Technical change and agricultural supply-demand analysis problems of measurement and problems of interpretation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

( [START_REF] Brown R-S | Estimating elasticities of substitution in a model of partial static equilibrium an application to U.S. agriculture[END_REF]Kulatilaka, 1985Hertel, 1987). Equally important fixities may exist among production outputs [START_REF] Thursby | [END_REF] Consequent1y the first objective of this study is to show the importance of taking into account the quasi-fixity of sorne inputs and/or outputs in the French sector for wheat and corn (section 4).

TOTAL FACTOR PRODUCTIVITY GROWTH AND TECHNICAL CHANGE :

THEORETICAL ANALYSIS AND EMPIRICAL IMPLICATIONS

Theoretical analysis

There is no single generally accepted way to measure productivity or productivity growth. Following Solow (1957) the more ccmmon procedure directly related to the structure of production begins with a production function representation of the input to output transformation process (Link, 1987) and total factor productivity is then defi~ed in terms of the efficiency with which inputs are transformed into "useful" output, assuming that homogeneous inputs produce a homogeneous output. More precisely, in the context of a production function, it is tradional to measure total factor productivity growth by the residual method, that is the growth in output quantity minus the growth in input quantities. In other words the multifactor productivity residual measure is linked to outward shifts in product long-run isoquant whereas the input effect, that is the effect measured by weighted growth rates of inputs, is associated with substitution effects along the isoquant. Alternatively and equivalently under certain regulari ty conditions which will be specified below, in the context of a total or long-run cost function, the dual total factor productivity measure is defined by the growth rate of average total cost minus the Divisia index of input priees this residual is linked to downward shifts in unit or average long-run cost curves.

In numerous empirical studies, the continuous growth rates are replaced by the annual diff erences in the logari thms of the variables and the shares used as weights are replaced by annual arithmetic averages. The resulting indexes are the Tornquist indexes of total factor productivity growth, primal and dual respectively (see, for example, Berndt andFuss, 1986 Hulten, 1986).

(TFP/TFP) = Y/Y -! l (Wl XII p y) (Xl IXI ) (PI) '" logY(t) -logY(t-l) -!l [Ml (t) + MI (t-l)]/2
(log Xl (t) -log Xl (t-l) (P.)

(TFP/TFP) = CT/CT-Y/Y -!l (Wl Xi ICT) (Wl IW1) (Dl) '" (logCT(t) -logCT(t-l)) -(logY(t) -log Y(t-l))- !l [SI (t)+SI (t-l)]/2. (log Wl (t)-log WI (t-l») (D.)
where Y ~0 is the output with price p ~0, X' = (X, , ... ,XN)

+1.3 -0.2 -0.03 quota 1evei -u +0.0 '0.2 -5.6 -2.6 quota levei -6.1 -0.3 o -5.9 -2.7
Assuming that dual and market priees are equal for the base period 1984, that is assuming that the milk market is in equilibrium in 1984, we can compute the dual price level of milk for each year and consequently we can calculate the bias induced by the milk quota system in measuring technical change by the traditional non-parametric total factor productivity index. As an example in 1984 the bias is equal to -0.031 % in France and -0.017 % in Germany. The bias, which depends not only on the gap between âual and market priees but also on the milk quota growth rate, increases wi th time since the difference between dual and observed milk priees increases too (see table 1).

Third, let us consider the case where returns to scale are constant, aIl output markets competitive and in equilibrium but some inputs quasi-fixed. In such a case, equation (c) reduces to ) 0 input X2 saving = 0 neutral < 0 input X2 using factor price ratio (w,/w2) constant.

(TFP/TFP)p = -Eco 1 + !' j [(w' j -w' j . (X'j /eD)]. (X'j lX'j
The previous definition can easily and equivalently in the two input case be transformed into a defini tion in terms of factor shares at constant factor price ratio. Furthermore the share approach generalizes immediatly to the many-input case.

The measure of bias for each factor proposed by Binswanger is given by,

Blt= ct _1_ S,
) 0 input X, using = 0 input X, neutral < 0 input X, saving factor price ratio (wl/wJ) constant.

where SI is the share of input Xl in total costs. Technical change biases are th en defined on the basis of a dual representation of the technology, assuming that there exists a long-run total cost function CT(Y, w, t) where ail inputs are variable. It is interesting to note that i f the long-run technology is not homothetic with respect to y, Both derivations are based on constant relative variable input priees as weIl as output and quasi-fixed input levels.

Furthermore the second defini tion implies also the constance of fixed input rentaI priees. BC R" and BC 0" are linked by the following equality, The signs of BI t and Be Olt may differ depending on the relative magnitude of this Il expansion" effect which can be ei ther posi ti ve or nega ti ve. In other words, technical change may be input X"l long-run saving and short-run disequilibrium using.

Be D 1 t = E C R 1 t -(C 0 t = E ç R 1 t -(c R
In the multioutput case, technical change biases may also be defined for outputs. Assuming that farmers are long-run profi t maximisers and using the property tha t maximisation of profits may be broken into two parts, maximisation of revenue Important fixiti8S exist among production outputs. The example that springs to mind immediately is that of cattle output in the agricultural sector [START_REF] Thursby | [END_REF].

When sorne outputs are considered as quasi-fixed, they appear as arguments in the restricted revenue function RR(pO ,YI ,X,t) and in the total disequilibrium revenue function RD(pO ,YI ,pl ,X,t). This last function represents the maximum attainable revenue given the vector of variable output priees pO, the vector of production inputs X, time t and with sorne outputs as quasi-fixed YI s . This function is defined by RD(pO ,yI ,X,t,p') = RR(pO ,Y' ,X,t) + l'splsY's. Consequently two short-run measures of technical change biases on the output side may also be defined, in terms of restricted revenue shares or in terms of disequilibrium revenue shares, respectively. The analysis which is ana1ogous to that proposed in the input bias case will not be developed in this paper (for more detai1s, see Boyle and Guyomard. 1989).

Problems of measurement

The A third specification of technical change is the factor augmenting form or, from the point of view of the dual, the priee diminishing form [START_REF] Berndt E-R | Concavity and the specification of technical progress in US manufacturing[END_REF]Wills, 1979).

In this case, input priees are written as : p'" in regressions such as :

SOME REMARKS ON THE USE OF TIME TREND AS

22

K YI = a + I3t + r 01 Z11 + el i=l (1)
where YI is the log of output, ZI = (Zl l , • 0 0 ' ZK 1 is a set of K exogenous variables (each taken in log), t is a linear time trend and el is a series of white noise (0, 0 2 . ) residualso

In -The first procedure is to assume that X is explained by a linear dependence on time, wi th the remaining variation in X being due to a stationary cyclical component El. In this case X, is modeled as :

Xt = a + bt + Et and ~(L)Et = 0(LI nt ( 2 )
where ~(L) and 0 (L) are polynomials in the lag opera tor L (Lk X, =Xl -k) of respective order p and q El is a stationary and invertible process and nt a whi te noise process (0, 0 2 n ) _

The most noteworthy characteristics of model (2), which Nelson and Plosser (1982) dub the Trend Stationary (TS) model, are that the trend incorporated in Xt is completely deterministic and tha t ail shocks to X are temporary or cyclical. Thus, the pa th of X in the long-run is completely deterministic and is not altered by current or past events. A related issue is that uncertainty on forcasting error is bounded since :

Var(X,-a -bt) = Var (El) = 0 2 • (3)
The second procedure maintains that X contains one unit where u, is a sta tionary ARMA process, Q (L) and r (L) are lag polynomials of respective order rand s, v, is a whi te noise process (0, 0 2 v). A particular case of the DS model is the random walk model in which u, does not follow an ARMA process but is instead an identically independent distributed (iid) white noise process (0, 02 u ) .

Since data are in logs, (Xl -X, -1 ) is the growth rate of X,. Furthermore the mean growth ra te b is theori tically the same as the coefficient on t in model ( 2). The great difference between OS and TS models lies in the fact that in the OS model the impact of a shock on X, embodied in the innovations v, is to shift the level of X upward or downward permanently.

In order to rapidely compare the properties of the OS and TS process, the first equation of model ( 4) is rewritten as

t X, = Xo + b t + r u, -J (5)
j=l by accumulating changes in X from any ini tial value, say Xo.

Although (5) has the same form as (1), we see that the intercept is no more a fixed parameter but rather depends on the initial value Xo. Furthermore the disturbance is no more stationary and the variance-covariance matrix depends on time.

