Intertemporal Risk Attitude Lecture 5

Transcription

1 Intertemporal Risk Attitude Lecture 5 Lectures 1-4 familiarized in a simplified 2 period setting with disentangling risk aversion and int substitutability the concept of intertemporal risk aversion Lecture 5 introduces a general model a general characterization of intertemporal risk aversion Lectures 6-8 will discuss axiomatic simplifications of the model a preference for the timing of resolution resolution discounting Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

2 Setup - The Choice Space setup Time: discrete with arbitrary finite planning horizon T X: (connected compact metric) space of goods ( ): Set of Borel probability measures on space (Prohorov metric) X T = X: Degenerate choices in period T P T = ( X T ) = (X): Uncertain choices in period T Recursively define X t 1 = X ( X t ) for all t {2,,T } (product metric) x t X t : degererate period t choice objects P t = ( X t ) for all t {1,,T } p t P t : general uncertain period t choice objects (lotteries) Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

3 Setup - The Choice Space setup Certain Choices: X t = X T t+1 denotes the (T t+1) fold Cartesian product x X t,x = (x t,x t+1,,x T ) = (x t,x t+1,,x T ): certain consumption path from period t to T Note: X t X t P t Notation: Given x X t define (x τ,x) = (x t,,x τ 1,x,x τ+1,,x T ) X t : consumption path coinciding with x in all but the i th period, in which it yields outcome x Preferences: t : Preference relation in period t on choice space P t Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

4 Axioms setup Axioms: 1-A3 (VNM axioms) As before, now for t on P t with t {1,,T } A4 (certainty separability) For all x,x X 1, x,x X and τ {1,,T } it holds that i) (x τ,x) 1 (x τ,x) (x τ,x ) 1 (x τ,x ) ii) If T = 2 additionally with x t,x t,x t X, t {1, 2}: (x 1,x 2 ) 1 (x 1,x 2) (x 1,x 2 ) 1 (x 1,x 2) (x 1,x 2) 1 (x 1,x 2) A5 (time consistency) For all t {1,,T 1}, x t X and p t+1,p t+1 P t+1 : (x t,p t+1 ) t (x t,p t+1) p t+1 t+1 p t+1 A0 (non-degeneracy) For all t {1,,T }: x X 1 and x X such that (x t,x) 1 x Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

5 Representation - Employed Concepts I Given a preference relation t on P t define: Induced preferences t on X by defining for all x,x X: x t x (x,x t+1,,x T ) t (x,x t+1,,x T ) x t+1,,x T X Because of certainty separability t x is a complete order on X The set of Bernoulli utility functions in period t: B t = {u t C 0 (X) : x t x u t (x) u t (x ) x,x X} Express preference over certain consumption in period t Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

6 Representation - Employed Concepts II Intertemporal Aggregation Rule, define g = (g t ) t {1,,T } : sequence of functions with g t C 0 (U t ) Each g t : U t IR weighs utility levels in U t IR [G t,g t ] =range(g t ) and G t = G t G t An intertemporal aggregation rule is a functional N g t : U t U t+1 IR with N g [ ] t (, ) = gt 1 θt g t ( ) + θ t θt+1 1 g t+1 ( ) + θ t θt+1 1 ϑ t where the normalization constants are defined as θ t = and ϑ t = G t+1g t G t+1 G t G t G t P T τ=t G τ Remark: G t = 0 ϑ t = 0 In stationary setting θ t relates to discount factor Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

7 Representation - Theorem Let ( t ) t {1,,T } on (P t ) t {1,,T } satisfy axiom A0 and let u (u t ) t {1,,T } satisfy u t B t (given Bernoulli utility!) Theorem 1: The sequence of preference relations ( t ) t {1,,T } satisfies A1-A5, if and only if, there exist strictly increasing and continuous functions f t : U t IR and g t : U t IR for t {1,,T } such that with defining recursively the aggregate welfare functions ũ t : Xt IR by ũ T (x T ) = u T (x T ) and for t {1,,T 1} ũ t (x t,p t+1 ) = N g ( t ut (x t ), M f t+1 (p t+1,ũ t+1 ) ) it holds for all t {1,,T } that p t t p t M f t (p t,ũ t ) M f t (p t,ũ t ) p t,p t P t Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

8 Representation - Gauges Use freedom to choose Bernoulli utility to simplify the representation ũ t (x t,p t+1 ) = N g ( t ut (x t ), M f t+1 (p t+1,ũ t+1 ) ) : Kreps-Porteus-form (f t = id) (von Neumann-Morgenstern u t ): ũ t (x t,p t+1 ) = N g ( ) t u vnm t (x t ), E pt+1 ũ t+1 Certainty-additive-form (g t = id) (certainty additive u t ): Normalizing Bernoulli utility to range [0, U t ] and absorbing θ t into f t ũ t (x t,p t+1 ) = u ca t (x t ) + M f t+1 (p t+1,ũ t+1 ) And only in a one commodity world: Epstein-Zin-form (u t = id) : ũ t (x t,p t+1 ) = N g t ( x t, M f t+1 (p t+1,ũ t+1 ) ) Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

