Christophe Ley

Yvik Swan

Thomas Verdebout

Optimal tests for the two-sample spherical location problem

Keywords: Directional statistics, local asymptotic normality, pseudo-FvML tests, rank-based inference, two-sample spherical location problem 62H11, 62H15, 62G10

We tackle the classical two-sample spherical location problem for directional data by having recourse to the Le Cam methodology, habitually used in classical "linear" multivariate analysis. More precisely we construct locally and asymptotically optimal (in the maximin sense) parametric tests, which we then turn into semi-parametric ones in two distinct ways. First, by using a studentization argument; this leads to so-called pseudo-FvML tests. Second, by resorting to the invariance principle; this leads to efficient rank-based tests. Within each construction, the semi-parametric tests inherit optimality under a given distribution (the FvML in the first case, any rotationally symmetric one in the second) from their parametric counterparts and also improve on the latter by being valid under the whole class of rotationally symmetric distributions. Asymptotic relative efficiencies are calculated and the finite-sample behavior of the proposed tests is investigated by means of a Monte Carlo simulation.

Introduction

Spherical or directional data naturally arise in a broad range of earth sciences such as geology and astrophysics (see, e.g., Watson 1983 or Mardia and[START_REF] Mardia | Directional Statistics[END_REF], as well as in studies of animal behavior (see [START_REF] Fisher | Statistical Analysis of Spherical Data[END_REF] or even in neuroscience (see [START_REF] Leong | Methods for spherical data analysis and visualization[END_REF]. Although this field of research is as old as mathematical statistics themselves and raises the same questions as the more classical "linear" statistics, its methodical and systematic study only started in the 1950s under the impetus of [START_REF] Fisher | Dispersion on a sphere[END_REF]'s pioneering work (see [START_REF] Mardia | Directional Statistics[END_REF]. It is now common practice to view directional data as realizations of random vectors X taking values on the surface of the unit hypersphere S k-1 := {v ∈ R k : v ′ v = 1}, k ≥ 2, the distribution of X depending only on its "angular distance" from a fixed point θ θ θ ∈ S k-1 (thus such spherical distributions belong to the category of statistical group models and enjoy the nice properties of that class of distributions; see Chang 2004 for details). This parameter θ θ θ, which can be viewed as a "north pole" (or "mean direction") for the problem under study, then is to be considered as a spherical location parameter.

In this paper, we investigate the two-sample spherical location testing problem H 0 : θ θ θ 1 = θ θ θ 2 against H 1 : θ θ θ 1 = θ θ θ 2 , where θ θ θ 1 ∈ S k-1 and θ θ θ 2 ∈ S k-1 are the respective spherical location parameters of two independent samples of i.i.d. observations with respective common distributions P 1 and P 2 , say.

Motivated by the fold test problem in palaeomagnetism (see McFadden and Jones 1981 and references therein), this problem has been extensively studied in the literature. Due to the difficulty of the task, most methods are either of parametric nature or restricted to small dimensions, or suffer from computational difficulties/slowness such as [START_REF] Wellner | Permutation tests for directional data[END_REF]'s permutation test or [START_REF] Beran | Nonparametric comparison of mean directions or mean axes[END_REF]'s bootstrap test. We refer the reader to the introduction section in [START_REF] Tsai | Asymptotically efficient two-sample rank tests for modal directions on spheres[END_REF] for a more complete description of the strengths and flaws of the different proposals. The paper [START_REF] Tsai | Asymptotically efficient two-sample rank tests for modal directions on spheres[END_REF] itself proposes a rank-based test for the two-sample problem. However, [START_REF] Tsai | Asymptotically efficient two-sample rank tests for modal directions on spheres[END_REF] considers the very restrictive case where the two samples share a common distribution, that is, P 1 = P 2 ; this is particularly uncomfortable when dealing with Fisher-von Mises-Langevin (FvML hereafter) distributions, where the additional concentration parameter κ > 0 then needs to be the same for both samples in order for the test to be valid. Thus, to the best of the authors' knowledge, the only computationally simple, efficient and asymptotically distribution-free test for the general null hypothesis H 0 above is the pseudo-FvML test given in [START_REF] Watson | Statistics on Spheres[END_REF]. The idea behind his test has the same flavor as the pseudo-Gaussian tests in the classical "linear" framework (see, for instance, [START_REF] Muirhead | Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations[END_REF]Waternaux 1980 or Hallin and[START_REF] Hallin | A general method for constructing pseudo-Gaussian tests[END_REF] for more information on pseudo-Gaussian procedures). More concretely, since the FvML distribution is considered as the spherical analogue of the Gaussian distribution (see Section 2 for an explanation of this fact), Watson has chosen the FvML as basis distribution and hence constructed his pseudo-FvML tests by "correcting" the FvML-likelihood ratio test, optimal under a couple (P 1 , P 2 ) of FvML distributions with fixed concentration parameters κ 1 and κ 2 , in such a way that the resulting test remains valid under a large class of distributions.

Our aim in the present paper consists in proposing new tests for the two-sample spherical location problem. More concretely, we aim to construct tests that are optimal (in the maximin sense) under a given pair of distributions (P 1 , P 2 ) but remain valid (in the sense that they meet the nominal level constraint) under a broad class of distributions, namely the rotationally symmetric distributions (introduced by [START_REF] Saw | A family of distributions on the m-sphere and some hypothesis tests[END_REF]; see Section 2 below for a definition). The backbone of our approach is the so-called Le Cam methodology (see [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF], as adapted to the spherical setup by [START_REF] Ley | Optimal R-estimation of a spherical location[END_REF]. Of utmost importance for our aims here is the uniform local asymptotic normality (ULAN) of a sequence of rotationally symmetric distributions established therein. This property allows us to construct optimal 2 parametric tests under the pair (P 1 , P 2 ). This optimality, however, is thwarted by the non-validity of the tests under any pair (Q 1 , Q 2 ) distinct from (P 1 , P 2 ). In order to palliate this problem, we have recourse to two classical tools: a studentization argument, which eventually leads to [START_REF] Watson | Statistics on Spheres[END_REF]'s pseudo-FvML tests, and the invariance principle, yielding optimal rank-based tests. Both tests are of semi-parametric nature.

The paper is organized as follows. In Section 2, we collect the main assumptions of the paper, summarize asymptotic results in the context of rotationally symmetric distributions and show how to construct the announced optimal parametric tests for the two-sample spherical location problem. We then extend the latter to pseudo-FvML tests in Section 3 and to rank-based tests in Section 4, and study their respective asymptotic behavior in each section. Asymptotic relative efficiencies are provided in Section 4. The theoretical results are corroborated via a Monte Carlo simulation in Section 5. Finally an appendix collects the proofs.

Main assumptions, notations and important preliminary results

Rotational symmetry and the particular rôle of the FvML distribution

Throughout, the two samples of data points X 11 , . . . , X 1n1 and X 21 , . . . , X 2n2 are assumed to belong to the unit sphere S k-1 of R k , k ≥ 2, and to satisfy Assumption A. (Rotational symmetry) X 11 , . . . , X 1n1 (resp., X 21 , . . . , X 2n2 ) are i.i.d. with common distribution P θ θ θ;f1 (resp., P θ θ θ;f2 ) characterized by a density (with respect to the usual surface area measure on spheres)

x → c k,fi f i (x ′ θ θ θ), x ∈ S k-1 , (2.1)
where θ θ θ ∈ S k-1 is a location parameter and f i : [-1, 1] → R + 0 is absolutely continuous and (strictly) monotone increasing, i = 1, 2. Then, if X has density (2.1), the density of X ′ θ θ θ is of the form

t → fi (t) := ω k c k,fi B( 1 2 , 1 2 (k -1)) f i (t)(1 -t 2 ) (k-3)/2 , -1 ≤ t ≤ 1,
where

ω k = 2π k/2 /Γ(k/2) is the surface area of S k-1 and B(•, •) is the beta function. The corresponding cumulative distribution function (cdf) is denoted by Fi (t), i = 1, 2.
The f i 's are called angular functions (because the distribution of each X ij depends only on the angle between it and the location θ θ θ ∈ S k-1 ). Throughout the rest of this paper, we denote by F 2 the collection of pairs of angular functions f := (f 1 , f 2 ). Although not necessary for the definition to make sense, monotonicity of f i ensures that surface areas in the vicinity of the location parameter θ θ θ are allocated a higher probability mass than more remote regions of the sphere. This property happens to be very appealing from the modelling point of view. Rotationally symmetric distributions also enjoy some appealing stochastic properties. Indeed, as shown in [START_REF] Watson | Statistics on Spheres[END_REF], for a random vector X distributed according to some P θ θ θ;fi as in Assumption A, not only is the multivariate sign vector S θ θ θ (X) := (X -

(X ′ θ θ θ)θ θ θ)/||X -(X ′ θ θ θ)θ θ θ|| uniformly distributed on S θ θ θ ⊥ := {v ∈ R k | v = 1, v ′ θ θ θ = 0}
but also the angular distance X ′ θ θ θ and the sign vector S θ θ θ (X) are stochastically independent.

The class of rotationally symmetric distributions contains several spherical distributions such as the linear, the logarithmic, the logistic (all three are provided in Section 4 below) or the wrapped normal distribution; for other examples and a more detailed description of the aforementioned ones, we refer to [START_REF] Duerinckx | Maximum likelihood characterization of rotationally symmetric distributions[END_REF]. By far the most popular and most used rotationally symmetric distribution is the FvML distribution (named, according to [START_REF] Watson | Statistics on Spheres[END_REF], after von Mises 1918[START_REF] Fisher | Dispersion on a sphere[END_REF], and Langevin 1905), whose density is of the form

f FvML(κ) (x; θ θ θ) = C k (κ) exp(κx ′ θ θ θ), x ∈ S k-1 ,
where κ > 0 is a concentration or dispersion parameter, θ θ θ ∈ S k-1 a location parameter and the normal-

ization constant C k (κ) is equal to C k (κ) = κ k/2-1 (2π) k/2 I k/2-1 (κ) , with I k/2-1 (κ)
:= n i=1 X i /|| n i=1 X i ||, X 1 , . . . , X n ∈ S k-1
, as the Maximum Likelihood Estimator (MLE) of its spherical location parameter, similarly as the Gaussian distribution can be characterized by the empirical mean n -1 n i=1 X i , X 1 , . . . , X n ∈ R k , as the MLE of its classical (linear) location parameter. More precisely, it has been shown that the Gaussian distribution is the only absolutely continuous (univariate and multivariate) distribution for which the sample mean is for all samples of fixed sample size n ≥ 3 the MLE of the location parameter (see [START_REF] Azzalini | On Gauss's characterization of the normal distribution[END_REF]

Le Cam optimal parametric tests for the two-sample spherical location problem

As stated in the Introduction, our first objective is to construct locally and asymptotically optimal parametric tests by having recourse to the Le Cam methodology. The main ingredient for this construction rests on the ULAN property of the parametric model P

(n) θ θ θ1;f1 | θ θ θ 1 ∈ S k-1 , P (n) θ θ θ2;f2 | θ θ θ 2 ∈ S k-1
for a fixed pair of angular functions (f 1 , f 2 ), where P

(n) θ θ θi;fi stands for the joint distribution of X i1 , . . . , X ini , i = 1, 2. We further denote by P (n) (θ θ θ1,θ θ θ2);f the joint law combining P (n) θ θ θ1;f1 and P (n) θ θ θ2;f2 . In order to be able to state our results, we need to impose a certain control on the respective sample sizes n 1 and n 2 , which will be achieved via the following

Assumption B. Letting n = n 1 + n 2 , both r (n) 1 := n 1 /n and r (n) 2 := n 2 /n converge to finite constants r 1 and r 2 respectively as n → ∞.
This condition explains why, in what precedes and in what follows, we simply use the superscript (n) for the different quantities at play and do not specify whether they are associated with n 1 or n 2 .

Informally, a sequence of rotationally symmetric models P

(n) (θ θ θ1,θ θ θ2);f | θ θ θ 1 , θ θ θ 2 ∈ S k-1 is ULAN if the logarithm of the likelihood ratio P (n) (θ θ θ (n) 1 +n -1/2 1 t (n) 1 ,θ θ θ (n) 2 +n -1/2 2 t (n) 2 );f /P (n) (θ θ θ (n) 1 ,θ θ θ (n) 2 );f allows a specific form of (probabilistic) Taylor expansion, with θ θ θ (n) 1 , θ θ θ (n) 2 ∈ S k-1 such that θ θ θ (n) i -θ θ θ i = O(n -1/2 ), i = 1, 2, and t (n) 1 , t (n) 2 ∈ R k bounded sequences of perturbations such that θ θ θ (n) i + n -1/2 i t (n) i
remains on the unit sphere for i = 1, 2. The latter condition means that each t

(n) i needs to satisfy 0 = (θ θ θ (n) i + n -1/2 i t (n) i ) ′ (θ θ θ (n) i + n -1/2 i t (n) i ) -1 = 2n -1/2 i (θ θ θ (n) i ) ′ t (n) i + n -1 i (t (n) i ) ′ t (n) i . (2.2)
Consequently, t

(n) i must be such that 2n

-1/2 i (θ θ θ (n) i ) ′ t (n) i + n -1 i (t (n) i ) ′ t (n) i = 0 or, equivalently, such that 2n -1/2 i (θ θ θ (n) i ) ′ t (n) i +o(n -1/2 i ) = 0. In other words, for θ θ θ (n) i +n -1/2 i t (n) i to remain in S k-1 , t (n) i must belong, up to a o(n -1/2 i
) quantity, to the tangent space to S k-1 at θ θ θ

(n)

i . Now, S k-1 is a non-linear manifold and as a consequence, establishing the ULAN property of a sequence of rotationally symmetric models is all but easy. [START_REF] Ley | Optimal R-estimation of a spherical location[END_REF] handle this difficulty by resorting to a natural re-parameterization in terms of spherical coordinates, for which it is possible to prove ULAN, subject to the following necessary condition.

