EC400 Part II, Math for Micro: Lecture 2 Leonardo Felli NAB.SZT 10 September 2010
Taylor’s Series For functions from R 1 to R 1 , the first order Taylor’s approximation is f ( a + h ) ≈ f ( a ) + f ′ ( a ) h The approximation holds in the following sense. Let R ( h ; a ) = f ( a + h ) − f ( a ) − f ′ ( a ) h By the definition of the derivative f ′ ( a ) , we have R ( h ; a ) → 0 as h h → 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 2 / 27
Taylor’s Series For functions from R 1 to R 1 , the first order Taylor’s approximation is f ( a + h ) ≈ f ( a ) + f ′ ( a ) h The approximation holds in the following sense. Let R ( h ; a ) = f ( a + h ) − f ( a ) − f ′ ( a ) h By the definition of the derivative f ′ ( a ) , we have R ( h ; a ) → 0 as h h → 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 2 / 27
Taylor’s Series For functions from R 1 to R 1 , the first order Taylor’s approximation is f ( a + h ) ≈ f ( a ) + f ′ ( a ) h The approximation holds in the following sense. Let R ( h ; a ) = f ( a + h ) − f ( a ) − f ′ ( a ) h By the definition of the derivative f ′ ( a ) , we have R ( h ; a ) → 0 as h h → 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 2 / 27
Geometrically, this is the formalization of the approximation of the graph of f ( x ) by its tangent line at ( a , f ( a )) . Analytically, it describes the best approximation of f by a polynomial of degree 1. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 3 / 27
Geometrically, this is the formalization of the approximation of the graph of f ( x ) by its tangent line at ( a , f ( a )) . Analytically, it describes the best approximation of f by a polynomial of degree 1. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 3 / 27
Definition The k th order Taylor polynomial of f at x = a is h 2 + ... + f [ k ] ( a ) P k ( a + h ) = f ( a ) + f ′ ( a ) h + f ′′ ( a ) h k 2! k ! where R k ( h ; a ) = f ( a + h ) − P k ( a + h ) and R k ( h ; a ) → 0 as h → 0 h Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 4 / 27
Example Consider the first and second order Taylor polynomial of the exponential function f ( x ) = e x at x = 0 . All the derivatives of f at x = 0 equal 1. Then: P 1 ( h ) = 1 + h 1 + h + h 2 P 2 ( h ) = 2 For h = 0 . 2 , then P 1 ( . 2) = 1 . 2 and P 2 ( . 2) = 1 . 22 compared with the actual value of e 1 / 5 which is 1.22140275816017. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 5 / 27
Example Consider the first and second order Taylor polynomial of the exponential function f ( x ) = e x at x = 0 . All the derivatives of f at x = 0 equal 1. Then: P 1 ( h ) = 1 + h 1 + h + h 2 P 2 ( h ) = 2 For h = 0 . 2 , then P 1 ( . 2) = 1 . 2 and P 2 ( . 2) = 1 . 22 compared with the actual value of e 1 / 5 which is 1.22140275816017. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 5 / 27
Taylor’s polynomial for functions of several variables First order Taylor polynomial: F ( a + h ) ≈ F ( a ) + ∂ F ( a ) h 1 + ... + ∂ F ( a ) h n ∂ x 1 ∂ x n where R 1 ( h ; a ) / || h || → 0 as h → 0 . Alternatively F ( a + h ) = F ( a ) + DF a · h + R 1 ( h ; a ) � ∂ F , . . . , ∂ F � where DF a = is the F Jacobian . ∂ x 1 ∂ x n Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 6 / 27
Taylor’s polynomial for functions of several variables First order Taylor polynomial: F ( a + h ) ≈ F ( a ) + ∂ F ( a ) h 1 + ... + ∂ F ( a ) h n ∂ x 1 ∂ x n where R 1 ( h ; a ) / || h || → 0 as h → 0 . Alternatively F ( a + h ) = F ( a ) + DF a · h + R 1 ( h ; a ) � ∂ F , . . . , ∂ F � where DF a = is the F Jacobian . ∂ x 1 ∂ x n Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 6 / 27
Second order Taylor polynomial: the analogue for f ′′ ( a ) h 2 is 2! 1 2 h T D 2 F a h , where D 2 F a is the Hessian matrix: ∂ 2 F ∂ 2 F ∂ 2 x 1 | x = a ... ∂ x n ∂ x 1 | x = a . . ... D 2 F a = . . . . . ∂ 2 F ∂ 2 F ∂ x 1 ∂ x n | x = a ... ∂ 2 x n | x = a The extension for order k then trivially follows. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 7 / 27
