1. Applied Economics for Business Management
Lecture outline:
Introduction to course
Math review
Introduction to consumer behavior

2. 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.

3. 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.

4. 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.

5. Math Review Example of (ii):
(consumer equilibrium for utility maximization)
(producer equilibrium
for profit maximization)

6. Math Review There are two general types of optimization problems:
- unconstrained optimization
- constrained optimization

7. Math Review Unconstrained optimization (i) Simplest case:
single argument functions
with one explanatory variable
(ii) General case:
multiple argument functions
with n explanatory variables

8. Math Review
Suppose you had these two single argument functions:

9. 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

10. 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)

11. Math Review First Derivative Test:
What is the function doing around the critical value?
relative max
relative min

12. 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

13. Math Review For the previous example: Critical value is a relative max
Critical value is a relative min

14. Math Review Procedure for optimizing a single argument function
(1) Given , find
(2) Set and solve for critical value(s)
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.

15. Math Review
Example:
Let
(critical value)

16. Math Review
Using the Second Derivative Test:
is a relative min

17. Math Review
Another example:
(critical values)

18. Math Review Second Derivative Test: is a relative max
is a relative min

19. Unconstrained optimization of multiple argument functions:
Let’s take an example:
Take partial derivatives and set equal to zero:

20. Unconstrained optimization of multiple argument functions:
Solving these two equations simultaneously:

21. Unconstrained optimization of multiple argument functions:
Distributing -10
Combining like terms
Subtracting -28 to both sides
Dividing both sides by -28

22. 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.

23. 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.

24. Math Review
Given
First order partial derivatives:

25. Math Review
Second order direct partial derivatives:

26. Math Review Second order cross partial derivatives:
If and are continuous functions, then by Young’s
Theorem
See Silberberg (pages 68 – 70) for proof.

27. Unconstrained optimization of multiple argument functions:
Returning to the example:
Recall the critical values were

28. Unconstrained optimization of multiple argument functions:
Also recall the following derivatives:

29. Unconstrained optimization of multiple argument functions:
Second order direct partials:

30. Unconstrained optimization of multiple argument functions:
Second order cross partials:
Shows that symmetry
condition holds

31. 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.
WHY?

32. 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.

33. 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

34. Unconstrained optimization of multiple argument functions:
So the cookbook procedure for optimizing multiple argument functions are:
(i) Take first order partial derivatives
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

35. Determinants Square matrix: Review of determinants:
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)

36. Determinants
Determinants are defined as follows:
(1 x 1) matrix

37. Determinants

38. Determinants 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

39. 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:

40. Determinants
Cofactor of is written in
terms of its assigned minor.

41. Determinants 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

42. Determinants La Place Transformation by any row or any column.
1st row
2nd row
3rd row

43. Determinants
By any column:
1st column
2nd column
3rd column

44. Determinants
Find |A|.
You can use the La Place Transformation by expanding on
any row or column.

45. Determinants
First, expand by 1st row:
Find cofactors:

46. Determinants
Finding cofactors, continued:

47. Determinants

48. Determinants
Or criss-cross method (Chiang)

49. Determinants
For your own practice, expand by the 3rd row:

50. Determinants

51. Determinants

52. 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:

53. 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

54. 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

55. 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

56. 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)

57. Second Order or Sufficient Condition
The second order or sufficient condition for a relative min is:

58. 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?

59. Second Order or Sufficient Condition
Use second order or sufficient condition:
represents a relative max