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

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Presentation on theme: "Frank Cowell: Consumption Uncertainty CONSUMPTION AND UNCERTAINTY MICROECONOMICS Principles and Analysis Frank Cowell 1 Almost essential Consumption: Basics."— Presentation transcript:

1 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

2 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

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

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

5 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

6 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

7 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

8 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

9 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

10 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

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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

32 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

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

34 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

35 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

36 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

37 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

38 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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