# What are the correct utility functions?

It is common to talk about utility functions. For example in a universe with only two goods, we might assume each person (or group of people) carries a function $$u(x,y)$$ in their heads. When offered some baskets $$B_j$$ containing $$x_j$$ amount of good 1 and $$y_j$$ amount of good 2 the person will select the basket $$B_k$$ that maximises $$u(x_k,y_k)$$.

I wonder has anyone ever tried to find what there utility functions look like experimentally? Of course they might vary from person to person, and depend on the type of goods. But if the functions are a realistic model it should be possible in principle to see what they look like. Has there been any attempts to do so?

The closest I can find are regression attempts, where data is gathered and then fit to some class of functions. For example linear regression tries to draw the straight line (hyperplane) that best fits the data. This is not what I am looking for however, since regression assumes the type of function in advance. I'm looking for something as simple as offering a bunch of baskets, plotting the data, and comparing to a bunch of different types of functions.

It would be really tough to find papers (other than structural models) that empirically determine a particular utility function. Because of their subjective nature, it is incredibly tough to obtain observable data to estimate them. So researchers study the primitive preference relations that utility functions represent. Arguably, the preference relation is what you want to study when talking about rationality and choice. Utility functions are useful insofar as it opens up a tool-box of real analysis tools which a binary relation doesnt typically allow.

So you should rather look for papers that test rationality from observable choice data. I have elaborated a bit on this in an answer to a related question: Click Here

• Thank you. I'll look up GARP's papers. Jul 26, 2020 at 12:25
• In the meantime I found this: researchgate.net/publication/… Jul 26, 2020 at 12:27

Good question!

I would like to address a more general point about utility functions, which is often overlooked. Suppose that such an experiment were conducted on a group of people. Each of them is offered a "basket" containing some combination of goods and it is somehow measured how the utility function for each person depends on each of the goods. The question is now:

By determining these functions, have we obtained any interesting/useful information at all?

There is a case to be made that the answer is "no!". To illustrate the point, suppose that we are looking only at one person and the utility they gain from owning bottles of water. Even if their utility function is somehow mapped out, it will only tell us how that particular person valued a particular type of bottled water at a particular time and place. It tells us nothing about how it looked in the past or how it will look in the future or how it changes if he goes outside the laboratory. It tells us nothing about how he might value mineral waver vs. tap water or how other people might behave differently. None of this changes if we introduce more goods (bicycles, hamburgers, etc.) or more test subjects.

In light of these caveats, you may wonder why the utility function is such a widely utilized concept. Its prevalence must be seen in the light of an ongoing trend that arose after WWII of "mathematizing" economics. Critics of this development argue that there is no reason to use complex mathematical formulas, statistical tools and regressions to say something that could be said in plain English. Why introduce an indeterminable, N-dimensional function and talk about optimizing it, when one could just say: "At any given time, people do things that satisfy their most urgent need in the best possible way!" and gain the exact same amount of insight? Some schools of thought actually take this statement as an axiom and use it to explain economic phenomena without introducing abstract math.

• The hope is to do a bunch of experiments and conclude that (in some special cases) people behave as though they have utility functions of the form (for example) $x_1 ^{a_1} \cdot \ldots \cdot x_n ^{a_n}$ where $x_1,\ldots, x_n$ are the amounts of eash resource and the parameters $a_1,a_2,\ldots, a_n >0$ vary from person to person. This would be an empirical result of course. Jul 21, 2020 at 21:02
• Then in order to make an educated guess at a new person's entire range of preferences we only need to offer some small number of baskets and solve the resulting equations for the parameters. Jul 21, 2020 at 21:05
• So far I cannot find even a single experiment. Granted this might be more psychology than economics, but I have asked the question on that forum too! Jul 21, 2020 at 21:06
• -1 You might wish to post a new question along these lines. But I can't see how what you have posted answers Daron's question. Jul 22, 2020 at 11:08