Intermediate Microeconomics 1

1
Intermediate Microeconomics 1

• Syllabus

1
2
1.Introducing the course

• This course is contained of 4 parts
• 1. The theory of consumer behavior
• 2. The theory of the firm
• 3. Market equilibrium
• 4. Monopoly , monopsony,
• monopolistic competition

2
2
3
1.Introducing the course

• The analyses are highly based on
• mathematics.
• The students will be responsible for problem
  solving.
• Discussing groups is recommended.

3
3
4
2.Students Activities

• a.Oral exam 15
• b.Mid-term exam 30
• c.Exercises 15
• d.Final exam 40

4
4
5
3.References

• a. The main text
• 1.J.M.Henderson R.E. Quandt , (1980) ,
• Microeconomic Theory
• b. Complementary texts
• 1. Eaton, B.C, Eaton, D.F.,(1995),
  Microeconomics
• 2. Griffiths,A S.Wall,(2000),
• Intermediate Microeconomics
• 3. Laidler,D. E. Saul, (1989) ,
• Introduction to Microeconomics
• 4. Nicholson,W,(2002),
• Microeconomic Theory

5
5
6
3.References

• 5. Varian,H.,(1993),
• Intermediate Microeconomics
• 6. Varian,H.,(1992),
• Microeconomic analysis

6
7
4.Description of the course

• Part 1
• Chapters 2 3 The theory of
• consumer behavior
• Utility maximization
• Demand function
• The Slutsky equation
• Duality theorem
• Risk and uncertainty

7
6
8
4.Description of the course

• Part 2
• Chapters 45 The theory of the firm
• Optimizing behavior
• Cost functions
• Input Demand
• CES production functions
• Linear programming

8
7
9
4.Description of the course

• Part 3
• Chapter 6 Market equilibrium
• 1. Demand supply functions
• 2. Commodity-Market equilibrium
• 3. Input-Market equilibrium
• 4. Stability of equilibrium

9
8
10
4.Description of the course

• Part 4
• Chapter 7 Monopoly , monopsony,
• monopolistic competition
• 1. Monopoly price determination
• applications
• 2. Monopsony
• 3. Monopolistic competition

10
9
11
Chapter 2
The theory of consumer behavior
Session One Session Two Session Three
11
10
12
Session One

• General goal
• Utility Maximization
• Detailed goals
• 1. Basic concepts
• 2. The first second order conditions
• for Utility maximization

12
11
13
1.Introduction Ses.1
Ch.2

• a. Utility function Definition
• b. Measuring the Utility
• 1.Cardinal theory (explanations)
• 2.Ordinal theory (explanations)
• -Rationality axioms

13
12
14
2. Basic concepts Ses.1
Ch.2

• a. 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

14
13
15
3. Utility Maximization Ses.1
Ch.2

• a. 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

15
14
16
EvaluationSes.1
Ch.2

• 1. Questions 2-1 to 2-6

16
15
17
Session Two

• General goal
• Demand functions
• Detailed goals
• 1. Ordinary Demand functions
• 2. Compensated Demand functions
• 3. Demand curves
• 4. Price income elasticities
• 5. Evaluation

17
16
18
1. Ordinary Demand Functions Ses.2
Ch.2

• a. Definition
• b. Mathematics
• c. Properties
• 1. Single valued for prices income
• 2. Homogeneous of degree zero
• d. Indirect utility function
• 1. Definition
• 2. Mathematics
• e. Example

18
17
Back
19
2. Compensated Demand Functions Ses.2
Ch.2

• a. Definition
• b. Mathematics
• c. Example

Back
19
19
20
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.5.5Nich.)
• c. Compensated Demand curve
• (fig.5.6Nich)
• d. Comparison of C.D.C and U.C.D.C
• (fig.5.7Nich) (fig.2.5)

20
20
Back
21
4. Price and income elasticities Ses.2
Ch.2

• a. 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
21
Back
22
Evaluation Ses.2
Ch.2

