economy by increasing the threshold equity level ẽ (r). Moreover, we have g (r; ) = F (ẽ; ) < 0, that is, an increase in lowers the proportion of imprudent intermediary (for any given value of the cut-o¤ ẽ (r)).

The total demand for funds aggregates the leverage choices of individual intermediaries, appropriately weighted by their shares in the economy. It is thus given by:

B d (r; ) = Z ẽ(
Equation [START_REF] Ioannidou | Monetary policy, risk-taking and pricing: evidence from a quasi-natural experiment[END_REF] shows that the interest rate will a¤ect the demand for loanable funds in two ways: …rst, it will a¤ect the demand for funding of every single intermediaries (the 'intensive' leverage margin); and second, by shifting the threshold ẽ (r), it will cause a discontinuous change in the leverage choice of some of them, from prudent to imprudent or the other way around (the 'extensive' leverage margin.) Substituting ( 15) and (19) into the latter expression, using [START_REF] Iannotta | Testing for opaqueness in the european banking industry: evidence from bond credit ratings[END_REF] and rearranging, the total demand for funds is found to be:

B d (r; ) = pR h 1 r (1 (1 p) g (r; )) E: (24)
In the (B; r) plane, the B d (r; ) curve lies to the right of the Bd (r; ) curve, its …rst-best counterpart. This is because, for any given value of r, the risk-shifting equilibrium includes a nonnegative fraction of imprudent intermediaries, whose demand for debt is larger than that of prudent intermediaries at any given interest rate r (see Figure 2

(b)).

There are two properties of the aggregate demand for loanable funds that are worth discussing at this stage. First, it is continuous and decreasing in the borrowing rate, i.e., B d r (r; ) = (1 (1 p) g (r; )) + (1 p) rg r (r; ) < 0:

Two factors contribute to make the demand for funds a downward-sloping function of r.

First, a lower interest rate raises the leverage of both prudent and imprudent intermediaries -see the optimal investment rules ( 14) and [START_REF] Fahri | Collective moral hazard, maturity mismatch and systemic bailouts[END_REF]. Second, a lower interest rate induces "marginal" intermediaries (those which are close to the cut-o¤ equity level ẽ in ( 21)) to switch from prudent to imprudent behaviour, and those experience a discontinuous increase in their leverage -again, by ( 14) and [START_REF] Fahri | Collective moral hazard, maturity mismatch and systemic bailouts[END_REF]. Hence, changes in the borrowing rate a¤ect the "intensive" (i.e., conditional on not switching behaviour) and "extensive" (i.e., the number of intermediaries who switch behaviour) leverage margins in the same direction.

The second relevant property of the curve is that, holding r constant, B d increases as the distribution of equity shifts leftwards. That is,

B d (r; ) = r (1 p) g (r; ) < 0:
This is because, as the equity level of intermediaries decreases, some of them switch from prudent to imprudent behaviour. As imprudent intermediaries choose higher leverage than prudent ones, this composition e¤ect translate into an upward shift in the aggregate demand for funds.

Aggregate supply of funds

The aggregate supply of funds depends on the expected return on loans, , which under risk shifting not only depends on the face borrowing rate but also on both the share of imprudent intermediaries and the probability that they go bankrupt. In state 1, which occurs with probability p, all intermediaries repay the face interest rate r to the lenders: prudent intermediaries because they are always able to, imprudent ones because their risky bets turned out to be successful. In state 2, which occurs with complementary probability, only prudent intermediaries, which are in number 1 g (r; ), are able to repay r. Imprudent intermediaries' bets, on the contrary, turn out to be unsuccessful, leaving lenders with no repayment at all. Summing up unit repayments across states and intermediaries types and the market for loanable funds must satisfy

B s (r; ; w) = B d (r; ) (28)
Since B d (r; ) is continuously decreasing in r while B s (r; ; w) is continuously increasing in r, the equilibrium is unique provided that it exists. Again, we are focusing on risk-shifting equilibria in which all intermediaries are leveraged, the conditions under which this is the case being summarised in the following lemma. To summarise, the equilibrium is well behaved provided that lenders'income, w; is suf-…ciently large. The existence conditions stated in Lemma 2 are slightly more stringent than those stated in Lemma 1, so the former also ensure the existence of the …rst-best outcome characterised in Section 2.3.

The equilibrium in the market for loanable funds is depicted in Figure 2(b). The intersection of the two curves gives the equilibrium contracted loan rate r, given the exogenous parameter set ( ; w). The loan rate in turn determines the equilibrium share of imprudent intermediaries g (r; ) (by equation ( 22)), as well as the equilibrium expected return on loans to intermediaries, (r; ) (by ( 25)). Note that despite di¤erences in the implied equilibrium interest rate in the two economies, the equilibrium amount of aggregate lending is the same.

Indeed, the interest rate in the …rst-best equilibrium satis…es pR h 1 r E = w u 0 1 (r) ;

while the expected rate of return in the risk-shifting equilibrium satis…es pR h 1 E = w u 0 1 ( ). This implies that = r (i.e., lenders'expected compensation for their loans in the same across the two equilibria), so that B s ( ; w) = Bs (r; w) (i.e., they lend the same amount). From ( 25) and the fact that = r, we …nd the interest rate premium generated by the presence of imprudent intermediaries to be:

r r = g (r; ) (1 p) 1 g (r; ) ;
which is positive and increasing in the both the number of such intermediaries, g (r; ), and the probability that they go bust, 1 p.

f(e;ε) e ( )       + - = 2

Impact of aggregate shocks

We may now state the main predictions of the model about how shifts in the underlying fundamentals (the supply of funds and the distribution of intermediaries'capital) a¤ect the three key equilibrium variables, r, (r; ) and g (r; ) : These predictions are summarised in the following propositions.

Proposition 1 (Lending boom). An exogenous increase in the supply of funds (i.e., dw > 0) i) lowers the equilibrium contracted rate, r, ii) lowers the expected return on loans, (r; ), and iii) raises the share of imprudent intermediaries in the economy, g (r; ).

Proposition 1 essentially states that easier …nancing conditions for intermediaries tend to fuel systemic risk by inducing an increasing number of intermediaries to take larger and riskier bets; conversely, tighter credit raises the interest rate and discipline banks'risk-taking behaviour. The e¤ect of the boom shift in the funds supply curve is depicted in Figure 3(a).

More speci…cally, the boom is associated with a rightward shift in the B s locus, whose direct e¤ect is to lower the equilibrium contracted loan rate. Holding constant, the new value of r is associated with a lower value of the equity cuto¤ ẽ (r) in ( 21); so that a increasing number of intermediaries turn from prudent to imprudent -i.e., g (r; ) rises. Both the lower value of r and the higher value of g (r; ) contribute to lower the expected return on loans, (r; ).

