Chapter 2 THE MATHEMATICS OF OPTIMIZATION. Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.

Transcription

1 Chapter THE MATHEMATICS OF OPTIMIZATION Copyright 005 by South-Western, a division of Thomson Learning. All rights reserved. 1

2 The Mathematics of Optimization Many economic theories begin with the assumption that an economic agent is seeking to find the optimal value of some function consumers seek to maimize utility firms seek to maimize profit This chapter introduces the mathematics common to these problems

3 Maimization of a Function of One Variable Simple eample: Manager of a firm wishes to maimize profits π π* π π f(q) f(q) Maimum profits of π* occur at q* q* Quantity 3

4 Maimization of a Function of One Variable The manager will likely try to vary q to see where the maimum profit occurs an increase from q 1 to q leads to a rise in π π π* π π f(q) π q > 0 π 1 q 1 q q* Quantity 4

5 Maimization of a Function of One Variable If output is increased beyond q*, profit will decline an increase from q* to q 3 leads to a drop in π π π* π f(q) π q < 0 π 3 q* q 3 Quantity 5

6 Derivatives( 导数 ) The derivative of π f(q) is the limit of π/ q for very small changes in q dπ dq df dq f( q1 + h) f( q1) lim h 0 h The value of this ratio depends on the value of q 1 6

7 Value of a Derivative at a Point The evaluation of the derivative at the point q q 1 can be denoted dπ dq q q 1 In our previous eample, dπ dq q q 1 > 0 dπ dq q q 3 < 0 dπ dq q q* 0 7

8 First Order Condition( 一阶条件 ) for a Maimum For a function of one variable to attain its maimum value at some point, the derivative at that point must be zero df dq q q* 0 8

9 Second Order Conditions The first order condition (dπ/dq) is a necessary condition( 必要条件 ) for a maimum, but it is not a sufficient condition ( 充分条件 ) π If the profit function was u-shaped, the first order condition would result in q* being chosen and π would be minimized π* q* Quantity 9

10 Second Order Conditions This must mean that, in order for q* to be the optimum, dπ dq > dπ 0 for q < q * and < 0 for q > q * dq Therefore, at q*, dπ/dq must be decreasing 10

11 Second Derivatives The derivative of a derivative is called a second derivative ( 二阶导数 ) The second derivative can be denoted by d π dq d f or dq or f "( q) 11

12 Second Order Condition The second order condition to represent a (local) maimum is d π dq q q* f "( q) q q* < 0 1

13 Rules for Finding Derivatives db 1. If b is a constant, then d 0 d[ bf ( )]. If b is a constant, then bf '( ) d 3.If b is constant, then d d b b b 1 d ln 1 4. d 13

14 Rules for Finding Derivatives da 5. d a lna for any constant a a special case of this rule is de /d e 14

15 Rules for Finding Derivatives Suppose that f() and g() are two functions of and f () and g () eist Then d[ f ( ) + g( )] 6. f '( ) + g'( ) d d[ f ( ) g( )] 7. f ( ) g'( ) + f '( ) g( ) d 15

16 16 Rules for Finding Derivatives 0 ) ( that provided )] ( [ ) '( ) ( ) ( ) '( ) ( ) ( 8. g g g f g f d g f d

17 Rules for Finding Derivatives If y f() and g(z) and if both f () and g () eist, then: 9. dy dz dy d d dz df d dg dz This is called the chain rule( 链式法则 ). The chain rule allows us to study how one variable (z) affects another variable (y) through its influence on some intermediate variable () 17

18 Rules for Finding Derivatives Some eamples of the chain rule include 10. de d a a de d( a) d( a) d e a a ae a d[ln( a)] d[ln( a)] d( a) 11. ln( a) a aln( a) d d( a) d d[ln( )] d[ln( )] d( ) 1 1. d d( ) d 18

19 Eample of Profit Maimization Suppose that the relationship between profit and output is π 1,000q - 5q The first order condition for a maimum is dπ/dq 1,000-10q 0 q* 100 Since the second derivative is always -10, q 100 is a global maimum 19

