Prerequisites Almost essential Consumption Basics CONSUMPTION AND UNCERTAINTY

Prerequisites Almost essential Consumption: Basics CONSUMPTION AND UNCERTAINTY MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Consumption Uncertainty 1

Overview Consumption: Uncertainty Modelling uncertainty Issues concerning the commodity space Preferences Expected utility The felicity function April 2018 Frank Cowell: Consumption Uncertainty 2

Uncertainty § Uncertainty extends consumer theory in interesting ways § New concepts • in the choice set • in concept of time § Fresh insights on consumer axioms • not all work in quite the same way § Further restrictions on the structure of utility functions April 2018 Frank Cowell: Consumption Uncertainty 3

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-of-the -world like this W={Rep, Dem} or perhaps like this: W={Rep, Dem, Independent} British example If the only uncertainty is about the weather then we might have states-ofthe-world like this § state-of-the-world § W a consumption bundle § pay-off (outcome) § x X § prospects § {x : W} § ex ante § before the realisation § ex post § after the realisation an array of bundles over the entire space W W={rain, sun} or perhaps like this: W={rain, drizzle, fog, sleet, hail…} April 2018 Frank Cowell: Consumption Uncertainty 4

* detail on slide can only be seen if you run the slideshow 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 to be decisions now) made here Only one realised state-of -the-world time at which the state-of the world is revealed April 2018 time Rainbow of possible statesof-the-world W Frank Cowell: Consumption Uncertainty 5

A simplified approach… § Assume the state-space is finite-dimensional § Then a simple diagrammatic approach can be used § This is 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 • W = {RED, BLUE} § The resulting diagram may look familiar April 2018 Frank Cowell: Consumption Uncertainty 6

The state-space diagram: #W=2 §The consumption space under uncertainty: 2 states x. BLUE §A prospect in the 1 good 2 -state case t c fe §The components of a prospect in the 2 -state case er p f · 45° O April 2018 P 0 §But this has no equivalent in choice under certainty payoff if BLUE occurs payoff if RED occurs so t ec ty p os ain r p rt ce x. RED Frank Cowell: Consumption Uncertainty 7

The state-space diagram: #W=3 x. BLUE §The idea generalises: here we have 3 states W = {RED, BLUE, GREEN} f so t c e sp nty o pr rtai ce • P §A prospect in the 1 -good 3 state case ct e f r pe N x GREE 0 O x. RED April 2018 Frank Cowell: Consumption Uncertainty 8

The modified commodity space § Can 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 3 x 2 = 6 “state-specific” goods… • …{a-r, a-s, b-r, b-s, c-r, c-s} § Can then invoke standard axioms over enlarged commodity space § But is more involved? April 2018 Frank Cowell: Consumption Uncertainty 9

Overview Consumption: Uncertainty Modelling uncertainty Extending the standard consumer axioms Preferences Expected utility The felicity function April 2018 Frank Cowell: Consumption Uncertainty 10

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 April 2018 Frank Cowell: Consumption Uncertainty 11

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 April 2018 Frank Cowell: Consumption Uncertainty 12

Ranking prospects x. BLUE §Greed: Prospect P 1 is preferred to P 0 §Contours of the preference map · P 1 · P 0 O April 2018 x. RED Frank Cowell: Consumption Uncertainty 13

Implications of Continuity x. BLUE §Pathological preference for certainty (violates of continuity) §Impose continuity §An arbitrary prospect P 0 §Find point E by continuity §Income x is the certainty equivalent of P 0 holes no holes · E x · P 0 O April 2018 x x. RED Frank Cowell: Consumption Uncertainty 14

Reinterpret quasiconcavity §Take an arbitrary prospect P 0 §Given continuous indifference curves… §…find the certainty-equivalent prospect E x. BLUE §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 · E · P 1 · P 0 O April 2018 x. RED Frank Cowell: Consumption Uncertainty 15

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… April 2018 Frank Cowell: Consumption Uncertainty 16

A change in perception §The prospect P 0 and certaintyequivalent prospect E (as before) x. BLUE §Suppose RED begins to seem less likely §Now prospect P 1 (not P 0) appears equivalent to E §Indifference curves after the change §This alters the slope of the ICs you need a bigger win to compensate · E · · P 0 P 1 O April 2018 x. RED Frank Cowell: Consumption Uncertainty 17

