Mathematical Psychics

Mainstream microeconomics assume the "end" of human behaviour is to make oneself as happy as possible. In economics-speak, this is to say consumers want to maximize utility. Utility is the catch-all term for happiness and/or benefits of any kind. The problem of utility maximization naturally leads to the question: how does one measure utility?

Difficulties arise immediately when we consider how we should define utility operationally, i.e. how do we define it in a way that allows it to be measured? But that leads to further questions of what we mean by measure? The following example highlights some of these difficulties.

A friend of mine recently observed that video games cost around $50 and playtime usually lasts over 20 hours, whereas movies cost around $10 and lasts about 2 hours. The cost of "fun per unit time" for video games would be around $2.50, whereas for movies it is around $5. (We just divide the cost by the hours of fun it is estimated to provide). He asked me, if the cost of fun per hour is higher for movies than for video games, then why do movies sell more than video games?

A quick look at the empirics shows that the observations are accurate at least for aggregate figures. Counting movies as $10 per viewing, there are significantly more movie views than game purchases, comparing the highest grossing films and best-selling video games. Of course, we cannot determine how these figures are aggregated and how they handle issues such as exchange rate of foreign currency and inflation in these quick statistics. A more fundamental issue is that friend's argument focuses on the substitution of movies for video games because one is cheaper in terms of fun per hour, whereas the raw number of sales says nothing about each consumer's decision to purchase one or the other. [1] The problem that I want to focus on, however, is that "fun" as derived from video games and "fun" as derived from movies are different notions altogether. By comparing $2.5 video game fun per hour to $5 movie fun per hour, we are assuming that video game fun and movie fun are on equal footing. For most people, they are not.

Despite the fact that different sources of utility like video games and movies are difficult to compare, they are all lumped into the category of utility. And we begin to see how measuring all the different sources of utility, and most importantly, combining them into a single measurement is a very tricky business. Indeed, economists over the years have struggled to operationalize the concept of utility into a measurable quantity, and consequently, the role that utility plays in consumer theory has evolved over time.

During the late 19th century, economists under the influence of utilitarianism were convinced that utility is an absolute physical quantity that can be measured, i.e., phrases like "video games give Mary 3 units of utility" carry a meaning independent of how many units of utility Mary assigns to other forms of consumption. This idea led to works such as Edgeworth's Mathematical Psychics, [2] which attempt to make rigourous how utility can be measured in an absolute sense.

Of course, no amount of mathematics can rectify the fundamentally problematic approach of assigning absolute significance to numbers that simply do not carry any physical meaning by themselves. Utility is similar to ideas like mass and length in physics, which can be defined easily enough as the amount of matter in an object and the distance between two objects, respectively, but is more subtle to operationalize into a measurable quantity. And similar to the physical counterparts, if one says the length of some object is 3, we do not have any idea how long the object is at all! 3 could be in units of feet, meters, Planck lengths or light-years. As soon as we specify units, however, we immediately have an idea of how long the object is.

How units help us is that we have defined what exactly a meter is in terms of a physical object or occurrence. Historically, a prototype meter stick was kept in storage by the International Bureau of Weights and Measures, but to reduce uncertainty, now the meter is fixed relative to the speed of light. Other fundamental quantities like mass and time are similarly either directly defined by some physical standard or is mathematically related to other quantities that are defined by physical standards. Therefore, measurements make sense only inasmuch as they are relative to some established standard.

Later interpretations of utility acknowledge the fact that it is meaningless to speak of x units of utility. Instead the substitution interpretation of utility became the standard. This is an attempt to put preferences of goods relative to each other. Therefore, even though units of video game fun per hour and movie fun per hour do not carry meaning alone, their quotient, video game fun per unit of movie fun does. Economists call this the marginal rate of substitution of video games for movies, which means the minimum number of hours of video game playing a consumer is willing to give up to gain one hour of movie-watching. (the minimum and consumer "willing" requirements together force the consumer to be indifferent before and after this substitution). [3] This is certainly an improvement over measuring absolute utility, for now at least the substitution of goods is observable.

With the solution to one problem, however, arises another. With substitution, we can reasonably say measurements of utility with respect to one consumer is well-defined, as long as the consumer's preferences are "nice". [4] But does substitution provide a well-defined way to compare utility between different consumers, i.e. can we compare one consumer who claims he is willing to give up 1 hour of video games to watch a movie with another consumer who claims she is willing to give up 8 hours for a movie?

I claim that we cannot because the two consumers' values of an hour of movie watching may be inherently different, which is to say the basis with respect to which the two consumers are measuring the value of playing video games may be different, so measurements relative to the two bases cannot be directly compared.

The physics analogy here is say two observers are asked to measure the distance between two points using two different meter sticks. The first meter stick is made slightly longer and the second slightly shorter, so the same physical distance is measured to be longer by the second observer than the first. To each observer, using his/her ruler as the basis of measurement, their measurements are accurate, yet the differences in the meter sticks prevent the two observers' measurements from agreeing.

