Zaka Rat

S Im

A L Ahelo

ON APPROXIMATE SYSTEM DYNAMIC1

Keywords: time series analysis; state space models, Hankel norm approximations, balanced realization, positive real lemma, unweighted principal components

In this paper concepts and techniques from system theory are used to obtain state-space (Markovian ) models of dynamic economic processes instead of the usual VARMA models. In this respect the concept of state is reviewed as are Hankel norm approximations,and balanced realizations for stochastic models. We clarify some aspects of the balancing method for state space modelling of observed time series. This method may fail to satisfy the so-called positive real condition for stochastic processes. We us a state variance factorization algorithm which does not require us to solve the algebraic Riccati equation. We relate the Aoki-Havenner method to the Arun -Kung method.

Introduction.

Linear system approximation has found important applications in model reduction, system identification, spectrum estimation, among others. We will consider the problem of constructing a state-space model (Markovian representation) for a stochastic process from a finite number of estimated covariance lags. The approach is to first obtain a high-order model which exactly matches the estimated covariance sequence, and then use balanced model reduction techniques to obtain a lower order model which approximates the given sequence. The balanced models can then be obtained from a realization algorithm.

The stochastic realization problem deals with the quest for a finite dimensional Markovian representation for a stochastic process from the known covariance information. If the covariance of the intervening random variables are axactly known, then we deal with the exact stochastic realization problem, which has received great attention [START_REF] Akaike | Canonical Correlation Analysis of Time Series and Use of an Information Criterion[END_REF], [START_REF] Faurre | Stochastic realization algorithm[END_REF]. For many applications the Markovian representation or state space model may be too complex owing to its high dimensionality, thus barring efficient computational management. This partially motivates the search for smaller dimensional Markovian realization which approximates the original one in some sense. The high dimensionality of the original state space model can, for instance, be caused by the incorporation of weakly coupled superfluous state components. These components may mask any tendencies hidden in the dynamics.

Another difficulty with the stochastic realization problem is the necessity of the exact covariance information. In most pratical situations, all one has available is an estimate of the covariances based on the real data (i.e., sample covariances). Not only would the noise fluctuations in the covariance structure lead to models of high dimensions, but what is more essential, the sample covariance sequence may not be positive-real. In such a case, the exact realization algorithm applied to inexact data may have no solution at all. In the second section we present the different representations of a times series model, a link with the power density spectral is used. Next we recall, in the third section, the stochastic realization problem as background. Finally in the fourth section we clarify some problematic aspects of the method proposed by [START_REF] Aoki | State space modeling of multiple time series[END_REF], we propose a factorization approach which is not used to solve the Riccati equation, and we relate the Aoki-Havenner algorithm to the unweighted principal components algorithm.

Different representations of a model 2.1. State-space representation.

Let yt be a weakly stationary multivariate stochastic process which has a zero mean. A state space time series model can be written in two matrix equations, the state equation and the observation equation: xt+i = Ax, + u, (1.1) y, = Cxt + v, (1.2) where x e Rn, ye Rm, A and C are matrices of appropriate dimensions, v, and u, are zero mean white noise with variance-covariance matrix (2)

( n \ v v . y \ VVs / Q s S R <5ts; Vt,
For large times the Kalman gain Bt and i2t reaches an asymptotic steady-state B = lim,^_ B, and

Q = l i m ^ Q,
In what follows, we consider the stationary case, thus the variance-covariance matrix associated with the system is:

* ^Be,^f B e0 f B" f Bl E = Q. . < e . y, e . J (3)
The innovation model is of considerable practical value because of its statistical properties. It can be seen that the innovation model is determined by the same dynamic parameters (A, C) as the initial model. However, the triple (Q, R, S) of the initial model is statistically equivalent to the double (B, Q.) of the innovation model (see [START_REF] Ratsimalahelo | Théorie de système et Séries Temporelles[END_REF]).

ARMA representation.

It is known that there is a one-to-one correspondence between stationary ARMA and state space models, any system (1) and ( 2) can be expressed in ARMA and conversely, cf [START_REF] Hannan | The statistical theory of linear systems[END_REF], [START_REF] Aoki | State space modeling of multiple time series[END_REF]. Let a general vector ARMA(p,q) a(z) y, = b(z) e, (4) where a(z) = LA, z'1 and b(z) = Z Bj z'1 are matrix polynomials in the lag operator z '1 of degrees p and q respectively, y,is vector of data, t = 1,...,T and et is a random error vector such that E(et) = 0 , E(£t£s ) = ES.S, I > 0. In the stationary case, the e, are the innovations if and only if det a(z) ^ 0 I z I < 1 (stability condition) and det b(z) * 0 I z I < 1 (miniphase condition). Then the ARMA model may be rewritten y, = g(z) e, with g(z) = a(z)'1 b(z) is a matrix of rational function, g(z) is said to be the transfer function from et to yt. If e, is the innovation sequence for y, again g(z) is analytic for I z I < 1, (g(z) is analytic within the unit disc) det g(z) * 0 I z I < 1. If we denote the covariance of process y, by Ns = E(ytyt+S ) then the power spectrum is given by the doubly infinite summation S ( z ) = X N ,zevaluated at z = exp(iw). Using the definition of the impulse response, y, = X g j £ H j=0 g(z) = ZgjZ'J, go = I. It can be easily seen that the covariance Ns is obtained by N , = Ë g , I g " S 2 0 j=0 N.s = Ns In z-transform language, this translates to S(z) = g(z)Sg(z') it is known spectral factorization, cf [START_REF] Hannan | The statistical theory of linear systems[END_REF]. Matrix T(z) = C(zl -A) 'B + I is termed the system transfer matrix. Its poles are the eigenvalues of A as det (zl -A) = 0 is a pole . It is a rational fraction matrix wherein the degree of denominator of each item of the matrix exceeds the degree of the corresponding numerator.

The covariance function, the transfer function and state

Spectral factorization.

The stochastic realization problem is closely connected with the spectral factorization Lemma 1. Given a system of the form (2), the spectral density function S(z) of y, is defined It may be asked what type of information is contained in the spectral density function S(z).The spectrum S(z) provides an external representation of the process yt, according to the matrix product expression, the spectrum S(z) is determined by the same set of parameters (A, B, C, Q) as the innovation model. S(z) is interpreted as an expression of the form S(z) = J(z) + J'(z ') (10) where J(z) = No/2 + C(zl -A) 'M is the transfer matrix of a linear dynamic system, J(z) is positive real. A link is then established with the problem of realisation of a transfer function.

Notice that, given a spectral density function of the form (6), if the triple (A, C, M) is a minimum system then the parameters A and C can be obtained from equation ( 6).

To have a minimum system we need the necessary and sufficient conditions provided by the following lemma.

Lemma 2. The stochastic dynamic system (2) admits a spectrum-minimal if and only if (A C) is observable and there are no solutions x * 0 to the following set of equations:

A'x = zx (C(zl -A)_IB + I) £2B'x = 0
The proof of this lemma is based on the relations (5). The second condition is closely related to the zeros of the transfer function T(z) = I + C(zl -A) 'B. The following corollary can thus be established.

Corollary 3. The stochastic linear dynamic system of the form (2) admits a minimum spectrum when (A, C, B£2) is minimum and if o(A) n {z '1 / z * 0, z is a zero of T(z)} = 0 . (a(A) denote the set of eigenvalues of the matrix A) Lemma 4. Given a system of the form (1), the spectral density function S(z) of y, is the form

S(z) = N0 + C(zl -A)'lM + M '(z"'l -A 'X'C'
where P, M and N0 are uniquely determined by P = APA' + Q M = APC' + S N0 = CPC' + R. The spectral density function S(z) may be expressed as

S(z) = [C(zl -A )'1 l £ Q s S' R ( z ' 'l -A )~'C I
The links between the different forms of representation for stochastic linear systems can be summarized in the following figure. In a stochastic process yt representations in ARMA form, in the form of (spectrum) transfer functions, or using the autocorrelation function Ns, are termed external descriptions while a state representation is termed an internal description.

Stochastic realization problem.

The realization problem consists in determining an internal representation from an external representation of a system such that the state model obtained has the same inputoutput behaviour as that specified by the external representation. Similarly for a stochastic system, realization consists in determining the quadruple (A, C, B, Q) or the quintuple (A, C, Q, R, S) from:

-the spectrum S(z) = N0 + C(zl -A) 'M + M'iz 'l -A') 'C ' -the autocorrelation function Ns = CAs lM s > 0.

Remarks.

i) The realization problem is also known as the spectrum factorization problem, Lindquist and Picci (1979), Caines (1987) i.e. it seeks a transfer function G(z) such that S(z) = G(z)G'(z ').

ii) Notice the similarity between the following problems:

-deterministic case, determining a triple (A, B, C) from Markov parameters U = CAs lB s > 0, -stochastic case, determining a triple (A, M, C) from the autocorrelation function Ns = CAs lM s > 0. An essential difference is that every triple (A, B, C) describes a corresponding input-output system, but that not every (A, M, C) describes a corresponding stochastic process. This is expressed by the well-known positive real lemma.

Given the covariance sequence, one forms the (infinite) Hankel matrix which is factorizable into an observability matrix O and a controllability matrix K The time sequence is rational if and only if this Hankel matrix has finite rank (say n). It follows then from the deterministic realization theory [START_REF] Faurre | Stochastic realization algorithm[END_REF] that the order of any minimal Markovian representation of y, is precisely n, and a triple (A, M, C) can be constructed such that N, N 2 N 3.

n 2 n 3 n 4 N 3. N 4 N 5 C CA CA2 AM A 2M . .] (12) H = O K Ns = CAS''M s > 0,
where in order to have a Markovian representation, (internal representation) the following needs to be satisfied P -A P A ' M -A P C N 0 -CPC > 0, P > 0 M'-C PA 1 this is known as the positive real lemma. Using the innovation model ( 2) and from eqs (5.1)-( 5.3) we have (B' I)' Q (B' I) > 0, P > 0 This positivity condition is equivalent to the spectral factorization theorem, (see section 2.4) cf [START_REF] Faurre | Opérateurs rationnels positifs[END_REF] Furthore this positive real condition is neglected in the balancing method as it does not arise for deterministic systems. The triple (A, M, C) together with No do not uniquely specify the covariances P, Q and matrix B. However, P completely specifies Q and B, and therefore characterizes the Markovian representation. Furthermore, note that any minimal realization of the covariance sequence is unique, modulo a similarity transformation. This classical algorithm gives no indication of how a good reduced-order model can be obtained if an approximate stochastic realization is desired.

