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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 …

If you look at the case $x_1=x_2=\ldots x_n=t>0$, you have $$F(x_1,x_2,\ldots,x_n)=t^{a_1}\cdot t^{a_2}\cdots t^{a_n}.$$ Verify that the function $t\mapsto t^{a_1}\cdot t^{a_2}\cdots t^{a_n}$ is not c …

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 …

Let $X$ be a nonempty set, $B(X)$ be the space of bounded functions from $X$ to $\mathbb{R}$, and $\|\cdot\|_\infty$ the sup-norm on $B(X)$. Write $f\leq g$ for $f,g\in B(X)$ if $f(x)\leq g(x)$ for al …

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 …

One can prove the existence of such optimal plans using the extreme value theorem of Weierstrass, but it requires some advanced math. Here is a toy version of the model without energy and emissions. B …

This is essentially a variation on the answer of denesp that requires slightly fewer assumptions. Assume there are $l$ commodities and $m$ agents. An allocation is then a point in $\mathbb{R}^{lm}_+$ …

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 …

You are basically on the right track but might have been carried away by not writing down function arguments. The profit function $\pi$ has arguments $p$ and $w$; don't drop them. In Hotelling's lemm …

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 …

Let $N=\max\{N_1,\ldots, N_J\}$ and $\sigma<0$. $$N=\left[N^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\leq\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\leq \left[ …

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 …

Yes. The proof is virtually the same. Let $x^*\in X$ be a Pareto dominated state. There must be some $x\in X$ such that $u_i(x)>u_i(x^*)$ for some agent $i$ and $u_j(x)\geq u_j(x^*)$ for every agent $ …

No, though coming up with a concrete counterexample is a messy business. Let $y_t=\rho^t$. Then $$\sum_{k=0}^\infty \rho^k y_{t+k}=\sum_{k=0}^\infty \rho^{2k+t}=\frac{\rho^t}{1-\rho^2}.$$ Let $c_t=1$ …