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Intermediate

Microeconomics 1

Syllabus

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1.Introducing the course

*This course is contained of 4 parts:

1. The theory o f consumer behavior

2. The theory of the f i rm

3. Market equ i l ib r ium

4. Monopo ly , monopsony, &monopol is t ic compet it ion

2

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1.Introducing the course

*The analyses are highly based on

mathematics.

*The students will be responsible for

problem solving.

*Discussing groups is recommended. 3

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2.Students Activities

a.Oral exam 15%b.Mid-term exam 30%

c.Exercises 15%

d.Final exam 40%

4

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3.References

a. The main text:

1.J.M.Henderson & R.E. Quandt , (1980) ,

Microeconom ic Theory

b. Complementary texts :1. Eaton, B.C,& Eaton, D.F.,(1995), “Microeconomics”

2. Griffiths,A & S.Wall,(2000),

Intermediate Microeconom ics

3. Laidler,D. & E. Saul, (1989) ,In t roduct ion to Microeconom ics

4. Nicholson,W,(2002),

Microeconom ic Theory

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3.References

5. Varian,H.,(1993),

Intermediate Microeconom ics

6. Varian,H.,(1992),

Microeconom ic analys is

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4.Description of the course

Part #1

Chapters 2 & 3 : The theory of

consumer behavior1. Utility maximization

2. Demand function

3. The Slutsky equation4. Duality theorem

5. Risk and uncertainty

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Part #2

Chapters 4&5 : The theory of the firm

1. Optimizing behavior

2. Cost functions3. Input Demand

4. CES production functions

5. Linear programming 78

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Part #3

Chap ter 6 : Market equilibrium

1. Demand & supply functions

2. Commodity-Market equilibrium3. Input-Market equilibrium

4. Stability of equilibrium89

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Part #4

Chap ter 7 : Monopoly , monopsony,

& monopolistic competition

1. Monopoly : price determination &

applications

2. Monopsony

3. Monopolistic competition9

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Chapter 2

10

Session One

Sess ion Two

Sess ion Three

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Session One

* General goal

Utility Maximization

*Detai led goals

1. Basic concepts2. The first & second order conditions

for Utility maximization11

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1.Introduction

Ses.1 Ch.2

a. Utility function: Definition

b. Measuring the Utility1.Cardinal theory (explanations)

2.Ordinal theory (explanations)

-Rationality axioms

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2. Basic concepts

Ses.1 Ch.2a. The nature of Utility function (explanation)

b. Indifference curves

1. Definition 2. Characteristics (fig.2-1 & 2-2)

c. The rate of commodity substitution

1. Definition 2. Mathematics

3. Economic interpretation

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3. Utility Maximization

Ses.1 Ch.2a. First & second order conditions

1. Mathematics: F.O.C & S.O.C

2. Economic interpretation of F.O.C

3. Example

b. The choice of a utility index (explanation)

c. Special cases: corner solution (fig.2-4)

1. Concave utility function

2. Economic bads

3. I.C are flatter than B.L 14

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Evaluation

Ses.1 Ch.2 1. Questions : 2-1 to 2-6

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Sess ion Two

*General goal

Demand functions

*Detai led goals

1. Ordinary Demand functions

2. Compensated Demand functions3. Demand curves

4. Price & income elasticities

5. Evaluation 16

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1. Ordinary Demand Functions Ses.2

Ch.2 a. Definition

b. Mathematics

c. Properties1. Single valued for prices & income

2. Homogeneous of degree zero

d. Indirect utility function

1. Definition

2. Mathematics 17

Back 18

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2. Compensated Demand Functions

Ses.2 Ch.2

a. Definition

b. Mathematics c. Example

19Back 19

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3. Demand curves: Graphical analysis

Ses.2 Ch.2 a. Substitution & income effects (review) of

price change: (fig.5.3 : Nicholson)

b. Ordinary Demand curve :(fig.5.5:Nich.)

c. Compensated Demand curve :

(fig.5.6:Nich)d. Comparison of C.D.C and U.C.D.C

(fig.5.7:Nich) & (fig.2.5)20

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4. Price and income elasticities

Ses.2 Ch.2a. Descriptions

1. Own Price elasticity

2. Cross Price elasticity 3. Income elasticity

b. Relationship among elasticities

1. Elasticity and total expenditure 2. Cournot aggregation

3. Engel aggregation 21

Back 21

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Evaluation

Ses.2 Ch.2

1. Questions : 2-7, 2-9

2. Questions : 7-6, 7-7 : Nicholson

22

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Sess ion Three

*General goal

Mathematical analysis of comparative

statics in the demand

*Detai led goals

1.Demand for income, income & leisure2. Slutsky equation

3. Substitutes & complements 23

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1.Introduction

Ses.3 Ch.2a. The inverse of a matrix

1. Definition

2. Calculation

3. Using adjoint matrix to find A-1

b. Simultaneous equation system1. Description

2. Solution 24

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2.Supply of Labor: Income & leisure

