The magnitude paradox

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.