The notion of state.

We first need to define some more notation, denote by

y; = 1 1 i ------------- i y,+. y.-i II (13)
the future and the past, respectively, of y,.

Let S (y,+) and S (yt ) be the Hilbert spaces obtained by taking the closed linear span of the random variables in yt+ and y,\ respectively. (In general, S (.) will be the Hilbert space generated by the closed linear span of elements of « . ».) The inner product on this space is the cross covariance. Orthogonal projection in these Hilbert spaces is equivalent to conditional expectation i.e. x \y = E(x \y) denotes the orthogonal projection of x onto S(y). This projection is given explicitly by

x \ y = E(xy) {E(yy')}'*x ( 14
)
where x and y are zero-mean random vectors.

Intuitively, the state of a linear system is a summary of the information in the past input history that is both necessary and sufficient to predict the future output. In the stochastic case, for the innovations representation, the state can also be interpreted as a as a summary of the past output history (instead of past input history) with regards to the prediction of the future output.

Construct a system of the following form xl+i = (A -BC) Xt + By, (15) e, = -Cx, + y, this is a formal inverse of the innovations representation and has all zeros within the unit circle i.e the eigenvalue of (A -BC) lie within the unit circle. Thus the state process x, as well as the input e, can be obtained causally from the output y, using the above filter. Let A» = A -BC is in linear system terminology the matrix of loop filter. The matrix A. is the dynamic matrix in the Kalman filter for (15).

The state transition equation

x, = [B A.B A .2 B A .3 B •• ] y. y.-i y«-2 (16)
x, = Sy,' The projection yt+ \ y,' defines the state space of a Markovian model of the form (2). Thus this projection is central to any stochastic realization algorithm. From ( 14), it can be seen that this projection is given in terms of the following matrices formed from the covariance sequence:

yt+\ y t-= H R -y (22)
where H = E[y,+(y,')'] and R = E[y,'(y,')'] are the Hankel and Toeplitz matrices formed from the covariance lags of the output. H is given from eq( 12) and R is given by 'N 0 N, n 3 . :

N, N0 N, . . R = E[yt-(ytT ] = n 2 N, N 0 • • (22)
Since x,= S y ' , in this case the state covariance matrix is P = E(x,x,') = SRS' A comparison eq (20) with y,+ \ y' = HR'ly indicates that H R 1 must be factorizable into O and S , and consequently it must have rank equal to the size of the state vector. Thus we have x, = Sy,' and y,+ \ y,' = H R y' = Ox,

The Hankel matrix can be factorised as H = OK, then the state vector as x, = Sy', with S = KR'1. Thus S is said to be a matrix aggregation. The state is considered by [START_REF] Aoki | State space modeling of multiple time series[END_REF] as sufficient statistics.

In this case the state covariance matrix of the innovations model is

P = SRS' = K(R'')'K' ( 24
)
The state is in fact the information interface between past and future and its dimension is equal to the order of the system. When the system has to be approximated by a lower order model, it is a question of compressing this information-interface into a lower dimensional which contains most of the information in the full-order state.

Stochastic balanced realization.

Balanced realizations originate in deterministic control theory, [START_REF] Moore | Principal component analysis in linear systems: controllability, observability, and model reduction[END_REF], this concept is well known in the literature. In words one may say that a balanced realization of a system has the property that the amount of controllability of a certain component of the state vector is equal to the amount of observability of this component. As shown by for instance [START_REF] Glover | All optimal Hankel norm approximations of linear multivariable systems and their L_ -error bound[END_REF] we can more or less consider the Gramians of a system as a tool to measure the controllability and observability of a realization.

Aoki-Havenner's Method.

For convenience we summarise [START_REF] Aoki | State space modeling of multiple time series[END_REF] method. We wish to predict future values of data vector y, from the past values. Let the upper bound on number of lags needed to model y, be p and the desired forecast horizon be f. Define the (mf x 1) and (mp x 1) future and past data vectors as y,+' = (y ,\ yt+, \ ..., y,+f.i') and y,'' = (y ,\ yt.i\...,y t.p+i) and let Ns = E(yt+Sy,'), s = 0, 1,2 ...the autocovariance sequence (Ns is a m x m matrix)

Step 1. Calculate the sample autocovariances Ns , s = 0, 1 , 2 , p+f Ns = T '1 £ y ,+s y,' where T is the sample size and arrange them in the Hankel matrix

N, N 2 • • N p " n 2 n 3 . • N p+1 1 JZ . N f+1 ■ • N p+f_,_
H has dimensions mf x mp.

The N0 and Np+f matrices do not appear in H but are used in estimating the A and B matrices.

Step 2. Obtain the singular value decomposition of the estimated Hankel matrix (HSV :

Hankel Singular Value) mf H = U D V = £ < j 1ui,v1 k = l
where G\ > <j2 ^ ^ <Tmf are the singular values of the matrix H and Uk, vk the corresponding eigenvectors. The matrices U = (ui,...,umf) and V = (vi,...,v",p) are orthogonal, and D = diag ( O i , . . . , < 7 mf) Choose the importantly nonzero singular values to be included in the approximation. Since the singular values are scale-dependent, the series should be scaled so that the covariances are all of approximately the same magnitude. By the well known Kronecker theorem, cf [START_REF] Kailath | Linear systems[END_REF], [START_REF] Aoki | State space modelling of time series[END_REF][START_REF] Aoki | State space modelling of time series[END_REF], [START_REF] Kung | Optimal Hankel-Norm Model Reduction: Multivariable Systems[END_REF] the rank of H is equal to the number of states (n) required to summarize the past history of the system. If the covariances that comprise H were the actual values rather than estimates , the largest n singular values would be nonzero with the remaining mf -n exactly zero, thus directly providing the order information critical to time-series modeling. Given the Hankel matrix with rank mf the problem is to find a Hankel matrix H with r( H ) = n < mf such that The Hankel norm lies between the conventional L2 and L" norm. These norms provide a formal measure of the degree to which the time-series model reproduces the sample autocovariances.

Step 3. Estimate the parameters A, C, M by least squares based on the singular value decomposition.

Note that the covariance matrix between yt+i+ and yt.i' is H = OK = OAK (24) the first submatrix row of H, denoted by Hi*, is H ,. = [ N , , N 2,...,NP] = C K (25) and the first submatrix column of H, denoted by H*i, is H»i = (N,, N2,...,Nf]' = OM (26) the system matrices A, C and M must be choosen to satisfy ( 24) through ( 26) These equations can be solved easily by using the singular value decomposition of the Hankel matrix. Compare the singular value decomposition Hn = UnDnVn' with the factorization in terms of O and K.(observability-controllability factorization) (UnDn,/2)(Dn1/2Vn') = OK Theoretically, the number of positive singular values is equal to the theoretical rank of the Hankel matrix. Choose a coordinate system in which O = UDI/2 and K = D 1/2V' in this coordinate system we have O 'O = KK' = D where O'O and KK' are, respectively, the observability and controllability gramians. Thus these system matrices are said to be balanced. Putting models in balanced representation or balanced form uniquely determines the system matrices, i.e., the model is uniquely identified. The word « balanced » in the above description comes from the fact that the representation balances two sources of errors: one related to the observability , i.e., error in reconstructing past state vector values from future observations, and the other errors related to reachability, i.e, the error due to past prediction errors. The solutions:

C = H ,*V D 1/2 (27) M = D 1/2U 'H., (28)
A = D ',/2U ' H V D 1/2
(29) Where V D'I/2 and D '1/2U 'are a singular value generalized inverse of O and K respectively so that 0

+ 0 = ( D 1/2U')(U D 1/2) = I K K + = (D 1/2V')(V D '1/2) = I
The balanced model representation uses the system matrices computed in eqs ( 27) -( 29). It has several useful properties. One is that system matrices A, C and M of any lower dimensional balanced model than a given one are obtained by merely taking a leading principal submatrix of the original dynamical matrix A and the corresponding submatrices C and M. Aoki and Havenner call this the nested property or orthogonality property . Another property is that these lower-dimensional models are all asymptotically stable if the original matrix A is asymptotically stable.

Step 4. Determination of the matrices B and Q requires solving a certain matrix Riccati equation. From eqs(5) note the relations P = APA' + BHB' and Q = N0 -CPC' with Q. > 0

From the definition of the matrix M we have B Q = M -APC'. From these we obtain the equation for P as P = APA' + (M -APC'XNo -CPC')'(M -APC')'

(30) The smallest positive definite solution of (30), which exists under certain technical conditions (see [START_REF] Faurre | Opérateurs rationnels positifs[END_REF]), is used to determine B and Q. Eq (30) can be solved by converting it into a (2nx2n

) symplectic matrix O <D = 'F -Q V F~'D Q 'F'1 -y 'D »F_l where V F = A -MN0''C, Q = C '

Hankel norm approximation.

The Hankel norm approximation problem was first shown to be solvable (in discrete time domain) by [START_REF] Adamyan | Analytic properties of Schidt pairs for Hankel operator and the generalized Schur-Takagi problem[END_REF] and the first reliable state space algorithm was developed by [START_REF] Glover | All optimal Hankel norm approximations of linear multivariable systems and their L_ -error bound[END_REF][START_REF] Glover | A tutorial on Hankel-norm Approximation[END_REF] (in the continuous time domain). Many other authors have contribued to this field, among which [START_REF] Kung | Optimal Hankel-Norm Model Reduction: Multivariable Systems[END_REF], [START_REF] Ball | Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case[END_REF].

Step 2 concerns the approximation of a high dimensional system by one containing fewer states, in such a way that norm of the difference between the corresponding Hankel operators is minimal. This involves the approximation of estimated Hankel matrix by Hankel matrix of smaller rank n. This « numerical rank » is determined by the number of « dominant » singular values of H. In this case the SVD can be pardoned as folllows:

H = [,J U( o B p .

where D contains the n dominant singular values and the singular vectors are partioned accordingly. Then the optimal (in Frobenius norm) rank n approximation to H is Hn = Un D" Vn' = OK Hn is not a Hankel matrix in general. Thus the singular value approximation of step 2 is simply not optimal in the sense of the Hankel norm for linear systems. This clarifies the incorrectness of the theorem on specification efficiency in Aoki and Havener (1991), as well as the corresponding reply of Havenner to comments of [START_REF] Deistler | Comment on state space modeling of multiple time series[END_REF] on this point. A possible modification of the algorithm would be to determine an optimal Hankel norm approximation error a"+i while the Hankel structure is preserved. For this modification the theorem on specification efficiency would hold true.

