Discounted utility model
Also known as: DU model, Exponential discounting, Discounted-utility model
The classical benchmark: future utility is discounted at a single constant rate per period.
What it means
The discounted utility model is the standard normative framework for intertemporal choice, in which an agent values a stream of outcomes by summing each period's utility weighted by a constant per-period discount factor. Its defining feature is exponential discounting, which implies a single, stationary discount rate and therefore time-consistent preferences: a plan optimal today remains optimal tomorrow. Introduced as a tractable analog to expected utility, it was proposed more for convenience than realism, and its author cautioned against taking it as descriptively accurate. Behavioral research has since documented systematic departures — present bias, the magnitude and sign effects, and preference reversals over time — that the constant-rate assumption cannot accommodate. The model remains the indispensable baseline against which these anomalies are defined and measured.
From measurement note to normative benchmark
Samuelson introduced the formula in a seven-page 1937 note whose real subject was measuring utility; discounting appeared almost as a by-product. He wrote it for tractability and warned explicitly that its assumptions were arbitrary and unlikely to describe how people actually behave. The model gained its authority two decades later, when Koopmans (1960) derived it from primitive axioms on preferences over consumption streams and showed that a handful of conditions force the exponential form. That axiomatic footing turned a convenient formula into the field's normative standard. Read correctly, it is not a claim about how people choose but a statement of how a fully consistent planner should choose, which is exactly why its descriptive failures are so informative.
What the constant rate assumes
The model's engine is stationarity: the trade-off between two dated outcomes depends only on the delay between them, not on when they sit relative to now. Preferring one apple today to two tomorrow commits you to preferring one apple in a year to two in a year and a day. Combined with additive separability across periods, stationarity forces a single geometric discount factor and, with it, time-consistent plans. Almost every documented anomaly in intertemporal choice is a failure of one of these assumptions, usually stationarity. This is why the model earns its keep less as a description of behavior than as the ruler that makes the deviations visible and gives them a precise magnitude.
Where the constant rate breaks
Thaler (1981) found implied discount rates falling sharply as delays lengthened, steep over weeks and shallow over years, the signature of hyperbolic rather than exponential discounting. Two further regularities resist any single rate. The magnitude effect: large sums are discounted less steeply than small ones. The sign effect: losses are discounted less than equivalent gains. Frederick, Loewenstein and O'Donoghue's (2002) review catalogued these and noted a further embarrassment. Estimated discount rates across studies span several orders of magnitude, from negative to many thousands of percent, suggesting the model's single parameter is absorbing a jumble of distinct forces, uncertainty, self-control, anticipation, liquidity, rather than one stable underlying preference for the present over the future.
Why it still anchors the field
Descriptive failure has not dislodged the model, because its role is comparative. Present bias, quasi-hyperbolic (beta-delta) discounting and preference reversals are all defined as departures from the exponential baseline; strip out the benchmark and the anomalies have nothing to be anomalous against. It also remains the workhorse of applied welfare analysis: corporate capital budgeting, pension design and the social discount rate used in climate and health policy all assume one constant rate, largely for tractability. Samuelson's original caution survives intact. A single rate that is convenient to compute with can quietly smuggle strong, testable and often false assumptions into a decision, so any conclusion that hinges on it should be checked against plausible alternative rates.
Examples
By the model, if you prefer $100 now to $110 in a year, you must also prefer $100 in five years to $110 in six — a consistency real choices routinely break.
Corporate finance runs on this model: every future year of a project's cash flow is discounted at one steady rate, so a decision made today would be made identically next quarter.
The model says a dieter who plans on Sunday to skip Friday's cake will still skip it on Friday. Real dieters reverse, and present bias is defined by exactly that gap.
Climate cost-benefit analysis runs on the model: whether future damages justify acting today turns almost entirely on the single constant rate chosen, the crux of the Stern-Nordhaus dispute.
A retirement calculator maximizing discounted utility at one rate implies the saver never regrets today's contribution level. Real enrollees repeatedly wish they had saved more, exposing the stationarity assumption.
First described in Paul Samuelson (1937).
Key references
- Frederick, S., Loewenstein, G., & O'Donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, 40(2), 351-401. doi.org/10.1257/002205102320161311
- Koopmans, T. C. (1960). Stationary ordinal utility and impatience. Econometrica, 28(2), 287-309. doi.org/10.2307/1907722
- Thaler, R. H. (1981). Some empirical evidence on dynamic inconsistency. Economics Letters, 8(3), 201-207. doi.org/10.1016/0165-1765(81)90067-7
- Samuelson, P. A. (1937). A note on measurement of utility. The Review of Economic Studies, 4(2), 155-161. doi.org/10.2307/2967612