Indifference Curve Analysis

Ordinal Approach or The concept of Scale of Preferences or

The Indifference Curve Technique

• Originated by Edgeworth in 1881 and Refined by Pareto in 1906.

• Application in the demand analysis at the hands of J.R. HICKS and R.G.D .Allen in 1934.

The Ordinal Approach In Utility Theory

• The ordinal theory suggests that utility is only relatively discernible but not quantifiable.

• U is the level of satisfaction than an amount of satisfaction.

• Utility is a series of assigned numbers to rank options by the consumer preference. The assigned numbers reveal what is more preferred but cannot tell how much the difference is.

• Utility can only be ranked by an order or a scale of preference to show the degree of willingness of a consumer.

• Hicks uses ‘Significance’ rather than ’Utility’.

Combinations between Apples

and Bananas

Level of Satisfaction

Derived

Ranking Order of

Preference

a) 12 Apples +

12 Bananas

Highest First

b) 10 Apples +

19 Bananas

Lesser than (a)

Second

c) 5 Apples

+

5 Bananas

Lesser than (b)

Third

Scale of Preferences - Characteristics

• Drawn by a consumer in his mind consciously or unconsciously.

• Based on subjective valuation of goods made by the customer on the basis of his liking, habits, tastes, desires, intensity of wants, etc.

• Independent of Price and consumer’s income.

• It represents Ordinal comparison of the level of satisfaction derived by the consumer from different combinations of goods.

• Scale of Preferences differs from person to person.• Scale of Preferences considers the significance of the

commodity in the context of their stocks.

Definitions• Indifference Schedule —An Indifference schedule is a

list of alternate combinations in the stocks of two goods which yield equal satisfaction to the consumer.

• Indifference Curve- An Indifference curve is the locus of points representing all the different combinations if the two goods (say X and Y) which yield equal utility or satisfaction to the consumer.

• Indifference Map- A graph showing a whole set of indifference curves is called an indifference map. All points on the same curve give equal level of satisfaction, but each point on higher curve gives higher level of satisfaction.

Indifference Curve Analysis

• The indifference curve analysis is a technique for explaining how choices between two alternatives are made.

Indifference ScheduleIndifference Schedule

CombinatCombinationion

ApplesApples MangoesMangoes

AA 1515 11

BB 1111 22

CC 88 33

DD 66 44

EE 55 55

Indifference Curve & Map

Indifference Curve Indifference Map

Y

XIC

0b’ b”

Y

X

IC

A

BC

0

a

a’

a”

b

ICIC

Hypothetical data for an IC Map

Combination of Goods (Units)

I II III

X Y X Y X Y

1 10 2 15 3 20

2 6 4 10 5 14

3 3 6 6 7 7

4 1 8 3 9 7

U1(IC1) U2(IC2) U3(IC3)

Third Order Preference

Second Order Preference

First Order Preference

QT

of

com

m Y

Y

X

U1

U3

U2

OQT of comm X

d

c

b

a

∙∙∙∙

ASSUMPTIONS• A consumer is interested in buying two goods in combination.

• He is able to rank his preferences & give a complete ordering of the scale of preferences.

• Non-satiation, i.e, the consume always prefers more quantities of goods to lesser quantities.

• He is rational and his choices are transitive.It means, if he prefers combination a to b and b to c, then he must also prefer a to c.

• Height of the IC indicates the level of satisfaction.

• IC are drawn as continuous curves assuming infinitesimal amount of changes in the combination of 2 goods i.e. perfect division of the goods under consideration.

The Indifference Curve Theory

Based on these assertions, Edgeworth F. Y. ( 1845 - 1926 ) first suggested the indifference curve to represent the level of preference a consumer has when two goods are consumed with different amount, but each combination of these two goods yields the same level of satisfaction.

The Properties of the Indifference Curve

• IC slopes downwards from left to right , i.e.

negatively sloped, indicating if X increases in combination X and Y , there should be a decrease in Y amount to be on the same level of satisfaction.

