EC400 Part II, Math for Micro: Lecture 2 Leonardo Felli NAB.SZT 10 - - PowerPoint PPT Presentation

EC400 Part II, Math for Micro: Lecture 2

Leonardo Felli

NAB.SZT

10 September 2010

Taylor’s Series

For functions from R1 to R1, 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) h → 0 as 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 R1 to R1, 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) h → 0 as 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 R1 to R1, 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) h → 0 as 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

• f 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

• f degree 1.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 3 / 27

Definition

The kth order Taylor polynomial of f at x = a is Pk(a + h) = f (a) + f ′(a)h + f ′′(a) 2! h2 + ... + f [k](a) k! hk where Rk(h; a) = f (a + h) − Pk(a + h) and Rk(h; a) h → 0 as h → 0

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) = ex at x = 0. All the derivatives of f at x = 0 equal 1. Then: P1(h) = 1 + h P2(h) = 1 + h + h2 2 For h = 0.2, then P1(.2) = 1.2 and P2(.2) = 1.22 compared with the actual value of e1/5 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) = ex at x = 0. All the derivatives of f at x = 0 equal 1. Then: P1(h) = 1 + h P2(h) = 1 + h + h2 2 For h = 0.2, then P1(.2) = 1.2 and P2(.2) = 1.22 compared with the actual value of e1/5 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 ∂x1 (a)h1 + ... + ∂F ∂xn (a)hn where R1(h; a)/||h|| → 0 as h → 0. Alternatively F(a + h) = F(a) + DFa · h + R1(h; a) where DFa = ∂F ∂x1 , . . . , ∂F ∂xn

• is the F Jacobian.

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 ∂x1 (a)h1 + ... + ∂F ∂xn (a)hn where R1(h; a)/||h|| → 0 as h → 0. Alternatively F(a + h) = F(a) + DFa · h + R1(h; a) where DFa = ∂F ∂x1 , . . . , ∂F ∂xn

• is the F Jacobian.

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) 2! h2 is 1 2hTD2Fah, where D2Fa is the Hessian matrix: D2Fa =    

∂2F ∂2x1 |x=a

...

∂2F ∂xn∂x1 |x=a

. . . ... . . .

∂2F ∂x1∂xn |x=a

...

∂2F ∂2xn |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) 2! h2 is 1 2hTD2Fah, where D2Fa is the Hessian matrix: D2Fa =    

∂2F ∂2x1 |x=a

...

∂2F ∂xn∂x1 |x=a

. . . ... . . .

∂2F ∂x1∂xn |x=a

...

∂2F ∂2xn |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 Rn. 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 Rn whose distance from x is less than r, that is B(x, r) = {y ∈ Rn; ||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 Rn. 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 Rn whose distance from x is less than r, that is B(x, r) = {y ∈ Rn; ||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 Rn. 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∗) ∂xi = 0 for i = 1, 2, ..., n.

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) = x3 − y3 + 9xy. Set ∂F ∂x = 3x2 + 9y = 0; ∂F ∂y = −3y2 + 9x = 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 Rn. 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,

• r

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,

• r

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,

• r

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

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

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

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

Functions of several variables

First order necessary conditions for a maximum in Rn: Suppose that f (x) is a real valued function for which all first partial derivatives of f (x) exist on a subset C ⊂ Rn. If x∗ is an interior point of C that is a local maximizer of f (x), then x∗ is a critical point of f (x), that is ∂f (x∗) ∂xi = 0 for i = 1, 2, ..., n.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 14 / 27

Functions of several variables

First order necessary conditions for a maximum in Rn: Suppose that f (x) is a real valued function for which all first partial derivatives of f (x) exist on a subset C ⊂ Rn. If x∗ is an interior point of C that is a local maximizer of f (x), then x∗ is a critical point of f (x), that is ∂f (x∗) ∂xi = 0 for i = 1, 2, ..., n.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 14 / 27

Can we say whether a critical point is a local maximum or a local minimum? For this we have to consider the Hessian, or the matrix of the second

• rder partial derivatives.

