Production and Consumption Decisions of Rural Households under
Price Risk: A Mean-Variance Approach
Vangelis Tzouvelekas∗
Dept of Ecnomics, School of Social Sciences, University of Crete
University Campus, 74100, Rethymno, Crete, Greece
Ph. 30.28310.77417; e-mail: [email protected]
December 4, 2011
Abstract
Relaxing the assumption of risk neutrality, the present paper develops a non-separable dual agricultural household model extending Lopez (1984) unified theoretical and econometric framework under a
non-linear mean variance approach suggested by Coyle (1999). The model has been applied in a sample of 296 UK cereal producers during the 2001-04 cropping period. The results suggest that indeed
farmers are risk-averse as price volatilities are affecting both production and consumption choices.
Keywords: agricultural household model, labor supply, price uncertainty, cereal producers, UK.
JEL Codes: J22, C33, D13, D81.
∗
The author wish to thank the participants of the 5th Conference on Research in Economic Theory and Econometrics (CRETE), Robert G. Chambers, Elie Appelbaum, Theofanis Mamuneas and Giannis Karagiannis for their useful
comments and suggestions that materially improved the paper. This research has received funding from the European
Communitys Sixth Framework Programme [FP 6/2005-08] under Grand Agreement Number 022653.
1
1
Introduction
Agricultural household models were first introduced to explain the counterintuitive empirical finding
that an increase in the price of crops did not significantly raise the marketed surplus in agricultural
sectors. Barnum and Squire (1979) and Rosenzweig (1980) in searching for an explanation of this
finding, ended up with a model in which production and consumption decisions are linked as the
deciding entity, i.e., rural household, is both a producer and a consumer. On the one hand, farmers
are choosing the allocation of labor and other variable inputs to crop production and on the other,
are choosing the allocation of income from farm profits and labor sales (i.e., off-farm work) to the
consumption of commodities and services including leisure. As long as perfect markets for all goods
and services, including labor, exist, the agricultural household is indifferent between consuming ownproduced and market-purchased goods. By consuming part of its own crop output, which alternatively
may be sold to at a given market price, household members implicitly purchase goods from themselves.
Similarly, by demanding leisure or allocating their time to household farm activities, they implicitly
buy their own time valued at the market wage rate.
This modeling framework applies to both developing and developed rural economies as long as
agricultural households are not commercialized consuming a very small share of their output or they
supply few, if any, of their own inputs. More importantly, it applies outside the farming sector, in every
small or medium enterprise owned by a single individual or family members. Based on these general
assertions, agricultural household models have been used extensively to analyze a wide range of policy
issues relating to agricultural development in both developed and developing countries. The early uses
were concerned primarily with farm price policy as the level at which agricultural terms of trade are
set has wide implications for both efficiency and equity in rural areas. Despite this early emphasis on
price policy, the uses of agricultural household models included applications to diverse topics such as
off-farm labor supply, technology policy, downstream growth, migration, income distribution, family
planning and critical environmental issues. Singh et al., (1986) provide an extensive literature review
of all relevant studies in agricultural household models worldwide.
However, all these theoretical models along with their empirical applications invariably presumes
separable consumption and production decisions so that the sole link between the two sides of the agricultural household model lies in the impact of farm profits, which are usually estimated independently,
on the household budget constraint. This particular type of interdependence called interchangeably
recursiveness or separability allows production decisions to be made independently of consumption
choices although the latter depends on the former through the budget constraint. Strauss (1986)
and de Janvry et al., (1991) suggest that the existence of perfectly competitive goods and services,
including labor, markets to be sufficient for the existence of recursiveness or separability. Further,
those models that are particularly directed towards analyzing household time allocation between on
and off-farm work they do not recognize that these two alternative occupations may confront different
utility connotations for the rural households. As a result they utilize a unique exogenous wage rate
implicitly assuming that household members are indifferent between alternative time allocations. If
this hypothesis is relaxed, then non-separable models are unable to adequately analyze agricultural
household behavior.
Lopez (1984) and later on Jacoby (1993), questioned the separability assumption on the basis
of non-homogeneous labor supply of on and off-farm work. First, Lopez (1984) formulated a model
analyzing the interdependence of production and labor supply decisions of agricultural households
that own and operate their farming business.1 His theoretical framework integrates output, input and
labor supply decisions and allows for differences in preferences between on and off-farm work. Imposing
constant returns to scale to obtain a shadow wage rate for on-farm work and using a cross-section
of Canadian aggregate data he jointly estimates output supply, variable-input demands, on-farm and
off-farm supply equations.2 On the other hand, Jacoby (1993) utilized a reduced form structural
1
Further, Lopez (1984) underlines the significant gains in explanatory power and efficiency in the empirical estimation
of his non-separable agricultural household model.
2
Thorton (2001) using data from Utah dairy farms model provides a variant of Lopez (1984) model using the marginal
product of on-farm labor as the shadow wage rate. However, his approach presumes a less-flexible Cobb-Douglas
functional specification for farm technology.
2
labor supply model dispensing also with the assumption of same utility connotations of different labor
allocation choices as well as of constant returns to scale inherent into Lopez (1984) theoretical model.
Still, however, existing non-separable household models presume that households members, those
that are making the production and consumption decisions, are risk neutral.3 This risk neutrality
assumption turns to be crucial for the recursiveness of the implied model. Barnum and Squire (1979)
pointed out that under risk averse behavior the separability assumption is not valid even under a
competitive market environment.4 If contingent claims markets exist or if household members are risk
neutral, separability exists under price uncertainty or production risk unless household members are
having differences in preferences between on and off-farm work as noted by Lopez (1984). However,
insurance markets may not exist especially for some kinds of uncertainty that rural households are
faced. Further, empirical evidence worldwide confirms that indeed households are rather risk averse
(Rosenzweig and Binswanger, 1993). If this is the case, independence does not hold and estimating
equations from certainty theory is not appropriate.
In the light of this, the present paper extent Lopez (1984) non-separable agricultural household
model to account for price uncertainty under the non-linear mean variance framework suggested by
Coyle (1999). The developed theoretical framework can be extended to account for other sources of risk
faced by agricultural household, while it can be applied in other case studies outside farming sector.
The model is applied to a panel of 296 UK cereal farms obtained from the Farm Business Survey for
UK. Considering European farming sector, the recent CAP reform towards abolishing price support
and making farming sector more market oriented is major shift in the way the EU provides income
support to farmers. Certainly the change in the way the EU supports farm incomes will directly affect
the production choices made by farmers. Similarly, the policy changes will affect the consumption
choices made by rural households and in particular their choices with respect the allocation of their
labor between on and off-farm activities. This is an important aspect given the pluriactivity of EU
agriculture, which is reflected in that (a) a significant proportion (often the whole) of the labor input
used in farming activities is supplied by the household members; (b) the returns from the family farm
operation constitute an important proportion of the households income available for consumption
purposes; and (c) an important source of household income is off-farm activities by rural households
members. Since farmers are risk averse, price variability is expected to play a crucial role in their
decisions.
2
2.1
A Model of Behavioral Choice under Price Risk
The Production Decision
Let assume that farmers are making their decisions facing random prices for the m crops they produce
where p̃qm = p̄qm + ε̃qm with p̄m being the respective mean price,
denoted by p̃q = {p̃q1 , . . . , p̃qm } ∈ <Ω×M
++
q
2
ε̃m ∼ N (0, σp ) is the random price coefficient the distribution of which is exogenous to farmer’s actions
and, <Ω denotes the random variable space formed by mapping the underlying random states, Ω, to reals. Farm production technology can be represented by the following closed, nonempty, production set:
n
o
T =
xv , xf , q, S : x, xf can produce q given S
(1)
where xv = {xv1 , . . . , xvj } ∈ <J+ is the vector of J variable inputs used in farm production, q =
S
{q1 , . . . , qm } ∈ <M
+ is the vector of the M crops produced on-farm, S = {S1 , . . . , Ss } ∈ <+ is the vector of farm policy variables and, xf ∈ <++ is the on-farm labor supply by household members which
3
A notable exception is the theoretical analysis by Finkelshtain and Chalfant (1991) and the three period model of
Huffman (2001). Still, both models are not empirically tractable.
4
Fabella (2001) exploring further this finding proved that under multiplicative risk allowing for risk increasing inputs
leads to non-separability of the agricultural household model. Only additive risk in production or revenues results
in separability regardless the risk attitudes of household members. However, this is still a special case of farming
technologies.
3
for simplicity is assumed to be homogeneous5 . Since the supply of on-farm labor is predetermined
by household members, it is considered as the only quasi-fixed factor of farm production. Finally,
we assume that long-run farm technology exhibits constant returns to scale in both variable inputs
and on-farm labor supply. Under these assumptions, farmer’s decision problem is to maximize his/her
expected utility of total farm income given the technological and policy constraints. The solution
of this maximization problem, defines the dual indirect utility function, that is the relation between
maximum feasible utility and exogenous variables, i.e.,
ṽ
p
q
v
f
W0 , p̃ , w , x , S
n
o
q
v v
v
f
p̃ q − w x : F x , q, x , S ≤ 0
= max ũ W0 + max
xv ,q
W̃
n
o
= max ũp W0 + π̃ p̃q , wv , xf , S
p
(2)
W̃
where wv = {w1v , . . . , wjv } ∈ <J++ are the variable-input prices, W0 ∈ <+ is the non-random initial
farm wealth (i.e., value of farm assets owned by household members) which following the relevant
literature under risk is assumed to affect production decisions (e.g., Chavas and Pope, 1985, Pope and
Just, 1991), Q = F xv , q, xf , S is a continuous convex transformation function and, W̃ is the total
farm income which is the sum of initial farm wealth and maximal restricted profits associated with
farm production, π̃ p̃q , wv , xf , S .
Since farm technology exhibits constant returns to scale, π̃ p̃q , wv , xf , S is positively linear homogeneous in on-farm labor supply the only quasi-fixed input considered. Thus, under a smooth farm
