# Inductive proofs in economics

Are there any thorems in economics which use proof by mathematical induction to prove them?

I've noticed that there is nothing (to my knowledge) from a standard PhD microeconomic theory sequence which uses this method and am wondering why this might be the case.

• Do you mean 1) induction as a scientific method, involving reasoning from specific observations to general conclusions, or 2) mathematical induction, involving showing that if a property holds for $n$ then it must also hold for $n+1$? The words "theorem" and "proof" tend to suggest (2). Jun 21, 2023 at 20:16
• @AdamBailey mathematical induction. These types of proofs are fun and I'm wondering if there are any applications in economics.
– EconJohn
Jun 22, 2023 at 2:12
• I'm not a theorist, so I don't know. From my micro theory courses, I recall the cases with 2 agents (firms) were often very different from cases with many agents, so I'm doubtful induction is relevant in most settings. Jun 22, 2023 at 10:57
• It could be helpful to specify mathematical induction in the title and the body. Jun 23, 2023 at 9:56
• There are quite a few proofs in MWG that rely at least partially on induction. For example, Zermelo's theorem and Prop 9.B.4. Jun 24, 2023 at 15:26

Mathematical induction is rarely relevant to economics due to the combination of three considerations:

1. Mathematical economics is very often concerned with variables taking continuous (or so large as to be effectively continuous) rather than integer values. Consider for example: output, demand and supply of commodities such as wheat or oil; GDP, consumption, investment, savings, imports and exports.
2. Where economics is concerned with integer variables, the focus of interest is often in identifying the optimum number (relative to some criterion) and not in showing that some proposition holds for any (or many) values. Consider for example: a luxury car firm assessing what number of cars produced in a period will maximise its profit; a planning authority deciding how many new homes it would be socially optimal to allow to be built on a given site.
3. Even in cases where it is desired to show that a proposition holds for any number, there are often simple non-inductive methods to prove such a proposition. For example, a firm producing discrete goods may be in a situation such that, given its fixed and variable costs, and given the demand curve it faces, it will not be profitable at any production volume. Given the cost and demand curves (and treating them as continuous despite the discrete nature of the goods), such a result would typically be established by deriving the profit function, using calculus to find its maximum, and then showing that profit is negative even at that maximum.

Having said that, there are, as Herr K points out, exceptional circumstances in which mathematical induction can usefully be applied in economics.