Ses.3 Ch.2 a. Time allocation model and utility

maximization

1. Mathematics

2. Graph: (fig. 13.9, 13.10 : Sexton)

b. Comparative statics for Labor Supply

1. Analysis

2. Graph:(fig.22.1 : Nicholson)

3. Example 25

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3. Substitution & income effects

Ses.3 Ch.2 a. The Slutsky equation

b. Slutsky equation & elasticities

c. Direct effects d. Cross effects

1. Slutsky equation

2. Compensated demand elasticities

3. Ordinary demand elasticities 2626

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3. Substitution & income effects

Ses.3 Ch.2 e. Substitutes & complements

1. Definition

2. Mathematics

3. Relationship between substitutes

and complements

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4. Generalization to n-variables

Ses.3 Ch.2 a. Optimization

b. Elasticity relations

27

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Evaluation

Ses.3 Ch.2 Questions : 2.8 to 2.12

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Fi 2 1 Q dt Ch 2

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Fig. 2-1: Quandt, Ch:2

Back

29

Back to the main page 30

Fi 2 2 Q dt Ch 2

http://d/ch.2-miyaneh%201.ppt


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Fig.2-2: Quandt, Ch:2

Back Back to the main page 31

Fig 2-4: Quandt Ch:2



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Back to the mane page

Fig.2-4: Quandt Ch:2

31Back to the explanation

32

Fig 5 3 Nicholson

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Fig.5-3: Nicholson

Explain Back

32

33

Explain 5 3: Nicholson

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Explain 5-3: Nicholson

Back to text Back to fig

33

S.E:

I.E : 21,

,

U U Y X

p

I

Y X MRS p

p

x

xy

y

x

(AB) , U=cte, (X*XB)

(BC) , (XBX**)

PX

T.E=S.E+I.E=X*XB+XBX**=X**X

34

Fig 5 5: Nicholson Ch 2

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Fig.5-5: Nicholson, Ch.2

Explain Back

34

35

Explain 5 5: Nicholson Ch 2

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Explain 5-5: Nicholson, Ch.2

Back to text Back to fig35

36

Fig 5-6 : Nicholson Ch 2

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Fig.5-6 : Nicholson, Ch.2

Explain Back

36

37

Explain 5-6: Nicholson Ch 2

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Explain 5-6: Nicholson, Ch.2


37

38

Fig 22-1: Nicholson Ch 2

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Fig. 22-1: Nicholson, Ch.2

Explain Back 38

39

Explain 22-1: Nicholson Ch 2

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Explain 22 1: Nicholson, Ch.2


39

40

Fig 13-9: Sexton Ch 2

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Fig.13-9: Sexton, Ch.2

Explain Back 40

41

Explain 13-9: Sexton Ch 2

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Explain 13 9: Sexton, Ch.2


41

42

Fig.13-10: Sexton, Ch.2

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Explain Back 42

Fig.13 10: Sexton, Ch.2

43

Explain 13-10: Sexton, Ch.2

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43

Explain 13 10: Sexton, Ch.2

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-All information pertaining to the

satisfaction that the consumer derives

from various quantities of commodities is

contained in his ”utility function” - He is going to maximize his satisfaction

derived from consuming commodities.

(he should be aware of the alternativesand should be able to evaluate them.)

Back to the main page

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Consider the utility is measurable. e.g. u(s) = log s ,du/ds=1/s

-The difference between utility numbers could be

compared &the comparison lead to : A Ps B twice as C PD. (Ua=45 , Ub=15 )

-The law of diminishing marginal utilityp=2

Buying if the lost utility is less than obtainedone. He buys 1 unit. if p=1.6 then he will buy 2.

Um=5

Cardinal theory: S.Jevons , L.Walras & A.Marshal (19th economists)

Unit

Coconut 1Coconut 2Coconut 3

Additional U

2097

Back 46

th20B h d i i hO di l h2

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century.th20: Bentham proposed it in theOrdinal theory .2

- Equivalent conclusions can be deduced from much weaker assumptions

- we can not indicate the amount of U in number , but we can only rank

the goods based on the utility obtained .i.e. if U(A) > U (B) , then A P B

: Rationality axioms

(i) Completeness: A P B , A I B , or B P A .

(ii) Full information about prices , goods, market condition.

(iii) Transitivity : A P B & B P C then A P C ( not choosing self

contradictory preferences )

- Rationality Requires that the consumer can rank his preferences.

- His utility function shows this ranking. i.e. if U (A) = 15 , U (B)=45 onecan only say that B is preferred to A , but it is meaningless to say B is

likely 3 times as strongly as A .

- So a monotonic transformation for utility function is justifiable .

- Max U = √x ≈ Max U = x Back 47

a.The nature of the util ity func tion

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y

1. Continuity of U.F U=f(q 1 , q 2 ) : continuous first &

second order partial derivatives.

2. Regular strictly quasi-concave function. Or

2f 12 f 1 f 2 – f 11 f 2 2 - f 22 f 1

2 > 0

f 11 f 2 2 – 2f 1 f 2 f 12 + f 22 f 1

2 < 0

[ we will see that using this assumption guarantee the

sufficiency of F.O.C ]

3. Partial derivatives are strictly positive : f 1 > 0 , f 2 > 0 :

q U (The consumer will always desires more of both

commodities.)

