1. ECON 1150, Spring 2013 Lecture 3: Optimization: One Choice Variable Necessary conditions Sufficient conditions Reference: Jacques, Chapter 4 Sydsaeter and Hammond, Chapter 8.

2. ECON 1150, Spring 2013 1. Optimization Problems Economic problems Consumers: Utility maximization Producers: Profit maximization Government: Welfare maximization

3. ECON 1150, Spring 2013 Maximization problem: max x f(x) f(x): Objective function with a domain D x: Choice variable x*: Solution of the maximization problem A function defined on D has a maximum point at x* if f(x)  f(x*) for all x  D. f(x*) is called the maximum value of the function.

4. ECON 1150, Spring 2013 Minimization problem: min x f(x) f(x): Objective function with a domain D x: Choice variable x*: Solution of the maximization problem A function defined on D has a minimum point at x* if f(x)  f(x*) for all x  D. f(x*) is called the minimum value of the function.

5. ECON 1150, Spring 2013 Example 3.1: Find possible maximum and minimum points for: a. f(x) = 3 – (x – 2) 2 ; b. g(x) =  (x – 5) – 100, x  5.

6. ECON 1150, Spring 2013 2. Necessary Condition for Extrema What are the maximum and minimum points of the following functions? y = 60x – 0.2x 2 y = x 3 – 12x 2 + 36x + 8

7. ECON 1150, Spring 2013 x y 0x* y* x1x1 x2x2 Maximum x 0 x > x* : dy / dx < 0 x = x* : dy / dx = 0 Characteristic of a maximum point

8. ECON 1150, Spring 2013 x y 0x* y* x1x1 x2x2 Minimum x < x* : dy / dx < 0 x > x* : dy / dx > 0 x = x* : dy / dx = 0 Characteristic of a minimum point

9. ECON 1150, Spring 2013 Theorem: (First-order condition for an extremum) Let y = f(x) be a differentiable function. If the function achieves a maximum or a minimum at the point x = x*, then dy / dx | x=x* = f’(x*) = 0 Stationary point : x* Stationary value : y* = f(x*)

10. ECON 1150, Spring 2013 Example 3.2: Find the stationary point of the function y = 60x – 0.2x 2. The first-order condition is a necessary, but not sufficient, condition.

11. ECON 1150, Spring 2013 Example 3.3: Find the stationary values of the function y = f(x) = x 3 – 12x 2 + 36x + 8.

12. ECON 1150, Spring 2013 3. Finding Global Extreme Points Possibilities of the nature of a function f(x) at x = c. f is differentiable at c and c is an interior point. f is differentiable at c and c is a boundary point. f is not differentiable at c.

13. ECON 1150, Spring 2013 3.1 Simple Method a.Find all stationary points of f(x) in (a,b) b.Evaluate f(x) at the end points a and d and at all stationary points c.The largest function value in (b) is the global maximum value in [a,b]. d.The smallest function value in (b) is the global minimum value in [a,b]. Consider a differentiable function f(x) in [a,b].

14. ECON 1150, Spring 2013 3.2 First-Derivative Test for Global Extreme Points x y 0x* y* x1x1 x2x2 x y 0x* y* x1x1 x2x2 Global maximumGlobal minimum

15. ECON 1150, Spring 2013 If f’(x)  0 for x  c and f’(x)  0 for x  c, then x = c is a global maximum point for f. If f’(x)  0 for x  c and f’(x)  0 for x  c, then x = c is a global minimum point for f. First-derivative Test

16. ECON 1150, Spring 2013 Example 3.4: Consider the function y = 60x – 0.2x 2. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values.

17. ECON 1150, Spring 2013 Example 3.5: y = f(x) = e 2x – 5e x + 4. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values. c. Examine lim x  f(x) and lim x  -  f(x).

18. ECON 1150, Spring 2013 3.3 Extreme Points for Concave and Convex Functions Let c be a stationary point for f. a.If f is a concave function, then c is a global maximum point for f. b.If f is a convex function, then c is a global minimum point for f.

19. ECON 1150, Spring 2013 Example 3.6: Show that f(x) = e x–1 – x. is a convex function and find its global minimum point. Example 3.7: The profit function of a firm is  (Q) = -19.068 + 1.1976Q – 0.07Q 1.5. Find the value of Q that maximizes profits.

