Prerequisites
Almost essential
Consumption: Basics
Frank Cowell: Microeconomics
November 2006
Consumption and Uncertainty
MICROECONOMICS
Principles and Analysis
Frank Cowell
Why look again at preferences...
Frank Cowell: Microeconomics
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Aggregation issues
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Modelling specific economic problems
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labour supply
savings
New concepts in the choice set
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restrictions on structure of preferences for consistency
over consumers
uncertainty
Uncertainty extends consumer theory in
interesting ways
Overview...
Frank Cowell: Microeconomics
Consumption:
Uncertainty
Modelling
uncertainty
Issues
concerning the
commodity space
Preferences
Expected utility
The felicity
function
Uncertainty
Frank Cowell: Microeconomics
New concepts
 Fresh insights on consumer axioms
 Further restrictions on the structure of utility
functions
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Story
Story
Concepts
Frank Cowell: Microeconomics
American example
If the only uncertainty is about who
will be in power for the next four
years then we might have states-ofthe-world like this

state-of-the-world

a consumption bundle
={Rep, Dem}
or perhaps like this:

pay-off (outcome)

={Rep, Dem, Independent}
x X
an array of bundles over
the entire space 
British example
If the only uncertainty is about the
weather then we might have states-ofthe-world like this


prospects

{x: }

ex ante

before the realisation

ex post

after the realisation
={rain,sun}
or perhaps like this:
={rain, drizzle,fog, sleet,hail...}
The ex-ante/ex-post distinction
Frank Cowell: Microeconomics
The time line
The "moment of truth"
The ex-ante view
The ex-post view
(too
Decisions
late to make
to be
decisions
made here
now)
Only one
realised stateof-the-world 
time at which the
state-of the world is
revealed
time
Rainbow of possible statesof-the-world 
A simplified approach...
Frank Cowell: Microeconomics
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Assume the state-space is finite-dimensional
Then a simple diagrammatic approach can be used
This can be made even easier if we suppose that
payoffs are scalars
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A special example:
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Consumption in state  is just x (a real number)
Take the case where #states=2
 = {RED,BLUE}
The resulting diagram may look familiar...
The state-space diagram: #=2
Frank Cowell: Microeconomics
The consumption space
under uncertainty: 2 states
xBLUE
A prospect in the 1good 2-state case
The components of a
prospect in the 2-state case
But this has no equivalent
in choice under certainty
payoff if
RED occurs

45°
O
P0
xRED
The state-space diagram: #=3
Frank Cowell: Microeconomics
The idea generalises: here we
have 3 states
xBLUE
 = {RED,BLUE,GREEN}
A prospect in the 1-good 3state case
•P
0
O
The modified commodity space
Frank Cowell: Microeconomics
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We could treat the states-of-the-world like characteristics
of goods.
We need to enlarge the commodity space appropriately.
Example:
 The set of physical goods is {apple,banana,cherry}.
 Set of states-of-the-world is {rain,sunshine}.
 We get 3x2 = 6 “state-specific” goods...
 ...{a-r,a-s,b-r,b-s,c-r,c-s}.
Can the invoke standard axioms over enlarged commodity
space.
But is more involved…?
Overview...
Frank Cowell: Microeconomics
Consumption:
Uncertainty
Modelling
uncertainty
Extending the
standard
consumer
axioms
Preferences
Expected utility
The felicity
function
What about preferences?
Frank Cowell: Microeconomics
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We have enlarged the commodity space.
It now consists of “state-specific” goods:
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For finite-dimensional state space it’s easy.
If there are # possible states then...
...instead of n goods we have n  # goods.
Some consumer theory carries over automatically.
Appropriate to apply standard preference axioms.
But they may require fresh interpretation.
A little
revision
Another look at preference axioms
Frank Cowell: Microeconomics
Completeness
 Transitivity
 Continuity
 Greed
 (Strict) Quasi-concavity
 Smoothness

