Introduction

Household behaviour under rationing, using duality theory results and the notion of virtual or shadow priees (Rorbarth, 1941) for the constrained goods, has been extensively analysed, e.g., recently by [START_REF] Neary | The theory of household behavior under rationing[END_REF], and [START_REF] Deaton | Economies and consumer behavior under rationing[END_REF].

Both studies have emphasised that "ail the properties of the rationed demand and supply functions may be expressed in terms of the properties of the unrationed functions, provided the latter are evaluated at the virtual priees" (Neary and Roberts,p. 26). Several empirical analyses of markets in disequilibrium have made use of this approach (in terms of virtual priees) to specifY and estimate demand systems in the case of binding non-negativity constraints (e.g., Lee andPitt, 1986, 1987).

In producer theory, Sakai (1974) has given analytical relationships between short-run and long-run supply and demand responses. [START_REF] Lau | A characterisation of the normalized restricted profit function[END_REF] has characterised the conditions under which supply and demand responses without quantity constraints on inputs or outputs can be derived from the responses estimated under sorne fixity, and vice-versa. Particularly, he has shown how the Le Chatelier's princip1e applies. These analytical developments have been the subject of numerous empirical studies applied to the problem of moving form short-run to long-run responses (see, e.g., [START_REF] Brown | Estimating elasticities of substitution in a model of partial static equilibrium : an application to U.S. agriculture[END_REF][START_REF] Kulatilaka | Tests on the validity of static equilibrium models[END_REF]. However, these papers do not use the concept of virtual priees but allow the long-run adjustment of quasi-fixed inputs to their optimallevels.

In this paper, we apply the concept of virtual or shadow priees to the analysis of producer behaviour under output and/or input rationing. We show how the comparative statics of endogenous variables in a regime of effective rationing (i.e., variable netput quantities and shadow priees of fixed netputs) can be characterised from the knowledge of supply and demand responses before the implementation of the constraints. The use of shadow priees makes the analytics simple and allows to derive the Hessian of the restricted profit function from the Hessian of the unrestricted for any level of the constraints, provided that the latter is calculated at the relevant shadow priees. Il makes it possible to characterise the structure of the new supply-demand system and of the shadow priees associated with the rationed outputs or inputs.

In section 2, shadow priees and disequilibrium gaps provide a simple manner to analyse various implications of quantity restrictions on supply and demand behaviour. In section 3, we develop the essentials of the comparative statics of producer behaviour when sorne netputs are constrained. Emphasis is placed on the comparison of the response parameters of unconstrained variables after sorne inputs or outputs have been rationed to the parameters without rationing. Attention is focused on cross-priee effects which are shown to move towards more substitution and more inferiority 1 when constraints are introduced, in the case of the "normal" technology of Sakai and even under weaker conditions. Section 3 deals with a dynarnic approach of the same problem applied to quasi-fixed factors; simple analytical relations between short-run and long-run responses are derived.

A theoretical framework to analyse producer behaviour under rationing

In order to characterise behaviour under constraints, we make use of the knowledge of supply and demand response without or before the implementation of the rationing. This is made easy under fairly general conditions as the comparative statics of unconstrained producer behaviour is better known (Sakai). As there is an evident symmetry between introducing constraints on the one hand and relaxing them on the other, the symmetrical expression relevant to the problem of moving from short-run to long-run response is aIso derived in our framework.

Notations and preliminary definitions

When ail priees are given and ail netputs free to adjust, the behaviour is described by the farniliar problem :

max[v'q;q eT]= Jil (v) q (2.1)
where q is the vector of (n+m) netput quantities, v' is the (transposed) vector of corresponding priees and Jil (v) the unconstrained profit function. The feasible set T is assumed non empty, c1osed, bounded from above and convex. Free disposai is a1so aUowed for. Furthermore, we assume that the unconstrained profit function is twice differentiable everywhere so that it is 10ca11y strongly convex everywhere in v. The solutions of the optimisation program (2.1) are the unconstrained netput responses :

(2.2) When sorne quantities are pegged, either by policy instruments (output quotas, input use regulations) or by market conditions (constrained outlets, inputs in fixed supply), the remaining variable netputs exhibit constrained response to priees. These responses can be derived from the constrained producer programme, i.e., from the constrained profit function.

1 Following Hugues (1981), an input xh will he said inferior with respect to an oulput Yi if the oulput priee increase decreases the MarshaIlian or long-run dernand of xh. By Young's theorem, Yi will he also said inferior with respectto xh sinee the input priee inerease inereases the MarshaIlian or long-run supply of Yi.