Notice that this is a symmetric matrix since cross-partial derivatives are equal (Clairant’s/Schwarz’s Theorem).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 15 / 27

Second order sufficient conditions for a local maximum in Rn Suppose that f (x) is a real valued function for which all first and second partial derivatives of f (x) exist on a subset C ⊂ Rn. Suppose that x∗ is a critical point of f . Then: If D2f (x∗) is negative definite, then x∗ is a strict local maximizer of f (x). If, instead, D2f (x∗) is positive definite, then x∗ is a strict local minimizer (minimizer) of f (x).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 16 / 27

Second order sufficient conditions for a local maximum in Rn Suppose that f (x) is a real valued function for which all first and second partial derivatives of f (x) exist on a subset C ⊂ Rn. Suppose that x∗ is a critical point of f . Then: If D2f (x∗) is negative definite, then x∗ is a strict local maximizer of f (x). If, instead, D2f (x∗) is positive definite, then x∗ is a strict local minimizer (minimizer) of f (x).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 16 / 27

Second order sufficient conditions for a local maximum in Rn Suppose that f (x) is a real valued function for which all first and second partial derivatives of f (x) exist on a subset C ⊂ Rn. Suppose that x∗ is a critical point of f . Then: If D2f (x∗) is negative definite, then x∗ is a strict local maximizer of f (x). If, instead, D2f (x∗) is positive definite, then x∗ is a strict local minimizer (minimizer) of f (x).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 16 / 27

It is also true that:

If x∗ is an interior point and a maximum of f , then D2f (x∗) is negative semi-definite If x∗ is an interior point and a minimum of f , then D2f (x∗) is positive semi-definite.

However, it is not true that if x∗ is a critical point, and D2f (x∗) is negative (positive) semi-definite, then x∗ is a local maximum. A counterexample is f (x) = x3, which has the property that D2f (0) is semidefinite, but x = 0 is not a maximum or minimum.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 17 / 27

It is also true that:

If x∗ is an interior point and a maximum of f , then D2f (x∗) is negative semi-definite If x∗ is an interior point and a minimum of f , then D2f (x∗) is positive semi-definite.

However, it is not true that if x∗ is a critical point, and D2f (x∗) is negative (positive) semi-definite, then x∗ is a local maximum. A counterexample is f (x) = x3, which has the property that D2f (0) is semidefinite, but x = 0 is not a maximum or minimum.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 17 / 27

Back to the Example

Consider again F(x, y) = x3 − y3 + 9xy the Hessian is: D2F(x, y) = 6x 9 9 −6y

• First order leading principle minor is 6x and the second order leading

principal minor is det

• D2F(x, y)
• = −36xy − 81.

At (0, 0) these two minors are 0 and −81 and hence the matrix is indefinite and this point is neither a local min or a local max (it is a saddle point). At (3, −3) these two minors are positive and hence it is a strict local min of F. Notice that it is not a global min (why?).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 18 / 27

Back to the Example

Consider again F(x, y) = x3 − y3 + 9xy the Hessian is: D2F(x, y) = 6x 9 9 −6y

• First order leading principle minor is 6x and the second order leading

principal minor is det

• D2F(x, y)
• = −36xy − 81.

At (0, 0) these two minors are 0 and −81 and hence the matrix is indefinite and this point is neither a local min or a local max (it is a saddle point). At (3, −3) these two minors are positive and hence it is a strict local min of F. Notice that it is not a global min (why?).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 18 / 27

Back to the Example

Consider again F(x, y) = x3 − y3 + 9xy the Hessian is: D2F(x, y) = 6x 9 9 −6y

• First order leading principle minor is 6x and the second order leading

principal minor is det

• D2F(x, y)
• = −36xy − 81.

