Louis De Mesnard
email: lmesnard@satie.u-bourgogne.fr

THE VARIATIONS OF TECHNICAL AND ALLOCATION COEFFICIENTS ARE THEY COMPARABLE REALLY ? (WHAT DO WE MEASURE WITH PROPORTIONAL AND BIPROPORTIONAL FILTERS?)

Keywords: Input-Output, Demand-driven, Supply-driven, Change, Biproportion, RAS

Two great alternative hypothesis are possible in the analysis input-output : the model may be demand-driven (Leontief) or supply-driven (Ghosh). To test the consistency of these hypotheses on the long term, this paper studies the interest of proportional filters (comparison of column or row coefficients) and the interest of the biproportional filter for the temporal comparison of input-output matrices. An application is proposed for France between 1980 and 1993. The result is the following : in the long period, there are more sectors supply-driven than demand-driven (i.e. row coefficients are less variable than column coefficients for the majority of the sectors).

Résumé. Deux grandes hypothèses alternatives sont possibles dans l'analyse input-output : le modèle peut être piloté par la demande (Leontief) ou piloté par l'offre (Ghosh). Pour tester la consistance des deux hypothèses sur le long terme, ce papier étudie l'intérêt des filtres proportionnels (comparaison des coefficients colonne et ligne) et du filtre biproportionnel pour la comparaison temporelle des matrices input-output. Une application est proposée pour la France entre 1980 et 1993. Le résultat est le suivant: dans la longue période, il y a plus de secteurs pilotés par l'offre que pilotés par la demande (c.a.d. que les coefficients-ligne sont moins variables que les coefficients-colonne pour la majorité des secteurs).

Introduction

If z t j is the intermediate flow from sector / to sector j, and if xj is the gross output of the buyer j, the technical coefficient is ay = ^ .In matrix terms, this is :

A = Zx" 1 (1)
where x is the diagonal matrix of gross outputs, Z is the n x n matrix of intermediate flows and A is the n x n matrix of the technical coefficients. The model is : [START_REF] Leontief | Quantitative Input-Output Relations In The Economie System Of The United States[END_REF] chosen a hypothesis, fixity of technical coefficients, in order to have a production function. This implies that the model is demand-driven : only shocks on demand have an impact over the total output.

x = A x + f
With Gosh [ 1958 ], the alternative hypothesis was made : the allocation coefficients are constant. If Xj is the output of the seller /, the allocation coefficient is by = Y~ . In matrix terms it is :

B^x'Z (2)
where B is the n x n matrix of allocation coefficients.

As shown by Miller and Blair [ 1985, p. 360 ], there is a simple relation between technical coefficients and allocation coefficients :

B = x" 1 A x (3)
Denote with a star the new vectors and matrices after a shock.

As the model of Leontief isx = Ax + fox = (I-A)" 1 f, then if there is an exogenous shock, x varies and thus B varies even if A is assumed to be fixed :

A*=A . Then.- B'^x* 1 Ax**B
We may write also : B* = x' 1 xBx*

x" 1 for A* = A (4)

And reciprocally, A* = x x" 1 A x x*" 1 for B* = B (4 f )

It is impossible to assume that technical coefficients and allocation coefficients are both fixed simultaneously. We must choose one hypothesis or the other because there is an incompatibility : this is a problem. The only case where both type of coefficients may be fixed is the trivial case of homotnetical variation of the gross output of all sectors l . That is, denoting k as the ratio of homothetical variation : x* = k x . Then :

The hypothesis of Leontief seems more natural, but only because it is "technical". In a certain way, the alternative may be acceptable in the real life. Advertising creates demand. New needs are created by innovation. When a firm has an exceeding capacity of production, it may lower its price to increase demand. This shows that supply side has an action over demand. In the neoclassical model, the equilibrium of the market depends to both demand and supply. In the other hand, Gruver [ 1989, p. 449] shows that the supply-driven model corresponds to a production function with perfectly substitutable factors. Moreover, Leontief himself wanted to build an application of the General Equilibrium in its model.

In [START_REF] Oosterhaven | On the Plausibility of the Supply-Driven Input-Output Model[END_REF][START_REF] Oosterhaven | The Supply-Driven Input-Output Model : a New Interpretation but Still Implausible[END_REF], [START_REF] Miller | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models: a Comment[END_REF], [START_REF] Gruver | A Comment on the Plausibility of Supply-Driven Input-Output Models[END_REF], Rose and Allison [ 1989 ], there is a discussion about the merits and the dismerits of the supply-driven model. Particularly, for [START_REF] Oosterhaven | On the Plausibility of the Supply-Driven Input-Output Model[END_REF], the supply-driven model is implausible. Without entering into this controversy, here we want to try to verify the real plausibility of the two models not in the short term as the controversy does, but in the long term. In the short term, coefficients are supposed to be fixed. The problem is different to the question of the stability in the long term : in this case, it is known that both types of coefficients are variables as shown by the empirical studies of [START_REF] Augustinovics | Methods of International and Intertemporal Comparison of Structure[END_REF], [START_REF] Giarratani | A Supply-Constrained Interindustry Model : Forecasting Performance and an Evaluation[END_REF], Helmstadter and Richtering [ 1982 ], [START_REF] Bon | Comparative Stability Analysis of Demand-Side and Supply-side Input-Output Models[END_REF],

de Mesnard [ 1990a and b ]. Oosterhaven [ 1988, p. 206 ] asserts that formula (4), that is to say "if technical coefficients are stable, then allocation coefficients are not stable", explain why these empirical studies about the comparative stability of coefficients are inconclusive. We do not agree with this view point because the word "inconclusive" is excessive and it is not adapted to these results : "a result is a result". In fact, there are three possibilities : technical coefficients are stable and allocation coefficients are stable, allocation coefficients are stable and technical coefficients are not, both types of coefficients are unstable. In each case, we have a result. Moreover, the arguments of Oosterhaven belong to the short term : does the hypothesis of fixity of allocation coefficients is realistic and does it have an acceptable theoretical basis? The question of the variability of coefficients is different : it belongs to the long term. It is another point of view. In the long term, the right question is : does allocation coefficients are more or less variable than technical coefficients?

In order to measure the variability of coefficients, we shall work with an original and unusual tool in this context : biproportion. In our study concerning the variation in the French productive structure for the period 1970[START_REF] Miller | Input-Output Analysis, Foundations and Extensions[END_REF][ de Mesnard 1990aand 1990b ] ], we do not found much more instability in the allocation coefficients than in technical coefficients. Yet, this work used biproportion, but the principal aim of this work was to measure change without favor the demand-side or the supply-side hypothesis and the conclusion about compared instability of technical coefficients and allocation coefficients was only a marginal result. Here, we want to do a specific study, completely explaining the method, and using more recent data.

II. The classic method : proportional filter

Classically, one may compare technical coefficients or allocation coefficients. We shall develop these two ways and then we shall prove that it is impossible to compare the results provided by one method with those provided by one another method.

Comparison of technical coefficients

If we compare two matrices of technical coefficients for the same country at two different dates, we can measure instability about technical coefficients (it is the founding job of Leontief, followed by Carter, etc.) and we put the demand-driven hypothesis, but we have no information over allocation coefficients because the growth of the total output of the sectors is differential (except in the trivial case of homothetical growth).

Consider two matrices of flow, Z and Z* .

We may want to compare technical coefficients ^ and -A-.

Now, we shall adopt a special but equivalent point of view : the above comparison is equivalent to compare r,y -j-and r* .

We see that the principle is to apply a simple proportion over the terms of the rows of the matrix Z (by the ratio of old gross output by new gross output such as the result have the same column margins than Z* ) and to compare the result to the matrix Z*.

In fact, when we compare matrices of technical coefficients, we confuse two phenomenon : the variation of the distribution of technical coefficients themselves and the variation of the other terms, essentially the added value, following the identity :

To study of the distribution of technical coefficients, it could be opportune to leave aside the study of the variation of the added value,

In matrix terms, we compare Z(SZ) (SZ*) to Z* .Note that it is equivalent to compare Z* <SZ)(SZT l and Z.

To do so, we may calculate the difference matrix Z(SZ)~l (SZ*)-Z* and then, we may calculate the absolute variability of columns Ic :

m m 2 Zij + Vj = xj <z> £ Uij + Vj• = 1 , /=1 7=1 W '
where Wj is the added value and Wj = comparing It is the norm of Frobenius of the column j and it is homogeneous to a flow because it is calculated in terms of unit of money.

A more significant indicator is the relative variability :

CJC, =

This number with no dimensions, a percentage, allows to compare variabilities of rows dividing the absolute variability of a row by the total of the row (answer to the question : "what is the variability for a sector buying one unit of money of intermediary goods?").

B. Comparison of allocation coefficients

Reciprocally, if we compare two matrices of allocation coefficients, we put the supply-driven hypothesis but we have no information about technical coefficients because the growth of the total input of the sectors is differential.

To compare allocation coefficients, we compare Y~ and -4-, what is equivalent to compare ^r-x] and , and we may separate the variation of the final demand to the variation in the distribution of In matrix terms, we compare (Z S) 1 (Z* S) Z to Z* .

We may calculate also the absolute variability of columns :

and the relative variability :

C. Comparability of these comparisons

Variances of technical coefficients and variances of allocation coefficients are not comparable, because the variance of the first type of coefficients depends on the margins of rows, and the The row coefficients of sectors 1 and 4 have changed, a little more for sector 1 than for sector 4 However, as

Z Z I Z <S Z> _1 <S Z*> ^/=ly=lV V > ij and as, EE ( f(ZS> _1 <Z* S> z) - \/=iy=iV v ' ij = 5.380 5.690 ,
the norm of Frobenius of the difference matrices are not equal, thus, we cannot compare with this method the variability of the column 4 with the variability of the row 4, and the variability of the row 1 with the variability of the column 1.

III. The biproportional method

To see if instability is more important for columns (and the Ghosh's hypothesis is more realistic) of if the instability is more important for rows (and the Leontief s hypothesis is more realistic), we want to measure the variability from one matrix to the other. Thus, it is necessary to have a method allowing to compare two matrices (for the same economy at two different dates) with the same margins. However, we cannot compare directly two input-output matrices, because margins of rows of both matrices are not identical and because margins of columns of both matrices are not identical : Z -Z* indicates as much the variability of margins as the variability of the coefficients. It is why we suggest a biproportional method, doing a transformation of one matrix in such a way that it has the same margins than the other, either for rows and columns : the way to perform it is biproportion 2 .