• They are convex to origin.

• They can’t intersect each other.

Indifference curves slope down from left to right.Consider 2 points A & B on the same indifference curve. At point B, you have more food than at point A. If the amount of clothing you had at point B was the same as or more than at point A (like at point C), you would not be indifferent between A and B (since more is better).So A & B could not be on the same indifference curve, which goes against our initial statement that they are. So you must have less clothing at B, which means than B lies below A and the indifference curve slopes downward.

clothing

food

A

B

C

Indifference curves to the northeast are preferred.

Point E is preferred to point A because it has more food & more clothing.Since you are indifferent between A & all points on IC1, E must be preferred to all points on IC1.Since you are indifferent between E and all points on IC2, all points on IC2 must be preferred to all points on IC1.

IC2

food

A

E

clothing

IC1

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reserved.

Indifference Curves: Shape

• The indifference curves are not likely to be vertical, horizontal, or upward sloping.

– A vertical or horizontal indifference curve holds the quantity of one of the goods constant, implying that the consumer is indifferent to getting more of one good without giving up any of the other good.

– An upward-sloping curve would mean that the consumer is indifferent between a combination of goods that provides less of everything and another that provides more of everything.

– Rational consumers usually prefer more to less.

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reserved.

Indifference Curve Shapes

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reserved.

Indifference Curves: Slope • The slope or steepness of indifference curves is

determined by consumer preferences.

– It reflects the amount of one good that a consumer must give up to get an additional unit of the other good while remaining equally satisfied.

– This relationship changes according to diminishing marginal utility—the more a consumer has of a good, the less the consumer values an additional value of that good. Convexity implies diminishing slope of the indifference curve.

Indifference curves can not intersect.Suppose indifference curves could intersect. Let the intersection of IC1 & IC2 be D.Then you must be indifferent between D & any other point A on IC1.Similarly, you must be indifferent between D & any other point B on IC2.

food

A

D

clothing

IC1

B IC2

By transitivity, you must be indifferent between A & B.But A & B are not on the same indifference curve, which they should be if you are indifferent between them.Then, our initial supposition that indifference curves could intersect must be wrong.

Indifference curves are convex[the slopes of IC’s fall as we move from left to right, or

we have a diminishing marginal rate of substitution (MRS)]

When we have lots of clothing & not much food (as near A & B), we are willing to give up a lot of clothing to get a little more food.When we have lots of food & not much clothing (as near C & D), we are willing to give up very little clothing to get a little more food.

food

A

C

clothing

B

D

Odd special cases that are not consistent with the properties of

Indifference curve listed previously.

Perfect Complements

You need exactly 4 tyres with 1 car body (ignoring the spare tyre).

Having more than 4 tyres with 1 car body doesn’t increase utility.

Also having more than 1 car body with only 4 tyres doesn’t increase utility either.

tyres

Car bodies

1 2

8

4

IC1

IC2

Perfect Substitutes

Consider two packs of paper; the mini-pack has 100 sheets & the jumbo pack has 500 sheets.No matter how many mini-packs or jumbo packs you have, you are always willing to trade 5 mini-packs for 1 jumbo pack.Since the rate at which you’re willing to trade is the slope of the IC, and that rate is constant, your IC’s have a constant slope.That means they are straight lines.

Mini-packs

Jumbo packs

1 2

10

5

IC1

IC2

“Neutral” Good

Your utility is unaffected by consumption of a neutral good.

Neutral good

Desired good

IC1 IC2 IC3

The Marginal Rate of Substitution

• Def: The MRS of Xof Y refers to the amount of Y that must be given up per unit of X gained by the consumer to keep the level if satisfaction unchanged.

• MRSxy= y/ x, where

• MRSxy = the MRs of X for Y

• Y = a small change in the quantity of Y

• X = a small change in the quantity of X

The slope of the indifference curve is the rate at which you are willing to trade off one good to get another good.

