# Usefulness of the Convexity Axiom

I'm asked to write an essay supporting the statement which says the convexity axiom has little economic content and should be eliminated from the economic models of consumer theory.

I'm supposed to find literature which supports my point with theoretical and empirical evidence but I've been searching for weeks and all I have found so far are papers which stress the importance of the convexity axiom. Perhaps you all can assist me in locating the material I'm looking for.

Regards

• It would help if you would explicitly define the convexity axiom you are referring to. – BB King Oct 27 '15 at 11:07
• It has nothing to do with the question, but I would be very curious to know what program tasks you with writing an essay about an axiom. – Giskard Oct 27 '15 at 11:57
• I have found this (pretty dumb) treatise that may be of help to you: samselikoff.com/writings/… Please in return do share what force is compelling people to write these. Grad school assignment? MA in what? Finance? – Giskard Oct 27 '15 at 15:14
• In case convexity of preferences is meant: Usually in consumer data we observe that individuals do not consume a little bit of everything but have a lot of zeroes in their consumption vector. This behavior does not fully exclude convex preferences but may be a step in the right direction. – HRSE Oct 28 '15 at 1:30
• You certainly drew a short straw. Of all the standard assumptions about consumer preferences (completeness, transitivity, local non-satiation, continuity), convexity is probably the one I'm most persuaded is relatively innocuous to assume. – Theoretical Economist Dec 7 '16 at 0:21

Consider a set of consumption bundles $X$, which is convex and let $\succeq$ denote 'at least as good as', then convex preferences may be defined mathematically as follows.

Convex preferences. For any $x,y,z\in X$ and $\lambda\in[0,1]$ we have that $$x\succeq z\text{ and }y\succeq z\Rightarrow \lambda x+(1-\lambda)y\succeq z.$$

Informally we may say that convex preferences captures the concept of averages being at least as good as extremes.

Critique 1. Below is an excerpt from the direct link where the author points out the fact that under certain circumstances, the assumption of convexity may not be adequately assumed as a property or preferences.

That is, [convex preferences says that] if an agent is indifferent between $x$ and $y$ and has convex preferences, then she will like $x/2+y/2$ at least as much as either $x$ or $y$. Of course, this doesn't always make sense. You might like a beer or wine, but not a mixture

However, it depends on the interpretation.

Whether convexity makes sense often depends on the interpretation of the goods space. For example, if the components of $x$ are rates of consumption, then a half-half mixture of beer and wine might mean drinking beer half the time and wine half the time. Convexity of preferences seems more plausible in that interpretation than in the previous one. Similarly, some find convexity easier to rationalize if the "goods" are more highly aggregated – for instance, if the goods are "food" and "clothing," than if goods are highly specific.

Critique 2. The link directs you to McCloskey's critique of economics. She criticizes how economists focus too much on assumptions and mathematical proofs and too little on empirical facts. In this sense, if microeconomics or mathematical economics is to be about general properties of consumer behaviour, then the assumption of convex preferences is not always suitable from a methodological perspective, as non-convexity may hold in a given economy. Also, following McCloskey, one may say that the assumption of convex preferences, while it may guarantee that certain mathematical propositions are true (i.e., one may deduce them from the assumption of convex preferences and simpler axioms), does not by itself say something about economic behavior. It surely implies certain economic behavior $\mathcal B$, but we cannot from the observation of such economic behavior $\mathcal B$ draw the conclusion that economical agents' preferences are such that they are convex over economical entities like goods.

Critique 3. Instead of assuming convex preferences, one may study the wants and sentiments of individuals, as Adam Smith does in his Theory of Moral Sentiments and later in his Wealth of Nations, or study cognitive biases, like Daniel Kahneman does in his best-selling Thinking, Fast and Slow. These two authors does not, to my knowledge, directly criticize the concept of convex preferences, but they show how economic analysis performed in another way may lead to other conclusions about the economy. If their conclusions do not agree with conclusions derived from axioms like that of convex preferences, then there is implicit criticism in the sense that there is a mismatch in theoretical conclusions.