EC400 Part II, Math for Micro: Lecture 1 Leonardo Felli NAB.SZT 9 - - PowerPoint PPT Presentation

EC400 Part II, Math for Micro: Lecture 1

Leonardo Felli

NAB.SZT

9 September 2010

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Course Outline

Lecture 1: Tools for optimization (Quadratic forms). Lecture 2: Tools for optimization (Taylor’s expansion) and Unconstrained optimization. Lecture 3: Concavity, convexity, quasi-concavity and economic applications. Lecture 4: Constrained Optimization I: Equality Constraints, Lagrange Theorem. Lecture 5: Constrained Optimization II: Inequality Constraints, Kuhn-Tucker Theorem. Lecture 6: Constrained Optimization III: The Maximum Value Function, Envelope Theorem, Implicit Function Theorem and Comparative Statics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Gill Wedlake, S.379, x6889, g.m.wedlake@lse.ac.uk Office Hours:

Thursday 9 September — 10:00-12:00 a.m. Friday 10 September — 10:00-12:00 p.m. Monday 13 September — 10:00-12:00 a.m. Tuesday 14 September — 10:00-12:00 a.m. Wednesday 15 September — 10:00-12:00 a.m. Thursday 16 September — 10:00-12:00 a.m. Friday 17 September — 10:00-12:00 a.m. Wednesday 22 September — 10:00-12:00 a.m.

• r by appointment (e-mail lfelli@econ.lse.ac.uk).

Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Gill Wedlake, S.379, x6889, g.m.wedlake@lse.ac.uk Office Hours:

Thursday 9 September — 10:00-12:00 a.m. Friday 10 September — 10:00-12:00 p.m. Monday 13 September — 10:00-12:00 a.m. Tuesday 14 September — 10:00-12:00 a.m. Wednesday 15 September — 10:00-12:00 a.m. Thursday 16 September — 10:00-12:00 a.m. Friday 17 September — 10:00-12:00 a.m. Wednesday 22 September — 10:00-12:00 a.m.

• r by appointment (e-mail lfelli@econ.lse.ac.uk).

Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Gill Wedlake, S.379, x6889, g.m.wedlake@lse.ac.uk Office Hours:

Thursday 9 September — 10:00-12:00 a.m. Friday 10 September — 10:00-12:00 p.m. Monday 13 September — 10:00-12:00 a.m. Tuesday 14 September — 10:00-12:00 a.m. Wednesday 15 September — 10:00-12:00 a.m. Thursday 16 September — 10:00-12:00 a.m. Friday 17 September — 10:00-12:00 a.m. Wednesday 22 September — 10:00-12:00 a.m.

• r by appointment (e-mail lfelli@econ.lse.ac.uk).

Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Gill Wedlake, S.379, x6889, g.m.wedlake@lse.ac.uk Office Hours:

Thursday 9 September — 10:00-12:00 a.m. Friday 10 September — 10:00-12:00 p.m. Monday 13 September — 10:00-12:00 a.m. Tuesday 14 September — 10:00-12:00 a.m. Wednesday 15 September — 10:00-12:00 a.m. Thursday 16 September — 10:00-12:00 a.m. Friday 17 September — 10:00-12:00 a.m. Wednesday 22 September — 10:00-12:00 a.m.

• r by appointment (e-mail lfelli@econ.lse.ac.uk).

Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne Strom Further Mathematics for Economic Analysis. Alpha C. Chiang Fundamental Methods of Mathematical Economics. Carl P. Simon and Lawrence E. Blume Mathematics for Economists. Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Akira Takayama Mathematical Economics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne Strom Further Mathematics for Economic Analysis. Alpha C. Chiang Fundamental Methods of Mathematical Economics. Carl P. Simon and Lawrence E. Blume Mathematics for Economists. Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Akira Takayama Mathematical Economics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne Strom Further Mathematics for Economic Analysis. Alpha C. Chiang Fundamental Methods of Mathematical Economics. Carl P. Simon and Lawrence E. Blume Mathematics for Economists. Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Akira Takayama Mathematical Economics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne Strom Further Mathematics for Economic Analysis. Alpha C. Chiang Fundamental Methods of Mathematical Economics. Carl P. Simon and Lawrence E. Blume Mathematics for Economists. Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Akira Takayama Mathematical Economics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne Strom Further Mathematics for Economic Analysis. Alpha C. Chiang Fundamental Methods of Mathematical Economics. Carl P. Simon and Lawrence E. Blume Mathematics for Economists. Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Akira Takayama Mathematical Economics.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27

What is a quadratic form?

