Marie Claire

Human Capital over the Life Cycle

« Human Capital over the Life Cycle: A European Perspective », Catherine Sofer (Ed.); Edward Elgar, Cheltenham and Northampton, x and 200 pages, Index.

Marie-Claire Villeval, Groupe d'Analyse et de Theorie Economique, CNRS -University Lumiere Lyon 2 -Ecole Normale Supérieure, Lyon (France), and Institute for the Study of Labor, Bonn (Germany).

Human Capital over the Life Cycle: A European Perspective identifies the problem of human capital creation as one of the fundamental issues in the development of knowledge-based economies. Of course, education policies matter in providing the skills required by innovation and competition. But this book also offers a stimulating analysis of the responsibility of these policies in the fight against social inequality that can be considered as a challenge for the achievement of sustainable growth. The value-added which the book contributes is imputable to its life-cycle perspective, considering education from its early stages. Long after psychologists, economists now consider that education in early childhood is a critical issue in the further development of human capital and on success in the labor market. Another interesting aspect of this book lies in its comparative dimension. The diversity of European education systems and labor markets helps in empirically identifying sources of performance and sources of failure. Therefore, if this book will rapidly find its way onto the shelves of education and labor economists, I am also convinced that it should occupy a high position on the desk of policy-makers because of its high policy relevance.

C. Sofer has collected 9 contributions from leading experts in Europe, presenting the original findings from a European research project on schooling, training and transitions. This book is organized in three broad sections. Part 1 introduces the economics of schooling systems. Part 2 analyzes the transitions from school to work. Part 3 relates human capital to work careers.

Part 1 begins with an introductory survey on the economics of early childhood education by W. Groot, H. Maassen van den Brink, S. Dobbelsteen and N. van Mierlo. Early intervention programs exert a positive influence on the cognitive and social development of the child but also on the family members' participation in the labor market and on the performance of siblings. These externalities justify policy interventions favoring a less unequal access of children to education. The question remains however unanswered as to the best mode of intervention. P. Dolton, R. Levacic and A. Vignoles's essay on the economic impact of schooling resources (Chapter 2) offers a clear overview of the methodological problems associated with the use of education production functions that contribute to explain the mixed evidence of a direct effect of school resources on student achievements. The originality of the approach developed by L. Levy-Garboua, N. Damoiselet, G. Lassibille and L. Navarro-Gomez (Chapter 3) consists of introducing both a policy trade-off for schooling systems between the provision of general education and of vocational training and a distinction between general ability and occupation-specific talents in the measure of performance. Interestingly, if differentiated schooling systems reach a higher performance level than uniform ones, the analysis pleads for a postponement of differentiation until after talents can be assessed accurately. One could however object that the choice of vocational training by students may be motivated by a rejection of general education and that the demand for education is influenced by the employment conditions.

An answer to the latter question is provided by Part 2. The analyses of transitions from school to work begins with a study of the influence of unemployment on school-leaving by M.J. San Segundo and B. Petrongolo (Chapter 4). The demand for post-compulsory education is influenced positively by youth unemployment but negatively by adult unemployment, revealing the importance of short-term vs. long-term opportunity costs. Its main determinant remains however the parents' level of education. The long-term implications of this result are all the more important as early career experiments strongly influence further success and failure in the labor market. This relationship is documented by the stimulating cross-country comparison carried out by D. Margolis, E. Plug, V. Simonnet and L. Vilhuber using long panel data sets (Chapter 5). No hope for a general theory of transition: if sorting models, tournament theory, job search models or human capital theory deliver differentiated predictions, each of them finds support in some countries, although no country can be described by a single theory. This result points to the necessity for the theorist to remain humble! In Chapter 6, S. Mendes and C. Sofer compare the efficiency of apprenticeship and vocational education in terms of transition to work. If the human capital acquired by apprentices is more valued by firms in the early stages of the working life, this relative advantage vanishes over time. This suggests that general human capital becomes more valued at later stages, when adaptability is at stake. Part 3 is introduced by a review of employer-provided training policies in Europe by P. Elias and R. Davies (Chapter 7). The countries trying to offset market failures by regulating compulsory continuing vocational training facilitate the training of less skilled employees; they remain however less efficient than systems based on a large access to initial training. It would be interesting to determine whether technological and organizational innovations have reduced the extent of market failures. In Chapter 8, J. Canada Vicinay and M. Sollogoub analyze the link between the nature of labor contracts and economic performance in stronglyregulated countries. While access to permanent employment is mainly driven by skill endowments, these regulations push companies to favor fixed-term contracts, under which the incentive to invest in human capital is low. This chapter illustrates the limitation of public intervention for the development of human capital. The last chapter--one of the best--consists of the analysis of the relationship between education and labor mobility by K. Salvanes. It shows that net job creation for highly educated women is higher than for highly educated men. If job destruction is declining in education, churning follows an inverse U-shape across education levels. These results are particularly interesting because they show that job flows and churning are distinct processes.

One could have expected more emphasis on the role of peer effects and social interactions on human capital, and on the behavioral effects of education. It is also time for a more systematic examination of which aspects of education efficiency and equity require intervention through public policy and which should be the responsibility of individuals. This book however constitutes a good state-of-the-art study in this field and paves the way for further research in this direction.


Christophe Dutang
email: christophe.dutang@univ-lemans.fr

Theoretical L-moments and TL-moments Using Combinatorial Identities and Finite Operators

Keywords: L-moments, Trimmed L-moments, Combinatorial identities, Finite-element operators, Uniform distribution, Exponential distribution

Moments have been traditionally used to characterize a probability distribution. Recently, L-moments and trimmed L-moments are appealing alternatives to the conventional moments. This paper focuses on the computation of theoretical L-moments and TL-moments and emphasizes the use of combinatorial identities. We are able to derive new closed-form formulas of L-moments and TL-moments for continuous probability distributions. Finally, closed-form formulas for the L-moments for the exponential distribution and the uniform distribution are also obtained.

Introduction

Linear moments (L-moments) were first introduced by [START_REF] Hosking | L-moments: Analysis and estimation of distributions using linear combinations of order statistics[END_REF] as new measures of the location, scale and shape of probability distributions. They are related to expected values of order statistics and analogous to the conventional moments (mean, variance, skewness, kurtosis and higher moments). Consider a random variable X and denote by X j,m the jth order statistic (that is the jth smallest variable of an i.i.d. sample of size m). The mth L-moment of X is defined as

λ m = 1 m m-1 j=0 (-1) j m -1 j E(X m-j,m ), (1)
where m ∈ N and m-1 j denotes the usual binomial coefficient, see e.g. Olver et al. (2010). For ease of notation, the dependence of λ m on the random variable X is not stressed. Throughout the paper, we assume that the expectation E(X m-j,m ) exists, otherwise the corresponding L-moment is infinite and the corresponding algebraic manipulation is useless.

L-moments can be computed for a large variety of probability distributions but may not exist for heavytailed distributions such as the Cauchy distribution which has an infinite mean (see e.g. [START_REF] Kotz | Continuous Multivariate Distributions[END_REF]).

Trimmed L-moments (so-called TL-moment) are a natural extension of L-moments introduced by [START_REF] Elamir | Trimmed L-moments[END_REF]. The TL-moments are of two types: either symmetric λ (t) m or asymmetric λ (s,t) m . They are defined as

λ (t) m = 1 m m-1 j=0 (-1) j m -1 j E(X m+t-j,m+2t ), (2)
λ (s,t) m = 1 m m-1 j=0
(-1) j m -1 j E(X m+s-j,m+s+t ).

(3)

The TL-moments have the advantage to exist for heavy-tailed distributions such as the generalized Pareto distribution, see e.g. [START_REF] Beirlant | Statistics of extremes: Theory and applications[END_REF], which makes them particularly attractive for modeling heavy tails distributions. Fitting distributions by matching empirical and theoretical TL-moments are particularly adapted for location and scale parameters. Indeed, the quantile function is by construction an affine function of location and scale leading to immediate estimation. Furthermore, shape parameters may also be obtained as a root of the difference equation. Table 1 list the L/TL-moments for some common two-parameter distributions. As for centered moments, λ 3 /λ 2 and λ

(1)

3 /λ

(1) 2 are measures of skewness respectively Lskewness and TL-skewness, while λ 4 /λ 2 and λ

(1) 4 /λ

(1) 2 are measures of kurtosis respectively L-kurtosis and TL-kurtosis. The listed two-parameter distributions assume fixed values of such L/TL-skewness and L/TLkurtosis. Therefore, looking for the distribution that best fit the data on such quantity is challenging. As stated above, TL-moments exist even if the corresponding L-moments do not, e.g. for the Cauchy distribution.

Distribution

Quantile Q(p) λ1 λ2 λ3/λ2 λ4/λ2 λ Table 1: L/TL-moments of common distributions.

Typical application of fitting distribution by equalizing theoretical TL-moments to empirical ones include loss distribution in insurance, traffic volumes on computer networks, return of stock indexes, see e.g. [START_REF] Karvanen | Estimation of quantile mixtures via L-moments and trimmed L-moments[END_REF], [START_REF] Hosking | Some theory and practical uses of trimmed L-moments[END_REF] and the references therein. This papers aims to provide a new point-of-view of the computation of theoretical TL-moments. Combinatorial methods are a powerful tool to simplify and to derive closed-form formulas or asymptotics of complex problems of finite or countable discrete structure, see e.g. [START_REF] Graham | Concrete Mathematics: A Foundation for Computer Science[END_REF], [START_REF] Rosen | Handbook of discrete and combinatorial mathematics[END_REF], [START_REF] Sprugnoli | An introduction to mathematical methods in combinatorics[END_REF] and [START_REF] Flajolet | Analytic combinatorics[END_REF]. In particular, combinatorial methods provide techniques with combinatorial sums of type (1), (2), and (3). In this paper, we derive new formulas for the theoretical TL-moments as well as two examples of TL-moments for the uniform and exponential distributions. Firstly, we use combinatorial identities and manipulations of binomial coefficients to derive simplified formulas of (1), ( 2), (3) as well as a general expression for TL-moments of the uniform distribution. Secondly, we work with finite operators to derive TL-moments of the exponential distribution.

The paper is structured as follows: Section 2 presents necessary tools for studying combinatorial identities as well as the identities needed in the subsequent sections. In Section 3, we provide new formulation of TLmoments. Section 4 and 5 apply these new formulations for the computation of TL-moments of uniform distribution and exponential distribution, respectively. Finally, Section 6 concludes.

Preliminaries

In this section, we present tools and derive a serie of propositions that will be used in subsequent sections. In the following, the Kronecker delta is denoted by δ i,n , the rising factorial by x n = x(x + 1) . . . (x + n -1) and the falling factorial by x n = x(x -1) . . . (x -n + 1), see e.g. Olver et al. (2010). We start with identities based on combinatorial arguments and, then we follow up with identities based on finite operators.

Identities used for TL moment proved by combinatorial arguments

From [START_REF] Graham | Concrete Mathematics: A Foundation for Computer Science[END_REF], we recall that the binomial coefficient is r k = r k /k!, for r ∈ R, k ∈ N. Base manipulations are given in Table 2. The so-called useful identities (Equations ( 16), ( 17), ( 18), ( 19)) are given in Appendix A. Let us start with two identities based on the well-known Vandermonde identity (16).

symmetry n k = n n-k n, k ∈ N absorption r k = r k r-1 k-1 = r-k+1 k r k-1 r ∈ R, k ∈ N sign change r k = (-1) k k-r-1 k r ∈ R, k ∈ N trinomial revision r m m k = r k r-k m-k r ∈ R, k ∈ N Table 2: Base manipulations of binomial coefficients. Proposition 1. Let n ≥ k ∈ N. n-k j=0 n j + k n -k j = 2n -k n -k .
Proof. Using the Vandermonde identity ( 16) with r ← n -k, m ← n, n ← n -k and the symmetry rule (Table 2), we get

n-k j=0 n j + k n -k j = n-k j=0 n n -k -j n -k j = n-k l=0 n l n -k n -k -l = n + n -k n -k = 2n -k n -k . Proposition 2. Let k, n ∈ N. n j=k j k n j 2 = n k 2n -k n -k .
Proof. Using the trinomial revision (Table 2) combined with Proposition 1, we get

n j=k j k n j 2 = n j=k n k n -k j -k n j = n k n j=k n -k j -k n j = n k n-k l=0 n -k l n l + k = n k 2n -k n -k .
We now focus on identities not linked to the Vandermonde identity. For the sake of completeness, we recall a result of (Graham et al. 1994, p. 185). The proof is also given in details as similar arguments will be used for the next propositions.

