Jhele rene advanced microeconomics

1

1

Advanced Microeconomic Theory

Book used :

“Advanced Microeconomic Theory” by

Jehle, G.A. and Reny, P.J.

Second Edition

You might also have a look at

• “Microeconomic Theory” by

Mas-Collel, Whinston and Green

• “Microeconomic Analysis” by

Varian

2


Outline of Part I

Chapter I: Consumer Theory:

• Preferences

• Utility functions

• The consumer’s problem

• Indirect utility and expenditure

• Properties of consumer demand

Chapter II: Advanced topics of consumer theory

• Duality

• Revealed preferences

• Risk and uncertainty

2

3


Chapter III: Theory of the firm

• Technology, production function

• Cost functions

• Profit maximization under perfect competition

Chapter IV: Partial Analysis

• Partial equilibrium

• monopoly

• (Duopoly -> part of game theory, Horst Raff does that)

4


Chapter V: General Equilibrium

• Definition

• Existence

• Efficiency

• Uniqueness and Stability

3

5

Chapter 1: Consumer Theory

1.1 Primitive Notions

4 building blocks:

• consumption set

• the feasible set

• the preference relation

• the behavioral assumptions

6


consumption set := set of all alternatives a consumer can conceive.

• We assume that the commodity space is infinitely divisible.

• n= dimension of the commodity space.

• Let represent the number of units of good

• is called a consumption bundle or consumption plan.

0, ≥∈ ii xIRx

n

n IRxx +∈= ),...,( 1x

.,,1, nii …=

4

7


Assumption 1.1

[Properties of the Consumption Set X ]

1. X is nonempty and

2. X is closed.

3. X is convex.

4.

(Diagram: Consumption Set)

X∈0

nIRX +⊆

8


Feasible set = set of all alternatives that are achievable for the consumer:

B can be for example the budget set.

But it can also be determined by institutional rules, e.g. rationing.

Note that different consumers usually have different feasible sets.

XB ⊂

5

9

1.2 Preferences and Utility

1.2.1 Preference relations

Consumer preferences are characterized axiomatically.

Formally, a preference relation is a binary relation on X

means “ is as least as good as “

Two axioms:

Axiom 1: Completeness: For all and in X

or

Axiom 2: Transitivity: For all and in X :

If and then

21xx ≻ 1

x2

x

1x

2x

21xx ≻ 12

xx ≻21, xx

3x

21xx ≻ 32

xx ≻ 31xx ≻

10


Definition 1.1 [preference relation]

A binary relation on X that satisfies the two axioms is called a preference relation.

Definition 1.2 [Strict preference relation]

“ is strictly better than “ if and only if

and not:

Definition 1.3 [Indifference relation]

if and only if and

21xx ≻

21xx ≻ 12

xx ≻

21~ xx

21xx ≻ 12

xx ≻

1x

2x

6

11


Definition 1.4 [Sets in X derived from the preference relation]

1.

is called the “as least as good set”

2.

is called the “no better set”

3.

is called the “worse than set”

4.

is called the “preferred to set”

5.

is called the “indifference set”

},|{)(00

xxXxxx ≻≻ ∈=

},|{)(00

xxXxxx ≻≺ ∈=

},|{)(00

xxXxxx ≻≺ ∈=

},|{)(00

xxXxxx ≻≻ ∈=

}~,|{)(~00

xxXxxx ∈=

12


Axiom 3: [Continuity]: For all

The “as least as good set”

and the “no better set”

are closed in

The continuity axioms rules out that sudden preference reversals occur.

Example for a non-continuous preference relation is the lexicographic preference relation.

+∈ nIRx

+n

IR

)(x≻

)(x≺

7

13


Axiom 4’: [Local Non-satiation]: For all

and all there exists some

such that

is a “ball” with radius around x0

This axiom rules out the possibility of having “thick zones of indifference”

+∈ nIRx

0

0>ε

+∩∈ nIRxBx )( 0

ε0

xx ≻

0( )B xε ε

14


Axiom 4: [Strict Monotonicity ]: For all

If then while if then

Where means for all i

and means for all i.

