Applied Economics for Business Management

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 overviewofoptimization.

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: • Number of rows and columns are equal • Review of determinants: • Associated with any square matrixA, 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. By any row: 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