Intertemporal Risk Attitude. Lecture 7. Kreps & Porteus Preference for Early or Late Resolution of Risk

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1 Intertemporal Risk Attitude Lecture 7 Kreps & Porteus Preference for Early or Late Resolution of Risk is an intrinsic preference for the timing of risk resolution is a general characteristic of recursive models can be explained by changing intertemporal risk aversion is, in difference to common belief, not necessary to disentangle Arrow Pratt risk aversion from intertemporal substitutability

2 Review of the General Representation Recall the general non-stationary representation (lecture 5): 1) Arbitrary Bernoulli utility For intertemporal aggregation define... g = (g t ) t {1,...,T } : sequence of functions weighing utility levels [G t,g t ] =range(g t ) and G t = G t G t normalization constants θ t = G t P T τ=t G τ and ϑ t = G t+1g t G t+1 G t G t...yielding the recursion relation for aggregate utility (R1): ũ t (x t,p t+1 ) = g 1 t [ ( θ t g t u(xt ) ) ( + θ t θ t+1 g t+1 M f t+1 (p t+1,ũ t+1 ) ) ] + θ t ϑ t θ t+1

3 Review of the General Representation 2) Certainty additive Bernoulli utility For g t = id aggregate utility equation simplifies to ũ t (x t,p t+1 ) = θ t u ca t (x t) + θ t θ t+1 M f t+1 (p t+1,ũ t+1 ) + θ t ϑ t θ t+1 θt 1 ũ t (x t, p t+1 ) = u ca t (x t ) + θt+1 1 f 1 θt 1 ũ t (x t, p t+1 ) = u ca t (x t ) + ft+1 1 E pt+1 ft+1 Further simplification t+1 E p t+1 f t+1 ũ t+1 + ϑ t ( θ 1 t+1ũt+1 Redefine ft (z) = f t (θ t z) and ũ t = θt 1 ũ t Normalize u ca t to range [0, ] implying ϑ t = 0 implies the representation ũ t(x t,p t+1 ) = u ca t (x t ) + M f t+1 (pt+1,ũ t+1) θ t+1 ) + ϑ t θ t+1 (R2) Note: 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.

4 Review of the General Representation Remarks with respect to representation ũ t(x t,p t+1 ) = u ca t (x t ) + M f t+1 (pt+1,ũ t+1 ) (R2) M f t (pt,ũ t ) = f t 1 E pt f t ( u ca t + M f t+1 (pt+1,ũ t+1 ) ) ( ) f t is defined on range(ũ t) = range(u t) θ t ft (z) = f t (θ t z) is concave/convex for representation (R2) iff f t gt 1 is concave/convex for arbitrary u for representation (R1) Define the intertemporal risk aversion measures RIRA * t(z) = f t (z) z and ft (z) AIRA* t(z) = f t analogous to AIRA t and RIRA t. (z). f t (z)

5 Timing Preference Definition: A decision maker prefers early resolution of risk... graphical 2 period intuition:...in period 1 for the fixed outcome x, iff 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 of Kreps & Porteus (1978):...in period t for the fixed outcome x t, iff 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).

6 Timing Preference Definition: A decision maker is timing indifferent... graphical 2 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).

7 Characterization Theorem Theorem 2: A decision maker prefers early [late] resolution of risk in period t for outcome x t, if and only if, in the preference representation (R1) the expression [ ] f t gt 1 θt g t u t (x t ) + θ t θt+1 1 g t+1 ft+1(z) 1 + θ t θt+1ϑ 1 t is convex [concave] in z f t (U t ). in the preference representation (R2) the expression [ ft u ca t (x t ) + ft+1 1 (z) ] is convex [concave] in z f t (range(ũ t)) = f t (U t ).

8 Characterization Theorem Intuition: A preference for early resolution of risk [ u ca t (x t ) + f t+1 1 (z) ] convex f t 1) Dismiss u ca (x t ) for the moment Recall f t f t+1 1 convex means f t more convex than f t+1 f t+1 more concave than f t Intertemporal risk aversion in period t + 1 higher than in period t Prefers early resolution of risk

9 Characterization Theorem Intuition: A preference for early resolution of risk [ u ca t (x t ) + f t+1 1 (z) ] convex f t 2) What about u ca t? If concavity of f t depends on welfare level, following can happen: Assume low expected welfare for period t + 1 high certain welfare in period t decreasing absolute intertemporal risk aversion in welfare Decision maker evaluates risk at high [low] welfare level if coin is flipped in period t [t + 1] Decision maker is less risk averse if evaluates risk in t Prefers evaluation/resolution of risk in period t

10 Timing Preference - Intuition A preference for early resolution of risk - Summary If intertemporal risk aversion changes over time - or over welfare and welfare changes over time - decision maker can have an intrinsic preference for early or late resolution of risk (uncertainty) The preference is intrinsic not instrumental! In fact: A decision make with an intrinsic timing preference gives up welfare in order to receive or postpone information that does not affect his payoffs. Intuition: In a recursive setting, risk aggregation takes place in the period when risk resolves If aggregation differs between periods then uncertainty is evaluated differently whenever risk over the same period resolves in different period

11 Timing Preference - Conclusion Possible perspectives Behavioral: People might exhibit preference for early or late resolution However, is it described well with a timing preference in the Kreps & Porteus (1978) sense? Rational or normative decision making: Indifference might be desirable as it implies that decision makers do not give up welfare to receive information which does not change outcomes Two indirect motivations in the literature to allow non-indifference: General recursive models with intrinsic timing preference allow to disentangle risk aversion from intertemporal substitutability to capture correlation aversion

12 Timing Indifference However: Only intertemporal risk aversion is needed to disentangle risk aversion from intertemporal substitutability to capture correlation aversion Not intrinsic timing preference! For a timing indifferent evaluation it will be sufficient to describe risk in the standard way: For a given temporal lottery p t P t, let p x t (X t ) describe the implied probability measure on consumption paths. (It is recursively obtained as a marginal distribution, somewhat complicated...) p x t is non-recursive, no probabilities over probabilities! p x t contains less information than p t (i.e. no timing information) 1

13 Timing Indifferent Representation Theorem 2: A sequence of binary relations ( t ) t {1,...,T } on (P t ) t {1,...,T } satisfies axioms A1-A5 and indifference to the timing of risk resolution, if and only if, there exist continuous functions u ca t : X IR, and a parameter ξ IR such that the function ũ t (x t ) = T τ=t uca t (x t τ) represents choice over certain consumption paths x t X t 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 }. Standard model: M exp0 (p x t,ũ t ) lim ξ 0 M expξ (p x t,ũ t ) =E p x t ũ t Adding certainty stationarity: u ca t (x t ) = β t 1 u(x t ) 1

14 Timing Indifference and IRA Moreover: Fix welfare of the worst outcome to zero in all periods (u ca t onto [0, ]) Fix a unit of welfare by fixing a unit outcome for some period (u ca t (x unit ) = 1) Then, the measures of relative and absolute intertemporal risk aversion are unique. The representation in theorem 2 is the special case of the representation (R2) where f t (z) = exp(ξz) for all t. Therefore it is RIRA t = ξ id and AIRA t = ξ. Thus, in the representation of (R2) absolute intertemporal risk aversion has to be constant over time and over welfare in order to yield timing indifference. (Recall intuition of preference for early resolution, makes sense?!). 1

15 References Kreps, D. M. & Porteus, E. L. (1978), Temporal resolution of uncertainty and dynamic choice theory, Econometrica 46(1),