Monotonicity is stronger than local non-satiation!

(Diagram: Non-Satiation/Monotonicity)

+∈ nIRxx

10,

,10 xx ≥ .10

xx ≻,10

xx ≻ ,10

xx >>

10xx ≥ 10

ii xx ≥

10xx >>

10

ii xx >

8

15


Axiom 5’: [Convexity ]:

If then for all

Axiom 5: [Strict Convexity ]:

If and then for all

(Diagram: Convexity)

10xx ≻ ]1,0[∈t110

)1( xxttx ≻−+

10xx ≻10

xx ≠110

)1( xxttx ≻−+)1,0(∈t

16


The marginal rate of substitution (MRS):

indicates how much a consumer is willing to give up of good 1, say, to obtain 1 more unit of good 2.

The slope of the indifference curve is exactly the MRS.

Often we observe: “a diminishing marginal rate of substitution” :

The less we have of one good, the less we are willing to give up of this good for more units of an other good.

The principle of “a diminishing marginal rate of substitution” and convexity of preferences are closely related.

9

17

1.2.2 The Utility Function

• Definition 1.5 [A utility function representing the preference relation _ ]

• A real-valued function

is called a utility function representing the preference relation _ , if for all :

_

IRIRun →+ :

≻ nIRxx

+∈ , 10

1010 x )( )( ≻xxuxu ⇔≥

≻

18


• Does a representing utility function always exists?

� Yes, if preference relation is rational and continuous [Debreu `54]

• It‘s easier to prove the result for monotonic preferences!

• Theorem 1.1: (Existence Theorem)

If a binary relation is complete, transitive, continuous, and strictly monotonic,

there exists a continuous function,

which represents .

, : IRIRun →+

≻

≻

10

19


• Proof: (to find at least one such function)

• Step 1: [Construction]:Define and consider

defined such that is satisfied. (P.1)

Note:

i.e. u(x)e is a point on a ray from 0 to e!

(Diagram: Constructing the Mapping u → next page)

nIRe +∈= )1,...,1(

})(,),({)(

times

��…

n

xuxuexu =×

xexu ~)( ×, : IRIRu

n →+

20


2x

1x

1

1

e

)(xu

)(xu

x

exu )(

11

21


• Step 2: [Existence of such a number ]

• Define:

If we find then

and satisfies (P.1).

By monotonicity:

By continuity of , A and B are closed.

Hence:

)(xu

}|0{ xettA ≻×≥≡}|0{ xettB ≺×≥≡

BAt ∩∈* ,~* xet

)(* xut =

ttBtBt

ttAtAt

≤′∀∈′⇒∈

≥′∀∈′⇒∈

],[ ∞= tA

],0[ tB =

≻

22


• Now completeness implies:

_ or _

• But this implies

i.e. there is at least one t satisfying (P.1).

xet ≻× xet ≺×

BAt ∪∈

[ ] [ ]∞∪=∪=⇒ + ,,0 ttBAIR

∅≠∩⇒≤⇒ BAtt

12

23


• Step 3: [Uniqueness of t]

Assume there are numbers with

and then, by transitivity,

and, by monotonicity,

• Step 4: [u(x) represents ]

Consider two bundles and their associated utility numbers

with

21, tt

xet ~1 ,~2 xet etet 21 ~

21 , xx)( ),( 21

xuxu

.~)( ,~)( 2211xexuxexu

.21 tt =

≻

24


• Then:

_

_ [by (P.1)]_ [Transitivity]

[Monotonicity]

_

• Step 5: [Continuity]

Theorem A1.6:

21xx ≻

⇔⇔⇔ exuxxexu )(~~)( 2211 ≻

exuexu )()( 21 ≻)()( 21

xuxu ≥≻ represents )(xu⇒

( ) { } interval.open an also is ),()(,|),(

image inverse the,),( intervalopen every for

ifonly and if continuous is :

1 baxuIRxxbau

IRba

IRIRu

n

n

∈∈=

∈

→

+−

+

13

25


• a < b.

