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What needs to be true about the demand function $z(y)$, where $y$ is income, for the income elasticity $\varepsilon = \frac{dz(y)}{dy}/\frac{z(y)}{y}$ to be always greater than one (so that the good is always a luxury)? In particular, does the demand function have to be convex in income? Would that be a sufficient condition?

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This may not be the answer you are looking for.

What you want may be technically feasible, but just barely. What I mean is that for any $\delta > 0$, you cannot have the income elasticity always be larger than $1 + \delta$. The simple reason is that it would mean that the amount spent on the good always increases faster than the income. Then eventually the amount spent on the good would just overtake the income, but this is, of course, a contradiction.

Income elasticity being exactly equal to one would be easy, e.g. demand derived from Cobb-Douglas.

You should also be able to construct a function $z(y)$ for which $$ \epsilon(y) = 1 + \frac{1}{y}, $$ so that $\epsilon(y) > 1$, although $$ \lim_{y\to \infty} \epsilon(y) = 1. $$ You would need to solve a differential equation: \begin{align*} 1 + \frac{1}{y} & = \frac{dz(y)}{dy}/\frac{z(y)}{y} \\ \\ z(y) \left(\frac{1}{y} + \frac{1}{y^2}\right) & = \frac{dz(y)}{dy}. \end{align*} Unfortunately I am not good at these, so I don't know what $z$ yields a solution, but I don't think this is what you had in mind anyway.

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