Proposition 3. Let k, n ∈ N. e n,k = n l=0 (-1) l 1 l + k n + l 2l 2l l = (k -1)!n! (k + n)! (-1) n k -1 n .
In particular, e n,k = 0 for n > k -1.

Proof. We apply the trinomial revision (Table 2)

e n,k = n l=0 (-1) l 1 l + k n l n + l l = n l=0 (-1) l 1 l + 1 n l n + l n l + 1 l + k .
We cannot directly simplify this expression except when k = 1. Different manipulations are needed using (20),

(-l -2) + 1 (-l -2) + 1 -(k -1) = j≥0 k -1 j / -l -2 j and -l -2 j = (-1) j j + l + 1 j . Thus l + 1 l + k = j≥0 k-1 j (-1) j j+l+1 j ⇒ l + 1 l + k 1 l + 1 n l = j≥0 k-1 j (-1) j n+1 l+1 j+l+1 j = j≥0 (k -1)!(n + 1)!(-1) j (k -1 -j)!(j + l + 1)!(n -l)!(n + 1)
.

Using the sign change rule n+l l (-1) l = -n-1 l , we obtain

e n,k = (k -1)!n! (k + n)! j≥0 n l=0 (-1) j k + n n + 1 + j n + 1 + j l + j + 1 -n -1 l .
Using (17), we deduce that n l=0 n+1+j l+j+1

-n-1 l = j n , leading by ( 18) to

e n,k = (k -1)!n! (k + n)! j≥0 (-1) j k + n n + 1 + j j n = - (k -1)!n! (k + n)! j<0 (-1) j k + n n + 1 + j j n
.

Letting i = n + 1 + j we derive e n,k = - (k -1)!n! (k + n)! i≤n (-1) i-n-1 k + n i i -n -1 n = (k -1)!n! (k + n)! i≤n (-1) i k + n i i -n -1 n (-1) n = (k -1)!n! (k + n)! i≤n (-1) i k + n i 2n -i n = (k -1)!n! (k + n)! (-1) n k -1 n ,
using the sign change rule and ( 19).

Similarly, we apply the same technique: trinomial revision, absorption and identities of Appendix A to derive a closed-form of the following sum.

Proposition 4. Let k, n ∈ N. ẽn,k = n l=0 (-1) l l! (l + k)! n + l 2l 2l l = n! (n + k)! k -1 n .
In particular, ẽn,k = 0 for n > k -1.

Proof. Consider the sequence

ẽn,k = n l=0 (-1) l l! (l + k)! 2l l n + l 2l = n l=0 (-1) l l! (l + k)! n l n + l l , by trinomial revision. Since l! (l + k)! n l = n! (n -l)!(l + k)! = n! (n + k)! n + k l + k , we deduce ẽn,k = n! (n + k)! n ls=0 (-1) l n + k l + k n + l l = n! (n + k)! l≥0 (-1) l n + k k + l n + l n .
By (18), we obtain l∈Z (-1)

l n + k k + l n + l n = (-1) n+2k n -k n -n -k = (-1) n n -k -k = 0. Thus using i = k + l, ẽn,k = n! (n + k)! (-1) l<0 (-1) l n + k k + l n + l n = n! (n + k)! (-1) i≤k-1 (-1) i-k n + k i n -k + i n .
By the sign change rule, we have

n-(k-1-i)-1 n (-1) n = k-1-i n . Hence by (19), ẽn,k = n! (n + k)! (-1) k+n+1 i≤k-1 (-1) i n + k i k -1 -i n = n! (n + k)! k -1 n .
Finally, we consider a sum related to Harmonic numbers. Let H n be Harmonic numbers defined H n = n k=1 1/k with the convention that H 0 = 0. Proposition 5. Let n ∈ N.

S n = n j=0 n j (-1) j (1 + j) 2 = H n+1 n + 1 .
In particular, S 0 = 1, S 1 = 3/4, S 2 = 11/18 and S 3 = 25/48.

Proof. Using the absorption rule, we get

S n = n j=0 n j (-1) j (1 + j) 2 = n j=0 n + 1 j + 1 (-1) j (1 + j)(n + 1) = 1 n + 1 n+1 j=1 n + 1 j (-1) j-1 j .
Indeed,

H n = n j=1 n j (-1) j-1 j
by Section 6.4 of [START_REF] Graham | Concrete Mathematics: A Foundation for Computer Science[END_REF]. Therefore S n = H n+1 /(n + 1).

Results proved by finite operators

Now, we present the finite operators E and ∆, first introduced by English mathematicians such as G. Boole, see e.g. [START_REF] Spiegel | Schaum's outline of Theory and Problems of Calculus of finite differences and difference equations[END_REF]. Forward operator E is defined as Ef (x) = f (x + 1) and difference operator ∆ is ∆f (x) = f (x + 1) -f (x). Base manipulations are given in Table 3.

linearity E[αf (x) + βg(x)] = αEf (x) + βEg(x) ∆[αf (x) + βg(x)] = α∆f (x) + β∆g(x) product E[f (x)g(x)] = Ef (x)Eg(x) ∆[f (x)g(x)] = ∆f (x)Eg(x) + f (x)∆g(x) exponentiation E n f (x) = f (x + n) ∆ n f (x) = n k=0 n k (-1) n-k f (x + k)
Table 3: Finite operators.

The two formulas for exponentiation can be used to derive closed-form of sums. Since we have

E n = n k=0 n k ∆ k , (4)
∆ n = n k=0 n k (-1) n-k E k . ( 5
)
If we are able to express ∆ n in a simpler way than Equation ( 5), then we can deduce 4) by identifying E k in Equation ( 5). Let us work on an illustrative example. From identities in Table 4, we have ∆ n [1/x] = (-1) n n!/(x(x + 1) . . . (x + n)). Multiplying by x, we get

n k=0 n k ∆ k as E n from Equation (
∆ n 1 x = n k=0 n k (-1) n-k x x + k = (-1) n n! (x + 1) . . . (x + n) .
Therefore, identifying terms of (5) in the previous equation yields by (4) to

n k=0 n k (-1) k k! (x + k) . . . (x + 1) = x x + n .
This binomial sum looked particularly terrible at the first shot, but simplifies largely thanks to the exponentiation identities of ∆ and E. Now, we consider the following function f (x) = p+x m H x , that will be used in Section 5. We want to apply the same technique by simplifying the nth-order finite difference of f and inverting exponentation identities in order to compute the sum

n k=0 n k (-1) k p + k m H k .
The computation of ∆ n f (x) requires some intermediate results which have been postponed in Appendix C. Firstly, we derive ∆ n f (x) for n ≤ m in Proposition 6 using Lemmas 15 and 16. Secondly, Proposition 7 for ∆ n f (x) for n > m is obtained using Lemmas 17 and 18.

In order to compute the ∆-differences of f , we use the function g defined below

g x; p, n m = x + p m x + p + n x + p x + n x -1
.

This helps in rewriting ∆f

∆ p + x m H x = p + x + 1 m 1 x + 1 - p + x m - p + x + 1 m H x = 1 m g x; p, 1 m -1 + p + x m -1 H x .
Therefore, further finite order differences use the differences of g. By manual inspection of the first three

finite difference ∆ 1 p+x m H x , ∆ 2 p+x m H x , ∆ 3 p+x m H
x , we guess Lemma 16 proved by recurrence. Choosing a specific x leads to the following proposition. This result is yet partial, since it is only valid for n ≤ m.

Proposition 6. Let m, p, n ∈ N such that n ≤ m. n k=0 n k (-1) k p + k m H k = (-1) n p m -n n j=1 (-1) j-1 (j -1)! (m -n + 1) j n j p + j j ,
with the convention that 0 1 cancels when n = 0.

Proof. By Lemma 16, we deduce that for

x ∈ R n k=0 n k (-1) n-k p + x + k m H x+k = p + x m -n H x + n j=1 (-1) j-1 (j -1)! (m -n + 1) j n j g x; p, j m -n .
In particular for x = 0,

n k=0 n k (-1) n-k p + k m H k = p m -n H 0 + n j=1 (-1) j-1 (j -1)! (m -n + 1) j n j g x; p, j m -n , where g 0; p, j m -n = p m -n p + j j j j -1 = p m -n p + j j .
That is

n k=0 n k (-1) n-k p + k m H k = p m -n n j=1 (-1) j-1 (j -1)! (m -n + 1) j n j p + j j ⇔ n k=0 n k (-1) k p + k m H k = (-1) n p m -n n j=1 (-1) j-1 (j -1)! (m -n + 1) j n j p + j j .
So far, we get half the job since we only have ∆ n f (and the corresponding sum) for n ≤ m. Let us study the ∆-difference for n = m. Using Proposition 6 for n = m yields to

∆ m p + x m H x = H x + m j=1
(-1) j-1 j m j g x; p, j 0 .

Finite difference of the first term is obtained by Sprugnoli ( 2006)

∆ i H x = (-1) i-1 i x + i i -1 . (6)
Thus, the computation of ∆ m+i f simply relies on the computation of ∆ i g (second term). For ease of notation, we introduce a new function h of x with parameters a, b, c, d defined as

h x; a, b c, d = (x + a) b (x + c) d .
By convention, when b or d equals 0, the corresponding term equals 1, and when c = d, d is omitted. This h function is directly linked to g by

g x; p, n 0 = x + p + n n x + n n -1 = h x; p + n, n n .
Therefore, finite difference ∆ m+i f uses the differences of h. Again by manual inspection of the first three finite difference ∆ m+1 p+x m H x , ∆ m+2 p+x m H x , ∆ m+3 p+x m H x , we guess Lemma 17 that we prove by recurrence. Using Lemma 17 and (6), we get the ∆-difference of f in Lemma 18. At last, we are able to derive a new formulation of the sum.

Proposition 7. Let i, m, p ∈ N such that i, m ≥ 1. We have m+i k=0 m + i k (-1) k p + k m H k = (-1) m+i (-1) i-1 i + (-1) m+i i j=1 j-1 l=0 (-1) j (l + 1) j(i + l + 1) m j j l + 1 p l + (-1) m+i (i -1)!p! (p + i)! m j=i+1 i l=1 (-1) j m j i l j -1 + l j p + j j + l .
By convention m j=i+1 cancels when i ≥ m.

Proof. Choosing x = 0 in Lemma 18, we get

h 0; p + j, j -i ∧ j j + i -k = (p + j) j-i∧j (j + i -k) j+i-k , 0 + i i -1
= 1, and

∆ m+i p + x m H x x=0 = (-1) i-1 i + m j=1 (-1) j-1 j m j (-1) i i∧j-1 k=0 i ∧ j i ∧ j -k × (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! (p + j) j-i∧j (j + i -k) j+i-k . Since for x = 0 ∆ m+i x + p m H x x=0 = m+i k=0 m + i k (-1) m+i-k p + k m H k ,
we obtain by simplifying (-1)

i m+i k=0 m + i k (-1) k p + k m H k = (-1) m+1 i + (-1) m+1 m j=1 (-1) j j m j i∧j-1 k=0 i ∧ j i ∧ j -k × (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! (p + j) j-i∧j (j + i -k) j+i-k .
When i < m, we can split the outer sum, say S j .

