Consumer preferences and utility - YMCA University of ... preferences and utility ... behaviour used...

Consumer preferences and utility

Modelling consumer preferences


How can we possibly model the decision of consumers ? What will they consume? How much of each good?

Actually, a very simple framework is enough ! This framework can explain a lot of the

behaviour of people on markets.


Last week’s “general rule”: A rational consumer will always choose

the best basket of goods amongst all the ones it can afford

But we need to clarify : What we mean by rational What we mean by best What we mean by afford

Today

Next week


The utility function as a measure of satisfaction

Indifference curves as a representation of preferences

The marginal rate of substitution

The Utility function

Historically, utility as a measure of satisfaction is grounded in utilitarianism

Jeremy Bentham (1748-1831): “It is the property of an object to produce pleasure, well-being or happiness”

Stanley Jevons (1835-1882): The father of the “marginalist revolution”, who generalised this concept to consumer behaviour


Cardinal utility assigns a value to the level of satisfaction associated with the consumption of a basket of goods.

Total utility is the sum of the satisfactions derived from the consumption of several goods.

Marginal utility is the increase in utility following the consumption of an extra unit of a good.

Beers consumed

Total Utility

Marginal Utility

0 0 0

1 10 10

2 15 5

3 18 3

4 19 1


The marginal utility of a good (mU ) measures the increase (or decrease) in total utility (∂U) following a small variation in the quantity consumed (∂x)

Remember last week’s lecture:

Marginal utility is the first derivative of the utility function.

It gives the slope of the utility function

UmUx

∂=∂


mU = 10

mU = 5

mU = 3

mU = 1


The marginal utility of the good (beers) gets smaller as the quantity consumed increases.

This phenomenon is called the law of diminishing marginal utility


The initial, historical approach to consumer behaviour used this concept of cardinal utility

However, this is a problematic concept: Is it possible to quantify the satisfaction derived from consuming

a good ? Is it possible for the quantities of utility derived from 2 different

goods to be compared ? More importantly, do consumers actually think that way when

they choose goods ???

This problem was solved by the introduction of ordinal utility More general, more realistic and more powerful


Ordinal utility is a representation of preferences

What is important is not the ability to quantify

« how much » utility is provided by a bundle, but the ability to rank bundles in order of increasing utility

This is much closer to the “real” behaviour of agents

1 2 11 22 1 2( ) ( ) if U( )y , y U( )x , x x , x y , y>

1 2 11 22 1 2( ) ( ) if U( )x , x U( )x , xy , y y , y<

1 2 11 22 1 2( ) ( ) if U( )y , y U( )x , x x , x y , y=


Some types of preferences cannot be represented by an ordinal utility function

Some simplifying assumptions have to be made Preferences are complete : Agents can always rank bundles (i.e. preferences

exist for all possible bundles) Preferences are transitive :

1 2 1 21

1

2

2

1

1

2

2

If ( ) ( ) and ( ) ( )(

x , xx , x

z ,y , y y) ( )

y zz

,z ,⇒


An example of non-transitive preferences

Your favourite childhood game:

Rock Paper

Scissors


You are exploring the desert, you get lost, and you run out of fuel and supplies. Luckily, another truck finds you and offers some help,

but you have to choose amongst some options. None is 100% satisfactory because they also have to

keep some resources to be able to get back.

How do you rank those options ?

Fuel Supplies Map Option 1 Full tank Very little Yes Option 2 Full tank Plentiful No Option 3 None Plentiful Yes


Such preferences cannot be represented by an ordinal utility function !! This is a first example of how consumer theory

simplifies a complex reality

Consumer theory (and economic theory in general) often “breaks down” in extreme situations People’s behaviour becomes governed by

different priorities

Indifference curves

Indifference curves represent preferences in “consumption space”

They are built from the ordinal utility function As seen above, an ordinal utility function can

represent preferences (under some conditions) The ranking of bundles in order of preference

corresponds to the ranking in order of increasing (or decreasing utility)

Good 1

Good 2

Indifference curves

Indifference curves

Utility function for a single good

Indifference curves

But how would you draw a utility function for the consumption of 2 goods ?

