Consumption and Uncertainty
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Transcript of Consumption and Uncertainty
Frank Cowell: Consumption Uncertainty
CONSUMPTION AND UNCERTAINTYMICROECONOMICSPrinciples and Analysis Frank Cowell
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Almost essential Consumption: Basics
Prerequisites
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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
Modelling uncertainty
Preferences
Expected utility
The felicity function
Consumption: Uncertainty
Issues concerning the commodity space
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Frank Cowell: Consumption Uncertainty
UncertaintyNew conceptsFresh insights on consumer axiomsFurther restrictions on the structure of utility functions
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Concepts
state-of-the-world
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w Î W American example
If the only uncertainty is about who will be in power for the next four years then we might have states-of-the-world like this
W={Rep, Dem}
or perhaps like this:
W={Rep, Dem, Independent}
Story
pay-off (outcome)
xw Î X
prospects {xw: w Î W}
an array of bundles over the entire space W
ex ante before the realisation
ex post after the realisation
a consumption bundle
British example
If the only uncertainty is about the weather then we might have states-of-the-world like this
W={rain,sun}
or perhaps like this:
W={rain, drizzle,fog, sleet,hail…}
Story
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The ex-ante/ex-post distinction
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time
time at which the state-of the world is revealed
Decisions to be made here
(too late to make decisions now)
The ex-ante view
The ex-post view
The "moment of truth"
The time line
Rainbow of possible states-of-the-world W
Only one realised state-of-the-world w
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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 w is just xw (a real number)
A special example:• Take the case where #states=2 • W = {RED,BLUE}
The resulting diagram may look familiar…
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The state-space diagram: #W=2
xBLUE
xREDO
The consumption space under uncertainty: 2 states
A prospect in the 1-good 2-state case
· P0
pay o
ff if
B
LUE
occ
urs
payoff if RED occurs
45°
The components of a prospect in the 2-state case But this has no equivalent in choice under certaintypros
pects o
f perf
ect
certai
nty
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Frank Cowell: Consumption Uncertainty
The state-space diagram: #W=3
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The idea generalises: here we have 3 states
xBLUE
xRED
xGREEN
O
prospects of perf
ect
certainty
W = {RED,BLUE,GREEN}
• P0
A prospect in the 1-good 3-state case
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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
Modelling uncertainty
Preferences
Expected utility
The felicity function
Consumption: Uncertainty
Extending the standard consumer axioms
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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 # W possible states then…• …instead of n goods we have n # W goods
Some consumer theory carries over automaticallyAppropriate to apply standard preference axiomsBut they may require fresh interpretation
A little revision
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Another look at preference axioms
CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothness
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to ensure existenceof indifference curves
to give shapeof indifference curves
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Ranking prospectsxBLUE
xREDO
Greed: Prospect P1 is preferred to P0
Contours of the preference map
· P1
· P0
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Implications of ContinuityxBLUE
xREDO
Pathological preference for certainty (violates of continuity)
· P0
x
x
Impose continuity
holesno holes
An arbitrary prospect P0
· E
Find point E by continuity Income x is the certainty equivalent of P0
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Reinterpret quasiconcavityxBLUE
xREDO
Take an arbitrary prospect P0 Given continuous indifference curves…
· P0
· E
…find the certainty-equivalent prospect E
Points in the interior of the line P0E represent mixtures of P0 and E
If U strictly quasiconcave P1 is preferred to P0
· P1
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More on preferences?We can easily interpret the standard axiomsBut 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 perceptionxBLUE
xREDO
The prospect P0 and certainty-equivalent prospect E (as before)
Suppose RED begins to seem less likely
· P0
· P1
· E
Now prospect P1 (not P0) appears equivalent to E
you need a bigger win to compensate
Indifference curves after the change
This alters the slope of the ICs
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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
Modelling uncertainty
Preferences
Expected utility
The felicity function
Consumption: Uncertainty
The foundation of a standard representation of utility
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A way forwardFor more results we need more structure on the
problemFurther restrictions on the structure of utility functionsWe do this by introducing extra axiomsThree 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: • identity of the states is unimportant
Independence: • induces an additively separable structure
Revealed likelihood: • induces coherent set of weights on states-of-the-world
A closer look
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1: State irrelevanceWhichever 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 #W > 2
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Independence axiom: illustration
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A case with 3 states-of-the-world
Compare prospects with the same payoff under GREEN
Ordering of these prospects should not depend on the size of the payoff under GREEN
xBLUE
xRED
O
xGREEN
What if we compare all of these points…?
Or all of these points…?
Or all of these?
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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 W0 and W1 be any two subsets of W Define two prospects:
• P0 := {x′ if wÎW0 and x if wW0}• P1 := {x′ if wÎW1 and x if wW1}
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
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1 apple 1 banana1 cherry 1 date
apple appleapple
apple
applebanana banana
apple apple appleapple bananabanana
bananaP2:P1:
States of the world (only one colour will occur)
Assume preferences over fruit Consider these two prospects
Choose a prospect: P1 or P2?
Another two prospects
Is your choice between P3 and P4 the same as between P1 and P2?
cherry cherrycherry
cherry
cherrydate date
cherry cherry cherrycherry datedate
dateP4:P3:
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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:
å pw u(xw)w ÎW
Properties of p and u in a moment. Consider the interpretation
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The vNM utility function
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å pw u(xw) wÎW
Components of vNM U-function
the cardinal utility or "felicity" function: independent of state w
payoff in state w
“revealed likelihood” weight on state w
additive form from independence axiom
Equivalently as an “expectation”
Eu(x)Defined with respect to the weights pw
The missing word : “probability”
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Implications of vNM structure (1)
xBLUE
xREDO
Slope where it crosses the 45º ray?
A typical IC
From the vNM structure So all ICs have same slope on 45º
ray
pRED – _____pBLUE
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Implications of vNM structure (2) xBLUE
xREDO
pRED – _____pBLUE
A given income prospect
From the vNM structure
E x
Mean income
· P0
· P1
· P
Extend line through P0 and P to P1
By quasiconcavity U() U(P0)
–
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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
Modelling uncertainty
Preferences
Expected utility
The felicity function
Consumption: Uncertainty
A concept of “cardinal utility”?
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The function uThe “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() 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
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u
xxBLUE xRED
If u is strictly concave then person is risk averse
If u is a straight line then person is risk-neutral
Payoffs in states BLUE and RED
Diagram plots utility level (u) against payoffs (x)
If u is strictly convex then person is a risk lover
u of the average of xBLUE
and xRED higher than the expected u of xBLUE and of xRED
u of the average of xBLUE
and xRED equals the expected u of xBLUE and of xRED
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Summary: basic concepts
Use an extension of standard consumer theory to model uncertainty• “state-space” approach
Can reinterpret the basic axiomsNeed extra axioms to make further progress
• Yields the vNM formThe felicity function gives us insight on risk aversion
Review
Review
Review
Review
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