In this regard, Uzawa (1968), Epstein (1983), and Shi and Epstein (1993) develop representations with time-varying discount factors that can also show time-varying aversion. As a reminder of the paper, time preferences refer to the structure of W (movement of discount factors) that includes aversion to time variability. Aversion to time variability is captured by the agent choosing discount factors to minimize the weighted sum of expected utility time indices.
In fact, the resistance to time variability is independent of the structure u(.), which can be concave or convex.17. We also confirm that the representation (7) satisfies the “local” (or “first-order”) resistance to time variability under steady action. However, without axiomatic reasoning, resistance to time variability can be identified on utility indices that depend on past consumption.
To derive a pure attitude toward uncertainty, we must neutralize the effects of time variability aversion on each event. Clearly, (14) captures time variability aversion more strongly because it is applied on a path-by-path basis. Furthermore, the "first-order" property of time variability aversion generates an asymmetric attitude toward gains and losses.
The main departure of our model from these theories is the introduction of an aversion to temporal variability with the "first-order" property. Under the appropriate parameters, both formulas include an aversion to “first-order” time variability; Shalev (1997) also captures the gain/loss asymmetry over time. However, without a formal derivation, we can extend resistance to time variability (i.e., the multiple discount factor model) to atemporal utility indices that depend on prior consumption levels.
Comparison among (9), (14), and Recursive Utility
The term u(ct− xt) represents an idea similar to that of Loewenstein and Thaler (1989), where a reference point is based on the previous level of consumption. However, unless we use loss aversion inu(ct−xt), (26) does not satisfy “first order” aversion to time variability, because increasing u is differentiable almost everywhere. This procedure only gains rationality if we introduce dynamic consistency where atemporal risk preferences from the last period become atemporal risk preferences from earlier periods.
However, when we evaluate a consumption sequence without uncertainty, intertemporal substitution necessarily depends not only on the attitude toward time variability, but also on the attitude toward risk. Then a binary relation of 'more aversion to time variability' is allowed only for pairs of preference relations that embody the same ranking of risk for one period. Moreover, in our model the ranking of uncertain future prospects depends on current consumption.
The agent first considers aversion to event-to-event temporal variability and then aggregates utilities under a subjective prior. In our model, the agent is informed about a certain event tree as an information structure. Therefore, the agent would prefer to consider consumption sequences as possibilities and forms a probability representation over these possibilities ex-ante at time t.
According to the above characteristics of both models, the model developed by Epstein and Zin (1989) is suitable for the situation in which the agent is faced with the possibility of early resolution of uncertainty. Furthermore, their model may include unexpected utility models, while our model allows only the expected utility model (although our model may include more prior models if we modify A11U-IA). Furthermore, the recursive utility model can evaluate objective time lotteries or state-space lotteries,35 while our model can only evaluate state-space lotteries.
In contrast, our model is suitable for the situation where an agent is informed about a particular information structure. This operation generates dependence on the ranking of uncertain future prospects on today's consumption and amplifies the effects of gain/loss asymmetry over time. Finally, our model provides a direct interpretation of time preferences through the discount factor model.
Comparison between (14) and Epstein and Schneider (2003)
We axiomatized the behavioral notion of aversion to temporal variability and derived a representation. Our model reaches the following conclusions: (i) Aversion to time variability is captured as a separate relationship from risk aversion through a multiple discounting factor model, and the axiomatic derivation provides a clear picture of the agent's motives. The representation is also modest; (ii) Given dynamic consistency and historical independence, the "first order" property of time variability aversion necessarily implies gain/loss asymmetry in discount factors; and (iii) in uncertainty, consideration of the intertemporal trade-off occurs before consideration of subjective risk.
According to this operation, discount factors depend on tomorrow's states and exhibit gain/loss asymmetry with respect to states by taking today's utility as an endogenous reference point; aversion to time variability effectively increases risk aversion over countries. We illustrate this with the following example: Suppose there are three periods and four states. Clearly, at time 0 the agent chooses consumption B, but after the agent finds that the agent is in (state 1, state 2) at time 1, the agent chooses consumption A.
More formally, in a neoclassical model of expected utility with a subjective prior, inconsistency emerges over time unless the agent's preferences are temporally additive. Since (9) is not time additive, the dynamically consistent utility function (9) for each state does not assert dynamic consistency. Then, from A1U to A6U as well as the results in the above paragraph, the preference relation on H depends only on which state act is assigned to each state ω ∈ Ω (in particular, hω = hω0 for all ω, ω0 ∈ Ω) ; thus the representation refers only to the specific lottery assigned to each state act.
This result shows that the preference relations in L at (1, ω0) can be represented by U(1,ω0), which is a positive affine transformation of U(0,ω). Repeating the same argument over all events in F1, A4U-CI is satisfied under º(1,ω0) for all. Repeating the same argument over all events in F1, A6U-TVA is satisfied under º(1,ω0) for all ω0 ∈Ω.
Repeatedly applying the same construction from time 2 to time T, under A7U-F, A8U-IHA, A9U-DC and A10U-SMS, we conclude that preference relations on H can be represented by V(t,ω), i.e. . at all identical to Vt of proposition 1 (t,ω)∈ T ×Ω, where a set of discount factors depends only on time.¥. In addition, both a set of efficient selection of discount factors for fω0 and a set for gω0 contain αt+1. 29) Furthermore, F(t,ω) is unique up to a positive affine transformation. Since V(t,ω)V (υ) is unique up to a positive affine transformation, given (iv) and (vii) of Lemma C.5, µ is uniquely determined.
Note that under fixed discount factors, V(0,ω) in f ∈ D is a weighted summation of U(f(t, ω)), where the weights are based on the subjective prior and discount factors. Finally, for Dt, the representation is equivalent to a state-by-state application of multiple discount factors of V(t,ω) weighted by a subjective priority.
