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I'm trying to show a result that involves utility and money (the latter is in dollars). I would like to know if it is safe to assume that the units of utility is dollars? After all, in auction scenarios, we seem to deduct the bid, which is in dollars from the total utility for the item, and maximize that difference.

Therefore, am I doing the right thing in saying that a utility gain of 10 units can be equated to a gain of 10 dollars?

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No, you can not really say that. Here is the reason:

Utility is generally used to describe preferences. Let's say my happiness is described by this utility of chocolates: $Y(X) = X$. So, two chocolates will give me two units of happiness. However, by definition of preferences, I am allowed to apply monotone transformations to them. Hence, you can just transform that function to $Y(X) = X^2$. Now two chocolates will give me 4 units of happiness.

Again, by definition, this is a completely valid transformation. However, now I am getting 4 dollars instead of $2 because I simply transformed the function. For obvious reasons, you can not do that.

Hence, utility is unitless.

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It depends on the type of utility. Generally speaking, utility is either considered to be an ordinal value (one that permits monotone / order-preserving transformations, such as squaring or taking the logarithm) or a cardinal value (one that only permits scaling by a constant factor, such as multiplying all values by three.)

Traditionally, the standard unit for cardinal utility is the "util" (which is a made-up unit created specifically for this purpose.) But you can substitute any other cardinal measure you like, including dollars.

Note: In economics, one more often sees reference to ordinal utility. In voting theory, you may find more references to cardinal utility.

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