theory of consumer behaviour -...
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Theory of Consumer Behaviour
1 Public Economy
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What is Consumer Behaviour?
• Suppose you earn 12,000 yen additionally
– How many lunch with 1,000 yen (x1) and how many movie with 2,000 yen (x2) you enjoy?
(x1, x2) = (10,1), (6,3), (3,4), (2,5), …
– Suppose the price of movie is 1,500 yen?
– Suppose the additional bonus is 10,000 yen?
2 Public Economy
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Consumer Behaviour
• Feature of Consumer Behaviour
• Consumption set (Budget constraint)
• Preference
• Utility
• Choice
• Demand
• Revealed preference
3 Public Economy
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Feature of Consumer Behaviour Economic Entity
(経済主体) Firm(企業), Consumer (家計), Government
Consumer Firm
Capital(資本), Labor(労働),Stock(株式)
Hire(賃料), Wage(賃金), Divided(配当)
Household’s income
Demand Supply
Price
Quantity
Goods Market (財・サービス市場)
Consumer = price taker (価格受容者) 4 Public Economy
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Budget Set (1)
• Constraint faced by consumer
– Budget Constraint (income is limited)
– Time Constraint (time is limited)
– Allocation Constraint
Possible to convert into monetary unit under the given wage rate
Combine to Budget Constraint
Generally, only the budget constraint is considered
5 Public Economy
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Budget Set (2)
Budget Constraint (without allocation constraint)
Budget Constraint (with allocation constraint)
x2
x1 0
n
i
ii Ixp1
n
i
ii
n IxpxRxB1
,0
Price Demand Income
x2
x1 0
n
i
ii
n xxIxpxRxB1
11,,0
n
i
ii Ixp1
11 xx
Budget Set (消費可能集合)
B B
6 Public Economy
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Preference (1)
• What is preference?
BA A is (strictly) preferred to B
(A is always chosen between A and B)
BA A is preferred to B, or indifferent between two (B is never chosen between A and B)
BA ~ A and B is indifferent (No difference between A and B)
7 Public Economy
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Preference (2)
• Assumption regarding to preference
1. Complete: (完備性or完全性)
2. Transitive : (推移性)
3. Reflexive : (連続性or反射性)
BAEither , BA or BA ~ is satisfied
BA and CB then CA
AA
8 Public Economy
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Preference (3) – indifference curve
x2
x1 0
y
x
Indifference curve (無差別曲線)
xyyx ~)( nRC
x2
x1 0
Question Are these two lines satisfy the three assumptions?
x
y
z
x, y and z would be indifferent, which is obviously inconsistent 9 Public Economy
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Preference (4) – indifference curve Question Are these two lines satisfy the three assumptions?
x2
x1 0
x
x2
x1 0
10 Public Economy
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Preference (5)
• Additional assumptions regarding to preference
1. Monotonicity (単調性):
2. Convexity (凸性)
),( 21 xxA
),( 21 xxB BAxx 11
Complete, Transitive and Reflexive
There exist Utility Function
x2
x1 0
x
Better bundles
Worse bundles
Average bundle
11 Public Economy
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Utility Function
• What is utility function? Definition
yxyx ,, nR
)()( yx uu
There exist RRu n :
that satisfies
Theorem
If the preference satisfies complete, transitive, reflexive and monotonicity, then there exist utility function that satisfies
yxyx ,, nR)()( yx uu
There exist RRu n :
that satisfies
12 Public Economy
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Utility function and Indifference curve
x2
x1 0
y
x 1)( uxu
3)( uxu
2)( uxu
x1
x2
u(x)
u1
u2
u3
Indifference curve is expressed as a contour line (等高線) of an utility function
13 Public Economy
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Example of utility function x2
0
21 log5.0log5.0)( xxxu
x1
14 Public Economy
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• Draw a graph and indifference curve of the following utility functions
211 log1.0log1.0)( xxxu
212 )( xxxu
2
2
2
13 )( xxxu
15 Public Economy
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Various Utility Function
• Ordinal utility (序数的効用)
– Cobb-Douglas type
– Linear
– Leontief type
– CES type
• Cardinal Utility (基数的効用)
2121 ),( xxxxu or
2121 lnln),( xxxxu
2121 ),( xxxxu
2121 ,min),( xxxxu
2121 ),( bxaxxxu
Only the order of the utilities is meaningful
The value of the utilities is also meaningful
Can we express the value of utility correctly?