production technology, the Clark-Wicksteed product exhaustion theorem applies and the quasi-rent to
the on-farm labor supply is given by π̃ p̃q , wv , xf , S = xf π̃ f (p̃q , wv , S). In other words, π̃ f (p̃q , wv , S)
is the shadow price of on-farm labor and the product of it’s shadow price with its use completely
exhaust quasi-rent from farming.
Then the utility maximization problem may be restated as:
n
o
= max ũp W0 + xf π̃ f (p̃q , wv , S)
ṽ p W0 , p̃q , wv , xf , S
W̃
o
n
f
f
q
v
f
= max x max W0 + π̃ (p̃ , w , S)
W̃
W˜f
= xf ṽ f W0f , p̃q , wv , S
(3)
where W̃ f = W̃0f + π̃ f (p̃q , wv , S) is the mean expected total farm income per unit of on-farm labor
which is the sum of initial farm wealth per unit of labor, W0f and farm restricted profits.
Assuming constant relative risk aversion preferences (CRRA), the certainty equivalent representation of the maximization problem in (3) can be expressed over the first two moments (i.e., mean and
variance) of the distribution of expected total farm income. Under these assumptions the maximization problem in (3) turns into the following certainty equivalent version:
p
q
v f
2
2
ṽ W0 , p̄ , w , x , S, σw = W0 + xf π̄ f − 0.5rw σw
(4)
2 = ṽ p /ṽ p is the Arrow-Pratt constant relative risk aversion coefficient with subwhere rw W̄ , σw
ww w
scripts denoting the partial derivatives of the indirect utility function evaluated at the mean value
of total farm income, W̄ is the mean expected total farm income which is the sum of initial farm
2 the variance of the total farm income. Since crop
wealth and farm profits, W̄ = W0 + xf π̄ f , and σw
prices remain the only random variable in the model, the variance of total
is simply the
farm income
2
2
2
2
variance of revenues realized from farm production, i.e., σw = σw q, σp with σp being the variance
5
We assume that the household does not demand nonfamily farm workers and operates its farm by self-employment.
The analysis that follows can be easily extended to that direction.
4
of crop prices. Further, ṽ p (.) is continuous and convex in it’s first two moments, while it is strictly
2 (Appelbaum and Ullah,
increasing in W̄ and strictly decreasing in it’s second argument, r = −0.5rw σw
6
1997). It’s properties can be summarized by the following proposition (see Appendix A for the proof):
2 in (4) is
PROPOSITION 1: Assuming that the indirect utility function ṽ p W0 , p̄q , wv , xf , S, σw
continuously differentiable on <++ , then the following properties hold:
2 = λṽ p W , p̄q , w v , xf , S, σ 2 for λ > 0 under constant relative risk
(a) ṽ p λW0 , λp̄q , λwv , xf , S, λσw
0
w
aversion.
2 (.) are differentiable and denoting partial derivatives with subscripts7
(b) assuming that rw (.) and σw
the following derivative properties hold:
(i)
q
qm W0 , p̄ , w
q
v
2
, σw
, xf , S
v
2
, σw
, xf , S
ṽp̄pm
∀m
= p
ṽW0
(ii)
−xj W0 , p̄ , w
(iii)
p
w 2
= 1 − 0.5rw̄
σw
ṽW
0
(iv)
2
ṽσpp = −0.5 rσw σw
+ rw σσ2p
(v)
ṽ pf
2
π̃ f p̄q , wv , σw
, S = px
ṽW0
=
p
ṽw
v
j
p
ṽW
0
∀j
(c) under DARA ṽ p (.) is quasi-convex in it’s elements.
(d) ṽ p (.) is weakly separable and the standard symmetry and reciprocity properties hold.
2.2
The Consumption Decision
On the consumption side we assume that agricultural households derive utility from consumption of a
numeraire aggregate marketed good, c, and leisure, `.8 Further household members are endowed with
T H units of time which can be consumed as leisure `, supplied to the market as labor, xo , or used
in farm operation xf . Following Lopez (1984) we further assume that members of farm household
confront different disutilities from working on and off farm (i.e., they supply heterogeneous farm and
nonfarm labor).
Under these assumptions farm households have the following monotonically increasing and strictly
concave utility function:
uc = uc c, `f , `o
(5)
where `f = T H − xf , `o = T H − xo is the total leisure consumed by household members.
The agricultural household income consists of the conditional random farm profits (i.e., crop prices
are random), the non-labor exogenous income which includes the returns obtained from financial and
real assets owned by the household and, the income obtained from off-farm employment by household
members. Hence, agricultural household income has the following form:
pc c = π̃ f xf + wo xo + I o
(6)
6
In fact, producer preferences defined in (4) are a special case of the translation invariant class preferences axiomatized
by Quiggin and Chambers (2004).
p
p
p
7
p
v
Derivatives are denoted with subscripts as follows: ṽp̄pm = ∂ṽ p /∂ p̄m , ṽW
= ∂ṽ p /∂W0 , ṽw
v = ∂ṽ /∂wj , ṽ f =
x
0
j
2
w
2
∂ṽ p /∂xf , ṽσpp = ∂ṽ p /∂σp2 , rσw = ∂rw /∂σw
, rw̄
= ∂rw /∂ w̄ and, σσ2 p = ∂σw
/∂σp2 .
8
Leisure is assumed to be a normal good.
5
where wo is the off-farm wage rate, I o is the exogenous non-labor income and, π̃ f is the conditional
random farm profits or the quasi-rent of on-farm labor supply defined in previous section.
Following Lopez (1984) reformulation and using (6) the agricultural household utility maximization problem may be formed now as:
n o
ṽ c pc , wo , π̃ f , Z̃ = max ũc c, `f , `o : Z̃ = pc c + π̃ f `f + wo `o
(7)
c,`f ,`o
where Z̃ = I˜H + I 0 with I˜H = T H π̃ f + wo being the random income obtained from the employment
of household members on the farm and off-farm activities. The above optimization problem has a linear
constraint, defined using non-negative prices and positive income, assuming that the time allocation
constraint is not binding. This implies that at any wage rate or conditional profits and commodity
prices, household members consume some leisure. Further it is defined over the non-negative orthant
of its elements which is sufficient to ensure continuity of the dual indirect utility function.9 Hence,
standard duality results apply and the solution of the above maximization problem defines the dual
indirect utility function in a straightforward manner. The indirect utility function defined above is
also continuous, quasi convex in its arguments, non-increasing in the price of aggregate market good,
non-decreasing in household income and homogeneous of degree zero in prices and household income.
Under price uncertainty we can express household indirect utility above in terms of its preferences
over moments of the distribution of the random variable. In the traditional univariate case, where only
income is random not due to random crop prices, the indirect utility function above can be reduced
to the familiar objective function defined over income or even final wealth alone. Nevertheless, since
the price of crop produced are in fact unknown when employment decisions are made, even under
flexibility of consumption-leisure choices, the joint distribution of income and shadow wage rate is
likely to affect the final choice. Therefore, we concentrate on risks associated with the relative price
of farm produce and in real household income, both of which are dependent on the distribution of the
random variable p̃q .
In a conventional sense the Arrow-Pratt risk premium measures the maximum amount that household members would pay to avoid income risk when everything else remained certain. If the price
of farm produced is the only random variable with only income stabilized, then the Arrow-Pratt risk
premium would not necessarily measure the willingness to pay by household members for insurance in
their real income. In such a situation the risk premium should measure the willingness to pay by the
agricultural household in order to avoid labor income risk when prices remain random (Finkelshtain
and Chalfant, 1991). In that case we have:
2
(8)
ṽ c pc , wo , π̃ f , I˜ = ṽ c pc , wo , π̄ f , Z̄ − Rz , σH
where π̄ f (.) and Z̄ are the mean profits and total household income, respectively, Rz is the risk pre2 is the
mium analogous to the Arrow-Pratt risk premium in the traditional univariate model and, σH
variance of total household income. For small risks using the Taylor series approximation the risk
premium is given by:
2
Rz = 0.5rz σz2 + rzπ σzπ
(9)
c /ṽ c and r zπ Z̄, π̄ f , σ 2
c
c
where rz Z̄, σz2 = ṽzz
z
zπ = ṽzπ̄ f /ṽz with subscripts denoting the partial derivatives of the indirect utility function evaluated at the mean value of total household income, σz2 is
2 is the covariance between household income and farms
the variance of household income and σzπ
conditional income obtained from farming activities. The first term in (9) is the Arrow-Pratt risk
premium expressed as the maximum amount the household is willing to pay to stabilize income when
prices are fixed. The second term captures the cost associated with the stochastic interaction between
9
Lopez (1984) has proved that the utility function defined in (7) is continuous from above and quasi concave in `f ,
` and c, non-decreasing in c and non-increasing in `f and `o .
o
6
the shadow price of on-farm labor and household income. If household income and output prices
are independent or if the utility function is additively separable, the risk premium would reduce to
the univariate case.10 Otherwise differs from the traditional Arrow-Pratt risk premium in magnitude
and possibly in sign. If conditional unit farm profits are the only random variable then the variance
2
H 2 H
of income and the covariance of income and farm
profits may be expressed as σz = T σπ T and
2
H
2
2
2
2
f
σzπ = T σπ , respectively with σπ = σπ q, σp , x being the variance of conditional farm profits and
σp2 the variance-covariance matrix of random crop prices (see the budget constraint in (6).
If we assume nonlinear mean-variance constant relative risk aversion (CRRA) preferences, the indirect utility function in (7) turns into the following certainty equivalent version:
2
2
ṽ c pc , wo , π̄ f , Z̄, σz2 , σzπ
= Z̄ − 0.5rz σz2 − rzπ σzπ
(10)
where Z̄ = pc c + π̄ f `f + wo `o is the mean expected household income which is the sum of exogenous
non-labor income and mean income obtained from on and off farm labor supply. The properties of the
dual indirect utility function of agricultural households can be summarized by the following proposition (see Appendix A for the proof):11
2
PROPOSITION 2: Assuming that the indirect utility function ṽ c pc , wo , π̄ f , Z̄, σz2 , σzπ
is continuously differentiable on <++ , then the following properties hold:
in (10)
2
2
(a) ṽ c λpc , λwo , λπ̄ f , λZ̄, λσz2 , λσzπ
= λṽ c pc , wo , π̄ f , Z̄, σz2 , σzπ
for λ > 0 under constant relative
risk aversion.
2 (.) are differentiable and denoting partial derivatives with
(b) assuming that rz (.), rzπ (.), σz2 (.) and σzπ
12
subscripts the following derivative properties hold:
"
#
2
ṽπ̄c + rπ̄zπ σzπ
c
o
f
2
2
f
(i)
` p , w , π̄ , Z̄, σz , σzπ = −
ṽzc
ṽ c
2
= wc
(ii)
`o pc , wo , π̄ f , Z̄, σz2 , σzπ
ṽz
ṽ c
p
2
(iii)
c pc , wo , π̄ f , Z̄, σz2 , σzπ
= c
ṽz
2
(iv)
ṽzc = 1 − 0.5 rzz σz2 + rzzπ σzπ
2
(v)
ṽσc p = − 0.5rσz σz2 + rz + rσzπ σzπ
+ rzπ σσ2p
(c) under DARA ṽ c (.) is quasi-convex in its elements.
(d) ṽ c (.) is weakly separable and the standard symmetry and reciprocity properties hold.
3
Empirical Model
To make the non-separable agricultural household model operational we need to impose a specific
parametric specification for both indirect utility functions in (4) and (10), respectively. Following
Coyle (1992; 1999), we rely on the normalized quadratic flexible functional form. Considering first
10
Independence is sufficient for the covariance to be equal to zero only for small risks. For large risks, independence
would not ensure equality but would imply that both Rz and the univariate Arrow-Pratt risk premium have the same
sign.
11
The quasi-concavity of (10) is sufficient to guarantee convexity of preferences.
12
c
Derivatives are denoted with subscripts as follows: ṽπ̄p = ∂ṽ c /∂ π̄ f , ṽzc = ∂ṽ c /∂ Z̄, ṽw
= ∂ṽ c /∂wo , ṽpc = ∂ṽ c /∂pc ,
c
c
2
z
z
2
zπ
zπ
f
zπ
zπ
2
z
z
zπ
zπ
ṽσp = ∂ṽ /∂σp , rσ = ∂r /∂σz , rπ̄ = ∂r /∂ π̄ , rσ = ∂r /∂σzπ , rz = ∂r /∂ Z̄, rz = ∂r /∂ Z̄, σσz p = ∂σz2 /∂σp2 and,
2
σσzπp = ∂σzπ
/∂σp2 .
7
(4), the normalized quadratic dual model under CRRA (i.e., λ = 1/wJ ) has the following general form
(Coyle, 1999):13
ṽ p (.)
xf wJv

