Course Overview and Overview
of Optimization in Ag Economics
Lecture 1
Course Outline

Static Optimization
 Overview of Optimality
 Review of Linear Algebra
 Optimality Conditions
 Algorithms
 Optimization on a Computer

Dynamic Optimization
 A Review of Dynamic Mathematics
 Calculus of Variations
 Optimal Control
 Applications of Optimal Control
Overview of Optimization

The Basic Microeconomic Problem
 Definition of Economics
 The Consumer’s Problem
Max U ( x)
x
s.t. px  Y

The Producer’s Problem
Max py  wx
s.t. y  F ( x)
Food and Diet Problem

Agricultural applications of the food and diet
problems include both human and animal diets.
 The food and diet research can be characterized
by two major focuses:
 Least cost combination of foods to meet
dietary needs. Stigler’s “Cost of
subsistence”.
 Least cost feed ration studies.

The basic application would involve
minimizing the cost of a diet subject to
some nutritional constraint:
min cx
x
s.t. Ax  b

c is a vector of prices for each food,
x is a vector of choice levels for each
food,
 A is a matrix of nutrients provided by
each food, and
 b is a vector of minimum nutritional
requirements.


More advanced formulations of the diet
problem have been developed in the guise
of the household production model.
 General form household production
problem:
max U  y 
y,x
s.t. y  F  x 
px  I
where F(x) denotes the production
relationship between purchased
foodstuffs and consumable goods (y).
 p is the price vector for purchased
foodstuffs, and
 I is income


A linear formulation of such a model can be
expressed as
max U  y 
y,x
s.t. y  Ax
px  I
 In addition to foodstuffs, x can be
augmented to include labor use.
Farm and Agribusiness
Management

Initially, linear programming was used to
find optimal crop mix. This work has
grown into large extension farm planning
efforts such as OK farms. These models
tend to be either general linear or integer
max cx
x
s.t. Ax  b
x could be a vector of possible crop
alternatives (wheat, cotton, and oats),
 c was a conformable vector of net returns from
each crop activity,
 A is a matrix of resource constraints

1
1
Land
1
A   .2 .3 .1   Labor
 25 100 10  Capital

b is the vector of resource constraints.

One way that risk may enter the farm
management model is by complicating the
objective function:
max E U  x  
x
s.t. Ax  b
where E[.] is the expected value operator,
U(.) is the utility function,
 A is the resource coefficients,
 b is the vector of resource constraints,
and
 x is the level of each activity.


Freund shows that given that preferences
are negative exponential and returns are
normally distributed, the expected utility
function becomes:
U  x    exp  x  

  E U  x    x  xx
x N  ,   
2

Therefore, the maximization problem
becomes a nonlinear optimization problem

max x  xx
x
2
s.t.
Ax  b

However, given that few closed form
conjugates exist, technologies have evolved
to allow direct optimization of more
generalized problems:

y
U  y 

y N  x, xx 
Farm Firm Development

The typical farm firm development model is
primarily interest in firm growth.
max  c1 c2
 A11
T
 12
s.t.  0


 0
c3
0
0
A22
0
T23
A33
0
0
 x1 
x 
 2
cn   x3 
 
 
 xn 
0   x1   b1 
0   x2  b2 
0   x3    b3 
   
   
Ann   xn  bn 

Focusing on the first two constraints
A11 x1  b1
T12 x1  A22 x2  b2  A22 x2  b2  T12 x1

So decisions made in year 1 could also
affect the resources available in year 2.

Again generalizing the model, a decision in
year 1 may have multiple possible
outcomes:
x 
11
max  c11
 A11
T
s.t.  12
T22

T32
p1c12
0
A12
0
0
0
0
A22
0
p2 c22
x 
p3c32   12 
 x22 
 
 x32 
0   x11   b1 
0   x12  b2 

0   x22   b3 
   
A32   x32  b4 
p1 denotes the probability of event 1
occurring, p2 is the probability of event 2
occurring, and p3 is the probability of
event 3 occurring.
 If event 1 occurs, the profit vector c12 and
T12 resources transfer to period 2.

A12 x12  b12  T12 x11
A22 x22  b22  T22 x11
A32 x32  b32  T32 x11
Production Response

Production response models have been used to
study the impact of some policy or external shock
to the sector.
 From the firm level, the effect of changing
fertilizer prices, labor availability, or support
prices on firm outputs, profits, and input
demands can be mapped out, much like the
duality approach to production.
The firm level effects are then aggregated
to the sector level.
 An example of this type of model is
CARD.
