Transcript Consumer preferences and utility

The rules of the game
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Lectures
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Seminars
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The marking: exams and exercises
The lectures
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12 weeks times 2 hours
Attendance to the lectures is compulsory
Make sure you do the reading each week
Prepare questions on lecture points or the
reading that seem unclear
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Do not hesitate to ask questions during the
lecture
The course outline and lecture slides will be
made available on the ENTG
The seminars
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A short seminar will be organised during
the first half hour of each lecture.
To go over the exercises for the week
 To clarify problematic points
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Make sure you prepare the exercises,
they are part of the learning process !
 The exercises to be prepared for each
week are given in the course outline
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Exams and marks
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The overall mark for the module is a
weighted average:
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2/3 given by the seminar marks
1/3 given by final exam
The final exam is composed of
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Multiple choice questions
Review questions
A standard exercise
An applied exercise
Exams and marks
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The seminar mark is composed of
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50% : 2 « galops d’essai » (mock exams)
30% : average exercise mark
20% : personal mark, that takes into account
participation, turnout, etc.
The average exercise mark (incentives!) :
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You are free to hand in exercises every week
The mark is the average of your best 6 results
If you hand in less than 6 exercises, then your
average takes in the extra 0’s needed to make up the
6 marks...
Consumer preferences
and utility
Modelling consumer preferences
Consumer preferences and utility
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How can we possibly model the decision
of consumers ?
What will they consume?
 How much of each good?
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Actually, a very simple framework is
enough !
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This framework can explain a lot of the
behaviour of people on markets.
Consumer preferences and utility
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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
Consumer preferences and utility
The utility function as a measure of
satisfaction
Indifference curves as a representation
of preferences
The marginal rate of substitution
The Utility function
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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
The Utility function
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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 Utility function
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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)
U
mU 
x
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Remember last week’s lecture:
 Marginal utility is the first derivative of the
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utility function.
It gives the slope of the utility function
The Utility function
mU = 1
mU = 3
mU = 5
mU = 10
The Utility function
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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 Utility function
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The initial, historical approach to consumer
behaviour used this concept of cardinal utility
However, this is a problematic concept:
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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
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More general, more realistic and more powerful
The Utility function
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Ordinal utility is a representation of preferences
(x1, x 2 )
( y1, y2 ) if
U(x1, x 2 )  U( y1, y2 )
( y1, y2 )
(x1, x 2 ) if
U(x1, x 2 )  U(y1, y2 )
(x1, x 2 ) ( y1, y2 ) if
U(x1, x 2 )  U( y1 , y2 )
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
The Utility function
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Some types of preferences cannot be
represented by an ordinal utility function
Some simplifying assumptions have to be
made
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Preferences are complete :
 Agents can always rank bundles (i.e. preferences
exist for all possible bundles)
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Preferences are transitive :
If (x1 , x 2 )
( y1 , y2 ) and ( y1 , y2 )
 (x1 , x 2 )
(z1 , z2 )
(z1 , z2 )
The Utility function
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An example of non-transitive preferences
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Your favourite childhood game:
Rock
Paper
Scissors
The Utility function
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Such preferences cannot be represented by
an ordinal utility function !!
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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
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People’s behaviour becomes governed by
different priorities
Consumer preferences and utility
The utility function as a measure of
satisfaction
Indifference curves as a representation
of preferences
The marginal rate of substitution
Indifference curves
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Indifference curves represent preferences in
“consumption space”
Good 1
Good 2
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They are built from the ordinal utility function
As seen above, an ordinal utility function can
represent preferences (under some conditions)
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The ranking of bundles in order of preference
corresponds to the ranking in order of increasing (or
decreasing utility)
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
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Indifference curves are a graphical (2-D)
representation of a 3-D utility function
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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
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The consumer is therefore indifferent to all these
baskets
Indifference curves
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Indifference curves further from the origin
correspond to higher levels of utility
Good 1
x1
X
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U(x1,x2) < U(y1,y2)
Y
y1

x2
y2
Good 2
Indifference curves
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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
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Some examples are in the exercise for next week
For a general theory of choice, economists like
“well-behaved” indifference curves
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2 more simplifying assumptions need to be made
Indifference curves
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Monotonicity (non-satiation)
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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
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Convexity (preference for variety)
Good 1
x1
X
A combination z of
extreme bundles x
and y is preferred to
x and y
Z
y1
Y
x2
y2
Good 2
Indifference curves
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Example of concave preferences
Good 1
The extreme bundles
x and y are preferred
to a combination z of
x and y
X
x1
Z
y1
Y
x2
y2
What can we say
about marginal
utility?
Good 2
Indifference curves
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“Well-behaved” indifference curves don’t
cross
Good 1
Let’s assume they can
x y and y
but z  x
z, so x
z
This violates monotonicity
(more is preferred to less)
Y
Z
X
Good 2
Consumer preferences and utility
The utility function as a measure of
satisfaction
Indifference curves as a representation
of preferences
The marginal rate of substitution
The marginal rate of substitution
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What is a rate of substitution ?
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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”
The marginal rate of substitution
Ice-cream
x1
X
 IC
Rate of Substitution =
 DVD
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 IC
(-)
Y
y1

x2
 DVD
y2
(+)
DVD
The marginal rate of substitution
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What is a marginal rate of substitution ?
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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
The marginal rate of substitution
∂IC
 IC
MRS =
 DVD
Ice-cream
∂DVD
x1
X

Y
y1

∂IC
∂DVD
x2
y2
DVD
The MRS is decreasing along the indifference curve
The marginal rate of substitution
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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
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This suggests a link with the idea of
decreasing marginal utility
Is there a way of clarifying this link ?
The marginal rate of substitution
Let’s “zoom in” on the indifference curve until it looks flat
Ice-cream
x1
X

Giving up ∂IC icecream causes a
loss of utility
Receiving ∂DVD
DVDs causes a gain
of utility
mUDVD  DVD
mUIC  IC
x2
DVD
Because we are still on the same indifference curve, loss=gain
The marginal rate of substitution
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The loss of utility from giving up one good equals
the gain from receiving the other good
mUIC  IC  mUDVD  DVD  0
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Or equivalently: mUIC  IC  mUDVD  DVD
Rearranging (dividing both sides by mUIC and
∂DVD):
mU DVD
IC
MRS 

DVD
mUIC
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The MRS is equal to the ratio of marginal utilities!
The marginal rate of substitution
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In general, with two goods x and y, we
have :
mU x
y
MRS 

x
mU y
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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