Assumption C. The Fisher information associated with the spherical location parameter is finite; this finiteness is ensured if, for i = 1, 2 and letting

ϕ fi := ḟi /f i ( ḟi is the a.e.-derivative of f i ), J k (f i ) := 1 -1 ϕ 2 fi (t)(1 -t 2 ) fi (t)dt < +∞.
After obtaining the ULAN property for this new parameterization, [START_REF] Ley | Optimal R-estimation of a spherical location[END_REF]

P (n) (θ θ θ1,θ θ θ2);f | θ θ θ 1 , θ θ θ 2 ∈ S k-1 is ULAN with central sequence ∆ ∆ ∆ (n) (θ θ θ1,θ θ θ2);f := (∆ ∆ ∆ (n) θ θ θ1;f1 ) ′ , (∆ ∆ ∆ (n) θ θ θ2;f2 ) ′ ′
, where

∆ ∆ ∆ (n) θ θ θi;fi := n -1/2 i ni j=1 ϕ fi (X ′ ij θ θ θ i )(1 -(X ′ ij θ θ θ i ) 2 ) 1/2 S θ θ θi (X ij ), i = 1, 2,
and Fisher information matrix

Γ Γ Γ (θ θ θ1,θ θ θ2);f := diag(Γ Γ Γ θ θ θ1;f1 , Γ Γ Γ θ θ θ2;f2 ) where Γ Γ Γ θ θ θi;fi := J k (f i ) k -1 (I k -θ θ θ i θ θ θ ′ i ), i = 1, 2.
More precisely, for any θ θ θ 1 , θ θ θ 2 ∈ S k-1 and any bounded sequences t

(n) 1 , t (n) 2
as in (2.2), we have

log    P (n) (θ θ θ (n) 1 +n -1/2 1 t (n) 1 ,θ θ θ (n) 2 +n -1/2 2 t (n) 2 );f P (n) (θ θ θ (n) 1 ,θ θ θ (n) 2 );f    = (t (n) ) ′ ∆ ∆ ∆ (n) (θ θ θ (n) 1 ,θ θ θ (n) 2 );f - 1 2 (t (n) ) ′ Γ Γ Γ (θ θ θ1,θ θ θ2);f t (n) + o P (1),
where t (n) := ((t

(n) 1 ) ′ , (t (n) 2 ) ′ ) ′ , and ∆ ∆ ∆ (n) (θ θ θ (n) 1 ,θ θ θ (n) 2 );f L → N 2(k-1) (0 0 0, Γ Γ Γ (θ θ θ1,θ θ θ2);f ), both under P (n) (θ θ θ1,θ θ θ2);f , as n → ∞.
Proposition 2.1 is a straightforward consequence of the result in [START_REF] Ley | Optimal R-estimation of a spherical location[END_REF], and hence the proof is omitted. With this in hand, constructing optimal f -parametric procedures (that is, under the pair of densities with respective specified angular functions f 1 and f 2 ) for testing H 0 : θ θ θ 1 = θ θ θ 2 against H 1 : θ θ θ 1 = θ θ θ 2 is plain sailing. Indeed, denoting by θ θ θ ∈ S k-1 the common null hypothesis value of the location parameter, the corresponding parametric test statistic is given by

(∆ ∆ ∆ (n) (θ θ θ,θ θ θ);f ) ′ (Γ Γ Γ (θ θ θ,θ θ θ);f ) -1 ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);f .
Evidently, the unknown value θ θ θ needs to be estimated, but we leave that issue to the subsequent sections.

Intuitively, this construction of optimal parametric tests follows from the fact that the second-order expansion of the log-likelihood ratio for the model P

(n) (θ θ θ1,θ θ θ2);f | θ θ θ 1 , θ θ θ 2 ∈ S k-1
strongly resembles the loglikelihood ratio for the classical Gaussian shift experiment, for which optimal procedures are well-known and are based on the corresponding first-order term. For more details and more formal explanations, we refer the reader to Le [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF], Section 11.9.

As mentioned in the Introduction, the optimal parametric tests suffer from the drawback of being only valid under the pair (f 1 , f 2 ). Since it is highly unrealistic in practice to assume that the underlying densities are known, these tests are useless for practitioners. The next two sections contain two distinct solutions allowing to set this problem right.

Pseudo-FvML tests.

For a given pair of FvML densities (φ κ1 , φ κ2 ) with respective concentration parameters κ 1 , κ 2 > 0 (where we do not assume κ 1 = κ 2 ), the quantities ϕ φκ i reduce to the constants κ i , i = 1, 2, and hence the central sequences for each sample take the form

∆ ∆ ∆ (n) θ θ θi;φκ i := κ i n -1/2 i ni j=1 (1 -(X ′ ij θ θ θ i ) 2 ) 1/2 S θ θ θi (X ij ) = κ i n -1/2 i ni j=1 (X ij -(X ′ ij θ θ θ i )θ θ θ i ) = κ i (I k -θ θ θ i θ θ θ ′ i )n -1/2 i ni j=1 X ij =: κ i (I k -θ θ θ i θ θ θ ′ i )n 1/2 i Xi = κ i (I k -θ θ θ i θ θ θ ′ i )n 1/2 i ( Xi -θ θ θ i ), i = 1, 2.
Optimal FvML-based procedures for the two-sample spherical location problem are then built upon

∆ ∆ ∆ (n) (θ θ θ,θ θ θ);(φκ 1 ,φκ 2 )
. We here again draw the reader's attention to the fact that this parametric test is only valid under the pair (φ κ1 , φ κ2 ) and becomes non-valid even if only the concentration parameters change.

In this section, this non-validity problem will be overcome in the following way. We will first study the asymptotic behavior of ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);(φκ 1 ,φκ 2 ) under any given pair g = (g 1 , g 2 ) ∈ F 2 and consider the newly obtained quadratic form in ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);(φκ 1 ,φκ 2 ) . Clearly, this quadratic form will now depend on the asymptotic variance of ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);(φκ 1 ,φκ 2 ) under g, hence again, for each g, we are confronted to an onlyfor-g-valid test statistic. The next and final step then consists in applying a studentization argument, meaning that we estimate this asymptotic variance quantity and study the asymptotic behavior of the new quadratic form under any pair of rotationally symmetric distributions. The final outcome of this procedure will be tests which happen to be optimal under any pair of FvML distributions (that is, for any values κ 1 , κ 2 > 0) and valid under the entire class of rotationally symmetric distributions; these tests are our so-called pseudo-FvML tests.

For the sake of readability, we adopt the notation φ for any pair (φ κ1 , φ κ2 ) and E f [•] for expectation under the angular function f . The following result characterizes, for a given pair of angular functions g ∈ F 2 , the asymptotic properties of the FvML-based central sequence ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);φ , both under P (n) (θ θ θ,θ θ θ);g and P

(n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n) 2 );g with t (n) 1 and t (n) 2
as in (2.2) for each sample.

= 1-E gi (X ′ ij θ θ θ) 2 for i = 1, 2, we have that ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);φ is (i) asymptotically normal under P (n)
(θ θ θ,θ θ θ);g with mean zero and covariance matrix

Γ Γ Γ * θ θ θ;g := diag Γ Γ Γ * θ θ θ;g1 , Γ Γ Γ * θ θ θ;g2 ,
where Γ Γ Γ * θ θ θ;gi :=

κ 2 i B k,g i k-1 (I k -θ θ θθ θ θ ′ ), i = 1, 2;
(ii) asymptotically normal under

P (n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n) 2 );g (t (n) 1 and t (n) 2
as in (2.2)) with mean Γ Γ Γ θ θ θ;φ,g t

(t := (t ′ 1 , t ′ 2 ) ′ with t 1 := lim n→∞ t (n) 1 and t 2 := lim n→∞ t (n) ii) for all θ θ θ ∈ S k-1 , Γ Γ Γ θ θ θ;φ,φ = Γ Γ Γ φ θ θ θ;φ .
The proof is provided in the Appendix. From these results, we can now deduce the new quadratic form to be used as a g-valid test statistic. This construction follows the ideas from [START_REF] Hallin | A general method for constructing pseudo-Gaussian tests[END_REF] where a very general theory for pseudo-Gaussian procedures is described. Defining

E k,gi := E gi X ′ ij θ θ θ , i = 1, 2
, and, for notational simplicity, D k,gi := E k,gi /B k,gi , i = 1, 2, and H φ,g := (r

(n) 1 D 2 k,g1 B k,g1 + r (n) 2 D 2 k,g2 B k,g2
), and letting

Ψ ⊥ θ θ θ;φ,g := (k -1)    1 κ 2 1 ( 1 B k,g 1 - r (n) 1 D 2 k,g 1 H φ,g )(I k -θ θ θθ θ θ ′ ) -1 κ1κ2 r (n) 1 r (n) 2 D k,g 1 D k,g 2 H φ,g (I k -θ θ θθ θ θ ′ ) -1 κ1κ2 r (n) 1 r (n) 2 D k,g 1 D k,g 2 H φ,g (I k -θ θ θθ θ θ ′ ) 1 κ 2 2 ( 1 B k,g 2 - r (n) 2 D 2 k,g 2 H φ,g )(I k -θ θ θθ θ θ ′ )    ,
the g-valid test statistic for the two-sample spherical location problem H 0 : θ θ θ 1 = θ θ θ 2 against H 1 : θ θ θ 1 = θ θ θ 2 corresponds to the quadratic form

Q (n) (g) := (∆ ∆ ∆ (n) ( θ θ θ, θ θ θ);φ ) ′ Ψ ⊥ θ θ θ;φ,g ∆ ∆ ∆ (n) ( θ θ θ, θ θ θ);φ .
At first sight, this construction might appear puzzling and the reader might wonder how the preceding results finally lead to this test statistic. It is easy to verify that the random quantity Q (n) (g) does not depend explicitly on the underlying concentrations κ 1 and κ 2 but still depends on the quantities B k,gi and E k,gi , i = 1, 2, still hampering the validity of the statistic outside of g. The last step in our construction thus consists in estimating these quantities. Consistent (via the Law of Large Numbers) estimators for each of them are provided by Bk,gi :

= 1 -n -1 i ni j=1 (X ′ ij θ θ θ) 2 and Êk,gi := n -1 i ni j=1 (X ′ ij θ θ θ), i = 1, 2.
For the sake of readability, we naturally also use the notations Dk,gi := Êk,gi / Bk,gi , i = 1, 2, and Ĥφ,g := (r

(n) 1 D2 k,g1 Bk,g1 +r (n) 2 D2 k,g2
Bk,g2 ). Straightforward calculations then show that our pseudo-FvML test statistic for the two-sample spherical location problem is

Q (n) = (k -1)    Dk,g1 Êk,g1 - r (n) 1 D2 k,g1
Ĥφ,g

  1 n 1 n1 i,j=1 X ′ 1i (I k -θ θ θ θ θ θ ′ )X 1j   + Dk,g2 Êk,g2 - r (n) 2 D2 k,g2
Ĥφ,g

  1 n 2 n2 i,j=1 X ′ 2i (I k -θ θ θ θ θ θ ′ )X 2j   -2 Dk,g1 Dk,g2 Ĥφ,g   1 n n1 i=1 X ′ 1i (I k -θ θ θ θ θ θ ′ ) n2 j=1 X 2j      ,
which no more depends on g. The following proposition finally yields the asymptotic properties of this quadratic form under the entire class of rotationally symmetric distributions, showing that the test is well valid under that broad set of distributions.

