A State-Dependent
Production Function: An
Economist’s Apology
Charles B. Moss
Food and
Resource
Economics
Department
1
Introduction
In this paper, I am using apology in a classical sense:




2
Apology comes from a Greek word apologia which means to
speak in defense of.
Some years ago, I read a book by G.S. Hardy titled A
Mathematician’s Apology in which the author tried to defend or
explain the way mathematicians view the world.
In this presentation, I want to defend or explain the way that
economists use production functions and to renew the
conversation on the estimation and use of production
functions.
7/17/2016
Food and Resource Economics Department
What is the Role of Economics?
Returning to the work of the Austrian Economist Ludwig
von Mises, economics is a science of human action.



Specifically, economics is the science of human action regarding
the allocation of goods and services.
In this historical approach, the most basic datum is the market
transaction – the quantity of any good purchased at a specific
price.



3
This market price is determined in part by what consumers are willing
to pay for any given quantity of goods and services – the demand.
The other half of the scissors is the quantity that producers are
willing to supply a given quantity of goods and services – the supply.
Production economics is primarily interested in the supply of
output.
7/17/2016
Food and Resource Economics Department
Two Approaches to Production Economics
Primal Approach




Specification dates back to
Wicksteed’s definition of the
production function.
Steps:


Estimate a production function
using input-output data.
Given these estimated
function, firm level supply
function and demand for each
input can be derived.
Dual Approach


Early work in the area dates
back to Hotelling, but its
recent popularity started with
the work of Shephard,
Diewert, and McFadden.
Steps:


4
7/17/2016
Assume that agents are making
optimizing decisions based on
a production technology they
know.
Estimate the optimizing
relationships directly (i.e., the
supply and derived demand
functions).
Food and Resource Economics Department
Typical Primal Estimation
Gather production data



This table comes from the
USDA Chemical Use
Survey.
Data and the economic
question is a significant
opportunity for
collaboration.
Specify the production
function.
Statistical estimation of
the function.


5
7/17/2016
Nit
Phos
Pot
Corn
127.0
60.0
90.0
140.0
202.0
104.0
120.0
110.0
88.0
24.0
90.0
61.0
150.0
69.0
120.0
138.0
200.0
0.0
0.0
150.0
153.3
52.6
126.6
102.0
139.0
35.0
90.0
160.0
150.0
60.0
120.0
115.0
160.0
40.0
50.0
165.0
180.0
37.0
120.0
140.0
160.0
30.0
60.0
135.0
182.7
76.8
127.7
160.0
Food and Resource Economics Department
Specifying the Production Function
Using the Cobb-Douglas Production Function

y  Ax1 x2 x3  ln  y   ln  A   ln  x1    ln  x2    ln  x3 

Estimating the coefficients using ordinary least squares
ln  y    0  1 ln  x1    2 ln  x2    3 ln  x3   
Solve for the economic relationships

max   pY x1 x2  w1 x1  w2 x2

Y
 pY   w1  0
x1
x1

Y
 pY 
 w2  0
x2
x2
6
7/17/2016
Food and Resource Economics Department
Deriving the Implication of the Primal

Economic results

Factor Demands
1 
*
1
x
*
2
x

 pY , w1 , w2   p
1
1  
Y
 pY , w1 , w2   p
1
1  
Y
 
 
 w1 

1  
Output Supply
 
Y *  pY , w1 , w2   pY
7
 w1 
 
 
1  
7/17/2016
1  
 
 
 w1 

 
 
 w2 
 w2 
 
 
1  

1  
 1
1  
 
 
 w2 

1  
Food and Resource Economics Department
Economic/Policy Questions Asked
Both the primal and the dual approach can be used to
answer questions such as:


What is the supply response to a change in input or output
prices?
The dual approach requires the assumption that the
researcher can observe people making optimal decisions.
Hence, it is difficult to address the impact of new
technologies (ex ante).
The approach may also obfuscate the impact of risk and
uncertainty on production.



8
7/17/2016
Food and Resource Economics Department
Production Function
My program in production economics focuses on how
individuals decide to employ factors of production (land,
labor and capital) in an effort to create production which
is offered to the market.


The essence of this question is again one of constraint. If we
envision a N x M space where there are N inputs and M
outputs there must be an constraint (or envelope) which limits
the combination of inputs and outputs which are feasible.


9
It may be possible for the producer to use 250 pounds of fertilizer
per acre to produce one bale of cotton;
However, it is impossible for that producer to choose to produce 5
bales of cotton per acre with the same 250 pounds of fertilizer.
7/17/2016
Food and Resource Economics Department
Graphical Definition of Production Function
Cotton
5
Feasible
1
250
10
7/17/2016
Nitrogen
Food and Resource Economics Department
The Technology Set

In general terms the production technology is mathematically
depicted as
x, y T

Economics theory suggests a set or conditions on this
technology which make the economic question interesting.

Economics requires the technology be defined so that the individual
can optimize some objective function (usually profit).




