Applied Economics for
Business Management
Lecture outline:
Introduction to course
Math review
Introduction to consumer behavior
Introduction

Applied economics for business management involves
investigating both consumer and producer behavior.

The course description states that the theory of the
consumer, firm and market is developed.

Often in agricultural economics programs, applied
economics is covered in two courses – a course in
applied production analysis and a course in applied
price analysis. We will try to cover both topics in one
course.
Introduction

The first half of the course will concentrate on consumer
behavior and the second half of the course will be on
producer behavior.

But before we start on consumer behavior, we will do a
quick math review.

This will not be a review of calculus per se, but will
provide an overview of optimization.
Math Review

The primary uses of mathematics in the study of
production and price analyses are two fold:
(i) to find extreme values of functions
e.g., maximum values of certain
functions (e.g., utility, profit, etc.)
and minimum values of certain
functions (e.g., costs, expenditures, etc.).
(ii) to study under which conditions
economic optima (maxima and
minima) hold.
Math Review
Example of (ii):


MU zi
pi

MU z j p j
MPPxi
ri

MPPx j r j
(consumer equilibrium
for utility maximization)
(producer equilibrium
for profit maximization)
Math Review
There are two general types of optimization
problems:
- unconstrained optimization
- constrained optimization
Math Review

Unconstrained optimization
(i) Simplest case:
single argument functions
with one explanatory variable
(ii) General case:
multiple argument functions
with n explanatory variables
Math Review

Suppose you had these two single
argument functions:
Math Review

What do we observe from these 2 graphs?
(i) the peaks and troughs occur where
the slope of the function is zero
(where critical points occur)
(ii) the slopes are positive and negative
to the left and right of the critical
points
Math Review

By using derivatives, we can solve for critical values and
determine if these critical values are relative maxima or
relative minima.
(critical value)
(critical value)
Math Review

First Derivative Test:
What is the function doing around the critical value?
relative max
relative min
Math Review

Second Derivative Test:
Another test to determine whether critical values are
relative maximum(s)/minimum(s)
(i) Relative maximum:
second derivative is negative or concave
down
(ii) Relative minimum:
second derivative is positive or concave up
Math Review

For the previous example:
Critical value
is a relative max
Critical value
is a relative min
Math Review

Procedure
for optimizing a single argument function
(1)
Given
, find
(2)
Set
(3)
Either use the First Derivative Test or
Second Derivative Test to verify whether
the critical value(s) are relative max or
relative min or neither.
and solve for critical value(s)
Math Review

Example:
Let
(critical value)
Math Review
Using the Second Derivative Test:
is a relative min
Math Review

Another example:
(critical values)
Math Review
Second Derivative Test:
is a relative max
is a relative min
Unconstrained optimization of multiple
argument functions:

Let’s take an example:
Take partial derivatives and set equal to zero:
Unconstrained optimization of multiple
argument functions:

Solving these two equations simultaneously:
Unconstrained optimization of multiple
argument functions:
Distributing -10
Combining like terms
Subtracting -28 to both sides
Dividing both sides by -28
Unconstrained optimization of multiple
argument functions:
So
and
or
are the critical
values.
However, we don’t know if these critical values
represent a relative max or relative min or neither.
Math Review

Before we investigate the second order or
sufficient condition for relative extrema,
we should briefly discuss the concept of
higher order partial derivatives and their
notation.
Math Review
Given
First order partial derivatives:
Math Review
Second order direct partial derivatives:
Math Review
Second order cross partial derivatives:
If and are continuous functions, then by Young’s
Theorem
.
See Silberberg (pages 68 – 70) for proof.
Unconstrained optimization of multiple
argument functions:

Returning to the example:
Recall the critical values were
.
Unconstrained optimization of multiple
argument functions:
Also recall the following derivatives:
Unconstrained optimization of multiple
argument functions:
Second order direct partials:
Unconstrained optimization of multiple
argument functions:
Second order cross partials:
Shows that symmetry
condition holds
Unconstrained optimization of multiple
argument functions:

Using the criteria for optimization with single argument
functions, we are tempted to conclude that if
and
critical values represent a relative max

Unfortunately, the second order conditions for multiple
argument functions is not that simple.

Because the sign of the second order direct partials
only insure an extremum in the
dimension or the
dimension, but not the
dimension.
Saddle Point

If the second order conditions rested
solely on the signs of the second order
direct partials, you could get cases such as
the saddle point.

See the example on saddle point:

The intersection of the ,
, and
shows a minimum in the
space
and a maximum in the
space.
Saddle Point

The point being:
the second order condition for multiple
argument functions is not so simple.

For this case, we have to set-up and then
evaluate a Hessian determinant.
Where
Unconstrained optimization of multiple
argument functions:

So the cookbook procedure for optimizing
multiple argument functions are:
(i) Take first order partial derivatives
(ii) Set first order partial derivatives equal to zero
and solve simultaneously for critical values
(iii) Take second order direct and cross
partial derivatives
(iv) Evaluate the Hessian determinant
Determinants

Square matrix:


Number of rows and columns are equal
Review of determinants:

Associated with any square matrix A, there is a scalar
quantity called the determinant of A and written:
or |A|

If A is n x n, then |A| is said to be of order n.
(So n is the dimension of the square matrix)
Determinants
Determinants are defined as follows:
(1 x 1) matrix
Determinants
Determinants
(iii) for n>2, the determinant of an n x n matrix
may be defined in terms of determinants of
(n - 1) x (n - 1) submatrices as follows:
(a) the minor
of an element
of
A is the determinant of the remaining
matrix by deleting the i th row and j th
column
Determinants
The minor of
(formed by deleting the 1st row and
2nd column) or the element 2 in the 1st row and
2nd column is:
Determinants
(b) Cofactor
of
is written in
terms of its assigned minor.
Determinants
(c) The determinant of an n x n matrix is
defined as the sum of the product of the
elements of any row or column of A and
their cofactors
Determinants
La Place Transformation by any row or any column.
By any row:
1st row
2nd row
3rd row
Determinants
By any column:
1st column
2nd column
3rd column
Determinants
Find |A|.
You can use the La Place Transformation by expanding on
any row or column.
Determinants
First, expand by 1st row:
Find cofactors:
Determinants
Finding cofactors, continued:
Determinants
Determinants
Or criss-cross method (Chiang)
Determinants
For your own practice, expand by the 3rd row:
Determinants
Determinants
Second Order or Sufficient Condition

Rules for second order or sufficient
condition for multiple argument
functions:


Let
Form Hessian determinant consisting of second order
direct and cross partials:
Second Order or Sufficient Condition

The first principal minor is defined by
deleting all rows and columns except the
first row and first column.

So,
First principal minor
Second Order or Sufficient Condition

The second principal minor is formed by
the first and second rows and columns
and deleting all other rows and columns

So,
Second principal minor
Second Order or Sufficient Condition

The third principal minor:

You can use the La Place Transformation procedure
or the criss-cross method shown by Chiang to solve
for
Second Order or Sufficient Condition

Fourth, fifth, etc. principal minors follow
this same pattern.

The second order or sufficient condition
for a relative max is:
(alternating signs)
Second Order or Sufficient Condition

The second order or sufficient condition
for a relative min is:
Second Order or Sufficient Condition

Recall the example:
Earlier we found the critical value to be
Is this critical value a relative max or min?
Second Order or Sufficient Condition
Use second order or sufficient condition:
represents a relative max