Indifference curves

Seen from above, the 3-D diagram looks like this...

Lines of constant utility

Indifference curves

This is the same “trick” as for this kind of diagram...

Indifference curves

Indifference curves are a graphical (2-D) representation of a 3-D utility function Just like the contour lines of a 2-D road map

represent the 3rd dimension (altitude)

A given indifference curve represents all the baskets of goods that provide the same utility to a consumer The consumer is therefore indifferent to all these

baskets

Indifference curves

Indifference curves further from the origin correspond to higher levels of utility

Good 1

Good 2

x1

x2

X U(x1,x2) < U(y1,y2)

y1

y2

Y

Indifference curves

Because they are derived from a utility function, indifference curves are a representation of preferences

However, at this point, indifference curves can still take a wide range of shapes Some examples are in the exercise for next week

For a general theory of choice, economists like

“well-behaved” indifference curves 2 more simplifying assumptions need to be made

Indifference curves

Monotonicity (non-satiation)

In other words, “more is always preferred to less”

Extra units of a good always increase utility, so consumers always prefer to have more of a good

The implication is that regardless of which indifference curve you are on, there always exists a higher one right next to it.

Indifference curves

Convexity (preference for variety)

Good 1

Good 2

y1

y2

Y

x1

x2

X A combination z of extreme bundles x and y is preferred to x and y Z

Indifference curves

Example of concave preferences Good 1

Good 2

y1

y2

Y

x1

x2

X The extreme bundles x and y are preferred to a combination z of x and y

Z

What can we say about marginal utility?

Indifference curves

“Well-behaved” indifference curves don’t cross

Good 1

Good 2

Y

X

Let’s assume they can

Z

x y y z and , x z so

but z x>This violates monotonicity (more is preferred to less)


What is a rate of substitution ? You currently have a bundle composed of 10

tubs of ice-cream and 3 DVDs. You want to keep your satisfaction the same How many tubs of ice-cream are you

prepared to give up to get some extra DVDs?

The rate at which you are prepared to exchange is known as the “rate of substitution”


Ice-cream

DVD

y1

y2

Y

x1

x2

X

∆ DVD (+)

∆ IC (-)

ICRate of Substitution = DVD∆

∆


What is a marginal rate of substitution ? Exactly the same idea, but this time we are

talking about a tiny change in your bundle (∂x) instead of a large change (∆x)

You have 10 tubs of ice-cream and 3 DVDs. How many tubs of ice-cream are you

prepared to give up to get ONE extra DVD ?

This means that the marginal rate of substitution is the slope of the indifference curve


Ice-cream

DVD

y1

y2

Y

x1

x2

X

ICMRS = DVD∂

∂

∂IC

∂DVD

∂IC

∂DVD

The MRS is decreasing along the indifference curve


So the marginal rate of substitution is the slope of the indifference curve

The amount of ice-cream you are willing

to give up for an extra DVD is lower the less ice-cream you have This suggests a link with the idea of

decreasing marginal utility

Is there a way of clarifying this link ?


Ice-cream

DVD

x1

x2

X

Let’s “zoom in” on the indifference curve until it looks flat

Giving up ∂IC ice-cream causes a loss of utility

ICmU IC× ∂

DVDmU DVD× ∂

Receiving ∂DVD DVDs causes a gain of utility

Because we are still on the same indifference curve, loss=gain


The loss of utility from giving up one good equals the gain from receiving the other good

Or equivalently:

Rearranging (dividing both sides by mUIC and ∂DVD):

The MRS is equal to the ratio of marginal utilities!

IC DVDmU IC mU DVD 0× ∂ + × ∂ =

IC DVDmU IC mU DVD× ∂ = − × ∂

DVD

IC

mUICMRSDVD mU∂

= = −∂


In general, with two goods x and y, we have :

Note: Economists typically “forget” about the minus sign and give the MRS as a positive number

This is result may seem a bit pointless, but

it will become clear when we examine consumer choice next week

x

y

mUyMRSx mU∂

= = −∂

Consumer preferences and utility - YMCA University of ... preferences and utility ... behaviour used...

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