We saw in section 3 that stochastic realization must satisfy the positivity condition, this is a basic problem in spectrum analysis cf [START_REF] Byrnes | Toward a solution of the minimal partial stochastic realization problem[END_REF]. This positivity constraint is neglected in Aoki-Havenner's method and it can cause problems for the algorithm. The estimates in step 3 may be such that the positivity condition is not satisfied, in which case no model can be determined. This can be checked in terms of the characteristic roots of a symplectic matrix arising in the algorithm.

Symplectic matrix and unit root

The eigenvalues of symplectic matrix O are not only symmetric with respect to the real axis but also with respect to the unit circle. Thus the algorithm of Vaughan fails only when 0 has a unit root i.e., an eigenvalue on the unit circle.

Symplectic Unit Root theorem. (Heij et al. (1992)) If has an eigenvalue at z = exp(iw), then either (A, C, M, N0) does not satisfy the positivity condition or spectrum is singular at z = exp(iw) As shown by Hannan and Poskitt (1988) on the unit circle the spectrum has a rank deficiency i.e. singularity, this is related to the existence of deterministic components in the process. For stationary ARMA processes singularity on the unit circle is equivalent to unit roots in the MA part, i.e., it is equivalent to non-invertibility. Heij et al (1992) are shown that the symplectic matrix 4> has a unit root if and only if the power density spectral S(z) has a zero on the unit circle. Then if this root is + 1 or -1 this corresponds to a deterministic component. If the unit root is not real then the positivity condition is violated because S(1)S(-1) < 0. This case causes problems for Aoki-Havenner's algorithm.

Factorization Approach.

Aoki-Havenner's algorithm is characterized by the fact that it requires the solution to an algebraic Riccati equation. We propose an algorithm which is representative of a class of algorithms which do not require a Riccati solution but only a matrix factorization to obtain the stochastic balanced realization. It will be recalled that the state vector of any innovation model must always be a linear function of y,' then we have x, = Sy,.f and y,+ \ y,.f = Hn R '1 y,.f = Ox, (34) (tr (A) denotes trace A) The solution to this optimization problem is: the n rows of S must be a basis for the space spanned by the n generalized eigenvectors corresponding to the n largest generalized eigenvalues of the matrix pencil (X.R -H'H). The approximation is based on the eigenvalues values of

H R ' H ' H R ' H ' = [U, U 2| Z] 0 0 z \ U,' U 2' ( 35
)
where Z] contains the n largest generalized eigenvalues i.e. the principal components, then the predictive efficience criterion is optimized by choosing S to be S = EU i'H R '1 (36) where E is any nxn invertible matrix. Different choices of E will correspond to different coordinate transformations of the state vector. This solution is called the unweighted principal components (UPC). The state vector will be in balanced coordinates by choosing E by Z { m S = E f 1/2U ,'HR-' (37) which implies that the observability matrix is O = U (I i 1/2 and the observability grammians, which is O 'O is equal to Si The state covariance is P = SRS' = Zi (38)

The UPC Algorithm.

Step 1. Perform an eigendecomposition of An advantage of UPC algorithm is that it can be applied to sample covariance sequence that are not positive definite. This condition often arises when the covariance sequence is estimated from a short data record.

To relate the UPC algorithm to the Aoki-Havenner's algorithm , it is necessary to work with the « square root »of (35) HR 1/2 = UZV' (where R 1 = R 1/2 R 1/2) which yields the following factorization of the Hankel matrix: H = UZV'R '1/2 = OK The resulting stochastic realization can be obtained from step 3 of the Aoki-Havenner's algorithm. Note that Step 4 of A-H's algorithm provides an explicit formula for £2, while the UPC algorithm does not.

Comparison to the Canonical Correlations Criterion and Akaike's Method. [START_REF] Arun | Generalized Principal Component Analysis and its Application of stochastic Processes[END_REF] point out that canonical realization algorithm is not well suited for model reduction due to the smallness of the canonical correlations and the fact that it works with unapproximated data. Akaike (1974) uses Hotelling's concept of canonical correlations coefficients. This method optimizes mutual information in the past and future observations. The method introduces canonical variables by normalizing the data vector by the inverse of square root of the covariance matrix of the data data vector. The selection of the state vector is based on the singular value decomposition of matrix R ',/2 H (R'l/2)', where H is the same Hankel matrix of the covariances of the future and past observations and the covariance matrix R is expressed as R = R 1/2(Rl/2)' where R l/2 is any full-rank factorization R. Thus, this Akaike's solution is different from Arun-Kung's solution because under perturbations, H will be full rank and the principal components of H R ''H \ R '1/2 H (R'l/2 )', and H will not be same. In addition the Akaike's method use the matrix R I/2 for normalize the data vector by Z* = R '^Y *. This indicate that the state vector component by Akaike's method do not do the best of predicting the future. It is inherent normalization in the canonical correlation criterion that causes it to ignore the strength of modes in the process. Following C.R. Rao the generalized principal component s analysis for studing the association between two randon vectors is different from Hotelling's canonical correlation analysis.

  s (the superscript (') indicates the transpose of a vector and matrix and Sts is the kronecker delta function). This model is in fact equivalent to the following innovation model x t+1 = Ax, + B ,e, y, = C x , + e , where x, = E(x, / y,_, ,y,_2>-••), is the optimal prediction estimate of xt, e, = yt -E(yt/yt.|,yt-2,...) is the innovation process of yt, B, = E (x,e't )E (e,e't )"' is Kalman gain, Q, = E(e,e\).

  transform of equation (2), we obtain: y(z) = [C(zl -A) 'B + I ]e(z) where y(z) et e(z) are respectively the z transform process y, and the innovation process e,.

  M and N0 are uniquely determined by eqs (5.1)-(5.3). The spectral density function S(z) may be expressed as a matrix product S(z) = T(z)flT(z') (7) with T(z) = [C(zl -A)'*B + 1], T(z) is the transfer function matrix. The spectrum is thus factorized as S(z) = G(z) G '(z_1) (8) with G(z) = T(z) £21/2. The poles G(z) which are equal to the eigenvalues of A (asymptotically stable by the stationary condition of yt) are within the unit disc, and the zeros are the eigenvalues of (A -BC). G(z) is analytical outside the unit circle Replacing z by elw gives S(eiw) = G(eiw) G '(e iw) = T(eiw) Q T(e iw) w € [0, 2n]. (9)

Fig 1 .

 1 Fig 1. The different representations of stochastic processes.

  Thus, for a deterministic model defined by x,+i = Ax, + Bu, y, = Cx, + Du, where u is a deterministic command. The realization problem consists in determining a quadruple (A, B, C, D) from: -the transfer function G(z) = C(zl -A)'*B + D -or Markov parameters Ls = CAs lB s > 0, Lo = D.

  Thus the state of the innovations model is completely reproduceable from the infinite past y,\ The state x, of the innovations model (minimum-phase model) is also the state of the Kalman filter which is an estimate of the state every other non-minimum phase model with the same parameters A and C matrices estimated from the infinite past y \ The state variance of the innovations model has to be smaller than the same A and C matrices. Thus for the innovation model, beginning with the solution of the difference equation for the states equation and substituting into the observation equation produces y+ = Ox, + Let+ (18)where e,+ is defined the same way as y,+ and L is a block lower triangular matrix containing the which means that the state of innovations model summarizes the past output history for predicting the future output.

  11 H -H || is minimum An (non-unique ) approximant H" of H with r(Hn) = n which minimizes the Hankel norm is Hn = UnDnVn' = ¿ < 7 |tu kv k k=l By the singular value decomposition theorem we have I I H -Hn I I h = C^n+1 The matrix Hn is in general not a Hankel matrix. If the spectral norm of the L2 is used to characterize the approximation we have II H -Hn 112 = [an+, + a n+2 + -+ a mp] l/2.

  No 'C and D = MNo 'M'. This is a symplectic matrix, i.e., <D 1 = J '<D'J with J = f o n v -I Oy =«► j -1 = -j = y Given No and A, M and C from eqs (27)-(29) then £2 and B are estimated by the following equations Q the sample autocovariance Ns s = 0, 1,2, ...f+p, form the covariance Hankel H 2) Determine approximate rank of Hankel matrix which specifies the number n of states, and using the singular value decomposition of the Hankel matrix Hn = UnD"Vn' 3) Factor Hn as Hn = (UnDn1/2)(Dn1/2Vn ) = OK and estimate the parameters A, M, C by least squares based on the singular value decomposition C = Hi*VD'1/2 M = D-,/2U'H., A = D 1/2U ' H VD 1/2 4) Solve the Ricatti equation for P and determine the parameters B and Q. B = (M -APC')(N0 -CPC')'1 £2 = No -C P C '

  Var [yt+ -yt+\ x] = R -HS'(SRS') 'SH' (33) thus we have min tr [R -HS'(SRS') 'SH'] = max tr [(SH'HS') (SRS') 1].

  the principal components. The aggregating matrix S is S = EU i'HR' where E is any nxn invertible matrix. In balanced coordinates S = IV ^U r H R '1 The observability matrix is O = U ,L f1/2 and the observability grammian O'O = E| Step 2. The parameter estimation step is taken from the deterministic algorithm A = 0 , + 0 2 where Oi denotes the first m rows of O and the superscript + stands for the pseudoinverse and O2 the last m rows of O. C = first row of O B = first column of S P = S, M = N0 -C I |C '

This paper was presented at the Second International Conference on Computing in Economics and Finance, Geneva, Switzerland,

26-28 June 1996

The aggregating matrix S is S = K R ' In the balanced realization form, the coordinate system is choosen so that O = U D I/2 and K = D i/2V \ hence the Hankel norm method summarizes past data in the state vector xt = D 1/2V'R ''yt' The aggregating matrix S = D I/2V'R '', given the model specification, as the aggregating matrix has full row rank the information in yt.f is preserved, and the states are sufficient statistics for information in the data. The state covariance is

We propose a method which is not used to solve the Riccati equation to find B and £2 estimate.

Proposition.

