Samuele Giraudo

Constructing combinatorial operads from monoids

Keywords: Operad, Monoid, Generalized Dyck path, Tree, Directed animal, Diassociative operad

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

Résumé. Nous introduisons une construction fonctorielle qui, à partir d'un monoïde, produit une opérade ensembliste. Nous obtenons de nouvelles opérades (symétriques ou non) comme sous-opérades ou quotients de l'opérade obtenue à partir du monoïde additif. Celles-ci mettent en jeu divers objets combinatoires familiers : fonctions de parking, mots tassés, arbres plans enracinés, chemins de Dyck généralisés, arbres de Schröder, chemins de Motzkin, compositions d'entiers, animaux dirigés, etc. Nous retrouvons également des opérades déjà connues : l'opérade magmatique, l'opérade commutative associative et l'opérade diassociative.

Introduction

Operads are algebraic structures introduced in the 1970s by Boardman and Vogt [START_REF] Boardman | Homotopy invariant algebraic structures on topological spaces[END_REF] and by May [START_REF] May | The geometry of iterated loop spaces[END_REF] in the context of algebraic topology. Informally, an operad is a structure containing operators with n inputs and 1 output, for all positive integer n. Two operators x and y can be composed at ith position by grafting the output of y on the ith input of x. The new operator thus obtained is denoted by x • i y. In an operad, one can also switch the inputs of an operator x by letting a permutation σ act to obtain a new operator denoted by x • σ. One of the main relishes of operads comes from the fact that they offer a general theory to study in an unifying way different types of algebras, such as associative algebras and Lie algebras.

In recent years, the importance of operads in combinatorics has continued to increase and several new operads were defined on combinatorial objects (see e.g., [START_REF] Chapoton | Operads and algebraic combinatorics of trees[END_REF][START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF][START_REF] Livernet | A rigidity theorem for pre-Lie algebras[END_REF]10]). The structure thereby added on combinatorial families enables to see these in a new light and offers original ways to solve some combinatorial problems. For example, the dendriform operad [10] is an operad on binary trees and plays an interesting role for the understanding of the Hopf algebra of Loday-Ronco of binary trees [START_REF] Hivert | The Algebra of Binary Search Trees[END_REF][START_REF] Loday | Hopf Algebra of the Planar Binary Trees[END_REF]. Besides, this operad is a key ingredient for the enumeration of intervals of the Tamari lattice [START_REF] Chapoton | Sur le nombre d'intervalles dans les treillis de Tamari[END_REF][START_REF] Chapoton | Operads and algebraic combinatorics of trees[END_REF]. There is also a very rich link connecting combinatorial Hopf algebra theory and operad theory: various constructions produce combinatorial Hopf algebras from operads [START_REF] Chapoton | Relating two Hopf algebras built from an operad[END_REF][START_REF] Loday | Algebraic operads[END_REF].

In this paper, we propose a new generic method to build combinatorial operads. The starting point is to pick a monoid M . We then consider the set of words whose letters are elements of M . The arity of such words are their length, the composition of two words is expressed from the product of M , and permutations act on words by permuting letters. In this way, we associate to any monoid M an operad denoted by TM . This construction is rich from a combinatorial point of view since it allows, by considering suboperads and quotients of TM , to get new operads on various combinatorial objects. This paper is organized as follows. In Section 2, we recall briefly the basics about set-operads. Section 3 is devoted to the definition of the construction associating an operad to a monoid and to establish its first properties. We show that this construction is a functor from the category of monoids to the category of operads that respects injections and surjections. Finally we apply this construction in Section 4 on various monoids and obtain several new combinatorial (symmetric or not) operads on the following combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, and segmented integer compositions. We conclude by building an operad isomorphic to the diassociative operad [10].

Preliminaries and notations

Permutations

Let us denote by [n] the set {1, . . . , n} and by S n the set of permutations of [n]. Let σ ∈ S n , ν ∈ S m , and i ∈ [n]. The substitution of ν into σ is the permutation B i (σ, ν) := σ 1 . . . σ i-1 ν 1 . . . ν m σ i+1 . . . σ n where σ j := σ j if σ j < σ i and σ j := σ j + m -1 otherwise, and ν j := ν j + σ i -1. For instance, one has B 4 (7415623, 231) = 941675823.

Operads

Recall that a set-operad, or an operad for short, is a set P := n≥1 P(n) together with substitution maps

• i : P(n) × P(m) → P(n + m -1), n, m ≥ 1, i ∈ [n], (1)
a distinguished element 1 ∈ P(1), the unit of P, and a symmetric group action

• : P(n) × S n → P(n), n ≥ 1. (2)
The above data has to satisfy the following relations:

(x • i y) • i+j-1 z = x • i (y • j z), x ∈ P(n), y ∈ P(m), z ∈ P(k), i ∈ [n], j ∈ [m], (3)
(x • i y) • j+m-1 z = (x • j z) • i y, x ∈ P(n), y ∈ P(m), z ∈ P(k), 1 ≤ i < j ≤ n, (4) 1 • 1 x = x = x • i 1, x ∈ P(n), i ∈ [n], (5)
(x • σ) • i (y • ν) = (x • σi y) • B i (σ, ν), x ∈ P(n), y ∈ P(m), σ ∈ S n , ν ∈ S m , i ∈ [n]. (6)
The arity of an element x of P(n) is n. Let Q be an operad. A map φ : P → Q is an operad morphism if it commutes with substitution maps and symmetric group action and maps elements of arity n of P to elements of arity n of Q. A non-symmetric operad is an operad without symmetric group action. The above definitions also work when P is a N-graded vector space. In this case, the substitution maps • i are linear maps, and the symmetric group action is linear on the left.

3 A combinatorial functor from monoids to operads

TM (n) := {(x 1 , . . . , x n ) : x i ∈ M for all i ∈ [n]} . (7)
We endow the set TM with maps

• i : TM (n) × TM (m) → TM (n + m -1), n, m ≥ 1, i ∈ [n], (8)
defined as follows: For all x ∈ TM (n), y ∈ TM (m), and i ∈ [n], we set

x • i y := (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y m , x i+1 , . . . , x n ). (9)
Let us also set 1 := (1) as a distinguished element of TM (1). We endow finally each set TM (n) with a right action of the symmetric group

• : TM (n) × S n → TM (n), n ≥ 1, (10)
defined as follows: For all x ∈ TM (n) and σ ∈ S n , we set

x • σ := (x σ1 , . . . , x σn ) . (11)
The elements of TM are words over M regarded as an alphabet. The arity of an element x of TM (n), denoted by |x|, is n. For the sake of readability, we shall denote in some cases an element (x 1 , . . . , x n ) of TM (n) by its word notation x 1 . . . x n . Proposition 3.1 If M is a monoid, then TM is a set-operad.

Proof: Let us respectively denote by • and 1 the product and the unit of M . First of all, thanks to ( 8) and ( 9), the maps • i are well-defined and are substitution maps of operads. Let us now show that TM satisfies (3), ( 4), ( 5), and [START_REF] Dvoretzky | A problem of arrangements[END_REF].

Let

x ∈ TM (n), y ∈ TM (m), z ∈ TM (k), i ∈ [n], and j ∈ [m]. We have, using associativity of •, (x • i y) • i+j-1 z = (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y m , x i+1 , . . . , x n ) • i+j-1 z = (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y j-1 , (x i • y j ) • z 1 , . . . , (x i • y j ) • z k , x i • y j+1 , . . . , x i • y m , x i+1 , . . . , x n ) = (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y j-1 , x i • (y j • z 1 ), . . . , x i • (y j • z k ), x i • y j+1 , . . . , x i • y m , x i+1 , . . . , x n ) = x • i (y 1 , . . . , y j-1 , y j • z 1 , . . . , y j • z k , y j+1 , . . . , y m ) = x • i (y • j z), (12)
showing that

• i satisfies (3). Let x ∈ TM (n), y ∈ TM (m), z ∈ TM (k), and i < j ∈ [n]. We have, (x • j z) • i y = (x 1 , . . . , x j-1 , x j • z 1 , . . . , x j • z k , x j+1 , . . . , x n ) • i y, = (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y m , x i+1 , . . . , x j-1 , x j • z 1 , . . . , x j • z k , x j+1 , . . . , x n ) = (x 1 , . . . , x i-1 , x i • y 1 , . . . , x i • y m , x i+1 , . . . , x n ) • j+m-1 z = (x • i y) • j+m-1 z, (13)
showing that • i satisfies (4).

The element 1 is the unit of TM . Indeed, we have 1 ∈ TM (1), and, for all x ∈ TM (n) and i ∈ [n],

x

• i 1 = (x 1 , . . . , x i-1 , x i • 1, x i+1 , . . . , x n ) = x, ( 14
)
since 1 is the unit for •, and

1 • 1 x = (1 • x 1 , . . . , 1 • x n ) = x, (15)
for the same reason. That shows [START_REF] Chapoton | Relating two Hopf algebras built from an operad[END_REF]. Finally, since the symmetric group S n acts by permuting the letters of a word (x 1 , . . . , x n ) of TM (n), the maps • i and the action • satisfy together [START_REF] Dvoretzky | A problem of arrangements[END_REF]. 2

Monoids morphisms to operads morphisms

Let M and N be two monoids and θ : M → N be a monoid morphism. Let us denote by Tθ the map

Tθ : TM → TN, (16)
• i y) = Tθ (x 1 , . . . , x i-1 , x i • M y 1 , . . . , x i • M y m , x i+1 , . . . , x n ) = (θ(x 1 ), . . . , θ(x i-1 ), θ(x i • M y 1 ), . . . , θ(x i • M y m ), θ(x i+1 ), . . . , θ(x n )) = (θ(x 1 ), . . . , θ(x i-1 ), θ(x i ) • N θ(y 1 ), . . . , θ(x i ) • N θ(y m ), θ(x i+1 ), . . . , θ(x n )) = (θ(x 1 ), . . . , θ(x n )) • i (θ(y 1 ), . . . , θ(y m )) = Tθ(x) • i Tθ(y). (18)
Moreover, since (1 M ) is by definition the unit of TM , we have

Tθ (1 M ) = (θ (1 M )) = (1 N ) . ( 19
)
Finally, since the symmetric group S n acts by permuting letters, we have for all x ∈ TM (n) and

σ ∈ S n , Tθ(x • σ) = Tθ(x) • σ.
The map Tθ satisfies the three required properties and hence, since by Proposition 3.1, TM and TN are operads, Tθ is an operad morphism.

Properties of the construction

Proposition 3.3 Let M and N be two monoids and θ : M → N be a monoid morphism. If θ is injective (resp. surjective), then Tθ is injective (resp. surjective).

