Income and cross elasticities, Engel curves and the type of good

The income elasticity of demand is the percentage change in the quantity bought divided by the percentage change in income, with prices held fixed. It is computed exactly like the price elasticity of Lesson 10, with m in the place of P.

Three exam calculations in the same form. A product’s quantity rises from 40,000 to 48,000 while income per head rises from Rs 30,000 to Rs 33,000: that is +20% on +10%, so η = 2 (CUET 2018). A person buys 8 units of sugar at income Rs 2,000 and 12 units at Rs 2,500: +50% on +25%, so η = 2 (UOH 2021). And in reverse: with η = 1.5, a 4% rise in income raises the quantity by 1.5 × 4 = 6% (UOH 2013).

A country’s GDP grows at 5% and its population at 2%. The income per head therefore rises by about 3%. With an income elasticity for food of 0.5, food demand per head grows by 1.5%, and the number of mouths adds 2%: total food demand grows by 3.5% (JNU 2013).

In increasing order of income elasticity: inferior goods, then necessities, then luxuries (UOH 2021). All of these statements are tested directly: a normal good has a positive elasticity, an inferior good has a negative one and a luxury has one above 1, not below (JAM 2026). If flour is inferior, then more income lowers its demand, the demand is negatively related to income, and the income elasticity is negative: all three are the same fact (JNU 2013). To say a good is inferior means its income elasticity is negative, not that it is bad or that the elasticity is above 1 (UOH 2017).

Take the three demands of UOH 2021: Q₁ = 120 − 3.5P₁ − 6P₂ + 14H and Q₂ = 100 − 2P₂ + 3P₃ + 1.1H have positive income coefficients and are normal, while Q₃ = 1500 − 0.5P₃ − 30H does not. For the electric-vehicle demand of JAM 2022, Q_E = 0.75 − 1.5P_E + 2.5P_F − 0.5P_B + 3.2I at P_E = 6.5, P_F = 4.5, P_B = 0.5 and I = 10, the quantity is 34 and so η = 3.2 × 10/34 = 0.94, while the cross elasticity with respect to the fossil-fuel price is 2.5 × 4.5/34 = 0.33.

Hold both prices fixed and raise income. Each income gives a budget line, and each budget line gives an optimum. Joining the optima across the (x, y) plane gives the income consumption curve (ICC), also called the income expansion path. Plotting the quantity of one good against income gives its Engel curve: income and the quantity of one good, nothing else (JAM 2025).

Their slopes carry the type of good:

• A normal good has an Engel curve that slopes up (more income, more of the good). An inferior good has one that slopes down. So the statements “inferior goods have upward-sloping Engel curves and normal goods downward-sloping ones” are both false (CUET 2023).
• For an inferior good, one good rises and the other falls along the ICC, so the ICC slopes down as well. For an inferior good both curves are negatively sloped (CUET 2020).
• A Giffen good is an extreme inferior good, so its Engel curve is negatively sloped (CUET 2022, UOH 2012 and 2015).

Orientation matters in one calculation. If income is on the vertical axis, the slope of the Engel curve is dm/dx, the reverse of how much extra x each rupee buys. For perfect complements min{x₁, x₂} with prices 1 and 2, the bundle costs 3 per pair, so x₂ = m/3 and the curve has income = 3x₂: slope 3 (JAM 2026).

Pick tastes and slide income. The dotted path on the left is the ICC and the red curve on the right is the Engel curve for x. The readouts give the income elasticity and say what kind of good x is at that income, which can change: the last set of tastes makes rice inferior over one range of income and normal over another.

• Cobb–Douglas (u = xy): x = m/2p_x, so the Engel curve is a straight line through the origin and η = 1. Linear Engel curves are the classic JNU statement (JNU 2017). Every homothetic taste has η = 1 for every good, since doubling income doubles the bundle.
• Quasi-linear (u = y + f(x)): the demand for x does not depend on income while she buys some y, so η_x = 0 and y absorbs every extra rupee. For u = y + √x, income elasticity of demand for x is 0, not 1 and not 0.5 (JAM 2025). For u = x₁ + 2√x₂ with the optimum at x₂ = 10, a 50% rise in income leaves x₂ at 10 (JAM 2021). For u = 2 ln x₁ + x₂, with x₁ = 10, doubling income leaves x₁ at 10 (JNU 2013). Whenever u = x₂ + u(x₁), the effect of income on the demand for good 1 is zero (JNU 2018).
• The catch with quasi-linear. For u = √x + y and a very small income she spends it all on x. So as M rises the demand for x first rises, then stops. The answer to “does x rise with M?” is “cannot say” (JNU 2016).
• Shifted Cobb–Douglas (Stone–Geary): with a subsistence amount of x (the shift is positive), x is a necessity, because the subsistence amount is a larger share of a small income. With a negative shift it is a luxury.
• Demand that does not change at all. Amit inherits Rs 10,000 and eats the same number of Gulab Jamuns. That cannot be homothetic tastes, which make demand proportional to income. It need not be a Giffen good, and the Engel curve being vertical is only known at that one point (JNU 2017).

Engel’s law: as income rises, the share of income spent on food falls (UOH 2010, 2016). Put in the textbook’s wider form, the share spent on primary products falls and the share spent on manufactured goods rises (UOH 2018). In the language of this lesson, food is a necessity: 0 < η < 1.

The second rule follows from the budget. Extra income has to go somewhere, so the shares of the extra income add up to one. If s_i is the share of income spent on good i:

If food takes half of the budget with η = 0.6, the other half of the budget must have a share-weighted average elasticity of (1 − 0.3)/0.5 = 1.4. The lab prints this sum for you: it equals 1 at every income, for every taste.

The cross-price elasticity of the demand for Y with respect to the price of X is the percentage change in Y divided by the percentage change in the price of X.

• Substitutes: a rise in the price of one raises the demand for the other. Cross elasticity positive (CUET 2026). If the quantity of rice rises 5% when the price of wheat rises 20%, the cross elasticity is 5/20 = 0.25 (JNU 2013, 2014).
• Complements: a rise in the price of one lowers the demand for the other. If the demand for Z goes up when the price of Y goes down, Z and Y are complements (CUET 2018, 2019; JNU 2013, 2014). For complements the cross elasticity is negative (UOH 2018).
• Perfect substitutes have an unbounded cross elasticity. When the Timex and Sofex watches are perfect substitutes, it is infinite (UOH 2020), and for identical products of firms in perfect competition too (UOH 2018). With equal prices she is indifferent, and a tiny price rise sends her all to the other.

Chains of relations: if X and Y are complements and Y and Z are substitutes, nothing follows about X and Z, but a rise in the price of Y raises the demand for Z (UOH 2013).

For Q = A·Y·P_X^β/P_Q^α the three elasticities are visible at once: −α for its own price, β for the price of X (a substitute if β > 0) and 1 for income (JNU 2021).