The estimates B and £2 are obtained by choosing the state vector as x, = D ^V 'R 'V i'. the state covariance matrix is P = F'F with F = R 'I/2VD1/2.

In this proposition the state covariance matrix is specified in terms of the factorization, thus no Riccati equation has to be solved, then P can be used directly to find B and £2. This method modified step 4 of Aoki-Havenner's method which can be summarized by In this approach the state covariance matrix is also specified in terms of factorization. Thus no Riccati equation has to be solved, but P can be used directly to determine B and £2.

Arun-Kung's Method.

The state vector can be introduced in yet another way. Recall from eq (22) that the state space of stochastic realization is spanned by H R 'y t. i . If H is of rank mp, then the dimension of the state space is mp. In order to obtain an n-th order model, n < mp , it is necessary to find a matrix S such that Syt' nearly spans the state space. [START_REF] Arun | Generalized Principal Component Analysis and its Application of stochastic Processes[END_REF] have suggested choosing S to minimize the following criterion:

Var [yt+ -yt+ \ xt]. Such a criterion is called the predictive efficiency criterion which was first used by [START_REF] Rao | The Use and Interpretation of Principal Component Analysis in Applied Research[END_REF] in multivariate statistics for the 2-vector problem. Noting that y+ \ x = HS'(SRS')'1 Sy,' and


MESNARD Louis De

Prof Louis De Mesnard

Industrial Organization with profit rate maximizing firms

Keywords: JEL classification. L20. D21, D41, D42, D43, L13 Coordination, Rate of Profit, Oligopoly, Behavior

We study the impact on industrial organization of the switching of objective function, from pure profit to profit rate maximization. The output level of firm is lower at optimum. This lead to a new conception of efficiency. Cases of no coordination are considered. In perfect competition, price signal disappears; factors remain paid at their marginal productivity, but modified. In imperfect competition, reaction functions may vanish even if collusion remains possible; limit of oligopoly remains perfect competition of profit rate; the paradox of Bertrand may remain; a new concept is studied: mixed duopoly, where firms can choose and change their objective.

I.

Introduction

Coordination among firms is one of the major themes of economics. It is the central idea of both signal theory and the theory of games.

• In perfect competition, each firm equalizes the price and their marginal cost to maximize its pure profit. The price signal allows coordination between the firm and the market.

• In imperfect competition with a small number of firms, as in duopoly-oligopoly, firms construct reaction functions in which conjectural variations play the central role, indicating the reaction of the firm(s) other than / with respect to the action of firm / d Qj (terms v, = q q with Q, as the output level of firm / and Qj as the output level of other firms). The intersect of the reaction functions determines the optimal output level of all firms. It is one of the starting points of the theory of games.

We will explore the consequences on coordination among firms of relinquishing the neoclassical objective function -the maximization of the pure profit -for another objective function, the maximization of the rate of profit, which is considered closer to actual business practice.

The variables that we will introduce are:

• P(Q), the inverse of the demand curve, decreasing with O.

• R(Q) = P(Q) Q > the income of the firm.

• ( V ( 0 ' the cost o f production (subscript g for "gross"). It includes all the costs (direct costs and opportunity costs) of variable factors, like labor, plus renting costs and writing off costs of fixed capital. It excludes interest on debts, remuneration and opportunity cost of the equity capital.

• K(Q), the positive capital (monetary, not physical) used in production.

• t , the constant and positive interest rate in the financial market, valid for both borrower and moneylender: as the financial market is assumed to be perfect, the same rate t is used for the debt of the firm and for the opportunity cost of the funds.

• K e(Q), the positive equity capital used in production.

Even if remuneration is not necessarily paid, equity has an opportunity cost t Ke(Q) sustained over each period at the fixed interest rate t.

• Kj(Q), the borrowed capital (the debts) used in production.

Interest is paid by the firm for each period: t Kj(Q) .

We obtain by hypothesis: K(0) = KC ((J) + Kj(Q)

• n e( 0 = R(Q) ~ ( ,e((?) , the production profit or gross profit.

• n ( 0 = n e((>)t Kj(Q) -t Ke(0) , the pure profit.

the economic rate o f profit denoted in the following the rate o f profit.

In the following, we will neglect to write the argument O in the writing of functions when there is no confusion.

The neoclassical microeconomic firm maximizes pure profit, subtracting the debts and the opportunity cost of the capital to the gross profit: max n . We shall examine another objective for firms: the maximization of the rate of profit, max 71,. As il = n y -/ K, these two objectives are independent to the financing decision i.e. to the division in equity capital and debts.

These two objectives are exactly equivalent concerning the condition of entry into the sector, i.e. when deciding whether the firm should or should not produce. A set of conditions of entry into the sector for the firm is as follows :

However, classical microeconomic theory chooses condition ( 1) and not condition (2). In fact, there are two possibilities for a firm : to maximize absolute satisfaction or to maximize relative satisfaction. If the entry conditions are equivalent, the question is: why has the subtractive form of the profit function, n , prevailed over the "divisive" form it? Why did firms historically choose to maximize pure profit? We may conjecture as follows .

In the pre-industrial age and from the beginning of the industrial revolution, the modem legal corporate form of firms was not widespread and as the banking system and the financial market were non-existent, capitalists disregarded debts and opportunity costs. Capitalists (or landowners) often had no alternative opportunity for investment 1 than their own business (or property): either they could leave their funds idle, or they could produce. In either case, they could count on their own funds only. Under the circumstances, maximization of profit was a valid objective even if the profit was only one cent for one million in gold coin. This is why the pre-classical authors, such as Bacon or Hobbes, naturally adopted the principle that best complied with the observed reality of the day, maximization of satisfaction, i.e. maximization of earnings, later modernized as maximization of profit.

Even if the classical authors emphasized the rate of profit (Smith, Ricardo, etc.), they considered the social rate of profit and the rate of profit of the sectors as Marx did later (see the law of 1 Imagine that a landowner finds a casket with one million in gold coin inside.

n > 0 » n e -/ K > 0 n e and it > t o -fr > t ( 1) (2 )
equalization of the sector rate of profit ), more than the rate of profit of the firm, and they did not think in terms of maximization of the rate of profit. Later, the neoclassical authors retained the principle of profit maximization, taking into account debts and opportunity costs to create the concept of pure profit2. As in General Equilibrium the profit of each firm is the income of the productive agent, it seemed natural to maximize the income of the agent. The maximization of pure profit did not change ways of thinking too much. Now, clearly, in the real world, a firm tries to maximize its return on capital, i.e. its rate of profit. This proves that it is necessary to explore an alternative theory of the rate of profit maximization. The microeconomic consequences of this have not been studied, although it is an old idea in the business world. We deliberately leave aside the improvements made to the neoclassical theory of the firm \ such as incomplete information, the theory of bounded rationality [START_REF] Simon | New Developments in the theory of the firm[END_REF], the theory of agency [START_REF] Grossman | An Analysis of the Principal-Agent Problem[END_REF], the theory of contracts and incentive [START_REF] Hart | The Market Mechanism as an Incentive Scheme[END_REF] ], managerial [ Baumol, ] and behaviorist theory of the firm [START_REF] Cyert R | A Behavioral Theory o f the Firm[END_REF] and separation of ownership and control [START_REF] Scherer | Industrial Market Structure and Economic Performance[END_REF], the theory of risk aversion [ Tirole 1989, p. 35 ] etc., to focus only on the question, of the consequences of profit rate maximization. Clearly these arguments may be applied also to this alternative theory and we agree with Tirole when he said [ Tirole 1989, p.

],

It will then be argued that even if managerial slack invalidates the profit-maximization hypothesis, the implications of this hypothesis for industrial organization need not to be erroneous.

applying this quotation to profit rate maximization.

However, it must be noted that even if many authors have well thought of the consequences of these theoretical improvements, not anyone has explored the consequences of profit rate maximization, as if pure profit maximization goes without saying.

Remarks.

1) The concept of profit remains unchanged from pure profit maximization to rate of profit maximization4 Even calculating pure profit is difficult because the opportunity costs of the capital are hard to know (this argument was presented historically by K.E. Boulding). Indeed, the alternative investment of the funds must be well defined, unique and known, in order to calculate the opportunity cost. Some authors treat the remuneration of equity as an ordinary cost [ WU 1989, p. 250 ]: it is excessive.

J

Remaining in what Simon calls the Theory of the Firm (F-theory), in contrast to Organization Theory (O-theory) [ Simon, 1952-53 ].

4

Remember that profit is due to an abstinence, as in Senior, or to an expectation, as in Marshall, or to a preference for the present as in Bohm-Bawerk or to an impatience, as in Fisher.

2) Many theories are based on profitability: the theory of investment choice (with the rate of actualization), the theory of capital demand (Keynes, among others, with the marginal efficiency of capital), etc.

3) Obviously, in the short-term, classical microeconomic theory assumes that capital is fixed in which case maximizing pure profit is equivalent to maximizing rate of profit. Classical theory argues that: in the short-run, capital is invariable while varying in the long-term. This is incorrect, because, in the short-term, if capital is fixed, the scale of production is given; thus it is necessary to introduce this constraint into the classical calculation of short-run equilibrium: the program is Max n under Q e [^¿>, Q , in which 0 and Q are the bounds of Q due to the physical capital invested, which generates the fixed costs. This is why, in what follows, we will not consider a long-term in which capital is variable as in [START_REF]La maximisation microéconomique du taux de profit[END_REF] (and the number of firms is also variable) because it is confusing, but a short term with an ex ante calculation, in which the amount of capital is free (while ex post, it is fixed) and the number of firms remains invariable Note that both profit rate maximization and pure profit maximization are concerned with this remark because both include capital in their formula.

4)

We may incidentally define many other types of profit ratios, such a s 6 n ? -1 Kci n r Ke which is the ratio of accounting profit over equity, or return on equity, which we term financial rate o f profit. This type of ratio is familiar to shareholders who use it to measure profitability. It is obvious that this ratio is maximized when Ke(Q) is minimized. However, it fails to discriminate between the production decision to the financing decision, because, with it, when trying to calculate the optimal output level, the manager also decides about financing, assuming an ex ante link between production and financing: clearly it is a function. That this link may be complicated is not the problem: the problem is that the link is exogenously determined and it may be unstable1.

With the rate of profit, we may separate the production decision and the financial decision, as we usually do with pure profit maximization: that is why we shall adopt this function in the rest of the paper even if it is far removed from the objective function of shareholders.

