Mathematical Economics: Lecture 8

Mathematical Economics:Lecture 8

Yu Ren

WISE, Xiamen University

October 17, 2012

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Chapter 13: Function of Several Variables

Outline

1 Chapter 13: Function of Several Variables

Yu Ren Mathematical Economics: Lecture 8

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New Section

Chapter 13: Functionof Several Variables


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Functions between Euclidean Spaces

A function f : A→ B a set A to a set B is a rulethat assigns to each object in A, one and onlyone object in B.

Domain: the set A of elements on which f isdefinedTarget: the set B in which f takes its valuesImage: y = f (x) ∈ BPreimage: the preimage of V isf−1(V ) = {a ∈ A : f (a) ∈ V}


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A function f : A→ B a set A to a set B is a rulethat assigns to each object in A, one and onlyone object in B.

Surjective: the whole target space of f isthe image of fInjective: f : A→ B is one-to-oneInverse function: f−1 : f (C)→ CComposition: f : A→ B and g : C → D.The composition of f with g,(g ◦ f )(x) = g(f (x)).


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Example

Example 13.1 the function f : R2 → R1 definedby f (x , y) = x2 + y2.the domain of f is all of R2,the target space of f is R1,the image of f is the set of all nonnegative realnumbers.


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Example

Example 13.2 The domain of the functiong(x) = 1/x is R1 - {0}, all real numbers except0.its image is R1 - {0}, too.


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Vocabulary

Onto: if for each element b ∈ B there is anelement a ∈ A s.t. b = f (a), in other words,if the whole target space of f is the image off , we say f maps A onto B or f is surjectiveinverse: f−1 : f (C)→ C


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Vocabulary

Onto: if for each element b ∈ B there is anelement a ∈ A s.t. b = f (a), in other words,if the whole target space of f is the image off , we say f maps A onto B or f is surjectiveinverse: f−1 : f (C)→ C


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Vocabulary

composition: let f : A→ B and g : C → Dbe two functions. Suppose that B, theimage of f , is a subset of C , the domain ofg. Then, the compostition of f with g,g ◦ f : A→ D is defined as the function(g ◦ f )(x) = g(f (x)) for all x ∈ AFigure 13.20


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Vocabulary: Example

Example 13.9 function f (x1, x2) = x21 + x2

2 , itstarget space is R1; its image is the set of allnonnegative real numbers. Since the two arenot the same, f is not onto. Neither is fone-to-one since f (1,0) = f (0,1) = 1.


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Vocabulary: Example

Example 13.10 the target space of the functionf (x) = 1

x is R1, but its image is R1 - 0. f is notonto, and is a one-to-one function.


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Vocabulary: Example

Example 13.11 consider the functions f and gfrom R1 to R1 defined by f (x) = 2x andg(x) = 2x − 1. For both of these, the domainand image are all of R1. Both are one-to-onemaps of R1 onto R1. Then their inverses aregiven by

f−1(y) =12

y and g−1(y) =12(y + 1).


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Vocabulary: Example

Example 13.12 the function h(x) = sin x2 is thecomposition of f (x) = x2 with g(x) = sin x .the function h(x) = (x + 4)3 is the compositionof f (x) = x + 4 with g(x) = x3.the function h(x) =

√x2 + y is the composition

of f (x , y) = x2 + y with g(z) =√

z.


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Utility Function

A utility function is a function which assigns anumber of u(x1, · · · , xk) to each commoditybundle.Some regularly used functions

Linear: q = a1x1 + a2x2

Cobb-Douglas: q = kxb11 xb2

2

Liontiff: q = min{x1c1, x2

c2}

CES: q = k(c1x−a1 + c2x−a

2 )−b/a


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Utility Function

A utility function is a function which assigns anumber of u(x1, · · · , xk) to each commoditybundle.Some regularly used functions

Linear: q = a1x1 + a2x2

Cobb-Douglas: q = kxb11 xb2

2

Liontiff: q = min{x1c1, x2

c2}

CES: q = k(c1x−a1 + c2x−a

2 )−b/a


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Level Curve

Level curve: for each (x , y) we onceevaluate f (x , y) to obtain z0. Then sketchthe locus in the xy − plane of all other (x , y)pairs for which f has the same value of z0.Example 13.3 Figure 13.7.Level sets for production functions arecalled isoquants. Figure 13.10.Level curves of a utility function are calledindifference curves.


