Marshallian Demand

How Marshallian demand works

Picture a consumer with a fixed income walking into a market with posted prices. She wants to leave with the bundle that makes her happiest — that reaches the highest indifference curve she can afford. The bundle she picks, as a function of the prices and her income, is Marshallian demand x(p, m).

The geometric construction is the standard tangency picture: draw the budget line p·x = m, then drop the family of indifference curves on the same axes. The optimal bundle is the unique point where the highest reachable indifference curve just touches the budget line — the tangency. Algebraically the tangency condition says MRS equals the price ratio, or equivalently MU_x/p_x = MU_y/p_y — the marginal utility per dollar is equalized across goods.

When prices change, the whole budget line pivots or shifts, and the optimum can land on a higher or lower indifference curve. Marshallian demand is the function that maps (prices, income) to that optimum.

Formal definition

Marshallian demand solves the utility-maximization problem:

x(p, m)  =  argmax     u(x)
            x ≥ 0
            subject to  p · x  ≤  m

The maximum value of the objective is the indirect utility function:

v(p, m)  =  max  u(x)  s.t. p·x ≤ m
         =  u(x(p, m))

Key properties of x(p, m):

• Homogeneous of degree zero. x(λp, λm) = x(p, m). Only relative prices and real income matter.
• Walras's Law. Under local non-satiation, p · x(p, m) = m — the consumer spends everything.
• Slutsky-decomposable. ∂x_i/∂p_j = ∂h_i/∂p_j − x_j · ∂x_i/∂m — the derivative splits into substitution and income pieces.
• Roy's identity. x_i(p, m) = −(∂v/∂p_i) / (∂v/∂m). One division recovers demand from indirect utility.
• Coincides with Hicksian at the optimum. x(p, m) = h(p, v(p, m)), and m = e(p, v(p, m)).

Worked example: Cobb-Douglas x(p, m)

Take U(x, y) = x^α · y^(1−α) with α = 0.4, p_x, p_y, m. Set up the Lagrangian:

L  =  x^0.4 · y^0.6  −  λ · (p_x · x + p_y · y − m)

FOCs:

∂L/∂x  =  0.4 · x^(−0.6) · y^(0.6)  −  λ · p_x  =  0
∂L/∂y  =  0.6 · x^(0.4)  · y^(−0.4)  −  λ · p_y  =  0

Divide:

(0.4 / 0.6) · (y / x)  =  p_x / p_y
y  =  (0.6 / 0.4) · (p_x / p_y) · x  =  1.5 · (p_x / p_y) · x

Substitute into the budget constraint:

p_x · x  +  p_y · 1.5 · (p_x / p_y) · x  =  m
p_x · x · (1 + 1.5)  =  m
x(p, m)  =  0.4 · m / p_x

Symmetrically y(p, m) = 0.6 · m / p_y. The Cobb-Douglas form has a famous closed form: expenditure shares equal the exponents. The consumer spends 40% of income on x and 60% on y, regardless of prices.

Check: own-price elasticity ∂x/∂p_x · p_x/x = (−0.4·m/p_x^2) · (p_x · p_x/(0.4·m)) = −1. Cobb-Douglas demand is unit elastic — a one-percent rise in p_x cuts x by exactly one percent. Income elasticity ∂x/∂m · m/x = (0.4/p_x) · (m/(0.4·m/p_x)) = 1. Income-elastic with elasticity 1: a one-percent rise in income raises x by one percent.

Marshallian vs Hicksian demand — side by side

	Marshallian x(p, m)	Hicksian h(p, u)
What's held constant	Income m	Utility u
Optimization problem	max u(x) s.t. p·x ≤ m	min p·x s.t. u(x) ≥ u
Value function	Indirect utility v(p, m)	Expenditure e(p, u)
Own-price slope	Sign ambiguous (Giffen possible)	Always ≤ 0
Effects in a price change	Substitution + income	Substitution only
Recovered from value function via	Roy's identity	Shephard's lemma
Directly observable?	Yes — prices, quantities, income	No — utility is unobservable

The two demand functions are duals. At the consumer's actual optimum they agree: x(p, m) = h(p, v(p, m)) and m = e(p, v(p, m)). Walking around in the space of prices and incomes, they trace different paths through quantity-space, with the gap equal to the income-effect correction in the Slutsky equation.

Roy's identity — Marshallian demand from indirect utility

Roy's identity recovers Marshallian demand directly from the indirect utility function v(p, m):

x_i(p, m)  =  −  (∂v / ∂p_i)  /  (∂v / ∂m)

The numerator is how indirect utility responds to a price rise — negative, because higher prices hurt. The denominator is how indirect utility responds to an income rise — positive, the marginal utility of income. Their ratio is the consumption level of good i. Roy's identity recovery: 1 division instead of full optimization — once you have v(p, m), every Marshallian demand pops out by differentiation.

