Report

Prerequisites Almost essential Consumption: Basics CONSUMPTION AND UNCERTAINTY MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Consumption Uncertainty 1 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 Frank Cowell: Consumption Uncertainty 2 Overview… Consumption: Uncertainty Modelling uncertainty Issues concerning the commodity space Preferences Expected utility The felicity function March 2012 Frank Cowell: Consumption Uncertainty 3 Uncertainty New concepts Fresh insights on consumer axioms Further restrictions on the structure of utility functions March 2012 Frank Cowell: Consumption Uncertainty 4 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-ofthe-world like this ={Rep, Dem} or perhaps like this: ={Rep, Dem, Independent} state-of-the-world a consumption bundle pay-off (outcome) x X prospects {x: } ex ante before the realisation ex post after the realisation British example If the only uncertainty is about the weather then we might have states-ofthe-world like this an array of bundles over the entire space ={rain,sun} or perhaps like this: ={rain, drizzle,fog, sleet,hail…} March 2012 Frank Cowell: Consumption Uncertainty 5 The ex-ante/ex-post distinction The time line The "moment of truth" The ex-ante view The ex-post view (too Decisions late to make to be decisions made here now) Only one realised stateof-the-world time at which the state-of the world is revealed March 2012 time Rainbow of possible statesof-the-world Frank Cowell: Consumption Uncertainty 6 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… March 2012 Frank Cowell: Consumption Uncertainty 7 The state-space diagram: #=2 The consumption space under uncertainty: 2 states xBLUE A prospect in the 1good 2-state case The components of a prospect in the 2-state case But this has no equivalent in choice under certainty payoff if RED occurs 45° O March 2012 P0 xRED Frank Cowell: Consumption Uncertainty 8 The state-space diagram: #=3 xBLUE The idea generalises: here we have 3 states = {RED,BLUE,GREEN} A prospect in the 1-good 3state case •P 0 O March 2012 Frank Cowell: Consumption Uncertainty 9 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 Frank Cowell: Consumption Uncertainty 10 Overview… Consumption: Uncertainty Modelling uncertainty Extending the standard consumer axioms Preferences Expected utility The felicity function March 2012 Frank Cowell: Consumption Uncertainty 11 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 A little revision March 2012 Frank Cowell: Consumption Uncertainty 12 Another look at preference axioms Completeness to ensure existence of indifference curves Transitivity Continuity Greed (Strict) Quasi-concavity to give shape of indifference curves Smoothness March 2012 Frank Cowell: Consumption Uncertainty 13 Ranking prospects xBLUE Greed: Prospect P1 is preferred to P0 Contours of the preference map P1 P0 O March 2012 xRED Frank Cowell: Consumption Uncertainty 14 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 holes no holes E x P0 O March 2012 x xRED Frank Cowell: Consumption Uncertainty 15 Reinterpret quasiconcavity Take an arbitrary prospect P0 Given continuous indifference curves… …find the certainty-equivalent prospect E xBLUE Points in the interior of the line P0E represent mixtures of P0 and E If U strictly quasiconcave P1 is preferred to P0 E P1 P0 O March 2012 xRED Frank Cowell: Consumption Uncertainty 16 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 Frank Cowell: Consumption Uncertainty 17 A change in perception The prospect P0 and certaintyequivalent 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 This alters the slope of the ICs you need a bigger win to compensate E P0 P1 O March 2012 xRED Frank Cowell: Consumption Uncertainty 18 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 Frank Cowell: Consumption Uncertainty 19 Overview… Consumption: Uncertainty Modelling uncertainty The foundation of a standard representation of utility Preferences Expected utility The felicity function March 2012 Frank Cowell: Consumption Uncertainty 20 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 Frank Cowell: Consumption Uncertainty 21 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 Frank Cowell: Consumption Uncertainty 22 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 Frank Cowell: Consumption Uncertainty 23 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 March 2012 Frank Cowell: Consumption Uncertainty 24 Independence axiom: illustration xBLUE A case with 3 states-of-theworld What if we compare all of these points…? Or all of these points…? Compare prospects with the same payoff under GREEN Ordering of these prospects should not depend on the size of the payoff under GREEN Or all of these? O March 2012 Frank Cowell: Consumption Uncertainty 25 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: • P0 := {x′ if 0 and x if 0} • P1 := {x′ if 1 and x if 1} 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 Frank Cowell: Consumption Uncertainty 26 Revealed likelihood: example Assume these preferences over fruit Consider these two prospects 1 apple < 1 banana 1 cherry < 1 date States of the world Choose a prospect: P1 or P2? Another two prospects Is your choice between P3 and P4 the same as between P1 and P2? (remember only one colour will occur) P1: apple P2: banana P3: P4: March 2012 apple apple apple banana banana banana banana apple apple apple apple apple cherry cherry cherry cherry cherry date date date date date cherry cherry cherry cherry Frank Cowell: Consumption Uncertainty 27 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: p u(x) Properties of p and u in a moment. Consider the interpretation March 2012 Frank Cowell: Consumption Uncertainty 28 The vNM utility function additive form from independence axiom payoff in state w p u(x) Identify components of the vNM utility function 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 Frank Cowell: Consumption Uncertainty 29 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 45º ray pRED – _____ pBLUE O March 2012 xRED Frank Cowell: Consumption Uncertainty 30 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 P0 O March 2012 pRED – _____ pBLUE _ By quasiconcavity U(P) U(P0) xRED Ex Frank Cowell: Consumption Uncertainty 31 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 March 2012 Frank Cowell: Consumption Uncertainty 32 Overview… Consumption: Uncertainty Modelling uncertainty A concept of “cardinal utility”? Preferences Expected utility The felicity function March 2012 Frank Cowell: Consumption Uncertainty 33 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 Frank Cowell: Consumption Uncertainty 34 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 Frank Cowell: Consumption Uncertainty 35 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 u of the average of xBLUE BLUE equals than the the and xRED RED higher expected u of xBLUE BLUE and of xRED RED If u is a straight line then person is risk-neutral If u is strictly convex then person is a risk lover xBLUE March 2012 xRED x Frank Cowell: Consumption Uncertainty 36 Summary: basic concepts Review Use an extension of standard consumer theory to model uncertainty Review Review • “state-space” approach Can reinterpret the basic axioms Need extra axioms to make further progress • Yields the vNM form Review The felicity function gives us insight on risk aversion March 2012 Frank Cowell: Consumption Uncertainty 37 What next? Introduce a probability model Formalise the concept of risk This is handled in Risk March 2012 Frank Cowell: Consumption Uncertainty 38