Another point is that whereas interval for future values of X finite dispersion of E', in (5)

with time :

the width of a confidence in (2) is limi ted by the i t increases wi thout bound

t Var (Xl -Xo -bt) = Var (r Ut-J) = t0 2 u j=l (6 )
(7 )

Econometrie Problems due to non-stationary disturbances

Model (l) implici tely assumes tha t the AC TC which can be written as where YT+m is the forecast of YT.m made at time T and ZIT.m the value taken for ZIT.m.

However if the AC TC is not a TS variable but instead a DS variable according ta :

K ACTC,= y,l 51 Z" = ACTC'-1 + ~+ e, i=l then first differencing of the relationship (1), that is, K (y, -y, -,) = ~+ l 51 . (Z, t -ZI' -1) + et i=1

would put it in the form suitable for estimation. This is due to the fact that if the AC TC is DS, then estimating a relationship such as (1) amounts to estimate a relationship similar ta :

K Yt= Yo + ~t + l 5, i=1 t ZI t + l e, -J j=1 (11 )
with non-stationary residuals. variables when none is present. This is due to the fact that an OLS es tima tion of 0' in (11) can be thought as regression of detrented Y on detrended Z according to :

K Y' t = r O t Z•. t + e*, i=l (12 )
where stars denote detrented variables.

in ( 11) is Equivalent to a regression variable and the error term are both and thus have the same autocorrelation However if Zt and et

m Xt = IJ + L dl. Xt -1 + pt + nt i=l ( 14
)
where nt is a sequence of normal independent random variables m (0, 02 n l and L dt" = 1 and p = 0 in the case of the DS model. i=l

By rearranging the terms Xt -t, the relationship ( 14) can be rewritten : The null hypothesis of a DS model is rejected for wheat and corn yields data but is not rejected for corn production.

m m Xt = IJ + (L dl 1 Xt -, + (-L dl 1. (X, -
Thus, wheat production and wheat yields data series both seem to incorporate a deterministic time trend. These results are compatible with the common idea that the increase in the French wheat production is mainly attributable to the regular improving of producing technics and to technical change. While corn yields seem to be distributed according to a TS process, corn production seems to be non stationary due to the presence of a unit root. Therefore, resul ts of regressions such as (1) where corn production is the endogenous variable can be subject to the spurious regression phenomenon and have to be considered with caution. Such a regression should be taken in difference before estimation.

APPLICATION OF TIME SERIES METHODS TO THE MEASURE OF STOCHASTIC AND DETERMINISTIC COMPONENTS OF TECHNICAL CHANGE

Definition of persistence

Let Xt be a given non-stationary logarithmic time series which can be made stationary with a first order difference.

The Wold (1938) theorem shows that (X, -Xt -1) can be written as an infinite moving average model according to :

Xt-Xt-1 = (:!.-L) Xt= M(L) w, (16 )
where M(L) = (l+M,L + M2L2+ ... ) is an infini te polynomial in the lag opera tor and w, is whi te noise (0, 0 2 w ) . The impact of a shock in period t on the growth rate in period t+k is Mk.

The impact of the shock on the level of X, is given by P(L) = (l-L)-l. M(L) which is equal to P,= 1 + 11, + .. + M, in period t+k. The total effect of the shock on the level of X, is given by the infinite sum of the moving average coefficients, which is

M(l) = lim PI= 1 + ! MI i->- i=l
The value of M( 1) is thus a measure of persistence. It indicates the extent to which the long run path of Xt deviates from its past trend after a once for a11 unit shock.

If Xl is a TS process, fluctuations in the series are stationary around a deterrninistic time trend and after a shock, the series a1ways returns to its trend in the long-run.

In this case, the persis tence i s thus zero (M (l) =0 There are severa1 approaches to es tirna ting persistence.

Two of thern. the Carnpbell-Mankiw and the Cochrane proceduces.

are briefly presented in this paper. The theory of persistence provides a useful method to measure the relative size of the randorn walk and stationary components. [START_REF] Beveridge | A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle[END_REF] show that any first-difference stationary process Xl can be represented as the sum of stationary or ternporary (Tt)

and random wa1k or permanent (Pt) components. It can be shown

  HG.120/i/!! :ns:it~t Natienal de la Recherche Agrono;ique Station ë'E=cno~ie et Scc~ologie RuralEs 65, rue de St-Brieuc 35042 RE~NES CEDEX m.NCE :,~_;:"J',. r:~,~llr( ----. ..__J 1 EC~...'J-..i 'li-::' ~t.'~P.:";:: 1 L-.._--'s",,-.""d",L:;e= 01.tiÉ.9U:: _ J TECHNICAL CHANGE AND AGRICULTURAL SUPPLY-DEMAND ANALYSIS PROBLEMS OF MEASUREMENT AND PROBLEMS OF INTERPRETATION H. Guy-omard. C. Ta véra 1989 meeting of the Gesellschaft für Wirtschafts und Sozialwissen-Schaften des Landbaues (GEWISOLA) -BRAUNSCHWEIG -GERMANY -Workshop on measuring the effects of Technical progress in agriculture (methods and application) paper is to explore various to the measurement of technical change and the measurement of total factor productivi ty 2 Principles of duali ty enable the economist to model the technology of a multiinput-multioutput technology by means of a transformation function or its dual cost, profit or revenue functions. The use of a flexible functional form which can be either the true function or a second order local approximation to this underlying function around a point of expansion does not impose a priori restrictions on many empirical studies are based on a restricted or short-run partial equilibrium framel;ork, fixing certain inputs such as capital, land and/or family labour.

  technical change biases when sorne inputs are treated as being quasi-fixed is explored on the basis of a restricted (or short-run) cost function the relevant concepts of biases are defined and related to different possible equilibria. This analysis is extended to technical change biases on the output side (section 2).sections 3 and 4 are devoted to econometric issues. In empirical works on technical change, one often encounters regression equations that include a linear time trend as a proxy for technical change. However, as was shown in highly misleading and can be subject to the spurious regression phenomenon. As a result, the estimated =oefficients of time and exogenous variables can widely overstate the size of autonomous and incorporated technical change. In order to avoid such problems, non-stationarity properties of time series data must be carefully examined. We show how standard estimation of regressions including a time trend can lead to erroneous conclusions on technical change if data series are not stationary around a function of time, but rather are stationary in first difference. Moreover, we sho'h ow the tests for stationarity in difference as opposed to stationarity around a trend line developed by Dickey and Fuller (1979, 1981) can be used to determine the appropriate transformation of time series data (section 3). Lastly we briefly review sorne recent developments on the analysis of persistence in time series process which can be used to measure the autonomous and stochastic component of different from standard analysis of technical change, and provide a complementary approach to the usual measure of technical change. As an example, these methods are applied to the comparison of the respective size of the autonomous and stochastic component of technical change

Õ

  the transposed vector of inputs with associated transposed price vector w'= (Wl, ••• ,WN) ~0, CT the total cost function, M, the income shares and SI the cost shares. Dots over variables indica te deri va ti ves wi th respect to time. P, (respectively Dl) is the primal (respectively dual) Divisia index of total factor producti vi ty growth, P2 (respectively D2) its Tornquist approximation. These residual measures of total factor producti vi ty are called non-parame tric insofar as P2 or 02 do not require an econometric estimation of the production function or of the cost function. Nevertheless, both measures are derived under the assumption of a competitive long-run equilibrium. More precisely firms seek to maximise long-run profit in the first case whereas they seek to minimise long-run cost in the second case. The assumption of long-run returns to scale is not necessary to develop formulae Pl and P2 if estimates of income shares are available. It is interesting to note that long-run cost minimisation is a weaker assumption implied by long-run profit maximisation. Nevertheless input and output markets must be competitive and in long-run Marshallian equilibrium in order to extend the dual representation of total factor productivity to the multioutput-multiinput case. How such measures, easy to compute, of total factor producti vi ty are rela ted to technical change measured by the rate at which the production function shifts ? Solow has shown that technical change and total factor productivity Ohta (1974) has shown that primal and dual non parame tric measures are nega ti ves of one another. Assuming a translog representation of either the production function or the cost function, Berndt and Jorgenson (1975), Diewert (1976) have proved that the assumption of neutral technical change is not necessary to have this equivalence. In other words, the non-parame tric measures of total factor productivity growth rates equal the rate of technical change (the rate at which the production function shifts or the rate at which the long-rull total cost function shifts) if the three following assumptions are verified -constant returns to scale -input and output markets are competitive -inputs and outputs are in long-run Marshallian equilibrium.