16 Public Economy
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Choice (選択)
• Consumers are assumed to choose most preferable bundle from their budget set
x2
x1 0
Budget Set B
The utility of the right upper indifference curve is higher
We should find a bundle whose utility is maximum among a given budget set
17 Public Economy
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Consumer Behaviour Model (消費者行動モデル)
),...,,(max 21 nx
xxxu
subject to
Ixpn
i
ii 1
nixi ,...,10
Monotonical utility function
x2
x1 0
Budget Set B
18 Public Economy
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Consumer Behaviour Model (消費者行動モデル)
),...,,(max 21 nx
xxxu
subject to
Ixpn
i
ii 1
x2
x1 0
Budget Set B
19 Public Economy
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First order condition (一階条件)
IxpxxxuxLn
i
iin
1
21 ),...,,(),(
iii pxuxL :0
IxpLn
i
ii 1
:0
20 Public Economy
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Marginal Rate of Substitution (MRS;限界代替率)
0 x1
x2
Dx1
Dx2
Slope = 1
2
x
x
D
D= marginal rate of substitution
MRS measures the rate at which the consumer is just willing to substitute one good for the other
21 Public Economy
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Graphic illustration of first order condition
0 x1
x2
marginal rate of substitution (限界代替率)
= market rate of substitution (市場代替率)
j
i
j
i
p
p
xu
xu
22 Public Economy
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Demand (需要)
• The consumer’s demand function give the optimal amount of each of the goods as a function of the prices and income faced by the consumer
Solution of the Consumer behaviour model
x1
p
Ipppx ni ,,...,, 21
(Marshallian) demand function (マーシャルの)需要関数
23 Public Economy
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Example
• Find demand function with Cobb-Douglas type utility function
max),( 2121 xxxxu
subject to
Ixpxp 2211
Hint: It is easier to solve if we assume Cobb-Douglas type utility function as 2121 lnln),( xxxxu
24 Public Economy
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Answer
• First order condition
• Rewrite above equations using MRS
IxpxpxbxaL 221121 lnln
01
11
p
x
a
x
L
02
22
p
x
b
x
L
02211
Ixpxp
L
2
1
2
1
p
p
xb
xa
02211 Ixpxp
25 Public Economy
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Answer (Cont)
• By solving those equations
1
211 ,,p
I
ba
aIppx
2
212 ,,p
I
ba
bIppx
26 Public Economy
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Homogeneous function
Definition
If a function f(x) satisfies following feature, then f is said to be homogeneous of degree k; (k次同次関数)
,,0 nRt x
xx fttf k
27 Public Economy
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Question
Assume Cobb-Douglas type utility function and proof following propositions
1. Demand function is homogeneous with degree 0
2. Demand function is monotonic decrease (単調
減少 ) with regard to price and monotonic increase (単調減少) with regard to income
28 Public Economy
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Answer
1.
2. It can easily be shown from above demand function
1
211 ,,p
I
ba
aIppx
Ippxtp
tI
ba
atItptpx ,,,, 211
1
211
29 Public Economy
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Indirect Utility Function (間接効用関数)
• A consumer’s indirect utility function v(p,I) gives the consumer’s maximal utility when faced with a price p and an amount income I. It represents the consumer’s preference over market conditions.
nn xxuIppv ,...,max,,..., 11x
subject to Ixpn
i
ii 1
Identity (恒等式)
IppxIppxuIppv nnnn ,,...,,...,,,...,,,..., 1111
30 Public Economy
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Example
1. Find indirect utility function whose utility function is Cobb-Douglas type
2. Proof following identity between indirect utility function and demand function
IIppv
pIppvIppx
n
inn
,,...,
,,...,,,...,
1
111
Roy’s identity (ロイの恒等式)
31 Public Economy
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Answer
1. From the identify,
2. (In case n=2)
We can get derivative of indirect utility function as following; and Then, if we differentiate the budget constraint ( ) with respect to p1 (with fixed I), we can get Also, by differentiating by I, we can get From those equations and the condition of utility maximisation ( ), we can obtain Q.E.D.
ba
ba
ba
baba
pp
I
ba
ba
p
I
ba
b
p
I
ba
aIppv
2121
21 ,,
1
2
21
1
11 p
x
x
u
p
x
x
u
p
v
I
x
x
u
I
x
x
u
I
v
2
2
1
1
Ixpxp 2211
02
22
1
111
p
xp
p
xpx
122
11
I
xp
I
xp
2121 ppxuxu
Iv
pvIppx
1
211 ,,
32 Public Economy
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Income change and Demand
• Superior good (上級財)
– Good whose demand increases as income increase
– Example: Luxury goods
• Intermediate good (中間財)
– Good whose demand is stable against income
– Example: Tissue paper
• Inferior good (下級財)
– Good whose demand decreases as income increase
– Example: Substitution of rice (such as potato)
33 Public Economy
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Price change and Demand
• Ordinal good (正常財)
– Good whose demand decrease as it’s price increase
– Example: Beer
• Giffen good (ギッフェン財)
– Good whose demand increase as it’s price increase
– Example: Substitution of rice (such as potato)
x2
x1 Price decrease Reduction in
demand for good 1
x2
x1 Price decrease
Ordinal good Giffen good
(Good 1)
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Another approach describing Consumer Behaviour
x2
x1 0
Budget Set B
x2
x1 0
Set satisfying a certain
utility level
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Expenditure Minimisation Problem (支出最小化問題)
• Expenditure function (支出関数)
• Hicksian demand (ヒックスの需要関数)
n
i
ii xp1
minx
subject to
uxxu n ,...,1
ue ,p
uh ,p
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Expenditure Function
Expenditure Minimisation Problem
n
i
iin xpuppe1
1 min,,...,x
subject to uxxu n ,...,1
Expenditure Function
First Order Condition jn
in
j
i
xxxu
xxxu
p
p
,...,
,...,
1
1
uxxu n ,...,1
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Hicksian Demand Function
• Solution of Expenditure Minimisation Problem
• Identity (恒等式)
uhi ,p
IxIvh ii ,,, ppp
uhIex ii ,,, ppp
IIve ,, pp
uuev ,, pp
x2
x1
Ivu ,p
h2
h1 IIve ,, pp
38 Public Economy
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Feature of Expenditure Function and Hicksian Demand Function
• Expenditure Function ( ) is homogenous of degree 1 with regard to p. Expenditure Function is increasing function with regard to p and u.
