EC400 Part II, Math for Micro: Lecture 1econ.lse.ac.uk/staff/lfelli/teach/EC400 Slides Lecture...

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) andUnconstrained optimization.

Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.

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 ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.

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

Course Outline

My coordinates: S.478, x7525, [email protected]

PA: Gill Wedlake, S.379, x6889, [email protected]

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.

or by appointment (e-mail [email protected]).

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

Office Hours:

Suggested Textbooks

Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther 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 andManagement.

Akira Takayama Mathematical Economics.

Suggested Textbooks

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 ofquadratic forms;

(iii) Economic optimization problems have a quadratic objective function,such as risk minimization problems in finance, where riskiness ismeasured by the quadratic variance of the returns from investments.

Among the functions of one variable, the simplest functions with aunique global extremum are the quadratic forms:

y = x2 and y = −x2.

The level curve of a general quadratic form in R2 is

a11x21 + a12x1x2 + a22x

22 = b

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

Among the functions of one variable, the simplest functions with aunique global extremum are the quadratic forms:

y = x2 and y = −x2.

The level curve of a general quadratic form in R2 is

a11x21 + a12x1x2 + a22x

22 = b

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

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

a11x21 + a12x1x2 + a22x

can be written (non uniquely) as:

(x1 x2

) ( a11 a120 a22

) (x1x2

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

a11x21 + a12x1x2 + a22x

can be written (non uniquely) as:

(x1 x2

) ( a11 a120 a22

) (x1x2

Each quadratic form can be represented as

Q(x) = xTA x

where A is a (unique) symmetric matrix:a11 a12/2 ... a1n/2a12/2 a22 ... a2n/2

......

. . ....

a1n/2 a2n/2 ... ann

Conversely, if A is a symmetric matrix, then the real valued functionQ(x) = xTA x, is a quadratic form.

Each quadratic form can be represented as

Q(x) = xTA x

where A is a (unique) symmetric matrix:a11 a12/2 ... a1n/2a12/2 a22 ... a2n/2

......

. . ....

a1n/2 a2n/2 ... ann

Conversely, if A is a symmetric matrix, then the real valued functionQ(x) = xTA x, is a quadratic form.

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, orneither.

Consider for example:y = ax2

then if a > 0, ax2 is non negative and equals 0 only when x = 0. Thisis positive definite, and x = 0 is a global minimizer.

If a < 0, then the function is negative definite.

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, orneither.

Consider for example:y = ax2

then if a > 0, ax2 is non negative and equals 0 only when x = 0. Thisis positive definite, and x = 0 is a global minimizer.

If a < 0, then the function is negative definite.

In two dimensions, for example:

x21 + x22

is positive definite, whereas

−x21 − x22

is negative definite, whereas

x21 − x22

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

Semi-definiteness of Quadratic Forms

There are two “intermediate” cases:

If the quadratic form is always non negative but also equals 0 for nonzero 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 pointsother than the origin is called negative semidefinite.

Semi-definiteness of Quadratic Forms

There are two “intermediate” cases:

If the quadratic form is always non negative but also equals 0 for nonzero 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 pointsother than the origin is called negative semidefinite.

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 6= 0 in Rn,

positive semi-definite if xTA x ≥ 0 for all x 6= 0 in Rn,

negative definite if xTA x < 0 for all x 6= 0 in Rn,

negative semi-definite if xTA x ≤ 0 for all x 6= 0 in Rn,

indefinite xTA x > 0 for some x 6= 0 in Rn and xTA x < 0 for somex 6= 0 in Rn.

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 6= 0 in Rn,

positive semi-definite if xTA x ≥ 0 for all x 6= 0 in Rn,

negative definite if xTA x < 0 for all x 6= 0 in Rn,

negative semi-definite if xTA x ≤ 0 for all x 6= 0 in Rn,

indefinite xTA x > 0 for some x 6= 0 in Rn and xTA x < 0 for somex 6= 0 in Rn.

Application (later this week)

A function y = f (x) of one variable is concave on some interval if itssecond derivative f ′′(x) ≤ 0 on that interval.

The generalization of this result to higher dimensions states that afunction is concave on some region if its matrix of second derivatives(Hessian matrix) is negative semi-definite for all x in the region.

Application (later this week)

A function y = f (x) of one variable is concave on some interval if itssecond derivative f ′′(x) ≤ 0 on that interval.

The generalization of this result to higher dimensions states that afunction is concave on some region if its matrix of second derivatives(Hessian matrix) is negative semi-definite for all x in the region.

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 a12a21 a22

det(A) = |A| = a11a22 − a12a21

The Determinant

Definition

The determinant of a matrix is a unique scalar associated with the matrix.

(a11 a12a21 a22

det(A) = |A| = a11a22 − a12a21

a11 a12 a13a21 a22 a23a31 a32 a33

det(A) = a11 det

(a22 a23a32 a33

)− a12 det

(a21 a23a31 a33

)+ a13 det

(a21 a22a31 a32

Definition

Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deletingn − k columns, say columns i1, i2, ..., in−k and the same n − k rows fromA, 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 orderprincipal minor of A.