Quadratic forms are useful because: (i) The simplest functions after linear ones; (ii) Conditions for optimization techniques are stated in terms of quadratic forms; (iii) Economic optimization problems have a quadratic objective function, such as risk minimization problems in finance, where riskiness is measured by the quadratic variance of the returns from investments.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27

What is a quadratic form?

Quadratic forms are useful because: (i) The simplest functions after linear ones; (ii) Conditions for optimization techniques are stated in terms of quadratic forms; (iii) Economic optimization problems have a quadratic objective function, such as risk minimization problems in finance, where riskiness is measured by the quadratic variance of the returns from investments.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27

What is a quadratic form?

Quadratic forms are useful because: (i) The simplest functions after linear ones; (ii) Conditions for optimization techniques are stated in terms of quadratic forms; (iii) Economic optimization problems have a quadratic objective function, such as risk minimization problems in finance, where riskiness is measured by the quadratic variance of the returns from investments.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27

Among the functions of one variable, the simplest functions with a unique global extremum are the quadratic forms: y = x2 and y = −x2. The level curve of a general quadratic form in R2 is a11x2

1 + a12x1x2 + a22x2 2 = b

and can take the form of an ellipse, a hyperbola, a pair of lines, a point, or possibly, the empty set.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 6 / 27

Among the functions of one variable, the simplest functions with a unique global extremum are the quadratic forms: y = x2 and y = −x2. The level curve of a general quadratic form in R2 is a11x2

1 + a12x1x2 + a22x2 2 = b

and can take the form of an ellipse, a hyperbola, a pair of lines, a point, or possibly, the empty set.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 6 / 27

Definition of a Quadratic Form

Definition

Definition: A quadratic form on Rn is a real valued function Q(x1, x2, ..., xn) =

• i≤j

aijxixj The general quadratic form of a11x2

1 + a12x1x2 + a22x2 2

can be written (non uniquely) as:

• x1

x2 a11 a12 a22 x1 x2

• Leonardo Felli (LSE, NAB.SZT)

EC400 Part II, Math for Micro: Lecture 1 9 September 2010 7 / 27

Definition of a Quadratic Form

Definition

Definition: A quadratic form on Rn is a real valued function Q(x1, x2, ..., xn) =

• i≤j

aijxixj The general quadratic form of a11x2

1 + a12x1x2 + a22x2 2

can be written (non uniquely) as:

• x1

x2 a11 a12 a22 x1 x2

• Leonardo Felli (LSE, NAB.SZT)

EC400 Part II, Math for Micro: Lecture 1 9 September 2010 7 / 27

Each quadratic form can be represented as Q(x) = xTA x where A is a (unique) symmetric matrix:      a11 a12/2 ... a1n/2 a12/2 a22 ... a2n/2 . . . . . . ... . . . a1n/2 a2n/2 ... ann      Conversely, if A is a symmetric matrix, then the real valued function Q(x) = xTA x, is a quadratic form.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 8 / 27

Each quadratic form can be represented as Q(x) = xTA x where A is a (unique) symmetric matrix:      a11 a12/2 ... a1n/2 a12/2 a22 ... a2n/2 . . . . . . ... . . . a1n/2 a2n/2 ... ann      Conversely, if A is a symmetric matrix, then the real valued function Q(x) = xTA x, is a quadratic form.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 8 / 27

Definiteness of Quadratic Forms

A quadratic form always takes the value 0 when x = 0. We focus on the question of whether x = 0 is a max, a min, or neither. Consider for example: y = ax2 then if a > 0, ax2 is non negative and equals 0 only when x = 0. This is positive definite, and x = 0 is a global minimizer. If a < 0, then the function is negative definite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 9 / 27