20 Functions of Several Variables Most goals of economic agents depend on several variables trade-offs ( 权衡, 两相取舍 ) must be made The dependence of one variable (y) on a series of other variables ( 1,,, n ) is denoted by f,,..., ) y ( 1 n 0

21 Partial Derivatives( 偏导数 ) The partial derivative of y with respect to 1 is denoted by y 1 f or 1 or f or f 1 1 It is understood that in calculating the partial derivative, all of the other s are held constant 1

22 A more formal definition of the partial derivative is Partial Derivatives h f h f f n n h n ),...,, ( ),...,, ( lim ,..., +

23 3 Calculating Partial Derivatives c b f f b a f f c b a f y and then, ), ( 1.If b a b a b a be f f ae f f e f y and then If., ), (

24 4 Calculating Partial Derivatives b f f a f f b a f y + and then 3.If, ln ln ), (

25 Partial Derivatives Partial derivatives are the mathematical epression of the ceteris paribus assumption show how changes in one variable affect some outcome when other influences are held constant 5

26 Partial Derivatives We must be concerned with how variables are measured if q represents the quantity of gasoline demanded (measured in billions of gallons) and p represents the price in dollars per gallon, then q/ p will measure the change in demand (in billiions of gallons per year) for a dollar per gallon change in price 6

27 Elasticity ( 弹性 ) Elasticities measure the proportional effect of a change in one variable on another unit free ( 与单位无关 ) The elasticity of y with respect to is e y, y y y y y y 7

28 Elasticity and Functional Form Suppose that In this case, y a + b + other terms e y, y y b y b a + b + e y, is not constant it is important to note the point at which the elasticity is to be computed 8

29 Elasticity and Functional Form Suppose that In this case, y a b y b 1 ey, ab b y a b 9

30 Elasticity and Functional Form Suppose that In this case, ln y ln a + b ln e y, y y b ln ln y Elasticities can be calculated through logarithmic differentiation 30

31 Second-Order Partial Derivatives The partial derivative of a partial derivative is called a second-order partial derivative ( f / j i ) j f i f ij 31

32 Young s Theorem Under general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter f f ij ji 3

33 Use of Second-Order Partials Second-order partials play an important role in many economic theories One of the most important is a variable s own second-order partial, f ii shows how the marginal influence of i on y( y/ i ) changes as the value of i increases a value of f ii < 0 indicates diminishing marginal effectiveness 33

34 Total Differential Suppose that y f( 1,,, n ) If all s are varied by a small amount, the total effect on y will be dy f f d1 + d f n d n dy f d + f d f n d n 34

35 First-Order Condition for a Maimum (or Minimum) A necessary condition for a maimum (or minimum) of the function f( 1,,, n ) is that dy 0 for any combination of small changes in the s The only way for this to be true is if f1 f... fn A point where this condition holds is called a critical point ( 奇点, 零点 ) 0 35

36 36 Finding a Maimum Suppose that y is a function of 1 and y - ( 1-1) - ( - ) + 10 y First-order conditions imply that y y OR 1 1 * *

37 Production Possibility Frontier Earlier eample: + y 5 Can be rewritten: f(,y) + y Because f 4 and f y y, the opportunity cost trade-off between and y is dy d f f y 4 y y 37

38 Implicit Function Theorem( 隐函数定理 ) It may not always be possible to solve implicit functions of the form g(,y)0 for unique eplicit functions of the form y f() mathematicians have derived the necessary conditions in many economic applications, these conditions are the same as the second-order conditions for a maimum (or minimum) 38

39 The Envelope Theorem( 包 络定理 ) The envelope theorem concerns how the optimal value for a particular function changes when a parameter of the function changes This is easiest to see by using an eample 39

40 The Envelope Theorem Suppose that y is a function of y - + a For different values of a, this function represents a family of inverted parabolas( 翻转抛物线 ) If a is assigned a specific value, then y becomes a function of only and the value of that maimizes y can be calculated 40

