theory of consumer behaviour -...

Theory of Consumer Behaviour

1 Public Economy

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

Consumer Behaviour

• Feature of Consumer Behaviour

• Consumption set (Budget constraint)

• Preference

• Utility

• Choice

• Demand

• Revealed preference

3 Public Economy

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

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

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

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

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

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

Preference (4) – indifference curve Question Are these two lines satisfy the three assumptions?

x2

x1 0

x

x2

x1 0

10 Public Economy

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

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

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

Example of utility function x2

0

21 log5.0log5.0)( xxxu

x1

14 Public Economy

• 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

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

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

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

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

First order condition (一階条件)

IxpxxxuxLn

i

iin

1

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

iii pxuxL :0

IxpLn

i

ii 1

:0

20 Public Economy

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

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

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

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

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

Answer (Cont)

• By solving those equations

1

211 ,,p

I

ba

aIppx

2

212 ,,p

I

ba

bIppx

26 Public Economy

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

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

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

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

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

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

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

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)

34 Public Economy

Another approach describing Consumer Behaviour

x2

x1 0

Budget Set B

x2

x1 0

Set satisfying a certain

utility level

35 Public Economy

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

36 Public Economy

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

37 Public Economy

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

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

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

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

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

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

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

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)


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?


Public Economy 47

0 x

p


Inverse of Demand Function

Producer’s Surplus (生産者余剰)

Public Economy 48

0 x

p


Supply Function

Social Surplus (社会的余剰)

Public Economy 49

0 x

p



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

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

Comparison between CV and EV

Public Economy 52

0

x2

x1

CV

0

x2

x1

EV

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

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

theory of consumer behaviour -...

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