Prerequisites Almost essential: Consumption and Uncertainty Probability Distributions RISK MICROECONOMICS Principles and Analysis Frank Cowell July 2017 1
Risk and uncertainty In dealing with uncertainty a lot can be done without introducing probability Now we introduce a specific probability model This could be some kind of exogenous mechanism Could just involve individual s perceptions Facilitates discussion of risk Introduces new way of modelling preferences July 2017 2
Probability What type of probability model? A number of reasonable versions: public observable public announced private objective private subjective How to represent probabilities in economic models? a few standard forms Lottery government policy? coin flip emerges from structure of preferences see the presentation Probability Distributions July 2017 3
Overview Risk Shape of the u-function and attitude to risk Risk comparisons Special Cases Lotteries July 2017 4
Risk aversion and the function u Given a probability model can discuss risk attitudes in terms of gambles Do this in terms of properties of felicity or cardinal utility function u Scale and origin of u are irrelevant But the curvature of u is important We can capture this in more than one way we will investigate the standard approaches then introduce two useful definitions July 2017 5
Risk aversion and choice Imagine a simple gamble Two payoffs with known probabilities: x RED with probability π RED x BLUE with probability π BLUE Expected value Ex = π RED x RED + π BLUE x BLUE A fair gamble : stake money is exactly Ex Would the person accept all fair gambles? Compare Eu(x) with u(ex) depends on shape of u July 2017 6
Attitudes to risk υ Risk-neutral u(x) Shape of u associated with risk attitude Neutrality: will just accept a fair gamble Aversion: will reject some better-than-fair gambles x BLUE Ex x RED x Loving: will accept some unfair gambles υ Risk-averse υ Risk-loving u(x) u(x) x BLUE E x x RED x x BLUE E x x RED x July 2017 7
Risk premium and risk aversion x BLUE A given income prospect Certainty equivalent income Slope gives probability ratio Mean income The risk premium P P 0 π RED π BLUE Risk premium: Amount that you would sacrifice to eliminate the risk Useful additional way of characterising risk attitude O ξ Ex x RED example July 2017 8
An example Two-state model Subjective probabilities (0.25, 0.75) Single-commodity payoff in each case July 2017 9
Risk premium: an example u(ex) Eu(x) u u(x RED ) u(ex) Eu(x) u(x BLUE ) amount you would sacrifice to eliminate the risk Utility values of two payoffs Expected payoff & U of expected payoff Expected utility and certainty-equivalent The risk premium again u(x) x BLUE ξ ξ Ex E x x RED x July 2017 10
Change the u-function u The utility function and risk premium as before Now let the utility function become flatter u(x RED ) u(x BLUE ) u(x BLUE ) Making the u-function less curved reduces the risk premium and vice versa More of this later x BLUE ξ ξ Ex x RED x July 2017 11
An index of risk aversion? Risk aversion associated with shape of u second derivative or curvature Could we summarise it in a simple index or measure? if so, we can characterise one person as more/less risk averse than another There is more than one way of doing this July 2017 12
Absolute risk aversion Definition of absolute risk aversion for scalar payoffs u xx (x) α(x) := ux (x) For risk-averse individuals α is positive For risk-neutral individuals α is zero Definition ensures that α is independent of the scale and the origin of u Multiply u by a positive constant add any other constant α remains unchanged July 2017 13
Relative risk aversion Definition of relative risk aversion for scalar payoffs: u xx (x) ρ(x) := x ux (x) Some basic properties of ρ are similar to those of α: positive for risk-averse individuals zero for risk-neutrality independent of the scale and the origin of u Obvious relation with absolute risk aversion: ρ(x) = x α(x) July 2017 14
Concavity and risk aversion utility υ lower risk aversion higher risk aversion û = φ(u) u(x) û(x) Draw the function u again Change preferences: φ is a concave function of u Risk aversion increases More concave u implies higher risk aversion payoff x now to the interpretations July 2017 15
Interpreting α and ρ Think of α as a measure of the concavity of u Risk premium is approximately ½ α(x) var(x) Likewise think of ρ as the elasticity of marginal u In both interpretations an increase in the curvature of u increases measured risk aversion Suppose risk preferences change u is replaced by û, where û = φ(u) and φ is strictly concave Then both α(x) and ρ(x) increase for all x An increase in α or ρ also associated with increased curvature of IC July 2017 16
