CAB/l0/D3/91
ECONOMIC BEHAVIOUR OF AGRICULTURAL
HOUSEHOLDS : IMPLICATIONS OF ASSUMING
PERFECT SUBSTITUTABILITY BETWEEN LABOURS
by
C. BENJAMIN, H. GUYOMARD {*}
I.N.R.A.
Station
d'Economie. et
Sociologie
Rurales
65, rue de St-Brieuc - 35042 Rennes Cédex (France)
de
Rennes
(*) incidentally, the order of the authors' names resulted from a coin flip.
March 1991
3
1.
INTRODUCTION
The neoclassical theory of the farm household, developed by, among others, Sen
(1966), Lopez (1980), Nakajima (1986) and Singh et al. (1986), emphasizes the
interdependences between household-utility maximisation and firm-objective function
optimisation. The main conclusion of theses studies is that the farm-household model
does not provide, in general and contrary to the non-family farm model, any prediction
with respect to output supply and input demand responses, i.e. that "the possibility of
downward sloping output supply or upward sloping input demand functions cannot be
ruled out" (Lopez p. 33). 50, the effect of an output i priee variation on output i supply is
indeterminate. The effect of this output i priee change on the equilibrium amount of
family labour supplied on farm is also indeterminate ; it can be decomposed into a
substitution effect and a wealth effect induced by the variation of farm returns, both
effects being more olten of contrary signs.
The objective of this paper is to examine the consequences on the farmhousehold model, and more particularly on comparative static properties of this model, of
assuming perfect substitutability between hired labour and on-farm family labour in
production processus on one hand and/or perfect substitutability between off-farm family
labour and on-farm family labour in utility function on the other hand. Consequently, the
farmer is assumed to allocate his time among the three possible uses, i.e. on-farm work,
off-farm work and leisure. He has also the possibility to hire labour at a given wage rate
which is not necessarily equal to the wage rate he may obtain working outside. The
theoretical model is presented in the second section. Comparative static results, in the
general case and under assumptions of perfect substitutability, are detailed in section
three. Our analysis includes the models of Nakajima and Dawson (1984) as particular
cases. We complete the paper with some conclusions and policy implications.
4
2. THE MODEL
The agricultural household. considered as an entity. seeks to maximise its utility
function which depends on its income and on the quantity of labour it supplies on farm
and off farm. subject to various constraints. and particularly to the time constraint and
to the budget constraint which incorporates on-farm or/and off-farm incomes. More
formally. the farm-household programme may be written as
[11
max U(LF.LO.M)
LF.LO.LH
subject to
+
+
M
~
"(p.v.LF.LH.Zl - WilLH
LF
+
LO
LF
~
0
(iii)
LO
~
0
(iv)
LH
~
0
(v)
M
~
+
LEI
~
waLD
(il
A
(ii)
T
(vi)
0
where p = (Pl .....PN) is the N dimensional priee vector of outputs Y. v
the M dimensional priee vector of variable inputs X. Z = (Zl •...•
Zr::)
=
(Y:!. ••.•• "t1) is
is the K dimensional
quantity vector of fixed inputs. Wil is the wage rate of hired labour LH and w 0 is the
wage rate of off-farm family labour LO. T are total number of hours that household
members have available for ail activities. i.e. leisure LEI. on-farm work LF and off-farm
work LO. The household's income M consists of three parts, off-farm income VIb LO, onfarm income "(p,v.LF.LH.Z) - WilLH and autonomous income A.
Ali output and variable input markets are perfectly competitive. Netput priees
and wage rates are taken exogenously by the farm household. except for on-farm family
labour "priee" which is, generally. an endogenous parameter determined within the farmhousehold complex. U(LF,LO.Ml satisfies the following regularity conditions (Diewert.
1974) : defined and continuous from above for LF. LO. M
~
0; quasi-convex in its
arguments ; non-decreasing in M and non-increasing in LF and LO. "(p,v.LH,LF.Zl
5
satisfies the following regularity conditions (Lopez) : non-negative ; linearly homogenous,
continuous and convex in priees; non-decreasing, continuous and concave in quantities.
Finally, we assume that labour is an essential input in production processus.
