Consumer behaviour: Ordinal approach Now, we go to the ordinal approach. Ordinalist cares only about...

Consumer behaviour: Ordinal approach

• Now, we go to the ordinal approach. • Ordinalist cares only about ranking. It doesn’t

matter if we sayU(A) = 20, U(B) = 10,

or U(A) = 25, U(B) = 12. Because in both cases, the U(A) > U(B).

• Any methods of “renaming” the utility figure is valid, provided that the order of the figures are retained. (monotonic transformation)


• To take out all the intensity implication of utility, ordinalist proposes to use “indifference curves”.

• Suppose you need only two goods X and Y. • U(X, Y) = U(20, 30) = 8. But you will be equally sa

tisfied if you have more X but fewer Y. U(40, 20) = 8.

• Joining all the consumption combinations that give you the same utility level → indifference curve on the (X, Y)-space.


•Indifferent between B, A, & D•E is preferred to U1

•U1 is preferred to H & G

X

10

20

30

40

10 20 30 40

Y50

U1

G

D

A

EH

B


• Properties of IC:

(1) Points on an IC gives the same utility to an consumer. The proof is trivial coz this is true by definition.


• Properties of IC (cont’d):(2) IC of two “goods” can

never be upward-sloping (mostly likely downward-sloping): At point A, you have more X and more Y than at point B, by “more is better”, U(A) > U(B).

Good Y

3 4Good X

A

B

6

4


• Properties of IC (cont’d):

(3) A higher IC represents a higher utility level: For a higher IC, it contains a point (such as A) that must be better than a point (such as B) for a lower IC. By “more is better”, A is better than B.

Good Y

4 6Good X

A

B5

4


• Properties of IC (cont’d):

(4) Two ICs never cross: U(A) > U(B) (more is better. U(A) = U(C) and U(B) = U(C) by the definition of IC. By transitivity, U(A) = U(B). Inconsistent.


Food

Clothing•U(A)≠U(B) (two ICs)•U(C)=U(A) & U(C)=U(B) (same IC)•U(A)=U(B) (transitivity) •Inconsistent.

U1

U1

U2

U2

C

A

B

•U(A)>U(B) (more is better)•U(C)=U(A) & U(C)=U(B) (same IC)•U(A)=U(B) (transitivity) •Inconsistent.


• The 4 properties of ICs are all derived from the 3 basic preference postulates.

• The set of ICs that reflects your preference is like a map: indifference map.

Indifference Map

U2

U3

X

Y

U1

ABD


• In cardinal approach, we said we need one more information in addition to the 3 basic postulates: diminishing MU.

• In ordinal approach, we can’t use the concept of MU: We don’t know the 8th bottle you consume generates fewer additional satisfaction than the 1st bottle or not.


• Instead, ordinalist believes preference is governed by another law: law of diminishing marginal rate of substitution (MRS).

• MRS: How much more of a good (e.g. Y) you would require to compensate for the loss of a unit of another good (e.g. X) so that you are equally well off?

X2 3 4 51

Y

2

4

6

8

10

12

14

16 A

B

D

EG

MRS = 6

• MRS: the slope of an IC, -∆Y/∆X.

• Why? Because along the IC, utility is the same. Reducing certain amount of X, utility is lowered. Increasing the quantity of Y, utility is increased. Along the IC, the gain and loss just offset. +∆Y can just compensate the utility loss due to -∆X.

• Each unit of reduction in X must be given an increase in Y by -∆Y/∆X to compensate the loss.

X2 3 4 51

Y

2

4

6

8

10

12

14

16 A

B

D

EG

+6

-1

-1

-1-1

+4

+2+1

MRS = 6

MRS = 2

• Diminishing MRS: the slope (absolute value) of IC is decreasing; it becomes flatter and flatter.

• Implication: when you have a lot of X, you need fewer Y to compensate for giving up a unit of X.

• IC with decreasing slope is called “convex” (to origin).

X2 3 4 51

Y

2

4

6

8

10

12

14

16 A

B

D

EG

+6

-1

-1

-1-1

+4

+2+1

MRS = 6

MRS = 2

• Diminishing MU provides a reason for diminishing MRS.

• When we have a lot of X, MUX is smaller, required compensation (∆Y) is smaller. But MUX is high when X is small, required compensation (∆Y) is large.


• Diminishing MRS also reflects people’s tendency to become fed up with a good when consuming more and more of it.

• But “MU” (a concept of cardinal U) is not needed (only changes in X and Y, no change in U).

• Alternatively, convex IC also means people prefer a balanced consumption combination to an unbalanced combination.

Convexity in IC

U2

C

X40 8020

Y

20

30

40

0

U1

A

B

•Balanced bundle: C•Unbalanced bundle: A, B•Along the line, X+Y is the same.


• With a convex IC, we can “solve” the consumer problem as the cardinal approach does.

• Budget constraint I = PXX + PYY is a straight line (budget line) on the (X, Y)-space:

- BL cuts vertical axis at I/PY.

- BL cuts horizontal axis at I/PX.

Budget Line

1Slope - -

2

X

Y

Y P

X P

10

20

A

B

D

E

G

(I/PY) = 40

X40 60 80 = (I/PX)20

10

20

30

0

Y


• What will happen to BL if income increases?

- Two end-points (I/PY, I/PX)

• What will happen to BL if PX falls?