4. The consumer’s U.F is not unique. Any single-valued

increasing function of q 1 & q 2 can serve U.F. Continue 48

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5. the U.F is defined with reference toconsumption during a specified period of time.

- Satisfaction depends on the length of time. - Variety in diets and diversification among the

commodities. U.F must not be defined for aperiod so short that the desire for varietycannot be satisfied.

- Tastes may change for too long a period.

[ Any intermediate period is satisfactory for thestatic theory of consumer behavior. ]


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Indifference curves

1. Definition

the locus of all commodity combination

from which the consumer derives the

same level of satisfaction form an

indifference curve.


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Indifference curves2. Characteris t ics

(i) Indifference map: a collection of indifferencecurves corresponding to different level of satisfaction.

(ii) The more is better: (fig.2-1)

(iii) No intersection: (fig.2-2)

(iv) Convex to origin :

* U.F is strictly quasi-concave I.C is convex.

In other word

If U0 = f(q10 , q2

0 ) = f( q 1(1) , q2

(1) )

U[λq10 + ( 1- λ )q1

(1) , λq20 + ( 1- λ )q2

(1) ] > U0

So I.C expresses q2 as a strictly quasi- concave

function of q1. ( Graph )


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1q

2q

0

2q

12q

0

1q1

1q

0

U

1U

))1(,)1(( 1

2

0

2

1

1

0

1 qqqqC

),( 0

2

0

1 qq A

),( 1

2

1

1 qq B

U(C)>U(A)=U(B)


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c. The rate of commodity

substitution1. Definition:

The rate of which a consumer

would be willing to substitute Q1

for Q2 per unit of Q1 in order to

maintain a given level of utility.


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c. The rate of commodity

substitution2. Mathematics:

, ,)( 12 q f q ),( 2111 qq f f ),( 2122 qq f f

2211 dq f dq f dU

2

1

2

1

1

22211 00

q

q

MU

MU

f

f

dq

dq RCS dq f dq f dU

2

2

21222111212112

12

21 )/()/(

f

f f f f f dqdq f f f

dq

qd

Cont inue

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3

2

2

12221212112

2

211

f

f f f f f f f f f f

22

2122212112112 )/()/(

f

f f f f f f f f f f

)2(1 2

1222112

2

2113

2

2

1

2

2

1

f f f f f f f f dq

qd

dq

dRCS

Since the U.F is regular strictly quasi-concave (by definition)

00(...) dq

dRCS RCS is diminishing along I.C


1

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c. The rate of commodity substitution

3. Econom ic interpretat ion :

dU = f 1dq1 + f 2 dq2 (1) : Total change in utility caused by

variations in q1 & q2 is approximately the change in q1

multiplied by the change in U resulting from a unitchange in q1 plus change in q2 multiplied by the change

in utility resulting from a unit change in q2 .

f 1 dq 1 ≈ resulting loss in U (dq1<0)

f 2 dq 2 ≈ resulting gain in U (dq2>0)

* (1) Is the equation of a plane tangent to the U.Fwhich is a 3 dimensional space.

Continue 56

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* Since ordinal utility:

1. f 1 dq 1 & f 2 dq 2 are not determinate numbers2.we can not recognize MU q1 & MUq2 by numbers.

* f 1

> 0 , f 2

>0 : an increase in q 1

(q 2

) will increasconsumer’s satisfaction level and move him to

higher indifference curve.

* RCS is the absolute value of the slope of I.C


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F.O.C Max U = f (q1 . q2)

s.t y0 = p1q1+ p2q2

F.O.CV = f (q1 , q 2) + λ (y0 – p1q1 – p2q2)

0

0

0

2211

0

22

2

11

1

q pq p yV

p f q

V

p f q

V

1. Mathematics

RCS p

p

f

f

2

1

2

1

Psychic rate of trade-off =

Mkt rate of trade-off

Interpretation

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λ p

f

p

f

2

2

1

1

2. Econom ic interpretat ion

(i) The rate at which satisfaction would increase if anadditional dollar were spent on a particular commodity

(ii) : Marginal utility of income

(iii) If f 1 /p1>f 2 /p2: More satisfaction gained by spending anadditional dollar on Q1 No utility maximized. Since it

is possible to increase utility by shifting some

expenditures from Q2 to Q1.

0


F.O.C

59

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- (n-m) last leading principle minor of boardered

Hessian should alternate in sign. The first with the

sign

S.O.C

2221

12112

0

f f p

f f p

p p

H

1)1( m

02

0

0)()(0

11

2

222

2

12112

11

2

221211212

1122112122221122

f p f p p p f

f p f p p p p f f p

f p f p p f p f p p H H

n=2 m=1 n-m=1

Dividing by2

2 p Cont inue 60

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02 22

2

12112

2

2112 f f f f f f f H

11222

2

2

1

2

1122 f f

p

p

p

p f

02

02

2

21122

2

12112

11222

2

2

1

2

112

f f f f f f f

f f f

f

f

f f Since P1 /P2=f 1 /f 2

Multiplied by

2

2 f or

Is satisfied by the assumption of regular strictly

quasi-concavity


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3. Example

Max U=q 1 q 2

s.T 100-2q 1 -5q 2 =0 (i)