= α0 + α w +
M
X
qw q
αm
p̄m +
J−1
X
m=1
+

M
X
q
αm
+ 0.5
m=1
+
J−1
X
M
X
qq q
αmh
p̄h +
αjv + 0.5
J−1
X
qv v
αmj
wj +
vv v
αjg
wg +
S
X
M X
M
X
va
αjs
Ss +
s=1

σw p 
αmρ
Vhρ W0f
m=1 ρ=1
qa
αms
Ss +
s=1
j=1
g=1
j=1
J−1
X
S
X
αswa Ss +
s=1
j=1
h=1

αjvw wjv +
S
X
M X
M
X

σq
p q
αmhρ
Vhρ
p̄m
(11)
h=1 ρ=1
M X
M
X

p v
σv
αjmρ
Vhρ
wj
h=1 ρ=1
where p̄qm are the normalized mean prices of the m, h = 1, . . . , M crops produced by agricultural
households, wjv are the normalized prices of the j, g = 1, . . . , J − 1 variable-inputs (j th input was used
as numeraire), W0f = W0 /xf is the normalized nonrandom initial farm assets per unit of on-farm
p
labor, Ss are the s = 1, . . . , S policy variables and, Vhρ
is the corresponding element of the normalized
variance-covariance matrix of the m = h, ρ crop prices. Symmetry and reciprocity property imply the
qq
qq
σq
vv = αvv , ασq
σv
σv
σw
σw
following restrictions in (11): αmh
= αhm
, αjg
gj
mhρ = αmρh , αjhρ = αjρh and, αhρ = αρh
∀h, ρ.
2 under CRRA is approximated using
Following Coyle (1999) the risk aversion coefficient rw W̄ , σw
the following quadratic relation which is less restrictive under the optimization hypothesis in (3):

!
!2 
1
W̄
W̄

rw = p ξ0w + ξ1w ln p
(12)
+ ξ2w ln p
2
2
2
σw
σw
σw
2 = q, xf , σ 2 with σ 2 = V p
where ξ w are the parameters to be estimated, W̄ = W0 + xf π̄ f and, σw
p
p
hρ
being the variance-covariance matrix of random m, ρ crop prices. Then, using (3) the conditional net
output supply, initial farm income and conditional profit equations are given by:
qm
=
xf
q
αm
+
M
X
qq q
αmh
p̄h
J−1
X
+
qv v
qw
αmj
wj + α m
W0f
j=1
h=1
S
X
+
qa
αms
Ss +
s=1
xj
− f =
x
αjv
+
M
X
qv q
αmj
p̄m
J−1
X
+
m=1
αw +
p
σq
Vhρ
αmhρ
R1w
(13)
h=1 ρ=1
vv v
αjg
wg + αjvw W0f
S
X
va
αjs
Ss +
s=1
M
X
,
g=1
+
p
2
0.5xf σw
−
=
W̄ f
M X
M
X
qw q
αm
p̄m
J−1
X
+
m=1
M X
M
X
,
p
σv
αjhρ
Vhρ
R1w
(14)
h=1 ρ=1
αjvw wjv
j=1
S
X
+
s=1
13
αswa Ss +
M X
M
X
,
σw p 
αhρ
Vhρ
R3w
(15)
h=1 ρ=1
The Hessian matrix of the normalized quadratic functional form is a matrix of constants and hence the curvature
conditions can hold globally.
8
0.5σp2
−
W̄ f
=

,
M
J−1
X
X
σq
σv
σw

αmhρ
p̄qm +
αjhρ
wjv + αhρ
W0f  R2w
m
j

π̄ f
(16)
= α0 +

M
X
q
αm
+
m=1
M
X
qq q
αmh
p̄h +
J−1
X
qv v
αmj
wj +
qa
αms
Ss +
s=1
j=1
h=1
S
X

σq
p q
αmhρ
Vhρ
p̄m
h=1 ρ=1

+
M X
M
X

,
J−1
J−1
S
M X
M
X
X
X
X
p  v
vv v
va
σv
αjv +
αjg
wg +
αjs
Ss +
αjhρ
Vhρ
wj
R1w
g=1
j=1
s=1
(17)
h=1 ρ=1
√
Rw σ 2
W̄
W̄
W̄
w
ξ0 +
1 + ln √ 2
+ ξ2 ln √ 2
2 + ln √ 2
, R1w = 1 − 0.5 3 W̄ w
where
=
σw
σw
σw
.
and, R3w = ξ1w + 2ξ2w ln √W̄ 2
R2w
√1 2
σw
ξ1w
σw
The derived demand equation for the jth variable input used as numeraire is obtained by applying
Hotelling’s lemma to (11) and using again the derivative property in Proposition 1, i.e.,