After partitioning the vector q (respectively v) into a variable netput vector qJ (resp. vJ) with dimension n J +rn J and a constrained netput vector q. (resp. vo) with dimension no +rn o , we get:

(2.3) and

(2.4)

The notion of disequilibrium profit function will also be useful in the developments below. It can be defined as the sum of the restricted profit and the value of fixed netputs at market prices, i.e. :

(25a) or, alternatively :

(2.5b) Using the sarne partition as before, (2.2) can be written as :

(2.6a) (2.6b)
From the definitions of ,f (vl'vo),n"(vJ,c[o) and , (J(vJ,ïfo'v o )' it follows that disequilibrium and unconstrained profits are equal when ïfo just happens to be the optimal vector q~(v J ' vo), i.e., (2.7)

Consequently, the unconstrained profit curve is the upper envelope to the farnily of disequilibrium profit functions and, at the tangency point, qY and q: coincide.

qr (v J , vo) = iJ,f (v J ' v o )! t?1 J =mr> (vl'q~(vI' v o ), v o )! t?1 1 =q: (vl'q~(vI' vo)) (2.8)
This relation is strictly valid only when ïfo, given VI and v o , is optimal. In order to characterise the constrained response !Tom the knowledge of ,f (vI' vo) at any point (vl'ïfo) in the constrained regime space, we make use of virtual prices introduced by Rorbarth. The vector of virtual prices TTo is defined as the system of prices which ensures that the unconstrained quantities q~, as functions of v, and TTo, will stay at level ?fo, i.e. :

(2.9)

Supply response belore and alter rationing

When rationed quantities ?fo are different from optimallevels q~(vJ' vo), equation (2.9) can be used to determine the virtual price vector TTo as a function of (v,,?fo), i.e., TTo(v,,?fo)' which will ensure that expression (2.8) holds at the point (v" TTo) in the price space, i.e. :

(2.10) Now, the comparative statics ofrationed responses q~(vi'?fo) with respect to vJ can be derived from q~ (v,.TTo) and expression (2.9). A local solution can be obtained by total differentiation of(2.9) and (2.10) and solving for dq~around (vi' TTo), i.e. :

(2. 11 a) and (2.1Ib) Then (2.11) can be solved for the endogenous variables in rationed regime (i.e., dqã nd dTT o ) with respect to the set of exogenous ones which are now dv, and dlJo' Omitting arguments (v,.TTo), solving (2.1 lb) for dTJo and plugging into (2.1 la), we obtain:

(2.12) System (2.12) allows to characterise the comparative statics under rationing constraints on the basis of the unconstrained supply and derived demand responses evaluated at the relevant constrained equilibrium point, i.e., at «(q~,?fo),(v l , TTo). The first row in (2.12) provides the comparative statics of q~in terms of VI and ?fo from the Hessian of the unconstrained profit function evaluated at the relevant point (vi' TTo). Similarly, the second row of (2.12) provides the comparative statics of rationed netput shadow prices based on the second partial derivatives of 1f (vi' TTo)' Note that a unique local solution to (2.11 b) for dTTo exists under our assumptions about the feasible production set, as the Hessian of the (nonnormalised) profit function is of order n+m-l [START_REF] Guesnerie | Modèles de l'économie publique, Monographies du séminaire d'économétrie[END_REF] and its principal minors of smaller order are therefore of full rank.

Supply response under rationing and de-rationing

There is a correspondence between (2.12) and a similar expression for qNv j Jio) and 17o(vl'li o ), but derived trom the restricted profit function. At the constrained equilibrium point (vl'lio) the vector 170 is also defined by the fo!lowing system obtained by first differentiation of (2.3) with respect to lio [START_REF] Lau | A characterisation of the normalized restricted profit function[END_REF]) :

(2.13) If the constrained profit function is known, the comparative statics of endogenous variables (ql' 170) in the rationed regime, around the point (vj,lio)' is given by :

(2.14) Clearly, given our assumptions, (2.12) and (2.14) are two alternative representations of the same behaviour under rationing which can be used alternatively depending on the available information. Note that this analysis is valid only when rationing remains effective throughout. By proper comparison of 17o(vj,lio) to market price vO' it is possible to describe the change trom the constrained regime defined either by (2.14) or by (2.12) to the unconstrained regime corresponding to (2.6), and vice versa.

Comparing (2.12) and (2.14) shows that :

(215)
The correction term in (2.15) is the "contraction effect" due to rationing. The Le Chatelier's principle applies as the diagonal elements of ~v differ trom those of 1f"v by terms

I l JI

of the contraction effects which are positive-definite quadratic forms. The response of an unconstrained output to a change of its own price is smaller with constraints put on sorne quantities qo. A similar result applies to the magnitude of input demand responses to their own pnces.

Producer response after relaxing constraints can also be derived trom the knowledge of the response under rationing, i.e., trom the sub-Hessians of the restricted profit function. By total differentiation of (2.1 0) and (2.13), one can retrieve the Hessian of the unrestricted profit function at the point ofrationing 17o(vj.lio), i.e. (2.14).

Solution of (2.14) for unrestricted responses q,u(v" 17o)andq~(v" 170) glves the relation between unconstrained netputs and prices at the disequilibrium point lio.