• What will happen to BL if PY falls?

Budget Line: Income Change

A increase inincome shifts

the budget lineoutward

X

Y

80=80/PX 160=160/PX40

20

160/PY=80

0

(I = $160)L2

(I = $80)

L1

L3

(I =$40)

A decrease inincome shifts

the budget lineinward

80/PY=40

Budget Line: Price Change

(PX = 1)

L1

An increase in theprice of X to

$2.00 changesthe slope of thebudget line and

rotates it inward.L3

(PX = 2)(PX’= 1/2)

L2

A decrease in theprice of X to$.50 changes

the slope of thebudget line and

rotates it outward.

40 X

Y

80=I/PX 160=I/PX’

40

I=80

U3

D

U2

C

X40 8020

Y

20

30

40

0

U1

A

B

• Slope of IC = Slope of BL

• MRS = -ΔY/ΔX = PX/PY

• A point on BL

• I = PXX + PYY

• Put BL and ICs together.

• At which point you can achieve the highest level of utility?

• The point where IC is tangent to BL.

• Some points above BL are not attainable.

• Some points below BL give you lower utility.


• This result is equivalent to the result in cardinal approach MUX/PX = MUY/PY.

• Why?

• The slope of BL is - PX/PY.

(Because I/PY I/PX = PX/PY)


• The slope of IC is MUX/MUY.

- A movement along IC: ∆X and ∆Y.

- This brings about utility change of MUX∆X + MUY∆Y.

- Utility does not change along IC: MUX∆X + MUY∆Y = 0

- Hence, the slope of IC:

∆Y/∆X = MRS = - MUX/MUY.


• At the point where BC is tangent to IC:

MRS = MUX/MUY = PX/PY

• Rearrange, this is exactly the result in cardinal approach MUX/PX = MUY/PY.

• Another condition is the budget constraint (I = PXX + PYY) – the best point must be on the budget line.

• Ordinal approach and cardinal approach is equivalent but ordinal approach doesn’t use MU and the controversial concept of cardinal utility.


• Another advantage of IC approach is: it can show up the relationship between two goods more clearly.

• Examples: Perfect substitutes, perfect complements.

Orange Juice(glasses)

Apple Juice

(glasses)

2 3 41

1

2

3

4

0

•Perfect substitutes: willing to trade X for Y at a fixed rate (not necessarily 1:1)

Right Shoes

LeftShoes

2 3 41

1

2

3

4

0

•Perfect complements: Can’t consume two goods at different ratios.

Relative Preference between Goods

These consumers

place a greater value on X than

Y

Y

X

Relative Preference between Goods

These consumers place a greater

value on Y than X

Y

X


• Other issues about IC analysis:• Trade-off between one good and one “bad”: - When both X and Y are “goods”, IC is

downward-sloping. - But when one of them is “bad”, it is upward-

sloping. - More X reduces utility, need more Y to

compensate.

IC for a Good and a Bad

Working Hours, L

Consumption, X Longer working time, higher value of consumption, constant utility

U1

U2

U3


• Trade-off between one good and one “bad” (cont’d):

- You won’t pay to purchase more “bad”. Rather, usually you are “paid” to induce you to accept more “bad”. E.g. you receive wage to accept a longer hours of work.

- Upward-sloping BL. PXX = wL.

- Consumer’s choice.

Optimal Choice of a Good and a Bad

X

Budget LinePXX = wL

U2

U1

U3

L

X*

L*


• Corner solutions:

- When consumer’s preference is “extreme”, e.g. prefer X strongly, the tangency point of IC and BL may not be the best choice.

- The best point is located at the corner.

- Our previous principle for best choice is no longer valid: MRS ≠PX/PY.

Corner Solution

Ice Cream (cup/month)

FrozenYogurt

(cupsmonthly)

B

A

U2 U3U1

A corner solutionexists at point B.

Orange Juice(glasses)

Apple Juice

(glasses)

2 3 41

1

2

3

4

0

U1

U4

U3

U2

BL

• Other examples of corner solutions:

• For perfect substitutes, optimal point is also at the corner.

• Why consume only one good?

P

Q Food ($)

OtherConsumption

($)

U2A

U1

B

•Old budget: PQ•A free gift of food PB•New budget PBC•B is corner solution•MRS ≠ PF/POC

C

• Other examples of corner solutions:

• Kinked budget line: You are given a free gift (food) of certain amount. But you can pay more to consume more (both food and other consumption).

Corner Solution: Price Change

Ice Cream (cup/month)

FrozenYogurt

(cupsmonthly)

B

A

U2 U3U1

A corner solutionexists at point B, even though price of Yogurt is different (AB, CB). Consumption changes only when price changes a lot (DB).

C

D

Interior Solution: Price Change

U3

D

U2

C

Food (units per week)40 8020

Clothing(units per

week)

20

30

40

0

U1

A

B


• Summary: - Cardinal approach and ordinal approach can

derive the same results for consumer behaviour.- Cardinal approach is more intuitive. - Ordinal approach sets aside the controversy

about measurement of utility. Thus, this approach is preferred by economists.

- In what follows, ordinal approach is used (you are required to be familiar with ordinal approach).

Consumer behaviour: Ordinal approach Now, we go to the ordinal approach. Ordinalist cares only about...

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