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If you have a single good, so that your commodity space is $\mathbb{R}$, then every increasing function is quasi-concave and even strictly quasi-concave. So any non-concave but increasing function fro …

This result is indeed a version of Berge's maximum theorem. If there is a continuous function $u:M\times H\to\mathbb{R}$ such that $x\preceq_e z$ if and only if $u(e,x)\leq u(e,z)$, one can derive the …

Not really. There are many compact metrizable topologies you can put on this space, but none that relate meaningfully to the structure of the problem. Let's look first at the case $A_1=[0,1]$ and $A_2 …

The Lagrangian is not really symmetric; something that is easier to see if you formulate it without the calculus implementation. First-order conditions for maxima and minima might look similar, but ma …

As Jesper Hypel has pointed out in a comment, the derivative need not exist. Actually, the function $a\mapsto x^*(a)$ need not even be nondecreasing. Here is an example: Let $g$ and $h$ be given by $g …

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$ is a connected sep …

No, the unique Nash equilibrium is the unique correlated equilibrium by a general property of two-player zero-sum games pointed out in: Forges, Françoise. "Correlated equilibrium in two-person z …

In general, it will not represent the same preferences. There seems to be confusion on what "monotonic transformation" means in this context. It does not have much to do with monotonic preferences. We …

Here is how one can show that Definition 1 implies Definition 2. We do the contrapositive, we show that if Definition 2 fails then Definition 1 will fail too. Suppose that $x\succ y$, but for every $\ …

I know a lot of examples of mathematical results that have been first developed in economics, mostly result in set-valued analysis and convex analysis. My ignorance of engineering and physics keeps me …

It is not that surprising if you have the right intuition, but let's make sure we consider it unsurprising for the right reasons. Roy's identity can be rewritten as $$x^*_{i}(\text{p},m)\frac{\partia …

If you have time and patience, "Foundations of Mathematical Economics" by Michael Carter is great. The book consists mostly of exercises that let you, broken down in manageable steps, prove many landm …

Clearly, mathematics could never cover the full richness of the human experience. …In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the …

No. And yes. For any set $X$ we have (by definition) $$X^k=\underbrace{X\times\cdots\times X}_{k\text{-times}}=\{(x_1,x_2,\ldots,x_k)\mid x_i\in X\text{ for }i=1,\ldots,k\}.$$ Now let, for example, $m …

As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems. So let $f:\mathbb{R}^n_+\to\mathbb{R} …