domain: phys info math

text:
Amir R Arab
email: amir.arab@phystech.edu

On diagonal quantum channels

Keywords: Cholesky decomposition, diagonal quantum channels, Kraus operators

In this paper we study diagonal quantum channels and their structure by proving some results and giving most applicable instances of them. Firstly, it is shown that action of every diagonal quantum channel on pure state from computational basis is a convex combination of pure states determined by some transition probabilities. Finally, by using the Cholesky decomposition it is presented an algorithmic method to find an explicit form for Kraus operators of diagonal quantum channels.

Introduction

Quantum information theory took shape as a self-consistent and then multidisciplinary area of research from nearly 30 years ago, while its origin can be traced back to the 1950-1960s, which was when the basic ideas of Shannon's information theory were developed. In quantum information theory, the notions of channel and its capacity, giving a measure of ultimate information-processing performance of the channel, play a central role. For a comprehensive introduction to quantum channels, see [START_REF] Holevo | Quantum Systems, Channels, Information A Mathematical Introduction[END_REF]. A quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. Quantum channels are the most general input-output relations which the framework of quantum mechanics allows for arbitrary inputs. Physically, they describe any transmission in space, e.g., through optical fibres, and/or evolution in time, as in quantum memories, from a general open-systems point of view. Mathematically, they are characterized by linear, completely positive maps acting, in the Schrödinger picture, on density operators in a trace-preserving manner.

Diagonal quantum channels have significant applications in communication and physics. There are some studies on different types of diagonal channels, for instance depolarizing channels [START_REF] King | The capacity of the quantum depolarizing channel[END_REF][START_REF] Amosov | Remark on the Additivity Conjecture for a Quantum Depolarizing Channel[END_REF][START_REF] Ch | Unital Quantum Channels -Convex Structure and Revivals of Birkhoff's Theorem[END_REF][START_REF] Amosov | On classical capacity of Weyl channels[END_REF], transpose depolarizing channels [START_REF] Datta | Additivity for transpose depolarizing channels[END_REF] and diagonal channels with constant Frobenius norm (depolarizing, transpose depolarizing, hybrid depolarizing classical, and hybrid transpose depolarizing classical) [START_REF] Sergeev | Generalizations of 2-Dimensional Diagonal Quantum Channels with Constant Frobenius Norm[END_REF] which are introduced in section 3 of this paper. There are also some works which classify diagonal channels in special cases, for instance see [START_REF] Ruskai | An analysis of completely-positive tracepreserving maps on 2 × 2 matrices[END_REF] .

Although we say "Kraus decomposition" it is not his result. It is only a corollary of the famous Stinespring theorem (1955) known e.g. to Choi [START_REF] Choi | Completely positive linear maps on complex matrices[END_REF]. Generally, there is no systematic way to find an explicit form for Kraus operators, and they can usually be found by trial and error method. There are some studies on finding the Kraus decomposition, for instance see [START_REF] Andersson | Finding the Kraus decomposition from a master equation and vice versa[END_REF][START_REF] Ivan | Operator-sum Representation for Bosonic Gaussian Channels[END_REF].

This paper is organized as follows. In section 2, we provide basic definitions and theorems on the Cholesky decomposition and quantum channels. In section 3, we introduce diagonal quantum channels and for instance we give four families of them as the most applicable diagonal channels. In section 4, we prove an interesting property of diagonal channels which says that action of every diagonal channel on pure state from computational basis is a convex combination of pure states. And finally in section 5, by using the Cholesky decomposition we present Kraus representation of diagonal channels.

Preliminaries

Cholesky decomposition

Definition 1. Let C n×n be the vector space of n × n matrices over

C. A Hermitian matrix A ∈ C n×n is called positive semi-definite (definite) , if for every nonzero x ∈ C n , x * Ax ≥ 0 (x * Ax > 0), where x * is conjugate transpose of x.
The Cholesky decomposition says: Every positive semi-definite matrix A can be decomposed as A = R * R, where R is an upper triangular matrix. When A is positive definite, the decomposition is unique [11, p.2]. The Cholesky decomposition goes back to the 1910 as it can be read from his hand-written manuscript deposited by his family in the Archives of the École poly-technique [12, p.91]. This decomposition has various applications such as in the Monte Carlo simulation, non-linear optimization, and linear least squares problems.

Quantum channels

Definition 2. Linear map Φ : C n×n → C l×l is called positive, if for every positive semi-definite matrix A ∈ C n×n , Φ(A) ∈ C l×l is so. Definition 3. For a positive integer m, linear map Φ :

C n×n → C l×l is called m- positive, if Φ ⊗ Id m : C n×n C m×m → C l×l C m×m is
C Φ = n i,j=1 E ij Φ(E ij ),
where {E ij } n i,j=1 is the standard basis for C n×n . C Φ is called Choi matrix of Φ.

To see a proof for these theorems refer [START_REF] Choi | Completely positive linear maps on complex matrices[END_REF].

Definition 5. Completely positive map Φ :

C n×n → C l×l is called quantum channel, if it is trace-preserving, i.e.

tr(Φ(A)) = tr(A).

Definition 6. A positive definite matrix A of trace one is called density matrix.

In fact requirement that density matrices are mapped to density matrices leads to the notion of a trace-preserving completely positive map. It is noncommutative analogue of a probability distribution, i.e. a vector whose coordinates are nonnegative and up to one. Trace-preserving property is equivalent to i K i K * i = I, where I is n × n identity matrix.

Diagonal channels

Let us construct an orthonormal basis of Hermitian matrices for C n×n with respect to the Hilbert-Schmidt inner product on C n×n , i.e. A|B = tr(A * B). For n = 2, the set of Pauli matrices {σ 0 , σ 1 , σ 2 , σ 3 } are Hermitian and form an orthogonal basis for C 2×2 , where

σ 0 = I, σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 .
Definition 7. For arbitrary n ≥ 3, the generalized Pauli matrices are defined in the following way:

σ 0 = I; σ 11 =     0 1 ... 0 1 0 ... 0 ... ... ... ... 0 0 ... 0     , σ 12 =      0 0 1 ... 0 0 0 0 ... 0 1 0 0 ... 0 ... ... ... ... ... 0 0 0 ... 0      , ..., σ 1N =     0 0 ... 0 ... ... ... ... 0 ... 0 1 0 ... 1 0     ; σ 21 =     0 -i ... 0 i 0 ... 0 ... ... ... ... 0 0 ... 0     , σ 22 =      0 0 -i ... 0 0 0 0 ... 0 i 0 0 ... 0 ... ... ... ... ... 0 0 0 ... 0      , ..., σ 2N =     0 0 ... 0 ... ... ... ... 0 ... 0 -i 0 ... i 0     ; σ 31 =     1 0 ... 0 0 -1 ... 0 ... ... ... ... 0 0 ... 0     , σ 32 =      1 0 0 ... 0 0 0 0 ... 0 0 0 -1 ... 0 ... ... ... ... ... 0 0 0 ... 0      , ..., σ 3N =     0 0 ... 0 ... ... ... ... 0 ... 1 0 0 ... 0 -1     ;
where N = n(n-1)

2

.

The generalized Pauli matrices are Hermitian and orthogonal, but they couldn't form a basis for C n×n . We remove matrices σ 31 , σ 32 , ..., σ 3N and add the following matrices:

A 1 =     1 0 ... 0 0 -1 ... 0 ... ... ... ... 0 0 ... 0     , A 2 =      1 0 0 ... 0 0 1 0 ... 0 0 0 -2 ... 0 ... ... ... ... ... 0 0 0 ... 0      , ..., A n-1 =     1 ... 0 0 ... ... ... ... 0 ... 1 0 0 ... 0 -(n -1)     .
We consider

β = σ 0 √ n , σ 11 √ 2 , ..., σ 1N √ 2 , σ 21 √ 2 , ..., σ 2N √ 2 , A 1 √ 2 , A 2 √ 6 , ..., A n-1 (n -1)n ,
β is an orthonormal basis of Hermitian matrices [START_REF] Sergeev | Generalizations of 2-Dimensional Diagonal Quantum Channels with Constant Frobenius Norm[END_REF]. ), where

Definition 8. Quantum channel Φ : C n×n → C n×n is called diagonal, if
-1 n-1 ≤ p ≤ 1 n+1 , 3. Φ = diag(1, -p, ..., -p N , -p, ..., -p N , p, ..., p n-1
), where

-1 2n-1 ≤ p ≤ 1 (n-1) 2 , 4. Φ = diag(1, -p, ..., -p N , p, ..., p N , p, ..., p n-1
), where

-1 n-1 ≤ p ≤ 1 n+1 .
Quantum channels 1, 2, 3, and 4 are called depolarizing, transpose depolarizing, hybrid depolarizing classical, and hybrid transpose depolarizing classical respectively. They are among most widely used channels in science and technology.

Transition probabilities

In this section we prove an interesting property of diagonal channels which is formulated in the following theorem.

Theorem 3. For every diagonal quantum channel Φ, there is a collection of transition probabilities {P kj } n j=1 , i.e. P kj ≥ 0, n j=1 P kj = 1 such that

Φ(|k k|) = n j=1 P kj |j j| (k = 1, 2, ..., n).
Proof. For convenience we write basis β in the following form: Let Φ be a diagonal channel and its matrix representation with respect to the basis β is

β = {e 0 ,
Φ = diag(1, r 1 , r 2 , ..., r N , s 1 , s 2 , ..., s N , t 1 , t 2 , ..., t n-1 ).
Suppose that {|j } is the standard basis for C n . We can write

E kk = |k k| = c 0 e 0 + 2 p=1 N q=1 c pq e pq + n-1 q=1 c 3q e 3q ,
and calculate the coefficients in the following way:

c 0 = e 0 |E kk = 1 √ n , c pq = e pq |E kk = 0, 1 ≤ p ≤ 2, 1 ≤ q ≤ N, c 3q = e 3q |E kk = 0, 1 ≤ q ≤ k -2, c 3(k-1) = e 3(k-1) |E kk = - k -1 k , c 3q = e 3q |E kk = 1 q(q + 1) , k ≤ q ≤ n -1.
Therefore for k = 1, 2, ..., n -1

E kk = 1 √ n e 0 - k -1 k e 3(k-1) + n-1 i=k 1 i(i + 1) e 3i , E nn = 1 √ n e 0 - n -1 n e 3(n-1) ,
and applying Φ we obtain

Φ(E kk ) = 1 √ n e 0 -t k-1 k -1 k e 3(k-1) + n-1 i=k t i i(i + 1) e 3i , (1)
Φ(E nn ) = 1 √ n e 0 -t n-1 n -1 n e 3(n-1) . (2)
It follows from ( 1)

[Φ(E kk )] jj =                          1 n - t k-1 k + n-1 i=k t i i(i + 1) 1 ≤ j ≤ k -1, (3)
1 n + (k -1)t k-1 k + n-1 i=k t i i(i + 1) j = k, (4)
1 n - t j-1 j + n-1 i=j t i i(i + 1) k + 1 ≤ j ≤ n. (5)
Taking into account (2) we get

[Φ(E nn )] jj =      1 n - t n-1 n 1 ≤ j ≤ n -1, (6)
1 n + (n -1)t n-1 n j = n. (7)
Now we consider the Choi matrix of Φ. We note that Φ(E kk ), (k = 1, 2, ..., n) are n × n diagonal matrices and their diagonals are given by ( 3)- [START_REF] Ruskai | An analysis of completely-positive tracepreserving maps on 2 × 2 matrices[END_REF]. By theorem 2, C Φ is positive semi-definite, then all of its diagonal entries are nonnegative. For 1 ≤ k, j ≤ n, we define P kj in the following manner:

P kj = [Φ(E kk )] jj .
Hence, P kj ≥ 0 and for every k ∈ {1, 2, ..., n}, the equality n j=1 P kj = 1 holds true. Therefore {P kj } are the desired transition probabilities.

Remark 1. Equality n j=1 P kj = 1 can be derived from the fact that Φ is tracepreserving.