•

• By continuity _ and _ are closed sets, hence their

complements and are open.

• The intersection is open. QED

)(ae≺

)()( beae ≺≻ ∩

( ) { }

)()(

(P.1) })({

tymonotonici }{

})({

image inverse def. ),()(,|),(1

beae

beexuaeIRx

bexaeIRx

bxuaIRx

baxuIRxxbau

n

n

n

n

≺≻

≺≺

≺≺

∩=

∈=

∈=

<<∈=

∈∈=

+

+

+

+−

)(ae≻ )(be≺

)(be≻

26


• Theorem 1.2: [Invariance of utility function to positive monotonic transformations]

Let _ be a preference relation on and let

represent it.

Then also represents _ if an only if

where is a strictly increasing function.IRIRf →:

≻

≻ )(xu

)(xv, ))(()( n

IRxxufxv +∈∀=

nIR+

14

27


• Definition: A function is (strictly) quasi-concave if the superior set

is (strictly) convex for all . (Diagram: Quasi-C)

• Theorem 1.3 [Properties of Preferences and utility functions]

Let _ be represented by

Then: 1.) is strictly increasing

_ is strictly monotonic.

2.) is (strictly) quasi-concave

_ is (strictly) convex.

IRIRfn →:

IRa ∈})(|{ axfx ≥

≻

≻

≻

⇔

⇔

)(xu

)(xu

IRIRun →+:

28


• Representing preferences by utility functions makes life easier.

• If necessary, we assume to be differentiable.

• marginal utility w.r.t. commodity i.

The marginal rate of substitution (MRS)

• Consider the utility function

• The indifference curve at

)(xu

=∂

∂

ix

u

),( 21 xxuu =

),( 1

2

1

1

1xxx =

15

29


• is given by

• MRS is given by

• The MRS says how much a consumer is willing

to give up of commodity 2 (or to pay in terms

of commodity 2) to get one more (small) unit of commodity 1.

• If preference are strictly convex, then MRS is strictly diminishing

( ) ( ).0

,,)(

!

2

2

1

2

1

11

1

1

2

1

11 =∂

∂+

∂

∂= dx

x

xxudx

x

xxuxdu

( )( ) .

,

,)(

2

1

2

1

1

1

1

2

1

1

1

21

12xxxu

xxxu

dx

dxxMRS

∂∂

∂∂=−=

30

A.1: Some Technical Additions

• Definition: [quasi-concavity]

is quasi-concave if and only if

Define

as the superior set for level

• Theorem A1.14:

is a quasi-concave function

if is a convex set for all .

IRDf → :

:, 21 Dxx ∈∀

})(,|{)( 00 yxfDxxyS ≥∈≡

)}(),(min{))1(( 2121 xfxfxttxf ≥−+ ]1,0[∈∀t

0y

IRDf →:

)( yS IRy ∈

16

31

1.3 The Consumer`s Problem

• Behavioral Assumption:

The consumer seeks such that _

for all

• Assumption 1.2 [Consumers Preferences]

Consumers preferences satisfy Axioms 1-5 (complete-ness, transitivity, continuity, strict monotonicity, strict convexity), hence can be represented by a continuous, strictly increasing, strictly quasi-concave utility function.

• Assumption: Market economy, i.e, consumers take prices as given, where

nIRBx +⊂∈* xx ≻*

.Bx ∈

.0),,( 1 >>= nppp …

32


• The budget set: given income

where

• The utility maximization problem:

(UM)

}||{ yxpIRxxBn ≤⋅∈= +

∑=

=⋅n

i

ii xpxp1

)( max xu ..ts yxp ≤⋅n

IRx +∈

,0≥y

17

33


• Note that if solves the problem, then

hence _

• Note also, since is compact, there is always a solution to (UM) by the Theorem of Weierstraß (A1.10).

