Transcript Document
Prerequisites
Almost essential
Consumption: Basics
CONSUMPTION AND
UNCERTAINTY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Consumption Uncertainty
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Why look again at preferences…
Aggregation issues
• restrictions on structure of preferences for consistency over consumers
Modelling specific economic problems
• labour supply
• savings
New concepts in the choice set
• uncertainty
Uncertainty extends consumer theory in interesting ways
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
Issues
concerning the
commodity space
Preferences
Expected utility
The felicity
function
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Uncertainty
New concepts
Fresh insights on consumer axioms
Further restrictions on the structure of utility functions
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Story
Concepts
Story
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
={Rep, Dem}
or perhaps like this:
={Rep, Dem, Independent}
state-of-the-world
a consumption bundle
pay-off
(outcome)
x X
prospects
{x: }
ex ante
before the realisation
ex post
after the realisation
British example
If the only uncertainty is about the
weather then we might have states-ofthe-world like this
an array of bundles over
the entire space
={rain,sun}
or perhaps like this:
={rain, drizzle,fog, sleet,hail…}
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The ex-ante/ex-post distinction
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
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time
Rainbow of possible statesof-the-world
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A simplified approach…
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
• Consumption in state is just x (a real number)
A special example:
• Take the case where #states=2
• = {RED,BLUE}
The resulting diagram may look familiar…
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The state-space diagram: #=2
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
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P0
xRED
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The state-space diagram: #=3
xBLUE
The idea generalises: here we
have 3 states
= {RED,BLUE,GREEN}
A prospect in the 1-good 3state case
•P
0
O
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The modified commodity space
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…?
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
Extending the standard
consumer axioms
Preferences
Expected utility
The felicity
function
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What about preferences?
We have enlarged the commodity space
It now consists of “state-specific” goods:
• 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
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Another look at preference axioms
Completeness
to ensure existence
of indifference curves
Transitivity
Continuity
Greed
(Strict) Quasi-concavity
to give shape
of indifference curves
Smoothness
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Ranking prospects
xBLUE
Greed: Prospect P1 is preferred to P0
Contours of the preference map
P1
P0
O
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xRED
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Implications of Continuity
Pathological preference for certainty (violates
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
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x
xRED
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Reinterpret quasiconcavity
Take an arbitrary prospect P0
Given continuous indifference curves…
…find the certainty-equivalent prospect E
xBLUE
Points in the interior of the line P0E
represent mixtures of P0 and E
If U strictly quasiconcave P1 is preferred to P0
E
P1
P0
O
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xRED
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More on preferences?
We can easily interpret the standard axioms
But what determines shape of the indifference map?
Two main points:
• Perceptions of the riskiness of the outcomes in any prospect
• Aversion to risk
pursue the
first of these…
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A change in perception
The prospect P0 and certaintyequivalent 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
This alters the slope of the ICs
you need a
bigger win to
compensate
E
P0 P1
O
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xRED
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A provisional summary
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?
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
The foundation of
a standard
representation of
utility
Preferences
Expected utility
The felicity
function
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A way forward
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
• There's a certain word that’s been carefully avoided so far
• Can you think what it might be…?
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Three key axioms…
State irrelevance:
• The identity of the states is unimportant
Independence:
• Induces an additively separable structure
Revealed likelihood:
• Induces a coherent set of weights on states-of-the-
world
A closer
look
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1: State irrelevance
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-the-world
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2: The independence axiom
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
Level at which payoff is fixed has no bearing on the orderings
over prospects where payoffs differ in other states of the world
We can see this by partitioning the state space for #> 2
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Independence axiom: illustration
xBLUE
A case with 3 states-of-theworld
What if we compare all
of these points…?
Or all of these
points…?
Compare prospects with the
same payoff under GREEN
Ordering of these prospects
should not depend on the size
of the payoff under GREEN
Or all of
these?
O
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3: The “revealed likelihood” axiom
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-the-world
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Revealed likelihood: example
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:
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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
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A key result
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
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The vNM utility function
additive form from
independence axiom
payoff in
state w
p u(x)
Identify components of the
vNM utility function
Can be expressed equivalently
as an “expectation”
The missing word was
“probability”
the cardinal utility or "felicity"
function: independent of state w
“revealed likelihood”
weight on state w
E u(x)
Defined with respect to the
weights pw
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Implications of vNM structure (1)
A typical IC
xBLUE
Slope where it crosses the 45º ray?
From the vNM structure
So all ICs have same slope on 45º ray
pRED
– _____
pBLUE
O
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xRED
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Implications of vNM structure (2)
xBLUE
A given income prospect
From the vNM structure
Mean income
Extend line through P0 and P to P1
P1
–
P
P0
O
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pRED
– _____
pBLUE
_
By quasiconcavity U(P) U(P0)
xRED
Ex
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The vNM paradigm: Summary
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
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
A concept of
“cardinal utility”?
Preferences
Expected utility
The felicity
function
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The function u
The “felicity function” u is central to the vNM
structure
• It’s an awkward name
• But perhaps slightly clearer than the alternative, “cardinal
utility function”
Scale and origin of u are irrelevant:
• 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
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Risk aversion and concavity of u
Use the interpretation of risk aversion as quasiconcavity
If individual is risk averse…
_
…then U(P) U(P0)
Given the vNM structure…
• u(Ex) pREDu(xRED) + pBLUEu(xBLUE)
• u(pREDxRED+pBLUExBLUE) pREDu(xRED) + pBLUEu(xBLUE)
So the function u is concave
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The “felicity” function
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
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xRED
x
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Summary: basic concepts
Review
Use an extension of standard consumer theory
to model uncertainty
Review
Review
• “state-space” approach
Can reinterpret the basic axioms
Need extra axioms to make further progress
• Yields the vNM form
Review
The felicity function gives us insight on risk
aversion
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What next?
Introduce a probability model
Formalise the concept of risk
This is handled in Risk
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