AEB 6182 Lecture VI
Professor Charles Moss
1
Farm Portfolio Problem: Part II
Lecture VI
I.
Hazell, P.B.R. “A Linear Alternative to Quadratic and Semivariance Programming for Farm
Planning Under Uncertainty.”
American Journal of Agricultural Economics
53(1971):5362.
A.
This article is the basis for the application of MOTAD (Minimize Total Absolute
Deviation) in agriculture.
1.
Hazell’s approach is two fold.
He first sets out to develop review expected
value/variance as a good methodology under certain assumptions.
2.
Then he raises two difficulties.
a.
The first difficulty is the availability of code to solve the quadratic
programming problem implied by EV.
b.
The second problem is the estimation problem.
Specifically, the
data required for EV are the mean and the variance matrix.
However, the variance matrix is an artifact of the assumption of
normality.
B.
The crux of the estimation problem is that the covariance terms in the EV
formulation are estimated by:
x x
s
c
g
c
g
j
k
hj
j
hk
k
h
s
k
n
j
n
1
1
1
1
1



=
=
=
∑
∑
∑
(
)(
)
where x
j
is the level of activity j in the portfolio,
c
hj
is the observed return on asset j
at time h, g
j
is the expected return on asset j, s is the number of observations, and n
is the number of assets.
This equality can be reformulated as:
s
2
1
1
2
1
1
1
=


=
=
=
∑
∑
∑
s
c
x
g x
j
j
n
j
j
j
n
h
s
.
Hazell suggests replacing this objective function with the mean absolute deviation
( 29
A
s
c
g
x
j
j
j
n
h
s
=

=
=
∑
∑
1
1
1
.
Thus, instead of minimizing the variance of the farm plan subject to an income
constraint, you can minimize the absolute deviation subject to an income constraint.
Another formulation for this objective function is to let each observation h be
represented by a single row
( 29
( 29
y
c
g
x
y
y
c
g
x
h
j
j
j
n
h
h
j
j
j
n
=


=

=
+

=
∑
∑
1
1
where y
h
is the deviation from average.
This deviation can be divided into positive
deviations from the average, y
h
+
, and negative deviations from the average, y
h

.
The
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 Fall '08
 Moss
 Optimization, Professor Charles Moss, Lecture VI Professor

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