Marshallian demand (also called ordinary demand or uncompensated demand) is the quantity of each good a consumer chooses when they maximize utility subject to a budget constraint, given prices and income.
Setup: utility maximization
Let (x) be a vector of quantities, (p) a vector of prices, and (m) income. The consumer’s problem is:
[ \max_{x \ge 0} u(x) \quad \text{s.t.} \quad p \cdot x \le m. ]
The solution (x(p,m)) is the Marshallian demand function: it tells you the chosen bundle at prices (p) and income (m).
Key properties (what economists use)
In standard consumer theory (with locally nonsatiated preferences), Marshallian demand has several useful properties:
- Adding-up (budget exhaustion): typically (p \cdot x(p,m) = m).
- Homogeneity of degree 0: (x(\lambda p, \lambda m) = x(p,m)) for any (\lambda > 0). Only relative prices and real income matter.
- Elasticities: price and income elasticities are computed directly from (x(p,m)) and are central for demand estimation and policy analysis.
Marshallian vs Hicksian demand (and the Slutsky equation)
Marshallian demand mixes two effects of a price change:
- Substitution effect: holding utility constant, how the consumer substitutes toward relatively cheaper goods.
- Income effect: the change in purchasing power caused by the price change.
The Hicksian (compensated) demand (h(p,u)) is defined by expenditure minimization at a target utility level (u):
[ \min_{x \ge 0} p \cdot x \quad \text{s.t.} \quad u(x) \ge u. ]
The Slutsky equation links Marshallian and Hicksian responses. In one common form:
[ \frac{\partial x_i(p,m)}{\partial p_j}
\frac{\partial h_i(p,u)}{\partial p_j} -; x_j(p,m),\frac{\partial x_i(p,m)}{\partial m}. ]
The first term is the (compensated) substitution effect; the second term is the income effect.
Worked example (Cobb-Douglas)
If preferences are (u(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}) with (0<\alpha<1), then Marshallian demand is:
[ x_1(p,m)=\frac{\alpha m}{p_1}, \qquad x_2(p,m)=\frac{(1-\alpha)m}{p_2}. ]
This makes the “income and price” dependence transparent: doubling income doubles demanded quantities, while a higher own-price reduces quantity inversely.
Practical interpretation
Marshallian demand is what you typically estimate from observed choices, because real consumers face actual budgets and actual incomes. It is the core object behind:
- demand curves and elasticities,
- tax and subsidy incidence analysis,
- welfare comparisons (often alongside compensated objects like the expenditure function).
Related Terms
- Budget Constraint
- Utility Maximization
- Indifference Curve
- Hicksian Demand
- Slutsky Equation
- Substitution Effect
- Income Effect
- Price Elasticity