Business Economics

Theory of Consumer BehaviorThomas & Maurice, Chapter 5

Herbert Stocker

herbert.stocker@uibk.ac.at

Institute of International Studies

University of Ramkhamhaeng

&

Department of Economics

University of Innsbruck

Simple Model Economy

Circular flow model

Two actors with different goals

HouseholdsBuy and consume goods and servicesOwn and sell factors of production

FirmsProduce and sell goods and servicesHire and use factors of production

Circular Flow Model

Households

QD

QS

Price

Quantity

Firms

LD

LS

Wage

Working Hours

Modeling Household Decisions

Plan of the chapter

1 Find a general way to describe what consumers(households) want ⇒ preferences

2 Map these preferences in a utility function

3 Describe the choices available (restrictions)⇒ budget constraint

4 Use a technique to perform the optimization:

→ Unconstrained vs. constrained maximization(see Th&M: Ch.3, Appendix)

Result: Demand curve of an individual household!

Consumer Choice

Preferences(exogeneous!)

Utility function

Budgetrestriction

Optimization

Decisions(Demand function)

The description of

Preferences

Preferences

Assumptions

Consumers obtain benefits (utility) from theconsumption of goods & services

Consumers have complete information aboutcharacteristics and availability of all goods &services

Consumers decide between different bundles ofgoods and services!

Preferences

Example for two different bundles:

Bundle P Bundle R

3 kg of rice2 shirts5 beer

1 trip to Paris0 trips to Rome

...10 ballpen

,

7 kg of rice3 shirts4 beer

0 trips to Paris1 trip to Rome

...5 ballpen

Bundles can be written as vectors!

Preferences

If a consumer chooses bundle P when bundle R isavailable, we can say that the consumer prefers bundleP to bundle R. We write

P ≻ R,

or

Bundle P Bundle R

3 kg of rice2 shirts5 beer

1 trip to Paris0 trips to Rome

...10 ballpen

≻

7 kg of rice3 shirts4 beer

0 trips to Paris1 trip to Rome

...5 ballpen

Preferences

All bundles of goods can be ranked based on theirability to provide utility:

P ≻ R: P-bundle is strictly preferred to theR-bundle

R ≻ P: R-bundle is strictly preferred to theP-bundle

P ∼ R: P-bundle is regarded as indifferent to theR-bundle

P � R: P-bundle is at least as good as (preferredto or indifferent to) the R-bundle

Preferences

Preferences are relationships between bundles!

Individuals choose between bundles containingdifferent quantities of goods

In the following we rely on only two goods(Theory also works with more than two goods)

⇒ Assumption: Consumers always prefer more of anygood to less: More is better!

Preferences & Indifference Curves

0

1

2

3

4

0 1 2 3 4

Wine

Bread

b E

b R

b P

More is better:Bundle R = (3, 3)is preferred to bundleE = (2, 2), whichis preferred overP = (1, 1)

More generally: Theconsumer prefers Eto all combinationsin the magenta box(e.g., P), while allthose in the yellowbox (e.g., R) arepreferred to E

Preferences & Indifference Curves

0

1

2

3

4

0 1 2 3 4

Wine

Bread

b E

b R

b P

b

b

b

b

A

B

C

D

What about bundles B & C?

Bundles A & D havemore of one goodbut less of another,compared to E

More informationabout consumerranking is needed!

Consumer might beindifferent betweenA, E and D

We can then connectthose points with anindifference curve

Preferences & Indifference Curves

Any bundle lying above (northeast of) anindifference curve (e.g. C) is preferred to anybundle lying on the indifference curve

Points on the curve are preferred to points below(southwest of) the curve (e.g. B)

Indifference curves slope downward to the right(negatively sloped)

Otherwise, they would violate the assumption thatmore is preferred to less: Bundles with more ofboth goods would be indifferent to baskets withless of both goods

Indifference curves and -map

To describe preferences for all combinations ofgoods/services, we use a set of indifference curves⇒ Indifference map

Each point represents a bundle

of different quantities of bread

and wine

Each indifference curve con-

nects the bundles among which

the consumer is indifferent

Preferredbundles

Wine

Bread

Notice: The higher the indifference curve, the higher an individual’s utility!