S j,i<j = i∧j-1 k=0 i ∧ j i ∧ j -k (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! (p + j) j-i∧j (j + i -k) j+i-k = i-1 k=0 i i -k (i -1)!j i-k p i-k (i -k -1)! (j + i -k -1)! (j + i -k -1)! (p + j) j-i (j + i -k) j+i-k = i-1 k=0 i i -k (i -1)! (j + i -k -1)! (j -1)!(i -k -1)! p! (p + k -i)!(j + i -k)! (p + j)! (p + i)! = j(i -1)!p! (p + i)! i-1 k=0 i k j + i -k -1 j p + j p + k -i = j(i -1)!p! (p + i)! i l=1 i l j -1 + l j p + j j + l .
Furthermore,

S j,i≥j = i∧j-1 k=0 i ∧ j i ∧ j -k (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! (p + j) j-i∧j (j + i -k) j+i-k = j-1 k=0 j j -k (j -1)!j j-k p j-k (j -k -1)! (j + i -k -1)! (j + j -k -1)! (p + j) j-j (j + i -k) j+i-k = j-1 k=0 j j -k (2j -k -1)!p! (j -k -1)!(p -j + k)! (j + i -k -1)! (2j -k -1)! 1 (j + i -k)! = j-1 k=0 j j -k p j -k -1 j -k (j + i -k) = j-1 l=0 j l + 1 p l l + 1 i + l + 1 . Therefore, m j=1 (-1) j j m j S j = m j=i+1 (-1) j j m j S j,i<j + i j=1 (-1) j j m j S j,i≥j = m j=i+1 (-1) j j m j j(i -1)!p! (p + i)! i l=1 i l j -1 + l j p + j j + l + i j=1 (-1) j j m j p j-1 l=0 j l + 1 p -1 l 1 i + l + 1 = (i -1)!p! (p + i)! m j=i+1 i l=1 (-1) j m j i l j -1 + l j p + j j + l + i j=1 j-1 l=0 (-1) j (l + 1) j(i + l + 1) m j j l + 1 p l .
Otherwise when i ≥ m, we always have i > j. Finally, we obtain

m+i k=0 m + i k (-1) k p + k m H k = (-1) m+i (-1) i-1 i + (-1) m+i i j=1 j-1 l=0 (-1) j (l + 1) j(i + l + 1) m j j l + 1 p l +(-1) m+i (i -1)!p! (p + i)! m j=i+1 i l=1 (-1) j m j i l j -1 + l j p + j j + l ,
where m j=i+1 cancels for i ≥ m.

Theoretical L and TL-moments

Computing theoretical L-moments (1) and TL-moments (2), (3) needs the expectation of the order statistic E(X j,n ). Let us recall a well known result of probability theory, see e.g. [START_REF] Johnson | Continuous Univariate Distributions[END_REF]. Consider a sample X 1 , . . . , X n of independent and identically distributed random variables X i ∼ X. We assume that the random variable X has a density function f and a cumulative distribution function F defined on (a, b) → R.

Possibly a and/or b can be infinite, e.g. a = -∞ and b = +∞ for a normal distribution. The jth order statistic X j,n has the following density

f Xj,n (x) = n! (j -1)!(n -j)! F (x) j-1 (1 -F (x)) n-j f (x). ( 7
)
By simple manipulations, the expectation can be written as

E(X j,n ) = j n j n-j k=0 n -j k (-1) n-j-k I F (n -k -1), (8)
where I F is defined as

I F (k) = b a xF (x) k f (x)dx = 1 0 Q(p)p k dp, (9)
with Q = F -1 the quantile function. Note that I F (0) = E(X). We now present the central result of this paper simplifying in a sense the computation of TL-moments.

Proposition 8. Let m, s, t ∈ N. The TL-moment of X can be expressed as

λ (s,t) m = m + t + s m m+t-1 l=0 (-1) m+l+1 I F (s + l) m + s + t -1 s + l m + s + l -1 l . ( 10
)
Proof. Using (3), ( 8), the absorption and the symmetry rules, we have

λ (s,t) m = 1 m m-1 j=0 (-1) j m -1 j E(X m+s-j,m+s+t ) (11) = 1 m m+t-1 k=0 (-1) t-k I F (s + m + t -1 -k) m-1 j=k-t m -1 j m + s + t m + s -j t + j k (m + s -j) (12) = m + s + t m m+t-1 l=0 (-1) l+m+1 I F (s + l) m-1 j=m-1-l m -1 j m + s + t -1 t + j t + j m + t -1 -l .( 13
)
Then using the trinomial revision, we get

λ (s,t) m = m + s + t m m+t-1 l=0 (-1) m+l+1 I F (s + l) m + t + s -1 s + l m-1 j=m-1-l m -1 j s + l l -m + j + 1 . Let S m,l = m-1 j=m-1-l m -1 j s + l l -m + j + 1 .
Changing the summation index and using the Vandermonde identity ( 16), we have

S m,l = l i=0 m -1 m -1 -i s + l l -m + (m -1 -i) + 1 = l i=0 m -1 i s + l l -i = m + s + l -1 l .
Therefore,

λ (s,t) m = m + s + t m m+t-1 l=0 (-1) m+l+1 I F (s + l) m + t + s -1 s + l m + s + l -1 l .
Hosking (2007) established links with shifted Legendre polynomials as

λ (s,t) m = (m -1)!(m + s + t)! (m + s -1)!(m + t -1)!m 1 0 u s (1 -u) t P (t,s) m-1 (u)Q(u)du,
where P (t,s) m is the mth Legendre polynomial, i.e.

P (t,s) m (u) = m j=0 (-1) m-j m + t j m + s s + j u j (1 -u) m-j .
The result (10) is in line with this representation. Indeed,

u s (1 -u) t P (t,s) m (u) = m j=0 (-1) m-j m + t j m + s s + j u s+j (1 -u) m+t-j = m j=0 (-1) m-j m + t j m + s s + j u s+j m+t-j k=0 m + t -j k (-u) m+t-j-k = m j=0 m+t-j k=0 (-1) t-k m + t j m + s s + j m + t -j k u m+t+s-k .
Hence inverting indexes j and k,

1 0 u s (1 -u) t P (t,s) m-1 (u)Q(u)du = m-1 j=0 m+t-1-j k=0 (-1) t-k m + t -1 j m + s -1 s + j m + t -1 -j k 1 0 u m+t+s-k-1 Q(u)du. = m+t-1 k=0 m+t-1-k j=0 (-1) t-k m + t -1 j m + s -1 s + j m + t -1 -j k I F (m + t + s -k -1) = m+t-1 l=0 l j=0 (-1) l-m+1 m + t -1 j m + s -1 s + j m + t -1 -j m + t -1 -l I F (s + l) = m+t-1 l=0 I F (s + l)(-1) l-m+1 l j=0 m + t -1 j m + s -1 s + j m + t -1 -j l -j . Since m + t -1 j m + s -1 s + j m + t -1 -j l -j = (m + t -1)!(m + s -1)! (m -1)!(s + l)!(m + t -1 -l)! m -1 j s + l l -j ,
and by ( 16)

l j=0 m -1 j s + l l -j = m + s + l -1 l ,
we get

1 0 u s (1 -u) t P (t,s) m-1 (u)Q(u)du = m+t-1 l=0 I F (s + l)(-1) l-m+1 (m + t -1)!(m + s -1)! (m -1)!(s + l)!(m + t -1 -l)! m + s + l -1 l = (m + t -1)!(m + s -1)! (m -1)!(m + s + t -1)! m+t-1 l=0 I F (s + l)(-1) l-m+1 m + s + t -1 s + l m + s + l -1 l .
Multiplying by (m-1)!(m+s+t)! (m+s-1)!(m+t-1)!m leads to Equation ( 10). Now, we turn our attention to the symmetric case when s = t. We can easily derive a new formulation of λ (t) m .

Proposition 9. Let m, t ∈ N. The TL-moment of X can be expressed as

λ (t) m = m + 2t m m+t-1 l=0 (-1) m+l+1 I F (t + l) m + 2t -1 t + l m + t + l -1 l . ( 14
)
Proof. Using (10) with t = s yields to

λ (t) m = λ (t,t) m = m + t + t m m+t-1 l=0 (-1) m+l+1 I F (t + l) m + t + t -1 t + l m + t + l -1 l = m + 2t m m+t-1 l=0 (-1) m+l+1 I F (t + l) m + 2t -1 t + l m + t + l -1 l .
This result is in line with Equation ( 7) of Elamir & Seheult ( 2003)

λ (t) m = m + 2t m m-1 j=0 (-1) j I Q (m, t, j) m + 2t -1 t + j m -1 j ,
where

I Q (m, t, j) = 1 0 Q(u)u m+t-j-1 (1 -u) t+j du.
In fact, we have

λ (t) m = m + 2t m m-1 j=0 (-1) j m + 2t -1 t + j m -1 j 1 0 Q(u)u m+t-j-1 (1 -u) t+j du = m + 2t m m-1 j=0 (-1) j m + 2t -1 t + j m -1 j 1 0 Q(u)u m+t-j-1 t+j k=0 t + j k 1 k (-u) t+j-k du = m + 2t m m-1 j=0 t+j k=0 (-1) t-k m + 2t -1 t + j m -1 j t + j k 1 0 Q(u)u m+2t-k-1 du = m + 2t m m-1 j=0 t+j k=0 (-1) t-k m + 2t -1 t + j m -1 j t + j k I F (m + 2t -k -1) = 1 m m-1 j=0 t+j k=0 (-1) t-k (m + 2t -j) m + 2t t + j m -1 j t + j k I F (m + 2t -k -1), using m+2t t+j = m+2t m+2t-j m+2t-1 t+j
. The last equation is exactly Equation ( 12) with s = t.

This result is also in line with [START_REF] Hosking | Some theory and practical uses of trimmed L-moments[END_REF] who establishes links with shifted Legendre polynomials as λ (s,t) m is valid for shifted Legendre polynomials. That is

λ (t) m = (m -1)!(m + 2t)! ((m + t -1)!) 2 m 1 0 m-1 j=0 (-1) m-1-j m + t -1 j m + t -1 t + j u t+j (1 -u) m+t-1-j Q(u)du.
Finally, we get a new formulation for the L-moment λ m .

Proposition 10. Let m ∈ N. The L-moment of X can be expressed as

λ m = m-1 l=0 (-1) m+l+1 I F (l) m -1 l m + l -1 l . ( 15
)
Proof. Using (10) with t = s = 0 yields to

λ m = λ (0,0) m = m + 0 + 0 m m+0-1 l=0 (-1) m+l+1 I F (0 + l) m + 0 + 0 -1 0 + l m + 0 + l -1 l = m-1 l=0 (-1) m+l+1 I F (l) m -1 l m + l -1 l .
A direct proof is also possible. Let m > 1, the L-moment is given by

λ m = 1 m m-1 j=0 (-1) j m -1 j E(X m-j,m ) = 1 m m-1 j=0 j k=0 j k m -1 j m j (-1) k I F (m -k -1)(m -j). Since m j = m m-j m-1 j
, we get

λ m = m-1 j=0 j k=0 j k m -1 j m -1 j (-1) k I F (m -k -1) = m-1 k=0 (-1) k I F (m -k -1) m-1 j=k m -1 j 2 j k .
Using Proposition (2), this yields to

λ m = m-1 k=0 (-1) k I F (m -k -1) m -1 k 2m -2 -k m -1 -k = m-1 l=0 (-1) m-1-l I F (l) m -1 l m + l -1 l .
Logically, this formula is still in line with [START_REF] Hosking | Some theory and practical uses of trimmed L-moments[END_REF] who establishes links with Legendre polynomials.

