Slutsky Equation

Slutsky equation (also termed the Slutsky identity or Slutsky decomposition) is the central analytical relationship in neoclassical consumer theory and business and economics theory, linking observable Marshallian (uncompensated) demand functions to unobservable Hicksian (utility-compensated) demand functions. First derived by Ukrainian-born mathematical statistician and economist Eugen Slutsky (Yevgeny Slutsky) in his 1915 Italian-language treatise "Sulla teoria del bilancio del consumatore" ("On the Theory of the Budget of the Consumer") and independently rediscovered and popularized in English by Sir John Hicks and R. G. D. Allen in 1934, the equation establishes that whenever the price p_j of a commodity changes, the resulting total response in the quantity demanded of good i decomposes additively into two distinct economic channels: (1) a substitution effect, measuring how the consumer rearranges consumption along a fixed indifference curve in response to altered relative prices, and (2) an income effect, measuring how the price change alters the real purchasing power of the consumer's nominal budget m.

Primal Marshallian vs. Dual Hicksian Demand

Consider a utility-maximizing consumer facing commodity price vector p = (p₁, p₂, …, p_n) ≫ 0 with nominal money income m > 0 and a continuous, locally non-satiated, quasi-concave utility function U(x) defined over consumption bundles x ∈ ℝ₊ⁿ:

1. Marshallian (Uncompensated) Demand x(p, m): Solves the primal utility-maximization problem max_x U(x) subject to the budget constraint p · x ≤ m, attaining maximum indirect utility u^* = v(p, m).
2. Hicksian (Compensated) Demand h(p, u): Solves the dual expenditure-minimization problem min_x p · x subject to attaining target utility U(x) ≥ u, requiring minimum monetary expenditure m^* = e(p, u).

At any interior optimum where u = v(p, m) and m = e(p, u), the bundle that maximizes utility for budget m is identical to the bundle that minimizes expenditure to achieve utility u. Consequently, the Hicksian and Marshallian demand functions are linked across all prices p by the fundamental duality identity:

Differentiating the Duality Identity

Differentiating both sides of the identity h_i(p, u) ≡ x_i(p, e(p, u)) partially with respect to the j-th commodity price p_j (while holding target utility u constant) via the multivariable chain rule yields:

By Shephard's lemma, the partial derivative of the expenditure function with respect to price p_j equals the Hicksian compensated demand for good j, which at the initial optimum equals the observed Marshallian quantity consumed x_j(p, m):

Substituting x_j(p, m) into the chain-rule expression and rearranging to solve for the observable Marshallian price derivative ∂x_i / ∂p_j produces the Slutsky equation:

For an own-price change (i = j), the Slutsky equation takes its canonical single-commodity form:

Intuitively, if a household currently consumes x_i = 10 loaves of bread per week and the price of bread p_i rises by Δp_i = 1 currency unit, the household needs Δm = x_i · Δp_i = 10 units of additional income just to afford its original bundle; thus a price increase of dp_i inflicts an effective real income loss of −x_i dp_i.

Multiplying the own-price Slutsky equation through by p_i / x_i and multiplying the last term by m / m converts the partial derivatives into dimensionless Slutsky elasticity form:

where:

• ε_i,i = (∂x_i / ∂p_i) · (p_i / x_i) is the uncompensated (Marshallian) own-price elasticity of demand;
• ε_i,i^c = (∂h_i / ∂p_i) · (p_i / h_i) ≤ 0 is the compensated (Hicksian) own-price elasticity of demand;
• s_i = (p_i · x_i) / m ∈ (0, 1) is the budget expenditure share devoted to good i; and
• η_i = (∂x_i / ∂m) · (m / x_i) is the income elasticity of demand for good i.

Because the expenditure function e(p, u) is globally concave in prices p, its second own-partial derivative is never positive: ∂h_i / ∂p_i = ∂²e / ∂p_i² ≤ 0 (the Law of Compensated Demand). Therefore, the sign of the total Marshallian slope ∂x_i / ∂p_i is governed by the interplay between the negative substitution effect and the income effect:

As the elasticity form ε_i,i = ε_i,i^c − s_i η_i makes transparent, a Giffen good (ε_i,i > 0) requires three simultaneous conditions that are rarely met in modern diversified economies: (1) the good must be strongly inferior (η_i < 0); (2) it must have very few close substitutes (small magnitude of ε_i,i^c); and (3) it must consume a very large fraction s_i of the household's total budget (as documented empirically by Robert Jensen and Nolan Miller in 2008 for staple rice and wheat consumption among ultra-poor households in Hunan and Gansu provinces).

In an n-commodity general equilibrium theory economy, the n × n matrix S(p, m) whose (i, j)-th entry is the compensated cross-price derivative:

is called the Slutsky substitution matrix. Because S_i,j(p, m) = ∂²e(p, u) / (∂p_i ∂p_j) is the Hessian matrix of the expenditure function e(p, u), utility maximization imposes three testable restrictions on observable Marshallian demand systems:

1. Symmetry (S_i,j = S_j,i): Although uncompensated cross-price derivatives are generally asymmetric (∂x_i / ∂p_j ≠ ∂x_j / ∂p_i whenever income elasticities or budget shares differ), the Slutsky-compensated cross-price effects are identical by Young's theorem: ∂h_i / ∂p_j = ∂h_j / ∂p_i. Goods with S_i,j > 0 are net (Hicksian) substitutes, whereas goods with S_i,j < 0 are net (Hicksian) complements.
2. Negative Semidefiniteness (v^⊤ S v ≤ 0): For any vector v ∈ ℝⁿ, the quadratic form ∑_i ∑_j v_i S_i,j v_j ≤ 0, implying S_i,i ≤ 0 for all i and S_i,i S_j,j − S_i,j² ≥ 0.
3. Singularity (S · p = 0): Because Hicksian demand h_i(p, u) is homogeneous of degree 0 in p, Euler's theorem implies ∑_j=1ⁿ S_i,j(p, m) · p_j = 0 for every row i.