Question: Is it possible that indifference curves cross?

Assumptions about preferences

Assumptions about preferences

complete: Any two bundles can be compared

reflexive: Any bundle is at least as good as itself

transitive: If Q ≻ R and R ≻ S , then Q ≻ S

⇒ Preferences can be represented in a utility

function

In addition, it is often useful to assume convexpreferences

Preferences

Convex preferences

bc

bc

Qy

Qx

“Averages are preferred toextremes”

i.e., goods are consumedtogether (e.g., bread andwine)

This is the ‘normal’ case!

Special Preferences

Concave Preferences

bc

bc

Qy

Qx

Goods are normally notconsumed together (e.g.,beer and wine)

→ time horizon!

→ corner solutions!

Special Case: Satiation (bliss point)

→ Not Monotonic!

Above the bliss point util-ity decreases, nobody willconsume there!

b

Bliss-PointQy

Qx

Special Preferences

Perfect Substitutese.g., U(Qx ,Qy) = Qx + Qy

Qy

Qx

Constant rate of trade-off be-tween two goods (e.g., red pen-cils and blue pencils)

Perfect ComplementsU(Qx ,Qy ) = min{Qx ,Qy}

Qy

Qx

Always consumed together infixed proportions (e.g., rightshoes and left shoes; coffee andcream)

The shape of indifference curves

Utility

Utility Function

A utility function assigns a number to each bundle ofgoods so that more preferred bundles get highernumbers, that is,

U(Qx ,Qy) > U(R1,R2)

if and only if

(Qx ,Qy) ≻ (R1,R2)

Utility

Two ways of viewing utility:Old way: measures how “satisfied” you are

not operational, many other problems

New way: summarizes preferences, i.e. theranking of bundles.

Utility functions are just a shorter and more elegantway to summarizes preferences.only the ordering of bundles counts, so this is a theoryof ordinal utilitygives a complete theory of demand; operational

Cobb-Douglas Utility Function

A very simple and ‘well be-haved’ utility function:

Cobb-Douglas Function

U = U(Qx ,Qy ) = QaxQ

by

(a and b are positive param-

eters determining the kind of

preferences)

Example:

U = Q0.3x Q0.7

y

0

2

4

6

8

10

Q1

0

2

4

6

8

10

Q2

0

2.5

5

7.5

10

U

0

2

4

6

8

10

Q1

Cobb-Douglas Utility Function

Indifference Curves (red) can also be drawn withutility functions → connect points with equal utility:

0

2

4

6

8

10

Q1

0

2

4

6

8

10

Q2

0

2

4

6

8

10

U

0

2

4

6

8

10

Q1

Cobb-Douglas Utility Function

Indifference Curves (red) are like contour lines:

02

4

6

8

10

Q1

0

2

4

6

8

10

Q2

02468

10

U

02

4

6

8

10

Q1

Special Preferences

Perfect Substitutes: e.g., U(Qx ,Qy ) = Qx + Qy

⇒ Constant rate of trade-off between the two goods

0

2

4

6

8

10

Q1

0

2

4

6

8

10

Q2

0

2

4

6

8

10

U

0

2

4

6

8

10

Q1

Qy

Qx

Special Preferences

Perfect Complements: U(Qx ,Qy) = min{Qx ,Qy}⇒ Always consumed together

0

2

4

6

8

10

Q1

0

2

4

6

8

10

Q2

0

2

4

6

8

10

U

0

2

4

6

8

10

Q1

Qy

Qx

Marginal Utility

Extra utility from some extra consumption of oneof the goods, holding the other good fixed

This is a derivative, but a special kind ofderivative, a partial derivative (∂)

This just means that you look at the derivative ofU(Qx ,Qy) keeping Qy fixed, treating it like aconstant

∂U

∂Qx

≡dU

dQx

∣∣∣∣dQy=0

Marginal Utility

Examples:

