Theory of Consumer Behavior - hsto.infoTheory of Consumer Behavior Herbert Stocker [email protected] Institute of International Studies University of Ramkhamhaeng & Department

Theory of Consumer Behavior

Herbert Stocker

[email protected]

Institute of International StudiesUniversity of Ramkhamhaeng

&Department of EconomicsUniversity of Innsbruck

Consumer Choice

“Economics is about making the best

of things.

In other words, it is about choice

subject to constraints.”

Layard / Walters (1978): Microeconomic Theory

Consumer Choice & Decisions

“Economists create good stories by being

simple, explicit, and plausible about three

things:

1 the actors involved,

2 their goals, and

3 the choices available to them.”(Fiona Scott Morton)

Households and Firms

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

1

Circular Flow Diagram

Households

QD

QSPrice

Quantity

Firms

LD

LS

Wage

Working Hours

Households and Firms

Two principles of economics:

Optimization principle: people choose

actions that are in their interest

Equilibrium principle: people’s actions

must eventually be consistent

with each other

Modeling of Decisionmaking

What we will do in this chapter:

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

2 Map these preferences in a utility function3 Describe the choices available (restrictions)⇒ budget constraint

4 Use a technique to perform the optimization(e.g. Lagrange Multiplier)

As a result we will get the demand curve of anindividual household!

Consumer Choice

Preferences(exogeneous!)

Utility function

Budget-restriction

Optimization

Decisions(Demand functions)

2

The description of

Preferences

Preferences

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

Assume consumers have complete informationabout characteristics 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

,


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 it’s natural to say that the consumer prefersbundle P to bundle R. We write P ≻ R , or

Bundle P Bundle R


1 trip to Paris0 trips to Rome

...10 ballpen

≻


0 trips to Paris1 trip to Rome

...5 ballpen

3

Preferences

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

P ≻ R means the P-bundle is strictly preferred tothe R-bundle.

R ≻ P means the R-bundle is strictly preferred tothe P-bundle.

P ∼ R means that the P-bundle is regarded asindifferent to the R-bundle,

P � R means the P-bundle is at least as good as(preferred to or indifferent to) the R-bundle.

Preferences

Preferences are relationships between bundles!

Preferences refer to the ranking of entire bundlesof goods, not to individual goods.

Individuals choose between bundles containingdifferent quantities of goods.

Theory works with more than two goods, but thenwe can’t draw pictures.

Therefore, we will restrict ourselves to two goods,bread and wine.

Generally, we will assume that consumers alwaysprefer more of any good to less; more is better!

Preferences & Indifference Curves

0 1 2 3 40

1

2

3

4

Win

e

Bread

b E

b R

b P

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

More generally: Theconsumer prefers E

to all combinationsin the magenta box(e.g.P), while allthose in the yellowbox (e.g. R) arepreferred to E .


0 1 2 3 40

1

2

3

4

Win

e

Bread

b E

b R

b P

b

b

b

b

A

B

C

D

Points such as A &D have more of onegood but less ofanother compared toE ; Need moreinformation aboutconsumer ranking!

Consumer maydecide they areindifferent betweenA, E and D.

We can then connectthose points with anindifference curve.4


Any bundle lying northeast of an indifferencecurve is preferred to any market basket that lieson the indifference curve.

Points on the curve are preferred to pointssouthwest of the curve.

Indifference curves slope downward to the right; Ifthey sloped upward, they would violate theassumption that more is preferred to less!

Some points that had more of both goods would beindifferent to a basket with less of both goods.

Indifference curves and -map

To describe preferences for all combinations ofgoods/services, we use a set of indifference curves - anindifference map.

Each point representsa bundle of differentquantities bread andwine.Each indifferencecurve in the mapconnects the bun-dles among whichthe consumer isindifferent.

Preferredbundles

Win

e

Bread

Indifference curves

Indifference curves graph the set of bundles thatare indifferent to some bundle.

Indifference curves are like contour lines on a map.

Note that indifference curves describing twodistinct levels of preference cannot cross (becausethey are like contour lines on a map; for a proofuse transitivity)

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 ;

Whenever these assumptions are fulfilled thepreferences can be represented in a utility

function.

Often, two additional assumptions are useful . . .

5

Well-behaved preferences

Monotonicity: more of either good is better;implies indifference curves have negative slope.

Convexity: averages are preferred to extremes;slope gets flatter as you move further to right(example of non-convex preferences?)

Preferredbundles

Qy

Qx

Preferences

Convex Preferences

bc

bc

Qy

Qx

“averages are preferred toextremes”

d.h. 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

6

Special Preferences

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

Qy

Qx

have a constant rate oftrade-off between the twogoods; e.g. red pencilsand blue pencils.

Perfect Complements:U(Qx , Qy ) =min{Qx , Qy}

Qy

Qx

always consumed to-gether, e.g. right shoesand left shoes; coffee andcream.

Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS)measures how the consumer is willing to trade offconsumption of good X for consumption of goodY.

The MRS is the slope along an indifference curve,keeping utility constant

MRS =∆Qy

∆Qx

(for dU = 0)

Sign: natural sign is negative, since indifferencecurves will generally have negative slope.


Diskrete:

∆Qy

∆Qy

∆Qy

∆Qx ∆Qx ∆Qx

Qy

Qx

Slope:∆Qy

∆Qx

Infinitesimal:

bc

bc

bc

Qy

Qx

Slope:dQy

dQx

Preferences: Example

Health-consciousconsumer:

Fib

re

Sugar

A ‘sweet tooth’consumer

Fib

re

Sugar

7


MRS measures marginal willingness to pay (whatthe consumer is willing to give up for oneadditional unit);

However, its irrespective of what the consumer isable to pay, therefore no demand yet!

