Elluminate – 7
• Among the conditions for the production function are positive
marginal product and concavity (positive derivatives and
negative definite or negative semi-definite hessian matrix)
• As a starting point, I generate a sample of 150 observations using a
quadratic production function.
A0
41.0569
A12
(31.9353)
A1
2.8010
(0.1012)
A13
(4.0867)
A2
15.4518
5.8894
A22
-0.6406
(0.1844)
-0.2386
(0.0228)
A23
(1.1206)
A11
0.0524
(0.1115)
(1.515)
A3
0.3610
0.0720
(0.0258)
A33
-0.2949
(0.0277)
 0.4264 


   0.1607 
0.0000948
• The idea is to resample the dataset, keeping the observations
which are concave.
• The alternative is to directly impose the constraint that the
eigenvalues are less than zero.
• The student of economic theory has been taught to write
O  f  L, C 
where L is the quantity of labor, C is a quantity of capital, and O is the rate of
output of commodities.
• He is instructed to assume all workers alike, and to measure L in man-hours of
labour;
• He is told something about the index-number problem involved in choosing a
unit of output; and
• Then is hurried on to the next question, in hope that he will forget to ask in
what units C is measured.
• Before ever he does ask, he has become a professor, and so sloppy
habits of thought are handed on from one generation to another.
• The question is certainly not an easy one to answer.
• Capital in existence at any moment may be treated simply as “part of the
environment in which labor works.” (Keynes – General Theory).
• We then have a production function in terms of labour alone.
• This is the right procedure for the short period within which supply of
concrete capital goods does not alter, but outside the short period it is a
very weak line to take, for it means that we cannot distinguish a change in
the stock of capital (which can be made over the long run by
accumulation) from a change in the weather (an act of God).
• When we know the future expected rate of output associated
with a certain capital good, and expect future prices and costs,
then, if we are given a rate of interest, we can value the capital
good as a discounted stream of future profit which it will earn.
• But to do so, we have to begin by taking the rate of interest as
given, whereas the main purpose of the production function is to
show how wages and the rate of interest (regarded as the
wages of capital) are determined.
• The idea or transition is that the amount of capital stock may
determine individual components about the production function.
• For example, assume that we have three inputs.
• The first two are variable inputs (labor and materials).
• The third is capital.
• Taking a second-order Taylor series expansion
 f
f  x1 , x2 , x3   f  x10 , x20 , x30   
 x1
1
 x1  x10
2
x2  x
0
2
f
x2
 x1  x10 
f  
0
  x2  x2  
x3 
 x3  x30 


 2 f

 x1x1
 2 f
0
x3  x3  
 x2 x1
 2 f

 x3x1
2 f
x1x2
2 f
x2x2
2 f
x3x2
2 f 

x1x3 
 x1  x10 
2
 f 
0
  x2  x2 
x2x3 
 x3  x30 



2
 f

x3x3 
• Next, allow the first two inputs to be variable while holding the
third input fixed
 f
f  x1 , x2 , x3   f  x , x , x   
 x1
0
1
0
2
1
 x1  x10
2
0
3
x2  x
0
2
f
x2
 x1  x10 
f  
0
x

x
 2 2
x3  
 0 


 2 f

 x1x1
 2 f
0  
 x2 x1
 2 f

 x3x1
2 f
x1x2
2 f
x2x2
2 f
x3x2
2 f 

x1x3 
0


x

x
1
1
2

 f 
0
  x2  x2 
x2x3 
 0 



2
 f

x3x3 
• Playing just a little fast and loose with the notation, we collapse
the point of approximation into the constant of the quadratic
function and then allow differences in the level of capital as a
shift variable
 f
f  x1 , x2 , x    0  f  x   
 x1
0
3
1
  x1
2
0
3
 2 f
 x x
1 1
x2   2
  f

 x2 x1
f   x1 

x2   x2 
2 f 
x1x2   x1 
x 
2

 f  2

x2 x2 
• Joan Robinson’s Point – Capital Levels could affect the marginal
physical product of the other inputs
 f
f  x1 , x2 , x3   f  x10 , x20 , x30   
 x1
1
 x1  x10
2
x2  x
0
2
f
x2
 x1  x10 
f  
0
x

x
 2 2
x3  
 x3  x30 


 2 f

 x1x1
 2 f
0
x3  x3  
 x2 x1
 2 f

 x3x1
2 f
x1x2
2 f
x2x2
2 f
x3x2
2 f 

x1x3 
 x1  x10 
2
 f 
0
  x2  x2 
x2x3 
 x3  x30 


2 f 

x3x3 
 f
f  x1 , x2 , x3   f  x , x , x   
 x1
0
1
0
2
0
3
0
f   x1  x1  1
  x1  x10

0
x2   x2  x2  2
 2 f
 x x
1 1
0 
x2  x2 
 2 f

 x2 x1
1

2 f
2 f
0
0
0
 kx3    x1  x1 
x3  x3    x2  x2 
x3  x30  


x1x3
x1x3
2

f
1
2 f
0
0
: k 
x3  x3    x3  x3 
x3  x30 


x3
2
x3x3
• Selection problem by farm typology?
2 f 
x1x2   x1  x10 

0
 2 f   x2  x2 

x2 x2 
• Differences in farm typology

 x1  1
 A11 A12   x1  
f  x1 , x2     z   0  1  2      x1 x2  
 x 
x
A
A
2
 2
 12
22   2  


 A11 A12   x1  
 x1  1
 1    z    0  1  2      x1 x2  
  

 x2  2
 A12 A22   x2  
exp bz 
y    z 
: y  0,1
1  exp bz 
• Rejoinder on Simulation of the firm under panel data (fixing the
capital input).