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Prerequisites Almost essential Consumption: Basics Frank Cowell: Microeconomics November 2006 Consumption and Uncertainty MICROECONOMICS Principles and Analysis Frank Cowell Why look again at preferences... Frank Cowell: Microeconomics Aggregation issues Modelling specific economic problems labour supply savings New concepts in the choice set restrictions on structure of preferences for consistency over consumers uncertainty Uncertainty extends consumer theory in interesting ways Overview... Frank Cowell: Microeconomics Consumption: Uncertainty Modelling uncertainty Issues concerning the commodity space Preferences Expected utility The felicity function Uncertainty Frank Cowell: Microeconomics New concepts Fresh insights on consumer axioms Further restrictions on the structure of utility functions Story Story Concepts Frank Cowell: Microeconomics 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 state-of-the-world a consumption bundle ={Rep, Dem} or perhaps like this: pay-off (outcome) ={Rep, Dem, Independent} x X an array of bundles over the entire space British example If the only uncertainty is about the weather then we might have states-ofthe-world like this prospects {x: } ex ante before the realisation ex post after the realisation ={rain,sun} or perhaps like this: ={rain, drizzle,fog, sleet,hail...} The ex-ante/ex-post distinction Frank Cowell: Microeconomics 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 time Rainbow of possible statesof-the-world A simplified approach... Frank Cowell: Microeconomics 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 A special example: Consumption in state is just x (a real number) Take the case where #states=2 = {RED,BLUE} The resulting diagram may look familiar... The state-space diagram: #=2 Frank Cowell: Microeconomics 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 P0 xRED The state-space diagram: #=3 Frank Cowell: Microeconomics The idea generalises: here we have 3 states xBLUE = {RED,BLUE,GREEN} A prospect in the 1-good 3state case •P 0 O The modified commodity space Frank Cowell: Microeconomics 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... Frank Cowell: Microeconomics Consumption: Uncertainty Modelling uncertainty Extending the standard consumer axioms Preferences Expected utility The felicity function What about preferences? Frank Cowell: Microeconomics 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 Another look at preference axioms Frank Cowell: Microeconomics Completeness Transitivity Continuity Greed (Strict) Quasi-concavity Smoothness to ensure existence of indifference curves to give shape of indifference curves Ranking prospects Frank Cowell: Microeconomics Greed: Prospect P1 is preferred to prospect P0 xBLUE Contours of the preference map. P1 P0 O xRED Implications of Continuity Frank Cowell: Microeconomics 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 O x xRED Reinterpret quasiconcavity Frank Cowell: Microeconomics Take an arbitrary prospect P0. xBLUE Given continuous indifference curves…. …find the certaintyequivalent prospect E Points in the interior of the line P0E represent mixtures of P0 and E. E If U is strictly quasiconcave P1 is strictly preferred to P0. P1 P0 O xRED More on preferences? Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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 . . P0 P1 O xRED This change alters the slope of the ICs. A provisional summary Frank Cowell: Microeconomics 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... Frank Cowell: Microeconomics Consumption: Uncertainty Modelling uncertainty The foundation of a standard representation of utility Preferences Expected utility The felicity function A way forward Frank Cowell: Microeconomics 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... Frank Cowell: Microeconomics State irrelevance: Independence: The identity of the states is unimportant Induces an additively separable structure Revealed likelihood: Induces a coherent set of weights on states-ofthe-world A closer look 1: State irrelevance Frank Cowell: Microeconomics 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-theworld. 2: The independence axiom Frank Cowell: Microeconomics 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 #> 2 Independence axiom: illustration Frank Cowell: Microeconomics A case with 3 states-of-theworld xBLUE Compare prospects with the same payoff under GREEN. What if we compare all of these points...? Or all of these points...? Or all of these? O Ordering of these prospects should not depend on the size of the payoff under GREEN. 3: The “revealed likelihood” axiom Frank Cowell: Microeconomics 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-theworld. Revealed likelihood: example Frank Cowell: Microeconomics 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: 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 A key result Frank Cowell: Microeconomics 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: pu(x) Properties of p and u in a moment. Consider the interpretation The vNM utility function Frank Cowell: Microeconomics additive form from independence axiom p u(x) payoff in state “revealed likelihood” weight on state the cardinal utility or "felicity" function: independent of state E u(x) Defined with respect to the weights p Identify components of the vNM utility function Can be expressed equivalently as an “expectation” The missing word was “probability” Implications of vNM structure (1) Frank Cowell: Microeconomics 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 O xRED Implications of vNM structure (2) Frank Cowell: Microeconomics A given income prospect From the vNM structure xBLUE Mean income Extend line through P0 and P to P1 . P1 – P P0 O pRED – _____ pBLUE xRED Ex By _ quasiconcavity U(P) U(P0) The vNM paradigm: Summary Frank Cowell: Microeconomics 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... Frank Cowell: Microeconomics Consumption: Uncertainty Modelling uncertainty A concept of “cardinal utility”? Preferences Expected utility The felicity function The function u Frank Cowell: Microeconomics The “felicity function” u is central to the vNM structure. Scale and origin of u are irrelevant: It’s an awkward name. But perhaps slightly clearer than the alternative, “cardinal utility function”. 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 Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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 xRED x Summary: basic concepts Frank Cowell: Microeconomics Review Review Review Review 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. What next? Frank Cowell: Microeconomics Introduce a probability model. Formalise the concept of risk. This is handled in Risk.