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Level Curve



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Level Curve



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Level Curve: Example

Example 13.3 f (x , y) = x2 + y2 start with (0, 1),f = 1⇒ the set {(x , y) : x2 + y2 = 1},a circle ofradius 1 about the origin in the plane.⇒the levelcurve f−1(1)


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Level Curve: Example

Example 13.3 f (x , y) = x2 + y2 choose anotherpoint (2, 1), f = 1⇒ the set{(x , y) : x2 + y2 = 5} ⇒the level curve f−1(5)we have drawn the level curvesf−1(1),f−1(5),f−1(4) and f−1(9)in Figure 13.7Every point on the plane lies on one and onlyone level curve of f . For z = x2 + y2, all the levelcurves are circles about the origin except thatf−1(0) is just the origin itself.


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Theorem 13.1 Let f : Rk → R1 be a linearfunction. Then ∃ a vector a ∈ Rk s.t.f (x) = ax for all x ∈ Rk .Theorem 13.2 Let f : Rk → Rm be a linearfunction. Then ∃ an m × k matrix A s.t.f (x) = Ax for all x ∈ Rk .


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Theorem 13.1 Let f : Rk → R1 be a linearfunction. Then ∃ a vector a ∈ Rk s.t.f (x) = ax for all x ∈ Rk .Theorem 13.2 Let f : Rk → Rm be a linearfunction. Then ∃ an m × k matrix A s.t.f (x) = Ax for all x ∈ Rk .


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Quadratic Forms: A quadratic form on Rk is areal-valued function of the formQ(x1, · · · , xk) =

∑ki ,j=1 aijxixj or

Qx1, · · · , xk) = x ′Ax , where A is a uniquesymmetric matrix. (Theorem 13.3)


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Monomial: f : Rk → R1 is a monomial iff (x1, · · · , xk) = cxa1

1 xa22 · · · x

akk .

Polynomial : a finite sum of monomial on Rk

Affine: f (x) = Ax + b. A : m × k , b:m-vector


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1 xa22 · · · x

akk .




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1 xa22 · · · x

akk .




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Monomial: Example

Example 13.6f (x1, x2) = −4x2

1 x2 is a monomial of degreethree;f (x1, x2, x3) = 3x2

1 x32 x3 is a monomial of

degree six;a constant function is a monomial of degreezero;each term of a linear function is a monomialof degree oneeach term of a quadratic form is amonomial of degree two.


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Affine: Example

Example 13.7 If f : Rk → Rm is polynomial ofdegree one, then each component of f has theform

fi(x) = ai · x + bi .

Therefore, f itself has the form: f (x) = A · x + b,for some m x k matrix A and some m-vector b.Such function is called an affine function.


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Continuous

Definition: f is continuous at x if xn → xthen f (xn)→ f (x).Theorem 13.4 Suppose f and g arecontinuous at x . Then f + g, f − g and f · gare all continuous at x .


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Continuous

Definition: f is continuous at x if xn → xthen f (xn)→ f (x).Theorem 13.4 Suppose f and g arecontinuous at x . Then f + g, f − g and f · gare all continuous at x .


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Continuous

Theorem 13.5 f : Rk → Rm. f is continuousif and only if fi : Rk → R1 is continuous.Theorem 13.7 f : Rk → Rm, g : Rm → Rn.Both f and g are continuous.g ◦ f : Rk → Rn is continuous.


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Continuous

Theorem 13.5 f : Rk → Rm. f is continuousif and only if fi : Rk → R1 is continuous.Theorem 13.7 f : Rk → Rm, g : Rm → Rn.Both f and g are continuous.g ◦ f : Rk → Rn is continuous.


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Continuous: Example

Example 13.8 If f is not continuous at x, there isa sequence {xn}∞n=1 which converges to x andfor which f (xn) does not converge to f (x).

f (x) ={

1, if x > 0,0, if x ≤ 0,

note that f (0) = 0, but f (x) = 1 for x arbitrarilyclose to 0 on the right hand side of 0. Thesequence {1/n} converges to 0, but f (1/n) = 1converges to 1, which is not f (0) = 0. seeFigure 13.18.


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