Consumption and Uncertainty

1. Prerequisites Almost essential Consumption: Basics Consumption and Uncertainty MICROECONOMICS Principles and Analysis Frank Cowell March 2012

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 March 2012

3. Overview… Consumption: Uncertainty Modelling uncertainty Issues concerning the commodity space Preferences Expected utility The felicity function March 2012

4. Uncertainty • New concepts • Fresh insights on consumer axioms • Further restrictions on the structure of utility functions March 2012

5. w Î W Story Concepts Story • state-of-the-world 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 March 2012

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 March 2012

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  is just xw (a real number) • A special example: • Take the case where #states=2 • W = {RED,BLUE} • The resulting diagram may look familiar… March 2012

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 March 2012

9. xGREEN xRED 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 • P0 O March 2012

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…? March 2012

11. Overview… Consumption: Uncertainty Modelling uncertainty Extending the standard consumer axioms Preferences Expected utility The felicity function March 2012

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 March 2012

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 March 2012

14. Ranking prospects xBLUE • Greed: Prospect P1 is preferred to P0 • Contours of the preference map • P1 • P0 xRED O March 2012

15. holes x x Implications of Continuity • Pathological preference for certainty (violates 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 March 2012

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 P0and E • If Ustrictly quasiconcaveP1 is preferred to P0 • E • P1 • P0 xRED O March 2012

17. More on preferences? • We can easily interpret the standard axioms • But what determines 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… March 2012

18. A change in perception • The prospect P0 and certainty-equivalent prospect E (as before) xBLUE • SupposeREDbegins to seem less likely • Now prospect P1 (not P0) appears equivalent to E • Indifference curves after the change • This alters the slope of the ICs you need a bigger win to compensate • E P0 P1 xRED O March 2012

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? March 2012

20. Overview… Consumption: Uncertainty Modelling uncertainty The foundation of a standard representation of utility Preferences Expected utility The felicity function March 2012

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 • There's a certain word that’s been carefully avoided so far • Can you think what it might be…? March 2012

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 March 2012

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 March 2012

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 • Level at which payoff 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 #W > 2 March 2012

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 March 2012

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 March 2012

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 March 2012

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: åpwu(xw) w ÎW Properties of p and u in a moment. Consider the interpretation March 2012

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 March 2012

30. pRED – _____ pBLUE Implications of vNM structure (1) • A typical IC xBLUE • Slope where it crosses the 45º ray? • From the vNM structure • So all ICs have same slope on 45ºray xRED O March 2012

31. pRED – _____ pBLUE Implications of vNM structure (2) xBLUE • A given income prospect • From the vNM structure • Mean income • Extend line through P0 and P to P1 • P1 – • P _ • By quasiconcavityU(P)  U(P0) • P0 xRED O Ex March 2012

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 axiomatisingvNM) • 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 March 2012

33. Overview… Consumption: Uncertainty Modelling uncertainty A concept of “cardinal utility”? Preferences Expected utility The felicity function March 2012

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 March 2012

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 March 2012

36. u of the average of xBLUEand 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 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 March 2012

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 March 2012

38. What next? • Introduce a probability model • Formalise the concept of risk • This is handled in Risk March 2012