Assuming an interior solution for M, the first-order, necessary conditions for a
maximum are
W - f./3 /(oU/oM) = orrlOLF
[2aJ
x - f./4
[2bJ
/(oU/oM) =
Wo
[2cl
orrlOLH = wh- f./5 MUIOM)
where f./3' f./4 and f./5 are the Lagrangean multipliers associated with contraints (iii), (iv)
and (v). respectively. We note W = -(oU/oLF)/(oU/oM) the marginal rate of substitution
of on-farm family labour for money and X = -(oUIOLOl/(oUIOM) the marginal rate of
substitution of off farm family labour for money.
From
the
different values
of Lagrangean multipliers,
six
cases can
be
distinguished :
i) in the first case lu3
=F 0, f./4 =F 0
and fJ5 = 0). household's members do not
work. They allocate their total available time to leisure and family labour supply is equal
to zero, i.e. LEI* =T, LF* =0 and LO* =0. The marginal rate of substitution of on-farm
family labour for money, i.e. (oU(O,O,M*)fOLFl/(oU(O,O,M* )fOM)
= W(O,O,M*). is
greater than the shadow priee of on-farm family labour orr(O)/oLF. The marginal rate of
substitution
of
off-farm
family
labour
for
money,
i.e.
(oU(O,O,M*)/oLOI/(oU(O,O,M*lIOM) = X(O,O,M*), is greater than the wage rate w
0
of off-farm family labour. The farm uses hired labour at a level Lf-t so that orr(U-t )/oLH
= wh·
6
ii) in the second case
tu:3 =l= 0, J14 =
0 and P5
=
0), household's members do
not work on farm and allocate their time between leisure and off-farm work so that X(a,
LO *, M*)
= Wo
and LEI*
= T - LO*.
Hired labour is used up to the level U..f where its
marginal productivity in value equals its priee.
Iii) and iv) in the third (J/3
and J/5
=l= Dl
= 0, J/4 =l= 0
and J15
= 0)
and fourth
tu:3 = 0, J/4 -+- 0
cases, household's members allocate their total working time on farm. The
shadow priee of on-farm family labour is equal to the marginal rate of substitution of onfarm family labour for money, i.e. 6rr<U:* lIOLF
= W{LF ,o,M*) and
LEt
= T - LF . Off-
farm employment is equal to zero and the firm may (case iii) or not (case Iv) use hired
labour.
v) and vi) in the fifth (P3
and J/5
=F 0)
= 0, J/4 = 0
and J15
= 0)
and sixth (P3
= 0, J/4 = 0
cases, marginal rates of substitution of on-farm and off-farm family labour
for money are equal to corresponding "priees", Le. W(LF ,Lo*
,M* 1
= 6rr(LF" l/6LF and
X(LF* ,LO* ,M*) = w o ' Again, the farm may (case v) or not (case vi) use hired labour.
It is interesting to examine the implications of assuming perfect substitutability
between LF and LH in production processus on one hand or/and perfect substitutability
between LF and LO in utility function on the other hand'. Perfect substitutability
between LF and LH implies that 6rrlOLF = h 6rrlOLH (h > 0) and perfect substitutability
= k 6UIOLO (k > 0) for M given 2 • Parameters h
So, if k = 1, household's members are indifferent
between LF and La implies that 6U/6LF
and k are constants of proportionality.
between on-farm and off-farm work ; if k < 1 {resp. > lI. they prefer (resp. do not
prefer) farm to off-farm work (Simpson and Kapitany, 1983). First-order conditions of
1 Perfect substitutabi l ity assurptions between LH and lF in production function and/or LF and LO in
utility function are often used to analyse, separateLy, production and conslITlption decisions of the
agricultural household. For eX8fi'fJle, if the tarm uses hired Labour and if householdls members do not
work outside, thè shadow priee of on·farm family Labour i5 exogenous and equal to the wage rate of hired
labour. Consequently, profit maximisation decisions may be determined using the wage rate of hired
Labour as the priee of alL labours used in production processus (for more details on the separability
assumption and its implications, see Lapez or Nakajima).
