Microeconomics: Utility and Demand

Consumer Behavior: Utility and Demand
Cardinal Utility
Consumer Surplus
Ordinal Utility
Indifference Curves
The Consumer’s Constrained Maximization Problem

1. Consumer Behavior (I):
Utility and Demand

Dr. Manuel Salas-Velasco
University of Granada, Spain

2. Consumer Behavior (I): Utility and
Demand

Introduction

3. How Do Consumers Make Their
Decisions?

i The theory of consumer behavior

TITANIC

IEEE Some assumptions:

e Consumer is unable to change the
prices of the goods (Pz, Py)

e Consumer's income: M

e Given prices, income and
individual tastes, the consumer's
goal is to maximize his/her utility
U=U(X, Y)

Dr. Manuel Salas-Velasco

4. How Well Do Economists Measure Utility?

We have two approaches to utility analysis in
economic theory:

e 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.

e Ordinal utility: consumers are assumed to rank
consumption bundles and choose among them.

Dr. Manuel Salas-Velasco

5. Consumer Behavior (1): Utility and
Demand

Cardinal Utility
CC

6. 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:

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).

Dr. Manuel Salas-Velasco

7. Total and Marginal Utility Schedules

Quantity (X), cans Total Utility (U), Marginal Utility
of coca-cola utils (MU), utils

a

8. Law of Diminishing Marginal Utility
AA

U=U(XY)
+ 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

AU 20-16
MU, === = =
u MU, = 4
OU
MU, ==
U=X MU, * 6x

Dr. Manuel Salas-Velasco

9. U (Utils per time peri

Utility Function (U)
eee

all]

0

1

try > y

4
X (Units per time period)

5

+ 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

Dr. Manuel Salas-Velasco

10. Marginal Utility Function (MU,)

—- MUx

MUx (Utils per unit time
period)

5 $ i
X (Units per time period)

Dr. Manuel Salas-Velasco

11. The Consumer’s Decision

Popcorn Py = Py = 1 dollar
(Y) Income = 5 dollars

U=U (x, Y)

Purchase:

(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

Dr. Manuel Salas-Velasco

12. Consumer Behavior (1): Utility and
Demand

Ordinal Utility
|

13. Ordinal Utility: Indifference Theory

e 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.

e 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).

Dr. Manuel Salas-Velasco

14. 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.

A Consumer’s Ordinal Preferences

Alternative bundles giving
a consumer equal utility

Good Y
(cups of
cold
lemonade)

Dr. Manuel Salas-Velasco

15. Preferences

An indifference curve

week, Y

Quantity of lemonade per

0 1 2 3 4 5 6

Quantity of ice-cream per week, X

Indifference Curves: A Way to Describe

* An indifferent curve shows
all combinations of goods that
yield the same satisfaction to

the consumer

The axiom of transitivity
A~B and B~C
ed E
U=U(X Y)
U =U (X,Y)

Dr. Manuel Salas-Velasco

16. Characteristics of Indifference
Curves

1. Indifference curves
generally possess
negative slopes

17. The Marginal Rate of Substitution

MRS = rate at which a consumer 5 107
is willing to substitute one good Q
for the other within his/her utility 3 8
function, while receiving the 5 >6
same level of utility. ES
2%
= 24
AY <4 25
MRS yx =—— en 2
YX TY MRS yx | 4 = 2
El
A negative MRS means that to S 0
increase consumption of one 0 1 2 3 4 5 6
product, the consumer is prepared Quantity of ice-cream per week,
to decrease consumption of a x

second product.
Dr. Manuel Salas-Velasco

18. The MRS measures the slope of the
indifference curve ...

Slope of the
indifference curve
(negative)

If consumption bundles are
continuous (or infinitely
divisible) then the MRS is a
ratio of marginal utilities.