While our analysis remains formal, several interpretations may be given to the shift in credit supply leading to easier …nancing conditions. According to [START_REF] Bernanke | The global saving glut and the U.S. current account de…cit[END_REF], for example, a supply-driven shift in funding occurred in the …rst half of the last decade due to recycled balance-of-payment surpluses from China and oil-exporting countries; in this interpretation, systemic risk in the U.S. was closely related to the "global imbalances"problem, which was itself rooted in the willingness of surplus countries to hoard wealth in the form of U.S. assets. Another view has it that exceptionally loose monetary policy leading to exceedingly low real interest rates in the wake of the 2001 recession in the U.S. would have given rise to a "risk-taking"channel of monetary policy, thereby fostering widespread systemic risk in the U.S. …nancial sector (see [START_REF] Taylor | Getting o¤ Track: How Government Actions and Interventions Caused, Prolonged and Worsened the Financial Crisis[END_REF], Adrian and Shin, 2010, as well as [START_REF] Altunbas | Does monetary policy a¤ect bank risk-taking? BIS Working Paper no 298[END_REF] for a survey and some evidence). 11 Be it the consequence of either or both, the model unambiguously predicts that falling interest rates raise risk-taking by an increasing number of banks and hence the economywide level of risk. Moreover, the model predicts that this increase in aggregate risk is rooted in changes in the portfolio choices of less capitalised intermediaries -i.e., those to the left of, but close to, the equity cuto¤ ẽ (r). This channel is consistent with the …ndings of Jiminez et al. ( 2010), who study the risk-taking behaviour of a panel of Spanish banks and …nd that falling short-term rates increase risk-taking low-capital banks (rather than the "average"bank.) Proposition 2 (Equity squeeze). A downward shift in the distribution of equity (i.e., d < 0) raises the equilibrium interest rate, r. If the elasticity of the credit supply with respect to is su¢ ciently high, then it also raises the share of imprudent intermediaries, g (r; ).

Proposition 2 re ‡ects the three e¤ects at work following a downward shifts in the distribution of equity. First, for a given value of the cut-o¤ ẽ, the shift directly increases the number of imprudent banks in the economy by lowering the stake of "marginal" intermediaries (i.e., those who are initially to the right of, but close to, ẽ); those intermediaries then discontinuously raise their leverage while engaging in imprudent behaviour (see [START_REF] Coval | Economic catastrophe bonds[END_REF] and ( 18)), thereby raising the demand for funds. Second, to the extent that this shift lowers the overall equity base of the intermediary sector, E, all intermediaries, which have a target portfolio size, seek to o¤set the loss in internal funding by external debt, again raising the economywide demand for funding. Both of these e¤ects shift the B d -curve rightwards and exert an upward pressure on the equilibrium borrowing rate, r. Third, this increase in the borrowing rate has a disciplining e¤ect on the intermediary sector by shifting the cut-o¤ equity level ẽ leftwards. Hence, while the e¤ect of the equity squeeze on the borrowing rate is not ambiguous, that on the share of imprudent intermediaries is. However, if the supply of funds is su¢ ciently elastic, the adjustment of the borrowing rate after the shock and its disciplining e¤ect will be limited, causing g (r; ) to rise. This situation is depicted in Figure 3(b). The initial distributional shift causes the g (r; ) curve to shift leftwards. The direct impact of higher risk (holding r …xed) is to lower the expected return on loans, (r; ), which in the (B; r) plane manifests itself as an exogenous reduction in lending (i.e., an inwards shift of the B s curve). Finally, the increase in the demand for funding causes the B d -curve to shift rightwards. If the supply of funds is su¢ ciently elastic (that is, the slope of the B s curve is su¢ ciently low), then the overall e¤ect of the three shifts is to raise the equilibrium value of g (r; )

Information about intermediaries'balance-sheets and endogenous market segmentation

Our analysis above emphasise intermediaries' balance-sheet opacity as a major source of systemic risk. The key mechanism is that the unobservability of balance sheets makes it possible for bad banks to pretend to be good banks, implying that clearing of the market for loanable funds operates in a single market and at a single interest rate. In order to make this channel as transparent as possible, we derived our results under the somewhat extreme assumption that all intermediaries look alike from the point of view of outside lenders. In reality, some public (and private) information about intermediaries'balance-sheet is available that may mitigate the opacity problem. In this Section, we extend our analysis to allow for (noisy) signals about intermediaries's capital, which naturally generates a di¤erentiation of the market for loanable funds -in as much a 'good' and 'bad' intermediaries can to some extent be recognised as such. For the sake of tractability we illustrate this possibility by means of a simple parametric example, but we conjecture that the properties that come out of this exercise hold much more generally.

Distributions, signal structure and parameters. We assume here that the unconditional distribution of intermediaires'inside equity is uniform with support [0; 1], so that

f (e) =
Note that the quality of the signal encompasses two limit cases. When = 1=2, the signal is uninformative and the two conditional distributions coincide with the unconditional one.

When ! 1, in the limit the signals exactly identify every intermediary as belonging to the upper or the lower halves of the distribution. Regarding the other deep parameters, we set p = w = 1=2; = 1, R h 1 = 4: Finally, we focus on the case where u (:) = 0, so that lenders only value terminal consumption and inelastically lend w to the intermediary sector.

The signals identify two categories of intermediaries, and hence two separate markets for loanable funds, each with their own face interest rate. In each market, the problem of an individual intermediary in is similar to that described in Section 3, except that they now take their own face borrowing rate r as given. Under the assumed parameters, an intermediary borrowing in market ; = b; g; behaves prudently if and only if

e ẽ (r ) = 2 3r 4 ; (31)
while the share of imprudent intermediaries in that market is given by: g (r ; ) = F ( ẽ (r )j ; ) ;

where F ( ẽ (r )j ; ) is determined by ( 29)- [START_REF] Taylor | Getting o¤ Track: How Government Actions and Interventions Caused, Prolonged and Worsened the Financial Crisis[END_REF].

We may now compute the demand for funds in each market by integrating intermediaries leverage choices as in [START_REF] Jimborean | Banks'…nancial conditions and the transmission of monetary policy: A FAVAR approach[END_REF]. Under our parameters, the demand for funds in market is

B d; (r ; ) = Z ẽ(r ) 0 b (r ; e) dF (ej ; ) + Z 1 ẽ(r ) b (r ; e) dF (ej s; ) = 2 r 1 1 2 g (r ; ) E ; (32)
where, from equations ( 29)-( 30),

E b = Z 1 0 ef (ej b; ) de = 3 2 4 ; E g = Z 1 0 ef (ej g; ) de = 1 + 2 4 :
Uninformative signals. Let us …rst solve for the equilibrium face interest rate and share of imprudent intermediaries in the uninformative case (i.e., = 1=2), which corresponds to the baseline model analysed in the previous Sections (since in this situation both conditional equity distributions coincide with the unconditional one, and we are back to the single-market case.) Under our parameters speci…cation, the (unique) face interest rate r is determined by the following equilibrium condition:

2 r 1 1 2 g r; 1 2 1 2 = 1 2 ; (33)
where the left hand side is the aggregate demand for loanable funds and the right hand side is the aggregate supply of loanable funds, w = 1=2. Using (25), we may then explicitly solve for the expected return on loans, which is given by: r; 1 2 = 1 :

By equation ( 25) again, the face interest rate must satisfy r (1 F (ẽ (r)) =2) = 1. Since the unconditional CDF is F (e) = e and ẽ (r) is given by ( 31), we get the equilibrium interest rate r = p 8=3 = 1:633; which in turns produces a share of imprudent intermediaries of g r;