It is called the marginal rate of substitution or MRS.

What is the MRS or slope of the IC?

Suppose points A & B are on the same indifference curve & therefore have the same utility level. Let’s break up the move from A to B into 2 parts.

AD: TU = C (MUC)

DB: TU = F (MUF)AB: 0 = TU = C (MUC) + F (MUF)

C (MUC) = – F (MUF)

C/F = – MUF / MUC

IC1

Food F

A

D

Clothing C

B

IC2

So along an indifference curve, the slope or MRS is the negative of the ratio of the marginal utilities (with the MU of the good on the horizontal axis in the numerator).

MRS = – MUX / MUY

For example,Suppose IC1 is the 90-util indifference curve & IC2 is the 96-util indifference curve. Point A is 7 units of food & 6 of clothing. B is 9 units of food & 5 of clothing.Since an additional unit of clothing gives you 6 more utils of satisfaction, the MU of clothing must be 6.Since an additional 2 units of food also give you 6 more utils of satisfaction, the MU of food must be 3.

So, MRS = – MUF / MUC = -3/6 = -0.5 .You’d give up 2 units of food to get 1 units of clothing.

IC1 =90

Food F

A

D

Clothing C

B

IC2 = 96

6

5

7 9

•Hicks replaces the law of DMU by the principle of Diminishing Marginal Rate of Substitution. •As the consumer increases the quantity of X then its MU decreases and % of substitution will be less as the point moves downwards on the IC curve

Comm Y Comm X MRS =

y / x

10 25 -

11 20 -5/1=-5

12 16 -4/1=-4

13 13 -3/1=-3

14 11 -2/1=-2

Budget Constraint Or Budget Line or Price Line

• DEF: The Budget line is the locus of points representing all the different combinations of the two goods that can be purchased by the consumer, given his money income and the prices of the two goods.

• What a consumer can actually buy depends on the income at his disposal and the prices of goods he wants to buy.

• Income and Price are 2 objective factors which form the budgetary constraint of the consumer.

• The consumption or purchase possibility of the consumer is restricted to the budget constraint.

Contd…• The slope of the budget line is called the marginal

rate of substitution in exchange = PX / PY.

• The concept of relative price is important because a rise in relative price would encourage the producer to put more resources in production. The concept also conveys the market information of relative scarcity of those resources.

• The budget line rotates when the relative price changes.

• The shift of the line means that either the income changes or there is a change in the price of both goods.

Alternate Purchase Possibilities

Units of Y

Units of X

A 5 0

4 2

3 4

2 6

1 8

B 0 10

Qt

of

Y

s

z

XO

Y

b

a

Qt of x

c

A

B

Given income = Rs. 50

If P of Y = Rs 10/ unit

If P of X = Rs 5/ unit

AB= Budget

( Price, Income) Line

Budget Constraint or Budget Line

This equation tells you what you can buy.For example, suppose you have $24, & there are

two goods. The price of the first good is $3 per unit & the price

of the second good is $4 per unit. So, if you buy X units of the first good for $3 each,

you spend 3X on that good.Similarly, if you buy Y units of the second good, you

spend 4Y on that good.Your total spending is 3X+4Y.If you spend all 24 dollars that you have, 3X+4Y=24.That equation is your budget constraint.

Example: Budget constraint for $24 of income, and $3 & $4 for the prices of the two goods.

X

Y

(0,6)

(8,0)

0

If you spent all $24 on the 1st good, you could buy 8 units.

If you spent all $24 on the 2nd good, you could buy 6 units.

So we have the intercepts of the budget constraint.

The slope of the line connecting these two points is Y/X = – 6/8 = – 3/4 = – 0.75 .

Let’s generalize. Keep in mind that income was $24 and the prices of the goods were $3 & $4. The equation of the budget constraint in our example was 3X + 4Y = 24.