Definiteness of Quadratic Forms

A quadratic form always takes the value 0 when x = 0. We focus on the question of whether x = 0 is a max, a min, or neither. Consider for example: y = ax2 then if a > 0, ax2 is non negative and equals 0 only when x = 0. This is positive definite, and x = 0 is a global minimizer. If a < 0, then the function is negative definite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 9 / 27

In two dimensions, for example: x2

1 + x2 2

is positive definite, whereas −x2

1 − x2 2

is negative definite, whereas x2

1 − x2 2

is indefinite, since it can take both positive and negative values.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 10 / 27

Semi-definiteness of Quadratic Forms

There are two “intermediate” cases: If the quadratic form is always non negative but also equals 0 for non zero x′s, is called positive semi-definite, for example: (x1 + x2)2 which can be 0 for points such that x1 = −x2. A quadratic form which is never positive but can be zero at points

• ther than the origin is called negative semidefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 11 / 27

Semi-definiteness of Quadratic Forms

There are two “intermediate” cases: If the quadratic form is always non negative but also equals 0 for non zero x′s, is called positive semi-definite, for example: (x1 + x2)2 which can be 0 for points such that x1 = −x2. A quadratic form which is never positive but can be zero at points

• ther than the origin is called negative semidefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 11 / 27

Definiteness and Semi-definiteness of Symmetric Matrixes

We apply the same terminology to the symmetric matrix A, such that: Q(x) = xTA x

Definition

Let A be an (n × n) symmetric matrix. Then A is: positive definite if xTA x > 0 for all x = 0 in Rn, positive semi-definite if xTA x ≥ 0 for all x = 0 in Rn, negative definite if xTA x < 0 for all x = 0 in Rn, negative semi-definite if xTA x ≤ 0 for all x = 0 in Rn, indefinite xTA x > 0 for some x = 0 in Rn and xTA x < 0 for some x = 0 in Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 12 / 27

Definiteness and Semi-definiteness of Symmetric Matrixes

We apply the same terminology to the symmetric matrix A, such that: Q(x) = xTA x

Definition

Let A be an (n × n) symmetric matrix. Then A is: positive definite if xTA x > 0 for all x = 0 in Rn, positive semi-definite if xTA x ≥ 0 for all x = 0 in Rn, negative definite if xTA x < 0 for all x = 0 in Rn, negative semi-definite if xTA x ≤ 0 for all x = 0 in Rn, indefinite xTA x > 0 for some x = 0 in Rn and xTA x < 0 for some x = 0 in Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 12 / 27

Application (later this week)

A function y = f (x) of one variable is concave on some interval if its second derivative f ′′(x) ≤ 0 on that interval. The generalization of this result to higher dimensions states that a function is concave on some region if its matrix of second derivatives (Hessian matrix) is negative semi-definite for all x in the region.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 13 / 27

Application (later this week)

A function y = f (x) of one variable is concave on some interval if its second derivative f ′′(x) ≤ 0 on that interval. The generalization of this result to higher dimensions states that a function is concave on some region if its matrix of second derivatives (Hessian matrix) is negative semi-definite for all x in the region.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 13 / 27

The Determinant

Definition

The determinant of a matrix is a unique scalar associated with the matrix. The determinant of a (2 × 2) matrix A = a11 a12 a21 a22

• is:

det(A) = |A| = a11a22 − a12a21

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 14 / 27

The Determinant

Definition

The determinant of a matrix is a unique scalar associated with the matrix. The determinant of a (2 × 2) matrix A = a11 a12 a21 a22

• is:

det(A) = |A| = a11a22 − a12a21

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 14 / 27

The determinant of a (3 × 3) matrix A =   a11 a12 a13 a21 a22 a23 a31 a32 a33  : det(A) = a11 det a22 a23 a32 a33

• − a12 det

a21 a23 a31 a33

• +

a13 det a21 a22 a31 a32

• .