Kraus representation for diagonal channel

Before we formulate the result of this section, we need to prove the following two lemmas. Lemma 1. Let κ = (x 1 , x 2 , ..., x n ) where x i 's are rows of n × n matrix K, then (K * E ij K) 1≤i,j≤n = (x * i x j ) 1≤i,j≤n = κ * κ. Proof. For m = 1, 2, ..., n, let x m = x m1 x m2 . . . x mn , then

K * E ij K = x * 1 x * 2 . . . x * n E ij         x 1 x 2 . . . x n         = ( n s=1 x is E si )E ij ( n t=1 x jt E jt ) = n s,t=1 x is x jt E st = x * i x j .
On the other hand, κ * κ = (x * i x j ) 1≤i,j≤n .

Lemma 2. Let Φ : C n×n → C n×n be a quantum channel, C Φ be its Choi matrix, and C Φ = R * R for some matrix R. If κ i 's are rows of R, and K i ' s are associated matrices to κ i 's in lemma 1 (1

≤ i ≤ n 2 ) then {K i } n 2
i=1 is a set of Kraus operators of Φ.

Proof. Choi matrix of Φ is in the following form:

C Φ = n i,j=1 E ij Φ(E ij ) = (Φ(E ij )) 1≤i,j≤n .
On the other hand,

C Φ = R * R = κ * 1 κ * 2 . . . κ * n 2         κ 1 κ 2 . . . κ n 2         = n 2 i=1 κ * i κ i . Then applying lemma 1, (Φ(E ij )) 1≤i,j≤n = n 2 l=1 (K * l E ij K l ) 1≤i,j≤n . Therefore for any A, Φ(A) = n 2 i=1 K * i AK i .
Now we are in a position to assert the main result of this section. We are going to find an explicit form for Kraus operators of channels 1-4 by using the Cholesky decomposition.

Here we solve this problem for channel 3 and for other channels it is solved similarly. ),

Kraus operators can be determined in the following explicit form:

K 1 =      √ a 0 0 ... 0 0 b 0 √ a 0 ... 0 ... ... ... ... 0 0 ... b 0 √ a 0      , K 2 =      0 1-p n ... 0 0 0 ... 0 ... ... ... ... 0 0 ... 0      , ..., K n =      0 0 ... 1-p n 0 0 ... 0 ... ... ... ... 0 0 ... 0      , K n+1 =      0 0 ... 0 1-p n 0 ... 0 ... ... ... ... 0 0 ... 0      , K n+2 =     0 0 ... 0 0 √ a 1 ... 0 ... ... ... ... 0 0 ... b 1 √ a 1     , ..., K 2n =      0 0 ... 0 0 0 ... 1-p n ... ... ... ... 0 0 ... 0      , ..., K n 2 -1 =      0 ... 0 0 0 ... 0 0 ... ... ... ... 0 ... 1-p n 0      , K n 2 =     0 0 ... 0 0 0 ... 0 ... ... ... ... 0 0 ... √ a n-1     ,
where

a m = (2p+ 1-p n )(1+ -p -pm+2p+ 1-p n ) for m = 1, 2, ..., n-1; b m = (2p+ 1-p n )( -p -pm+2p+ 1-p n )
for m = 1, 2, ..., n -2; a 0 = p + 1-p n , and b 0 = -p.

Remark 2. The statement and proof of theorem 4 is based on the condition: (p + 1-p n )( 1-p n ) = 0 (in fact the entries of main diagonal of Choi matrix be positive). In case at least one of them is 0, matrix R in Cholesky decomposition of C Φ is not necessarily unique and then Kraus operators which are obtained in this way are so.

Proof. It follows from (1) that for

t 1 = t 2 = ... = t n-1 = p, Φ(E 11 ) = 1 √ n e 0 + n-1 i=1 p i(i + 1) e 3i =     α 0 ... 0 0 β ... 0 ... ... ... ... 0 0 ... β     , where α = p + 1-p n , β = 1-p n .
Similarly we can obtain Φ(E kk ) for k = 2, ..., n. We note that

E 12 = 1 √ 2 e 11 + i √ 2 e 21 , hence Φ(E 12 ) = -p √ 2 e 11 + -pi √ 2 e 21 =     0 γ ... 0 0 0 ... 0 ... ... ... ... 0 0 ... 0     ,
where γ = -p. Similarly we can obtain Φ(E kj ) for 1 ≤ k = j ≤ n. Therefore Choi matrix of Φ has the following form:

C Φ =                        α
                       , Let C Φ = R t R be Cholesky decomposition of C Φ .
We divide matrices into 4 blocks as the following: , and its associated K 1 in lemma 2 as the following:

C Φ = α A 10 A t 10 A 20 = r 0 0 R t 10 R t 20 r 0 R 10 0 R 20 , where r 0 = √ α, R 10 = 1 r 0 A 10 , A 20 -R t 10 R 10 = R t 20 R 20 ,
K 1 =      √ α 0 ... 0 0 γ √ α ... 0 ... ... ... ... 0 0 ... γ √ α      . Now we consider A 20 -R t 10 R 10 : A 20 -R t 10 R 10 =                        β
. 0 α -γ 2 α                       
, we note that by applying algorithm α , γ are replaced with α -γ 2 α , γ -γ 2 α respectively.

A 20 -R t 10 R 10 = β A 11 A t 11 A 21 = r 1 0 R t 11 R t 21 r 1 R 11 0 R 21 , where R 20 = r 1 R 11 0 R 21 , r 1 = √ β, R 11 = 0 1×(n 2 -2)
, and R 21 is an upper triangular matrix. We let κ 2 = 0 √ β 0 ... 0 and its associated K 2 in lemma 2 as the following:

K 2 =     0 √ β ... 0 0 0 ... 0 ... ... ... ... 0 0 ... 0     .
By repeating algorithm, similarly we obtain Now we reach the following matrix:

K 3 =     0 0 √ β ... 0
                          α -γ 2
a m = (2p + 1 -p n )(1 + -p -pm + 2p + 1-p n ), 1 ≤ m ≤ n -1; b m = (2p + 1 -p n )( -p -pm + 2p + 1-p n ), 1 ≤ m ≤ n -2.

  e 11 , ..., e 1N , e 21 , ..., e 2N , e 31 , ..., e 3(n-1) }.

Theorem 4 .

 4 For hybrid depolarizing classical quantum channel Φ = diag(1, -p, ..., -p N , -p, ..., -p N , p, ..., p n-1

  0 ... 0 0 γ ... 0 ... 0 0 ... γ 0 β ... 0 0 0 ... 0 ... 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... β 0 0 ... 0 ... 0 0 ... 0 0 0 ... 0 β 0 ... 0 ... 0 0 ... 0 γ 0 ... 0 0 α ... 0 ... 0 0 ... γ ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... 0 0 0 ... β ... 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... 0 0 0 ... 0 ... β 0 ... 0 0 0 ... 0 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 ... β 0 γ 0 ... 0 0 γ ... 0 ... 0 ... 0 α

  and R 20 is an upper triangular matrix. Then R 10 = 1 √ α 0 ... 0 γ 0 ... 0 γ 0 ... 0 γ ,

  0 ... 0 0 0 ... 0 ... 0 0 ... 0 0 β ... 0 0 0 ... 0 ... 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... β 0 0 ... 0 ... 0 0 ... 0 0 0 ... 0 β 0 ... 0 ... 0 0 ... 0 0 0 ... 0 0 α -γ 2 α ... 0 ... 0 0 ... γ -γ 2 α ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... 0 0 0 ... β ... 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... 0 0 0 ... 0 ... β 0 ... 0 0 0 ... 0 0 0 ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 ... β 0 0 0 ... 0 0 γ -γ 2 α ... 0 ... 0 ..

α 0 ... 0 0 0 γ -γ 2 α. 2 m- 1 a m- 1 , 2 m- 1 a

 0221121 ... 0 ... 0 0 ... γ -γ 2 . 0 0 0 α -γ 2 α ... 0 ... 0 0 ... γ -γ 2 α ... ... ... ... ... ... ... ... ... ... ... ... ... 0 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 ... β 0 γ -By repeating this algorithm, we see that β's are not changed, but α's, γ's will be changed in the following way: a m = a m-1 -b for m = 1, 2, ..., n -1 and b m = b m-1 -b m-1 , for m = 1, 2, ..., n -2, where a 0 = α, b 0 = γ. Therefore a m -b m = a m-1 -b m-1 = ... = a 0 -b 0 = 2p + 1-p n and then b m = b m-1 -

  a positive map, where Id m : C m×m → C m×m is the identity map and is the tensor (Kronecker) product.

	Theorem 1. Φ : C n×n → C l×l is completely positive map if and only if there exist
	n × l matrices K i such that
	Φ(A) =	K * i AK i ,
	i
	K i 's are called Kraus operators of Φ. Kraus operators are not unique.
	Theorem 2. Φ : C n×n → C l×l is completely positive map if and only if the following
	matrix is positive semi-definite:
	Definition 4. Linear map Φ : C n×n → C l×l is called completely positive, if for every
	positive integer m, Φ is m-positive.
	For example, Φ(A) = A t ( t is transposition) is positive but not completely positive.
	There are two fundamental theorems which classify completely positive maps:

Acknowledgement . Advice given by Professor Grigori G.Amosov has been a great help in writing this paper. I would like to offer my special thanks to him.


Stefano Bosi Equippe
email: stefano.bosi@orange.fr

Kazuo Nishimura
email: nishimura@kier.kyoto-u.ac.jp

Alain Venditti
email: alain.venditti@univmed.fr

Multiple equilibria in two-sector monetary economies: an interplay between preferences and the timing for money

Keywords: Money-in-the-utility-function, Indeterminacy, Sunspot equilibria. Journal of Economic Literature Classification Numbers: C61, E32

 formulation. First, we show that for all the three types of preferences, local indeterminacy easily arises under the CIA timing with a low enough interest rate elasticity of money demand. Second, we show that with the CAM timing, determinacy always holds under separable preferences, but local indeterminacy can arise in the case of GHH and KPR preferences. We thus prove that compared to aggregate models, two-sector models provide new rooms for local indeterminacy when non-separable standard preferences are considered.

Introduction

A wide-spread perception among economists is that macroeconomic businesscycle fluctuations are not driven only by shocks on technologies or preferences, but also by changes in expectations about the fundamentals. The 2008 financial crisis is a good illustration to show how coordination failures in the monetary market and self-fulfilling prophecies can have dramatic effects either on the stock or the goods market, and account for the observed macroeconomic fluctuations to a large extent.

A major strand of the literature focussing on fluctuations derived from agents' beliefs is based on the concept of sunspot equilibria that dates back to the early works of Shell [START_REF] Shell | Monnaie et Allocation Intertemporelle[END_REF], Azariadis [START_REF] Azariadis | Self Fulfilling Prophecies[END_REF] and Cass and Shell [START_REF] Cass | Do Sunspots Matter ?[END_REF]. As shown by Woodford [START_REF] Woodford | Stationary Sunspot Equilibria. The Case of Small Fluctuations Around a Deterministic Steady State[END_REF], the existence of sunspot equilibria is closely related to the equilibrium indeterminacy under perfect foresight, i.e. the existence of a continuum of equilibrium paths converging toward one steady state from the same initial value of the state variable.

It is well-known that equilibrium indeterminacy can arise in monetary models in which the central bank follows an exogenous money growth rule. 1Transactions need money and a standard approach consists in assuming that agents face borrowing constraints, demanding money and bearing the opportunity cost of a nominal interest rate. 2 In order to capture the transactions motive for money holding, the Money-In-the-Utility-Function (MIUF) approach is widely used in the literature as it generalizes either the transactions cost or the shopping time models as shown by Feenstra [START_REF] Feenstra | Functional Equivalence Between Liquidity Costs and the Utility of Money[END_REF]. 3 However, the timing of transactions matters depending on the way real balances enter the utility function. On the one hand, with the Cash-In-Advance (CIA) timing, the cash available to buy goods is money in the consumers' hands after they leave the bond market but before they enter the goods market. Under this timing and a Leontief utility function, the MIUF model reduces to the Clower [START_REF] Clower | A Reconsideration of the Microfoundations of Monetary Theory[END_REF] constraint. 4 Within aggregate models, local indeterminacy easily arises under a CIA timing and requires a sufficiently weak elasticity of intertemporal substitution, together with dominant income effects, i.e. a low enough interest elasticity of money demand. 5 On the other hand, the traditional Sidrauski [START_REF] Sidrauski | Rational Choice and Patterns of Growth in a Monetary Economy[END_REF] view implies that end-ofperiod real balances enter the utility function. 6 As agents hold money for transactions after leaving the goods market, this timing is called Cash-Afterthe-Market (CAM), and whithin an aggregate economy with production, it implies that indeterminacy never arises. 7 The aim of our paper is to explore the robustness of those conclusions within a monetary two-sector economy.