In the first part of the story, we shall see that results d'verge when we seek to maximize pure profit or the rate of profit: "max IT' diverges from "max 7t". In a second part of the paper, we will Even if it is possible to maximize the profit rate under constraints of capital.

6

As in [ de Mesnard 1992], 7

Even if Modigliani and Miller theorem proves that the value of the firm is independent of the interest rate [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF]and even if Jensen and Meckling prove that there is an optimal financial structure in the financial theory because of agency cost [START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs, and Ownership Structure[END_REF]: agency cost spent by the moneylender to control the firm may prevent it to reduce the part of shareholders; also bankrupt cost imply an optimal financial structure.

prove that coordination among firms may fail when firms maximize a profit ratio instead of the pure profitx 9

II. The optimal output level maximizing the rate of profit

We shall study a single firm in a market as a general case for the firm (i.e. a monopoly).

Proposition 1. Consider the monopoly with function of capital. The firm produces up to the point where marginal profit is equal to average profit multiplied by the relative elasticity of capital to output level. FL Proof Consider the rate of profit k = -p -. Its maximization gives: K n' = n,, < = > n' = eK -¡o in which euo is the relative elasticity of the capital to the output level

and n e = -j y is the average gross profit. ■

We have also:

n ' = 7i K!
As the optimum of the pure profit maximizing monopoly is given by: n £ = t K'

Thus rt has the role that t plays in the condition of maximization of pure profit. In a certain sense, the difference between the classical theory and the new theory is weak: where in the classical case, cost of capital is calculated with the exogenous fixed rate t, in the profit rate maximization, the cost of capital is calculated with the endogenous variable rate n: there is no evaluation of the cost of capital in the formula n = and everything is as if the evaluation was made at the intern rate n while in the formula II --1 the evaluation of the capital is made directly at the extern rate /. It is logical: if n > t at the optimum, then the firm uses less capital and produces less because it evaluates the cost of its capital at a higher am ount10 However, most part of the following results may be retrieved with the financial rate of profit. 9

The exposé is volountary similar to first degree manuals, to focus on the specificty of the aaproach.

10

With the financial rate of profit, only cost of equity is calculated at the rate 7t . It could seem more satisfying, (independently of the difficulties posed by the ratio) but we must remember that the opportunity cost of each dollar of equity is equal to the cost of each dollar of Proposition 1'. The monopoly with a fixed coefficient of capital k produces up to the point where marginal profit is equal to average profit (or to the point where average profit is at a maximum).

Proof. We have : K = k O with k > 0 , and then en/o = 1 Hence, n ' = 0 ^ . ■ As claimed in [START_REF]La maximisation microéconomique du taux de profit[END_REF], this corresponds to the observed behavior of firms: a firm produces and invests so that the profit contributed by an extra unit is greater than or at least equal to the average profit yielded by other units.

Proposition 1". If the cost per unit is a constant c and if there is a fixed coefficient of capital then the optimal output level of the profit rate maximizing monopoly is indeterminate.

Remember that in this case the optimum for a pure profit maximizing monopoly does not exist excepted in the case p = c + t k.

Proof. The proof is obvious because n = --,-6-^ is a constant." k Proposition 2. The optimum is independent of the level of fixed costs and of the constant term in the demand equation. Thus, if demand increases uniformly, or if fixed costs increase, a monopoly maintains an unchanged optimum when it maximizes the rate of profit.

Proof. Suppose Cg = Cg + f ( 0 in which Cg are fixed costs and I) = 1) () + g(Q) . Then n ' does not include the terms ( £ and Do ■ Remark.

1) This equilibrium is not the point of long-term equilibrium of the traditional monopoly (i.e. R' = C(;// + 1 K ! ,i ) and does not correspond to the point where average cost is minimum, even if capital is variable.

2) The criteria of efficiency is changed: an efficient system is no more a system where "marginal cost equals marginal revenue", it becomes a system where "maximum average profit maximum" (or more generally "marginal profit equals average profit multiplied by elasticity of capital"). For example, in health economics, if we use the new criteria, less treatments will be provided. In education economics, less teaching will be provided. Etc. In all sectors of economic life, the new criteria of efficiency leads to a limited service but also less expenditures. This consequence of the new criteria of the rate of profit may seem strange or revolting. However, the new criteria (the profit rate maximization) is not more doubtful than the old one (the pure profit maximization). ■ Denoting (J11 as the classical pure profit equilibrium and Q% as the rate of profit equilibrium, we obtain the following graph:

debts: in both cases, it is t.

C Ik

Figure 1. Short-term monopoly with fixed coefficient of capital

We may compare in the general case the optimum of the rate of profit with the classical microeconomic optimum (pure profit); we obtain the following fundamental theorem.

Theorem 1. Consider the more probable case K'(QK) > 0 , in which Q K is the optimal quantity that maximizes the rate of profit 7 i . Compared with the maximization of pure profit, maximization of the rate of profit gives a lower optimal output level (respectively higher) when the rate of profit is higher (respectively lower) than the interest rate at the optimum of the rate of profit.

• n(QK) > t => (max 7i) occurs before (max FI)

• H Q X) < 1 => (max k) occurs after (max n ): it is less probable because in this case the firm leaves the sector.

The results are reversed if K'(QK) < 0.

Proof. The first order condition of optimality 11 for the maximization of the pure profit n is FI' -( K' = 0. The first order condition of optimality for the maximization of the rate of profit n is F I ' -7i £ ' = ().

Consider the curve f* = n 'n K'. Assume that FI' is decreasing and write f l' as a function of f*: n '= ft + [ :r -/] K ' . Denote Q u as the point of maximization of pure profit, such as

11

It would be necessary to study second order conditions. However, the mathematical aspects (convexity, etc.) are similar in pure profit and in profit rate maximization. n'((> 1,) = 0 and denote O* as the solution of the maximization of the rate of profit, such that

U Q ' ) = o .
1) Suppose that K' > 0 and n(QK) > I. Then, n '((;*) > f ^* ) = 0 . Near (J11 , curve f* is under curve O '. Thus, Q u must be greater than Q*. At the limit, if n ' is vertical near (7", n ' will cut the x-axis at Q K and then (Jn = Q* . Remark that it in fact sufficient that Arf(QK) > 0.

Figure 2. Mathematical comparison of optima

2) Suppose that K' > 0 and n(()K) < t. Then, n '^* ) < f^ = 0. Then, near to 0 K , curve f* is over curve n ' . Thus, O 11 must be smaller than Q* .■ Following this theorem, the monopolistic firm produces less, so at a higher price, when it maximizes the profit rate. Thus the monopoly is harsher with profit rate maximization. Now, we will prove that coordination among firms could fail when firms maximize the rate of profit instead of pure profit.

III. Perfect competition

A. Price signal

Consider a single firm in a market in short-run perfect competition (the number of firms is invariable). We apply here theorem 1 above, assuming that the price is independent of Q , i.e. p = p .

Proposition 3. In perfect competition at the optimum, we have p -( '* = euo [ p -C g J -

Proposition 3'. In perfect competition with a fixed coefficient of capital or with a relative elasticity of capital over output equal to 1, the firm produces up to the point where the average gross cost Cg is minimum.

Proof. We have : e^/o = 1 Thus C ' = Cg at optimum. ■

Figure 3. Short-term equilibrium of the firm in perfect competition with coefficient of capital

In this figure, the optimum of pure profit is given by p = C'e + i Kf .

Remarks.

1) The optimum does not coincide with the classical point of long-term equilibrium of the firm which is situated at the minimum of Cg + t K. However, let's not get confused: we are not in long-term, but only ex ante, as said before. The number of firms remains fixed in our calculation, whereas the number of firms is variable in the long-run.

On the other hand, in the long term, p decreases in both cases because news firms enter the sector. The movement stops at the same point when individual pure profit is nil for a pure profit maximizing firm and when the profit rate is equal to ( for a profit rate maximizing firm. This point is the entry point in the sector, corresponding to the minimum of Cg1 + 1 Ku .

2) The figure 3 suggests that if t k > p then (Ju is before Q K. It is right, but impossible, because in this case n < 0 .

We deduce from proposition 3 the following conclusions:

1) The optimum does not depend on price; thus, there is not coordination in the market.

Each firm is alone, doing nothing with the price. Price is no more a signal. We need no more additional theories to explain that the output is independent to the price (like cost plus pricing, U-curves of cost, etc.)

2) As a corollary, the output level does not vary if price varies (as long as the profit remains positive: when it becomes negative, the firm leaves the sector as usual). This corresponds to observed behavior. Consequently, output level does not matter with taxation or other manipulations of prices.

3) The individual supply curve is vertical, even in perfect competition. The global supply curve is completely inelastic and the market is in global equilibrium, following a classical pattern 12:

Figure 4. Global equilibrium in short-run perfect competition

This proves that a global equilibrium is compatible with no coordination by price, unlike good sense tends to believe. It must be noted that, unlike in the classical pure profit maximization (where price is determined by the intercept of the global curve of demand with the global curve of supply), it is not the global equilibrium which determines individual equilibrium of the firm, but it is the individual equilibrium which determines global equilibrium by adding up: price is only a consequence of the global equilibrium without impact on individual level (except for the amount of earnings of the firm but not for its optimal output).

Moreover, if demand curve is also completely inelastic, there may be no equilibrium; if it is only very inelastic, price may be very variable to slipping of demand (without any effect on the optimal output of firm, obviously). Note that if price decrease, some firms may go bankrupt and leave the sector, in this case global supply decreases *\

12

In [ de Mesnard 1992 ], it is claimed that, as price is no longer a signal for the output level of firms, the optimum does not automatically clear the market in the short-term (as assumed in the neoclassical theory). This is false: the market is cleared in the sense that price is fully adaptive, as the quantity remains fixed.

13

Individual supply is equal to O' if firm remains, or to zero if it leaves.

4) As firms are smaller, there could be more firms; thus interaction among firms could be lower and competition greater (even if the concept of competition becomes particular).

5) Accordingly with proposition 1", if the cost per unit is a constant c with a fixed coefficient of capital, 7i = ---is a constant then the level of output is undetermined whereas pure profit has no optimum except if p = c + t k.