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 15 / 27

Definition

Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deleting n − k columns, say columns i1, i2, ..., in−k and the same n − k rows from A, i1, i2, ..., in−k , is called a kth order principal submatrix of A.

Definition

The determinant of a k × k principal submatrix is called a kth order principal minor of A.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 16 / 27

Definition

Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deleting n − k columns, say columns i1, i2, ..., in−k and the same n − k rows from A, i1, i2, ..., in−k , is called a kth order principal submatrix of A.

Definition

The determinant of a k × k principal submatrix is called a kth order principal minor of A.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 16 / 27

Example

Consider a general (3 × 3) matrix A. There is one third order principal minor: det(A). There are three second ordered principal minors:

• a11

a12 a21 a22

• ,
• a11

a13 a31 a33

• ,
• a22

a23 a32 a33

• There are three first order principal minors: a11, a22 and a33.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27

Example

Consider a general (3 × 3) matrix A. There is one third order principal minor: det(A). There are three second ordered principal minors:

• a11

a12 a21 a22

• ,
• a11

a13 a31 a33

• ,
• a22

a23 a32 a33

• There are three first order principal minors: a11, a22 and a33.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27

Example

Consider a general (3 × 3) matrix A. There is one third order principal minor: det(A). There are three second ordered principal minors:

• a11

a12 a21 a22

• ,
• a11

a13 a31 a33

• ,
• a22

a23 a32 a33

• There are three first order principal minors: a11, a22 and a33.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27

Definition

Let A be an (n × n) matrix. The kth order principal submatrix of A

• btained by deleting the last n − k rows and columns from A is called the

kth order leading principal submatrix of A denoted Ak.

Definition

The determinant of the kth order leading principal submatrix is called the kth order leading principal minor of A denoted |Ak|.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 18 / 27

Definition

Let A be an (n × n) matrix. The kth order principal submatrix of A

• btained by deleting the last n − k rows and columns from A is called the

kth order leading principal submatrix of A denoted Ak.

Definition

The determinant of the kth order leading principal submatrix is called the kth order leading principal minor of A denoted |Ak|.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 18 / 27

Testing Definiteness

Let A be an (n × n) symmetric matrix. Then: A is positive definite if and only if all its n leading principal minors are strictly positive. A is negative definite if and only if all its n leading principal minors alternate in sign as follows: |A1| < 0, |A2| > 0, |A3| < 0 . . . The kth order leading principal minor should have the same sign of (−1)k.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 19 / 27

Testing Definiteness

Let A be an (n × n) symmetric matrix. Then: A is positive definite if and only if all its n leading principal minors are strictly positive. A is negative definite if and only if all its n leading principal minors alternate in sign as follows: |A1| < 0, |A2| > 0, |A3| < 0 . . . The kth order leading principal minor should have the same sign of (−1)k.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 19 / 27

A is positive semi-definite if and only if every principal minor of A is non negative. A is negative semi-definite if and only if every principal minor of odd

• rder is non positive and every principal minor of even order is non

negative: |A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 20 / 27

A is positive semi-definite if and only if every principal minor of A is non negative. A is negative semi-definite if and only if every principal minor of odd

• rder is non positive and every principal minor of even order is non

negative: |A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 20 / 27

Special Case: Diagonal Matrixes

Consider the following (3 × 3) matrix A, a diagonal one. A =   a1 a2 a3   A also corresponds to the simplest quadratic forms: a1x2

1 + a2x2 2 + a3x2 3.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 21 / 27

Special Case: Diagonal Matrixes

Consider the following (3 × 3) matrix A, a diagonal one. A =   a1 a2 a3   A also corresponds to the simplest quadratic forms: a1x2

1 + a2x2 2 + a3x2 3.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 21 / 27

This quadratic form will then be: positive (negative) definite if and only if all the ais are positive (negative). It will be positive semi-definite if and only if all the ais are non negative and negative semi-definite if and only if all the ais are non positive. If there are two ais of opposite sign, it will be indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27

This quadratic form will then be: positive (negative) definite if and only if all the ais are positive (negative). It will be positive semi-definite if and only if all the ais are non negative and negative semi-definite if and only if all the ais are non positive. If there are two ais of opposite sign, it will be indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27