to ensure existence
of indifference curves
to give shape
of indifference curves
Ranking prospects
Frank Cowell: Microeconomics
Greed: Prospect P1 is
preferred to prospect P0
xBLUE
Contours of the
preference map.
 P1
 P0
O
xRED
Implications of Continuity
Frank Cowell: Microeconomics
A pathological preference for
certainty (violation of continuity)
xBLUE
Impose continuity
An arbitrary prospect P0
Find point E by continuity
Income x is the certainty
equivalent of P0
holes
no
holes
 E
x
 P0
O
x
xRED
Reinterpret quasiconcavity
Frank Cowell: Microeconomics
Take an arbitrary
prospect P0.
xBLUE
Given continuous
indifference curves….
…find the certaintyequivalent prospect E
Points in the interior of
the line P0E represent
mixtures of P0 and E.
 E
If U is strictly quasiconcave
P1 is strictly preferred to P0.
 P1
 P0
O
xRED
More on preferences?
Frank Cowell: Microeconomics
We can easily interpret the standard axioms.
 But what determines the shape of the
indifference map?
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Two main points:
 Perceptions of the riskiness of the outcomes in
any prospect
 Aversion to risk
pursue the
first of these...
A change in perception
Frank Cowell: Microeconomics
The prospect P0 and
certainty-equivalent
prospect E (as before)
xBLUE
Suppose RED begins to
seem less likely
Now prospect P1 (not P0)
appears equivalent to E
Indifference curves after
the change
you need a
bigger win to
compensate
 E
 . .
P0 P1
O
xRED
This change alters
the slope of the ICs.
A provisional summary
Frank Cowell: Microeconomics
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In modelling uncertainty we can:
...distinguish goods by state-of-the-world as well
as by physical characteristics etc.
...extend consumer axioms to this classification of
goods.
...from indifference curves get the concept of
“certainty equivalent”.
... model changes in perceptions of uncertainty
about future prospects.
But can we do more?
Overview...
Frank Cowell: Microeconomics
Consumption:
Uncertainty
Modelling
uncertainty
The foundation of
a standard
representation of
utility
Preferences
Expected utility
The felicity
function
A way forward
Frank Cowell: Microeconomics
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For more results we need more structure on the
problem.
Further restrictions on the structure of utility
functions.
We do this by introducing extra axioms.
Three more to clarify the consumer's attitude to
uncertain prospects.


By the way, there's a certain that’s been carefully
avoided so far.
Can you think what it might be...?
Three key axioms...
Frank Cowell: Microeconomics
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State irrelevance:
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Independence:
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The identity of the states is unimportant
Induces an additively separable structure
Revealed likelihood:
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Induces a coherent set of weights on states-ofthe-world
A closer
look
1: State irrelevance
Frank Cowell: Microeconomics
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Whichever state is realised has no intrinsic value
to the person

There is no pleasure or displeasure derived from
the state-of-the-world per se.
Relabelling the states-of-the-world does not affect
utility.
All that matters is the payoff in each state-of-theworld.
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2: The independence axiom
Frank Cowell: Microeconomics
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Let P(z) and P′(z) be any two distinct prospects such that
the payoff in state-of-the-world  is z.
 x = x′ = z.
If U(P(z)) ≥ U(P′(z)) for some z then U(P(z)) ≥ U(P′(z))
for all z
One and only one state-of-the-world can occur.
So, assume that the payoff in one state is fixed for all
prospects.
The level at which the payoff is fixed should have no
bearing on the orderings over prospects whose payoffs can
differ in other states of the world.
We can see this by partitioning the state space for #> 2
Independence axiom: illustration
Frank Cowell: Microeconomics
A case with 3 states-of-theworld
xBLUE
 Compare prospects with the
same payoff under GREEN.
What if we compare all
of these points...?
Or all of these
points...?
Or all of
these?
O
Ordering of these prospects
should not depend on the size
of the payoff under GREEN.
3: The “revealed likelihood” axiom
Frank Cowell: Microeconomics
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Let x and x′ be two payoffs such that x is weakly preferred
to x′.
Let 0 and 1 be any two subsets of .
Define two prospects:
 P0 := {x′ if 0 and x if 0}
 P1 := {x′ if 1 and x if 1}
If U(P1)≥U(P0) for some such x and x′ then U(P1)≥U(P0)
for all such x and x′
Induces a consistent pattern over subsets of states-of-theworld.
Revealed likelihood: example
Frank Cowell: Microeconomics
Assume these preferences
over fruit
Consider these two prospects
1 apple < 1 banana
1 cherry
< 1 date
States of the world
Choose a prospect: P1 or P2?
Another two prospects
Is your choice between P3 and
P4 the same as between P1 and
P2?
(remember only one colour
will occur)
P1: apple
P2: banana
P3:
P4:
apple
apple
apple
banana banana
banana banana
apple
apple
apple apple
apple
cherry
cherry
cherry
cherry
cherry
date
date
date
date
date
cherry
cherry
cherry
cherry
A key result
Frank Cowell: Microeconomics
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We now have a result that is of central importance to the
analysis of uncertainty.
Introducing the three new axioms:
 State irrelevance
 Independence
 Revealed likelihood
...implies that preferences must be representable in the
form of a von Neumann-Morgenstern utility function:
 pu(x)

