The magnitude paradox
研究发现相对风险厌恶随结果幅度增加而上升，但耐心却随幅度增加而下降，这无法用同一效用函数解释。本文提出双速模型，将风险与时间贴现领域有机联系，并通过高赌注实验室实验验证该模型优于其他竞争模型。
Abstract There is strong evidence indicating that relative risk aversion increases with outcome magnitude whereas impatience decreases with outcome magnitude. This finding seems paradoxical because it cannot be captured by the same utility function: Increasing relative risk aversion requires decreasing elasticity of the utility function, whereas decreasing impatience requires increasing elasticity with respect to outcome magnitudes. We develop a model that organically links the domains of risk taking and time discounting. The resulting two-speeds model generates a magnitude-dependent discount function such that increasing relative risk aversion is not only compatible with the magnitude effect in discounting but actually predicts magnitude-dependent discount weights. Moreover, we conduct a high-stakes laboratory experiment that reproduces the magnitude paradox and enables us to structurally estimate competing models of magnitude-dependent discounting. Both the Bayesian Information Criterion and the Akaike Information Criterion favor our two-speeds model. The results suggest that participants behave as if they apply a heuristic procedure: the value of a future reward consists of a certain percentage of the reward, irrespective of the length of delay, plus a delay-dependent component.