Theory of Consumer Behavior
Herbert Stocker
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
,
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 it’s natural to say that the consumer prefersbundle P 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
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 .
Preferences & Indifference Curves
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
Preferences & Indifference Curves
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.
Marginal Rate of Substitution (MRS)
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
Marginal Rate of Substitution (MRS)
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
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
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
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 40
1
2
3Qy
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 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)
s.t.: M = PxQx + PyQy
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
Demand and Changes in Income
Income-consumptioncurve: normal good
Clo
thes
Food
Income-Consumption
Curve
Engel Curve: normalgood
Inco
me
Food
EngelCurve
17
Demand and Changes in Income
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
s.t.: M = PxQx + PyQy
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}
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 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
s.t. M = PxQx + PyQy
(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
s.t. M = PxQx + PyQy
(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