Agriculture Reference
In-Depth Information
where
is the constant coefficient of absolute risk aversion. Since the problem is symmetric
across periods, and since by assumption
R
Var(θ
1
)
=
Var(θ
2
)
, the optimal solution requires
s
1
=
s
2
, and
e
1
=
e
2
. This means that the contract has identical incentives (
s
) and work effort
e
(
) in each period.
Case II: No Commitment over Time.
Now consider the case in which two one-period
contracts are made sequentially, and the landowner learns about the relative contribution of
nature by observing the output in the first period. In this case there is no commitment to
the terms of the contract over the two periods. Following Milgrom and Roberts (1992), we
assume there is a positive correlation between the values of the random inputs (
θ
1
and
θ
2
)
in the two periods; that is, a high value of
θ
2
is likely. This means
that the landowner can use observed performance in the first period to get an estimate
θ
1
means a high value of
θ
2
of
θ
2
). In turn, this estimate (
θ
2
) can be used to obtain
a better estimate of the farmer's actual effort in the second period, (
the random input in the second period, (
e
2
).
13
θ
2
=
γ(e
1
+
θ
1
)
is an adaptive
expectation of the landowner, used to estimate nature's contribution to farm output in the
second period.
14
The output that the landowner uses to pay the farmer is now adjusted by
the estimate of nature from the first period, and becomes
Q
2
=
Q
2
−
θ
2
=
e
2
+
θ
2
−
θ
2
. The
landowner now uses this information to adjust the contract so that the farmer's compensation
(gross income) over the two periods becomes
15
Let the landowner's estimate of
θ
2
be given by
, where
γ
Q
2
].
Y
=
[
β
1
+
s
1
Q
1
]
+
[
β
2
+
s
2
(7.3)
Q
2
=
e
2
+
θ
2
−
θ
2
, and
θ
2
=
γ(e
1
+
θ
1
)
After substituting—
Q
1
=
e
1
+
θ
1
,
—and collecting
terms, the compensation function can be rewritten as
Y
=
β
1
+
(s
1
−
γs
2
)(e
1
+
θ
1
)
+
β
2
+
s
2
(e
2
+
θ
2
)
[
]
[
].
(7.4)
It is clear that the effective share coefficient on first-period output (
Q
1
) is not the nominal
contract amount (
s
1
) paid under the case of commitment, but a smaller net amount
(s
1
−
γs
2
)
s
1
,but
greater effort in the first period leads to a reduction in compensation for second-period
effort by
. In terms of first-period compensation, the direct return to additional effort is
γs
2
. This means the share in the first period
(s
1
−
γs
2
)
is lower than the share in
s
2
). This model does not imply the contract would have a complicated
formula whereby the share in the first period would be, for instance, 60 percent minus some
fraction of the share in the second period. We would simply observe two sequential share
contracts that might give the first farmer 50 percent of the crop in the first year, and then
observe a second farmer getting 60 percent of the crop in the second year.
This upward adjustment is the
ratchet effect.
The incentives for farmer effort are
increased—or become more high-powered—from period 1 to period 2. In other words,
the second period (