41 The Envelope Theorem Optimal Values of and y for alternative values of a Value of a Value of * Value of y* / 1/ / 9/ / 5/

42 The Envelope Theorem y* As a increases, the maimal value for y (y*) increases The relationship between a and y is quadratic a 4

43 The Envelope Theorem Suppose we are interested in how y* changes as a changes There are two ways we can do this calculate the slope of y directly hold constant at its optimal value and calculate y/ a directly 43

44 The Envelope Theorem To calculate the slope of the function, we must solve for the optimal value of for any value of a Substituting, we get dy/d - + a 0 * a/ y* -(*) + a(*) -(a/) + a(a/) y* -a /4 + a / a /4 44

45 The Envelope Theorem Therefore, dy*/da a/4 a/ * But, we can save time by using the envelope theorem for small changes in a, dy*/da can be computed by holding at * and calculating y/ a directly from y 45

46 The Envelope Theorem Holding * y/ a y/ a * a/ This is the same result found earlier 46

47 The Envelope Theorem The envelope theorem states that the change in the optimal value of a function with respect to a parameter of that function can be found by partially differentiating the objective function while holding (or several s) at its optimal value dy * da y a { * ( a)} 47

48 The Envelope Theorem The envelope theorem can be etended to the case where y is a function of several variables y f( 1, n,a) Finding an optimal value for y would consist of solving n first-order equations y/ i 0 (i 1,,n) 48

49 The Envelope Theorem Optimal values for theses s would be determined that are a function of a 1 * 1 *(a) * *(a)... n * n *(a) 49

50 The Envelope Theorem Substituting into the original objective function yields an epression for the optimal value of y (y*) y* f [ 1 *(a), *(a),, n *(a),a] Differentiating yields dy da f d1 f d da da * 1 + f n d da n + f a 50

51 The Envelope Theorem Because of first-order conditions, all terms ecept f/ a are equal to zero if the s are at their optimal values Therefore, dy * da f a { * ( a)} 51

52 Constrained Maimization What if not all values for the s are feasible? the values of may all have to be positive a consumer s choices are limited by the amount of purchasing power available One method used to solve constrained maimization problems is the Lagrangian multiplier method 5

53 Lagrangian Multiplier Method Suppose that we wish to find the values of 1,,, n that maimize y f( 1,,, n ) subject to a constraint that permits only certain values of the s to be used g( 1,,, n ) 0 53

54 Lagrangian Multiplier Method The Lagrangian multiplier method starts with setting up the epression L f( 1,,, n ) + λg( 1,,, n ) where λ is an additional variable called a Lagrangian multiplier When the constraint holds, L f because g( 1,,, n ) 0 54

55 Lagrangian Multiplier Method First-Order Conditions L/ 1 f 1 + λg 1 0 L/ f + λg 0.. L/ n f n + λg n 0 L/ λ g( 1,,, n ) 0 55

56 Lagrangian Multiplier Method The first-order conditions can generally be solved for 1,,, n and λ The solution will have two properties: the s will obey the constraint these s will make the value of L (and therefore f) as large as possible 56

57 Lagrangian Multiplier Method The Lagrangian multiplier (λ) has an important economic interpretation The first-order conditions imply that f 1 /-g 1 f /-g f n /-g n λ the numerators above measure the marginal benefit that one more unit of i will have for the function f the denominators reflect the added burden on the constraint of using more i 57

58 Lagrangian Multiplier Method At the optimal choices for the s, the ratio of the marginal benefit of increasing i to the marginal cost of increasing i should be the same for every λ is the common cost-benefit ratio for all of the s λ marginal benefit of marginal cost of i i 58

59 Lagrangian Multiplier Method If the constraint was relaed slightly, it would not matter which is changed The Lagrangian multiplier provides a measure of how the relaation in the constraint will affect the value of y λ provides a shadow price ( 影子价格 ) to the constraint 59