Proposition 3.3 Let Assumptions A, B and C hold and let θ θ θ be an estimator of the common value θ θ θ such that Assumption D holds. Then,

(i) Q (n) is asymptotically chi-square with k -1 degrees of freedom under θ θ θ∈S k-1 g∈F 2 P (n) (θ θ θ,θ θ θ);g ; (ii) Q (n)
is asymptotically non-central chi-square with k -1 degrees of freedom and non-centrality parameter

l (θ θ θ,θ θ θ),t;φ,g := t ′ Γ Γ Γ θ θ θ;φ,g Ψ ⊥ θ θ θ;φ,g Γ Γ Γ θ θ θ;φ,g t under P (n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n)
2 );g

, where t

(n) 1 and t

(n) 2 are as in (2.2) and t :=

(t ′ 1 , t ′ 2 ) ′ with t 1 := lim n→∞ t (n) 1 and t 2 := lim n→∞ t (n) 2 .
From Part (i) we deduce that our pseudo-FvML tests for the two-sample spherical location problem, denoted by φ (n) , reject the null hypothesis H 0 : θ θ θ 1 = θ θ θ 2 in favor of H 1 : θ θ θ 1 = θ θ θ 2 at asymptotic level α as soon as Q (n) exceeds the α-upper quantile of a chi-square distribution with k -1 degrees of freedom. It is easy to verify as in [START_REF] Watson | Statistics on Spheres[END_REF] that it is asymptotically equivalent (the difference is a o P (1) quantity) to the FvML likelihood ratio test in the FvML case and therefore keeps its optimality properties in that configuration. Thus, our pseudo-FvML tests, although the construction is different, coincide with Watson's proposal. However, our work here is of intrinsic interest since we provide more insight on how to correct the optimal FvML test and, contrarily to [START_REF] Watson | Statistics on Spheres[END_REF], we investigate in detail the impact of the replacement of the unknown quantities by the corresponding estimators.

Rank-based tests.

The pseudo-FvML test constructed in the previous section is valid under any pair of (non-necessarily equal) rotationally symmetric distributions and retains the optimality properties of the FvML likelihood ratio test in the FvML case. In this section, we start from any given pair f ∈ F 2 and our objective is to turn the f -parametric tests into tests which are still valid under any pair of (non-necessarily equal) rotationally symmetric distributions but which are optimal under the pair f . To obtain such a test, we have recourse here to the second of the aforementioned tools to turn our parametric tests into semiparametric ones: the invariance principle. This principle advocates that, if the sub-model identified by the null hypothesis is invariant under the action of a group of transformations G T , one should exclusively use procedures whose outcome does not change along the orbits of that group G T . This is the case if and only if these procedures are measurable with respect to the maximal invariant associated with G T . The invariance principle is accompanied by a nice and appealing corollary for our purposes here: provided that the group G T is a generating group for H 0 , the invariant procedures are distribution-free under the null. In view of all this, our strategy in this section is the following: determine the correct group of transformations G T , re-express our f -parametric tests in terms of the corresponding maximal invariant and study the asymptotic properties of the resulting test statistic, after replacement of all the unknown quantities by consistent estimators, as in the previous section.

Invariance with respect to "common rotations" is crucial in this context. More precisely, letting

O ∈ SO k := {A ∈ R k×k , A ′ A = I k , det(A) = 1}
, the null hypothesis is unquestionably invariant with respect to a transformation of the form g O : X 11 , . . . , X 1n1 , X 21 , . . . , X 2n2 → OX 11 , . . . , OX 1n1 , OX 21 , . . . , OX 2n2 .

However, this group is not generating for H 0 as it does not take into account the underlying angular functions f , which are an infinite-dimensional nuisance under H 0 . This group is actually rather generating for θ θ θ∈S k-1 P (n) (θ θ θ,θ θ θ);f with fixed f . Now, denote as in the previous section the common value of θ θ θ 1 and θ θ θ 2 under the null as θ θ θ. Then

X ij = (X ′ ij θ θ θ)θ θ θ + 1 -(X ′ ij θ θ θ) 2 S θ θ θ (X ij
) for all j = 1, . . . , n i and i = 1, 2. Let (ii) is asymptotically normal under P (n) (θ θ θ,θ θ θ);g with mean zero and covariance matrix

Γ Γ Γ θ θ θ;K f := diag J k (K f1 ) k -1 (I k -θ θ θθ θ θ ′ ), J k (K f2 ) k -1 (I k -θ θ θθ θ θ ′ ) , where J k (K fi ) := 1 0 K 2 fi (u)du.
(iii) is asymptotically normal under

P (n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n) 2 );g (t (n) 1 and t (n) 2
as in (2.2)) with mean

Γ Γ Γ θ θ θ;K f ,g t (t := (t ′ 1 , t ′ 2 ) ′ with t 1 := lim n→∞ t (n) 1
and t 2 := lim n→∞ t

(n)

2 ) and covariance matrix

Γ Γ Γ θ θ θ;K f ,g := diag J k (K f1 , g 1 ) k -1 (I k -θ θ θθ θ θ ′ ), J k (K f2 , g 2 ) k -1 (I k -θ θ θθ θ θ ′ ) ,
where

J k (K fi , g i ) := 1 0 K fi (u)K gi (u)du for i = 1, 2.
(iv) satisfies, under P

(θ θ θ,θ θ θ);g as n → ∞, the asymptotic linearity property

∆ ∆ ∆ (n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n) 2 );K f -∆ ∆ ∆ (n) (θ θ θ,θ θ θ);K f = -Γ Γ Γ θ θ θ;K f ,g t (n) + o P (1), for t (n) = ((t (n) 1 ) ′ , (t (n) 2 ) ′ ) ′ with t (n) 1 and t (n) 2
as in (2.2).

Now, as for the pseudo-FvML test, our rank-based procedures are not complete since we still have to estimate the common value θ θ θ of θ θ θ 1 and θ θ θ 2 under H 0 . Therefore, we will assume the existence of an estimator θ θ θ satisfying Assumption D; as explained in Section 3, such an estimator is easy to construct.

In order to deal with these rank-based test statistics, we however need to strengthen Assumption D into Assumption D'. Besides root-n consistency under P

(θ θ θ,θ θ θ);g for any g ∈ F 2 , the estimator θ θ θ ∈ S k-1 is further locally and asymptotically discrete, meaning that it only takes a bounded number of distinct values in θ θ θ-centered balls with O(n -1/2 ) radius.

This discretization condition is a purely technical requirement (see pages 125 and 188 of Le Cam and

Yang 2000 for a discussion), with little practical implications (in fixed-n practice, such discretizations are irrelevant as the radius can be taken arbitrarily large). We will therefore tacitly assume that θ θ θ is locally discrete throughout this section. Following Lemma 4.4 in [START_REF] Kreiss | On adaptive estimation in stationary ARMA processes[END_REF], the local discreteness allows to replace in Part (iv) of Proposition 4.1 the non-random perturbations θ θ θ + n

-1/2 i t (n) i , i = 1, 2, (of θ θ θ) by θ
θ θ (see also [START_REF] Hallin | One-step R-estimation in linear models with stable errors[END_REF]. Based on the asymptotic result of Proposition 4.1 and letting

H K f ,g := r (n) 1 J 2 k (K f 1 ,g1) J k (K f 1 ) + r (n) 2 J 2 k (K f 2 ,g2) J k (K f 2 ) and Ψ ⊥ θ θ θ;K f ,g := (k-1)     ( 1 J k (K f 1 ) - r (n) 1 J 2 k (K f 1 ,g1) J 2 k (K f 1 )H K f ,g )(I k -θ θ θθ θ θ ′ ) - r (n) 1 r (n) 2 J k (K f 1 ,g1)J k (K f 2 ,g2) J k (K f 1 )J k (K f 2 )H K f ,g (I k -θ θ θθ θ θ ′ ) - r (n) 1 r (n) 2 J k (K f 1 ,g1)J k (K f 2 ,g2) J k (K f 1 )J k (K f 2 )H K f ,g (I k -θ θ θθ θ θ ′ ) ( 1 J k (K f 2 ) - r (n) 2 J 2 k (K f 2 ,g2) J 2 k (K f 2 )H K f ,g )(I k -θ θ θθ θ θ ′ )     ,
the g-valid rank-based test statistic (built in a similar way as the pseudo-FvML statistic, see Section 3) we propose for the two-sample spherical location problem H 0 : θ θ θ 1 = θ θ θ 2 against H 1 : θ θ θ 1 = θ θ θ 2 corresponds As in [START_REF] Ley | Optimal R-estimation of a spherical location[END_REF], consistent estimators of J -1 k (K f1 , g 1 ) and J -1 k (K f2 , g 2 ) (and therefore readily of J k (K f1 , g 1 ) and J k (K f2 , g 2 )) can be obtained by taking respectively ρ1 := inf{ρ > 0 :

h (n) 1 (ρ) < 0} and ρ2 := inf{ρ > 0 : h (n) 2 (ρ) < 0}.
Denoting by Ĵk (K f1 , g 1 ) and Ĵk (K f2 , g 2 ) the resulting estimators, ĤK f ,g :=

r (n) 1 Ĵ 2 k (K f 1 ,g1) J k (K f 1 ) + r (n) 2 Ĵ 2 k (K f 2 ,g2) J k (K f 2 )
and letting

U ij := K fi Rij ni+1 S θ θ θ (X ij ), i = 1, 2, ( Rij naturally stands for the rank of X ′ ij θ θ θ among X ′ i1 θ θ θ, . . . , X ′ ini θ θ θ), the proposed rank test φ (n) K f
rejects the null hypothesis of homogeneity of the locations when

Q (n) K f := (k -1)    1 J k (K f1 ) - r (n) 1 Ĵ 2 k (K f1 , g 1 ) J 2 k (K f1 ) ĤK f ,g   1 n 1 n1 i,j=1 U ′ 1i (I k -θ θ θ θ θ θ ′ )U 1j   + 1 J k (K f2 ) - r (n) 2 Ĵ 2 k (K f2 , g 2 ) J 2 k (K f2 ) ĤK f ,g   1 n 2 n2 i,j=1 U ′ 2i (I k -θ θ θ θ θ θ ′ )U 2j   -2 Ĵk (K f1 , g 1 ) Ĵk (K f2 , g 2 ) J k (K f1 )J k (K f2 ) ĤK f ,g   1 n n1 i=1 U ′ 1i (I k -θ θ θ θ θ θ ′ ) n2 j=1 U 2j     
exceeds the α-upper quantile of the chi-square distribution with k -1 degrees of freedom. This asymptotic behavior under the null as well as the asymptotic distribution of Q

i) Q (n) K f is asymptotically chi-square with k -1 degrees of freedom under θ θ θ∈S k-1 g∈F 2 {P (n) (θ θ θ,θ θ θ);g }; (ii) Q (n) K f
is asymptotically non-central chi-square, still with k -1 degrees of freedom, but with noncentrality parameter

l (θ θ θ,θ θ θ),t;K f ,g := t ′ Γ Γ Γ θ θ θ;K f ,g Ψ ⊥ θ θ θ;K f ,g Γ Γ Γ θ θ θ;K f ,g t under P (n) (θ θ θ+n -1/2 1 t (n) 1 ,θ θ θ+n -1/2 2 t (n)
2 );g

, where t Thanks to Proposition 4.1, the proof of this result follows along the same lines as that of Proposition 3.3 and is therefore omitted. Exactly as the pseudo-FvML tests φ (n) , our rank-based tests φ

f = (K f1 , K f2 ) with K fi (u) = ϕ fi ( F -1 i (u))(1 -( F -1 i (u)) 2 ) 1/2 , i = 1, 2, the rank- based central sequence ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);K f is asymptotically equivalent to the parametric central sequence ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);f under P (n) (θ θ θ,θ θ θ);f . Therefore, the test φ (n) K f based on the central sequence ∆ ∆ ∆ (n) (θ θ θ,θ θ θ);K f
keeps the optimality properties of the f -parametric test for any f ∈ F 2 . Thus, while the pseudo-FvML tests are logically FvML-based and only enjoy optimality under FvML densities, one can construct f -optimal rank-based tests on basis of any pair f ∈ F 2 . This, in particular, provides practitioners with much more flexibility than was previously available.

We conclude this section by comparing the optimal pseudo-FvML test φ (n) with optimal rank-based tests φ

(n) K f
for several choices of f ∈ F 2 by means of Pitman's asymptotic relative efficiency (ARE). Letting ARE (θ θ θ,θ θ θ);g (φ

(n) 1 , φ (n) 2 ) denote the ARE of a test φ (n) 1 with respect to another test φ (n) 2 under P (n) (θ θ θ,θ θ θ);g , we have that ARE (θ θ θ,θ θ θ);g (φ (n) K f , φ (n) ) = l (θ θ θ,θ θ θ),t;K f ,g /l (θ θ θ,θ θ θ),t;φ,g .
In the homogeneous case g = (g 1 , g 1 ) (the angular density is the same for both samples) and if the same score function-namely, K f1 -is used for the two rankings (the test is therefore denoted by φ

(n) K f 1
), the ratio in (4) simplifies into ARE (θ θ θ,θ θ θ);g (φ

(n) K f 1 /φ (n) ) = J 2 k (K f1 , g 1 ) J k (K f1 )D 2 k,g1 B k,g1
f logis(a,b) (t) := a exp(-b arccos(t)) (1 + a exp(-b arccos(t))) 2 .
The constants a and b are chosen so that all the above functions are true angular functions satisfying Assumption A. The score functions associated with these angular functions are denoted by K lin(a) for f lin(a) , K log(a) for f log(a) and K logis(a,b) for f logis(a,b) . For the FvML distribution with concentration κ, the score function will be denoted by K φκ .

Inspection of Table 1 confirms the theoretical results. As expected, the pseudo-FvML test φ (n) dominates the rank-based tests under FvML densities, whereas rank-based tests mostly outperform the pseudo-FvML test under other densities, especially so when they are based on the score function associated with the underlying density (in which case the rank-based tests are optimal).