Proof: Assume that θ is injective and that there are two elements x and y of TM such that Tθ(x) = Tθ(y). Then, Tθ(x) = (θ(x 1 ), . . . , θ(x n )) = (θ(y 1 ), . . . , θ(y n )) = Tθ(y), (20)

implying θ(x i ) = θ(y i ) for all i ∈ [n].
Since θ is injective, we have x i = y i for all i ∈ [n] and thus, x = y. Hence, since Tθ is, by Proposition 3.2, an operad morphism, it also is an injective operad morphism. Assume that θ is surjective and let y be an element of TN (n). Since θ is surjective, there are some elements x i of M such that θ(x i ) = y i for all i ∈ [n]. We have

Tθ(x 1 , . . . , x n ) = (θ(x 1 ), . . . , θ(x n )) = (y 1 , . . . , y n ). (21)
Hence, since (x 1 , . . . , x n ) is by definition an element of TM (n), and since Tθ is, by Proposition 3.2, an operad morphism, it also is a surjective operad morphism. 2

Theorem 3.4 The construction T is a functor from the category of monoids with monoid morphisms to the category of set-operads with operad morphisms. Moreover, T respects injections and surjections.

Proof: By Proposition 3.1, T constructs a set-operad from a monoid, and by Proposition 3.2, an operad morphism from a monoid morphism.

Let M be a monoid, θ : M → M be the identity morphism on M , and x be an element of TM (n). We have Tθ(x) = (θ(x 1 ), . . . , θ(

x n )) = (x 1 , . . . , x n ) = x, (22)
showing that Tθ is the identity morphism on the operad TM . Let (L, • L ), (M, • M ), and (N, • N ) be three monoids, θ : L → M and ω : M → N be two monoid morphisms, and x be an element of TL(n). We have

T(ω • θ)(x) = (ω (θ(x 1 )) , . . . , ω (θ(x n ))) = Tω (θ(x 1 ), . . . , θ(x n )) = Tω (Tθ (x 1 , . . . , x n )) = (Tω • Tθ)(x), (23)
showing that T is compatible with map composition. Hence, T is a functor, and by Proposition 3.3, T also respects injections and surjections. 2

4 Some operads obtained by the construction Through this Section, we consider examples of applications of the functor T. We shall mainly consider, given a monoid M , some suboperads of TM , symmetric or not, and generated by a finite subset of TM . We shall denote by N the additive monoid of integers, and for all ≥ 1, by N the quotient of N consisting in the set {0, 1, . . . , -1} with the addition modulo as the operation of N . Note that since T is a functor that respects surjective maps (see Theorem 3.4), TN is a quotient operad of TN.

The operads constructed in this Section fit into the diagram of non-symmetric operads represented in Figure 1. Table 1 summarizes some information about these operads. FCat (1 ) FCat (2 ) FCat (3 )

SComp DA PRT Motz Comp FCat (0 )

Endofunctions, parking functions, packed words, and permutations

Neither the set of endofunctions nor the set of parking functions, packed words, and permutations are suboperads of TN. Indeed, one has the following counterexample:

12 • 2 12 = 134, (24)
and, even if 12 is a permutation, 134 is not an endofunction. Therefore, let us call a word u a twisted endofunction (resp. parking function, packed word, permutation) if the word (u 1 + 1, u 2 + 1, . . . , u n + 1) is an endofunction (resp. parking function, packed word, permutation). For example, the word 2300 is a twisted endofunction since 3411 is an endofunction. Let us denote by End (resp. PF , PW , Per ) the set of endofunctions (resp. parking functions, packed words, permutations). Under this reformulation, one has the following result: For example, we have in End the following substitution:

2123 • 2 30313 = 24142423, (25)
and the following application of the symmetric group action: Note that End is not a finitely generated operad. Indeed, the twisted endofunctions u := u 1 . . . u n satisfying u i := n -1 for all i ∈ [n] cannot be obtained by substitutions involving elements of End of arity smaller than n. Similarly, PF is not a finitely generated operad since the twisted parking functions u := u 1 . . . u n satisfying u i := 0 for all i ∈ [n -1] and u n := n -1 cannot be obtained by substitutions involving elements of PF of arity smaller than n. However, the operad PW is a finitely generated operad: Proposition 4.2 The operad PW is the suboperad of TN generated, as a symmetric operad, by the elements 00 and 01.

Let K be a field and let us from now consider that PW is an operad in the category of K-vector spaces, i.e., PW is the free K-vector space over the set of twisted packed words with substitution maps and the right symmetric group action extended by linearity.

Let I be the free K-vector space over the set of twisted packed words having multiple occurrences of a same letter. Proposition 4.3 The vector space I is an operadic ideal of PW . Moreover, the operadic quotient Per := PW / I is the free vector space over the set of twisted permutations.

One has, for all twisted permutations x and y, the following expression for the substitution maps in Per :

x • i y = x • i y if x i = max x, 0 K otherwise, (27)
where 0 K is the null vector of Per and the map • i in the right member of ( 27) is the substitution map of PW .

Planar rooted trees

Let PRT be the non-symmetric suboperad of TN generated by 01. One has the following characterization of the elements of PRT :

Proposition 4.4
The elements of PRT are exactly the words x on the alphabet N that satisfy x 1 = 0 and

1 ≤ x i+1 ≤ x i + 1 for all i ∈ [|x| -1].
Proposition 4.4 implies that we can regard the elements of arity n of PRT as planar rooted trees with n nodes. Indeed, there is a bijection between words of PRT and such trees. Given a planar rooted tree T , one computes an element of PRT by labelling each node x of T by its depth and then, by reading its labels following a depth-first traversal of T . Figure 2(a) shows an example of this bijection. This bijection offers an alternative way to compute the substitution in PRT : Proposition 4.5 Let S and T be two planar rooted trees and s be the ith visited node of S following its depth-first traversal. The substitution S • i T in PRT amounts to graft the subtrees of the root of T as leftmost sons of s. Proposition 4.6 also says that PRT is isomorphic to the magmatic operad. Hence, PRT is a realization of the magmatic operad. Moreover, PRT can be seen as a planar version of the operad NAP of Livernet [START_REF] Livernet | A rigidity theorem for pre-Lie algebras[END_REF].

Generalized Dyck paths

Let k ≥ 0 be an integer and FCat (k ) be the non-symmetric suboperad of TN generated by 00, 01, . . . , 0k.

One has the following characterization of the elements of FCat (k ) :

Proposition 4.7 The elements of FCat (k ) are exactly the words x on the alphabet N that satisfy x 1 = 0 and 0 ≤

x i+1 ≤ x i + k for all i ∈ [|x| -1].
Let us recall that a k-Dyck path of length n is a path in N 2 connecting the points (0, 0) and ((k + 1)n, 0) and consisting in n up steps (1, k) and kn down steps (1, -1). It is well-known that k-Dyck paths are enumerated by Fuss-Catalan numbers [START_REF] Dvoretzky | A problem of arrangements[END_REF]. Proposition 4.7 implies that we can regard the elements of arity n of FCat (k ) as k-Dyck paths of length n. Indeed, there is a bijection between words of FCat (k ) and such paths. Given a k-Dyck path P , one computes an element of FCat (k ) by writing, from left to right, the ordinate of the starting point of each up step of P . Figure 3 shows an example of this bijection. Note that the operad FCat (0 ) is the commutative associative operad. Next Theorem elucidates the structure of FCat (1 ) :

Directed animals

Let DA be the non-symmetric suboperad of TN 3 generated by 00 and 01. Since FCat (1 ) is the nonsymmetric suboperad of TN generated by 00 and 01, and since TN 3 is a quotient of TN, DA is a quotient of FCat (1 ) .

From now, we shall represent by -1 the element 2 of N 3 . With this encoding, let Recall that a directed animal is a subset A of N 2 such that (0, 0) ∈ A and (i, j) ∈ A with i ≥ 1 or j ≥ 1 implies (i -1, j) ∈ A or (i, j -1) ∈ A. Using a bijection of Gouyou-Beauchamps and Viennot [START_REF] Gouyou-Beauchamps | Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem[END_REF] between directed animals of size n and prefixes of Motzkin paths of length n -1, one obtains, by Proposition 4.16, a bijection between the elements of DA of arity n and directed animals of size n. Hence, DA is a non-symmetric operad on directed animals.

φ : {-1, 0, 1} n → {-1, 0, 1} n-1 , (47)

Segmented integer compositions

Let SComp be the non-symmetric suboperad of TN 3 generated by 00, 01, and 02. Since FCat (2 ) is the non-symmetric suboperad of TN generated by 00, 01, and 02, and since TN 3 is a quotient of TN, SComp is a quotient of FCat (2 ) . One has the following characterization of the elements of SComp: Proposition 4.17 The elements of SComp are exactly the words on the alphabet {0, 1, 2} that begin by 0.

Recall that a segmented integer composition of n is a sequence (S 1 , . . . , S ) of integers compositions such that S 1 is an integer composition of n 1 , . . . , S is an integer composition of n , and n 1 +• • •+n = n. Proposition 4.17 implies that we can regard the elements of arity n of SComp as segmented integer compositions of n. Indeed, there is a bijection between words of SComp and segmented compositions since there are 3 n-1 segmented compositions of n and also, by the above Proposition, 3 n-1 elements of SComp of arity n.

The diassociative operad

Let M be the submonoid of the multiplicative monoid restricted to the set {0, 1}. Let D be the nonsymmetric suboperad of TM generated by 01 and 10. One has the following characterization of the elements of D: Proposition 4.18 The elements of D are exactly the words on the alphabet {0, 1} that contain exactly one 1.

Recall that the diassociative operad [10] Dias is the non-symmetric operad generated by two elements and of arity 2, subject only to the relations

• 1 = • 2 = • 2 , ( 49
)
• 2 = • 1 = • 1 , (50)
• 1 = • 2 . (51)

3. 1

 1 The construction 3.1.1 Monoids to operads Let (M, •, 1) be a monoid. Let us denote by TM the set TM := n≥1 TM (n), where for all n ≥ 1,

  defined for all (x 1 , . . . , x n ) ∈ TM (n) by Tθ (x 1 , . . . , x n ) := (θ(x 1 ), . . . , θ(x n )) . (17) Proposition 3.2 If M and N are two monoids and θ : M → N is a monoid morphism, then the map Tθ : TM → TN is an operad morphism. Proof: Let us respectively denote by • M (resp. • N ) and 1 M (resp. 1 N ) the product and the unit of M (resp. N ). Let x ∈ TM (n), y ∈ TM (m), and i ∈ [n]. Since θ is a monoid morphism, we have Tθ(x

Figure 1 :

 1 Figure 1: The diagram of non-symmetric suboperads and quotients of TN. Arrows (resp. ) are injective (resp. surjective) non-symmetric operad morphisms.

Proposition 4 . 1

 41 The sets End , PF , and PW are suboperads of TN.

2 =

 2 A planar rooted tree and the corresponding element of PRT .• (b) A substitution in PRT .

Figure 2 :

 2 Figure 2: Interpretation of the elements and substitution of the non-symmetric operad PRT in terms of planar rooted trees.