X

Y

(0,6)

(8,0)

0

So the budget constraint is p1X + p2Y = I

Solving for Y in terms of X, p2Y = I – p1X,

or Y = I /p2 – (p1/p2)X

So from our slope-intercept form, we see that the intercept is I /p2, and the slope is –p1/p2 .

The intercept is income divided by the price of the good on the vertical axis.

The slope is the negative of the ratio of the prices, with the price of the good on the horizontal axis in the numerator.

We have the intercept I /p2, & the slope is –p1/p2 .

What if income increased? The slope would stay the same & the budget constraint would shift out parallel to the original one.Suppose in our example with income of 24 & prices of 3 & 4, income increased to 36. Our new y-intercept will be 36/4 =9 & the new X-intercept will be 36/3=12.

X

Y

(0,6)

(8,0)

0

(0,9)

(12,0)

Suppose the price of the good on the X-axis increased.

If we bought only the good whose price increased, we could afford less of it.

If we bought only the other good, our purchases would be unchanged.

So the budget constraint would pivot inward about the Y-intercept.

X

Y

(0,6)

(8,0)

0 (6,0)

For example, if the price increased from $3 to $4, our $24 would only buy 6 units.

Similarly, if the price of the good on the Y-axis increased, the budget constraint would pivot in about the X-intercept.

X

Y

(0,6)

(8,0)

0

(0,4)

Suppose the price of the 2nd good increased from $4 to $6. If you bought only that good, with your $24, your $24 would only buy 4 units of it.

Changes in Income.

Px, Py constant

X

A3

O B3B1 B2

A1

A2

Co

mm

of

Y

Comm of X

Changes in

Price of Y

Changes in

Price of X

Comm of X

Comm of X

Y

A1

O XB3B1 B2

A2

A1

OB

Y

Changes in

money income,

Prices

and the BL Co

mm

of

YC

om

m o

f Y

Let’s combine our indifference curves & budget constraint to determine our utility maximizing point.

Point A doesn’t maximize our utility & it doesn’t spend all our income. (It’s below the budget constraint.)

X

Y

0

IC1

IC2

IC3

A

Points B & C spend all our income but they don’t maximize our utility. We can reach a higher indifference curve.

X

Y

0

IC1

IC2

IC3

B

C

Point D is unattainable. We can’t reach it with our budget.

X

Y

0

IC1

IC2

IC3

D

Point E is our utility-maximizing point.

We can’t do any better than at E.

Notice that our utility is maximized at the point of tangency between the budget constraint & the indifference curve.

X

Y

0

IC1

IC2

IC3

E

Assumptions of the Consumer Equilibrium

• Consumer• Has fixed amount of money income.• Intends to buy combination of 2 goods, X and Y.• Has definite tastes and preferences.• Hence has definite scale of preferences. Expressed

through ICM.• S of P remains same through out the analysis.• Is rational and mazimizes his satisfaction

• Each of the goods X and Y is homogenous (identical characteristics) and divisible, so various combinations of these goods can be sold.

The consumer Equilibrium

Point e is the

equilibrium

point given the

Budget line.Satisfaction

is max

when the MRS

of x for

y is just equal

to the price

of x to the

price of y.Qt. of comm X

Qt.

of

co

mm

Y

b

a

eM

N

Consumer Optimum or Equilibrium

• In mathematics, the slopes of the indifference curve and the budget line are the same.

• Slope of the budget line = M R S in exchange = PX / PY

• Slope of the indifference curve = M R S in consumption = Y / X

• In equilibrium, PX / PY = Y / X

What happens to consumption when income rises?

For normal goods, consumption increases.For inferior goods, consumption decreases.What does this look like on our graph?

Two Normal Goods

As income increases, the budget constraint shifts out & we are able to reach higher & higher IC’s.

The points of tangency are at higher & higher levels of consumption of both goods.

X

Y

IC1

IC2

IC3

C

B

A

Y3

Y2

Y1

X1 X2 X3

Income-Consumption Curve

The curve that traces out these points is called the income-consumption curve.