20. ECON 1150, Spring 2013 4. Identifying Local Extreme Points 0 x y a b c d

21. ECON 1150, Spring 2013 Let a < c < b. a. If f’(x)  0 for a < x < c and f’(x)  0 for c < x < b, then x = c is a local maximum point for f. b. If f’(x)  0 for a < x < c and f’(x)  0 for c < x < b, then x = c is a local minimum point for f. 4.1 First-derivative Test for Local Extreme Points

22. ECON 1150, Spring 2013 Example 3.8: y = f(x) = x 3 – 12x 2 + 36x + 8. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values. c. Examine lim x  f(x) and lim x  -  f(x).

23. ECON 1150, Spring 2013 Example 3.9: Classify the stationary points of the following functions.

24. ECON 1150, Spring 2013 4.2 Second-Derivative Test The nature of a stationary point: Decreasing slope  Local maximum x y 0x* y* x1x1 x2x2 x dy/dx y* 0 x* x1x1 x2x2

25. ECON 1150, Spring 2013 The nature of a stationary point: Increasing slope  Local minimum dy/dx x 0x*x1x1 x2x2 x y 0 y* x1x1 x2x2

26. ECON 1150, Spring 2013 The nature of a stationary point: Point of inflection  Stationary slope

27. ECON 1150, Spring 2013 Second order condition: Let y = f(x) be a differentiable function and f’(c) = 0. f”(c) < 0  Local maximum f”(c) > 0  Local minimum f”(c) = 0  No conclusion

28. ECON 1150, Spring 2013 Example 3.10: Identify the nature of the stationary points of the following functions: a. y = 4x 2 – 5x + 10; b. y = x 3 – 3x 2 + 2; c. y = 0.5x 4 – 3x 3 + 2x 2.

29. ECON 1150, Spring 2013 0 X Y 0 X Y 4.3 Point of Inflection ab

30. ECON 1150, Spring 2013 Let f be a twice differentiable function. a. If c is an inflection point for f, then f”(c) = 0. b. If f”(c) = 0 and f” changes sign around c, then c is an inflection point for f. Test for inflection points:

31. ECON 1150, Spring 2013 Example 3.11: y = 16x – 4x 3 + x 4. dy / dx = 16 – 12x 2 + 4x 3. At x = 2, dy/dx = 0. However, the point at x = 2 is neither a maximum nor a minimum. Point of inflection

32. ECON 1150, Spring 2013 Example 3.12: Find possible inflection points for the following functions, a. f(x) = x 6 – 10x 4. b. f(x) = x 4.

33. ECON 1150, Spring 2013 4.4 From Local to Global a.Find all local maximum points of f(x) in [a,b] b.Evaluate f(x) at the end points a and b and at all local maximum points c.The largest function value in (b) is the global maximum value in [a,b]. Consider a differentiable function f(x) in [a,b].

34. ECON 1150, Spring 2013 5. Curve Sketching Find the domain of the function Find the x- and y- intercepts Locate stationary points and values Classify stationary points Locate other points of inflection, if any Show behavior near points where the function is not defined Show behavior as x tends to positive and negative infinity

35. ECON 1150, Spring 2013 Example 3.13: Sketch the graphs of the following functions by hand, analyzing all important features. a. y = x 3 – 12x; b. y = (x – 3)  x; c. y = (1/x) – (1/x 2 ).

36. ECON 1150, Spring 2013 6. Profit Maximization Total revenue: TR(q)  Marginal revenue: MR(q) Total cost: TC(q)  Marginal cost: MC(q) Profit:  (q) = TR(q) – TC(q) Principles of Economics: MC = MR MC curve cuts MR curve from below.

37. ECON 1150, Spring 2013 Calculus First-order condition: I.e., MR – MC = 0 Thus the marginal condition for profit maximization is just the first-order condition.

38. ECON 1150, Spring 2013 Second-order condition: Calculus At profit-maximization, the slope of the MR curve is smaller than the slope of the MC curve.

39. ECON 1150, Spring 2013 6.1 A Competitive Firm Example 3.14: Given (a) perfect competition; (b) market price p; (c) the total cost of a firm is TC(q) = 0.5q 3 – 2q 2 + 3q + 2. If p = 3, find the maximum profit of the firm.