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

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Frank Cowell: Consumption Uncertainty 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 2 March 2012

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

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Frank Cowell: Consumption Uncertainty Uncertainty New concepts Fresh insights on consumer axioms Further restrictions on the structure of utility functions 4 March 2012

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Frank Cowell: Consumption Uncertainty Concepts state-of-the-world 5 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 ={Rep, Dem} or perhaps like this: ={Rep, Dem, Independent} Story pay-off (outcome) x X prospects { x : } an array of bundles over the entire space ex ante before the realisation ex post after the realisation a consumption bundle British example If the only uncertainty is about the weather then we might have states-of- the-world like this ={rain,sun} or perhaps like this: ={rain, drizzle,fog, sleet,hail…} Story March 2012

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Frank Cowell: Consumption Uncertainty The ex-ante/ex-post distinction 6 time time at which the state-of the world is revealed Decisions to be made here (too late to make decisions now) The ex-ante view The ex-post view The "moment of truth" The time line Rainbow of possible states- of-the-world Only one realised state- of-the-world March 2012

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Frank Cowell: Consumption Uncertainty 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 x (a real number) A special example: Take the case where #states=2 = {RED,BLUE} The resulting diagram may look familiar… 7 March 2012

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Frank Cowell: Consumption Uncertainty The state-space diagram: # 8 x BLUE x RED O The consumption space under uncertainty: 2 states A prospect in the 1- good 2-state case P 0 payoff if BLUE occurs payoff if RED occurs 45° The components of a prospect in the 2-state case But this has no equivalent in choice under certainty prospects of perfect certainty March 2012

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Frank Cowell: Consumption Uncertainty The state-space diagram: # =3 9 The idea generalises: here we have 3 states x BLUE x RED x GREEN O prospects of perfect certainty = {RED,BLUE,GREEN} P 0 A prospect in the 1-good 3- state case March 2012

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Frank Cowell: Consumption Uncertainty 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…? 10 March 2012

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

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Frank Cowell: Consumption Uncertainty 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 # possible states then… …instead of n goods we have n # goods Some consumer theory carries over automatically Appropriate to apply standard preference axioms But they may require fresh interpretation 12 A little revision March 2012

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Frank Cowell: Consumption Uncertainty Another look at preference axioms Completeness Transitivity Continuity Greed (Strict) Quasi-concavity Smoothness 13 to ensure existence of indifference curves to give shape of indifference curves March 2012

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Frank Cowell: Consumption Uncertainty Ranking prospects 14 x BLUE x RED O Greed: Prospect P 1 is preferred to P 0 Contours of the preference map P 1 P 0 March 2012

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Frank Cowell: Consumption Uncertainty Implications of Continuity 15 x BLUE x RED O Pathological preference for certainty (violates of continuity) P 0 Impose continuity holes no holes An arbitrary prospect P 0 E Find point E by continuity Income is the certainty equivalent of P 0 March 2012

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Frank Cowell: Consumption Uncertainty Reinterpret quasiconcavity 16 x BLUE x RED O Take an arbitrary prospect P 0 Given continuous indifference curves… P 0 E …find the certainty-equivalent prospect E Points in the interior of the line P 0 E represent mixtures of P 0 and E If U strictly quasiconcave P 1 is preferred to P 0 P 1 March 2012

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Frank Cowell: Consumption Uncertainty 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 17 pursue the first of these… March 2012

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Frank Cowell: Consumption Uncertainty A change in perception 18 x BLUE x RED O The prospect P 0 and certainty- equivalent prospect E (as before) Suppose RED begins to seem less likely P0 P0 P1 P1 E Now prospect P 1 (not P 0 ) appears equivalent to E you need a bigger win to compensate you need a bigger win to compensate Indifference curves after the change This alters the slope of the ICs March 2012

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Frank Cowell: Consumption Uncertainty 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? 19 March 2012

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

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Frank Cowell: Consumption Uncertainty 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…? 21 March 2012