The main motivation to consider a two-sector model is that local indeterminacy has been shown to occur much more easily in such a framework than within aggregate models. The shifting of resources between the productions of the two goods provides indeed new mechanisms through the implied variations of prices. This is true at the same time in real models with productive externalities, 8 and in monetary models with fractional CIA constraint. 9 In this paper, we investigate whether two-sector models also provide new rooms for local indeterminacy when a general MIUF formulation is considered.

While the production side is based on the consideration of general technologies, the side of preferences is defined on the ground of three different specifications of the utility function which are commonly used in the business cycle and growth literature: i) an additively separable formulation, ii) a King, Plosser and Rebelo [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF] (KPR) formulation which is compatible with balanced growth, and finally iii) a Greenwood, Hercovitz and Huffman [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] (GHH) formulation for which the marginal rate of substitution between consumption and real money balances depends on the latter only. The CIA and CAM timings are systematically compared throughout the paper. Considering the CIA timing, we first generalize the conclusions obtained by Bosi, Magris and Venditti [START_REF] Bosi | Multiple Equilibria in a Cash-in-Advance Two-Sector Economy[END_REF] in a two-sector economy with CIA constraint : local indeterminacy requires a low enough interest rate elasticity of money demand. 10 Second, and contrary to the aggregate formulation, we highlight the possible existence of local indeterminacy in two-sector monetary economies with a CAM timing depending on the utility function. While we show that the determinacy result is robust under additively separable preferences, we prove that this is no longer the case with non-separable preferences by providing simple sufficient conditions for local indeterminacy in the case of GHH [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] and KPR [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF] utility functions. Explicit examples with CES technologies illustrate these theoretical findings. Summing up, the novelty of the paper not only rests on a more general (bi-sectoral) MIUF approach to the CIA timing where indeterminacy more likely arises, but also on the occurrence of indeterminacy under a CAM timing (never shown in literature) with heterogeneous technologies and non-separable preferences.

The paper is organized as follows. The next section sets up the basic model, defines the intertemporal equilibrium and proves the existence of a steady state. In Section 3, we present our main results by focussing successively on additively separable, GHH and KPR preferences. Section 5 illustrates our conclusions through a simple example. Section 6 contains concluding comments and all the proofs are in a final Appendix.

The model 2.1 Technology

The basic model is a two-sector optimal growth model. We assume that there are two commodities with one consumption good y 0 and one capital good y. The labor supply is assumed to be inelastic. Total labor is normalised to 1 and each good is produced with a constant returns to scale technology:

y 0 = f 0 (k 0 , l 0 ), y = f 1 (k 1 , l 1 )
with k 0 + k 1 ≤ k, k being the total stock of capital, and l 0 + l 1 ≤ 1.

Assumption 1. Each production function f i : R 2 + → R + , i = 0, 1, is C 2 , increasing in each argument, concave, homogeneous of degree one and such that for any x > 0,

f i 1 (0, x) = f i 2 (x, 0) = +∞, f i 1 (+∞, x) = f i 2 (x, +∞) = 0.
By definition we have y ≤ f 1 (k, 1). The monotonicity properties and the Inada conditions in Assumption 1 imply that there exists k > 0 solution of k -f 1 (k, 1) = 0 such that f 1 (k, 1) > k when k < k, and f 1 (k, 1) < k when k > k. It follows that it is not possible to maintain stocks beyond k. The set of admissible pairs (k, y) is thus defined as follows

D = (k, y) ∈ R 2 + | 0 ≤ k ≤ k, 0 ≤ y ≤ f 1 (k, 1) (1)
There are two representative firms, one for each sector. For any given (k, y), profit maximization is equivalent to solving the following problem of optimal allocation of productive factors between the two sectors:

T (k, y) = max k 0 ,k 1 ,l 0 ,l 1 f 0 (k 0 , l 0 ) s.t. y ≤ f 1 (k 1 , l 1 ) k 0 + k 1 ≤ k l 0 + l 1 ≤ 1 k 0 , k 1 , l 0 , l 1 ≥ 0 (2)
The social production function T (k, y) describes the frontier of the production possibility set associated with interior temporary equilibria such that (k, y) ∈ D, and gives the maximal output of the consumption good. Under Assumption 1, for any (k, y) ∈ D, T (k, y) can be shown to be concave and twice continuously differentiable. 11 Denoting w the wage rate, r the gross rental rate of capital and p the price of investment good, all in terms of the price of the consumption good, we formulate the aggregate profit maximization problem as follows max

(k,y)∈D T (k, y, ) + py -rk -w (3)
and we derive that for any (k, y) ∈ intD, with intD denoting the interior of the set D, the first order derivatives of the social production function give

T 1 (k, y) = r, T 2 (k, y) = -p (4)
Buiding on the constant returns to scale, it follows that the wage rate satisfies T (k, y) + py = rk + w.

Preferences

The economy is populated by a large number of identical infinitely-lived agents. We will assume without loss of generality that the total population is constant. In each period t ≥ 0, each consumer receives income from capital, labor, money and lump-sum transfers from the government. His expenditures concern the consumption good, the investment good and money for the next period t + 1. Let us then denote by M t the stock of money at time t. In each period t ≥ 0, the representative agent is subject to the budget constraint

qt c t + pt y t + M t+1 = rt k t + ŵt + M t + τt ( 5
)
where qt is the price of the consumption good, pt is the price of the investment good, rt is the rental rate of capital, ŵt is the wage rate and τ is the nominal lump-sum transfers issued by the government. Choosing the consumption good as the numéraire, i.e. dividing equation ( 5) by qt , gives the following formulation for the budget constraint

c t + p t y t + q t M t+1 = r t k t + w t + q t M t + τ t (6)
with p t = pt /q t , q t = 1/q t , r t = rt /q t , w t = ŵt /q t and τ t = τt /q t . Money in real terms is thus given by q t M t . The per-period utility function u(c t , q t M t ) depends on consumption c t and real money balances denoted now q t M t . In the literature, there are indeed two typical choices for the formulation of the cash that facilitates trading at time t. The first one, used for instance by Woodford [START_REF] Woodford | Monetary Policy and Price Level Determinacy in a Cash-in-Advance Economy[END_REF], corresponds to the cash the household has in advance of goods-market trading, namely

M t = M t (7)
and is called Cash-In-Advance (CIA) timing. The second formulation, used for instance by Matsuyama [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF], corresponds to the cash left to the household after leaving the goods market, namely

M t = M t+1 (8)
and is called Cash-After-the-Market (CAM) timing.

Considering that M t = M t or M t+1 , the per-period utility function satisfies the following basic restrictions: Assumption 2. u(c, qM) is C 2 , increasing in each argument, concave and satisfies the boundary conditions lim qM→0 u 2 (x, qM)/u 1 (x, qM) = +∞, lim qM→+∞ u 2 (x, qM)/u 1 (x, qM) = 0 for any x > 0.

We also introduce a standard normality assumption between consumption c and real money balances qM Assumption 3. Consumption c and real money balances qM are normal goods.

The consumption of the representative agent at time t is given by c t = T (k t , y t ), and the capital accumulation equation is such that

k t+1 = y t + (1 -δ)k t (9)
with δ ∈ [0, 1] the rate of depreciation of capital. The intertemporal maximisation program of the representative agent is then as follows max

{ct,k t+1 ,M t+1 } +∞ t=0 +∞ t=0 β t u(c t , q t M t ) s.t. c t + p t [k t+1 -(1 -δ)k t ] + q t M t+1 = r t k t + w t + q t M t + τ t k 0 , M 0 given
where β ∈ (0, 1] denotes the discount factor, and M t = M t or M t+1 .

Intertemporal equilibrium

The generalised Lagrangian is

L = +∞ t=0 β t u(c t , q t M t ) + +∞ t=0 β t λ t [r t k t + w t + q t M t + τ t -c t -p t [k t+1 -(1 -δ)k t ] -q t M t+1 ]
with λ the non-negative Lagrange multiplier associated with constraint [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF].

With both timings, the first order conditions with respect to k t+1 is identical and given by

λ t p t = βλ t+1 [r t+1 + (1 -δ)p t+1 ] (10)
where r t and p t are defined by ( 4). However, the first order conditions with respect to c t and M t depend on the timing and are given by

λ t = u 1 (c t , q t M t ) (11)
λ t q t = βq t+1 [λ t+1 + u 2 (c t+1 , q t+1 M t+1 )] (12)
with the CIA timing, or

λ t = u 1 (c t , q t M t+1 ) ( 13
)
βλ t+1 q t+1 = q t [λ t -u 2 (c t , q t M t+1 )] (14)
with the CAM timing.

By manipulating the first order conditions ( 10)-( 14), we obtain the Euler equation and the arbitrage condition between consumption and real balances

u 1 (c t , q t M t )p t = βu 1 (c t+1 , q t+1 M t+1 )[r t+1 + (1 -δ)p t+1 ] u 1 (c t , q t M t )q t = βq t+1 [u 1 (c t+1 , q t+1 M t+1 ) + u 2 (c t+1 , q t+1 M t+1 )] (15)
with the CIA timing, or

u 1 (c t , q t M t+1 )p t = βu 1 (c t+1 , q t+1 M t+2 )[r t+1 + (1 -δ)p t+1 ] βu 1 (c t+1 , q t+1 M t+2 )q t+1 = q t [u 1 (c t , q t M t+1 ) -u 2 (c t , q t M t+1 )] (16)
with the CAM timing.

In order to complete the definition of the intertemporal equilibrium, we need to specify the supply of money. We assume that the government follows a simple monetary rule: in each period it issues lump-sum transfers of money balances at the constant rate µ -1 > 0, so that in period t the supply of money M s t satisfies M s t = µ t M s 0 , with M s 0 = M 0 the initial amount of money balances. Thus nominal transfers are given by τ t = (µ -1)q t M s t . Using the notations m t = q t M t and z t = q t M t+1 , we finally get from ( 15)-( 16) and from the equilibrium on the monetary market

M s t = M t : u 1 (c t , m t )p t = βu 1 (c t+1 , m t+1 )[r t+1 + (1 -δ)p t+1 ] u 1 (c t , m t )m t = β µ [u 1 (c t+1 , m t+1 ) + u 2 (c t+1 , m t+1 )]m t+1 (17)
with the CIA timing, or

u 1 (c t , z t )p t = βu 1 (c t+1 , z t+1 )[r t+1 + (1 -δ)p t+1 ] β µ u 1 (c t+1 , z t+1 )z t+1 = z t [u 1 (c t , z t ) -u 2 (c t , z t )] (18)
with the CAM timing. 17) and ( 18) are two systems of implicit difference equations of order 2 in the capital stock k and order 1 in real money balances m or z.

Taking into account c t = T (k t , y t ), r t = T 1 (k t , y t ), p t = -T 2 (k t , y t ) and y t = k t+1 -(1 -δ)k t , we conclude that (
Any optimal plan for the representative household that satisfies the Euler equations ( 17) or ( 18) and the corresponding following transversality condition lim

t→+∞ β t u 1 (c t , x t )p t k t+1 = 0 (19) lim t→+∞ β t u 1 (c t , x t )x t = 0 (20)
with x t = m t or z t , is called an equilibrium path.

