# Consumption and Uncertainty

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

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

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

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

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

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

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...

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

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

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…?

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

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

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

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

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

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

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...

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

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?

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

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...?

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

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.

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

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

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.

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:

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

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

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

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

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.

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

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.

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.

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

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

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