This quadratic form will then be: positive (negative) definite if and only if all the ais are positive (negative). It will be positive semi-definite if and only if all the ais are non negative and negative semi-definite if and only if all the ais are non positive. If there are two ais of opposite sign, it will be indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27

Special Case: (2 × 2) Matrixes

Consider the (2 × 2) symmetric matrix: A = a b b c

• The associated quadratic form is:

Q(x1, x2) = xTA x = (x1, x2) a b b c x1 x2

• =

ax2

1 + 2bx1x2 + cx2 2

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 23 / 27

Special Case: (2 × 2) Matrixes

Consider the (2 × 2) symmetric matrix: A = a b b c

• The associated quadratic form is:

Q(x1, x2) = xTA x = (x1, x2) a b b c x1 x2

• =

ax2

1 + 2bx1x2 + cx2 2

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 23 / 27

If a = 0, then Q cannot be negative or positive definite since Q(1, 0) = 0. Assume that a = 0 and add and subtract b2x2

2/a to get:

Q(x1, x2) = ax2

1 + 2bx1x2 + cx2 2 + b2

a x2

2 − b2

a x2

2

= a(x2

1 + 2bx1x2

a + b2 a2 x2

2) − b2

a x2

2 + cx2 2

= a(x1 + b ax2)2 + (ac − b2) a x2

2

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 24 / 27

If a = 0, then Q cannot be negative or positive definite since Q(1, 0) = 0. Assume that a = 0 and add and subtract b2x2

2/a to get:

Q(x1, x2) = ax2

1 + 2bx1x2 + cx2 2 + b2

a x2

2 − b2

a x2

2

= a(x2

1 + 2bx1x2

a + b2 a2 x2

2) − b2

a x2

2 + cx2 2

= a(x1 + b ax2)2 + (ac − b2) a x2

2

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 24 / 27

If both coefficients, a and (ac − b2)/a are positive, then Q will never be negative. It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0 and x2 = 0. In other words, if |a| > 0 and det(A) =

• a

b b c

• > 0

then Q(x1, x2) is positive definite. Conversely, in order for Q to be positive definite, we need both a and det(A) =

• ac − b2

to be positive.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27

If both coefficients, a and (ac − b2)/a are positive, then Q will never be negative. It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0 and x2 = 0. In other words, if |a| > 0 and det(A) =

• a

b b c

• > 0

then Q(x1, x2) is positive definite. Conversely, in order for Q to be positive definite, we need both a and det(A) =

• ac − b2

to be positive.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27

If both coefficients, a and (ac − b2)/a are positive, then Q will never be negative. It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0 and x2 = 0. In other words, if |a| > 0 and det(A) =

• a

b b c

• > 0

then Q(x1, x2) is positive definite. Conversely, in order for Q to be positive definite, we need both a and det(A) =

• ac − b2

to be positive.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27

Similarly, Q will be negative definite if and only if both coefficients are negative, which occurs if and only if a < 0 and

• ac − b2

> 0. That is, when the leading principal minors alternative in sign. If

• ac − b2

< 0. then the two coefficients will have opposite signs and Q(x1, x2) will be indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 26 / 27

Similarly, Q will be negative definite if and only if both coefficients are negative, which occurs if and only if a < 0 and

• ac − b2

> 0. That is, when the leading principal minors alternative in sign. If

• ac − b2

< 0. then the two coefficients will have opposite signs and Q(x1, x2) will be indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 26 / 27

Examples of a (2 × 2) matrixes

Consider A = 2 3 3 7

• . Since |A1| = 2 and |A2| = 5, A is positive

definite. Consider B = 2 4 4 7

• . Since |B1| = 2 and |B2| = −2, B is

indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 27 / 27

Examples of a (2 × 2) matrixes

Consider A = 2 3 3 7

• . Since |A1| = 2 and |A2| = 5, A is positive

definite. Consider B = 2 4 4 7

• . Since |B1| = 2 and |B2| = −2, B is

indefinite.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 27 / 27