Overview of Optimization in
Ag Economics
Lecture 2
Interregional and Spatial
Economics
Leibnitz’s Rule
V r  
V r  
B r 
 f  x, r  dx
A r 
V  r 
r
 f  B r , r 
B  r 
r
 f  Ar , r 
A  r 
r
B r 


A r 
f  x, r 
r
dx
In equilibrium
x
max
x
x
 p  z  dz   p  z  dz
d
0
s
0
pd(z) is the consumer’s inverse demand
curve and ps(z) is the producer’s supply
curve.
Applying Leibnitz’s Rule
p
d
 z  p  z  0
s
x1
x2
xT
xT , x1 , x2
 p  z  dz   p  z  dz   p  z  dz  tx
s.t.
x1  x2  xT
max
d
1
0
d
2
0
s
1
0
S
d
s

 p1  x1   p  x1  x2   0
x1
S
 p2d  x2   p s  x1  x2   t  0
x2
Econometrics and Statistical
Applications
Historically, econometrics relied on
closed form solutions made possible
by linear models of normally
distributed random variables.
More complicated models that
introduce factors such as concavity
constraints and nonnormality do not
imply closed form solutions.
Concavity constrained cost functions:
Basic cost function formulation:
min wx

x
  C  w, y 
s.t. F  x, y   0 
C  w, y    0  w  1 ww  y  1 yy  wy
2
2
In this formulation, both the A and B
matrices are symmetric by Young’s
theorem. In addition, if the
optimization conditions are met (as
we will describe in this course) the
cost function is concave in input
prices, implying that the A matrix is
negative semi-definite.
Fitting the cost function:
We estimated this cost function using
concentrated maximum likelihood.
Specifically, the residual vector based on
any set of parameter estimates is




ci   0  wi  1 wiwi  yi  1 yiyi  wiyi
2
2


 x1i   1  11w1i  12 w2i  13 w3i  11 y1i   21 y2i   31 y31  
ei     

 x2i    2   21w1i   22 w2i   23 w3i  12 y1i   22 y2i   32 y31  
 x     w   w   w   y   y   y  
3
31 1i
32 2 i
33 3i
13 1i
23 2 i
33 31 
 3i
These parameters are estimated by
maximizing
T
max  ln    

2
1 T
s.t.       ei    ei   
T i 1
While this formulation is complex, it can be
estimated using iterative generalized least
squares without resorting to complex
mathematical programming algorithms.
While this formulation is complex, it can
be estimated using iterative generalized
least squares without resorting to
complex mathematical programming
algorithms.
However, one approach to estimating a
concavity constrained const function is
to estimated the Cholesky decomposition
of the A matrix. Specifically,
0  11 12 13 
 11 0
  12  22 0   0  22  23 
 13  23  33   0
0  33 
1112
1113
 1111


 1112 1212   22 22
1213   22 23

 1113 1213   22 23 1313   23 23   33 33 
Policy Analysis–Reduction of
Price Floor
Good 1
Good 2
Good 1
Good 2
General Equilibrium
Modeling the interaction between the two
markets involves moving from a partial
equilibrium analysis to a general equilibrium
analysis.
Early work on general equilibrium analysis
involves the concept of a Walrasian equilibrium.
The primary idea of the Walrasian equilibrium
was the concept that some price vector could
be found for any endowment that equated the
supply and demand, or resulted in zero excess
demand:
x i  p   D  p, W   S  p, w 
pi x i  p   Wi   0
x i  p   Wi  0
xI(p) is the excess demand for good i, it is a
function of the price vector. Demand is
determined by the initial endowment of goods
W.
pi(xi(p)-Wi) is the complementary slackness
conditions. This condition implies that either
the price of the ith good is zero, or its excess
demand is zero.
Nonparametric Efficiency
Analysis
Economic efficiency of farms and
agribusinesses has been analyzed by first
estimating a parametric cost or profit
function as developed in Featherstone and
Moss.
However, the efficiency results in these
studies are conditioned on the choice of
structural form used to estimate the
parametric production function as well as
the distributional assumptions used in
estimation.
An alternative approach is to allow
the most efficient firms to form an
efficient technological envelope.
Assuming that a firm produces a
vector of m output from n inputs, a
vector of outputs, , could be produced
using some combination of outputs
from firms in the sample.
Data Envelope Analysis
n
min c   ci z i
*
z
n
st
y
i 1
i 1
z  y j  1, m
*
j
ij i
n
z
i 1
i
1