See the annex for a remind about biproportion.

A. First step

We do a biproportion of one matrix over the other, for example K(Z, Z*) or K{Z\Z) \

We see that we have two series of computations : if Z is previous to Z* , the first calculation is direct and the second calculation is reverse. Unfortunately, results will not be identical because biproportion is not a linear operator. To be rigorous, it is necessary to do both calculations and to compare results.

Remark. There is an important difference between the biproportional approach and the classical approach : in the classical approach, coefficients are calculated with the gross outputs x and not with the margins of the matrix Z , whereas in the biproportional approach, the projection of one matrix is made over the margins of the second matrix, and the gross output plays no role.

Second step

We calculate of the difference between the projected matrix and the matrix which provided the margins :

K(Z, Z*)-Z* for the direct calculation,
or K(Z\Z) -Z for the reverse calculation.

The difference matrix correspond to the "structural change" between Z and Z\ that is to say the change when all effects due to the variation of the size of sectors are removed. Remember that AT(Z,Z*) and Z* have the same row and column margins, and reciprocally for K(Z\Z) and Z . Thus, both row and column of the difference matrix Af(Z,Z*)-Z* are nil (reciprocally for A r (Z*,Z)-Z ). In the difference matrix, we read directly the terms (/,y) that have change the more.

C-Third step

We calculate the variation in unit of money of rows and columns, denoted L# and Lr respectively, that is to say a calculation of the norm of Frobenius for the row vectors and for the column vectors over the difference matrix :

1«, = l|(^(Z,Z-),-4) 2 and Z CJ = j|(/:(Z,Z*),-4.) 2

3

As said above, we choose to take as margins the sums of the terms of the rows or columns, and not the gross productions of the sectors : this is a difference with the simple approches by technical or allocation coefficients where the denominator is the gross production of the sectors This norm, homogeneous to a flow, measures the absolute variability of the supplying sector (row vector) or buying sector (column vector).

Remark. For each calculation, the difference matrix is unique, thus the sum of the squares of the norms of rows is equal to the sum of the squares of the norms of columns. It is not the case with the naïve approach of technical coefficients matrix or allocation coefficients matrix : the difference matrix is not the same with technical coefficients than with allocation coefficient (that is to say A* -A * B* -B ). These two approaches are irreconcilable, when they are with biproportion. And we have for the direct calculation :

I (i«,) 2 = IS (*(z, z*)" -J ' = I f i« ( whereas, E *E generally. /=1 V/=i v y/ J /=i V/=i v y/

J

and reciprocally for the reverse calculation. In other words, for each calculation, we have a unique difference matrix, allowing to compare result of variability for columns (technical coefficients) and for rows (allocation coefficients) 4 .

D. Fourth step

We calculate the relative variabilities OR, and oc ; . The relative variability is the ratio of the absolute variability of the supplying sector (row vector) to the corresponding row margin (and reciprocally for column vectors). It allows to compare variabilities of sectors, expressing it in percentage. They are :

]|(*<z,z%-4) 2 J|(k(z,z-)(/ --) 2

GR, =

and GCj = -J for the direct calculation, or, 4 Do not confuse between the two matrices of difference for technical and allocation coefficients, and the two matrices caused by the existence of two calculations, direct and reverse. Also, do not confuse between the method and ANOVA : in ANOVA, calculations are made over one unique matrix (there is no time); here we compare two matrices over time.

£(A-(Z\Z) y -_-*

and or, =

!!(a-(z\z) (/ -_-;

&R, =

for the reverse calculation.

E. Interpretation of the results of the biproportional method

1.

Rules of interpretation

We have four typical cases, depending on the sectors :

• If a sector is more variable in row than in column, i.e. a/*, > oc, »0 , it is demand-driven.

• If a sector is more variable in column than in row, i.e. 0«<r#, < ar 7 , it is supply-driven.

• If a sector is not variable in row and in column, i.e. CR, = or, -> 0 , we can say nothing, except that the sector is very stable.

• If it is so variable in row and in column, i.e. c Ri = GCJ » 0 , we can say nothing, except that the sector is strongly unstable.

These cases may be combining : some sector may be in the first case, some other in another case, and so on. So, we may make a typology of sectors, without any a prion hypothesis about the direction of the economy, demand-driven or supply-driven. These simple comparisons of relative variabilities are sufficient: in practice, it is not necessary to build some complicated indices.

Example 1 (following).

The biproportional projection is :

It is clear that this matrix is, in a certain way, intermediary between the proportional projected matrices Z <S Z)" 1 <S Z*> and (Z S> _1 <Z* S> Z , except for the terms (1,4) , (4,1) , (1,1) , and (4,4). Not surprisingly, the results are not very clear because for sectors 1 and 4 both row and column have been changed of a comparable amount.

K(& z-) = 3.

The case of proportional changes

Following However, all this is not true if not all coefficients are identical.

There is an apparent paradox. If all technical coefficients of Z and Z* are identical, or if all column coefficients (that is to say zr^-) of Z and Z* are identical, then all variabilities are nil.

This seems surprising because one may think that row variabilities must be not nil because the allocation coefficients (or the row coefficients) of Z and Z* are not identical. There, the method says : "technical coefficients do not vary and the rest of the variations follow from this fact directly". Obviously, the same result exists if all allocation or row coefficients of Z and Z* are identical.

In other words, the method disregard the proportional variations of the flows, that is the variations which belong to the following types : z]j = Zij ^~ (column fixity), or z]j = z 0 ^ (row fixity) and even r* = z tj ? J* (x 2 fixity).

Thus, if we find that all variabilities (for column and row) are nil, we must think that either Z = Z*, or Z* follows from Z by a proportional variation : in this case, it is simple, by an elementary calculation, to see if it is either all technical coefficients either all allocation coefficients which are stable. In other cases, both technical coefficients (or column coefficients) and allocation coefficients (or row coefficients) have varied.

This apparent paradox is very important to understand what the method do. The biproportional method filters the "autonomous" variations of the terms of the flow matrices. In the above case, the technical coefficients or the column coefficients remain fixed and this force the allocation coefficient or the row coefficient to vary : every thing is as if the variation would be not "voluntary" and as if it would be a "forced" variation. Thus, the method disregard this type of regularity. In fact, the method consider only the variations of the coefficients which are independent to margins. What it measure is the rest of the variations, what we call "autonomous".

Remark. Suppose that we search the input-output table which is the nearest to the old matrix of technical coefficients with respect to Z\ Then we obtain the same result because the theorem of ineffectiveness of separability 5 :

K(A, Z*) = K(Z x'\ Z*) = K(Z, Z*) . Example 2.
We take the same matrix Z than in example 1 , but we chose another matrix Z* Z = 2 15 6 4 4 12 3 6 3 1 5 4 2 4 and Z* = 4.000 1.333 6.818 4.615 8.000 5.333 1.364 1.538 6.000 8.000 4.091 0.769 10.000 5.333 2.727 3.077

We have K(Z, Z*) = Z* and ÀT(Z\ Z) = Z; all variabilities are nil with the biproportional filter. Why? Because Z and Z* have the same column coefficients (we denote by À and À* the matrices of column coefficients) : The S c are 1.049 , 0.402 , 0.968 , 2.364 and the o c are 3.75% , 2.01% , 6.45% , 15.36% .

A = A* = 0.
The Sfl are 2.340 , 0.680 , 0.470 , 1.276 and the G R are 10.56% , 4.19% , 2.49% , 6.03%.

Beyond the general fixity of column coefficients, the row of sector 4 change less than its column, the column of sector 1 change less than its row; for sectors 2 and 3 , the results are less clear even if the row of sector 2 change less than its column and even if the column of sector 3 change less than its row. Similar results will be found for the reverse calculation :

K(Z", Z) =
1.533 0.757 4.217 7.493 4.168 4.119 1.147 1.567 3.004 5.937 3.306 0.753 5.296 4.187 2.331 3.186

The Ir are 0.578 , 0.334 ,0.915 , 1.77 and the or are 4.13% , 2.23% , 8.32% , 13.63% .

The 1 R are 1.767 , 0.501 , 0.398 , 0.945 and the a R are 12.62% , 4.55% , 3.06% , 6.31% Following the above remark, K(X, Z" j = K(Z, Z"). Then it is unuseful to search to conserve the fixity of column coefficients even if it is known that Z** come from Z* which has the same column coefficients than Z : the result would be the same. In other words, no information is lost if one works on Z instead of À . We retrieve the apparent paradox. Only the column coefficients of sector 4 have changed between Z* and Z" or between Z and Z**, as shown by the simple proportional analysis : only the first row have changed compared to the preceding matrix (Z S) -1 <Z* S) Z : with the proportional method applied to rows, we detect that the row coefficients of sector 1 have changed in Z" compared to Z* , but we cannot say anything about the direct comparison between Z and Z" .

IV. Application and results

We shall work on French input-output tables, comparing the years 1980 and 1993 . We work on the prices of 1980 to remove price effects.

In the following tables, the first column indicates the relative variation or for the columns (technical coefficients), the second column the relative variation GR for the rows (allocation coefficients) and the third column, denoted as the "gap", the simple difference between the first two columns. If the gap is positive, the allocation coefficients are more stable than the technical coefficients. However, the level of the gap is not the only parameter : the level of the relative variations gets an importance.

Note that "Trade" and "Non Marketable Services" have not a row, thus there is not allocation coefficients for these sectors. It remains 34 sectors with both types of coefficients.

A. Proportional filter

The norm of Frobenius of the difference matrix for column coefficients is 462341 billion of Francs; for the row coefficients, it is 425011 billions of Francs . This and the following table 1 shows that both type of coefficients are variable. Remember that we must not compare the variations of column and row coefficients.

However, we may note the considerable variation of both coefficients for "Services of Financial Institutions", the important variation of column coefficients of "Electricity, Gas and Water" and of "Telecommunications and Mail" and the important variation of row coefficients for "Domestic Equipment Goods for Households".