»

Quantity of lemonade per
week, Y
wo

U=U(X, Y) 0 1 2 3 4 5 6

Quantity of ice-cream per week,
x

au = ax Var
ax “ay
dY MU,

0=MU, dX + MU, dY x MU,

Dr. Manuel Salas-Velasco

19. Characteristics of Indifference
Curves

2. Indifference curves
cannot cross

20. Indifference Curves That Cross

indifferent indifferent

A i= B I= D

ee

— This contradicts the assumption that A is
preferred to D

A

Dr. Manuel Salas-Velasco

21. Characteristics of Indifference
Curves

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

22. An Indifference Curve Map

Y

0, > U, > 8,

(Units
per time + Consumer wishes to maximize
period) utility, he wishes to reach the

highest attainable indifference
suis. ook AN curve

(Units per time period)
Dr. Manuel Salas-Velasco

23. What Choices Is an Individual
Consumer Able to Make?

Budget Constraint
CC

24. The Budget Constraint

Good Y (cups of cold

lemonade) Good X (cons of ice-cream)

Price Quantity | Expend. Price Quantity | Expend.

1 Il 10 | 2 | o | o
8
6
4
2
0

8

1 2
1 6 2
1 4 2
1 2 2
1 0 2

1
2
8
4

(5)

Dr. Manuel Salas-Velasco

25. The Budget Line

* 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

Quantity of lemonade per week, Y

0 il 2 3 4 5

Quantity of ice-cream per week, X

Dr. Manuel Salas-Velasco

26. The Budget Line

Point f M _10

. A =—=5 M = 10; Py = 2; Py=1
horizontal intercept P, 2
Point a M 10
vertical intercept B= = 10 .
5
AY _-2_ 2
Slope: Ay ı 5
6
Relative price ratio ate = =e =-2 =
B 1 2
The equation for the budget line: 0
0 1 2 3 4 5
Y =10-2X Quantity of X

Dr. Manuel Salas-Velasco

27. The Consumer’s Utility Maximizing

Choice
AA
« The consumer's 12

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 0 1 2 4 5 6

Quantity of ice-tréam per week, X

o

Quantity of lemonade per week, Y

MU, = Fe or Mur = MU, The condition for utility
MU, PB Py P, maximization

Dr. Manuel Salas-Velasco

28. The Consumer’s Constrained
Maximization Problem

P, = price of good X
Py = price of good Y
M = consumer’s income

+ 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:

PxX+P,Y=M __ |the constraint

Dr. Manuel Salas-Velasco

29. The Consumer’s Constrained
Maximization Problem

Maximize: U = U(X, Y) the objective function
Subject to: PX +P,Y=M the constraint

« In order to solve such a problem, we will use the
Lagrangian Multiplier Method

To do so, we first set the constraint function equal to zero:
M-P, X-PyY=0

We then multiply this form by lambda to form the Lagrangian function:

L =U(X Y) +1 (M-PyX-Py Y)

Dr. Manuel Salas-Velasco

30. The Consumer’s Constrained
Maximization Problem

B- GX, YN
@=U(% Y)+2(M-P,X-P, Y)

EMP, XP YO

Dr. Manuel Salas-Velasco

31. The Consumer’s Constrained
Maximization Problem

SU
EXA ou -AP,=0 x=> qe x
EX ax Py

BE OU jp 9 m._> er
er or P,

Therefore:
ous ayy The condition for utility
Yt mb» maximization

Py B

Dr. Manuel Salas-Velasco

32. The Consumer’s Constrained
Maximization Problem. Example

Maximize: U = X05 Y0.5 (Cobb-Douglas Indifference curve)
Subject to: 4 X + Y = 800

(where: Py = 4; Py = 1: M = 800)

L = XS Y0S + 1 (800 - 4X - Y)

Dr. Manuel Salas-Velasco

33. The Consumer’s Constrained
Maximization Problem. Example

G = XS YS +7 (800 — 4X — Y)

—0.5 470.5
22 =05¥ 57% _41-0 > 2 „05x Y
OX 4
280500 —a=0 ==> 1-05X%7

OL
Fee. => Y=300-4X

Dr. Manuel Salas-Velasco

34. The Consumer’s Constrained
Maximization Problem. Example

0.5 470.5

1-05X Y A=0.5 Xx y 05 Y = 800 -4X
4

0.5 x Los xs yes

0.5 xe yo?

x'y=4 Yea, Y=ax
X
Y = 800 - 4X; 4X = 800 - 4X; 8X = 800; X = 100 units; Y = 400 units

U = X95 Y5 = (100) (400)°5 = 200 Lambda (lagrangian multiplier) measures
the change in utility due to a one dollar
Lambda = 0.5 (100)%5 (400)°5 = 0.25 change in the consumer’s utility

Dr. Manuel Salas-Velasco