2 Derivatives are calculated at the equilibrium, i.e. wh en profit and utility functions are maximised.
1
7
the farm-household programme may be then written as
a"(LF* ,LH*)IOLF = ha" (LF ,LH* )/OLH
• •
= h (wh - JlsI(OU(LF • ,LO ,M )IOM))
= W(LF* ,LO* ,M*) - Jl3I(OU(LF* ,LQ* ,M* )IOM)
= k X(LF* ,LQ* ,M*) - Jl3I(OU(LF* ,LO* ,M* )/aM)
*,M
* )/OM)) - Jl3I(OU(Lt-- *
* )I6M)
= k (wo +Jl4I(OU(LF*
,LO
,LO*,M
Let us now assume that LF* ,LO* and LH* are different from zero. In such a
case, first-order conditions imply that hWh
=
kwo . If hWh is not equal to klJ\b, one
labour level, at least, is equal to zero. More precisely, we have the following rules (if h =
k = 1, rule iii) corresponds to corollary 3.1. and rule il corresponds to corollary 3.2. of
Lopez (p.23)).
i) if hWh
> kw o , household's members work on farm and off-farm if and only if
the firm does not use hired labour.
iil if hWh > kw o ' the firm uses hired labour and family labour if and only if
household's members do not work outside the farm.
iiil if hWh < kwo, household's members do not work on farm and allocate their
total working time to off-farm work.
3.
COMPARATIVE STATIC RESULTS
Let us now consider the effects of an increase in the priee of an output on
labour supplies and demands, assuming that on-farm family labour is not an inferior input
with respect to this product. This example will allow us to show the implications of
assuming perfect substitutability between LH and LF on one hand or/an between LO and
LF on the other hand. The effects are analysed formally and illustrated diagrammaticaly.
i) First, let us assume that LF* , LH* and LO* are different from zero.
8
Since household's members work on farm and off farm, and the firm farm uses
hired labour, we have
w
= olTfoLF
[3aJ
x = Wo
[3b]
= Wh
olTfoLH
[3cl
Nevertheless, in such a case the farm income function may be written as
lT(p,v,LF,LH * ,Z)· Wh LH * = ITT(p,v,LF,Wh,Z)
[4]
Therefore, olTlOLF = oITTIOLF, and we have
W
x
= olTTIOLF
=
W
[5al
[5bl
o
·olTTfowh
= LH *
(p,v,LF,Wh,Z)
[5cl
Equations [5a] and [5bl determine optimal levels of on·farm work LF* and off·
farm work Lü * . Equation [5c] defines the demand function of hired labour LH* .
Differentiating the equations [5al and [5bl partially with respect to the price p of
one output, we obtain the following system
OW/OLF+OW/OM.onT/OLF-0 2 nT/OLFOLF
[ oX/oLF + oX/oM.onT/oLF
o2nT/oLFoP - Ow/oM.onT/oP]
[ - oX/oM.onT/op
wo]
oW/ oLO+OW/ oM.
[OLF* / op]
oX/oLO+oX/oM.W o oLO*/op =
[6]
9
Solving and simplifying [6], we obtain
oLF*/op
=
l/det ([o2rrT/oLFop-ow/oM.orrT/op] [oX/oLO+oX/oM.w o ] -
[-oX/oM.orrT/op] [oW/oLo+oW/oM.woJI
[7J
oLO * top =
1/det([-o2rrT/oLFop+oW/oM.orrT/op] [oX/oLF+oX/oM.orrT/oLFJ 1 +
[-ox/oM.orrT/oPJ[ow/oLF+ow/oM.orrT/oLF-0 2 rrT/oLFoLFJI
with det
[8J
[oW/oLF+ow/oM.orrT/oLF-0 2 rrT/oLFoLFJ[oX/oLO+oX/oM'Wo J
- [oX/oLF+oX/oM.orrT/oLFJ[oW/oLO+oW/oM.woJ
=
Second-arder, necessary and sufficient for maximising U, conditions imply that
det > 0 3 . Consequently, we have
oLF*/op > < 0
[9]
oLO*/op > < 0
[101
ln the general case, the effect of an increase in the price p of an output on LF *
and LO * is then indeterminate. This result is weil known (Lapez p.33). In the particular
case where the firm does not use hired labour and household's members do not work
outside, system [7] and [8] reduces ta
= ([02 rrI6LFop-c5WI6M.orrl6pl/[c5WI6LF + c5WI6M.orr/oLF - 02
oLF*l6p
•
and oLF /op
rr/oLFoLFI
> < O.
