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Prerequisites
Almost essential
Consumption: Basics
CONSUMPTION AND
UNCERTAINTY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Consumption Uncertainty
1
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
March 2012
Frank Cowell: Consumption Uncertainty
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
Issues
concerning the
commodity space
Preferences
Expected utility
The felicity
function
March 2012
Frank Cowell: Consumption Uncertainty
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Uncertainty
 New concepts
 Fresh insights on consumer axioms
 Further restrictions on the structure of utility functions
March 2012
Frank Cowell: Consumption Uncertainty
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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…}
March 2012
Frank Cowell: Consumption Uncertainty
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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
March 2012
time
Rainbow of possible statesof-the-world 
Frank Cowell: Consumption Uncertainty
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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…
March 2012
Frank Cowell: Consumption Uncertainty
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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
March 2012
P0
xRED
Frank Cowell: Consumption Uncertainty
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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
March 2012
Frank Cowell: Consumption Uncertainty
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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…?
March 2012
Frank Cowell: Consumption Uncertainty
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
Extending the standard
consumer axioms
Preferences
Expected utility
The felicity
function
March 2012
Frank Cowell: Consumption Uncertainty
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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
March 2012
Frank Cowell: Consumption Uncertainty
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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
March 2012
Frank Cowell: Consumption Uncertainty
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Ranking prospects
xBLUE
Greed: Prospect P1 is preferred to P0
Contours of the preference map
 P1
 P0
O
March 2012
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
March 2012
x
xRED
Frank Cowell: Consumption Uncertainty
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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
March 2012
xRED
Frank Cowell: Consumption Uncertainty
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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…
March 2012
Frank Cowell: Consumption Uncertainty
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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
March 2012
xRED
Frank Cowell: Consumption Uncertainty
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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?
March 2012
Frank Cowell: Consumption Uncertainty
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Overview…
Consumption:
Uncertainty
Modelling
uncertainty
The foundation of
a standard
representation of
utility
Preferences
Expected utility
The felicity
function
March 2012
Frank Cowell: Consumption Uncertainty
20
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…?
March 2012
Frank Cowell: Consumption Uncertainty
21
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
March 2012
Frank Cowell: Consumption Uncertainty
22
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
March 2012
Frank Cowell: Consumption Uncertainty
23
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
March 2012
Frank Cowell: Consumption Uncertainty
24
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
March 2012
Frank Cowell: Consumption Uncertainty
25
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
March 2012
Frank Cowell: Consumption Uncertainty
26
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:
March 2012
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
Frank Cowell: Consumption Uncertainty
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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
March 2012
Frank Cowell: Consumption Uncertainty
28
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
March 2012
Frank Cowell: Consumption Uncertainty
29
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
March 2012
xRED
Frank Cowell: Consumption Uncertainty
30
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
March 2012
pRED
– _____
pBLUE
_
 By quasiconcavity U(P)  U(P0)
xRED
Ex
Frank Cowell: Consumption Uncertainty
31
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
March 2012
Frank Cowell: Consumption Uncertainty
32
Overview…
Consumption:
Uncertainty
Modelling
uncertainty
A concept of
“cardinal utility”?
Preferences
Expected utility
The felicity
function
March 2012
Frank Cowell: Consumption Uncertainty
33
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
March 2012
Frank Cowell: Consumption Uncertainty
34
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
March 2012
Frank Cowell: Consumption Uncertainty
35
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
March 2012
xRED
x
Frank Cowell: Consumption Uncertainty
36
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
March 2012
Frank Cowell: Consumption Uncertainty
37
What next?
 Introduce a probability model
 Formalise the concept of risk
 This is handled in Risk
March 2012
Frank Cowell: Consumption Uncertainty
38

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