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” • get concept of certainty preference • model changes in perceptions of uncertainty about future prospects § But can we do more? April 2018 Frank Cowell: Consumption Uncertainty 18

Overview Consumption: Uncertainty Modelling uncertainty The foundation of a standard representation of utility Preferences Expected utility The felicity function April 2018 Frank Cowell: Consumption Uncertainty 19

A way forward § For more results we need more structure on the problem • further restrictions on the structure of utility functions • do this by introducing extra axioms § Three more axioms to clarify the consumer's attitude to uncertain prospects § Looking ahead to our main result • There's a certain word that’s been carefully avoided so far • Can you think what it might be? April 2018 Frank Cowell: Consumption Uncertainty 20

Three key axioms… § State irrelevance: • identity of the states is unimportant § Independence: • induces an additively separable structure § Revealed likelihood: • induces coherent set of weights on states-of-the-world A closer look April 2018 Frank Cowell: Consumption Uncertainty 21

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 April 2018 Frank Cowell: Consumption Uncertainty 22

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 z 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 April 2018 Frank Cowell: Consumption Uncertainty 23

Independence axiom: illustration x. BLUE §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 payoff under GREEN Or all of these points…? N x GREE Or all of these? O x. RED April 2018 Frank Cowell: Consumption Uncertainty 24

3: The “revealed likelihood” axiom § Let x and x′ be two payoffs such that x is weakly preferred to x′ § Let W 0 and W 1 be any two subsets of W § Define two prospects: • P 0 : = {x′ if W 0 and x if W 0} • P 1 : = {x′ if W 1 and x if W 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 April 2018 Frank Cowell: Consumption Uncertainty 25

Revealed likelihood: example §Assume preferences over fruit §Consider these two prospects §Choose a prospect: P 1 or P 2? §Another two prospects States of the world (only one colour will occur) P 1: apple P 2: banana P 3 : P 4 : April 2018 apple banana apple cherry cherry date date cherry banana §Is your choice between P 3 and P 4 the same as between P 1 and P 2? banana apple Frank Cowell: Consumption Uncertainty 26

A key result § We now have a result that is of central importance to the analysis of uncertainty § Assume three new axioms: • State irrelevance • Independence • Revealed likelihood § Then preferences must be representable in the form of a von Neumann-Morgenstern utility function: å p u(x ) W Properties of p and u in a moment. Consider the interpretation April 2018 Frank Cowell: Consumption Uncertainty 27

The v. NM utility function additive form from independence axiom §Components of v. NM U-function payoff in state å p u(x ) W “revealed likelihood” weight on state §Equivalently as an “expectation” the cardinal utility or "felicity" function: independent of state §The missing word so far: “probability” Eu(x) April 2018 Defined with respect to the weights p Frank Cowell: Consumption Uncertainty 28

Implications of v. NM structure (1) x. BLUE §A typical IC §Slope where it crosses the 45º ray? §From the v. NM structure §So all ICs have same slope on 45º ray p. RED – _____ p. BLUE O April 2018 x. RED Frank Cowell: Consumption Uncertainty 29

Implications of v. NM structure (2) x. BLUE §A given income prospect §v. NM structure: slope is given §Mean income, Ex §Extend line through P 0 and P to P 1 · P 1 – · P 0 O April 2018 Ex p. RED – _____ p. BLUE x. RED Frank Cowell: Consumption Uncertainty 30

The v. NM 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 v. NM) § 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 April 2018 Frank Cowell: Consumption Uncertainty 31

Overview Consumption: Uncertainty Modelling uncertainty A concept of “cardinal utility”? Preferences Expected utility The felicity function April 2018 Frank Cowell: Consumption Uncertainty 32

The function u § The “Felicity function” u is central to the v. NM 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 April 2018 Frank Cowell: Consumption Uncertainty 33

Risk aversion and concavity of u § April 2018 Frank Cowell: Consumption Uncertainty 34

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 x. BLUE equals the and x. RED higher than the expected u of x. BLUE and of x. RED §If u is a straight line then person is risk-neutral §If u is strictly convex then person is a risk lover x. BLUE April 2018 x. RED x Frank Cowell: Consumption Uncertainty 35

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 v. NM form § The felicity function gives us insight on risk aversion April 2018 Frank Cowell: Consumption Uncertainty 36

What next? § Introduce a probability model § Formalise the concept of risk § This is handled in Risk April 2018 Frank Cowell: Consumption Uncertainty 37