its value once by (100% + r%). So if I hold the investment for n days, its value will have multiplied itself by this amount n times, making that value (100% + r%)n of what it was at the start – that is, (1 + r)n times what it was at the start. So to figure out how much I would need to start with today to get y dollars n days from now, I need to divide y dollars by [1 + r]n." The hyperbolic discounter, however, thinks "The savings plan adds to its value, in each day, r percent. Therefore, after n days, it adds to its value r×n percent [There lies the hyperbolic discounter's error]. So to figure out how much I would need to start with today to get y dollars n days from now, I need to divide y dollars by [1 + n×r]." As n becomes very large, the value of (1 + r)n becomes much larger than the value of [1 + n×r], with the effect that the value of y / (1 + r)n becomes much smaller than the value of y/[1 + n×r]. Therefore, the minimum value of x (the number of dollars in the immediate choice) that suffices to be greater than that amount will be much smaller than the hyperbolic discounter thinks, with the result that they will perceive x-values in the range from y/(1 + r )n to y/[1 + n×r] inclusive as being too small and, as a result, irrationally turn those alternatives down when they are in fact the better investment. Formal model Hyperbolic discounting is mathematically described as where g(D) is the discount factor that multiplies the value of the reward, D is the delay in the reward, and k is a parameter governing the degree of discounting (for example, the interest rate). This is compared with the formula for exponential discounting: Comparison If is an exponential discounting function and a hyperbolic function (with D the number of weeks of delay), then the exponential discounting a week later from "now" (D=0) is , and the exponential discounting a week from week D is , which means they are the same. For g(D), , which is the same as for f, while . From this one can see that the two types of discounting are the same "now", but when D is much greater than 1,