B. Optimal input combination and distribution

Consider n factors, used in quantity q,, at a price /?,. The function of production is Q(q\,...,qn).

n

The costs of production are ('(c/i,...,q") = Y j P , </,. The capital K(q\,...,q") is not considered as /=1 a factor, even if the cost of the capital (interest of debts and opportunity cost of equity) is a part of the total cost (at the interest rate /). Recall. A pure profit maximizing firm remunerates it at its marginal productivity minus (in value) the marginal amount of capital required by this factor evaluated at the rate /.

n n n ( i i v , ? » ) = p C % i v , ? » ) -S p / i i -/ % i v , i n ) = ^? ...... /= 1 ,= i denoting Q(q i = Q(q i,.
.., q") -t K(q i,..., qn) as the net output. The classical distribution rule for homogeneous production function with degree 1 is the following:

n n 2 i Q, = Q ^ P i y i = p Q■ i= I i= l
Proposition 4. A profit rate maximizing firm remunerates each factor at its marginal productivity (in value) minus the marginal amount of capital required by this factor evaluated at the rate n.

Proof. It is very simple.

pQ{<i\,-,qn)-ŸéPi<li

71 ( i / i , . -'=l n(qi,...,q") will be noted n further.

Then,

! = » o P , = p Q ' , -* k !
Remark. The variable rate n plays the role of the fix interest rate /. ■ The last condition can be noted:

Pi = PQ'>
denoting Q' = (j' -4 K' as the modified net marginal productivity of the factor /. Thus, we can say that each factor is remunerated at its modified net marginal productivity (in value) with profit rate maximization: things are not so different... And as said before, in the profit rate maximization the cost of capital is calculated at the rate n where it is calculated at the rate t in the pure profit maximization.

Proposition 4'. If ti > /, which is normal situation, then the remuneration of each factor is lower with the profit rate maximization than with the pure profit maximization: each factor is no more remunerated at its net marginal productivity. Consequently, if the price of each factor is exogenous, a lower amount of each factor is used when the firm maximizes its profit rate (remember that in this case the output is lower:

Q K < Q [l).
Proof. If n > I, we have the following graph: Corollary. If Q\ < Q\ for all / then the net output is not exhausted:

E T.Pi Hi <pQ- /=i /■ = i
In fact denoting Q(q\, = 0(q\, n K ( q \ , as the modified net product, it is the modified net product that is exhausted:

S <ii Q ', = Q = > i p i q ' = p Q -

IV. Imperfect competition

A. Duopoly and vanishing of reaction functions

Let's look at the case of duopoly (competition in quantities). The result will be generalized to oligopoly later. The demand for the sector with two firms 1 and 2 is D( Ql + Q 1). The inverse demand is p = D~]( Ql + Q2) assumed to be decreasing. The total revenues of each firm are R'(Ql

+ Q 2) = Q' D ~\Q X + Q1).
The production costs of each firm are C'g(Q '). The total capital (equity plus debt) of each firm is K'(Q). The pure profits of each firm are n

' ( £ ? \ Q 2) = R '( Q \ Q 2) -C'g(£>')-1 £ '(£ ')• The Proflt rates are: * '( g 1, g 2) -C k g ') K \ Q ') a n X Q \ Q 2) d Q l
obtain the following system:

d D -\ Q l + Q 2) A Q ' , Q 2) =
For the classical maximization of pure profit, we solve the equations = 0 . We

d Q x d D -' ( Q l + Q 2) d Q 2 Q \ + d -\ Q ' + C?2) -C'g'iQ1) -1 = 0 Q 2 +D~l( Ql + Q2) -C2 / ( Q 2) -1 K 2,(Q2) = 0
Generally, except special cases 14, the system has two reaction functions: the solution is found by the intersect of two curves (see the example later for a very simple linear demand function). These reaction functions allow coordination between firms, by way of conjectural variations 15

dQ> denoted v' = d Q ' 14
There is the question of uniqueness [START_REF] Gaudet | Uniqueness of Cournot Equilibrium: New Results From Old Methods[END_REF], existence or convergence of equilibrium [START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF], and for example the question of discontinuity in reaction functions involving a non-existent equilibrium [ Roberts and Sonnenchein, 1977 ].

15

We do not enter in the debate about the interest of reaction functions and the fact that they are essentially concerned with competition in price (with an auctioneer). From Edgeworth to Kreps and Scheinkman, it is known that competition in price and competition in quantities are linked [START_REF] Edgeworth | La theoria puria del monopolio[END_REF][START_REF] Kreps | Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes[END_REF][START_REF] Krishna | Dynamic Duopoly: Prices and Quantity[END_REF] (see [ Tirole, 1989, Chap 5 ] for a complete discussion). However, it must be noticed that profit rate maximization is compatible obviously with a model including capacity constraints (recalling note 1 on the remark in introduction, it is possible to maximize profit rate with bounds on the Now, assume that each firm maximizes its rate of profit. The necessary conditions of optimality are:

d n ' ( Q \ Q 2) ô Q ' = 0 for all /. d D -\ Q x+ Q 2) Ô Q 1 q 1 + D-I (Q 1 + Q 2 ) ( 1 _ eKWQl ) = C " ( 0 1 ) -Ct' (Q 1 ) eKM Q 0 Q2 + D -'® ' + Q'-) (1 -< * ,* ,) -C*«?=) -C>(Q2) e K,,Q, S Q ! C'(Q') d C U Q ')
in which C ' g(Q ') = ---, and C'^Q') = --, and =

¿ c m _ £ m _ d Q Q
In a dynamic perspective l6, the classical iterative process is: firm 1 announces that it will produce quantity Q l (t_ 1}. Firm 2 calculates its own output level Q\t) that maximizes its profit, taking the output level of firm 1 as fixed, i.e., dQ\,_X)= 0. Then, firm 1 recalculates Q l (l+ n maximizing its profit, with Q}t) fixed, i.e., d Q2 (t) = 0. The following system to be solved is the expression of the two reaction functions :

d * { Q U , Q l > ) SQ\ = 0 for all / (H-l)
It is easy to find counter-examples in which the reaction functions disappear with maximization of the rate of profit, while they occur with the maximization of pure profit. Generally, reaction functions disappear if the first equation excludes Q2, and the second equation excludes Q x .

Proposition 5. Whether reaction functions can be canceled depends only on the parameters of the demand function and on the elasticity of equity capital with respect to the output level, not on the cost function.

Proof. In the above system of reaction functions, the right-hand side of the equation of equilibrium of firm i does not depend on the level of output Q> of the other firm j , but the left-hand side does." This is an interesting result because there is a reaction function in all other models of duopoly, even though it is known that duopoly does not always have a stable solution (and even if capital).

16

As pointed out by Tirole, the dynamic perspective is the only valid approach for Nash game equilibrium, because in a static perspective, firm choose its action before observing the other and there is no reaction [ Tirole 1989, p. 208 ]. However, it is not here a repeated game. dynamics may be complicated or chaotic [START_REF]Exotic Phenomena in Games and Duopoly Models[END_REF][START_REF] Piatecki | Sources du chaos dans les interactions stratégiques[END_REF] or even this solution can be non-unique or cannot exist Here are some counter-examples of non-existence when maximizing profit rate. Proposition 6. In duopoly, the reaction functions vanish if the inverse function o f demand is linear: D~X ( Q X , Q 2) = d -a Q x -b Q2 , and if the elasticity of equity capital with respect to the output level is equal to 1 for each firm: e^,/Q, = 1 for all i (this occurs when there is a coefficient of capital: K'(Q') = k Q ' for any / ) .

Proof-With these conditions, the system of reaction functions becomes:

-a Q x -b v x Q x = Cr X ( Q X ) -C X ( Q X ) e^io' I -a Q 2 -b v2 Q 2 = a 2(Q2) -C2(Q2) J '
As each equation is independent to the other, there are no more reaction functions. Thus, Q x does not depend on Q 2 and reciprocally. ■

When reaction functions are canceled, that is when the solution is not found by the intersect of two functions, each firm acts independently of the other to find the optimal solution (even it has an opinion about the other firm). In a dynamic perspective, each firm never observes the other firm and there is no iterative adjustment ("immediate" solution). Finally, there is no coordination among firms and no game. We can say that duopoly is degenerated. Nash equilibrium exists anyway but it is special. Note that, as there is no reaction function, their slope does not matter in this case thus it does not matter to know if firms are strategic substitutes or strategic complements to the mind of [START_REF] Bulow | Multimarket Oligopoly: Strategic Substitutes and Complements[END_REF].

Example. The above proposition corresponds to a very common case, as in Milleron's example [ 1979, pp. 117-122 ], adapted to introduce pure profit. Data are:

P = d -( Q X + Q 2), R \Q X + Q 2) = d Q -(Q ')2 -Q> Q> for all / Ci g{Qi) = c i + {Q') 2 for all z , K \Q ') = k Q' for all / .
Thus, if both firms maximize the rate of profit in a general duopoly, the system to be solved is not a Cramer system:

~Ql+ d (1 + v 1) = Q l (M) - - < 1) -Q l) (1 + v2) = Q\t) -■£- Qu) Q *= /------: for all i X 2 + v '
Here, each firm does not consider demand or capital parameters (nor interest rate) but only its fixed costs and conjectural variations when determining its output level at equilibrium (moreover, decisions are not identical because costs are not equal). This solution could be compared to the equivalent of the general duopoly maximizing pure profit:

f (4 + v1)Q(,+d + Qu) = d -t k |
The solution of this Cramer system is:

Qi = . ( ^-^) Q + y/)
for a ll, 15 + 4v' + 4v2 + v v Note that i is assumed to know the conjectural variation of j : it is not realistic.

When v 1 = v2 = v we obtain Q \ ^ for all / . 5 + v For the original Cournot duopoly [START_REF] Cournot | Recherches sur les principes mathématiques de la théorie des richesses[END_REF], the equation is solved assuming that when either one firm modifies its output level, the other firm would not react and change its own output level ( v 1 = v2 = 0):

I gi Q'* = J ~2 f°r
i* for the rate of profit: it behaves than a profit rate maximizing monopoly.

It could be compared to the corresponding duopoly of Coumot-Nash maximizing pure profit: d -t k Q* = ---for all i (which is Milleron's result assuming that k = 0).

These results prove that, unlike in monopoly or perfect competition, a profit rate maximizing firm in duopoly may have a greater output than a pure profit maximizing firm. ■

Where demand is not linear, reaction functions may also disappear:

Proposition 7. Reaction functions may be canceled in duopoly if the inverse demand function is a simple quadratic function :

D~X ( Q X , Q 2) = d -a Q x Q2
and if the elasticity of capital versus output level is equal to 2 for each firm : eK,/Q, = 2 for all /.