60 Lagrangian Multiplier Method A high value of λ indicates that y could be increased substantially by relaing the constraint each has a high cost-benefit ratio A low value of λ indicates that there is not much to be gained by relaing the constraint λ0 implies that the constraint is not binding 60

61 Duality ( 对偶性 ) Any constrained maimization problem has associated with it a dual problem in constrained minimization that focuses attention on the constraints in the original problem 61

62 Duality Individuals maimize utility subject to a budget constraint dual problem: individuals minimize the ependiture needed to achieve a given level of utility Firms minimize the cost of inputs to produce a given level of output dual problem: firms maimize output for a given cost of inputs purchased 6

63 Constrained Maimization Suppose a farmer had a certain length of fence (P) and wished to enclose the largest possible rectangular shape Let be the length of one side Let y be the length of the other side Problem: choose and y so as to maimize the area (A y) subject to the constraint that the perimeter is fied at P + y 63

64 Constrained Maimization Setting up the Lagrangian multiplier L y + λ(p - - y) The first-order conditions for a maimum are L/ y - λ 0 L/ y - λ 0 L/ λ P - - y 0 64

65 Constrained Maimization Since y/ / λ, must be equal to y the field should be square and y should be chosen so that the ratio of marginal benefits to marginal costs should be the same Since y and y λ, we can use the constraint to show that y P/4 λ P/8 65

66 Constrained Maimization Interpretation of the Lagrangian multiplier if the farmer was interested in knowing how much more field could be fenced by adding an etra yard of fence, λ suggests that he could find out by dividing the present perimeter (P) by 8 thus, the Lagrangian multiplier provides information about the implicit value of the constraint 66

67 Constrained Maimization Dual problem: choose and y to minimize the amount of fence required to surround the field minimize P + y subject to A y Setting up the Lagrangian: L D + y + λ D (A - y) 67

68 Constrained Maimization First-order conditions: Solving, we get L D / - λ D y 0 L D / y - λ D 0 L D / λ D A - y 0 y A 1/ The Lagrangian multiplier (λ D ) A -1/ 68

69 Envelope Theorem & Constrained Maimization Suppose that we want to maimize y f( 1,, n ;a) subject to the constraint g( 1,, n ;a) 0 One way to solve would be to set up the Lagrangian epression and solve the firstorder conditions 69

70 Envelope Theorem & Constrained Maimization Alternatively, it can be shown that dy*/da L/ a( 1 *,, n *;a) The change in the maimal value of y that results when a changes can be found by partially differentiating L and evaluating the partial derivative at the optimal point 70

71 Inequality Constraints In some economic problems the constraints need not hold eactly For eample, suppose we seek to maimize y f( 1, ) subject to g( 1, ) 0, 1 0, and 0 71

72 Inequality Constraints One way to solve this problem is to introduce three new variables (a, b, and c) that convert the inequalities into equalities To ensure that the inequalities continue to hold, we will square these new variables to ensure that their values are positive 7

73 Inequality Constraints g( 1, ) - a 0; 1 - b 0; and - c 0 Any solution that obeys these three equality constraints will also obey the inequality constraints 73

74 Inequality Constraints We can set up the Lagrangian L f( 1, ) + λ 1 [g( 1, ) - a ] + λ [ 1 - b ] + λ 3 [ - c ] This will lead to eight first-order conditions 74

75 Inequality Constraints L/ 1 f 1 + λ 1 g 1 + λ 0 L/ f 1 + λ 1 g + λ 3 0 L/ a -aλ 1 0 L/ b -bλ 0 L/ c -cλ 3 0 L/ λ 1 g( 1, ) - a 0 L/ λ 1 - b 0 L/ λ 3 - c 0 75

76 Inequality Constraints According to the third condition, either a or λ 1 0 if a 0, the constraint g( 1, ) holds eactly if λ 1 0, the availability of some slackness of the constraint implies that its value to the objective function is 0 Similar complemetary slackness ( 互补松弛 )relationships also hold for 1 and λ 1 L/ λ