• Since is convex and is strictly quasi-concave the solution is unique and lies on the boundary of B.

• Solution is function of y and p and called

Marshallian demand function

*x

Bxxuxu ∈∀≥ )()( *

Bxxx ∈∀ * ≻B

B u

),(*

ypxx ii =

34


Budget set,

in the case of two commodities

1x

2x

B

α

2/ py

1/ py

},|{ yxpIRxxB n ≤⋅∈= +

2

1tanp

p−=α

18

35


2x

1x

*

1x

*x

*

2x

The solution of the consumer’s

Utility-maximization problem.

2/ py

1/ py

36


0

2

0py

),,(00

2

0

12 yppx

0

2

0

1 pp

2x

1x

The consumer’s problem and

consumer demand behavior (1)

0

2

1

1 pp

),,(00

2

1

12 yppx

),,(00

2

0

11 yppx ),,(00

2

1

11 yppx

19

37


The consumer’s problem and

consumer demand behavior (2)

0

1p

1p

1x

1

1p

),,(00

2

0

11 yppx ),,(00

2

1

11 yppx

),,(00

211 yppx

38


• including differentiability, to solve the problem, we employ the Kuhn-Tucker-method (note: x>>0,λ≥0)Lagrangian:

• FOCs:

• Since u is monotonic, (1) holds with equality and (2) is redundant. (Diagram: Kuhn-Tucker)

][)(),( xpyxuxL ⋅−+= λλ

(2) .0][

(1) ,0

(0) ,,1,0)(

**

*

**

=−

≥−

==−∂

∂=

∂

∂

pxy

px y

nipx

xu

x

Li

i

i

i

λ

λ …

20

39

(0) implies

Theorem 1.4 [Sufficiency of F.O.C.’s]

If is continuous and quasi-concave and

, then the solution to (0), (1) solves the consumer’s problem.


)(xu

0),( >>yp

k

j

xu

xu

jkp

pMRS

k

i ==∂∂

∂∂

40

1.4 Indirect Utility & Expenditure

1.4.1 The Indirect utility function

• Direct utility function: represents direct utility from the consumption of commodity bundles

• Indirect utility: gives the utility depending on prices and income after the utility maximization process has been carried out

• Definition:

• Hence,

. s.t. )(max),( yx pxuypvnIRx

≤⋅=+∈

.)),((),( ypxuypv =

21

41

1.4.1 Indirect Utility Function

21 /tan pp−=α

2x

1x

2/ py

),( ypυ

1/ py

α

42


• Theorem 1.6 [Properties of the indirect utility function]

If is continuous and strictly increasing on ,

then is

1. Continuous on

2. Homogeneous of degree zero in

3. Strictly increasing in

4. Decreasing in

5. Quasi-convex in

6. Roy’s identity: if is differentiable at and then:

)(xu nIR+

).,( yp

,p

,+++ × IRIRn

),,( yp

,y

),( ypv

,),(),(

),(

0000

00

y

yp

p

yp

iiypx

∂

∂

∂

∂

−=υ

υ

.,...,1 ni =

),( ypv ),( 00yp

0),( 00 ≠ypvy

22

43




then is

1. Continuous on



4. Decreasing in

5. Quasi-convex in


)(xu nIR+

).,( yp

,p

,+++ × IRIRn

),,( yp

,y

),( ypv

,),(),(

),(

0000

00

y

yp

p

yp

iiypx

∂

∂

∂

∂

−=υ

υ

.,...,1 ni =

),( ypv ),( 00yp

0),( 00 ≠ypvy

44


Homogeneity of the indirect utility function in prices and income

2121// pptptp −=−

2x

1x

22pytpty =

),(),( ypvtytpv =

11pytpty =

),(),( ypvtytpv =

23

45


• Proof of some of the properties:

1. Follows from Theorem of Maximum (A2.4).

2.

