The Half-Life Tax

An interactive model of AI agent economics. Agent cost per successful outcome scales with task length, while human cost scales linearly. The shape of that scaling depends on the survival model: exponential (Ord's constant hazard rate) or Weibull with κ<1 (Hamilton's declining hazard rate). Based on Ord's half-life analysis, METR's empirical data, and Hamilton's Weibull reanalysis.

Survival model

How Does Success Decay?

The original model assumes a constant hazard rate (exponential decay). Hamilton's reanalysis of the METR data suggests the hazard rate may decrease over time, following a Weibull distribution with κ≈0.6-0.9 for SOTA models (and ≈0.37 for humans). Crucially, κ does not correlate with model size: scaling reduces the base hazard rate (λ) but does not change the shape of the decay. The two models fit the available data about equally well, but diverge dramatically at the tails.

Constant hazard rate. The probability of failing in the next step is the same regardless of how far you have got. Success decays as S(t) = 0.5^(t/T50). Expected attempts grow exponentially. This is the worst case for long tasks.
Parameters

Model Inputs

Adjust these to match your scenario. The model recomputes instantly.

$0.22
Average cost of one model call including context. $0.02 for cheap models, $0.50+ for frontier with long context.
80
Agent actions per hour of equivalent human work. 50 for complex reasoning, 120+ for routine tasks.
5h
Task length at which the agent succeeds 50% of the time. METR data: ~2.5-5h for frontier models (2025).
$150
Salary + benefits + overhead. $100-200 typical for skilled knowledge workers.
0.70
κ=1 is exponential (constant hazard). κ<1 means the hazard rate decreases over time. Hamilton finds SOTA models cluster κ≈0.6-0.9; humans ≈0.37.
0%
Human time to review each agent attempt. Under the Weibull model with fatter tails, this may become the binding constraint rather than raw agent cost.
// Key relationship (Exponential)
P(success) = 0.5 ^ (task_hours / half_life)
E[attempts] = 1 / P(success) = 2 ^ (task_hours / half_life)
Agent cost = steps × $/step × 2 ^ (task_hours / half_life)
Human cost = hourly_rate × task_hours  // linear
Results

Agent vs Human Cost by Task Length

The highlighted row marks where agent cost per success overtakes human cost. Note how the shape of the divergence depends on the survival model.

Logarithmic scale. Under the exponential model the divergence is dramatic; under the Weibull model (k<1) it is much gentler.
Task length Steps $/attempt P(success) E[attempts] Agent cost Human cost Ratio
Ratio = (agent cost + verification cost) / human cost. Values below 1 favour the agent; above 1 favour the human.
Sensitivity analysis

Half-life vs Cost per Step

For a fixed task length, how does the agent-to-human cost ratio change as you vary half-life and cost per step? Under the exponential model the half-life dominates because it acts on the exponent. Under Weibull (k<1), cost per step matters relatively more.

8h
Agent cheaper Human cheaper
Each cell shows (agent cost + verification) / human cost. Moving vertically (T₅₀) changes the ratio far more than moving horizontally ($/step). T₅₀ enters the exponent; cost per step is a linear multiplier.
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