• 1. Questions 2-7, 2-9
• 2. Questions 7-6, 7-7 Nicholson

22
22
Back
23
Session Three

• General goal
• Mathematical analysis of comparative
• statics in the demand
• Detailed goals
• 1.Demand for income, income leisure
• 2. Slutsky equation
• 3. Substitutes complements

23
23
24
1.Introduction Ses.3
Ch.2

• a. The inverse of a matrix
• 1. Definition
• 2. Calculation
• 3. Using adjoint matrix to find A-1
• b. Simultaneous equation system
• 1. Description
• 2. Solution

24
24
25
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
25
26
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

26
26
27
3. Substitution income effects Ses.3
Ch.2

• e. Substitutes complements
• 1. Definition
• 2. Mathematics
• 3. Relationship between substitutes
  and complements

27
28
4. Generalization to n-variables Ses.3
Ch.2

• a. Optimization
• b. Elasticity relations

28
27
29
EvaluationSes.3 Ch.2

• Questions 2.8 to 2.12

29
28
30
Fig. 2-1 Quandt, Ch2
30
29
Back
Back to the main page
31
Fig.2-2 Quandt, Ch2
31
Back
Back to the main page
32
Fig.2-4 Quandt Ch2
Back to the mane page
Back to the explanation
32
31
33
Fig.5-3 Nicholson
33
32
Back Explain
34
Explain 5-3 Nicholson
S.E I.E
(AB) , Ucte, (XXB) (BC) , (XBX)
PX
T.ES.EI.EXXBXBXXX
34
33
Back to fig Back to text
35
Fig.5-5 Nicholson, Ch.2
35
34
Back Explain
36
Explain 5-5 Nicholson, Ch.2
Back to fig Back to text
36
35
37
Fig.5-6 Nicholson, Ch.2
37
36
Back Explain
38
Explain 5-6 Nicholson, Ch.2
Back to fig Back to text
38
37
39
Fig. 22-1 Nicholson, Ch.2
39
38
Back Explain
40
Explain 22-1 Nicholson, Ch.2
Back to fig Back to text
40
39
41
Fig.13-9 Sexton, Ch.2
41
40
Back Explain
42
Explain 13-9 Sexton, Ch.2
Back to fig Back to text
42
41
43
Fig.13-10 Sexton, Ch.2
43
42
Back Explain
44
Explain 13-10 Sexton, Ch.2
Back to fig Back to text
44
43
45
-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
alternatives and should be able to evaluate
them.)
Back to the main page
45
46
Cardinal theory S.Jevons , L.Walras A.Marshal
(19th economists)

• Consider the utility is measurable. e.g. u(s)
  log s , du/ds1/s
• The difference between utility numbers could be
  compared the comparison lead to A Ps B twice
  as C P D. (Ua45 , Ub15 )
• The law of diminishing marginal utility
• p2
• Buying if the lost utility is less than
  obtained one. He buys 1
  unit. if p1.6 then he will buy 2.
• Um5

46
Back
47

• 2. Ordinal theory Bentham proposed it in the
  20th century.
• - 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) gt 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 one
• can 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 vx Max U x
  Back

47
48

• The nature of the utility function
• 1. Continuity of U.F Uf(q1 , q2 ) continuous
  first
• second order partial derivatives.
• 2. Regular strictly quasi-concave function. Or
• 2f12f1f2
  f11f22 - f22f12 gt 0
• f11
  f22 2f1f2f12 f22f12 lt 0
• we will see that using this assumption
  guarantee the
• sufficiency of F.O.C
• 3. Partial derivatives are strictly positive f1
  gt 0 , f2 gt 0
• q U (The consumer will always desires
  more of both
• commodities.)
• 4. The consumers U.F is not unique. Any
  single-valued increasing function of q1 q2 can
  serve U.F. Continue
  

48
49

5. the U.F is defined with reference to
consumption 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 a period so short that the desire
for variety cannot be satisfied. -
Tastes may change for too long a period. Any
intermediate period is satisfactory for the
static theory of consumer behavior. Back to the
main page
49
50
Indifference curves