3. In order to prove increasingness, we assume that vis strictly positive and differentiable, where (p,y)>>0 and that is differentiable with

),(

] .. )(max[

] .. )(max[),(

ypv

yxptsxu

tyxpttsxutytpv

=

≤⋅=

≤⋅⋅=

)(⋅u .00/)( >>∀>∂∂ xxxu i

46

A: The Envelope Theorem

• Here we apply the Envelope Theorem:As is increasing, the budget constraint is binding.Hence,

Lagrangian:

Let solve the problem. Hence, there must besome such that

As and

Envelope Theorem:

. . )(max),( ypxs.txuypvn

Rx

==+∈

QED .0),(),( *

**

>=∂

∂=

∂

∂λ

λ

y

xL

y

ypv

).()(),( pxyxuxL −+= λλ

)(⋅u

0),(* >>= ypxx

IR∈*λ .)(),( *

***

i

ii

px

xu

x

xLλ

λ−

∂

∂=

∂

∂

ip .00/)( ** >⇒>∂∂ λixxu

24

47




then is

1. Continuous on



4. Decreasing in

5. Quasi-convex in


)(xu nIR+

).,( yp

,p

,+++ × IRIRn

),,( yp

,y

),( ypv

,),(),(

),(

0000

00

y

yp

p

yp

iiypx

∂

∂

∂

∂

−=υ

υ

.,...,1 ni =

),( ypv ),( 00yp

0),( 00 ≠ypvy

48


• Excursion: The Envelope Theorem

Consider the maximization problem

x is a vector of choice variables and a is a vector of exogenous parameters.

Suppose, that for each a the solution is unique and denoted by x(a).We define the maximum-value function:

0. xand 0),( .s.t ),(max ≥=axgaxfx

.0 and 0),( s.t. ),(max)( ≥== xaxgaxfaMx

25

49


• Then (Envelope Theorem):

where

• Proof: see the Appendix of Jehiel and Reny, p. 506-507.

• The Envelope Theorem says that if you change the exogenous parameters of a maximized function, • then the variation of that function is completely determined by

the direct effect,

• and you can neglect indirect effects, because it is already

chosen optimally.

)(),(

)(

aaxjj a

L

a

aM

λ∂

∂=

∂

∂

).,( ),( axgaxfL λ+=

50


6. Proof of Roy’s identity:

Employing the envelope theorem:

Hence,

Accordingly:

QED

. and ** λλ =∂

∂=

∂

∂−=

∂

∂=

∂

∂

y

L

y

vx

p

L

p

vi

ii

][)( xpyxuL ⋅−+= λ

⇒ *

*

**

),(

),()(

ii

y

yp

p

yp

xx

i =−

−=−∂

∂

∂

∂

λ

λυ

υ

26

51




then is

1. Continuous on



4. Decreasing in

5. Quasi-convex in


)(xu nIR+

).,( yp

,p

,+++ × IRIRn

),,( yp

,y

),( ypv

,),(),(

),(

0000

00

y

yp

p

yp

iiypx

∂

∂

∂

∂

−=υ

υ

.,...,1 ni =

),( ypv ),( 00yp

0),( 00 ≠ypvy

52

1.4.2 The Expenditure Function

• Question: What is the minimum level of expenditure to achieve a certain utility level?

• Look at:

• For different this equation generates iso-expenditure curves.

• Definition: the expenditure function is defined as

2211 xpxpe +=e

.)( s.t. min),( uxuxpupen

IRx

≥⋅≡+∈

27

53

• The solution is called Hicksian demand

or compensated demand.

