mathematical economics: lecture 15 · math chapter 20: homogeneous and homothetic functions new...

Mathematical Economics:Lecture 15

Yu Ren

WISE, Xiamen University

November 19, 2012

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Chapter 20: Homogeneous and Homothetic Functions

Outline

1 Chapter 20: Homogeneous and HomotheticFunctions

Yu Ren Mathematical Economics: Lecture 15

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New Section

Chapter 20:Homogeneous and

Homothetic Functions


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Definitions

Definition: For any scalar k , a real-valuedfunction f (x1, x2, · · · , xn) is homogenous ofdegree k if f (tx1, · · · , txn) = tk f (x1, · · · , xn)for all x1, · · · , xn and all t > 0focus on homogenous functions defined onthe positive orthant Rn

+


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Example

Example 20.1

(a) x21 x2 + 3x1x2

2 + x32

(b) x71 x2x2

3 + 5x61 x4

2 − x52 x5

3

(c) 4x21 x3

2 − 5x1x22

(d) z = a1x1 + a2x2 + · · · anxn

(e) z =∑

aijxixj


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Example 20.2

Example 20.2 Replace x1, x2, x3 by tx1, tx2, tx3respectively in Example 20.1a and 20.1b yields

(tx1)2(tx2) + 3(tx1)(tx2)

2 + (tx2)3

= t2x21 tx2 + 3tx1t2x2

2 + t3x32

= t3(x21 x2 + 3x1x2

2 + x32 )


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Example 20.2

(tx1)7(tx2)(tx3)

2 + 5(tx1)6(tx2)

4 − (tx2)5(tx3)

5

= t10(x71 x2x2

3 + 5x61 x4

2 − x52 x5

3 ).

However, no such relationship exists forExample 20.1c.


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Example 20.3

Example 20.3 The function

f1(x1, x2) = 30x1/21 x3/2

2 − 2x31 x−1

2

is homogeneous of degree two. The function

f2(x1, x2) = x1/21 x1/4

2 + x21 x−5/4

2

is homogeneous of degree three-quarters. Thefractional exponents in these two examples giveone reason for making the restriction t > 0 inthe definition of homogeneous.


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Example 20.3

Example 20.3 The function

f3(x1, x2) =x7

1 − 3x21 x5

2

x41 + 2x2

1 x22 + x4

2

is homogeneous of degree three(= 7− 4).


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Example 20.4

Example 20.4 However, the onlyhomogeneous functions of one variable arethe functions of the form z = axk , where kis any real number.To prove this statement, let z = f (x) be anarbitrary homogeneous function of onevariable. Let a ≡ f (1) and let x be arbitrary.Then,

f (x) = f (x · 1) = xk f (1) = axk .


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Definitions

Homogenous function in Economics:constant returns to scale↔ homogenous ofdegree one;increasing returns to scale↔ k > 1;decreasing returns to scale↔ k < 1Cobb-Douglas function: q = Axa1

1 xa22 · · · x

ann


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Properties

Theorem 20.1 Let z = f (x) be a C1 function onan open cone in Rn. If f is homogeneous ofdegree k , its first order partial derivatives arehomogeneous of degree k − 1.


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Properties

Theorem 20.2 Let q = f (x) be a C1

homogeneous function on the positive orthant.The tangent planes to the level sets of f haveconstant slope along each ray from the origin.Figure 20.2 and Figure 20.3 (income expansionpath)


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Properties

Theorem 20.3 Let U(x) be a utility function onRn

+ that is homogeneous of degree k . Then, (i)the MRS is constant along rays from the origin.(ii) income expansion paths are rays from theorigin. (iii) the corresponding demand dependslinearly on income (iv) the income elasticity ofdemand is identically 1.


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Properties

Let q = f (x) be a production function on Rn, thatis homogeneous of degree k . Then (i) themarginal rate of technical substitution (MRTS) isconstant along rays from the origin (ii) thecorresponding cost function is homogeneous ofdegree 1/k : C(q) = bq1/k


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Calculus criterion

Theorem 20.4 Let f (x) be a C1 homogeneousfunction of degree k on Rn

+. ThenX TOf (x) = kf (x)


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Calculus criterion

Theorem 20.5 Suppose that f (x1, · · · , xn) is aC1 function on the positive orthant Rn

+. Supposethat X TOf (x) = kf (x) for all X ∈ Rn

+. Then f ishomogeneous of degree k .


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Properties

Homogenizing a functionTheorem 20.6: Let f be a real-valuedfunction defined on a cone C in Rn. Let kbe an integer. Define a new functionF (x1, x2, · · · , xm, z) = zk f (x1

z ,x2z . · · · ,

xnz ).

Then F is a homogeneous function ofdegree k. And F (x ,1) = f (x).


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Properties

Theorem 20.7: Suppose (x , z)→ f (x , z) is afunction that is homogenous of degree k on aset C × R+ for some cone C in Rn and thatF (x ,1) = f (x) for all x.ThenF (x1, x2, · · · , xm, z) = zk f (x1

z ,x2z . · · · ,

xnz )


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Example 20.5

Example 20.5 If f (x) = xa on R+, then itshomogenization of degree one is

F (x , y) = y ·(

xy

)a

= xay1−a


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Example 20.6

Example 20.6 If f is the nonhomogeneousfunction x → x − ax2, then itshomogenization of degree one is

F (x , y) = y · f(

xy

)= y

[(xy

)− a

(xy

)2]

= x − a(

x2

y

).


