Hicksian Demand

How Hicksian demand works

Picture a consumer sitting on an indifference curve. Her budget — for now — doesn't matter; what matters is that she's just satisfied enough, no more. Now ask: what's the cheapest way for her to stay on that curve when the market prices change? That cheapest-way bundle, as a function of prices p and the target utility u, is Hicksian demand h(p, u).

Geometrically the answer is a tangency. Sweep a family of parallel lines with slope −p_x/p_y across the indifference curve U = u; the one tangent to the curve gives the cheapest bundle, because pushing the line any closer to the origin would leave the indifference curve unreachable, while pushing it farther means spending more than necessary. The tangency point — where MRS = p_x/p_y on the fixed indifference curve — is the Hicksian bundle.

Hicksian demand is sometimes called compensated demand because, relative to a baseline situation, the consumer is "compensated" (or de-compensated) with just enough income that they end up on the original indifference curve at the new prices. The compensation cancels the income-effect channel, leaving only substitution.

Formal definition

Hicksian demand solves the expenditure-minimization problem:

h(p, u)  =  argmin     p · x
            x ≥ 0
            subject to  u(x) ≥ u

The minimum value of the objective is the expenditure function:

e(p, u)  =  min  p · x  s.t. u(x) ≥ u
         =  p · h(p, u)

Key properties of h(p, u):

• Homogeneous of degree zero in p. Doubling all prices doesn't change relative prices, and the consumer still picks the same bundle to reach the same utility — so h(λp, u) = h(p, u) for any λ > 0.
• Symmetric Slutsky matrix. The cross-partials of h are symmetric: ∂h_i/∂p_j = ∂h_j/∂p_i. Equivalent to Young's theorem on the second partials of e(p, u).
• Negative semi-definite Slutsky matrix. Implies own-price slopes are non-positive: ∂h_i/∂p_i ≤ 0. This is the rigorous Law of Demand for compensated demand.
• Generated by Shephard's lemma. The expenditure function delivers the demand via h_i(p, u) = ∂e(p, u)/∂p_i — derivative replaces minimization.

Worked example: Cobb-Douglas h(p, u)

Take U(x, y) = x^α · y^(1−α) with α = 0.5. Prices p_x, p_y, target utility u. To find h(p, u), solve:

minimize  p_x · x + p_y · y
subject to  x^0.5 · y^0.5 = u

Set up the Lagrangian and take FOCs:

p_x  =  λ · 0.5 · x^(−0.5) · y^(0.5)
p_y  =  λ · 0.5 · x^(0.5)  · y^(−0.5)

Dividing the two:

p_x / p_y  =  y / x      →      y  =  x · p_x / p_y

Substitute into the utility constraint x^0.5 · y^0.5 = u:

x^0.5 · (x · p_x / p_y)^0.5  =  u
x · (p_x / p_y)^0.5          =  u
h_x(p, u)                     =  u · (p_y / p_x)^0.5
h_y(p, u)                     =  u · (p_x / p_y)^0.5

The Hicksian demand for x rises with p_y and falls with p_x — both signs match intuition. The expenditure function is:

e(p, u)  =  p_x · h_x + p_y · h_y
         =  p_x · u · (p_y/p_x)^0.5  +  p_y · u · (p_x/p_y)^0.5
         =  2u · (p_x · p_y)^0.5

Now check Shephard's lemma: ∂e/∂p_x = 2u · 0.5 · (p_y/p_x)^0.5 · 1 = u · (p_y/p_x)^0.5 = h_x. The derivative of the expenditure function in p_x returns the Hicksian demand for x exactly — no separate optimization required.

Hicksian vs Marshallian — side by side

	Hicksian h(p, u)	Marshallian x(p, m)
What's held constant	Utility u	Income m
Optimization problem	min p·x s.t. u(x) ≥ u	max u(x) s.t. p·x ≤ m
Value function	Expenditure e(p, u)	Indirect utility v(p, m)
Own-price slope	Always ≤ 0	Sign ambiguous (Giffen possible)
Effects in a price change	Substitution only	Substitution + income
Observable directly?	No (utility unobservable)	Yes (prices, quantities, income)
Used for	Welfare; Slutsky decomposition; duality	Empirical demand; standard "demand curve"

The two demands coincide at the consumer's actual optimum: h(p, v(p, m)) = x(p, m) and e(p, v(p, m)) = m. Away from the actual optimum they diverge, and the divergence is precisely the income effect.

