Proving the marginal value theorem
The marginal value theorem holds true under three fairly mild conditions:
The fixed cost is larger than zero.
The reward function increases with .
The slope of the reward function decreases with (i.e. is a diminishing returns function).
We wish to prove that the instantaneous reward rate equals the average reward rate when is maximal. To achieve this, we need to find the value of when is maximal. To find the maximal average reward rate, we make use of the fact that its slope is zero at a maximum.
The average reward rate is defined as
and its derivative is
where (by definition)
is the instantaneous reward rate, and where
Substituting Equations 3 and 4 into Equation 2,
is the average reward rate, so that Equation 5 becomes
At a maximum, this is equal to zero,
Finally, multiplying both sides by , and re-arranging yields
This proves that the average reward rate is maximal when the instantaneous reward rate equals the average reward rate.