Figure 2 (Proposition 4 . 6

 246 Figure 2(b) shows an example of substitution in PRT . Proposition 4.6 The non-symmetric operad PRT is isomorphic to the free non-symmetric operad generated by one element of arity 2.

Figure 3 : A 2 -

 32 Figure 3: A 2-Dyck path and the corresponding element of the non-symmetric operad FCat(2 ) .

  be the map defined for all a, b ∈ {-1, 0, -1} by φ(a) := and φ(a • b • u) := (ba) • φ(u). (48) For example, the element x := 011220201 of DA is represented by the word x := 011 -1 -10 -101 and we have φ(x ) = 10101 -111. Proposition 4.16 By interpreting letters -1 (resp. 0, 1) as down (resp. stationary, up) steps, the map φ induces a bijection between the elements of DA of arity n and the prefixes of Motzkin paths of length n-1.

Table 1 :

 1 Generators, first dimensions, and combinatorial objects involved in the non-symmetric suboperads and quotients of TN.

	11210 • 23514 = 12011.	(26)

1365-8050 c 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Acknowledgements

The author would like to thank Florent Hivert and Jean-Christophe Novelli for their advice during the preparation of this paper. This work is based on computer exploration and the author used, for this purpose, the open-source mathematical software Sage [15] and one of its extensions, Sage-Combinat [14].

Theorem 4.8 The non-symmetric operad FCat (1 ) is the free non-symmetric operad generated by two elements and of arity 2, subject to the three relations

Schr öder trees

Let Schr be the non-symmetric suboperad of TN generated by 00, 01, and 10. One has the following characterization of the elements of Schr :

Proposition 4.9 The elements of Schr are exactly the words x on the alphabet N that have at least one occurrence of 0 and, for all letter b ≥ 1 of x, there exists a letter a := b -1 such that x has a factor aub or bua where u is a word consisting in letters c satisfying c ≥ b.

Recall that a Schröder tree is a planar rooted tree such that no node has exactly one child. The leaves of a Schröder tree are the nodes without children. We call sector of a Schröder tree T a triple (x, i, j) consisting in a node x and two adjacent edges i and j, where i is immediately on the left of j. Proposition 4.9 implies that we can regard the elements of arity n of Schr as Schröder trees with n leaves. Indeed, there is a bijection between words of Schr and such trees. Given a Schröder tree T , one computes an element of Schr by labelling each sector (x, i, j) of T by the depth of x and then, by reading the labels from left to right. Figure 4 shows an example of this bijection.

←→ 1132002122 Let us respectively denote by , , and the generators 00, 01, and 10 of Schr .

Proposition 4.10 The generators , , and of Schr are subject, in degree 2, to the seven relations

Motzkin paths

Let Motz be the non-symmetric suboperad of TN generated by 00 and 010. Since 00 and 01 generate FCat (1 ) and since 010 = 00 • 1 01, Motz is a non-symmetric suboperad of FCat (1 ) . One has the following characterization of the elements of Motz :

The elements of Motz are exactly the words x on the alphabet N that begin and start by 0 and such that |x i -

Recall that a Motzkin path of length n is a path in N 2 connecting the points (0, 0) and (n, 0), and consisting in up steps (1, 1), down steps (1, -1), and stationary steps (1, 0). Proposition 4.11 implies that we can regard the elements of arity n of Motz as Motzkin paths of length n -1. Indeed there is a bijection between words of Motz and such paths. Given a Motzkin path P , one computes an element of Motz by writing for each point p of P the ordinate of p, and then, by reading these labels from left to right. Figure 5 shows an example of this bijection.

←→ 001123221010 Let us respectively denote by and the generators 00 and 010 of Motz .

Proposition 4.12 The generators and of Motz are subject, in degree 2, to the four relations

Integer compositions

Let Comp be the non-symmetric suboperad of TN 2 generated by 00 and 01. Since FCat (1 ) is the nonsymmetric suboperad of TN generated by 00 and 01, and since TN 2 is a quotient of TN, Comp is a quotient of FCat (1 ) . One has the following characterization of the elements of Comp : Proposition 4. [START_REF] May | The geometry of iterated loop spaces[END_REF] The elements of Comp are exactly the words on the alphabet {0, 1} that begin by 0.

Proposition 4.13 implies that we can regard the elements of arity n of Comp as integer compositions of n. Indeed, there is a bijection between words of Comp and integer compositions. Given a composition C := (C 1 , C 2 , . . . , C ), one computes the following element of Comp:

Encoding integer compositions by ribbon diagrams offers an alternative way to compute the substitution in Comp: Theorem 4.15 The non-symmetric operad Comp is the free non-symmetric operad generated by two elements and of arity 2, subject to the four relations


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Alain Pholo Bala

Social capital as an engine of growth: multisectoral modelling and implications

Keywords: social capital, human capital, economic growth, imbalance effects. JEL Classification: C61, E20, E22, E24, O41

We propose a multisector endogenous growth model incorporating social capital.

Social capital only serves as input in the production of human capital and it involves a cost in terms of the final good. We show that in contrast to existing alternative specifications, this setting assures that social capital enhances productivity gains by playing the role of a timing belt driving the transmission and propagation of all productivity shocks throughout society whatever the sectoral origin of the shocks.

Further econometric work is conducted in order to estimate the contribution of social capital to human capital formation. We find that depending on the measure of social capital considered, the elasticity of human capital to social capital varies from 6% to 10%. Finally we investigate the short-term dynamics and imbalance effects properties of the models depending on the value of this elasticity (taking the Lucas-Uzawa model as a limit case). In particular, it's shown that when the substitutability of social capital to human capital increases, the economy is better equipped to surmount initial imbalances as individuals may allocate more working time in the final goods sector without impeding economic growth.

Introduction

The concept of social capital has recently received rising acceptance in economics research.

It has been pointed as a potential source of economic growth [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF]; Sequeira and Ferreira-Lopes, 2008). While some researchers still express some reluctance to consider it as a "capital", 1 an increasing number of economists now admit that social capital shares at least some similarities with physical and human capital namely its intertemporal dimension and its ability to generate a stream of future benefits [START_REF] Chou | Three simple models of social capital and economic growth[END_REF]. Like other sociological concepts, social capital encompasses several different meanings. According to [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF], trust, cooperative norms, and associations within groups represent the essence of the definition of that concept. Others characterize it in terms of cultural values such as compassion, altruism and tolerance, while others emphasize institutions and the quality and quantity of "associational" life. Some researchers find it practical to separate the sources of social capital (primarily, social networks) from their consequences (which can be positive or negative, depending on the circumstances), such as trust, tolerance and cooperation [START_REF] Chou | Three simple models of social capital and economic growth[END_REF].

Several empirical papers have tried to assess the impact of different dimensions of social capital on economic growth. Indicators of social capital has been shown to affect local financial development as well as general economic growth in Italy [START_REF] Helliwell | Economic Growth and Social Capital in Italy[END_REF]. [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF] establish a causal relationship between trust and growth.

Controlling for initial income per head, a human capital variable, and the relative price of investment goods, they find that a level of trust that is 10 percentage points higher implies an annual growth rate higher by 0.8 percentage points. However, they do not find a very robust association (Sequeira and Ferreira-Lopes, 2006). [START_REF] Temple | Social Capability and Economic Growth[END_REF] use an index composed by several measures of social capital. They found those measures useful for predicting economic growth. Several other empirical studies estimate a robust relationship between social capital and growth but with a wide interval of point estimates (e.g. Beugelsdijk et al., 2004;[START_REF] Whiteley | Economic Growth and Social Capital[END_REF][START_REF] Rupasinga | Social Capital and Economic Growth: A County-Level Analysis[END_REF]. Figure 1 depicts the relationship between economic growth and various dimensions of social capital. It indicates that the economic growth rate rises with the different social capital measures.

By which channel social capital affects economic growth? According to [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF], social capital impacts economic performance by improving cooperation and confidence between economic agents. Economic activities that require some agents to rely on the future actions of others are realized at lower cost in higher trust environments [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF]. Individuals in high trust societies are also likely to divert fewer resources to protecting themselves -through tax payments, bribes, or private security services and equipment -from unlawful (criminal) violations of their property rights. Trust may also favor innovation. When entrepreneurs devote more time to monitoring possible malfeasance by partners, employees, and suppliers, they have less time to devote to innovation in new products or processes [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF].

Nevertheless, in this contribution we choose to focus on another channel through which social capital impacts economic growth: the influence of social capital in human capital accumulation. To do, so we assume that human and social capital are both necessary inputs for human capital accumulation and that social capital does not play any role in the production of goods and services. The literature on social capital has acknowledged the fact that trusting societies, in addition to have stronger incentives to innovate and to accumulate physical capital, are also likely to have higher returns to accumulation of human capital. Where trust improves access to credit for the poor, enrollment in secondary education -which, unlike primary education, has a high cost in forgone income -may be higher [START_REF] Galor | Income Distribution and Macroeconomics[END_REF]; [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF]. The issue of the interaction between human and social capital and their joint effects on economic growth, though less developed by the literature, is very important [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF]. [START_REF] Glaeser | An Economic Approach to Social Capital[END_REF] found a strong empirical relationship between human capital and membership of a given social organization (the proxy used to measure social capital).

The impact of civil society, and social networks on the education and raising of children has long been recognized [START_REF] Chou | Three simple models of social capital and economic growth[END_REF]. This has been well encapsulated in the old African proverb, "It takes a village to raise a child". [START_REF] Coleman | Foundations of Social Theory[END_REF] argues that "social capital is the set of resources that inhere in family relations and in community social organization.

Those resources are useful for the cognitive or social development of a child or young person". [START_REF] Teachman | Social Capital and the Generation of Human Capital[END_REF] pointed out that social capital is important in the creation of human capital. These last authors found that a wide range of social capital indicators determine school continuation [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF]. [START_REF] Becker | Human Capital: A Theoretical and Empirical Analysis with Special Reference to Education[END_REF], the great pioneer in the study of the economics of human capital, also recognized the importance of family relations on human capital accumulation. He acknowledges that "no discussion of human capital can omit the influence of families on the knowledge, skills, values, and habits of children".

Empirical evidence backs the alleged interaction between human and social capital. Us-ing the number of siblings as a proxy for social capital, 2 Coleman (1988) provides empirical evidence that the presence of social capital within the family is important in determining whether a child drops out of school. Coleman also uncovers that social capital outside the family has a significant impact on the dropping out decision. Students who have changed schools because their parents moved are more likely to drop out than others. For families that often changed locations, the social relations that constitute social capital are broken at each move. Furthermore, he finds that dropout rates are lower in religiously based private schools than in public or secular private schools. He also discovered that whether parents devote time to religious activities affects human capital accumulation in their offspring. Grootaert and Van-Bastelaer (2002) provide evidence that community participation in parent teacher associations in Burkina Faso is associated with substantially higher rates of school attendance. Other relevant empirical findings are provided by [START_REF] Costa | Understanding the Decline in Social Capital, 1952-1998[END_REF], where the rise in women's labor force participation rates explains the observed decline in social capital produced within the home, and by [START_REF] Glaeser | Measuring Trust[END_REF], who find that people who invest in human capital also invest in social capital.

