Intertemporal Risk Aversion, Stationarity, and Discounting

Traeger, CES ifo 10 p. 1 Intertemporal Risk Aversion, Stationarity, and Discounting Christian Traeger Department of Agricultural & Resource Economics, UC Berkeley Introduce a more general preference representation & risk attitude Add axioms characterizing what to keep from standard model Derive implications for social discounting

Traeger, CES ifo 10 p. Motivation Motivation: Climate Change & Optimal Greenhouse Gas Abatement Crucial modeling determinants: Uncertainty Pure time preference (impatience)

Traeger, CES ifo 10 p. Motivation Motivation: Climate Change & Optimal Greenhouse Gas Abatement Crucial modeling determinants: Uncertainty Pure time preference (impatience) Example: Nordhaus DICE-007 model Changing rate of pure time preference: 1.5% 0% Social Cost of Carbon 1st commitment period: $40 $95 Paper points out that the discounted expected utility (standard) model contains an implicit assumption of risk neutrality

Traeger, CES ifo 10 p. 3 Outline Outline: Upfront summary Some related literature General representation under Certainty stationarity and additivity + von Neumann-Morgenstern axioms Intertemporal risk attitude: Axiomatic and functional characterization Add axioms of Risk stationarity + indifference to the timing of risk resolution Results on discounting Conclusions

Upfront Summary Traeger, CES ifo 10 p. 4

Traeger, CES ifo 10 p. 5 IES vs Risk, Related Literature Epstein & Zin (1989,E): Disentangle Arrow Pratt risk aversion from intertemporal elasticity of substitution

Traeger, CES ifo 10 p. 5 IES vs Risk, Related Literature Epstein & Zin (1989,E): Disentangle Arrow Pratt risk aversion from intertemporal elasticity of substitution The time and uncertainty additive (expected utility) standard model: curvature: aversion to int. subst. Time additive : t {}}{ u t (x t ) Expected utility: E u }{{} t (x t ) curvature: Arrow Pratt risk aversion } E t curvature:?! {}}{ u t (x t ) Remark: For stationary preferences: u t = β t u β = 1 1+ρ : utility discount factor ρ: pure rate of time preference

Traeger, CES ifo 10 p. 5 IES vs Risk, Related Literature Epstein & Zin (1989,E): Disentangle Arrow Pratt risk aversion from intertemporal elasticity of substitution The time and uncertainty additive (expected utility) standard model: curvature: aversion to int. subst. Time additive : t {}}{ u t (x t ) Expected utility: E u }{{} t (x t ) curvature: Arrow Pratt risk aversion } E t curvature:?! {}}{ u t (x t ) As opposed to the above authors: General preferences satisfying vnm axioms Introduce axiomatic definition of (multi-commodity) intertemporal risk aversion Analyze axiomatic consequences for discounting

Traeger, CES ifo 10 p. 6 Related Literature Koopmans (1960,E): The great wizard... Kreps & Porteus (1978,E): Axiomatic extension of Koopmans s (1960) recursive (non-time-additive) model to uncertainty Introduce the concept of temporal lotteries (which I employ) Introduce intrinsic preference for timing of uncertainty resolution This paper: Start from certainty additive framework Can preserve linearity over time But requires non-linear risk aggregation Discuss intertemporal risk aversion, stationarity, discounting Remark: Kreps & Porteus s (1978) timing preference can be explained by a change intertemporal risk attitude over time

Traeger, CES ifo 10 p. 7 Setup - The Choice Space Time: discrete, arbitrary finite planning horizon T X : Space of goods (outcomes) (connected compact metric) ( ): Set of Borel probability measures on space (Prohorov metric) Choice space: Example General 1 3 3 x 1 1 x 1 ( X P T 1 T 1 1 4 1 4 1 5 3 5 P T T x 1 x x 3 x 4 x 5 ) (X) (recursive definition) : Temporal Lottery (rather than (X X)) : Choice Spaces in Periods T 1, T : Preferences in Periods T 1, T

Traeger, CES ifo 10 p. 8 Setup - The Choice Space Certain Choices: x = (x t,x t+1,...,x T ): Certain consumption path from t to T X t : Space of all consumption paths from t to T