Direct projection

Firstly, note that we retrieve some of the biggest relative variations found by the proportional filter, but there is in addition the "Solid Mineral Combustibles and Coke" and "Marketable Services to Private Individuals". However, our purpose is not to search the biggest relative variations but to compare the relative variations of both type of coefficients. For example, "Electricity, Gas and Water" and "Insurances" have such big differences : for these sectors, technical coefficients are much more unstable than allocation coefficients.

Then, about the comparison of the relative variations of column and row coefficients, over 34 sectors, for 20 sectors, allocation coefficients are more stable than technical coefficients, and this is the contrary for the remaining 14 sectors.

The 17 sectors which are clearly more supply-driven than demand-driven are, by decreasing order :

Comparison of the two ways of projection

The two ways of projection, direct and reverse, give a pretty good convergence. Compared to the direct projection, there is a clear "discordance" only for : Even if we remove these discordant sectors, there are more sectors which are supply-driven than sectors which are demand-driven.

V. Conclusion

To compare technical coefficients and allocation coefficients in an input-output table, in the first part of the paper, we have presented the classical methods based on a proportional filter to detect what coefficients have changed : they are equivalent to the comparison of two matrices of flow with either the same column margins, either the same row margins. We prove that the change of technical coefficients is not comparable to the change of allocation coefficients with this type of methods.

Then, we have explored the possibilities of the biproportional filter, after recalling the principle of it, which consist into a biproportional projection of one matrix over another matrix, and a calculation of the variation between the projected matrix and the another matrix.

This method allows comparison between the change in column and row coefficients. However, it discard the proportional change in flows, that is to say the change which conserve the stability either of all column coefficients either of all row coefficients : if for an input-output table all the column coefficients or all the row coefficients remains fixed, then the method does not found a variability.

As an application, we work with French data to detect if the input-output model is demand-driven or supply-driven, comparing two tables, 1980 and 1993 .

In the long period considered here, instability is not concentrated over one single type of coefficients : roughly, an amount of half the sectors are more stable for technical coefficients and an another half are more stable for allocation coefficients, even if there are more sectors that ATf U, U J = U where U has the same margins than U )•

B. Some discussions about biproportion

The biproportional approach suggested here is in fact a generalization of the relative Joint stability defined by [START_REF] Chen | The joint stability of Input-Output Production and Allocation Coefficients[END_REF] 7 . Consider an old situation and a new situation where the technical coefficient has varied; there is relative joint stability if :

«y = a

X

where "xty-is the ratio of the new output of / over old output of /

and a]j is the new technical coefficient and or,y is the ancient technical coefficient.

Thus, a*. = ^f-\Xj J -fi.

* X,-

and the flow r,y is corrected by the ratio of the new to the old output of the supplier /. This is the term of correction in the first step of a RAS method: it is know that if we correct alternatively rows and column in order to respect new margins, we make a RAS-projection 8 , that is to say, a biproportional projection in accordance with the theorem of [START_REF] De Mesnard | Unicity of Biproportion[END_REF] .

Thus the definition of the relative joint stability, may be generalized as the relative joint global stability using biproportional projection : if AT(Z,Z*) = Z* then there is a relative joint global stability for the direct projection (respectively K(Z\Z) = Z for the reverse projection).

As well pointed out by [START_REF] Miller | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models: a Comment[END_REF], Deman made an important countersense in its paper [ 1988, p. 815 ]. Deman writes the relation which insures that allocation coefficients rest fixed after a perturbation 9 : 7 See also Rose and Allison [ 1989 ].

8

In the RAS method, the terms of rows are corrected by the ratio of the new margins over the old margins; then the column margins are recalculated; if these new columns margins are not equal to old margins, the terms of columns are corrected by the ratio of these new margins over the old. If it does not hold, a new correction is made in rows, etc. 9 It is a pity because Deman confuses between scalars and matrix notations. We use here consistent notations. Denoting R = x* x" 1 and S = xx H , Deman gets : A* = R A S where R and S are diagonal matrices. Thus, Deman likens this formula to the RAS procedure, that is to say to a biproportional formula. According to the section where we defined biproportion, we are sorry to say that the point of view of Deman is completely wrong because there are no margins in its reasoning. The fact to premultiply and postmultiply a matrix by diagonal matrices does not make a biproportion. What Deman forgot is that the premultiplying and postmultiplying diagonal matrices are not ordinary : they allow margins to hold. In Deman's equation A is only proportional to A*, not biproportional.

Note that, the projections of technical coefficients is not equal generally to the technical coefficients of the projection :AT(A, A*) = AT^Zx _1 , Z* (x*) ! ) = K{Z, Z* (X*) ! ) is not genarally equal to K(Z, Z*) (x*) 1 . This comes from the following theorem : Theorem 2. K(V 9 Q M T) is not equal generally to Q K(U, M) T .

Proof. It is simple : the margins of Q M T are not generally equal to the margins of Q K(V, M) r , because if there exists one term (/,/) such as m tJ * K(V, M) f y , then YJ ~ij Gty * 2 AT(U, M) f y coy and 2 "</ Y/ * £ ^(U, M) /y y, generally (where the co ; and y, are the j / ' ' terms of the diagonal matrix Q and r respectively). •

  (Z',Z), -_-;J and S<-= |£ (/(Z\Z) (/ for the reverse calculation.

  theorem 1 in annex, if all technical coefficients of Z and Z* are identical (reciprocally, if all allocation coefficients are identical), then Z* = Z Q (reciprocally Z* = <t> Z ) and then AT(Z,Z*) = Z* and K(Z\Z) = Z . Consequently, the variabilities are nil. Note that we have in this case K(A,A*) = A\ K(A\\) = A and £(B,B') = B\ £(B',B) = B.

Theorem 1 .

 1 K(H<t> V Q) = <PV Q Proof. It is obvious : K(U,<t> II Q) = K(<t> U Q,0 II Q) from preceding theorem, thus, AXU,OUQ) = <DUQ.,because K(\J, U) = U (it is a particular case of the Theorem 1 in[START_REF] De Mesnard | Unicity of Biproportion[END_REF] which says

  as B -x" 1 Z and B* = x* H Z* , then B -B* => x 1 Z = x* H Z* oZ* = x x" 1 Z (it is equation 15 of Deman). As for technical coefficients A-Zx" 1 => Z = A x and A* = Z* x*~l => Z* = A* x* (it is equation 16 of Deman), then equation 15 of Deman becomes : A* x* = x* x" 1 A x <=> A* = x* x" 1 A x x* H .

y -.y £(-/y --,yj ^2-,. -,. -;/) /=1

  are :4.155 ,3.887. The relative variabilities a R are : 20.78% , 19.44% .

	variance of the second type of coefficients The proportional row projection is	depends on the margins	of columns.	In other	words,
	the column	margins	of the difference	matrix	Z	(S	Z>~'	(S	Z*)	-	Z*	are nil (and row margins are
						1.429 7.143 8.571
	not nil), where	it is the row margins of the difference	matrix	(Z	S)~l	(Z*	S)	Z -	Z*	which	are nil
	(and (generalize Euclidean distance) is not equal to those of the second generally column margins are not nil); and the norm of Frobenius of the 4 4 1 2 <ZS>"' <Z* S)Z = 6 1	first :	difference	matrix
					' 5.333 2.667 5. JJJ
	jit((z(Sz>-	<sz->) -.-;) 2 0.857 0.429 2.143 -•3.429 * jii((<zs>-	<z-S)z)^.-;) :
	or, (Z S> _1 <Z* S>Z-Z* =	0 0	0 0		0 0			0 0
					-3.333		0.667 1.333
	2 -.;=l		y	;=1	y=l v			y
	Thus a direct comparison between the results of an analysis in technical coefficients and an
	analysis in allocation coefficients is not mathematically correct. It seems to be more suitable to
	fix all margins, in order to eliminate their effects when we measure the variance of columns or
	rows.
	Note that we may build an hybrid index, dividing each of the difference matrix by the norm of
	Frobenius of the matrix, but this is not very "clean" and it has no justification.
	Example 1.
		2 15 6			2 1 5 12
	Z =	4 4 12 3 6 3 1	and Z* =	4 4 12 3 6 3 1	(coefficients (1,4) and (4,1) are doubled).
		5 4 2 4			10 4 2 4
	The proportional column projection is :
					2.714 1 5 8.769
	Z<SZ>"' <sz*> =	5.429 4 1 2.923 4.071 6 3 1.462
					6.786 4 2 5.846
						0.714 0 0 -3.231
	z<sz> _1 <SZ*)-Z* =	1.429 0 0 0.923 1.071 0 0 0.462
					-3.214 0 0 1.846
	The absolute variabilities Sc are :3.746 , 3.862 .
	The relative variabilities CTC are : 19.71%, 20.32%.
	We see that the column coefficients of sectors 1 and 4 have changed, a little more for the sector
	4 than for the sector 1 .

LR

  TheSr are 2.96, 0.92, 0.87, 2.94; the G C are 15.6% , 6.1% , 7.9% , 15.5%.The I* are 2.72, 0.80, 0.93, 3.18; the G R are 13.6% , 7.3% , 7.1% , 15.9%.

	318 1.276 5.672 9.734
	4.529 3.483 0.774 2.214
	3.664 5.635 2.506 1.194
	7.488 4.606 2.048 5.858
		1.318 0.275 0.672 -2.266
	K(Z 9 Z*)-Z* =	0.529 -0.517 -0.226 0.214 0.664 -0.365 -0.494 0.194
		-2.512 0.606 0.048 1.856
	See annex.

