Some quick notes on basic undergrad microeconomics from a mathematical perspective.
Basic consumer and producer theory is all about constrained optimization, using basic calculus. (Lagrange multipliers, etc.)
“marginal” means “local” – ie, computed using derivatives and nearby infinitesimal behavior.
Constraints are typically linear (budgets, etc). Functions to optimize are highly nonlinear. Choice of function to optimize depends on model.
Common restrictions on these functions: monotonicity, convexity, quasi-convexity. Functions are (of course) smooth except for finitely many discontinuities.
A surprising number of intuitive qualitative results (monopoly behavior, taxation effects, etc) fall out of these simple models.
Quantitative results at this level are almost all trivial.
To build more sophisticated models with incomplete information, multiple organized actors, adversarial competition, etc, either need to fall back on an “infinite-agent approximation” (perfect free market, perfect information, etc) or move into more advanced theory.
Game theory becomes highly intertwined with these more advanced models.
Using basic microeconomics to tackle large scale real problems (housing markets, labor markets, capital markets) leads to quick results but makes me very nervous.