Mathematical Economics: Lecture 2

Transcription

1 Mathematical Economics: Lecture 2 Yu Ren WISE, Xiamen University September 25, 2012

2 Outline 1

3 Number Line The number line, origin (Figure 2.1 Page 11)

4 Number Line Interval (a, b) = {x R 1 : a < x < b} [a, b] = {x R 1 : a x b} (a, b] = {x R 1 : a < x b} [a, b) = {x R 1 : a x < b}

5 Function on R 1 Function on R 1 : a rule which assigns a number in R 1 to a number in R 1 Example in the book: f (x) = x + 1

6 Function on R 1 Function on R 1 : a rule which assigns a number in R 1 to a number in R 1 Example in the book: f (x) = x + 1

7 Function on R 1 x = 0 f (x) = 0.5 x = 3.25 f (x) = 0.98 x = π f (x) = 2 Is f (x) a function about x?

8 Function on R 1 Domain: the set of numbers at which the function is defined Example: f (x) = 1 x

9 Function on R 1 Domain: the set of numbers at which the function is defined Example: f (x) = 1 x

10 Function on R 1 x: independent variable, exogenous variable y = f (x): dependent variable, endogenous variable

11 Function Categories monomials: f 1 (x) = 4x 3, f 1 (x) = 5x 2, f 1 (x) = 2x f (x) = ax b b: positive integer

12 Function Categories monomials: f 1 (x) = 4x 3, f 1 (x) = 5x 2, f 1 (x) = 2x f (x) = ax b b: positive integer

13 Function Categories polynomials: f (x) = 4x 3 + 5x 2 + 2x + 1 f (x) = a 1 x b 1 + a 2x b a nx b n b i : positive integer

14 Function Categories polynomials: f (x) = 4x 3 + 5x 2 + 2x + 1 f (x) = a 1 x b 1 + a 2x b a nx b n b i : positive integer

15 Function Categories rational function f (x) = polynomial polynomial,

16 Function Categories exponential function f (x) = a x,

17 Function Categories trigonometric function f (x) = F (sin(x), cos(x), ),

18 Function Categories increasing function, decreasing function

19 Minimum local minimum: a minimum within a local neighborhood For example: if x 1 is a local minimum of f (x), then f (x 1 ) is the lowest value within (x 1 ɛ, x 1 + ɛ), where ɛ is a small positive number

20 Minimum local minimum: a minimum within a local neighborhood For example: if x 1 is a local minimum of f (x), then f (x 1 ) is the lowest value within (x 1 ɛ, x 1 + ɛ), where ɛ is a small positive number

21 Minimum global minimum: a minimum within the whole domain For example: if x 1 is a local minimum of f (x) and x 2 can be another local minimum of f (x), but only one of them can be the global minimum.

22 Minimum global minimum: a minimum within the whole domain For example: if x 1 is a local minimum of f (x) and x 2 can be another local minimum of f (x), but only one of them can be the global minimum.

23 Linear Function linear function: f (x) = mx + b m is the slope, b is the intercept

24 Linear Function Example 2.2 The slope of the line joining the points (4,6) and (0,7) is m = = 1 4.

25 Economic Interpretation: Marginal Slope in economics: marginal cost, marginal utility, marginal product marginal cost is the change in total cost that arises when the quantity produced changes by one unit. marginal product is the extra output produced by one more unit of an input

26 Economic Interpretation: Marginal Slope in economics: marginal cost, marginal utility, marginal product marginal cost is the change in total cost that arises when the quantity produced changes by one unit. marginal product is the extra output produced by one more unit of an input

27 Economic Interpretation: Marginal Slope in economics: marginal cost, marginal utility, marginal product marginal cost is the change in total cost that arises when the quantity produced changes by one unit. marginal product is the extra output produced by one more unit of an input

28 Slope of Nonlinear Function nonlinear function derivative: f (x 0 ) = lim h 0 f (x 0 +h) f (x 0 ) h

29 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

30 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

31 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

32 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

33 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

34 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

35 Slope of Nonlinear Function computing rules: Theorem 2.4: (f ± g) (x 0 ) = f (x 0 ) ± g (x 0 ) kf (x 0 ) = k(f (x 0 )) (f ( ) g) (x 0 ) = f (x 0 )g(x 0 ) + f (x 0 )g (x 0 ) f (x0 ) = f (x 0 )g(x 0 ) f (x 0 )g (x 0 ) g g(x 0 ) 2 (f (x) n ) = n(f (x)) n 1 f (x) (x k ) = kx k 1

36 Example Example 2.5: For f (x) = x 2, in order to prove that f (3) = 6, we need to show that (3 + h n ) h n 6, as h n 0 for every sequence {h n } which approaches zero.