From all this empirical literature, we may conclude that there are complementarities between social and human capital in human capital accumulation and a positive correlation between both possibly caused by double-sided causality [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF].

Relation to literature

To date, only few papers have jointly considered social and human capital in models of economic growth. This has been done in [START_REF] Bisin | Social Capital, Modernization and Growth, ASSET Conference[END_REF] in an OLG framework.

They are concerned with the different roles human and social capital have in rural and urban societies. Sequeira and Ferreira-Lopes (2008) consider a three sector endogenous growth model where human capital and social capital are complementary in the production of the final good and substitutes in the production of each other. They assume that there is an opportunity cost of accumulating social capital in terms of human capital and foregone earnings.

In his formal three sector model of bonding social capital, [START_REF] Chou | Three simple models of social capital and economic growth[END_REF] departs from [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF] in two respects: firstly, he assumes that social capital has no direct effect on final goods production but impacts it only indirectly through human capital accumulation. Secondly, he considers that human and social capital are complementary in the production of each other. We will follow [START_REF] Chou | Three simple models of social capital and economic growth[END_REF] in two respects.

Firstly, as we believe that social capital matters essentially in the education sector, we assume that social capital has no direct effect on final goods production. Secondly, as we aim to remain consistent with the aforementioned evidence that human and social capital are complementary in human capital accumulation, we depart from Sequeira and Ferreira-Lopes (2008) assumption that human and social capital are substitutes in human capital accumulation. However, conversely to [START_REF] Chou | Three simple models of social capital and economic growth[END_REF] and [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF] we do not consider that accumulation of social capital requires a specific technology. We rather assume that the output of the final goods sector can be used on a one-for-one basis for consumption, investment in physical capital and investment in social capital. Such an assumption implies that accumulating social capital implies an opportunity cost only in terms of consumption and physical capital accumulation. By doing so, we aim to capture two main features of social capital.

1. Essentially, our multisectoral modelling attempts at reflecting the main property of social capital as advocated by its early proponents: "...that social networks have value. Just as a screwdriver (physical capital) or a university education (human capital) can increase productivity (both individual and collective), so do social contacts affect the productivity of individuals and groups..." [START_REF] Putnam | Bowling Alone: The Collapse and Revival of American Community[END_REF]. That is social capital enhances productivity gains by playing the role of a timing belt driving the transmission and propagation of productivity improvements throughout society. By modelling investment in social capital in terms of the final good, and not in terms of specific human capital, we can obtain the latter outcomes. In particular, we will show that any productivity shock in any sector, including the final good sector, will affect all the sectors and the long-run growth rate of the economy, in contrast to the benchmark Lucas-Uzawa model where only productivity shocks in the education sector matter for long-term growth. If social investment were integrally built with human capital as in Sequeira and Ferreira-Lopes (2008), we would obtain exactly the same shock propagation mechanism as in the Lucas-Uzawa model. This is demonstrated explicitly in our Appendix E.

2. An immediate implication of our modelling of social capital is that it should be larger in richer countries. Though such an implication could be discussed on many grounds, specially given the variety of ways to measure social capital, we believe that it does not go at odds with cross-country evidence as reflected in Figure 1: using 2 different measures of social capital (norms of civic behavior and trust indicators) in 1995 and 2000, we show that social capital measures grow in average with real GDP per capita.

Main findings

In addition to the contributions to modelling social capital in multisectoral growth setting as explained above, several findings can be put forward.

1. First of all, we theoretically show that the impact of social capital on long-term growth (in the range of admissible parameterizations) as measured by the elasticity of human capital with respect to social capital in the education sector is ambiguous. One effect is obvious: since social capital (increased through final good expenditures) adds to human capital as a growth engine, long-term growth should be increased thanks to this additional channel. Furthermore, one would conclude that according to the latter reasoning, the more important is social capital in the education sector, the larger the long-term growth attainable. However, this property is not true for all values of the elasticity of human capital to social capital: in particular when this elasticity is close to zero (or equivalently, if we are close enough to the benchmark Lucas-Uzawa model), the long-term growth rate first decreases. It only raises when this elasticity becomes large enough. Therefore, there is another mechanism counter-balancing the one just mentioned, the interaction of both being responsible of that non-monotonic pattern.

This opposite mechanism may be explained as follows: as the elasticity rises, the education sector relies less on human capital and more on social capital, which leads to having a smaller share of human capital in the education sector (and more in the final good sector), ultimately pushing long-term growth down.

2. Because the previous finding points at the necessity to estimate the elasticity of human capital with respect to social capital in the education sector, we do undertake this challenge. Given data limitation concerning social capital, we could only run simple econometric regressions using the World Values Survey for social capital measures and GDP per capita growth (PPP-adjusted) within the period 1980-2000. Our main finding in this respect is that the elasticity of human capital to social capital varies from 6% to 10% depending on the measure of social capital selected.

3. Last but not least, we provide with a complete study of the dynamic implications of social capital. To this end, we carefully calibrate the model and simulate the resulting dynamic systems. Two sets of exercises are considered: technological shocks and imbalance effects analysis (that is dynamics induced when initial conditions are not equal to the corresponding long-term values). For the exercises to be insightful, we consider three different structures depending on the value of elasticity of human capital with respect to social capital: the limit Lucas-Uzawa case (without social capital), the "realistic" parameterization using the result of the priori econometric step, and a last fictive case where social capital is as important as human capital in the education sector. It's shown that this elasticity parameter plays a crucial role in the short-term dynamics and imbalance effects generated by the model. In particular, it's shown that when the substitutability of social capital to human capital increases, the economy is better equipped to surmount initial imbalances as individuals may allocate more working time in the final goods sector without impeding economic growth.

4. In the case of a "realistic calibration", the weight of social capital in human capital accumulation is low. Then, the only way to foster economic growth is through increases of the human to social capital ratio. Such a parameter region is consistent with a finding of [START_REF] Putnam | Bowling Alone: The Collapse and Revival of American Community[END_REF] according to which social capital has been declining in the US, whereas the economy has been growing. This trend is in contrast to what happened in the first half of the century [START_REF] Sequeira | An endogenous growth model with human and social capital interactions[END_REF].

The paper is organized as follows. The second section describes the model. The third section presents some numerical simulations. Then, section 4 provides some concluding remarks.

The model

We present here our endogenous growth model with many identical infinitely lived agents and two sectors. The final goods sector production technology relies on a Cobb-Douglas production function using two types of input: physical and human capital. The second sector is devoted to human capital accumulation thanks to a Cobb-Douglas human capital technology, using human and social capital as outputs. The output of the final goods sector can be used either for consumption, investment in physical capital or for investment in social capital. Therefore, in our model each agent faces a trade-off between devoting human capital to final goods production and to human capital accumulation, and between allocating final goods production to either consumption, investments in physical and social capital.

Our formal model implies the following assumptions: (1) the building or accumulation of social capital requires resources to be diverted from final goods production; (2) social capital decays over time without new "investment" in social capital; (3) social capital (5) human capital is an important input in final goods production.

Production of final goods and capital accumulation 2.1.1 First sector: final goods production, physical and social capital accumulation

The final good sector produces a homogeneous good that is used either to consume or to invest in either physical or social capital. The investment in social capital may increase social interaction. It is detrimental to the physical capital accumulation since it reduces the amount of resources devoted to the physical capital investment. Therefore, it is potentially harmful to the growth of final good production. Moreover, it implies an opportunity cost in terms of foregone consumption. However, as shown in the next subsection, those adverse effects may be compensated by its positive impact on human capital growth. Individuals allocate a fraction u(t) of their time to the production of final goods. Under the Cobb-Douglas technology, the production function takes the following form:

Y (t) = A (K (t)) α (u (t) H (t)) 1-α = C (t) + I K (t) + I S (t) (1)
The remaining fraction of time is allocated to human capital accumulation. Equation [START_REF] Alesina | The Determinants of Trust[END_REF] shows that production of the final goods enable current consumption, and investment in either physical and social capital. Physical and social capital laws of accumulation are respectively:

K (t + 1) = I K (t) -δK (t) = Y (t) -C (t) -I S (t) + (1 -δ) K (t) (2)
S (t + 1) = I S (t) + (1 -δ) S (t) (3)
We consider that all forms of capital depreciate at the same rate δ.

Second sector: human capital accumulation

Individuals allocate the complementary fraction of their time, i.e. 1 -u (t), to the accumulation of human capital

H (t + 1) = B ((1 -u (t)) H (t)) β (S (t)) 1-β + (1 -δ) H (t) (4)
The law of motion depicted in ( 4) is consistent with the assumption that final goods sector is more intensive in physical capital while the education sector is more intensive in human and social capital. Social and human capital are to a certain extent complementary in the educational sector production function. This captures the aforementioned observation that social capital is important in the creation of human capital and that both interact in human capital accumulation. Further, one may ensure the usual requirement that the final good sector is less intensive in human capital than the education sector (putting social capital aside). This requires the restriction:

1 -α < β,
to hold. We shall have it in mind throughout this paper but we also consider theoretically possible situations where the education sector output relies more on social capital than on "pure" human capital. This possibility is certainly consistent with a much less academic view of human capital. In this case, the elasticity parameter β might be below 1 -α.

Eventually, this model exhibits an asymmetry that is quite standard in two sectors endogenous growth model (Mulligan and Sala-i-Martin, 1992): we have on one hand physical and social capital whose accumulations are perfect substitute for consumption, and human capital on the other hand whose accumulation proceeds from a different technology.

Firms

The final good sector produces a composite good that is used either to consume or to invest in physical capital or in social capital. Following the classical Ramsey, Cass and Koopmans model (Barro and Sala-i-Martin, 1995), we make the standard assumption that firms produce final goods, pay wages for human capital input and make rental payments for physical capital input. Given the Cobb-Douglas production function already described in (1), the discounted profits are given by:

Π (t) = ∞ t=0 [Y (t) -r (t) K (t) -w Y (t) (u (t) H (t))] R (t) (5)
where

R (0) = 1 and R (t) = t τ =0 1 1 + r (τ )
is the discount factor at time t.