Steady state

A steady state is defined as

k t = k * , y t = y * = δk * , c t = c * , p t = p * = -T 2 (k * , δk * ), r t = r * = T 1 (k * , δk * ) and m t = m * or z t = z * for all t.
Along a stationary solution, we have indeed a decreasing path for the price of money balances q t+1 /q t = 1/µ which is associated with the expansion of money supply M t+1 /M t = µ so that at the steady state real money balances m t or z t remain constant. From the dynamical system ( 17) and the fact that c = T (k, δk), the stationary levels of capital k * and real money balances m * or z * are obtained as solutions of

r p = -T 1 (k,δk) T 2 (k,δk) = β -1 -(1 -δ) (21)
and

u 2 (T (k,δk),m) u 1 (T (k,δk),m) = µ β -1 (22)
with the CIA timing, or

u 2 (T (k,δk),z) u 1 (T (k,δk),z) = 1 -β µ ( 23
)
with the CAM timing.

As in the standard one-sector Ramsey model, we get the following result:

Proposition 1 . Under Assumptions 1-2, there exists a unique steady state (k * , m * ) or (k * , z * ) solution of equations ( 21)-( 22) or ( 21)-( 23).

The steady states (k * , m * ) and (k * , z * ) are called the Modified Golden Rule.

Main results

In order to study the local determinacy properties of the steady state, we need to compute the characteristic roots associated with the linearized Euler equations around the Modified Golden Rule. At any time t we have one predeterminate variable, the current capital stock k t , and two forward variables, the capital stock of the following period k t+1 and the current real money balances m t or z t . Local indeterminacy will be obtained if at least two characteristic roots are inside the unit circle. Note from equations ( 21)-( 22) that the steady-state (k * , x * ), with x * = m * or z * , is defined as follows: the capital stock k * is only determined by the technological characteristics, the discount factor β and the rate of depreciation of capital δ. The properties of preferences only influence the stationary value of real money balances x * = m * or z * . Then for some given technologies and a given pair (β, δ), the value of k * is fixed and it is easy to consider variations of preferences in order to study the occurrence of bifurcations. This will be our strategy of proof in the sequel of the paper.

We are looking for the simplest sufficient conditions for the occurrence of local indeterminacy. In order to derive economically interpretable results, we introduce, following Benhabib and Nishimura [START_REF] Benhabib | Competitive Equilibrium Cycles[END_REF], the relative capital intensity difference across sectors such that

b = l 1 y k 1 l 1 -k 0 l 0 (24)
The sign of b is positive if and only if the investment good is capital intensive. Note also that the demands for input k 0 , k 1 , l 0 , l 1 when evaluated at the steady state are functions of the discount factor β and the rate of depreciation of capital δ. It follows that the capital intensity difference satisfies b = b(β, δ). We also define the elasticities of substitution of consumption and real money balances evaluated at the steady state

12 cc = -ξ -1 cc = -(u * 11 c * /u * 1 ) -1 > 0, xc = -ξ -1 xc = -(u * 21 c * /u * 2 ) -1 (25) xx = -ξ -1 xx = -(u * 22 x * /u * 2 ) -1 > 0, cx = -ξ -1 cx = -(u * 12 x * /u * 1 ) -1 (26) with x = m or z, u * i = u m i or u z i , u * ij = u m ij or u z ij , i, j = 1, 2
, and the following elasticities of the consumption good's output and the rental rate with respect to the capital stock, all evaluated at the steady state

ε ck = T * 1 k * /T * > 0, ε rk = T * 11 k * /T * 1 < 0 ( 27
)
We will explore the existence of local indeterminacy with CIA and CAM timings considering successively different specifications for the utility function which are widely used in the Real Business Cycle literature. We will then show that the occurrence of business cycles based on self-fulfilling prophecies depends on an interplay between preferences and the timing for money.

Additively separable preferences

Consider the standard case of additively separable preferences.

Assumption 4. ξ cm = ξ mc = ξ cz = ξ zc = 0
Let us first assume the CIA timing. We introduce the following critical value for the elasticity of intertemporal substitution in consumption:

¯ cc = ε ck 2β(1+β)θ 2 ε rk [1+(2-δ)b][β+b[1+(1-δ)β]] (28) with θ = [1 -β(1 -δ)] -1 .
Theorem 1. With the CIA timing, let Assumptions 1-4 hold. Then the modified golden rule (k * , m * ) is locally indeterminate if and only if the elasticity of the money demand satisfies mm < (µ -β)/2µ and one of the following conditions is satisfied:

i) the investment good is capital intensive around (k * , m * ); ii) the consumption good is capital intensive around (k * , m * ) with b ∈ (-∞, -1/(2 -δ)] ∪ [-β/(1 + (1 -δ)β), 0]; iii) the consumption good is capital intensive around (k * , m * ) with b ∈ (-1/(2 -δ), -β/[1 + (1 -δ)β]) and cc < ¯ cc .
The existence of local indeterminacy with the CIA timing is easily obtained as it does not require particular restriction on the capital intensity difference across sectors. Notice however that it requires a low enough interest rate elasticity of money demand as in the aggregate formulation. 13 Let us now consider the CAM timing. Contrary to the previous timing, local indeterminacy is always ruled out. With additively separable preferences we obtain the same conclusion as in the case of aggregate models in which b = 0: with the CAM timing, the equilibrium is unique and endogenous fluctuations based on self-fulfilling prophecies cannot occur. Note however that within aggregate models, this conclusion holds for any utility function. We then inquire whether this conclusion still holds for non-separable specifications. We will show that this is not the case by considering two different standard specifications of preferences. 14

GHH preferences

Consider a Greenwood-Hercovitz-Huffman [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] (GHH) formulation such that u(c, qM

) = U (c + G(qM)) (29)
with U (.) and G(.) some increasing and strictly concave functions. Such a specification implies that the marginal rate of substitution between consumption and real money balances depends on the latter only as

u 2 (c,qM) u 1 (c,qM) = G (qM) Let us denote G xx = -(ξ G xx ) -1 = -(G (x)x/G (x)) -1
> 0 the elasticity of the function G(x). Using ( 22) and [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF], we derive at the steady state

G (m * ) = µ-β β ( 30
)
with the CIA timing, or

G (z * ) = µ-β µ ( 31
)
13 Theorem 1 generalizes the conclusions obtained by Bosi, Magris and Venditti [START_REF] Bosi | Multiple Equilibria in a Cash-in-Advance Two-Sector Economy[END_REF] in a two-sector economy with CIA constraint.

14 See Carlstrom and Fuerst [START_REF] Carlstrom | Money Growth Rules and Price Level Determinacy[END_REF].

with the CAM timing. Let v = c * /m * denote the steady state velocity of money and z * = µm * . We easily get from ( 26) that ξ cc = ξ xc for x = m or z and

ξ cm = ξ cc µ-β βv , ξ mm = ξ cc µ-β βv + ξ G mm
with the CIA timing, or

ξ cz = ξ cc µ-β v , ξ zz = ξ cc µ-β v + ξ G zz
with the CAM timing.

Consider first the CIA timing. We introduce the following critical value for the elasticity of intertemporal substitution in consumption:

¯ cc = βθ 2 + µ-β βv ε rk ε ck b[1+(1-δ)b] ε rk ε ck b[1+(1-δ)b] > 0 (32)
Considering a particular interval of values for the capital intensity difference across sectors b we get:

Theorem 3. With the CIA timing, let Assumptions 1-3 hold, b ∈ (-1/(1 - δ), -1/(2-δ)
) and consider ¯ cc as defined by (32). Then the modified golden rule (k * , m * ) is locally indeterminate if cc < ¯ cc and G mm ≤ (µ -β)/2µ. It is worth noting that local indeterminacy can arise for a broader range of values for b but complex additional restrictions on technologies involving the elasticities ε ck and ε rk are necessary. Therefore, as our aim is to provide the simplest sufficient conditions for local indeterminacy, we focus on a specific range of values for b. Note also that the condition on the interest rate elasticity of money demand is the same as in the case of additively separable preferences. It is worth pointing out finally that negative values for b are compatible with recent empirical evidences. Building on aggregate Input-Output tables, Takahashi et al. [START_REF] Takahashi | Measuring Capital Intensity in the Postwar Japanese Economy: a Two-Sector Approach[END_REF] have shown that over the last 30 years the OECD countries have been characterized by a consumption good sector which is more capital intensive than the investment good sector.

Consider now the CAM timing. We introduce the following critical value for the elasticity of intertemporal substitution in consumption:

˜ cc ≡ 2 h µ-β v(1-δ) 2 ε rk ε ck +β(1+β)θ 2 " 2-µ-β β ξ G zz "i ε rk ε ck (1-δ) 2 " 2-µ-β β ξ G zz " > 0 (33)
Again, our strategy is to focus on the simplest sufficient conditions for the occurrence of local indeterminacy. We then consider a specific set of values for the capital intensity difference across sectors b and some condition on the elasticities ε ck and ε rk , although more general conclusions could be obtained at the price of additional restrictions on the technologies.

Assumption 5. b ∈ (-1/(1-δ)-, -1/(1-δ)+ ) with > 0 small enough, and lim δ→1

ε rk ε ck (1-δ) 2 = -∞.
Note from the capital accumulation equation ( 9) that the first part of Assumption 5 implies that around the steady state only small fluctuations of the capital stock and thus prices are considered. 15 Theorem 4. With the CAM timing, let Assumptions 1-3 and 5 hold. Then there exist δ ∈ (0, 1) and ˜ cc > 0 as defined by (33) such that the modified golden rule (k * , z * ) is locally indeterminate if δ ∈ (δ, 1) and cc < ˜ cc .

The conclusion provided by Theorem 4 is important as it shows that contrary to the aggregate formulation with b = 0, there exists some non-separable specification for preferences which allows for the existence of local indeterminacy when the consumption good sector is sufficiently capital intensive with respect to the investment good sector. Theorem 4 therefore shows that uniqueness no longer holds with the CAM timing if two sectors which are sufficiently asymmetric are considered.

KPR preferences

Consider a King-Plosser-Rebelo [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF] (KPR) formulation such that

u(c, x) = [cg(x)] 1-φ 1-φ (34)
which is compatible with balanced growth. Let us define h(x) = g (x)/g(x) and

ψ(x) = xh(x), η(x) = xh (x) h(x) (35)
Beside Assumption 3, we introduce the following restrictions: Assumption 6. g(x) is a positive increasing function, φ ≥ 0, η(x) ≤ -ψ(x)(1 -φ) and η(x) ≤ ψ(x)(1 -1/φ). Moreover lim x→0 h(x) = +∞ and lim x→+∞ h(x) = 0 Assumption 6 implies that Assumption 2 holds. 16 Note that ψ(x) > 0 can be interpreted as the elasticity of the utility of real money balances and η(x) < 0 is linked to the interest rate elasticity of real money demand. Using ( 22) and ( 23) and considering that z * = µm * and v * = c * /m * , we derive at the steady state ψ(m * ) = µ-β βv * (36) 15 We have indeed under Assumption 5: dkt+1/dkt = 1 -δ + dyt/dkt = 1 -δ + 1/b ≈ 0. 16 See Hintermaier [START_REF] Hintermaier | Lower Bounds on Externalities in Sunspot Models[END_REF][START_REF] Hintermaier | On the Minimum Degree of Returns to Scale in Sunspot Models of Business Cycles[END_REF] and Pintus [START_REF] Pintus | Local Determinacy with Non-Separable Utility[END_REF].

with the CIA timing, or

ψ(z * ) = µ-β v * (37)
with the CAM timing. We then easily get from (26) that ξ cc = -φ, ξ xc = 1 + ξ cc for x = m or z and

ξ cm = (1 + ξ cc ) µ-β
βv * , ξ mm = ξ cm + η(m * ) with the CIA timing, or

ξ cz = (1 + ξ cc ) µ-β v * , ξ zz = ξ cz + η(z * ) with the CAM timing.
Consider first the CIA timing and let η = η(m * ). We proceed as in the case of GHH preferences and focus on a particular interval of values for b.