T37 Services of Financial Institutions 85,51% 80,22% T38 Non Marketable Services 5,36%

		variation of
		column	variation of row
		coefficients	coefficients
	T01 Agriculture, Sylviculture, Fishing	7,61%	4,06%
	T02 Meat and Milk Products	5,63%	5,04%
	T03 Other Agricultural and Food Products	5,42%	3,46%
	T04 Solid Mineral Combustibles and Coke	9,83%	7,37%
	T05 Oil Products, Natural Gas	7,51%	9,24%
	T06 Electricity, Gas and Water	33,61%	7,30%
	T07 Mining and Ferrous Metals	5,14%	6,41%
	T08 Mining and non Ferrous Metals	2,18%	15,46%
	T09 Building Materials, Varied Minerals	6,09%	1,55%
	T10 Glass	7,61%	7,26%
	T11 Basic Chemicals, Synthesized Fibers	3,88%	8,78%
	T12 Parachemestry, Pharmaceuticals	4,74%	14,06%
	T13 Smelting Works, Metal Works	4,51%	3,83%
	T14 Mechanical Construction	7,51%	5,08%
	T15A Electric Professional Engineering	4,41%	6,09%
	T15B Domestic Equipment Goods for Households	10,62%	23,33%
	T16 Motor Cars for Land Transport	7,70%	5,41%
	T17 Shipping, Aeronautics., Weapons	13,35%	13,34%
	T18 Textile Industry, Clothing Industry	5,35%	9,00%
	T19 Leather and Shoe Industries	7,89%	10,71%
	T20 Wood, Furnitures, Varied Industries	3,97%	10,96%
	T21 Paper, Cardboard	3,35%	3,54%
	T22 Press and Publishing	6,44%	4,66%
	T23 Rubber, Transformation of Plastics	8,72%	7,04%
	T24 Building Trade, Civil and Agricultural Engineering	6,32%	7,52%
	T25-8 Trade	5,47%
	T29 Car Trade and Repair Services	7,38%	6,82%
	T30 Hotels, Cafes, Restaurant	4,36%	3,42%
	T31 Transports	5,50%	5,00%
	T32 Telecommunications and Mail	26,89%	7,82%
	T33 Marketable Services to Firms	5,72%	4,96%
	T34 Marketable Services to Private Individuals	5,11%	6,89%
	T35 Hiring, Leasing for Housing	8,60%	4,07%
	T36 Insurances	11,14%	7,02%

Table 1 . Proportional projection : relative variations 6. Biproportional filter 1.

 1

57% T34 Marketable Services to Private Individuals 15,92% 15,34% 0,58% T35 Hiring, Leasing for Housing 12,70% 22,59% -9,89% T36 Insurances 13,74% 3,56% 10,19% T37 Services of Financial Institutions 34,48% 33,27% 1,21% T38 Non Marketable Services 6,20%

  For the following three sectors, row coefficients are more stable than column coefficients, but not so much :

	variation of
	column coefficients 5,10% T34 Marketable Services to Private Individuals. T03 Other Agricultural and Food Products T30 Hotels, Cafes, Restaurant T01 Agriculture, Sylviculture, Fishing T02 Meat and Milk Products 3,51%	variation of row coefficients 2,21% 2,42%	gap 2,89% 1,09%
	T03 Other Agricultural and Food Products	6,93%	6,70%	0,24%
	The 10 sectors that are clearly more demand-driven than supply-driven are : T04 Solid Mineral Combustibles and Coke 14,27% 23,58%	-9,31%
	T05 Oil Products, Natural Gas T24 Building Trade, Civil and Agricultural Engineering 3,95% T06 Electricity, Gas and Water 27,83%	5,87% 5,70%	-1,92% 22,13%
	T29 Car Trade and Repair Services T07 Mining and Ferrous Metals	6,20%	3,17%	3,02%
	T35 Hiring, Leasing for Housing T08 Mining and non Ferrous Metals	3,49%	11,77%	-8,28%
	T33 Marketable Services to Firms T09 Building Materials, Varied Minerals	9,41%	4,66%	4,76%
	T10 Glass T15B Domestic Equipment Goods for Households T11 Basic Chemicals, Synthesized Fibers T04 Solid Mineral Combustibles and Coke T12 Parachemestry, Pharmaceuticals T08 Mining and Non Ferrous Metals T13 Smelting Works, Metal Works	9,84% 8,03% 7,41% 8,58%	2,78% 2,63% 8,92% 2,50%	7,07% 5,39% -1,51% 6,08%
	T32 Telecommunications and Mail T14 Mechanical Construction	9,72%	6,54%	3,18%
	T22 Press and Publishing T15A Electric Professional Engineering	9,16%	4,68%	4,48%
	T21 Paper, Cardboard. T15B Domestic Equipment Goods for Households	8,86%	18,40%	-9,54%
	T16 Motor Cars for Land Transport	9,08%	2,28%	6,80%
	T17 Shipping, Aeronautics., Weapons Column coefficients are just a little more stable for the following four sectors : 14,91% 16,63% T18 Textile Industry, Clothing Industry 8,17% 2,18%	-1,72% 6,00%
	T05 Oil Products, Natural Gas T19 Leather and Shoe Industries T06 Electricity, Gas and Water T36 Insurances T10 Glass T16 Motor Cars for Land Transport T20 Wood, Furnitures, Varied Industries T17 Shipping, Aeronautics., Weapons T21 Paper, Cardboard T12 Parachemestry, Pharmaceuticals T22 Press and Publishing T20 Wood, Furnitures, Varied Industries. T23 Rubber, Transformation of Plastics	6,38% 4,66% 9,09% 6,72% 6,16%	3,88% 5,51% 13,47% 12,59% 4,18%	2,50% -0,85% -4,39% -5,87% 1,98%
	T24 Building Trade, Civil and Agricultural Engineering T09 Smelting Works, Metal Works T25-8 Trade T18 Textile Industry, Clothing Industry T29 Car Trade and Repair Services Til Basic Chemicals, Synthesized Fibers T30 Hotels, Cafes, Restaurant	18,52% 13,14% 20,42% 4,41%	35,85% 30,71% 4,32%	-17,33% -10,29% 0,10%
	T09 Building Materials, Varied Minerals T31 Transports	8,66%	4,15%	4,50%
	T31 Transports T32 Telecommunications and Mail	25,61%	33,69%	-8,08%
	T15A Electric Professional Engineering T33 Marketable Services to Firms	5,18%	14,74%	-9,
	T14 Mechanical Construction
	T07 Mining and Ferrous Metals
	TOI Agriculture, Sylviculture, Fishing
	T19 Leather and Shoe Industries
	T23 Rubber, Transformation of Plastics
	T37 Services of Financial Institutions
	T02 Meat and Milk Products.

Table 2 . Direct biproportional projection 1980 over 1993 : relative variations 2.

 2

	Reverse projection
	Over 34 sectors, 17 are more stable for allocation coefficients and 17 are more stable for
	technical coefficients.
	The 14 sectors which are more supply-driven than demand-driven are :
	T06 Electricity, Gas and Water (above all)
	T32 Telecommunications and Mail
	T36 Insurances
	T35 Hiring, Leasing for Housing
	T31 Transports
	T09 Building Materials, Varied Minerals
	T16 Motor Cars for Land Transport
	T01 Agriculture, Sylviculture, Fishing
	Tl 1 Basic Chemicals, Synthesized Fibers
	T13 Smelting Works, Metal Works
	T07 Mining and Ferrous Metals
	T19 Leather and Shoe Industries
	T10 Glass
	T30 Hotels, Cafes, Restaurant
	Row coefficients are just a little more stable than technical coefficients for three sectors :
	T23 Rubber, Transformation of Plastics
	T18 Textile Industry, Clothing Industry
	T21 Paper, Cardboard
	The 11 sectors which are more demand-driven than supply-driven are :
	T37 Services of Financial Institutions (above all)
	T15B Domestic Equipment Goods for Households
	T08 Mining and non Ferrous Metals
	T12 Parachemestry, Pharmaceuticals
	T24 Building Trade, Civil and Agricultural Engineering
	T04 Solid Mineral Combustibles and Coke
	T03 Other Agricultural and Food Products
	T17 Shipping, Aeronautics., Weapons
	T05 Oil Products, Natural Gas
	T14 Mechanical Construction
	T29 Car Trade and Repair Services

83% T23 Rubber, Transformation of Plastics 4,56% 4,10% 0,46% T24 Building Trade, Civil and Agricultural Engineering 2,88% 7,58% -4,71% T25-8 Trade 3,35% T29 Car Trade and Repair Services

		3,96%	5,68%	-1,72%
	T30 Hotels, Cafes, Restaurant	2,48%	1,65%	0,83%
	T31 Transports	5,37%	1,95%	3,42%
	T32 Telecommunications and Mail	17,03%	8,37%	8,66%
	T33 Marketable Services to Firms	2,74%	3,47%	-0,73%
	T34 Marketable Services to Private Individuals	3,68%	4,53%	-0,85%
	T35 Hiring, Leasing for Housing	10,76%	5,49%	5,27%
	T36 Insurances	10,85%	4,01%	6,84%
	T37 Services of Financial Institutions	49,61%	68,09%	-18,48%
	T38 Non Marketable Services	4,86%

Table 3 . Reverse biproportional projection 1993 to 1980 : relative variations 3.

 3

Do not confuse between mathematical algorithms and computing algorithm : for a same mathematical algorithm, there may be several computing algorithm. For example, there are many mathematical methods to calculate n (method of Archimede, the formula of Machin using arctan, the modular equation of Ramanujan, etc.), and for each one, there are several way to do the computation.

Rose, A. and T. Allison. 1989. "On the Plausibility of the Supply-Driven Input-Output Model : Empirical Evidence on Joint Stability", Journal of Regional Science, 29, 451-458. Torre, A. 1993. "Sur la signification théorique du modèle d'offre multisectoriel", Revue Economique, 5, 44, 951-970.

VII. Annex : biproportion

We shall remind what is exactly biproportion and then sum up some discussions about it.

A. Definition of biproportion

Unfortunately, [START_REF] Deman | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models[END_REF], [START_REF] Gruver | A Comment on the Plausibility of Supply-Driven Input-Output Models[END_REF], and later Torre [ 1993 ] describe equation (3) as "biproportional". It is a pity, because it is confusing with another concept, very different: biproportion to the sense of [START_REF] Bacharach | Biproportional Matrices and Input-Output Change[END_REF].

The real biproportion, to the sense of Bacharach, is :

V is biproportional to a nxmmatrix U, with respect to the margins of a n x m matrix M , denoted V = K(U, M) if: V = PUQ where P and Q are n x n and m x m diagonal matrices, such as the matrix V have the same margins than the given matrix M , that is to say S' V = S 7 M and V S = M S , where S is the sum vector. All matrices II and M must have no negative terms, thus matrix V have no negative terms.