Sketch of proof. Take the identity v(p, m) = u(x(p, m)) and differentiate with respect to p_i:

∂v/∂p_i  =  Σ_j  (∂u/∂x_j) · ∂x_j/∂p_i

At the optimum, ∂u/∂x_j = λ · p_j where λ is the Lagrange multiplier — the marginal utility of income. So:

∂v/∂p_i  =  λ · Σ_j  p_j · ∂x_j/∂p_i

Differentiate Walras's law (p·x = m) with respect to p_i: Σ_j p_j · ∂x_j/∂p_i = −x_i. So ∂v/∂p_i = −λ · x_i. Similarly differentiate the same identity with respect to m: ∂v/∂m = λ. Divide and the λ cancels: x_i = −(∂v/∂p_i)/(∂v/∂m). Done.

The Slutsky equation

The Slutsky equation decomposes the Marshallian derivative into a Hicksian (substitution) piece and an income-effect piece:

∂x_i / ∂p_j   =   ∂h_i / ∂p_j     −     x_j · ∂x_i / ∂m
                  ≤ 0 (own price)        sign = sign(∂x_i/∂m)

The first term, the Hicksian own-price slope, is always non-positive (Slutsky matrix is negative semi-definite). The second term, the income effect, has sign determined by ∂x_i/∂m — positive for normal goods, negative for inferior. For a normal good both pieces push the same way and Marshallian demand slopes down. For an inferior good they oppose; the substitution piece usually wins. For a Giffen good — inferior with strong income channel — the income piece wins and demand slopes up.

For the Cobb-Douglas example, x = 0.4·m/p_x. Direct partial: ∂x/∂p_x = −0.4·m/p_x^2. By Slutsky: the Hicksian piece is what's left after subtracting the income-effect correction. Computing both pieces from the closed form (Hicksian for Cobb-Douglas with α = 0.4 is h_x = u·(α p_y / (1−α) p_x)^(1−α) · ((1−α)/α)) and combining shows the Slutsky equation holds identically — a useful sanity check on the algebra.

From individual to market demand

Market demand is the horizontal sum of individual Marshallian demands at each price:

X(p)  =  Σ_h  x_h(p, m_h)

Individual demand curves vary with each consumer's income; market demand at a given p depends on the entire income distribution. A famous result (the Sonnenschein-Mantel-Debreu theorem) shows that aggregate excess demand can take essentially arbitrary shapes — even with all consumers maximizing rationally, market demand need not inherit the negativity, monotonicity, or symmetry of individual demands. Market demand is well-behaved only under additional restrictions like identical homothetic preferences.

Applications

• Empirical demand estimation. The Almost Ideal Demand System (Deaton-Muellbauer 1980), Translog, and Linear Expenditure System are parametric forms of Marshallian demand fit to scanner data, household budget surveys, and trade flows. Estimated price and income elasticities feed welfare analysis, tax incidence, and antitrust merger simulation.
• Tax incidence. Who bears the burden of an excise tax depends on the elasticities of Marshallian supply and demand — the more inelastic side absorbs more. The 1929 Marshallian-Pigou tax incidence framework remains the workhorse of public finance.
• Antitrust merger simulation. The HHI shift and unilateral-effects upper bound from a horizontal merger depend on the cross-price elasticities of Marshallian demand for the merging products. UPP and GUPPI tests used by FTC and DOJ rely directly on these elasticities.
• Welfare cost of a price change. The Marshallian-surplus integral ∫ x(p, m) dp approximates the exact Hicksian welfare measure. For goods with small income elasticities (most luxury or narrow categories), the approximation is excellent; for staples or broad categories the gap is consequential.
• International trade. Armington elasticities — the Marshallian substitution between domestic and imported varieties of a good — drive most quantitative trade models. CGE simulations of tariff impacts hinge on these numbers.
• Cost-of-living adjustments. Chained CPI indices that approximate Hicksian compensation are derived from Marshallian demand observations using estimated elasticities.

Common pitfalls

• Confusing Marshallian and Hicksian elasticities in welfare work. Compensating and equivalent variation use Hicksian elasticities; Marshallian elasticities approximate them but mix in income effects. For narrow goods the gap is small; for staples it can swap the sign of policy conclusions. Hausman (1981) shows how to recover Hicksian elasticities from Marshallian estimates.
• Forgetting homogeneity of degree zero. Theory demands x(λp, λm) = x(p, m); empirical demand systems estimated without this restriction can produce nonsense like "pure inflation changes consumption shares". Modern systems (AIDS, Translog) impose it.
• Misreading negative income elasticity as Giffen behavior. Inferior goods have negative income elasticity but their Marshallian demand almost always still slopes down — the substitution effect dominates. Giffen behavior requires the income effect to win, which needs a very large budget share and strong inferiority simultaneously.
• Treating market demand as if it inherited individual properties. Sonnenschein-Mantel-Debreu warns that aggregate demand can be arbitrarily ill-behaved even with rational individuals. Symmetry, monotonicity, and the Weak Axiom of Revealed Preference all can fail at the market level.
• Confusing Marshallian demand x(p, m) with demand at a single price q(p). Marshallian demand is a function of all prices and income, not just own price. A demand curve in (p, q) space is a slice of x(p, m) holding cross prices and income constant.
• Ignoring the corner-solution case. The tangency formulation assumes an interior optimum. If p_x is very high (or the consumer's preference for x is weak), the optimum can be at x = 0 — a corner solution. Most demand-system software handles this, but back-of-envelope FOC calculations can produce nonsensical negative quantities if applied uncritically.