Theorem 5. With the CIA timing, let Assumptions 1-3 and 6 hold, b ∈ (-1/(1-δ), -1/(2-δ)) and η ≤ -2µ/(µ-β). Then there exists ¯ cc < 0 such that the modified golden rule

(k * , m * ) is locally indeterminate if cc < ¯ cc .
Again it is worth to mention that local indeterminacy can arise for a broader range of values of b but complex additional restrictions on technologies involving the elasticities ε ck and ε rk are necessary. Note also that the condition on the elasticity η is the same as in the cases of additively separable and GHH preferences since it implies to consider a low enough interest rate elasticity of real money demand.

Consider now the CAM timing and let η = η(z * ). We introduce the following critical value for the elasticity of intertemporal substitution in consumption:

˜ cc ≡ min β-(µ-β)(η-µ-β v * ) (µ-β) 2 v * , 2 µ-β v * 2+ µ-β v * " 2-ηv * β " > 0 ( 38
)
We also proceed as in the case of GHH preferences and consider a specific set of values for b which is far enough from zero and which implies that around the steady state, the fluctuations of prices are small. Theorem 6. With the CAM timing, let Assumptions 1-3, 5 and 6 hold.

Then there exist δ ∈ (0, 1) and ˜ cc > 0 as defined by ( 38) such that the modified golden rule (k * , z * ) is locally indeterminate if δ ∈ (δ, 1) and cc = 1/φ < ˜ cc .

Theorem 6 confirms the conclusion given by Theorem 4 in the case of a GHH utility function: it proves that there exist standard non-separable specifications of preferences for which local indeterminacy arises with both formulations of the timing for money. These results are based on a capital intensity difference across sectors which is sufficiently negative and show that two-sector models lead to a greater flexibility in terms of local indeterminacy of equilibrium paths than aggregate models.

Example

We do not consider additively separable preferences as we have shown that the main conclusions are similar to those obtained with an aggregate formulation. Moreover, we focus on the CAM timing as the proof of existence of local indeterminacy in this case is the main contribution of our paper. Assume that preferences are given either by a GHH utility function

u GHH (c, x) = 1 1-ϕ c + ζ x 1-ν 1-ν 1-ϕ (39)
or a KPR utility function

u KP R (c, x) = 1 1-φ c x 1+ρ 1+ρ 1-φ (40)
with ϕ, ν, φ ≥ 0, ρ > -1 and ζ > 0. Consider also the following CES technologies:

f 0 (k 0 , l 0 ) = χk -σ 0 + (1 -χ)l -σ 0 -1/σ = y 0 f 1 (k 1 , l 1 ) = min{k 1 /γ, l 1 } = y
with χ ∈ (0, 1), σ > -1 and γ > 0. Since at the equilibrium y = k 1 /γ = l 1 , we get k 0 = k -γy, l 0 = 1 -y and thus the social production function

T (k, y) = χ(k -γy) -σ + (1 -χ)(1 -y) -σ -1/σ
Assuming that βθ -γ > 0, the steady state for capital is

k * = 1-χ χ 1 βθ-γ 1/(1+σ) (1 -γδ) + δ -1
and since y * = δk * , we easily derive

ε * rk ε * ck = -(1 + σ) 1-χ χ 1/(1+σ) (βθ -γ) σ/(1+σ) < 0 b = γ -χ(βθ-γ) 1-χ 1/(1+σ)
As a result we have b = -1/(1 -δ) if and only if

1-χ χ 1 βθ-γ 1/(1+σ) = 1-δ 1+γ(1-δ)
Obviously, the previous equality holds if χ = χ as given by χ = 1 + (βθ -γ)

1-δ 1+γ(1-δ) 1+σ -1
Moreover, when χ = χ we have

1- χ χ 1/(1+σ) = (1-δ)(βθ-γ) 1/(1+σ) 1+γ(1-δ)
and thus we get

ε * rk ε * ck = -(1 + σ) (1-δ)(βθ-γ) 1+γ(1-δ) so that ε rk ε ck (1-δ) 2 = -(1+σ)(βθ-γ) (1-δ)[1+γ(1-δ)]
As θ ∈ [0, 1] for any δ ∈ [0, 1], Assumption 5 is satisfied. We also derive

k * = 1 + γ(1 -δ) which leads to c * = T (k * , δk * ) = (1 -γδ) [1+γ(1-δ)] 1+σ +(βθ-γ)(1-δ) 1+σ θ 1/σ (41)
Consider first the case of GHH preferences. Solving the first order condition ( 23) using (39) gives at the steady state

z * = µm * = µ-β ζµ -1/ν
and we derive from (41) the stationary velocity of money

v * ≡ c * /m * = µ µ-β ζµ 1/ν (1 -γδ) [1+γ(1-δ)] 1+σ +(βθ-γ)(1-δ) 1+σ θ 1/σ It follows from (39) that cc = 1 ϕ + µ-β ϕ(1-ν)v *
Moreover, as ξ G xx = -ν for all x > 0, we derive from (33) the following critical value for the elasticity of intertemporal substitution in consumption:

˜ cc ≡ 2β µ-β v * [2β+(µ-β)ν] -(1+β)θ 2 (1-δ)[1+γ(1-δ)] (1+σ)(βθ-γ)
The critical bound δ ∈ [0, 1) is obtained as the value of δ such that the expression between brackets on the right-hand-side is greater or equal to zero. 17 Assuming δ ∈ (δ, 1), we conclude that there exists ϕ > 0 such that

17 Note that δ = 0 if µ-β 2β+(µ-β)ν ≥ (1+β)(1+γ)µ " µ-β ζµ " 1/ν [(1-β)[(1+γ) 1+σ -γ]+β] 1/σ (1+σ)(1-β)[β-γ(1-β)]
cc < ˜ cc , and thus local indeterminacy arises, if ϕ > ϕ.

Consider now the case of KPR preferences. We derive from (40) that ψ(x) = 1 + ρ and η(x) = -1 for all x > 0. Assumption 6 then requires 1 ≥ (1 + ρ)(1 -φ) and 1 ≥ (1 + ρ)(1 -φ)/φ. Solving the first order condition [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF] with z = µm gives the stationary velocity of money

v * ≡ c * /m * = µ-β 1+ρ
Using (38) we derive the following critical value for the elasticity of intertemporal substitution in consumption:

˜ cc ≡ 2β(1+ρ) µ+β+2β(1+ρ)
Therefore, there exists φ > 0 such that Assumption 6 holds and

cc ≡ 1 φ < ˜ cc if φ > φ.
Under this restriction we derive the critical bound δ ∈ (0, 1) as the value of δ such that the following equation is satisfied

βθ-γ (1-δ)[1+γ(1-δ)]βθ 2 = 2 1+β 1+σ (1-φ)(1+ρ) µ-β µ+β -φ (1-φ)(1+ρ) 2β µ+β +1
We then conclude that local indeterminacy arises if φ > φ and δ ∈ (δ, 1).

Concluding comments

We have considered a two-sector MIUF model with general technologies and preferences defined on the basis of three popular specifications of the utility function in the business cycle and growth literature: i) an additively separable formulation, ii) a KPR formulation, and iii) a GHH formulation. The CIA and CAM timings have been systematically compared throughout the paper. First, we have shown that for all the three different specifications of preferences, local indeterminacy with a CIA timing requires a low enough interest rate elasticity of money demand. Second, and contrary to the aggregate formulation, we have proved the possible existence of local indeterminacy with a CAM timing depending on the utility function. While we have shown that determinacy always holds under separable preferences, we have provided simple sufficient conditions for local indeterminacy in the case of GHH [START_REF] Greenwood | Investment, Capacity Utilization and the Real Business Cycle[END_REF] and KPR [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF] preferences. Explicit examples with CES technologies have also been given to illustrate these theoretical findings. We have thus proved that compared to aggregate models, two-sector models provide new rooms for local indeterminacy when non-separable standard preferences are considered.

with θ = [1 -β(1 -δ)] -1 .
The second derivatives of the function T (k, y) will be evaluated at k * using the following notation: T * ij = T ij (k * , δk * ). Using equations ( 25)-( 27) and ( 42)-(46), and denoting dκ t = dk t+1 , total differenciation of the Euler equations [START_REF] Grandmont | On the Role of Money and the Existence of a Monetary Equilibrium[END_REF] gives after tedious but straightforward computations the following linear system

  0 -βA m 1 A m 2 0 βA m 3 A m 4 1 0 0     dk t+1 dκ t+1 dm t+1   =   A m 1 A m 5 A m 2 -ξ cc θ ξ cc βθ + A m 3 -c * m * T * 1 (1 + ξ cm ) 0 1 0     dk t dκ t dm t   where A m 1 = βθ 2 ξ cc + ε rk ε ck b[1 + (1 -δ)b] A m 2 = θ T * 1 ξ mc (µ -β) A m 3 = θ µ [ξ cc β + ξ mc (µ -β)] A m 4 = -c * m * T * 1 1 + 1 µ ξ cm β + ξ mm (µ -β) A m 5 = -β(1 + β)θ 2 ξ cc -ε rk ε ck b 2 + β[1 + (1 -δ)b] 2 > 0 with the CIA timing, or    A z 1 -βA z 2 ξcz θε ck ξ cc -βξ cc 1+ξcz θε ck 1 0 0      dk t+1 /k dκ t+1 /κ dz t+1 /z   =    A z 2 -βA z 3 ξcz θε ck A z 4 -βA z 4 1+A z 5 θε ck 0 1 0      dk t /k dκ t /κ dz t /z   where A z 1 = ξ cc + ε rk ε ck 1+(1-δ)b θ 2 < 0 A z 2 = ξ cc + ε rk ε ck b βθ 1+(1-δ)b θ A z 3 = ξ cc + ε rk ε ck b βθ 2 < 0 A z 4 = µ β ξ cc + 1 -µ β ξ zc A z 5 = µ β ξ cz + 1 -µ β ξ zz with the CAM timing. P(1) P(0) = µ-β β (ξzz-ξcz) (1-δb)(β-θ -1 b) βθ 2 ε rk ε ck ξcc+(1+ξcz) 1+(1-δ)b θ b βθ ε rk ε ck -µ-β β h ξccξzz-ξczξzc+(ξzz-ξcz) 1+(1-δ)b θ b βθ ε rk ε ck i P(-1) P(0) = h 2(1+ξcz)-µ-β β (ξzz-ξcz) ih 1+(1-δ)b θ + b θ ih 1+(1-δ)b θ + b βθ i ε rk ε ck ξcc+(1+ξcz) 1+(1-δ)b θ b βθ ε rk ε ck -µ-β β h ξccξzz-ξczξzc+(ξzz-ξcz) 1+(1-δ)b θ b βθ ε rk ε ck i + 2(1+β) h 2ξcc-µ-β β (ξccξzz-ξczξzc) i ξcc+(1+ξcz) 1+(1-δ)b θ b βθ ε rk ε ck -µ-β β h ξccξzz-ξczξzc+(ξzz-ξcz) 1+(1-δ)b θ b βθ ε rk ε ck i (49)
with the CAM timing.

Proof of Theorem 1

Under Assumption 4 we derive from Lemma 1 that

D = 1 β h 1+ξmm µ-β µ i , S = 1 β - A m 5 A m 1 D, T = βD - A m 5 βA m 1
and thus the characteristic polynomial rewrites as

P(λ) = λ 2 + λ A m 5 βA m 1 + 1 β (λ -βD) = λ 2 -λ β(1+β)θ 2 ξcc+ ε rk ε ck [b 2 +β[1+(1-δ)b] 2 ] β h βθ 2 ξcc+ ε rk ε ck b[1+(1-δ)b] i + 1 β (λ -βD) ≡ P(λ) (λ -βD) (50)
Therefore, when the utility is additively separable, the characteristic roots are separated. Note that P(λ) corresponds to the second-degree characteristic polynomial obtained within a standard two-sector optimal growth model without money (see Bosi, Magris and Venditti [START_REF] Bosi | Competitive Equilibrium Cycles with Endogenous Labor[END_REF]). It follows that one characteristic root is within the unit circle while the other is outside in the following cases:

i) b ∈ (-∞, -1/(2 -δ)] ∪ [-β/(1 + (1 -δ)β), βθ), ii) b ∈ (-1/(2 -δ), -β/[1 + (1 -δ)β]) and ξ cc < - ε rk ε ck [1+(2-δ)b][β+b[1+(1-δ)β]] 2β(1+β)θ 2 ≡ ξcc
The third characteristic root is obviously derived from (50) and depends on money as

λ = βD = 1 1+ξmm µ-β µ
Therefore, λ ∈ (-1, 0) if and only if ξ mm < -2µ/(µ -β). Let ¯ cc = -ξ-1 cc and ¯ mm = (µ -β)/(2µ). The final results follow from ( 25) and (26).