• 1. Definition
• the locus of all commodity combination from
  which the consumer derives the same level of
  satisfaction form an indifference curve.
• Back to the main page

50
51
Indifference curves

• 2. Characteristics
• (i) Indifference map a collection of
  indifference
  curves 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 , q20 ) f( q 1(1) , q2(1)
  )
• U?q10 ( 1- ? )q1(1) , ?q20 (
  1- ? )q2(1) gt U0
• So I.C expresses q2 as a strictly quasi-
  concave
• function of q1. (Graph)

51
Back to the main page
52
52
U(C)gtU(A)U(B)
Back to the main page
53
c. The rate of commodity substitution

• 1. 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.

Back to the main page
53
54

c. The rate of commodity
substitution

• 2. Mathematics
• ,
  ,

54
Continue
55
Since the U.F is regular strictly quasi-concave
(by definition)
RCS is diminishing along I.C
55
1
Back to the main page
56
c. The rate of commodity substitution

• 3. Economic interpretation
• dU f1dq1 f2dq2 (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
  unit
• change in q1 plus change in q2 multiplied by the
  change
• in utility resulting from a unit change in q2.
• f1dq1 resulting loss in U (dq1lt0)
• f2dq2 resulting gain in U (dq2gt0)
• (1) Is the equation of a plane tangent to the
  U.F which is a 3 dimensional space.
• 
  

56
Continue
57
Since ordinal utility
1. f1dq1 f2dq2 are not
determinate numbers
2.we can not recognize
MUq1 MUq2 by numbers. f1 gt 0 , f2gt0 an
increase in q1(q2) will increase consumers
satisfaction level and move him to higher
indifference curve. RCS is the absolute value
of the slope of I.C
Back to the main page
57
58
F.O.C
1. Mathematics

• Max U f (q1 . q2)
• s.t y0 p1q1 p2q2
• F.O.C
• V f (q1 , q 2) ? (y0 p1q1 p2q2)
• 
  
• 
  

58
Psychic rate of trade-off Mkt rate of trade-off
Interpretation
59
F.O.C
2. Economic interpretation

• The rate at which satisfaction would increase if
  an additional dollar were spent on a particular
  commodity
• (ii) Marginal utility of income
• (iii) If f1/p1gtf2/p2 More satisfaction gained
  by spending an additional dollar on Q1 No
  utility maximized. Since it is possible to
  increase utility by shifting some expenditures
  from Q2 to Q1.

59
Back to the main page
60
S.O.C
n2 m1 n-m1
- (n-m) last leading principle minor of boardered
Hessian should alternate in sign. The first with
the sign
60
Continue
Dividing by
61
Since P1/P2f1/f2
Multiplied by
or
Is satisfied by the assumption of regular
strictly quasi-concavity
61
Back to the main page
62
3. Example

• Max Uq1q2
• s.T 100-2q1-5q20 (i)
• RCSf1/f2q2/q1 F.O.C q2/q1p1/p2
  2q15q2
• q15/2q2 (ii) (i) , (ii)
• S.O.C

62
Continue
63
3. Example
I.C is convex Rectangular hyperbula
Back to the main page
63
64
b. The choice of utility index

• Ordinal utility
• No need to have cardinal significance for the
  numbers which the utility function assigns to the
  alternative commodity combinations i.e.
• if U (A) gt U (B) A3 or
  A 400
• B2
  B 2
• If a particular set of numbers associated with
  
  various
  combinations of Q1 Q2 is a utility
  index, any positive monotonic
  transformation of it is also a utility index.
• 
  Continue

64
65
F(U) is a positive monotonic transformation

of U If F (U1) gt F (U0)
whenever U1 gt U0 e.g. U x F(U) x2 , U
x F(U) ln x order presenting
transformation F(U) gt 0 If Uf (q1,q2) then
WF(U)F f(q1, q2)
65
Continue
66
Max U Max W
Proof If max f (q1 ,q2) s.t B.L
we find (q10 , q20 ) If (q1(1) ,
q2(1) ) Another bundle satisfying B.L then by
assumption f (q10 ,q20) gt f (q1(1) , q2(1) ) By
definition of monotonicity W (q10, q20)
Ff(q10 ,q20) gt Ff(q1(1),q2(1) ) w (q11,q21)
W (q1 , q2) is Max by commodity bundle
(q10 , q20)
66
Back to the main page
67
1- Concave utility function (I.C) (fig 2-4a)
U x2 y2