11
: x  RN , y  RM
The technology should be bounded (so that an infinite amount of output
cannot be produced from a finite bundle of inputs),
Concave (so that a unique optimal exists),
Inputs should be weakly essential (so that a positive quantity of a least one
input is required), and
Continuous.
7/17/2016
Food and Resource Economics Department
The Production Mapping

Given that the production technology meets these
criteria a production map (or production function) can be
defined which depicts the level of outputs resulting from
the application of any fixed set of inputs
f  x, y  : R  R

N


12

M
This formulation is consistent with the objection that
production scientists have levied against simplified economic
applications.
Life is complicated so reducing the input space could negate
the economic implications of the production function.
May 27, 2009
Food and Resource Economics Department

To address some of these shortcomings this analysis
starts with a production function where combinations of
controllable inputs (pounds of nitrogen applied to each
acre) are combined with uncontrollable inputs (such as
rainfall, which I will use as a stochastic variable such as
rainfall) to produce output.

This transformation can be written as
f : X   R
1

13
7/17/2016
Food and Resource Economics Department

Approximating this production function with a secondorder Taylor series expansion:
f  x,    f  x0 , 0    x
  x   x0  


  f  x0 , 0      
   0  

1   x   x0   2
        x
2    0  
  x   x0  


  f  x0 , 0      
   0  
 O 2  x0 , 0 
14
7/17/2016
Food and Resource Economics Department
f  x,    g  x,    h  x,  
1
 x  x0  Axx  x0 , 0  x  x0 
2
h  x,      x0 , 0   0    x  x0  A x  x0 , 0   0  
g  x,     0   x  x0 , 0  x  x0  
1
  0  A  x0 , 0   0   O 2  x0 , 0 
2
1


   x0 , 0    x  x0  A x  x0 , 0     0  A  x0 , 0     0 
2


 O 2  x0 , 0 
15
7/17/2016
Food and Resource Economics Department

As a starting point, we formulate a quadratic production
function where production is a function of two
controllable inputs and one uncontrollable input.
 0.15   x1 
 x1   0.001 0.0005 0.002  x1 

   1  
 
f  x1 , x2 ,    1.5   0.25   x2    x2   0.0005 0.0015 0.001  x2 
 0.10     2     0.002

0.001 0.009 

 
 
  
16
7/17/2016
Food and Resource Economics Department
 0.1163   x1  1  x1   0.001 0.0005   x1 
f  x1 , x2 , 16.83  1.4582  
 x  x  
 x 
0.2332
0.0005

0.0015
2

  2
 2 
 2 
 0.15   x1  1  x1   0.001 0.0005   x1 
f  x1 , x2 , 0.00   1.50  
 x  x  
 x 
0.25
0.0005

0.0015
2

  2
 2 
 2 
 0.1837   x1  1  x1   0.001 0.0005   x1 
f  x1 , x2 , 16.83  1.9083  
 x  x  
 
 0.2668   2  2  2   0.0005 0.0015   x2 
17
7/17/2016
Food and Resource Economics Department

To estimate the state-dependent production function, I
use a quantile regression approach:
P Yi  y   F  y  xi  
where P(Yi < y) denotes the probability of the observed
variable (Yi) less than some target value (y), F(.) is a known
cumulative probability density function, xi are observed
independent variables, and  is a vector of estimated
coefficients.
18
7/17/2016
Food and Resource Economics Department

Koenker and Bassett demonstrate that the regression
relationship at the th quantile can be estimated by solving

mink    yi  xi    1    yi  xi 
 R
ii: yi  xi  
 ii: yi  xi  
19
7/17/2016



Food and Resource Economics Department

To examine the possibility of this specification, I
formulated a stochastic production function consistent
with the general specification above
 0.15   x1 
 x1   0.001 0.0005 0.002  x1 

   1  
 
f  x1 , x2 ,    1.5   0.25   x2    x2   0.0005 0.0015 0.001  x2 
 0.10     2     0.002
  
0.001

0.009

 
 
 
Next, I generate a dataset assuming that the stochastic
factor of production () is distributed normally with
mean of zero and a variance of 400.
20
7/17/2016
Food and Resource Economics Department

In addition, I also considered a negative exponential error
term
z     exp    ,    ,0

Finally, I applied the specification to wheat production on
the Great Plains
21
7/17/2016
Food and Resource Economics Department
Results for Great Plains
22
0.20 Quantile
0.50 Quantile
0.80 Quantile
Intercept
19.713
9.647
32.906
Nitrogen
0.278
0.325
0.484
Phosphorous
-0.544
0.975
-1.217
Nitrogen2
-0.0057
-0.0059
-0.0039
Phosphorous2
-0.0048
-0.0131
-0.0089
Nit*Phos
0.0151
0.0149
0.0066
Missouri
7.727
8.028
9.106
Nebraska
10.241
10.665
11.181
Kansas
13.504
11.486
13.233
7/17/2016
Food and Resource Economics Department
Wheat bushels acre
35
30
25
20
15
Nitrogen pounds acre
10
23
20
30
7/17/2016
40
50
Food and Resource Economics Department