1. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 1 1

2. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 2 Optimization Techniques Methods for maximizing or minimizing an objective function Examples –Consumers maximize utility by purchasing an optimal combination of goods –Firms maximize profit by producing and selling an optimal quantity of goods –Firms minimize their cost of production by using an optimal combination of inputs

3. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 3 Expressing Economic Relationships Equations: TR = 100Q - 10Q 2 Tables: Graphs:

4. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 4 Total, Average, and Marginal Revenue TR = PQ AR = TR/Q MR =  TR/  Q

5. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 5

6. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 6

7. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 7 Total Revenue Average and Marginal Revenue

8. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 8 Total, Average, and Marginal Cost AC = TC/Q MC =  TC/  Q

9. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 9

10. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 10

11. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 11

12. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 12 Geometric Relationships The slope of a tangent to a total curve at a point is equal to the marginal value at that point The slope of a ray from the origin to a point on a total curve is equal to the average value at that point

13. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 13 Geometric Relationships A marginal value is positive, zero, and negative, respectively, when a total curve slopes upward, is horizontal, and slopes downward A marginal value is above, equal to, and below an average value, respectively, when the slope of the average curve is positive, zero, and negative

14. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 14 Profit Maximization

15. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 15

16. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 16

17. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 17 Steps in Optimization Define an objective mathematically as a function of one or more choice variables Define one or more constraints on the values of the objective function and/or the choice variables Determine the values of the choice variables that maximize or minimize the objective function while satisfying all of the constraints

18. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 18

19. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 19 New Management Tools Benchmarking Total Quality Management Reengineering The Learning Organization

20. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 20 Other Management Tools Broadbanding Direct Business Model Networking Performance Management

21. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 21 Other Management Tools Pricing Power Small-World Model Strategic Development Virtual Integration Virtual Management

22. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 22 Chapter 2 Appendix

23. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 23 Concept of the Derivative The derivative of Y with respect to X is equal to the limit of the ratio  Y/  X as  X approaches zero.

24. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 24

25. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 25

26. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 26 Rules of Differentiation Constant Function Rule: The derivative of a constant, Y = f(X) = a, is zero for all values of a (the constant).

27. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 27 Rules of Differentiation Power Function Rule: The derivative of a power function, where a and b are constants, is defined as follows.

28. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 28 Rules of Differentiation Sum-and-Differences Rule: The derivative of the sum or difference of two functions, U and V, is defined as follows.

29. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 29 Rules of Differentiation Product Rule: The derivative of the product of two functions, U and V, is defined as follows.

30. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 30 Rules of Differentiation Quotient Rule: The derivative of the ratio of two functions, U and V, is defined as follows.

31. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 31 Rules of Differentiation Chain Rule: The derivative of a function that is a function of X is defined as follows.

32. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 32

33. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 33 Optimization with Calculus Find X such that dY/dX = 0 Second derivative rules: If d 2 Y/dX 2 > 0, then X is a minimum. If d 2 Y/dX 2 < 0, then X is a maximum.

34. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 34 Univariate Optimization Given objective function Y = f(X) Find X such that dY/dX = 0 Second derivative rules: If d 2 Y/dX 2 > 0, then X is a minimum. If d 2 Y/dX 2 < 0, then X is a maximum.

35. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 35 Example 1 Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue: TR = 100Q – 10Q 2 dTR/dQ = 100 – 20Q = 0 Q* = 5 and d 2 TR/dQ 2 = -20 < 0

36. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 36 Example 2 Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue: TR = 45Q – 0.5Q 2 dTR/dQ = 45 – Q = 0 Q* = 45 and d 2 TR/dQ 2 = -1 < 0

37. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 37 Example 3 Given the following marginal cost function (MC), determine the quantity of output that will minimize MC: MC = 3Q 2 – 16Q + 57 dMC/dQ = 6Q - 16 = 0 Q* = 2.67 and d 2 MC/dQ 2 = 6 > 0

38. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 38 Example 4 Given –TR = 45Q – 0.5Q 2 –TC = Q 3 – 8Q 2 + 57Q + 2 Determine Q that maximizes profit ( π): –π = 45Q – 0.5Q 2 – (Q 3 – 8Q 2 + 57Q + 2)

39. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 39 Example 4: Solution Method 1 –d π /dQ = 45 – Q - 3Q 2 + 16Q – 57 = 0 –-12 + 15Q - 3Q 2 = 0 Method 2 –MR = dTR/dQ = 45 – Q –MC = dTC/dQ = 3Q 2 - 16Q + 57 –Set MR = MC: 45 – Q = 3Q 2 - 16Q + 57 Use quadratic formula: Q* = 4

40. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 40 Quadratic Formula Write the equation in the following form: aX 2 + bX + c = 0 The solutions have the following form:

41. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 41 Multivariate Optimization Objective function Y = f(X 1, X 2,...,X k ) Find all X i such that ∂ Y/ ∂ X i = 0 Partial derivative: –∂ Y/ ∂ X i = dY/dX i while all X j (where j ≠ i) are held constant

42. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 42 Example 5 Determine the values of X and Y that maximize the following profit function: –π = 80X – 2X 2 – XY – 3Y 2 + 100Y Solution –∂π / ∂ X = 80 – 4X – Y = 0 –∂π / ∂ Y = -X – 6Y + 100 = 0 –Solve simultaneously –X = 16.52 and Y = 13.92

43. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 43

44. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 44 Constrained Optimization Substitution Method –Substitute constraints into the objective function and then maximize the objective function Lagrangian Method –Form the Lagrangian function by adding the Lagrangian variables and constraints to the objective function and then maximize the Lagrangian function

45. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 45 Example 6 Use the substitution method to maximize the following profit function: –π = 80X – 2X 2 – XY – 3Y 2 + 100Y Subject to the following constraint: –X + Y = 12

46. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 46 Example 6: Solution Substitute X = 12 – Y into profit: –π = 80(12 – Y) – 2(12 – Y) 2 – (12 – Y)Y – 3Y 2 + 100Y –π = – 4Y 2 + 56Y + 672 Solve as univariate function: –d π /dY = – 8Y + 56 = 0 –Y = 7 and X = 5

47. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 47 Example 7 Use the Lagrangian method to maximize the following profit function: –π = 80X – 2X 2 – XY – 3Y 2 + 100Y Subject to the following constraint: –X + Y = 12

48. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 48 Example 7: Solution Form the Lagrangian function –L = 80X – 2X 2 – XY – 3Y 2 + 100Y + (X + Y – 12) Find the partial derivatives and solve simultaneously –dL/dX = 80 – 4X –Y + = 0 –dL/dY = – X – 6Y + 100 + = 0 –dL/d = X + Y – 12 = 0 Solution: X = 5, Y = 7, and = -53

49. C opyright  2007 by Oxford University Press, Inc. PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.Slide 49 Interpretation of the Lagrangian Multiplier, Lambda,, is the derivative of the optimal value of the objective function with respect to the constraint –In Example 7, = -53, so a one-unit increase in the value of the constraint (from -12 to -11) will cause profit to decrease by approximately 53 units –Actual decrease is 66.5 units