microeconomics, elasticityWhat is the correlation between elasticity and calling a good a “luxury” or inferior? Do we need to deal with price-demand or income elasticity?
Also, in what case (i.e. income elasticity = o?) can we call a good quasilinear? I recently answered a question in which the demand equation was: $x(p_x,p_y)= (4p_y/p_x)^2$ and asked to find the income elasticity of demand for x. I was a little baffled, but following the definition of income elasticity, I found it to be 0, so I concluded it was quasilinear, was I correct? If so, why?
If demand for a good $i$ can be represented as a function $x_i(\cdot)$ of income for some consumer, we say $i$ is a luxury good if for income levels $0<m\leq m'$, one has $\frac{x_i(m)}{m}\leq\frac{x_i(m')}{m'}$, that is $\frac{x_i(\cdot)}{\cdot}$ is an increasing function. A sufficient condition for this is that $x_i$ is differentiable and $\bigg(\frac{x_i(m)}{m}\bigg)'>0$ for positive income levels $m$. Applying the quotient rule, this is equivalent to $$\frac{x'_i(m)-mx_i(m)}{m^2}>0$$ $$\frac{x'_i(m)}{m^2}>\frac{x_i(m)}{m}$$ $$\frac{x'_i(m)}{m}>x_i(m)$$ $$x'_i(m)\frac{m}{x_i(m)}>1.$$ The left side of the inequality is simply the income elasticity of demand for good $i$.
This is a bit more complicated, since quasi-linearity is a property of utility functions or preferences, not demand. Since the demand function you gave does not depend on income, the income elasticity is indeed $0$. This holds true for quasi-linear utility functions at interior solutions. If you don't allow for $y$ to be negative, there are always boundary solutions for $m$ small enough.
Another, more demaning question is, is whether preferences giving rise to constant interior demand for a good are quasi-linear (under some regularity conditions). I dont know the answer to this question, I guess one might obtain a positie nswer by using Gorman's theorem.
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