Can A Utility Function Take On Negative Values?
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?
Tony BuiTony Bui
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):
A preference relation $\succeq$ is defined as follows:
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 $x^1 \succeq x^3$.
jmbejarajmbejara
$\begingroup$ Nice answer! Solved my question. $\endgroup$ – Jinhua Wang May 19 '19 at 15:20
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 $U(x) = \log(x)$.
NickJNickJ
$\begingroup$ "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 $\gamma > 1$). $\endgroup$ – jmbejara Mar 13 '15 at 17:19
$\begingroup$ 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. $\endgroup$ – NickJ Apr 17 '15 at 14:15
$\begingroup$ "there's a long history of interpeting the utility function as representing something real" Yes. But to be specific, this is only the case with "cardinal utliity" as opposed to "ordinal utility." But even VnM utliity functions (which are cardinal and are the utility functions that are most commonly used) are defined up to affine transformations. Thus, the sign of the utility is in some sense still arbitrary. $\endgroup$ – jmbejara Apr 22 '15 at 22:09
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)
snoramsnoram
Not the answer you're looking for? Browse other questions tagged utility or ask your own question.
Algebraic approach towards convexity
When can one safely talk about decreasing marginal utility?
Does risk aversion cause diminishing marginal utility, or vice versa?
What are good dataset alternatives to estimate value functions?
How to show that a homothetic utility function has demand functions which are linear in income
performance as utility function / price?
Ruling out boundary solutions in Utility Maximization
Existence of maximum utility with two bads
Purpose of a monotonic transformations in utility functions
How is the Euler Equation for Consumption derived from from intertemporal budget constraint and lifetime utility function in basic macroeconomics
Weakly monotone preferences with singleton indifference curves: do any of them admit a utility representation?