ARE(φ

(n) K f 1 /φ (n) ) Underlying density φ (n) K φ 2 φ (n) K φ 6 φ (n) K lin(2) φ (n) K lin(4) φ (n) K log(2.5) φ (n) K logis(1,1) φ (n) K logis(2,1)
FvML( 1 with respect to the pseudo-FvML test φ (n) under various three-dimensional rotationally symmetric densities.

Simulation results

In this section, we perform a Monte Carlo study to compare the small-sample behavior of the pseudo-FvML test φ (n) and various rank-based tests φ

(n) K f
. For this purpose, we generated M = 2, 500 replications of four pairs of mutually independent samples (with respective sizes n 1 = 100 and n 2 = 150) of threedimensional rotationally symmetric random vectors ε ε ε ℓ;iji , ℓ = 1, 2, 3, 4, j i = 1, . . . , n i , i = 1, 2, with FvML densities and linear densities: the ε ε ε 1;1j1 's have a FvML(15) distribution and the ε ε ε 1;2j2 's have a FvML(2) distribution; the ε ε ε 2;1j1 's have a Lin(2) distribution and the ε ε ε 2;2j2 's have a Lin(1.1) distribution; the ε ε ε 3;1j1 's have a FvML(15) distribution and the ε ε ε 3;2j2 's have a Lin(1.1) distribution and finally the ε ε ε 4;1j1 's have a Lin(2) distribution and the ε ε ε 4;2j2 's have a FvML(2) distribution.

The rotationally symmetric vectors ε ε ε ℓ;iji 's have all been generated with a common spherical location Clearly, the spherical locations of the X ℓ;1j1 's and the X ℓ;2j2;0 's coincide while the spherical location of the X ℓ;2j2;ξ 's, ξ = 1, 2, 3, is different from the spherical location of the X ℓ;1j1 's characterizing alternatives .0504 .1180 .3660 .6916

Table 2: Rejection frequencies (out of M = 2, 500 replications), under the null and under increasingly distant alternatives, of the pseudo-FvML test φ (n) and various rank-based tests φ (n) (K φ 15 ,K φ 2 ) (based on FvML(15) and FvML(2) scores), φ (n) (K Lin(2) ,K Lin(1.1) ) (based on Lin(2) and Lin(1.1) scores), φ (n) (K Lin(2) ,K φ 2 ) (based on Lin(2) and FvML(2) scores) and φ for more details) we know that

E gi 1 -(X ′ ij θ θ θ) 2 (S θ θ θ (X ij )) ′ = E gi 1 -(X ′ ij θ θ θ) 2 E gi [(S θ θ θ (X ij )) ′ ] = 0 0 0 ′ .
This concludes Part (i) of the proposition. Regarding Part (ii), let X be a random vector distributed according to an FvML distribution with concentration κ. Then, writing c for the normalization constant, a simple integration by parts yields

C k,φκ = κ E φκ [1 -(X ′ θ θ θ) 2 ] = κ c 1 -1
(1u 2 ) exp(κu)(1u 2 ) (k-3)/2 du = κ c u exp(κu)(1u 2 ) (k-3)/2 du

  for the latest version of this result first expressed in Gauss 1809), and exactly the same result is known to hold true for the FvML distribution among spherical distributions (see Duerinckx and Ley 2012, where the earlier findings of von Mises 1918 for dimension k = 2, Arnold 1941 and Breitenberger 1963 for dimension k = 3 and Bingham and Mardia 1975 for any dimension have been generalized to yield the latter statement) 1 .

Proposition 3. 1

 1 Let Assumptions A, B and C hold. Then, letting B k,gi :

  Let Assumptions A, B, C and E hold and let θ θ θ be an estimator of the common value θ θ θ such that Assumption D' holds. Then

(

(

  -necessarily equal) pair of rotationally symmetric densities. Furthermore, as shown in Part (i) of Proposition 4.1, for K

  ′ . Then, each replication of the ε ε ε ℓ;iji 's was transformed into    X ℓ;1j1 = ε ε ε ℓ;1j1 , ℓ = 1, 2, 3, 4, j 1 = 1, . . . , n 1 X ℓ;2j2;ξ = O ξ ε ε ε ℓ;2j2 , ℓ = 1, 2, 3, 4, j 2 = 1, . . . , n 2 , ξ = 0

  15 ,K Lin(1.1) ) (based on FvML(15) and Lin(1.1) scores). Sample sizes are n1 = 100 and n2 = 150.which is o P (1) from the boundedness of θ θ θ and since from[START_REF] Watson | Statistics on Spheres[END_REF] (see the proof of Proposition 3.1

  the modified Bessel function of the first kind and of order k/2 -1. In what follows, we shall replace f FvML(κ) with the lighter notation φ κ . As already mentioned in the Introduction, the

	FvML is considered as the spherical analogue of the Gaussian distribution for purposes of mathematical
	statistics (see Schaeben 1992 for a discussion on analogues of the Gaussian distribution). This analogy is
	mainly due to the fact that the FvML distribution can be characterized by the empirical spherical mean
	θ θ θ Mean

  Thisfinally yields in the property given in Proposition 2.2 of[START_REF] Ley | Optimal R-estimation of a spherical location[END_REF]. Here, we obviously need a slightly different version of that result since we are dealing with the two-sample problem. Fortunately, the inner-sample independence and the mutual independence between the two samples entails that we can readily write down the ULAN property in the present setup.

	Proposition 2.1 [ULAN for the two-sample problem] Let Assumptions A, B and C hold. Then the model

use a lemma from

[START_REF] Hallin | Optimal rank-based testing for principal components[END_REF]

explaining how to transpose ULAN from one parameterization to another.

Table 1 :

 1 Asymptotic relative efficiencies of (homogeneous) rank-based tests φ

	)	0.9744 0.8787	0.9813	0.9979	0.9027	0.9321	0.7364
	FvML(2)	1	0.9556	0.9978	0.9586	0.9749	0.9823	0.8480
	FvML(6)	0.9555	1	0.9381	0.8517	0.9768	0.9911	0.9280
	Lin(2)	1.0539 0.9909	1.0562	1.0215	1.0212	1.0247	0.8796
	Lin(4)	0.9709 0.8627	0.9795	1.0128	0.8856	0.9231	0.7097
	Log(2.5)	1.1610 1.1633	1.1514	1.0413	1.1908	1.1625	1.0951
	Log(4)	1.0182 0.9216	1.0261	1.0347	0.9503	0.9741	0.7851
	Logis(1,1)	1.0768 1.0865	1.0635	0.9991	1.0701	1.0962	0.9778
	Logis(2,1)	1.3182 1.4426	1.2946	1.0893	1.4294	1.3865	1.5544
						(n)
						K f 1

It is interesting in this context to note that Gauss, in his manuscript "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" of 1809, has defined the famous distribution named after him by searching for the probability law for which the sample mean is always the MLE of the location parameter, and that von Mises, in 1918, aiming at constructing a circular analogue of the Gaussian distribution, started precisely from this classical MLE characterization.

) and covariance matrix Γ Γ Γ * θ θ θ;g , where, puttingC k,gi := E gi [(1 -(X ′ ij θ θ θ) 2 )ϕ gi (X ′ ij θ θ θ)] for i = 1, 2,

(n) (θ θ θ,θ θ θ);f under P (n) (θ θ θ,θ θ θ);f .

Acknowledgements

The research of Christophe Ley is supported by a Mandat de Chargé de Recherche from the Fonds National de la Recherche Scientifique, Communauté française de Belgique. Arnold, K. J. (1941). On spherical probability distributions. PhD thesis, Massachusetts Institute of Technology.

to the null hypothesis of common spherical locations. Rejection frequencies based on the asymptotic chi-square critical values at nominal level 5% are reported in Table 2 below. The inspection of the latter reveals expected results:

(i) The pseudo-FvML test and all the rank-based tests are valid under heterogeneous densities. They reach the 5% nominal level constraint under any considered pair of densities.

(ii) The comparison of the empirical powers reveals that when based on scores associated with the underlying distributions, the rank-based test performs nicely. The pseudo-FvML test is clearly optimal in the FvML case.

Appendix A.

Proof of Proposition 3.1. From [START_REF] Watson | Statistics on Spheres[END_REF] (and the beginning of Section 2) we know that, under P (n) (θ θ θ,θ θ θ);g , the sign vectors S θ θ θ (X ij ) are independent of the scalar products

for i = 1, 2 and for all j = 1, . . . , n i . These results readily allow to obtain Part (i) by applying the multivariate central limit theorem, while Part (ii) follows from the ULAN structure of the model in Proposition 2.1 and Le Cam's third Lemma.

Proof of Proposition 3.2. We start by proving Part (i). First note that easy computations yield (for


Rodrigue Mendez
email: rodrigue.mendez@free.fr.

Predatory Lending

Keywords: D03, D18, D49, D86 hyperbolic discounting, time inconsistency, sophistication, partial naïveté, exploitative contracts, credit cards

This paper studies the equilibrium predatory practices that may arise when the borrowers have behavioral weaknesses. Rational lenders offer short term contracts that can be renewed at the cost of paying a penalty fee. We show how the optimal contracts depend on the degree of naïveté of the time inconsistent customers. Penalty fees have a dual role : they increase market share by providing a useful commitment device to time-inconsistent but otherwise rational borrowers ; they are also a source of revenue from the semi-naïve borrowers who understand the need for commitment but fail to forecast their future time discount factor. We also show that perfect competition does not eliminate predatory practices, since the equilibrium contract entails a subsidized (below marginal cost) short-term loan that can only be profitable if a fraction of the borrowers end up paying the penalty fee.

Introduction

The paper focuses on the equilibrium predatory practices that may arise when the borrowers have behavioral weaknesses. Our main assumption is that the borrowers have the (δ, β, β) quasi-hyperbolic preferences introduced by O' Donoghue andRabin (1999a, 2001). This assumption has two components. First, the quasi-hyperbolic preferences imply that the borrowers have a self-control problem (their inability to resist the urges of immediate consumption), which may lead to time-inconsistent choices, since their preferences are time-varying. Second, the borrowers are not fully aware of the dynamics of their preferences (they tend to overestimate their future discount factor). This second hypothesis, which we label the naïveté hypothesis will turn out to be a crucial element of the analysis.

The lending institutions are fully rational. They are aware of the behavioral weaknesses of their customers and will devise contracts that take advantage of those weaknesses. The purpose of the paper is to make a systematic exploration of those contracts, which we label exploitative contracts. We assume that the lending institutions offer short-terms lending contracts that can be renewed, at a cost. The contracts have therefore two main characteristics : the interest rate R and the roll-over fee F .

We show that the optimal exploitative contracts depend crucially on the degree of naïveté of the borrowers. Sophisticated individuals (those who are fully aware of the dynamics of their preferences) cannot be fooled. They are offered contract with positive fees, where the fees are used as a commitment device (that allows them to implement their optimal consumption path). This is no longer the case with naïve borrowers. Fully naïves don't understand the commitment value of the fees, and therefore focus only on the interest rate (since they don't expect to pay the fee). The optimal contract is one with high interest rates and zero fees. Partially naïves understand the commitment value of the fee, but can nevertheless be fooled, since they overestimate their future discount factor. The optimal contract is one with high interest rates and a positive fee, that the borrowers end-up paying in some configurations. Deceitful contracts thus only arise with partially naïves borrowers in the monopoly case.

Most authors argue that competition should eliminate the predatory practices. We show that this is not the case in our framework. Perfect competition allows the sophisticated consumers to take the whole surplus (the equilibrium interest rate falls to marginal cost and the firm's profits are zero at equilibrium). But predatory contracts do persists as long as a there is a small fraction of naïves. The optimal equilibrium contract is clearly based on deceit, since the firms offer short term loans at discount rates (rates below marginal cost) that can only be profitable if a fraction of customers roll over the loan and thus end up paying the penalty fee they did not expect to pay. Perfect competition implies that the lending firms do not profit from their predatory practices in equilibrium (profits are null in equilibrium). The main beneficiaries of the deceitful contracts are the sophisticated borrowers who benefit from the discount rates, and use the penalty fee as a commitment device.

This paper is related to the growing literature on contracting with boundedly rational agents which started with the seminal contribution of Della [START_REF] Della Vigna | Contract design and self-control: Theory and evidence[END_REF] (henceforth referred as D&M). Their model is a three period version of the (δ, β, β) model of O'Donoghue andRabin (1999a, 2001). They show that the optimal two-part contract depends on the nature of the good provided by the firm. They distinguish two types of goods : leisure good, which provides current benefits, but have future costs and investment good, that have current costs and future benefits. A monopoly that sells a leisure good should choose a low entry cost and a high usage cost (i.e. above marginal cost) to take advantage of the naïve's tendency to underestimate their future usage. If the firm sells an investment good, the optimal pricing policy is the converse : a high fixed cost and a low usage cost (below marginal cost), in order to exploit the semi-naïves tendency tendency to overestimate their future usage as well as their demand for commitment (which is provided by the high entry cost). [START_REF] Heidhues | Futile attempts at self-control[END_REF] focus on over-commitment. Semi-naïves individuals who want to avoid the future consumption of an harmful good (a leisure good using D&M terminology), have the opportunity to buy a costly commitment technology. But most of them end up buying the good anyway as they underestimate their lack of self-control. The authors show that higher sophistication (greater awareness of the self control problem) is often worse than full naïveté, since it increases the spending on commitment without preventing the consumption of the harmful good. [START_REF] Heidhues | Exploiting naivete about self-control in the credit market[END_REF] studies exploitative contracts in the credit card and mortgage market. Even though they assume full competition, their results are similar to D&M : the equilibrium contracts have low teaser rates for short term loans and large penalty fees for those who fall behind on their payment schedule.

