Investopedia article - What is the utility function and how is it calculated?


Article describes ordinal versus cardinal utility:

When economists measure the preferences of consumers, it's referred to ordinal utility. In other words, the order in which consumers choose one product over another can establish that consumers assign a higher value to the first product. Ordinal utility measures how consumers rank one product versus another.

Economists take the utility-function concept one step farther by assigning a numerical value to the products that consumers choose or choose not to consume. Assigning a value of utility is called cardinal utility, and the metric used to it is called utils.

Article includes the following statement under limitations and benefits of utility function:

Of course, in reality, economists can't assign a true numerical value to a consumer's level of satisfaction from a preference or choice. Also, pinpointing the reason for the purchase can be difficult if there are many variables being considered.

Is this an accurate description of economic reasoning? Utility functions are models constructed by a particular economist who cannot assign a true numerical value?

To further develop my question, if it is not clear, in a scientific measurement there is a procedure which defines a standard unit of measure. Instruments are calibrated in terms of these procedures and then calibrated instruments are applied to assign the numbers with units in the models. Is there a standard procedure for assigning numbers to specify utils or is it constructed when the economist assigns numbers arbitrarily to the ordinal set?

Another article - Constructing a utility function:


Unfortunately, establishing the existence of a utility function does not produce a systematic way to construct it. In most circumstances, one is forced to design $ {\cal U}$ by a trial-and-error process that involves repeatedly checking the preferences. In the vast majority of applications, people create utility and cost functions without regard to the implications discussed in this section. Thus, undesirable conclusions may be reached in practice. Therefore, it is important not to be too confident about the quality of an optimal decision rule.


1 Answer 1


I think the Investopedia article does not use clear terminology which leads to confusion.

In Economics, utility can be split across two categories as ordinal and cardinal utility.

In the case of ordinal utility assigning numeric values is meaningless. For example, if A is proffered to B, assigning 200 utils to A and 100 utils to B is meaningless because that would imply that A is twice as good as B, but the ordinal utility can only measure preferences, not intensity.

In the case of cardinal utility, which measures both preference ordering and their intensity it is completely valid to assign some value like $u(A)=200$ utils or $u(B)=100$ utils. In this case, the numerical values can be assigned and they are, mathematically, 'true' numerical values.

However, from my reading of the Investopedia article, they seem to be referring to the measurability problem. That is an old discussion in economic theory that revolves around whether utility can be actually measured in some way (i.e. can we somehow scan a person's brain to measure precisely utility).

Now from a purely practical perspective whether the utility is measurable or not we definitely cannot measure in any direct way presently. Right now we can only observe people's choices and their revealed preference which is only solid basis for ordinal utility, not cardinal utility. However, this does not mean that utility would be immeasurable. In fact, famously Von Neumann and Morgenstern (1944) in their Theory of Games and Economic Behavior argued that utility even if we cannot measure it yet is measurable. The authors argued that utility can be viewed the same way as temperature in physics. Temperature also originally could not be measured beyond just saying it is getting colder or warmer but developments in thermodynamic theory later allowed us not just to cardinally measure temperature, but even allowed us to recognize that there is actually some absolute zero and absolute units temperature (presently even Celsius scale is actually a derived measure in terms of Kelvins).

Consequently, in the view of Von Neumann and Morgenstern utility should also be measurable and their view is quite influential although not accepted by all. If utility is measurable then there is a 'true' underlying measurable utility value to any choice. Whether we actually have instruments to do the measurements or not is irrelevant for the problem.

However, from my reading of the Investopedia article I think that what Investopedia refers to is the present practical inability to directly measure utility. I cannot see any indication from the article that it is taking stance on the measurability problem itself. Economists definitely don't go around hooking jumper cables to peoples heads and directly measuring utility from their choices (yet). Consequently, it is an accurate statement from a present-day practical perspective but not necessarily valid statement from the perspective of measurability problem.

Response to second question after edit:

The issue here is that true numerical value does not necessarily depend on measurement. As explained in this article from Science Campus site:

The measurement value (which is sometimes referred to simply as the measurement) is the value given by a measuring instrument and the true value is the actual value of the property being measured.

Of course, there are no instruments to measure utility, however that does not mean that there are no true numerical values for it. In fact, as argued by Von Neumann and Morgenstern (1944) in cardinal utility there will be some mapping of the underlaying preferences on the set of real numbers. Hence, cardinal utility actually does represent the 'true' values of utility (within the vNM framework at least).

However, Investopedia is correct in pointing out that economists do not measure it when constructing utility because (presently) there is no practically viable way that would in real life make it possible to create such mapping from the unobservable preferences onto real numbers. But just because we are not able to practically make such mapping does mean it cannot exist (again in past people were unable to map the true values of temperature onto real numbers - in fact for the most part of the human history such mapping was impossible until technology and science advanced enough to make it possible).

Furthermore, of course there are economist who are pure ordinalist but from my reading of the Investopedia article they are not taking pure ordinalist position rather they mean that we are simply unable to measure utility in practical sense.

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    $\begingroup$ Am I conflating or are you waffling? I think you have given a good answer except for the waffling when it comes to the distinction between constructing a utility curve based on working assumptions without measurements and taking actual scientific measurements. I understand that the revealed preferences are observations used to construct the utility functions from further working assumptions. $\endgroup$ Commented Oct 18, 2020 at 19:54
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    $\begingroup$ I gave your answer +1 vote and will likely select it as correct. For clarity I argue as follows. The concept of "true numerical value" has several potential meanings in this exchange: (1) all numerical values are true values; or (2) no numerical value assigned by a scientific measurement method can be taken as the true value (due to measurement error and to arbitrary choice of units); or (3) utility curves are constructed without the use of scientific measurements so the economist cannot assign a true numeric value. I would say (3) is accurate utility is not a "true" measured value. $\endgroup$ Commented Oct 18, 2020 at 20:17
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    $\begingroup$ So it appears that utility functions are made up or constructed from some systematic assumptions that cannot be verified by systematic measurements. In other words several economists might observe the same ordinal utility but are likely to construct distinct numerical models for a utility function. It would be like a scientist or engineer saying we can't measure temperature yet but if we could the curves in this experimental observation should look something like this constructed model. $\endgroup$ Commented Oct 18, 2020 at 23:47
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    $\begingroup$ I agree since I stated that ordinal relations or revealed preference is subject to independent verification or duplication using some standard measurement methods. I am convinced that ordinal utility means constructing an ordered set of preferences as tuples or similar logical structures; and that utility functions as math models do not assign number values via a systematic measurement procedure. I do not know how economists apply these models in a context where one can say "this model fails if (apply logic)". I want to comprehend reasoning methods in engineering, economics, and law. $\endgroup$ Commented Oct 19, 2020 at 1:20
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    $\begingroup$ @SystemTheory SE format is tailored to more narrow questions. That’s why we have too broad as a valid option to vote to close. I think originally your Q was nice and narrow but you keep expanding it and making it more broad and less fit for SE format - I think that’s why someone decided to downvote your Q (I have high rep to see vote totals). I won’t be changing my vote retroactively and I still find your Q interesting but expanding it to question about understanding the reasoning of all methodology underpinning utility will most likely just invite negative reactions and poor answers $\endgroup$
    – 1muflon1
    Commented Oct 19, 2020 at 10:27

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