Proof of Theorem 2

Under Assumption 4 we derive from Lemma 1 that

D = 1- " µ-β β " ξzz β , S = 1 β + βD A z 1 βA z 2 + A z 3 A z 2 , T = βD + A z 1 βA z 2 + A z 3 A z 2
and thus the characteristic polynomial rewrites as

P(λ) = λ 2 -λ A z 1 βA z 2 + A z 3 A z 2 + 1 β (λ -βD) = P(λ) (λ -βD) (51)
with P(λ) as defined by (50). As in the CIA timing the characteristic roots are separated, and the first two characteristic roots have the same properties as in the proof of Theorem 1. The third root satisfies

λ = βD = 1 -µ-β β ξ zz > 1
The result follows from (26).

Proof of Theorem 3

When the utility function assumes the GHH formulation, we have with the CIA timing

ξ cc = ξ mc , ξ cm = ξ cc µ-β βv , ξ mm = ξ cc µ-β βv + ξ G mm
It follows from (48) that P(1)

P(0) = µ-β µ ε rk ε ck (1-δb)(β-θ -1 b)ξ G mm ξcc " βθ 2 + µ-β βv ε rk ε ck b[1+(1-δ)b] " + ε rk ε ck b[1+(1-δ)b] P(-1) P(0) = 2ξcc h µ-β βv ε rk ε ck [1+(2-δ)b][β+b[1+(1-δ)β]]+β(1+β)θ 2 " 2+ µ-β µ ξ G mm "i ξcc " βθ 2 + µ-β βv ε rk ε ck b[1+(1-δ)b] " + ε rk ε ck b[1+(1-δ)b] + ε rk ε ck [1+(2-δ)b][β+b[1+(1-δ)β]] " 2+ µ-β µ ξ G mm " ξcc " βθ 2 + µ-β βv ε rk ε ck b[1+(1-δ)b] " + ε rk ε ck b[1+(1-δ)b]
Local indeterminacy will be obtained if P(1)/P(0) < 0 and P(-1)/P(0) < 0. Assume that b ∈ (-1/(1 -δ), -1/(2 -δ)), and consider first P(1)/P(0). As the numerator is positive and b[1+(1-δ)b] < 0, the denominator is negative, and thus P(1)/P(0) < 0, if

ξ cc < - ε rk ε ck b[1+(1-δ)b] βθ 2 + µ-β βv ε rk ε ck b[1+(1-δ)b]
≡ ξcc Consider then P(-1)/P(0). As the denominator is negative, P(-1)/P(0) < 0 if the numerator is positive. Since

[1 + (2 -δ)b] [β + b[1 + (1 -δ)β]] > 0, we get a positive numerator if 2 + µ-β µ ξ G mm ≤ 0
The result follows.

Proof of Theorem 4

When the utility function assumes the GHH formulation, we have with the CAM timing

ξ cc = ξ zc , ξ cz = ξ cc µ-β v , ξ zz = ξ cc µ-β v + ξ G zz
It follows from (49) that P( 1)

P(0) = µ-β β ε rk ε ck (1-δb)(β-θ -1 b)ξ G zz ξcc " βθ 2 + µ-β v ε rk ε ck b[1+(1-δ)b]-µ-β β βθ 2 ξ G zz " + ε rk ε ck b[1+(1-δ)b] " 1-µ-β β ξ G zz " P(-1) P(0) = 2ξcc h µ-β v ε rk ε ck [1+(2-δ)b][β+b[1+(1-δ)β]]+β(1+β)θ 2 " 2-µ-β β ξ G zz "i ξcc " βθ 2 + µ-β v ε rk ε ck b[1+(1-δ)b]-µ-β β βθ 2 ξ G zz " + ε rk ε ck b[1+(1-δ)b] " 1-µ-β β ξ G zz " + ε rk ε ck [1+(2-δ)b][β+b[1+(1-δ)β]] " 2-µ-β β ξ G zz " ξcc " βθ 2 + µ-β v ε rk ε ck b[1+(1-δ)b]-µ-β β βθ 2 ξ G zz " + ε rk ε ck b[1+(1-δ)b] " 1-µ-β β ξ G zz "
Local indeterminacy occurs if P(1)/P(0) < 0 and P(-1)/P(0) < 0. Assume then that 1 + (1 -δ)b = 0, i.e. b = -1/(1 -δ). After simplifiying we get

P(1) P(0) = µ-β β(1-δ) 2 ε rk ε ck ξ G zz βθ 2 ξcc " 1-µ-β β ξ G zz " P(-1) P(0) = 2ξcc h µ-β v(1-δ) 2 ε rk ε ck +β(1+β)θ 2 " 2-µ-β β ξ G zz "i + ε rk ε ck (1-δ) 2 " 2-µ-β β ξ G zz " βθ 2 ξcc " 1-µ-β β ξ G zz "
Consider first P(1)/P(0). The numerator is positive while the denominator is negative, so that P(1)/P(0) < 0. Consider then P(-1)/P(0). As the denominator is negative, P(-1)/P(0) < 0 if the numerator is positive. A necessary condition for this is given by the following condition

µ-β v(1-δ) 2 ε rk ε ck + β(1 + β)θ 2 2 -µ-β β ξ G zz < 0 Assume that lim δ→1 ε rk ε ck (1-δ) 2 = -∞
Then there exists δ ∈ (0, 1) such that when δ ∈ (δ, 1), this inequality is satisfied. We conclude that when δ ∈ (δ, 1), the numerator is positive if

ξ cc < - ε rk ε ck (1-δ) 2 " 2-µ-β β ξ G zz " 2 h µ-β v(1-δ) 2 ε rk ε ck +β(1+β)θ 2 " 2-µ-β β ξ G zz "i ≡ ξcc
The result follows.

Proof of Theorem 5

When the utility function assumes the KPR formulation, we have with the CIA timing ≡ ξ 1 cc Consider then P(-1)/P(0). As the denominator is negative, P(-1)/P(0) < 0 if the numerator is positive. Consider the term between brackets in the numerator which is multiplied to ξ cc . Since [1 + (2 -δ)b] [β+b[1+(1-δ)β]] > 0, a sufficient condition to get a negative expression is 2 + µ-β µ η ≤ 0 If this inequality holds we have lim ξcc→-∞ P(-1)/P(0) = -∞. Therefore, there exists ξ 2 cc < 0 such that P(-1)/P(0) < 0 when ξ cc < ξ 2 cc . The result follows by taking ξcc = min{ξ 1 cc , ξ 2 cc }.

ξ cc = -φ, ξ zc = 1 + ξ cc , ξ cz = (1 + ξ cc ) µ-β βv , ξ zz = ξ cz + η with η = η(m * ) < 0. It follows from (48) that P(1) P(0) = µ-β µ ε rk ε ck (1-δb)(β-θ -1 b)η ξcc " βθ 2 + µ-β

Proof of Theorem 6

When the utility function assumes the KPR formulation, we have with the CAM timing ξ cc = -φ, ξ zc = 1 + ξ cc , ξ cz = (1 + ξ cc ) µ-β v , ξ zz = ξ cz + η with η = η(z * ) < 0. It follows from (49) that

P(0) = ε rk ε ck (1-δ) 2 h 2+ µ-β v " 2-ηv β " +2ξcc µ-β v i +2β(1+β)θ 2  ξcc h 2-µ-β β (η-µ-β v ) i + (µ-β) 2 βv ff ξccθ 2 [β-(µ-β)(η-µ-β v )]+ (µ-β) 2 v θ 2
Consider first P(1)/P(0). Since the numerator is positive, P(1)/P(0) < 0 if the denominator is negative, i.e. if

ξ cc < - (µ-β) 2 v β-(µ-β)(η-µ-β v )
≡ ξ 1 cc Consider then P(-1)/P(0). As the denominator is negative, P(-1)/P(0) < 0 if the numerator is positive. Assume that the term between brackets which is pre-multiplied by the ratio ε rk /ε ck is negative, namely

2 + µ-β v 2 -ηv β + 2ξ cc µ-β v < 0 ⇔ ξ cc < - 2+ µ-β v " 2-ηv β " 2 µ-β v ≡ ξ 2 cc
Assume also that lim δ→1 ε rk ε ck (1-δ) 2 = -∞ Under these two conditions we get lim δ→1 P(-1)/P(0) = -∞. Therefore, there exists δ ∈ (0, 1) such that P(-1)/P(0) < 0 for any δ ∈ (δ, 1). The result follows by taking ξcc = min{ξ 1 cc , ξ 2 cc }.

Theorem 2 .

 2 With the CAM timing, let Assumptions 1-4 hold. Then the modified golden rule (k * , z * ) is always locally determinate.

  Local indeterminacy will be obtained if P(1)/P(0) < 0 and P(-1)/P(0) < 0. Assume that b ∈ (-1/(1 -δ), -1/(2 -δ)), and consider first P(1)/P(0). As the numerator is positive and b[1+(1-δ)b] < 0, the denominator is negative, and thus P(1)/P(0) < 0, if ξ cc < -

2 - 2 2 2 v θ 2 P

 22222 δ)b][β+b[1+(1-δ)β]]+β(1+β)θ 2 h 2-δ)b][β+b[1+(1-δ)β]] h 2+ (1-δ)b]-(µ-β)(η-Local indeterminacy will be obtained if P(1)/P(0) < 0 and P(-1)/P(0) < 0. Assume then that 1+ (1 -δ)b = 0, i.e. b = -1/(1 -δ). Straightforward simplifications give [β-(µ-β)(η-µ-β v )]+ (µ-β)

See Michener and Ravikumar[START_REF] Michener | Chaotic Dynamics in a Cash-In-Advance Economy[END_REF] for a survey.

See also Brock[START_REF] Brock | Money and Growth: The Case of Long Run Perfect Foresight[END_REF].

See Benhabib and Nishimura[START_REF] Benhabib | Stability of Equilibrium in Dynamic Models of Capital Theory[END_REF].

Note that the normality Assumption 3 implies ξcc -ξxc ≤ 0 and ξxx -ξcx ≤ 0.

Appendix

Proof of Proposition 1

Consider in a first step equation [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF]. Note that the steady state value of k only depends on the characteristics of the technologies and is independent of the per-period utility function. The proof of Theorem 3.1 in Becker and Tsyganov [START_REF] Becker | Ramsey Equilibrium in a Two-Sector Model with Heterogeneous Households[END_REF] restricted to the case of one homogeneous agent applies so that there exists a unique k * solution of [START_REF] King | Production, Growth and Business Cycles: II. New Directions[END_REF].

Consider in a second step equations ( 22) and ( 23) evaluated at k * . We then write these two conditions as:

The normality Assumption 3 implies that for any x > 0, Θ (x) < 0. This monotonicity property together with the boundary conditions in Assumption 2 finally guarantee the existence and uniqueness of a solution m * or z * of equation ( 22) or ( 23).

The characteristic polynomial

In order to derive a tractable formulation for the third-degree polynomial associated with the dynamical system [START_REF] Grandmont | On the Role of Money and the Existence of a Monetary Equilibrium[END_REF], we need to compute the second derivatives of T (k, y). Let b be the capital intensity difference across sectors as defined by [START_REF] Michener | Chaotic Dynamics in a Cash-In-Advance Economy[END_REF]. We know from Benhabib and Nishimura [START_REF] Benhabib | Competitive Equilibrium Cycles[END_REF] that

with T 11 < 0 implied by the concavity of T (k, y).