The subject of [START_REF] Gottlieb | Competition over Time-Inconsistent consumers[END_REF] is competition over time-inconsistent consumers. They point that the impact of competition on the pricing of leisure goods depends crucially on the customer's switching costs. D&M pricing (lump-sum transfers to bait customers and above marginal cost usage price) requires exclusive contracts. When those contracts are not feasible, firms choose marginal cost pricing (and zero lump-sum transfers).

Others significant contributions are [START_REF] Eliaz | Contracting with diversely naive agents[END_REF] and [START_REF] Gabaix | Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets[END_REF]. Both show how firms may profit by devising contracts that discriminate between sophisticated and naïve customers. In the Eliaz & Spiegler setting, the agents differ in their ability to forecast the change in their future tastes. The optimal menu of contract provides perfect commitment devices for the sophisticated consumers, and exploitative contracts for the naïve. Gabaix & Laibson analyze the issue of informational shrouding. Firms offer a base good and and add-ons. They show how firms may exploit myopic consumers through marketing schemes that offers cheap base goods and shroud high-priced add-ons, and how sophisticated consumers profit from those schemes. They also show that competition does not eliminate shrouding.

Empirical evidence on the pricing of investment good is provided by Della [START_REF] Della Vigna | Paying not to go to the gym[END_REF] in the context of the health club markets. Della Vigna and Malmendier (2004) also gives (second hand) evidence of the pricing of leisure goods with switching costs (credit cards, mobile phones). [START_REF] Gottlieb | Competition over Time-Inconsistent consumers[END_REF] provides evidence of the pricing on leisure goods with switching costs in competitive industries (Tobacco and and alcohol industries). [START_REF] Gabaix | Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets[END_REF] provide evidence on shrouding in the retail banking industry and in the printer market. First hand evidence on the practices of the credit card industry can be found in [START_REF] Ausubel | The failure of competition in the credit card market[END_REF] and [START_REF] Shui | Time inconsistency in the credit card market[END_REF] A full presentation of the payday loan market can be found in [START_REF] Stegman | Payday lending[END_REF] and [START_REF] Agarwal | Payday loans and credit cards: New liquidity and credit scoring puzzles[END_REF].

The paper proceeds as follows. Section 2 presents the model. Sections 3 and 4 analyze the optimal pricing strategy under monopolistic conditions. The form and the impact of the contract depend crucially on the degree of naïveté of the borrowers. Section 3 establishes the optimal contract for sophisticated borrowers, whereas section 4 show how firms can take advantage of the borrower's naïveté. Section 5 analyzes the effect of competition on the equilibrium pricing strategies. Section 6.1 studies the impact of predatory lending on the welfare of the borrowers under monopoly and competition. Section 7 concludes. All proofs omitted in the main text are given in the appendix.

The Model

Intertemporal Preferences

We assume that the borrowers have (δ, β, β) quasi-hyperbolic intertemporal preferences At time t, the present value of the flow of future utilities (u s ) s≥t is :

u t + β ∑ s≥t+1 δ s-t u s (1)
where δ ≤ 1 is the long-run discount factor and β ≤ 1 is the short-run one. β < 1 results from a self-control problem (the inability to resist to the urges of immediate consumption) Quasi-hyperbolic preferences (henceforth referred as q-hyper pref) imply that discounting is time-varying, since the discount factor between t + 1 and t + 2 is δ at time t and βδ at time t + 1. Time consistency thus requires β = 1. β can therefore be considered as a measure of the degree of time inconsistency of the agent. The parameter β ≤ β ≤ 1 is the expected short-run discount factor. A naïve agent overestimate his future discount factor (he expects -erroneously -to have the discount function 1, βδ, βδ 2 , .. in all future periods). An agent with β = 1 is said to be fully naïve, whereas an agent with β = β is a sophisticated one. β -β is therefore a measure of the degree of the agent's naïveté (the tendency to believe that his lack of self-control is temporary)

For the sake of simplicity we assume from now on that : (i) individuals live three periods ; (ii) their instantaneous utility is linear ; and (iii) the long-run discount factor is δ = 1.

The lending institution

The lending institution is a fully rational profit maximizing monopoly. The firm has zero fixed and marginal costs (the lending institution can borrow at zero cost from the central bank, her discount rate is thus zero). The firm proposes two types of borrowing contracts : a short-run contract and a long run one.

• The short-run contract (henceforth referred as SR contract). The firm lends for one period (at interest rate R), and gives the individual the possibility to rollover the debt for one period more. The option to rollover is costly. An individual that chooses to rollover must pay a penalty F on the outstanding debt. The total cost of borrowing is therefore R for the borrowers that repay in time and R(R + F ) for the borrowers that choose to rollover. Profit is therefore R -1 per unit lent in the former case, and R(R + F ) -1 in the latter case (recall that the firm's discount rate is zero).

• The long-run contract (LR). The firm lends for two periods at interest rate R (per period), which yield a profit equal to R 2 -1.

The firm is fully aware of the self-control and naïveté problems. Contracts are designed to exploit them.

The distribution of (β, β)

The borrowers are heterogenous. There is a continuum of agents of size one. We assume that the short-run discount factor β are uniformly distributed on the support [β 0 , 1], where β 0 is the short-run discount factor of the most impatient borrower.

We also assume that the expected short-run discount factor β is an increasing function of β, with :

β(β) = { β if β < β β if β ≥ β (2)
This formulation implies that all individuals with β < β are naïves, whereas all individual with β ≥ β are fully sophisticated. The naïveté and the lack of self-control are correlated since β -β is decreasing in β.

Contracting with sophisticated borrowers

We assume in this section that all the borrowers are fully sophisticated ( β = β 0 ). We first study the optimal contract when penalties are not allowed. Then we show that penalties increase both the profit of the firm and the welfare of the borrowers.

3.1 The optimal contract when penalties are not allowed At time t = 1, the net benefit of borrowing one unit is (1 -βR) if the agent repays at time t = 2, and (1 -βR 2 ) if the agent repays at time t = 3. One-period borrowing is thus better for the agent (the converse is true for the firm, who earns R -1 with the short term contract and R 2 -1 with the long term one).

All agents with β ≤ 1/R wish to borrow for one period. Yet, sophisticated agents know that their intertemporal preferences will change at time t = 2. The net benefit of deferring the repayment, which is R -R = 0 at time t = 1, will be, at time t = 2, (1 -βR). Therefore all agents who choose to borrow for one period at time t = 1 will rollover at time t = 2, and end up paying R 2 at time t = 3. Since two-period borrowing is profitable only for agents with β ≤ 1/R 2 , all agents with β > 1/R 2 will choose not to borrow. Figure 1 represents the choice of the individuals in the [β 0 , 1] space. This implies that the interest rate chosen by the firm must not exceed 1/ √ β 0 (since no one borrows when β 0 R 2 > 1), in which case the firm earns R 2 -1 per unit lent to each agent in the interval [β 0 , 1/R 2 ]. Assuming that the agents can borrow at most one unit1 , the firm's profit function Π(R, F ) writes :

1/R 1/R 1 β 0 2-period β 2 Contract
Π(R, 0) = { (R 2 -1)G(1/R 2 ) if 1 ≤ R ≤ 1/ √ β 0 0 if R > 1/ √ β 0 (3)
where

G(β) = (β -β 0 )/(1 -β 0 ) is the cumulative distribution function of β. It is easy to show that the profit function is concave in the interval [1, 1/ √ β 0 ]. The maximum profit is reached when R = R * 0 ≡ (1/β 0 ) 1/4 .

The optimal contract with penalties

Introducing penalties will increase both the firm's profit and the average welfare of the borrowers.

The reason is that penalties are used as a commitment device by the borrowers. Recall that all borrowers with β ∈]1/R 2 , 1/R[ wish to borrow (since 1-βR > 0) but choose not to since they know that they will rollover, and rollover is not profitable for them. A penalty changes the net benefit of deferring repayment at t = 2, which is now 1 -β(R + F ), which implies that rollover is no longer profitable for all agents with β > 1/(R + F ). Knowing this, all agents with β ∈ [1/(R + F ), 1/R] choose to borrow for one period. The behavior of the other agents is not modified. Figure 2 represents the choice of the sophisticated borrowers in the [β 0 , 1] space when F is positive. Let's compute the profit, assuming that the interest rate is

1/R 1/R 1 β 0 2-period 1-period 1/(R+F) β 2 Contract Contract
R = R * 0 and that 1/(R * 0 ) 2 ≤ 1/(R * 0 + F ).
The profit of the firm is the sum of the profit made on the 2-period contracts Π(R * 0 , 0) and the profit made on the one-period contract.

Π(R * 0 , F ) = Π(R * 0 , 0) + (R -1) [G(1/R * 0 ) -G(1/(R * 0 + F ))] (4)
which is clearly an increasing function of F . The maximum is reached when 1

/(R * 0 ) 2 = 1/(R * 0 + F ) ⇒ F * 0 = R * 0 (R * 0 -1)
. This results from the fact that choosing a penalty above

F * 0 allows the individuals in the interval [1/(R * 0 + F ), 1/(R * 0 ) 2 ]
to replace their 2-period contract by one-period ones, which clearly reduces the profit of the firm by (R * 0 )2 -R * 0 per contract.

To compute the optimal (R S , F S ) we must write the profit function. Assuming that

F < 1/β 0 -1/ √ β 0 , the profit function writes : Π(R, F ) =              (R 2 -1)G ( 1 R+F ) + (R -1) [ G ( 1 R ) -G ( 1 R+F )] if 1≤R<R F (R 2 -1)G ( 1 R 2 ) + (R -1) [ G ( 1 R ) -G ( 1 R+F )] if R F ≤R< 1 √ β 0 (R -1) [ G ( 1 R ) -G ( 1 R+F )] if 1 √ β 0 ≤R< 1 β 0 -F (R -1)G ( 1 R ) if 1 β 0 -F ≤R≤ 1 β 0 (5)
where

R F = (1 + √ 1 + 4F )/2 is the unique positive solution of the equation 1/R 2 = 1/(R + F ). 2
Proof. It is obvious that no one borrows for R > 1/β 0 . One period contracts are chosen by all individuals when 1/β 0 -F ≤ R ≤ 1/β 0 . This results from the fact that they are profitable for all individuals (since βR < 1, ∀β) and that the borrowers know that they will not rollover (since β(R + F ) > 1, ∀β). The profit is (R -1) times the number of contracts G(1/R). Note that 2-period contracts are nor an alternative since βR 2 > 1, ∀β. One-period contact are still the only alternative when 1/ √ β 0 ≤ R ≤ 1/β 0 -F , but they will be chosen only by a fraction of the individuals, since all the agents will β such that β(R + F ) ≤ 1 know that they will rollover. The profit is therefor R -1 times the number of contracts

G(1/R) -G(1/(R + F )). Choosing R ≤ 1/
√ β 0 allows the individual to choose between one-period and 2-period contracts. One-period contracts will be chosen by all those who don't expect to rollover (those with β such that β(R + F ) > 1), the others will chose either a 2-period contract (if βR 2 < 1) or to abstain from borrowing

(if 1/R 2 < β < 1/(R + F )). The no-borrowing zone disappears when 1/R 2 > 1(R + F ) ⇐⇒ R > R F . Thus the profit is R 2 -1 times the number of 2-period contracts (G(1/R 2 ) when R ≥ R F , and G(1/(R + F )) when R < R F ) plus R -1 times the number of one-period contracts G(1/R) -G(1/(R + F )).
As we can see in figure 3, the maximum profit is reached for R S = R F . The penalty is set at a level that : (i) ensures all agents choose either a one-period contract or a two-period contract ; (ii) avoids the cannibalization of 2-period contracts by one-period contracts. In the appendix A.2, we show that the optimal (R S , F S ) couple is :3

0 F = 0 F ≥ 1/β 0 -1 F = F S R F R Π(R, F S ) 1/β 0 -F S 1/β 0 1 1/ √ β 0
R S = ( 1 β 0 ) 1/3 (6) F S = R S (R S -1) = ( 1 β 0 ) 1/3 ( 1 - ( 1 β 0 ) 1/3 ) (7)

Contracting with naïve borrowers

We first we study the full naïveté case ( β = 1). Then we look at the partial naïveté case β 0 < β < 1

Full naïveté

In this case the potential borrowers don't expect to rollover (since β(R + F ) = R + F > 1, ∀R > 1, F ≥ 0). Thus all the agents who wish to borrow one period (those with β ≤ 1/R) will do so. Of course, all agents with β < 1/(R + F ) will rollover and pay the penalty. Now the firm must balance the additional income generated by the penalty fees and the loss of some 2-period borrowers due to the penalty (recall that the agents in the [1/(R + F ), 1/R] interval do not rollover). Figure 4 represents the choices of the individuals when β 0 < 1/(R + F ) : The profit function of the firm writes :

1/R 1 β 0 1-period Contract 1-period 1/(R+F) β + Rollover Contract
Π(R, F ) = { (R(R+F )-1)G( 1 R+F )+(R-1)[G( 1 R )-G( 1 R+F )] if 1≤R< 1 β 0 -F (R-1)G( 1 R ) if R≥ 1 β 0 -F (8)
Proof. All agent with β ≤ 1/R borrow for one period. If R ≥ 1/β 0 -F the penalty is big enough to deter rollover for all agents (β(R + F ) > 1, ∀β). The profit is (R-1) times the number of contracts G(1/R). This is no longer the case for R < 1/β 0 -F . The agents whose β < 1/(R+F ) will rollover and pay the penalty. The profit is (R(R+F)-1) times the number individual who rollover

G(1/(R + F )) plus R -1 times the number of individuals who don't G(1/R) -G(1/(R + F )).
It is easy to show that (i) the firm chooses a contract with zero penalty ( F = 0); and (ii) the optimal interest rate R N is such that 1/ √ β 0 < R N < 1/β 0 (see the appendix for the proof). The figure 5 shows the optimal solution.