Traeger, CES ifo 10 p. 8 Setup - The Choice Space Certain Choices: x = (x t,x t+1,...,x T ): Certain consumption path from t to T X t : Space of all consumption paths from t to 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 τ th period, in which it yields outcome x. Notation: For a (compact metric) space Y define C 0 (Y): Set of continuous functions Y IR

Traeger, CES ifo 10 p. 9 Setup - Axioms Axioms: 1-A3 (vnm axioms) Standard, applied for t on P t with t {1,...,T} (weak order) t is transitive and complete (independence) p,q,r P t : p t q λp+(1 λ)r t λq +(1 λ)r λ [0,1] (continuity) p P t : The sets {q P t : q t p} and {q P t : p t q} are closed in P t

Traeger, CES ifo 10 p. 9 Setup - Axioms Axioms: 1-A3 (vnm axioms) Standard, applied 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) Minor modification for T = (e.g. Thomson condition)

Traeger, CES ifo 10 p. 10 Certainty Stationarity Definition: A decision maker s preferences are called certainty stationary... graphical period illustration:... iff for all x,x,x,x X x x x 1 x x x general definition:...iff for all x,x X and x X: (x,x) 1 (x,x) x x

Traeger, CES ifo 10 p. 11 Representation - Definition Uncertainty Aggregation Rule For f : IR IR continuous and strictly increasing and some compact metric space Y define M f : (Y) C 0 (Y) IR M f (p,u) = f 1[ Y f u dp] It satisfies M f (y,u) = u(y) y Y (degenerate lottery) M f (p,u) = E p u for f = id (expected value)

Traeger, CES ifo 10 p. 1 Representation - The Certainty Stationary Representation Theorem 1: A preference relation ( t ) t {1,...,T} on (P t ) t {1,...,T} satisfies the vnm axioms A1-3, certainty separability A4, and time consistency A5, if and only if, there exists β IR ++, a continuous function u : X IR, and strictly increasing and continuous functions f t : IR IR for t {1,...,T} such that defining ũ T (x T ) = u(x T ) and recursively ũ t 1 (x t 1,p t ) = u(x t 1 )+βm f t (p t,ũ t ) preferences t are represented by M f t (p t,ũ t ) for all t {1,...,T}.

Traeger, CES ifo 10 p. 13 Intertemporal Risk Aversion formal Let x,x be two consumption paths of length T. Example, T = 4: x =(,,, ) x =(,,, ) Let x x denote a strict preference for x over x. Let denote indifference. Define for x and x the consumption paths x high : collects better outcomes of every period x low : collects inferior outcomes of every period

Traeger, CES ifo 10 p. 13 Intertemporal Risk Aversion formal Let x,x be two consumption paths of length T. Example, T = 4: x =(,,, ) x high =(,,, ) x =(,,, ) x low =(,,, ) Let x x denote a strict preference for x over x. Let denote indifference. Define for x and x the consumption paths x high : collects better outcomes of every period x low : collects inferior outcomes of every period

Traeger, CES ifo 10 p. 14 A Question of Preference general Assume you d be indifferent between (,,, ) (,,, )

Traeger, CES ifo 10 p. 14 A Question of Preference general Assume you d be indifferent between (,,, ) (,,, ) If not, please mentally adjust the corners of the mouth of the red frowny to reach indifference. What preference do you have in the following choice? (,,, ) certain path vs. 1 1 (,,, ) (,,, ) coin toss lottery

Traeger, CES ifo 10 p. 14 A Question of Preference general Assume you d be indifferent between (,,, ) (,,, ) If not, please mentally adjust the corners of the mouth of the red frowny to reach indifference. What preference do you have in the following choice? (,,, ) 1 1 (,,, ) (,,, ) certain path coin toss lottery STANDARD MODEL E t βt u(x t )

Traeger, CES ifo 10 p. 14 A Question of Preference general Assume you d be indifferent between (,,, ) (,,, ) If not, please mentally adjust the corners of the mouth of the red frowny to reach indifference. What preference do you have in the following choice? (,,, ) 1 1 (,,, ) (,,, ) certain path coin toss lottery INTERTEMPORAL RISK AVERSE DM

Traeger, CES ifo 10 p. 15 Intertemporal Risk Aversion - Functional Characterization Theorem : In the representation of theorem 1, a decision maker is weakly intertemporal risk averse in period t < T, if and only if, f t is concave.