(2.16)

But, wlùle the Le Chatelier effect is easily seen, what happens to cross-price effects is much less obvious as noted by Mosclùni (1988), particularly when one uses the restricted profit function where cross effeets between quantities of the type ~"" are not easy to interpret. The situation is much easier when one starts with the unconstrained function Tf (vI,V o )' in particular when the unconstrained technology is "normal" (Sakai, pages 272-273).

The consequences of rationing on substitutability between inputs and outputs

Own price effects of rationing follow directly from conditions of convexity, i.e., the Le Chatelier effeet shown by [START_REF] Lau | A characterisation of the normalized restricted profit function[END_REF] and subsequent authors, the negative effect of reducing the quota on its shadow price (Mosclùni, 1988), and the mirror positive effect of reducing a fixed factor quantity on its shadow price.

As regards cross effects, the comparative statics of supply behaviour under rationing involves at least two broad issues, i) the impact of the level of rationed netputs on output supply, input demand and on shadow prices of rationed netputs (problem 1), and ii) the consequences of rationing on cross-price effects between the netputs remaining unconstrained (problem 2).

The latter issue has received 1ess attention than the Le Chatelier effect, but it can be important in the assessment of fiscal or trade policies in a multioutput sector subject to sorne rationing. [START_REF] Madden | A generalisation ofHicksian q substitutes and complements with application to demand rationing[END_REF] has recently dealt with tlùs problem on Hicksian demand functions, and shown with a partition of the Hessian of responses under rationing similar to system (2.12), that, if two goods are substitutes before, they are also substitutes after addition of one rationing (Madden, theorem 4, p. 1502). He also notes that "substitutes are likely to remain substitutes, but complements may "more easily" change and become substitutes".

In the case of producer theory, a corresponding result can be obtained under a fairly weak assumption that cross effects between two netputs and the price of the netput to be rationed have the same sign before the imposition of rationing.

Rationing under "pairwise similarity"

A pair of netputs q, and q, will be said to be similar with respect to a tlùrd netput % if the cross-price elasticities of q~and q~with respect to V o have the same sign, that is, with

usual notation, if with, (3.\) r={i.h} .. s~{j.k} .. iJ = 1• ...• n+m .. h,k ~1. ...• n+ m.
There is not a single definition of sirnilarity as (3.1) when it is expressed in terms of the elements of the Hessian of the profit function. For purpose of c1arity, unconstrained outputs (Yi =qi ; i=I, ...•n) with prices Pi and inputs (x. =-q. ; h=I.....m) with prices w. are again identified. With these notations, quantities of output or inputs are positive. Inequalities in (3.1) can then be written in terms of the Hessian of 7P (p, w). Table 3.1 summarises these expressions for various pairs of outputs and/or inputs.

Table 3.1. Sirnilarity oftwo netputs q, and q, with respect to a third qo : alternative formulations alternative formulations of the pairwise sirnilarity of q, and q, W.r.t. qo q, and q, are two outputs Yi q, is an output Yi and q, is an q, and q, are two inputs -x.

and Yj input -x. and -x.

u u > 0 U U >0 c:.Vll• e~iVO ~0 EY/'-'o' &Yil/o - Ey1v(J' Ex"v o - 0'~1 i}yo' 0'f 1 i}yo ? 0 0'i U 1 i}yo' ac~1 i}yo ? 0 ac~1 i}yo' ac~1 i}yo ? 0 (,r;,.,.(-<.)? 0) ((-<.,)(-<.)?O) <"O' <v o 2= 0 <"o'<v o :sO <"O' <"0 ~0
For example, two inputs x. and x. are said to be sirnilar with respect to input X o if both are substitutes with respect to X o or if both are complements with respect to x D • Two outputs Yi and Y j are said to be sirnilar with respect to X o ifthis factor is superior (or inferior) in the production ofboth outputs. Output Yi and input x. are said to be sirnilar with respect to input X o if input X o is superior (inferior) in the production of Yi and inputs x. and x D are complements (substitutes). Two netputs q, and q, can be sirnilar with respect to q. at point E corresponding to 7P (v) and not at the point E* corresponding to 7P (v'). They could be similar with respect to q. under one set of constraints but not under another.

Consider now the case when either one quota or one input is rationed. The sub-Hessian ~,., in (2.15) reduces then to a positive scalar ;f.". Under the assumption of sirnilarity of each pair of goods with respect to the rationed netput, the following properties concerning the cross effects at the rationed equilibrium can now be derived : Property J. Under the similarity assumption, a binding constraint on an output or an input makes the unrestricted outputs and unrestricted inputs more substitutable. This follows from the last row m table 3.1, as under similarity and convexity, n:;,(~rl ~is non negative and, ' Z:R = n: = 7f~n:;,(~rI ~5'7f~when i and j are outputs, Property 2. Under the similarity assumption, a binding constraint on an output or an input makes the unrestricted outputs more regressive and equivalently the unrestricted inputs more inferior (the same argument as in property 1 applies).