The logic of the no-penalty result is straightforward. Introducing a penalty F leads to a gain of F on the G(1/(R + F )) individuals that continue to rollover and a loss of R 2 -R on the G(1/R) -G(1/(R + F )) individuals that no longer rollover. The net gain is :

F × G ( 1 R + F ) -R(R -1) [ G ( 1 R ) -G ( 1 R + F )] = F 1 R+F -β 0 1 -β 0 -R(R -1) 1 R -1 R+F 1 -β 0 = - 1 1 -β 0 F 2 R + F < 0
which is always negative. Adding a penalty is therefore never profitable when the borrowers are fully naïve.

0 1 1/β 0 R 1/ √ β 0 Π(R, 0) R N

Partial naïveté

There are three equilibrium configurations in the partial naïveté case, which we label : (i) nearly sophisticated ; (ii) sophisticated naïve ; and (iii) nearly naïve. In the following we characterize each configuration (full proofs are given in appendix C).

Configuration 1 : Nearly sophisticated

This is the equilibrium configuration when β is low ( β ∈ [β 0 , β1 [)4 . As can be seen in figure 6, this configuration is such that β < 1/R 2 = 1/(R + F ). Thus all the agents with β ≤ 1/(R + F ) (rightly) expects to rollover. Recall that the expected discount factor is β(β) = β ≥ β for the sophisticated agents and β(β) = β for the naïve ones, which implies that the expected discount factor is small enough ( β(β) < 1/(R + F ), ∀β < 1/(R + F )) to send the right signal to the naïve borrowers (do not sign the one-period contract since you will rollover!). Therefore all agents whose β > 1/(R + F ) choose the one-period contract, whereas those with β ≤ 1/(R + F ) choose the 2-period contract5 No penalty is ever paid.

The configuration 1 equilibrium is an "honest" equilibrium where no one is fooled, and the penalty fee F is used as a commitment device for both the naïve and sophisticated borrowers. The equilibrium (R, F ) pair is the same as the one chosen when all individuals are sophisticated (R . As can be seen in figure 7, this configuration is such that β = 1/(R + F ) > 1/R 2 . Therefore all borrowers choose the one-period contracts (no one expects to rollover since β ≥ β, ∀β). This is the right decision for the sophisticated agents, since they will not rollover (β ≥ 1/(R + F ), ∀β ≥ β), and the wrong decision for the naïve ones (since they will rollover and pay the penalty). The penalty has therefore a dual role in configuration 2. The penalty is used as a commitment device for the sophisticated agents (who would choose to abstain from borrowing if F = 0), and as a bait for the naïve ones. Note that the optimal behavior for the naïve agents would be either to borrow for two periods (for those with β < 1/R 2 ) or to abstain from borrowing (for those with β ≥ 1/R 2 ).

= R S = (1/β 0 ) 1/3 , F = F S = R S (R S -1)). 1/R 1 β 0 2-periods 1-period 1/(R+F)=1/R 2 β Contract Contract β _ Naïve Sophisticated
1/R 1 β 0 1-period Contract 1-period 1/(R+F)= β + Rollover Contract Naïve Sophisticated 1/R 2 β _
We show in appendix C that the optimal interest rate and penalty fee are : 7

R SN = √ β β 0 -β(1 -β) (9)
F SN = 1 β -R SN = 1 β - √ β β 0 -β(1 - β ( 10
)
The figure 8 represents the profit as a function of R when F = F SN .

6 The value of β2 is given in appendix C 7 The subscript SN stands for sophisticated naïve.

0 R 1/ √ β 0 1/β 0 Π(R, F S N ) 1/β 0 -F S N 1/ β -F S N

Configuration 3 : Nearly naïve

This is the equilibrium configuration when β is high ( β ∈ [ β2 , 1]). As can be seen in figure 9, this configuration is such that β ≥ 1/R. The optimal interest rate is such that : (i) sophisticated agents do not borrow ; and (ii) all naïve agents borrow for one period and then rollover in the second period. As in fully naive case, the choice of the optimal penalty fee must balance the additional income generated by the fee and the loss of some two-period contracts caused by this fee. We show in appendix C that, as in the fully naïve case the optimal penalty fee is zero.8 .

1 β 0 1-period Contract 1/R β + Rollover Naïve Sophisticated 1/R 2 _ β
β ∈ [ β2 , 1/R N [, is such that : 9 R N N = 1/ β (11) F N N = 0 (12) Configuration 3.2, which is the equilibrium configuration for β ∈ [1/R N , 1], is such that : R N N = R N (13) F N N = 0 (14)
where R N is the interest rate chosen by the firm when consumers are fully naïve (section section 4.1).

4.3

The impact of naïveté on the optimal strategies and the profit per customer

The figures 10, 11 and 12 plot the optimal interest rate, the optimal fee and the profit per customer as a function of the index of naïveté β when β 0 = 2/2. This section studies the impact of full competition on the equilibrium pricing strategies. We assume the following : (i) entry is unrestricted and costless, (ii) there are no capacity constraints, and (ii) firms compete by announcing a (R, F ) couple. We show that the impact of competition depends crucially on the degree of naïveté of the borrowers. We first study the polar cases (full naïveté and full sophistication), then we solve the intermediate case (β 0 < β < 1).

β R R S N R N R N S R S

Fully sophisticated versus fully naïves

Lets's examine the impact of competition when the borrowers are fully sophisticated. Competition is a two dimensions game, since firms may increase their market share either by reducing R or by increasing F . However, it is easy to show that choosing a slightly lower interest rate than the other firms is always the dominant strategy. Suppose for instance that there are two identical firms and let R 2 > 1 and F 2 > 0 be the interest rate and the fee chosen by firm 2. Firm 1 may increase its profit by choosing any

F 1 > F 2 , as long as (1/R 2 ) 2 < 1/(R 2 + F 2 ) (Firm 1 takes all the consumers in the interval [1/(R 2 + F 1 ), 1/(R 2 + F 2 )]
, which yields a net profit equal to (R -1) per consumer)10 . But decreasing marginally the interest rate (choosing R 1 = R 2 -ϵ) is a much better strategy, since it allows the firm to take the whole market (profits are multiplied by a factor slightly less than 2). Of course, the same logic applies for firm 2. Therefore the equilibrium is such that R 1 = R 2 = 1, which implies that the fees are no longer needed. The same reasoning applies for any number of firms. Firm's profits are null at equilibrium and all the surplus is taken by the consumers. Furthermore, time inconsistency is no longer a problem, since borrowing at period 2 at interest rate = 1 entails no loss from the point of view of the period one self.

Has competition the same beneficial effects when borrowers are fully naïve? Naïve borrowers are not aware of the benefits and dangers of the penalty fees, since they don't realize their time inconsistency problem. They, therefore, do not take into account the fee when comparing the borrowing contracts. The only relevant variable in the (R, F ) contract is the the interest rate, which implies that the firms are free to set the fee at the level they see fit. This open the way for a new strategy based on deceit.

Suppose, for instance, that all the other firms set R = 1 (the equilibrium level when agents are sophisticated). A positive profit can be made with the following strategy : offer a one period contract with a "discount" interest rate R < 1 and the possibility to rollover at cost R + F11 . All agents will take the contract, and then a fraction G(1/(R + F )) will rollover (those whose preferences are such that β ≤ 1/(R + F )). The firm will loose R -1 per contract on the one period borrowers (whose mass is 1 -G(1/(R + F )) and gain R(R + F ) -1 per contract on the G(1/(R + F )) individuals who will rollover. The expected profit of the firm thus writes :

Π(R, F ) = (R(R + F ) -1)G ( 1 R + F ) + (R -1) ( 1 -G ( 1 R + F )) (15)
which is positive for some R < 1 and F > 0 (see the end of appendix D.1). Of course, all other firms will offer the same contract. Then free entry implies, again, that the equilibrium will be a zero-profit one. A symmetric equilibrium will therefore be characterized by the following three conditions : (i) all firms choose the same interest rate R ; (ii) each firm chooses the penalty fee F so as to maximize her expected profit given R ; (iii) expected profits are null. In the appendix D.1, we show that these three conditions imply that the equilibrium (R, F ) are : 16)

R c N = 1 + √ β 0 2 (
F c N = 1 √ β 0 - 1 + √ β 0 2 (17)
where the superscript c stands for competition, and the subscript N for fully naïve.

The equilibrium with partially naïve agents

Recall that the agent's population is divided in two groups : the sophisticated ones (those with β ≥ β) and the partially naïves (those with β < β). Both groups are aware of their time inconsistency problem and, therefore, do realize the commitment value of the fee. The big difference between the two groups is that the partially naïve can be fooled by an appropriate contracts, whereas the sophisticated cannot.

The deceitful contract has the same structure as before : a one period lending contract with a "discount" interest rate R < 1, and the possibility to rollover at cost R + F . The key condition is that R and F must verify :

β ≥ 1 R + F (18)
Condition ( 18) implies that nobody expect to roll-over, since the expected discount factor is β(β) ≥ β ≥ 1/(R + F ). When the firms offer this kind of contract, the game is very much the same as the one played with fully naïve agents. The firms are free to set the level of the fee, since all agents consider that the fee is big enough to deter rollover (and nobody expects to pay the fee). Therefore firms compete by offering the lowest interest rate. Free entry implies a zero-profit equilibrium.

A symmetric equilibrium will therefore be characterized by the following three conditions : (i) all firms choose the same interest rate R ; (ii) each firm chooses the penalty fee F so as to maximize her expected profit given R subject to condition (18); (iii) expected profits are null. It is easy to show (see appendix D.2) that there are two equilibrium configuration.

The constrained equilibrium arises when the share of partially naïve agents is low ( β < √ β 0 ). F is such that all the partially naïve rollover R + F = 1/ β. Then, the zero profit condition yields :

R c P N = β(1 -β 0 ) β(1 -β 0 ) + (1 -β)( β -β 0 ) ( 19
)
F c P N = 1 β - β(1 -β 0 ) β(1 -β 0 ) + (1 -β)( β -β 0 ) ( 20
)
where the subscript P N stands for partially naïve.

The unconstrained equilibrium arises when the share of partially naïve agents is high ( β ≥ √ β 0 ). The firms make the same choice as in the fully naïve case. Therefore : 13) and ( 14) show the equilibrium F and R as a function of the index of naïveté β. Both the fee and the interest rate decrease with β (this is a general result, ∂R c P N /∂ β < 0 and ∂F c P N /∂ β < 0, ∀ β ∈]β 0 , 1]). Incumbent firms react to the increase of the share of the naïves G( β), by cutting the penalty fee F , which reduces the profit par naïve customer but increases the market share of the exploitative contracts. The latter effect dominates as long as 1/(R + F ) ≤ √ β 0 (recall that R + F = 1/ √ β 0 maximizes the expected profit for fully naïve borrowers). Positive profits lead to the entry of new firms, which brings down the interest rate.

R c P N = R c N ( 21
)
F c P N = 1 √ β 0 -R c N ≡ F c N (22) and R c P N +F c P N = 1/ √ β 0 > 1/ β. Figures (
Note that the pricing policy engenders a cross-subsidy from the naïves to the sophisticated individuals (and some of the naïves too when β ≥ √ β 0 ). Each one of the G(1/(R +F )) naïves that roll-over "pays" R(R +F )-1 to the 1-G(1/(R +F )) individuals that manage to stick to the one period loan. Note also that, somewhat counterintuitively, the intensity of exploitation (as measured by the penalty fee) is globally decreasing with the index of naïveté, and is maximal when the share of naïves id close to zero. Evaluating the borrower's welfare is difficult since preferences are time-varying. One solution, proposed by O' Donoghue and Rabin (1999a,b), is to distinguish between the long-run welfare and the actual welfare. Note that the intertemporal utility of the agent can be rewritten as :

u t + β ∑ s≥t+1 δ s-t u s = (1 -β)u t + βV LR t ( 23
)
where V LR t = ∑ s≥t δ s-t u s stands as the long run utility of the agent. With this formulation the actual intertemporal utility of the individual is an average of his short-run utility (u t ) and his long-run utility (V LR t ). Rabin asserts that V LR t , which is time-consistent, is the relevant measure of the agent's welfare.

