Agriculture Reference
In-Depth Information
B.1
The Solution to the Partnership Problem
Assuming the first-order approach is satisfied, noting that
h
=
h(e
,
k
,
q
−
1
)
and
e
=
at
, the following two first-order
necessary conditions define the solution to equation (9.5):
∂h
∂k
−
r(N)
T
∂h
∂e
t
N
T
α
∂t
t
∂k
+
N
−
w
=
0
(B.1)
t
=
1
T
T
N
α
−
1
T
α
∂h
∂e
t
N
T
α
∂r
∂N
∂t
t
∂N
∂h(α)
∂e
t
−
w
+
N
−
w
+
w
−
k
=
0
(B.2)
t
=
1
t
=
1
Equations (B.1) and (B.2) illustrate the trade-offs in a partnership farm. Equation (B.1) defines the conditions
for the optimal level of capital and can be discussed as two parts. In part 1, the first bracketed term is simply the net
marginal product of capital. Part 2 is the total indirect effect of capital choices on task effort and comprises three
terms. The last bracketed term is the effect of a change in capital stock on task effort, summed over all tasks, while
the first bracketed term is the size of the distortion in effort. The sign of
depends on whether capital and
task effort are complements or substitutes (see section B.2). Term 3 multiplies all of this by the number of partners
(N)
(∂t
t
/∂k)
to get a total effect. Equation (B.2) defines the conditions for the optimal number of partners. The marginal
benefit of adding another partner comprises an increase in task specialization (part 1) and a fall in marginal capital
costs (part 3). The marginal cost is the total indirect effect of reduced farm task effort that results from an increase
in the number of residual claimants (part 2).
Part 1 comprises two terms. The first term in brackets is the specialization effect of changing the number of
partners (holding the number of tasks constant), which is then summed over all the tasks in the stage (term 2). Part
3 comprises two independent terms, the direct effect on capital costs and the addition of off-farm income. Part 2
parallels the second part in equation (B.1) and, accordingly, comprises three terms. The only difference is that the
distortion on effort is multiplied by the effect of partnership size on task effort. This effect can be shown to be
nonpositive (see section B.3); the effect is negative except in the case in which specialization effects are at their
maximum
(α
=
1
)
, thus eliminating the effect altogether.
B.2
Effect of Partnership Capital on a Farm Partner's Task Effort
We want to evaluate the partial derivative
∂t
t
/∂k
, so first create an identity from equation (9.6), assuming
homogeneous partners, to get
N
α
−
1
T
α
∂h
∂t
t
(t
t
())
≡
w
.
(B.3)
Differentiate equation (B.3) with respect to
k
and solve to get
∂
2
h
∂t
t
∂k
−
∂t
t
∂k
=
∂
.
(B.4)
2
h
2
t
∂t
The denominator in equation (B.4) is negative by assumption, but the sign of the numerator depends on whether
capital and effort are substitutes or complements (or independent). Thus, sign of
∂t
t
/∂k
is positive (negative) if
k
and
t
are complements (substitutes).