At (0, 0) these two minors are 0 and −81 and hence the matrix is indefinite and this point is neither a local min or a local max (it is a saddle point). At (3, −3) these two minors are positive and hence it is a strict local min of F. Notice that it is not a global min (why?).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 18 / 27

Back to the Example

Consider again F(x, y) = x3 − y3 + 9xy the Hessian is: D2F(x, y) = 6x 9 9 −6y

• First order leading principle minor is 6x and the second order leading

principal minor is det

• D2F(x, y)
• = −36xy − 81.

At (0, 0) these two minors are 0 and −81 and hence the matrix is indefinite and this point is neither a local min or a local max (it is a saddle point). At (3, −3) these two minors are positive and hence it is a strict local min of F. Notice that it is not a global min (why?).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 18 / 27

Back to the Example

Consider again F(x, y) = x3 − y3 + 9xy the Hessian is: D2F(x, y) = 6x 9 9 −6y

• First order leading principle minor is 6x and the second order leading

principal minor is det

• D2F(x, y)
• = −36xy − 81.

At (0, 0) these two minors are 0 and −81 and hence the matrix is indefinite and this point is neither a local min or a local max (it is a saddle point). At (3, −3) these two minors are positive and hence it is a strict local min of F. Notice that it is not a global min (why?).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 18 / 27

Sketch of the Proof

Consider: F(x∗ + h) = F(x∗) + DF(x∗)h + 1 2hTD2F(x∗) h + R(h) Ignore R(h) and set DF(x∗) = 0. Then F(x∗ + h) − F(x∗) ≈ 1 2 hTD2F(x∗) h If D2F(x∗) is negative definite, then for all small enough h = 0, the right hand side is negative. Then F(x∗ + h) < F(x∗) for small enough h. Hence, x∗ is a strict local maximizer of F.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 19 / 27

Concavity and Convexity

Definition

A real valued function f defined on a convex subset U of Rn is concave, if for all x, y in U and for all t ∈ [0, 1] : f (tx + (1 − t)y) ≥ t f (x) + (1 − t) f (y)

Definition

A real valued function g defined on a convex subset U of Rn is convex, if for all x, y in U and for all t ∈ [0, 1] : g(tx + (1 − t)y) ≤ t g(x) + (1 − t) g(y)

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 20 / 27

Concavity and Convexity

Definition

A real valued function f defined on a convex subset U of Rn is concave, if for all x, y in U and for all t ∈ [0, 1] : f (tx + (1 − t)y) ≥ t f (x) + (1 − t) f (y)

Definition

A real valued function g defined on a convex subset U of Rn is convex, if for all x, y in U and for all t ∈ [0, 1] : g(tx + (1 − t)y) ≤ t g(x) + (1 − t) g(y)

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 20 / 27

Notice: f is concave if and only if −f is convex. Notice: linear functions are convex and concave.

Definition

A set U is convex if for all x ∈ U and y ∈ U, then for all t ∈ [0, 1] : tx + (1 − t)y ∈ U Concave and convex functions need to have convex sets as their

• domain. Otherwise, we cannot use the conditions above.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 21 / 27

Notice: f is concave if and only if −f is convex. Notice: linear functions are convex and concave.

Definition

A set U is convex if for all x ∈ U and y ∈ U, then for all t ∈ [0, 1] : tx + (1 − t)y ∈ U Concave and convex functions need to have convex sets as their

• domain. Otherwise, we cannot use the conditions above.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 21 / 27

Notice: f is concave if and only if −f is convex. Notice: linear functions are convex and concave.

Definition

A set U is convex if for all x ∈ U and y ∈ U, then for all t ∈ [0, 1] : tx + (1 − t)y ∈ U Concave and convex functions need to have convex sets as their

• domain. Otherwise, we cannot use the conditions above.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 21 / 27

Notice: f is concave if and only if −f is convex. Notice: linear functions are convex and concave.