In the following, we denote as biproportion the biproportion in the sense of Bacharach. The method RAS is a biproportional method. There is potentially an infinity of mathematical algorithms to calculate biproportion. For example, the following :

Pi = m

for every /, and q } = for every / /•=i or in matrix terms : P = (<U Q S,,,)" 1 M S,") and Q = (S 7 M <S 7 P U)~l) .

The solution of this algorithm is found iteratively. The properties of convergence and existence of the solution of this algorithm are known [ deMesnard 1990a and b ]. It is also the case for another biproportional algorithm, RAS [START_REF] Bacharach | Biproportional Matrices and Input-Output Change[END_REF] ], [START_REF] Bachem | On the RAS-Algorithm[END_REF].

To measure the speed of convergence, systematic studies must be done with computers , but with real input-output tables (34x36), it converges fastly in 10 or 20 iterations [deMesnard 1990a and b ].

As there are many mathematical possible algorithm, one may think that the solutions of each algorithm differ. However, in [ de Mesnard 1994 ], we proved that all types of biproportion give the same solution however be the mathematical algorithm 6 , those of RAS, those above, or another. This theorem says that if you consider a biproportion of U over the margins of M made with the above algorithm : V = K(V, M) = P U Q ; and if you consider another biproportion of U over the margins of M with another unknown algorithm : X 7 = A^(U,M) = P 7 II Q 7 , then, V 7 = V . This important result give a real theoretical status to biproportion, which becomes so "unique" than proportion in the world of scalars and vectors. Also, the matrix V is the nearest to the matrix U with respect to the margins of M in the sense of the theory of information, the theory of interactions, the gravitation, and so on [ de Mesnard 1990 ] . For example, with the theory of information, biproportion minimizes n m y.. This theorem indicates that the terms P and Q are not identified in the sense of econometrics : you can multiply P by <I> and Q by Q without change the result. A consequence is :


Il y avait déjà longtemps que le comte M osca était de retour à Parme, comme premier ministre, plus puissant que jamais... Les prisons de Parme étaient vides, le comte immensément riche, Ernest V adoré de ses sujets qui comparaient son gouvemment à celui des grands-ducs de Toscane (Stendhal).

Introduction

That in democracies more inequality leads to more redistribution is an implication of Allan Meltzer and Scott Richard's well-known model ( 1981).1 That, in turn, more redistribution leads to less growth is a generally accepted proposition. That "inequality is harmful for growth" (Persson and Tabellini, 1994) is thus the predictable result of the introduction of policy-making à la Meltzer and Richard into the theory of growth. The small literature in which such introduction has been attempted includes contributions by Alberto Alesina, Giuseppe Bertola, Roberto Perotti, Thomsten Persson, Dani Rodrik, Gilles Saint-Paul, Guido Tabellini and Thierry Verdier. Short surveys are provided by [START_REF] Perotti | Income Distribution, Politics, and Growth[END_REF], Persson and Tabellini (1992b) and [START_REF] Verdier | Models of Political Economy of Growth: A Short Survey[END_REF]. The proposition that inequality of income or wealth, measured at one point of time, has a negative influence on subsequent growth is derived by all these authors with the exception of Saint-Paul and [START_REF] Saint-Paul | Education, Democracy and Growth[END_REF], and some empirical support for it is displayed in [START_REF] Alesina | Distribution, Political Conflict, and Economic Growth: A Simple Theory and Some Empirical Evidence[END_REF]Rodrik (1992, 1994) and in [START_REF] Persson | Growth, Distribution, and Politics[END_REF]Tabellini (1992a, 1994).

For these two pairs of contributors -and, with qualifications, in some of the articles of the others [START_REF] Bertola | Factor Shares and Savings in Endogenous Growth[END_REF]Perotti, 1993) -the causality is the following: inequality of income or wealth generates, through a political mechanism identical or analogous to the one proposed by Meltzer and Richard, redistributive policies (e.g. taxes and transfers) that are unfavourable to investment (tangible investment and/or investment in human capital), hence to economic growth (see Figure 1). In the simplest models (Persson and Tabellini, 1992a), collective policy-making consists in fixing the rate of a linear income tax and redistributing the proceeds of that tax equally to all individuals. The only issue is the tax rate, voter preferences are single-peaked, and the median voter sets the policy according to his or her own cost-benefit calculation. This is modelled as the maximisation of a utility function whose form ensures that the income distribution and consequently the identity of the median voter remain the same in the future.2 The larger the distance between the median income and the average income, the higher the tax rate and larger the transfer which the median voter chooses. The higher the tax rate, the lower the after tax returp of individual investments and thus the level of aggregate investment. The lower the latter, the smaller the rate of growth. In the somewhat more complicated model of Alesina and Rodrik (1994), among other differences, the proceeds of taxes are used for the production of public goods favourable to growth as well as for transfers. But the basic logic remains the same: the net effect of taxes (and underlying that, of inequality) on investment is negative.

[Insert Figure 1 about here]

As noted, [START_REF] Alesina | Distribution, Political Conflict, and Economic Growth: A Simple Theory and Some Empirical Evidence[END_REF]Rodrik (1992, 1994) and [START_REF] Persson | Growth, Distribution, and Politics[END_REF]Tabellini (1992, 1994) present empirical work which, they argue, supports the predictions of their models. But, whatever confirmation they can claim to find in the data only applies to the reduced-form relation between inequality at one point of time and subsequent growth. To say the least, the regressions do not confirm the intermediary relations which are necessary components of the causal chain spelled out above. Indeed, the findings reported in Perotti (1994) are that a more equal distribution of income leads to more (not less) transfers and that more transfers results in more (not less) investment.

Both Perotti (1994) and Thomas Piketty (1995) consider this evidence as casting serious doubts on the approach summarised above -especially since the negative relation between inequality and investment or growth can be explained by other factors or mechanisms than the ones borrowed from Meltzer and Richard.3 2 There is thus something in the approach discussed here which reminds one o f Tocqueville's famous distinction between the effects o f equality and o f the attempts to reduce inequality. Jon Elster (1991, p. 279) writes: "From a political perspective, the distinction between transitional effects and equilibrium effects is perhaps the central idea in Democracy in America. Tocqueville wanted his compatriots to understand the distinction between the often disruptive effects of equalization and the much less dangerous effects o f equality". However, in the approach discussed in the text, redistributive policies do not as a rule produce more equality.

One approach which does not bring us too far astray from the foregoing involves "socio-political instability"a composite variable which is increased by inequality in the distribution of income and which affects negatively

There is an additional reason not to claim too much for what has been achieved so far.4

On the basis of empirical work, it is a moot point whether the nature of the political regime plays a role in the relationship. Is democracy an important variable or condition? Or, in spite of their finding their inspiration in Meltzer and Richard, is the reference to it in the models merely convenient as an explanatory device? [START_REF] Persson | Growth, Distribution, and Politics[END_REF]Tabellini (1992a, 1994), as well as [START_REF] Alesina | Distribution, Political Conflict, and Economic Growth: A Simple Theory and Some Empirical Evidence[END_REF], find the democratic character of a country to be empirically significant, whereas Alesina and Rodrik (1994) do not any more, and thus argue that the reference to it in their theoretical reasoning should not be interpreted literally.

If it is clear to all that we are still at a very early stage of the reflection on the relationship between inequality and growth and about the way political variables, and especially the democratic nature of the political regime, might play a role in it. This has a pleasant consequence. It allows us to explore new possibilities or lines of research in as informal and unempirical a way as we wish, provided we do not claim too much for whatever such explorations or speculations may lead up to.

In the remainder of this paper, I will indulge in such speculation and develop an argument whose starting point is that growth, especially relative growth, often enters directly as an argument in the preference functions of voters. More exactly, I will argue that voters become concerned with economic growth when they perceive it to be, or fear it to become, in their country, abnormally low. This may very well lead, depending on the circumstances, to the rate o f investment. Others focus exclusively on non-political variables or relationships such as demand or capital markets. For the sake o f testability, Perotti interprets these various approaches as alternatives, but it is conceivable that they work together in the real world. As a consequence, even if the reduced-form, negative relation between inequality and growth is considered as established, it is not compelling with regard to models or explanations which are avowedly partial or ceteris paribus. This applies, for example, to the minority position, so to say, o f Saint-Paul and [START_REF] Saint-Paul | Education, Democracy and Growth[END_REF]. In their model also, following Meltzer and Richard, the median-income voter sets a tax rate, but the proceeds of the tax are used for public education instead of transfers. Public education is a form o f investment and thus favourable to growth. As a consequence, greater income inequality tends to affect growth positively, contrary to what is argued by the other authors. That the data answer otherwise (it seem s) would be a serious objection to Saint-Paul and Verdier's story only if this story were posited as the whole one. A s argued elsewhere [START_REF] Mingat | Méthodologie économique[END_REF], Salmon, 1994), contributions like Saint-Paul and Verdier's should be inteqireled as implicit counter-arguments (addressed at the suggestion that inequality is always, under democratic dccision-making, unfavourable to growth). 4 T o be fair, it must be stressed that all this work is very recent, with a lot more in process (see some references in Perotti, 1994, and[START_REF] Verdier | Models of Political Economy of Growth: A Short Survey[END_REF].

redistributive concerns (among others) being subordinated to the single policy objective of fostering growth. However, in this story, there will be considerable variation across countries. This variation will be related to differences in the sensitivity of public opinion to growth comparisons and in the nature of the political regime. We will in particular consider whether democracy is likely to be a significant variable in the cross-section explanation of growth.

Figure 2 provides a first view of this alternative hypothesis.

[Insert Figure 2 about here ] 2. From insufficient growth to discontent: Similarity and variation across countries With the magnitude of growth rates experienced since the beginning of the 19th century in an increasing number of countries, economic growth has become, almost "objectively", of the utmost importance -arguably more important than any other economic or social issue except extreme deprivation or poverty (non-economic issues such as peace or freedom are another matter). One reason for this "quasi-objective" importance, in a world whose economy is generally growing, is that, for a given country, to experience no growth or only a rate of growth much smaller than that of otherwise similar countries means falling behind these other countries rapidly -not at all the same situation as if growth were slow or non-existent everywhere.