RCS=f 1 /f 2 =q 2 /q 1 F.O.C q 2 /q 1 =p 1 /p 2 2q 1 =5q 2

q 1 =5/2q 2 (ii) (i) , (ii)

S.O.C: 25

10

1

2

q

q

01010)02(5)50(2

015

102

520

2

H

Cont inue 62

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3. Example

02

1

2

1

q

q

q

RCS

I.C is convex Rectangular hyperbula


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b. The choice of utility index

Ordinal ut i l i ty :

* No need to have cardinal significance for the numberswhich the utility function assigns to the alternative

commodity combinations i.e.if U (A) > U (B) A:3 or A = 400

B:2 B = 2

* If a particular set of numbers associated withvarious combinations of Q1 & Q2 is a utilityindex, any positive monotonic transformation of it isalso a utility index.

Cont inue

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*F(U) is a positive monotonic transformation

of U If F (U 1

) > F (U 0

) whenever U 1

> U 0

e.g. U = x F(U) = x 2 , U = x F(U) = ln x

[order presenting transformation F ’ (U) > 0 ]

*If U=f (q 1 ,q 2 ) then W=F(U)=F [f(q 1 , q 2 )]

Continue 65

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Max U = Max W

Proof :

If max f (q1 ,q2)

s.t B.L we find (q10 , q2

0 )

If (q1

(1)

, q2

(1)

) : Another bundle satisfying B.L then byassumption

f (q10 ,q2

0) > f (q1(1) , q2

(1) )

By definition of monotonicity :W (q1

0 , q20) = F[f(q1

0 ,q20)] > F[f(q1

(1) ,q2(1) ) = w (q1

1 ,q21)

W = (q1 , q2) is Max by commodity bundle (q10 , q2

0)


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1- Concave utility function (I.C) :(fig 2-4a)

U= x2 + y2

-F.O.C shows local minimum since S.O.C isnot satisfied for maximum. RCS is increasingalong I.C. U.F is not quasi-concave.

- y0 /p1 or y0 /p2 will be chosen dependingon whether f(y0 /p1) >< f(y0 /p2)

- Only one good should be consumed tohave higher U.


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2- Economic bads (fig.3-8 Nich,92)

- U = αx – βy , y U then y is an

economic bad

- X is the locus of Max utility

(corner solution)


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3- I.C are flatter than B.L

(fig 2-4.b) , ( fig 4-4 : Nicholson )

- Kuhn-tucker condition is valid &U.F is strictly concave or has a

positive monotonictransformation Kuhn-Tucker issufficient for U.Max.

Continue 69

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- Max U= f (q1 , q 2)

S.t y0 – p1q1 – p2q2 ≥ 0 , q1 ≥0 , q2≥ 0

Solut ion

0

0

0

2211

22

1

1

q pq p y F

p f q

F

p f q

F

0)(,

0)(,

0)(,

2211

0

2

1

q pq p y

p f q

p f q

Cont inue 70

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If

If


p f

p f

U by q1

U by q1

71



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Definition

It gives the quantity of a commodity

that he will buy as a function of

commodity prices and his income.They are obtained from utility

maximization.


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Marshalian D.C

Or Uncompensated D.C

Mathematics

Max U=f 1(q1,q2)

s.t y0=p1q1+p2q2

q1*=f 1(p1,p2,y

0)

q2*=f 2(p1,p2,y

0)

Original

problem


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1. Sing le value for pr ices and income

-When the utility function is strict quasi-concave,

a single commodity combination corresponds to

a given set of prices and income.

-If the utility function were quasi-concave but notstrictly quasi-concave, the indifference curves wouldposses straight-line portions, and maxima would notneed to be unique. In this case more than one valueof the quantity demanded may correspond to a given

price, and the demand relationship is called ademand correspondence rather than demandfunction


2 Homogeno s o f degree ero in

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2. Homogenous o f degree zero in

pr ice and income

f(kp1,kp2,ky0)=kf(p1,p2,y0)=g , k=0

Max U=f(q1,q2)

s.t ky0=kp1q1+kp2q2

F.O.C: V=f(q1,q2)+ [ky0-kp1q1-kp2q2] 011

1

kp f

q

V

022

2

kp f q

V

02211

0

qkpqkpky

V

2

1

2

1

p

p

f

f

02211

0 q pq p y

(II)

(I)

Cont inue 75

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(I) ,(II) Demand function for the price-income

set (kp1,kp2,ky0) is derived from the same

equations as for the price-income set (p1,p2,y0

).It can be shown that S.O.C is also satisfied in

this manner.


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1.Definition

The maximum utility which is derived from

original problem and is a function of prices

and income.