Definition

A set U is convex if for all x ∈ U and y ∈ U, then for all t ∈ [0, 1] : tx + (1 − t)y ∈ U Concave and convex functions need to have convex sets as their

• domain. Otherwise, we cannot use the conditions above.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 21 / 27

Theorem

Let f be a continuous and differentiable function on a convex subset U of Rn. Then f is concave on U if and only if for all x, y in U : f (y) − f (x) ≤ Df (x) (y − x) = ∂f (x) ∂x1 (y1 − x1) + ... + ∂f (x) ∂xn (yn − xn)

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 22 / 27

Proof for R1

Since f is concave, then t f (y) + (1 − t) f (x) ≤ f (ty + (1 − t)x) ⇔ t(f (y) − f (x)) + f (x) ≤ f (x + t(y − x)) ⇔ f (y) − f (x) ≤ f (x + t(y − x)) − f (x) t ⇔ f (y) − f (x) ≤ f (x + h) − f (x) h (y − x) For h = t(y − x) the limit when h → 0 gives: f (y) − f (x) ≤ f ′(x)(y − x).

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 23 / 27

Theorem

If f is a continuous and differentiable concave function on a convex set U and if x0 ∈ U, then Df (x0)(y − x0) ≤ 0 implies f (y) ≤ f (x0), and if this holds for all y ∈ U, then x0 is a global maximizer of f .

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 24 / 27

Theorem

Let f be a continuous twice differentiable function whose domain is a convex open subset U of Rn. If f is a concave function on U and Df (x0) = 0 for some x0, then x0 is a global maximum of f on U.

Theorem

A continuous twice differentiable function f on an open convex subset U

• f Rn is concave on U if and only if the Hessian D2f (x) is negative

semi-definite for all x in U. The function f is a convex function if and only if D2f (x) is positive semi-definite for all x in U.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 25 / 27

Theorem

Let f be a continuous twice differentiable function whose domain is a convex open subset U of Rn. If f is a concave function on U and Df (x0) = 0 for some x0, then x0 is a global maximum of f on U.

Theorem

A continuous twice differentiable function f on an open convex subset U

• f Rn is concave on U if and only if the Hessian D2f (x) is negative

semi-definite for all x in U. The function f is a convex function if and only if D2f (x) is positive semi-definite for all x in U.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 25 / 27

Second order sufficient conditions for global maximum (minimum) in Rn : Suppose that x∗ is a critical point of a function f (x) with continuous first and second order partial derivatives on Rn. Then x∗ is: A global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) semi-definite on Rn. A strict global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) definite on Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 26 / 27

Second order sufficient conditions for global maximum (minimum) in Rn : Suppose that x∗ is a critical point of a function f (x) with continuous first and second order partial derivatives on Rn. Then x∗ is: A global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) semi-definite on Rn. A strict global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) definite on Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 26 / 27

Second order sufficient conditions for global maximum (minimum) in Rn : Suppose that x∗ is a critical point of a function f (x) with continuous first and second order partial derivatives on Rn. Then x∗ is: A global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) semi-definite on Rn. A strict global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) definite on Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 26 / 27

Why?

The property that critical points of concave functions are global maximizers is an important one in economic theory. For example, many economic principles, such as marginal rate of substitution equals the price ratio, or marginal revenue equals marginal cost are simply the first order necessary conditions of the corresponding maximization problem as we will see. Ideally, such a rule would be sufficient when the objective function is concave.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 27 / 27

Why?

The property that critical points of concave functions are global maximizers is an important one in economic theory. For example, many economic principles, such as marginal rate of substitution equals the price ratio, or marginal revenue equals marginal cost are simply the first order necessary conditions of the corresponding maximization problem as we will see. Ideally, such a rule would be sufficient when the objective function is concave.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 27 / 27

Why?

The property that critical points of concave functions are global maximizers is an important one in economic theory. For example, many economic principles, such as marginal rate of substitution equals the price ratio, or marginal revenue equals marginal cost are simply the first order necessary conditions of the corresponding maximization problem as we will see. Ideally, such a rule would be sufficient when the objective function is concave.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 2 10 September 2010 27 / 27