Agriculture Reference
In-Depth Information
B.3
Effect of Partnership Size on a Farm Partner's Task Effort
∂t
t
/∂N
To evaluate the partial derivative
, differentiate equation (B.3) with respect to
N
and solve to get
−
(α
−
1
)
N
2
h
∂t
t
∂N
∂
∂t
t
∂N
=
∂
≤
0.
(B.5)
2
h
2
∂t
t
The sign of the second term on the right-hand size is negative, but the sign of the first term depends on the
value of
α
∈
(
0, 1
)
.If
α
=
1 (maximum specialization potential), then changes in the number of partners
(N)
have
no effect on task effort
(t)
. However, if 0
≤
α<
1, then an increase in the number of partners will decrease task
effort.
B.4
The Solution to the Factory-Corporate Farm Problem
t
FC
t
k
FC
that solves the following first-order
The solution to equation (9.7) is given by the optimal input choices
and
necessary conditions:
(α/)(N
h
)
α
−
1
T
N
T
α
α
∂h
∂t
t
(t
FC
,
k
FC
)
≡¯
wt
=
1,
...
,
T
t
t
+
(B.6)
t
∂h
∂k
(t
t
,
k
C
)
≡
r
min
(B.7)
B.5
The Relationship between Farm Capital and Farm Organization
Prediction 9.10 states that family farms will employ less capital than partnerships and that partnership will employ
less capital than corporate farms. Although it appears to be a straightforward implication of our assumption that
the capital costs function,
r
=
r(N)
, is decreasing and convex in
N
, the prediction also depends on the relationship
between task effort
(t)
and capital
(k)
and on the specialization coefficient
(a)
. Consider a move from a family
farm to a partnership, which means that
N
increases. The main prediction is that, ceteris paribus,
k
must increase,
but an increase in
N
(inherent in a shift from a family farm to a partnership) must decrease the level of task effort
(t)
. When
N
increases, the marginal product of effort shifts outward because of specialization gains. The marginal
product also shifts inward because of shirking. Under our maintained assumptions—
α
∈
(
0, 1
)
and
N
≤
T
— the
outward shift (specialization gains) can never be larger than the inward shift (moral hazard losses). In general,
the inward shift will be larger so that an increase in
N
will reduce
t
. Only when
α
=
1 do the two effects exactly
offset each other. To see this, note that a worker's full marginal product depends on (a) his ownership share of
the output, or 1
/N
; and (b) the specialization coefficient,
a
=
(N/T )
α
. Now define
to be the marginal product
shifter where
1
N
N
T
α
N
α
−
1
T
α
.
=
=
(B.8)
It is easy to see from equation (B.8) that
∈
(
0, 1
)
, which means that an increase in the number of owners
(N)
can never increase task effort
(t)
. Given that
t
must decline from an increase in
N
, there are three possible cases
to consider in order to determine the final choice of capital.
1. Capital
(k)
and effort
(t)
are independent
(h
tk
=
0
)
.