(

  the counterpart of equation [g] established in the case of output disequilibrium. So the same reasoning applies. The magnitude and sign of the difference between often characterized as neutral or biased. Based on original Hick's definition and assuming a two input -one output linearly homogeneous technology, technical change is said to be neutral if it leaves unchanged the.marginal product of input X, to that of input X2. However, as noted byBlackorby, Lovell and Thursby (1976), "to compare situations before and after technical change, something must be he Id constant. Exactly what is to be held constant has been the subject of sorne debate and constitutes the crux of the issue at hand",Kennedy and Thirlwall (1972) among others argue that factor endowments must be he Id constant at least at the macro level and consequently technical change effects must be measured along a ray where factor proportions remain unchanged. At the firm level and also at the macro level in a sector like agriculture where enterprises are more often ) biases and neutrality are defined expansion path, that is in terms of the proportional the input ratio holding factor price ratio constant.

  constant not only relative factor prices but also output levels. Following Sato (1970), the bias Bl1 can be interpreted using the following decomposition B, 1 = OSI/Ot . l/S, 1Y, Wl /Wl = 010g(WIXI (Y,w,t)/CT(Y,w,t})/ot = olog Xl (Y,w,t)/ot -ologCT(Y,w,t)/ot = ( 1 t -terl Consequently the bias is the difference of two effects the percentage change in demand for the input Xl minus the average percent age variation in inputs. The sign of this second effect is known unambiguously if technical change occurs (Ec Tl < 0) Then a technical change which is input Xl saving decreases expenditure on that factor because the reduction in XI from a change in t is greater th an average.

-:

  on the output side in terms of longrun revenue output shares holding relative output priees and input levels constant. More specifically, the total or longrun revenue function is defined by R(p,X) = MaxYlLrprYr total profit (-X,Y)eT) is the output possibility set, Y the with priees p' = (Pl, ... ,P.) ~O. The is then defined by n(p,w)= Maxr.x(LrprYr = R(p,X(p,w)) -CT(w,Y(p,w)), where T LI Wl Xl the production possibility set. Consequently a measure of bias for each output is given by , Brl = ct _1_ Sr input levels, relative output priees. where Sr is the total revenue share of output Yr. A positive value of Br t implies that technical change output Yr (or relatively rth output using), value of Br 1 implies that technical change output Yr (or relatively rth output saving) definition can be illustrated geometrically. The curve F (t) (f igure 1) represents the product transformation frontier for a single hypothetical industry, which produces only two outputs y, and Y2 given the levels of inputs. The slope of the tangent to this transformation curve indicates relative commodity prices given prices p, and P2 0 the firm faces an isorevenue line PP':R = P,Y. + P2Y2 which has slopep, /P2. Consequentlyo with relative output prices depicted oY'2). The direct effect of technical change would result in a shift of the product transformation curve. The curves F, 0 F20 F3 and F. represent four alternative possibilities. For technologies F, 0 F2 and F3 0 the proportional "using" in outputs is equal to PaP,/OPa 0 but the output using biases differ. In the first case (technology F,), technical change uses both outputs in the same proportion as before the new tangency point is Band technical change will be said to be neutral with respect to both outputs (B, t = B2 t = 0). On contrary 0 if the new equilibrium is at point C on curve F2 0 the output Y2 level remains unchanged but technical change is output y, using. By the same wayo on curve FJ at point Do technical change is output Y2 using but neutral with respect to Y.. These three possibili ties are limi ting cases insofar as technical change• is defined as neutral and/or purely output using. If the technical change effects are more favorable to Y2 relative to y, 0 the new curve will be skewed toward Y2 50 that the new equilibrium point is E where Y2 > y' 2 and y, < Y' 1 in such a case 0 technical change is output Y2 using and output y. saving. In other words 0 technical change is biased toward output Y2 and against output y, . ?igure 1. Neutralicy and tiases cf technical change in output spac; a: :~nstaJt fact~! lE.els and :the input-bias case : Brt = oSr lot. 1/Sr 1 X, pr Ips s = olog (pr Yr (p, t , X) 1R (p , t , X) ) lot = i5logYr (p, t,X) li5t -i5logR(p,t,X)/i5t bias Brt is the difference of two effects : the percentage change in supply for the output Yr minus the average percentage variation in outputs. The sign of this second effect is positive if technical change occurs. Then a technical change which is biased toward output Yr increases revenue on that output because the rise in Yr from a change in t is greater than average. An alternative interpretation may be proposed using the following derivation Br 1 = 1/Sr (i5'logR /ologprot)=(otRI/élogPr).1/Sr and consequently Brl measures also the effect on total revenue increase from a change in pr.

  li long-run trans!og cost function four inputs: !, N, 5, CI one output: Y nen-neutra! technical chac,e Incdel Il -static lonq-run e,ui!ibriun -d!namic long-run equi!ibr:u! I!u~tivariate adjustenen,} : ;odel 1 corresponds to: ti~e trend represéntatÎon of cecJoical chançe, model II 1S estilated assucinq that the different paraceters, var! !inearl! ~ith time. K• capital, N• famil! and hired labeur, S :: land, CI :: rail !:lateria!s. •"oi:j can be desaqregated in th!ee compcnents : A :: animal teed, ? :: f=!ti~izers, 0 :: other r~. lat2ri~ls, SS : hire~l~bour.

  the autonomous component of technical change)

  equation (ll, the effect of the Stochastic Component of Technical Change (SCTC) is captured through the effect of shocks affecting the economy or the sector under consideration 0 Recently several papers on nonstationarity have shown that results from the estimation of equation (ll are strongly subject to the spurious regression phenomenon and have to be taken with caution if YI and the Zlt ; i=l, o'' ,K; are non-stationaryo 3.1. Some remarks on non-stationarity Many economic time series are characterized by nonstationarities such as a changing mean and variance over time 1 • There is obviously an unlimited number of possibilities that can account for non-stationary behaviour but lots of . Iü applied ëJtirical ~nal!sis, Jnly ~eak staticn~r:~ï is re~uir:d ~hich is ~qui7alent t~E!Xtl : :cnstant lnd Co. Il! ,1,",1 = !: il S = 0 and = 0 il S +D. authors have recently reexamined two widely used procedures to model changing means over time. Let Xt be a logarithmic nonstationary time series.

2

  root 2 and differencing of the data is therefore the correct way to remove non-stationarities _ The adequate model, whichNelson and Plosser (1982) cali the Difference Stationary (Let Xl f~ll:~a stcchascic difference 2~uation :!cdel of the fcr~: (l-ctL .. -3IltjXt=(>dlL .. -èlLfJ Il, -"here ~t is : su::: vi ::d ranèol sh~cks .i:h :::::D ~::ld nri~jj:: :Z 'il. :H : ~e:'!ct: ~::~p.:x nria:l: --d ("1 .:;;.. ~::unc•:• . . . ;17'-11-. •"-0' -" t; ; .... "fI:'It .. f ;'''}-' 'l'-h 1 1-1 . .1,.. • 11-" :..: ... n~J. ..:r :.. .J.'__I li wS-.. 't_ t _ • 1_ J .~: ... 11 •• \,IllI -J ~.. lU -., \,u:n U 15 ca Cl,i • unit foot cf ~{Il : O.