,
M
M X
M
J−1
p
p
p
p
X
X
X
∂ṽ
∂ṽ
∂ṽ
∂ṽ
xJ
w
wj
wj
wj
p
 R1w
− f = ṽ p −
p̄qm q j −
− W0f
−
Vhρ
(18)
p
f
∂wj
x
∂
p̄
∂V
m
∂W0
hρ
m=1
ρ=1
j=1
h=1
with the associated parameters obtained using the homogeneity property.
p
Denoting with xi = {qm , xj } the netput vector and, pj = {p̄qm , wjv , W0 , Ss , Vhρ
} the exogenous variables in (11), the crop supply and variable-inputs demand elasticities are calculated from the following:
i
h
p
εpij = εp∗
−
ε
ij
Rj
(19)
∗
∗
∗
where , εp∗
ij = (∂xi /∂pj ) (pj /xi ) with xi (.) being the term into brackets in the crop supply and
variable-inputs demand equations in (11) and, εpRj = (∂R1w /∂pj ) (pj /R1w ).
Turning now to the consumption side of farm households, the normalized quadratic indirect utility
function (i.e., λ = 1/pc ) takes the following form (Pinkse and Slade, 2004):
ṽ c (.)
pc

= β0 + β I + β f I π̄ f + β oI wo +
M X
M
X

Iσ f  ¯
βhρ
Vhρ I +
β f + β f o wo
(20)
h=1 ρ=1
+
β f f π̄ f +
M X
M
X


fσ f  f
βhρ
Vhρ π̄ + β o + β oo wo +
h=1 ρ=1
M X
M
X

oσ f  o
βhρ
Vhρ w
(21)
h=1 ρ=1
where π̄ f and wo are the normalized prices of on-farm and off-farm labor supply, I¯ is the normalized
p
total farm income and, Vhρ
is the corresponding element of the normalized variance-covariance matrix
of the m, ρ crop prices. Symmetry and reciprocity property imply the following restrictions in (20):
fσ
oσ , β oσ = β Iσ and β Iσ = β f σ .
β of = β f o , βhρ
= βρh
hρ
ρh
hρ
ρh
Following Coyle (1999) again we specify a quadratic functional form for the risk aversion coeffi2 ) which under CRRA preferences results in the following relations:
cients rz (Z̄, σz2 ) and rzπ (Z̄, π̄ f , σzπ

rz =
rzπ =
!
!2 
1
Z̄
Z̄

p ξ0z + ξ1z ln p
+ ξ2z ln p
2
2
σz
σz
σz2
"
!
!#
f
f
1
Z̄
π̄
Z̄
π̄
p
ξ0zπ + ξ1zπ ln p
+ ξ2zπ ln p
2
2
2
σzπ
σzπ
σzπ
9
(22)
(23)
where ξ z and ξ zπ are the parameters
to be estimated, Z̄ = T H π̄ f + wo + I 0 , σz2 = T H σπ2 T H and
2 = T H σ 2 with σ 2 = q, σ 2 and σ 2 = V p being the variance-covariance matrix of m = h, ρ random
σzπ
π
π
p
p
hρ
crop prices. Then using the homogeneity property the system of the demand equations for on-farm
and off-farm labor supply and initial household income equations are given from:
T
H
−x
f
= −
f
β +β
ff
π̄ + β w + β I¯ +
f
fo
o
fI
M X
M
X
!,
fσ p
βhρ
Vhρ
−
R1z
R3z
(24)
h=1 ρ
TH
− xo = −
β o + β f o π̄ f + β oo wo + β oI I¯ +
M X
M
X
!,
oσ p
βhρ
Vhρ
R3z
(25)
h=1 ρ
−
0.5
p
2
σzπ
Z̄
=
M X
M
X
β I + β f I π̄ f + β oI wo +
!,
Iσ p
βhρ
Vhρ
R2z
(26)
h=1 ρ
.
0.5T H σp2
fσ f
Iσ ¯
oσ o
=
βhρ
I + βhρ
π̄ + βhρ
w
(R4z + R5z )
(27)
Z̄
√ 2 σzπ
Z̄ π̄ f
Z̄
Z̄ π̄ f
z
zπ
zπ
z
H
z
z
zπ
zπ
√
√
√
where R1 = π̄f
ξ1 + 2ξ2 ln 2 , R2 = T
ξ1 + 2ξ2 ln 2 + ξ1 + 2ξ2 ln 2 , R3z =
σzπ
σ
σ
z zπ
√ 2
σ
1 − 0.5 Z̄zπ R2z , R4z = √1 2 ξ0z + ξ1z 0.5 + ln √Z̄ 2
+ ξ2z ln √Z̄ 2
1 + ln √Z̄ 2
and, R5z =
σz
σz
σz
σz
f
f
f
Z̄
π̄
Z̄
π̄
Z̄
π̄
1
zπ
zπ
zπ
√ 2 f ξ0 + ξ1 0.5 + ln √ 2
+ ξ2 ln √ 2
1 + ln √ 2
−
σzπ π̄
σzπ
σzπ
σzπ
Again the derived demand equation for the aggregated marketed good used as numeraire is obtained from (20) using the derivative property in Proposition 2 as:

,
M X
M
c
c
c
c
X
∂ṽ
∂ṽ
∂ṽ
∂ṽ
c
p
p
p 
p
p
c = ṽ c − π̄ f f − wo o − Z̄
−
Vhρ
R3z
p
∂w
∂ π̄
∂Vhρ
∂ Z̄
(28)
h=1 ρ=1
with the associated parameters obtained using the homogeneity
property.
p
f
o
f
c
Denoting with xi = {x , x , c} and noting that xi π̄ , wo , p , Z̄, Vhρ
, Z̄ = T H [π̄ f + wo ] + I 0 and
p
π̄ f p̄q , wv , S, Vhρ
the on- and off-farm labor supply and aggregate marketed good demand elasticities
with respect to all exogenous variables are calculated from the following:
εciwo
= eciwo + Swz o eciz
εciI 0
= SIz0 eciz ,
εcipc
= (∂xi /∂pc ) (xi /pc )
εciV p
hρ
εcij
where Swz o = T H wo
=
eciV p
hρ
=
εpπ̄j
+
εpπ̄V p
hρ
[eciπ̄
+
(29)
Sπ̄z eciz ]
∀h, ρ
[eciπ̄ + Sπ̄z eciz ] ∀j = {p̄q , wv , S}
Z̄ , SIzo = I o Z̄ , Sπ̄z = T H π̄ f Z̄ , eciz = (∂xi /∂ Z̄)(Z̄/xi ), εpπ̄j and εpπ̄V p
hρ
defined appropriately in (19) when i = π̄ f and,
c
ecij = ec∗
ij − εRj
(30)
∗ /∂p ) (p /x∗ ), εc
z /∂p ) (p /Rz ) and, p = π̄ f , w , pc , Z̄, V p
∗
where ec∗
=
(∂x
=
(∂R
j
j
j
j
j
o
3
3
ij
i
i
Rj
hρ with xi (.)
being the term into brackets in the labor supply and aggregate marketed good demand equations in
(24)-(28).
10
4
Data and Estimation Strategy
The data used for the estimation of the agricultural household model are from the Farm Business
Survey for UK and refer to 296 farms observed during the 2001-04 cropping period. The database
provides information on all the price and quantity variables required to estimate econometrically the
model. All farms in the sample are based exclusively on household members for their operation
producing all crop outputs considered (i.e., no hired labor). Further, all sample participants exhibit
non-zero hours of work off-farm implying the absence of corner solutions either for outputs produced
or for off-farm working hours.
Three outputs were considered (i.e., wheat, barley and oilseeds) for which arable CAP regime
ensures different levels of area payments and three variable inputs (i.e., seeds, chemical fertilizers
and land). The survey also includes data on the number of hours of off-farm work for household
members, the number of hours worked on-farm, the off-farm wage rate and, the households non-labor
income. Total short-run profits have been computed as the sum of total gross sales and total CAP
production aid minus total variable costs divided by hours worked on-farm. Non-labor household
income was measured as the asset income generated from off-farm investments, assuming a 6 percent
rate of return. Household off-farm wage rate was calculated as the weighted average of individual wage
rates (husband, wife and spouses) with hours of work off-farm used as weights. Initial farm wealth has
been approximated by the value of farm equity also provided by the survey. For the price of aggregate
marketed good, we use a regional-specific consumer price index published by the UK Department of
Environment, Food and Rural Affairs (formerly MAFF).
Given the policy regime under the recent CAP reform and following Sckokai and Moro (2006),
three policy variables were included into the model: the total decoupled area payments, the set-aside
payments and the set-aside percentage.14 Finally, we have used managers education level and age as
separate explanatory variables in both the production (i.e., factors affecting managerial ability) and
consumption equations (i.e., taste shifters). Summary statistics of the variables are presented in Table
1. All prices are real prices, i.e., nominal prices, deflated by the producer price index also published
by the UK Department of Environment, Food and Rural Affairs (DEFRA, 2002).
In order to generate the variance-covariance matrix for crop prices we used the adaptive expectation hypothesis of Chavas and Holt (1990; 1996). Specifically, the untruncated crop price at time t is
defined as the market price of the previous year plus the sample mean of the past difference between
observed prices and nave expected prices (i.e., prices in the previous period) as:15
E pqmit+1 = pqmit + E pqmit − pqmit−1
(31)
Similarly untruncated price variance is computed by taking a weighted sum of the squared deviations
of past prices from their expected prices as:
V
ar (pqmit )
=
3
X
h
ζj pqmit−j
− Et−j−1
pqmit−j
i
(32)
j=1
where ζj are the three lag weights defined as 0.50, 0.33 and 0.17.16 Individual price covariances were
estimated in a similar way. In computing the above individual farm lagged prices for every year in
the panel was used. The above approach actually presumes that price risk is measured as the squared
deviation of past prices from their expected values with declining weights.
14
Area payments are awarded to all farmers producing cereals oilseeds an protein crops. The payment varies by region
and crop and is computed multiplying a fixed per ton amount by a regional historical yield and then by the acreage
declared each year by farmers sowing eligible crops. In order to access full amount, the total land allocated to program
crops cannot exceed a fixed national base acreage. Set-aside payments are relevant to large producers who are obliged to
divert from production a fixed percentage of the land allocated to program crops. The set-aside percentage is established
each year by the EU Commission based on expected market conditions and varies across crops and regions.
15
Chavas and Holt (1990) in their appendix and Sckokai and Moro (2006) provide more information for the calculation
of truncated price distributions.
16
The same values have been used in all previous studies adopting Chavas and Holt (1990) approach assuming three
lags for current price determination.
11
Equations in (13)-(??) and (24)-(27) define a system of twenty-two simultaneous equations (thirteen for the production and nine for the consumption decisions) that can be econometrically estimated
jointly using FIML techniques after appending a suitable econometric error structure. The on-farm
labor equation obtained from the last derivative property in Proposition 1 was also included in the
system of equations to correct for the endogeneity problem of shadow wage of on-farm labor supply
used as an explanatory variable in the consumption side.17 The associated likelihood function was
maximized using Berndt, Hall, Hall and Hausman (BHHH) algorithm. The FIML estimator has the
same asymptotic properties as the three-stage least squares estimator and with normally distributed
disturbances it is asymptotically efficient (Hausman, 1975). The price of land and aggregate marketed
good (i.e., the regional CPIs) were used as numeraire in imposing linear homogeneity in (11) and (20),
respectively.
5
Empirical Results
The parameter estimates of the non-separable agricultural household model along with the corresponding asymptotic standard errors obtained by the joint estimation of the consumption and production
sides are presented in Tables 2, 3 and 4. As it is shown in these Tables the majority of the consumption
and production coefficients are statistically significant at least at the 5 per cent level. The asymptotic
standard errors for land input and aggregate marketed good used as numeraire were estimated using
block re-sampling techniques which entails grouping the data randomly in a number of blocks of farms
and re-estimating the model leaving out each time one of the blocks of observations and then computing the corresponding standard errors (Politis and Romano 1994).18 The generalized R-squared is fair
high, R̃2 = 0.8742 indicating a satisfactory fit of the agricultural household model. The same applied
for the raw-moment R-squared values for individual equations that are reported in Tables 2 and 4.
The Hessian matrices of the first and second-order partial derivatives of both indirect utility
functions indicate that regularity conditions are met at the point of approximation (i.e., non-decreasing
in crop prices and on-farm labor and non-increasing in variable inputs, leisure and marketed goods).
Further, checking individual data points, all farms in the sample exhibit the correct sign for the own
price supply and demand elasticities in both the production and consumption equations.
Statistical testing, using the conventional LR-test, indicate that all variance and covariance coefficients are not jointly equal to zero rejecting thus the hypothesis of risk-neutrality (the value of the
LR-test was 123.56, well above the corresponding critical value of the chi-squared distribution at a 5
per cent significance level with 60 d.f.). The estimated CRRA coefficients in relations (12), (22) and
(23), also reported in Tables 2 and 3, show that the hypothesis of risk-averse behavior is maintained for
sample participants. All parameter estimates are statistical significant at any conventional statistical
level. Using these estimates we have calculated the relative risk aversion coefficients for both sides
of agricultural household model that appear in Table 5.19 The average farm values are 4.384 for rw ,
0.0656 for rIπ and 0.1539 for rI . From the point estimates it is also revealed that the UK farmers are
less sensitive in changes of household income than they are for farm wealth.