• F.O.C shows local minimum since S.O.C is not
  satisfied for maximum. RCS is increasing along
  I.C. U.F is not quasi-concave.
• y0/p1 or y0/p2 will be chosen depending
  on whether f(y0/p1) gtlt
  f(y0/p2)
• Only one good should be consumed to have higher
  U.

67
Back to the main page
68
2- Economic bads (fig.3-8 Nich,92)

• U ax ßy , y U then y is an economic
  bad
• X is the locus of Max utility (corner
  solution)

Back to the main page
68
69
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 monotonic
  transformation Kuhn-Tucker is sufficient for
  U.Max.
• 
  

69
Continue
70

• Max U f (q1 , q 2)
• S.t y0 p1q1 p2q2 0 , q1 0 , q2 0

Solution
70
Continue
71

• If
• If

U by q1
U by q1
71
Back to the main page
72
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.

72
Back to the main page
73
Mathematics

• Max Uf1(q1,q2)
• s.t y0p1q1p2q2
• q1f1(p1,p2,y0)
• q2f2(p1,p2,y0)

Original problem
Marshalian D.C Or Uncompensated D.C
Back to the main page
73
74
1. Single value for prices 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
  not strictly quasi-concave, the indifference
  curves would posses straight-line portions, and
  maxima would not need to be unique. In this case
  more than one value of the quantity demanded may
  correspond to a given price, and the demand
  relationship is called a demand correspondence
  rather than demand function

74
Back to the main page
75
2. Homogenous of degree zero in price and income

• f(kp1,kp2,ky0)kf(p1,p2,y0)g , k0
• Max Uf(q1,q2)
• s.t ky0kp1q1kp2q2
• F.O.C Vf(q1,q2) ky0-kp1q1-kp2q2

(I)
(II)
75
Continue
76

• (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.

Back to the main page
76
77
1.Definition

• The maximum utility which is derived from
  original problem and is a function of prices and
  income.

Back to the main page
77
78
2.Mathematics

• UVU(q1,q2)Uf1(p1,p2,y0),f2(p1,p2,y0)U(
  p1,p2,y0)

Back to the main page
78
79
Example

• Uq1q2 , y0p1q1p2q2
• F.O.C

79
80
Example

• S.O.C

is a maximum point
I.U.F
80
Back to the main page
81
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 income necessary to achieve the initial
  utility.

81
Back to the main page
82
Mathematics

• Min Ep1q1p2q2
• s.t U0f(q1,q2)
• q1F(p1,p2,U0)
• q2F(p1,p2,U0)

Dual problem
C.D.C
Back to the main page
82
83
Example

• Uq1q2 , Ep1q1p2q2
• Zp1q1p2q2 (U0-q1q2)
• F.O.C

83
84
Example
Back to the main page
84
85
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.

luxury goods necessities giffen normal
goods
85
Back to the main page
86
2. Cross price elasticity

• It relates the proportionate change in one
  quantity to the proportionate change in the other
  price.

gt0 or lt0
Back to the main page
86
87
Income elasticity
lt , gt or 0
Back to the main page
87
88
1. Elasticity and total expenditure

• Consumers expenditure on Q1 is p1q1.

88
Back to the main page
89
2. Cournot aggregation
Yp1q1p2q2 if dY0dp20 then
The proportion of total expenditure for goods
the share of every commodity in consumers income.
89
90
2. Cournot aggregation
Summation of own price elasticity
90
91
2. Cournot aggregation

• Knowing the own price elasticity, we can evaluate
  cross price elasticity.
• If
• If
• If

The above conditions hold for O.D.F. For C.D.F we
have
U(q1,q2)
, if dU0 then
91
92
2. Cournot aggregation
Since f1/f2p1/p2
92
93
2. Cournot aggregation
compensated price elasticities
Back to the main page
93
94
3. Engel aggregation
Engle curves
94
Continue
95
3. Engel aggregation

• 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.