KK

  ACTC, = y,l 51 ZI' = a + ~t + e, i=l follows a TS process. In this case the ACTC is assumed deterministic. Forecasts made with such a model are thus based on the hypothesis that only the SCTC can be altered by a given policy in the long-run since 25 YT+m= a + l 5i ZIT+m + ~.(T+m) i=l ( 8 )

  o' by OLS in 1evels is subject to the spurious regression phenomenon. That is, conventional t and R2 tests are biased in favour of indicating a relationship between the

  errors will be to inflate the variance of the OLS coefficients. In this respect, the precision of the estimate of O• will be greatly overstated if seriaI correlation in the regression errors is ignored conventional standard errors and t statistics will mislead by overstating significance of coefficient estimates. 3 Chan-H~11a and Crd 119771 ShOi that ~ben the trus ~cdel of a time series is ra~dom walk (or ocre generally a OS :nodal!, the use cf a Boen determi::istic time :renè to ali:nin:ta a suspected tre:lè .ill proèuce large spurious positive autcc:rrelation in the ~:r5t :ew lags.

  the permanent component shifts from period to period in response to current innovation. Beveridge and Nelson cal1 Pt the stochastic trend of the series Xl. In other words the the ory of persistence indicates which part of the process is TS and which part is DS. of the variance ratio V(k) is a measure of persistence : V = lim V(k) .k->F or the two polar cases, both measures of persistence produce the same number V = M{l) = 0 if X, is a pure TS and V=M (1) =1 if Xl is a pure DS model. In the more genera1 case, that is when Xl includes a permanent and a stationary component, the two measures are not the same. However they are not independent since it can be shown that M(l)= [V/(1-R2)]l/2 where R2 is defined as [1 -Var (w)/Var(XI-XI-I)]. Application of Cochrane and Campbell-Mankiw procedures When data series on technical change are available, the the ory of persistence can be used to measure the relative sizes of the AC TC and SCTC. However data series on technical change are rarely available or are subject to lots of problems (see Section 1 and 2). In this sub-section, we assume that along the long-run or equilibrium path, production grows at the same rate as technical change. Thus the measure of persistence in production is a proxy for pesistence in technical change. The Cochrane and Campbell-Mankiw methods are thus applied to the previously defined French data series for production and yields for wheat and corn. The Cochrane measure in estimated with k = 30 which amounts to assume that temporary shocks have disappeared after 30 years. Empirical

34A

  shown by Equation (23) the two measures take different values depending on the size of R'. However results are fully compatible with those obtained with the Dickey-Fuller tests ~, and ~3. Wheat production, wheat yields and corn yields include rather small permanent component Irespectively 0.12, 0.08 and 0.12 as measured by V*k and 0.41, 0.32 and 0.41 as measured by MIl) ). This is in harmony wi th ~2 and ~3 previous resul ts tha t these series are TS. The greatest part of technical change seems to be deterministic in the French wheat and corn sector. In these sectors, technical change is thus only little affected by temporary shocks. A given shock has only a minor impact on the long-run behaviour of technical change in those sectors. Both the permanent and transitory components of corn production are important. This is consistent with estimated ~2and ~3 statistics which showed that this series is DS. This apparent contradiction between results obtained with acre ages for corn during the period of analysis" while, at the opposite, planted acreages for wheat are rather constant aIl along the period. Thus, taken as a whole, resul ts in table 2 seem consistent wi th the idea that the greatest part of technical change is deterministic and autonomous in the French wheat and corn sectors with the smallest SeTe in the wheat sector.

  and technical change measurement has taken place in the last decade, sorne important problems, which have been reviewed in this paper, have not been treated in a completely satisfactory manner. As an example. the existence of the dairy quota in the Ee requires further analysis in order to correctly measure the impact of this policy instrument on economic theory and time series analysis can contribute to a better understanàing of the sources of variations in the patterns of proàuctivity growth and technical change. ~Planted acrea;es :o~C~:J In::eaS2 ~ich a :elatÎïelï ~:~5tant ~:o~tJ :ate UJ:il 1~73, tje5 dêc~:as= in a ~b:~tic ny. expression into [cl we obtain, (TFP/TFP)p = -olog CD/ot + IO r Rr . yo r /yo r + Ils Rs . Y1 s !Yl s yo r /yo r + L' s pl s YI s /CD . YI s /Y' s ] + L') (W') -W' J) X' j /CD . Xl J /XI j [d] TAVERA C., 1989, Composantes stochastiques et déterministes du progrès technique les enseignements de la mesure de la persistance. INRA-OP nOCTAL8911, 25 p. TAVERA C., 1989, Testing for stationarity literature. INRA-OP 30 p. a survey of recent TH!RTLE C-G., RUTTAN V-W., 1987, The in the generation and diffusion of , 1979, Technical change in the US primary metals industry. Journal of Econometries, vol. 10, p. 85-98. WOLD ed. , H., 1938, The analysis of stationary Uppsala: Almquist and Wicksell. time series. 1st

  EC Tl + !s (ps Ys IRT -ps Ys ICT). Ys lYs

					8
	Nevertheless when aIl markets are in long-run Marshallian
	(TFP/TFP)p= ror Rr YOr/yor + Ps Rs Y's /Y's -roi 5, xo,/Xo, equilibrium and if returns to scale are constant, equations (TFP/TFP)p = -[g]
	[cl and [dl collapse to the usual expression ; -P J SJ X' J /X' J . TFP/TFF = -olog CT/ct ; since in such a case The direction of the bias induced by a non-marginal
	tarification of	outputs	depends	on the gap between market
	Technical change is defined by -clog CD/ct, that is the CT = CD, p' s = p' s for aIl s, w ' J = W' J for aIl j, ou tpu t shares ps Ys IRT and "marginal" output shares ps Ys 1 CT,
	rate at which the disequilibrium total cost function CD ~= 1 and CT = RT. In order to analyse the consequences of the where ps = 6CT/6Ys is the marginal cost of output Ys. As an
	shifts. Then, i t can be shown tha t technical change and the simplifying assumptions allowing to show the equivalence example, let us consider the case of the dairy quota which is
	traditional total factor productivity non-parame tric measure between total factor producti vi ty growth ra te and technical binding since its implemantation in 1984 in EEC. Guyomard,
	When one are related by the following equation (see annex nOl) . of these assumptions is violated, change, we will consider successively three particular cases. simple Mahé, Tavéra and Trochet (1988) consider that from 1984 to
	1986 the shadow priee of milk decreases by 5 to 6 percent per corrections can be applied to relate the growth rate of total factor productivity non-parame tric indexes to technical change. Assuming that some inputs are quasi -fixed, [c] reduces to we First, i f aIl markets (inputs and outputs) are in year. More precisely, us ing a theoretical model developed by -clog CD/ct = 10r (pOr YOr/CD -pOr YOr/ RTl YOr/ Y•r equilibrium but if returns to scale are not constant, equation Mahé and Guyomard (1989) which links the endogeneous dual
	+ l's (pis Y's/CD-pis Y's/RT) priee of milk pM s to (TFP/TFP)p = -leTt + Ir pr Yr (1/RT-1/CT). Yr/Yr its determinants Y' s /Y' s variable input priees, quasi-fixed input levels and priees, (output and [el
	milk-quota level and technical change) it is possible to
	If returns calculate the dual milk priee growth rate for each year. The to scale are decreasing, RT > CT and
	+ (TFP/TFP). therefore (TFP /TFP) p < -le Tt. Decreasing returns to scale [cl results for France and Germany are presented in table 1.
	will	yield negative scale effects and the residual measure
	principal source of fixity is the lack of mobility of self-Equation [cl may be also written as . !able 1. Observed and dual lilk pri~e gro.th rate. lilk quota growth rate; France and G,rcany, 1334 tJ of total factor productivity will be underestimated as a 1988. _. lationai priees iprovisionai resultsl
	employed farm labour which is enhanced by high unemployment in measure of technical change. For increasing returns to scale,
	other economic sectors (Brown and Christensen, 1981 ; Guyomard and Vermersch, 1989). At the farm level, . (TFP/TFP). = -clog CD/ct FR.INCE the bias is inverted. In the special case of a multiinput-Jual prieel Jarket priee/ available monooutput technology or if outputs are separable with respect 1984 -2.2 14.1
	agricultural land is often +(l-~) (~)-l [IOr pOr YOr/CD.Y.r/Y.r+Psp'sY's/CD.Y's/Y's] fixed over short to medium 1985 -4.1 14.1 to inputs, the previous equation may be written as 1986 -1.8 13.0
	adjustment periods (Shumway, Pope and Nash. 1984). At the macro-economic level, land can be considered as a fixed factor even over long adjustment periods (Mahe and Rainelli, 1987). A + 1°r Rr Y' r /yo r + P s Rs 1981 -lU 11.3 (TFP/TFP)p = -leTt + (1 -cCT/cy. Y/CT). y/y 1988 -5.8 +2. 5 Y' s /Y' s = -1 e T t + (1 -~-1 . YIY [f] mmv
	same partition applies to output vector Y -(Y•, Y') in order -(~)-l ëual ~rice 1 market prieel [Er pOr Y•r /CD. YOr/yor + lis pis Y's/CD. Y's/Y's] 1934 -3.3 -2.6 This equation shows that if the long-run cost function is to take into account the quasi-fixity (cattle) or the fixity (production quotas) of some outputs and also in order to take into account the possibility of a non-marginal tarification of + li, (W'J-W'f) X'] /CD. X'j /X'j . y/y is zero in this case, 1988 -3.2 (TFP/TFP)p = -EeTt. Under 1937 -8.6 [dl 1986 -].5 linearly homogeneous with respect to Y,~=1 and the term(1-~-l) 1985 -1.0 -1.1
	certain outputs. increas ing (decreas ing) Furthermore, returns we do not assume long-run to scale, (TFP/TFP)p is
	returns to scale. 1.2. Consequences for empirical studies greater (smaller) th an -EeTt.
	Total costs are variable costs plus fixed costs, that is, Equations [cl and [dl show that when sorne netputs are in Second, consider the case where returns to scale are
	CD (Y• ,Yi. w• ,w ' . X' ,t) -CR (Y. ,Y' ,w• ,X' ,t) + 1:' J w ' j disequilibrium (quasi-fixities and/or non-marginal tarifica-X' j . The constant, aIl inputs variable (no quasi-fixity on the input
	calculated rate at which total factor productivity primaI tions) the non-parame tric measure of total factor productivity side) but some output markets in disequilibrium. Equation [cl
	measure changes is always equal to TFP/TFP does not equal the rate of technical change. becomes, assuming that aIl outputs may be in disequilibrium.