These estimates though vary considerable according to the age and size distribution of farms in the
sample. As it is shown from the middle and lower panel of Table 5, large farms and younger farmers are
more willing to bear risk against price volatility. For an 1 per cent increase in farm equity, the marginal
utility of farm wealth decreases more rapidly at lower levels of farm equity (small farms) than for higher
levels (large farms). Small farms are more risk averse with rw , rIπ and rI being 6.9883, 0.1221 and
0.2631, respectively; for medium farms risk aversion coefficients become lower (4.7652, 0.0598 and
0.1234) and for large farms become very low although rw is still above unity. The same applies for
younger farmers who exhibit a higher marginal utility of initial farm equity. Specifically, farmers
aged less than 40 years have risk aversion coefficients rw = 5.8741, rIπ = 0.0766 and rI = 0.1735
17
Thorton (2001) uses an iterative three stages least squares estimation procedure in order to correct for the endogeneity
bias using farmers age, education and no of children as instruments.
18
In our case we use groups of ten farms each time.
19
The reported coefficients are in line with prior expectations and the findings reported in the literature. According
to Chavas and Holt (1990) and Rosenzweig and Binswanger (1993), estimates of relative risk aversion coefficients in
agriculture vary from 0 to over 7.5 with a median around 1.
12
considerably higher than the respective mean values for their older counterparts. Young farmers
with less experience seems to be more sensitive to crop price volatility. Finally, concerning the time
development of the relative risk aversion coefficients there is no clear pattern that emerges. In all five
years the mean values of rw , rIπ and rI are close to their mean period estimates. This applies also
for the age and size distribution of farms in the sample. The short time interval of the dataset does
not allow to draw safe conclusions concerning the temporal pattern of risk aversion behavior.
The Consumption Side
The on- and off-farm labor supply and aggregate marketed good demand elasticities are reported
(at sample means) in Table 6. The estimated mean values calculated using (29) are all consistent
with theoretical expectations. The demand equation for the aggregate marketed good is downward
sloping, while the supply of off-farm labor is upward sloping. Concerning on-farm labor supply, point
estimates reported in Table 6, indicate that it is upward sloping with respect to all three crop prices
with the highest impact arising from the price of wheat (0.1129 for wheat, 0.0554 for barley and 0.0379
for oilseeds). Given that farms in the sample are devoting the majority of their resources in wheat
cultivation, they are more responsive to changes in wheat prices.
Further on farm-labor supply, is downward sloping with respect to all three variable input prices
with similar magnitude: 0.0207 with respect to the price of seeds input, 0.0215 to that of fertilizers
and 0.0168 to that of land input. Increases in the cost of farm production decreases quasi profits
turning household resources to off-farm employment. All policy variables decrease the supply of labor
from household members in farming activities. The impact of area payments is significant although
inelastic indicating that decoupled payments introduced by the last amendment of CAP regime are
not fully decoupled (the relevant point estimate is -0.0175). Autonomous income transfers to farmers
decrease their incentive to work on-farm. Similarly, set-aside payments and set-aside percentage are
also affecting negatively the supply of on-farm work. Although the effect is strongly inelastic it
underlines also the non-neutral effect of policy changes in farming activities.
Off-farm wage rate as it is expected theoretically, decreases the incentive of household members to
work on-farm indicating the substitutability among the alternative work opportunities. Specifically,
the mean cross-price supply elasticity is -0.1348. The same negative effect, -0.0163 on the average, is
observed for non-labor income changes. However, the supply of on-farm labor increases as the price
of aggregate marketed good increases as more income is required for market purchases.
Concerning off-farm labor supply, the own-price elasticity is 0.1533 indicating that members of
rural households are not so responsive to changes in off-farm wage rate. Concerning cross-price effects
the mean estimates reported in Table 6 suggest that an increase in all three crop prices decreases the
supply of off-farm labor again with the price of wheat having the highest responsiveness. Farmers
realize that farming is getting more profitable and they switch their labor endowments into farming
activities reducing off-farm employment. The increase in variable input prices, on the other hand,
increases the supply of off-farm labor as farming turns to be less profitable for household members.
This is also true for all policy variables but for different reasoning: 0.0184 for area payments, 0.0118
for set-aside payments and 0.0008 for the set-aside percentage. As decoupled income transfers increase
for farmers the opportunity cost of their off-farm employment increases.
Concerning the demand for the aggregate marketed good, the own-price effect is negative, -0.0363,
although of a small magnitude. The use of the regional-specific consumer price index as a proxy of
the price of marketed good may be partly responsible for this low elasticity value. Regarding the
cross-price effects, they are all positive except for those with respect to variable-input prices (increase
in farm profits increase the demand for purchased goods from household members).
Finally, as Table 8 indicates, rural households seems to respond positively in price volatility increasing the supply of on-farm labor and reducing at the same time their supply of labor in off-farm
activities as a way to hedge against price risk. All the on-farm labor supply variance elasticities reported in that Table are positive, whereas those for off-farm labor supply exhibit the opposite sign.
The highest effects for both types of labor employment are arising from the variance of oilseeds prices
and the covariance between wheat and oilseeds prices.
13
The Production Side
The crop supply supply and variable-inputs demand elasticities evaluated at the mean values of the
variables are also presented in Table 7. All the estimated crop supply and input-demand functions
are consistent with theory as evidenced by these elasticities. Specifically, each variable-input demand
equation (i.e., seeds, fertilizers, land) is downward slopping in its own price, and the supply elasticities
for the three crops (i.e., wheat, barley, oilseeds) considered is positive at a low magnitude though.
Starting from the own-price supply elasticities these are 0.1821 for wheat, 0.4882 for barley (the
highest mean value) and 0.4177 for oilseeds. The low point estimate for wheat is also due to the fact
that farmers are specialized mostly on wheat cultivation and hence less responsive to it’s price change.
All three crops are found to be gross substitutes as all cross-price supply elasticities are negative with
oilseeds being more elastic to other crop price changes (again wheat is less price responsive). The
estimated elasticities of supply with respect to the price of variable inputs are all negative, with the
highest impact arising from the price of land input that is more elastic in all three crops.
The supply elasticities with respect to initial farm wealth are all positive as expected and of
close magnitude (they range between 0.1183 and 0.1449). Finally, the supply responses to the three
policy variables are all negative. The highest effect seems to arise from area payments indicating that
decoupled payments are affecting significantly crop production. The corresponding point estimate
is -0.5828 for barley, -0.4803 for wheat and -0.2626 for oilseeds. The corresponding values for the
other two policy variables are considerably lower with the set-aside percentage having a low and
a rather insignificant impact. Although these responses are inelastic, they also confirm empirically
that the CAP reform tools are not fully decoupled affecting production decisions by British farmers.
The payment responsiveness is always lower than the corresponding price effects indicating that crop
production is indirectly influenced by income support measures through the land allocation decisions.
Concerning variable-input demand elasticities all appear to be of plausible magnitudes, with the
those with respect to the price of seeds being somewhat higher than those with respect to the prices of
fertilizers or land. Specifically, own-price elasticity of seeds is -0.2024 whereas the cross-price elasticity
of fertilizers with respect to seeds price is the highest -0.2844 followed by that of land input -0.2037.
All variable-inputs are nonregressive (normal) in the sense that profit maximizing input demand
