"It should immediately be clear from this definition of “represent” that if one function, U(· ), represents the preference ordering, then any positive monotonic (i.e. rank order-respecting) transformation of U(· ) will also represent the ordering. The reason neoclassicals stress this seemingly trivial mathematical detail is that it (should) prevent any possible significance from being assigned to a “util.” For example, if U(a) = 20, while U(b) = 10, we can conclude is that a @ b, i.e. that the individual considers bundle a to be at least as good as bundle b [Bob used the symbol "@" to indicate the preference relation in this piece]. But we may not conclude that bundle a offers ten more “utils,” nor can we say that bundle a is “twice as good” as bundle b. Why not? Because our representation theorem tells us that the function V(· ), defined, say, as the square of the function U(· ), will also represent the preference relation. So if we used this monotonic transformation, we would have V(a) = 400 and V(b) = 100. It’s still true that the number assigned to the first bundle is higher, and thus we still conclude that bundle a is (ordinally) preferred by the individual. But now the utility assigned to the first bundle exceeds by three hundred that assigned to the second bundle, and it is no longer double the quantity."
- Bob Murphy, 2000
7 minutes ago