Proof. The left hand-side of the equation urm 1 becomes:

-2 a Q x Q2 -a v x ( Qx)2 + d -d e Ki,Qi + e Ki/Qi a Q x Q2 .

The terms Q 2 vanish if e^'/o' = 2 . ■ However, it is false to say that the elasticity condition seems to be closely tied to the degree of the demand function. As a counter-example, see the following proposition:

Proposition 8. Reaction functions are not canceled if the quadratic inverse demand function includes a linear term:

D ~\ Q \ Q 2) = d -a Q 1 -b Q1 -f Q l Q2 .
Proof. The terms including Q 2 on the left hand-side of the equation for firm 1 are -/£ ? ' Q2 ( 2 -e Ki/Q> ) -b Q 2 (\ - i/o1 ) -This does not vanish whatever the level of output. ■

B. Collusion

One question remains. Does collusion is possible with profit rate maximization? If yes, a certain form of coordination remains possible. Classically, the collusion equilibrium is found by the maximization of the sum of the pure profit functions: max Tl(Q{ + Q2),

where 11(0' + Q 2) = n ' ( ^') + U 2(Q2) -D ~\ Q ] + Q 2) ( Q x + Q 2) -C ] g( Q 1) -C2(Q2) -t [K'(Q') + K 2(£>2)] , • d n ( Q ' + Q 2) n , " solving -------: ------= 0 for all i.

dQ'

With the profit rate maximization, we calculate

K \ Q ' ) + K \Q > ) n,ax + Q >)
where , ( 0 ' + 0 >) . ~<*&■>-<% ?») and we solve

+ Q ) = o

for all /. dQ1

To simplify calculations, we choose to work on the above example.

n ((?' +

Q 2) = ( d -Q l -Q2) ( Q l + Q 2) -( c 1 + (£>')2) -[ c 2 + (Q2) 2) -t k ( Q l + Q2)
Thus, for the pure profit maximization, we solve:

i (4 + 2 v 1) £>* + (2 + 4 v 1) Q2 = (1 + v 1) { d -1 k) 1 I (2 + 4 v 2) Q l +(4 + 2 v2) = (1 + v2) ( d -t k) J
Classically, it is assumed that v 1 = v2 = v when there is collusion. Thus, the collusion equilibrium is:

Q x = Q 2 = ---.
For the profit rate maximization, we solve:

With v 1 = v2 = v, the solution is Q l = Q 2 = J c ^L and collusion remains possible. Again, demand and cost parameters play no role (and naturally conjectural variations also).

C. Oligopoly

In the above example, we shall study the case of oligopoly.

Proposition 9. The limit of a profit rate maximizing oligopoly remains perfect competition, but perfect competition with profit rate.

Proof. For a general oligopoly with n firms, it is easy to verify that the solution of the profit rate d 0 s maximizing oligopoly is Q = for all z, where v* = -~r

Perfect competition with n profit rate maximizing firms is identical, assuming that v = 0:

6 Q ' i.e.
-> t J--> oo (assuming that c ' = c and v' = v for all z) and quantity increases continuously, even if Q -» 0 for all z (with v d 2 n = --: the number of firms is limited). for all z ... 4 + v" i.e. Q't = for all / if v' = v for all / and if n -> oo, global quantity becomes stationary (even if Q -►()), (?'« -> • ti -t k, with p -> t k (the number of firms may increase indefinitely): it is perfect competition."

D. Competition in prices

The solution of Bertrand's model [START_REF] Bertrand | Théorie Mathématique de la Richesse Sociale[END_REF] adapted to a profit rate maximizing duopoly may differ strongly to the solution of the Bertrand's pure profit maximizing duopoly in prices. We will take up the data classically used for the Bertrand's model. Proposition 10. With the same hypothesis than classical duopoly of Bertrand (fixed cost per unit of output, discontinuous demand), if there is a coefficient of capital then the solution of the symmetric duopoly in price does not correspond to perfect competition: price is not equal to marginal cost.

Proof. Consider the demand to firm /: D '( p \ p 2) = <

D(p') ifp ' < p> ±D(p') i f p ' = pi 0 if p' > pi

and consider that the cost is c by unity of output. Thus, the pure profit function of firm / is

W ( P l,P2) = (p' ~c ) D ' ( p \ p 2) -1 k D '( p \ p 2) and its profit rate function is: , , (pl -c) D '( p x, p 2) (p' -c ) n (P >P~) --

^-= k wh&tever are relative values o f p x and p 2.

The classical Bertrand solution for a pure profit maximizing duopoly is perfect competition because each firm has interest to lower its price in order to take all the market; thus P x = p 2 = c + 1 k with U ' { p \ p 2) = 0 for both firms. The corresponding solution for a profit rate maximizing duopoly would be n'( p l, p 2) = t with p x = p 1 = c + tk. However, the problem is complicated. First, the solution p l = p 2 for this duopoly is obviously infinite with a profit rate also infinite because profit rate has no maximum (it is a straight increasing line) and the situation of long run perfect competition n'( p l, p 2) = t is never realized. Second, starting from p x -p 2, the same reasoning than Bertrand does with the pure profit maximizing duopoly is applicable here: one firm may lower its price just a little to take all the market but, as this does not increase its rate of profit, the firm does hot adopt this behavior, prices do not fall down to c + t k and there is not perfect competition."

The most important conclusion of the Bertrand's model, which is named the paradox of Bertrand, is called into question. For Bertrand, the behavior o f two firms which compete with prices is necessarily competitive; this is seen as the forerunner of Baumol-Panzar-Willig approach [ Baumol, Panzar and Willig, 1986 ]. With profit rate maximization, both firms may earn a profit rate superior to interest rate by increasing price indefinitely.

A disturbing aspect of this conclusion is that the problematic of coordination is reversed. The paradox of Bertrand may be interpreted as a default of coordination among firms which causes (as in the Prisoner Paradox and in non-cooperative games) a less favorable solution compared to the cooperative one, while with profit rate, the solution is always favorable (even if the technical cause is the same than in case of perfect competition and duopoly in quantities: demand plays a minor or nil role in the solution).

Yet, the hypothesis of coefficient of capital seems improper regarding to the form of demand function: it may be more suitable to suppose that capital is proportional to the global demand because firm expects to cover entire demand.

Proposition 11. With the same hypothesis than classical duopoly of Bertrand (fixed cost per unit of output, discontinuous demand), if the amount of capital required by a firm, when both prices are equal, is always equal to the amount of capital required by this firm when its price is the lower (i.e. K '(Q) = k D(p') whatever p ' ), then the equilibrium is competitive in pure profit maximizing and in profit rate maximizing: p l = p 2 -c .

Proof. When p l = p 2, the profit rate is n ' ( p \ p 2) = --^ ^ ^ ^ and the optimal profit rate for firm 1 is also ^ ^: it is lower than the optimal profit rate when p l < p 2 which is ^ ^ (remind that if p l > p 2 then n ]( p l, p 2) ---= 0 ).

* k D \ ( p l)

Thus each firm may want to lower its price, and there is a movement toward p x = p 2 = c + 2 1 k which is the competitive solution (71' = t). In the other hand, pure profit maximization gives also perfect competition, p ' -c + 2 t k, with a nil pure profit. ■

The paradox of Bertrand remains in this case.

E. Mixed duopoly

Both firms may maximize either pure profit or profit rate. Denote the optimal output for firm 1 when both firms maximize profit rat" / J ;noted n for firm /) and nQ* the optimal output for firm / when both firms maximize pure profit (denoted II', for firm /). Denote the optimal output for firm i when it maximizes its profit rate (denoted v/7t ») whereas simultaneously firm j maximizes its pure profit (denoted A/n i ) with an optimal output o f 1 LV /( / . . We have four cases, that is for optimal outputs: Optimal output 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate xQl and XQ\ * 'Q l and m ,Q l 1 maximizes its pure profit m 'Q l and ^Q i n£>i a°d uQ l and for optimal objective function:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate 1 Tti and 7i* 2 l/7ti and un l 1 maximizes its pure profit ? A/n i and A/7i* 4 111 and n*

The second table looks like a game matrix but it is not a game because we have not comparable earnings in it 17. In an ordinary game, to find Nash equilibria, the reasoning is the following: if firm 2 chooses strategy a, what is the best reply for firm 1, a or b, and then, if firm 2 chooses strategy b, what is the best reply for firm 1, a or A? The evaluation of each strategy is made looking at the value of a fixed objective function. There is a Nash equilibria if both players choose the same case of the matrix. If the reply is the same in both cases, for example firm 1 must chose strategy a in both cases whatever the choice of firm 2, then this strategy is always played by firm 1.

However, in our table, the objective function changes 18! We cannot compute a maximin equilibrium as usual, even if some computations are possible. Cases 1 and 4 are pure, cases 2 and 3 are mixed l9. A firm cannot compare its profit rate with its pure profit. Cases 1 and 4 corresponding to similar choices of objective functions for both firms are not at all comparable. Cases 1 and 2 and cases 3 and 4 are not comparable for firm 2, while cases 1 and 3 and cases 2 and 4 are not comparable for firm 1: firm 1 can compare case 1 with case 2, and case 3 with case 4 and firm 2 can compare case 1 with case 3 and case 2 with case 4; that is to say, firm i can compare IT, with * W IT* or n' , with A/7 t'». It can only compare the values of its objective function when the other firm changes its objective function (even it cannot force it to change) 20 ; for example, when firm 2 chooses to maximize its pure profit, firm 1 cannot compare l/7ti and n i .

The choice between "be a profit rate maximizing firm" or "be a pure profit maximizing firm" may not be seen as an ordinary game. Only in evolutionary games players may chose a strategy in order to satisfy a meta-objective: for example, survive [START_REF] Maynard | Evolution and the Theory o f Games[END_REF]. Here, if we suppose that players may chose their objective function, we must have a meta-objective, like survive. And it is not sure that choosing the objective function is realistic for a firm. We see here the interest of econometric studies to determine what is the objective function of real firms.

18

The term "multiobjective duopoly" is not suitable because each firm has only one objective at the same time. That is why we prefer "mixed duopoly".

19

Do not confuse with pure strategies and mixed (i.e. probabilistic) strategies.

