Strotz Meets Allais: Diminishing Impatience and the Certainty Effect: Comment
The American Economic Review
The main theorem of Yoram Halevy (2008, Theorem 1, p. 1150) states that a rank-dependent expected utility maximizer exhibits diminishing impatience (i.e., quasi-hyperbolic discounting) if and only if the elasticity of the probability weighting function of the decision maker is increasing. An implication of such a result is the equivalence between diminishing impatience and the common ratio effect (CRE). Claim 1 of this comment shows that the "only if " part of Theorem 1 in Halevy (2008) is
... levy (2008) is false. One might wonder whether, even though the theorem is false, the main implication relating diminishing impatience with the CRE is true. Claim 2 shows that diminishing impatience does not imply the CRE. Given the correction, two natural questions arise when considering a joint model, such as Halevy's (2008), that relates decision making under risk and intertemporal decision making: (i) "Is there a behavioral property in decision making under risk that is equivalent to diminishing impatience?"; (ii) "Is there a behavioral property of intertemporal decision making that is equivalent to the CRE?" Claim 3 answers these questions: it shows that diminishing impatience is equivalent to the certainty effect (CE) and that strong diminishing impatience (i.e., hyperbolic discounting) is equivalent to the CRE. An implication of Claim 3 is that under any additional assumptions which would make the "only if " part of Theorem 1 in Halevy (2008) true, diminishing impatience becomes equivalent to strong diminishing impatience, and, similarly, the CRE also becomes equivalent to the CE. Thus, whatever assumption is added to make the "only if " part of Theorem 1 true, it must confound the conceptually clear and empirically robust distinction between quasi-hyperbolic discounting and hyperbolic discounting on the one hand, and also between the CRE and the CE on the other hand.