U = Qx + Qy ⇒ MUx ≡∂U

∂Qx

= 1

U = QaxQ

1−ay ⇒ MUx ≡

∂U

∂Qx

= aQa−1x Q1−a

y

U = QaxQ

1−ay ⇒ MUy ≡

∂U

∂Qy

= (1− a)QaxQ

−ay

Marginal Rate of Substitution (MRS)

Marginal Rate of Substitution (MRS):Measures how a consumer is willing to trade offconsumption of good X for consumption of Y

⇒ Slope along an indifference curve, keeping utilityconstant

MRSxy ≡ −dQy

dQx

=MUx

MUy

≡∂U∂Qx

∂U∂Qy

(for dU = 0)

Sign: Generally, indifference curves have anegative slope

Marginal Rate of Substitution (MRS)

Discrete

∆Qy

∆Qy

∆Qy

∆Qx ∆Qx ∆Qx

Qy

Qx

Slope:∆Qy

∆Qx

Infinitesimal

bc

bc

bc

Qy

Qx

Slope:dQy

dQx

MRS diminishes along an indifference curve!

Derivation of MRS

Consider U = U(Qx ,Qy)

Totally differentiating this utility function yields

dU =∂U

∂Qx

dQx +∂U

∂Qy

dQy = 0

Re-arranging this expression gives

−dQy

dQx

=∂U∂Qx

∂U∂Qy

≡ MRSxy �

What we can afford

The Budget Constraint

Budget Constraint

The Budget Constraint

M = PxQx + PyQy

shows for given prices Px and Py all combinations ofQx and Qy a household with given income can afford

No lending and no borrowing

Rewriting gives

Qy =M

Py

−Px

Py

Qx Slope:dQy

dQx

= −Px

Py

Budget Constraint

MPy

∆Qx

∆Qy

M = PxQx + PyQy

Qy =MPy

− Px

PyQx

∆Qy

∆Qx= −Px

Py

Qy

Qx

Changes in income and in prices changes the

shape of the budget line!

Budget Constraint

The price ratio Px/Py shows how many units ofthe second good can be obtained on the market

for one unit of the first good.

Example: when QB is the quantity of bread,and QW the quantity of winethen PB/PW gives the price of one unit bread inunits of wine.

Example:

PB

PW

=

2 Eurokg Bread4 Eurolt Wine

=2 Euro

4 Euro

lt Wine

kg Bread=

0.5 lt Wine

kg Bread

Budget Constraint

012345

0 1 2 3 4 5 6 7 8 9 10

Qy

Qx

α

β

−dQx

dQy

=Py

Px

= tan β = 2

M = PxQx + PyQy

Qy =M

Py

−Px

Py

Qx

−dQy

dQx

=Px

Py

= tanα = 0, 5

one unit of Qx costs 0.5units of Qy (= tanα)!

or, one unit of Qy costs2 units of Qx (= tan β).

Changes in the Budget Line

What happens when all prices and the incomemultiply? (e.g. inflation)

Multiply all prices and income with a constant t:

tM = tPxQx + tPyQy

but this is the same as the initial budget constraint

M = PxQx + PyQy

therefore “a perfectly balanced inflation doesn’tchange consumption possibilities”!

Changes in the Budget Line

What happens when all prices double, but theincome remains constant?

Multiply all prices with a constant t:

M = tPxQx + tPyQy

this is the same as

M

t= PxQx + PyQy

therefore it makes no difference whether all pricesdouble or income is halved, multiplying all prices by aconstant t is just like dividing income by t.

Changes in the Budget Line

What happens when a specific tax is levied onQx?

A specific tax (quantity tax) T raises the price of Qx

to Px + T , d.h. the budget line becomes steeper.

What happens when a ad-valorem subsidy s ispaid on Qx?

the budget line becomes

M = (1− s)PxQx + PyQy

i.e. Qx becomes cheaper, the budget line flatter!

Changes in the Budget Line

What happens when the consumer gets one unit of Qx

for free?