The representative firm chooses physical capital and human capital in order to maximize its discounted profits taking prices as given and subject to its technological constraint:

max {K(t) ∞ t=0 ,(u(t)H(t)) ∞ t=0 } Π (t) (6)
Because the firm rents capital and labor services and do not face any adjustment costs, there are no intertemporal elements in the firm's optimization problem. This implies that the problem of maximizing the present value of profits reduces to a problem of maximizing profits in each period without considering the outcomes in other periods as in the Ramsey, Cass and Koopmans model (Barro and Sala-i-Martin, 1995). Therefore, the first-order conditions characterizing an interior maximum for Π (t) are the following:

r (t) = αA (K (t) /u (t) H (t)) α-1 (7)
w

Y (t) = (1 -α) A (K (t) / (u (t) H (t))) α (8)
(7) and ( 8) indicate that the firm chooses the ratio of physical to human capital in order to equate the rental price of physical capital (i.e. the interest rate) to the marginal product of capital and the wage rate to the marginal product of labor. This implies for the firm zero profit in each period since factor payments exhaust total output.

Households behavior

We consider a closed economy inhabited by a constant population normalized to one. This population is composed by identical infinitely-lived households that maximize the following intertemporal utility function:

∞ t=0 C (t) 1-σ -1 1 -σ ρ t , ρ > 0, σ > 0 (9)
subject to the household flow budget constraint and human and social capital laws of accumulation ( 4) and (3):

A (t + 1) = r (t) A (t) + w Y (t) u (t) H (t) -C (t) -I S (t) + (1 -δ) A (t) (10)
H (t + 1) = B ((1 -u (t)) H (t)) β (S (t)) 1-β + (1 -δ) H (t) (11)
S (t + 1) = I S (t) + (1 -δ) S (t) ( 12
)
where ρ is a psychological discount factor that is inversely related to the rate of time preference. 3 The representative household must end up with 0 net debt. Therefore, since the economy is closed we have A(t) = K(t). This implies that the household flow budget constraint [START_REF] Boucekkine | Imbalance effects in the Lucas model: An analytical exploration[END_REF] reduces to expression (2), the law of motion of physical capital.

The first-order necessary conditions for this problem are the following:

C (t + 1) C (t) σ = ρ (1 + r (t + 1) -δ) (13)
u (t) = u (t + 1) K (t) H (t + 1) K (t + 1) H (t) 1 + r (t + 1) -δ 1 + w H (t + 1) -δ 1 -u (t) 1 -u (t + 1) S (t + 1) H (t) S (t) H (t + 1) (1-β) 1 α (14) K (t) S (t) = α 1 -α β 1 -β u (t) 1 -u (t) (15)
where w H (t) = Bβ (S (t) / (1 -u (t)) H (t)) 1-β is the marginal productivity of human capital in the educational sector. Equations ( 13) and ( 14) describe respectively the optimal consumption and time allocation to final goods consumption. Equation [START_REF] Delhey | Predicting Cross-National Levels of Social Trust: Global Pattern or Nordic Exceptionalism?[END_REF] gives the physical to social capital ratio at equilibrium.

Equilibrium

We now characterize the equilibrium of this economy. This is done in the following proposition.

Proposition 1 (Equilibrium) Given the initial conditions K (-1), H (-1), and S (-1), an equilibrium is a path {Y (t) ; I K (t) ; K (t) ; C (t) ; u (t) ; H (t) ; I S (t) ; S (t) ; r (t) ; w H (t)} t≥0 that satisfies the following conditions:

C (t + 1) C (t) σ = ρ (1 + r (t + 1) -δ) (16)
u (t) = u (t + 1) K (t) H (t + 1) K (t + 1) H (t) 1 + r (t + 1) -δ 1 + w H (t + 1) -δ 1 -u (t) 1 -u (t + 1) S (t + 1) H (t) S (t) H (t + 1) (1-β) 1 α (17) r (t) = αA (K (t) /H (t)) α-1 u (t) 1-α (18)
w

H (t) = Bβ (S (t) / (1 -u (t)) H (t)) 1-β (19) K (t) S (t) = α 1 -α β 1 -β u (t) 1 -u (t) (20)
Y (t) = A (K (t)) α (u (t) H (t)) 1-α (21)
Y (t) = C (t) + I K (t) + I S (t) (22)
K (t + 1) = I K (t) + (1 -δ) K (t) (23)
H (t + 1) = B ((1 -u (t)) H (t)) β (S (t)) 1-β + (1 -δ) H (t) (24)
S (t + 1) = I S (t) + (1 -δ) S (t) (25)
Equations ( 23), ( 24) and ( 25) are the accumulation rules of respectively physical, human and social capital. Equations ( 16)-( 25), together with the usual transversality conditions: lim

T →∞ K (T ) ρ T = 0 (26) lim T →∞ H (T ) ρ T = 0 (27) lim T →∞ S (T ) ρ T = 0, (28)
are sufficient for an optimum.

The proof of proposition 1 is given in Appendix A. In this proof, the household intertemporal optimization problem depicted by ( 9), ( 10), ( 11) and ( 12) is solved explicitly.

Solving this problem implies the optimization of the following expression of the current value Hamiltonian:

J = (C (t)) 1-σ -1 1 -σ ρ t + λ 1 (t + 1) AK (t) α (u (t) H (t)) 1-α -C (t) -I S (t) -δK (t) + λ 2 (t + 1) B ((1 -u (t)) H (t)) β S (t) 1-β -δH (t) + λ 3 (t + 1) (I S (t) -δS (t))
We can draw two standard results from this optimization: firstly, household 's intertemporal optimization with respect to consumption and investment in social capital (equations ( 45) and ( 46)) implies that

C (t) -σ = λ 1 (t + 1) = λ 3 (t + 1) .
This entails the fact that on the margin goods must be equally valuable in their three uses: consumption, physical capital accumulation and social capital accumulation. Secondly, the household 's intertemporal optimization with respect to the control variable u (t) (equation ( 47)) yields:

p (t + 1) = λ 2 (t + 1) λ 1 (t + 1) = A (1 -α) (K (t) / (u (t) H (t))) α Bβ (S (t) / ((1 -u (t)) H (t))) 1-β ⇔ ( 29
)
λ 2 (t + 1) Bβ (S (t) / ((1 -u (t)) H (t))) 1-β = λ 1 (t + 1) A (1 -α) (K (t) / (u (t) H (t))) α (30)
where p (t + 1) denotes the shadow price of human capital in terms of final goods.

(29) ensures that the marginal productivity of human capital is equalized across sectors.

(30) expresses the condition that time is on the margin equally valuable in its two uses, production and human capital accumulation. Therefore, this is equivalent to equalization of returns of human capital across sectors.

Balanced growth paths

We now come to the study of balanced growth path (BGP) regimes. As usual, a balanced growth path is a particular solution to the equilibrium dynamics system displayed above where all variables grow at a constant rate except r (t), w H (t) and u (t) which should be constant along this path. For human capital we have

H (t + 1) = H (t) (1 + γ H ). The growth rates of the variables Y (t), I K (t), K (t), C (t), I S (t), S (t) are respectively γ Y , γ I K , γ K , γ C , γ K , γ C , γ I S ,
and γ S . We first show that in our model with social capital all growing variables along the BGP should have the same growth path.

Proposition 2 If H (t) grows at a rate γ * H > 0, then all the other variables I K (t), K (t), C (t), I S (t), S (t) grow at strictly positive rates with:

γ * Y = γ * H = γ * K = γ * I K = γ * C = γ * S = γ * I S (31)
The proof of this proposition is given in Appendix B.

Next we have to determine γ H . To this end we need to impose restrictions on the longrun levels. Computing these restrictions from the dynamic system ( 16)-( 25) we end up with 8 equations for 9 unknowns Ȳ ; K; C; ū; H; ĪK ; ĪS ; S; γ H . Therefore, the system in terms of levels is undetermined, which is a usual property of endogenous growth models. However, it is possible to stationarize this system to get rid of this indeterminacy. To do so we rewrite the dynamic system ( 13)-( 25) as a function of the following six stationary variables: Lucas-Uzawa case. This is discussed in the following section.

y (t) = Y (t) /H (t), k (t) = K (t) /H (t), i k (t) = I K (t) /H (t), c (t) = C (t) /H (t

Steady state growth rate: the role of social capital

Solving the aforementioned stationarized system, and after many tedious computations (see Appendix C), it is possible to identify a closed-form solution to the steady state growth rate. Specifically, one gets the following result.

Proposition 3 (Existence and uniqueness)

Let B > 1 Ψ 1-β 1 ρ + δ -1 2-(α+β) 1-α
. Then, an unique positive steady state growth rate exists and is characterized by the following stable and positive long-run value:

γ * H = ρ 1 -δ + B 1 1-β Ψ (1-α)(1-β) 2-(α+β) 1 σ -1 ( 32
)
with Ψ = (Aα)

1 1-α (1-α) α (1-β) β β 1 1-β
Proposition 3 suggests that provided that the education sector is productive enough, there exists a unique steady state growth rate. Because the sectors are heavily inter-related, the expression of the growth rate is much more complicated than the counterpart in the benchmark Lucas-Uzawa model. But in both cases, the growth rate is positive provided productivity in the education sector is large enough. In the model with social capital, the growth rate is a complicated function of many parameters, including the technology parameters in the final good sector: this makes a significant difference with the Lucas-Uzawa model (see algebraic details below). This is a desirable property of social capital which essential role is to facilitate the connection between the different activity sectors. As modeled here, social capital is the vehicle through which productivity improvements in the final good sector may also have a positive impact in the education sector: the resulting positive wealth effect is likely to increase investment in social capital, therefore boosting growth in the education sector.

We may perform comparative statics to check out the impact of the model's parameters on the balanced growth path. Unfortunately, the expressions involved are so complex that it turns out to be impossible to obtain the comparative statics analytically except for the psychological discount factor (standard negative effects of impatience on the long-term growth rate). The same can be claimed on the other expressions obtained in the BGP, like the equilibrium allocation of human capital to the final good sector, the ratios physical to human capital and social to human capital respectively:

u * = 1 - β (γ * + δ) (1 + γ * ) σ 1 ρ -1 + δ (33)
k * = u * Aα (1 + γ * ) σ 1 ρ -1 + δ 1 1-α (34)
s * = (γ * + δ)    (1 + γ * ) σ 1 ρ -1 + δ β Bβ β    1 1-β . ( 35
)
We shall obtain the comparative statics numerically in Section 3 once the model conveniently calibrated. At this stage, we move for comparison with the benchmark Lucas-Uzawa model.

Comparison with the Lucas-Uzawa' model

With β → 1, there is no payoff of accumulating social capital since it contributes neither to physical capital nor to human capital accumulation. Therefore, the model reduces to

Lucas-Uzawa' framework without externality as there is no social capital accumulation and the production functions from (1) and ( 4) simplify to:

Y (t) = A (K (t)) α (u (t) H (t)) 1-α = C (t) + I K (t) (36)
It can be readily verified that the dynamic optimization of Lucas-Uzawa' model yields the following expressions of the steady state growth rate, the share of human capital allocated to physical capital accumulation and the physical to human capital ratio:

γ * lu = (ρ (1 -δ + B)) 1 σ -1 (38) u * lu = 1 - γ * lu + δ B ( 39
)
k * lu = u * lu Aα B 1 1-α (40)
In contrast to our model with social capital, the steady state growth rate is unaffected by parameters of the final goods production function such as the productivity constant A (Mulligan and Sala-i-Martin, 1992) and the physical capital elasticity α. In our model productivity shocks in any of the two sectors raise the economy's ability to accumulate more social capital, therefore fostering human capital accumulation and ultimately economic growth. In the classical Lucas-Uzawa's model, only direct productivity shocks in the education sector can do the job.

