Agriculture Reference
In-Depth Information
t
t
()
=
t
t
(N
,
t
n
is
,
T
,
α
,
w
,
L
,
k
,
q
−
1
(d))
and solves the following first-order necessary
conditions:
N
α
−
1
T
α
∂h
∂t
j
(t
t
())
≡
wt
=
1,
...
,
T/N
.
(9.6)
Equation 9.6 shows that the number of partners does not affect the marginal rate of substitu-
tion between tasks on the farm but does affect the amount of each partner's effort on the farm.
Thus, as the number of partners increases, each partner spends less time on the farm, and
this translates into less time spent on each farm task. Note that when potential specialization
gains are greatest
, equation (9.6) reduces to equation (9.3) and the partner's choice
of time spent on each task is identical to that of the family farmer because
(α
=
1
)
a
=
1
/T
for each
task. On the other hand, if specialization has no value
, then equation 9.6 reduces
to a classic Marshallian sharecropping first-order condition because
(α
=
0
)
for each task,
and acts like a tax on task effort. The lesson is that as the potential gains from specialization
increase (higher
a
=
1
/N
α
), the incentives inherent in partnerships become more valuable.
Taking this optimizing behavior into account, the partners' joint problem is to maximize
expected profit by choosing the level of capital and the number of partners, subject to
each partner's incentive compatibility (
IC
) and individual rationality (
IR
) constraints and
the total time constraints of the partners. Because we assume that partners have identical
endowments, the effective effort term for each task is [
]
α
t
N/T
. Similarly, each partner earns
1
. Substituting this constraint directly into
−
T
t
=
off-farm income equal to
wm
=
w
1
t
t
the objective function gives
1
N
T
α
T
N
P
=
h
max
k
t
t
,
k
;
Q
−
1
(d)
−
r(N)k
+
Nw
−
1
t
t
(9.7)
,
t
=
t
)
t
=
t
t
()
=
π
p
subject to
(
IC
argmax
t
=
1,
...
,
T
)π
P
≥
V
(
IR
,
V
where
is the reservation income level for each partner.
The solution to equation (9.7) is derived in appendix B, but the main implications are
illustrated graphically in figure 9.1. Simply put, adding a partner yields a return from
increased task specialization and lower capital cost. At the same time, adding a partner
generates additional costs, in terms of decreased farm effort, from greater moral hazard.
The partnership farm will also have greater capital levels than will the family farm (see
appendix B). Depending on the relative size of these various effects, a partnership farm
may or may not be more valuable than a family farm.