1 2 = 2 3 4 r = 0:775 (34)
Informative signals. We …rst note that even when > 1=2 and markets are di¤erentiated, by no-arbitrage and given lenders'risk neutrality, the expected return on loans in the two markets must be identical, i.e., we must have g (r g ; ) = b r b ; ~ r g ; r b ; . Second, by [START_REF] Jimenez | Hazardous times for monetary policy: What do tewenty-three million bank loans say about the e¤ects of monetary policy on credit risk-taking?[END_REF] this common expected rate of return satis…es ~ r g ; r b ; = r 1 1 2 g (r ; ) ; = b; g:

Using ( 32) and ( 35), we may express the demand for loanable funds in market as

B d; (r ; ) = 2 ~ r g ; r b ; E : (36)
The marginal density of the signal is Pr ( = h) = Pr ( = l) = 1=2), so that total equity is 29)-(30) determine the shares of imprudent intermediaries in each market. This implies that these shares are given by:

E b + E g =2 = E
g b r b ; = F ẽ r b b; ẽ(r)>1=2 = 2 1 + 2 (1 ) 2 3r b 4 ; (37)
g g (r g ; ) = F ( ẽ (r g )j g; ) ẽ(r)>1=2 = 1 2 + 2 2 3r g 4 :

Finally, in both markets we have (r ; ) = r (1 g r b ; =2). Using (37)-( 38), the fact that (r ; ) = 1; = g; b and rearranging, we …nd that r b ; r g solve:

1 = r b 1 2 + 3 (1 ) r s 4 ; 1 = r g 1 2 + 3 4 r s :
Then, with r b ; r g known, we may compute the shares of imprudent intermediaries in each market from (37)-( 38), and that in the whole economy (g b r b ; + g g (r g ; ))=2: When = 1=2, we have r b = r g = r = p 8=3 (the uninformative limit studied above.) As rises above = 1=2 -i.e., the signal becomes more and more informative-, r b goes up and r g goes down -since low-versus high-equity intermediaries are more and more identi…ed as such. Figure 4 plots the two face interest rates as a function of ; as well as the shares of imprudent intermediaries in the two markets and the implied proportion of such intermediaries economywide. In this example, the more informative the signal, the higher the share of imprudent intermediaries in the economy. To understand why this is the case, compare the uninformative case ( = 1=2; r = 1:633 and g = 0:775) to the polar opposite ( ! 1=2.) Relative to the former, in the latter i) low-equity intermediaries (i.e., those for whom e < 1=2)

are perfectly identi…ed as excess risk takers but are charged accordingly (i.e., r = 2, g = 1, so that = 2 (1 1=2) = 1:), and ii), high-equity intermediaries enjoy lower face interest rates, which induces some of them to behave imprudently (while they would be prudent when charged the high face rate that prevails in the uninformative case.) 6 Impact of capital requirements

In this section, we explore the e¤ect on systemic risk of imposing capital constraints on intermediaries'behaviour. For tractability, we carry out our analysis under the same parametric speci…cation as in the previous section (i.e., f (:) is uniformly distributed over [0; 1] ; p = w = 1=2; = 1, R h 1 = 4, and the supply of loanable funds is inelastic at w = 1=2.) We consider two simple forms of capital ratios: a 'naïve'capital ratio based exclusively on balance-sheet size, and a risk-based capital ratio that ties the stringency of the ratio to the level of portfolio diversi…cation achieved by the intermediary. As we show, the former may turn out to raise than lower aggregate risk-taking, due to the impact of constrained …rms on the equilibrium interest rate. However, in our parametric example risk-based capital ratio are e¤ective at curbing systemic risk (assuming that they are feasible.)

Asset size-based capital ratios. We …rst consider the impact of a simple (naïve) capital ratio prescribing that intermediaries must hold as initial equity at least some pre-speci…ed fraction 2 (0; 1) of total assets. That is, we impose

e P x i : (39)
The constraint (39) might be binding or not, depending on the equity level of each individual intermediary. In order to keep the analysis concise, we focus on the (realistic) case where is (i) su¢ ciently high for (39) to be binding at least for some intermediaries, given the equity distribution f (e); and (ii) su¢ ciently low for (39) not to be binding for intermediaries that would spontaneously choose the prudent portfolio in the absence of a capital constraint (essentially because those are su¢ ciently capitalised in the …rst place.) In short, we focus on the case where (39) may limit the leverage and investment of some (but not necessarily all) of the intermediaries that would behave imprudently in the absence of the constraint. Under our parametric equity distribution, this amounts to assuming that is positive but small.

We …rst show that the constraint may be e¤ective at limiting the leverage of imprudent intermediaries -i.e., when (39) is binding-, but not at inducing portfolio diversi…cation by those intermediaries. Second, we show that by limiting the leverage of low-equity intermediaries, the capital ratio exerts a downward pressure on the equilibrium face interest rate, which induces some of the originally prudent intermediaries to become imprudent. In con-sequence, a capital ratio purely based on size may ultimately raise, rather than lowers, the share of imprudent intermediaries in the economy.

Impact of the ratio on (low-equity) intermediaries. From (39), the leverage of an intermediary facing a binding capital constraint is b (e) = P

x i e = e On the liability side, an intermediary facing a binding capital constraint chooses a lower level of leverage than it would otherwise. On the asset side, does the constraint alter its portfolio choice? The answer is no. To see this, compare the values of a prudent and an imprudent intermediary with total assets given by P x i = e= (i.e., the intermediary is constrained.) From our analysis in Section 3 and under our parameters speci…cation, the former and the latter are given by, respectively Loanable funds equilibrium. Under our maintained assumption that is small, we have ê (r) < ẽ (r), so that the capital constraint may only be binding for originally imprudent intermediaries. Then, the demand for loanable funds by the intermediary sector is given by:

Ṽ (e) = re + e ( 2
B d (r; ) = Z ê(r
Finally, since intermediaries facing a binding capital constraint choose the imprudent portfolio (x 1 ; x2 ) = (e= ; 0), the share of imprudent intermediaries in the economy is given by: Z ẽ(r)

0 de = ẽ (r ( )) = 2 3 4 r ( ) ;
which is increasing in . To summarise, simple capital ratios based on balance-sheet size are (in our example) ine¤ective at limiting systemic risk. Quite on the contrary, by lowering the equilibrium face interest rate, the capital constraint worsens the risk-taking channel and induces imprudent behaviour by those intermediaries that would otherwise behave prudently.

Risk-based capital ratios One key reason for the ine¤ectiveness of simple capital ratios is that even though the ratio does limit some of the intermediaries'borrowing, it does not curb their risk-taking incentives on the asset side. Suppose now that the regulator (but not an outside lender) is able to observe the riskiness of intermediaries'portfolio and to set the capital ratio accordingly. For example, assume that the capital ratio e=

P x i is 2 (0; 1)
for a prudent intermediary, but ~ > for an imprudent intermediary. where r ( ) is given by (44). The latter expression is decreasing in , implying that a riskbased capital requirement is e¤ective at reducing economywide risk-taking (again, within our parametric example.)

Concluding remarks

In this paper, we have analysed the portfolio and leverage choices of limited-liability intermediaries and their implications for the level of aggregate risk and the way it responds to changes in economic conditions. The novelty of our framework relative to earlier analysis of intermediaries'risk-shifting behaviour is twofold. First, we emphasise the disciplining role of shareholders' inside equity stake and the heterogeneities that it implies for their equilibrium balance sheets -both on the asset and liability sides. Second, and relatedly, we explicitly model changes in economywide risk-shifting along the "extensive margin" -i.e., due to changes in the number of intermediaries endogenously choosing to expose themselves to the risk of default-, in addition to the usual intensive margin -i.e., related to changes in their individual balance-sheet choices.