• Thus:


)),(),...,,(),,(( 21 upxupxupxxhn

hhh =

),(),( upxpupeh⋅=

54


• Finding the lowest level of expenditure to a achieve utility level u

21 / pp−

1x

2* / pe

23 / pe

13 / pe

1* / pe 1

1/ pe 1

2 / pe

hx

u

u

),(2 upxh

),(1 upxh

2x

28

55


The Hicksian demand for good 1 as a function of price:

Fig.1.16 (a)

1x

2x

02

01 / pp−),,( 0

2012 uppx

h

u

),,( 02

112 uppx

h

),,( 02

111 uppx

h),,( 0

2011 uppx

h

02

11 / pp−

56


1x

1p

01p

11p

),,( 02

111 uppx

h),,( 0

2011 uppx

h

),,( 0211 uppx

h

The Hicksian demand for good 1 as a function of price:

Fig.1.16 (b)

29

57


Theorem 1.7: (Properties of the expenditure function)

If is strictly increasing and continuous,

then is

a) Zero for the lowest utility level in U.

b) Continuous on its domain .

c) For all strictly increasing and unbounded above in .

d) Increasing in .

e) Homogeneous of degree 1 in .

)(⋅u

),( upe

UIRn ×++

0>>pu

p

p

58


f) Concave in .

g) If, in addition, is strictly quasiconcave, we have Shephard's Lemma:

is differentiable in at and

p

p),( upe

.,,1 ,),(),( 00

00

niupxp

upe h

i

i

…==∂

∂

)(⋅u

,0),,( 00 >>pupo

30

59

• Proof of f):

Assume and are two price vectors

We have to show:

Let minimize

Let minimize

Let minimize


1p

10 ≤≤ t

2p

21 )1( pttppt −+=

1x

),(),()1(),( 21upeupetupte

t≤−+

),( 1upe

),( 2upe

),( upet

2x

*x

),( *upe

2p t

p 1p

60


• Then it must hold: that achieve u.

• In particular,

• Multiplying by and , respectively, adding up:

• by definition:

QED

xpxp

xpxp

222

111

≤

≤

t

),(),()1(),( 21upeupetupte

t≤−+

**212211 ])1([)1( xpxpttpxptxtp t=−+≤−+

)1( t−

x∀

)1,0(∈∀t⇒

.*222

*111

xpxp

xpxp

≤

≤

31

61

uupep =)),(,(υ

1.4.3 Relation between Indirect

Utility and Expenditure Function

• We observe that

and

• Theorem 1.8 [Relations between indirect utility and

expenditure functions]

Let and be continuous and strictly increasing. Then for

1.)

2.)

The proof is a bit technical, and can be found in the book.

yypvpe ≤)),(,(

uupepv ≥)),(,( .),( UIRupn ×∈∀ +

),( ypv ),( upeUuyp ∈≥>> ,0 ,0

yypvpe =)),(,(

0),( >>∀ yp

62



• Observe that for fixed :

• Theorem 1.9: [Duality between Marshallian and

Hicksian Demand Function]

a)

b)

p

⇔uupepv =)),(,(

):(),( 1upvupe

−=

yupve =)),((

):(),( 1ypeypv

−=

)),(,(),( ypvpxypxh

ii =

⇔

)),(,(),( upepxupx i

h

i =

32

63



• Proof of a):

Let and

Then

By Theorem 1.8:

or

But this means solves .

Hence and so

.

),( 000ypxx = )( 00

xuu =000000 )(),((),( uxuypxuypv ===

0)( s.t. }min{ uxuxp ≥⋅

0000 )),(,( yypvpe =

0x

),( 000upxx

h=

)),(,(),( 00000ypvpxypx

h=

000 ),( yupe =

64



• Proof of a):

Let and

Then

By Theorem 1.8:

or

But this means solves .

Hence and so

.

),( 000ypxx = )( 00

xuu =000000 )(),((),( uxuypxuypv ===

0)( s.t. }min{ uxuxp ≥⋅

0000 )),(,( yypvpe =

0x

),( 000upxx

h=

)),(,(),( 00000ypvpxypx

h=

000 ),( yupe =

33

65



1x

2/ py

11

)),(,(

p

ypvpe

p

y=

hx

)),(,(),( upepvypvu ==

u

*

2x

*

1x

2xIllustration of Theorems 1.8

and 1.9

66



1x

2/ py

hx

)),(,(),( 11 ypvpxypxh

=

u

*

2x

),()),(,(),( 111

*

1 ypxypvpxupxxhh

===

1pIllustration of Theorems 1.8

and 1.9

)),(,(),( 11 upepxupxh

=

34

67

1.5 Properties of Consumer Demand

1.5.1 Relative Prices and Relative Income

• Note that real price ratios have dimension units/unit:

• Real income:

• For the utility maximizing consumer only relative prices and real income matter.