0

1

2

3

0 1 2 3 4

Qy

Qx

Changes in the Budget Line

What happens when the consumer gets the second twounits of Qx for half the price of the first two units?

0

1

2

3

4

0 1 2 3 4 5 6

Qy

Qx

Combining preferences and budget constraint

Optimal Choice

Decisions (in a neoclassical perspective)

Preferences(exogeneous!)

Utility function

Budgetrestriction

Optimization

Decisions(Demand function)

Decisions: neoclassical point of view

Preferences

U = U(Qx ,Qy)

M = PxQx + PyQy

max : U(Qx ,Qy )s.t. M = PxQx + PyQy

L = U(Qx ,Qy) + λ[M − PxQx − PyQy ]

Q∗x = Qx(Px ,Py ,M),

Q∗y = Qy (Px ,Py ,M)

Consumer ChoiceCobb-Douglas utility function and linear budgetconstraint

0

2

4

6

8

10

Q1

0

2

4

6

8

10

Q2

0

2

4

6

8

10

U

0

2

4

6

8

10

Q1

Optimization

Problem

maxQx ,Qy

U(Qx ,Qy)

s.t.: M = PxQx + PyQy

Two possibilities for optimization

Substitution method (rather awkward)

Lagrange method (simple and elegant)

Lagrange Method

Joseph Louis Lagrange (1736 - 1813):

an Italian-French mathe-matician and astronomerwho made important con-tributions to all fieldsof analysis and numbertheory was arguably thegreatest mathematician ofthe 18th century.

Developed a simplemethod for constrained

optimization.

Lagrange Method

Step 1: Problem

maxQx ,Qy

U(Qx ,Qy)

s.t.: M = PxQx + PyQy

Step 2: Lagrange function(goal function plus Lagrange multiplier λ timesthe restriction in implicit form)

L = U(Qx ,Qy) + λ [M − PxQx − PyQy ]︸ ︷︷ ︸

=0

Lagrange Method

Step 3: Set partial derivatives of the Lagrangefunction with respect to the endogenous(decision) variables Qx and Qy as well as theLagrange multiplier λ equal to zero:

∂L

∂Qx

=∂U

∂Qx

− λPx!= 0

∂L

∂Qy

=∂U

∂Qy

− λPy!= 0

∂L

∂λ= M − PxQx − PyQy

!= 0

⇒ First order conditions (FOC)

Lagrange Method

Step 4: Solve the equation system for theendogenous variables Qx , Qy and λ

Q∗x = Qx(Px ,Py ,M), Q∗

y = Qy (Px ,Py ,M)

λ∗ = λ(Px ,Py ,M)

The solutions to Q∗x and Q∗

y are the demand

functions for an individual household

Describe the optimal decisions of an householdunder given restrictions

Lagrange Method: Example

U = U(Qx ,Qy ) = QxQy

With M = PxQx + PyQy , we have

L = QxQy + λ[M − PxQx − PyQy ]

The FOC read as

∂L

∂Qx

= Qy − λPx!= 0 (1)

∂L

∂Qy

= Qx − λPy!= 0 (2)

∂L

∂λ= M − PxQx − PyQy

!= 0 (3)

Lagrange Method: Example

From (1) and (2), we have

Qy

Qx

=Px

Py

or Qy =Px

Py

Qx

Inserting Qy in (3) gives

M − PxQx − Py

(Px

Py

)

Qx = 0

Solving for Qx we obtain demand for X

Q∗x =

M

2Px

Similarly, demand for Y is Q∗y = M

2Py

Optimal Choice

The FOC allow some more insights in the problem ofoptimal consumer choice

L = U(Qx ,Qy) + λ [M − PxQx − PyQy ]

∂L

∂Qx

=∂U

∂Qx

− λPx!= 0

∂L

∂Qy

=∂U

∂Qy

− λPy!= 0

∂L

∂λ= M − PxQx − PyQy

!= 0

⇒ λ =∂U∂Qx

Px

=

∂U∂Qy

Py

orMUx

Px

=MUy

Py

Example

Suppose, M = 140. All income is spend on 20 units of X withPX = 4, and on 30 units of Y with PY = 2:140 = 4 · 20 + 2 · 30

