# Does concavity of the utility function has any bite?

A utility function in general has only ordinal meaning, any monotone transformation preserves the order isomorphism of the underlying preference ordering. However, there are several econometrics papers, that estimate demand with concavity shape restrictions on the utility. Is this justified? My first thought is that yes, that the key idea is that there exists a concave utility representation, and this generates the dataset, however it worries me the identification issue since there is infinitely many of these utilities not all of them concave. In short, does it make sense to require concavity restrictions on the utility function in econometric work?

This post shows clearly why in the world of "standard" ordinal utility, concavity of a utility function cannot obtain an economically meaningful interpretation, although it may be useful as a mathematical property.

But "standard" ordinal utility is not compatible with Econometrics, because Econometrics deal inherently with situations where there exists uncertainty, and in a framework with uncertainty we move from "fully ordinal utility" to Expected Utility theory, where properties like concavity have economically meaningful content -they express attitude towards risk (as well as "preference intensity", as this laborious post shows).

The concavity assumption/restriction is made because there is universal consensus that the vast majority of people exhibit risk aversion in their economic behavior.

• I was about to answer with a link to your previous answer on this topic until I saw that you already did it. Nice post. I do, however, believe that the statement that "'standard' ordinal utility is not compatible with econometrics" is perhaps too strong. For example, perhaps you are not considering the discrete choice literature (see, e.g., Daniel McFadden's work). Apr 21 '15 at 3:07
• @jmbejara Thanks for pointing that out. I am revisiting discrete choice lit. and will come back. Nevertheless, the justification of why the concavity assumption is made in Econometrics stands (from its positive angle): researchers have in mind expected utility theory. Apr 21 '15 at 11:59
• But still there is one issue, take a preference relation that has a non convex upper contour set, then they do not admit a concave utility function so convexity of the upper contour set has empirical content. By extension the existence of a concave utility representation means that the the preference has a convex upper contour set (i.e., the preference is convex). Apr 21 '15 at 13:51
• @user157623 Are you asking whether we can remain totally agnostic about preferences when constructing an econometric model? I don't think we can. Apr 21 '15 at 14:09
• No, what I ask is that if it is meaningful to impose a shape restriction when estimating demand functions in the form of concavity of the utility function. I understand your explanation of expected utility but in the previous comment I suggested that the convexity of preferences has as an implication that There Is a concave utility representation. By the contrapositive if there is no concave utility representation then the preferences are in fact not convex, thus making meaningful to impose such restrictions from the point of view of empirical relevance. Apr 21 '15 at 15:32

I think that the way marginal utility is frequently used is that there is an assumed unit of measurement. For example if in the utility function $$U(x,y) = v(x) + y$$ $y$ denotes income spent on other goods then the unit of measurement becomes money, as one unit will always increase the utility by one. In this case $MU_x(x,y)$ is actually $MRS(x,y)$ because $MU_y(x,y) = 1$, so the two are equal. The concavity of $v(x)$ will mean that the preferences w.r.t. $x$ and $y$ are convex. This remains unchanged if you perform monotone transformations.

I am afraid I cannot give advice on econometric applications.

• This assumption is common in applied work because we do not observe the whole consumption bundle, in that sense I think you are right but I think that in that case we are testing quasilinear utility not a general u Apr 20 '15 at 23:28