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3

I assume the notation $\mathbb R^2_+$ refers to $[0,\infty)^2$. Note that the set on which a function is defined need not be the same set on which a function is differentiable. In particular, it's typical that differentiability requirements are imposed on open sets (see e.g. the fundamental theorem of calculus). This is because defining differentiability on ...

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No that would not be correct definition of elasticity. First, mathematically in multivariate function elasticity is defined as follows: $$EL_x =\frac{ f_x '(x,y)}{f(x,y)}x$$ or in your case it would be: $$\frac{ \partial \ln [w(age,Y,T,Mar)]}{\partial age} \frac{age}{\\ln [w(age,Y,T,Mar)]}$$ However, even if you would plug in the expressions in this ...

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This problem is quite specific to economics. The correct statement is: Proposition If $u(\cdot)$ is quasiconcave, strictly increasing, and continuous, then $\forall x$, there exists $p \gg 0$ and $w \geq 0$ such that $x \in x^*(p, w)$, where $x^*(p, w)$ is the Marshallian demand correspondence. Proof Quasiconcavity of $u$ means the upper-contour set \$\...

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