Second order Taylor polynomial: the analogue for f ′′ ( a ) h 2 is 2! 1 2 h T D 2 F a h , where D 2 F a is the Hessian matrix: ∂ 2 F ∂ 2 F ∂ 2 x 1 | x = a ... ∂ x n ∂ x 1 | x = a . . ... D 2 F a = . . . . . ∂ 2 F ∂ 2 F ∂ x 1 ∂ x n | x = a ... ∂ 2 x n | x = a The extension for order k then trivially follows. Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 7 / 27
Definition Extreme Points Definition Suppose that f ( x ) is a real valued function defined on a subset C of R n . A point x ∗ ∈ C is: A global maximizer for f ( x ) on C if f ( x ∗ ) ≥ f ( x ) for all x ∈ C . A strict global maximizer for f ( x ) on C if f ( x ∗ ) > f ( x ) for all x ∈ C such that x � = x ∗ . Definition The ball B ( x , r ) centred at x of radius r is the set of all vectors y in R n whose distance from x is less than r , that is B ( x , r ) = { y ∈ R n ; || y − x || < r } . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 8 / 27
Definition Extreme Points Definition Suppose that f ( x ) is a real valued function defined on a subset C of R n . A point x ∗ ∈ C is: A global maximizer for f ( x ) on C if f ( x ∗ ) ≥ f ( x ) for all x ∈ C . A strict global maximizer for f ( x ) on C if f ( x ∗ ) > f ( x ) for all x ∈ C such that x � = x ∗ . Definition The ball B ( x , r ) centred at x of radius r is the set of all vectors y in R n whose distance from x is less than r , that is B ( x , r ) = { y ∈ R n ; || y − x || < r } . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 8 / 27
Definition Suppose that f ( x ) is a real valued function defined on a subset C of R n . A point x ∗ ∈ C is: A local maximizer for f ( x ) if there is a strictly positive number δ such that f ( x ∗ ) ≥ f ( x ) for all x ∈ B ( x ∗ , δ ) ⊂ C . A strict local maximizer for f ( x ) if there is a strictly positive number δ such that f ( x ∗ ) > f ( x ) for all x ∈ B ( x ∗ , δ ) ⊂ C and x � = x ∗ . A critical point for f ( x ) if the first partial derivative of f ( x ) exists at x ∗ and ∂ f ( x ∗ ) = 0 for i = 1 , 2 , ..., n . ∂ x i Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 9 / 27
Example Consider the function F ( x , y ) = x 3 − y 3 + 9 xy . Set ∂ F ∂ F ∂ x = 3 x 2 + 9 y = 0; ∂ y = − 3 y 2 + 9 x = 0 . The critical points are (0 , 0) and (3 , − 3) . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 10 / 27
Do extreme points exist? Theorem (Extreme Value Theorem) Suppose that f ( x ) is a continuous function defined on C, which is a compact (closed and bounded) subset of R n . Then there exists an point x ∗ in C , at which f has a maximum, and there exists a point x ∗ in C , at which f has a minimum. Thus, f ( x ∗ ) ≤ f ( x ) ≤ f ( x ∗ ) for all x ∈ C . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 11 / 27
Functions of one variable Necessary condition for maximum in R : Suppose that f ( x ) is a differentiable function on an interval I . If x ∗ is a local maximizer of f ( x ) , then either x ∗ is an end point of I , or f ′ ( x ∗ ) = 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 12 / 27
Functions of one variable Necessary condition for maximum in R : Suppose that f ( x ) is a differentiable function on an interval I . If x ∗ is a local maximizer of f ( x ) , then either x ∗ is an end point of I , or f ′ ( x ∗ ) = 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 12 / 27
Functions of one variable Necessary condition for maximum in R : Suppose that f ( x ) is a differentiable function on an interval I . If x ∗ is a local maximizer of f ( x ) , then either x ∗ is an end point of I , or f ′ ( x ∗ ) = 0 . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 12 / 27
Second order sufficient condition for a maximum in R : Suppose that f ( x ) , f ′ ( x ) , f ′′ ( x ) are all continuous on an interval in I and that x ∗ is a critical point of f ( x ) . If f ′′ ( x ) ≤ 0 for all x ∈ I , then x ∗ is a global maximizer of f ( x ) on I . If f ′′ ( x ) < 0 for all x ∈ I for x ∗ � = x , then x ∗ is a strict global maximizer of f ( x ) on I . If f ′′ ( x ∗ ) < 0 then x ∗ is a strict local maximizer of f ( x ) on I . Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 13 / 27
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