Application to the uniform distribution

Let us consider the uniform distribution U(a, b) whose quantile and distribution functions are given by

Q(p) = a + (b -a)p, F (x) = x -a b -a 1 1 [a,b] (x) + 1 1 ]b,∞[ (x),
for p ∈ [0, 1] and x ∈ R. Therefore, the integral function I F defined in ( 9) is

I F (k) = 1 0 (a + (b -a)u)u k du = au k+1 k + 1 + (b -a) u k+2 k + 2 1 0 = a k + 1 + b -a k + 2 .

L-moments

Using preliminary calculus of Section 1, we get a result generalizing Table 1 of [START_REF] Hosking | L-moments: Analysis and estimation of distributions using linear combinations of order statistics[END_REF].

Proposition 11. For uniform distribution U(a, b), the L-moments are given by

λ m = δ m,1 a + b 2 + δ m,2 b -a 6 .
In particular, λ m = 0 for m > 2.

Proof. Using Equation ( 15), we get

λ m = m-1 l=0 (-1) m+l+1 a l + 1 + b -a l + 2 m -1 l m + l -1 l = a(-1) m+1 m-1 l=0 (-1) l 1 l + 1 m -1 l m + l -1 l + (b -a)(-1) m+1 m-1 l=0 (-1) l 1 l + 2 m -1 l m + l -1 l . Using m -1 l m + l -1 l = m + l -1 2l 2l l , we obtain λ m = a(-1) m+1 e m-1,1 + (b -a)(-1) m+1 e m-1,2 .
Therefore by Proposition 3

λ m = a 1 m 0 m -1 +(b-a) 1 m(m + 1) 1 m -1 = aδ m,1 +(b-a) 1 2 δ m,1 +(b-a) 1 6 δ m,2 = δ m,1 a + b 2 +δ m,2 b -a 6 .

Symmetric TL moments

Using preliminary calculus of Section 1, we generalized the previous proposition.

Proposition 12. For uniform distribution U(a, b), the symmetric TL-moments are given by

λ (t) m = (a + b)(t + 1) 2t + 2 δ m,1 + (b -a) 2(2t + 3) δ m,2 .
In particular, λ

m = 0 for m > 2, and with t = 0, we get back to Proposition 11.

Proof. Using Equation ( 14), we get

λ (t) m = m + 2t m m+t-1 l=0 (-1) m+l+1 a t + l + 1 + b -a t + l + 2 m + 2t -1 t + l m + t + l -1 l = a(m + 2t) m (-1) m+1 m+t-1 l=0 (-1) l t + l + 1 m + 2t -1 t + l m + t + l -1 l + (b -a)(m + 2t) m (-1) m+1 m+t-1 l=0 (-1) l t + l + 2 m + 2t -1 t + l m + t + l -1 l . Since m + 2t -1 t + l m + t + l -1 l = m + t -1 + l 2l 2l l (m + 2t -1)! (m + t -1)! l! (t + l)! ,
we get

λ (t) m = (m + 2t -1)! (m + t -1)! a(m + 2t) m (-1) m+1 m+t-1 l=0 (-1) l t + l + 1 m + t -1 + l 2l 2l l l! (t + l)! + (m + 2t -1)! (m + t -1)! (b -a)(m + 2t) m (-1) m+1 m+t-1 l=0 (-1) l t + l + 2 m + t -1 + l 2l 2l l l! (t + l)! = (m + 2t)! (m + t -1)! a m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 1)! + (m + 2t)! (m + t -1)! (b -a) m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l!(t + l + 1) (t + l + 2)! = (m + 2t)! (m + t -1)! a m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 1)! + (m + 2t)! (m + t -1)! (b -a) m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 1)! - (m + 2t)! (m + t -1)! (b -a) m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 2)! = (m + 2t)! (m + t -1)! b m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 1)! - (m + 2t)! (m + t -1)! (b -a) m (-1) m+1 m+t-1 l=0 m + t -1 + l 2l 2l l (-1) l l! (t + l + 2)! , using l!(t+l+1) (t+l+2)! = l!(t+l+2) (t+l+2)! - l! (t+l+2)! .
Using Proposition 4, we get

λ (t) m = (m + 2t)! (m + t -1)! b m (-1) m+1 ẽm+t-1,t+1 - (m + 2t)! (m + t -1)! (b -a) m (-1) m+1 ẽm+t-1,t+2 = b m (-1) m+1 t m + t -1 - b -a m(m + 2t + 1) (-1) m+1 t + 1 m + t -1 . Furthermore, t m+t-1 = tδ m,0 + δ m,1 , and t+1 m+t-1 = t(t + 1)δ m,0 + (t + 1)δ m,1 + δ m,2 . So λ (t) m = b m (-1) m+1 δ m,1 - (b -a) m(m + 2t + 1) (-1) m+1 ((t + 1)δ m,1 + δ m,2 ) = bδ m,1 - (b -a) 2t + 2 (t + 1)δ m,1 - (b -a) 2(2t + 3) (-1)δ m,2 = δ m,1 1 2t + 2 (b(2t + 2) -(b -a)(t + 1)) + b -a) 2(2t + 3) δ m,2 = (a + b)(t + 1) 2t + 2 δ m,1 + (b -a) 2(2t + 3) δ m,2 .

Application to the exponential distribution

Let us consider the exponential distribution E(λ) whose quantile and distribution functions are

Q(p) = -log(1 -p)/λ, F (x) = (1 -e -λx )1 1 R+ (x).
This enables to compute I F by Proposition 5

I F (k) = ∞ 0 x(1 -e -λx ) k λe -λx dx = k j=0 k j (-1) j ∞ 0 λxe -λx(1+j) dx = k j=0 k j (-1) j ∞ 0 t 1 + j e -t dt λ(1 + j) = 1 λ k j=0 k j (-1) j (1 + j) 2 = H k+1 λ(k + 1)
.

L moments

Proposition 13. For an exponential distribution E(λ), the L-moments are given by

λ m = δ m1 1 λ + (1 -δ m1 ) 1 m(m -1)λ , for m ∈ N .
Proof. Let us consider the simplest case (m = 1).

λ 1 = (-1) 1+0+1 H 1 λ 0 0 0 0 = 1 λ .
In the following, we assume m ≥ 2. Using Equation ( 15) and the absorption rule, we get

λ m = m-1 l=0 (-1) m+l+1 H l+1 λ(l + 1) m -1 l m + l -1 l = (-1) m mλ m-1 l=0 (-1) l+1 H l+1 m l + 1 m + l -1 m -1 = (-1) m mλ m l=1 (-1) l H l m l m + l -2 m -1 = (-1) m mλ m l=0 m l (-1) l H l m -2 + l m -1 , since both H 0 = m-2 m-1 = 0. Using Proposition 7 with m ← m -1, p ← m -2, i ← 1, i ∧ j = 1, m l=0 m l (-1) l H l m -2 + l m -1 = (-1) m-1+1 1 + (-1) m-1+1 m-1 j=1 (-1) j j m -1 j × 1-1 k=0 1 1 -k (1 -1)!j 1-k (m -2) 1-k (1 -k -1)! (j + 1 -k -1)! (j + 1 -k -1)! (m -2 + j) j-1 (j + 1 -k) j+1-k = (-1) m + (-1) m m-1 j=1 (-1) j j m -1 j j(m -2) (m -2 + j) j-1 (j + 1)! = (-1) m + (-1) m 1 m -1 m-1 j=1 (-1) j m -1 j (m -2 + j) . . . (m -2) (j + 1)! = (-1) m + (-1) m 1 m -1 m-1 j=1 (-1) j m -1 j m -2 + j m -3 .
Hence,

λ m = 1 mλ + 1 m(m -1)λ m-1 j=1 (-1) j m -1 j m -2 + j m -3 . Using (19) with l ← m -1, m ← 0, s ← m -2, n ← m -3, we have for m > 1 m-1 j=1 (-1) j m -1 j m -2 + j m -3 = +∞ j=0 (-1) j m -1 j m -2 + j m -3 -(-1) 0 m -1 0 m -2 + 0 m -3 = (-1) m+1 m -2 -2 -(m -2) = 0 + 2 -m.
Thus,

λ m = δ m1 1 λ + (1 -δ m1 ) 1 mλ + 2 -m m(m -1)λ = δ m1 1 λ + (1 -δ m1 ) 1 m(m -1)λ .
This result is in line with [START_REF] Hosking | Regional Frequency Analysis[END_REF], where the first four L-moments are given λ 1 = 1/λ, λ 2 = 1/(2λ), λ 3 = 1/(6λ), λ 4 = 1/(12λ).

symmetric TL-moments

Proposition 14. For an exponential distribution E(λ), the L-moments are given by

λ (t) m = 1 λm(t + 1) + (-1) t λm t!p! (m + t -1)! m+t-1 j=t+2 t+1 l=1 (-1) j m + t -1 j t + 1 l j -1 + l j m -2 + j j + l + (-1) t λm t+1 j=1 j-1 l=0
(-1) j (l + 1) j(i + l + 1)

m + t -1 j j l + 1 m -2 l ,
for m ∈ N .

Proof. Using Equation ( 14), we get

λ (t) m = m + 2t m m+t-1 l=0 (-1) m+l+1 H t+l+1 λ(t + l + 1) m + 2t -1 t + l m + t + l -1 l = (-1) m λm m+t-1 l=0 (-1) l+1 H t+l+1 m + 2t t + l + 1 m + t + l -1 l = (-1) m λm m+2t j=t+1 (-1) j-t H j m + 2t j m -2 + j m + t -1 = (-1) m+t λm m+2t j=0 m + 2t j (-1) j H j m -2 + j m + t -1 .
since m-2+j m+t-1 = 0 for j < t + 1. We now use Proposition 7 with m ← m + t -1, i ← t + 1, p ← m -2. We get for m ≥ 2

S m+2t = m+2t k=0 m + 2t k (-1) k m -2 + k m + t -1 H k = (-1) m+2t (-1) t t + 1 + (-1) m+2t t+1 j=1 j-1 l=0
(-1) j (l + 1) j(i + l + 1)

m + t -1 j j l + 1 m -2 l +(-1) m+2t t!p! (m + t -1)! m+t-1 j=t+2 t+1 l=1 (-1) j m + t -1 j t + 1 l j -1 + l j m -2 + j j + l .
Multiplying by (-1) m+t /(λm) leads to the desired result.

Conclusion

This paper proposes a new direction to compute closed-form formulas of theoretical TL-moments. Proposition 8 provides a general formulation of TL-moments for any continuous probability distribution. This formulation has been applied on two particular distributions: the uniform and the exponential distributions (Propositions 11,12,13 and 14). Other distributions such as Normal or Cauchy could be studied in this way.

In future research, we plan to study two directions: the computation of sample TL-moments that used a double combinatorial sum, and the computation of TL-moments of other distributions possibly multivariate distributions.

A Known combinatorial identities

We give below known identities, see e.g (Graham et al. 1994, Chap. 5). The Vandermonde Identity is

r l=0 m l n r -l = m + n r . (16) Let s ∈ R, m, n ∈ Z and l ∈ N k∈Z l m + k s n + k = l + s l -m + n . ( 17
) Let s ∈ R, m, n ∈ Z and l ∈ N k∈Z l m + k s + k n (-1) k = (-1) l+m s -m n -l . ( 18
) Let s ∈ R, m, n ∈ Z and l ∈ N k∈Z l -k m s k -n (-1) k = (-1) l+m s -m -1 l -m -n . ( 19
) Let m, n ∈ Z such that n = m -1 n + 1 n + 1 -m = m j / n j . (20)

B Finite calculus

Let us study the powerful tool of finite element analysis : E, ∆.

