Agriculture Reference
In-Depth Information
In this case an increase in
N
unambiguously leads to an increase in
k
because the decrease in from the change
in
N
has no effect on the marginal product of
k
.
2. Capital
(k)
and effort
(t)
are substitutes
(h
tk
<
0
)
.
In this case an increase in
N
unambiguously leads to an increase in
k
because the decrease in
t
from the change
in
N
shifts out the marginal product of capital, thus adding to the effect of lower capital costs.
3. Capital
(k)
and effort
(t)
are complements
(h
tk
>
0
)
.
In this case an increase in
N
can possibly lead to a decrease in
k
because the decrease in
t
from the change in
N
shifts in the marginal product of capital, thus countering the effect of lower capital costs. Only if the complement
effect is strong enough to exceed the effect of reduced capital costs,
r
(N)
, can a partnership optimally employ
less capital than a family farm. When this is the case, however, the value of the family farm will always exceed the
value of the partnership because the deadweight loss (compared to first-best) for the partnership will be greatest
since the partnership cannot attain greater efficiency in task effort and is less efficient in capital employment.
Thus, even though it is possible for the model to generate a partnership with less capital than a family farm, such
a partnership will never be the wealth-maximizing choice of farm organization. The same analysis holds for the
comparison between corporate and family or partnership farms, although a sufficiently low effective wage
( w)
could, in principle, lead to greater task effort under corporate organization.
B.6
Comparative Statics of Farm Organizations
The value function for the family farm is
1
T
α
1
T
α
t
1
,
t
T
V
F
=
h
k
f
;
max
k
f
+
wm
f
.
...
,
q
−
1
(d)
−
r
(B.9)
The value function for the partnership farm is
N
P
T
α
N
P
T
α
V
P
=
h
t
1
,
...
t
T
,
k
P
;
q
−
1
(d)
−
r(N
P
)k
P
+
wm
P
.
(B.10)
Changes in
α
/T)
t
=
1
∂h/∂t
t
(
·
)t
t
<
V
α
=
(
/T)
α
ln(
1.
0
2.
V
αα
=
(
1
/T)
α
[
ln(
1
/T)
]
2
t
=
1
∂h/∂t
t
(
·
)t
t
>
0
3.
1
1
V
α
=
(N/T )
α
ln(N/T )
j
=
1
∂h/∂t
j
(
·
)t
t
So when
α
is close to zero the first term approaches one, and the derivative is smaller in absolute terms than
V
α
.
As
α
increases in size, however, the first term also decreases, but more slowly than for
V
α
. Hence, the whole
derivative can be larger than
V
α
in absolute terms.
Changes in T (assuming that
α
=
1
)
/T)
t
=
1
∂h/∂t
t
(
·
)t
t
<
V
T
=
(
−
1.
1
0
)
t
=
1
∂h/∂t
t
(
·
)t
t
>
0
2.
V
TT
=
(
1
/T
2