With V LR t as the welfare index any type of borrowing is harmful in a monopoly setting since R > 1 (recall that δ = 1 and u t is linear). In this sense all contracts that lead some agents to borrow either for one period or two periods are exploitative in a monopoly setting. The aggregate welfare losses are just equal to the monopoly profits, which are an increasing function of the index of naïveté β (this is due to the fact that the net global surplus is equal to zero). However the individual losses depend on a non trivial way on the borrower's impatience β and naïveté β(β). When all the individuals are sophisticated, the more impatient (those who take a two-period loan) loose 1 -(R S ) 2 each, whereas the patient borrowers (those who take a one-period loan) loose 1 -R S each12 . When all the individuals are naïves, all those who borrow take a two-period loan which leads to a loss equal to 1-(R N ) 2 . This loss is clearly greater that the losses incurred by any sophisticated borrower, weather patient or impatient, since R N > R S . Yet the partially naïves may loose more than the fully naïves. The patient ones (those who borrow for one period) pay a higher interest rate (R SN > R N ), and the impatient ones end up paying a two-period loan plus a penalty fee, which leads to a loss equal to 1 -R SN (R SN + F SN ) < 1 -(R N ) 2 (her loss is greater than the loss sustained by a fully naïve that take a two-period loan). Partial naïveté is therefore worse than full naïveté for those who borrow.

Perfect competition drives the profits to zero. This has a positive impact on the average borrower's welfare, which is now equal to zero (recall that the net global surplus is zero). However the impact of competition on individuals crucially depends on their naïveté. Sophisticated individuals (those who manage to borrow for one period) can't loose from competition. When everybody is sophisticated ( β = β 0 ), the interest rate is R = 1, which implies that all individuals have a surplus equal to zero. When some of the individuals are naïve, the sophisticated earn a positive surplus since R < 1. Note that this surplus rises with the index of naïveté β (since R is a decreasing function of β). The gains of the sophisticated are the losses of the naïves (those who borrow for two periods and therefore pay the penalty fee). It must be noted that the losses of the naïves 1 -R(R + F ) are, generally, lower under competition. Let's show this when all individuals are fully naïve. Under monopoly, all individuals in the [β 0 , 1/R N ] interval borrow for one period and then rollover, which lead to a loss equal to 1 -(R N ) 2 (there is no penalty fee). Under perfect competition, the same applies for all the individuals in the [β 0 , √ β 0 ] interval, which lead to a loss equal to 1

-R c N (R c N + F c N ). The welfare loss under competition is lower since R c N (R c N + F c N ) = R c N / √ β 0 < 1/ √ β 0 < R N < (R N )
2 . Yet some naïves are made worse off by competition : those who do not borrow under monopoly (those whose 15 show the extent of two period borrowing under monopoly and perfect competition. The same logic applies under partial naïveté. The naïves that roll-over under monopoly and competition benefit from competition, since they pay lower rates (including the penalty fee). But the naïves who do not borrow under monopoly are made worse-off by competition. 13

β ∈]1/R N , √ β 0 ]). Figure

Welfare from the point of view of self one

Using the long run utility as a measure of welfare has one big drawback : it may lead to violate the individual's choice at each point of time. This certainly can be considered as hard paternalism. [START_REF] Sunstein | Libertarian paternalism is not an oxymoron[END_REF] preconize a softer approach, which they call libertarian paternalism 14 . The purpose of libertarian paternalism is to help the individual to make freely the "right" choices. This implies that another welfare measure must be chosen. Utility at time one seems to be the natural choice, since it represents the wills of at least one of the multiple selves of the individual and is not too far from the long-run utility 15 . Note that the credit market is no longer useless, as it fulfills the "legitimate" need for credit of self one. Lending for one period creates a surplus equal to 1 -δ, which is shared between the bank and the borrower.

Sophisticated agents cannot loose from borrowing. Their need for commitment devices is partially fulfilled under monopoly (they would prefer a higher penalty fee), and entirely fulfilled under competition (since the optimal contract deters them from renewing the loan). Competition clearly benefits them since they take the whole surplus of the exchange (plus a transfer from the naïves).

Naïves may loose from borrowing under monopoly, since they over-estimate the benefits of the loan. For instance when all the borrowers are fully naïve, of all the individuals who borrow and then roll-over the loan, all those whose β ∈](1/R N ) 2 , 1/R N ] have a welfare loss. This is no longer the case under perfect competition. The individuals that roll-over the loan have a welfare gain equal to 1

-βR c N (R c N + F c N ) = 1 -βR c N / √ β 0 > 0 (since β < √ β 0 ).
In fact all naïves are made better off by competition, even those who do not 13 There are in fact two cases. When the index of naïveté is low ( β ≤ √ β0), the same population borrows and roll-over under monopoly and competition (those whose β ∈ [β0, β]). Therefore nobody looses from competition. When the index of naïveté is high ( β > √ β0), a fraction of the individuals that borrow and roll-over under competition do not so under monopoly (those whose β ∈]1/RN , √ β0]). This is the population that looses from competition.

14 [START_REF] Camerer | Regulation for conservatives: Behavioral economics and the case for asymmetric paternalism[END_REF] use the term "asymmetric paternalism", others prefer the slightly ironic "soft paternalism". 15 We mean, of course, self one real utility (the one he will derive by his effective consumption path), as opposed to his expected utility.

roll-over under monopoly 16 . These results extend to the partially naïve case 17 .

Conclusion

This paper studies the equilibrium predatory practices that may arise when the borrowers have behavioral weaknesses. Rational lenders offer short term contracts that can be renewed at the cost of paying a penalty fee. We show how the optimal contracts depend on the degree of naïveté of the time inconsistent customers. Penalty fees have a dual role : they increase market share by providing a useful commitment device to time-inconsistent but otherwise rational borrowers ; they are also a source of revenue from the semi-naïve borrowers who understand the need for commitment but fail to forecast their future time discount factor. We also show that perfect competition does not eliminate predatory practices, since the equilibrium contract entails a subsidized (below marginal cost) short-term loan that can only be profitable if a fraction of the borrowers end up paying the penalty fee.

16 Proof. The welfare gain for the fully naïve is, under monopoly : 1 -β(RN ) 2 if β < 1/RN and 0 if β ≥ 1/RN ; and under perfect competition 1

-βR c N / √ β0 if β < √ β0, and 1 -βR c N if β ≥ √ β0. 1/RN < √ β0, therefore we have three cases: (i) β ∈ [β0, 1/RN [, 1 -β(RN ) 2 < 1 -βR c N / √ β0 since RN > 1/ √ β0 > 1 > R c N ; (ii) β ∈ [1/RN , √ β0[,0 < 1 -βR c N / √ β0, since β < √ β0 and R c N < 1; (iii) β ∈ [ √ β0, 1],0 < 1 -βR c N , since R c N < 1. 17 Proof. If β ≥ √ β0... Π ′′ 1 (R) < 0 implies that Π 1 (R) has a unique maximum and Π ′ 1 (1/ √ β 0 ) = √ β 0 -1 β 0 -1 < 0 =⇒ R * < 1/ √ β 0 . R * < 1/ √ β 0 implies that 1/(R * + F ) < (1/R * ) 2 (since R F > 1/ √ β 0 )
. Such a configuration can't be optimal since the profit can be increased by reducing F (which allows the firm to replace one-period contracts by the 2-period ones). This shows that the optimal F is such that F < 1/β 0 -√ 1/β 0 .

A.2 The optimum

Taking into account that the optimal F is such that F = R F (R F -1), we can write the profit function as :

Π S (F ) = ( R 2 F -1 ) G ( 1 R 2 F ) + (R F -1) ( G ( 1 R F ) -G ( 1 R F + F )) = R F (R F -1) G ( 1 R 2 F ) + (R F -1) G ( 1 R F ) = 1 1 -β 0 ( F × ( 1 R 2 F -β 0 ) + (R F -1) ( 1 R F -β 0 )) Using R ′ F = dR F dF = 1 2R F -1 and F = R F (R F -1)
we compute the first and second derivatives :

Π ′ S (F ) = 1 1 -β 0 ( 1 R 2 F -β 0 -2F R ′ F R 3 F + R ′ F R 2 F -β 0 R ′ F ) = 2 1 -β 0 × R 2 F -β 0 R 4 F -F R 3 F (2R F -1) = 2 1 -β 0 × 1 -β 0 R 3 F R 2 F (2R F -1) Π ′′ S (F ) = 2R ′ F 1 -β 0 × β 0 R 3 F -6R F + 2 R 3 F (2R F -1) 2 = - 2 1 -β 0 × 3 R 3 F (2R F -1) 2 < 0 Since Π ′′ S (F ) < 0 the maximum of the function Π S is attained when Π ′ S (F ) = 0 ⇔ R F = (1/β 0 ) 1/3 .

B Fully naïve borrowers

Since F N = 0, R N is the maximum of the function :

Π N = Π(R, 0) = (R 2 -1)G(1/R) = (R 2 -1)(1/R -β 0 ) 1 -β 0 in the interval [1, 1/β 0 ]. The function Π N (R) is concave (since Π ′′ N (R) = -2 β 0 +1/R 3 1-β 0 ). The maximum is the solution of Π ′ N (R) = 1 1 -β 0 ( 1 -2β 0 R + 1 R 2 ) = 0
which yields a cubic equation (-

2β 0 R 3 +R 2 +1 = 0). Since Π ′ N (1) = 2 > 0 and Π ′ N (1/β 0 ) = -(1 + β 0 ) < 0, this equation has a unique solution in the interval [1, 1/β 0 ]. Π ′ N (1/ √ β 0 ) = (1 - √ β 0 ) 2 /(1 -β 0 ) > 0 implies that 1/ √ β 0 < R N < 1/β 0 .

C.2 The profit maximizing strategy when the borrowers are partially naïve

It is easy to see that the profit function is not convex, and has several discontinuities (the profit jumps between region 2,3 and 4 and between region 2 and region 5). Our method is first to compute the sign of the partial derivatives in each region in order to locate the local maxima, and then compare those maxima. The arrows in figures 16, 17 and 18 show the "directions" that the firm must follow to increase her profit.

We distinguish three cases.

Case 1:

β 0 ≤ β < √ β 0 and R SN = √ β β 0 -β(1-β) < 1 β
The optimum profit strategy are :

   (S) R = ( 1 β 0 ) 1/3 ≡ R S and Z = ( 1 β 0 ) 2/3 if β 0 ≤ β < β1 (SN) R = √ β β 0 -β(1-β) ≡ R SN and Z = 1/ β if β1 ≤ β < β2
where β1 < β2 < 1 are given at the end of the subsection.

Proof.

First we show that the optimal strategy in this configuration must be either (S) or (SN ) (see figure 16). Let's begin with region 1. The profit function does not depends on Z. The derivative with respect to R is :

(1 -β 0 ) ∂Π 1 (R, Z) ∂R = 1 R 2 -β 0 ( < > ) 0 if R ( > < ) 1 √ β 0
The local maximum is therefore attained when R = 1/ √ β 0 (∀Z ≥ 1/β 0 ). But the global optimum cannot lie in region 1 since Π

2 (R, Z) -Π 1 (R, Z) = (RZ -1)G(1/Z) > 0 ∀Z > 1.
Let's move to region 2. The partial derivatives of the profit function are :

(

1 -β 0 ) ∂Π 2 (R, Z) ∂Z = R ( 1 Z 2 -β 0 ) < 0 since Z > 1 β > 1 √ β 0 (1 -β 0 ) ∂Π 2 (R, Z) ∂R = (Z -1) ( 1 Z -β 0 ) + 1 R 2 -β 0 ( < = > ) 0 if R ( > = < ) ϕ(Z) where ϕ(Z) = √ Z 1-Z(1-β 0 Z
) is a continuous function that crosses the frontier with region 3 in R = R SN and the frontier with region 1 in R = 1/ √ β 0 . Since ∂Π 2 /∂Z < 0, the local maximum lies in the western frontier of region 2 (Z = 1/ β)18 , whereas the sign of ∂Π 2 /∂R implies that the optimal interest rate is R SN = ϕ(1/ β) (see figure 16). Now we examine region 3. The partial derivatives are :

(

1 -β 0 ) ∂Π 3 (R, Z) ∂Z = R -1 Z 2 > 0 (1 -β 0 ) ∂Π 3 (R, Z) ∂R = 1 R 2 - 1 Z < 0 since R > √ Z
R and Z are therefore pushed toward the frontiers with region 4 and 2 (see figure 16). Note that the profit function jumps upward when the frontier between region 3 and region 2 is crossed, since :

Π 2 (R, 1/ β) -Π 3 (R, 1/ β) = β -β 0 1 -β 0 ( R β -1 ) > 0
This implies that the optimum can't lie in region 3. Now let's examine region 4 and 5. The partial derivatives are :

(

1 -β 0 ) ∂Π 4 (R, Z) ∂Z = R -1 Z 2 > 0 (1 -β 0 ) ∂Π 4 (R, Z) ∂R = 2 ( 1 R 3 -β 0 ) + 1 R 2 - 1 Z ( < > ) 0 if Z ( < > ) ψ(R) (1 -β 0 ) ∂Π 5 (R, Z) ∂Z = - R(R -1) Z 2 < 0 (1 -β 0 ) ∂Π 5 (R, Z) ∂R = (2R -1) ( 1 Z -β 0 ) + 1 R 2 - 1 Z > 0 since R ≤ √ Z where ψ(R) = ( 2 ( 1 R 3 -β 0 ) + 1 R 2 ) -1 is an increasing function of R which cross the R = Z line in Z = Z 1 ∈]1, (1/β 0 ) 2/3
[ and the frontier between region 4 and 5 in R = (1/β 0 ) 1/3 and Z = (1/β 0 ) 2/3 .