[Insert Table 1 about here]

A comparison of the first two columns of Table 1 shows what growth has done to income per head in a number of countries over a period of time as short as forty-three years.

In 1870, the income per head of Spain was quite respectable, close to the income per head of Germany, higher than that of Finland, Norway, or Sweden, about two thirds of the income per head of the United States, already one of the highest. Calculations of this kind are obviously precarious. But, the enviable position occupied by Spain around 1870 is not a statistical aberration. As explained by Bradford De [START_REF] Long | [END_REF]Long ( , p. 1142)), it is confirmed by other indicators (for instance about four thousand miles of railroad having been built by 1877).

Forty-three years later, income per head in Spain was about half the income per head of France and Germany and about the third of the income per head of the United States. Spain was not part of the first world anymore, if I jnay venture this anachronism.5 Somewhat arbitrarily, one can say that Spain had recovered its place in the first world by 1979 while Portugal, Argentina and Chile -also part of it in 1870 -had not.

It is unlikely that public opinion in Spain realised what was happening. There were no rates of growth available, and anyhow economics was not in the news or in the minds then to the extent that it is now. That this situation has changed profoundly is clear from everyday experience. According to The Economist (13 August 1994, p. 52), in countries such as Malaysia, Singapore or Indonesia, "the latest GDP figures invariably get banner headlines". In these countries, the article goes on, "like shareholders in a rising market, the locals like to monitor their countries' ascent up the world GDP league table". I do not claim that our locals, in Europe or North America, are similarly obsessed with growth rates. Unemployment or inflation -the variables typically cited in the literature on the effect of economic variables on voting -probably get more attention. But differences in growth have cumulative effects (arguably more so than inflation). What happened to the Spanish economy between 1870 and 1913 may have remained unnoticed at the time. I submit that an evolution of much smaller amplitude would not today.

It should be stressed that, contrary to the assumption made in Meltzer and Richard's model and models borrowing their structure from it, the foregoing discussion about what people may be concerned with is based on the assumption that their policy preferences and evaluations are of a "sociotropic" rather than "pocket-book" kind [START_REF] Lewis-Beck | Economics and Electoral Behavior in France[END_REF].

According to Martin Paldam (1991, p. 14), "there seems to be a fairly general agreement... that what people react to is their perception of the general economic conditions, and not their own economic grievances" (the classical reference on this matter is [START_REF] Kinder | Economic Discontent and Political Behavior: The Role of Personal Grievances and Collective Economic Judgments in Congressional Voting[END_REF]. The reasons which voters may have to behave as Paldam and others claim that they do may involve some altruism. But a high level of uncertainty about the idiosyncratic effects of policies may do just as well.

[Insert Table 2 about here]

On the basis of the figures of Table 1, everybody would agree, I think, that the Spanish economy did badly between 1870 and 1914, but this is an easy case. In most other cases, things are not so clear. I make nonetheless the assumption that experts can determine, and even measure "growth performance gaps", thanks to comparisons with what obtains in comparable countries, or by more sophisticated means. That this is unrealistic is obvious. But to what degree? An example of the capacity which I have in mind is the interesting article of Steve [START_REF] Dowrick | OECD Comparative Economic Growth ^SO BS: Catch-Up and Convergence[END_REF] from which I derived Table 2. The authors start for each period from the OECD average growth rate of per capita GDP and indicate how each country deviates from that average (the numbers in parentheses in Table 2). If there is a tendency for countries to converge (see [START_REF] Sala-I-Martin | Cross-sectional Regressions and the Empirics of Economic Growth[END_REF], and thus for countries which have a lower productivity to catch-up, countries such as Canada, the United States or Switzerland, which have a high productivity should not be expected to grow as fast as countries which start with a much lower productivity level such as Ireland, Italy or Japan.

Hence the adjusted deviations written in bold types in Table 2. By idealising somewhat from that example, I will assume that experts can say when the relative performance of a country with regard to economic growth is negative and measure the corresponding gap. This growth performance gap, deemed "objective", is measured on the horizontal axis of Figure 3. I assume now that objective gaps cause discontent among voters and/or in public opinion; and that for a given country, the larger the gap, the higher the level of discontent. This is a reduced form relation, whose strength and shape depend on two sets of underlying mechanisms or forces. A first set pertains to information and concerns the way objective gaps translate into perceived gaps. We do not have to specify how this works to predict the outcome: the larger the objective gap, the more (in a given country) it is perceived by voters and/or public opinion. The second set relates to how a perceived gap causes discontent to build up. Again, we can safely assume that the outcome is unambiguous: for a given country, the larger the perceived gap, the larger the level of discontent.

In Figure 3, discontent (D) being measured along the vertical axis, the overall (reduced form) relation for a given country has an S-shape, as illustrated by curve Vj. In the case of this particular curve, the two underlying mechanisms operate with maximum vigour. Some sectors of opinion perceive the performance gap as soon as it appears and even a small perceived gap creates some discontent. Then, unitary increases in the performance gap have increasing effects on the amount of discontent (the curve becomes steeper). Other people, less perceptive, discover the gap as it gets bigger, conversations and the media focus on it to an increasing degree, etc. When discontent caused by underperformance is very widespread, however, unitary increases in underperformance have decreasing effects on discontent. The reasons for this are that most people who could be informed about the existence of a gap have already been informed, that the discontent of informed people is already so high that the gap becoming even more serious does not increase their level of discontent very much, and that a fraction of the population are left more or less unconcerned by the performance gap whatever its level. The process considered here is not a pure diffusion process but it does include a process of information diffusion as one of its components. Thus, it is not surprising that the shape of the curves is of a kind which is typically found when diffusion processes are involved.

[Insert Figure 3 6 Finally the population may not be as concerned with growth as is the case elsewhere. These various factors can interact. Anyhow, at some level of the performance gap, discontent will appear and, if the gap becomes bigger, it will spread to a large part of the population. Thus curve V2 has the same general shape as curve Vj. I make the obviously simplifying but convenient assumption that the only difference is that a given amount of discontent is caused by a performance gap which is larger by a fixed amount. This is reflected in Figure 3 by curve V2 being derived from curve Vj by a straightforward horizontal translation. The same applies to curve V3, which corresponds to mechanisms even weaker than is the case for curve V2.

From discontent to political or policy change: the role of political regimes.

A given level of discontent will not have the same effect under a system of majority rule allowing rapid alternation in office of competing political parties and in which, thus, officeholding is highly "contestable", under a system of proportional representation with strong political parties of the kind observed for instance, until recently, in Italy (called " parti tocracy" by mahy Italians), under an undemocratic or authoritarian regime which, however, tolerates some expression of dissenting opinion, and under a totalitarian dictatorship in which no such expression is tolerated. The effect of discontent on the likelihood of political or policy change may thus serve as an indicator of the nature of the political regime.

^ 1 have discussed all these points more in detail 111 previous work [START_REF] Salmon | Decentralisation as an Incentive Scheme[END_REF](Salmon, , 1991)).

In Figure 3, the horizontal lines Rj indicate the levels of discontent which create a political crisis and endanger the incumbent politicians or political parties, or even the existing political regimes or systems. Each line reflects the nature of the political system or of a state of the political system. As soon as the level of discontent reaches it, it is assumed that this is a sufficient condition for political change -or drastic policy change with the same politiciansto take place. If, following Karl Popper (1945, p. 121), we define democracy as a system by which ordinary people can get rid of what they think as bad government "without bloodshed"

and "by the way of general elections", line Ri, labelled "high contestability" in Figure 1, corresponds to democracy at a high degree. Line R2 corresponds to "partitocracy", say, of the kind alleged to have existed in Italy -somewhat less democratic under Popperian definitions.

Line R3 corresponds to relatively mildly undemocratic regimes of the kind observed in many third-world countries. Finally, line R4 corresponds to totalitarian dictatorships.

From insufficient growth to potential political or policy change

At a point of time, each country has two structural attributes, which can be called, for convenience, "growth gap sensitivity" and "nature of political regime". Each country can thus be located at the intersection point of a V; curve (the first attribute) and a Rj horizontal line (the second). This gives us the maximum objective growth performance gap which can be tolerated in the country.

Let me illustrate this by commenting on some such points in Figure 3.

. Point A reflects a situation in which office-holding is highly contestable and in which objective gaps cause discontent swiftly. A small objective performance gap (Ga) would cause an amount of discontent Di which, although very small, would be sufficient to provoke political change. Thus, the government is highly constrained by performance competition.

. Point B reflects a situation in which office-holding is not very contestable (the "partitocracy" situation, say), while objective gaps cause discontent swiftly. As a result of the first characteristic, only a relatively high level of discontent (D2) would provoke political change, but, as a result of the second characteristic, this high level of discontent could be brought about by an objective performance gap (Gb) still relatively small (although higher, of course, than in the previous case).

. In the case of point C. an authoritarian jegime could survive a very high amount of discontent (up to D3), but, as a result of a high sensitivity of the population to growth comparisons, discontent could nonetheless reach its limit with a performance gap (Gc) still relatively modest.

. Point U corresponds to a situation in which office-holding is extremely contestable but in which the sensitivity of the population to growth comparisons is mediocre. A very small amount of discontent (Di) would provoke political change but the performance gap (Gu)

would have to be relatively important to cause that amount of discontent.

. Finally, point S reflects the situation in a totalitarian regime of the worst kind. Very little information from abroad trickles in and the regime could survive a very high level of discontent, while for such a level to obtain, the performance gap would have to be considerable (Gs).

Figure 3 suggests an hypothesis which I formulate now but on which I do not want to insist too much in this paper. It is that the sensitivity of public opinion to comparisons in growth rates is more important or constraining than the political regime or system (totalitarian dictatorships excepted) for explaining observed growth. This is reflected in the shape of the Vi curves, implying that moving along a Vi curve would not change very much the maximum tolerable performance gap while moving from one Vi curve to another would have a much more Significant effect on this gap.

