# Indifference curves and equivalence classes

Last week in class, my professor described indifference curves in the context of modelling consumer choice with multiple goods. He defined a partial-ordering operator $$\prec$$ and an equivalence operator $$\sim$$ to manipulate tuples that satisfied the budget constraint problem. So far, so good.

But then we went on to explore the behavior of indifference curves and I noticed that the $$\sim$$ was assumed to be reflexive, symmetric, and transitive. We learnt that indifference curves never intersect and that there is a completeness property that says that for all the tuples in our domain space belong to an indifference curve.

At this point, indifference curves looks like textbook equivalence classes to me, but I would like confirmation and perhaps some context from an economics perspective.

First of all, why not call them "equivalence classes" or openly saying that "indifference curves are equivalence classes". Second, does that mean that if I prove something about equivalence classes, then it automatically transfers to indifference curves in the context of this economic model?

• Yes, indifference curves are exactly the equivalence classes corresponding to the equivalence relation $\sim$. – Michael Greinecker Jan 18 '19 at 13:44

Yes, indifference is an equivalent relation.

That was covered in other answers and comments. I'm here to point out that preference relations are not partial orders. A partial order has the property that

$$a\leq b, b\leq a \implies a=b$$

This not true precisely because of indifference. Rather, preference is typically assumed to be complete and transitive which means it is a total order on the quotient $$X/\sim$$

Michael Greinecker's comment basically answers the second question, but not to leave the question unanswered, let me try to answer your first question.

In most economics courses, we don't get mathematical training more than calculus and linear algebra. That means we don't know what an equivalence class is. So, for economists, explaining that indifference curves are equivalence classes won't mean anything.

• I think your last sentence is perhaps a bit too pessimistic about economists' understanding of mathematics in general. Surely many economists may not immediately realize that indifference curves are what mathematicians call equivalence classes. But I doubt that most would still fail to see the connection after you explain what an equivalence class is to them. – Herr K. Apr 8 '19 at 1:43