Discounted Utility

What is discounted utility?

Discounted utility is a way to compare outcomes that arrive at different times. The model assigns a time weight to each outcome’s utility, or subjective value, then adds the weighted values together.

Discounted utility model and formula

In the standard exponential model, an outcome t periods away has weight δ^t. The symbol δ is the per-period discount factor. When it lies between zero and one, a larger factor gives the future more weight: 0.9 retains more of an outcome’s utility than 0.5.

A discount rate runs in the opposite direction. A higher rate gives the future less weight, with δ = 1/(1 + r). Confusing the factor with the rate reverses the meaning.

For a finite sequence, the discounted utility model can be written U = Σ_t=0^T δ^tu(x_t): add each outcome’s utility after applying its time weight.

A worked example

Suppose one option gives 80 units of utility now and the other gives 100 units two periods from now, with no other differences. If δ = 0.9, the delayed option has present weighted utility 0.9² × 100 = 81. The model therefore favors 81 over 80. If δ = 0.8, the same delayed option becomes 0.8² × 100 = 64, so the immediate option is preferred. These are assumed utility units, not a claim that a dollar always provides one unit of utility.

Why it matters

Saving and spending decisions involve trade-offs between what we want now and what we want later. Discounted utility gives those trade-offs a mathematical form. Utility is subjective, so it need not rise dollar for dollar with money.

As Frederick, Loewenstein, and O’Donoghue explain, the standard exponential model predicts time-consistent preferences under its assumptions. When people reverse their preferences as a reward gets closer, that is a challenge for this model to explain.

For the broader tendency to value delayed outcomes less, see temporal discounting.

Investment and health choices also involve outcomes at different times, but risk, constraints and how outcomes are valued matter alongside delay. For example, someone weighing exercise today against a hoped-for future health benefit faces immediate effort and delayed value. A time weight can represent one part of that trade-off; pain, available time, uncertainty about benefits, and enjoyment of the activity can also affect the choice. Neither one skipped session nor a discount factor estimated in another task explains every such choice. Hyperbolic discounting offers a different time-weighting pattern that can accommodate preference reversals.