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Consumer Preferences

Intermediate Microeconomic Theory ECN 312

From Last Class

• Price determines quantity demanded.

• Determinants of demand: – Income – Population – Preferences – Prices and availability of other goods

• Goal is to understand market demand. We’ll do this piece by piece.

From Last Class

Preferences Budget (Income)

Choice (Individual Demand)

Population

Market Demand

From Last Class

Preferences Budget (Income)

Choice (Individual Demand)

Population

Market Demand

Modelling Consumer Choice

• We are going to describe the consumer’s choice problem (and solution) using mathematical tools

• We think that this is the best way to describe behavior

• It also allow us to use 20,000 year of knowledge in mathematical solutions (google Ishango bone)

Modelling Consumer Choice

Modelling Consumer Choice What would

happen if I asked Ronnie about Pythagoras’ Theorem?

Modelling Consumer Choice Do you think that

Ronnie knows anything about the sides of a

triangle?

Consumers’ Choice Problem

• The solution to a consumer’s choice problem is a bundle of goods that is preferred to all other feasible (affordable) bundles.

• A bundle (or basket) of goods is a collection of items that a person might consume.

– 2 Notre Dame t-shirt, 1 Nissan Leaf

– 1 slice of pizza, 1 Pepsi

– 1 slice of pizza today, 1 slice of pizza tomorrow

Assumptions on Bundles

• We make two key assumptions on bundles:

– Non-negativity: the components of a bundle always have non-negative (≥ 0) quantity.

– Divisibility: The components of a bundle can take on any non-negative value.

Assumptions on Bundles

Consider two goods: x and y

Assumptions 1 & 2 imply: y

Assumptions on Bundles

Consider two goods: x and y

Assumptions 1 & 2 imply: y

Every (x,y) in this

quadrant is

possible

• We make three key assumptions on preferences:

– Completeness: Every pair of bundles can be ranked. Either I prefer A to B, B to A, or I am indifferent.

– Transitivity: Rankings are rational. If I prefer A to B and B to C, then I prefer A to C.

– Non-satiation: More is better than less.

Assumptions on Preferences

Consider two goods: x and y

Assumption 3 implies: y

Assumptions on Preferences

Consider two goods: x and y

Assumption 3 implies: y

Worse

Assumptions on Preferences

Consider two goods: x and y

Assumption 3 implies: y

Worse

Assumptions on Preferences

Consider two goods: x and y

Assumption 3 implies: y

Worse

Better

Assumptions on Preferences

Consider two goods: x and y

Assumption 3 implies: y

Worse

Better

Indifference Curves

• An indifference curve shows all combinations of two goods that yield the same level of satisfaction/happiness/utility to a person.

• Different people may have different looking indifference curves for the same pair of goods.

• Indifference curves between different pairs of goods will generally look different for the same person.

Indifference Curves y

Indifference curves are like the lines on a topographical

map – a third dimension projected onto a two

dimensional graph

Aside….

Indifference Curves y

Indifference Curves

1 3 4

Indifference Curves

1 3 4

Which bundle is preferred?

Indifference Curves

1 3 4

Which bundle is preferred?

But A has the most of y….

Indifference Curves

• Rules:

– Indifference curves cannot cross.

– Indifference curves slope downwards (for `goods’)

Indifference Curves

• Rules:

– Indifference curves cannot cross.

– Indifference curves slope downwards (for `goods’)

What about pollution, disease, hunger,

homework?

Indifference Curves

• Rules:

– Indifference curves cannot cross.

– Indifference curves slope downwards (for `goods’)

What about pollution, disease, hunger,

homework?

pizza

1 hour doing homework

Indifference Curves

• Rules:

– Indifference curves cannot cross.

– Indifference curves slope downwards (for `goods’)

What about pollution, disease, hunger,

homework?

1 hour doing homework

pizza

Indifference Curves

• Rules:

– Indifference curves cannot cross.

– Indifference curves slope downwards (for `goods’)

What about pollution, disease, hunger,

homework?

1 hour free time

pizza

Marginal Rate of Substitution

• How much of good y would you give up for another unit of good x?

• Mathematically, MRS = (-1)(slope of I.C.)

Since our indifference curves have a negative slope, our MRS will be

positive.

Marginal Rate of Substitution

MRS

Marginal Rate of Substitution

MRS

MRS = (-1)(slope) ≈ (-1)(rise/run)

Marginal Rate of Substitution

• Diminishing marginal rate of substitution

• As we move down an indifference curve, MRS typically falls.

– Averages are better than extremes.

– Indifference curves are usually convex.