The effect of an increase in the price of an output on optimal quantity of family
labour on farm is always indeterminate. This situation may be illustrated by Figure 1.
(insert Figure 1).
3 Fo, more details, see annex.
10
Initial equilibrium point corresponds to point E* where firm demand curve for
family labour orrlOLF and on-farm family labour supply curve -(oUIOLF)/(oU/oM) intersect.
An increase in the priee of an output shifts the demand curve to the right and the supply
curve to the left, so that the new equilibrium point may be El
(LF l
< LF
*)
or
E2 (LF 2 > LF*). Note that the supply curve is a function of LF and M, but since M in
turn is a function of LF, the supply curve is ultimately a function of only LF (remember
that off-farm family labour is equal to zero). This case corresponds to the "basic" model
of Nakajima (p.21) and to the case of family-Iabour-only farms of Dawson (p.14).
ii) Let us now assume that LF*
[8]
are
always
verified
under the
=!= 0,
LO*
condition
(p,v,LF* ,wh'Z), Consequently 4, oLF* top
=!= 0
to
and LH* = O. Equations [7] and
use
rr(p,v,U:* ,Z)
and
not
rrT
> < 0 and oLO* IOp > < O. Nevertheless, if
we assume that on-farm family labour and off-farm family labour are perfect substitutes
in utility function, Le. if oU/oLF = k oU/oLO, [7J and [8] reduce to
oLF* top = 1/det ([0 2 rr/oLFop][OXIOLO + oXIOM. 'Ab ])
[ 11]
oLO* top = 1/det ([-oXIOM.orrlOp][-o 2 rr/oLFoLFHo 2 rr/oLFopJ(OX/oLF + oXIOM.w
0
])
[12]
with det
= [_02 rr/oLFoLF][oXIOLO +oXIOM."'b J.
Finally
oLF* top = [0 2 rrIOLFop]/ [-0 2 rr/oLFoLF] > 0 and oLd' top <
Furthermore, if LU
•
= LF • + LO •
o.
represents the total family labour level, we
have
oLU* top
= oLF * top + oLO* top
= 1/det ([02 rrIOLFop][(1-k)OX/oLO] + [-oXIOMorrlOp][o 2 rr/oLFoLF])
Therefore, if k
2:
1 oLU* top < 0 and if k < 1
4 Agaln, we assume that 62T/ôLF ôp
>
o.
ouf IOp > < o.
[13J
11
ln the case where the firm does not use hired labour but household's members
work on farm and off farm, and if both family labours are perfect substitutes in utility
function, an increase in the price p of an output increases Lr* ,decreases
decreases LU* if k ;;: 1 and increases or decreases LI} if k
= k oU/oLO).
uj,
< 1 (k is defined by oUIOLF
This result is iIIustrated in the particular case where k
=
1.
(insert Figure 2)
First-order conditions reduce to
orr(LF * )/6LF
= Wo
[14a]
-(oU(LU* ,M*)IOLU)/(oU(LU* ,M* 1/6M) =
[14bI
Wo
Equation [14a) defines labour demand LF• by the firm and equation [14bl
determines the total labour supply by household's members. From the point of view of
the household, the relevant decision variable is total family labour supply and not onfarm and off-farm supplies. In other words, the household labour supply schedule is
unique. Initial equilibrium values are LU* , LF and LO*
LH*
= 0).
= LI! - Lr* (remember that
An increase in the price of an output shifts the demand curve to the right and
the supply curve to the left 5. Consequently LF• increases, LO•
and•
LU decrease.
iii) Let us now assume that LF*
=1=
0, LO* = 0 and U;*
=1=
O. In such a case,
we have
oLF/6p = ([02 rrTIOLFop]-[OWIOM-orrTIOpJ)/(OWIOLF + OW/oM.orrT/oLF-02 rrTIOLF6LFI > < 0
[ 15]
Nevertheless, if hired labour and on-farm family labour are perfect substitutes in
production processus, i.e. if orrTIOLF
oLF* /op
= hlNh, [15) reduces to
= (-[OW/oM.orrT/6pJ)/([OWIOLF + OWIOM.6rrT/oLFJ)
5 Remember that we have assumed that 6a./6LF6p
>
O.