Proposition 4 (Impact of A and α on γ H ) In contrast to the Lucas-Uzawa special case, the BGP growth rate γ * H is sensitive to productivity shocks in the final goods sector and to changes in the physical capital elasticity.

The same remark can be done on the share of human capital in the final good sector and physical to human capital ratios along the BGP. Comparison of equations ( 38) and (39) with equations ( 33) and (34) speaks by itself. Only the productivity parameter in the education sector, B is relevant in the long-run for the latter magnitudes in the Lucas-Uzawa case while the presence of social capital in our model provides the necessary vehicle for technology improvements in the final good sector to matter in the long-run for these magnitudes. In this sense, our modelling exemplifies the role of social capital in the development process.

Note that this result is sensitive to the assumption made about the cost of social capital accumulation. Assuming that social capital accumulation implies an opportunity cost in terms of foregone consumption or physical capital investment entails a direct link between the two sectors which materialized in the expression of the steady state growth rate. An alternative modelling strategy implying that the cost of human capital accumulation is incurred in terms of human capital would entail a thoroughly different result as shown in Appendix E. Indeed, as can be seen in (84) the expression of the growth rate in this alternative model does not depend of the parameters of the final goods production function A and α.

BGP growth rate and factor intensity of social capital in the education sector

An interesting side product of our BGP analysis is the evolution of the growth rate when the education sector becomes more and more intensive in social capital, that's when β decreases. The analysis of the function form of γ * H shows that it is continuous in β on the interval ] 0, 1 [ . Moreover, we have:

lim β→0 γ * H = ρ (1 -α)α α 1-α B A 1 1-α 1-α 2-α -δ + 1 1 σ -1 lim β→1 γ * H = (ρ (1 -δ + B)) 1 σ -1 = γ * lu A careful
β) = 0 should admit at least one solution in that interval. Therefore, the steady state growth rate is a nonmonotonic function of the elasticity of human capital in the education sector. One can easily find numerical examples when the first derivative is always increasing, i.e. d 2 γ H d 2 β > 0 for β ∈ ] 0, 1 [ . In such cases, the human capital steady growth rate displays an inverted-U shape curve. 4Three comments are in order here. First of all, it is important to notice that having the education sector more intensive in social capital is good for long-run growth: in our model, as social capital is produced from the final good and not from human capital (reflecting the hypothesis that it builds more on time diverted from production than on specific human capital), the economy has two ways to stimulate growth, either through social capital or human capital, instead of one in the Lucas-Uzawa model. The more important is social capital in the education sector, the larger the long-term growth attainable. Second, this property is not true when β is close to one: when one starts departing from the Lucas-Uzawa model, the growth rate first decreases. It only increases (when β keeps decreasing) when β is low enough. As a consequence, there must be another mechanism counter-balancing the one mentioned just above, the interaction of both being responsible of the non-monotonic picture encountered. A potential opposite mechanism is the following: as β goes down, the education sector relies less on human capital (and more on social capital), therefore the share of human capital in this sector is likely to decrease, 5 which causes the growth rate to drop. In the neighborhood of the Lucas-Uzawa model, that when β is not too distant from 1, the latter mechanism dominates and the BGP growth rate drops when β goes down. As β continues to decreases, this mechanism gets dominated by the first one (the availability of a second powerful growth engine, social capital). Such a monotonic pattern may also be observed in the model presented in Appendix E (cfr Figure 7), where the cost of social capital accumulation is expressed in terms of human capital. Last but not least, even under the restriction 1 -α < β, ensuring that the education sector is more intensive in "pure" human capital than the final good sector, the non-monotonicity property still holds. 63 Numerical exercises existing empirical studies, it is quite impossible to calibrate β in that way. Indeed, one can hardly find in the literature information about the elasticity of either social or human capital in the education sector. To circumvent that difficulty, we perform a structural estimation of ( 4), the law of accumulation of human capital.

Estimation of the elasticity of social capital

To simplify our specification, we assume full depreciation of human capital, that is δ = 1 and obtain

g H (t) = B (1 -u (t)) β S (t) H (t) 1-β (41)
where g H (t) = 1 + γ H (t) is the growth factor. Taking the logs of both sides of (41), assuming that B = Be ǫ and that the economies are in the steady state so that Proposition

(2) implies γ * Y = γ * H = γ * K = γ * I K = γ * C = γ * S = γ * I S = γ *
, we may write:

log (g * ) = log B + βlog (1 -u * ) + (1 -β) log S H * + ǫ (42)
The specification (42) implies that the sum of the coefficients of the regressors is equal to 1. To be consistent with our theoretical model we may assume that this restriction holds.

In that case, the estimation of ( 42) is equivalent to estimating:

log g * H S = log B + βlog (1 -u * ) H S + ǫ (43)
Therefore, we may estimate β through a simple regression model by regressing the logarithm of the product of the steady state growth rate and the human to social capital ratio on a constant and the logarithm of the product of the fraction of time the human capital factor devotes to educational sector and the inverse of the normalized social capital. 7The steady state growth rate is proxied by data on GDP per-capita growth adjusted for purchasing power parity (PPP, expressed in constant 2000 US Dollars). Those growth rates are computed on the basis on data on real GDP per capita from the Penn World Tables 6.2 between 1980 and 2000 (More information on the data sources is provided on Appendix F). Since yearly data on GDP per capita may incorporate short-run disturbances, real GDP per capita rates averaged over five-year periods and growth rates are computed.

More precisely, we compute growth rates for the following period: 1980-1985, 1985-1990, 1990-1995 and 1995-2000.

Data for social capital are obtained from the World Values Surveys. Other associations or activities like political parties, labor unions or professional organizations are discarded since they seem to refer predominantly to organizations generally oriented towards redistributive goals at the exclusive benefit of their members.

Following [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF], the strength of the indicator of norms of civic behavior is evaluated from responses to question about whether each of the following behaviors:"claiming government benefits to which you are not entitled", "avoiding a fare on public transport", "cheating on taxes if you have a chance", "keeping money that you have found", "failing to report damage you've done accidentally to a parked vehicle" can always be justified, never be justified or something in between.

There are several ways to construct an indicator of social capital: we may either consider separately measures of trust, of norms and of participation in networks or we may combine different measures of social capital in an unique social capital index. This unique social capital index can be built from the different measures of social through principal component analysis. Then, we retain the first principal component which account for the highest share of the total variance of a set of social capital variables. The problem with principal components is that they take negative values which are not convenient for logarithmic transformation. We circumvent this difficulty by considering a monotonic transformation of the first principal component: the Cumulative Normal Distribution Function. This allows us to obtain a social capital index with values between 0 and 1.

Our model needs an approximation for the time spent by individuals to build up human capital accumulation, (1 -u * ). For this, we use the ratio of the average years of schooling to life expectancy). For average years of schooling data we take [START_REF] Barro | International Data on Educational Attainment: Updates and Implications[END_REF]' data about the educational attainment of the total population aged 15 and over. Data for life expectancy are taken from the World Bank. We consider data for 1980, 1985, 1990[START_REF] Solow | But Verify, The NewRepublic[END_REF], and 2000. Following Földvari and Van Leeuwen (2009), we consider that (1 -u * ) is roughly equal to the share of time allocated to education and learning. Thus, "dividing this by the life expectancy yields the share of the representative agent's life that is devoted to human capital formation by means of education" [START_REF] Földvari | An alternative interpretation of 'average years of education' in growth regressions[END_REF].

We get the following results: when social capital is measured as trust, or as a combined measure of trust and civic norms, we obtain an estimate of β, of roughly 100%. Such results appear as a confirmation of the Lucas-Uzawa model where human capital is the only factor that plays a role in its own accumulation. They seem to suggest that social capital do not impact human capital formation, although this measure of social capital have been shown

to impact positively the growth rate of the economy (Knack and Keefer, 1998;Temple, 2000;Fukuyama, 2002).

When social capital is measured by the indicator of norms and civic behavior, we obtain a value of the elasticity of human capital in the education of 94% which implies an elasticity of social capital of roughly 6%. Those ways of measuring social capital allow to enough degrees of freedoms in the estimation (68 observations for trust, 62 observations for norms and for the combined measure of trust and norms). However, considering only norms and trust alone does not seem to be the most intuitive way to capture social capital. While norms and trust are pertinent dimensions of social, they are not sufficient to capture all the aspects of the polymorphous concept of social capital. Considering associational activity may allow us to broaden the perception of that concept. Yet, this implies a severe drawback:

the degree of freedoms decreases sharply.

Measuring social capital exclusively as associational activity implies a value of β of 0.89 and a number of used observations egal to 29. Combining trust, norms and associational activity in a single indicator, we obtain 0.90 as the OLS estimate of the elasticity of human capital in the education sector. In the simulation, we will consider 0.90 as our value of β. This implies an elasticity of social capital in the educational sector of 10%. There is obviously an issue of endogeneity with this OLS estimate. Indeed, one may expect that the level of output growth rate may affect the choice of the inputs of the educational sector. To tackle that problem, we use values of regressors measured at the beginning of the period on which the growth rates are evaluated [START_REF] Knack | Does Social Capital Have an Economic Payoff? A Cross-Country Investigation[END_REF].

To end this subsection, let us just indicate that with β = 0.9 and with the other values of parameters indicated in table 1, we are in the neighborhood of the Lucas-Uzawa model (cfr. Figure 2). Therefore, in this parameters region the second mechanism mentioned in subsection 2.8 is at play: as β goes up, the education sector relies more on human capital and less on social capital. Thus, the share of human capital in this sector as well as the human to social capital ratio rise, which cause the growth rate to increase. Such a pattern is consistent with Putnam (2000)'s finding according to which social capital has been declining in the US, although the economy has been growing.

Numerical comparative statics of the BGPs

As mentioned in Section 2.6, the comparative statics of the BGPs cannot be obtained analytically except for the psychological discount factor. For this parameter, we find the standard and expected result that the higher is the psychological discount factor ρ (i.e. the lower the rate of time preference or equivalently the more people value future consumption), the higher is the long-run growth rate. Table 2 includes the numerical comparative statics with respect to other parameters and for other variables than the BGP growth rate. The computations are performed on the baseline described above.

Table 2: Comparative statics: general model.