Ec = c ∆c = 0 Ex m = (x + 1) m ∆x m = m-1 k=0 m k x k Ex m = (x + 1) m ∆x m = mx m-1 E 1 x+m = 1 x+m+1 ∆ 1 x+m = 1 (x+m)(x+m+1) E n 1 x+m = 1 x+m+n ∆ n 1 x+m = (-1) n n! (x+m)...(x+m+n) E x m = x+1 m ∆ x m = x m-1 E n x m = x+n m ∆ n x m = x m-n E x m -1 = x+1 m -1 ∆ x m -1 = -m m+1 x+1 m+1 -1 E n x m -1 = x+n m -1 ∆ n x m -1 = (-1) n m m+n x+n m+n -1
Table 4: Known finite operations.

By elementary manipulations, we have

∆(f (x)g(x)h(x)) = ∆(f (x))E(g(x))E(h(x)) + f (x)∆(g(x))E(h(x)) + f (x)g(x)∆(h(x)). (21)
C Lemmas for Section 2

We introduce two auxiliary functions g and h in order to deal with the computation of ∆f .

Lemma 15. Let us define the auxiliary function g of x with parameters p, m, n as

g x; p, n m = x + p m (x + p + 1) . . . (x + p + n) (x + 1) . . . (x + n) = x + p m x + p + n x + p x + n x -1 , for m ∈ N. We have for all m ≥ 1 ∆g x; p, n m = g x; p, n m -1 n + m m -g x; p, n + 1 m -1 n m .
Proof. We have to compute the ∆-difference of p+x m and n+x n -1 in order to use (21). From Table 4, we have

∆ x + p m = x + p m - , ∆ x + p + n n = x + p + n n -1 , ∆ x + n n -1 = - n n + 1 x + n + 1 n + 1 -1
.

Using (21), we get

∆g x; p, n m = x + p m -1 x + p + n + 1 n x + n + 1 n -1 + x + p m x + p + n n -1 x + n + 1 n -1 - x + p m x + p + n n n n + 1 x + n + 1 n + 1 -1
.

Using the absorption rule (Table 2), we have

x+p+n+1 n = x+p+n+1 n x+p+n n-1 , x+p m = x+p+1-m m x+p m-1 , x+n+1 n+1 -1 = n+1 x+n+1 x+n n -1 . Thus ∆g x; p, n m = x + p m -1 x + p + n n -1 x + n + 1 n -1 x + p + n + 1 n + x + p m -1 x + p + n n -1 x + n + 1 n -1 x + p + 1 -m m - x + p m -1 x + p + n n x + n + 1 n + 1 -1 x + p + 1 -m m n n + 1 = x + p m -1 x + p + n n -1 x + n + 1 n -1 (x + p + 1)(n + m) nm - x + p m -1 x + p + n n x + n + 1 n + 1 -1 x + p + 1 -m m n n + 1 = x + p m -1 x + p + n n x + n n -1 n + m m x + 1 x + n + 1 - x + p m -1 x + p + n n x + n + 1 n + 1 -1 x + p + n + 1 -m -n m n n + 1 =
x + p m -1

x + p + n n

x + n n x + p + n + 1 n + 1

x + n + 1 n + 1 -1 n m .
Replacing the corresponding term by g(x; ) leads to the result.

Lemma 16. Let p, n, m ∈ N such that 0 ≤ n ≤ m, we have

∆ n p + x m H x = p + x m -n H x + n j=1
(-1) j-1 (j -1)! (m -n + 1) j n j g x; p, j m -n ,

with the convention that the sum 0 1 cancels. The coefficients n j (j -1)! are related to the number of permutations of the symmetric group that are pure j-cycles, see e.g. https:// oeis.org/ A111492 . Proof. Let us prove (22) by recurrence. For n = 1, the proposition is verified. Indeed, ∆ p + x m H x = p + x m -1 H x + (-1) 0 (0)! (m -1 + 1) 1 1 1 g x; p, 1 m -1 .

Assume the formula ( 22) valid for n ∈ N.

∆ n+1 p + x m H x = ∆ p + x m -n H x + n j=1
(-1) j-1 (j -1)! (m -n + 1) j n j ∆g x; p, j m -n .

The first term gives

∆ p + x m -n H x = p + x m -n -1 H x + 1 m -n g x; p, 1 m -n -1 .
In the sum, we use n + 1 1 g x; p, 1 m -n -1 .

Hence,

∆ n+1 p + x m H x = p + x m -n -1 H x + n+1 j=1
(-1) j-1 (j -1)! (m -n) j n + 1 j g x; p, j m -n -1 .

Thus, the recurrence ( 22) is valid for n ∈ N .

Lemma 18. Let p, m, i ∈ N such that 1 ≤ i, we have

∆ m+i p + x m H x = (-1) i-1 i x + i i -1 + m j=1
(-1) j-1 j m j (-1) i i∧j-1 k=0 i ∧ j i ∧ j -k × (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! h x; p + j, j -i ∧ j j + i -k .

Proof. We have for i ≥ 1

∆ m+i p + x m H x = ∆ i H x + m j=1
(-1) j-1 j m j ∆ i h x; p + j, j j .

Using Lemma 17, ∆ i h x; p + j, j j = (-1) i i∧j-1 k=0 i ∧ j i ∧ j -k (i ∧ j -1)!j i∧j-k p i∧j-k (i ∧ j -k -1)! (j + i -k -1)! (j + i ∧ j -k -1)! h x; p + j, j -i ∧ j j + i -k .

Using (6) leads to the desired result.

Acknowledgements

We thank Vladimir Fock for constructive comments on earlier versions of this paper when the author started this work in the University of Strasbourg. The author thanks the Editor and the Referee for useful remarks and suggestions.

Lemma 17. Let p, n, i ∈ N such that 1 ≤ i, we have

with i ∧ n = min(i, n). The coefficients i i-k (i-1)! (i-k-1)! are related to triangular arrays, see e.g. https:// oeis. org/ A089231 . Proof. By standard calculations, we have

Hence by ( 23)

For i = 1, we obtain

The last expression is valid by ( 24). Now, assume the property correct for i ≥ 1. Let us compute ∆ i+1 h (x; .)

By (23), we have

Consider first the case i < n. This simplifies to

we get for i < n,

So for i < n, splitting the sum and removing respectively the first term (k = 1) and the last term (k = i -1)

of the two sums gives

Since the summand simplifies to

The terms removed when splitting the sum are respectively (up to (-1) i+1 )

Hence for i < n

Consider now the case i ≥ n. By Table 4, we get

So for i ≥ n,

Therefore, the property is valid for all i ∈ N .


Claude Montmarquette
email: claude.montmarquette@cirano.qc.ca

Jean-Louis Rullière
email: rulliere@gate.cnrs.fr

Marie-Claire Villeval
email: villeval@gate.cnrs.fr

Romain Zeiliger
email: zeiliger@gate.cnrs.fr

Laurent Volpi

Muriel Prat

Redesigning Teams and Incentives in a Merger: An Experiment with Managers and Students

Keywords: C81, C92, J33, M52 Executive and team-based compensation, subject pool effects, real task experiment, incentives, sorting, mergers

After a merger, company officials face the challenge of making compensation schemes uniform and of redesigning teams with managers from companies with different incentives, work habits and recruiting methods. In this paper, we investigate the relationship between executive pay and performance after a merger by dissociating the respective influence of shifts, which occur in both compensation incentives and team composition. The results of a real task experiment conducted with managers within a large pharmaceutical company not only show that changes in compensation incentives affect performance but also suggest that the sorting effect of incentives in the previous companies impact cooperation and efficiency after the merger. Replicating this experiment with students showed differences in strategy rather than in substance between the two groups of subjects with managers appearing performance driven while students are more cost driven.

Introduction

There is strong evidence of the existence of heterogeneity among the compensation packages applied by firms within the same industry. It is not surprising to find that after a merger, difficulties can arise because of the different compensation policies of the newly merged firms and that new consolidated policies need to be designed. Furthermore, downsizing and the reorganization of production entail a reshuffling of teams, which affect executives from the companies involved in the merger. In order to promote social cohesion, mergers usually lead to the harmonization of company statutes so that all executives are paid according to the same compensation schemes. But within new teams comprised of executives of the merged companies, performance also depends on the willingness of individuals to cooperate with others. This willingness to cooperate may be affected by the heterogeneity of past compensation practices, work habits and non-market interactions.

Many pitfalls can hamper an empirical analysis of the relationship between new executive pay packages and executive performance after a merger. The impact of new compensation packages may differ from one employee to another depending on the short, medium and long-term influence of preceding modes of compensation (see [START_REF] Nalbantian | Productivity under Group Incentives: An Experimental Study[END_REF]. Thus, assessing the impact of new compensation schemes on executive performance after a merger requires controlling for the possibility of the long-term impacts of the compensation packages used before the merger. Furthermore, unbiased estimates of the relationship between pay and performance require disentangling the effects of shifts in direct incentives from the effects of the emergence of a new group culture founded on a variety of previous corporate cultures [START_REF] Kreps | Corporate Culture and Economic Theory[END_REF]. Individuals coming from a variety of previous cultures, with different norms of fairness and social comparison, could be expected to behave differently in the new company. Previous cultures can affect the efficiency of a new unified compensation policy particularly in the short run. Experimental methods can help in circumventing part of these difficulties through the comparison of various treatments in a controlled environment. This point has been successfully made in the context of a merger by Weber and Camerer (2003) who show that performance decreases following the merger of two laboratory firms because of failure in coordination.

In this paper, we design an experiment to analyze the relationship between executive compensation schemes and performance after a merger. Our laboratory experiment was conducted with 36 managers of a large pharmaceutical company created by the recent merger of two companies, one French and one German. It was replicated with 72 students in Lyon and Montreal. Thus, our paper also contributes to the literature on subject pool effects by comparing decisions made by student-subjects and manager-subjects. Studies by [START_REF] Dyer | A Comparison of Naive and Experienced Bidders in Common Value Offer Auctions: A Laboratory Analysis[END_REF], [START_REF] Carpenter | Comparing Students to Workers: The Effects of Social Framing on Behavior in Distribution Games Mimeo[END_REF] have underlined differences (risk attitudes, fairness) and similarities (winner's curse) across subject pools. [START_REF] Hannan | Partial Gift Exchange in an Experimental Labor Market: Impact of Subject Population Differences, Productivity Differences, and Effort Requests on Behavior[END_REF] show that MBA effort levels are greater than those of undergraduate students and [START_REF] Fehr | The Hidden Costs and Returns of Incentives -Trust and Trustworthiness among CEOs[END_REF] observe that CEO's exhibit more trustful behavior. How can one explain that in some games expertise seems to influence behavior whereas in others it has no significant influence [START_REF] Cooper | Gaming against managers in Incentive Systems: Experimental Results with Chinese Students and Chinese Managers[END_REF] emphasize the relationship between context and expertise. Expertise can improve the relative efficiency of manager-subjects in experiments only when they are able to recognize the similarity between the laboratory context and their field experience. Here, we could expect that conducting the experiment within the company context and performing an abstract task which reproduces characteristics of executive teamwork, should induce a greater level of performance from managers. Our paper is also innovative since it examines these effects in a team context with individual and team rewards.

The task required of participants consisted of searching for the highest value of a multiplepeaked function in a two-dimensional space. This task has a cognitive component since intense concentration is required because of uncertainty and time pressure and there is a monetary cost linked to the chosen speed of progression. This real-task experiment adds to the limited number of experimental papers [START_REF] Dickinson | An Experimental Examination of Labor Supply and Work Intensities[END_REF][START_REF] Sillamaa | How Work Effort Responds to Wage Taxation: An Experimental Test of a Zero Top Marginal Tax Rate[END_REF][START_REF] Van Dijk | Incentives Systems in a Real Effort[END_REF][START_REF] Falk | Clean Evidence on Peer Pressure[END_REF][START_REF] Gneezy | Performance in Competitive Environments: Gender Differences[END_REF], which study rewards and behavior in a real work setting. Many experiments, which require subjects to choose an effort level not related to a real effort confirm that monetary incentives do matter, but assume the equivalence between intention of contribution and effort, and between disutility of effort and money. Our experiment also differs from naturally occurring experiments (for example, [START_REF] Lazear | Performance Pay and Productivity[END_REF] or field experiments [START_REF] Erev | Constructive Intergroup Competition as Solution to the Free Rider Problem: A Field Experiment[END_REF] in which subjects perform a task in a real work environment. In our study, manager-subjects undertake tasks, which reproduce aspects of a manager's job under a familiar structure of incentives. They make their decisions in an artificial environment of anonymous interactions, according to instructions using neutral wording and without field referents.

