1. Consumer Behavior
MA (Economics), University of
Kelaniya 2012
Athula Ranasinghe

2. Three Approaches
•
Cardinal Approach
•
Ordinal Approach
•
Revealed Preference Approach

3. Cardinal Approach
•
Economic rationality
•
Utility can be numerically measured
•
Marginal utility of money is constant
•
Law of diminishing marginal utility
•
Independent utility
•
Introspection method (based on own
feelings)

4. Illustrative Example
MU
1
= 1000
- 20
X1
MU
2
= 500
- 5X
2
M
= 20,
P1
= 2
,
P2
= 1

5. Illustrative example
•
Problem:
•
To allocate given income between two
commodities to maximize total utility.
•
Solution:
•
Allocate income between two
commodities such that •per-rupeeMarginal Utility from two commodity willequal•.

6. Illustrative example
1MU
= 2MU
1P
2P
1000
1
2
20X- = 500
1
5X- 2
X
2
= 12X

7. Illustrative example
•
Any commodity combination
satisfying the above condition will
maximize utility.
•
However, his freedom of choice
according to the solution given aboveis limited by income constraint.
•
Therefore, we have to find the
commodity combination satisfying
both conditions.

8. Illustrative example: Budget
constraint
M
= p
X
+ p
X
11
22
20
= 2X1
+ X
2
20
= 2X1
+ X
2
20
= 2X1
+ 2X1
20
= 4X1
X1
= 5
,
X
2
= 10

9. Ceteris-paribus price
change
•
Now assume that price of commodity1 drops to Rs. 1.
•
This affects the budget constraint.
20
= X1
+ X
2
Increases the
20
= X1
+ 2X1
demand of both
20
= 3X1
commodities. Why?
X1
= 7
,
X
2
= 14

10. Demand function
p
2
1
Q
5 7

11. Increase income
•
Suppose all other factors remain
constant but income increases.
•
This affects demand through budgetconstraint.
•
The new budget constraint after
income doubles is given below.
40
= 2X1
+ X
2
Substituting this to first condition
40
= 2X1
+ 2X1
40
= 4X1
X1
= 10

12. Doubling all factors
•
Suppose now that all factors
determining demand are doubled.
(Income and all prices).
•
Then, the budget constraint is
40
= 4X1
+ 2X
2
•
Note that this will not affect the
demand

13. What we learnt
•
Negative relationship between
demand and price.
•
Positive cross-price effect?
•
Positive income effect.
•
Homogenous of degree zero.

14. Ordinal Analysis
•
Utility cannot be quantified.
•
Commodity baskets can be ranked based onpreference.
•
Preferences are
•
Complete
•
Reflexive
•
Transitive
•
Continue
•
Strong monotonicity
•
Law of diminishing rate
•
Diminishing marginal rate of substitution

15. Utility function
•
Index to map commodity space to utility space.
•
Indifference curve (Mathematical derivation andDiscussion)
•
Consumer choice under unlimited options

16. Budget constraint
•
Consumer•s ability to purchase (graphical)
•
Equilibrium: (ability and willingness)
•
Mathematical approach
•
Marshallian Demand curve
•
Hicksian demand curve
•
Indirect utility
•
Expenditure function
•
Welfare change

17. Graphical Method
•
Indifference map represents whatconsumer wants to do.
•
If he/she can have a free choice go tothe highest indifference curve.
•
Budget constraint represents whatthe consumer can do.
•
In the equilibrium, consumer choosesthe best commodity combinationwithin his/her
ability,

18. X1
Commodity Space and
Indifference Map
X2
O

19. X1
Commodity Space and
Budget Constraint
X2
O

20. X1
Commodity Space and
Consumer Equilibrium:
Desire, Ability match
X2
O

21. Mathematical Derivation of
Equilibrium
At equilibrium, slope of the indifference
curve is equal to the slope of budget
constraint.
Slope of indifference
curve;
U
= (X1
,
X
2
)
dU
= MU
dX
+ MU
dX
= 0
11
22
dX
MU
1
= - 2
= MRS
dX
2
MU1
1,2

22. M
= p
X
+ p
X
11
22
dM
= p
dX
+ p
X
11
22
dX1
p2
= dX
2
p1

23. Equilibrium
1
2
MU
MU
=
1
2
p
p
Re-arranging the
terms
2
2
p
MU
=
2
1
p
MU
Same condition
derived from
cardinal analysis.

24. Demand curves
Let
,
U
= X1
a X
2
b be
utility
function
.
U
MU1
=a
X1
U
MU2
= b
X
2

25. Slope of indifference
curve
Equilibrium condition
X
X
b
a
X1
b
a
X
2
p
2
p
2
1
=
X1
Demand Functions
M
X
2
p
1
M

26. 1
ö .
÷÷ ø
ö .
÷÷ ø
b
b
+
+
ß
a
a
a
æ .
çç è
æ .
çç è
=
=
p

27. Indirect
Utility: a ß
éæ a ö M
ù éæ b ö M
ù
U
=.. ..
÷÷çç÷÷çç
êúê ú
a+ b p
a+ b p
ëè ø 1
û ëè ø 2
û
Solve this for M (for given U to obtain expenditure function.
Expenditure function measures the “minimum income required
to attain a given level of utility.
Given the utility level, the minimum expenditure is a function of
prices.
It is a homogenous of degree one function of prices. When all
the prices are doubled, the minimum expenditure required to
attain the given level of utility will be double the initial income.

