Simultaneity and Other
“Simple” Problems
Lecture VI
Simultaneity and Estimation of
the Production Function

The above discussion (and estimates) makes
the experimental plot design assumption
regarding the data.


Specifically, I essentially assumed that the data
are being generated from some sort of
experimental design so that the errors are truly
random.
If the data are actually the result of farm level
decisions, the data are endogenous.

Hock, Irving. “Simultaneous Equation Bias in
the Context of the Cobb-Douglas Production
Function.” Econometrica 26(4)(Oct 1958):
566-78.

The basic firm-level model is that we have an
empirical model under the assumption of:


A Cobb-Douglas production function, and
Competition.
Q
X 0  K0  X
q 1
aq
q
 X0 
X 0
P0
 P0  aq
 Pq

 Xq 
X q


 X 0 P0
aq 
 X q Pq


Y0
  aq  1
Yq


Klein demonstrates that the best linear unbiased
estimate of aq is
 Yqi 
aˆq    
i 1  Y0 i 
I

1
I
In this approach the “average” firm is defined to
be the optimal firm.

As an alternative
 X0 
P0  a0
 Rq Pq

 Xq 


where Rq is “some constant and the investigator
wishes to test whether it is equal to one.”

“The firm sets the value of the marginal product equal
to the price augmented by the effect of any restrictions
that exist.”

“In this formulation, Rq can of course vary among
firms; but, for a sample of firms, the investigator
would be interested in testing whether the average Rq
is equal to one.”
Two models of simultaneity:

Model 1: Production disturbance not
transmitted to the “independent” variables.


“If the disturbance in the production equation
affects only the output and is not transmitted to
the other variables in the system, then there is no
simultaneous equation bias. Single equation
estimates are consistent.”
For example if inputs are fixed or are
predetermined .

Model 2: Production disturbance transmitted
to the “independent variables.”

“Simultaneous equation bias arises when
disturbances in the production relations affect the
observed values of all variables, and, as a result,
single equation estimates are not consistent.”
Empirical setup
Q
X 0  K0  X U
q 1
aq
q
aq P0
Xq 
X 0Vq
Rq Pq
q  1, 2,
Q
Q
x0  k0   aq xq  u
q 1
xq  kq  x0  vq
x0  ln  X 0 
xk  ln  X q 
 aq P0 

kq  ln 

 Rq Pq 

q  1, 2,
Q
Indirect Least Square Solution

Kmenta, J. “Some Properties of Alternative
Estimates of the Cobb-Douglas Production
Function.” Econometrica 32(1/2)(Jan-Apr
1964): 183–8.

Simplifying the general system of equations
x0i  k0  a1 x1i  a2 x2i  v0i
x1i  k1  x0i  v1i
x2i  k2  x0i  v2i

Transforming the estimation problem to
x0i  b0  b1  x1i  x0i   b2  x2i  x0i   ei
yields estimates of b1 and b2 that are consistent.
Note by the definitions
x1i  k1  x0i  v1i  x1i  x0i  k1  v1i
x2i  k2  x0i  v2i  x2i  x0i  k2  v2i

Working through the mechanics
x0i  b0  b1 x1i  b1 x0i  b2 x2i  b2 x0i  ei
1  b1  b2  x0i  b0  b1 x1i  b2 x2i  ei
b0
b1
b2
1
x0i 

x1i 
x2i 
ei
1  b1  b2 1  b1  b2
1  b1  b2
1  b1  b2
bˆr
ar 
1  bˆ  bˆ
1
2
r  1, 2
Zeros in the Cobb-Douglas Functional
Form

Moss, Charles B. “Estimation of the CobbDouglas with Zero Input Levels:
Bootstrapping and Substitution.” Applied
Economics Letters 7(10)(Oct 2000): 677–9.

Zeros raise several difficulties in estimating the
Cobb-Douglas production function.


On the theoretical side, the presence of a zero-level
input is that it violates weak necessity of inputs.
On the empirical side, how do you take the natural
logarithm of zero.
Two assumptions:

The existence of zeros is the result of
measurement error.


Agronomically, production of a crop is
impossible without some level of each fertilizer.
Thus, production occurs based on some true level
of each nutrient available to the plant
x  xi   i
*
i


In fact we can think of the soil as a sponge that
contains a variety of nutrients that we can
augment by applying fertilizer. The actual level of
fertilizer used by the crop could then be a
function of what we add, the weather (i.e., if
adequate moisture is not present the crop does not
use the full potential), etc.
Second, a production function that does not
admit zero input could represent a
misspecification.

In this paper, I consider two techniques for
adjusting the zero observations.


First, I redraw from the sample averaging the
result until the pseudo sample contains no zeros.
Second, I substitute a small non-zero number for
those observations that contain zeros (i.e., 0.1,
0.01, 0.001).

The goodness of fit for each procedure is then
compared using a Strobel measure of information
 si 
I   si ln  
i 1
 si 
N
Where si is the theoretically appropriate budget
share for each input and si is the budget share
estimated using each empirical approximation of
zero.
Linear Response Plateau
Y
N