Another look at indifference curves u and π determine the shape of IC Alf and Bill differ in risk aversion Alf, Charlie differ in subj probability Alf x BLUE Alf Same πs but different us x BLUE Same us but different πs Bill Charlie O x RED O x RED July 2017 17
Overview Risk CARA and CRRA Risk comparisons Special Cases Lotteries July 2017 18
Special utility functions? Sometimes convenient to use special assumptions about risk Constant ARA Constant RRA By definition ρ(x) = x α(x) Differentiate w.r.t. x: dρ(x) dα(x) = α(x) + x dx dx So one could have, for example: constant ARA and increasing RRA constant RRA and decreasing ARA or, of course, decreasing ARA and increasing RRA July 2017 19
Special case 1: CARA We take a special case of risk preferences Assume that α(x) = α for all x Felicity function must take the form 1 u(x) := e α αx Constant Absolute Risk Aversion This induces a distinctive pattern of indifference curves July 2017 20
Constant Absolute RA x BLUE Case where α = ½ Slope of IC is same along 45 ray (standard vnm) For CARA slope of IC is same along any 45 line O x RED July 2017 21
CARA: changing α x BLUE Case where α = ½ (as before) Change ARA to α = 2 O x RED July 2017 22
Special case 2: CRRA Another important special case of risk preferences Assume that ρ(x) = ρ for all ρ Felicity function must take the form 1 u(x) := x 1 ρ 1 ρ Constant Relative Risk Aversion Again induces a distinctive pattern of indifference curves July 2017 23
Constant Relative RA x BLUE Case where ρ = 2 Slope of IC is same along 45 ray (standard vnm) For CRRA slope of IC is same along any ray ICs are homothetic O x RED July 2017 24
CRRA: changing ρ x BLUE Case where ρ = 2 (as before) Change RRA to ρ = ½ O x RED July 2017 25
CRRA: changing π x BLUE Case where ρ = 2 (as before) Increase probability of state RED O x RED July 2017 26
Overview Risk Probability distributions as objects of choice Risk comparisons Special Cases Lotteries July 2017 27
Lotteries Consider lottery as a particular type of uncertain prospect Take an explicit probability model Assume a finite number ϖ of states-of-the-world Associated with each state ω are: A known payoff x ω, A known probability π ω 0 The lottery is the probability distribution over the prizes x ω, ω=1,2,,ϖ The probability distribution is just the vector π:= (π 1,, π 2,,, π ϖ ) Of course, π 1 + π 2 + +π ϖ = 1 What about preferences? July 2017 28
The probability diagram: #Ω=2 Probability of state RED π BLUE (0,1) Probability of state BLUE Cases of perfect certainty Cases where 0 < π < 1 The case (0.75, 0.25) π RED +π BLUE = 1 (0, 0.25) Only points on the purple line make sense This is an 1-dimensional example of a simplex (0.75, 0) (1,0) π RED July 2017 29
The probability diagram: #Ω=3 π BLUE (0,0,1) Third axis corresponds to probability of state GREEN There are now three cases of perfect certainty Cases where 0 < π < 1 The case (0.5, 0.25, 0.25) π RED + π GREEN + π BLUE = 1 (0, 0, 0.25) (0, 0.25, 0) (0,1,0) Only points on the purple triangle make sense This is a 2-dimensional example of a simplex 0 (0.5, 0, 0) (1,0,0) π RED July 2017 30
Probability diagram #Ω=3 (contd.) (0,0,1) All the essential information is in the simplex Display as a plane diagram The equi-probable case The case (0.5, 0.25, 0.25) (1/3,1/3,1/3) (0.5, 0.25, 0.25). (1,0,0) (0,1,0) July 2017 31
Preferences over lotteries Take the probability distributions as objects of choice Imagine a set of lotteries π, π', π", Each lottery π has same payoff structure State-of-the-world ω has payoff x ω and probability π ω or π ω ' or π ω " depending on which lottery We need an alternative axiomatisation for choice amongst lotteries π July 2017 32
Axioms on preferences Transitivity over lotteries If π π' and π' π" then π π" Independence of lotteries If π π' and λ (0,1) then λπ + [1 λ]π" λπ' + [1 λ] π" Continuity over lotteries If π π' π" then there are numbers λ and µ such that λπ + [1 λ]π" π' π' µπ + [1 µ]π" July 2017 33
Basic result Take the axioms transitivity, independence, continuity Imply that preferences must be representable in the form of a von Neumann-Morgenstern utility function: π ω u(x ω ) ω Ω or equivalently: π ω υ ω ω Ω where υ ω := u(x ω ) So we can also see the EU model as a weighted sum of πs July 2017 34
π-indifference curves (0,0,1) Indifference curves over probabilities Effect of an increase in the size of υ BLUE. (1,0,0) (0,1,0) July 2017 35
What next? Simple trading model under uncertainty Consumer choice under uncertainty Models of asset holding Models of insurance This is in the presentation Risk Taking July 2017 36