Question

How can I prove that "if the consumer's preferences are homothetic, then Engel Curves are straight lines"?

Answer 1

Answer :

If preferences are homothetic, it means that when income is scaled up or down by t>0;the demanded bundle,(x1;x2) scales up or down by the same amount, i.e., the new bundle is(tx1;tx2)

Derivation of Engel Curves From Income Consumption Lines :

Income consumption lines are the locus of points representing the equilibrium purchase patterns as income changes, holding preferences and relative prices constant. Engel Curves are the locus of all points representing the quantities demanded of the goods at various levels of income, when prices and preferences are held constant (Fig. 4.11 for normal goods and Fig. 4.12 for inferior goods).

Normal Good :

Inferior Good :

The first part of Fig. 4.13 shows the equilibrium purchase pattern of a household at two levels of income, Y₁ and Y₂, holding preferences and relative prices constant. The second part shows how the quantity of fiber-rich diet changes with income; the incomes Y₁ and Y₂ are represented by the respective budget lines. AB is the income consumption line, and CD is the analogous Engel curve. Engel curves begin at origin and intersect the income axis: with zero income, the quantity demanded for any good must be zero, as the household is unable to afford any purchases. At low incomes, the demand for particular diets (such as a fiber-rich diet) may also be zero (Fig. 4.13).

Figure 4.13. Deriving an Engel curve from the income consumption line.