Optimal Foraging
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Chapter 5:
Marginal Value Theorem
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As in any optimality model, you arrive at the predictions of the marginal value theorem by following through the logic of its constraints. Imagine you're a hummingbird that's just left a flower patch. Time is ticking relentlessly, and as it does you move steadily rightward along the x-axis. Your energy gain is shown by any rises in the y-axis as you confront each constraint in turn: Travel time. It takes a while to get from one patch to another, and during that time, you don't get any food. Gain curve. Once you're in a patch, you get more and more energy. But gradually, for every unit of time you spend, your gain in energy gets less and less. That might happen for several reasons, for example because you're depleting the patch or because the food gets harder to find. Remember what the object of the game is: to get as much energy per time spent as you can. On the graph, that means for every unit on the x-axis, you'll want to go as high on the y-axis as you can. In other words, wherever you end up on the gain curve, you want the line A-B to be as steep as possible. |  | Early Gain At first, the line A-B gets steeper and steeper. Don't quit the patch now; you're on a roll.
|  | Staying Too Long But after a while, the line A-B starts falling again. That means even though you're still getting more energy, your energy gain relative to your time spent is getting smaller.
|  | Optimal Gain So the solution is to leave at the point a little earlier, where line A-B is at its maximum, which is where the line just barely touches the curve, i.e., where it's tangent to it. Those of you who are familiar with calculus will recognize this as the point where the instantaneous slope, or derivative, of the gain curve equals the maximum possible slope of a line from the origin tangential to the curve. The economics term for derivative is marginal value; hence the name of the marginal value theorem.
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