< 0
(16)
12
Furthermore, we have
6LH* /op = _02 "T/oWhoP - 02 "T/olNh olF.olF /op > 0
al* /op = olH* /op
+
OU:* /op = -a 2 "T/0Vl.h op
Therefore, if h 2: 1 al* I6p > 0 and if h
+
[17]
11-hlolF /op
[181
< 1 ol* /op > < O.
ln the case where household's members do not work outside but the firm uses
hired labour and family labour, and if both labours are perfect substitutes in production
processus, an increase in the priee p of an output decreases l f , increases
U.J ,
increases l * if h 2: 1 and increases or decreases [' if h < 1 (h is defined by o"l6lF =
ho"/olH). Again, this result is iIIustrated in the particular case where h = 1.
(insert Figure 3)
The relevant decision variable, from the point of view of the firm, is total labour
demand and not hired labour and family labour demands, i.e. the farm demand schedule
is unique. First-order conditions may be then written as
*
o"ll l/ol = Wh
[19al
-loUllF* ,M)/olFl/loUllF* ,M* l/oM) = lNh
[19b]
An increase in the priee of an output shifts the supply curve to the left and the
*
lH* increase. This
demand curve to the right. Consequently, lF* decreases and land
particular case corresponds to "Iabour-hiring farms" in Dawson's study Ip.141.
iv) Finally, let us assume that lF*
9= 0, lO* 9= 0
and lH*
9= 0
. If hired labour
and on-farm family labour are perfect substitutes in production processus and if on-farm
and off-farm family labours are perfect substitutes in utility function, it is easy to show
lstarting from equations [7] and [8]) that OU:* /op and olO* /op are not defined in that
case. The equilibrium is not unique. In particular, when h
=k =
1, relevant variable are
total labour demand by the firm (o"ll* l/ol = 1Nh) and total labour supply by the
13
household (- oU(LU* ,M* lIâLUI/(oU(LU* ,M* )/OM)
= wo ).
Consequently, L* and
uf
are
uniquely determined, but not LF* , LH* and LO* .
Let us assume that the perfect substitutability assumption is verified only in
utility function. In such a case, we have
oLF* lop
= 1/det ([02 rrTIâLFop)[OXIâLO + oXIâM. VIb Il
oLO*lop
= 1/det ([-OXIâM.orrTop][-o
[20]
2rrTIâLFoLF]-[,f rrTloLFop] [oXIâLF+ oXloM.w 0
]
[21 J
with det
= _[02 rrTloLFoLF][OXloLO +OXIâM.VIb]
Equations [20] and [21] correspond to equations [11] and [12] except that we
use rrT and not rr. Consequently, oLF* lâp > 0 and oLO* lâp < 0, i.e. the effect of an
increase in the priee of an output on LF and LO is determinated.
Let us now assume that the perfect substitutability assumption is verified only in
production processus. We obtain
= 1/det (orrTlâp.[-OWIâM.oXloLO +OXIâM.OWIâLOll
[22]
oLO * lâp = 1/det (orrTlop. [-OXIâM.OW10LF + OW IâM.OXIâLFIl
[23]
OLF* lâp
with det
= [OWloLF +OWIâM.WJ1][OXIâLO +oXIâM.VIb] [oXIâLF +oXIâM.WJ1][OWIâLO +OWIâM.wo]
Consequently, LF* lâp >
< 0 and LO * lop > < O. The effect of an increase in
the priee of an output on LF and LO is indeterminated in such a case.
4. CONCLU DING REMARKS
A general model of the farm household economic behaviour has been developed
that can be used to examine, in particular, labour allocation decisions. More precisely,
14
the farm household is facing two types of decisions on labour use : the first concerns
the supply of family labour, i.e. the allocation of time among leisure or non-work
activities. on-farm work and off-farm work ; the second concerns the demand of labour
by the firm farm and this demand can be satisfied by family labour only, by hired labour
only, on by both types of labours.
Under the assumption that wage rates are given exogenously to the farm
household, we highlight the importance and the consequences of assuming perfect
substitutability between hired labour and on-farm family labour in production processus
and/or perfect substitutability between on-farm family labour and off-farm family labour
in utility function. This importance is illustrated by analysing the effects of an output
priee change on the subjective equilibrium of the farm household, and more specifically
on labour allocations, for family-Iabour-only farms, for labour-hiring farms, for outsidelabour-supplying farms and for labour-hiring and outside-Iabour supplying farms.
