Gamma Discounting Calculator

An agent uncertain about its own discount rate values the far future by the expectation of the discount factor, not the expectation of the rate. Set a two-point prior over ρ and watch the certainty-equivalent rate ρ_eff decline toward the patient scenario as the horizon lengthens.

⚠ Decision-theory model
Uncertainty about patience is not the same as patience. Weitzman's gamma-discounting result (Weitzman 1998, 2001) shows that a mixture of possible discount rates is certainty-equivalent to a rate that declines with horizon toward the minimum plausible rate in the prior. An agent does not need to know it is patient, only that it might be (Swanson 2026, "The Economics of Inward Migration", §7).
Two-point prior over ρ
1.0×10⁻³
0.50
1.0×10⁻⁶
1.0×10⁵ yr
Certainty-equivalent rate
Naive expectation E[ρ] = w₁ρ₁+(1−w₁)ρ₂
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Gamma-discounted ρ_eff(t)
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Implied half-life ln 2 / ρ_eff
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ρ_eff(t) versus horizon
Certainty-equivalent rate declines from the weighted mean toward min(ρ₁,ρ₂) as t grows. Dashed reference: the patience threshold ρ* ≈ 2.0×10⁻⁴ yr⁻¹ at Paper D's crossover fiducials.

The formula

Under exponential discounting with a discount rate ρ known with certainty, the value of a unit payoff at time t is e^(−ρt). If instead an agent is uncertain over its own ρ, drawn from some prior distribution, the correct valuation is the expectation of the discount factor, E[e^(−ρt)], not the discount factor at the expected rate. Converting that expectation back into an equivalent constant rate gives the certainty-equivalent discount rate

ρ_eff(t) = −(1/t) ln E[e^(−ρt)]

For the two-point prior used here, E[e^(−ρt)] = w₁e^(−ρ₁t) + (1−w₁)e^(−ρ₂t).

Why it declines with horizon

At short t, both exponential terms are close to 1 and ρ_eff ≈ E[ρ], the naive weighted mean. At long t, the term with the larger ρ decays to zero far faster than the term with the smaller ρ, so the expectation is dominated entirely by the patient scenario, and ρ_eff falls toward min(ρ₁,ρ₂) regardless of how little prior weight it carried. An agent uncertain whether its own discount rate is 10⁻³ or 10⁻⁶ per year has ρ_eff of order 10⁻⁶ by the time t reaches the ω Cen transit time (~10⁵ yr), far below the migration threshold derived in the Patience Threshold Calculator.

Why this strengthens the migration case

The crossover condition of Paper D requires ρ+λ below a threshold of order a few×10⁻⁴ yr⁻¹. A lineage need not be confident it discounts the far future at that rate; it only needs to be uncertain whether it might. Migration therefore requires uncertainty about patience, not patience itself, which is a substantially weaker and more plausible condition to hold of an arbitrary long-lived technological lineage.

Limits

The two-point prior is a minimal illustration, not a claim about the true distribution of ρ across agents; a continuous prior produces the same qualitative decline by the same mechanism (Weitzman 1998, 2001). Hyperbolic discounting (Laibson 1997) is a related but distinct phenomenon, time-inconsistent rather than time-declining under certainty, and is not modeled here.

v1.0 — 2026-07-23 · Code MIT · Prose CC BY 4.0 · Swanson 2026, "The Economics of Inward Migration" ("Paper D") §7; Weitzman 1998 (J. Environ. Econ. Manage. 36:201), 2001 (Am. Econ. Rev. 91:260)