77 Inequality Constraints These results are sometimes called Kuhn-Tucker conditions they show that solutions to optimization problems involving inequality constraints will differ from similar problems involving equality constraints in rather simple ways we cannot go wrong by working primarily with constraints involving equalities 77

78 Second Order Conditions - Functions of One Variable Let y f() A necessary condition for a maimum is that dy/d f () 0 To ensure that the point is a maimum, y must be decreasing for movements away from it 78

79 Second Order Conditions - Functions of One Variable The total differential measures the change in y dy f () d To be at a maimum, dy must be decreasing for small increases in To see the changes in dy, we must use the second derivative of y 79

80 Second Order Conditions - Functions of One Variable d[ f '( ) d] d d y d f "( ) d d f "( ) d Note that d y < 0 implies that f ()d < 0 Since d must be positive, f () < 0 This means that the function f must have a concave( 凹 ) shape at the critical point 80

81 Second Order Conditions - Functions of Two Variables Suppose that y f( 1, ) First order conditions for a maimum are y/ 1 f 1 0 y/ f 0 To ensure that the point is a maimum, y must diminish for movements in any direction away from the critical point 81

82 Second Order Conditions - Functions of Two Variables The slope in the 1 direction (f 1 ) must be diminishing at the critical point The slope in the direction (f ) must be diminishing at the critical point But, conditions must also be placed on the cross-partial derivative (f 1 f 1 ) to ensure that dy is decreasing for all movements through the critical point 8

83 Second Order Conditions - Functions of Two Variables The total differential of y is given by dy f 1 d 1 + f d The differential of that function is d y (f 11 d 1 + f 1 d )d 1 + (f 1 d 1 + f d )d d y f 11 d 1 + f 1 d d 1 + f 1 d 1 d + f d By Young s theorem, f 1 f 1 and d y f 11 d 1 + f 1 d 1 d + f d 83

84 Second Order Conditions - Functions of Two Variables d y f 11 d 1 + f 1 d 1 d + f d For this equation to be unambiguously negative for any change in the s, f 11 and f must be negative If d 0, then d y f 11 d 1 for d y < 0, f 11 < 0 If d 1 0, then d y f d for d y < 0, f < 0 84

85 Second Order Conditions - Functions of Two Variables d y f 11 d 1 + f 1 d 1 d + f d If neither d 1 nor d is zero, then d y will be unambiguously negative only if f 11 f - f 1 > 0 the second partial derivatives (f 11 and f ) must be sufficiently negative so that they outweigh any possible perverse effects from the crosspartial derivatives (f 1 f 1 ) 85

86 Constrained Maimization Suppose we want to choose 1 and to maimize y f( 1, ) subject to the linear constraint c - b b 0 We can set up the Lagrangian L f( 1, ) + λ(c - b b ) 86

87 Constrained Maimization The first-order conditions are f 1 - λb 1 0 f - λb 0 c - b b 0 To ensure we have a maimum, we must use the second total differential d y f 11 d 1 + f 1 d 1 d + f d 87

88 Constrained Maimization Only the values of 1 and that satisfy the constraint can be considered valid alternatives to the critical point Thus, we must calculate the total differential of the constraint -b 1 d 1 - b d 0 d -(b 1 /b )d 1 These are the allowable relative changes in 1 and 88

89 Constrained Maimization Because the first-order conditions imply that f 1 /f b 1 /b, we can substitute and get Since d -(f 1 /f ) d 1 d y f 11 d 1 + f 1 d 1 d + f d we can substitute for d and get d y f 11 d 1 - f 1 (f 1 /f )d 1 + f (f 1 /f )d 1 89

90 Constrained Maimization Combining terms and rearranging d y f 11 f - f 1 f 1 f + f f 1 [d 1 / f ] Therefore, for d y < 0, it must be true that f 11 f - f 1 f 1 f + f f 1 < 0 This equation characterizes a set of functions termed quasi-concave( 拟凹 ) functions any two points within the set can be joined by a line contained completely in the set 90

