A mathematical approach to economics

Solution 1:

Be careful of what you wish for... Bernt Stigun's book "Towards a Formal Science of Economics: The Axiomatic Method in Economics" probably goes above and beyond of what you had in mind when asking for rigor. He starts with mathematical logic (first-order languages)...

http://www.amazon.com/Toward-Formal-Science-Economics-Econometrics/dp/0262192845/

Also I have a vague impression some research in "epistemic game theory" may use the axiomatic approach but I might be wrong.

Solution 2:

Gerard Debreu's Theory of Value provides a mathematically rigorous analysis of the theory of general equilibrium in economics. It is considered a classic by many economists and won the author a Nobel Prize in Economics.

Here's the link to the pdf of the monograph: http://cowles.econ.yale.edu/P/cm/m17/m17-all.pdf.

Solution 3:

You might be interested in Economics for mathematicians by Cassels. It is more focused on what mathematicians find interesting, rather than what economists find useful.

Solution 4:

Axiomatic Theory of Economics, 1999, by Victor Aguilar

This is a short 26-page simplified exposition written at the undergraduate level:

http://axiomaticeconomics.com/Axiomatic_Economics_Article_by_Aguilar.pdf