28. Expenditure function and Hicksian Demand function. First derivativeof the
expenditure function will be the Hicksian demand function.
Two measures of welfare change:
-Compensating Variation (CV): This measures the minimum incomecompensation
required for a consumer to be unaffected after pricechange (policy implemented).
-Equivalent Variation (EV): This measures the minimum income shouldbe withdrawn
from the consumer to be indifferent between before and
after price change (policy change).
Price Indices and Welfare Changes:
-Two price indices, Laspreyer (base year quantity) and Paasche(current year
quantity).
-Laspreyer measures the minimum income compensation required fora consumer to
consume the same commodity basket that he/sheconsumed before price change.
Conceptually this is consistent with
CV. However, this over estimates CV. (Why?)
-Paasche measures the minimum income required for a consumer to
consume the commodity basket that he/she would consume afterprice change.
Consistent with EV. Paasche index underestimates the
EV (Why?)

29. Illustrative numerical
example
“
Assume a = 0.45 and ß = 0.35.
“
Questions.
“
Identify the degree of homogeneity of thisfunction?
“
What is the meaning of it?
“
Can the utility function be homogeneous ofdegree one or above? Give reasons for
your
answer.
“
Resulted demand functions are; 0.45
ö M
æ 0.35
ö M
M
æ M
X1
= ç ÷ = 0.5625
and
X
2
= ç ÷ = 0.4375
è 0.45
+ 0.35
ø p1
p1
è 0.45
+ 0.35
ø p2
p2

30. Illustrative example
“
Calculate price and income
elasticities of the above demand
functions.
“
What the degree of homogeneity ofthese demand functions. Explain
reasons for that.

31. Illustrative example
“
Substitute them into the utilityfunction to derive indirect utility.
0.35
M
p
1
M
p
1
U
U
ß
a
ú
.
2
p
.
.
M
÷÷
.
M

32. 2
p
ö .
÷÷
.
ß
a
ß
0.4375
+
æ .
çç
.
êëù úû
.
.
çç
.
÷÷
.
0.45
.
ö .
÷÷
.
a
.
ß
a

33. 0.5625
+
.
ççè=
æ .
çç
.
ê
.
=
0.45
0.35
p
p
1
2
U
M
0.80
0.57796
=

34. Illustrative example:
indirect utility
•
Indirect utility function is
homogeneous of degree zero. Why isthat?
•
When the indirect utility function issolved for M (for given level of
utility), expenditure function is
derived.

35. Illustrative example:
expenditure function
U
0.80
0
0.45
0.35
0.45
0.35
M
=
p
p
= 1.73
p
pU
12
120
0.57796
0.56
0.44
1.25
M
= 1.98
p
pU
1
2
0
•
Expenditure function is homogeneous ofdegree 1 with respect to prices. What is
the meaning of it?
•
First derivative of expenditure function
with respect to own price is Hicksian
Compensated demand curve. Why?

36. Illustrative example:
Compensated demand
0.56
0.44
1.25
M
= 1.98
p
pU
1
2
0
¶M
-0.44
0.44
1.25
= (1.98)(0.56)
p
pU
1'2
0
¶p1
•
Calculate own price-elasticity of the
Compensated demand curve and comparethat with the own price elasticity of
Marshallian demand curve.

37. Numerical example
•
Assume that M= 1000, p1 = 1 and p2
= 1.
•
Use the utility and demand functions
derived in previous slides.
•
Inserting these into the two demandfunctions; 1,000
X
= 0.5625
= 563
1
1
1,000
X
2
= 0.4375
= 437
1

38. Numerical example
•
Insert X1 and X2 to the utilityfunction;
0.45
0.35
U
= ( 563
) ( 437
) = 145
0
•
Insert U0 and prices to calculate
expenditure function
0.56
0.35
1.25
M
= 1.98(1)
(1)
(145)
= (1.98)(504)
= 1,000

39. Numerical example
•
Now assume that all other factors remain
constant but p1 increases from 1 to 2.
Using the expenditure function derived
above, expenditure requires to attain theinitial level of satisfaction U0 can be
0.56
0.44
1.25
calculated. M
= 1.98
p
pU
1
2
0
p
= 2,
p
= 1,
U
= 145
12
0
Then
,
0.56
0.44
1.25
M
= 1.98(2)
(1)
(145)
= Rs
.1,474

40. Measuring welfare change
•
When price of commodity increasesfrom 1 to 2, consumer needs
additional Rs. 474 to enjoy the initial
level of utility.
0
,
0
2
.
0
1
•
This is called •Compensating
M
p
) M
( p
Variation: CV•.
0
,
0
2
.
1
1
(
)
CV
U
U

41. =
p
p

42. CV and Laspreper Price
Index
•
LPI (Base year basket).
•
How much a consumer needs to
purchase the commodity basket that
he/she purchased before price
change.
•
In this example, consumer needs 2(563)
+ 1(437) = Rs. 1,563 to buy the initialcommodity basket after price changed.
This is LPI.
•
Note that LPI is an over estimate of CV.
Why?