A important property of the model's equilibrium is that it jointly determines the (common) borrowing rate faced by intermediaries and the level of aggregate risk in the economy, due to the endogenous sorting of intermediaries along the equity dimension. Unsurprisingly, intermediaries with low shareholders'stake are more likely to behave imprudently than those with high inside equity stake. More interestingly, the sorting of intermediaries his itself affected by the interest rate, with falling interest associated with a rising number of imprudent intermediaries and aggregate risk. For this reason, exogenous factors that a¤ect the market for loanable funds (e.g., international capital ‡ows) have a direct impact on the level of risk where r 0 > r. For the two curves to cross, the following conditions must be satis…ed:

B d (r 0 ; ) = pR h E > B s (r 0 ; ; w) = w u 0 1 = 0; B d ( r; ) = r (1 p) g (( r; )) E < B s ( r; ; w) = w u 0 1 ( r (1 (1 p) g ( r; ))) :

Solving both conditions for w, we obtain:

w > u 0 1 pR h E= ; w > r (1 p) g (( r; )) E + u 0 1 ( r (1 (1 p) g ( r; ))) ;

which is the second condition of Lemma 2.

Proof of Proposition 1. First, substitute (25) into ( 24) so as to write [START_REF] Obstfeld | Global imbalances and the …nancial crisis: products of common causes[END_REF] as follows:

B s ( (r; ) ; w) + (r; ) = p R h From (25), we have (r; ) = r (1 p) g (r; ) > 0 (i.e., an increase in raises the return on loans by inducing more intermediaries to adopt a prudent behaviour), while E = @ @ R emax 0 ef (e; ) di 0. Hence, the numerator in the ratio is positive. Since the denominator is also positive, we have dr=d < 0, i.e., a downward shift in the economywide equity base raises the equilibrium interest rate (as intermediaries seek to substitute equity for debt).

The equilibrium share of imprudent intermediaries is g (r; ) ; and from above we have dg (r; ) d = g r (r; ) dr d + g (r; ) :

The direct e¤ect of on g (r; ), as measured by g (r; ) and holding r constant, is negative -i.e., a leftward shift in the distribution of equity raises the number of imprudent banks. The ambiguity in the overall response of g (r; ) comes from the reaction of the equilibrium interest rate. We have just seen that dr=d < 0, and we know from ( 22) that g r (r; ), hence g r (r; ) dr=d > 0 -i.e., a leftward shift in the distribution of equity raises the aggregate demand for funds, which raises the interest rate and ultimately disciplines bank behaviour.

The second part of the proposition can be established by continuity with the limiting case, where B s ( (r; ) ; w) is in…nitely elastic to changes in (r; ). In this case, (r; ) is constant, so that total di¤erencing it and using [START_REF] Jimenez | Hazardous times for monetary policy: What do tewenty-three million bank loans say about the e¤ects of monetary policy on credit risk-taking?[END_REF], we obtain: Since g (r; ) < 0 while (1 p) 1 > 1 g (r; ), the numerator is necessarily negative.

By [START_REF] Landier | Going for broke: New Century Financial Corporation[END_REF], the denominator is positive, so dg (r; ) =d < 0. By continuity, if the supply of funds is su¢ ciently elastic then a leftward shift in the distribution of equity raises g (r; ) :

B. Isoelastic cost function

Our results in Sections 2 to 4 have been derived under the simplifying assumption that the cost function is quadratic. We now show that most of our results (and, in particular, those pertaining to the impact of aggregate shocks) do hold under the more general isoelastic case, where the cost function takes the form:

c ( P x i ) = ( P x i ) 1+ (1 + ) ; ; > 0; (48)
where indexes the elasticity of the cost with respect to the size of the balance sheet and a level parameter.

Intermediaries'behaviour and value. From (9), under (48) the optimal portfolio of a prudent intermediary with inside equity e is given by:

P x i = pR h 1 r 1= ; b = pR h 1 r 1= e: (49)
Given this balance-sheet structure, the value of a prudent intermediary is

V (e) = re + ( P x i ) pR h

:

Similarly, using [START_REF] Dubecq | Fuzzy capital requirement, risk shifting, and the risk-taking channel of monetary policy[END_REF] we …nd the optimal portfolio of an imprudent intermediary under (48) to be: The threshold ẽ (r) is now a complicated function of r, but we show that it is strictly decreasing in r for all > 0, as in the baseline model with quadratic investment cost. From the de…nition of ẽ (r) in the latter inequality, we …nd that ẽ0 (r) < 0 if and only if

x 1 = p R h
R h 1 r R h 1 + r < R h 1 pr R h 1 + pr : (51)
Let us now de…ne the function H : 0; R h 1 ! R + as follows:

H (x) = R h 1 x R h 1 + x :
Since pr < r, a su¢ cient condition for (51) to hold for all > 0 is that H 0 (x) < 0, or equivalently that h 0 (x) < 0; where h (x) = ln H (x) : We …nd

h 0 (x) = 1 R h 1 x + R h 1 + x = x (1 + ) R h 1 x R h 1 + x < 0;
implying that ẽ0 (r) < 0 for all > 0.

Figure 1 :

 1 Figure 1: Ranked forecast errors of an AR(2) of the idiosyncratic component of the capital/asset ratio for 90 French banks. These are obtained in two stages. First, the capital-asset ratio of each of the 90 banks is regressed on their …rst three time-series common static factors

1 r

 1 Figure 1(a) below. Equilibrium. To complete the characterisation of the …rst-best outcome, we must compute the equilibrium interest rate r that results from the equality of the aggregate demand and supply for loanable funds. Since P xi = pR h , it follows that the leverage of an intermediary with inside equity e and facing the interest rate r is given by b (r; e) = pR h 1 r e. The aggregate demand for funds is obtained by summing up the demands for debt by all intermediaries, i.e., Bd (r; ) = Z emax 0 b (r; e) dF (e; ) = pR h 1 r E:

Lemma 1 . 1 e

 11 Assume that i. pR h 1 > e max and ii. w > e max E + u 0 1 pR h max = : Then, the …rst-best equilibrium is unique and such that b (r; e) > 0 for all e 2 [0; e max ].

Figure 1 :

 1 Figure 1: Intermediaries'optimal portfolios (a) and value (b).

r) 0 b

 0 (r; e) dF (e; ) + Z emax ẽ(r) b (r; e) dF (e; ) :

Lemma 2 .= pR h 1 e

 21 Assume that i. pR h 1 > e max and ii.w > max u 0 1 pR h E= ; r (1 p) g (( r; )) E + u 0 1 ( r (1 (1 p) g ( r; ))) ;where r max = : Then, the equilibrium with risk shifting is unique and such that b (e) and b (e) are positive fall all e 2 [0; e max ].

2 .

 2 Share of imprudent intermediaries (a) and loanable funds equilibrium (b).

Figure 3 .

 3 Figure 3. Impact of a lending boom (a) and a equity squeeze (b) on the interest rate, r, and the share of imprudent intermediaries, g (r; ).

