Relation of Engel-curve to income elasticity of demand; is the slope of the Engel-curve equal to the elasticity of income?

Question

I learnt that $\frac{\Delta x}{\Delta m} \gt 0$ for normal goods, $\frac{\Delta x}{\Delta m} \lt 0$ for inferior goods, $\frac{\Delta x}{\Delta m} \gt 1$ for luxury goods and $0 \lt \frac{\Delta x}{\Delta m} \lt 1$ for necessary goods (where x is the amount of units of some good and m is the income).

Now, looking at the Engel-curve for homothetic preferences (i.e. Cobb-Douglas, perfect substitutes/perfect complements), I understand $\frac{\Delta x}{\Delta m} = 1$ for homothetic goods (btw is that a term, "homothetic goods"?).

Since the slope of the Engel-curve is $\frac{\Delta m}{\Delta x_1}$, I'm wondering if $\left(\frac{\Delta m}{\Delta x_1}\right)^{-1}$ is a way to determine the elasticity of demand, which, from my understanding, deals with the exact same values to categorise goods.

So I'm wondering if $\left(\frac{\Delta m}{\Delta x_1}\right)^{-1}$ is equal to the elasticity of income or if there's a way to determine the elasticity of income from the Engel-curve.

Answer 1

Income elasticity of demand

Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is then given by: $$ \varepsilon^y_q = \frac{\partial q}{\partial y} \frac{y}{q} $$ It measures the percentage point change in demand $q(y)$, due to a 1$\%$ increase in income, $y$.

If $\varepsilon^y_q \ge 0$ the good is normal, if $\varepsilon^y_q < 0$, it is inferior. If $\varepsilon^y_q > 1$ it is a luxury good, while if $\varepsilon^y_q < 1$ the good is called a necessity. The demand for luxury goods increases more than proportionally compared to income. Necessary goods increase less than proportionally.

The advantage of using the elasticity $\varepsilon^y_q$, instead of the slop $\frac{\Delta q}{\Delta y}$ is that it is unit-independent We can see this as follows: $$ \varepsilon^y_q = \underbrace{\dfrac{\partial q}{\partial y}}_{\dfrac{kilos}{Euros}} \underbrace{\dfrac{1}{q}}_{\dfrac{1}{kilos}} \underbrace{y}_{\dfrac{Euros}{1}} $$

This means that $\varepsilon^y_q$ does not change if we express the goods or the income in different units. For example the elasticity $\varepsilon^y_q$ does not change whether you express income in Euros or Dollars. Also, the elasticity does not change whether you express demand in kilos or pounds. As such, elasticities can easily be compared across countries and over different time periods. The slope $\frac{\Delta q}{\Delta y}$ does not have this advantage.

Homothetic preferences

If preferences are homothetic, the demand function is linear in income: $$ q(y) = c y, $$ where $c$ is a constant. In fact, substituting $y = 1$ into this equation gives: $$ q(1) = c, $$ so $c$ is the unit income demand (the amount that you would buy if you would have 1 Euro). This means that we can also write: $$ q(y) = q(1) y. $$ Engel curves are straight lines through the origin and with slope $q(1)$: $$ \frac{\partial q(y)}{\partial y} = q(1). $$ As such: $$ \varepsilon^y_q = \frac{\partial q}{\partial y}\frac{y}{q(y)} = \frac{q(1)y}{q(y)} = \frac{q(y)}{q(y)} = 1. $$ So demand curves arising from homothetic preferences have unit income elasticity.

Answer 2

1. Homothetic goods is not a widely used term (as far as I know).

2. It is not true that when preferences are homothetic $$ \frac{\Delta x}{\Delta m} = 1 $$ always holds. Instead $$ \frac{\Delta x}{\Delta m} = \text{constant} \cdot \frac{1}{p_x} $$ (where the constant is the share of income spent on $x$)
   or $$ \frac{\Delta x}{\Delta m} \frac{m}{x} = 1 $$ hold in this case.

3. The slope of the Engel-curve $\frac{\text{d} x}{\text{d} m}$ and income-elasticity are related, as the slope appears in the (point) income elasticity formula: $$ \eta_x(m) = \frac{\text{d} x}{\text{d} m}\frac{m}{x}. $$