Definition

Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deletingn − k columns, say columns i1, i2, ..., in−k and the same n − k rows fromA, 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 orderprincipal minor of A.

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 a12a21 a22

∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33

∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33

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

Example

∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33

∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33

Example

∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33

∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33

Definition

Let A be an (n × n) matrix. The kth order principal submatrix of Aobtained by deleting the last n − k rows and columns from A is called thekth order leading principal submatrix of A denoted Ak .

Definition

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

Definition

Let A be an (n × n) matrix. The kth order principal submatrix of Aobtained by deleting the last n − k rows and columns from A is called thekth order leading principal submatrix of A denoted Ak .

Definition

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

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 arestrictly positive.

A is negative definite if and only if all its n leading principal minorsalternate in sign as follows:

|A1| < 0, |A2| > 0, |A3| < 0 . . .

The kth order leading principal minor should have the same sign of(−1)k .

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 arestrictly positive.

A is negative definite if and only if all its n leading principal minorsalternate in sign as follows:

|A1| < 0, |A2| > 0, |A3| < 0 . . .

The kth order leading principal minor should have the same sign of(−1)k .

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

A is negative semi-definite if and only if every principal minor of oddorder is non positive and every principal minor of even order is nonnegative:

|A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .

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

A is negative semi-definite if and only if every principal minor of oddorder is non positive and every principal minor of even order is nonnegative:

|A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .

Special Case: Diagonal Matrixes

Consider the following (3× 3) matrix A, a diagonal one.

a1 0 00 a2 00 0 a3

A also corresponds to the simplest quadratic forms:

a1x21 + a2x

22 + a3x

Special Case: Diagonal Matrixes

Consider the following (3× 3) matrix A, a diagonal one.

a1 0 00 a2 00 0 a3

A also corresponds to the simplest quadratic forms:

a1x21 + a2x

22 + a3x

This quadratic form will then be:

positive (negative) definite if and only if all the ai s are positive(negative).

It will be positive semi-definite if and only if all the ai s are nonnegative and negative semi-definite if and only if all the ai s are nonpositive.

If there are two ai s of opposite sign, it will be indefinite.

Special Case: (2× 2) Matrixes

Consider the (2× 2) symmetric matrix:

(a bb c

The associated quadratic form is:

Q(x1, x2) = xTA x = (x1, x2)

(a bb c

)(x1x2

ax21 + 2bx1x2 + cx22

Special Case: (2× 2) Matrixes

Consider the (2× 2) symmetric matrix:

(a bb c

The associated quadratic form is:

Q(x1, x2) = xTA x = (x1, x2)

(a bb c

)(x1x2

ax21 + 2bx1x2 + cx22

If a = 0, then Q cannot be negative or positive definite sinceQ(1, 0) = 0.

Assume that a 6= 0 and add and subtract b2x22/a to get:

Q(x1, x2) = ax21 + 2bx1x2 + cx22 +b2

ax22 −

= a(x21 +2bx1x2

a2x22 )− b2

ax22 + cx22

= a(x1 +b

ax2)2 +

(ac − b2)

If a = 0, then Q cannot be negative or positive definite sinceQ(1, 0) = 0.

Assume that a 6= 0 and add and subtract b2x22/a to get:

Q(x1, x2) = ax21 + 2bx1x2 + cx22 +b2

ax22 −

= a(x21 +2bx1x2

a2x22 )− b2

ax22 + cx22

= a(x1 +b

ax2)2 +

(ac − b2)

If both coefficients, a and (ac − b2)/a are positive, then Q will neverbe negative.

It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0and x2 = 0. In other words, if

|a| > 0 and det(A) =

∣∣∣∣ a bb c

∣∣∣∣ > 0

then Q(x1, x2) is positive definite.

Conversely, in order for Q to be positive definite, we need both a anddet(A) =

(ac − b2

)to be positive.

∣∣∣∣ a bb c

∣∣∣∣ > 0

(ac − b2

)to be positive.

∣∣∣∣ a bb c

∣∣∣∣ > 0

(ac − b2

)to be positive.

Similarly, Q will be negative definite if and only if both coefficientsare negative, which occurs if and only if

a < 0 and(ac − b2

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.

Similarly, Q will be negative definite if and only if both coefficientsare negative, which occurs if and only if

a < 0 and(ac − b2

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.

Examples of a (2× 2) matrixes

Consider A =

(2 33 7

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

definite.

Consider B =

(2 44 7

). Since |B1| = 2 and |B2| = −2, B is

indefinite.

Examples of a (2× 2) matrixes

Consider A =

(2 33 7

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

definite.

Consider B =

(2 44 7

). Since |B1| = 2 and |B2| = −2, B is

indefinite.

EC400 Part II, Math for Micro: Lecture 1econ.lse.ac.uk/staff/lfelli/teach/EC400 Slides Lecture...

Documents

Transcript of EC400 Part II, Math for Micro: Lecture 1econ.lse.ac.uk/staff/lfelli/teach/EC400 Slides Lecture...