Our design involves two parts: in the first part, teams are homogeneous and are paid according to the rules in effect before the merger of the pharmaceutical firms. In the second part, the teams remain homogenous or are formed randomly with participants originating from each of the two merged companies or from both schools; all are paid according to the rules in use after the merger. These treatments are aimed at disentangling the effect of the shift in team composition and the impact of the shift in incentive schemes.

Our main results indicate that there is a pure effect on performance of the shift in incentives after the merger. They show that the past matters in as much as some managers reduce their effort when they are potentially mixed with managers from the other incoming firm. This may be the result of sorting effect of previous incentives schemes: paying executives under different rules has probably contributed to the creation of attitudes towards cooperation in teams. Lastly, we find evidence that manager-subjects and student-subjects differ more in strategy than in substance, with managers being oriented more to the maximization of performance while students focus more on cost minimization.

In Section 2, the design of the experiment is outlined. Section 3 presents the experimental procedures. The empirical results are discussed in Section 4. In section 5, we conclude.

Experimental Design

The experiment consists of two parts. It reconstitutes in the first part the pay structure of the firms before they merge and in the second part, the pay structure prevailing in the merger.

The nature of the task to be performed during the experiment remains the same, thus allowing an analysis of the consequences on performance of both changes in the payment structure and of team composition. In this section, we present the design of the task, the structure of payment schemes, the treatments and information conditions.

Task design and behavioral heuristics

One original aspect of this experiment lies in the design of a task, which mimics some aspects of the content of a manager's job (concentration, variability, adjustment of means to targets and ability to cope with uncertainty under time pressure). The challenge was to be able to discriminate the impact of a cognitive effort on the outcome from that of ability.

The task consists in searching for the highest value of an increasing function built in a twodimensional space defined vertically by height (H) and horizontally by distance (D) from the origin, with

[ ] 0,100 H ∈ , [ ] 0,300 D ∈
and with

( )

Max Hf = D . It is common knowledge that the curve can have single or multiple peaks. When the period starts, the box in which the curve will appear is fully black. During a one-minute period, the subject progressively uncovers the curve on his computer screen starting at the origin, by clicking a button. i The subject progresses by discrete steps on the horizontal axis. She can stop her progression at any moment. The outcome achieved by a subject in a period is given by the maximum height reached on the curve, which depends notably on the number of moves.

The monetary cost associated with the task is represented by the choice of the speed of progression, i.e. the work pace. Two buttons are available to the subjects: a "1-step button" used to uncover the curves at a regular speed and is cost free; a "2-step button" that doubles the speed but which costs 0.4 point. The subject can switch speeds at will. This design allows a control over the subject's tasks cost and makes possible an analysis of efficiency.

It is impossible to reach maximum height during the one-minute period allowed by using the regular speed only, with the exception of one of the 13 randomly occurring curves. The subjects do not know this information. A subject's outcome depends on the use of two-step moves and on chance since there is uncertainty about the distances from the origins to the peaks. Therefore, the cognitive dimension of the task relates to the uncertainty about the shape of the curve, to time pressure and to the decision to use the 1-step or 2-step button.

Unlike van Dijk, Sonnemans and van Winden's (2001) real-task experiment, ours involves no algorithm enabling the discovery of a peak at minimum cost while under time constrains; no one benefits from previous learning. At each point on the curve, the subject cannot infer from the already uncovered part of the curve the slope of the curve at the next point. The subject cannot discover one single algorithm but must build heuristic grounded on a continuous trade-off between the speed to choose and the time left. For example, if a subject has already used many extra cost moves and the curve remains flat, paying each additional 2-step move requires a continuous trade-off between its marginal cost and its expected marginal revenue. Compared to a traditional experiment where effort values are chosen from a payoff table with a unique trade-off between cost and outcome, our task involves an unknown optimal way of reaching the maximum height at a minimum cost since subjects do not know the locations of peaks. The property of duality is not checked: the minimization of cost does not correspond to the maximization of the objective. The multiplicity of heuristics and uncertainty concerning the location of peaks prevent any theoretical prediction. ii

In short, there are two aspects to the task chosen in this experiment: the decision to use the costless or costly way to progress and the speed of progression, where both can be measured. However, the cognitive problem of how to reach a higher altitude cannot be measured. For the rest of the paper, the altitude reached is the observable outcome of the cognitive effort. This output and the cost of progression are the variables to be explained.

Payment schemes

Another point of originality lies in the design of compensation schemes that combine fixed pay and performance-pay, the latter involving both individual and team incentives. The game involves teams consisting of three subjects who have to uncover the same curve.

Subjects are not allowed to communicate with teammates and are not informed of their simultaneous progression. Individual earnings from a task in a given period are calculated from the sum of three elements whose amount and relative proportion depend on the stage of the game and on the treatment. Specifically, earnings are defined as: i FIT

α ααα π =++, with { } , X Y α =
. X and Y correspond to the two firms before they merge, and for simplicity X and Y subjects keep their respective labels after the merger in order to track their origin.

F α is a fixed-wage earned by subject i when her individual outcome reaches a first threshold, , defined by the height reached. This threshold can always be achieved with no cost steps in the time allowed, but this information is not given to subjects. An employer would consider a distance achieved below this benchmark as professional misconduct. from the origin and the greater the distance between the first and the second thresholds, the more difficult it becomes to reach additional rewards.

Experimental treatments and information conditions

The experiment aims at identifying the separate influences of changes in incentives and in team composition. To measure the impact of changes in payment schemes, a session was designed having two parts of 10 periods each, with a random order of presentation of 13 payoff curves. In the first part, used as a benchmark, we reproduce initial payment schemes that were used before the merger; in the second part the payment scheme in use after the merger is applied. In the first part, we team X and Y subjects separately, each playing under the payment scheme used in their initial company. Members of X teams may receive a fixed wage, an individual bonus and a team reward. Earnings for members of Y teams are derived from a fixed wage and a team reward only (see Table 1). The proportion of potential total earnings from the fixed wage is higher for Y subjects than for X subjects, but the same performance is required from all subjects to trigger their fixed payment and team reward. In the second part of the session, the payment scheme is the one used after the merger and is the same for all subjects. It includes a fixed wage, an individual bonus and a team reward. Compared to the first part, Y subjects may now receive an individual bonus and X subjects have seen an increase in their fixed wage. In avoiding a decrease in the absolute level of any pay component of earnings between the two parts of the session, the maximum earnings all subjects can obtain is now 120 ECU instead of 100.

To measure the impact of team composition on performance, a new matching treatment was introduced in two sessions. This treatment is referred as the mixed-treatment, where teams of 3 randomly associated subjects are formed potentially involving both X and Y subjects.

The fixed-treatment used in the first part of all sessions, where X and Y subjects are teamed exclusively with X and Y subjects, respectively, is maintained in the second part in one session. The fixed-treatment serves as a benchmark against which the effect of team composition after the merger is tested. A stranger matching protocol is always used. In the second part of the session, only team composition is changed and not the size of the teams. iii

[Insert Table 1 about here]

All subjects, except those playing the fixed treatment, knew of the existence of two categories of subjects in equal numbers in the room, but they were kept unaware of the meaning of labels X and Y. Subjects learned their own identity by reading the instruction sheet and were notified that they would keep the same identity throughout the session. The instruction sheet for the first 10 periods mentioned that they were matched with two other subjects belonging to the same category as themselves and that the composition of groups would within the same category change with each new period. They were informed that they would never know the identity and the payoff of their successive teammates. Subjects knew the description of the task to be performed and the payoff structure applicable to their category. The participants were not given information about the payoff structure of the other category of subjects; were aware that the same task was to be achieved by the three members of their group and had no current information about the progression of their teammates on the curve. During play, their screen indicates the time currently left, the cumulated cost of their 2-step decisions and the height of the different thresholds reached.

At the end of each period, a historic table gave each subject feedback on their own outcome, the outcome (cumulative height) achieved by their team, total cost, amount of compensation obtained, by source, and total payoff net of costs.

In the second part of the session, subjects were informed about the payoff structure of the first part for each of the two categories. The new payoff structure common to all subjects was explained. In one session, an equal number of subjects from both categories continued to play under the fixed treatment for periods 11 to 20. In two sessions, subjects drawn in equal numbers from X and Y were informed that during the remaining 10 periods they could be matched randomly with members of the other category. iv

Experimental Procedures

The experiment was first conducted with managers of the pharmaceutical company about two years after the merger and it was later replicated with students in engineering and business from Lyon and Montreal.

36 volunteer executives were recruited by email with average annual earnings of 69,000 euros, consisting of 18 managers from each incoming firm. To limit uncontrolled peer group effects, sessions were designed such that the participants represented a large diversity of departments and came from different geographical locations. v The experiment was conducted at the company headquarter in Paris on the same day to limit the dissemination of information. In the first two sessions, 12 participants (6X, 6Y) were subjected to the mixed-treatment protocol. In a third session, 12 participants played under the fixedtreatment protocol. On average, a session lasted 75 minutes including practice periods.

Transactions were conducted in Experimental Currency Units, with ECU convertible to Euros at the rate 150 ECU = 4.5 €. A show-up fee of 8 € was added. On average, a subject earned 51.45 € (S.D.=3.75). Subjects were paid a few days later with vouchers.

Upon arrival, each manager had to register and was invited to draw a ticket from an envelope to assign him or her a computer. In fact, company specific envelopes were presented to subjects according to their originating company, but subjects were unaware of this allocation rule. Subjects discovered a set of written instructions for the first part of the session under their keyboard. As the payment schemes differed among X and Y participants, the experimenter did not read the instructions aloud. vi Instructions were phrased in neutral terms with queries answered in private. Three practice rounds were run before the experiment began. At the end of the first part, the game stopped and further instructions for the second part were distributed, without any questions allowed.

This experiment was replicated with 72 student subjects, in the experimental laboratories of Subjects were immediately paid in cash in a separate room.

Experimental results

Overall statistics

In Figures 1 to 3, we present histograms of average output achieved, average rate of freeriders and average return to output level plotted against treatment, subject pool and the group of periods (1-10 or 11-20). Figure 1 shows that the reaction to incentives differs according to the subject pools: the average output level is higher for managers than for students throughout the experiment. For the managers, output increases in the second part of the experiment under the merged company's compensation package. This increase is particularly noticeable in the fixed treatment sessions, where the only change introduced relates to the increase in monetary incentives; when both incentives and team composition are changed, the increase in managers' average output is not significant.

[Insert Figures 1, 2, and 3 about here]

These different output levels may be related to differences in the proportion of free riders in the two pools. Because of the team reward in the payment scheme, our experiment can be connected with the Voluntary Contributions Mechanism in public good games. It is interesting to verify whether the co-existence of a fixed wage, an individual bonus and a team reward in the same compensation scheme changes the behavior towards cooperation as typically observed in these games. Transposing the strict definition of free riding in the VCM to our experiment means that a free rider is here defined in the following way. In part 1, X is qualified as a free rider in period t when her output is 60 ≤ , i.e. when the subject does not contribute to the team outcome beyond the level that triggers her individual bonus.