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Example 20.7

Example 20.7 In a two-factorconstant-return-to-scale production process, aneconometrician estimates that when the secondfactor is held constant, the production for thefirst factor is f1(x1) = xa

1 for some a ∈ (0,1).Then, the complete production function wouldbe the Cobb-Douglas production functionF (x1, x2) = xa

1 x1−a2 , as we computed in Example

20.5. If units are chosen so that x2 = 1 duringthe estimation of f1, then the estimated functionis the restriction f1(x1) = F (x1,1).


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Example 20.7

The marginal product of the hidden factor x2when x2 = 1 is

∂F∂x2

(x1,1) = (1−a)xa1 x−a

2

∣∣∣∣ x2 = 1 = (1− a)f (x1)

in the specially chosen units of x2 for whichf1(x1) = F (x1,1).


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Homothetic Function

Definition: A function v : Rn+ → R is called

homothetic if it is a monotone transformation ofa homogeneous function.


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Homothetic Function

Definition: If X ,Y ∈ Rn X ≥ Y if xi ≥ yi fori = 1, · · · ,n; X > Y if xi > yi for i = 1, · · · ,n. Afunction u : Rn

+ → R is monotone if for allX ,Y ∈ Rn X ≥ Y ⇒ U(X ) ≥ U(Y ), is strictlymonotone if for all X ,Y ∈ Rn X > Y ⇒U(X ) > U(Y ),


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Homothetic Function

Example 20.13The two functions at thebeginning of this section,

v(x , y) = x3y3 + xy , and w(x , y) = xy + 1,

are homothetic functions with u(x , y) = xy andwith g1(z) = z3 + z and g2(z) = z + 1,respectively. The five examples in Example 20.1are homothetic functions.


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Homothetic Function

Theorem 20.8 Let u;Rn+ → R be a strictly

monotone function. Then u is homothetic if andonly if for all X and Y in Rn

+, U(X ) ≥ U(Y )⇔U(αX ) ≥ U(αY ) for all α > 0


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Homothetic Function

Theorem 20.9 Let u be a C1 function on Rn+. If

u is homothetic, then the slopes of the tangentplanes to the level sets of u are constant alongrays from the origin; in other words, for every i , jand for every X ∈ Rn

∂U∂xi

(tX )∂U∂xj

(tX )=

∂U∂xi

(X )∂U∂xj

(X )


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Homothetic Function

Theorem 20.10 Let u be a C1 function on Rn+. If

∂U∂xi

(tX )∂U∂xj

(tX )=

∂U∂xi

(X )∂U∂xj

(X )

holds for all X ∈ Rn+ all t > 0, and all i , j , then u

is homothetic


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Cardinal and Ordinal Utility

A property of utility function is calledOrdinal if it depends only on the shape andlocation of a consumer’s indifference sets.A property of utility function is calledCardinal if it depends on the actual amountof utility that the function assigns to eachindifference sets.A characteristic of function is called ordinalif every monotonic transformation of afunction with this characteristic still has thischaracteristic. Cardinal properties are notpreserved by monotonic transformation.


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Example 20.9

Example 20.9The function

3z + 2, z2, ez , ln z

are all monotonic transformations of R++, theset of all positive scalers. Consequently, theutility functions

3xy +2, (xy)2, (xy)3+xy , exy , ln xy = ln x + ln y

are monotonic transformations of the utilityfunction u(x , y) = xy .


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Example 20.10

Example 20.10 Consider the class of utilityfunctions on R2

+ that aremonomials−polynomials with only one term; forexample, the polynomial u(x , y) = x2y . Theutility function v(x , y) = x2y + 1 is a monotonictransformation of u. As we discussed above,both u and v have the same indifference curves.However, v is not monomial. So, beingmonomial is a cardinal property. We should beuncomfortable with any theorem which onlyholds for monomial utility functions.


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Example 20.11

Example 20.11 A utility function u(x1, x2) ismonotone in x1 if for each fixed x2, u is anincreasing function x1. If u id differentiable, wecould write this property as ∂u

∂x1> 0. Intuitively,

monotonicity in x1 means that increasingconsumption of commodity one increases utility;in other words, commodity one is a good.


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Example 20.11

This property depends only on the shape andlocation of the level sets of u and on thedirection of higher utility. Therefore, it is anordinal property. Analytically, if g(z) is amonotonic transformation with g′ > 0, then bythe Chain Rule

∂

∂x1[g(u(x1, x2))] = g′(u(x1, x2)) ·

∂u∂x1

(x1, x2) > 0.


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Example 20.12

Example 20.12 Because of their preferences forordinal concepts over cardinal concepts,economists would much rather work with themarginal rate of substitution (MRS) than with themarginal utility (MU) of any given utility function,because MU is a cardinal concept. For example,if v = 2u

∂v∂x1

(x∗1 , x∗2) = 2

∂u∂x1

(x∗1 , x∗2).


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Example 20.12Thus, equivalent utility functions have differentmarginal utilities at the same bundle. On theother hand, MRS is an ordinal concept. Let v bea general monotonic transformation ofu : v(x , y) = g(u(x , y)). The MRS for v at

∂v∂x (x

∗, y∗)∂v∂y (x

∗, y∗)=

∂∂x g(u(x∗, y∗))∂∂y g(u(x∗, y∗))

=∂u∂x (x

∗, y∗)∂u∂y (x

∗, y∗)

the MRS for u at (x∗, y∗).Yu Ren Mathematical Economics: Lecture 15

mathematical economics: lecture 15 · math chapter 20: homogeneous and homothetic functions new...

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