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

Consumer Behavior (I): Utility and 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


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

UMU X

4

23

1620

XMU

XMUΣU X

UMU X

• 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 (

Uti

ls p

er t

ime p

erio

d)

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)

MU

x (

Uti

ls p

er

un

it t

ime

peri

od

)

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 Dr. Manuel Salas-Velasco11

Ordinal Utility

Consumer Behavior (I): Utility and 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

Qu

an

tity

of

lem

on

ad

e p

er

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

Qu

an

tity

of

lem

on

ad

e p

er

week,

Y

X

YMRSYX

4

1

4

YXMRS

A

B

CB’

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.

bc

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 …

YdY

UXd

X

UUd

0

2

4

6

8

10

0 1 2 3 4 5 6

Quantity of ice-cream per week,

X

Qu

an

tity

of

lem

on

ad

e p

er

week,

Y

YdMUXdMU YX 0

X

YMRSYX

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

XYX

MU

MUMRS



2. Indifference curves

cannot cross


Indifference Curves That Cross

2U

A B D

indifferent indifferent

indifferent

X

Y

A

DB

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.



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


Qu

an

tity

of

lem

on

ad

e p

er

we

ek

, 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

Qu

an

tity

of

Y

a

b

c

d

e

f

M = 10; PX = 2; PY = 1 52

10

XP

MPoint f

horizontal intercept

Point a

vertical intercept 101

10

YP

M

Slope: 21

2

X

Y

21

2

Y

X

P

P

The equation for the budget line:

XP

P

P

MY

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


Qu

an

tity

of le

mo

na

de

pe

r w

ee

k, 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

MUor

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

XU

Step 3. Solving for lambda from the two first-order conditions:

YP

YU

Therefore:

YX P

YU

P

XU

Y

Y

X

X

P

MU

P

MU

The condition for utility

maximization


The Consumer’s Constrained 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

5.05.05.0 YX

Y = 800 – 4X



4

5.0 5.05.0 YX

5.05.05.0 YX

XYX

YYX

YX

YX

YXYX

4;4;4

45.0

5.0

5.04

5.0

1

5.05.0

5.05.0

5.05.05.05.0

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 utilityDr. Manuel Salas-Velasco34

microeconomics: utility and demand

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