The same condition applies for part 2. In part 1, Y is qualified as a free-rider when her output in period t is , i.e. when she does not contribute to the team outcome beyond the level that secures her fixed wage, and in part 2 when her output level is . In a typical public good game, whereas each subject has a dominant strategy not to contribute, the level of contribution is initially positive and declines with repetition (see [START_REF] Ledyard | Public Goods: A Survey of Experimental Research[END_REF]. Figure 2 shows the rate of free riders by subject-pool and by treatment for parts 1 and 2. It indicates that there is a low proportion of free-riders compared to traditional results of public good games, probably because of both the coexistence of individual and collective payments and the impossibility of calculating the marginal per capita return of investing in the team outcome.

40 ≤

≤

vii The proportion of free riders is lower among managers than among students in both parts. viii This proportion is lower when the threshold that triggers the greater individual reward is low. It increases in part 2 in both subject pools, particularly in the mixed sessions, and this cannot be explained by a restart effect at the 11 th period of the game as will be shown later. This is in line with the declining cooperation over time usually found in public good experiments and could be compared with the differences observed between stranger and partner-matching protocols (see Keser and van Winden, 2000). Figure 3 displays the average return to output level, i.e. the average output achieved divided by the average cost level. On average, managers achieve a lower level of efficiency than students. The former perform a greater output but at a higher cost. In part 2, the average level of efficiency rises slightly in both subject pools.

However, overall statistics are uncontrolled for time, difficulty of the curves and individual effects. Regression analyses controlling for these dimensions are thus required to identify the determinants of two endogenous variables: output levels and costs. In Table 2, we present the definition and descriptive statistics of variables used in these regressions.

[Insert Table2 about here] Exogenous variables are the period, the category of subjects, the mode of compensation, the composition of groups and an index of difficulty for each curve. The "lagged output achieved of the other group members" variable assesses whether subjects modulate their outputs to what their teammates did in the previous period. Interaction variables involving the Y subjects are created to test whether X and Y subjects behave differently. Coefficient estimates of the variable "mode of compensation in part 1" report the decisions of the X subjects in part 1 relative to their decisions under the mode of compensation in part 2 (element of the constant term). With the coefficients of the "mixed session" variable, we further distinguish the decisions of the X subjects in the mixed sessions of part 2 relative to the fixed sessions of part 2. The decisions of the Y subjects in part 1 are the sum of the coefficients of the variables "Y subject", "mode of compensation in part 1" and "Y subject and mode of compensation in part 1". This last variable is needed as the modes of compensation differ in part 1 between X and Y subjects. Summing-up, coefficient estimates of variables "Y subject", "mixed session" and "Y subject and mixed session" give the decisions of Y subjects in the mixed sessions in part 2. The coefficient of the "Y subject" variable shows the decisions of "Y subjects" in the fixed sessions in part 2.

The index of difficulty and the period variables enter regressions with interacting variables and nonlinear forms. Instead of including a dummy for each period to control for time effects, we distinguish the variable "logarithm of the period" from the "periods 1 to 20" variable in order to separate two potentially adverse effects. While the former may account for elements such as boredom that could push output level down over time, the latter may account for learning of the task that could increase output over time. Lastly, demographic variables such as gender and age were entered to control for their potential impact but since none proved to be significant, they were subsequently removed.

Econometric results

Column 1 of Tables 3 and3a display the results for the students and managers, respectively, on the subjects' output achieved. Column 2 of both tables report the results on the determinants of cost levels. The cost incurred corresponds to the number of occasion the 2-steps button was used to perform the task. All econometric estimates are obtained with a linear one-way random effects model. ix

[Insert Tables 3 and 3a about here]

For the students considered as a benchmark, few variables significantly influence outputs, except periods and the difficulty of the task. In contrast, the cost levels are influenced by more elements. A change in monetary incentives in the second part of the experiment makes students react by increasing their cost levels but not their output. An increase in the output levels of their teammates during the preceding period incites them to increase their costs, particularly for the X subjects. Lastly, costs decline over time and the relationship between the difficulty of the curves and the costs indicates a reverse U-shape.

In contrast, we observe for managers a significant and substantial increase in outputs by both X and Y subjects in part 2 relative to part 1. The change in incentives after the merger increases outputs by almost 12 points. The change in the composition of teams exerts no significant influence on output. Note however the negative but statistically insignificant coefficient of the interaction variable "Y subject and Mixed": Y subjects, knowing that they may be interacting with X subjects, have a tendency to lower their output levels. Despite the stranger-matching protocol, Y subjects are also influenced by the behavior of their teammates in the preceding period: an increase in teammate outputs motivates subjects to increase their own output. It should also be noted that, if we observe a negative first period effect on output achieved, there is no restart effect at the 11 th period. The difficulty of the curves and the number of periods affect outputs in a nonlinear way.

The change in composition of the teams in part 2 has no significant impact on effort decisions, however, it exerts a determinant impact on the cost levels chosen by the subjects.

Ceteris paribus, X subjects increase their costs in a mixed session by 6.23 units relative to a fixed session, while Y subjects substantially reduce theirs by 8.38 units when they know they may be teamed with X subjects. x The relationship between the difficulty of the curves and the costs indicates a reverse U-shape as with the students. Lastly, there is a positive first period effect on cost levels, but costs decline more linearly as the experiment evolves.

Discussion

This section will focus on the reactions to the shifts in incentives, in team composition and in the difficulty of the task. Our results provide clear evidence of differences in reactions to a change in monetary incentives between the subject-pools. The new compensation scheme in the second part does increase the output levels of X and Y managers. In contrast, the same monetary incentives, controlling for other variables, incite students to increase not their outputs but their costs to accomplish the different tasks imposed on them. Among managers, Y participants substantially decrease their costs in part 2. Another difference is the reaction to lagged output levels of the other group members. Y managers increase their outputs if the output levels of their previous teammates increase. Student participants modify their cost levels to the previous output levels of their group members.

These results show that the change in incentives influences both subject-pools, with managers appearing to be objective driven while students are cost driven. This is not that surprising as managers in their professional life are evaluated, remunerated and promoted by meeting their objectives whatever the cost they must incur to do so (long working hours for example). For most students, a large part of the return of their academic effort is from lowering their cost (time devoted to studying a specific matter) in order to obtain good grades. Note that we were able to observe this kind of results because we used a real costly task. If the student-subjects appear to be more money maximizing than managers in the laboratory, this cannot be attributed to the differences in the opportunity costs to participate, since this was taken into account in the conversion rates.

The changing composition of teams also influences the behavior of subjects. Within the same category (X or Y), most subjects are influenced by the behavior of their preceding teammates although teams are reshuffled each period. This might suggest that the subjects refer to their category as a whole. However, the linear one-way effects models recognized that subjects are heterogeneous; our results suggest that subject-pools and categories are also heterogeneous. Both X and Y student-subjects are influenced by the output of the subjects they were previously teamed with but they are not influenced by the merging between categories in the second part of the experiment; thus, they do not refer to a specific category but to the whole set of subjects. By contrast, after the merger, the decisions of the category of managers who are more sensitive to the outputs of previous teammates (the Y managers) change behavior knowing that they may be interacting with subjects from the other category; they become less cooperative and they substantially reduce their costs and to a lesser degree their outputs. On the contrary, X managers who are not influenced by the behavior of previous teammates react positively to the merging of the categories in the same teams. Groups of reference are not the same across subject-pools.

In traditional experimental economics literature, reaction to others' decision is usually characterized by a reciprocity concept (see [START_REF] Fehr | Psychological Foundations of Incentives[END_REF]. Since our experiment is run with randomly re-matched subjects at each period, this can also be explained by learning and conditional cooperation: subjects learn the behavior of other subjects in the same category or in the same room and constantly update their beliefs. But more puzzling is the reaction to the mixing of categories after the merger. Three explanations could be evoked. A first explanation might be that the merger changes preferences by introducing new incentives. This explanation cannot help since we measure the impact of a shift in team composition other things equal, i.e. we control for changes in incentives. The psychological concept of "in-group/out-group" [START_REF] Tajfel | Social Categorization and Intergroup Behavior[END_REF]) might affect the cooperative behavior of participants. This explanation cannot directly help here since students and managers do not react in the same way. If managers were able to transfer their experience of the merger into the laboratory whereas the students were not able to do so, this psychological effect might play a role for the managers. But it does not explain why X and Y managers did not react similarly. A more convincing explanation refers to a sorting effect of incentives as documented by [START_REF] Lazear | Performance Pay and Productivity[END_REF]. The incoming companies that merged might differ in recruiting different profiles of managers, then developing different cultures, because of their various incentive schemes. The pre-existence of an individual bonus may contribute to focusing on one's own performance instead on the other's output.

The reaction to the difficulty of the task is similar for managers and students. The U-shape curve suggests that more difficult tasks may actually elicit, to some extent, more output. This job challenge effect is present even in the later stages of the experiment (see the "index of difficulty and period" crossed variable), whereas the logarithm of period exerts a negative effect on the output level (that could be interpreted as boredom or tiredness). This result is consistent with the psychological literature showing that challenging goals lead to higher performance than easy goals [START_REF] Locke | Goal Setting and Task Performance: 1969-1980[END_REF]. The job challenge may even be at the initiative of the subject herself (for example if she uses targets like reaching the second peak). The relationship between the difficulty of the curves and the costs supported by all participants indicates a reverse U-shape. If the task is too difficult, subjects increase their outputs but without resorting to costly 2-step moves. This result reinforces our preceding analysis: an increased difficulty does not discourage output under the condition that subjects can save on their costs. Lastly, there is a positive first period effect on the cost levels (significant for the managers), but costs decline more linearly as the experiment evolves. This is possibly due to a learning effect on the task, on the other's behavior, and on the best moment to use the costly moves. Students appear to learn more than managers as they play the game since they not only decrease their costs but also increase their outputs (see variable "Periods 1 to 20"). xi

Summary and conclusion

Executive behavior with respect to performance, motivation and cooperation is a major element in the success or failure of a merger between companies. Traditionally, economists have suggested looking for an adapted compensation policy to facilitate cooperation within groups of individuals coming from different corporate cultures. The aim of this paper is to check whether a harmonization of compensation packages is sufficient to motivate all managers to cooperate to the same extent. A laboratory experiment has been run involving managers of two large pharmaceutical companies that recently went through a merger. The experimental design has introduced various compensation schemes, including an incentive scheme combining individual and team incentives that were implemented in the context of a real task. As in most mergers, these manager-subjects have experienced the redesigning of both compensation schemes and team composition in their newly merged company. The protocol reproduced the pre-and post-merger situation in terms of both compensation and team composition. To complement this experiment, a replication with a subject pool consisting of French and Canadian students was conducted that can serve as a benchmark.

The results show that financial incentives do work in improving output achieved, in accordance with standard results [START_REF] Prendergast | The Provision of Incentives in Firms[END_REF]. However, the unified incentives are not entirely sufficient to create cooperation among heterogeneous groups, as already experimentally observed [START_REF] Meidinger | Does Team-Based Compensation Give Rise to Problems when Agents Vary in Their Ability[END_REF]. Past matters. In contrast with [START_REF] Nalbantian | Productivity under Group Incentives: An Experimental Study[END_REF], it matters more in terms of shifting team composition than in terms of incentives, since the change from the two pre-merger incentive schemes to the unified one increases output in the same proportion. Individuals coming from different corporate cultures, likely with different fairness norms and social comparison behavior, tend to react differently in the mixed treatment part of our experiment. This may result notably from a sorting effect, attributable to various selection policies in the originating firms: companies with different incentive policies probably attract different types of managers. This suggests that shifting team composition may limit, at least in the short run, the efficiency of a new unified compensation policy, if not taken into account. Merging cultures requires more time than merging incentives and deserves special attention. Our experiment also shows that introducing complex tasks is not necessarily detrimental to performance and cooperation. The concept of job challenge is perhaps more important to soliciting greater effort among employees than is usually suggested in current literature.