91 Concave and Quasi- Concave Functions The differences between concave and quasi-concave functions can be illustrated with the function y f( 1, ) ( 1 ) k where the s take on only positive values and k can take on a variety of positive values 91

92 Concave and Quasi- Concave Functions No matter what value k takes, this function is quasi-concave Whether or not the function is concave depends on the value of k if k < 0.5, the function is concave if k > 0.5, the function is conve 9

93 Homogeneous( 齐次 ) Functions A function f( 1,, n ) is said to be homogeneous of degree k if f(t 1,t, t n ) t k f( 1,, n ) when a function is homogeneous of degree one, a doubling of all of its arguments doubles the value of the function itself when a function is homogeneous of degree zero, a doubling of all of its arguments leaves the value of the function unchanged 93

94 Homogeneous Functions If a function is homogeneous of degree k, the partial derivatives of the function will be homogeneous of degree k-1 94

95 Euler s Theorem If we differentiate the definition for homogeneity with respect to the proportionality factor t, we get kt k-1 f( 1,, n ) 1 f 1 (t 1,,t n ) + + n f n ( 1,, n ) This relationship is called Euler s theorem 95

96 Euler s Theorem Euler s theorem shows that, for homogeneous functions, there is a definite relationship between the values of the function and the values of its partial derivatives 96

97 Homothetic( 同位 ) Functions A homothetic function is one that is formed by taking a monotonic transformation of a homogeneous function they do not possess the homogeneity properties of their underlying functions 97

98 Homothetic Functions For both homogeneous and homothetic functions, the implicit trade-offs among the variables in the function depend only on the ratios of those variables, not on their absolute values 98

99 Homothetic Functions Suppose we are eamining the simple, two variable implicit function f(,y) 0 The implicit trade-off between and y for a two-variable function is dy/d -f /f y If we assume f is homogeneous of degree k, its partial derivatives will be homogeneous of degree k-1 99

100 Homothetic Functions The implicit trade-off between and y is dy d If t 1/y, t t f f k 1 k 1 y ( t, ty ) ( t, ty ) f f y ( t, ty ) ( t, ty ) dy d F' f F' f y y y,1,1 f f y y y,1,1 100

101 Homothetic Functions The trade-off is unaffected by the monotonic transformation and remains a function only of the ratio to y 101

102 Important Points to Note: Using mathematics provides a convenient, short-hand way for economists to develop their models implications of various economic assumptions can be studied in a simplified setting through the use of such mathematical tools 10

103 Important Points to Note: Derivatives are often used in economics because economists are interested in how marginal changes in one variable affect another partial derivatives incorporate the ceteris paribus assumption used in most economic models 103

104 Important Points to Note: The mathematics of optimization is an important tool for the development of models that assume that economic agents rationally pursue some goal the first-order condition for a maimum requires that all partial derivatives equal zero 104

105 Important Points to Note: Most economic optimization problems involve constraints on the choices that agents can make the first-order conditions for a maimum suggest that each activity be operated at a level at which the ratio of the marginal benefit of the activity to its marginal cost 105

106 Important Points to Note: The Lagrangian multiplier is used to help solve constrained maimization problems the Lagrangian multiplier can be interpreted as the implicit value (shadow price) of the constraint 106

107 Important Points to Note: The implicit function theorem illustrates the dependence of the choices that result from an optimization problem on the parameters of that problem 107

108 Important Points to Note: The envelope theorem eamines how optimal choices will change as the problem s parameters change Some optimization problems may involve constraints that are inequalities rather than equalities 108

109 Important Points to Note: First-order conditions are necessary but not sufficient for ensuring a maimum or minimum second-order conditions that describe the curvature( 曲度 ) of the function must be checked 109

110 Important Points to Note: Certain types of functions occur in many economic problems quasi-concave functions obey the second-order conditions of constrained maimum or minimum problems when the constraints are linear homothetic functions have the property that implicit trade-offs among the variables depend only on the ratios of these variables 110