# Do claims in economics require proofs?

My professor stated four axioms in class

• Scarcity
• Rationality (aka purposive behavior)
• Stable Preferences
• Equilibrium

I don't understand what these axioms are for. I am used to axioms with distinct mathematical properties (eg definition of a vector space) and these axioms don't have that. I had an exam question that said

True/False/Uncertain. The Law of Demand is a direct result of diminishing marginal utility.

and I was supposed to use the scarcity axiom. But what does it mean to show this claim is false? The principle of scarcity is just that "resources are scarce". That's not mathematical. How can I prove the above statement is false using it?

My Question:

How can I tell when a question requires a proof versus merely conceptual reasoning / explaining? What is the difference between the two?

• I guess that is something only your professor can answer. It is possible to do consumer theory in a mathematically rigorous way. Apparently, that is not the approach taken by your professor in the course. – Michael Greinecker May 13 '15 at 0:02
• A bad question, no they are not axioms. The law of compensated demand is a consequence of the weak axiom of revealed preference. – user157623 May 13 '15 at 0:43
• It's a very good question. The answer is essentially, not yet. There is a great deal of current economic theory, especially that of scarcity and equilibrium which rests on faulty assumptions, and this becomes glaringly obvious if you take the trouble to go back to the original papers. As a student you unfortunately will have to suspend disbelief in order to pass the course, but as a researcher you should regard this as a huge opportunity. – Lumi May 13 '15 at 16:57
• A good source to see the theory developed in a genuinely axiomatic fashion is "microeconomic theory" by Mas Collel, Whinston, and Green. That is the standard graduate econ theory text. – Ubiquitous May 13 '15 at 21:18
• See this similar question about math and economics: economics.stackexchange.com/questions/511/… – jmbejara May 13 '15 at 21:26

These assumptions actually have distinct mathematical ground, but your professor likely decided to leave this out for (of course bad) pedagogic reasons.

• scarcity: society has insufficient productive resources to fulfil all human wants and needs. Scarcity is the motive behind economics. If it is cheaper for everybody to get a given good from the ground than to produce it, then they are not economic goods, they don't have a market value and are not treated by introduction classes in economics. This can be stated as: for all actor, for all (studied) goods, the cost of acquiring the good is strictly positive.

• Rationality: people adopt the best actions to achieve their goals. Usually, this is translated in maths by introducing a utility function from the possession set to a measuring set (usually $\mathbb{R}$), and stating people always act in order to maximize their utility.

• Stable Preferences: Using the same utility function as above, this states that this function does not depend of time. This implies that if you prefer having a kilogram of soup than a kilogram of raw potatoes today, it will still be the case tomorrow.

• Equilibrium: Here I have to guess, but given your example I suppose you're studying equilibrium on the market. I guess what your teacher meant is that you study the economy at a time when the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers.

All these definitions are not guaranteed to match with those implied by your teacher, but are consistent and widely taught in economics. When not otherwise specified, economical answers require referring to a consistent framework of axioms from which mathematics should be able to guide you to the solution you're trying to state. Different models can show different solution, but one model should give only one solution (which can be composite: "people buy more or less goods at the end of the month" is a valid solution, but "people buy more goods at the end of the month" and "people buy less goods at the end of the month" should not be obtained from the same data with the same model...).

However, economic practice does not require you to re-demonstrate any known result (mostly, the results which you know a name for can be used as starting points for your answers). If you're not confident with "explaining", you need to be able to think about a one-line mathematical justification for each explanation you give, and thus you will ensure your explanations give you the expected marks.

Without going into rigorous formulations in the language of the logic discipline: outside the mathematics discipline, we can think of an axiom as something that is accepted as fact without rigorous proof, usually because it appears self-evident, being observed countless times, in many different historical, social, cultural etc contexts. What is the difference from an assumption?

Well, by uttering the word "assumption", one already acknowledges that its generality is much less than that of an axiom. It may be an assumption that appears to hold widely, but not with the universality that an axiom carries. So changing assumptions is "fair game" -we try out different assumptions, to see what happens. On the other hand, ignoring an axiom, although certainly not "forbidden", casts a shadow of doubt on any results that will be obtained thus.

Under this view, three out of four of what your professor stated as "axioms", can indeed be seen as such, in my opinion and in the loose sense of the word we are applying here.

SCARCITY: The economics discipline would not exist without Scarcity: both efficiency and distributional issues would be non-existent in the long-run, in the absence of scarcity.

PURPOSIVE BEHAVIOR: In such abstract formulation, I could make it a theorem stemming from the axiom of Scarcity and the biological facts about the human body. But it is more convenient to treat it as an "axiom" (again, in the rather loose sense we are using the word here). Note that I avoided the word "rationality" because it has also a strict use in economics and then it means something different all together (we call preferences "rational" if they are complete and transitive, that's all -nothing to do with purposive behavior).

EQUILIBRIUM Why an axiom? Because it essentially reflects survival (biological or other), in an almost analytical sense (i.e. devoid of meaning). I understand "equilibrium" as a tendency, not as an always-holding situation. Also, an older answer of mine applies.

But in no way should we treat "stable preferences" as an "axiom"- on the contrary, it is an observed fact that preferences can be state-dependent, but also and more widely observed, time-varying (are your preferences the same when young, compared to when middle-aged? I don't think so). And there is a lot of economic research going on with non-stable preferences (the simplest I can think of is a time-varying pure rate of time-preference in growth models). Here we have a clear case of an "assumption".

• A number of experiments question purposive behavior among human beings, so I think your définition of axiom is just too strong for ecpnomics – VicAche May 14 '15 at 10:49
• @VicAche In what sense do they question it? In the sense that it appears that in some instances , a human does not exhibit purposive behavior, or in the sense that a human lives his life exhibiting predominantly non-purposive behavior? And even if the latter, how many humans, amongst the billions that exist? I wrote that purposive behavior essentially is a "Theorem" stemming from the biological survival drive embedded in all living things. Also, in Social Sciences no concept (like "axiom" or "law") has the absolute character that it has elsewhere, like in Physics and Mathematics. – Alecos Papadopoulos May 14 '15 at 11:01
• Well you should mention this in defining axiom. "usually because it appears self-evident, being observed countless times, in many different historical, social, cultural etc contexts"... Nope. In economics an axiom is an assumption that is very popular, nothing more. – VicAche May 14 '15 at 11:18
• For the second part, most tested human being do not make full use of a dominant situation in a Nash game. en.wikipedia.org/wiki/… – VicAche May 14 '15 at 11:19
• @VicAche On your first comment, this is not a description of an axiom, but of a prejudice. Of course, one can argue that there are no axioms in Economics, only prejudices. On your second comment, I do not subscribe to the view that in order to increase my probability of survival, someone else's survival probability must necessarily decrease. – Alecos Papadopoulos May 14 '15 at 12:44

The behavior of any theoretical agent requires a traditional proof. These are the only ones fit which such a proof can be done, and serve as comparative benchmarks or predictive models.

Explaining, and/or empirical evidence is needed for any real agent, because they can't be "solved".