Results from the student-subject pool differ in strategy more than in substance, allowing confirmation of the external validity of laboratory experiments. In contrast to the managers, students react to an increase in monetary incentives by accepting more costs to complete a given task rather than increasing their output levels. They are cost driven whereas managers appear to be objective-driven. This means that the managers are also more cooperative and free ride less than the student-subjects. Our results corroborate the interpretation of [START_REF] Cooper | Gaming against managers in Incentive Systems: Experimental Results with Chinese Students and Chinese Managers[END_REF] in that when they are able to recognize the similarity between the laboratory context and their field experience, manager-subjects may choose different strategic options than inexperienced subjects. Moreover, it may indicate that if students are more inclined to minimize costs than experts in the laboratory, when one observes the existence of otherregarding preferences in traditional experiments involving student-subjects, one may deduce that this deviation from the equilibrium is likely more developed in real settings. ). If the subject's team reaches a cumulated height of 240, the subject receives a payoff of 100, corresponding to the sum of the fixed wage, the individual bonus and the team reward (

). Part 2 and the teams of Y subjects should be interpreted in a similar manner. (1 df, prob value = .000000). a) significant at 5% level; b) significant at 10% level. Experiment. European Economic Review 45, 187-214. Weber, R.A., Camerer, C.F., 2003. An Experimental Approach to the Study Culture. Management Science. In Press.

Notes

i The experiment was computerized using the REGATE program. All curves used in this experiment and the instructions can be found at the following web site: http://platypus.gate.ec-lyon.fr/groupware/g_gate/Raventis.htm.

ii Once the curve is known, it is possible to find the best strategy to reach the highest peak at the lowest cost. A referee's suggestion to compare the subject's behavior with this optimal strategy appears worthwhile to pursue to study the learning process and the cognitive effort of the participants.

iii The negative impact on efficiency of the increasing size of teams in a merger has been documented by [START_REF] Camerer | Creating "expectational assets" in the laboratory: "weakest-link" coordination games[END_REF]. iv It would have been interesting to test whether belonging to the majority category or the minority category of a team would influence individual behavior within teams. However, this would have required collecting a far greater number of observations. For this reason we did not inform subjects about the detailed composition of their teams.

v Overall, participants came from 4 different sites and 22 departments. x From the calculation: [8.06 -(8.06 + 6.23 -14.61)]. The first term is the coefficient of the variable for Y participants in part 2 when groups are fixed. The second term represents the value of the coefficients for Y participants in part 2 when groups are mixed.

xi Despite their different strategies, the similarity of students' and managers' average net payoffs is striking. However, with more output achieved by the managers, the average surplus of a hypothetical firm (height times a constant minus wage costs) is higher when employing managers-subjects rather than students-subjects. The net payoffs are higher under the Y payment scheme than under the X payment scheme before the merger. After the merger, net payoffs are higher when teams, whether composed of students or managers, are homogeneous. In contrast, the hypothetical firm's performance is greater under the X payment scheme before the merger and with mixed teams after the merger.

  a team reward obtained when the sum of individual outcomes by the team of 3 subjects reaches a third threshold with . A subject may earn this reward even though she does not reach an output level greater than the level giving her the fixed wage or the individual bonus. This creates an incentive to free ride. of the game, a new curve is randomly drawn, whose shape determines the extent of uncertainty faced by the subjects. The analysis of performance and costs must control for the degree of difficulty of the curves. Because of the structure of the compensation package, the difficulty of a curve depends on the location of the various thresholds. An index of difficulty is calculated as : d = (D 1 ) 2 + (D 2 -D 1 ) 2 + (D 3 -D 2 ) with D 1 being the abscissa at the origin of the first threshold, D 2 the abscissa of the second threshold and D 3 the abscissa of the maximum height. The more distant the first threshold is

  GATE-Lyon and at the LUBC3E-CIRANO in Montreal. Three sessions were organized over two days. The two mixed-treatment sessions were played on-line, the REGATE software enabling the on-line reshuffling of groups between Montreal and Lyon subjects.During each mixed-treatment session, 12 French subjects and 12 Canadian subjects interacted as subjects X and Y without being informed that they were playing with subjects abroad. The third session (fixed-treatment) involved 12 students at each location. Sessions were conducted under the same conditions as sessions with manager-subjects, except that the ECU were convertible at the rate 150 ECU = 1 Euro for the French subjects and at the rate 150 ECU = 1.5 Canadian Dollars for the Canadian subjects. A show-up fee of 4 Euros or 6 Canadian Dollars was added. On average, a subject earned 12.89 € (S.D.=0.80 €).

Figure 2 .Figure 3 .

 23 Figure 1. Average output by subject pool and by treatment

vi

  Reading instructions aloud guarantees that rules are common knowledge. However the section of instructions related to different payment schemes of the X and Y subjects must remain unknown until the end of the first part of the session. Reading aloud only other sections of the instructions would have focused undue attention on the question of compensation. it E i vii This is evocative of the result found by[START_REF] Charness | How Robust is Laboratory Gift Exchange?[END_REF] that in a gift-exchange game, there is less reciprocity when the payoffs for different actions are made clearer.viii The observation that managers free-ride less than students is reminiscent of the[START_REF] Hannan | Partial Gift Exchange in an Experimental Labor Market: Impact of Subject Population Differences, Productivity Differences, and Effort Requests on Behavior[END_REF] result that MBA students (which are more like managers) do more giftexchange than other students. ix Let measure individual i's output level in period t, explained by a vector of observable variables z it , the corresponding parameter vector δ, a random individual component η and a random variable it

Table 1 --Payment schemes in ECU

 1 For the teams of X subjects the table should be read as follows. In the first part of the experiment, if a X subject realizes an outcome lower than the first threshold of 40, he receives no payoff. If he reaches a height between the first (40) and the second threshold (60), he receives only a fixed wage of 40 (). If the subject reaches the second threshold, he receives a payoff of 60, consisting of the sum of the fixed wage and the individual bonus (

				Part 1 -All treatments	Part 2 -Fixed treatment	Part 2 -Mixed treatment
	Group		Teams of X	Teams of Y	Teams of X	Teams of Y	Teams of X and Y
	Composition	subjects	subjects	subjects	subjects	subjects
	Height reached
	40 ≤< i H < 40 i H 60 100 60 i H ≤≤	0 40 60	0 60 60 100	0 60 80 120	0 60 80 120	0 60 80 120
	3 1 ∑	i H ≥	240	100
	Note:

Table 2 --Variables and descriptive statistics

 2

	Variable	Definition	Mean (Standard deviation)
	Endogenous variables		Managers	Students
	Output achieved	Output levels reached by the subjects	78.04	73.9
			(20.96)	(20.680)
	Cost	Costs of outputs supported by the subjects	16.60	12.75
			(13.19)	(11.39)
	Exogenous Variables
	1 st period effect	1 if the 1 st period;	0.05	0.05
		0 otherwise	0.95	0.95
	11 th period effect	1 if the 11 th period ;	0.05	0.05
		0 otherwise	0.95	0.95
	Y subject	1 if the individual is a Y subject;	0.50	0.50
		0 otherwise (X subject)	0.50	0.50
	Mixed session	1 if X and Y subjects can interact;	0.33	0.33
		0 otherwise	0.67	0.67
	Mixed session and	1 if Y is involved in a mixed session;	0.16	0.16
	Y subject	0 otherwise	0.84	0.84
	Mode of compensation in	1 if part 1 (periods 1 to 10);	0.50	0.50
	part 1	0 if part 2 (periods 11 to 20)	0.50	0.50
	Mode of compensation in	1 if a Y subject in part 1;	0.25	0.25
	part 1 and Y subject	0 otherwise	0.25	0.25
	Lagged output levels of the	Output of the other members of the group in the	154.88*	146.49*
	other members of the group	preceding period	(32.07)	(34.67)
	Lagged output levels of the other members of the group and Y	Output of the other members of the group in the preceding period and Y	155.15** (32.37)	147.58** (35.04)
	Index of difficulty	Index of difficulty of the curve/100	225.27	230.96
			(143.10)	(143.36)
	Index of difficulty squared Index of difficulty squared	71798.18	73882.02
			(72847.8)	(72.466.04)
	Index of difficulty and	Interaction of index of difficulty and period	2498.6	2581.17
	period		(2482.7)	(2512.12)
	Period	Period number from 1 to 20	10.5	10.5
	Logarithm of period	Logarithm of period	2.12	2.12
	* 1 st and 11 th periods excluded. ** For Y subjects only and 1 st and 11 th periods excluded.

Table 3 --Students: determinants of output levels and costs Variable Output

 3

				Cost
		Panel Random*	Panel Random**
		Coefficient	t-ratio	Coefficient	t-ratio
	Constant	135.9 a	26.5	11.3 a	4.21
	1 st period	-43.09 a	-10.9	0.671	0.362
	11 th period	-4.858	-1.46	0.387	0.303
	Y subject	3.045	0.858	-1.9608	-0.831
	Mixed	-0.7015	-0.324	-1.322	-1.24
	Y and Mixed	1.908	0.625	1.912	1.27
	Mode of compensation in	-2.740	-1.02	-5.27 a	-4.10
	part 1
	Y and mode of compensation	2.896	1.14	1.410	1.14
	in part 1
	Output levels of other group	-0.01724	-1.17	0.02892 a	4.18
	members
	Y and output levels of other	-0.01738	-1.23	-0.01186 b	-1.79
	group members
	Index of difficulty	-0.3076 a	-24.2	0.03958 a	6.63
	Index of difficulty squared	0.0004981 a	21.6	-0.00003859 a	-3.57
	Index of difficulty and period	0.001846 a	2.73	0.0007233 a	2.28
	Periods 1 to 20	1.0409 a	2.50	-0.6201 a	-3.17
	Logarithm of period	-18.20 a	-7.35	0.4660	0.968
	Adjusted R 2	0.3569		0.163
	Number of observations	1440		1440
	* Linear one-way random effects model. Lagrange multiplier test versus OLS = 433.71
	(1 df, prob value = 0.00000).** Linear one-way random effects model. Lagrange multiplier test
	versus OLS = 2761.10
	(1 df, prob value = .000000)
	a) significant at 5% level; b) significant at 10% level.

Table 3a --Managers: determinants of output levels and costs Variable Output

 3a Linear one-way random effects model. Lagrange multiplier test versus OLS = 12.06 (1 df, prob value = .000515). ** Linear one-way random effects model. Lagrange multiplier test versus OLS = 114.41

				Cost
		Panel Random*	Panel Random**
		Coefficient	t-ratio	Coefficient	t-ratio
	Constant	125.32 a	13.3	6.197	1.13
	1 st period	-11.80	-1.59	10.25 a	2.42
	11 th period	-1.539	-0.289	0.967	0.318
	Y subject	-3.030	-0.526	8.06 a	2.16
	Mixed	1.500	0.414	6.23 a	2.83
	Y and Mixed	-7.194	-1.41	-14.61 a	-4.69
	Mode of compensation in	-11.791 a	-2.47	-1.805	-0.653
	part 1
	Y and mode of compensation	-4.559	-1.04	-10.09 a	-3.88
	in part 1
	Lagged output levels of other	0.03469	1.251	0.02427	1.53
	group members
	Y and output levels of other	0.05420 a	2.33	0.0195	1.37
	group members
	Index of difficulty	-0.2132 a	-9.41	0.05503 a	4.27
	Index of difficulty squared	0.0003381 a	8.37	-0.00007559 a	-3.29
	Index of difficulty and period	0.003394 a	2.82	0.001122 b	1.64
	Periods 1 to 20	-1.354 b	-1.83	-1.037 a	-2.47
	Logarithm of period	-7.768 b	-1.76	2.799	1.11
	Adjusted R 2	0.1562		0.1792
	Number of observations	720		720
	*

The final version of the paper has benefited from comments and criticisms provided by the associate editor and the referees. Financial support from the Region of Rhône-