As shown previously, the steady states (k * , m * ) and (k * , z * ) are characterized by equations ( 21)- [START_REF] Matsuyama | Sunspot Equilibria (Rational Bubbles) in a Model of Money-in-the-Utility-Function[END_REF] which can be written as follows

with

In the following we will also consider a second formulation for equation (43)

Assuming that with both timings the matrix on the left-hand-side is nonsingular, the following lemma provides a formulation for the characteristic polynomial:

with the CIA timing, or

with the CAM timing.

Proof : Considering that ξ cm β(c * /m * ) = ξ mc (µ -β) and ξ cz µ(c * /z * ) = ξ zc (µ -β), tedious but straightforward computations give the result.

In the sequel, the local stability analysis will be derived considering the characteristic polynomial evaluated at λ = 1, λ = 0 and λ = -1. We easily compute from Lemma 1:

with the CIA timing, and


M. D'Ottavio O Polit

M Ottavio

P Vidal

Plate and shell finite element for piezoelectric patch modelling

Keywords: Composite structures, finite element, high-order model, Zig-Zag function, piezoelectric patch analysis

 for the mechanical part.

.

This FE is evaluated on some standard piezo-electric plate/shell tests including sensor and actuator configurations. Tests concerning bimorph piezoelectric beam/plate/shell are presented in order to assess the high-order kinematics and the ZigZag effect. The role of electrode segmentation, i.e. the size of equipotential surfaces, on the electro-mechanical response has been also considered.

INTRODUCTION

Research and development concerning high-performance structures are very intense since some decades. Structural health monitoring, active vibration damping, and energy harvesting are some examples of possible applications of a multifunctional structural component. Piezoelectric materials permit to convert mechanical and electrical energy at frequency ranges that are most interesting for technical applications such as vibration damping and rapid shape adaptation [START_REF] Chopra | Review of state of art of smart structures and integrated systems[END_REF]. Development of theoretical and numerical models for this kind of structures is very important and active. For this purpose and in the framework of two-dimensional plate/shell models, different choices can be made for the mechanical approximation and the following classification is classically admitted for the variation in the thickness direction: (i) Equivalent Single Layer (ESL) models, in which the number of unknowns is independent of the layer number; (ii) Layer-Wise (LW) descriptions, for which the number of unknowns and, thus, the computational cost increases with the number of layers. While most developments employ an ESL description for the mechanical behavior, and particularly the First order Shear Deformation Theory (FSDT), a Layer-Wise description is necessary for the piezoelectric approximation to impose electric boundary conditions at each piezoelectric layer interfaces, i.e., the electrodes, within the stack. Inside each piezoelectric layer, the electric potential can be linear, quadratic or higher and a comparison has been proposed in [START_REF] Polit | Electric potential approximations for an eight node plate finite element[END_REF].

A review of different approaches is available in [START_REF] Gopinathan | A review and critique of theories for piezoelectric laminates[END_REF][START_REF] Gibson | A review of recent research on mechanics of multifunctional composite materials and structures[END_REF] and in the framework of the Carrera Unified Formulation (CUF) in [START_REF] Carrera | Hierarchic multilayered plate elements for coupled multifield problems of piezo-electric adaptive structures: formulation and numerical assessment[END_REF]. For the FE approximations, a recent review limited to shell models is also given in [START_REF] Vidal | An efficient shell finite element for the static response of piezoelectric laminates[END_REF].

The limitation of the FSDT model is related to the constant transverse displacement hypothesis, inducing no thickness change and the use of the reduced 2D constitutive law. The use of the full 3D constitutive law is an important feature for a consistent representation of complex physical interactions like multi-field coupling. Furthermore, accurate modeling of thick structures needs the transverse normal stress and the 3D constitutive law.

Therefore, a high-order model is chosen with sinus function for the in-plane displacements and quadratic assumption along the thickness of the transverse deflection. Thus, the 3D constitutive law is retained and a parabolic distribution of the transverse shear strains and a non linear variation of the transverse normal strain are recovered. In order to introduce transverse strain discontinuities required to fulfill the interlaminar equilibrium, Murakami's Zig-Zag function (MZZF) [START_REF] Murakami | Laminated composite plate theory with improved in-plane response[END_REF] is superimposed to the high-order ESL kinematics for the 3 displacement components. Note that MZZF does not depend on the constitutive coefficients and is, hence, attractively simple in conjunction with three-dimensional constitutive laws including multi-field coupling. Based on this kinematics, an 8-node plate Finite Element (FE) is proposed, free of numerical illness such as transverse shear and Poisson lockings, oscillation and spurious mechanics [START_REF] Polit | Robust c 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mecanical analysis[END_REF]. The approximation of the electric potential must be able to model piezoelectric patches, and a constant value is considered on each elementary domain while a cubic variation in each layer is used, based on the polynomial expansion given in [START_REF] Jiang | A new finite element model for piezothermoelastic composite beam[END_REF][START_REF] Beheshti-Aval | A refined sinus finite element model for the analysis of piezoelectric-laminated beams[END_REF].

The paper is organized as follows: Section 2 describes the plate problem, the approximations for the displacement and the electric potential and the system to be solved. The resulting FE are evaluated in Section 3 for a composite plate, in order to talk about the electrode segmentation modelling.

2 Description of the plate problem

Governing equations

Let us consider a plate occupying the domain

V = Ω × [-e 2 ≤ z ≤ e 2 ]
in a Cartesian coordinate system (x 1 , x 2 , x 3 = z). The plate is defined by an arbitrary surface Ω in the (x 1 , x 2 ) plane, located at the midplane for z = 0, and by a constant thickness e.

The displacement is denoted ⃗ u(x 1 , x 2 , z) and the electric potential is φ(x 1 , x 2 , z). ε i j (x 1 , x 2 , z) and ⃗ E(x 1 , x 2 , z) are the strain tensor components and the electric field vector, respectively, deduced from primal variables by the geometric relations. Furthermore, σ i j (x 1 , x 2 , z) and ⃗ D(x 1 , x 2 , z) are the conjugated fluxes (stress tensor components and dielectric displacement vector, respectively) obtained from the constitutive equations given in the next subsection.

Constitutive relation

The 3D constitutive equation for a linear piezoelectric material is given by the following set of coupled equations [START_REF]IEEE Standard on Piezoelectricity[END_REF] for a layer (k):

[σ (k) ] = [C (k) ] [ε (k) ] -[e (k) ] T [E (k) ] (1a) [D (k) ] = [e (k) ] [ε (k) ] + [ε (k) ] [E (k) ] (1b)
where we denote by [C] the matrix of elastic stiffness coefficients taken at constant electric field, by [e] the matrix of piezoelectric stress coefficients and by [ε] the matrix of electric permittivity coefficients taken O. Polit, M. D'Ottavio and P. Vidal at constant strain. The explicit form of these matrices can be found in [START_REF] Vidal | An efficient shell finite element for the static response of piezoelectric laminates[END_REF] for an orthotropic piezoelectric layer polarized along the thickness direction z. Eq. (1a) expresses the piezoelectric converse effect for actuator applications, whereas Eq. (1b) represents the piezoelectric direct effect which is exploited in sensor applications. Note that the constitutive law is expressed in the local reference frame associated to each layer.

The weak form of the boundary value problem

The classical piezoelectric variational formulation of [START_REF] Tiersen | Linear Piezoelectric Plate Vibrations[END_REF] is employed in which the primary field variables are the "generalized displacements", i.e. the displacement field, and the electrostatic potential. Using a matrix notation and for admissible virtual displacements ⃗ u * and electric potential φ * (virtual quantities are denoted by an asterisk), the variational principle is given by:

V ρ[u * ] T [ ü ] dV = - V [ε(u * )] T [σ(u , φ)] dV + V [u * ] T [ f ] dV + ∂V F [u * ] T [F] d∂V + V [E(φ * )] T [D(u , φ)] dV - V q φ * dV - ∂V Q Q φ * d∂V (2)
where [ f ] is the body force vector, [F] the surface force vector applied on ∂V F , q the volume charge density, Q the surface charge density supplied on ∂V Q and ρ is the mass density. Finally, ε(u * ) and E(φ * ) are the virtual strain and virtual electric field that satisfy the compatibility gradient equations. In the remainder of this article we will refer only to static problems, for which the left-hand side term is set to zero. Furthermore, body forces and volume charge densities will be discarded ([ f ] = [0]; q = 0).

The mechanical part 2.2.1 The displacement field

Based on the sinus model, see [START_REF] Touratier | An efficient standard plate theory[END_REF], a new plate model which takes into account the transverse normal stress is presented in this section. This extension is based on following developments

• various models for beams, plates and shells based on the refined sinus theory, see [START_REF] Touratier | An efficient standard plate theory[END_REF][START_REF] Ganapathi | Flexural loss factors of sandwich and laminated composite beams using linear and nonlinear dynamic analysis[END_REF][START_REF] Polit | A multilayered/sandwich triangular finite element applied to linear and nonlinear analyses[END_REF][START_REF] Dau | C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures[END_REF][START_REF] Vidal | A thermomechanical finite element for the analysis of rectangular laminated beams[END_REF][START_REF] Vidal | A family of sinus finite elements for the analysis of rectangular laminated beams[END_REF];

• our previous paper on a 7 parameter model for thermo-mechanical analysis [START_REF] Polit | Robust c 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mecanical analysis[END_REF].

In the framework of ESL approach, the kinematics of our model is assumed to have the following particular form

⎧ ⎨ ⎩ U 1 (x α , z) = u 0 1 (x α ) + z u 1 1 (x α ) + f (z) u f 1 (x α ) U 2 (x α , z) = u 0 2 (x α ) + z u 1 2 (x α ) + f (z) u f 2 (x α ) U 3 (x α , z) = u 0 3 (x α ) + z u 1 3 (x α ) + z 2 u 2 3 (x α ) (3)
where α ∈ {1, 2} and i ∈ {1, 2, 3}. In Eq. ( 3), the superscript is associated to the expansion order in z while the subscript is related to the component of the displacement. Thus, u 0 i are the displacements of a point of the reference surface while (u 1 α , u f α ) are measures for rotations of the normal transverse fiber about the axis (0, x α ). The functions u α 3 permit to have a non-constant deflection for the transverse fiber and allow to have non zero transverse normal stretch. Furthermore, the quadratic assumption for the transverse displacement avoids the occurrence of Poisson (or thickness) locking, see [START_REF] Polit | Robust c 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mecanical analysis[END_REF].

In the context of the sinus model, we have

f (z) = e π sin πz e (4)
It must be noticed that the classical homogeneous sinus model [START_REF] Touratier | An efficient standard plate theory[END_REF] can be recovered from Eq. ( 3) assuming u 1 α = -u 0 3,α , and neglecting the unknown functions u α 3 .

The Murakami's Zig-Zag terms

In order to evaluate the influence of Zig-Zag terms [START_REF] Murakami | Laminated composite plate theory with improved in-plane response[END_REF] in a high-order ESL model, the following displacement per layer (k) are added to Eq. ( 3):

⎧ ⎨ ⎩ U 1 (k) (x α , z) = Z (k) (z) u z 1 (x α ) U 2 (k) (x α , z) = Z (k) (z) u z 2 (x α ) U 3 (k) (x α , z) = Z (k) (z) u z 3 (x α ) (5)
with

Z (k) (z) = (-1) k ζ k (z) and ζ k (z) = 2 e k z - 1 2 (z k + z k+1 ) (6)
where e k is the thickness of the k th layer while (z k , z k+1 ) are the bottom and top coordinates of this layer.

It is obvious that Z (k) (z) is a piecewise linear function with bi-unit amplitude for all the layers as we have

ζ k (z) ∈ [-1, 1]
. Note that, despite the Z (k) (z) function depends on the layer index inside the stack, the amplitudes u z i are unique for the whole laminate, i.e., the ESL framework is still preserved.