  1, F (e) = e and E = 1=2. Outside lenders receive the following symmetric binary signal about every intermediary: if e 1=2, then = g ('good') with probability 2 [1=2; 1) and = b ('bad') w.p. 1 . Symmetrically, if e < 1=2, then = b with probability and = g w.p. 1 . Under these assumptions, the marginal density of the signals is simply Pr ( = h) = Pr ( = l) = 1=2. From Bayes' rule, the observation of the signal produces the following two conditional distributions and cumulative density functions (both of which are indexed by the signal quality ): f (ej g;

1 =

 1 while the total demand for loanable funds is (B d; (r ; ) + B d; (r ; ))=2. The latter must sum up to the aggregate supply of loanable funds w = 1=2; so from (36) we get ~ r g ; r b ; = In can be shown (by contradiction) that for all 2 [1=2; 1) and = b; g, we always have ẽ (r ) > 1=2, so the upper halves of the conditional cumulative distribution functions in (

Figure 4 .

 4 Figure 4. Face interest rates and shares of imprudent intermediaries in market = b; g; as a function of the precision of the signal, :

  e) > Ṽ (e), an intermediary facing a binding constraint always chooses the imprudent portfolio (x 1 ; x2 ) = (e= ; 0) :

; ) 0 b

 0 (e) dF (e) + Z ẽ(r) ê(r; ) b (r; e) dF (e) + Z 1 ẽ(r) b (r; e) dF (e) : Using (40)-(41), the fact that dF (e) = de and rearranging, we may rewrite the latter expression as: rate that clears the market equates B d (r; ) with the aggregate supply of funds, w = 1=2. When = 0 (our baseline scenario), we again have r = p 8=3 (the solution to 3=2 3r 2 =8 = 1=2) and g (r; ) = 2 3r=4 = 0:775 (see equation (34) above.) As rises and constrains the leverage choices of more and more intermediaries, the aggregate demand curve B d (r; ) shifts down. Given the vertical loan supply curve B s = 1=2, the equilibrium face interest rate r must go down. Solving the equation B d (r; ) = 1=2 for r, we indeed obtain the decreasing interest rate function r

Proof of Proposition 2 .

 2 di¤erentiating (45) whilst holding constant, we …nd:dr dw = B s w ( (r; ) ; w) + B s ( (r; ) ; w) r (r; ) < 0;(46)which in turn implies, by the properties of (r; ) and g (r; ): Total di¤erentiating (45) again but now holding w rather than constant, we …nd dr d = + B s ( (r; ) ; w) (r; ) + E + B s ( (r; ) ; w) r (r; ) :

r

  (r; ) dr + (r; ) d = [1 (1 p) g (r; ) (1 p) rg r (r; )] dr r (1 p) g (r; ) d = 0; or dr d = rg (r; ) (1 p) 1 g (r; ) rg r (r; ) : Finally, substituting this latter expression into (47) gives: dg (r; ) d = g (r; ) (1 p) 1 g (r; ) (1 p) 1 g (r; ) rg r (r; ) :

1 r 1 :1=( 1

 111 1= ; x 2 = 0; b = p R h value V (e) = p re + x 1 R hEquity threshold. An intermediary with inside equity e chooses the prudent investment strategy if and only if V (e) V (e), that is, if and only if e

  A su¢ ciently large value of ~ (relative to ) acts as a deterrent and induces prudent behaviour by constrained intermediaries, so that rather than ~ e¤ectively applies. Under this regulatory arrangement, the aggregate demand for loanable funds is as in (43), and consequently the equilibrium face interest rate as in (44). However, since intermediaries

	facing a binding constraint now behave prudently, the share of imprudent intermediaries in
	the economy is now
	Z ẽ(r) ê(r; )	de = ẽ (r) ê (r; ) = 2	3r ( ) 4		2	1 2	r ( ) ;
												Incorporating this
	risk-based capital ratios into the values of being prudent or imprudent in (42), we …nd that
	a constrained intermediary prefers to be prudent if and only if
	re +	e pR h 1	r	c	e > p re +	e ~	R h 1	r		c	e ~	:

See Jimenez et al. (2010) for direct evidence that falling short-term interest rates favour risk-taking by banks that are at the lower end of the capital distribution.

According to[START_REF] Landier | Going for broke: New Century Financial Corporation[END_REF], rising interest rate were conducive to more risk shifting, while many authors (e.g.[START_REF] Diamond | The credit crisis: conjectures about causes and remedies[END_REF] suggests that low interest rates favor leverage and excess risk-taking.However, two e¤ects of interest rates on intermediaries'risk taking must be distinguished. First, holding the equity stake of bankowners …xed, low interest rates may indeed favor high leverage and risk taking. However,

This paper focuses on agency problems between the intermediary's owner-manager and its creditors, and hence abstracts from incorporating inobservability and con ‡ict of interest between the owners and the managers. See[START_REF] Acharya | Caught between Scylla and Charybdis? Regulating bank leverage when there is rent seeking and risk shifting[END_REF] for a model of risk-shifting that explicitely incorporates both dimensions.

We assume for simplicity that intermediaries are completely identical from the point of view of the lender. Our result carry over in a set-up with partially segmented market involving di¤erent groups of intermediaries, with the members of each group facing the same interest rate. What matters for our results is the presence of an unbserved residual heterogeneity in interminedaries'equity stake.

As argued by[START_REF] Obstfeld | Global imbalances and the …nancial crisis: products of common causes[END_REF], these two views are likely more complementary than substitutes.

rearranging, we …nd the ex ante gross return on loans to be:

(r; ) = pr + (1 p) (1 g (r; )) r

= r (1 (1 p) g (r; )) :

Note that this ex ante return is strictly increasing in the face interest rate, i.e., r (r; ) = 1 (1 p) g (r; ) (1 p) rg r (r; ) > 0:

The increasingness of (r; ) with respect to r occurs for two reasons. First, a higher face interest rate increases intermediaries'repayment if they do not default (the 1 (1 p) g (r; ) > 0 part of ( 26)). Second, a higher face interest rate favours prudent rather than risky behaviour by raising the threshold ẽ, and hence by lowering the probability of default on a loan unit (the (1 p) g r (r; ) > 0 part). It follows that for ( ; w) given the loan supply function is a nondecreasing, continuous function of r, which we may express as:

Let us brie ‡y summarise the properties of the aggregate supply curve, before we analyse the equilibrium in the market for loanable funds. First, B s (r; ; w) is strictly increasing in r; holding ( ; w) constant; this follows from (5), the strictly concavity of u(:), and the strict monotonicity of w.r.t. r (see [START_REF] Landier | Going for broke: New Century Financial Corporation[END_REF]). Second, from (5) it is strictly increasing in w, holding r and constant. Third, it is increasing in , holding r and w constant. The reason for this is that a higher overall level of equity in the economy raises the number of prudent intermediaries (i.e., g (r; ) < 0), and hence the expected return on loans (see [START_REF] Jimenez | Hazardous times for monetary policy: What do tewenty-three million bank loans say about the e¤ects of monetary policy on credit risk-taking?[END_REF]).

In the (B; r) plane, the B s (r; ; w) curve lies to the left of its …rst-best analogue, Bs (r; ; w) :

This is because in the equilibrium with risk shifting lenders expect a nonnegative fraction of intermediaries to go bankrupt if state s 2 occurs. Hence, any given value of the face interest rate r is associated with a lower expected return in the risk-shifting equilibrium than in the …rst-best -and hence with a lower supply of loanable funds (see Figure 2(b)).

