# Is GARP trivially satisfied with only 1 good?

As with all revealed preference work, when the number of goods is greater than 1, then GARP (Generalised Axiom of Revealed Preference) is not always trivially satisfied. However, is it always the case that GARP is trivially satisfied when bundles only contain one good?

My take on it is that it is possible to violate GARP when the consideration set is only 1 good. However, empirically speaking, it should never be observed, as it really does not make sense from any rational point of view (ceteris paribus). So theoretically, it is possible to violate GARP in a 1 good economy, but this is something we would never imagine observing.

(I've been having this discussion with a supervisor, and he seems to believe that GARP should never be violated with one good as it makes absolutely no sense for it to occur, despite my argument that it is theoretically possible)

• What does "one good" mean exactly? GARP is a statement made with regard to choice data of the form $(\mathbf p,\mathbf x,y)$, i.e. consumption bundle $\mathbf x$ is chosen when prices are $\mathbf p$ and income is $y$. Does "one good" simply mean that $\mathbf p$ and $\mathbf x$ are scalars? In other words, are people still allowed to choose different quantities of the only good available at different prices? At the very least, are people allowed to not choose (or choose the empty set)? – Herr K. Apr 25 '19 at 1:43
• It depends what you mean by "trivially". By Afriat's theorem (trivial?) GARP is equivalent to the consumer making each choice as if was optimal (utility-maximizing). – Fizz Apr 25 '19 at 2:43

## 2 Answers

I assume by "GARP" you mean "Generalized Axiom of Revealed Preference". One can take GARP to be satisfied with one good, but it's a rather vacuous type of "satisfied"; with one good the concept "preference" don't really have any meaning. You can't have any preferences if there's only one option.

• I think Chris refers to having bundles that contain a single good, say apples. There is not only one option since you can have various quantities of apples, but there are no other goods. – Regio May 1 '19 at 1:36
• @GabrielMartínez GARP is based on an idea of a nondecreasing utility function. If you're not even assuming that, then you are far afield of standard utility theory. – Acccumulation May 1 '19 at 2:14
• When did I assume that the utility was not monotonic? I actually think that Chris is overlooking that the axioms of revealed preferences assume that the budget line is binding. It was not clear to me from your answer that you were addressing that point. – Regio May 1 '19 at 2:46

According to Kreps (2013, p.70):

In words, GARP is violated if any bundle chosen at given prices and income costs less at those prices than the level of income.

Hence with one good (say orange), we can construct the following simple choice that violates GARP: $$\begin{equation} (p,x,y)=(\2,0,\10). \end{equation}$$ In other words, we observe that someone chooses not to purchase the only good available when he could easily afford $$5$$ units of it. By definition, a bundle $$x^i$$ is revealed to be strictly preferred to bundle $$x^j$$ if $$p^ix^j (i.e. if the cost of $$x^j$$ at price $$p^i$$ is strictly below income $$y^i$$). But according to this definition, the above choice suggests that $$x$$ is revealed to be strictly preferred to itself (note that $$i=j$$ here since there is only one observation), which violates GARP.

• Thank you very much, I very much appreciate the example and the reference. – Chris Apr 30 '19 at 15:57
• This example assumed that preferences are not monotonic or, at least non-locally-satiable. If preferences are not monotonic (i.e. if the consumer does not use all their income), it doesn't really matter if there is one or more goods in the economy, it will be super easy to find examples that violate GARP. Not sure if that was your question since the answer does not seem to rely on the fact that there is a single good. – Regio May 1 '19 at 1:33
• @GabrielMartínez: OP asked whether GARP is trivially satisfied with one good. I gave an example with one good that shows GARP is not satisfied, trivial or otherwise. How does this not answer the question? Also, not all non-monotonic preferences violate GARP. For example, preference over a good $x_1$ and a bad $x_2$ is not monotonic, but could be consistent with GARP if one spends the entire income on $x_1$ and nothing on $x_2$. – Herr K. May 1 '19 at 2:21
• That is right, I said, it is easy to find examples where GARP is violated, not that it will always be. The example with one bad and one good satisfies local non-satiation which is sufficient to ensure that the budget line is binding. My point was that the axioms of revealed preferences (GARP included) assume that the consumer is spending all her income; when this is not true, it is super easy to find violations of the axioms. – Regio May 1 '19 at 2:49
• Thanks for the discussion, I believe both of you are actually arguing the same correct point! Many thanks. I take the point about the assumption of binding budget constraints, and when choices are made that are not binding, then its easy to find violations. I don't believe GARP itself actually implies budget constraints are binding. A sequence of choices can be found to satisfy GARP that are not on the budget line i.e. it should always be possible to find a chain of preferences that isn't a cycle. Unless we invoke Afriat's theorem, not sure we can say much more about binding constraints. – Chris May 2 '19 at 17:26