Delay discounting

In behavioral economics, hyperbolic discounting refers to the empirical finding that people more often prefer smaller payoffs to larger payoffs when smaller payoffs come sooner in time relative to larger payoffs than when all the payoffs are either distant or proximal in time, in which case they tend to prefer the larger. The phenomenon of hyperbolic discounting was first discovered and the term first used by Richard Herrnstein in experiments involving pigeons and food (Chung and Herrnstein, 1967) and later reproduced with human subjects.

For instance, when offered the choice between $50 now and $100 a year from now, most people will choose the immediate $50. However, given the choice between $50 in five years or $100 in six years most people will choose $100 in six years. In addition, given the choice between $50 today or $100 tomorrow, most people will choose $100 tomorrow.

Notice that whether discounting future gains is logically correct or not, and at what rate such gains should be discounted, depends greatly on circumstances. Many examples exist in the financial world, for example, where it is logically reasonable to assume that there is an implicit risk that the reward will not be available at the future date, and furthermore that this risk increases with time. Consider: Paying $50 dollars for your dinner today or delaying payment for sixty years but paying $100,000. In this case the restaurateur would be reasonable to discount the promised future value as there is significant risk that it might not be paid (possibly due to your death, his death, etc).

In order to explain the hyperbolic discounting phenomenon, it was hypothesized that the discount function with regards to time is shaped like a hyperbola. In other words, people will "discount" in order to get the payoff sooner (over short horizons) at a higher rate, but at a relatively low rate over long horizons.

This pattern of discounting is dynamically inconsistent, and therefore inconsistent with standard models of rational choice, since the rate of discount between time t and t+1 will be low at time t-1, when t is the near future, but high at time t when t is the present and time t+1 the near future. Nevertheless, it appears to be descriptively accurate.

More recently these observations about discount functions have been used to study saving for retirement, borrowing on credit cards, and procrastination. However, hyperbolic discounting has been most frequently used to explain addiction.

The functional equation for hyperbolic discounting is as follows: $$v=V/(1+kD)$$

Where v is the discounted value of the reward, V is the undiscounted value of the reward, D is the delay in the reward, and k is the degree of discounting.

The degree of discounting is vitally important in describing hyperbolic discounting, especially in the discounting of specific rewards such as money. The discounting of monetary rewards varies across age groups due to the varying rate of k (Green, Frye, and Myerson, 1994). k depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward (Lowenstein and Prelac, 1992; Raineri and Rachlin, 1993).