Marginal Value Theorem, explained.
The marginal value theorem studies when a forager should leave a diminishing-return patch, comparing marginal gain with the environment's long-run reward rate under specified assumptions.
Why it happens
A patch yields R(t)=20(1−exp(−t/5)). Harvest one time unit at a time, then leave and add travel time to complete a cycle. Completed reward divided by completed time is tracked. The benchmark maximises R(t)/(t+travel) over integer stays from one to thirty.
The marginal value theorem studies when a forager should leave a diminishing-return patch, comparing marginal gain with the environment's long-run reward rate under specified assumptions.
Read the result
Compare complete cycles, including travel. A large patch total may still have a poor reward-per-time rate after a long unproductive stay.
A worked example
Leave a thinning berry patch
Harvest several time units and observe that each additional unit adds less reward under the exponential patch-yield curve.
Leave and travel so the model banks both the patch reward and its full stay-plus-travel duration.
Compare completed-cycle rates with the best integer stay, then raise travel time and inspect how the benchmark changes.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
A patch yields R(t)=20(1−exp(−t/5)). Harvest one time unit at a time, then leave and add travel time to complete a cycle. Completed reward divided by completed time is tracked. The benchmark maximises R(t)/(t+travel) over integer stays from one to thirty.
Where this idea is useful
A practical use
Leaving too soon wastes travel overhead; staying too long spends time on increasingly small rewards.
A common misconception
“Leave as soon as the next reward is smaller than the first.”
The comparison is with the attainable long-run rate including travel, not the initial reward. High travel cost can justify a longer stay in a depleting patch.
What this explanation leaves out
- Patches are identical, known and immediately replenished between cycles. The integer benchmark is a discretisation, not the exact continuous tangency solution. Predation, variable patches and learning about yield are absent.
Is the displayed optimum the continuous marginal-value solution?
It is the best integer stay from one to thirty under the displayed reward function. It approximates the continuous problem rather than solving an unrestricted biological case.
What switching cost should be included before deciding that the next small gain is no longer worth the time?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.