Further, assume that MUx = 20 and MUy = 16

MUx

Px= 20

4= 5 < 8 = 16

2=

MUy

Py

It is optimal for the consumer to spend more money on Y :

Spending one EUR on Y increases utility by 8 units

Reducing consumption of X by one unit induces a utility loss of only 5units

Note: Budget remains constant at 140

⇒ Consumer maximizes utility if income is allocated in a way that

the marginal utility per money unit spent on each good is

identical

Optimal Choice

Since on an indifference curve utility is constant bydefinition it follows

dU = 0 = MUxdQx +MUydQy

Hence,

MRS = −dQy

dQx

=MUx

MUy

Therefore:

Px

Py

=∂U∂Qx

∂U∂Qy

≡MUx

MUy

= −dQy

dQx

≡ MRSxy

Optimal Choice

0

2

4

68

10

x1

0

2

4

68

10 x2

0

2

4

6

8

10

0

2

4

68

10

x1

0

2

4

68

10 x2

MRS = dQy

dQx

= MUx

MUy

= −Px

Py

A

Good X (Qx) →

←

Good

Y(Q

y )

Uti

lity

(U)

Indifferencecurves

Budget-

constraint

Utility function

Optimal Choice

Condition for optimality: MRS = Price ratio

Qy

Qx

Slope:dQy

dQx

∣∣∣dU=0

Slope: −Px

Py

∣∣∣dM=0

bc

bcbc

bc

Income-

Consumption-Curve

Optimal Choice

Implications of MRS condition:

Why do we care that MRS = − price ratio?

If everyone faces the same prices, then everyonehas the same local trade-off between the twogoods. This is independent of income and tastes.

Since everyone locally values the trade-off thesame, we can make policy judgments. Is it worthsacrificing one good to get more of the other?Prices serve as a guide to relative marginalvaluations!

MRS condition: Recap

The MRS is an indicator for the willingness to pay

A budget constraint shows the ability to pay

When we combine the MRS with the ability to

pay, i.e., the budget constraint, we can derivedemand

Demand and Changes in Income

Demand and Changes in Income

Income-consumptioncurve: Normal good

Clothes

Food

Income-Consumption

Curve

Engel Curve: Normalgood

Income

Food

EngelCurve

Demand and Changes in Income

Inferior good

Beefsteak

Hamburger

Income-Consumption

Curve

Engel Curve: Inferiorgood

Income

Hamburger

EngelCurve

b

normal

inferior

Demand and Changes in Price

→ Demand Curves

Cobb-Douglas Preferences

0

1

2

3

4

0 1 2 3 4

Qy

Qx

0

1

2

3

4

0 1 2 3 4

Px

Qx

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

maxQx ,Qy

U(Qx ,Qy ) = QxQy

s.t.: M = PxQx + PyQy

Budget constraint

for M = 4, Py = 1 : ⇒ 4 = PxQx + 1Qy

Qy = 4− PxQx

Px = 4, 4, 2, 1,33, 1, 0,8, 0,5, Solu-tion:

Q∗

x =M

2Px

=2

Px

This is the usual demand curveQD

x = Q(Px ,M)

Special Cases

The usual methods for maximization (e.g., Lagrangemethod) are not applicable when preferences areconcave or indifference curves are not differentiable inthe relevant point (e.g., kinky, linear, . . . )

Examples:

Perfect Substitutes (⇒ corner solution)

Perfect Complements

An analytical solution is in these cases more difficult(i.e., Kuhn-Tucker conditions)

Perfect Substitutes

0

1

2

3

4

0 1 2 3 4

Qy

Qx

0

1

2

3

4

0 1 2 3 4

Px

Qx

bcbc

bc bc bc

maxQx ,Qy

U(Qx ,Qy ) = Qx + Qy

s.t. M = PxQx + PyQy

[Graph: M = 3 und Py = 1]

MRS = −dQy

dQx

= 1,Px

Py

= 33/2, 1, 3/4, 3/5

Q∗

x =

0 wennPx > Py[

0, MPx

]

if Px = Py ,

MPx

if Px ≤ Py .