The sign of the derivatives imply that the local maximum lies in the frontier between region 4 and 5. It is easy to check that the local optimum is such that : R = (1/β 0 ) 1/3 = R S and Z = (1/β 0 ) 2/3 = Z S (see figure 16) 19 . Now we show that strategy (S) is chosen for low β whereas strategy (SN) is optimal otherwise. Let's compute the optimal profit for both strategies :

Π S = Π 4 (R S , Z S ) = 1 -β 1/3 0 1 -β 0 ( 2 -β 1/3 0 -β 2/3 0 ) Π SN = Π 2 (R SN , Z SN ) = 1 1 -β 0 ( 1 + β 0 - 2 R SN ) Thus Π S > Π SN when : R SN = √ β β 0 -β(1 -β) > 2 3β 1/3 0 -1 which is verified for all β ∈ [β 0 , β1 [, where β1 = -b- √ b 2 -4ac 2 (with a = 1, b = -(1+(3β 1/3 0 - 1) 2 /4) and c = β 0 ) is the unique positive solution of the equation : β2 - ( 1 + 1 4 ( 3β 1/3 0 -1 ) 2 ) β + β 0 = 0
Now we compute the upper bound β2 . The strategy (SN) is chosen when β ≥ β1 and :

R SN ≤ Z SN = 1/ β ⇐⇒ √ β β 0 -β(1 -β) ≤ 1/ β
β2 therefore solves the following cubic equation :

Υ( β) = β3 -β2 + β -β 0 = 0
It is easy to check that equation Υ( = 0 has a unique solution in the interval

[β 1 , 1]. 20 19 The optimum strategy is RS = argmax Π4(R, R 2 ) and ZS = R 2 . 20 Υ(β0) = β 2 0 (β0 -1) < 0, Υ(1) = 1 -β0 > 0 and Υ ′ ( β) = 3 β2 -2 β + 1 > 0, ∀ β ∈ [β0, 1] Case 2: β 0 ≤ β < √ β 0 , R SN = √ β β 0 -β(1-β) ≥ 1 β and R N < 1/ β
The optimum profit strategy is :

(NS) R = 1/ β = R N S and Z = 1/ β = Z N S for β2 ≤ β < β3
where β3 is given at the end of the subsection.

Proof.

As before (see figure 17) there is a local optimum at the frontier between region 4 and 5 (R = R S and Z = Z S ). We shall prove that this optimum can't be the global one

The other local optimum lies either at the western frontier or at the northern frontier of region 2 (since ∂Π 2 /∂Z < 0). The profit maximizing strategy at the western frontier (the one such that Z = 1/ β) is R = Z = 1/ β since ∂Π 2 (R, 1/ β)/∂R > 0. This strategy is also the profit maximizing strategy at the northern frontier since R N < 1/ β ⇒ dΠ 2 (R, R)/dR < 0∀R ≥ 1/ β. 21 Therefore β3 = 1/R N (since R = Z = 1/ β is no longer the profit maximizing strategy when R N ≥ 1/ β).

Let's show that the (NS) strategy is better than the (S) strategy when β ∈ [ β2 , β3 [. The profit of the (NS) strategy writes : As seen in figure 18, there is a local optimum at the frontier between region 4 and 5 and another at the northern frontier of region 2 (since dΠ 2 /dR > 0). The profit function in the latter case writes:

Π N S = Π 2 ( 1/ β, 1/ β) = β -β 0 1 -β 0 ( 1 β2 -
Π N = Π 2 (R, R) = ( R 2 -1 ) G ( 1/R 2 )
which is also the profit when the agents are fully naïve. The unique maximum of Π N is attained when dΠ N /dR = 0 ⇔ R = R N (see section B for the full proof).

21 Recall that RN is the profit maximizing strategy for fully naïve agents.

22 Let dΠNS/d β = Ω( β). Simple algebra yields Ω( β2) = - The equilibrium is such that : (i) all firms choose the same R ; (ii) each firm chooses F so as to maximize her expected profit given R ; (iii) expected profits are null. Assuming that there is a continuum of firms of size m (the variable m is unimportant), condition (ii) implies :

F * (R) = arg max F Π i (R, F ) = 1 m [ (R(R + F -1)G ( 1 R + F ) + (R -1)
]

where Π i (R, F ), the profit of firm i, is a concave function of F . Thus :

∂Π i (R, F ) ∂F = 1 m R 1 -β 0 ( 1 (R + F ) 2 -β 0 ) = 0 =⇒ F * (R) = 1 √ β 0 -R
The zero profit condition (iii) then writes :

Π i (R, F * (R)) = 1 m [ 2R 1 + √ β 0 -1 ] = 0 which yield R c N = (1 + √ β 0 )/2 and F c N = F * (R c N )
. We use these calculations to show that the expected profit of a firm that chooses R < 1 and F > 0 when all other firms choose R = 1 is positive for some F . Let R = 1 -ϵ and F * (1 -ϵ) be the interest rate and the fee chosen by the firm. Her profit writes :

Π i (1 -ϵ, F * (1 -ϵ)) = [ 2(1 -ϵ) 1 + √ β 0 -1 ] which is positive ∀ ϵ < (1 - √ β 0 )/2.

D.2 Partially naïve borrowers

The fee is chosen so as to maximize the expected profit per consumer subject to condition (18) :

F * * (R) = arg max F Π i (R, F ) = 1 m [ (R(R + F -1)G ( 1 R + F ) + (R -1) ] s.t. R+F ≥ 1/ β
This program has two solutions : constrained and unconstrained. The unconstrained solution is the one given in appendix D.1. It arises when R + F * = 1/ √ β 0 ≥ 1/ β. The constrained solution is such that : R + F * * = 1/ β and ∂Π ∂F < 0 (which is a consequence of R + F * * = 1/ √ β 0 < 1/ β). Then the profit writes :

Π i (R, F * * (R)) = R ( 1 β -1 ) β -β 0 1 -β 0 + R -1
The zero profit condition Π i (R, F * * (R)) = 0 yield the equilibrium value R c P N and F c P N = 1/ β -R c P N .

Figure 1 :

 1 Figure 1: The Choices of sophisticated borrowers (F = 0)

Figure 2 :

 2 Figure 2: The Choices of sophisticated borrowers (F > 0)

Figure 3 :

 3 Figure 3: The profit function with sophisticated borrowers for F = F S

Figure 4 :

 4 Figure 4: The Choices of fully naïve borrowers

Figure 5 :

 5 Figure 5: The profit function with fully naïve borrowers

Figure 6 :

 6 Figure 6: Configuration 1 : Nearly sophisticated

Figure 7 :

 7 Figure 7: Configuration 2 : Sophisticated naïve

Figure 8 :

 8 Figure 8: The profit function with sophisticated naïves.

Figure 9 :

 9 Figure 9: Configuration 3 : Nearly naïve There are two sub-configurations, depending on the value of β. Configuration 3.1, which is the equilibrium configuration for β ∈ [ β2 , 1/R N [, is such that : 9

Figure 10 :FFigure 12 :

 1012 Figure 10: The optimal interest rate as a function of the index of naïveté β

Figure 13 :Figure 14 :

 1314 Figure 13: The equilibrium interest rate under competition as a function of the index of naïveté β 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Figure 15 :

 15 Figure 15: Exploitative contracts under monopoly and perfect competition with fully naïve agents

)

  It is easy to show that dΠN S /d β > 0 if β2 ≤ β < 1/R N 22 . This and Π N S = Π SN > Π S for β = β2 imply that Π N S > Π S ∀ β ∈ [ β2 , β3 [. Case 3: R N ≥ 1/ β 23The optimum profit strategy is :(N) R = R N and Z = Z N for β3 ≤ β ≤ 1Proof.

Figure 16 :Figure 17 :Figure 18

 161718 Figure 16: A mapping of the profit function when β 0 < β < √ β 0 and R SN < 1/ β

This assumption is not essential. All we need is that all agents borrow the same amount.

The restriction F < 1/β0 -1/ √ β0 ensures that RF lies in the interval [1, 1/ √ β0]. We show in appendix A.1 that the optimal F lies in the interval [0, 1/β0 -1/ √ β0].

The subscript S stands for sophisticated.

The value of β1 is given in appendix C

The firm chooses R such that 1/R 2 = 1/(R + F ) since 1/R 2 < 1/(R + F ) leads to the loss of valuable consumers and 1/R 2 > 1/(R + F ) leads to the cannibalization of 2-period contracts by the less profitable one-period contracts.

Note that the borrowers can be considered as fully naïve (even if they have on average an expected β closer to their true β, they act exactly as if they were fully naïves in equilibrium)

The subscript N N stands for nearly naïve.

This applies as long as 1/(R2 + F1) ≥ (1/R2) 2 . Increasing further R1 leads to a loss for firm 1, due to the cannibalization of the 2-period contracts by the 1-period ones.

The firm may also propose a portfolio of two contracts, where contract 1 is the standard contract at rate R = 1 (with costless rollover) and contract 2 is the discount contract (R < 1), with a penalty F in case of rollover. The outcome will be the same, since the expected surplus from contract 1 (1 -β) is lower than the expected surplus for contract 2 (1 -βR) for all naïves.

Note that there is always a fraction a of the potential borrowers (the very patient) that choose to abstain from borrowing.

The local maximum can't lie in the line R = Z since ∂Π2/∂R < 0 when R = Z.

A Sophisticated borrowers

A.1 The profit function when

In this case, the profit function writes :

Proof. All individuals choose one-period contract when 1/β 0 -F ≤ R ≤ 1/β 0 , since they are profitable (βR < 1, ∀β) and the individuals know that they will not rollover (β(R + F ) > 1, ∀β). The profit is therefore (R -1) times the number of contracts

. From this we know that : (i) some individuals will roll-over (those such that β < 1/(R + F )); (ii) 2-period contracts are profitable for those individuals (since 1/(R + F ) < 1/R 2 ). All the other individuals choose oneperiod contracts. The profit is therefore (R 2 -1) times the number of 2-period contracts G(1/(R+F )) plus (R-1) times the number of one-period contracts G(1/R)-G(1/(R+F )).

Let's show that the optimal F cannot be greater than

The profit function

For the sake of simplicity we write the profit function as a function of R and Z = R+F . There are five different regions (see figures 16 and 17 for a mapping in the (R, Z) space). Hereupon we show that the profit function writes, when β 0 < β < √ β 0 (figure 16 and17)

(25) and, for β 0 < √ β 0 ≤ β (figure 18):

All individuals choose the one period contract, since they don't expect to rollover ( β(β) ≥ β > 1/Z, ∀β). No one rollovers since β ≥ β 0 > 1/Z. The profit is (R -1) times the number of contracts G(1/R).

Region 2:

As before all individuals choose the one period contract. But some individuals will rollover (those with β < 1/Z). The profit is (R-1) times the number of one period contracts G(1/R)-G(1/Z) plus RZ -1 times the number of individual who rollover G(1/Z).

Region 3: Z < 1 β and 1

uals with β > 1/Z choose (rightly) the one period contracts since β(β) = β > 1/Z, ∀β > 1/Z. All the other individuals expect to rollover, and thus abstain from borrowing (twoperiod contracts are not profitable since β > 1/R 2 , ∀β). The profit is (R -1) times the number of one period contracts G(1/R) -G(1/Z).

as in the previous case. All the individuals with β < 1/Z rightly expects to rollover. They either abstain from borrowing or choose a two period contract (which is profitable for those with β < 1/R 2 ). The profit is (R -1) times the number of one period contracts G(1/R) -G(1/Z) plus R 2 -1 times the number of two period contracts G(1/R 2 ).

Region 5: Z < 1 β and R ≤ √ Z =⇒ β 0 < β < 1/Z < 1/R 2 < 1/R. Same logic as in the previous case. All the individuals with β < 1/Z choose a two period contract. The profit is (R -1) times the number of one period contracts G(1/R) -G(1/Z) plus R 2 -1 times the number of two period contracts G(1/Z).