95
Back to the mane page
96
1. Definition

• If A , B are two rectangular matrices and we have
  A.BB.AIn , then BA-1 is called the inverse
  of A.

96
Back to the main page
97
2. Calculation
If and then
We determine bij
using n equations. Example
97
98
2. Calculation
Example
98
Back to the main page
99
3. Using adjoint matrix to find A-1

• Assertion It is
  symetric.
• Calculate co-factor matrix
• Calculate adjoint matrix
• Example

99
100
3. Using adjoint matrix to find A-1

• Example

100
Back to the main page
101
1. Description
101
AXB
Back to the main page
102
2. Solution

• i- Using the inverse of matrix

102
Back to the main page
103
2. Solution

• ii- Cramers approach (rule)

103
Back to the main page
104
1. Mathematics

• Consumers satisfaction depends on income and
  leisure Ug(L,Y).where L leisure and Y labor
  income
• Time constraint TLW where W amount of work
  
• Income constraint where r wage
  rate WT-L
• Optimization Max U(T-W , rW) or

YrW LT-W
Max g(L,Y) s.t Y-r(T-L)0
Methods
104
105
1. Mathematics

• Method 1
• Fg(L,Y)?Y-r(T-L)
• F.O.C F1g1r ?0
• F2g2 ?0
• F?Y-r(T-L)0
• r opportunity cost of leisure
• Result Wf(r,T) supply of labor or
  (uncompensated) demand
  for income

g1/g2-dY/dLr MRSLYMUL/MUY
105
Continue
106
1. Mathematics

• Method 2
• Max U(T-W , rW)
• F.O.C
• S.O.C

106
Continue
107
1. Mathematics
107
Back to the main page
108
1. Analysis

• r

T.ES.EI.EABBCAC Graph
Fig.22.1 Nicholson
108
109
Y
r
SL
SL
r2T
Y2
B
C
r2
r1T
U2
Y1
A
U1
r1
L
W
L2
L1
L3
T
T-L2
T-L1
T-L3
Back to the main page
109
110
3. Example
YrW LWT
U48LLY-L2 ,

• Approach 1
• MRSr

Supply of labor (Demand for y)
110
Back to the main page
111
3. Example

• Approach 2
• U48(T-W)(T-W)rW-(T-W)2

Supply of labor (Demand for y)
111
Back to the main page
112
a. The Slutsky equation

• 1. Comparative statics To find
• (p and y are exogenous factors)
• 2. To maximize Uf(q1,q2) subject to
  y0-p1q1-p2q20

F.O.C Vf(q1,q2)?(y0-p1q1-p2q2) V1f1-p1?
V2f2-p2 ? V ? y0-p1q1-p2q20
(I)
112
Continue
113
a. The Slutsky equation

• Step 1 total differentiation of (I) allowing all
  variables vary
  simultaneously
• f11dq1 f12dq2-p1d? ?dp1
• f21dq1 f22dq2-p2d? ?dp2
• -p1dq1-p2dq2 -dyq1dp1q2dp2
• A system of 3 equations .
• Solution requires that right-hand side be
  constant

(II)
Step 2
113
114
a. The Slutsky equation

• Step 2 Solution of the system

114
Continue
115
a. The Slutsky equation
(III)
Step 3
115
116
a. The Slutsky equation

• Step 3 Calculation of substitution and income
  effect.

