Slutsky Equation

How the Slutsky equation works

When a price changes, the consumer's demand shifts for two distinct reasons. First, the relative price of the good has changed — even if the consumer had the same purchasing power as before, she would now substitute toward whichever good has become cheaper. Second, the consumer's real purchasing power has changed: the same nominal income now buys a different basket. The two channels — substitution and income — both contribute to the observed shift in quantity demanded, and they can be told apart algebraically.

The Slutsky equation is the precise way to do it. For any pair of goods i and j:

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

       │              │                          │
       │              │                          └── income effect
       │              └─── Hicksian (compensated) substitution effect
       └─── Marshallian (ordinary) price derivative

Read left to right: the total observed response of demand for i to a price change in j splits into a pure substitution effect (the Hicksian derivative, computed at constant utility) and an income-effect correction (the consumption level of j times the income elasticity of i, suitably scaled).

Formal definition

The Slutsky equation in own-price form (i = j):

∂x_i / ∂p_i   =   ∂h_i / ∂p_i     −     x_i · ∂x_i / ∂m
                  ≤ 0 always              sign depends on
                                          normal vs inferior

And in cross-price form (i ≠ j):

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

Key properties of the Slutsky matrix S, where S_ij = ∂h_i/∂p_j:

• Symmetric. S_ij = S_ji. Hicksian cross-derivatives are symmetric because they are second partials of the expenditure function (Young's theorem).
• Negative semi-definite. z^T S z ≤ 0 for all z. Implies the diagonal entries S_ii ≤ 0 — the rigorous Law of Compensated Demand.
• Singular. Sp = 0 — the Slutsky matrix has a one-dimensional null space spanned by the price vector. A consequence of homogeneity of degree zero in (p, m).

The three properties — symmetry, negativity, singularity — are exactly the testable restrictions imposed by rational-choice theory on observed demand systems. Adding Walras's Law (p·x = m), they yield the famous Slutsky restrictions: adding-up, homogeneity, symmetry, negativity.

Derivation in two lines

Start with the duality identity that connects Marshallian and Hicksian demands at the consumer's actual optimum:

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

This just says: if you're given the income e(p, u) needed to reach utility u at prices p, the cost-minimizing bundle equals the utility-maximizing bundle. Differentiate both sides with respect to p_j (chain rule on the right):

∂h_i/∂p_j  =  ∂x_i/∂p_j  +  ∂x_i/∂m · ∂e/∂p_j

By Shephard's lemma, ∂e/∂p_j = h_j. At the optimum, h_j = x_j. So:

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

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

That's the Slutsky equation. Two lines, given Shephard's lemma. The whole machinery of consumer duality is encoded in this short derivation.

Worked example: a typical food study

Suppose a household survey gives the following estimates for ground beef:

• Marshallian own-price elasticity: ε_p = −1.0
• Income elasticity: ε_m = 0.6
• Budget share of beef: w = 0.05 (5% of income)

Convert to the Slutsky terms. The Marshallian own-price derivative ∂x/∂p · p/x = ε_p = −1.0. The income-effect correction in elasticity form is w · ε_m = 0.05 × 0.6 = 0.03. By Slutsky in elasticity form:

ε_p  =  ε_p^h  −  w · ε_m
−1.0 =  ε_p^h  −  0.03
ε_p^h  =  −0.97

The compensated (Hicksian) elasticity is −0.97 — almost identical to the Marshallian. Why? Because beef is a small share of income (5%), so the income effect is tiny relative to substitution. Slutsky decomposition: 70% substitution + 30% income (typical CPS food study) — that ratio applies to staples with larger budget shares like grain or housing, not to narrow categories like beef.

Now suppose the same calculation for housing, with w = 0.25 (25% of income), ε_m = 0.8, ε_p = −0.5:

ε_p^h  =  ε_p  +  w · ε_m
       =  −0.5  +  0.25 × 0.8
       =  −0.5  +  0.2
       =  −0.3

For housing, the Marshallian elasticity (−0.5) and the Hicksian (−0.3) differ noticeably — 40% of the observed Marshallian elasticity is the income-effect correction. Welfare analysis using Marshallian elasticities directly would overstate the substitution channel's importance by a comparable margin.

The sign table

Type of good	∂x/∂m (income elasticity sign)	Income effect on price-rise demand	Total ∂x/∂p	Marshallian demand slope
Normal good	> 0	Demand falls (real income drops, normal good follows)	Falls	Downward
Inferior good	< 0 (small)	Demand rises (inferior good rises when real income falls)	Falls (sub wins)	Downward
Giffen good	< 0 (large), big budget share	Demand rises (income channel huge)	Rises (income wins)	Upward

Giffen requires (1) inferior good, (2) strong income elasticity, (3) large budget share. All three rarely co-occur outside subsistence-level economies and staple grains. The Jensen-Miller 2008 QJE study in rural China remains the cleanest empirical case.

Slutsky vs Hicks compensation

The Slutsky equation is the same under both compensation conventions to first order at the original optimum. The two differ in their finite-displacement versions:

	Hicks compensation	Slutsky compensation
What's held fixed	Utility u(x) = U₁	Original bundle x₁ affordable
Compensated bundle	Hicksian h(p_new, u₁)	Marshallian x(p_new, p_new · x₁)
Compensation income	e(p_new, u₁) − e(p_old, u₁)	p_new · x₁ − p_old · x₁
Direction of slack	—	Overcompensates (richer than Hicks)
Where it's used	Welfare theory; CV, EV	Empirical demand; CPI bias estimation
Observable from market data?	No (needs utility function)	Yes (just bundle and prices)

For small price changes the two coincide and the Slutsky equation describes both. For finite changes Slutsky over-compensates relative to Hicks because at the new prices the consumer would substitute toward the cheaper good and reach a higher utility than U₁. This gap is the source of the Laspeyres-vs-true-COL bias estimated by the Boskin Commission.