parti tion the input vector X into a subvector X• of variable inputs and a subvector X' of quasi-fixed inputs. Indeed, the hypothesis that aIl inputs instantaneously adjust to their long-run equilibrium levels seems restrictive, especially for the agricul tural technology since certain factors cannot be freely varied within the single period of observation. The

Table 2 !echni:aI ch:nge biases on the input sièe. !French açriculture, period 1959-19841

 2

	= P11-exp (-

h a,. . log p' k + al y .log Y = a, + l:k al. log Pk + a, y .log

  c) A related issue is that correspondingly, R2 will exagerate the extend to which movement of the data is actually accounted for by time and exogenous Z variables. Using a Monte Carlo experiment, Nelson and Kang show that time and a random walk will tipically explain about 50 % of the variation in a random walk that is in fact unrelated to either. .3. Testing for the incorporation of a linear time trend

	Several tests for stationarity have recently been proposed
	in the	literature on	time series models.	The	tests	for
	stationari ty in differences as opposed to s ta tionari ty around
	a trend line developed by Dickey and Fuller (1981) and largely
	applied in papers such as Nelson-Plosser (1982) seem to be a
	d) Estimating a relationship similar to (11) leads to spurious useful preliminary to analysis of non-stationary series. The
	sample Dickey-Fuller likelihood ratio statistic is based on the autocorre1ations of residuals which exponentialy
	decline as it is the case in a first order autoregressive hypothesis that stationary time series can always be written
	process. as an autoregressive (P.R) model. In order to test for the If the investigator believes the regression
	disturbances to be stationary then he can use the value of hypothesis that the series Xt follows a TS model with a m t h
	autocorrelation order AR process against the alternative that Xt follows a DS at 1ag one f, as an estimate of the
	autoregressive model wi th a (m-1) t h order AR process, the testing procedure coefficient in the following transformed
	regression equation : is to nest the TS and DS models in a more general model as
	follows :
	k (Yt -fl y,-tl = o. (l-ftl + 13 • (t-fdt-1)) + r 01. (Zll -f, Z, t -tl
							i=l
	Regression (13) would be properly specified if f, were set
	a t uni ty. Only in this case (which is equi valent to taking
	first differences of equation (1)) residuals (et-et-l) would
	be random. However as was pointed out in Nelson-Kang the
	empirical standard deviation of fi is only 0.064 around the
	mean of 0.852 and sample values of f, are thus rarely close to
	unity.	The	problem	of	non-random	and	non-stationary
	disturbances is still present in (13). It can be shown that
	the problem of spurious relationship of Y to time is partly
	a1leviated by the transformation but it is still very strong.
	Lastly, con tinued i tera tion of the Cochrane-Orcut t procedure
	improves	the	properties	of	estimates	but	only	first-
	differencing is the correct and adequate procedure.

3

inèicat:s re:ection of the null Jl~ochesis :t the 5 \ le1ellc~itical 7alu:s ~r: 5.1} ror 't and 5.73 for

  Ils 2 and Ils 3 are developed by Dickey and Fuller. Ils 2 is the usual regression F test of the null hypothesis Ho Ils 3 is the usual regression F test of the null hypothesis Ho: (~, D" pl = (~, 1, 0) against the alternative that Xl is TS. A1though

	both ~2	and ~3	are conventional F statistics, the usual
	significance values	are inappropriate if unit roots	are
	present and must be replaced by the test statistics given in
	Dickey-Ful1er (1981) -table V (test ~2) and table VI (test
	~3 ) •
		The Dickey-Fuller testing procedure is applied to French
	data for production and yields for wheat and corn. Data are
	annual and for the period 1940-1988. Tests are based on
	estimates of Equation (15). AIl series are taken in logarithms
	and the order of autoregressive process is determined after
	examination	of	the		partial	autocorrelation	function	of
	deviations from trend. Empirical results are reported in Table
	3 .
	Table J. 9ickeï-~uller tasts /1 and;,	,hen and corn production and yields, Franch agriculture, 1940-
	1988
		?roàuction			11:10s
		.h:èt		~orn			.heàt	corn
	1	1		2			1	1
	p a,	12.315 -1.:87		1.20, 0.835			2.371 -0.116	1.089 O. J74
	p	D.ell		0.015			0.019	Q.029
	Il "	~4. ~5t 35.01 1	3.08 ua			21.13' 29. ,6'	7.80 i 1l.J0•
	;li .
								1 -Xt -21 + ....
		i=l				i=2
		.... +(-dml. (Xt-m-, -Xt-ml + pt + nt
					m
	= IJ + Dl Xt -, + L DI. (Xt • ,-1 -Xt -,) + pt + n,	(151
					i=2
		In equation (151 the null hypothesis of a DS model is
	equivalent to Dl = 1		and	p = O. Two likelihood ratio tests

(IJ, Dl, p) = (0,1,0) against the alternative that X, is TS while t

  estimates of V*k and MIl) are reported in table 4.

		k	Production	Yields
		----	1-i	li (l J	vt k	MIll
	ïheat	-10	0.16 0.47	0.14	0.42
		15	o.10 O. J8	0.09	0.35
		" " 25	0.12 0.41 0.11 O. J9	O. !J 0.12	0.41 O. J9
		1" .V	0.12 0.4!	0.08	o.32
	Corn	10	0.72 0.93	0.J2	0.67
		15	0.55 0.81	0.21	0.54
		20 2:	J.46 o.75 O. 1J 0.71	0.19 0.19	0.52 0.51
		30	O. J1 0.61	0.12	O.H

'!able 4. Cochrane and Campbell-Mankiw !easures of persistance Product

Methods to estimate persistence

The first approach is based on ARMA methods and is proposed by [START_REF] Campbell | Permanent and transitory macroeconomic fluctuations[END_REF]. Let X, be the log of the economic time series under consideration. The change in Xt is first written as a stationary ARMA process The analysis presented in this annex is similar to that developed by Morrison (1986) or Slade (1986), whieh in turn is heavily derivative of the pionering work of Berndt andFuss (1981, 1986). The formula we propose is more genera1 insofar as the technology is mu1tiinput-mu1tioutput with some outputs as "quasi-fixed". The basie notations are the following yo is the veetor of variable outputs with typieal priee element given by po.; Y' is the veetor of quasi-fixed outputs wi th typiea1 priee elemen t gi ven by p' s .

~i s the veetor of variable inputs (wo, of quasi-fixed inputs (w' J )

CD (yo , Y' , WO , w' , X' , t) is the disequi1ibrum total cost funetion (Berndt and Fuss, 1981) that is the total eost of produeing Y when sorne inputs are fixed CR (yo , Y' , Wo , X' t) is the eorresponding restrieted or variable eost funetion.

(TFP/TFP). = P. R•. yo./Y. + Ps Rs. Y's/Ys-P, S,.Xo,/X,r'j SI Xj/Xj, where R., Rs are long-run revenue output shares and S" Sj disequilibrium total eost input shares.

A logarithmie differentia1 of CD ( . ) ean be written as, d log CD/dt = ologCD/ot


Hervé ,;UYOMARD Dominique Vermersch

t;CONOMIE RURALE L--_ _B"'I""'B!J.QUE ECONOMY AND TECHNOLOGY OF THE FRENCH CEREAL SECTOR A DUAL APPROACH

INTRODUCTION

Ernpiriçal an21'tsis of agricultural tcchnc:'..:.•gy 1.5 t..:,,") •')ftën c~rri~d out within 3 long run static equilibri~m Tiamework : m~~: ~f the studi~s assume 3 putty-putCY ~rGducti~n technolc~y in whic~! maximL:ationJ Tully adjL:st ::c their ootimE:.!. 1:-'/-=:1... ( i i )

if y " 0 then x " 0 or z " O.