If axioms are fulfilled (i.e. preferences arecomplete, reflexive and transitive) preferences canbe expressed more elegantly with a utility function.

Utility

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

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)

8

Utility Function

Utility functions are not unique:

if U(Qx , Qy is a utility function that representssome preferences, and f (U) is any increasingfunction, then f (U(Qx , Qy) represents the samepreferences, becauseU(Qx , Qy) > U(R1, R2) only iff [U(Qx , Qy)] > f [U(R1, R2)],

so if U(Qx , Qy) is a utility function then anypositive monotonic transformation of it is also autility function that represents the samepreferences.

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


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


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

9

Special Preferences

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

e.g. red pencils and blue pencils;have a constant rate of trade-off between the twogoods.

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, e.g. right shoes and leftshoes; coffee and cream).

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 = Qax Q

1−ay ⇒ MUx ≡

∂U

∂Qx

= aQa−1x Q1−a

y

U = QaxQ

1−ay ⇒ MUy ≡

∂U

∂Qy

= (1−a)QaxQ−ay

10

Marginal Utility & MRS

Note that marginal utility depends on which utilityfunction you choose to represent preferences:if you multiply utility times 2, you multiplymarginal utility times 2, but thus it is not anoperational concept.

However, MU is closely related to the Marginal

Rate of Substitution (MRS), which is anoperational concept.

Marginal Utility & MRS

With calculus one can show that the MRS is theratio of marginal utilities:

MRS ≡ −dQy

dQx

=MUx

MUy

≡∂U∂Qx

∂U∂Qy

The MRS is an indicator for the willingness to pay.

A budget constraint will show the ability to pay.

When we combine the MRS with the ability to

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

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.

Assume he spends all money.

Rewriting:

Qy =M

Py

−Px

Py

Qx Slope:dQy

dQx

= −Px

Py

11

Budget Constraint

M = PxQx + PyQy

MPy

∆Qx

∆Qy

M = PxQx + PyQy

Qy = MPy− Px

PyQx

∆Qy

∆Qx= −Px

Py

Qy

Qx

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

0 1 2 3 4 5 6 7 8 9 10012345

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 (= tan α)!

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”!

12


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.


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!


What happens when the consumer gets one unit of Qx

for free?

0 1 2 3 40

1

2

3Qy

Qx


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 5 60

1

2

3

4Qy

Qx

13

Combining preferences and budget constraint . . .

Optimal Choice

Desicions (in a neoclassical perspective)

Preferences(exogeneous!)

Utility function

Budget-restriction

Optimization

Decisions(Demand functions)

Desicions: 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

14

Optimization

Problem:

maxQx ,Qy

U(Qx , Qy)

s.t.: M = PxQx + PyQy

Two Possibilities:

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

1. Step: Problem

maxQx ,Qy

U(Qx , Qy)


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

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

=0

Lagrange Method

3. Step: Set partial derivatives of the Lagrangefunction with respect to the endogenenous(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

15

Lagrange Method

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

Q∗x = Qx(Px , Py , M), Q∗y = Qy (Px , Py , M)

λ∗ = λ(Px , Py , M)

These solutions are the demand functions foran individual household and describe theoptimal decisions of an household under givenrestrictions.

Additionally, the first order conditions allow somemore insights in the problem of optimal consumerchoice . . .

Optimal 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

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

≡ MRS

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

16

Optimal Choice

Condition for optimality: MRS = Price ratio

Qy

Qx

Slope:dQy

dQx

∣∣∣dU=0

Slope: −Px

Py

∣∣∣dM=0

bc

bc

bc

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!

Demand and Changes in Income


Income-consumptioncurve: normal good

Clo

thes

Food

Income-Consumption

Curve

Engel Curve: normalgood

Inco

me

Food

EngelCurve

17


Inferior good:

Bee

fste

ak

Hamburger

Income-Consumption

Curve

Engel Curve: inferior good

Inco

me

Hamburger

EngelCurve

b

nor

mal

infe

rior Demand and Changes in Price

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


Budget constraintfor 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

Special Cases

The usual methods for maximization (e.g. Lagrangemethod) is 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(Kuhn-Tucker conditions!

18

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}


[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 curve.a horizontal price-consumption curve.the 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

19

Consumer Choice

The theory of consumer choice addresses the followingquestions:

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 thepurchasing power is held constant

Hicks-decomposition: what happens with demand,when relative prices change, but the utility isheld constant

Slutsky-decomposition

0 1 2 3 40

1

2

3

4

Qy

Qx

max U = QxQx


(for M = 4 und Py = 1)

bc bc

bc

SE EE

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 (=EE): constantprice ratio, but purchasingpower increases!

Hicks-Decomposition

0 1 2 3 40

1

2

3

4

Qy

Qx

max U = QxQx


(mit M = 4 und Py = 1)

bc bc

bc

SE EE

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 (=EE): constantprice ratio, but higher income!

20

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

EE

GE

Although Qx be-comes cheaper less

of Qx is demanded!

Market Demand

Market Demand

Market Demand (D): is the horizontal sum ofindividual demands.

D1 = Q1x (Px , Py , M

1)+Q2x (Px , Py , M

2)+· · ·QNx (Px , Py , M

N)

(the subscript denotes the good, the superscript the consumerder i ; N is the totalnumber of consumers.)

21

Market Demand

Attention: Quantities can never be negative, onlyzero!

0 1 2 3 4 5 60

1

2

3

1.5

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

Thanx!

Any questions?

22

Documents

Theory of Consumer Behavior - hsto.infoTheory of Consumer Behavior Herbert Stocker [email protected] Institute of International Studies University of Ramkhamhaeng & Department