Lecture 3: Optimization : One Choice Variable

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

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

3. Maximization problem: maxx 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. ECON 1150, Spring 2013

4. Minimization problem: minx 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. ECON 1150, Spring 2013

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

6. 2. Necessary Condition for Extrema What are the maximum and minimum points of the following functions? y = 60x – 0.2x2 y = x3 – 12x2 + 36x + 8 ECON 1150, Spring 2013

7. y y* x 0 x1 x* x2 Characteristic of a maximum point Maximum x < x* : dy / dx > 0 x > x* : dy / dx < 0 x = x* : dy / dx = 0 ECON 1150, Spring 2013

8. y y* x 0 x1 x* x2 Characteristic of a minimum point Minimum x < x* : dy / dx < 0 x > x* : dy / dx > 0 x = x* : dy / dx = 0 ECON 1150, Spring 2013

9. 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*) ECON 1150, Spring 2013

10. Example 3.2: Find the stationary point of the function y = 60x – 0.2x2. The first-order condition is a necessary, but not sufficient, condition. ECON 1150, Spring 2013

11. Example 3.3: Find the stationary values of the function y = f(x) = x3 – 12x2 + 36x + 8. ECON 1150, Spring 2013

12. 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. ECON 1150, Spring 2013

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

14. y y y* y* x x 0 0 x1 x1 x* x* x2 x2 3.2 First-Derivative Test for Global Extreme Points Global maximum Global minimum ECON 1150, Spring 2013

15. First-derivative Test • 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. ECON 1150, Spring 2013

16. Example 3.4: Consider the function y = 60x – 0.2x2. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values. ECON 1150, Spring 2013

17. Example 3.5: y = f(x) = e2x – 5ex + 4. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values. c. Examine limx f(x) and limx-f(x). ECON 1150, Spring 2013

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

19. Example 3.6: Show that f(x) = ex–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 – 0.07Q1.5. Find the value of Q that maximizes profits. ECON 1150, Spring 2013

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

21. 4.1 First-derivative Test for Local Extreme Points 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. ECON 1150, Spring 2013

22. Example 3.8: y = f(x) = x3 – 12x2 + 36x + 8. a. Find f’(x). b. Find the intervals where f increases and decreases and determine possible extreme points and values. c. Examine limx f(x) and limx-f(x). ECON 1150, Spring 2013

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

24. y dy/dx y* y* x2 x x* 0 x1 x 0 x1 x* x2 4.2 Second-Derivative Test The nature of a stationary point: Decreasing slope  Local maximum ECON 1150, Spring 2013

25. y dy/dx y* x x1 x* x2 0 x 0 x1 x* x2 The nature of a stationary point: Increasing slope  Local minimum ECON 1150, Spring 2013

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

27. 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 ECON 1150, Spring 2013

28. Example 3.10: Identify the nature of the stationary points of the following functions: • y = 4x2 – 5x + 10; • y = x3 – 3x2 + 2; • y = 0.5x4 – 3x3 + 2x2. ECON 1150, Spring 2013

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

30. Test for inflection points: 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. ECON 1150, Spring 2013

31. Point of inflection Example 3.11: y = 16x – 4x3 + x4. dy / dx = 16 – 12x2 + 4x3. At x = 2, dy/dx = 0. However, the point at x = 2 is neither a maximum nor a minimum. ECON 1150, Spring 2013

32. Example 3.12: Find possible inflection points for the following functions, a. f(x) = x6 – 10x4. b. f(x) = x4. ECON 1150, Spring 2013

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

34. 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 ECON 1150, Spring 2013

35. Example 3.13: Sketch the graphs of the following functions by hand, analyzing all important features. • a. y = x3 – 12x; • y = (x – 3)x; • y = (1/x) – (1/x2). ECON 1150, Spring 2013

36. 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. ECON 1150, Spring 2013

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

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

39. 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) = 0.5q3– 2q2 + 3q + 2. If p = 3, find the maximum profit of the firm. ECON 1150, Spring 2013