Microeconomics: Utility and Demand

Dr. Manuel Salas-Velasco
University of Granada, Spain
Consumer Behavior (I):
Utility and Demand
1

Consumer Behavior (I): Utility and
Demand
Introduction
Dr. Manuel Salas-Velasco2

How Do Consumers Make Their
Decisions?
 Consumer is unable to change the
prices of the goods (PX, PY)
 Consumer’s income: M
 Given prices, income and
individual tastes, the consumer’s
goal is to maximize his/her utility
U = U (X, Y)
The theory of consumer behavior
Some assumptions:

How Well Do Economists Measure Utility?
 Cardinal utility: assumes that the consumer has the
ability to accurately measure the level of utility he/she
derives from consuming a particular combination of
goods, and assign an number to it.
 Ordinal utility: consumers are assumed to rank
consumption bundles and choose among them.
We have two approaches to utility analysis in
economic theory:

Cardinal Utility
Demand

Cardinal Utility
Let’s assume that there are only two consumption goods, good
X (cans of coca-cola) and good Y (cups of popcorn). The utility
function can be expressed as:
U = U (X, Y)
We are going to focus on the relationship between utility and the
consumption levels of only one of the goods:
)YX,(UU 
We are interested in the full satisfaction (or total utility, U) resulting
from the consumption of different units of coke when the value of
the good Y is held constant (e.g. Y = 1; one cup of popcorn).

Total and Marginal Utility Schedules
Quantity (X), cans
of coca-cola
Total Utility (U),
utils
Marginal Utility
(MUX), utils
0 0
1 10 10
2 16 6
3 20 4
4 22 2
5 22 0
6 20 -2
Dr. Manuel Salas-Velasco
7

X U MUX
0 0
1 10 10
2 16 6
3 20 4
4 22 2
5 22 0
6 20 -2
)YX,(UU 
X
U
MUX


 4
23
1620



XMU
XMUΣU  X
U
MUX



• This law states that as additional units
of a good are consumed, while holding
the consumption of all other goods
constant, the resulting increments in
utility will diminish
Law of Diminishing Marginal Utility

Utility Function (U)
0
2
4
6
8
10
12
14
16
18
20
22
24
0 1 2 3 4 5 6
X (Units per time period)
U(Utilspertimeperiod)
U
S • The slope of the utility
curve is the marginal utility
• Utility rises at a decreasing
rate with respect to
increases in the
consumption levels of good,
until S
• After X = 5, utility actually
begins to decline with any
further increases in X

Marginal Utility Function (MUX)
-2
0
2
4
6
8
10
12
0 1 2 3 4 5 6
X (Units per time period)
MUx(Utilsperunittime
period)
MUx

The Consumer’s Decision
Coke
(X)
Popcorn
(Y)
MUX MUY
0
1 10 6
2 6 4
3 4 2
4 2 0
Y
Y
X
X
P
MU
P
MU

Purchase:
U = U (X, Y)
Income = 5 dollars
PX = PY = 1 dollar
(X, Y, X, Y, X)
• The consumer will allocate
expenditure so that the utility gained
from the last dollar spent on each
product is equal

Ordinal Utility
Demand

Ordinal Utility: Indifference Theory
 The second approach to study the theory of
demand assumes that consumer can always say
which of two consumption bundles he/she prefers
without having to say by how much he/she prefers it:
consumers are assumed to rank consumption
bundles and choose among them.
 Let’s assume that there are only two consumption
goods, X and Y; each consumption bundle contains
x units of X and y units of Y: (x, y).

A Consumer’s Ordinal Preferences
 Let’s suppose that: X = cons of ice-
cream; Y = cups of cold lemonade.
 Consider three consumption bundles
(units per week):
• A (2, 8)
 B (3, 4)
 C (5, 2)
 Suppose that each bundle gives the
consumer equal satisfaction or utility:
the consumer is indifferent between
the three bundles of goods.
Alternative bundles giving
a consumer equal utility
Good Y
(cups of
cold
lemonade)
Good X
(cons of
ice-
cream)
A 8 2
B 4 3
C 2 5

Indifference Curves: A Way to Describe
Preferences
An indifference curve
0
2
4
6
8
10
0 1 2 3 4 5 6
Quantity of ice-cream per week, X
Quantityoflemonadeper
week,Y
A
B
C
• An indifferent curve shows
all combinations of goods that
yield the same satisfaction to
the consumer
),( YXUU 
U = U (X, Y)
The axiom of transitivity

Characteristics of Indifference
Curves
1. Indifference curves
generally possess
negative slopes

The Marginal Rate of Substitution
0
2
4
6
8
10
0 1 2 3 4 5 6
Quantity of ice-cream per week,
X
week,Y
X
Y
MRSYX


 4
1
4


YXMRS
A
B
C
B’
MRS = rate at which a consumer
is willing to substitute one good
for the other within his/her utility
function, while receiving the
same level of utility.
b
c
A negative MRS means that to
increase consumption of one
product, the consumer is prepared
to decrease consumption of a
second product.