The similarity properties defined at the unconstrained equilibrium does not allow generally to sign the contraction effect in the case of no + m o ~2 rationed netputs. The contraction effect relating the off-diagonal elements of ~y in (2.15) is :

JJ (3.4)
When s ~r, similarity between netputs rand s with respect to each constrained netput at the unrationed equilibrium is not sufficient to determine the sign of expression (3.4) since off-diagonal elements of ( ~y ri have in general an indeterminate sign. The diagonal terms of " (~y rI are non negative by convexity. The sign of expression (3.4) is then defined if off-" diagonal elements of (7f y y rI are non negative too. Technologies such that (~y ri is a non-(} 0 (} (} negative matrix are worth considering.

Notice that in a procedure by steps, i.e., by implementing constraints one at a time, outputs and inputs tend to become more substitutable and outputs tend to become more inferior with respect to inputs as long as the similarity property is verified at each subequilibrium. Since there is apparently no reason why two netputs cannot be similar under one set of constraints but no longer similar under another, technologies which verii» the similarity property at each equilibrium are worth considering. Two interesting cases will be detailed, i.e., the case where outputs only are rationed (no;:: 2 and m o = 0) and the case where inputs only are rationed (no = 0 and m o ;:: 2).

Let us examine first two situations where off-diagonal terms of ( ,r:v r' are ail non .. negative, i) ,r:v is a diagonal matrix and, ii) ,r:, . is an M-matrix and its off-diagonal elements /1 () /1 /1 are ail non-positive 2 In both cases, the inverse of ,r:, includes non-negative elements only .. and, under similarity assumption of rand s with respect to each rationed netput, the contraction effect (3.4) is then non-positive and more substitutability follows !Tom rationing.

i) The matrix ,r:v is diagonal in the case where rationed netputs are outputs only, if

.. and only if the technology is non joint in input quantities. Consequently, binding constraints on outputs makes unrestricted outputs more substitutes and regressive and inputs more substitutes, under the similarity assumption of variable netputs with respect to each rationed output. In that case, the supply of a variable output does not depend either on other variable output prices or constrained output quantities as the off-diagonal elements of ,r:v are zero in ..

(3.4). Similarly, if rationed netputs are inputs only, the matrix ,r:v is diagonal if and only if the ..

technology is non-joint in output quantities. In that case, binding constraints on inputs makes unrestricted inputs more inferior and outputs more substitutable. The demand of a variable input is independent of other variable input prices and of constrained input quantities.

ii) Consider now the case where ,r:v is an M-matrix. When rationed outputs (inputs)

..

are already substitutes and outputs are non inferior with respect to inputs in the unconstrained regime. The tendency towards more substitutability and more inferiority is then verified, still under the similarity assumption. In particular, outputs i and j (inputs h and k) will be stronger substitutes the more restrictions are implemented.

Rationing under a normal technology

Moschini has studied the effect of the quota level on optimal unrestricted outputs and inputs in the context of two set of technologies : the normal technology as defined by Sakai (cost complementarity) and the case of cost substitutability. Here, we consider mainly the change in cross-price effects between variable netputs after rationing, starting !Tom a normal technology at the unconstrained Marshallian equilibrium.

A multioutput-multiinput technology has been said to be normal by Sakai if the following conditions on the cost and revenue functions are satisfied : CY'YJ $ 0 for any couple of outputs i and), i ~) (cost complementarity), (3.5)

Cw,.yJ ~0 for any input h and output) (non regressivity of outputs), (3.6) R.,x, ~0 for any couple of inputs h and k, h ~k (input complementarity), (3.7)

R px ~0 for any output i and input k (non inferiority of inputs).

, ,

where CÔ',w) is the long-run or total cost function, R(P,x) is the long-run or total revenue function, y is the vector of outputs (with corresponding vector price p) and x is the vector of inputs (with corresponding vector price w).

Properties (3.5) to (3.8) imply the following restrictions on the technology at the longrun Marshallian equilibrium: gross substitution among inputs and outputs is ruled out and regressive (or inferiority) relationships between inputs and outputs are also ruled out. When the arguments of the profit function are p's for outputs and w's for inputs, the following implications hold

tf"Pl ~0 for any outputs i and}.

<,w, ~0 for any inputs h and k.

ffl Plw = <p S; 0 for any pair of output i and input h.

, , ,

(3.10)

(3.11)
First note that Moschini's remark on the impact of quota leveI on unconstrained supply and demand can be readily extended to input rationing and to the shadow price response.

Property 3 a. When the technology is "normal", tightening the quota constraint will reduce both the supply of variable outputs and the demand for variable inputs.

This follows first ITom the convexity which implies a fall in the shadow price associated with the quota and ITom the normality assumption which implies that the (shadow) price fall of the rationed netput leads to a decrease in the supply of complement outputs and in the demand of non inferior inputs.

Property 3 b. Under a "normal" technology, tightening a fixed factor constraint decreases variable input demands and variable output supply.