Labor-supply Slutsky equation

Treat leisure L as a consumption good and the wage w as its price. The consumer is endowed with total time T and earns w·(T − L) in labor income plus non-labor income Y. The Slutsky equation, adapted for the endowment:

∂L / ∂w   =   ∂L^h / ∂w     +     (T − L) · ∂L / ∂Y
              ≤ 0                 ≥ 0 if leisure is normal

              (substitution:           (income:
               leisure more             more hours-worth of
               expensive →              wage income → richer →
               less leisure)            more leisure)

The sign of ∂L/∂w is ambiguous — sometimes substitution wins (more wage → more work), sometimes income wins (more wage → less work). When income wins, the labor-supply curve bends backward: high earners work less than middle earners. Empirical labor-supply elasticities estimated around 0.0–0.2 for prime-age men and 0.3–0.6 for prime-age women drive the modern optimal-tax literature.

Related theorems and identities

• Shephard's lemma. ∂e(p, u)/∂p_i = h_i(p, u). The Slutsky equation depends on this — used in the chain-rule step of the derivation.
• Roy's identity. x_i(p, m) = −(∂v/∂p_i)/(∂v/∂m). The Marshallian counterpart of Shephard on the indirect-utility side.
• Walras's Law. p · x(p, m) = m at any optimum with local non-satiation. Implies the adding-up restriction on the Slutsky matrix.
• Homogeneity of degree zero. x(λp, λm) = x(p, m). Implies the Slutsky matrix is singular (Sp = 0).
• Cournot, Engel, Slutsky aggregations. Sums of Slutsky terms across goods that follow from adding-up. Used to impose consistency on estimated demand systems.
• Antonelli matrix. The inverse Slutsky construction — the matrix of inverse-demand price derivatives, with analogous symmetry and negativity properties under the dual setup.

Applications

• Welfare cost of taxation. Harberger triangles (deadweight loss of a tax) are computed from compensated (Hicksian) elasticities, recovered from observed Marshallian elasticities via the Slutsky equation. Without the decomposition, deadweight-loss estimates conflate the distortion channel with the income channel.
• Optimal taxation (Ramsey, Mirrlees). Ramsey's inverse-elasticity commodity-tax rule and Mirrlees-Saez optimal income tax all hinge on compensated elasticities. The Slutsky equation is the bridge between observable behavior and these theoretical objects.
• Elasticity of taxable income (Feldstein, Chetty, Saez). The Saez optimal-top-rate formula uses the compensated elasticity of taxable income — Slutsky-decomposed from observed responses to tax-rate changes. Compensated ETI near 0.25 implies Laffer-curve peaks around 60–70%.
• Cost-of-living index bias. The Boskin Commission (1996) and BLS follow-ups concluded the CPI overstates inflation by 0.3–1.1 percentage points/year, partly because Laspeyres (Slutsky-compensated) indices overshoot the true Hicks-compensated cost-of-living change.
• Antitrust merger simulation. Upward Pricing Pressure (UPP) and GUPPI calculations used by FTC and DOJ depend on cross-price elasticities of Marshallian demand; Slutsky symmetry is checked as a consistency test on estimated demand systems.
• Carbon and gasoline taxes. The welfare cost is the compensated-elasticity Harberger triangle; the equity question (regressivity) is the income-effect channel. Slutsky decomposition lets the optimal-tax designer separately address the two: lump-sum rebates cancel income effects without dulling the substitution incentive.

Common pitfalls

• Sign confusion in the second term. The minus sign in ∂x_i/∂p_j = ∂h_i/∂p_j − x_j · ∂x_i/∂m is essential. For normal goods (∂x_i/∂m > 0), the second term is negative, reinforcing the negative first term: total slope is steeper than the substitution piece alone. For inferior goods (∂x_i/∂m < 0), the second term is positive, partially canceling the first.
• Reading ∂x_i/∂m from time-series data. Income elasticities for the Slutsky equation are conceptually cross-sectional: across consumers with different incomes at fixed prices. Time-series increases in income usually come with price changes, so naive time-series estimates conflate the two and bias the decomposition.
• Applying the equation to endowment problems without correction. If the consumer's income depends on prices (labor supply, asset-holding), the standard Slutsky equation gets an extra endowment-revaluation term. Ignoring it overstates the substitution effect of a wage rise.
• Confusing first-order Slutsky with finite-difference Slutsky. The equation is a derivative identity, not a finite-difference statement. For a large price change, integrating the equation along the price path gives the cumulative substitution and income contributions, with the integrand evolving over the path.
• Forgetting that Marshallian symmetry generally fails. Hicksian cross-derivatives are symmetric (∂h_i/∂p_j = ∂h_j/∂p_i). Marshallian cross-derivatives are generally not: ∂x_i/∂p_j ≠ ∂x_j/∂p_i unless income elasticities of i and j are equal. The asymmetry is the Slutsky-decomposable income-effect correction.
• Treating Slutsky equation as if it applies to revenue-equivalent tax changes. The decomposition assumes nominal income is held constant; a tax that rebates revenue (lump-sum or otherwise) introduces a separate income channel that the standard formula doesn't account for. Optimal-tax welfare analysis is careful to net these channels.