The MRS measures the slope of the
indifference curve …
Yd
Y
U
Xd
X
U
Ud






0
2
4
6
8
10
0 1 2 3 4 5 6
Quantity of ice-cream per week,
X
week,Y
YdMUXdMU YX 0
X
Y
MRSYX



A
B
C
U = U (X, Y)
If consumption bundles are
continuous (or infinitely
divisible) then the MRS is a
ratio of marginal utilities.
Slope of the
indifference curve
(negative)
Y
X
MU
MU
Xd
Yd

Y
X
YX
MU
MU
MRS 

Curves
2. Indifference curves
cannot cross

Indifference Curves That Cross
2U
A B D
indifferent indifferent
indifferent
X
Y
A
D
B
1U
This contradicts the assumption that A is
preferred to D

3. The farther the curve is
form the origin, the
higher is the level of
utility it represents.
Curves

An Indifference Curve Map
1U
3U
2U
X
Y
(Units per time period)
(Units
per time
period)
• Consumer wishes to maximize
utility, he wishes to reach the
highest attainable indifference
curve
3U 2U 1U> >

What Choices Is an Individual
Consumer Able to Make?
Budget Constraint

The Budget Constraint
Good Y (cups of cold
lemonade)
Good X (cons of ice-cream)
Total
expend.
Price Quantity Expend. Price Quantity Expend.
a 1 10 10 2 0 0 10
b 1 8 8 2 1 2 10
c 1 6 6 2 2 4 10
d 1 4 4 2 3 6 10
e 1 2 2 2 4 8 10
f 1 0 0 2 5 10 10

The Budget Line
0
2
4
6
8
10
12
0 1 2 3 4 5
Quantityoflemonadeperweek,Y
a
b
c
d
e
f
• The budget line
shows all
combinations of
products that are
available to the
consumer given
his money income
and the prices of
the goods that
he/she purchases

The Budget Line
0
2
4
6
8
10
12
0 1 2 3 4 5
Quantity of X
QuantityofY
a
b
c
d
e
f
M = 10; PX = 2; PY = 15
2
10

XP
MPoint f
horizontal intercept
Point a
vertical intercept 10
1
10

YP
M
Slope: 2
1
2





X
Y
2
1
2

Y
X
P
P
The equation for the budget line:
X
P
P
P
M
Y
Y
X
Y

XY 210 
Relative price ratio

The Consumer’s Utility Maximizing
Choice
0
2
4
6
8
10
12
0 1 2 3 4 5 6
Quantityoflemonadeperweek,Y
E
• The consumer’s
utility is maximized at
the point (E) where an
indifference curve is
tangent to the budget
line
• At that point, the
consumer’s marginal
rate of substitution for
the two goods is equal
to the relative prices
of the two goods
Y
Y
X
X
Y
X
Y
X
P
MU
P
MU
or
P
P
MU
MU
 The condition for utility
maximization

The Consumer’s Constrained
Maximization Problem
Maximize U = U (X, Y) the objective function
PX = price of good X
PY = price of good Y
M = consumer’s income
Subject to:
the constraintPX X + PY Y = M
• A basic assumption of the theory of consumer behavior is that
rational consumers seek to maximize their total utility, subject to
predetermined prices of the goods and their money income
• Formally, we can express this concept of consumer choice as a
constrained optimization problem

Maximize: U = U (X, Y) the objective function
Subject to: the constraintPX X + PY Y = M
Step 1. Set up the Lagrangian function:
To do so, we first set the constraint function equal to zero:
M – PX X – PY Y = 0
We then multiply this form by lambda to form the Lagrangian function:
• In order to solve such a problem, we will use the
Lagrangian Multiplier Method

Step 2. Determine the first-order conditions:

XP
X
U



Step 3. Solving for lambda from the two first-order conditions:
YP
Y
U



Therefore:
YX P
Y
U
P
X
U





Y
Y
X
X
P
MU
P
MU

The condition for utility
maximization

Maximization Problem. Example
Maximize: U = X0.5 Y0.5 (Cobb-Douglas Indifference curve)
Subject to: 4 X + Y = 800
(where: PX = 4; PY = 1: M = 800)
Step 1. Set up the Lagrangian function:

Step 2. Determine the first-order conditions:
4
5.0 5.05.0
YX 

5.05.0
5.0 
 YX
Y = 800 – 4X

4
5.0 5.05.0
YX 
 5.05.0
5.0 
 YX
XY
X
Y
YX
YX
YX
YX
YX
4;4;4
4
5.0
5.0
5.0
4
5.0
1
5.05.0
5.05.0
5.05.0
5.05.0








Y = 800 – 4X
Y = 800 – 4X; 4X = 800 – 4X; 8X = 800; X = 100 units; Y = 400 units
U = X0.5 Y0.5 = (100)0.5 (400)0.5 = 200
Lambda = 0.5 (100)0.5 (400)-0.5 = 0.25
Lambda (lagrangian multiplier) measures
the change in utility due to a one dollar
change in the consumer’s utility

Microeconomics: Utility and Demand

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