Cost Functions and the Estimation of Flexible Functional Forms

Lecture XVIII

Flexible Functional Forms

The crux of the dual approach is then to estimate a manifestation of behavior that economist know something about. Thus, instead of estimating production functions

that are purely physical forms that economist have little expertise in developing, we could estimate the cost function that represents cost minimizing behavior.

We then would be able to determine whether the properties of these cost functions are consistent with our hypotheses about technology.

However, it is often the direct implications of the cost minimizing behavior that we are interested in: How will farmers react to changes in agricultural

prices through commodity programs?

;C y wp p

y

What is the impact of a change in input prices (say in an increase in fuel prices) on agricultural output?

Thus, the dual cost function results are usually sufficient for most question facing agricultural economists.

;C y w wp

y

Given that we are interested in estimating the cost function directly, the next question involves how to specify the cost function? One approach to the estimation of cost functions

would then be to hypothesize a primal production function and derive the theoretically consistent specification for the cost function based on this primal.

However, this approach would appear too restrictive.

Thus, economists have typically turned to flexible functional forms that allow for a wide variety of technologies.

A basic approach to the specification of a cost function is to assume that an unspecified function exists, and then derive a closed form approximation of the function. One typical approach from optimization theory

involves the Taylor series expansion:

General Form of Univariate Taylor Expansion

0 0

0

22

0 0 02

01

1

2

1

!

x x x x

ii

ii x x

f x f xf x f x x x x x

x x

f xx x

i x

0 0 0

2 332 *

0 0 02 3

*0

1 1

2 6

for some ,

x x x x x x

f x f x f xf x f x x x x x x x

x x x

x x x

The real problem is that we don’t know the value of x*. As a result, we approximate this term with a residual:

Given this approach, we can conjecture the relative size of the approximation error based on the relative size of the third derivative of the cost function.

0 0

22

0 0 0 02

1

2x x x x

f x f xf x f x x x x x x x

x x

Extending this result to vector space, we have:

0 0 0 0 0 0 01

2x xxf x f x f x x x x x f x x x x x

Given that the cost function is a function of input prices w and output levels y, we could then stack the two into a single vector and derive the flexible functional form:

This form is typically referred to as the quadratic cost function. It is a second-order Taylor series expansion to an unknown cost function.

01 1, 2 2C w y w w Aw y y By w y

Following the general concept (and ignoring for the moment the error of approximation), Shephard’s lemma can be applied to this cost specification:

*,,i i i i

i

C w yx w y A w y

w

Flexible Cost System

0

1 1 11 1 12 2 13 3 14 4 11 1 21 2

2 2 12 1 22 2 23 3 24 4 21 1 22 2

3 3 13 1 23 3 33 3 34 4 31 1 32 2

1 1, 2 2C w y w w Aw y y By w y

x A w A w A w A w y y

x A w A w A w A w y y

x A w A w A w A w y y

Why have I imposed symmetry? Why am I only estimating three demand curves?

One generalization of the Taylor series approach involves a transformation of variables. Specifically, if we assume:

xg x

The cost function can be expressed as:

This formulation complicates the Shephard’s lemma results slightly:

0

1, 2

12

g C w y g w g w Ag w g y

g y B g y g w g y

1

1

1 1

*

,,

,

,, , ,

C w yC w y

C w yw w

w

ww

C w yC w y C w y C w y

xw w w w

Note that letting go to zero implies

0

1

*

0

lim ln

,lim

,i i

i

xx

C w y w xx s

w C w y

0 Translog

1 Quadratic

.5 Leontief

Fourier Expansion

In a univariate sense, any function can be approximated by a series of sine and cosine curves:

where the I are different periodicities.

01

2 2sin cos

N

i ii i i

x xf x

Extending this representation to a multivariate formulation:

where i is a constant for periodicity and k

is referred to as an Elementary Multi-Index:

0,

sin cosi i i ii

f x k x k x

*

1

:N

ii

k k k K

1

2

3

4

1

2

w

w

wz

w

y

y

1 2 6

1 0 0

0 1 0

0 0 0; ;

0 0 0

0 0 0

0 0 1

k k k

K=1

K=2

7 8 9 12 13 14

2 1 1 1 0 0

0 1 0 0 2 1

0 0 1 0 0 1; ; ; ; ;

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 1 0 0

k k k k k k

This representation minimizes the Sobolev Norm, which says that it does a better job approximating the derivatives of the function. In fact, it represents up to the kth derivative of the function.

Note that if the cost function is specified as a multivariate Fourier expansion, the system of demand equations can be defined by Shephard’s lemma.

Estimation of Cost Systems

Regardless of the function form, cost functions are typically estimated as systems of equations using Seemingly Unrelated Regression, Iterated Seemingly Unrelated Regression, or Maximum Likelihood.

I prefer the use of Maximum Likelihood based on a concentrated likelihood function.

1,

1,

1 *

1

*

1

1

,

,

,

1ln ln

2 2 2

1:

i i i

i i ii

n i i i

T

i ii

N

i ii

i i

C C w y

x x w y

x x w y

T TL L

T

L