Market clearing

In equilibrium, the total demand for funds by the intermediary sector must equal the total supply of funds provided by outside lenders. In other words, the face interest rate that clears generated by the …nancial sector. Similarly, exogenous changes in the distribution of intermediaries' capital a¤ect the equilibrium interest rate, aggregate risk, and the return that ultimate lenders can expect from entrusting the …nancial sector with their funds.

While we have focused on two speci…c …nancial fragility channels (the risk-taking channel of low interest rates and the gambling-for-resurrection channel of falling equity), our model could be elaborated further to analyse the impact on intermediaries risk taking of other changes in macroeconomic conditions. For example, it is frequently argued that booms are times of low risk aversion, thereby a¤ecting investors'portfolio choices (e.g., Bernanke and Kuttner, 2005; Campbell and Cochrane, 1999). Analysing the impact of changes in risk aversion for intermediaries' risk taking would require departing from the risk neutral assumption, with nontrivial implications for both intermediaries' choices and the implied aggregate welfare.

Appendix A. Proofs

Proof of Lemma 1. From ( 15), the best capitalised intermediary (i.e., that for whom e = e max ) has positive leverage if and only if the equilibrium interest rate is not to low, i.e., r < r = pR h 1 e max = ; were inequality i. not to hold, then no positive equilibrium interest rate would satisfy this property. Moreover, Bs (r; w) > 0 provided that r > r = u 0 (w) :

Hence, the relevant domain of r is (r; r), and we need r < r, that is, u 0 (w) < pR h 1 e max = :

For the two curves to cross, we need Bd (r; w) = pR h 1 u 0 (w) E > Bs (r; w) = 0 and Bd ( r; w) = e max E < Bs ( r; w) = w u 0 1 pR h 1 e max = : Assumption ii. ensures that both are satis…ed (and hence that r < r).

Proof of Lemma 2. As before we need r < r = pR h 1 e max = for all intermediaries (including the best capitalised amongst the prudent ones) to be leveraged, so inequality i. is required for this to be possible at positive interest rate levels. Lending is positive whenever B s (r; ; w) = w u 0 1 ( ) > 0, that is, using [START_REF] Jimenez | Hazardous times for monetary policy: What do tewenty-three million bank loans say about the e¤ects of monetary policy on credit risk-taking?[END_REF], whenever = r (1 (1 p) g (r; )) > = u 0 (w) :

Letting r 0 solve r 0 (1 (1 p) g (r 0 ; )) = u 0 (w), the relevant domain of r is now (r 0 ; r), Loanable funds equilibrium. Integrating the demand for funds by all intermediaries as in [START_REF] Jimborean | Banks'…nancial conditions and the transmission of monetary policy: A FAVAR approach[END_REF] but with b and b given by ( 49


Sabrina Texier

M David

Sophie Cliquet

Maxime Davy

Cultural age impacts Plectosporium alismatis propagule yields and subsequent desiccation and UV-radiation tolerance

Keywords: Plectosporium alismatis, mycoherbicide, age, UV tolerance, conidia, chlamydospores

HAL is

SUMMARY

The effect of cultural age was studied on yields, desiccation tolerance and resistance to ultraviolet radiation of Plectosporium alismatis, a potential mycoherbistat of aquatic weeds in Australian rice fields.

P. alismatis was grown in a liquid basal medium supplemented with malt extract and sodium nitrate and harvested after 7, 14

or 21 days incubation. Although chlamydospore yields harvested from 14-day-old liquid cultures were significantly higher (29.2 x 10 5 chlamydospores mL -1 ) than chlamydospore yields harvested from 7-day-old liquid cultures (1.07 x 10 5 chlamydospores mL -1 ) or from chlamydospore yields harvested from 21-day-old liquid cultures, the germination of freshlyharvested chlamydospores from 7-day-old cultures (72.7%) was significantly reduced when propagules were grown for 14 days (55.3%). When exposed to UV-radiation, conidia and chlamydospores harvested from 14-day-old cultures germinated at a lower rate (<20%) than conidia and chlamydospores harvested from 7-day-old cultures (>40%). When conidia and chlamydospores were dried and subsequently exposed to UV, less than 30% of propagules harvested from 7-day-old germinated, whereas less than 10% of propagules harvested from 14-day-old cultures germinated. A 3 way analysis of variance including cultural age, UV exposure and type of propagules confirmed that the cultural age had more impact on the germination of [START_REF] Cother | Pathogenicity of Rhynchosporium alimatis and its potential as a mycoherbicide on several weed species in the Alismataceae[END_REF] as an alternative to the utilization of Londax ® , a bensulfuron herbicide which is likely to have contributed to the development of herbicide resistant weeds (Graham, Prat, Pratley, Slater, & Baines, 1996).

For weed control, P. alismatis will be applied inundatively and its effect is likely to be static rather than cidal [START_REF] Crump | Clarifying the nomenclature in microbial weed control[END_REF]. Consequently, the appropriate term for P. alismatis is mycoherbistat rather than mycoherbicide. Our current studies aim at growing P. alismatis in submerged cultures with the goal of producing propagules with the most fitted potential for the development of a mycoherbistat.

In previous work, we showed that P. alismatis was able to produce high yields of conidia in a casamino-acids, glucose based medium [START_REF] Cliquet | Impact of nutritional conditions on yields, germination rate and shelf-life of Plectosporium alismatis conidia and chlamydospores as potential candidates for the development of a mycoherbicide of weeds in rice crops[END_REF].

We also showed that P. alismatis was able to produce chlamydospores in a malt extract, sodium nitrate medium after 7-day incubation [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF]. Based on this medium, nutritional studies led to the development of a liquid medium containing appropriate malt extract and sodium nitrate concentrations for optimal chlamydospore yields.

Furthermore, recent studies demonstrated that 10%

chlamydospores produced in this malt-extract, sodium nitrate based medium were able to germinate after 4-month storage at 25°C, while conidia produced under the same culture conditions showed poor survival (0% after 2-month storage [START_REF] Cliquet | Impact of nutritional conditions on yields, germination rate and shelf-life of Plectosporium alismatis conidia and chlamydospores as potential candidates for the development of a mycoherbicide of weeds in rice crops[END_REF]. Propagules are required to survive drying in order to maintain the viability of a dry preparation (Jackson & Schisler, 2002). Chlamydospores are therefore promising candidates for the development of a mycoherbistat. However, under our cultural conditions, P. alismatis chlamydospore yields obtained in submerged cultures were significantly lower than conidia yields. Since we had already defined the optimal nutritional conditions for high chlamydospore yields, [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF]), our work focused on non-nutritional conditions that may further increase chlamydospore yields. Because long incubation periods (2-3 weeks) are generally required to reach high levels of chlamydospores (Hebbar, Lewis, Poch, and Lumsden, 1996), we investigated how cultural age may impact chlamydospore yields in the present study.

In general, modifications of cultural conditions are known to have significant consequences on fungal morphology and spore qualities. Numerous studies have been conducted on the impact of nutritional conditions on fungal desiccation tolerance (Jackson, Cliquet & Iten, 2003;Jackson & Schisler, 1992).

