The paper surveys over twenty models of delay discounting (alsoknown as temporal discounting, time preference, timediscounting), that psychologists and economists have put forward toexplain the way people actually trade off time and money. Using little more thanthe basic algebra of powers and logarithms, I show how the models are derived,what assumptions they are based upon, and how different models relate to eachother. Rather than concentrate only on discount functionsthemselves, I show how discount functions may be manipulated to isolaterate parameters for each model. This approach, consistentlyapplied, helps focus attention on the three main components in any discountingmodel: subjectively perceived money; subjectively perceived time; and how theseelements are combined. We group models by the number of parameters that have tobe estimated, which means our exposition follows a trajectory of increasingcomplexity to the models. However, as the story unfolds it becomes clear thatmost models fall into a smaller number of families. We also show how new modelsmay be constructed by combining elements of different models.
The surveyed models are: Exponential; Hyperbolic; Arithmetic; Hyperboloid (Green& Myerson, Rachlin); Loewenstein and Prelec Generalized Hyperboloid;quasi-Hyperbolic (also known as β-δ discounting);Benhabib et al’s fixed cost; Benhabib et al’s Exponential /Hyperbolic / quasi-Hyperbolic; Read’s discounting fractions;Roelofsma’s exponential time; Scholten and Read’sdiscounting-by-intervals (DBI); Ebert and Prelec’s constant sensitivity(CS); Bleichrodt et al.’s constant absolute decreasing impatience (CADI);Bleichrodt et al.’s constant relative decreasing impatience (CRDI);Green, Myerson, and Macaux’s hyperboloid over intervals models;Killeen’s additive utility; size-sensitive additive utility; Yi, Landes,and Bickel’s memory trace models; McClure et al.’s twoexponentials; and Scholten and Read’s trade-off model.
For a convenient overview, a single “cheat sheet” table capturesthe notation and essential mathematics behind all but one of the models.