• Identity (恒等式)
ue ,p
n
i
ii uhpue1
,, pp
i
ip
ueuh
,,
pp
39 Public Economy
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Income Effect and Substitution Effect (所得効果と代替効果)
• The effect of changing the price of goods = Income Effect + Substitution Effect
40 Public Economy 0 x1
x2
Pivot
Shift Substitution
effect Income effect
Pivot (which represents the price of goods 1 change but income stays fix ) gives the substitution effect, and Shift (which represents the change of income) gives the income effect
Reduce the price
of goods 1
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Income Effect and Substitution Effect (所得効果と代替効果)
• Substitution effect… the change in demand due to the change in rate of exchange between two goods
• Income effect … the change in demand due to having more purchasing power
Public Economy 41
If the price of good 1 decreases, the price of good 2 increases relatively
If the price of good 1 decrease, the substantive income will increase
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SLUTSKY Equation (スルツキー方程式)
• Equation representing the relationship between Income Effect and Substitution Effect (i.e. The effect of changing the price of goods = Income Effect + Substitution Effect)
Public Economy 42
I
IxIx
p
Ivh
p
Ix ij
j
i
j
i
,,
,,, pp
ppp
Substitution Effect (with a given utility)
Income Effect
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Proof
From the identify of the relationship between the Hicksian demand function and the Marshallian demand function, we can have
By substituting the 2nd equation to the 1st equation and then differentiate by pj, we can get
Furthermore, since we can get
Public Economy 43
Ivevhvex ii ,,),(,, pppp
j
i
j
i
j
i
p
h
p
e
I
x
p
x
vexvhp
ejj
j
,,, ppp
I
xx
p
h
p
x ij
j
i
j
i
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Relationship between Each Function
44 Public Economy
Utility maximisation
Expenditure minimisation
Ixi ,p uhi ,p
Iv ,p ue ,p
Marshallian demand function
Hicksian demand function
Indirect utility function
Expenditure function
SLUTSKY Equation
Roy’s identity
IIv
pIvIx i
,
,,1
p
pp
i
ip
ueuh
,,
pp
I
xx
p
h
p
x ij
j
i
j
i
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Consumer’s Surplus (消費者余剰)
• Evaluation of Benefit
• Consumer’s Surplus
– The difference between the maximum price a consumer is willing to pay and the actual price they do pay
Public Economy 45
Cost-benefit analysis (費用便益分析)
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Consumer’s Surplus (消費者余剰)
Suppose you have three computers and your friends are willing to pay following amount of money to get the computer. How much would you charge to your computer?
Public Economy 46
Mr. A : ¥130 thousands Mr. B : ¥20 thousands Ms. C: ¥70 thousands Ms. D: ¥100 thousands
Answer: ¥70 thousands !
70
A D C B
How much would A and D
earn?
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Consumer’s Surplus (消費者余剰)
Public Economy 47
0 x
p
Consumer’s Surplus (消費者余剰)
Inverse of Demand Function
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Producer’s Surplus (生産者余剰)
Public Economy 48
0 x
p
Producer’s Surplus (生産者余剰)
Supply Function
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Social Surplus (社会的余剰)
Public Economy 49
0 x
p
Consumer’s Surplus (消費者余剰)
Producer’s Surplus (生産者余剰)
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Compensating Variations (補償変分)
Public Economy 50
0
x2
x1 Increase p1
Compensate for increase p1
CV
CV represents for the change in income necessary to restore the consumer to his/her original indifference curve
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Equivalent Variations (等価変分)
Public Economy 51
0
x2
x1 Increase p1
Pay back income EV
EV represents for the amount of income to be taken away from the consumer before the price change to leave him/her as well off as (s)he would be after the price change
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Comparison between CV and EV
Public Economy 52
0
x2
x1
CV
0
x2
x1
EV
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Comparison between CV and EV
• Basically, |CV|≠|EV|
– The amount of money that the consumer would have to pay to compensate him/her for a price change would be different from the amount of money that the consumer would be willing to pay to avoid a price change
• However, |CV|=|EV| in the case of quasilinear utility (準線形効用)
– where the indifference curves are parallel
Public Economy 53
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In case of Quasilinear Utility
Public Economy 54
0
x2
x1
x2
0 x1
Utility difference
• Utility difference is the same regardless of initial solution