Consumption and Uncertainty

1. 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... 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. state-of-the-world w Î W 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} a consumption bundle • xw Î X • pay-off (outcome) 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...} • {xw: w Î W} • prospects • before the realisation • ex ante • after the realisation • ex post

6. Only one realised state-of-the-world w Rainbow of possible states-of-the-world W The ex-ante/ex-post distinction • The time line • The "moment of truth" • The ex-ante view • The ex-post view (too late to make decisions now) Decisions to be made here 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. payoff if RED occurs payoff if BLUE occurs 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 • P0 45° xRED O

9. xGREEN xRED The state-space diagram: #W=3 • The idea generalises: here we have 3 states xBLUE W= {RED,BLUE,GREEN} • A prospect in the 1-good 3-state case prospects of perfect certainty • 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... 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 • Greed: Prospect P1 is preferred to prospect P0 xBLUE • Contours of the preference map. • P1 • P0 xRED O

15. holes x x 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 no holes • E • P0 xRED O

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 Uisstrictly quasiconcave P1 is strictly preferred to P0. • E • P1 • P0 xRED O

17. 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 • 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... 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: • 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 allz • 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 • A case with 3 states-of-the-world xBLUE • 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 tox′. • 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. P3: cherry cherry cherry cherry cherry date date P4: cherry cherry cherry cherry date date date 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: apple apple apple apple banana banana banana

28. 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: åpw 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 åpwu(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. pRED – _____ pBLUE 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. xRED O

31. pRED – _____ pBLUE Implications of vNM structure (2) • A given income prospect • From the vNM structure xBLUE • Mean income • Extend line through P0 and P to P1 . • P1 – • P • 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... 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. uof the average of xBLUE and xRED equals the expected u of xBLUE and of xRED uof the average of xBLUE and xRED higher than the expected u of xBLUE and of xRED The “felicity” function • Diagram plots utility level (u) against payoffs (x). u • Payoffs in states BLUE and RED. • 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.