Marginal Rate of Substitution

1 1

MRSA

MRSB

Marginal Rate of Substitution

1 1

MRSA

MRSB

A: More y than x B: More x than y

MRSA > MRSB

Marginal Rate of Substitution

• Why can’t indifference curves cross? y

Marginal Rate of Substitution

• Why can’t indifference curves cross? y

Marginal Rate of Substitution

• Lets look at two consumers with different tastes y

ICAnn

ICBob

Marginal Rate of Substitution

• Lets look at two consumers with different tastes y

ICAnn

ICBob

Who has stronger preferences for X

(relative to Y)?

Indifference Curves

• Extreme Case #1 – Perfect Substitutes

– No balance necessary. Extremes are as good as averages.

– Constant MRS, ICs have constant slope

– Examples:

Indifference Curves

• Extreme Case #1 – Perfect Substitutes

– No balance necessary. Extremes are as good as averages.

– Constant MRS, ICs have constant slope

– Examples:

• Coke and Pepsi

• Blue pens and black pens

• Margarine and butter

• Equal and Sweet’n Low

Perfect Substitutes Blue Pens

Black Pens

4 62

Perfect Substitutes Blue Pens

Black Pens

4 62

You are indifferent between Blue and

Black Pens at all quantities….

Indifference Curves

• Extreme Case #2 – Perfect Complements

– These goods are used in fixed proportions.

– MRS is very sensitive to proportion of goods.

– ICs are kinked.

– Examples:

Indifference Curves

• Extreme Case #2 – Perfect Complements

– These goods are used in fixed proportions.

– MRS is very sensitive to proportion of goods.

– ICs are kinked.

– Examples:

• Left shoes and Right shoes

• Peanut butter and Jelly

• Apple pie and Ice cream

• Coffee and Milk (coffee and cream, coffee and sugar)

Perfect Complements Coffee

Milk1 2 3

I like 2 parts coffee to 1 part milk

Perfect Complements Left Shoes

Right Shoes1 2 3

Example – Auto Prices

• See “A Different Beat: Toyota Raises Prices While Detroit Cuts Deeply” New York Times

• In the summer of 2005: – US firms introduced “employee discounts” for all

– GM introduced first, other US firms followed

– Toyota (and Honda) did not

• (One) Economic question: Why does Ford care about GM’s pricing while Toyota does not?

Example – Auto Prices GM

Ford

Example – Auto Prices GM

Ford

GM and Ford may not be perfect substitutes, but

they’re pretty substitutable (relatively)

Example – Auto Prices GM

Toyota

Example – Auto Prices GM

Toyota

GM and Toyota are relatively less substitutable

Utility

• We can do all our analysis of consumers’ choices with indifference curves (graphically)

• We can also express consumers’ tastes with a utility function (mathematically)

• Utility function: a rule for translating bundles into a numerical value for “happiness”

Utility

• Utility functions allow us to state consumer choice as an optimization problem and use calculus.

• Rules: – A utility function U associates a total utility number

with each possible bundle. Example: U(x,y) = 20

– All bundles on an IC have the same level of utility.

– Preferred ICs have higher levels of utility.

Utility

• Each indifference curve is a level set of a utility function

• A level set is set of input values such that the function takes on some constant value

• When there are two inputs, a level set may also be called a level curve, a contour line, or an isocurve

Utility y

U(x,y) = 30

U(x,y) = 20

U(x,y) = 10

Utility • Suppose the only products in the economy are

food (F) and clothing (C). Possible utility functions are:

– U(F,C) = 2F + C

– U(F,C) = 3F½C¼

• What is the difference between these utility functions?

Utility • Suppose the only products in the economy are

food (F) and clothing (C). Possible utility functions are:

– U(F,C) = 2F + C

– U(F,C) = 3F½C¼

• What is the difference between these utility functions?

– They imply different preferences over food and clothing.

Utility Important:

• Numerical values of utility levels are unimportant. All that matters is that better ICs have higher levels of U.

• Utility functions provide ordinal information, not cardinal.

Utility

• Bundle A is (F,C) = (2, 3). B is (F,C) = (3, 2)

– If U = 2F + C, then UA = 7 and UB = 8

– If U = 4F + 2C + 1, then UA = 15 and UB = 17

• Both versions of the utility function rank A and B (and all other bundles!) in the same order.

• Both imply: Choose B

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(0,0)

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(0,0) U = 0

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(0,2)

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(0,2) U = 0

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(1,1)

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(1,1) U = 1

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(1,2)

Utility

Cobb-Douglas utility functions

• These have the form U(X,Y) = aXY.

• We will see these frequently because of their convenient properties.