• In other words: there is no money illusion.

• In reality, of course, consumers are not always free of money illusion.

i

jj

ij

i

p

p

j

i

unit

unit

EUR

unit

unit

EUR

unit / EUR

unit / EUR=⋅==

jjp

y

j

of units of units / EUR

EUR==

68

1.5.1 Relative Prices and Relative Income1.5.1 Relative Prices and Relative Income1.5.1 Relative Prices and Relative Income1.5.1 Relative Prices and Relative Income

Theorem 1.10: [Homogeneity and Budget Balancedness]

If the consumer’s preference relation is continuous, strictly monotonic, and strictly convex, the consumer’s demand satisfies budged balancedness:

and is homogeneous of degree 0.

⋅==⇒ −

nn

n

n p

y

p

p

p

pxtytpxypx ,1,,...,),(),( 11

yypxp =⋅ ),(

35

69

1.5.2 Income and Substitution Effects

• The Hicksian decomposition of a price change. (a)

2x

1x01x

0u

02x

02

01 / pp−

70



2x

1x01x

0u

02x

02

01 / pp−

0

2

1

1 / pp−

36

71



2x

1x11x

12x

01x

0u

1u

02x

02

01 / pp−

0

2

1

1 / pp−

72


The Hicksian decomposition of a price change. (a)

2x

1x11x

12x

01x

sx1

0u

1u

02x

sx2

}SE

��

SE IE

}}

TE

IE

02

01 / pp−

02

11 / pp−

0

2

1

1 / pp−

��

TE

37

73


The Hicksian decomposition of a price change. (b)

1p

1x11x

01x

sx1

01p

11p

),,( 0

21 yppx

),,( 0

211 uppxh

)),(,,(0

211 upeppx=

��

SE IE

TE

��

��

TE

��

1x∆

1p∆ { )),(,,(0

211 ypvppxh

=

74


Different possible responses of quantity demanded to a change in price.

2x 2x2x

1x 1x1x0

1x11x

11

01 xx =

01x1

1x

)(a )(c)(b

38

75


Theorem 1.11: [The Slutsky-Equation]

Let be the Marshallian demand.

Let . Then

�� effect income

effect onsubstituti

*

Effect totalTE

),(),(

),(),(

y

ypxypx

p

upx

p

ypx ij

j

h

i

j

i

∂

∂−

∂

∂=

∂

∂

)),((*ypxuu =

),( ypx

nji ,...,1, =

76


• Proof: Start with Hicksian-demand:

Shepard's Lemma:

Rearranging:

�� j

i

j

i

j

hi

p

upe

y

upepx

p

upepx

p

upx

∂

∂⋅

∂

∂+

∂

∂=

∂

∂ ),(),(,(),(,(),( ****

)),(,(),( **

��y

i

h

i upepxupx =

),(),( *

ypxp

upej

j

=∂

∂

y

ypxypx

p

upx

p

upx ij

j

hi

j

i

∂

∂⋅−

∂

∂=

∂

∂ ),(),(

),(),( **

39

77


The own price effect:

Theorem 1.12: [Negative own-substitution-effect]

Proof: Shepard's Lemma:

because expenditure

function is concave.

y

ypxypx

p

upx

p

ypx ii

i

hi

i

i

∂

∂⋅−

∂

∂=

∂

∂ ),(),(

),(),( *

),(),(

upxp

upe hi

i

=∂

∂

i

hi

jp

upx

p

upe

∂

∂=

∂

∂≥

),(

)(

),(0

2

2

0≤∂

∂

i

hi

p

x

⇒

78


Theorem 1.13 [Law of Demand]

A decrease in the own price of a normal good will cause quantity demanded to increase.

