# Economics and Euclidean geometry

I am looking for economics papers which use concepts, techniques and theorems from geometry. I am mainly interested in Euclidean geometry, the kind of material that is taught on high-schools (e.g. proving theorems about triangles, squares, circles, etc.). But, other kinds of geometry are also interesting.

I looked at some papers about land economics, thinking that it geometry can be useful there (after all, the word "geometry" means "measurement of land"). But, the papers I found used mostly advanced measure theory and topology. I am looking for more basic geometric methods.

• There is an academic economist from Cyprus that has mapped basic aspects of Euclidean Geometry to some of the foundations of Economics theory. For the moment I cannot remember a specific reference but I will look it up. Commented Jun 26, 2015 at 20:08
• There is a theory of spatial voting with Euclidean preferences. See here for the paper that started it all. Some of the literature gets rathert echnical though. Commented Jun 27, 2015 at 14:07
• There a paper by Lee Smolin (a physicist) that applies gauge theory to economics; gauge theory is geometric, but its advanced geometry, so doesn't fit the 'basic geometric methods' requirement. Commented Sep 21, 2017 at 12:46

Another example is Economies with Public Goods: An Elementary Geometric Exposition by William Thomson, which relies extensively on the geometry of equilateral triangles.

In general, William Thomson is known for making a heavy and interesting usage of geometry in his papers. See for instance a recent working paper of his : Compromising between the proportional and constrained equal awards rules (although this one might be further from basic high-school euclidean geometry than what you're looking for).

Other somewhat similar examples include papers relying on the geometry of simplices. An example I came across recently is

but there are many others, starting with the exposition of the expected utility theorem in Mas-Collel, Whinston and Green.

I am also tempted to point at the following recent paper, although it definitely uses geometric tools which go far beyond high-school euclidean geometry:

I don't have a specific paper for it (any intro to econ textbook would do), but I would also add the classical graphical measure of consumer and producer surpluses in ECON 101. (Especially for linear demand and consumer curves where is simplifies to measuring the area of a triangle. Can't do much more "measurement of land" than that)

UPDATE 19-9-2017

Got it. I remember browsing the pages of this book. But it was in Greek, and it does not appear to have been published also in English:

Euclidean Economics: A Mathematical Method for Construction and Application of Economics Models

"The book establishes that economics can be studied and utilised on the basis of a minimum of fundamental hypotheses and the general laws of mathematics. Starting from 12 major propositions, it proves that closed exchange networks are replicated by systems of 2n first-order non-homogeneous linear differential equations, which simulate household, business, bank and government receipts and payments. Such networks are defined in the (R2n, t) vector space, where n is the number of sectors or markets and countries or currencies, while t is the elapsed time. Uniqueness, stability and controllability of the steady state as well as takings and outgoings are functions of behaviour coefficients and policy parameters. Annual sales of services, products and securities divided by average salaries, prices and values yield the man hours, output items and liquidity units traded. Demanded quantities do not coincide with available quantities. Potential inflation or deflation is proportional to the weighted deficit or surplus of market goods. Digital animation of the endogenous variables is possible on the basis of historical, projected or hypothetical data about the exogenous variables."

Author: Sophocles Michaelides
Language: Greek
Publisher: Mesogios Publications – Ellinika Grammata
ISBN: 9963-9176-0-7
Year: 2006

The summary does not state it clearly, but the "12 major propositions" are Euclidean, as I recall.

• Wow this is really cool.
– EconJohn
Commented Sep 20, 2017 at 4:04

Harry Johnson frequently applied euclidean geometry to trade theory and growth theory. Have a look at his many papers and books. For a more or less randomly chosen starting point, try his "Trade and Growth: A Geometrical Exposition".

There are some papers of Stefan Behringer from University of Bonn.

Optimal Harvesting of a Spatial Renewable Resource published in Journal of Economic Dynamics and Control

As the title indicates, there is a geometrical application, at least there is the notion of space.

Another paper that I remember is "Spatial dynamics and convergence: the spatial AK model" by Boucekkine et al. published in Journal of Economic Theory.

I think you can reach another strand of literature on space and mainly on geometric technics used in economics.

Here are two problems (not papers, though the results are probably used in many papers)

1. In the case of a three asset Markowitz portfolio optimization problem the feasible set of portfolios in $\mu$ / $\sigma$ space can be shown to be an ellipse.
2. When a demand curve is a straight line the marginal revenue curve bisects the demand curve.

One random example is this paper which uses similar triangles in the analysis of tax burdens. I think at least that bit of the paper (section 2) could be used in the high school context:

Hines, Hlinko, & Lubke (1995). "From each according to his surplus: Equi-proportionate sharing of commodity tax burdens" (ScienceDirect link, gated)