Example: Food and clothing, with U = FC.

What are the utility levels for the following bundles?

(1,2) U = 2

Cobb-Douglas Utility

U = FC

Food

Clothing

U = 30

U = 20

U = 10

Cobb-Douglas Utility

U = FC

Food

Clothing

But what if U = 2FC?

Cobb-Douglas Utility

U = 2FC

Food

Clothing

But what if U = 2FC?

Essentially nothing…

U = 60

U = 40

U = 20

Cobb-Douglas Utility

U = FC

Food

Clothing

But what if U = F2C?

U = 20

U = 10

Cobb-Douglas Utility

U = FC

Food

Clothing

But what if U = F2C?

Indifference curves get steeper.

U = 30 U = 20

U = 10

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(0,0)

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(0,0) U = 0

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(1,3)

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(1,3) U = 4

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(3,1)

Utility

Perfect substitutes

• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N

What are the utility levels for the following bundles?

(3,1) U = 4

Perfect SubstitutesNapkins

Paper Towels1 3

U = 4

U = 7

U = N + P

Perfect SubstitutesNapkins

Paper Towels1 3

U = 4

U = 7

But what happens if U = 2N + 2P?

U = N + P

Perfect SubstitutesNapkins

Paper Towels1 3

U = 8

U = 14

U = 2N + 2P

Perfect SubstitutesNapkins

Paper Towels1 3

U = 8

U = 14

U = 2N + 2P

But what happens if U = 2N + P?

Perfect SubstitutesNapkins

Paper Towels1 3

U = 5

U = 7

U = 2N + P

Utility

Perfect complements

• These have the form U(X,Y) = c∙min(aX, bY)

Example: Peanut butter and jelly, with U = min(P,J)

What are the utility levels for the following bundles?

(3,1)

Utility

Perfect complements

• These have the form U(X,Y) = c∙min(aX, bY)

Example: Peanut butter and jelly, with U = min(P,J)

What are the utility levels for the following bundles?

(3,1) U = 1

Utility

Perfect complements

• These have the form U(X,Y) = c∙min(aX, bY)

Example: Peanut butter and jelly, with U = min(P,J)

What are the utility levels for the following bundles?

(2,4)

Utility

Perfect complements

• These have the form U(X,Y) = c∙min(aX, bY)

Example: Peanut butter and jelly, with U = min(P,J)

What are the utility levels for the following bundles?

(2,4) U = 2

Perfect Complements J

P1 2 3

U = min(P,J)

U = 3

U = 2

U = 1

Perfect Complements J

P1 2 3

U = min(P,J)

But what happens if

U = 2*min(P,J) or

U = min(2P,2J) ?

U = 3

U = 2

U = 1

Perfect Complements J

P1 2 3

U = 2*min(P,J)

U = 6

U = 4

U = 2

Perfect Complements J

P1 2 3

U = 2*min(P,J)

U = 6

U = 4

U = 2

We saw already what

happens if U = min(P,2J)

Marginal Utility

• What is the additional benefit of one more X, conditional on your current bundle?

• What is the utility gain from a small change in X, while the amount of Y is held fixed?

(for well-behaved preferences)

– U always increases with X (U never decreases)

– Increase in U is greatest when quantity of X is small

Marginal Utility

U(X,YA)

X0 1 7 8

Total Utility Curve

Marginal Utility

U(X,YA)

X0 1 7 8

U(X,YB)

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = X + Y

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = X + Y MUX = 1

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = 4X + Y

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = 4X + Y MUX = 4

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = 4X*Y

Marginal Utility

• Change in utility from a small change in quantity of one good in a consumption bundle.

• MU of X is the derivative of U with respect to X, treating Y as a constant.

U = 4X*Y MUX = 4Y

Marginal Utility

• Importance of MU to microeconomics

– main goal is to understand decision-making

– When should you decide to do anything???

Marginal Utility and ICs We can define MUY in the same way we defined MUX

MRSXY = MUX MUY

Marginal Utility and ICs We can define MUY in the same way we defined MUX

MRSXY = MUX MUY

∙ ∙

1 1

MRSA

MRSB

Marginal Utility and ICs We can define MUY in the same way we defined MUX

MRSXY = MUX MUY

∙ ∙

1 1

MRSA

MRSB

Why is MRSA

>MRSB?

Marginal Rate of Substitution

• MRSXY = MUX/MUY

– Cobb Douglas: U = XaYb MRS =

– Perfect Substitutes: U = aX + bY MRS =

– Perfect Complements: U = min(aX,bY) MRS = 0 or ∞

aY bX

a b