elasticities with respect to all output prices are positive (this was also manifested by the negative sign
of all supply elasticities with respect to input prices). The highest response was found for the price of
wheat in all three variable-inputs demand. All policy variables (i.e., area and set-aside payments and
set aside percentage) are affecting negatively variable-input demands. Again decoupled area payments
seems to reduce significantly the incentive to produce for the members of rural households. Finally,
the variable-input demand elasticities with respect to the value of farm equity were all found negative
indicating that wealthier farms are demanding less variable inputs for their farm production.
Both supply and variable-input demand elasticities with respect to price variances are small in
magnitude but higher that those of labor supply (see lower panel in Table 8). Increasing price risk was
found to have a negative own-effect in the supply of all three crops with the highest being for oilseeds.
Supply of wheat crop increases as the price volatility of barley and oilseeds prices increases, while the
same applies for oilseeds supply when the price volatility of wheat crop raises. For all variable-input
demands the cross-effect of price variances is negative with the highest impact coming from the price
of oilseeds followed by that of wheat. The cross-effects though exhibit different signs over crops and
inputs indicating some substitution effects among farmers aimed to hedge against price volatility.
6
Conclusions
Relaxing the assumption of risk neutrality, the present paper developed a non-separable dual agricultural household model extending Lopez (1984) unified theoretical and econometric framework under a
non-linear mean variance approach suggested by Coyle (1999). Risk neutrality is a crucial assumption
or the recursiveness of the agricultural household models. Barnum and Squire (1979) underlined that
under risk averse behavior the separability assumption between production and consumption decisions
of rural households is not valid even under a competitive market environment. The proposed model
can accommodate risk aversion behavior and price uncertainty in a manner tractable for empirical
14
research and it has been applied in a sample of 296 UK cereal producers during the 2001-04 cropping
period.
The econometric model under CRRA and price risk is non-linear in coefficients but the degree of
non-linearity is relatively minor and does not obstacle the econometric estimation with standard ML
techniques. The empirical results suggest that indeed British cereal farmers are risk averse underlying
the need to accommodate both production and consumption decisions in a unified estimation framework. The degree of risk aversion depends on the age and size distribution of the farms with older
and big farmers more willing to bear risk against price volatility. Finally, the price effects on household labor supply are strong and significant and it should be taken into consideration in analyzing
household behavior.
15
References
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16
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31-55.
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17
A
Appendix
PROOF OF PROPOSITION 1: To prove proposition (1a) note that under CRRA scaling utility level
by λ must multiply consumer objective with λ otherwise the optimal consumption decisions would
change. Therefore it should hold for the risk aversion coefficients that r λµv , λ2 σv2 = λ−1 r µv , σv2
for every λ > 0. Hence, multiplying (4) by λ > 0, i.e.,
2 2
2 2
2
2
xf λW̄ − 0.5rw λW̄ , λ2 σw
λ σw = xf λW̄ − 0.5λ−1 rw W̄ , σw
λ σw
2
f
w
2
= λx W̄ − 0.5r W̄ , σw σw
(A.1)
2 = λṽ p W , p̄q , w w , xf , S, σ 2 and therefore the unique optimal
Hence, ṽ p λW0 , λp̄q , λwv , xf , S, λσw
0
w
solutions of the optimization problem in (3) remain unchanged.
To prove (1b), apply the envelope theorem to the optimization problem in (3). Then we get:
∂ṽ p
∂ p̄m
= xf
= qm
∂ṽ p
∂wjv
∂ṽ p
∂W0
∂ṽ p
∂σp2
∂ṽ p
∂xf
f
∂ π̃ f
w 2 ∂ π̃
− 0.5rw̄
σw
∂ p̄m
∂ p̄m
w 2
1 − 0.5rw̄ σw
f
∂ π̃ f
w 2 ∂ π̃
−
0.5r
σ
w̄ w
∂wjv
∂wjv
w 2
= −xj 1 − 0.5rw̄
σw
w 2
= 1 − 0.5rw̄
σw
= −xf
= xf
∂ π̃ f
w 2
1 − 0.5rw̄
σw
∂ p̄m
(A.2)
!
= −xf
∂ π̃ f
w 2
1 − 0.5rw̄
σw
v
∂wj
(A.3)
(A.4)
2
2
= −0.5 rσw σw
+ rw σσ2p = −0.5 rσw σw
+ rw σσ2p
w 2
w 2
= π̄ f − π̄ f 0.5rw̄
σw = π̄ f 1 − 0.5rw̄
σw
(A.5)
2 , r w = ∂r w /∂ w̄ and, σ 2 = ∂σ 2 /∂σ 2 while the last equality in (A.2) and (A.3) is
where rσw = ∂rw /∂σw
p
w
σp
w̄
due to the assumption of CRTS farm technology as π̃(.) = xf π̃ f . Substituting (A.4) into (A.2), (A.3)
and (A.6) establishes properties (1bi), (1bii) and (1bv), respectively. Expressions (A.4) and (A.5) are
exactly relationships (1biii) and (1biv) in the proposition.
Properties (1c) and (1d) can be proved using Hatta (1980) primal-dual approach of the optimization problem in (3) which implies that the Hessian matrix obtained from the indirect utility function
is positive semidefinite and thus is is convex with respect to prices (lemma 2, theorem 5 and equation 17 therein) and by adapting the proof of quasiconvexity and symmetry of the standard indirect
utility function in consumer theory (Epstein, 1985). The curvature property in (1c) actually places
restrictions on the first- and second-order partial derivatives of ṽ p and rw which is analogous with the
risk-neutral profit function. Instead property (1d) more clearly indicate that optimization imposes
restrictions on the Hessian matrix and thus a flexible approximation of ṽ p is enough to represent
curvature restrictions under the maximization hypothesis.
PROOF OF PROPOSITION 2: To prove proposition (2a) we multiply (10) by λ > 0 which under
CRRA should result into:
2
2
λZ̄ − 0.5rz λZ̄, λ2 σz2 λ2 σz2 − rzπ λZ̄, λπ̄ f , λ2 σzπ
λ2 σzπ
(A.6)
2
2
= λZ̄ − 0.5λ−1 rz z̄, σz2 λ2 σz2 − λ−1 rzπ Z̄, π̄ f , σzπ
λ2 σzπ
2
2
= λ Z̄ − 0.5rz Z̄, σz2 σz2 − rzπ Z̄, π̄ f , σzπ
σzπ
18
2
2 .
Therefore, it holds that ṽ c λpc , λwo , λπ̄ f , λZ̄, λ2 σz2 , λ2 σzπ
= λṽ c pc , wo , π̄ f , Z̄, σz2 , σzπ
To prove (2b), apply the envelope theorem to the optimization problem in (7) and we get:
∂ṽ c
∂ π̄ f
∂ṽ c
∂wo
∂ṽ c
∂pc
∂ṽ c
∂ Z̄
∂ṽ c
∂σπ
2
2
= −`f + 0.5 `f rzz σz2 + `f rzzπ σzπ
+ rπ̄zπf σzπ
2
2
= −`f 1 − 0.5 rzz σz2 + rzzπ σzπ
+ rπ̄zπf σzπ
2
2
= −`o + 0.5 `o rzz σz2 + `o rzzπ σzπ
= −`o 1 − 0.5 rzz σz2 + rzzπ σzπ
(A.7)
2
2
= −c + 0.5 rzz σz2 + rzzπ σzπ
= −c 1 − 0.5 rzz σz2 + rzzπ σzπ
(A.9)
2
= 1 − 0.5 rzz σz2 + rzzπ σzπ
(A.8)
(A.10)
2
= −σπ2 T H 0.5rσz σz2 + rz T H + rσzπ σzπ
+ rzπ
(A.11)
2 , r z = ∂r z /∂ Z̄, r zπ = ∂r zπ /∂ Z̄, σ z =
where rσz = ∂rz /∂σz2 , rπ̄zπf = ∂rzπ /∂ π̄ f , rσzπ = ∂rzπ /∂σzπ
z
z
σπ
2
2
zπ
2
2
∂σz /∂σπ and, σσπ = ∂σzπ /∂σπ .
Substituting (A.10) into (A.7), (A.8) and (A.9) establishes properties (2bi), (2bii) and (2biii),
respectively. Expressions (A.10) and (A.11) are exactly relationships (2biv) and (2bv) in the proposition. Again properties (2c) and (2d) can be proved using using the same approach with Proposition 1.
19
Tables
Table 1: Summary Statistics of the Variables
Variable
Consumer price index (CPI)
Household non-labor income (GBP)
Off-farm labor (hrs)
Off-farm labor income (GBP)
Off-farm wage rate (GBP/hr)
On-farm labor (hrs)
On-farm wage rate (GBP/hr)
Wheat production (tones)
Wheat price (GBP/ton)
Barley (tones)
Barley price (GBP/ton)
Oilseeds (tones)
Oilseeds price (GBP/ton)
Seeds (kgs)
Seeds price
Fertilizers (kgs)
Fertilizer price (GBP/kg)
Land (ha)
Land price (GBP/ha)
Initial farm assets (GBP)
Area payments (GBP)
Set-aside payments (GBP)
Set-aside percentage (%)
20
Mean
1.570
3,918
1,274
13,460
9.43
2,916
14.89
493.3
72.4
166.3
73.8
56.2
154.8
330.2
0.24
3,027
3.30
173
45.1
16,541
26,492
2,598
0.33
Max
1.775
8,241
5,625
76,500
26.79
9,857
45.15
6,000
105.3
2,959
109.3
294.3
221.0
5,212.1
0.35
20,848
4.66
1,587
57.6
59,874
152,471
11,245
0.79
Min
1.346
854
50
300
1.15
200
6.61
302.3
44.2
405.2
50.0
102.9
110.0
100.8
0.11
373
2.14
18
22.7
1,621
3,836
1,141
0.10
StDev
0.122
2,148
742
12,609
4.77
2,191
5.81
246.3
10.0
140.5
9.9
44.8
21.6
268.5
0.05
2,937
0.54
186
11.8
12,487
21,487
2,055
0.13
Table 2: Parameter Estimates of the Consumption-Side Equations
Variable
Constant
Demand Functions
0.2129
T H − xf
(0.0855)
0.2106
T H − xo
(0.0684)
Aggregate
0.5745
Marketed Good
(0.2098)
0.0623
Household Income
(0.0241)
p
Vwb
-0.0261
T H − xf
(0.0121)
0.0047
T H − x0
(0.0110)
Aggregate
0.0210
Marketed Good
(0.0112)
-0.3191
Household Income
(0.1200)
Constant
Risk Aversion Functions
0.0038
rI
(0.0012)
0.0344
rIπ
(0.0066)
π̄ f
wo
pc
Z̄
p
Vww
-0.5461
(0.1719)
p
Vwo
-0.0236
(0.0122)
-0.0049
(0.0122)
0.0242
(0.0123)
-0.4191
(0.1557)
0.5
Z̄/ σz2
0.0514
(0.1041)
-0.4075
(0.0762)
Vbop
-0.0096
(0.0275)
0.0149
(0.0071)
-0.0047
(0.0021)
0.1509
(0.4141)
Z̄ 2 /(σz2 )
0.2546
(0.1234)
0.1282
(0.0394)
-0.3160
(0.1230)
Vbbp
0.0623
(0.0304)
-0.0517
(0.0190)
-0.0114
(0.0034)
-0.2658
(0.0985)
2 )0.5
Z̄/(σzπ
0.2154
(0.1121)
0.2141
(0.0946)
0.2957
(0.1123)
p
Voo
0.0020
(0.0383)
0.0155
(0.0262)
-0.0324
(0.0123)
0.0660
(0.0279)
f
2 )0.5
π̄ /(σzπ
0.0013
(0.0003)
0.0352
(0.0170)
-0.0677
(0.0231)
0.0684
(0.0214)
R̄2
2 )
Z̄ π̄ f /(σzπ