(i)
(ii)
116
Continue
117
a. The Slutsky equation

• Substitution effect Price rise is accompanied
  by increase in the income dU0
  f1dq1f2dq20 since f1/f2p1/p2
  p1dq1p2dq20 Last equation of (II)
  ,-dyq1dp1q2dp20
• (iii)
• (i)

117
Slope of O.D.C
S.E (Slope of C.D.C)
I.E (slope of Engel curve)
Back to the main page
118
b. Slutsky equation and elasticities
Price elasticity of O.D.C
Price elasticity of C.D.C
Income elasticity
118
Continue
119
b. Slutsky equation and elasticities

• is more negative than if gt0
• C.D.C is steeper than O.C.D

119
Back to the main page
120
c. Direct effects

• 1. Marginal utility of money
• In F.O.C

We prove that () confirms the result of (II)
120
Continue
121
c. Direct effects
Assume dp1dp20
If U.F is strictly concave (MUy is
increasing whit y ) but since only strictly
quasi-concave
2
121
122
c. Direct effects

• 2. The sign of S.E

-S.E is always negative -C.D.C is always downward
sloping
3
122
123
c. Direct effects

• 3. Inferior, normal and giffen good

4
123
124
c. Direct effects

• 4. Example

Uq1q2 y0-p1q1-p2q20 Fq1q2?(y0-p1q1-p2q
2) F.O.C
124
Continue
125
c. Direct effects

• 4. Example
• Total differentiation

125
Continue
126
c. Direct effects

• 4. Example
• Cramers rule

If y100, p12, p25 ?5
126
Back to the main page
127
d. Cross effects

• 1.The Slutsky equation
• The Slutsky equation and its elasticity
  representation can be extended to account changes
  in the demand for one commodity resulting from
  changes in the price of the other.

(1)
(2)
127
Continue
128
d. Cross effects

• The sign of the cross-substitution effects are
  not known in general.
• Let Sij?Dji/D and Sji?Dij/D (cross S.E)
• Since D is a symmetric determinant, D12D21, then
  SijSji

2
128
Back to the main page
129
d. Cross effects

• 2. Compensated demand elasticities
• - Assertion
• - Proof
• p1D11p2D210
• 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.

3
129
Back to the main page
130
d. Cross effects

• 3. Ordinary demand elasticities
• Assertion
• Proof By (2)
• The income elasticity of demand for a commodity
• equals the negative of the sum of ordinary price
• elasticities.

130
Back to the main page
131
e. Substitutes complements

• 1.Definition
• - 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
132
e. Substitutes complements

• 2. Mathematics
• - Cross substitutes (If the total cross effect is
  positive.)
• - Cross complements
• - Net substitutes
• - Net complements

3
132
Back to the main page
133
e. Substitutes complements

• 3. Relationship between substitutes and
  complements
• (i) All commodities can not be complements for
  each other.
• Proof
• Summation

133
Continue
134
e. Substitutes complements
Since it is an equation in
terms of alien cofactors S11p1S12p20.
Since S11lt0 S12 must be positive Q1and
Q2 are necessarily substitutes.
(ii)
134
Back to the main page
135
e. Substitutes complements

• (ii) Gross and net substitutability and
  complementarity
• - Assertion In the 2-good case it is
  possible to be substitutes in terms of Sij (net)
  and at the same time gross complements.
• Example
• Max Uq1q2-q2
• S.T y-p1q1-p2q20
• F.O.C
• F1q2-p1?0
• F1q1-1-p2?0
• F3y-p1q1-p2q20

135
2
Continue
136
e. Substitutes complements
136
137
4. Generalization to n-variables

• a. Optimization

Max
s.t
F.O.C
n1 equation (n qs and ?)
S.O.C
137
138
4. Generalization to n-variables

• S.O.C
• Boardered Hessian determinants must alternate in
  sign .
• Convexity of indifference curves can be extended
  to indifference hypersurfaces in n-dimensions.
• The satisfaction of the S.O.C is ensured by the
  regular strict quasi-concavity of the U.F

138
Back to the main page
139
4. Generalization to n-variables

• b. Elasticity relations
• 
  
• 
  
• 
  

Cournot aggregation Compensated price
elasticities Engel aggregation Sum of
compensated demand elasticities
139
Sum of ordinary demand elasticities
Back to the main page
140
Fig.2.5 Quant, ch.2
Back
141
fig.5.7 Nicholson, ch.2
Back