( i i i) f or a 11 (x, y, z) E: T, if x ( ro and z ( 00 t hen y ( '"

(iv)

There is free disposaI of inputs and outputs i.e., for al1

( x • Y. z) E: T, the production plan ( x ' , y' , z ) such that ( x

2: X Z " Z Y -y) is possible, i.e ( x , y z' ) E: T (1) , ( v)
The set X(z, y) = [ x ( x , y, z) -Tl is strictly convex.

A few comments can be made upon these assumptions. Properties At last ~assumpt ion (v) i s a basic hypothesis ta ensure the dual approach in production theory.

The restricted cost iunction i5 then defined by

CR (p , Y, z) = Min [p' x x x x X E X(y, z)] Cl)
homogeneous,

CR (p y z) = Min [p' . x X E X(y o'z)] x ' 0' x x ~M in [p' . x X E X(Y1'z)]. x x or Min x [p' _ x x ") ] = CR (p , x Yi' z)
By the same way, let us consider. ) is obtained by

X ( y , z 1) end t ha t CR Ip y, 2 2 ) = Min [p' . x x ~X (y, 2 2 ) ) x x x < Min [p' • x x -X ( y, 2 1
) )

x x and r1in [p' x

x Ë X(y, Z1) = CR (D y, Z1) x x'
x Subseouently, '.le l""i11 go back on that resu1t. In addition, under the convexity hypothesis (v) • s x = ' -x + ( l -\ ) x ' .

But, by free disposal of inputs~t his equation

x -X (y, z) => X (y, z) i s canve x .

50, the t'No assumotions of convexity trom bêlol"J of the section X(y,2) and free disposaI imply that the section X(y,z) i~convex.

We !.oz 2' '-yy and variable input demand is defined 3S the oarti51 derivative of the restricted cost funetion with resp~ct to priee

"-2Y "-yy ]. Figure 1. Lack of fr~e disposal " a " --+-_._-- a " , " ) " o , 1 o , 2 v ) € T el 1 ( :.; 1 l 0 .... it.h x > x 1 1 X(yl Figure Non-convexity of " ' 1 ! 1 1 ---1 r ------------7 } 1 " The support.. funcl.ion C(y. c X ( v ) .
Ô CR (p , y, 2) / Ô p x n : X (p , n y, 2 )
The constant outputs and quasi-fixed factors priee elasticiti~s of demand can be computed frem the variabl~c ost function

aIl x tn = 1ñ l'='9 ~-
where y and 2 ,9"-''''::' is the inverse oi the revenue share of outl:,ut

if MVC = ù CR 1 ù y = p r r y r th-::-n (à log CR 1 à log Y r -1 = l 1 l P . y y r r 1 CR) = 1 1
The restricte~c ost function allows also ta test the validity of sèParabi li ty (between inputs 1 outputs, -inputs and outputs), jointness_ hc,mogeneity of degree I-J in aIl inputs or in variablç, inputs only.

The çhange in single 3hort run variable factor demand can be decomposed into four effects as shown in the following eouations obtained bY total derivation of input x demand eQuation n (figure

x (p x ' z, y, A) n R M '* d x = L: (à x / a y J. d y + L: (a x / a 2 ) . d z n n r r n m m r= 1 m=l N + L: (a n'=l x / a p n x n d p x n + (a x n / a A) d A (9a) '* .;j log x = n R )' r =l (a log x n / a log y ) d log ' "' y r M + L: (a lc.g x / a 109 2 ) d log 2 E 2 m=l n m m N + L: (a log x / a 109 p ) d log ~E 3 n=l n x n' x n + (a log x / a log Al d log A E 4 (9b) n d log R L: (a r=l log . d log Y r with M + L (él log (x./x.) 1 a log z ) .j log z E 2 m==1 1 J m m N +
L (à log (x./x .) / à 109 P

-• J 1 1 ! 1 , ! cr + " 0 - " 1 1 1 ) 1 1 , 1 _ M X ....

""

;' -w~-t -;, ). . . . "

"y,.,

•••• -. 1 '1 ~1 , 1 1 1 1 , -t 0 - " 1 1 1 1 1 1 1 1 , , , ' , , " '1 " ,, , " " "
' ,' 1 • <.

-" max , .

p. Hicksian eoullibria in than, equal te, or less than unitv only if the shadol.J priee (respectively the optimal level) of the quasi-fixed factor is great.:=r than, equal to, or less than the market priee (respectivelY the observed level).

-h -h CR(z )-p•z)(4-) z -m ;.tin (CR(v ) -tp. zJ z z ( 5 ) p ) • ) ( 5 = »-p.
• • CR ( Y ) / v + p z / y = 0 z a • Bv • • ... CR' Y ) / , CR ( Y ) P z = 0 " B • )/Ov ,a • )/B,,) • • • or CR < '1 = [ 1 CR ( Y ( z/ y ) ( C R ( Y ) / v t..hen • (8 • a" ) • CR ( , CR< v ) / z CR' y ) p z 0 z (B • B z) ... CR ( Y ) / z + p z 0 z [ ( a • Bz) ... CR< ~' ) / + p 1 " 0 " • 1 n l he m quasi-fi xed fact.ors cas e. " i s defi ned bv 2: ,B • B CR ( v ) / " ) + p " 0 m " m m= 1 m
The two last measures embody identical information. i. t=. th-= di~ference between the long run Hicksian equilibrium and short run Hicksian eouilibriLlm in the cost soace. The first measure is equal to the ratio of Ivng r-...:n returr:s to scale CRTSLP.) t•:o medium run returns to scale, as definê"d in section '2 (RTS MR ). The sec6nd measure is th~dual cBPacity utilization rete DlJC D . Morrison (1985) shows that many ~Quivalent definitions ,~f this utilization degree are possible. For simpiicity we Ch00S~t he measure defi~ed as the ratio of shadow costs on total COSt5.

A caSè study Tor th-:-french C"êreal sêctor

Our pur::.ose, in this sectioQ, i5 simply t•:o illustrate som€: x.f CR = c.

1 1 4 + L j=l d .. (Ln pj) lJ (15)
i = 1, 2, 3, 4.

Symmetry and additivity constraints ~nsure the theoretical restriction of ~om0geneity of de9r~e o~e in inputs priees.

The set oi equations ( 14) and (15J will be used te estimate t he parame t er s cd CR, f rom wt-. i -:::h the shor t run elasticities of demand will b~d erived Hicksian priee M. P.

1 M .) j 1 M. 1 (d .. + M1 1 1 1 M. 1 'Vi ,

212-214J; we call compute the long run

Hicksian priee elasticities of demand, not only for a variable factor but also for a ouasi-fixed factor. By the S3me way, a long run mëasure of returns to scale can be deri••,ted. We. oetain trie , , =: 3.69

E~~(h) = cCi, h, k) 1 B (h, k) E LT(h) = M ( ) , B ( . ) hk k 9 kh + M h -M k , h,k M h -n 1 CR (y, 7
where h corre3Ponds to family labor and k ta land. chaD. 12., 1982.

  ;:.ultul-al sect'or sinee certain inputs cann•:,t be freely varied within the single perit:•d •yf observati•:•n. Tt is con'/enient to refer ta this semiDuttY-Putty model as one ai short run static equilibrium : only the variable inputs adjust instantaneously to a change in relative priees the remaining inputs are fi xed can achieve che same long run èouilibrium. ConseQuent1y the first objectiv~of this paper i5 to provide an exhaustive crlaracterization of a short run êQuilibrium. This one is derived from a restrict~d cost function. The second goal of this research is to sho.... that this short run equi1ibrium frame ....ork allows the characterization of the different eouilibria (short run and long run Hicksian equilibria short run and long run Marshallian eauilibria), under sufficient conditions of curvature restrictionsT he paper i5 divided into six parts. Section 1 establis~les the existence and properties of the restricted cost i~nction, as weIl as the duality results bét'....een te•.=-hnology and restricted cost minimizat ion economic behavior. Sect ion 2 de'le I-;:)P5 the mode 1 OT the firm' s choice of techniaue in this optimization frameworkS ect ion 3 shows ho..... the di fferent possible equi 1 ibria can be derived from the kno.... 1edge of the restricted cost function. model is then apolied to tt"".e fr~nch cereal sector in section 5. The conclusions are outlined in section 6 they must be se en as permitting a better unterstanding of Drevious studi

  x = 0 and z = 0 ~y = 0) (iii) is a boundedness hYPoth~sis it ensures the existence of the corresporlding production function for everv fini te value of x and z.

