1. Consumption and Uncertainty
Prerequisites
Almost essential
Consumption: Basics
Consumption and Uncertainty
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

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

3. Overview... Issues concerning the commodity space
Consumption: Uncertainty
Modelling uncertainty
Issues concerning the commodity space
Preferences
Expected utility
The felicity function

4. Uncertainty New concepts Fresh insights on consumer axioms
Further restrictions on the structure of utility functions

5. Concepts w Î W xw Î X {xw: w Î W} before the realisation
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-of-the-world like this
W={Rep, Dem}
or perhaps like this:
W={Rep, Dem, Independent}
state-of-the-world
w Î W
a consumption bundle
pay-off (outcome)
xw Î X
an array of bundles over the entire space W
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...}
prospects
{xw: w Î W}
ex ante
before the realisation
ex post
after the realisation

6. The ex-ante/ex-post distinction
The time line
The "moment of truth"
The ex-ante view
The ex-post view
Rainbow of possible states-of-the-world W
(too late to make decisions now)
Decisions to be made here
Only one realised state-of-the-world w
time
time at which the state-of the world is revealed

7. 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...

8. The state-space diagram: #W=2
The consumption space under uncertainty: 2 states
xBLUE
A prospect in the 1-good 2-state case
The components of a prospect in the 2-state case
prospects of perfect certainty
But this has no equivalent in choice under certainty
payoff if
RED occurs
BLUE occurs
payoff if P0
45°
xRED O

9. The state-space diagram: #W=3
xBLUE
The idea generalises: here we have 3 states
W = {RED,BLUE,GREEN}
A prospect in the 1-good 3-state case
prospects of perfect certainty
xRED
xGREEN • P0 O

10. 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…?

11. Overview... Extending the standard consumer axioms
Consumption: Uncertainty
Modelling uncertainty
Extending the standard consumer axioms
Preferences
Expected utility
The felicity function

12. 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 automatically.
Appropriate to apply standard preference axioms.
But they may require fresh interpretation.
A little revision

13. Another look at preference axioms
Completeness
Transitivity
Continuity
Greed
(Strict) Quasi-concavity
Smoothness
to ensure existence
of indifference curves
to give shape
of indifference curves

14. Ranking prospects P1 P0 xBLUE xRED O
Greed: Prospect P1 is preferred to prospect P0
xBLUE
Contours of the preference map. P1 P0
xRED O

15. Implications of Continuity
A pathological preference for certainty (violation 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
xRED O x

16. Reinterpret quasiconcavity
Take an arbitrary prospect P0.
xBLUE
Given continuous indifference curves….
…find the certainty-equivalent prospect E
Points in the interior of the line P0E represent mixtures of P0 and E.
If U is strictly quasiconcave P1 is strictly preferred to P0. E P1 P0
xRED O

17. pursue the first of these...
More on preferences?
We can easily interpret the standard axioms.
But what determines the 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...

18. A change in perception E . . P0 P1
The prospect P0 and certainty-equivalent 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
you need a
bigger win to compensate E
This change alters the slope of the ICs. . P0 . P1
xRED O

19. 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?

20. Overview... The foundation of a standard representation of utility
Consumption: Uncertainty
Modelling uncertainty
The foundation of a standard representation of utility
Preferences
Expected utility
The felicity function

21. 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.
By the way, there's a certain that’s been carefully avoided so far.
Can you think what it might be...?

22. Three key axioms... State irrelevance: Independence:
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

23. 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.

24. 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.
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 the state space for #W > 2

25. Independence axiom: illustration
xBLUE
A case with 3 states-of-the-world
Compare prospects with the same payoff under GREEN.
What if we compare all of these points...?
Ordering of these prospects should not depend on the size of the payoff under GREEN.
Or all of these points...?
xGREEN
Or all of these? O
xRED

26. 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 wW0 and x if wW0}
P1 := {x′ if wW1 and x if wW1}
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.

27. Revealed likelihood: example
Assume these preferences over fruit
1 apple < 1 banana
1 cherry < 1 date
Consider these two prospects
Choose a prospect: P1 or P2?
Another two prospects
States of the world (remember only one colour will occur)
Is your choice between P3 and P4 the same as between P1 and P2?
P1:
apple
apple
apple
apple
apple
banana
banana
P2:
banana
banana
banana
apple
apple
apple
apple
cherry
date
P4:
P3:

28. A key result å pw u(xw) w ÎW
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

29. The vNM utility function
additive form from independence axiom
Identify components of the vNM utility function
payoff in state w
å pw u(xw)
wÎW
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

30. Implications of vNM structure (1)
A typical IC
What is the slope where it crosses the 45º ray?
xBLUE
From the vNM structure
So all ICs must have same slope at the 45º ray.
pRED
– _____
pBLUE
xRED O

31. Implications of vNM structure (2)
A given income prospect
xBLUE
From the vNM structure
Mean income
Extend line through P0 and P to P1 . P1 – P
pRED
– _____
pBLUE
By quasiconcavity U(P)  U(P0) _ P0
xRED O Ex

32. 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.

33. Overview... A concept of “cardinal utility”? Consumption: Uncertainty
Modelling uncertainty
A concept of “cardinal utility”?
Preferences
Expected utility
The felicity function

34. 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.

35. 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.

36. The “felicity” function
Diagram plots utility level (u) against payoffs (x). u
Payoffs in states BLUE and RED.
u of the average of xBLUE and xRED equals the expected u of xBLUE and of xRED
u of the average of xBLUE and xRED higher than the expected u of xBLUE and of xRED
If u is strictly concave then person is risk averse
If u is a straight line then person is risk-neutral
If u is strictly convex then person is a risk lover x
xBLUE
xRED

37. Summary: basic concepts
Use an extension of standard consumer theory to model uncertainty
“state-space” approach
Can reinterpret the basic axioms.
Need extra axioms to make further progress.
Yields the vNM form.
The felicity function gives us insight on risk aversion.
Review
Review
Review
Review

38. What next? Introduce a probability model.
Formalise the concept of risk.
This is handled in Risk.