If we do not assume perfect substitutability, the effect of an increase in a
product priee on labour levels is indeterminate in ail cases. For labour-hiring farms, and if
hired labour and on-farm family labour are perfect substitutes in production processus,
an increase in a product priee leads to a decrease in family labour and to an increase in
hired labour, but the effect on total labour may be indeterminate (more precisely, "l• /Op
> 0 if h
~
•
1 and "l /Op
> < 0 if h < 1, where h is defined by ,,"/OlF = h ""/OlH). For
outside-Iabour-supplying farms, and if off-farm family labour and on-farm family labour
are perfect substitutes in utility function, an increase in a product priee leads to an
increase in on-farm family labour and a decrease in off-farm family labour, but the effect
•
on total labour may be indeterminate ("lU /"p < 0 if k
~
•
1 and "lU /Op > < 0 if k <
1. where k is defined by "U/"lF = k "U/olO). For labour-hiring and outside-Iabour
supplying farms, and if we assume perfect substitutability in production processus only,
the effect on an output priee increase on hired labour, on-farm family labour and off-farm
family labour is indeterminate ; if we assume perfect substitutability in utility function
15
only, an increase in an output priee leads to an increase in on-farm family labour and a
decrease in off-farm family labour.
These illustrations show that policy implications must be derived with caution.
"Most agricultural policies subsidise product priees.
For labour-hiring farms, the
implications are clear: product priee subsidies, which seek to increase farm incomes,
increase total labour, even though family labour falls" (Dawson, page 14). Of course,
this proposition is based on the assumption that hired labour and on-farm family labour
are perfect substitutes, with h = 1, in production processus. Nevertheless, we have
shown that this proposition does not necessarily hold if, il h < 1, i.e. the marginal
productivity in value of on-farm family labour is lower than the marginal productivity in
value of hired labour, and iil household's members work outside, even if on-farm family
labour and off-farm family labour are perfect substitutes in utility function.
16
Figure 1. The effect of an increase.in the pric..e of an output on family labour level on
farm (case where LF =l= 0, Lü = 0 and LH = 0)
Priees
--- --- ---- ---'---- -- 1
/
/
_............
-.............
..................... ..................E1
..........
.............
/
?-
/
/
SU /SLF
SU/SM
/
"
""
"
/
/
/
/
/
/
/
..............
.....
/
.....A..
. . . ,E2 /
....
....
"
/"
/"
/"
/"
/"
--
Srr
SLF
Quantities
17
Figure 2. The effect of an increase in the priee of an output on labour levels (case where
LF and Lü are perfect substitutes in utility function with k = 1 and LH =
0)
Priees
8rr
8U /8LU
8LF
8U/8M
" "-
1
"
1
1
1
"",
,
1
1
\
1
\
1
\
,
,\
l
:\
: \
: \
: \
/
/
/
/
/
\
1
/
l',
,
,'
,
'
:
'
/
/
LU·
Quantities
18
Figure 3. The effect of an increase in the priee of an output on labour levels (case where
LH and LF are perfect substitutes in production processus with h = 1 and
La = 0)
Priees
su / SLF
on
SU/SM
...
1
/
/
"-
oL
"' "-
/
/
\
/
\
/
\
:\
:\
: \
: \
:, \\
/'
1 :
l
l
l
'\
\
/
l
"-
,'
,
,',
1
,
,,,
,
/
/
/
•
L
.---LH
..
1
Quantities
19
ANNEX
Second-order conditions for maximising U (when LF* ,LH* and LO* are different
from zero) imply that the following inequalities are verified
d/dLF(dU/dLF)
d/dLO(dU/dLO)
<
0
< 0
d/dLF(dU/dLF) . d/dLO(dU/dLO) - d/dLF(dU/dLO) . d/dLO(dU/dLF)
>
0
The above inequalities can be rewritten as
[OWIOLF
+
OW/t5M . o"T/t5LF - 0 2 "IOLFoLF]
[6XIOLO
+
6XIOM . w o ]
[OWIOLF
+
OWIOM . 6"TIOLF - 6 2 "IOLF6LF][6XIOLO
6"TIOLF][OWlOLO
+
>
>
0
0
OWIOM .