Properties of p and u in a moment. Consider the interpretation
The vNM utility function
Frank Cowell: Microeconomics
additive form from
independence axiom
 p u(x)
payoff in
state 

“revealed likelihood”
weight on state 
the cardinal utility or "felicity"
function: independent of state 
E u(x)
Defined with respect to the
weights p
Identify components of the
vNM utility function
Can be expressed equivalently
as an “expectation”
The missing word was
“probability”
Implications of vNM structure (1)
Frank Cowell: Microeconomics
A typical IC
What is the slope where it
crosses the 45º ray?
xBLUE
From the vNM structure
So all ICs must have same
slope at the 45º ray.
pRED
– _____
pBLUE
O
xRED
Implications of vNM structure (2)
Frank Cowell: Microeconomics
A given income prospect
From the vNM structure
xBLUE
Mean income
Extend line through P0 and
P to P1 .
 P1
–
 P
 P0
O
pRED
– _____
pBLUE
xRED
Ex
 By
_ quasiconcavity
U(P)  U(P0)
The vNM paradigm: Summary
Frank Cowell: Microeconomics
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To make choice under uncertainty manageable it is helpful
to impose more structure on the utility function.
We have introduced three extra axioms.
This leads to the von-Neumann-Morgenstern structure
(there are other ways of axiomatising vNM).
This structure means utility can be seen as a weighted sum
of “felicity” (cardinal utility).
The weights can be taken as subjective probabilities.
Imposes structure on the shape of the indifference curves.
Overview...
Frank Cowell: Microeconomics
Consumption:
Uncertainty
Modelling
uncertainty
A concept of
“cardinal utility”?
Preferences
Expected utility
The felicity
function
The function u
Frank Cowell: Microeconomics
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The “felicity function” u is central to the vNM
structure.
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Scale and origin of u are irrelevant:
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It’s an awkward name.
But perhaps slightly clearer than the alternative,
“cardinal utility function”.
Check this by multiplying u by any positive constant…
… and then add any constant.
But shape of u is important.
Illustrate this in the case where payoff is a scalar.
Risk aversion and concavity of u
Frank Cowell: Microeconomics
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Use the interpretation of risk aversion as quasiconcavity.
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If individual is risk averse...
_
...then U(P)  U(P0).
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Given the vNM structure...
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u(Ex)  pREDu(xRED) + pBLUEu(xBLUE)
u(pREDxRED+pBLUExBLUE)  pREDu(xRED) + pBLUEu(xBLUE)

So the function u is concave.
The “felicity” function
Frank Cowell: Microeconomics
Diagram plots utility level
(u) against payoffs (x).
u
Payoffs in states BLUE
and RED.
If u is strictly concave then
person is risk averse
u of the average of xBLUE
BLUE
equals than
the the
and xRED
RED higher
expected u of xBLUE
BLUE and of
xRED
RED
If u is a straight line then
person is risk-neutral
If u is strictly convex then
person is a risk lover
xBLUE
xRED
x
Summary: basic concepts
Frank Cowell: Microeconomics
Review
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Review
Review
Review
Use an extension of standard consumer
theory to model uncertainty
“state-space” approach
Can reinterpret the basic axioms.
 Need extra axioms to make further progress.


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Yields the vNM form.
The felicity function gives us insight on risk
aversion.
What next?
Frank Cowell: Microeconomics
Introduce a probability model.
 Formalise the concept of risk.
 This is handled in Risk.