γ * H u * k * s * A + + + + B + + + - α - + + + β - - + - ρ + - - + δ - - - + σ -1 + + -
Baseline 0.0200 0.4335 1.7893 0.6062

Notes: 1 Holds if ρ 1 -δ + B 1 1-β Ψ (1-α)(1-β) 2-(α+β)
> 1. The red signs are displayed when it is impossible to uniquely sign the derivatives through an analytic inspection. Then, they are determined from the evaluation of the derivatives of the steady state values arrayed vertically with respect to the parameters arrayed horizontally.

Some comments are in order. In first place, and as suggested in Section 6, technological shocks in the final good sector do foster long-term growth in our model with social capital.

An increase in A does raise the BGP share of human capital in the final good sector and the ratios physical to human capital, and social to human capital as well, which is the intuitive outcome. An increase in B, that's a productivity boom in the education sector, has the same qualitative properties except that it raises human capital more than social capital, which is again intuitively acceptable. The obtained impact of B-shocks on the ratio physical to human capital is standard (see the Lucas-Uzawa case below). Second, a decrease in β is found to increase growth: we are in a parametric region where making the education sector more intensive in social capital triggers long-term growth. Note also that decreasing β increases the share of human capital in the final good sector. This corroborates our interpretation of the non-monotonicity feature in Section 2.8, and more precisely, our identification of the reverse mechanism playing through the human capital share decision variable. In this sense, the obtained numerical comparative statics are completely consistent with the intuitions presented so far.

The benchmark Lucas-Uzawa case In order to highlight the role of social capital in the findings above, we present here the counterpart comparative statics on the Lucas-Uzawa model.

Table 3: Comparative statics: Lucas-Uzawa model.

γ * H u * k * A 0 0 + B + + - α 0 0 + 3 ρ + - - δ - -2 -2 σ -1 + 1 + 1
Baseline 0.0255 0.3846 1.3791

Notes: 1 Holds if ρ (1 -δ + B) > 1. 2 Holds if ρ (1 -δ + B) > 1 and σ > 1. 3 Holds if 1 -α + α log αA B > 0.
As in table 2 the red signs are determined from the evaluation of the derivatives of the steady state values arrayed vertically with respect to the parameters arrayed horizontally evaluated with the following baseline parameters: δ=0.05, ρ=0.98, σ=2, α=0.3, A=1, B=0.12273.

Comparison of the two tables confirm the main and essential difference between the two models: while technology shocks in the final good sector do not play any role in the BGP in the Lucas-Uzawa model, they do matter when social capital is modelled. Other than this, the comparative statics of the two models are qualitatively similar.

Productivity shocks

By propositions 3 and 4, we know that productivity parameters impact positively the growth rate of the economy. This is one of the important differences between our model and the standard Lucas-Uzawa model where productivity parameters of the final good sector have no impact in the long run growth. In what follow, we analyze how an economy responds to shocks to the parameters, A (the productivity in the final good sector), and B (the productivity in the education sector). All the shocks considered are permanent (from t = 0) and have an intensity equal to 1%. Shocks to these parameters can arise due to changes in education policy, to education subsidies, to changes in policy regimes, to innovation, etc...

Productivity shocks in the goods and services sector

In response to a productivity shock in the final goods sector, standardized physical capital increases to take advantage of the increased efficiency of the productive sector (Figure 4).

Depending on the value of the human capital elasticity in the education sector, the share of human capital allocated in the goods and services sector may initially increase (for β = 1 or β = 0.9, Figure 8 in Appendix G) or decrease (for β = 0.5, Figure 9 in Appendix G). In the first case this implies a reallocation of human capital from the education to the productive sector, while this would entail a reassignment of human capital from the final goods to the educative sector in the second case.

As the marginal productivity of human capital in the education sector rises with social capital, human and social capitals are complements in human capital accumulation.

Therefore, the demand for social capital in the education sector decreases in the first case, while it raises in the second case (Figure 4). But as the education sector is the driver of economic growth, it also entails a smaller human capital accumulation, an initial decline of the economic growth rate in the first case (Figure 9 in Appendix G) and an initial increase in the second case (Figure 10 in Appendix G). In the first case, the resulting paucity of human capital induces agents to devote less time in the productive sector. This entails a reverse reallocation of human capital from the productive to the education sector and a subsequent rise of the demand of social capital and of the economic growth rate. A reverse mechanism occurs in the second case.

The intensity of the elasticity of human capital in the education sector is of fundamental importance in the way the economy adjust in case of productivity shocks. For values of β close to one, human and social capital are less substitutable as inputs of the education sector. Therefore, in case of a raise of A the reallocation of human capital from the education to the final goods sector entails an initial drop-off of the growth rate. With β << 1, the substitutability between the two inputs of the education sector increases. 9 Therefore, the reallocation of human capital to the final goods sector can be accommodated by an increase of the social to human capital ratio. As the education sector is more efficient, workers reallocate their working time in its favor (Figure 11 in Appendix H). Since human and physical capital are complements, this lowers the physical to human capital ratio (Figure 5).

Human capital and social capital are complements as well, therefore standardized social capital also increases in a first step (Figure 5). But, the reallocation of working time in favor of the education sector ends up by causing a shortage of physical capital investment. To avoid detrimental effects on output and economic growth rate, agents opt to subsequently spend more time in the productive sector.

In the case β << 1, once again the substitutability between social and human capital increases. Therefore, agents take advantage of the income effect generated by the increased productivity of the education sector by increasing their working time in the final goods sector (Figure 12). As human and social capital are complements, this induces an initial decrease of the social to human capital ratio (Figure 5). This has initially a detrimental effect on the human capital growth rate. However, it subsequently increases as agents increase their working time in the education sector.

Imbalance effects

The analysis of imbalance effects represent an important line of research in endogenous growth models with human capital. They are due either to the relative abundance of physical capital10 or inversely to the relative abundance of human capital. The most important result that can be derived in analyzing the imbalance effects in the Lucas-Uzawa model is that a shortage of human capital motivates an allocation of resources to production of goods rather than education. This will decrease the accumulation of human capital, lowering the economy's growth rate (Boucekkine and Ruiz-Tamarit, 1995). Thus, the model predicts that an economy should experience faster recovery after an event that destroys physical capital, than if it had destroyed human capital. It also suggests that the economies which are growing faster are those with higher ratios of human capital to physical capital.

Contrary to the one sector model with the same technology to produce physical and human capital, the two-sector model does not give rise to symmetric U-shaped imbalance effects. The rational behind this finding is quite simple: since the education sector is more intensive in human capital, its operation cost is larger in case of a shortfall of human capital because of the induced higher wage. This motivates people to allocate human capital to the final good sector, rather to the education sector (Boucekkine and Ruiz-Tamarit, 2004 and 2008, Boucekkine et al., 2008).

Imbalances effects are depicted in Figure 6. Figure 6 shows that the relationship between the human capital growth rate and the physical to human capital ratio is always monotonic.

However, depending on the human capital elasticity β, it can have a negative (when β = 0.5) or a positive slope (when β = 0.9 or β = 1). Indeed, as stated before when the elasticity of human capital in the education sector is high, social capital is less substitutable to human capital. Therefore, individuals allocate more working time in the education sector which allows a higher accumulation of human capital and therefore a higher economic growth rate.

Consequently, since there is less human capital available in the final goods sector, firms use proportionally more physical capital which explains the positive relationship between the economic growth rate and the physical to human capital ratio.

A contrario, when β = 0.5, the substitutability between social and human capital increases, therefore individuals may allocate more working time in the final goods sector without impeding economic growth. Therefore, lower physical to human capital ratio coexist with higher economic growth rates as suggests by the negative slope of the curve in

Conclusion

In this paper, we build an endogenous two-sector model where the interaction between human and social capital drive the accumulation of human capital. First of all, we choose a multi-sector modelling such that social capital plays the advocated role of timing belt propagating shocks through the macroeconomy. As a result, and in contrast to the seminal Lucas-Uzawa model, the steady state growth rate depends on productivity parameters of ALL the sectors, not only those arising from the education sector. Assuming that any investment in social capital implies an opportunity cost in terms of foregone physical capital accumulation and consumption creates a direct link between the two sectors which materializes in the expression of the steady state growth rate.

Three types of findings are obtained in this paper. First of all, it is shown that the presence of social capital has an ambiguous effect on long-term growth. Indeed, we obtain a U-shape pattern for the steady state growth rate with respect to social capital elasticity in the education sector. When the education sector is intensive in social capital, the latter may act as a substitute of human capital. This allows a higher allocation of human capital in the final goods sector and therefore enable the economy to achieve higher output, higher consumption level, and higher investment in social capital which recursively may sustain a higher economic growth rate. However, in the neighborhood of the Lucas-Uzawa model, when the social capital elasticity is much lower, the education sector relies less on social capital. Therefore, in such a case the only way to foster economic growth is through increases of the human to social capital ratio. Such a parameter region is consistent with a strong empirical result uncovered by [START_REF] Putnam | Bowling Alone: The Collapse and Revival of American Community[END_REF] on the US economy: the concommittance of a growing economy with a declining social to human capital ratio. Second, we try to provide with some estimate for the "weight" of social capital in the process of human capital formation. Our main finding in this respect is that the elasticity of human capital to social capital varies from 6% to 10% depending on the measure of social capital selected.

Last but not least, through some numerical examples, we show that the magnitude of social capital elasticity may have a strong impact on transitory dynamics. A higher social capital elasticity may induce a decreasing pattern of the steady state growth with respect to the physical to human capital ratio, while lower values may entail an increasing pattern.

It goes without saying that our analysis has the advantages and limits of multi-sector endogenous growth models, which build on stylized laws of motion for aggregate variables.

The process of social capital formation is probably much trickier at the micro level. However we believe that our model highlights in a transparent and accurate way the role of social capital in the growth process, and this role is indeed ambiguous. We have uncovered the sources of this ambiguity and provided a numerical assessment of the impact of social capital on long-term growth and short-term dynamics using available data on social capital measures. More work is needed on the microfoundations of social capital and on its measurement.

Appendix A: Proof of Proposition 1

The households' intertemporal optimization problem yields the following discrete time current value Hamiltonian: J = (C (t)) (52) and (53) implies that γ * K = γ * H . As γ * K and γ * H are constant we may conclude from (56) and (58) that K and I K on one hand and S and I S on another hand grow at the same rate. Therefore, we have the following relationship:

γ * Y = γ * H = γ * K = γ * I K = γ * C = γ * S =
As before after solving the corresponding stationarized system, we are able to identify a closed-form solution to the steady state growth rate:

γ * H = ρ 1 -δ + Bβ β ((1 -β) C) 1-β 1 2-β 1 σ -1 (84)
Conversely to expression [START_REF] Teachman | Social Capital and the Generation of Human Capital[END_REF], this function does not depend on parameters A and α.