Derivation of Shephard's lemma

Shephard's lemma — h_i = ∂e/∂p_i — is the most elegant result in duality theory, falling out of the envelope theorem in one line. Here's the proof.

The expenditure function is e(p, u) = min_x p·x subject to u(x) ≥ u. Let x*(p) = h(p, u) be the minimizer. Then:

e(p, u)  =  p · x*(p)  =  Σ_j  p_j · h_j(p, u)

Differentiate with respect to p_i, using the chain rule:

∂e/∂p_i  =  h_i  +  Σ_j  p_j · ∂h_j/∂p_i

The second term vanishes at the optimum. Why? Because x* satisfies the first-order conditions p_j = λ · ∂u/∂x_j, so:

Σ_j  p_j · ∂h_j/∂p_i  =  λ · Σ_j  (∂u/∂x_j) · ∂h_j/∂p_i
                       =  λ · ∂[u(h(p, u))]/∂p_i
                       =  λ · 0   (utility held constant)
                       =  0

So ∂e/∂p_i = h_i. Shephard: ∂e/∂p_i = h_i directly — derivative replaces minimization. You don't have to redo the Lagrangian — once you have e(p, u), every Hicksian demand pops out by differentiation.

The Slutsky equation, derived from Hicksian demand

Hicksian demand is the foundation of the Slutsky decomposition. Start with the identity that relates the two demands at the actual optimum:

h_i(p, u)  =  x_i(p, e(p, u))     for all p

This just says: if you're given the right income to hit u, the cost-minimizing bundle equals the utility-maximizing bundle. Differentiate both sides with respect to p_j:

∂h_i/∂p_j  =  ∂x_i/∂p_j  +  ∂x_i/∂m · ∂e/∂p_j
            =  ∂x_i/∂p_j  +  ∂x_i/∂m · h_j         (Shephard: ∂e/∂p_j = h_j)

At the optimum h_j = x_j, so:

∂x_i/∂p_j  =  ∂h_i/∂p_j  −  x_j · ∂x_i/∂m

That's the Slutsky equation. The Marshallian price derivative on the left equals the Hicksian substitution term minus the income-effect correction. Slutsky decomposition: 70% substitution + 30% income (typical CPS food study) — meaning two-thirds of an observed price elasticity for food is the compensated piece, one-third the income-effect correction.

Welfare measurement with Hicksian demand

The point of compensated demand isn't just elegance; it makes welfare arithmetic exact. For a price change from p^0 to p^1, the compensating variation (CV) — the income that must be taken away after the change to leave the consumer at the original utility u^0 — is the integral of Hicksian demand:

CV  =  e(p^1, u^0)  −  e(p^0, u^0)
     =  ∫_{p^0}^{p^1}  h(p, u^0)  dp     (line integral)

The equivalent variation (EV) is the same construction at the new utility:

EV  =  e(p^1, u^1)  −  e(p^0, u^1)
     =  ∫_{p^0}^{p^1}  h(p, u^1)  dp

Both formulas are exact measures of welfare change because they integrate compensated demand. The Marshallian consumer-surplus integral ∫ x(p, m) dp is only approximate: it ignores the income effect that arises as the price walks from p^0 to p^1. For most goods the approximation is close but for staples or luxuries with strong income elasticities the gap can matter.

