The Consumer Uncertainty(伦敦经济学院高级微观讲义)

This topic Welfare Aggregation again...??
The Basics (Again)
The Consumer
Opportunities and Preferences
Optimisation and Comp. Statics
Welfare
Aggregation
Modelling Uncertainty

state-of-the-world



pay-offs (outcomes)
prospects ex ante
x( X
[x(



ex post
British example American example If the only uncertainty is about before the realisation the weather then we for might have who will be in power four the have states-of-the-world states-of-the-world like ... like or years next four then years we might then we have might each consists of an after realisation this: perhaps like this:the states-of-the-world have states-of-the-world like this: like array of bundles over the ={rain,sun} ={rain,overcast,drizzle,mist, ={Gore,Bush} ... or perhaps like this: entire space fog, sleet,hail...} ={Gore,Bush,Nader????}
The ex ante view... The ex-ante/ex-post distinction: The "moment of truth"
Decisions to be made here
time
Rainbow of time at which the possible states-ofstate-of the world is concept This important the-world revealed can be thought of in
terms of a simple diagram
The ex post view...
(it's too late to make decisions now)
payoffs from decisions
time
Only one realised state-of-the-world
A simplified approach...
Expected Utility
Risk Aversion
Uncertainty
New concepts Fresh insights on consumer axioms
Further restrictions on the structure of utility functions
Concepts



The Basics (Again)
The Consumer
Opportunities and Preferences
Optimisation and Comp. Statics
Welfare
Aggregation
Modelling Modelling Uncertainty Uncertainty
Let's
}
shape
Greed: Prospect B is preferred to Contours of the preference map Prospect A
x(BLUE)

B
x(RED)
A
A pathological preference certainty If we invoke continuity then for an important ...the certainty equivalent (violation of continuity) concept becomes available...
0
What about preferences?


We could treat the states-of-the-world like characteristics of goods.
Then invoke standard axioms over this extended space of goods. So for example {apple,banana,cherry} and OK. Time for=a {rain,sunshine} gives us 3x2 6 spot "statespecific" goods: of revision {a-r,as,b-r,b-s,c-r,c-s}.
x(BLUE)
The ordering of points is identical for each plane
0
Example for Independence Axiom
Red A A' 1 2
Blue 6 3
Green 10 10
The independence axiom

One and only one state-of-the-world can occur. So if the payoff in one state is fixed... the level at which the payoff is fixed should have no bearing on the orderings over prospects whose payoffs can differ in other states of the world
We can see this by partitioning up the state space for #> 2:
What if we compare prospects that haveThe the same payoff under GREEN? independence axiom
theconsumeruncertainty伦敦经济学院高级微观讲义经济学院thethe微观经济高级微观
Microeconomics
The Consumer: Uncertainty
Overview...
The Consumer
Opportunities and Preferences
Optimisation and Comparative Statics

If the state-space is finite then a simple diagrammatic approach can work This can be made even easier if we suppose that payoffs are scalars. The resulting diagram may look strangely familiar...
x(BLUE)

E

A
x(RED)
But what determines the shape of the indifference map?
Perceptions of the riskiness of the outcomes in
any prospect
Aversion to risk
(Two main points...) let's pursue the first of these...

look
1: State irrelevance
Relabelling the states-of-theworld does not affect utility
All that matters is the payoff in each state-ofthe-world
2: The independence axiom
Three key axioms...
State irrelevance: The identity of the states is unimportant

Independence: Induces an additively separable structure

Revealed likelihood: Time on for a closer Induces a coherent set of weights states-of-the-world
After the change B (not A) is suppose RED begins to seem likely Given this situation... Indifference curves after the less change equivalent to E
x(BLUE)


The components of aspace prospect in The consumption under A prospect in the 1-commodity 2-state the 2-state uncertainty: 2 states casecase
x(BLUE) payoff if RED occurs
Expected Utility
Risk Aversion
For more results we need more structure
Three extra axioms to clarify the consumer's attitude to uncertain prospects
Further restrictions on the structure of utility functions
x(BLUE)
no holes holes
E
prospect A is seen as equivalent to a “dead cert” payoff of x


x
x
A
x(RED)
Quasi-concavity: mixtures with certainty equivalent are preferred
change in perception rotates the contours
you need a bigger win to compensate


E
A

B
x(RED)
A provisional summary on 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
Another look at preference axioms


Completeness to ensure Transitivity existence Continuity of indifference
curves
Greed look at these in the 2-state (Strict) Quasi-concavity example... Smoothness

A
= {RED,BLUE}
45
o x(RED)
This has no equivalent in the ordinary 2-good case...
x(BLUE)
x(RED)
The idea generalises: here we have 3 states
x(BLUE)
= {RED,BLUE,GREEN}