Limitations, Aggregation, and Constraints
Lecture X
I.
Limitations to Flexible Functional Forms
A.
The limitations of Flexible Functional Forms, particularly with respect to the
limitations imposed by the Taylor series expansion varieties can be
demonstrated in several ways.
1.
Chambers demonstrates the limitations of the functional forms based
on limitations in imposing separability.
2.
These arguments are similar to arguments related to imposing
separability on various demand systems (i.e. the AIDS models).
B.
I prefer to demonstrate the limitations to Flexible Function Forms by resorting
to the basic notions behind the Taylor Series expansion on which it is based.
Specifically, as presented last time:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
0
0
0
2
2
0
0
2
0
1
1
2
1
!
x
x
x
x
i
i
i
i
x
x
f
x
f
x
f
x
f
x
x
x
x
x
x
x
f
x
x
x
i
x
=
=
∞
=
=
∂
∂
=
+
−
+
−
+
∂
∂
∂
=
−
∂
∑
L
0
1.
Thus, for an infinite expansion, or for a true
n
th
order polynomial, this
expression is exact.
Otherwise, we know that:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
[
]
0
0
0
2
3
3
2
*
0
0
0
2
3
*
0
1
1
2
6
for some
,
x
x
x
x
x
x
f
x
f
x
f
x
f
x
f
x
x
x
x
x
x
x
x
x
x
x
x x
=
=
=
∂
∂
∂
=
+
−
+
−
+
−
∂
∂
∂
∈
2.
Focusing on the “residual term”
(
)
(
)
(
)
[
]
0
3
3
*
*
0
3
1
for some
,
6
x
x
f
x
x
x
x
x
x x
x
ε
=
∂
=
−
∂
∈
1
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AEB 6184 – Production Economics
Lecture XIX
Professor Charles Moss
Fall 2005
a.
As long as the third derivative of the function is nonzero at the
point of approximation, we know that the Flexible Functional
Form has a “specification” or “approximation” error.
b.
Further, if we bring this concept together with our typical
notions of sampling theory, this approximation error may
confound the estimation of parameters.
C.
Finally, there is a problem with the estimation of a functional form and the
point of approximation. Implicitly, if one estimates the quadratic cost
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 Fall '09
 Staff
 Economics, Howard Staunton, Professor Charles Moss, Economics Professor Charles, Lecture XIX Fall

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