0.0010
(0.0007)
-
0.0033
(0.0012)
-
0.0002
(0.0001)
0.0169
(0.0065)
0.0255
(0.0103)
0.6421
0.6011
0.7121
where xf are the on-farm labor hours, xo are the off-farm labor hours, c is the aggregate marketed good, T H is the
total available time for household members, π̄ f are the profits per hour of on-farm labor, wo the off-farm hour wage
rate, pc the price of aggregated marketed good, Z̄ is the mean total household income with Z̄ = T H (π̄ f + wo ) + I 0
p
p
2
with I 0 being the non-labor income, σz2 = (T H )2 (q 0 Vhρ
q) and σzπ
= T H (q 0 Vhρ
q) is the variance of unit profits and
p
the covariance of unit profits and household income, respectively,with Vhρ
being the variance-covariance matrix
of the m = h, ρ crop prices (i.e., wheat, barley and oilseeds). Parameter estimates for the demand equation for
marketed good where obtained from the homogeneity restrictions, while the corresponding standard errors using
block re-sampling techniques (Politis and Romano, 1994).
21
22
-
-0.1328
(0.0598)
-0.1286
(0.0620)
0.1451
(0.0311)
-0.1432
(0.1525)
1.0512
(0.3151)
-
po
pb
-0.1100
(0.0495)
-
-0.1525
(0.0798)
-0.3507
(0.1337)
-0.0552
(0.1401)
ws
-0.2405
(0.0953)
-1.2543
(0.2578)
-
-0.1025
(0.1259)
-0.0372
(0.2872)
0.0272
(0.0122)
wf
-0.0750
(0.0231)
-0.1810
(0.0546)
-0.7947
(0.1239)
-
-0.3469
(0.1023)
-0.2169
(0.0938)
-0.0168
(0.0092)
wl
0.0563
(0.0140)
0.0106
(0.0027)
3.9116
(0.6563)
-
0.0104
(0.0018)
0.0029
(0.0007)
0.0011
(0.0002)
W0
-0.0579
(0.0244)
-0.0784
(0.0358)
-0.1690
(0.0737)
-0.2710
(0.0452)
-0.0165
(0.0040)
-0.0144
(0.0052)
-0.0034
(0.0007)
Sa
-0.0057
(0.0181)
-0.0129
(0.0049)
-0.0051
(0.0023)
-0.0102
(0.0145)
-0.0092
(0.0020)
-0.0027
(0.0015)
-0.0015
(0.0499)
Ss
-0.0996
(0.0258)
-0.1841
(0.0774)
-0.2286
(0.0938)
-0.0054
(0.0080)
-0.2711
(0.0509)
-0.1251
(0.0410)
-0.1214
(0.0610)
Sτ
where pw , pb and po are the prices for wheat, barley and oilseeds, respectively, ws , wf and wl are the prices for seeds, fertilizers and, land, respectively, W0f is
the initial farm wealth per unit of on-farm labor, Sa are the total area payments, Ss are the set aside payments and, Sτ is the set aside percentage. Parameter
estimates for the demand equation for land where obtained from the homogeneity restrictions, while the corresponding standard errors using block re-sampling
techniques (Politis and Romano, 1994).
Constant
pw
Crop Supply Functions
0.8088
1.1854
Wheat
(0.2031) (0.2002)
0.7695
Barley
(0.2439)
2.7453
Oilseeds
(0.6578)
Variable-Input Demand Functions
-0.5282
Seeds
(0.2522)
-0.6025
Fertilizers
(0.3082)
-2.1930
Land
(0.8791)
0.1082
Farm Wealth
(0.0253)
-
Table 3: Parameter Estimates of the Production-Side Equations
Table 4: Parameter Estimates of the Production-Side Equations (continued)
p
p
Vww
Vwb
Crop Supply Functions
-0.2311
-0.0307
Wheat
(0.1061)
(0.0667)
-0.1032
0.9851
Barley
(0.2296)
(0.1520)
0.1138
0.1410
Oilseeds
(0.2612)
(0.1755)
Variable-Input Demand Functions
0.1302
0.3076
Seeds
(0.0412)
(0.0828)
0.3896
0.2615
Fertilizers
(0.1562)
(0.1023)
0.5532
-0.7442
Land
(0.1233)
(0.2310)
-0.8525
-0.9202
Farm Wealth
(0.2122)
(0.9155)
2 )0.5
Constant W̄ /(σw
Risk Aversion Function
0.0122
0.0172
rw
(0.0013)
(0.0066)
p
Vwo
Vbop
Vbbp
p
Voo
-0.0811
(0.0257)
-0.2021
(0.1053)
-0.0104
(0.0052)
-0.0420
(0.0975)
-0.3041
(0.2407)
0.0659
(0.2737)
0.0202
(0.0105)
-0.4029
(0.1231)
-0.0532
(0.0251)
0.0468
(0.1208)
-0.1503
(0.3287)
-0.0427
(0.0207)
0.1064
(0.0822)
0.1519
(0.1041)
0.8773
(0.2313)
-0.8420
(0.2146)
2
W̄ 2 /σw
-0.0159
(0.1236)
0.4131
(0.1602)
-0.1171
(0.0982)
0.0002
(0.0002)
0.1290
(0.0413)
0.2906
(0.1080)
0.5365
(0.2134)
-0.5203
(0.2021)
0.2702
(0.1068)
0.3254
(0.1300)
0.5257
(0.1873)
-0.9753
(0.1458)
R̄2
0.4566
0.5987
0.6321
0.5443
0.5876
0.4453
0.0139
(0.0035)
p
p
2
where W̄ = W0 + π̄ is the total farm income σw
= (q 0 Vhρ
q)/(xf )2 is the variance of farm profits with Vhρ
being the variance-covariance matrix of the m = h, ρ crop prices (i.e., wheat, barley and oilseeds). Parameter
estimates for the variance equation for land input where obtained from the homogeneity restrictions, while
the corresponding standard errors using block re-sampling techniques (Politis and Romano, 1994).
Table 5: Estimated Relative Risk Aversion Coefficients
Total Sample
rzπ
0.0656
(0.0014)
rz
0.1539
(0.0168)
rw
4.3840
(0.5287)
0.0766
(0.0117)
0.0546
(0.0110)
0.1735
(0.0247)
0.1343
(0.0383)
5.8741
(0.5369)
2.8932
(0.4987)
0.1221
(0.0114)
0.0598
(0.0127)
0.0152
(0.0087)
0.2631
(0.0332)
0.1234
(0.0414)
0.0752
(0.0231)
6.9883
(0.6241)
4.7652
(0.5698)
1.3983
(0.2341)
Age Distribution
Young Farmers (<40 years)
Older farmers (>40 years)
Size Distribution
Small farms (<80 ha)
Medium farms (80-300 ha)
Large farms (>300 ha)
Standard errors were computed using block re-sampling techniques (Politis
and Romano, 1994).
23
24
0.0243
(0.0073)
-0.1348
(0.0065)
0.1533
(0.0564)
0.0073
(0.0021)
-0.0163
(0.0061)
-0.0113
(0.0034)
Io
-0.0363
(0.0105)
0.0305
(0.0113)
0.0175
(0.0041)
pc
0.0301
(0.0098)
0.1129
(0.0367)
-0.1187
(0.0511)
pw
0.0142
(0.0056)
0.0534
(0.0132)
-0.0561
(0.0135)
pb
0.0101
(0.0033)
0.0379
(0.0102)
-0.0398
(0.0101)
po
-0.0055
(0.0021)
-0.0207
(0.0057)
0.0218
(0.0065)
ws
-0.0057
(0.0022)
-0.0215
(0.0098)
0.0226
(0.0087)
wf
-0.0045
(0.0018)
-0.0168
(0.0072)
0.0177
(0.0056)
wl
0.0047
(0.0013)
-0.0175
(0.0043)
0.0184
(0.0078)
Sa
0.0030
(0.0011)
-0.0112
(0.0032)
0.0118
0.0034)
Ss
0.0002
(0.0001)
-0.0008
(0.0003)
0.0008
(0.0003)
Sτ
where wo is the off-farm wage rate, I 0 is the non-labor income, pc is the price of aggregate marketed good, pw , pb and po are the prices for wheat, barley and oilseeds, respectively,
ws , wf and wl are the prices for seeds, fertilizers and, land, respectively, Sa are the total area payments, Ss are the set aside payments and, Sτ is the set aside percentage. The
corresponding standard errors using block re-sampling techniques (Politis and Romano, 1994).
Demand
Marketed Good
Off-Farm
Labor Supply
On-Farm
wo
Table 6: On- and Off-Farm Labor Supply and Aggregate Marketed Good Demand Elasticities
25
ws
-0.0752
(0.0265)
-0.0642
(0.0234)
-0.0299
(0.0109)
-0.2024
(0.0981)
-0.2844
(0.1023)
-0.2037
(0.0782)
po
-0.1308
(0.0551)
-0.1252
(0.0453)
0.4177
(0.1467)
0.0541
(0.0204)
0.0291
(0.0108)
0.0157
(0.0065)
-0.1003
(0.0412)
-0.1191
(0.0446)
-0.1800
(0.0654)
-0.0488
(0.0156)
-0.0368
(0.0102)
-0.0333
(0.0112)
wf
-0.1048
(0.0387)
-0.1541
(0.0612)
-0.1081
(0.0342)
-0.1658
(0.0653)
-0.4101
(0.1781)
-0.0705
(0.0276)
wl
-0.0932
(0.0430)
-0.1198
(0.0401)
-0.1545
(0.0476)
0.1449
(0.0652)
0.1183
(0.0342)
0.1317
(0.0554)
W0
-0.0986
(0.0409)
-0.0730
(0.0321)
-0.0687
(0.0276)
-0.4803
(0.1345)
-0.5828
(0.1563)
-0.2626
(0.1093)
Sa
-0.0915
(0.0443)
-0.0361
(0.0103)
-0.0198
(0.0078)
-0.0507
(0.0231)
-0.0446
(0.0131)
-0.0720
(0.0276)
Ss
-0.0021
(0.0009)
-0.0042
(0.0016)
-0.0023
(0.0007)
-0.0019
(0.0008)
-0.0026
(0.0011)
-0.0075
(0.0027)
Sτ
where pw , pb and po are the prices for wheat, barley and oilseeds, respectively, ws , wf and wl are the prices for seeds, fertilizers and, land,
respectively, W0f is the initial farm wealth, Sa are the total area payments, Ss are the set aside payments and, Sτ is the set aside percentage.
The corresponding standard errors using block re-sampling techniques (Politis and Romano, 1994).
pw
pb
Crop Supply Functions
Wheat
0.1821
-0.1570
(0.0771) (0.0431)
Barley
-0.0653
0.4882
(0.0234) (0.1355)
Oilseeds
-0.1790
-0.1766
(0.0764) (0.0654)
Variable-Input Demand Functions
Seeds
0.0682
0.0174
(0.0304) (0.0081)
Fertilizers 0.0513
0.0190
(0.0224) (0.0076)
Land
0.1517
0.0967
(0.0573) (0.0374)
Table 7: Crop Supply and Variable-Inputs Demand Elasticities
Table 8: On- and Off-Farm Labor Supply, Aggregate Marketed Good Demand, Crop Supply
and Variable-Inputs Demand Variance Elasticities
p
Vww
p
Vwb
p
Vwo
V p bo
Vbbp
p
Voo
0.0027
(0.0010)
-0.0028
(0.0011)
0.0045
(0.0021)
-0.0047
(0.0021)
0.0075
(0.0030)
-0.0079
(0.0027)
0.0055
(0.0021)
-0.0058
(0.0021)
0.0023
(0.0008)
-0.0024
(0.0009)
0.0076
(0.0023)
-0.0080
(0.0024)
0.0007
(0.0004)
0.0012
(0.0007)
0.0020
(0.0011)
0.0015
(0.0007)
0.0006
(0.0003)
0.0020
(0.0009)
-0.0327
(0.0114)
Barley
-0.0712
(0.0254)
Oilseeds
0.0080
(0.0032)
Variable-Input Demand Functions
Seeds
-0.0276
(0.0165)
Fertilizers
-0.0204
(0.0098)
Land
-0.0196
(0.0081)
-0.0217
(0.0087)
0.0172
(0.0071)
0.0050
(0.0026)
-0.0574
(0.0178)
-0.0848
(0.0309)
-0.0601
(0.0298)
-0.0297
(0.0108)
-0.0638
(0.0265)
0.0047
(0.0023)
0.0273
(0.0111)
-0.0564
(0.0243)
-0.0836
(0.0356)
0.0331
(0.0121)
-0.0631
(0.0276)
-0.0604
(0.0287)
-0.0109
(0.0039)
-0.0183
(0.0763)
0.0026
(0.0011)
-0.0753
(0.0342)
-0.0091
(0.0041)
-0.0956
(0.0308)
0.0011
(0.0006)
-0.0193
(0.0076)
0.0021
(0.0008)
-0.0061
(0.0028)
-0.0111
(0.0043)
-0.0190
(0.0081)
-0.0382
(0.0176)
-0.0341
(0.0103)
-0.0149
(0.0066)
Labor Supply
On-Farm
Off-Farm
Demand
Aggregate Marketed Good
Crop Supply Functions
Wheat
p
where Vhρ
is the variance-covariance matrix of the m = h, ρ crop prices (i.e., wheat, barley and oilseeds). The
unconditional elasticities with respect to the price of aggregate marketed good is the same with the conditional
ones. The corresponding standard errors using block re-sampling techniques (Politis and Romano, 1994).
26