However, literature is scarce on the impact of P. alismatis cultural age on fungal attributes, except for a few articles (Hall, Peterkin, Ali & Lopez, 1994 ;[START_REF] Bardin | Inoculum production and long-term conservation methods for cucurbits and tomato powdery mildews[END_REF] and, to our knowledge, no study has been reported on the impact of cultural age on fungal desiccation tolerance.

Ultra Violet (UV) radiation is an additional environmental stressfull factor for fungal species [START_REF] Moody | Variation in the responses of litter and phylloplane fungi to UV-B radiation (290-315 nm)[END_REF]. While UV-C radiation (200-280 nm) does not reach the ground level due to oxygen and ozone [START_REF] Madronich | Changes in ultraviolet radiation reaching the earth's surface[END_REF] 2004).

Inoculum production

A frozen suspension from stock culture at -80°C was inoculated onto a PDA plate, incubated for 2-3 weeks at 25°C until profuse sporulation occurred and renewed every month.

Sub-cultures on PDA were produced from the sporulated plate (one serial transfer).

Conidia for use as aqueous conidial suspension were produced by inoculating PDA sub-culture onto PDA and growing these inoculated PDA plates at 25°C for 4 days or 7 days.

Four-day-old plates were gently washed with 3 mL distilled water and the conidial suspension inoculated into shake flasks. Conidia harvested from 4-day old and 7-day-old plates were tested for UV tolerance.

Medium composition and growth conditions

The basal mineral composition for P. alismatis growth was derived from a Czapex-Dox composition which contained: K 2 HPO 4 , 1.0 g; MgSO 4 .7H 2 0, 1.0 g; KCl, 0.5 g; Fe 2 SO 4 .7H 2 O, 0.018 g; deionized water: 1 L. For chlamydospore production, malt extract (Difco, 8.8 gL -1 ) and sodium nitrate (Sigma Chemical, St. Louis, MO, USA, 5.74 g L -1 ) were used [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF].

Flasks containing the malt extract sodium nitrate medium (100 mL medium in 250 mL baffled flasks) were inoculated with conidia (4 x 10 3 conidia mL -1 ). Cultures were placed at 150 rpm, 25°C on a rotary shaker incubator (Infors HT Bottmingen, Switzerland).

The pH of cultures was maintained at 7 ± 0.5 during growth by addition of 1 N NaOH or 1 N HCl.

Four replicate flasks were harvested after 7, 14 or 21 days incubation, respectively.

Determination of propagule yields and dry weights

Cultures were vacuum-filtered on cellulose filter papers (110mm diameter, Whatman plc, Brentford, UK) to remove the spent medium.

Filtered cultures were rinsed with 50 mL deionized water and allowed to dry on the bench top for 12 h until constant weight. Dry mats were weighed and suspended in 21.5 ml distilled water. The suspension was fragmented in a Potter homogeniser (Fisher Scientific Bioblock, Illkirch, France).

Propagule counts were performed using a haemocytometer.

Preparation of propagule suspension for UV tolerance and desiccation tolerance studies

Conidial suspensions from solid media Conidia harvested from 4-day-old or 7-day-old PDA plates were gently washed with sterile distilled water. Aqueous

suspensions were filtered through 2 cheese-cloth layers to remove any mycelium fragment, placed on a cellulose filter and washed with sterile distilled water in order to rinse and concentrate conidial suspensions to 5 ± 2 x 10 6 conidia mL -1 .

Conidial and chlamydospore suspensions from liquid media Liquid fungal cultures from duplicate flasks were poured onto 2 cheese-cloth layers in order to retain most of the mycelium.

Microscopic observation showed that no mycelium remained in suspensions and that conidia were not aggregated.

Conidial suspensions were placed on a cellulose filter, rinsed and concentrated with sterile distilled water to 5 ± 2 x 10 6 conidia mL -1 . Mycelium remaining on cheese-cloth was rinsed to remove most conidia, and homogenised to release chlamydospores. The homogenised suspension was filtered on cellulose filter to rinse and concentrate the aqueous suspension to 5 ± 2 x 10 6 chlamydospores mL -1 .

Chlamydospore and Conidial Germination

Drops of the propagule suspension were placed on four 2-cm square pieces of cellophane on the surface of water agar plates (granulated agar, Difco, 20 g L -1 ). Plates were incubated at 25°C as previously described [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF], although incubation time was reduced from 12h to 8 h to prevent any possible microcycle conidiation that may occur as previously mentioned [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF]Ghajar et al, 2006a). Germination was evaluated microscopically after staining cellophane pieces with lactophenol cotton blue.

Drying experiments

Conidia and chlamydospore suspensions harvested from liquid media (5 x 10 6 propagules mL -1 ) were filtered on autoclaved cellulose filters and placed at room temperature until constant weights (12h). Suspensions were prepared by gently washing dry filters with sterile distilled water and spore germination evaluated using the method described above.

Exposure of conidia and chlamydospores to UV radiation

Irradiation experiments were conducted in a dark cabinet. A 312 nm-UV-lamp (Vilber Loumat, Marne la Vallée, France) supplied 95% radiation ranged in wavelengths from 260 to 380 nm with a peak at 312 nm. The UV-A and UV-B irradiances were measured using a UVX radiometer (UVP, LLC, Upland, CA, USA) and a UV 31 and a UV 36 sensors. The total irradiance was 2.30 W m-2 at a distance of 30 cm of the lamp. No germination was observed after direct exposure to UV radiation (Fig. 1).

Plates (Greiner Bio-one 34/15) were covered with their plastic lid and exposed to UV radiation. To determine the transmission of UV radiation, pieces of plastic lids were examined in a spectrophotometer (Hitachi 2000, France). The plastic lid formed a useful cut-off filter, transmitting less than 1% of wavelengths <280 nm, corresponding to UV-C radiation.

Transmission of UV-B was maintained (280-315 nm) as well as UV-A transmission (315-400 nm). The total irradiance was 1.64 W m-2 , 80% of which produced by UV-B in the range 280-340 nm.

Germination increased from 30 to 70%, depending on exposure time (Fig. 1). Accordingly, wavelengths < 315 nm have been reported to reduce conidium germination, with the most marked effect with wavelengths < 290nm (Ghajar et al., 2006a).

A 25 min exposure time to UV radiation (UV-B dose = 2 kJm -2 ;

UV-A dose = 0.4 kJm -2 ) was selected for further experiments on the impact of UV on solid-or liquid-cultures. This UV-B dose is in the lower range of UV-B levels (0 to 26 kJm -2 ) recorded after exposure to full-spectrum sunlight at different times of the day (Richmond, NSW, Australia, 18/03/02, Ghajar et al.

2006a)

Statistical analysis

The propagule yield experiment was repeated once.

RESULTS AND DISCUSSION

Propagule yields harvested from liquid cultures

Maximal chlamydospore and conidia yields were reached at 14 days incubation (29.2 x 10 5 chlamydospores mL -1 and 7.8 x 10 6 conidia mL -1 (Table 1). Similarly, other studies on Fusarium oxysporum growth kinetics showed that chlamydospores were formed on day 5 and that chlamydospore yields reached a peak 10-14 days after inoculation (Hebbar et al., 1996;[START_REF] Elzein | Influence of agricultural byproducts in liquid culture on chlamydospore production by the potential mycoherbicide Fusarium oxysporum Foxy 2[END_REF]. In a previous time course experiment (0-10 d incubation) in which propagule yields were recorded each day [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF], maximum chlamydospore yields were obtained after 72 h incubation and remained constant until day The germination of freshly-harvested, or dry propagules harvested at 7 days or 14 days incubation and exposed to UV radiation was examined.