Let me illustrate this by a comparison of two countries, Italy and Britain. In a previous paper (1991), I have argued that all Western European continental countries, and particularly, the six founding members of the European Community, have been for a long time and still are in a situation which favours comparisons of performance. In the same paper, I have argued also that Britain, for historical reasons (having won the war, starting from a relatively high level, being culturally insulated, etc.) was not until the 1970s's in such a situation. This means that, in Figure 3 , 1 would put Italy, say, on the Vi curve, and Britain in the 1960's and 70's, say, on the V2 curve. On the other hand, Britain has a political, and in particular electoral, system which is considered to make its government highly contestable, while Italy has had until recently the already referred to so-called "partitocracy" system which made the Italian government much less contestable. This implies putting Britain, say, on a Ri line and Italy, say, on a R2 line. If we consider both structural attributes, Italy would be in B, and Britain in U. But, then, the maximum performance gap tolerable for Italy (Gb) would be much smaller than it would be for Britain (Gu). And, in fact, as argued in the same paper, the Thatcher revolution came as a consequence of a particularly serious performance gap over an extended period of time (see also Table 2). If Italy were to moves leftwards on the V 1 curve, as seems now plausible as a result of the change in the electoral system, the foregoing discussion implies that the effect of this move on the tolerable performance gap of the country would be limited (from Gb to Ga)-But, let me repeat, this hypothesis is even more tentative, if that is possible, than the rest of the paper.

5. Implications: From potential political or policy change to actual policy-making.

What are the implications of the foregoing hypothesis on the questions raised in the introduction about the relationship between democracy, redistribution and growth? Before trying to spell out these implications, let me formulate two remarks. First, so far I have referred to discontent without being too specific about what this discontent is directed at. The models discussed in the introduction, like the model of Meltzer and Richard, are "direct democracy" models, in the sense that no government is mentioned and policy-making reflects without bias the preferences of the median voter. Even as an idealisation, this kind of approach to democracy is often misleading (Salmon, 1993). Here, however, I must admit that one might wish to use the vocabulary of direct democracy and assume that the median voters, when dissatisfied, are dissatisfied mainly with their past decisions (I use the term "median voter" for convenience). This is related to the line adopted, for instance, by [START_REF] Cohen | What Caused the Rise of Conservatism: A French View[END_REF], 1994) to explain policy reversals in France and elsewhere in the early 1980s's. In his very interesting analysis, median voters progressively discover new realities and change their minds. Especially given what we observed in France in 1982-83, I must admit that, in important respects, what counts is the policy reversal itself, and the fact that it was supported by a majority of voters, not whether it was operated by the politicians elected in the first place or by their opponents. Although I use the vocabulary of representative democracy and refer to politicians as well as to voters, the just noted possible equivalence should certainly be kept in mind.

Whether underperformance is ascribed by voters to office-holders or to their own mistaken policy preferences, it causes discontent with current policies and arrangements which, when reaching or moving near the threshold lines identified in Figure 3, creates a compelling incentive for policy change. That is (and this is my second remark): provided of course that domestic policies are responsible for economic growth, or are considered as such by voters. More exactly, the logic of the foregoing discussion is that domestic policies are assessed in the light of growth comparisons and this makes sense if part of total growth can being ascribed to an element common to all countries (or a subset, e.g. Western European countries). The average growth rate of GDP per capita of OECD countries fell from about 4 percent over the period 1960-73 to about 1.5 percent over the period 1973-85 (Table 2). It was argued everywhere that all countries were affected by a common factor. The foregoing discussion concentrates on the actual or feared underperformance of individual countries visà-vis the average (or some standard of the kind), not on the evolution of the average (this statement is qualified below).

How difficult is it for a government to remain within the tolerance limit? Much depends on what happens abroad. If we introduce uncertainty as to policy outcomes, and if governments are risk-averse, it may be the case that each one will tend to adopt policies which have a high probability of resulting in a growth rate well above whai is needed to avoid excessive discontent. But this means that all policies which are detrimental to growth will tend to be discarded. This includes redistribution of the kind considered by Alesina and Rodrik or by Persson and Tabellini, not of the growth-enhancing kind considered by Saint-Paul and Verdier though. Thus the first implication of the hypothesis developed in this paper is that, in each country, redistribution will be accepted only if it is favourable to growth or if that country is far enough of, in terms of Figure 3, of its threshold point (itself dependent of factors analysed previously). This may explain transfers being correlated positively with investment and growth, as was found, we noted above, by Perotti (1994).

A second implication is that, at the international level, several equilibria are likely to be possible. Performance competition may be very intense; we may have a kind of "rate race"; and the consequences on income distribution may be severe. Or, we may be in a situation in which the best placed countries use their discretion margin in a way which lowers growth and makes life easier for the underperformers. Thus an average rate of growth at the international level which is low according to comparisons in time (for instance between 1973 and 1985 as showed in Table 2) may be the outcome (unlikely in the particular case), not of a common exogenous factor as suggested above, but of a coordination game which allows all governments, under the leadership of some, to pursue policies inimical to growth, in the domain of redistribution notably. In that case, as predicted in the models discussed at the beginning of this paper, and contrary to the result that I have just recalled, it might be the case that a positive correlation between redistribution (and high taxes) and low growth could be found.

If performance gaps are perceived only on the basis of comparisons within a subset of countries, the foregoing analysis could lead to multiple equilibria in space rather than in time, so to say. For instance, European countries, under the leadership, say, of Germany might adopt with regard to growth a standard less demanding than the one adopted by East Asian countries. As a consequence, more redistribution could be indulged in Europe than in East

Asia.

-14-

The third implication of the foregoing analysis is more a set of assumptions formulated somewhat differently than a real implication. The way Figure 3 has been drawn embodies the assumptions that the degree of democracy is a continuous variable and that, ceteris paribus, it has a positive influence on growth (more democracy, in the Popperian sense, reducing the tolerable growth gap), but also the assumptions that there is no discontinuity between democracies and mildly undemocratic regimes (the discontinuity being rather between these regimes and the totalitarian dictatorships) and that the degree of democracy may have less influence on growth (via the tolerable growth gap) than sensitivity to growth performance comparisons.7 This may help explain some of the hesitations found in the empirical literature about the significance of democracy as an explanatory variable or condition.

Concluding remarks

Contrary to Figure 1, Figure 2 should be read from right to left. The foregoing discussion suggests a relationship between democratic politics, growth and redistribution which is at least sometimes the opposite of that underlying the models summarised in the first section. In the modem world, when endangered, growth is too important to be left to the petty calculations of a median voter à la Meltzer and Richard. More generally, redistribution objectives or considerations often seem compelling because the reasoning is developed under the assumption of certainty or perfect information (or of rational expectations), and thus as if we were already on a utility frontier of some sort (Salmon, 1993). In the real world, we are very far from the frontier, there is plenty of scope for Pareto improvements which benefit large numbers if not all, the effects on income distribution of various policies are extremely uncertain. This explains also that redistribution is less and growth more important than assumed in many analyses.

All this is tentative, of course. Among the many problems I am aware of, let me mention one which seems to me particularly intriguing. To introduce it, let me note how puzzling, to say the least, is the time dimension implicit in the hypothesis developed in the preceding sections. Typically, politicians care_most about their current popularity and the results of the next election. But policies have effects on growth which are delayed and voters cannot be assumed to be competent enough to anticipate these effects. The usual solution of that kind of puzzle is party reputation and I agree that it is important.8 1 think, however that even in a perspective of reputation building or maintaining, an actor which I did not mention until now must be given a role. This actor is the bureauracy, and more precisely the subset of it referred to in France as the administrations économiques et financières. Let me conclude this paper with a brief discussion of what should have been, ideally, a major component of it.

The rigorous, relatively far-sighted and not very popular policy adopted by France between 1983 and 1993 was not invented by the President , the Prime Ministers or the political parties in power (the Socialist Party most of the time).9 Nor is it very enlightening for our purpose to observe that it was eventually accepted by the "median voter". It came out progressively, I suggest, from the reflections of that part of the bureaucracy which is concerned in the French system with economic and budgetary policies CBudget, Trésor, Banque de France, Direction de la Prévision, INSEE, etc.). In France, every Ministre de 8 The reputation concerns of individual politicians may be also essential (Salmon, 1993). 9 The objectives o f this policy are interpreted by Patrice Vial (1993, p. 48), an insider in many respects, as follows: "While it can be argued that the French experience has developed as a pragmatic construction rather than from a well-defined ex ante concept, the ultimate objective has never changed; it has consistently been to restore long-term, and not just short-term competitiveness -and this was understood as the capacity to sustain permanently adequate growth with a balanced current account in the context o f a stable exchange rate against the DM Let me also quote Vial's analysis o f the policy implications:

"Central to the strategy, therefore, was the idea that it was necessary to promote or enforce structural change and reforms in the basic relations governing, among other things, the pricesetting mechanism, wage settlement procedures and the pattern of income distribution. One of the important features differentiating France from Germany in the 1973-82 period has been that wages had consistently increased faster than labour productivity, leading to low margins, high indebtedness and a global impoverishment o f the enterprises. To reverse this tendency was considered a priority. Clearly this meant a diminution in the relative price of labour." Note the reference to permanently adequate growth, comparisons with Germany, the need of anti-wage redistribution, all this concerning policies adopted over almost all the period by a left-wing government. _______ [1994], "Is Inequality Harmful for Growth?". American Economic Review 84. no. 3 (June), pp. 600-21.

Piketty, Thomas [1995]. "Inégalités et redistribution: développements théoriques récents".

Revue d'Economie Politique 104, no. 6 (November-December), pp. 769-800.

  about here] Curve V2 corresponds to a situation in which the mechanisms underlying the reduced form relation are weaker than is the case with curve This difference may stem from obstacles to assessing the growth performance of the country by comparisons with what obtains in other countries. The situation of the country is felt to be special in one way or another, for instance the economy to be more subject to risks of the kind called in the tournament literature idiosyncratic than to risks common with the economy of the countries with which comparisons could be made. The difference can also be the result of geographical, linguistic or other obstacles to information flows from abroad. It can arise from the population being relatively unwilling, for instance for cultural or historical reasons, to engage into comparisons.

For a critical discussion of Meltzer and Richard, see, e.g., Mueller (1989), Brosio and Márchese (J993), Winer and Rutherford (1993).