Traeger, CES ifo 10 p. 15 Intertemporal Risk Aversion - Functional Characterization Theorem : In the representation of theorem 1, a decision maker is intertemporal risk seeking in period t < T, if and only if, f t is convex.

Traeger, CES ifo 10 p. 15 Intertemporal Risk Aversion - Functional Characterization Theorem : In the representation of theorem 1, a decision maker is intertemporal risk neutral in period t < T, if and only if, f t is linear. Time additive expected utility standard model

Traeger, CES ifo 10 p. 15 Intertemporal Risk Aversion - Functional Characterization Theorem : In the representation of theorem 1, a decision maker is strictly intertemporal risk averse in period t < T, if and only if, f t is strictly concave. Interpretation: f t measures risk aversion with respect to utility gains and losses M f t (p t,ũ t ) = f t 1 ( E pt f t [ u(xt )+βm f t+1 (p t+1,ũ t+1 ) ]) Define the measure of absolute intertemporal risk aversion as AIRA t (z) = d dz f t(z). d dz f t(z)

Traeger, CES ifo 10 p. 16 Risk Stationarity Definition: A decision maker s preferences are called risk stationary... graphical period illustration:... iff for all x,x,x,x X 1 1 x x x x x 1 x 1 1 x x x general definition:...iff for all t {1,...,T 1} and x X: 1 (x,x)+ 1 (x,x) t (x,x) 1 x+ 1 x t+1 x for all x,x,x X t+1.

Traeger, CES ifo 10 p. 17 Timing Preference Definition: A decision maker is timing indifferent... graphical period intuition:...iff for all outcomes x X and for all x,x X,λ [0,1] λ x x λ x 1 x 1 λ x x 1 λ x. Uncertainty resolves ( biased coin toss takes place ) in first versus second period general definition:...iff for all t {1,...T 1} and all x t X, and for all p t+1,p t+1 P t+1 and all λ [0,1] holds λ(x t,p t+1 )+(1 λ)(x t,p t+1) t (x t,λp t+1 +(1 λ)p t+1).

Traeger, CES ifo 10 p. 18 Stationary Representation int Define p x t (X t ) as the non-recursive lottery obtained from p t by integrating out the information on timing of risk resolution

Traeger, CES ifo 10 p. 18 Stationary Representation int Define p x t (X t ) as the non-recursive lottery obtained from p t Theorem 3: A sequence of binary relations ( t ) t {1,...,T} on (P t ) t {1,...,T} satisfies i) vnm axioms, additive separability, time consistency ii) indifference to the timing of risk resolution iii) certainty stationarity iv) strict intertemporal risk aversion

Traeger, CES ifo 10 p. 18 Stationary Representation int Define p x t (X t ) as the non-recursive lottery obtained from p t Theorem 3: A sequence of binary relations ( t ) t {1,...,T} on (P t ) t {1,...,T} satisfies i) vnm axioms, additive separability, time consistency ii) indifference to the timing of risk resolution iii) risk stationarity iv) strict intertemporal risk aversion if and only if, there exists a continuous function u : X IR, a discount factor β = 1, and ξ < 0 such that the function ũ t (x t ) = T τ=t u(xt τ) represent choice over certain consumption paths and M expξ (p x t,ũ t )= 1 ξ ln[ dp x t exp[ξ ũ t (x t ) ] ] represents choice over lotteries in period t {1,...,T}.

