<p>This may not be the answer you are looking for.</p>

<p>What you want may be technically feasible, but just barely. What I mean is that for any <span class="math-container">$\delta &gt; 0$</span>, you cannot have the income elasticity always be larger than <span class="math-container">$1 + \delta$</span>. The simple reason is that it would mean that the amount spent on the good <em>always</em> 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.</p>

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

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

CC BY-SA 4.0
2018-09-28T22:59:25.250Z
2018-09-29T09:28:04.440Z
2018-09-29T09:28:04.440Z
Giskard
<p>What needs to be true about the demand function <span class="math-container">$z(y)$</span>, where <span class="math-container">$y$</span> is income, for the income elasticity <span class="math-container">$\varepsilon = \frac{dz(y)}{dy}/\frac{z(y)}{y}$</span> 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? </p>

<consumer-theory><elasticity>
Income Elasticity and Convexity of Demand in Income
CorNap
# Income Elasticity and Convexity of Demand in Income

(Asked by: username_0 on 2018-09-28)

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?

**Tags:** consumer-theory, elasticity


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## 1 answers

### Answer by username_1 on 2018-09-28. Score: 1

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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