The fix to the physics case of measuring length is obvious: give everyone a standardized ruler! Unfortunately, this implementation doesn't translate well into economics because the equivalent would be to use a neural-scrambler to set all consumers' values of movies to be exactly equal. Manipulation of preferences to achieve mathematically nice results is really not what economics is about.

Surprisingly, even the physics fix of standardizing the rulers used by the two observers is not enough. One of the consequences of special relativity is that there is no universal rest frame, so there is in fact no physical grounds to believe a single set of measurements should be taken as the absolute standard in physics either! [5] Physicists deal with this problem by exploiting invariant quantities, i.e. quantities that are independent of the observers' state, such as Einstein's postulate that the speed of light is the same in all inertial frames. Along with invariant quantities are transformation rules that give a mathematical connection between quantities made by observers in different states. By applying the correct transformation rules, one can in fact compare observations made by observers in two different locations, traveling at different speeds relative to each other, etc.

Economists cannot emulate the physicists' solution. As far as I know, there are simply no invariant quantities that economists can take advantage of. (Setting everyone's value of movies as the same amounts to arbitrarily setting an economic invariant, but then again, the keyword is arbitrarily). Without an invariant, it is not possible to derive transformation laws that connect consumer A's value of a movie with consumer B's value of a movie, and cross-person substitution rates remain painfully incomparable.

This inherent incomparability is usually silently neglected in economic works. For example, every time that the utility functions of individuals are aggregated to derive the social welfare function (i.e. the well-being/utility of the society as a whole) we are assuming that utilities of different people can be added, which would require that utilities be comparable in the first place!

Perhaps the incomparability is just a philosophical nitpick and doesn't really affect the end result of our work? Examples of carelessly comparing measurements made in different reference frames in physics, and the host of paradoxes they lead to, should convince us otherwise. [6] On the other hand, relativistic effects in physics only become apparent in real life when we deal with objects traveling near the speed of light, so for most phenomena on earth, classical mechanics still holds well enough. Perhaps no difficulties arise in economics currently because we are not dealing with situations that approach some yet-unknown limit.

Economics and utility theory have a "can't live with it; can't live without it" relationship. The shaky foundations of utility measurement is one of the main criticisms toward modern economic theory, and economists are always searching for new ways to make the concept of utility or preference more rigourous. Barring some miraculous invariance relation, the current trend in experimental economics and neuroeconomics essentially boils down to the need to find more systematic ways to characterize preferences. Yet, no matter how we look at preferences, economics cannot do away with utility because "the point" of economics is to allocate scarce resources such that utility, happiness, fun (no matter what word you use for the concept) is maximized. Without considering utility, the enterprise of resource allocation, and hence economics, will be for naught.


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[1] The aggregate statistics merely give the number of video games versus movies sold, but does not shed any insight on the degree of overlap between the video game and movie market. It is possible that there is great overlap between the two markets, suggesting that consumers actually tend to buy both goods together as opposed to choosing one over the other as friend's claim tacitly assumes. Even if the market overlap is small, the fact that both goods bring in decent numbers is sales suggests that there are probably better reasons for the divide in the markets than the substitution effect. For example, it may be that younger generations play more games, and older generations watch more movies.
[2] No joke, I seriously it said Mathematical Physics when I first happened upon the title. 
[3] Below I provide the mathematics behind how utility gives rise to the marginal rate of substitution:
Let x be hours of video game playing and y be hours of movie watching, and U be the utility of the consumer. Assuming a consumer has well-defined utility that is affected by movie watching and video game playing, so the utility U is a function of x and y, i.e., utility is denoted U(x,y). The first-order total differential is

dU = (∂U/∂x) dx + (∂U/∂y) dy.

Setting dU = 0 and rearranging we get 

- dy / dx = (∂U/∂x) / (∂U/∂y)

The right-hand side is the quotient of the marginal utilities of video games and watching movies, which is the ratio of the raw units of utility gained from playing an extra hour of video games and from watching an extra hour of movies. (In this context, U takes on real numbers, but should not be interpreted as an existing, acceptable measurement of utility in the sense that the numbers that U take on have a physical meaning. Instead, one should think of U(x,y) as the image of an order-isomorphism U, between the totally ordered set of the consumers' preferences to the real line). We see this is equal to the left-hand side, which is the hours of movies the consumer is willing to forgo (hence the negative sign) for one hour of video game playing -- the left-hand side is certainly a physical quantity. Since we assumed dU = 0, we know this equality holds when the consumer is indifferent before and after the substitution.
[4] By "nice" consumer preferences, we specifically mean preferences that can be cast into a total ordering, i.e. reflexive, antisymmetric, transitive and complete preferences. Of particular note is that these properties force the non-existence of cycles in preferences, e,g. situations where a consumer prefers x to y, y to z, but z to x. 
[5] Special relativity is covered in almost every introductory mechanics or electricity and magnetism text. Kleppner & Kolenkow's An Introduction to Mechanics, and Griffiths' Introduction to Electrodynamics both offer illuminating discussions of the implications of special relativity on measurement. 
[6] Length contraction, time dilation are good examples. Of particular note is the relativity of simultaneity, which leads to paradoxes that seem to defy our notion of causality if we do not treat it carefully!