Perfect Complements

0

1

2

3

4

0 1 2 3 4

Qy

Qx

0

1

2

3

4

0 1 2 3 4

Px

Qx

bc

bc

bcbc

maxQx ,Qy

U(Qx ,Qy ) = min{Qx ,Qy}

s.t. M = PxQx + PyQy

[Graph: M = 4, Py = 1]Lagrange not applicable!!!Insert efficiency-condition Qx = Qy in bud-get constraint:

M = (Px + Py )Qx

Q∗

x =M

Px + Py

Preferences and Demand

The kind of assumed preferences determines theproperties of the demand functions!For example, Cobb-Douglas preferences imply

a linear income-consumption curvea horizontal price-consumption curvethe price elasticity of demand is always −1the income elasticity of demand is always +1cross price elasticities are always zeroexpenditure shares are always constant

Effects of Price Changes

Slutsky- and Hicks Decomposition

Consumer Choices: Examples

The theory of consumer choice, inter alia, addressesthe following questions:

What happens with labor supply when wagesincrease?

Do people save more when interest rates go up?

Do the poor prefer to receive cash or in-kindtransfers?

Do all demand curves slope downward?

Price Changes

A fall in the price of a good has two effects:

First, relative prices change

Second, the purchasing power changes

Slutsky-decomposition: What happens withdemand, when relative prices change, but the

purchasing power is held constant?

Hicks-decomposition: What happens withdemand, when relative prices change, but the

utility is held constant?

Slutsky-decomposition

0

1

2

3

4

0 1 2 3 4

Qy

Qx

max U = QxQx

s.t. M = PxQx + PyQy

(for M = 4 and Py = 1)

bc bc

bc

SE IE

Optimal decision when Px = 4:Qx = 0.5, Qy = 2

Optimal decision when Px = 1:Qx = 2, Qy = 2

Slutsky Substitution Effect(=SE): new price ratio, butconstant purchasing power!

Income effect (=IE): constantprice ratio, but purchasingpower increases!

Hicks-Decomposition

0

1

2

3

4

0 1 2 3 4

Qy

Qx

max U = QxQx

s.t. M = PxQx + PyQy

(for M = 4 and Py = 1)

bc bc

bc

SE IE

Optimal decision when Px = 4:Qx = 0.5, Qy = 2

Optimal decision when Px = 1:Qx = 2, Qy = 2

Hicks Substitution Effect(=SE): new price ratio, butconstant utility!

Income Effect (=IE): constantprice ratio, but higher income!

Substitution- and Income Effects

When preferences are convex the substitutioneffect can never be positive!

The income effect can either be positive ornegative

If the income effect is negative ⇒ inferior goods

If the income effect is negative and larger as thesubstitution effect ⇒ Giffen-good

Giffen-Good

Qy

Qx

bc

bc

bcbc

SE

IE

TE

Although Qx be-comes cheaper lessof Qx is demanded!

Market Demand

Market Demand

Market Demand, D, is the horizontal sum ofindividual demands

Dx = Q1x (Px ,Py ,M

1) + Q2x (Px ,Py ,M

2)

+ · · ·QNx (Px ,Py ,M

N)

Note: The subscript denotes good x and y , the superscript consumer i ∀i = 1, 2, ...,N

Market Demand

Attention: Quantities can never be negative, onlyzero!

0

1

2

3

0 1 2 3 4 5 61.5

P

Q2.5 5.5

The market demand function

has kinks!Q

d1= 1 − P

Qd2

= 3 − 1.5P

Qd3

= 1.5 − 0.5P

D =

0 forP ≥ 3

1.5 − 0.5P for 2 ≤ P ≤ 3

4.5 − 2P for 1 ≤ P ≤ 2

5.5 − 3P for 0 ≤ P ≤ 1