Wo
+
6XIOM . "'6 ] - [t5XIOLF
J> 0
Consequently, [7] and [8] can be rewritten as
•
6LF IOp
=
1/det
ll02"TIOLFop][6X/oLO
+ [OXI6M . 6rrTlOp]
+
6XIOM .
[OW/6LO
- [OWIOM. 6"TlOp][6XIOLO
+
+
Wo
1
OWIOM . w o ]
6X/6M . w o ])
[a 1 ]
+
oX/6M .
20
•
oLD IOp =
1/det ([-02ITTIOLF op][oXIOLF
+
+
oXIOM . oITTIOLF]
[-oXIOM . oITT/op] IOWIOLF
+
OWIOM . oITTIOLF - 0 2 ITTIOLFoLF]
+ [OW/oM . oITT/op][oX/oLF + oXIOM . oITTIOLFll
la 2 ]
Furthermore, we assume that
OW/oLF > 0 and oX/oLF > 0
OWIOLO
> 0 and oX/oLO > 0
OW/oLM
> 0 and OXIOLM > 0
The two first right-hand side terms of [a 1] are then positive and the third right-
hand side term of la 1] is negative. Consequently
•
oLF IOp <
> 0
Finally, we have also
oLD"/op < > O.
21
REFERENCES
DAWSON P.J. (1984) "Labour on the family farm: a the ory and some policy
implications". Journal of Agricultural Economies, January, vol. XXXV, nO 1, 119.
DIEWERT W.E. (1974) n Applications of duality theory". In frontiers of quantitative
economics, vol. Il, M.D. Intriligator, and D.A. Kenrick, eds. London, NorthHolland.
LOPEZ R.E. (1980) "Economie behaviour of self-employed farm producers". Ph. D.
thesis, University of British Columbia, 175 p.
NAKAJIMA C. (1986) "Subjective equilibrium the ory of the farm household". In
development in Agricultural Economics 3, Elsevier, Amsterdam, 302 p.
SEN A.K. (1966) " Peasant and dualism with or without surplus labour". Journal of
Political Economy, n074, 425-450.
SIMPSON W. and KAPITANY M. (1983) "The off-farm work behavior of farm
operators". American Journal of Agricultural Economies, n065, november, 801805.
SINGH 1., SQUIRE L. and STRAUSS J. (1986) "Agricultural household models
extensions, applications and policy'·. Published for the World Bank, The John
Hopkins University Press, Baltimore and London, 335 p.
22
DOCUMENTS DE TRAVAIL
Mars 1991
90-01
L'IMPACT DE LA PROPOSITION AMERICAINE AU GATT SUR LES
AGRICULTURES DE LA CEE ET DES USA. Hervé Guyomard, Louis P.
Mahé et Christophe Tavéra (1990).
90-02
AGRICULTURE IN THE GATT: A QUANTITATIVE ASSESSMENT OF THE US
1989 PROPOSAL. Hervé Guyomard, Louis P. Mahé et Christophe
Tavéra (1990).
90-03
EC-US AGRICULTURAL TRADE RELATIONS: DO POLITICAL COMPROMISES
EXIST. Louis P. Mahé and Terry L. Roe (1990).
90-04
ANALYSE MICRO-ECONOMIQUE DE L'EXPLOITATION AGRICOLE. Catherine
Benjamin (1990).
90-05
PSE, AMS AND THE CREDIT FOR SUPPLY MANAGEMENT POLICIES IN THE
GATT NEGOTIATIONS : (application to the EC case). Hervé Guyomard,
Louis P. Mahé (1990).
90-06
COMPLETING THE EUROPEAN INTERNAL MARKET AND INDIRECT TAX
HARMONIZATION IN THE AGRICULTURAL SECTOR. Hervé Guyomard,
Louis P. Mahé (1990).
90-07
ALIMENTATION ANIMALE ET DYNAMIQUE DES PRIX DES MATIERES
PREMIERES SUR LE MARCHE FRANCAIS. Yves Dronne, Christophe
Tavéra (1990).
91-01
ECONOMIC BEHAVIOR OF AGRICULTURAL HOUSEHOLDS : IMPLICATIONS
OF ASSUMING PERFECT SUBSTITUTABILITY BETWEEN labours.
Catherine Benjamin, Hervé Guyomard (1991).