If a decrease of the own price causes a decrease in quantity demanded, the good must be inferior.

40

79


Theorem 1.14: [Symmetric Substitution Terms]

Let be Hicks-demand. Then

Proof:i

h

j

j

h

i

p

upx

p

upx

∂

∂=

∂

∂ ),(),(

),( upxh

j

hi

jiijj

hi

p

upx

pp

upe

pp

upe

p

upx

∂

∂=

∂∂

∂=

∂∂

∂=

∂

∂ ),(),(),(),( 22

80


Theorem 1.15: [Negative semidefinite Substitution Matrix]

The matrix

is negative semi definite.

∂

∂

∂

∂

∂

∂

∂

∂

=

n

hn

hn

n

hh

p

upx

p

upx

p

upx

p

upx

up

),(),(

),(),(

),(

1

1

1

1

⋯

⋮⋱⋮

⋯

σ

41

81


Theorem 1.16:

The matrix with

is symmetric and negative semidefinite.

Result is useful for empirical tests!

Note that the right hand side of (*) is observable!

y

ypxypx

p

ypxupS i

j

j

iij

∂

∂⋅+

∂

∂=

),(),(

),(),(

),( ypS

(*)

82

1.5.3 Some elasticity relations

• Define:

= income elasticity

= price elasticity

income share on good .

),(

),(

ypx

y

y

ypx

i

ii ⋅

∂

∂=η

),(

),(),(

ypx

p

p

ypxup

i

j

j

iij ⋅

∂

∂=ε

, 0≥isy

ypxps ii

i

),(=

i

11

=∑=

n

i

is

42

83

• Theorem 1.17: [Aggregation of Consumer Demand]

a) (Engel aggregation)

b) (Cournot aggregation)

Proof of a):

Differentiate w.r.t. :


11

=⋅∑=

n

i

iis η

j

n

i

iji ss −=⋅∑=1

ε

�

∑∑∑===

⋅=⋅∂

∂=

∂

∂=

n

i

ii

n

i i

i

s

iiin

i

i sx

y

y

x

y

xp

y

xp

ii

111

1 η

η ��

),( ypxpy ⋅=

y

(*)

84


Proof of b)

Differentiate (*) w.r.t. :

Multiply by and rearrange:yp j /

jp

∑=

+∂

∂=

n

i

j

j

ii x

p

xp

1

0

∑ ∑= =

−

∑

∂

∂=

∂

∂=

−

=

n

i

s

n

i i

j

j

iiij

j

ii

s

jj

n

i

iji

j

x

p

p

x

y

xpp

p

x

y

p

y

xp

1 1

1

��

ε

43

85

A useful diagram

Relationships between the UMP and the EMP.

),( ypx ),( uph

),( ypυ ),( upe)),(,(),( upepupe υ=

EquationSlutsky

Identity

sRoy'),(

),(

upe

uph

p∇

=

)),(,(),( yppeyp υυ =

s)derivative(for

)),(,(),( upepxph =υ

)),(,(),( ypphypx υ=

3.E.1) ons(Propositi

PROBLEMS DUAL""EMP The UMPThe

Lemma

sShepard'

86

Slutsky vs. Hicksian compensation

• Hicksian versus Slutskywealth compensation.

1x

2x

),( wpx

onCompensatiSlutsky

onCompensatiHicksian

uxu =)(

p′wpB ,wpB ,′

44

87

A.2: Some Technical Additions

• Definition: [Negative definite]

A matrix is called negative semi-definite, if

:

A matrix is called negative definite if

:

A matrix is positive semi-definite if:

is negative (semi-) definite

AnIRz ∈∀

A−

A

0≤AzzT

0≠∀z0<Azz

T

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Jhele rene advanced microeconomics