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2.Mathematics

U * =V=U * (q* 1,q

* 2 )=U * [f 1(p1,p2 ,y 0 ),f 2 (p1,p2 ,y 0 )]=U * (p1,p2 ,y 0 )


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Example

U=q1q2 , y0=p1q1+p2q2

F.O.C:

0

0

0

][

2211

0

21

12

2211

0

21

q pq p y Z

pqq

Z

pqq

Z

q pq p yqq Z

2

1

1

2

p

p

q

q

2

0

2

1

0

1

0

11

2

22

p

yq

p

yq

E yq p

79

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Example

S.O.C:

2

1

3

2

1

21

3

02)()(

01

10

0

q

q E pp p p p p H

p

p

p p

H

is a maximum point

)2

)(2

(2

0

1

0*

p

y

p

yU

21

0*

4

2

p p

yU I.U.F


8/13/2019 Ch.2 Miyaneh 1


Definition

It gives the quantities of the commodities

that the consumer will buy as a function of

commodity prices and given utility . i.e it

shows those combinations of consumption

bundles for which his utility is constant

(using some public compensation like taxes

and subsidies). Whit the minimum incomenecessary to achieve the initial utility.

Back to the main page81

8/13/2019 Ch.2 Miyaneh 1


Mathematics

Min E=p1q1+p2q2

s.t U0=f(q1,q2)

q1=F(p1,p2,U0)

q2=F(p1,p2,U0)

C.D.C

Dual problem


82

8/13/2019 Ch.2 Miyaneh 1


Example

U=q1q2 , E=p1q1+p2q2

Z=p1q1+p2q2+ (U0-q1q2)

F.O.C:

0

0

021

qqU Z

q pq

Z

q pq

Z

0

21

2

1

2

121

0

2

112

1

2

2

1

0 U p pq

p

pqqU

p

pqq

q

q

p

p

83

8/13/2019 Ch.2 Miyaneh 1


Example

2

1

0

2

1

2

0

1 ,

p

pU q

p

pU q


84

8/13/2019 Ch.2 Miyaneh 1


1. Own price elasticity

Proportionate rate of change of q1 divided by

the proportionate rate of change of its own price

with p2 and y0 constant.

1

1

1

1

1

111

ln

ln

p

q

q

p

p

q

0

0

1

1

11

11

11

11

: luxury goods

: necessities

: giffen

: normal goods Back to the main page 85

8/13/2019 Ch.2 Miyaneh 1


2. Cross price elasticity

It relates the proportionate change in one

quantity to the proportionate change in the

other price.

1

2

2

1

1

221

ln

ln

p

q

q

p

p

q

>0 or <0


86

8/13/2019 Ch.2 Miyaneh 1


Income elasticity

y

y p pQ

q

y

y

q

q

y

y

q

,,

ln

ln 11

1

1

1

1 < , > or =0


87

1 El ti it d t t l

8/13/2019 Ch.2 Miyaneh 1


1. Elasticity and total

expenditure

Consumer’s expenditure on Q1 is p1q1.

01

01

01

111)(

1

1111

1

1111

1111

111111

1

1

1

11

1

111

p

q p p

q p

q p

qq

p

q

q

pq

p

q pq

p

pq


8/13/2019 Ch.2 Miyaneh 1


2. Cournot aggregation

0

2220

111

0

11

2

1

1

2

0

22

1

1

1

1

0

11

1

1

22

1

11

112211

221111

,

0

y

q p

y

q p

y

q p

q

p

p

q

y

q p

q

p

p

q

y

q p

qdp

q p

dp

q p

dpqdq pdq p

dq pdpqdq p

Y=p1q1+p2q2 if dY0=dp2=0 then

The proportion of total expenditure for

goods; the share of every commodity

in consumer’s income.

89

8/13/2019 Ch.2 Miyaneh 1



1212111

Summat ion of own pr ice elast ic ity

90

8/13/2019 Ch.2 Miyaneh 1



Knowing the own price elasticity, we can

evaluate cross price elasticity.

If

If

If 01

01

01

2111

2111

2111

The above conditions hold for O.D.F. ForC.D.F we have :

U=(q1,q2) , if dU=0 then:

91

8/13/2019 Ch.2 Miyaneh 1



0

00

0

0

22

2

1

1

2

0

11

1

1

1

1

221121

2

1

2211

yq p

q p

dpdq

yq p

q p

dpdq

dq pdq pdqdq p

p

dq f dq f Since f 1 /f 2=p1 /p2

92

8/13/2019 Ch.2 Miyaneh 1



02211

21 , : compensated price elasticities


8/13/2019 Ch.2 Miyaneh 1


3. Engel aggregation

2

0

0

2

0

22

1

0

0

1

0

11

02

201

1

22112211

212211

1

1

:)(),(0

,

p

y

y

q

y

q p

p

y

y

q

y

q p

yq p

yq p

y f q y f qdydq pdq pdy

cte p pq pq p y

Engle curves

Continue94

8/13/2019 Ch.2 Miyaneh 1


3. Engel aggregation

12211

- The sum of income elasticities weighted by total

expenditure proportion equals unity.

- Two commodities in the basket can not be inferior.

- Income elasticities can not be derived for C.D.F .

Since income is not an argument of these functions.

Back to the mane page 95

8/13/2019 Ch.2 Miyaneh 1


1. Definition

If A , B are two rectangular matrices

and we have A.B=B.A=In , then B=A-1 is

called the inverse of A.