Complementary between inputs and non regressivity of outputs, together with the rise in the shadow price of the rationed input, imply this result.

Property 4. The effect of unconstrained netput prices (v,) on the shadow price of the rationed netputs (rM is the negative of the effect of the ration level ifo on the unconstrained netput (g,) (by symmetry of2.12).

As regards cross-price effects between unconstrained netputs, the normal technology allows us to establish stronger results than the similarity condition. First note that the "normal" technology ensures that ail pairs of inputs and outputs are similar. Such a technology could be called strongly similar. The results of the one rationing case therefore extend to the normal technology, hence rationing favours substitutability.

In the normal technology case, it tums out that rationing more than one input or several outputs in the same time strengthens the tendency toward substitution without ambiguity.

Consider first the case where ail outputs are rationed at once. The restricted profit function then relates directly to the cost function. Outputs and inputs are again distinguished and we have:

1? (y, w) = -C(y, w)
hence, 1?.,.,(y,w) = -C.,., (y,w) The unrestricted profit function 7? (p, w) may be written as : 7? (p, w) =py(w,p)-C(w,y(w,p» Following Sakai procedure, we have then :

(3.12)

(3.13) (3.14) (3.15)
With consideration of (3.13) and (2.15), it is c1ear that the contraction effect and the expansion effects are identical, i.e. :

(3.16)

The advantage of (3.16) is that the signs of its off-diagonal elements unambiguously follow from the normality assumption. C», being an M-matrix, its inverse is non negative.

Normality implies the non negativity of C"". Hence, ail off-diagonal elements in (3.16) are positive as weil, and rationing will therefore both reduce the magnitude of the own price elasticity (Le Chatelier) and reduce the complementary (increase the substitution) between inputs.

A similar argument can be applied to constraining ail inputs at once, using relation (3.7) and (3.8) on the revenue function, to establish the impact of input rationing on output substitutability.

When only a subset Yo of outputs are constrained by quotas, a proper partition of 7? (.),1?(.) and C(.), and corresponding Hessian matrices, leads to similar expressions which rely on the equivalence between the expansion matrix based on the cost function (with known signs) and the contraction effect based on the unrestricted profit function. From (3.17) a similar expression to (3.15) is derived (Appendix 1), and 1f_ -n"_ which has also nonnegative e1ements everywhere under the normality assumption, hence the tendency toward more substitutability (Iess complementary). These results do not encompass ail cases, however, since, in the examination of the simultaneous imposition of input and output rations, a similar result has not been found.

Empiricall11ustration

The previous theoretical framework is illustrated by the following empirical model based on a restricted translog profit function for the EC agricultural sector. The data set spans the period 1960-1984, i.e., before the implementation of the dairy quota system. The estimated model is based on four outputs (grains, other vegetable products, milk, and other animal products), three variable inputs (animal feed, fertilisers and other raw materials), and two quasi-fixed inputs (labour and capital). Technical change effects are captured by adding a time trend variable to each share equation. Symmetry and linear homogeneity of the profit function with respect to variable input prices is imposed. Ali regressors are normalised to 1980 = 1 and the estimator employed is Zellner's procedure modified in order to impose convexity in prices of the restricted profit function at the expansion point (i.e., 1980). Convexity in prices IS imposed by using the Cholesky decomposition of any definite positive matrix (Lau, 1978).

Tables 3.2 and 3.3 iIIustrate the implications of imposing one rationing on both the shadow price behaviour relative to unconstrained netput prices (and the quota Ievel) and the supply-demand response of variable netputs to their market prices. The quota was imposed on milk which exhibits, before rationing, significant cross-price effects with other netputs. Table 3.2 suggests that the estimated technology is (given pre-existing exogenous leveIs of sorne inputs, i.e., land, capital and labour) normal as complementary prevails among outputs and inputs. However, this structure is seriously modified when an important output as milk is constrained. The Le Chatelier effect is verified but moreover, as expected from the analysis developed above, complementarity between outputs is reduced and substitution between variable inputs now prevails. Less superiority of inputs is also obtained and fertiliser inputs flip to an inferior situation as their demand now depends negatively on variable output prices. The latter result depends on the particular estimated value of unconstrained response of milk supply to its own price and to the price offertiliser which turn out to be large.