For two normal goods, the curve slopes upward.

It may be convex (as drawn here), concave, or linear.

X

Y

IC1

IC2

IC3

C

B

A

Y3

Y2

Y1

X1 X2 X3

We can also look at consumption levels of two goods when the price of one of them changes.

Suppose there is an increase in the price of the 1st good (the good on the X-axis).

The budget constraint pivots inward.

Here we see X drop & Y increase.

In this case, our 2 goods are substitutes.

X

Y

X3 X2 X1

Y3

Y2

Y1

If we connect the points, we have the price consumption curve.

It shows the utility-maximizing points when the price of a good changes.

X

Y

X3 X2 X1

Y3

Y2

Y1

If we look at the price of a good & the amount of it consumed, we have the demand curve

for our particular individual.

As the price decreases the quantity demanded increases & vice versa.

X

P

X1 X2 X3

P1

P2

P3

We can separate the effect of a change in the price of a good on its consumption level into two parts: the income effect & the substitution effect.

Suppose the price of the first good increases.The budget constraint was originally the blue line and we were at A consuming quantities XA & YA. After the price change, the budget constraint is the red line, and we’re at B consuming XB & YB .

X

Y

ABYB

YA

XB XA

We first want to capture the effect of the price change without the effect of the change in income.

X

Y

H

ABYH

YB

YA

XB XH XA

We draw a line parallel to the new budget constraint and tangent to the old indifference curve.

This will reflect the new relative prices, but since we are tangent to the old indifference curve we are just as well off as initially.

Under those circumstances we would be at point H (for hypothetical).

Since the 1st good is now relatively more expensive compared to the 2nd, we will substitute, increasing Y & decreasing X.

The movement from A to H is the substitution effect.

X

Y

H

ABYH

YB

YA

XB XH XA

As a result of the increase in the relative price of the 1st good, we reduce our consumption of it and consume more of the other good.

Now we move from H to B

XX

Y

H

ABYH

YB

YA

XB XH XA

Our purchasing power has been reduced by the price change. That results in the income effect.In our graph, we now hold the relative prices constant at the new level, but income has fallen. Our budget constraint has shifted inward. If both goods are normal, as a result of the change in income, we reduce our consumption of both goods, and X & Y fall.This is the income effect of the price change.

Total Effect of Price Increase

X

Y

H

ABYH

YB

YA

XB XH XA

The total effect is to move from A to B.X has fallen.Both the substitution & income effects led to a drop in X.Y has increased in this case.The substitution effect increased consumption of the 2nd good, but the income effect reduced it by less than the substitution effect increased it.

Let’s do a price decrease.

X

Y

H

B

A

YB

YA

YH

XA XH XB

The budget constraint moves from the blue line to the red line.We draw a line parallel to the new budget constraint and tangent to the old indifference curve.H is the tangency of the hypothetical budget constraint with the old indifference curve.The substitution effect is the movement from A to H. We substitute increasing X & decreasing Y.

The movement from H to B is the income effect.

X

Y

H

B

A

YB

YA

YH

As a result of the higher income (greater purchasing power), we consume more of both goods, if they are normal goods.

XA XH XB

Total Effect

X

Y

H

B

A

YB

YA

YH

The total effect is to move from A to B.X has increased.Both the substitution & income effects led to an increase in X.Y has also increased in this case.The substitution effect decreased consumption of the 2nd good, but the income effect increased it by more than the substitution effect decreased it.

XA XH XB

Income and Substitution Effects, in words

The income effect is the result of the change in purchasing power.

If the price of a normal good increases, you feel poorer, and the income effect is to consume less.

If the price of a normal good decreases, you feel richer, and the income effect is to consume more.

The substitution effect is the result of a change in relative prices.

If the price of a good increases, the substitution effect is to consume less of it & more of the other goods that are now relatively cheaper.

If the price decreases, the substitution effect is to consume more of it & less of the goods that are now relatively more expensive.