Related theorems and identities

• Shephard's lemma. h_i = ∂e/∂p_i. Differentiate the expenditure function in any price and you get the corresponding Hicksian demand. Foundation of duality theory.
• Slutsky symmetry. ∂h_i/∂p_j = ∂h_j/∂p_i. The Slutsky matrix is symmetric, a direct consequence of Young's theorem on the second derivatives of e(p, u).
• Slutsky negative semi-definiteness. The matrix S = [∂h_i/∂p_j] is negative semi-definite; equivalently, e(p, u) is concave in p. On the diagonal, ∂h_i/∂p_i ≤ 0 — the rigorous Law of Demand.
• Roy's identity. Recovers Marshallian demand from indirect utility: x_i(p, m) = −(∂v/∂p_i)/(∂v/∂m). The Marshallian counterpart of Shephard.
• Hotelling's lemma. On the producer side, ∂π(p)/∂p_i = y_i(p) — the analogue of Shephard for profit-maximizing supply.
• Adding-up (Cournot, Engel). Σ_i p_i · h_i = e(p, u). Differentiating gives constraints among the Slutsky elements used to check empirical demand systems for consistency.

Applications of Hicksian demand

• Cost-of-living indices. The Konüs (true) cost-of-living index between two periods is e(p^1, u)/e(p^0, u) — a ratio of expenditure functions at fixed utility. The Boskin Commission used this framework to argue that the U.S. CPI overstates inflation by 0.3–1.1 percentage points per year because Laspeyres (fixed-bundle) indices miss compensated substitution.
• Welfare cost of taxation. The deadweight loss of a commodity tax is computed by integrating the Hicksian demand from the pre-tax to the post-tax price. The famous Harberger triangle is a small-Δp linearization of this integral.
• Optimal taxation (Ramsey). Ramsey's inverse-elasticity rule for commodity taxes is derived from Hicksian elasticities: tax goods with low compensated own-price elasticity more heavily. The compensated elasticity isolates the distortion channel from the income-redistribution channel.
• Carbon pricing welfare analysis. The deadweight burden of a carbon tax — and the optimal rebate — relies on Hicksian elasticities of energy demand. Marshallian elasticities would mix in income effects, double-counting the burden if the rebate is lump-sum.
• Empirical demand systems. The Almost Ideal Demand System (AIDS, Deaton-Muellbauer 1980) writes budget shares as functions of log prices and log expenditure, with the Hicksian elasticities recoverable from estimated parameters via the Slutsky equation. Industry standard for food-, energy-, and luxury-demand studies.
• Insurance and risk. Hicksian state-contingent demand for insurance is the building block of optimal insurance contracts; the compensated framework underlies the Mossin theorem on full insurance under actuarially fair premiums.

Common pitfalls

• Treating Hicksian demand as directly observable. It isn't — utility is unobservable, so a quantity indexed by it is too. What's observable is Marshallian demand x(p, m); Hicksian demand is recovered indirectly via the Slutsky equation or by estimating a structural demand system.
• Confusing Hicks with Slutsky compensation. Hicks compensation keeps utility constant; Slutsky compensation keeps the original bundle just affordable. The two coincide to first order at the original prices but diverge for finite price changes. The Hicksian demand of consumer-theory textbooks is the Hicks-compensated quantity; "Slutsky-compensated demand" in empirical work uses the bundle-affordable rule.
• Forgetting that homogeneity of degree zero is a consequence, not an axiom. Hicksian demand satisfies h(λp, u) = h(p, u) because the consumer's choice depends only on relative prices when utility is fixed — derived from the underlying preferences, not imposed.
• Drawing Hicksian and Marshallian curves identically. They differ by the income effect: Marshallian is steeper for normal goods (income reinforces substitution), Hicksian is the steepest possible for inferior goods (income partially cancels substitution). Plot them on the same axes to see the gap visually.
• Applying Shephard's lemma without verifying differentiability. e(p, u) is differentiable in p almost everywhere but can fail at kinks in preferences (perfect complements). In those cases the Hicksian "demand" is a set-valued correspondence, not a function; Shephard's lemma generalizes via the subdifferential.
• Ignoring the duality identity h_i = x_i at the optimum. Many students see the two demands and treat them as separate objects, missing that they coincide whenever the income equals the optimal expenditure. The whole machinery hangs on h(p, e(p, u)/p) = x(p, m) — keep the link visible.