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Can someone provide a rigorous definition of a utility function? I had thought that a utility function only needs to the preserve the order of preferences. Thus a utility function can take on negative values as long as it preserves the order of preferences. Others have told me that a utility function cannot take on negative values. Is this a condition of a rigorous definition of a utility function?

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3 Answers 3

up vote 5 down vote accepted

A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function.

Also, there are different decision theory frameworks that allow the utility function to be transformed. You alluded to something like this in your question. In the traditional framework without uncertainty, the utility function is defined up to a monotonic transformation. Under certain kinds of uncertainty, we get Von Neumann–Morgenstern utility functions that are unique up to affine transformations. You can read more about this elsewhere. For now, the consider the following definition of a utility function. It is taken from Advanced Microeconomic Theory by Jehle and Reny (3rd edition):

Definition of a utility function

A preference relation $\succeq$ is defined as follows:

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where the axioms references are these:

Axiom 1: Completeness. For all $x^1$ and $x^2$ in $X$, either $x^1 \succeq x^2$ or $x^2 \succeq x^1$.

Axiom 2: Transitivity. For any three elements $x^1$, $x^2$, and $x^3$ in $X$, if $x^1 \succeq x^2$ and $x^2 \succeq x^3$, then $x^1 \succeq x^3$.

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Here is one possible rigorous definition of a utility function:

Let $X$ be a set of alternatives. Let $\succeq$ be a preference relation over those alternatives. $U: X \to \mathbb{R}$ is a utility function means that $U(x) \geq U(y) \iff x \succeq y$.

Then if for example $X$ is 'amounts of money you might be given', and $x \succeq y$ only if $x \geq y$, then some possible utility functions are $U(x) = x, U(x) = e^x, U(x) = \log(x)$...

Some of these are negative.

One could of course require that $U(x) > 0$. Maybe this makes it easier to swallow as an interpretation of the 'well-being' of an individual. But that would rule out many commonly used utility functions, such as $U(x) = x$ or $U(x) = \log(x)$.

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3  
"Maybe this makes it easier to swallow as an interpretation of the 'well-being' of an individual." But of course it's just best to think of a utility function as a representation of the preference relation $ \succeq$ and nothing more. Some utility functions are always non-positive (e.g., the CRRA type $C^{1-\gamma}/(1-\gamma)$ where $\gamma > 1$). –  jmbejara Mar 13 at 17:19
    
I wouldn't say it's 'best'. For welfare analysis or for thinking about things like attitudes toward risk we sometimes give more meaning to the utility function than as merely representing a preference relation. One could debate the philosophy of it but there's a long history of interpreting the utility function as representing something real. –  NickJ Apr 17 at 14:15

Like jmbejara says generally in economics utility is not measured in anything but preference relations, so it's called ordinal utility (which contrasts cardinal utility). So a bundle giving utility of -1 is preferred to any bundle giving less than -1. The number -1 doesn't tell us anything else.

"Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles (combinations) of goods can be ordered such that one is considered by an individual to be worse than, equal to, or better than the other. This contrasts with cardinal utility theory, which generally treats utility as something whose numerical value is meaningful in its own right." (source: http://en.wikipedia.org/wiki/Ordinal_utility)

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