8/13/2019 Ch.2 Miyaneh 1


8/13/2019 Ch.2 Miyaneh 1


2. Calculation

35

47

3

4

075

143

5

7

175

043

10

01

7575

4343

1

A B

t

z

y x

y x

y

x

t z

t z

t z y x

t z y x

Example:


98

8/13/2019 Ch.2 Miyaneh 1



Assertion : It is symetric.

Calculate co-factor matrix:Calculate adjoint matrix:

Example:

18

213

321132

1

1

11

A A

AadjA

AcofA

adjA A

A

ji

ji

ij

ji

99

8/13/2019 Ch.2 Miyaneh 1



Example:

nnnn

n

n

D D D

D D D

D D D

cofA A

adjAcofA

21

22221

11211

1

175517

751

18

1

175

517

751

157

715

571


1 Description

8/13/2019 Ch.2 Miyaneh 1


1. Description

mnmnmm

n

n

mnmnmm

nn

nn

b

b

b

x

x

x

aaa

aaa

aaa

or

b xa xa xa

b xa xa xab xa xa xa

2

1

2

1

21

22221

11211

2211

22222121

11212111

AX=B Back to the main page

101

8/13/2019 Ch.2 Miyaneh 1


2. Solution

i - Using the inverse of matr ix :


2

3

2

3

4

5

12

11

3

1

12

11

3

1

12

11

.11

21

.3

4

5

12

11

42

5.

y

x

adjcof

y

xor

y x

y x g e

nm If

x

BXX

102

2 Solution

8/13/2019 Ch.2 Miyaneh 1


2. Solutioni i- Cramer’s approach (rule):

2

3

2

3

3

63

9

642

51

914

15

,3

42

5.

,,

21

2

2

32

223222

113121

1

1

1

y

x

y

x

Aand A A

y x

y x g e

A

A x

A

A x

aaab

aaab

aaab

A A

A x

n

n

nnnnn

n

n


103

8/13/2019 Ch.2 Miyaneh 1


1. Mathematics

Consumer’s satisfaction depends on incomeand leisure U=g(L,Y).[where L: leisure and Y:labor income]

Time constraint : T=L+W [where W: amount ofwork]

Income constraint : [where r = wagerate & W=T-L]

Optimization : Max U(T-W , rW) or

Y=rW

L=T-W

Max g(L,Y)

s.t Y-r(T-L)=0

Methods

104

8/13/2019 Ch.2 Miyaneh 1


1. Mathematics

Method 1:

F=g(L,Y)+λ[Y-r(T-L)]

F.O.C: F 1 =g 1 +r λ=0

F 2 =g 2 + λ=0

F λ=Y-r(T-L)=0

[r : opportu ni ty cos t of le isu re]

Result : W=f(r,T) supply of labor or

(uncompensated) demand for income

g1/g2=-dY/dL=r

=MRS LY =MU L /MU Y

Cont inue

105

8/13/2019 Ch.2 Miyaneh 1


1. Mathematics

Method 2:

Max U(T-W , rW)

F.O.C:

S.O.C:

r g

g r g g

W

Y

Y

U

L

U

dW

dU

2

121 0

02 2

2212112

2

r g r g g dW

U d

Cont inue

106

8/13/2019 Ch.2 Miyaneh 1


1. Mathematics


020

0)()(

1

10

22

2

121111211222

2

11211222

2221

12112

g r rg g g rg rg g r

g rg g rg r

g g

g g r

r

H

107

8/13/2019 Ch.2 Miyaneh 1


1. Analysis

r)(&)(,:. U W Y L MRS E S

)&(:. W U LY E I

T.E=S.E+I.E=AB+BC=AC

Graph

Fig.22.1: Nichols on

108

8/13/2019 Ch.2 Miyaneh 1


A

r 2T

Y2 B

L

U1

U2

T

C

L2 L1 L3

Y1

r 1T

Y

T-L2 T-L1 T-L3 W

r

r 2

r 1

S’L SL


109



8/13/2019 Ch.2 Miyaneh 1


)1(2

48)2(

2248

)(248248

r

r T

W rW rT W T rW

W T rW r L

LrW

Y=rW

L+W=T

3. Example

Approach 1:

MRS=r


U=48L+LY-L2 ,

Supply of labor

(Demand for y)110

8/13/2019 Ch.2 Miyaneh 1


3. Example

Approach 2:

U=48(T-W)+(T-W)rW-(T-W)2


0)1(222:..

)1(2

48)2(

022248:..