  =strict-=•j (or variabl-a) eost functi.•,:.Jn. eorresponding to the oro'~ramm of minimi=ing the cast of sorne subset of the input

  measure of partial returns ta 3•:ale. i.e. a log CR 1 a log 2 , ) 1 L: (a log cRia log y ) L: (a log CT 1 a log y ) Under the ne ..... assumDtion of marginal variable cost pricing for outputs.

For-

  example, by applying the theoretical results established by Hall (1973) in the case of a static total cost function, it is easy to show the following statements If the production frontië'r i5 separable fnto a function of outputs and a function of input..=. fsufficient conditi,)n f:::The short run technology is nonJoint in input quantities if the restricted cast function may be writt~l' as the sum of independent cast functions for each kind of output

  factors effect(s) E 3 = total short run substitution effect(a) E 4 = technical change, research or develoDment èXDenditures, .... effect (a).

  Figure 3.

  The term a CR 1 a z is the reduction in variable costs fo," a m unit increase in the level of the quasi-fi.ed factor z

  ~y must be Dosi t i ve aef ini te (in a domain which includes the observed and short run Marshellian levels of ta inter the unccmpensat~d short run demand fram the short run Hicksian equilibrium. be positive definit.: (in a dümain ..... hich ~z ~y includes the observed and long run Marshallian levels of z and m m = 1, ... , M and r = 1, .. , R) to infer th~uncompensated run equilibrium fram the comoensated short run equilibrium. f irm' s tecr-,nology ''::.3n C' O:-reDresent~d bY a restricted cost iunction with only one output and one Quasi-fixed factor the generalization to the R outputs and M ouasi-fixed factors case i5 straightforwsrd. 2û If the observë-d level of the . ë'. minimization of the total cost functi•)n. the-n short run and long run Hicksian -:Quilibria coïncide. In a dual way, in the priee spaçe, signific3nt departures between the sctual and the shadow priees of the ~uasi-fixed factor can characterize the disequilibrium. measures of the possible disequilibrium ean he oroposed. The first-one is the primai capacity utilization rat~~DUC P defined as actual output v over capacity (or potential) output y 1 y This capaci ty output y i5 simply defined as the output at which the s~lort and long run average total cast curves are tangent to one another (Berndt and Morrison. 1981 Morrison, 1ge5, 1936) (figur~5in the case oi constant long run return3 to scale. one output and one ~uasi-iixed factor). The ratio OUC P captur-es.

Figure

  Figure 5. Capacity output y CoU

  previous theoreti•.::a2. issues in an econometriç model in arder--to characterize the structure oi the french céreal sector. Data concern a cross-section of farms and are detailed in Guyomard af,d Vermersch (1988). !-Je soeciiy a trans1eo9 iunctiona1 -form for the restricted cast function with an additive error

  negative shadow priee involve,s a l.3ck of fre-=:disposal. This last statement is untenable regarding sorne usual inputs as land or capital for instance. 50, €xcept the lack of free disposaI, an economic interpretation of a negative shadow priee i5 risky. Dur point of vie'"" i5 that trJÏs issue is c-rovide.j bY wrong signs of parameters which are probably due to multicolline3rity problems. Se-=:ondlv, most acplied studies tak~o nly 0ne ouasi-fixed factor into account. ~•Je hsvç' develüped thë analYtical necessa;specifie caSê of a translç.g restricted function an•j two quasi-fix-ed factors. Howê"ver, the gene:sliz3tiün te the m auas-:-fixe-d factors .:::aSé is not straightf•:>rward. Tr-IUS, trl':-optimal •::-ost share of each quasi-fixed input are not eXPlicitelv and at,alvtically defined. Finally our pap~r shows the importance of t3kin~i nto account th~q uasi-fixity of sorne inputs. The research also illustrat~s ths ensitivity of fin•jings ta the sOëciti:ation of ~quilibria. Sa. if the farrrlër.s 3re-sh•:.rt rur: p,-ofit maximisers (model of Hig•~ins, ation to U. S. agriculture, 1947 to 1974". Berndt et (ed. ), 1931. o::haD. 10. Handbook of 1"1atf""lematical economiçs, vol. II,

Until now, the free disposal hvpothesis is Quiet plausible in t~le cereal sector.

								y can a1so oroduce
	y	furthermore,		if		(x, z) can produce y, it can produce als0
	every y	such that y
	( 1)	x	~x**	x	n	<::x	n	n = 1

ASSuffiPtion (i'''''-) is a monotoni.:;ity assume-tion if (x, z) can trlen (x'. Z ), .....,ith x' ~x and z S Y.

end this characterization of the rds~ricted cost function with a discussion about the differentiabili•ty assumption of CR (p , y, z). Indeed, in many economic applic3:tions, ~'articularly x comparative statics, it is cünvenie-nt to kno\.J that thê obj~ctive function has a differential in input priees fo;-aIl positive inDut priees vectors for that reason. we imposed the strict convexitv di f fere:lt iable in of X (y ~zJ whierl imolies that input pr ices CR ( p , Y, x furthermore~2 ) 1s recent cont inuoLisl y works show that this last involvement is an eQuival-::nce under certain assuffiPtic.ns upon the t-echnology (see, for examDle, r1ac Fadden, 1978 SaUo, 1983). Differentiability of the cost fuction is intimately related to the well-known Shephard's lemma

	a CRI ap x n	= x	n	(px' y,z)	n = .L.	N	(2)
	In addition, as it has been noted by Mac F8d•jen (1978, l, P. 15),

the cost function has first and second derivatives with respect to input oriees for almost all positive input pric-e-s. This implies that the matrix of partial derivatives of inputs with respect ta input priees (when they ë'xist) is negative semi-definite. 3ymmetric and of rank N-l~5o, the following Hessian matrix

	a p x n	a p	x , n	]	n = 1 n'= 1	N N

is negative semi-definite, symmetric and of rank N-l.

			Finally, under assumptions	of differentiebility of CR (px'
	Y, 2) with respect to the fixed factors 2 m	and te the outputs Yr'
	it is convenient to define the total Hessian matrix
	L	r °l	"-l ;:	p p x x 2P X	Z "-"-	p ., x 22	;: "-"-	p y x 2Y

yi:' yz yy X At shvrt run Hicksian eouilibrium it is not necessary to impose curvature restrictions on the submatrices ~)

dual way, if the firm optimizes not onlv with respect ta the variable inputs but also l".d tri respect to the outputs and is in short run Marshallian eQuilibrium, then, bv Hotelling's lemma, the shadow priees (respectivelY the short run uncompensated levaIs) eoual the markets priees (resoectivelv the observed levels) of the output3.

	In a * a CR [p ? Y 1 la y = P (p x' -, r Y r	x	z, p ) Y	(12a)
			a CR [p, ~-m (p y -,	x'	z, p	y	) J 1 a	y	r	= p	y	r	(12b)
	FinallY,		if
								•				• p.
	long U nd-er	run l he	ret.urns t.o scole. assumptions of one at.	the z -h out..pul.	po in t. v. 0 ne	qU'1sl-fixed	fact.or	and
		a SRAC / a z 0
	i f	z >	-h z	(figure 4 ) ,		t..hen t..he	slope	of	t.he	tangent.	t.o	t.. he SRAC
	curve in z is posit.ive and so p	z	>	Pz	In Q	svmmetrical	.av.	i f
	h nt"gl1ti ve and SiO p z t..hen t.he slope of t.he z < • < P z z	t.<lngent	t.o t.he 3RAC curve ln z	i s

where SRAC is t..he short.. run t..ot..a! overùge cost. # l/y <a CR / a z

• P ) = 0 z • • P z o

the firm optimizes with r~spect te the variable inputs, the Quasi-fixed factors and the outputs and is in long run