It is noticeable that the structure of price response after imposing just one rationing can be altered so deeply. We assume that a multivariate partial adjustment process describes the movement of the finn toward the optimal target : (4.3)

By combining (4.1), (4.2) and (4.3), a system of observable equations is obtained. For qOt , we get :

(4.4)

Tlùs generalised adjustment scheme permits disequilibrium in one quasi-fixed factor market to affect the demand for another quasi-fixed input. The actual levels of quasi-fixed inputs are a weighted average of the optimallevels at time t and past observed levels at time t-1, where the adjustment matrices M and (1 -M) serve as weights. Through substitution, we can see that equation (4.4) can be rewritten as:

w qOt = L(1 -M)' Mq~_i +(1 -MrqOt_w ;:0 w = L(1 -Ml M[ d.>lt-i + n;,'OV Ot -i ]+(1 -MrqOt-w j:::.O
In equilibrium, q~-i = qg. Therefore, in the long-run : = 2:(1 -Ml Mq~+(1 -Mrqot_;

=0

Stability of the adjustment scheme requires that the characteristics roots of the matrix (1 -M) be within the unit circle. Furthermore the adjustment path is monotonic if the characteristic roots are real positive numbers and oscillates otherwise (Nadiri and Rosen, 1969). From (4.6), short-run, interim and long-run price elasticities of q. can be derived.

For observed qJt , there is a virtual price vector TTot which corresponds to the observed vector qOt and which explains the optimal level of unconstrained netputs given the quasi-fixity offactors. First, solving (4.2 b) for TTot and using (4.4), we obtain :

The observed sequence for unconstrained netputs q" given the interim leve! of quasi fixed factors, may now be derived, i.e. :

qJt = 7I/}vJt + 71/0 TTot =7I/}v" + 7I/0[r ,r,;"F 1 ( M -1)71:;>Jt + (,r,;"F J M,r,;"v o , + ( ,r,;"F) (I -M )qOt-l]
Finally : qJt = [ 7I/J + 71/0 ( ,r,;"F J (M -1)tfaJ]vu + 7I/0( ,r,;"F' M,r,;"v ot -7I/0(tfaoF I ( M -1)qOt-1

(4.8)
In (4.8), it can be checked that if M =1, the first line of system (4.1) is retrieved and actual path matches optimal path. Short-run, interim and long-run price elasticities of unconstrained goods q} can be derived from (4.8) in a similar way to (4.5).

Equations (4.4) and (4.8) provide ana!ytical forms which are estimable with proper specification of the error terms. It can be seen that in (4.8) two variables appear in simultaneously, name!y the observed quantities of the quasi-fixed inputs and their market renta! prices. This is necessary here as it appears clearly that observed quantities are neither in shortrun nor long-run equilibrium, but converging to the latter.

Clearly (4.8) shows that by failing to specif)t the netput interactions created by the quasi-fixities in the estimated mode!, there is little chance of obtaining consistent estimates of short-or long-run responses. These parameters depend on the speed of adjustment and there is no way judge in a relevant way the magnitude of elasticities without a time frame in mind, as weil known. As for the technology and particularly for input and output substitution, complementarity and normality relationships, the speed of adjustment and therefore the time frame is also a necessary element of information to be specified clear!y. Another advantage of expression (4.8) is that the estimation is made on the long-run parameters directly, that convexity condition can be imposed on this long run structure, and that the relation between short-run, interim and long-run responses is more transparent than with the J-function proposed by Epstein.

Empirical illustration

The dynamic model is ilIustrated by an application for the French agricultural sector based on a simple model : one aggregate output, one variable input (raw materials), one quasifixed input (capital) and two fixed inputs (labour and capital). The dynamics of this simplified model is described by Figure 4.1 for the output (panel a), for the variable input (panel b) and for the quasi-fixed factor (panel c). only regarding capital as land and labour are supposed to be given. Table 4.1b includes the short-run elasticities when capital does not adjust, and the response of the shadow price of capital to exogenous variables. The Le Chatelier effect is verified, but not the expected tendency toward substitution or more inferiority in the short-run when the quasi-fixed factor constrains adjustment.

The reason arises from the substitution between capital and intermediate consumption in the "capital-unconstrained" technology, which is therefore not normal so that the contraction effect increases superiority of the variable factor c instead of reducing it, as would be the case if capital and intermediate consumption had been complements. Note also that, in the table 4.1 a, output and intermediate consumption are not pairwise similar with respect to capital, hence the similarity criterion is, in this one rationing case, sufficient to predict the outcome of imposing a constraint on cross effects. This result ilIustrates the previous analysis which predicted that, even while imposing constraints on several inputs or (but not and) on several outputs on a normal technology favours substitution and inferiority, further constraints may produce the opposite effect when substitution already exists due to previously imposed constraints. In the one-rationing increment case, the similarity a1lows the outcome to be predicted. Tables 4.2. to 4.4. illustrate the cumulative elasticities of output supply and factor demand for successive lags after the shock on prices of the output and of inputs. The first rows of these tables exhibit parameter values close to those of table 4.1 b, but slightly different because of sorne limited instantaneous adjustment of capital. The last rows are identical to the corresponding rows in table 4.1a, i.e., the long-run response parameters. Long-run own price effect is larger than short run and superiority of the variable factor is increased. Cumulative elasticities = L Olog z, / Olog V'_i Z = (y, C, k) v = (p, w, wk ). p = priee of output; w = priee ofvariabJe input; wk = user cost of quasi-fixed capital

Conclusion

The main implications of strict fixities like production quotas or input rationing is to alter the behaviour of unconstrained supply and derived demand responses to prices.