The electric part

On each elementary domain Ω e , the electric potential is assumed to be constant. Therefore, no variation with respect to x α is considered. A cubic layerwise (LW) description is used across the thickness, according to the approximation introduced in [START_REF] Jiang | A new finite element model for piezothermoelastic composite beam[END_REF] and used in [START_REF] Beheshti-Aval | A refined sinus finite element model for the analysis of piezoelectric-laminated beams[END_REF]: In each layer (k), the electric potential distribution is described by the normal electric field components at the bottom and top surfaces, denoted E 3b and E 3t , respectively, and the potential difference ∆φ between top and bottom surface. Using these dofs, the approximation of the electric potential can be written as:

φ (k) = F φ Cst (k) φ q e (k) φ (7)
where the following definitions have been introduced:

F φ = 1 z z 2 z 3 ; q e (k) φ T = E 3b (k) ∆φ (k) E 3t (k) (8)
and Cst φ (k) is a (4 × 3) matrix containing constant coefficients. The electric field vector in each layer

E (k)
is then obtained as:

E (k) = ⎡ ⎢ ⎣ 0 0 -φ (k) ,3 ⎤ ⎥ ⎦ = [F E ] Cst E (k) q φ e(k) with [F E ] = 1 z z 2 (9)
where Cst E (k) is a (3 × 3) matrix. So, the adopted approximation yields a quadratic transverse electric field across the thickness of each layer.

2. [START_REF] Polit | High-order plate finite elements for smart structure analysis[END_REF] The system to be solved

The FE approximations are not detailed here for the sake of brevity and can be found elsewhere [START_REF] Vidal | An efficient shell finite element for the static response of piezoelectric laminates[END_REF][START_REF] Polit | Robust c 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mecanical analysis[END_REF]. The eight-node quadrilateral finite element is used and classical FE approximation is used for the geometry. A special treatment is used to control the transverse shear locking by using a dedicated interpolation for γ α3 0 according to the methodology presented in [START_REF] Polit | Robust c 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mecanical analysis[END_REF]. Note that since the electric potential is assumed to be constant on each elementary domain, there is no need to introduce any FE approximation for this field.

The elementary matrices are then deduced considering the bi-dimensional weak form obtained from Eq. ( 2). Assembling each elementary contribution in the global reference frame, the following discrete form of the coupled piezoelectric system is obtained:

[K uu ] [K uφ ] [K uφ ] T [K φφ ] [q u ] [q φ ] = [L u ] [L φ ] (10)
where [K uu ], [K φφ ] and [K uφ ] are the global stiffness, dielectric and piezoelectric matrices of the plate, respectively. The mechanical dofs are in the vector [q u ], while the electrical dofs are in the vector [q φ ] and we have at the elementary level

[q u e ] = u 0 1 u 0 2 u 0 3 u 1 1 u 1 2 u 1 3 u f 1 u f 2 u 2 3 u z 1 u z 2 u z 3 i=1,8 q φ e = E 3b (k) ∆φ (k) E 3t (k) k=1,Nl (11)
with Nl the number of layers. From the mechanical point of view, the Zig-Zag dofs can be activated or not and comparisons will be presented in the next section dedicated to numerical evaluations using no Zig-Zag dof (P9), using only in-plane Zig-Zag dof u z α (P9Z) and with all the Zig-Zag dof u z i (P9ZZ). Finally, the three models use 9, 11 or 12 kinematical unknown functions and the associated FE has 72, 88, 96 mechanical dofs per element, respectively. Since the electric potential is assumed constant on each FE, the assembly involves only the approximation along the thickness. A piezoelectric patch comprising several FEs can be defined through the imposition of the equipotential condition between electrodes: the same ∆φ is imposed on the piezoelectric layer for all elements belonging to the same patch. From the numerical point of view, this is accomplished through linear homogeneous and non-homogeneous Multi-Point Constraints (MPC) using penalty function method.

In Eq. ( 10), the load vectors [L u ] and [L φ ] represent the external loading from applied forces and prescribed charges, respectively. Essential boundary conditions (i.e., prescribed displacements and electric potentials) are imposed numerically by a penalty technique. The coupled system is then solved by the classical static condensation procedure for the electrical dof:

[q φ ] = [K φφ ] -1 ([L φ ] -[K uφ ] T [q u ]) (12a)
which yields the following purely mechanical system with a modified equivalent stiffness matrix:

[K uu ] -[K uφ ][K φφ ] -1 [K uφ ] T [q u ] = [L u ] -[K uφ ][K φφ ] -1 [L φ ] (12b)
In the remainder of the paper, [L φ ] = [0] will be considered.

O. Polit, M. D'Ottavio and P. Vidal

Numerical results: clamped composite plate

This test is interesting to evaluate the segmentation of a piezoelectric layer using different number of patches. It considers the sensory response of a cantilever, rectangular, hybrid sandwich plate with one edge clamped and other edges free subjected to a uniform pressure load according to the following data [START_REF] Kapuria | Static electromechanical response of smart composite/sandwich plates using an efficient finite element with physical and electric nodes[END_REF]: This test can be used to compare the effect of the electrode segmentation: using the proposed FE meshes, the electrode sensor surface is subdivided in 1, 10, 40, 160 patches through the imposition of equipotential conditions.

geometry
u 3 = u 3 10 2 E 2 (core) / e S 4 p 0 φ = φ 10 4 E 2 (core) d 0 / e S
Tab. 2 presents the convergence properties of the P9 FE when only one patch is used for each piezoelectric layer. In the open circuit configuration, the equipotential condition enforces ∆φ in each layer to be the same for all elements. In the closed circuit condition, ∆φ = 0 for all elements (equipotential condition) and the maximum electric potential value inside the piezoelectric layers is reported. Note that the cubic approximation for φ allows to recover the electric potential induced by the local bending of the piezoelectric layers. The maximum displacement is located at the center of the tip edge (a, b/2) while the maximum induced electric potential (φ max for closed circuit) is in the FE close to the clamped edge. For the open circuit, the convergence is very fast for both displacement and electric potential difference -the percentage of error with respect to the 20 × 8 mesh results is indicated in parentheses. For the closed circuit, an asymptotic value is recovered for the displacement whereas φ max still increases with mesh refinement; this is due to the decreasing size of the FE at the clamp, which increasingly localizes the bending deformation of the piezoelectric layers.

The distribution of the electric potential across the thickness is illustrated for the two electric boundary conditions in Fig. 1. The non-linear variation across the bottom and top piezoelectric layers is visible on both graphs and increases with mesh refinement. Due to the equipotential condition, it has no influence on the ∆Φ max for the open circuit (left) but it is directly related to the Φ max for the closed circuit (right). This non-linear effect measures the local bending deformation of the piezoelectric layers and is accordingly larger at the clamped edge than at the free tip of the plate.

Finally, the effect of electrode segmentation is discussed in Tab. [START_REF] Murakami | Laminated composite plate theory with improved in-plane response[END_REF] for the open circuit boundary condition. The corresponding distribution along the length of the plate of the electric potential difference is illustrated in Fig. 2 for all considered meshes and electrode segmentations. The induced potential difference ∆φ reported in Tab. 3 is taken at the electrode next to the clamped edge, i.e., where it has its maximum value. Comparing the results for 10 patches obtained with two different meshes allows to appreciate the role of the mesh refinement: a finer mesh increases the tip displacement and enhances the sensed voltage. The same mesh with 10 × 4 elements is then used with two different number of electric patches, i.e., with different sizes of the equipotential surfaces: a very small influence on the tip deflection can be seen, with u 3 slightly larger when the equipotential surfaces are larger; on the contrary, the electric potential induced at the clamped edge is clearly higher the smaller the electrode. Finally, considering the 20 × 8 with 160 patches (i.e., one electrode per FE), the tip displacement is slightly higher due to the refined mesh and the sensed voltage is further increased due to both, the larger deflection and the reduced electrode's size at the clamp.

Conclusion

This paper has presented a new family of FE for piezoelectric composite plates. A high-order ESL kinematic model is considered that includes the sinus function for the in-plane displacements, a quadratic polynomial expansion for the transverse displacement and Zig-Zag functions for introducing slope discontinuities at layers' interfaces for both, the in-plane and the transverse displacements. A LW cubic approximation is used for the electric potential in order to capture its non-linear distribution induced by the local bending of the piezoelectric layer; the elementary domain is considered as an equipotential surface (constant electric potential).

  rectangular plate a× b with a = 10 mm, a/b = 2 and total thickness e = 1 mm (length-tothickness ratio S= 10) materials seven-layers plate (pz, 0 • , 90 • , core, 90 • , 0 • , pz); the outer piezoelectric layers of thickness 0.1 e are made out of PZT-5A material; each layer constituting the laminated skins has a thickness 0.04 e and the core thickness equals 0.64 e. The material properties for PZT-5A, skin layers and core are given in Tab. 1 boundary conditions clamped at x 1 = 0; uniform pressure load p 0 ; piezoelectric layers can be in open circuit (∆φ let free) or in closed circuit (∆φ = 0) mesh three regular meshes 5 × 2, 10 × 4 and 20 × 8 are used results and locations transverse displacement at the tip and piezoelectric potential in the element close to the clamped edge. They are made non-dimensional according to

O

  . Polit, M. D'Ottavio and P. Vidal

Figure 1 :

 1 Figure 1: Clamped composite plate: electric potential distribution across the thickness for the OC (left) and CC (right) electric boundary conditions.

Figure 2 :

 2 Figure 2: Clamped composite plate in OC configuration: distribution along the length of the sensed voltage at the top (left) and bottom (right) piezoelectric layer for different meshes and electrode segmentation.

Table 1 :

 1 2 p 0 with d 0 = 374 10 -12 CN -1 Material properties employed in the considered problems.

	Prop.	PZT-5A skin	core	Prop.	PZT-5A skin core
	E 1 [GPa] E 2 [GPa] E 3 [GPa] ν 23 ν 13 ν 12 G 23 [GPa] G 13 [GPa]	61.0 61.0 53.2 0.38 0.38 0.35 21.1 21.1	172.5 0.276 e 15 [C/m 2 ] 6.9 0.276 e 24 [C/m 2 ] 6.9 3.45 e 31 [C/m 2 ] 0.25 0.02 e 32 [C/m 2 ] 0.25 0.02 e 33 [C/m 2 ] 0.25 0.25 ε 11 [nF] 1.38 .414 ε 22 [nF] 3.45 .414 ε 33 [nF]	12.3 12.3 -7.2 -7.2 15.1 15.3 15.3 15.	0 0 0 0 0 ε 0 ε 0 ε 0	0 0 0 0 0 ε 0 ε 0 ε 0
	G 12 [GPa]	22.6	3.45 0.1104

Table 2 :

 2 Clamped composite plate with only one electric patch: convergence results for open circuit (OC) and closed circuit (CC).

			O. Polit, M. D'Ottavio and P. Vidal
	elec BC mesh OC u 3	5 × 2 22.649 (-1.9) 22.967 (-0.5) 23.089 10 × 4 20 × 8
	CC	∆φ bottom -23.003 (-2.7) -23.408 (-1.0) -23.649 24.473 (-2.6) 24.886 (-1.0) 25.129 ∆φ top 23.370 (-1.9) 23.698 (-0.3) 23.833 u 3 φ Max -3.307 -5.212 -7.295

Table 3 :

 3 Clamped composite plate in OC configuration: evaluations for different FE meshes and equipotential surface sizes.

	100	(Top layer)			mesh 5x2; 10 electrd mesh 20x8; 1 electrd	0		(Bottom layer)
	80				mesh 10x4; 10 electrd	20
					mesh 10x4; 40 electrd
					mesh 20x8; 160 electrd
	60					40
	40					60
											mesh 20x8; 1 electrd
											mesh 5x2; 10 electrd
	20					80					mesh 10x4; 10 electrd
											mesh 10x4; 40 electrd
											mesh 20x8; 160 electrd
	0					100
	0	2	4	6	8	10	0	2	4	6	8	10
			x1						x1

The element has been validated through linear static case studies for both sensor and actuator configurations as well as homogeneous and laminated plates. The role of electrode segmentation, i.e., the size of equipotential surfaces, on the electro-mechanical response has been also evaluated. The results are in good agreement for actuator and sensor configurations for thin to very thick cases. The proposed P9ZZ FE, using only 12 mechanical dof per node, is very accurate, simple to use, without any numerical problem and could be used for a large range of plate problems involving piezoelectric patches or layers. Future works are pointed towards the extension of this model to piezoelectric shell structures.