¹⁹See, for example, Skiba and Tobacman (2008), who estimate two-week discount rates of 21 percent, and Allcott, Kim, Taubinsky, and Zinman (2021), who estimate an initial period discount factor between 0.74 and 0.83 in a model where a period is eight weeks long. Both of these papers use quasi-hyperbolic preferences, so the estimates are not directly comparable with parameters in our model. Nevertheless, they support the point that very high discount rates are necessary to model the part of the population that takes out payday loans at very high interest rates.