The cross-commodity relationships, i. e. complementarity and substitutability, which are often considered as basic features of the underlying technology, can therefore be seen to depend highly on the economic environment and on the time perspective where they are observed. The observable technology is a1ways in a temporary stage between the short and long run and cannot be characterised without making a c1ear reference to existing strict or quasi fixities or to the time lag after the shock affecting exogenous variables of the firm's environment.

In that perspective, it is c1ear that concepts of jointness, technical progress bias, economies of scale and economies of scope, must also be looked at with reference to a time frame and to the degree of constraint in the environment. The widespread use of flexible forms for the technology in empirical studies where estimation is performed without dynamics is therefore quite exposed to capturing an indeterminate concept of technology with respect to time frame, with an undefined mix oftechnological constraints and speed of adjustment.

The normality definition, which is useful in order to avoid ambiguous effects of rationing, does not seem to be too demanding, at least when it is applied on the long run unconstrained supply system. It should be verified in many situations where the aggregation structure over outputs and inputs is defined with a fairly homogenous degree of detail. Clearly, when very c10sely related outputs are kept in parallel with highly aggregated groups, substitutability and complementarity among inputs or outputs may coexist even in the long run, and the normality condition will be violated, but separability conditions could then be used in the analysis.

From a policy point of view, the preceding analysis suggests that the efficiency of public intervention will be more likely to run into problems in the presence of strict or quasifixity of quantities. The effect of constraints is to reduce the response of the system to the traditional price incentives. By the Le Chatelier effect and the tendency toward substitutability and inferiority, unconstrained outputs react less to their own prices and less negatively to prices of inputs, but less posivitely (or more negatively) to the prices of other outputs. The supply system could then be broadly characterised by its smaller reaction to its environment and a higher degree of internai interaction. Policy instruments which apply to, say, oruy one output or one input, are than likely to induce spill-over effects on other goods. The shift of the budget problem of the EC from the grain to the oilseed sector in recent years is a case in point.

A similar implication applies to fast technical progress on a constrained output (as the beef industry in EC is likely to show due to the quota on dairy production).

Policy refonns will tend to shift problems rather than to solve them, if they do not tackle the whole sector at once. In agriculture, where existing distortions are large and widespread, problems of second-best limitations of partial policy refonns are more likely to occur.

The above analysis also suggests that the issue of policy reform, in a sector like agriculture with dec1ining outlets and steady pace of technical change, adjusting downward production capabilities raises specifie problems, because internaI cross relations are exacerbated by the quasi-fixities of factors and increasingly of sorne products.

  -tXj~= -~= -7f". + 71/071/,.-1 7f~:l? -7f". when h and k are inputs.(3.3) ~.

Figure 4 . 1 .

 41 Figure 4. 1. The time structure of priee elasticities with respect to output, variable input and quasi-fixed input priees.

Table 3 .

 3 2. Price e1asticities before imposing the dairy quota

	1	q\v	MIL	VEG	CER	ANI	FEED	FER	DIVI
		MIL	0.239	0.223	0.212	0.201	.{),401	.{).195	.{).280
		VEG	0.146	0.239	0.248	0.227	.{).375	.{).110	.{).375
		CER	0.343	0.613	0.547	0.649	.{).922	.{).227	-1.003
		ANI	0.105	0.182	0.210	0.309	.{).546	.{).008	.{).251
		FEED	0.393	0.562	0.559	1.023	-1.998	.{).081	.{),457
		FER	0.636	0.549	0,458	0.052	.{).271	-1.024	.{),401
		DM	0.616	0.667	0.301	0.516	.{).501	.{).132	-1.467

Table 3

 3 -fixed factors, by making use of the previous approach based on shadow priees. Now, if in fact qo cannot adjust immediately to the optimal level, the actually observed quantities qlt and %t are produced by the same model where virtual priees 170t are substituted for observed priees VOt :

		.3. The complete system ofsupply-demand e1asticities after imposing the dairy quota
	1 a\v	MILla)	VEG	CER	ANI	FEED	FER	DIVI
	MIL (1])	4.184	.{).933	.{).887	.{).841	1.678	0.816	1.172
	VEG	0.611	0.103	0.118	0.104	.{).130	0.009	.{).204
	CER	1.435	0.293	0.243	0.361	.{).347	0.053	.{).601
	ANI	0,439	0.084	0.117	0.221	.{).370	0.078	.{).128
	FEED	1.644	0.195	0.210	0.692	-1.339	0.240	0.003
	FER	2.661	.{).044	.{).I06	.{),483	0.796	.{).505	0.344
	DlVI	2.577	0.092	.{).245	.{).002	0.533	0.371	.{).745

quasi

Table 4 .

 4 1. gives the short-run and "long-run" elasticities of the illustrative dynamic linear model based on a quadratic profit function and on partial adjustment of capital to its optimal level after a price shock. Inputs are superior with respect to output, but substitution between capital and intermediate consumption prevails in this technology which is "Iong-run"

Table 4 .