	Marshallian	èQuilibrium~then	the	shadaw	priees	(of	the
	Quasi-fixed factors and the outputs) eQual the observed priees.
	Therefore,	trIe long

run Marshallian situation. not necessarily achieved by the obs.erved techno'logy, can be 3Pproached by three €-quivalent W3Y!? either by profit maximizeti0n with respéct ta z

	and	to	y,	while	holding	the	variable	in~•uts	at	their	cost
	minimizing lev~ls	or t,

y orof i t max imizat ion wi th respect to y gi ven :2 and trlen by cost minimizat ion wi th r,=spect to z, holding the variable inputs at their cost minimizing levels or by cost minimization with respect to z 9iven y and then by profit maximization with resD~ct to y, the variable inputs held at thèir cast minimizing levels :

	Ther.efore, i t i5 possible to infer the di fferent theoret ical Table 1. Characterization of different theor-::-tical
	possible equilibria (short run and long run Hicksian equilibria, ë~uilibrium5 rrom the knowledge of the only restricted cost function
	short run and long run Marshallian eQuilibria), given knowledge of short run environment, i. ê:. from the restrictèd cost function CR (p x z, y).
	(cf. table il.
					min~mizC1t..ion of t.he rE'''iLr-ict.ed cost..	funcl.ion
														CR	l p		z.	y)			(1)
																		"
								3HORT-RU~HIC~SIA~EQUILIBRIU~SRHEJ
						/
	sufficient.. condit.ion													SlJfficient. condit.ion
	St.C'Î."::L con~•~Xlt:.. ... of CR in z									SLC'LCl.	conve-xit.y of CR in v
	'1in [CRe. z		+p' z		zl		(2)											~1QX •	[P'" -CR<	) J	( 3 )
	minimizat.ion of t.he Lot.ol									maximizat.ion of t.he
	cost.. funct.iol)																	l'est.ri ct.ed pro fi t, fllncLlon
			ç T ( P	X	P	z			V' )															TH: ( p	x	z, p ) •
	LONG-RU~HIC~SlA'		EQUILIBRIU[				1		SH0RT-RUN ~ARSHAL[.I~~EOUILIBRIUM
				L R H E J												'II
	sufflcient. condit.ion		nlox (pv-CR< z. y	)-P	z		z]			1
	xm(p n	x	,0 ,1:' z	y	)	= X	n		(p	x	, Z	m (p	x '	p ), p :z ~y y	m (p	x'	P z'	P	Y	)
							= x		(p	--n (p , z		, P , y	m	CP		~,	p	)	) , m (p y	, p , p	)
										n			x					x		' "				x		Z	Y	X	2	Y
							= x	n	(p	x'	z	m (p	x	, p	z'	p	y	)	y	(p	x	, 2	m (p	X	P z'	p	Y	) ,	p	Y	)
																										(13 )

[SR'[EJ! suffictent. condit.ion of st.rict.. convexit.v of CR in Z Qnd in ... s,•,rict.. con\"exil.v of CR in v (st.rict.. con\•exit.v of CT in v ) sufficie-nt.. condition strict. convexit..v of CR in z <strict.. concaviLv of "R in z)

  Suff ic ient •::ondi t i,,:'ns to use the framework Dr>:::-sented in table l

	are the following
	\' '-'zz	must							includes
		max •	[ p.	CT] , , ,	"-	:J	{ "		V	) m:?' ( m
				mo,lXlm.1 zut.1 on or t 1102'	t.vf..ul	profiL funct.ion
							"T { P	x	P	z
				LO~G-RU~~ARSHALL[A~EOUrLrB~IUM	(LR~IE!

z] m -CR...... (

j refer to a variable factor h and k refer ta a quasi-fixed factor. The parame ter ~stimat~s for the final form of the model 5re ..=.hown in table 2 together lNith their estimat~d standard êrrors. For each point. the estimated cost shares are ~'ositive and the concavity of the restricted C'ost function in inl:,uts prieEs i5

  ')

	Tabl~2.	Param~ter estimates
								Standard
		Parame ter		Estimate	error
	verified	at			aver8.;;!e.	If		anc:	represent
	respectively tt-,e l eve l s of family labor 8 0 12• 14.6	land. 0.035
	average, al	the fitted ~ost function is convex in 2 h 0.379 0.121	and concave in
		the "2	wrong sign of parameter f k 0.055	i5 probably derived from 0.117
	multicollinearity between cl	Zk and y, the 0.109	lev.;,l of output. 0.004.	However,
	f k	is reestimated e 2	bY	a	production 0.306	function 0.011	model	r•"l y ing	y
	to Zk (the level of the quasi-fixed factor), 0.139 0.013 -r, solve the optimal level (the level of output) c 3 in order to 2 k in this case, the dU 0.027 0.015
	: -P h -Zh multicollinearity problems are replaced by simultaneity problems. P , -0.026 0.007 d 12 x ,j13 -0.022 0.011
	M k	-n 1eR ( y , =-Pk . zk d..,~0 .107 ;t') _L P x ' d 23 0.002		0.016 0.018
		d~, L ..		-0.084.		0.018
		d 3 4.		-0.081		0.031
		d 33			0.101		0.033
		d 44			0.14.5		0.039
		f 1			-0.137		0.086
		f 2				0.730		0.14.4.
		9 11		-O.OSO		0.136
		9 12			0.011		0.204.
		9 22		-0.223		0.311
		k 11			0.004.		0.100
		k	12		-0.002		0.009
		k 21			0.035		0.029
		k 22			O. 0';•4.		0.025
		k~H		-0.095		0.033
		k 32			0.009		0.030

i and

Estimatas of shûrt-run Hicksian priee elasticities of demand, evaluated at sampl-:-average, are shown in table 3 l,.Jith their estimated t r2tios. ExceDt Tor hired le:bor, aIl the astimaté'd for energy and capital are the most price-rasponsive siGnificant

	substitution	DOS.3ibilities	between	capital	and	hired	labür,
	,:apital and fé'rtilizars are found.	At the sample average,
		this	figure	may		be viewed as	too high but
	:nulticollinearity problams or •~mitted factors

can distùrt this result. Neverthele5s, the result i5 consistent with other measures of returns to scale based on a tc,tal cost Tunction where aIl inputs

  are variable[Boutitie and al., 1987]

that there are scale economies. both studies show

	Table	3.	Short-run	Hicksian	priee	elasticities
								an,j	(h: f.3mily labor, k	land)
		priee	fu>S"l and	capital	hired	fertili-
						oil		labor	zers
	Quantity
	fuel and	-0.638	-0.075	-0.065	0.628
		oil			(4..63)	(1. 24.)	(0.6.2)	(4..53)
	capital		-0.026	-0.34.4.	O. 14.5	0.173
					( 1. 24. )	(6.681	(3.91)	(2.98)
	hired labor		-	0.051 rO.621	0.321 (3.91)	-	0.136 (0.56)	0.135 (O.~)
	rertilizers		0.153	-0.118	-ü.04~-0.23
					(4..5.31	(2.98)	(0.6)	l2.65)

of dê'mand. ~veluated at sample 8vera'.Je,

Assuming that observed levels of Quasi-fixed factors, are long run Hicksian levels. estimates of long run priee elasticities of demand are shawn in table 4 these are •::onsistent 1,.,11 th Le Chatelier orinciole. Substitution Dossibilities appear between family Labor and hired labor, family labor and capital. land and energy, land and capital~land and f~rtili2er3.

Table 4 . Long-run Hicksian pric~e lasticities of demand, evaluated

 4

	at sample average zh and zk (h : family labor. k	capital)
	priee	fuel and capital oil	hired labor	fertili-family zers labor	land
	Quantity
	fuel and	-0.706	-0.047	-0.16	0.307	0.119	0.492
	oil
	capital	-0.0168 -o. •:"25	-0.065	-0.0:22	0.130	0.268
	hired labor	0.129	-0.144	0.496	-0.337	-0.961	-0.14.3
	fertilizers	0.075	-0.015	-0.105	-0.634	-0.039	0.718
	family labor 0.034	0.106	0.355	0.047	-1.09	0.064
	land	0.170	0.262	-0.063	1.025	0.763	-2.16

Table 5 shows the long run Hicksian priee elasticities in the C8se where onlv fixity of land is relaxed of 5ubstituability with land also 2Dpear the previous relations the last ~lasticities can't b~compared. in the view of the Le Chatelier principle, with the prçvious short run Hieksian priee -= last ici t ies because, in caen case, the point of aDDroximatieon is differ-:nt. Finall" in the case where fixity of land i5 relaxed, th~measure oi returns to scale givè'

 5

	ECH	LT (z r, •	(z~) )

Table 5 . Long-run Hicksian priee elasticities

 5

	of	demand,