What Shifts Demand sorted goods into normal and inferior by how demand responds to income. With a utility function we can trace that response exactly: hold prices fixed, raise income, and follow the best bundle from Cobb-Douglas Demand or any other demand.
Key idea
As income rises the optimum traces the income offer curve; plotting $x$ against income $m$ gives the Engel curve. The income elasticity $$\eta = \frac{dx}{dm}\cdot\frac{m}{x}$$ classifies the good at that income: inferior if $\eta < 0$, normal if $\eta > 0$, a necessity if $0 < \eta < 1$ and a luxury if $\eta > 1$.
Why it works
Why does 1 separate necessities from luxuries? Look at the budget share $s = px/m$. Differentiate with respect to $m$ (price fixed): $$\frac{ds}{dm} = \frac{p}{m}\frac{dx}{dm} - \frac{px}{m^2} = \frac{px}{m^2}\left(\eta - 1\right).$$ So the share of income spent on the good rises with income exactly when $\eta > 1$ and falls when $\eta < 1$. Food is the classic necessity: richer households spend more on food, but a smaller share of their income (Engel's law, from Ernst Engel's 1857 study of family budgets).
Worked example: A linear Engel curve
Demand for bus rides is $x = 10 + 0.05m$ at fixed prices ($m$ in dollars per month). Find $\eta$ at $m = 200$.
- $\dfrac{dx}{dm} = 0.05$ and $x = 10 + 0.05(200) = 20$.
- $\eta = 0.05\cdot\dfrac{200}{20} = 0.5$.
- Between 0 and 1: a necessity at this income. A 10% raise brings about 5% more rides.
Answer: $\eta = 0.5$.
Worked example: Normal, then inferior
Demand for store-brand pasta is $x = 8m - 0.1m^2$, with $m$ in thousands of dollars a year. Where does it turn inferior, and what is $\eta$ at $m = 60$?
- $\dfrac{dx}{dm} = 8 - 0.2m$, which is zero at $m = 40$. Below 40 thousand dollars the good is normal; above it, inferior.
- At $m = 60$: $\dfrac{dx}{dm} = -4$ and $x = 480 - 360 = 120$.
- $\eta = -4\cdot\dfrac{60}{120} = -2$: inferior at this income.
Answer: Inferior above $m = 40$; $\eta = -2$ at $m = 60$.
Common mistake
Treating "inferior" as a fixed label of a good. Whether a good is inferior depends on the person and on where she is on her Engel curve: the pasta above is normal for a low-income household and inferior for a richer one. Check: compute $dx/dm$ at the income in question, not at some other income, and say "inferior at this income."
An income change shifts the budget line in parallel (prices are fixed), so it moves the consumer along her income offer curve. On a market graph it shows up as a shift of the demand curve: right for a normal good, left for an inferior one.