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What is Marginal Utility?
Marginal utility is defined as the increase in utility as a result of consuming one more unit of the good. In other -- algebraic -- terms,
MU = ^U/^Q dU/dQ
where U is utility and Q is the quantity of the good. In the numerical example, the MU of water is 1 and the MU of diamonds is 15, so the consumer is willing to pay 15 times as much for a diamond as for a gallon of water. (This example is, of course, not "realistic.") Despite that, the person gets about 67000 times as much total utility from water as from a diamond.
Economists call the part of utility that is gained or lost (in the decision to buy one more unit) the marginal utility.