Proof of the Law of Diminishing Marginal Utility (optional material)
Suppose that the units of a good G can be allocated to uses U1, U2, . . . , Un, where the Ui are arranged in order of descending value to the given acting individual—i. e., U1 is the most valued use, and Un is the least valued. Since the units are instances of the same good, any one unit can potentially serve any one of the n uses. Since G (like all goods) is scarce, the individual's stock m cannot exceed the number of uses to which it can be allocated; that is, m n.

The individual always seeks to maximize his or her subjective value. Consequently, if he or she has only one unit, it will be allocated to the highest use U1. Similarly, a stock of (m - 1) units will be allocated to uses U1 through Um-1, while a stock of m units will be allocated to U1 through Um. In a stock of m units, the marginal utility of the last unit is therefore Um. Since the Ui are arranged in descending utility sequence, that marginal utility decreases as the size m of the stock increases.

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