Properties of demand and aggregation

Take any consumer who maximises a well-behaved utility function. The demands x(p, m) she generates are not arbitrary functions of prices and income. Three properties hold whatever her tastes:

1. Homogeneity of degree zero. Multiply all prices and income by the same number and demand does not change.
2. Walras’ law. She spends all her income: p·x(p, m) = m.
3. Slutsky symmetry and negativity. The substitution terms form a symmetric matrix whose diagonal is never positive.

Each comes from something already in the module: the first from the budget line of Lesson 4, the second from non-satiation, the third from the expenditure function of Lesson 14. Together they are the testable restrictions the theory puts on observed demand.

Multiply every price and income by a factor a > 0. The budget line has the same intercepts m/p and the same slope, so it is the same line, and she makes the same choice.

That is how UGC NET 2023 described it: demand is homogeneous of degree zero if X(aP, aW) = X(P, W), not if it equals aX(P, W); and if price and wealth both change in the same proportion the individual’s choice does not change. A consumer whose choice did change would suffer money illusion.

By Euler’s theorem, a function homogeneous of degree zero has the elasticities of its arguments summing to zero:

That answers the opening question: zero (UGC NET 2024). It also lets you fill in a missing elasticity: if the own-price elasticity is −0.8 and the cross elasticity is +0.3, the income elasticity must be +0.5. And it tests a given demand function: Q = A p_x⁻² p_y^1.5 m^β is homogeneous of degree zero only if −2 + 1.5 + β = 0, so β = 0.5.

Choose tastes and scale prices and income together: the black and dashed budget lines are the same line and the bundle does not move. Scale prices alone and a different (red) line appears with a different bundle. The cards underneath check Walras’ law and print the three elasticities with their sum.

With non-satiation she spends everything: p_xx + p_yy = m. This is Walras’ law, and the statement “Walras’ law says the consumer fully expends her wealth” is exactly that (UGC NET 2023). Differentiate it with respect to income and you get the adding-up rule of Lesson 11: budget-share weighted income elasticities sum to one.

Two familiar results follow at once.

• Not every good can be inferior. If every η were negative the sum could not be 1 (UOH 2017). At least one good is normal, and if one is a necessity another is a luxury.
• If one of two goods is Giffen, the other is normal. A Giffen good is inferior (η < 0), so its term in the sum is negative and the other good’s term must be more than 1. With a share below 1, that needs η > 1: the other good is a luxury, so certainly normal (JNU 2011).

Collect the substitution terms ∂h_i/∂p_j of Lesson 14 into a matrix. Since h_i = ∂e/∂p_i, the matrix is the matrix of second derivatives of e, which is symmetric (the effect of p_y on the compensated demand for x equals the effect of p_x on the compensated demand for y) and negative semidefinite because e is concave. The diagonal terms, the own substitution effects, are never positive: the statement of Lesson 12.

Homogeneity then makes each row, weighted by the prices, add to zero: raise every price in the same proportion and the compensated demands do not move. These properties are all one can say in general; any further shape of demand comes from the particular utility.

A market demand curve adds the individual demands at each price. Because it adds quantities at the same price, the curves are added horizontally (UOH 2018). Two people with x_A = 30 − 3p and x_B = 18 − p have market demand 48 − 4p up to p = 10; above 10, A buys nothing and the market is just B: 18 − p, to p = 18. The market curve has a kink where the first person drops out.

But does the market behave like a single consumer? Not by right. Individual demands depend on each person’s income, so market demand depends on how income is shared, not only on its total. Move income from one person to another and the market curve can move. The market behaves like one consumer in two cases:

• There are no income effects. If each person’s demand ignores income (quasi-linear tastes, with each buying some of the other good), the total only depends on the prices (UOH 2010).
• Gorman form. More generally, parallel, straight wealth expansion paths at every price vector require preferences with an indirect utility of the Gorman form. Two cases that satisfy it: every consumer has the same homothetic preferences, or everyone has quasi-linear preferences in the same good. It is not enough that preferences be heterogeneous, or that wealth be equally shared (UGC NET 2025).

The lab shows both sides. With identical Cobb–Douglas tastes the market curve does not budge when you move income between the two people. With different tastes, or one Cobb–Douglas and one quasi-linear consumer, it does.