The Knee of a CurveUseful Clue but Incomplete Support

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[abstract based on the first two paragraphs]

In our studies of military systems, we typically look at the relationship between resources expended and expected results, the latter often given in terms of a measure of effectiveness (MOE). Frequently we have, as in Figure 1, what Gen. Jasper Welch (Welch, 1977) referred to as a Pay More-Do Better curve. Very often, as in Figure 1, inspection of the curve leads us to select a point (x, y) where the curve's direction is changing rapidly from large slopes to small slopes. We say that the curve has a knee at (x, y), or at least in its vicinity. Moreover, it is often inferred that x is a good limit on the resources worth expending on the given system. To the right of x lies the region of diminishing returnswhat Alain Enthoven used to call the flat of the curve (Tucker, 1966). Rather than expend more on the given system (i.e., where x > x), it may be more rewarding to devote resources to other uses.

In a general sense the above inference has much merit, and the knee of a curve may be a useful clue, but it is incomplete in itself. We need to embed the given resource expenditure problem in a larger resource allocation problem, in order to compare returns as shown in Figure 1 with the returns available from other uses of resources. There is nothing new in this observationit has been a generation since Charles Hitch wrote his eloquent words on suboptimizing (Hitch, 1953) and many more generations since economists first discoursed on diminishing returns.
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