I borrow the expression from Carlos Diaz-Alejandro (cited by DeLong, p. 1142), who was referring to what h;»p|x;ned to Argentina at a later date

Countries such as Malaysia, Singapore and Indonesia are not democracies but, following the observation of The Economist cited in Section 2, they are indeed sensitive to growth comparisons. That they grow fast is what we should ex|>ect from the assumptions embodied in Figure 3.

l'Economie et des Finances, after a while, expresses views which -it is difficult not to suspect -reflect exactly the views which have come to dominate in these bureaux. Obviously, bureaux are much more able than voters to decide whether a policy is oriented toward long term objectives. By adopting policies which have such orientation, political office-holders build up their reputation with the bureaux. What are, here, the motivations of the bureaucrats?

Why are politicians interested in this kind of reputation? How does this interact with their concern with public opinion and voters? These questions are difficult to deal with and may be viewed as somewhat unusual from the perspective of the rational choice theory of politics.

But I think that their exploration will have to be undertaken for an improved understanding of policy-making in contemporary representative democracies. Selected from Table 6 of [START_REF] Dowrick | OECD Comparative Economic Growth ^SO BS: Catch-Up and Convergence[END_REF]Nguyen (1989, pp. 1026-27). The numbers in parentheses are the unadjusted deviations. An example of the calculation: For the period 1950-1960, the average annual per capita GDP growth in the OECD countries is 3.12 percent; with regard to that, the deviation of the average GDP per capita of the United States is equal to -1.79 per cent. To that (negative) figure 0.22 is substracted for "cyclical bias" and 1.75 is added for "catch-up" (or, rather, the absence thereof)-As a result of these corrections the adjusted deviation is equal to -0.26. For the periods 1960-73 and 1973-85, the same kind of calculation yields, for the US, an adjusted deviation which is equal to zero and is positive, respectively.

Models of the relation between inequality, voting, and growth


The linear request problem Benoit R. Kloeckner * † † Université Paris-Est, Laboratoire d'Analyse et de Matématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-94010, Créteil, France

June 23, 2017

We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yields a solution. This allows us to work in any dimension and dispense from any dynamical hypothesis. In particular, we don't need to assume hyperbolicity to obtain a solution, although expansion moreover ensures the existence of an infinite-dimensional space of solutions.

Let M be a compact n-manifold and T : M → M be a smooth map, seen as a discretetime dynamical system. Assume that there is a positive smooth invariant measure ω, also seen as a nowhere vanishing n-form.

The linear response theory(see e.g. [START_REF] Ruelle | A review of linear response theory for general differentiable dynamical systems[END_REF][START_REF] Baladi | Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps[END_REF]) is typically concerned with the following question: if (T t ) t∈(-ε,ε) is a family of maps with T 0 = T , differentiable at t = 0, such that each T t preserves a smooth measure ω t , can we differentiate ω t with respect to t and express .

ω 0 := d dt t=0 ω t in term of
.

T 0 := d dt t=0 T t ? A recent article by Galatolo and Pollicott [START_REF] Galatolo | Controlling the statistical properties of expanding maps[END_REF] proposes to study the opposite direction; they roughly ask two questions (and provide some answers for expanding maps of the circle), the second of which we voluntarily keep imprecise: Question 1. Given a smooth function ρ : M → R of vanishing integral, 1 can we find a perturbation (T t ) t∈(-ε,ε) of T , differentiable at t = 0 and preserving a family of smooth measure (ω t ) t∈(-ε,ε) such that . ω 0 = ρω? Can we then express the possible values of . T 0 in term of ρ? Question 2. When the previous question has a positive answer, can we find an "optimal" value of .

T 0 ?

The goal of this note is to observe that Question 1 has a positive answer in a very general setting, without dynamical hypothesis and in every dimension. This follows from well-known facts in differential geometry, and the only faint bit of novelty is in the observation that one can restrict to conjugate deformations, of the form

T t = ϕ t • T • ϕ -t
where (ϕ t ) t∈(-ε,ε) is the flow of a vector field.

Let us first define properly what it means for a family (T t ) t∈(-ε,ε) to be differentiable at t = 0. Galatolo and Pollicott work on the circle and implicitly use its parallelism (all tangent spaces can be identified), which is not possible on a general manifold. Pointwise, we want to ask that for each x ∈ M, the curve (T t (x)) t∈(-ε,ε) is differentiable at t = 0, and we want the derivative of this curve to depend smoothly on x. We want to stress that .

T 0 (x) := d dt t=0 T t (x) is an element of T T (x) M, not of T x M, and
.

T 0 is thus not a vector field. We will thus consider the set Γ T (M) of smooth maps Z : M → T M such that for all x ∈ M, Z x ∈ T T (x) M, and say that a family

(T t ) t∈(-ε,ε) of smooth maps M → M is differentiable at t = 0 if . T 0 (x) is defined for all x and . T 0 ∈ Γ T (M).
For example, if X is a smooth vector field on M, we can consider its flow (ϕ t ) t∈R and the family T t := ϕ t • T • ϕ -t . Then a direct computation shows that (T t ) t∈R is differentiable at 0:

ϕ t • T • ϕ -t (x) = ϕ t • T x -tX x + o(t) = ϕ t T (x) -tD x T (X x ) + o(t) = T (x) -tD x T (X x ) + tX T (x) + o(t) so that . T 0 (x) = -D x T (X x ) + X T (x)
, which we also write .

T 0 = -DT (X) + X T . This is naturally an element of Γ T (M), as it should.

Observe that since T preserves ω, T t then preserves

ω t := ϕ t * ω. But d dt t=0 ϕ t * ω = -L X ω
where L denotes the Lie derivative. To answer Question 1 by restricting to such deformations by conjugacy, we thus only have to check that for all ρ such M ρω = 0, there is a vector field X such that L X ω = -ρω. This is well-known, see e.g. the proof of Moser's theorem in [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF] (Theorem 5.1.27 page 195), but let us recall the argument for the sake of completeness.

Since ω is a n-form, the Cartan formula reads L X ω = d(i X ω) where i X ω is the (n -1)-form ω(X; •; • • • ; •) obtained by contraction with X. Since M is compact, its top-dimensional cohomology is 1-dimensional, generated by the class of ω (or any volume form), and every n-form of vanishing total integral is exact. This means that there exist a (n -1) form θ such that dθ = -ρω. Now, ω being a volume form it is non-degenerate, which means that any (n -1)-form can be obtained by contracting ω with a well-chosen vector field; in particular, there must exist a vector field X such that i X ω = θ. Putting all this together, we get the desired conclusion:

.

ω t = -L X ω = -d(i X ω) = -dθ = ρω.
We conclude: Theorem 3. Let T : M → M be a smooth map acting on a compact smooth Riemannian manifold, preserving a smooth volume form ω. Let ρ : M → R be a smooth function such that M ρω = 0.

There exist a deformation (T t ) (-ε,ε) of T that is differentiable at t = 0 and preserving smooth volume forms ω t , such that .

ω 0 = ρω. Moreover one can ask all T t to be smoothly conjugate to T . Now, observe that in the above θ is not unique: one can add to it any closed (n -1) form, and will obtain a different suitable X. If n ≥ 2, the space of closed (n -1) form is infinite-dimensional (it contains all exact forms dα where α is a (n -2)-form); but if M = S 1 , there are no (n -2)-forms and the space of closed (n -1)-form is the space of functions with vanishing derivative, i.e. of constant functions.

This lack of uniqueness points, in dimension at least 2, to the existence of many solutions to Question 1 even when restricting to conjugate deformations; however, since we consider perturbations only at their first order, we have to determine whether the space of possible . T 0 is indeed as large as the set of vector fields X generating the conjugacy.

We thus consider smooth vector fields X and Y on M, such that L X ω = L Y ω = -ρω and -DT (X) + X T = -DT (Y ) + Y T . Does this imply that X = Y ? The answer strongly depends on T . The first equation means that X -Y preserves ω, while the second equation can be rewritten as

X T -Y T = DT (X -Y ),
i.e. the vector field X -Y is T -invariant. The difference between the spaces of possible values for X and for . T 0 above is therefore determined by the space of T -invariant vector fields preserving ω. There is no general rule: there could be no non-zero such vector fields (e.g. if T is expanding) or there could be plenty (e.g. if T is the identity, which of course is a very degenerate case: we get . T 0 = 0). Depending on this, there could be an infinite-dimensional space of possible values for . T 0 in Theorem 3, or a unique one, or anything in between.

We conclude with a few remarks.

Remark 4. Sometimes one consider perturbations of the form .

T 0 (x) = X T (x) for some vector field X. This is equivalent to considering . T 0 ∈ Γ T (M) if T is invertible, but is less general otherwise as the images of any x, y ∈ M such that T (x) = T (y) would be asked to be perturbed identically.

Remark 5. Using Moser's theorem [START_REF] Moser | On the volume elements on a manifold[END_REF] instead of its first-order version, one sees that for all volume form ω ′ , there is a conjugate of T that preserves ω ′ . Remark 6. It could be asked what happens in lower regularity, e.g. C k . We do not enter into these details, since the strategy would be the very same, only keeping track of the available regularity for the various objects ω, θ, X, etc.

Remark 7. One can wonder whether the observation here can shed some light on Question 2. If one measures optimality by fixing a Riemannian metric on M and trying to minimize . T 0 L 2 , as Galatolo an Pollicott do in [START_REF] Galatolo | Controlling the statistical properties of expanding maps[END_REF], then in general the optimal first-order perturbation is not of the form obtained here by conjugacy. This is easily seen in the case of the doubling map on the circle.

One can also restrict to conjugate perturbations, and try to minimize X L 2 where X is the vector field generating the conjugating diffeomorphisms. Then the problem directly relates to solving the modified Poisson equation

∇ • (η∇u) = -ρ
where η, ρ are the densities of the invariant measure and of its perturbation with respect to the Riemannian volume and u is the unknown (then ∇u is the optimal vector field X). This is an elliptic PDE, whose theory is well established and that admits smooth solutions. The first version of this note used this to obtain Theorem 3, but not insisting on finding a gradient vector field can be done even more easily, as above. I wish to thank Anatole Katok for interesting criticism on the first version of this note, pointing to this simplification.

The condition M ρω = 0 is needed whenever invariant measures are normalized to have fixed total mass.

* Supported by the Agence Nationale de la Recherche, grant ANR-11-JS01-0011.