 4 1a. Matrix oflong-run elasticities

		D	w	wk
	y	0.446	-0.319	-0.127
	c	0.737	-0.762	0.025
	k	1.381	0.115	-1.496
	Table 4.1 b. Matrix of short-run elasticities (zero adjustment of capital to target)
		p	w	K
	y	0.329	-0.329	0.085
	c	0.760	-0.760	-0.017
	h	0.923	0.077	-0.668
	Lambda	0.194

Table 4 .

 4 3. Cumulative elasticities of variable inputs (c, intermediate consumption) Table 4.4. Cumulative elasticities of capital demand (k)

	1 0	P 0.352	w -0.327	wk -0.025
	1	0.370	-0.326	-0.044
	2	0.385	-0.324	-0.061
	3 4	0.397 0.406	-0.323 -0.323	-0.073 -0.084
	5	0.414	-0.322	-0.092
	6	0.420	-0.321	-0.099
	7	0.425	-0.321	-0.104
	8	0.429	-0.321	-0.109
	9	0.433	-0.320	-0.112
	10	0.435	-0.320	-0.115
	11	0.437	-0.320	-0.117
	12	0.439	-0.320	-0.119
	13	0.441	-0.320	-0.121
	14	0.442	-0.320	-0.122
	15	0.443	-0.320	-0.123
	16	0.443	-0.319	-0.124
	17	0.444	-0.319	-0.124
	18	0.444	-0.319	-0.125
	19	0.445	-0.319	-0.125
	20	0.445	-0.319	-0.126
	21	0.445	-0.319	-0.126
	22	0.445	-0.319	-0.126
	23	0.446	-0.319	-0.126
	24	0.446	-0.319	-0.126
	25	0.446	-0.319	-0.127
	26	0.446	-0.319	-0.127
	27	0.446	-0.319	-0.127
	28	0.446	-0.319	-0.127
	29	0.446	-0.319	-0.127
	30	0.446	-0.319	-0.127
	31	0.446	-0.319	-0.127
	32	0.446	-0.319	-0.127
	33	0.446	-0.319	-0.127
	34	0.446	-0.319	-0.127
	35	0.446	-0.319	-0.127
	{~810gcJ810gVt-i ; v=(P,W,Wk)}

Situation il, which is a special case of ii), is detailed because it corresponds to particular technologies if outputs (respectively) only are rationed.

The dynamics of adjustment of fixed quantities and the observable technology

In the previous section the nature of cross effects in the behavioural equations was shown to depend on the extent of binding constraints on producer profit-maximising decisions.

The underlying dual technology is also clearly dependent on these constraints of fixity. As the producer cannot adjust immediately to price changes, the firm is never observed in an equilibrium either short-run or long-run, but somewhere in between on a transitory path toward il. Then the speed of adjustment is the key factor in the interpretation of the observed technology in econometric work where for example the adjustment process is not formally included but implicitly assumed. This section attempts to built on the disequilibrium framework From these two definitions, a similar decomposition of. 7f_. and 1?_ , as in section 2, glves:

The contraction effect in (A3) is the negative of the expansion effect in (A4).

Twice differentiation of (Al) and (A2) gives :

(AS)

[ ~o] [dPo] '::; = (7f):; where (7f) is the familiar Hessian. An expression identical to (A6) can be derived trom differentiation of the profit maximisation conditions (as arguments cannot be output prices, notations are simplified) applied to the cost function.

(AS) (A.S) can be solved to find the system where oruy YI vector can adjust to optimal Jevel wlûle Yo remains fixed. The second row is solved for d y ,d y , and plugged into the first and tlûrd rows to express 1]0 and x in terms of the exogenous variables (Yo.1]1' w), dyl =C I -: {d1]1 -CI~O -Clwdw}, hence, (A. 9)

COPI~l

C-I

II

CWICl~1

The expansion effect in (A.4) can now be written in terms of the sub-Hessian of the cost function, as (A.9) and (A.6) are identical : ~. = C wO -CwIC;/C IO is a positive matrix under normality conditions, Cll being an M-matrix (CwO~0 .. Cwl~0 .. C I -/ ~0 .. -CIO> 0 ). The same is true for ~,w. as for ~.y. = Cao -COPI~ICIO' it is an M-matrix because Cao is an Mmatrix and COIC;/CIO is positive. Hence, (~oY' r is positive. Therefore, the expansion effect subtracted from 7?_ makes the unrestricted inputs more complement after output rationing relaxation and similarly in (A.3), ff'_;? 7f_, and imposing a constraint increases cross effects and favours substitution. The Le Chatelier principle is also embodied. A similar approach can be used for constraints on inputs, with the same tendency toward substitution between outputs. derived from normality conditions on the revenue function.


Bonafous

Influence of nitrogen availability on growth and development of tomato plants until fruit-setting