9 Recall: Risk Aversion & Intertemporal Substitutability N g t x t, M ft+1 (p t+1, ũ t+ Analysis of Epstein-Zin-form M f t,f t : describe atemporal (Arrow-Pratt) risk aversion N g t,g t : describe intertemporal substitutability BUT: f t and g t depend on good under observation measure scale of good level of other consumption the choice of Bernoulli utility functions in Theorem 1 HOWEVER: The functions f t gt 1 are invariant under different choices of Bernoulli utility, and uniquely determined by the preferences t up to affine indeterminacies of f t and g t Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p

10 Intertemporal Risk Aversion - Axiom For IRA Axiom in general setting: Define for x,x X t consumption paths x high (x,x ),x low (x,x ) X t, (x high (x,x )) τ = { x τ if x τ τ x τ x τ if x τ τ x τ, (x low (x,x )) τ = { x τ if x τ τ x τ x τ if x τ τ x τ (1) for τ {t,,t } x high (x,x ): collects better outcomes of every period x low (x,x ): collects inferior outcomes of every period Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

11 Intertemporal Risk Aversion - Axiom A decision maker is called strictly intertemporal risk averse in period t < T iff for all x,x X t satisfying x t x τ {t,,t } sth x τ τ x τ it holds that x x high (x,x ) x low (x,x ) Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

12 Intertemporal Risk Aversion - Functional Characterization Theorem 2: In the representation of theorem 1, a decision maker is strictly intertemporal risk averse in period t < T, if and only if, f t gt 1 is strictly concave Define the measures of relative intertemporal risk aversion as: RIRA t (z) = f t gt 1 (z) f t g 1 t (z) z, and the measures of absolute intertemporal risk aversion as: AIRA t (z) = f t gt 1 (z) f t g 1 t (z) For a particular point in consumption space define AIRA t [ x t ] = AIRA t (z) z=gt ũ t ( x t ) Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

13 Intertemporal Risk Aversion - Functional Characterization Let the preferences = ( t ) t {1,,T } be represented in the sense of theorem 1 with twice differentiable functions f t g 1 t Theorem 2: a) For all t {1,,T } choose x t X and fix g t u t ( x t ) = 0 Then, the risk measures RIRA t are determined uniquely by the preferences and the point x t in consumption space They are independent of the choice of the Bernoulli utility functions b) For some arbitrary period t {1,,T } choose two outcomes x t, `x t X with x t t `x t, and fix g t u t ( x t ) g t u t (`x t ) = 1 Then, the risk measures AIRA t are determined uniquely by the preferences and the point x t in consumption space They are independent of the choice of the Bernoulli utility functions Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

14 Intertemporal Risk Aversion - Functional Characterization Interpretation: The risk measures RIRA t and AIRA t depend on a welfare measure scale, where welfare is determined by certain intertemporal trade-offs The intertemporal trade-off welfare scale is only determined up to affine transformations Thus, that the measures (recall discussion on measure scale dependence!) the measures RIRA t are uniquely defined if the zero welfare level is fixed the measures AIRA t are uniquely defined if the unit of welfare is fixed This intertemporal trade-off welfare measure scale for IRA is given by g t u t (X) = u ca t (X) Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

15 Intertemporal Risk Aversion - Functional Characterization The certainty additive form is particular convenient for interpretation: u ca t measures welfare in period t (determined up to affine trafos) f t measures intertemporal risk aversion in period t as risk aversion with respect to welfare gains and losses Fixing u ca t (x 0 t) = 0 and u ca t (x 1 t ) = 1 fixes measure scale uniquely Summarizing functional characterization of IRA: Intertemporal Risk Aversion is measured with respect to a measure scale defined up to affine transformations by preferences Therefore: The measures RIRA t and AIRA t are independent of the measure scale for goods, and the good under observation However: For a numerical measure of Intertemporal Risk Aversion unit and/or zero level of the measure scale have to be fixed Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1

16 Good and measure scale dependence of risk measures back Space of goods coconut quality Φ 2 3 7? 4 Φ Φ Φ 1 Coordinate space (space of numbers) IR n coconut quantity Robinson s preferences Function f on IR Is there a risk attitude measure that is independent of coordinate system Φ and the particular curvature good? describes risk attitude Ie that describes attitude with respect to risk rather than Φ: Coordinate with respect systemto coconuts, arbitrary litchisif differs or coordinate money? for litchis Φ 3 arbitrary litchi quantity Lecture 5 on Intertemporal Risk Attitude, Christian Traeger, Berkeley 07 p 1