Germination of propagules freshly-harvested from liquid cultures

In our experimental conditions, 85 % freshly-harvested conidia germinated after 8 h incubation regardless of cultural age (Figure 2A). In a previous work, we reported that 4 days of growth were required for conidia to reach 80% germination, corresponding to the time needed to produce new conidia in large numbers, this germination remaining constant until 10 days of growth [START_REF] Cliquet | Conidial and chlamydospore production of Rhynchosporium alismatis in submerged culture[END_REF]. Since 80% of P. alismatis conidia germinate after 14 days of incubation, we may conclude that P. alismatis germination, under our growth conditions, is not affected by cultural age in a range 5-14 days. According to Hall et al. (1994), the impact of cultural age on conidial germination appears to be strain-dependant, some fungal spores from 2-3 old cultures germinating more rapidly (40-60% germination) than those taken from 14-old cultures, (10% germination) probably as a consequence of a first-formed conidial effect. Additionnal experiments with various P. alismatis isolates are needed to specify whether differences in germination of conidia harvested from various incubation periods are strain-dependant.

The germination of chlamydospores harvested from 14-day-old cultures was significantly reduced (55.3%) compared to the germination of chlamydospores harvested from 7-day-old cultures (72.7%). Similarly, 71% of 8-day-old chlamydospores, 60% of 3-month-old chlamydospores, and 34% of 6-month-old chlamydospores germinated [START_REF] Lanoiselet | Production, germination and infectivity of chlamydospores of Rhynchosporium alismatis[END_REF], indicating how time of harvest may impact fungal germination. In our experimental conditions, most chlamydospores are produced in intercalary chains inside hyphal aggregates in which substrate and oxygen diffusion is likely limiting, as reported in a large number of fermentation studies on filamentous fungi (Gibbs, Seviour & Schmid, 2000).

Stress due to lack of substrate and oxygen may increase with incubation length and substrate depletion, thus affecting the physiological state of chlamydospores.

Germination of UV-exposed propagules freshly harvested from liquid cultures

The germination of conidia and chlamydospores harvested from 7-day-old cultures following UV exposure decreased, (40% and 50% germination, respectively) whereas only 10 to 20% of UV exposed propagules harvested from 14-day-old cultures germinated (Fig. 2B).

The adverse impact of UV radiation on Plectosporium alismatis germination has been reported (Ghajar et al., 2006a). As germination proceeds, some of the most basic pathways involved are those concerned with the synthesis of DNA, RNA, and protein (Garraway & Evans, 1984). A major effect of UV-B radiation on fungi is direct damage to DNA, with possible production of reactive oxygen, especially hydrogen peroxide [START_REF] Friedberg | Production of chlamydospores of the nematode-trapping Duddingtonia flagrans in shake flask culture[END_REF], generating probably a delay in protein synthesis and oxidative stress (Rangel et al., 2006).

A 3-way analysis of variance ( , 1992).

Germination of UV-exposed conidia freshly harvested from solid cultures (PDA)

While conidia harvested at 4 days or 7 days had similar germination rates (respectively 93.5 ± 2.2 and 95.8 ± 0.8 % germination), germination of conidia harvested from 7-day-old cultures decreased dramatically after UV exposure (3.6 ± 0.6) compared to germination of conidia harvested from 4-day-old cultures (27.6 ± 2)(data not shown). These results confirm that the cultural age has a significant impact on UV tolerance and should be further considered for enhancement of UV tolerance together with adjuvant addition as proposed (Ghajar et al., 2006b).

Germination of propagules harvested from liquid cultures and dried

Conidia and chamydospores produced in our liquid culture, malt-extract sodium nitrate medium, and harvested either at 7 day or at 14 days incubation were dried. As previously recorded for freshly-harvested propagules, the germination of dry propagules after U-V exposure decreased as cultural age increased (Fig. 3). The age of culture had a significant impact (Table 2) on dry propagule germination rate (P = 4.10 -

4

), regardless of the type of dry propagules considered (P = 0.48).

Generally, studies on propagule survival examine the impact of environmental factors on a one factor-at-a-time basis.

  Duplicate 200-µL droplets of each propagule suspension were placed in a Petri dish (89 mm diameter) (Greiner Bio-one 34/15 with vents, Courtaboeuf, France) at pre-specified position at a distance of 30 cm under the UV-lamp, and droplets were simultaneously placed in a dark cabinet with no lamp (controls). In order to prevent dehydration, distilled water was placed underneath the plates. Temperature and relative humidity in the dark cabinets with or without the UV-lamp were recorded (Testo probe 175, Forbach, France) and remained constant (21°C, RH = 75%) during the experiment.In order to evaluate the impact of the UV-lamp on conidial germination, suspensions obtained from 4-day-old PDA cultures were exposed for 0, 15, 20 or 25 min.At the end of UV exposure, 100-µL were taken from the 200

10.

  Increasing culture duration to 2-3 weeks is therefore required for time course chlamydospore yields determination, as generally reported in studies on chlamydospore production(Gardner, Wiebe, Gillepsie, & Trinci, 2000); Shabana,Muller- Strover & Sauerborn, 2003).

  Figure 1.

  Figure 2.

Table 2

 2

	) clearly indicates that

Table 2 .

 2 The effect of type of propagules, cultural age, UV exposure and factor interactions on the germination rate of

	As

Table 1 .

 1 The impact of culture age on conidial and chlamydospore yields and germination by Plectosporium alismatis a P. alismatis was grown in 8.8 g L -1 malt extract and 5.74 g L -1 sodium nitrate in submerged culture at 150 rpm and 25°C. b Germination of fresh propagules was evaluated after 8 h incubation on cellophane squares placed on water agar at 25°C c Production of chlamydospores during growth was expressed as total chlamydospore counts. A count represents either a chain of chlamydospores, a single-celled chlamydospore or a double-celled chlamydospore. d Pairs of treatments with a letter into brackets in common do not differ significantly (P<0.05) based on a pair-wise LSD test (c) into brackets = standard error; (d) letters followed by…LSD 5% (homogenous variances checked) 07/09/2012 Texier et al.

Table 2 .

 2 The effect of type of propagules, cultural age, UV exposure and factor interactions on the germination rate of Plectosporium alismatis expressed as a 3-way analysis of variance 07/09/2012 Texier et al.

		Freshly-harvested propagules	Dried propagules
	Main effect and interaction
		F-ratio	Probability	F-ratio	Probability
	A: type of propagules	0.44	0.51	0.49	0.48
	(conidia or chlamydospores)
	B: UV exposure	172.8	0.00001	58.8	0.00001
	(transmission of UV-B and UV-A)
	C: Age of culture (7 or 14 days)	38.6	0.00001	14.4	0.0004
	AB: Propagules x UV	18.3	0.0001	0.89	0.34
	AC: Propagules x age	2.2	0.14	0.04	0.84
	BC: UV x Age	10.1	0.002	0.17	0.68
	ABC:Propagule x UV	0.51	0.47	9.2	0.0036
	x Age

ACKNOWLEDGMENTS

We are thankful to E. Cother who sent a freeze-dried strain of P. alismatis. This work was supported by French Minister of Research.