Consumption and Uncertainty

Frank Cowell: Consumption Uncertainty

CONSUMPTION AND UNCERTAINTYMICROECONOMICSPrinciples and Analysis Frank Cowell

1

Almost essential Consumption: Basics

Prerequisites

July 2015

Frank Cowell: Consumption Uncertainty 2

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

July 2015


Overview

Modelling uncertainty

Preferences

Expected utility

The felicity function

Consumption: Uncertainty

Issues concerning the commodity space

July 2015


UncertaintyNew conceptsFresh insights on consumer axiomsFurther restrictions on the structure of utility functions

4 July 2015


Concepts

state-of-the-world

5

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

July 2015


The ex-ante/ex-post distinction

6

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

July 2015


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…

July 2015


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

July 2015


The state-space diagram: #W=3

9

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

July 2015


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…?

July 2015


Overview


Preferences

Expected utility



Extending the standard consumer axioms

July 2015


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

July 2015


Another look at preference axioms

CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothness

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to ensure existenceof indifference curves

to give shapeof indifference curves

July 2015


Ranking prospectsxBLUE

xREDO

Greed: Prospect P1 is preferred to P0

Contours of the preference map

· P1

· P0

July 2015


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

July 2015


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

July 2015


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…

July 2015


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

July 2015


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?

19 July 2015


Overview


Preferences

Expected utility



The foundation of a standard representation of utility

July 2015


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…?

July 2015


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

July 2015


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

23 July 2015


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

24 July 2015


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?

July 2015


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

26 July 2015


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:

July 2015


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

July 2015


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”

July 2015


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

July 2015


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)

–

July 2015


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

32 July 2015


Overview


Preferences

Expected utility



A concept of “cardinal utility”?

July 2015


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

July 2015


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

July 2015


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

July 2015


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

July 2015


What next? Introduce a probability modelFormalise the concept of riskThis is handled in Risk

38 July 2015

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Consumption and Uncertainty