37 Example For any h, (3 + h) h = 9 + 6h + h2 9 h which clearly converges to 6 as h 0. f (3) = 6. = h(6 + h) h = 6+h

38 Differentiability and Continuity f (x differentiable: lim 0 +h) f (x 0 ) h 0 h exists and same h Is f (x) = x differentiable at x 0 = 0?

39 Differentiability and Continuity f (x differentiable: lim 0 +h) f (x 0 ) h 0 h exists and same h Is f (x) = x differentiable at x 0 = 0?

40 Differentiability and Continuity lim h 0 f (x 0 ) = lim h 0 + f (x 0 ) left limit: lim h 0 f (x 0 ) right limit: lim h 0 + f (x 0 )

41 Differentiability and Continuity lim h 0 f (x 0 ) = lim h 0 + f (x 0 ) left limit: lim h 0 f (x 0 ) right limit: lim h 0 + f (x 0 )

42 Differentiability and Continuity continuous: f (x 0 ) = lim h 0 f (x 0 + h) when the limit of x 0 exists, f may or may not be continuous at x 0 f (x) = x when (, 3) (3, + ) and f (3) = 9. f (x) is not continuous at 3.

43 Differentiability and Continuity continuous: f (x 0 ) = lim h 0 f (x 0 + h) when the limit of x 0 exists, f may or may not be continuous at x 0 f (x) = x when (, 3) (3, + ) and f (3) = 9. f (x) is not continuous at 3.

44 Differentiability and Continuity continuous: f (x 0 ) = lim h 0 f (x 0 + h) when the limit of x 0 exists, f may or may not be continuous at x 0 f (x) = x when (, 3) (3, + ) and f (3) = 9. f (x) is not continuous at 3.

45 Higher-Order Derivatives f (x 0 ) or d dx ( df dx )(x 0) = d 2 f dx 2 (x 0 ) Example 2.9: the derivative of the function f (x) = x 3 + 3x 2 + 3x + 1 is f (x) = 3x 2 + 6x + 3 f (x) = 6x + 6

46 Higher-Order Derivatives f (x 0 ) or d dx ( df dx )(x 0) = d 2 f dx 2 (x 0 ) Example 2.9: the derivative of the function f (x) = x 3 + 3x 2 + 3x + 1 is f (x) = 3x 2 + 6x + 3 f (x) = 6x + 6

47 Higher-Order Derivatives f (x 0 ) or d dx ( df dx )(x 0) = d 2 f dx 2 (x 0 ) Example 2.9: the derivative of the function f (x) = x 3 + 3x 2 + 3x + 1 is f (x) = 3x 2 + 6x + 3 f (x) = 6x + 6

48 Higher-Order Derivatives twice continuously differentiable: If f has a second derivative everywhere, then f is a well defined function of x. And if f is a continuous function of x, then f is twice continuously differentiable, denoted by C 2 C 1, C n, C

49 Higher-Order Derivatives twice continuously differentiable: If f has a second derivative everywhere, then f is a well defined function of x. And if f is a continuous function of x, then f is twice continuously differentiable, denoted by C 2 C 1, C n, C

50 Approximation by Differentials f (x 0 ) = dy dx y x y = f (x 0 + x) f (x 0 ) f (x 0 ) x f (x 0 + x) f (x 0 ) + f (x 0 ) x Figure 2.15 (page 37)

51 Approximation by Differentials f (x 0 ) = dy dx y x y = f (x 0 + x) f (x 0 ) f (x 0 ) x f (x 0 + x) f (x 0 ) + f (x 0 ) x Figure 2.15 (page 37)

52 Approximation by Differentials f (x 0 ) = dy dx y x y = f (x 0 + x) f (x 0 ) f (x 0 ) x f (x 0 + x) f (x 0 ) + f (x 0 ) x Figure 2.15 (page 37)

53 Approximation by Differentials f (x 0 ) = dy dx y x y = f (x 0 + x) f (x 0 ) f (x 0 ) x f (x 0 + x) f (x 0 ) + f (x 0 ) x Figure 2.15 (page 37)

54 Example Example 2.11 Production function F(x) = 1 2 x 100 units of labor input x its output is 5 units. The derivative of the production F at x = 100, F (100) = /2 = 1 40 = 0.025,

55 Example F (100) is a good measure of the marginal product of labor the actual increase in output is F (101) F (100) = , pretty close to