20

As a small comparison, it is false to say that rugby is better than soccer because players Even if a firm maximizes its profit rate, it can calculates the corresponding pure profit, thus a firm can compare its old pure profit IT* with the pure profit, denoted n ( A V") = xX 1 Kl V V", that it obtains when it maximizes its rate of profit A V , (^'K 1 , is the optimal amount of capital of firm / when this firm maximizes its profit rate and the other firm maximizes the pure profit ). This argument is perfectly right, but fI(A V ,) < IT, (equality is the particular case where firm i just enters in the sector: see theorem 1) because n ', is yet a maximum! The same reasoning can be v/n i made with ji', and 7r(A/IT") = 1U/ ~y, and for other configurations. Thus, the firm which abandon one objective to another always looses in terms of the first objective (in other words: there is never pareto-optimality between the two objectives). The real question is: does it freezes the movement? The answer is not, because we think here in terms of objective, not simply in terms of decisions: the problem is not only to take a decision or another decision, it is to switch the objective function. This theme is not much studied in economic theory: when an agent thinks about switching, it do not calculates the value of both objective functions. Example: if a man thinks about the possibility to work hard to do a great career or to be happy, he knows that he will be less happy with a great career (to much work) or he will have a poor career if he his happy (to much tranquillity); anyway, he chooses or switches.

We will say that a mixed case is dominated for a firm if, starting from a pure case to go on this mixed case, this firm is loosing (when the comparison is possible). Suppose that starting from case 1, firm 2 changes its objective function (choosing pure profit instead of profit rate) and suppose that Mni < rci and suppose that starting from case 4, firm 1 changes its objective function (choosing profit rate instead of pure profit) and suppose that A/n 2 < T il. Then case 2 is dominated21.

For example, to see if case 2 is dominated 22, we compute X1ni and v/n 2, and we solve the system:

d n \ Q \ Q 2) d Q x ' d n 2 (Q'>Q2) _ Q > d Q 2
may earn more points in rugby than in soccer, it is another game and if a player leave out from rugby to go playing soccer, he may have regrets but not for the quantity of points, even if a rugby team may notice that its results have change in good or in bad when this player leave rugby to go playing soccer.

21

The same analysis can be made with a more familiar objective function, Baumol's maximization of turnover; for example, with three firms, each one with a different objective.

22

Reversing indexes gives the conditions to know if case 3 is dominated.

Q l d P '

\ Q l + Q 2) d Q 2 Q2 + D ~\ Q 1 + Q2) -C 2 / ( Q 2) -1 K 2\ Q 2) = 0
In the case of the above example, we solve:

~C?(/+nO + v') = (?(<+1) -' thus *-KiQl = 2 + v 1 and lulQ l = Qo+\) (?!_.)+ (4 + v2) Q 2 u) = d -t k d -t k -M Ql 4 + v2
We see that the profit rate maximizing firm determines its output independently to the other firm, when it is not the case of the second firm; there is a unilateral link between firm 1 and firm 2: firm 1 is dominant in a certain sense.

Remember that *Qi = ----for both firms and 11Q1 , = ^ 1 ^ for both firms. 2 + v' 5 + v Suppose that both firms are initially profit rate maximizing (case 1) and then firm 2 becomes pure profit maximizing (case 2): in this case, only firm 2 is affected for its output level. If *Q\ < nQl (i.e. if firm 1 has the same behavior than in monopoly or perfect competition), as ^Q i = nQl , we have:

m iQ 2 = d -t k -*Ql 4 + v2 > n Q l lf u7ti
> ji i then firm I my want that firm 2 becomes pure profit maximizing. Does firm 2 have lost anything in this operation? Not, because it is assumed to be equally satisfied with its new objective function as it was with the old one: firm 2 plays another game, not the same than before.

• Suppose that both firms are initially pure profit maximizing (case 4) and then firm 1 becomes profit rate maximizing (case 2): in this case, both firms are affected. Moreover, if wn 2 > n 2 then firm 2 may want that firm 1 becomes profit rate maximizing.

As we said, there is no game strictly speaking. However, there are cases that are wished by firms.

If a mixed case, 2 or 4, is dominated for both firms, starting from pure cases, no firm wants that the other firm changes its objective function: the mixed cases are never wished. In this situation, there is a potential conflict because, starting from a mixed case, each firm wants that the other firm changes its objective function, or there is a potential jamming because each firm waits the change of the other firm 23. For example:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit I maximizes its profit rate 1 Til = 0.15 and tc, = 0.20 2 v/7r' =0.13 and A/n 2 = 130 1 maximizes its pure profit 3 wn i = 80 and Mnl = 0.17 4 n ! = 1 0 0 a n d n 2 = 150

Conversely, pure cases 1 and 4 can be dominated for both firms: starting from mixed case, no firm wants a change in the objective function of the other firm. For example:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate 1 7ii = 0.15 and ni = 0.20 2 v/7ci =0.18 and v/n 2 = 170 1 maximizes its pure profit 3 v/n i = 110 and uni = 0.22 4 n i = 100 and n 2 = 150 Some complex movements may arise. Suppose that case 2 is dominated for firm 1 but not for firm 2. For example:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate 1 7ii = 0.15 and ni = 0.20 2 A/7ii =0.13 and A/n 2 = 170 1 maximizes its pure profit 3 iV /n i = 110 and Xin l = 0.17 4 n i = 1 0 0 a n d n 2 = 150 A complex movement is wished: starting from case 4, firm 2 wishes that firm 1 changes its objective function going in case 2, and there, assuming that firm 1 executes the movement (remember that firm 1 cannot compare case 4 and case 2), firm 1 wishes that firm 2 changes its objective function going in case 1; if we assume that firm 2 executes also the movement (firm 2 cannot compare case 2 and case 1), then both firms abandon the pure profit maximization to adopt profit rate maximization. Conversely, the opposite movement is not wished because, starting from case 1, firm 1 wishes that firm 2 keep on profit rate maximization (i.e. wishes that firm 2 do not go on case 2), and from case 2, firm 2 wishes that firm 1 remains profit rate maximizing.

Does a hoped movement is realized? This is the question of incentives. We may imagine that a firm, /, which wishes that the other firm j do a movement pay for it (for example, firm / pays to firm j , following Nash's rule, the half of what it earns by this movement): logically, firm j must accept the deal24.

It is known that in a classical framework, waiting strategies may lead to war of attrition [ Tirole 1989, p. 380], 24

Remember that even if the pure profit is lower when a firm maximizes its profit rate, the Clearly, the complex wished movement can start from a mixed case toward a mixed case, for instance from case 3 to case 2 by way of case 4. For example:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate 1 Jii = 0.12 and jt2 = 0.20 2 uni = 0.13 and u n ; = 170 1 maximizes its pure profit 3 un i = 90 and ujr; = 0.22 4 n i = 1 0 0 a n d n 2 = 150

There can be a cyclical movement, like: case 4 case 2 ->• case 1 case 3 case 4. This type of cycle in wished movement has no end, but we must remember that we said above about the incentives required to obtain a real movement. For example:

Optimal objective function 2 maximizes its profit rate 2 maximizes its pure profit 1 maximizes its profit rate 1 = 0.15 and 7i, = 0.20 2 u 7ii =0.13 and v/n2 = 170 1 maximizes its pure profit 3 A /n ' = 90 and v /tt; = 0.22 4 n ' = 100 and n 2 = 150

V. Conclusion

We have examined an alternative objective for the firm, the maximization of the rate of profit over capital instead of the maximization of pure profit. Both objectives are independent of financing decisions. Even if the conditions of entry into the sector are similar for both objectives, the optima differ. We have proved that the optimal output level of firms is lower when they adopt this objective. Thus, we have a conception of efficiency different to the classical one.

We have proved that coordination among firms may fail when the objective of firms is changed from maximizing pure profit to maximizing rate of profit. We have found some major instances in which each firm acts independently of the other. In competition with a fixed coefficient of capital, the price signal plays no role: firms are not coordinated even if they are smaller than in the classical model but there is a global equilibrium with no inelastic supply curve. If we adopt a special definition of marginal productivity (replacing interest rate by profit rate), factors may remain paid at their modified marginal productivity. In imperfect competition, in duopoly in quantities, reaction functions may disappear in some common cases, such as with a linear inverse demand function with a coefficient of capital: there is no more game among firms; however, collusion remains possible; so, firms do not consider other firms to determine their firm remains indifferent between both objective function: when the firm is profit rate maximizing, it engages less capital in production, the rest of capital may be engaged elsewhere out of this duopoly.

optimal output. In oligopoly in quantities equilibrium may not correspond to perfect competition. For competition in prices, things are more complex.

As a general (even surprising) rule, demand plays no role in the determination of equilibrium in perfect competition in any case and in imperfect competition in some cases. Does it mean that there is nothing at all in this economy? Not, because the classical concept of equilibrium remains. Remark that these results about non-coordination are independent of the rate of profit chosen: if the financial rate of profit replaces the rate of profit, they remain.

These simple cases may be seen as counter-examples of non-coordination in the standard classical model there is always coordination 25. Even if these cases are not absolutely general, there are sufficiently eloquent to prove the fragility of the behavior classically claimed for firms. Conversely, coordination appears to be an artifact, that is a consequence of the choice made about the objective function. On the other hand, it could be said that pure profit maximization is the best objective because results are "well behaved" while it is not the case with profit rate maximization; this argument is not suitable because a result is a result even it is not attractive and we cannot push back profit rate maximization only because its results are embarrassing. We must test these theories, and it could be suited to do an econometric test to see how many firms choose to maximize the rate of profit instead of the pure profit. One question again: what could be a general competitive equilibrium for profit rate maximizing producers26?

Even if some cases of coordination failures are studied in the literature; see for example [ Heller, 1986 ] [ Bagwell and[START_REF] Bagwell | Advertising and Coordination[END_REF].

26

The starting point would be the section above about factors. In this context, we must define the counterpart of the profit rate for consumers and other non-productive agents, defining the effort to buy the good and then making a ratio satisfaction / effort.

Then

  the marginal productivity ••>#») factor z K\ denoting the marginal amount of capital required by factor / and O' = O' -■= K' denoting the net marginal productivity of the factor i.

Figure 5 .

 5 Figure 5. Marginal remuneration of factorsQ* denotes the quantity of factor / used by firm when it maximizes the profit rate Q)1 denotes the quantity of factor i used by firm when it maximizes the pure profit.