2

2

r r dW

U d C OS

r

r T W

T W rW dW dU C O F

Supply of labor

(Demand for y)

111

S

8/13/2019 Ch.2 Miyaneh 1


a. The Slutsky equation

1. Comparative statics : To find

(“ p” and “y ” are exogenous factors)

2. To maximize U=f(q1,q2 ) subject to y 0

-p1q1-p2 q2 =0

F.O.C:

V=f(q1,q2)+λ(y0-p1q1-p2q2)

V1=f 1-p1λ

V2=f 2-p2 λ

V λ= y0-p1q1-p2q2=0

(I)

Cont inue

112

Th Sl k i

8/13/2019 Ch.2 Miyaneh 1



Step 1: total differentiation of (I) allowing all

variables vary simultaneously:

f 11

dq1

+ f 12

dq2

-p1

dλ = λdp1

f 21dq1+ f 22dq2-p2dλ = λdp2

-p1dq1-p2dq2 = -dy+q1dp1+q2dp2

A system of 3 equations . Solution requires that right-hand side be constant

(II)

Step 2

113

Th Sl t k ti

8/13/2019 Ch.2 Miyaneh 1



Step 2: Solution of the system:

D

D

A

Adqrule sCramer

dpqdpqdy

dp

dp

d

dq

dq

f f p

f f p

p p

Cofactor D D H matrixt Coefficien

f f p

f f p

p p

H

111

0

0

:'

2211

2

1

2

1

22212

12111

21

112

22212

12111

21

2

Cont inue

114

Th Sl t k ti

8/13/2019 Ch.2 Miyaneh 1


Step 3


D

dpqdpqdy Ddp Ddp Ddq

D

dpqdpqdy Ddp Ddp D

dq

dpqdpqdy D Ddp Ddp f f dpqdpqdy

f f dp

p pdp

D

)(2

)(

)()()(

221132222112

2211312211111

221131212111

22212211

12112

211

1

(III)

115

Th Sl t k ti

8/13/2019 Ch.2 Miyaneh 1



Step 3: Calculation of substitution and income

effect.

D

D

y

q

dpdp yq

D

Dq

D

D

p

q

dydp p

q

effect Total

311

21

311

11

1

1

21

1

)01(?

)0(?:

(i)

(ii)

Cont inue

116

Th Sl t k ti

8/13/2019 Ch.2 Miyaneh 1



Substitution effect : Price rise is accompanied by

increase in the income : dU=0 f 1dq1+f 2dq2=0

since f 1 /f 2=p1 /p2 p1dq1+p2dq2=0 Last

equation of (II) ,-dy+q1dp1+q2dp2=0

(iii)

(i):

D

D

p

q

cteu

11

1

1

y

qq

p

q

p

q

U

1

1

1

1

1

1

Slope of

O.D.CS.E (Slope of C.D.C) I.E (slope of Engel curve)


117

b Sl t k ti d l ti iti

8/13/2019 Ch.2 Miyaneh 1


:

:

:

1

11

11

111111

1

111

1

1

1

1

1

1

1

1

y yq p yqq

q p pq

q p pq

U

b. Slutsky equation and elasticities

Price elasticit y o f O.D.C

Price elastici ty o f C.D.C

Income elast ic i ty

Cont inue

118

b Sl t k ti d l ti iti

8/13/2019 Ch.2 Miyaneh 1


- is more negative than if >0

- C.D.C is steeper than O.C.D

b. Slutsky equation and elasticities

11 1 11


Di t ff t

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

1. Marg inal ut i l i ty of money: In F.O.C :

221121 ,)0( p f p f dpdpwhile y

U

)()(

)()(

22

11

22

11

m MU y

q p y

q p

y

q f

y

q f

y

U

221121 ,),( dq f dq f duqq f U

We prove that (*) confirms the result of (II)

Cont inue

120

Di t ff t

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

D

dpqdpqdy Ddp Ddp Dd

)( 221133223113

D

f f f

D

D

y

2

12221133

)(of signof sign0 2

122211 f f f y

D

Assume: dp1=dp2=0

0

y

0

or

y

If U.F is strictly concave (MUy is increasing whit y )but since only strictly quasi-concave

2

121

Di t ff t

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

2. The s ign o f S.E:

0

.

2

211

11

1

1

p D

D

D

p

q E S

U

-S.E is always negative

-C.D.C is always

downward sloping

3

122

Di t ff t

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

3. Infer ior, no rmal and gi f fen good :

E S E I yqGiffen

D E T E I E S y

q Inferior

y

q Normal

..,0:

0...,0:

0:

1

1

1

4

123

Di t ff t

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

4. Example:

0

0

0

2211

0

3

212

121

q pq p y F

pq F

pq F

2

2

11

21

2

2

2

p

yq

p

y

q

p p

y

U=q1q2 y0-p1q1-p2q2=0 F=q1q2+λ(y0-p1q1-p2q2)

F.O.C:

Cont inue

124

c Direct effects

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

22112211

221

112

dpqdpqdydq pdq p

dpd pdqdpd pdq

0

01

10

21

2

1

p p

p

p

D

4. Example:

Total differentiatio n:

231

2121

2211

p D

p p D

p D

Cont inue

125

c Direct effects

8/13/2019 Ch.2 Miyaneh 1


c. Direct effects

4. Example:Cramer’s rule:

D

dpqdpqdy Ddp Ddp Ddq

)( 221131221111

If y=100, p1=2, p2=5 λ=5

1

1

1

2

21

21

21

2

2311

11

1

1

2222 p

q

p

p

p p

pq

p p

p

D

Dq

D

D

p

q

25.6.

25.6.