However, as shown in Figure 7 for baseline parameters δ=0.05, ρ=0.98, σ=2, α=0.3, A=1, B=0.12273 and C=1.07105, it displays an inverted-U shape as in Figure 2.

Figure 1 :

 1 Figure 1: Relationship between social capital and economic development

Figure 2 :Figure 3 :

 23 Figure 2: Human capital's growth rate as a function of β.

9 Figure 4 :

 94 Figure 4: Effects of productivity shocks on k and s.

Figure 5 :

 5 Figure 5: Effects of productivity shocks on k and s.

Figure 6 corresponding to β = 0. 5 .

 5 Figure 6 corresponding to β = 0.5.

Figure 6 :

 6 Figure 6: Comparison of different Physical to Human capital ratio imbalance effects for different values of the elasticity of the human capital in the education sector (β = 1 (Lucas-Uzawa model), β = 0.9, β = 0.5).

( 1 -

 1 β) S (t) -β -δ (50) Our boundaries conditions are standard. They imply the initial values K (-1), H (-1), S (-1) and the following transversality conditions:

Figure 9 :Figure 10 :Figure 12 :Figure 13 :

 9101213 Figure 9: Effects of A productivity shocks on u for β = 0.5 and on γ H for β = 1.

  ), s (t) = S (t) /H (t), i s (t) = I S (t) /H (t), u (t) and γ H . The stationarized dynamic system is given in Appendix C. With such a stationnarized system, we are able to discuss the existence and uniqueness of the steady state growth rate, and its main determinants as well. As we shall see hereafter, the introduction of social capital crucially changes the comparative statics of the steady state growth rate relative to the Lucas-Uzawa benchmark. In particular, (permanent) technological shocks in the final good sector affect the growth rate in the presence of social capital (as modeled in our paper) while it definitely does not in the

  examination of the steady state growth rate first and second derivatives yields

	the following results:
	lim β→0	dγ H dβ	= -∞,	lim β→1	dγ H dβ	= +∞,	lim β→0	d 2 γ H d 2 β	= +∞,	and	lim β→1	d 2 γ H d 2 β	= +∞
	Since dγ H

dβ (β) is also continuous in β on ] 0, 1 [ , then the equation dγ H dβ

Table 1 )

 1 Let us consider the following calibration of the model. A first set of parameters is fixed a priori to what we view as reasonable values given the available empirical evidence (see . The rate of depreciation of all the forms of capital is set to 5%. The psychological discount factor is 0.98. The absolute value of the elasticity of marginal utility is 2. A, the total factor productivity of the goods and services sector, is normalized to 1. B, the productivity parameter of the education sector is set to 0.12273, in order to obtain a growth rate of 2%. While the values of most of the parameters are calibrated on the basis of the

Table 1

 1

	: Parameters
	Parameter	Symbol	Value
	Rate of depreciation of capital	δ	0.05
	Psychological discount factor	ρ	0.98
	Absolute value of the elasticity of marginal utility	σ	2
	Physical capital share in the final sector	α	0.3
	Total productivity in the final sector	A	1
	Total productivity in the education sector	B	0.12273
	Human capital share in the education sector	β	0.9

8

  According to previous researches on social capital at macro level, social capital can be measured through different indicators: the levels of generalized trust, associational activity and norms of civic behavior. Trust is coded from WVS data as the percentage of respondents answering that most people can be trusted when asked "Generally speaking, would you say that most people can be trusted, or that you can't be too careful in dealing with people?"(Inglehart

	et al., 2000; Knack and Keefer, 1997; Paxton, 1999; Paxton, 2002; Uslaner, 1999; Alesina
	and La Ferrara, 2000; Putnam, 2000; Whiteley, 2000; Zak and Knack, 2001; Delhey and
	Newton, 2005).
	Associational activity is the percentage of people involved in the following organizations
	or activities: social welfare services for elderly; handicapped or deprived people; education,
	arts, music or cultural activities; local community action on issues like poverty, employment,
	housing, racial equality; third world development or human rights; youth work; religious
	or church organizations; sports or recreation; Peace movement; Voluntary organizations
	concerned with health.

  1-σ -1 1 -σ ρ t + λ 1 (t + 1) AK (t) α (u (t) H (t)) 1-α -C (t) -I S (t) -δK (t) + λ 2 (t + 1) B ((1 -u (t)) H (t)) β S (t) 1-β -δH (t) + λ 3 (t + 1) (I S (t) -δS (t))(44)where λ 1 (t), λ 2 (t), λ 3 (t) respectively denote are the K (t), H (t) and S (t) co-state variables. We consider that the households fully internalize the benefit of social capital. the necessary first order conditions are:

	∂J ∂C (t)	= 0 ⇒ C (t) -σ = λ 1 (t + 1)	(45)
	∂J ∂I S (t)	= 0 ⇒ λ 1 (t + 1) = λ 3 (t + 1)	(46)
	∂J ∂u (t)	= 0 ⇒	λ 1 (t + 1) λ 2 (t + 1)	=	AK (t) (48)
	∂J ∂H (t)	= -(λ 2 (t + 1) -λ 2 (t)) ⇒
		λ 2 (t) λ 2 (t + 1)	= 1 +	λ 1 (t + 1) λ 2 (t + 1)	AK (t) (49)
	∂J ∂S (t)	= -(λ 3 (t + 1) -λ 3 (t)) ⇒
		λ 3 (t) λ 3 (t + 1)	= 1 +	λ 2 (t + 1) λ 3 (t + 1)

α (1 -α) u (t) -α H (t) 1-α Bβ (1 -u (t)) β-1 H (t) β S (t) 1-β (47) ∂J ∂K (t) = -(λ 1 (t + 1) -λ 1 (t)) ⇒ λ 1 (t) λ 1 (t + 1) = 1 + AαK (t) α-1 (u (t) H (t)) 1-α -δ α u (t) 1-α (1 -α) H (t) -α +Bβ (1 -u (t)) β H (t) β-1 S (t) 1-β -δ B ((1 -u (t)) H (t)) β

  γ * I S = γ *

	C (t + 1) C (t)	σ	= ρ (1 + r (t + 1) -δ)			(74)
	u (t) u (t + 1)	=	K (t) H (t + 1) K (t + 1) H (t)	1 + r (t + 1) -δ 1 + w H (t + 1) -δ
				(1 -u (t) -l (t)) (1 -u (t + 1) -l (t + 1))	S (t + 1) H (t) S (t) H (t + 1)	(1-β)	1 α	(75)
	r (t) = αA (K (t) /u (t) H (t)) α-1			(76)
	w H (t) = Bβ (S (t) / (1 -u (t) -l (t)) H (t)) 1-β		(77)
	1 -u (t) -l (t) 1 -u (t -1) -l (t -1)	=	S (t) H (t -1) S (t -1) H (t)	1 + C (1-β) (β) 1 + w H (t) -δ (1-u(t)-l(t))H(t) S(t)	-δ	1 1-β	(78)
	Y (t) = A (K (t)) α (u (t) H (t)) 1-α			(79)

Y (t) = C (t) + I K (t) + I S (t)

(80)

K (t + 1) = I K (t) + (1 -δ) K (t)

(81)

H (t + 1) = B ((1 -u (t) -l (t)) H (t)) β (S (t)) 1-β + (1 -δ) H (t) (82) S (t + 1) = Cl (t) + (1 -δ) S (t)

For instance,[START_REF] Solow | But Verify, The NewRepublic[END_REF] suggests that social capital may hardly be considered as capital since the measurement of its stock "seems very far away".

The number of siblings measures the dilution of adult attention to a child.

The precise expression of the psychological discount factor ρ in terms of the rate of time preference ξ is the following ρ = 1 1+ξ .

Figures 2 and 3 illustrate such an example with the following baseline parameters: δ=0.05, ρ=0.98, σ=2, α=0.3, A=1, B=0.12273.

This specific effect is corroborated in the next numerical section.

In Figure2, α = 0.3, non-monotonicity arises in the β-interval, ]0.7 1].

It is obvious that it would have been better to use more sophisticated methods, such that those proposed by Panel data econometrics, to estimate the human capital elasticity. But, the significant amount of missing data in the social capital indicators prevents it.

A property of the Cobb-Douglas production function of the education sector is that the inputs involved are complements since no human capital accumulation is possible when the stock of either of them is zero. But, at the same time they are substitutable since human capital accumulation may be kept constant while the decrease in the stock of one of the imput is compensated by the increase of the stock of the other. The substitutability between human and social capital increases when β << 1.

or equivalently to the shortage of human capital.

After some manipulation on equation (45), we obtain the following Euler equation on the consumption control variable:

Equating the growth rates of λ 1 (t) and λ 3 (t) (cfr. equations ( 46), ( 48) and ( 50)), we obtain after further manipulations the following relationship between K (t) and S (t):

From (47) giving the ratio of costates variables λ 1 (t) and λ 2 (t), we can compute the ratio of their growth rates; this yields the second Euler equation on u (t):

. The other conditions of the equilibrium are merely accumulation rules of physical, human and social capital or final goods equilibrium conditions.

Appendix B: Proof of Proposition 2

If a balanced growth path should satisfy the eight Equations ( 16)- [START_REF] Mulligan | Transitional dynamics in two-sector models of endogenous growth[END_REF], then one should have the following eight restrictions among the various growth rates:

Appendix C: Proof of Proposition 3

At the steady state the growth rate of the costate variables λ 1 (t) and λ 2 (t) are equal. From this equality, we derive that the marginal productivity of K(t) in the final goods sector, r * and the margin productivity of H(t) in the education sector are equal in the steady state

Then, from ( 51) and (57) we may derive the steady state value of time allocation of human capital to the final goods sector:

Replacing this expression in ( 51) and (57), we obtain the following steady state values of stationnarized physical and social capital:

Replacing the expression of the ratio of (60) and (61) in the leftside of (51) yields, after some straightforward computations, [START_REF] Teachman | Social Capital and the Generation of Human Capital[END_REF] the expression of the steady state growth rate.

Appendix D: The stationarized dynamic system

The dynamic system ( 16)-( 25) can be rewritten in terms of stationary variables as:

Appendix E: Alternative model

In our model, we assume that, while increasing social interaction, investment in social capital implies an opportunity cost in terms of foregone physical capital accumulation and consumption. There is an alternative way to model the cost of social capital accumulation;

we may assume that individuals devote a fraction l (t) of their time to build their social networks. Such an assumption implies the following laws of motion for human and social capital accumulation:

With the intertemporal utility function [START_REF] Boucekkine | Special functions for the study of economic dynamics: the case of the Lucas-Uzawa model[END_REF] and the physical capital accumulation law [START_REF] Boucekkine | Imbalance effects in the Lucas model: An analytical exploration[END_REF], they imply the following equilibrium conditions:

Appendix F: Data and descriptives

The dataset used in the empirical subsection consists of 74 countries: Appendix H: Effects of productivity shocks in the education sector on u (t) and γ H (t).