Traeger, CES ifo 10 p. 18 Stationary Representation int Define p x t (X t ) as the non-recursive lottery obtained from p t Theorem 3: A sequence of binary relations ( t ) t {1,...,T} on (P t ) t {1,...,T} satisfies... all axioms including strict intertemporal risk aversion... if and only if, there exists a continuous function u : X IR, a discount factor β = 1, and ξ < 0 such that the function ũ t (x t ) = T τ=t u(xt τ) represent choice over certain consumption paths and M expξ (p x t,ũ t )= 1 ξ ln[ dp x t exp[ξ ũ t (x t ) ] ] represents choice over lotteries in period t {1,...,T}. Note: Absolute intertemporal risk aversion is AIRA t = ξ.

Traeger, CES ifo 10 p. 18 Stationary Representation int Define p x t (X t ) as the non-recursive lottery obtained from p t Theorem 3: A sequence of binary relations ( t ) t {1,...,T} on (P t ) t {1,...,T} satisfies... all axioms including strict intertemporal risk aversion... if and only if, there exists a continuous function u : X IR, a discount factor β = 1, and ξ < 0 such that the function ũ t (x t ) = T τ=t u(xt τ) represent choice over certain consumption paths and M expξ (p x t,ũ t )= 1 ξ ln[ dp x t exp[ξ ũ t (x t ) ] ] represents choice over lotteries in period t {1,...,T}. Note: Absolute intertemporal risk aversion is AIRA t = ξ. Note: M exp0 (p x t,ũ t ) lim ξ 0 M expξ (p x t,ũ t ) = E p x t T τ=t βτ 1 u(x t τ)

Traeger, CES ifo 10 p. 19 Intuition Timing indifference: Compares risk resolving in different periods as viewed from the present Makes intertemporal risk aversion constant over time, when measured w.r.t. present value utility gains and losses Risk Stationarity Compares risky scenarios moving from period to period Makes intertemporal risk aversion constant over time, when measured w.r.t. current value utility gains and losses Intertemporal risk aversion can only be constant over time in present and in current value, if it is zero intertemporal risk neutral standard model both values coincide nontrivially no pure time preference

Traeger, CES ifo 10 p. 0 Discounting for Reasons of Increasing Uncertainty An intertemporal risk averse decision maker discounts for reasons of uncertainty: Consider a representation in the sense of theorem 3 with β = 1 Assume for simplicity that risk is independent between periods: p x t = p 1 p... p T where p t (X) describes risk over x t Then the evaluation simplifies to M expξ (p x t,ũ t ) = M expξ (p 1,u)+M expξ (p,u)+...+m expξ (p T,u) ( ) Let expected per period utility be constant over time: Eu(x t ) = ū for all t {1,...,T} Let uncertainty over utility increase over time: Strictly more weight on the tails of the induced utility lottery Then the summands in equation ( ) decrease over time, i.e. M expξ (p t+1,u) < M expξ (p t,u) conclusions evidence

Traeger, CES ifo 10 p. 1 Back to Climate Change Relating these findings to climate change evaluation implies: The Stern review argues for a zero rate of pure time preference based on ethical arguments These ethical arguments are critizised as a British utilitarian perspective We found: also simple constraints on decision making under uncertainty can lead to a zero rate of pure time preference Pay more attention to the long-run: The more we know about climate change, the more attention we should also pay to its long-term effects Pay more attention to reducing risk Related empirics

Traeger, CES ifo 10 p. Conclusions Conclusions: intertemporal risk aversion: is supported by standard axioms and captures risk aversion with respect to utility gains and losses

Traeger, CES ifo 10 p. Conclusions Conclusions: intertemporal risk aversion: is supported by standard axioms and captures risk aversion with respect to utility gains and losses risk stationarity + indifference to the timing of risk resolution have no bite in the int. risk neutral standard model restrict the pure rate of time preference to zero in the intertemporal risk averse model

Traeger, CES ifo 10 p. 41 References Epstein, L. G. & Zin, S. E. (1989), Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica 57(4), 937 69. Epstein, L. G. & Zin, S. E. (1991), Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis, Journal of Political Economy 99(), 63 86. Koopmans, T. C. (1960), Stationary ordinal utility and impatience, Econometrica 8(), 87 309. Kreps, D. M. & Porteus, E. L. (1978), Temporal resolution of uncertainty and dynamic choice theory, Econometrica 46(1), 185 00.