5.121

1

EI

E S

p

q


126

d Cross effects

8/13/2019 Ch.2 Miyaneh 1


d. Cross effects

1.The Sluts ky equation :

The Slutsky equation and its elasticity

representation can be extended to account

changes in the demand for one commodityresulting from changes in the price of the

other.

i jijij

cte price

i j

U j

ii j

ji

j

i

y

qq

q

q

D

Dq

D

D

p

q

3

(2)

(1)

Cont inue

127

d Cross effects

8/13/2019 Ch.2 Miyaneh 1


d. Cross effects

The sign of the cross-substitution effects

are not known in general.

Let Sij=λD ji /D and S ji=λDij /D (cross S.E) Since D is a symmetric determinant, D12=D21,

then Sij=S ji

2


d Cross effects



8/13/2019 Ch.2 Miyaneh 1


d. Cross effects

2. Compensated demand elast ic i t ies:

- Assertion:

- Proof:

p 1 D 11 +p 2 D 21 =0

Since the cofactors of the elements of the first

column of the determinant are multiplied by the

negative of the elements in the last column.

12111211 0 or

0)(

1

212111

1

221

1

1111211

Dq

D p D p

q

p D

q

p

D

D

3


d Cross effects



8/13/2019 Ch.2 Miyaneh 1


d. Cross effects

3. Ordinary demand elast ic i t ies:

Assertion:

Proof:By (2):

*The income elasticity of demand for a commodity

equals the negative of the sum of ordinary price

elasticities.

12112111211

11211

)()()(

)(


e Substitutes & complements



8/13/2019 Ch.2 Miyaneh 1


e. Substitutes & complements

1.Definit ion:

- Substitutes: Two commodities which can satisfy

the same need of the consumer.

- Complements: They are consumed jointly in

order to satisfy some particular need.

2

131


8/13/2019 Ch.2 Miyaneh 1



2. Mathematics : - Cross substitutes (If the total cross effect is

positive.):

- Cross complements:

- Net substitutes:

- Net complements:

0

i

j

p

q

0

i

j

p

q

0,0 21

2

1

D

D

p

q

p

q

U i

j

0,0 21

2

1

D

D

p

q

p

q

U i

j


3

132


8/13/2019 Ch.2 Miyaneh 1



3. Relat ionship between subs t i tutes and

complements :

(i) All commodities can not be complements for each

other.Proof:

Summat ion :

equationSlutskyCross p D

Dq

D

D

p

q

y D

D

y

qequationSlutsky p

D

Dq

D

D

p

q

231

221

2

1

3111

311

11

1

1

y D

D p

D

Dq p

D

D p

D

Dq p

D

D 312

3122

211

3111

11

Cont inue 133


8/13/2019 Ch.2 Miyaneh 1


Since it is an equation in terms of alien cofactors

S11p1+S12p2=0.

Since S11<0 S12 must be positive Q1and Q2 are necessarilysubstitutes.


0

).(,01

1

31221111

221131221111

E S Cross D

D

S D p D p D D

q pq p y D p D p D D

ji

ij


(ii)

134


8/13/2019 Ch.2 Miyaneh 1


y p p

p y

p

p

p

p p yq

p

p y p

pq

p

p yq p yq p

pq p yq pq p y

1

1

1

2

1

1

112

1

12

1

11111

1111111

:22

21

2

1,

22

0201


(ii) Gross and net substitutability and complementarity- Assertion: In the 2-good case it is possible to be substitutesin terms of Sij (net) and at the same time gross complements.

Example:

Max U=q1q2-q2

S.T y-p1q1-p2q2=0F.O.C:

F1=q2-p1λ=0

F1=q1-1-p2λ=0

F3=y-p1q1-p2q2=0

2

112

2

1

1

2 1

1 p

q pq

p

p

q

q

Cont inue

1352


8/13/2019 Ch.2 Miyaneh 1



022

:

02

1:

21

21

2

112

21

2

1

21

2

p p

p p

p

qS

D

D

p

q Net

p p

qGross

U

U

136

4 Generalization to n-variables

8/13/2019 Ch.2 Miyaneh 1



a. Optim ization :

ni p f q

V

q p yqqq f V

q p y

qqq f U

ii

i

n

i

iin

n

iii

n

,,2,10

0),,(

0

),,(

1

21

1

21

Max

s.t

F.O.C:

[n+1 equation (n qs and λ)]

S.O.C

137


8/13/2019 Ch.2 Miyaneh 1



S.O.C:

Boardered Hessian determinants must alternate insign .

Convexity of indifference curves can be extendedto indifference hypersurfaces in n-dimensions.

The satisfaction of the S.O.C is ensured by theregular strict quasi-concavity of the U.F



8/13/2019 Ch.2 Miyaneh 1



b. Elast ic i ty relat ion s:

:

:0

:1

:0

:

1

1

1

1

1

n

j

ij

n

j

ij

n

j

ji

n

iiji

ijiji Cournot aggregat ion

Compensated pr ice elast ic i t ies

Engel aggregat ion

Sum of compensated demand

elasticit ies

Sum o f ordinary demand elast ic i t ies


139

Fig.2.5: Quant, ch.2

8/13/2019 Ch.2 Miyaneh 1


Back

8/13/2019 Ch.2 Miyaneh 1


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