Agriculture Reference
In-Depth Information
Mother Nature: Seasonal and Uncertain Production
To the farmer, a season is a distinct period of the year during which a stage of agriculture
(such as planting and harvesting) is optimally undertaken. For example, for spring wheat
grown on the northern Great Plains, the month-long planting season usually begins in April
and the harvest season is primarily restricted to August. This broad definition of a season,
however, hides some important features of nature that directly influence the incentives
inherent in agricultural production. To uncover these features, we now model seasonality as
a collection of parameters: (1)
C
, the number of times per year the entire production
cycle
can be completed; (2)
S
, the number of
stages
in the process; (3)
T
, the total number of
tasks
in a given stage; and 4)
the
length
of the stages. Crop seasons (stages) are ultimately linked
to biological processes (such as birth, planting, flowering, and mating) that depend on such
variables as day length, temperature, and rainfall that vary across nature's seasons. Annual
crops like wheat and corn have
L
1, while irrigated vegetables in Southern California
that generate several harvests may have
C
=
C
=
5. A continuously harvested or nonseasonal
crop would have
indicates how often a stage and its tasks
are repeated during the year. Note that tree crops may be annual even though the plant is
perennial. Trees for timber represent a case where the crop cycle equals a small fraction.
The first modification of our model is to recognize that farm production is cumulative and
to use a stage production function that depends on natural parameters and specialization.
8
Let
C
=
365. Among other things,
C
be the final consumer product (such as bacon or bread) derived from a cumulative
production process with
Q
discrete stage's of production. The output in each stage is an input
into the next stage's production function, so that
S
. Hence,
the farmer in our model takes the output from a previous stage as an input into the next stage
and makes an optimal effort choice that depends, in part, on what nature did in the prior stage.
At each stage the output depends, not only on the previous stage output, but also on farmer
effort
Q
=
Q
s
=
h(Q
s
−
1
(Q
s
−
2
(...)))
(e)
, a capital input
(k)
, and a random stage-specific natural shock
(θ)
determined by
th
such natural forces as pests and weather. In particular, for the
s
stage, the stage-specific
2
s
random input of nature
θ
s
is distributed with mean 0 and variance
σ
. Consequently, the
production function for a single stage is
Q
s
=
h
s
(e
s
,
k
s
;
Q
s
−
1
)
+
θ
s
, where inputs
e
and
k
have positive and diminishing marginal productivity, and these marginal products are
increasing in
Q
s
−
1
.
Because there are many tasks within a given stage, we define
t
stn
as the effort (in hours) in
th
stage, on the
th
task, performed by the
th
worker. Tasks are indexed by
the
s
t
n
t
=
1,
...
,
T
;
stages are indexed by
s
=
1,
...
,
S
; and workers are indexed by
n
=
1,
...
,
N
. Let
T
be the
number of tasks for a given stage and assume that
is exogenous to the farm, determined
by nature and technology. Tasks are well-defined jobs that take place during a stage, such
as operating a combine or a grain truck during wheat harvest. Tasks may be mostly mental
T
