Hyperbolic Discounting

Exponential discounting and preference reversals

The discounted utility model proposed by Paul Samuelson (1937) assumes exponential discounting, in which future rewards are discounted at the same rate each period. A choice between today and tomorrow then yields the same answer as a choice between a year from now and a year and a day from now, so preferences are consistent over time. Robert Strotz (1955) showed that if discount rates are not constant, people will reverse their own earlier plans. George Ainslie (1975) found in animal and human experiments that discount curves fall steeply at short delays and flatten afterwards, a hyperbolic shape, and Mazur (1987) proposed a hyperbolic function in which value is proportional to 1/(1+kD). With such curves people choose $100 today over $110 tomorrow yet prefer $110 in a year and a day over $100 in a year, a preference reversal. Thaler (1981) offered survey evidence that annualized discount rates are very high for short horizons.

The quasi-hyperbolic model and types of agents

David Laibson (1997) proposed the tractable quasi-hyperbolic or beta–delta model, which applies an extra discount factor β (less than one) between now and every later period and standard exponential discounting δ thereafter, capturing present bias, a special preference for immediate gratification. Laibson used it to explain why people hold illiquid assets, golden eggs, as commitment devices while carrying credit card debt. O'Donoghue and Rabin (1999) distinguished naive agents, unaware of their future present bias, from sophisticated agents who anticipate it, showing that naifs tend to bring pleasant tasks forward and keep postponing unpleasant ones. In a brain imaging study, McClure et al. (2004) reported that limbic regions respond more to immediate rewards and prefrontal regions to delayed ones. Frederick, Loewenstein and O'Donoghue (2002) cautioned that estimated discount rates vary enormously across studies.

Evidence and applications

DellaVigna and Malmendier (2006) found that gym members on monthly contracts attended so rarely that they paid more than the per-visit price would have cost, which they explained by present bias and overconfidence about future self-control. In a field experiment in the Philippines, Ashraf, Karlan and Yin (2006) offered a commitment savings product restricting withdrawals; many took it up and their savings rose. Thaler and Benartzi's (2004) Save More Tomorrow program, in which employees commit in advance to save part of future pay raises, sharply increased retirement saving rates. Hyperbolic discounting complements the standard concept of present value and discount rates and, together with nudges and mental accounting, is widely used to understand long-run decisions about saving, health behavior and study plans.

FAQ

What is hyperbolic discounting in simple terms?

Rewards available right now feel much bigger and even a short delay sharply cuts their value, while people are comparatively patient when comparing two dates far in the future.

How does it differ from exponential discounting?

Exponential discounting applies the same rate every period, so plans stay consistent; hyperbolic discounting discounts near delays more heavily, so people later reverse plans they made in advance.

Sources reviewed: 2026-10-26

References

1. Samuelson, P. A. (1937). A note on measurement of utility. Review of Economic Studies, 4(2), 155–161.
2. Strotz, R. H. (1955). Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23(3), 165–180.
3. Ainslie, G. (1975). Specious reward: A behavioral theory of impulsiveness and impulse control. Psychological Bulletin, 82(4), 463–496.
4. Thaler, R. (1981). Some empirical evidence on dynamic inconsistency. Economics Letters, 8(3), 201–207.
5. Mazur, J. E. (1987). An adjusting procedure for studying delayed reinforcement. In M. L. Commons, J. E. Mazur, J. A. Nevin, & H. Rachlin (Eds.), Quantitative Analyses of Behavior (Vol. 5, pp. 55–73). Hillsdale, NJ: Erlbaum.
6. Loewenstein, G., & Prelec, D. (1992). Anomalies in intertemporal choice: Evidence and an interpretation. Quarterly Journal of Economics, 107(2), 573–597.
7. Laibson, D. (1997). Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, 112(2), 443–477.
8. O'Donoghue, T., & Rabin, M. (1999). Doing it now or later. American Economic Review, 89(1), 103–124.
9. Frederick, S., Loewenstein, G., & O'Donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, 40(2), 351–401.
10. McClure, S. M., Laibson, D. I., Loewenstein, G., & Cohen, J. D. (2004). Separate neural systems value immediate and delayed monetary rewards. Science, 306(5695), 503–507.
11. Thaler, R. H., & Benartzi, S. (2004). Save More Tomorrow: Using behavioral economics to increase employee saving. Journal of Political Economy, 112(S1), S164–S187.
12. DellaVigna, S., & Malmendier, U. (2006). Paying not to go to the gym. American Economic Review, 96(3), 694–719.
13. Ashraf, N., Karlan, D., & Yin, W. (2006). Tying Odysseus to the mast: Evidence from a commitment savings product in the Philippines. Quarterly Journal of Economics, 121(2), 635–672.