consumption and uncertainty - london school of economics

Frank Cowell: Consumption Uncertainty

CONSUMPTION AND UNCERTAINTYMICROECONOMICSPrinciples and AnalysisFrank Cowell

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Almost essential Consumption: Basics

Prerequisites

April 2018


Overview

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Modelling uncertainty

Preferences

Expected utility

The felicity function

Consumption: Uncertainty

Issues concerning the commodity space

April 2018


UncertaintyUncertainty extends consumer theory in interesting ways

New concepts• in the choice set• in concept of time

Fresh insights on consumer axioms• not all work in quite the same way

Further restrictions on the structure of utility functions

3April 2018


Concepts

state-of-the-world

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ω ∈ Ω 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

Ω=Rep, Dem

or perhaps like this:

Ω=Rep, Dem, Independent

Story

pay-off (outcome)

xω ∈ X

prospects xω: ω ∈ Ω

an array of bundles over the entire space Ω

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

Ω=rain,sun

or perhaps like this:

Ω=rain, drizzle,fog, sleet,hail…

Story

April 2018


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 Ω

Only onerealised state-of-the-world ω

April 2018

* detail on slide can only be seen if you run the slideshow


A simplified approach…Assume the state-space is finite-dimensional Then a simple diagrammatic approach can be used This is easier if we suppose that payoffs are scalars

• Consumption in state ω is just xω (a real number)

A special example:• Take the case where #states=2• Ω = RED,BLUE

The resulting diagram may look familiar

6April 2018


The state-space diagram: #Ω=2

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xBLUE

xREDO

The consumption space under uncertainty: 2 statesA prospect in the 1-good 2-state case

• P0

payoff if RED occurs

45°

The components of a prospect in the 2-state case

But this has no equivalent in choice under certainty

April 2018


The state-space diagram: #Ω=3

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The idea generalises: here we have 3 states

xBLUE

O

Ω = RED,BLUE,GREEN

•P0

A prospect in the 1-good 3-state case

April 2018


The modified commodity space Can treat the states-of-the-world like characteristics of goodsWe 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 then invoke standard axioms over enlarged commodity space But is more involved?

9April 2018


Overview

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Preferences

Expected utility



Extending the standard consumer axioms

April 2018


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 # Ω possible states then…• …instead of n goods we have n × # Ω goods

Some consumer theory carries over automaticallyAppropriate to apply standard preference axiomsBut they may require fresh interpretation

11April 2018


Another look at preference axioms

CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothness

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to ensure existenceof indifference curves

to give shapeof indifference curves

April 2018


Ranking prospects

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xBLUE

xREDO

Greed: Prospect P1 is preferred to P0

Contours of the preference map

• P1

• P0

April 2018


Implications of Continuity

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xBLUE

xREDO

Pathological preference for certainty (violates of continuity)

• P0

ξ

ξ

Impose continuity

holesno holes

An arbitrary prospect P0

• E

Find point E by continuityIncome ξ is the certainty equivalent of P0

April 2018


Reinterpret quasiconcavity

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xBLUE

xREDO

Take an arbitrary prospect P0Given continuous indifference curves…

• P0

• E

…find the certainty-equivalent prospect E

Points in the interior of the line P0Erepresent mixtures of P0 and EIf U strictly quasiconcave P1 is preferred to P0

• P1

April 2018


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

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pursue the first of these…

April 2018


A change in perception

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xBLUE

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

April 2018


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”• get concept of certainty preference• model changes in perceptions of uncertainty about future prospects

But can we do more?

18April 2018


Overview

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Preferences

Expected utility



The foundation of a standard representation of utility

April 2018


A way forward For more results we need more structure on the problem

• further restrictions on the structure of utility functions• do this by introducing extra axioms

Three more axioms to clarify the consumer's attitude to uncertain prospects

Looking ahead to our main result• There's a certain word that’s been carefully avoided so far• Can you think what it might be?

20April 2018


Three key axioms…

State irrelevance: • identity of the states is unimportantIndependence:

• induces an additively separable structureRevealed likelihood:

• induces coherent set of weights on states-of-the-world

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A closer look

April 2018


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

22April 2018


2: The independence axiom Let P(z) and P′(z) be any two distinct prospects such that the

payoff in state-of-the-world ϖ is z• xϖ = xϖ′ = z

If U(P(z)) ≥ U(P′(z)) for some z then U(P(z)) ≥ U(P′(z)) for all zOne 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 z 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 #Ω > 2

23April 2018


Independence axiom: illustration

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A case with 3 states-of-the-world

Compare prospects with the same payoff under GREENOrdering of these prospects should not depend on payoff under GREEN

xBLUE

O

What if we compare all of these points…?

Or all of these points…?

Or all of these?

April 2018


3: The “revealed likelihood” axiom

Let x and x′ be two payoffs such that x is weakly preferred to x′ Let Ω0 and Ω1 be any two subsets of ΩDefine two prospects:

• P0 := x′ if ω∈Ω0 and x if ω∉Ω0• P1 := x′ if ω∈Ω1 and x if ω∉Ω1

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

25April 2018


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 fruitConsider these two prospectsChoose a prospect: P1 or P2?

Another two prospects

Is your choice between P3 and P4the same as between P1 and P2?

cherry cherrycherry

cherry

cherrydate date

cherry cherry cherrycherry datedate

dateP4:P3:

April 2018


A key resultWe now have a result that is of central importance to the

analysis of uncertaintyAssume the three new axioms:

• State irrelevance• Independence• Revealed likelihood

Then preferences must be representable in the form of a von Neumann-Morgenstern utility function:

∑ πω u(xω)ω ∈Ω

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Properties of π and u in a moment. Consider the interpretation

April 2018


The vNM utility function

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∑ πω u(xω)ω∈Ω

Components of vNM U-function

the cardinal utility or "felicity" function: independent of state ω

payoff in state ω

“revealed likelihood” weight on state ω

additive form from independence axiom

Equivalently as an “expectation”

Eu(x)Defined with respect to the weights πω

The missing word so far: “probability”

April 2018


Implications of vNM structure (1)

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xBLUE

xREDO

Slope where it crosses the 45º ray?A typical IC

From the vNM structureSo all ICs have same slope on 45º ray

πRED– _____πBLUE

April 2018


Implications of vNM structure (2)

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xBLUE

xREDO

πRED– _____πBLUE

A given income prospectvNM structure: slope is given

Ex

Mean income, Ex

• P0

• P1

• P

Extend line through P0 and P to P1

By quasiconcavity U( 𝑃𝑃) ≥ U(P0)

–

April 2018


The vNM paradigm: Summary To make choice under uncertainty manageable it is helpful to

impose more structure on the utility functionWe 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

31April 2018


Overview

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Preferences

Expected utility



A concept of “cardinal utility”?

April 2018


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

33April 2018


Risk aversion and concavity of uUse the interpretation of risk aversion as quasiconcavity If individual is risk averse then U( 𝑃𝑃) ≥ U(P0)

Given the vNM structure…• u(Ex) ≥ πREDu(xRED) + πBLUEu(xBLUE)• u(πREDxRED+πBLUExBLUE) ≥ πREDu(xRED) + πBLUEu(xBLUE)

So the function u is concave

34April 2018


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 BLUEand 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 xBLUEand xRED equals the expected u of xBLUE and of xRED

April 2018


Summary: basic conceptsUse 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 form

The felicity function gives us insight on risk aversion

36April 2018


What next?

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

37April 2018

consumption and uncertainty - london school of economics

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