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Frank Cowell: Consumption Uncertainty 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 22 A closer look March 2012

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Frank Cowell: Consumption Uncertainty 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 23 March 2012

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Frank Cowell: Consumption Uncertainty 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 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 > 2 24 March 2012

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Frank Cowell: Consumption Uncertainty Independence axiom: illustration 25 A case with 3 states-of-the- world Compare prospects with the same payoff under GREEN Ordering of these prospects should not depend on the size of the payoff under GREEN x BLUE x RED O x GREEN What if we compare all of these points…? Or all of these points…? Or all of these? March 2012

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Frank Cowell: Consumption Uncertainty 3: The “revealed likelihood” axiom Let x and x′ be two payoffs such that x is weakly preferred to x′ Let 0 and 1 be any two subsets of Define two prospects: P 0 := {x′ if 0 and x if 0 } P 1 := {x′ if 1 and x if 1 } If U(P 1 )≥U(P 0 ) for some such x and x′ then U(P 1 )≥U(P 0 ) for all such x and x′ Induces a consistent pattern over subsets of states-of-the-world 26 March 2012

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Frank Cowell: Consumption Uncertainty Revealed likelihood: example 27 1 apple < 1 banana 1 cherry < 1 date apple banana apple banana P2:P2: P1:P1: States of the world (remember only one colour will occur) Assume these preferences over fruit Consider these two prospects Choose a prospect: P 1 or P 2 ? Another two prospects Is your choice between P 3 and P 4 the same as between P 1 and P 2 ? cherry date cherry date P4:P4: P3:P3: March 2012

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Frank Cowell: Consumption Uncertainty 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: u x 28 Properties of and u in a moment. Consider the interpretation March 2012

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Frank Cowell: Consumption Uncertainty The vNM utility function 29 u x u x Identify components of the vNM utility function the cardinal utility or "felicity" function: independent of state w payoff in state w “revealed likelihood” weight on state w additive form from independence axiom Can be expressed equivalently as an “expectation” E u(x) Defined with respect to the weights p w The missing word was “probability” March 2012

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Frank Cowell: Consumption Uncertainty Implications of vNM structure (1) 30 x BLUE x RED O Slope where it crosses the 45º ray? A typical IC From the vNM structure So all ICs have same slope on 45º ray RED – _____ BLUE RED – _____ BLUE March 2012

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Frank Cowell: Consumption Uncertainty Implications of vNM structure (2) 31 x BLUE x RED O RED – _____ BLUE RED – _____ BLUE A given income prospect From the vNM structure ExEx Mean income P 0 P 1 P Extend line through P 0 and P to P 1 By quasiconcavity U(P) U(P 0 ) – _ March 2012

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Frank Cowell: Consumption Uncertainty 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 32 March 2012

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

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Frank Cowell: Consumption Uncertainty 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 34 March 2012

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Frank Cowell: Consumption Uncertainty Risk aversion and concavity of u Use the interpretation of risk aversion as quasiconcavity If individual is risk averse… _ …then U(P) U(P 0 ) Given the vNM structure… u( E x) RED u(x RED ) + BLUE u(x BLUE ) u( RED x RED + BLUE x BLUE ) RED u(x RED ) + BLUE u(x BLUE ) So the function u is concave 35 March 2012

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Frank Cowell: Consumption Uncertainty The “felicity” function 36 u x x BLUE x RED If u is strictly concave then person is risk averse If u is a straight line then person is risk-neutral Payoffs in states BLUE and RED Diagram plots utility level (u) against payoffs (x) If u is strictly convex then person is a risk lover u of the average of x BLUE and x RED higher than the expected u of x BLUE and of x RED u of the average of x BLUE and x RED equals the expected u of x BLUE and of x RED March 2012

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Frank Cowell: Consumption Uncertainty 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 37 Review March 2012

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Frank Cowell: Consumption Uncertainty What next? Introduce a probability model Formalise the concept of risk This is handled in Risk 38 March 2012

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