By Hurwicz and Uzawa's (1971) Integrability Theorem, any system of continuously differentiable functions x(p, m) satisfying budget exhaustion (p · x(p, m) = m), homogeneity of degree zero, and a symmetric negative semidefinite Slutsky matrix S(p, m) can be integrated back to recover a well-behaved underlying utility function U(x) that rationalizes the observed behavior.

When a household begins not with fixed cash income m but with an initial physical endowment vector ω = (ω₁, …, ω_n) so that nominal income is endogenous (m = p · ω), a rise in p_j also increases the market value of the household's endowment by ω_j. Replacing −x_j with (ω_j − x_j) yields the Endowment Slutsky Equation:

In labor supply theory, a worker possesses a time endowment T (for example, 24 hours per day), consumes leisure L ≤ T at an opportunity cost equal to the real wage rate w, and supplies labor hours H = T − L (so the worker is a net seller of time: T − L = H > 0). Applying the Endowment Slutsky Equation to leisure demand L with respect to the wage w gives:

Here, a higher wage w makes leisure more expensive, inducing a positive substitution effect on labor supply (−∂L^c / ∂w > 0). However, because leisure is a normal good (∂L / ∂m > 0) and the worker is a net seller of labor (H > 0), a higher wage makes the worker wealthier, inducing a negative income effect on labor supply (−H · ∂L / ∂m < 0) as the wealthier worker chooses to consume more leisure. At low wages where H is small, the substitution effect dominates and labor supply slopes upward; at high wages where H is large, the income effect can outweigh the substitution effect, producing the classic backward-bending labor supply curve.

What is the difference between Hicksian compensation and Slutsky compensation?

In Hicksian compensation (the partial derivative ∂h_i / ∂p_j used in the continuous Slutsky equation), income is adjusted after a price change to keep the consumer on their original indifference curve (constant utility u₀). In Slutsky's original 1915 discrete compensation, income is adjusted by Δm = Δp · x⁰ so the consumer can just afford their original consumption bundle x⁰ (constant purchasing power). For infinitesimal price changes (dp_j → 0), Hicksian and Slutsky compensation converge to the exact same derivative because the budget line is tangent to the indifference curve at x⁰.

Why are Hicksian cross-price effects always symmetric even when Marshallian cross-price effects are not?

Suppose good 1 is coffee (budget share s₁ = 0.01) and good 2 is housing (budget share s₂ = 0.40). A 10% rise in housing rent p₂ inflicts a large real income loss that reduces coffee consumption (∂x₁ / ∂p₂ < 0), whereas a 10% rise in coffee price p₁ has a negligible income effect on housing demand (∂x₂ / ∂p₁ ≈ 0). Once the asymmetric income effects −x_j(∂x_i / ∂m) are stripped out via the Slutsky equation, the pure compensated substitution effects S_1,2 = ∂²e / (∂p₁ ∂p₂) = S_2,1 are equal by Young's theorem.

Why is Every Giffen Good an Inferior Good, but Not Every Inferior Good is a Giffen Good?

By the Slutsky equation ∂x_i / ∂p_i = (∂h_i / ∂p_i) − x_i(∂x_i / ∂m), the substitution effect ∂h_i / ∂p_i is always negative or zero. For the total price derivative ∂x_i / ∂p_i to be positive (a Giffen good), the second term −x_i(∂x_i / ∂m) must be positive—which requires ∂x_i / ∂m < 0 (an inferior good)—and must also be larger in absolute magnitude than the negative substitution effect. Most inferior goods (such as intercity bus tickets or instant noodles) have small budget shares s_i and close substitutes, so the negative substitution effect dominates and their demand curves still slope downward.

1. Slutsky, E. E. (1915). "Sulla teoria del bilancio del consumatore." Giornale degli Economisti e Rivista di Statistica, 51(1), 1–26 (English translation in Stigler, G. J., & Boulding, K. E., Eds., 1952, Readings in Price Theory, pp. 27–56, Richard D. Irwin). https://www.jstor.org/stable/23225031
2. Hicks, J. R., & Allen, R. G. D. (1934). "A Reconsideration of the Theory of Value." Economica, 1(1), 52–76 and 1(2), 196–219. https://doi.org/10.2307/2548574
3. Samuelson, P. A. (1947). Foundations of Economic Analysis. Harvard University Press. https://www.hup.harvard.edu/books/9780674313019
4. Hurwicz, L., & Uzawa, H. (1971). "On the Integrability of Demand Functions." In J. S. Chipman et al. (Eds.), Preferences, Utility, and Demand (pp. 114–148). Harcourt Brace Jovanovich. https://www.jstor.org/stable/1913028
5. Jensen, R. T., & Miller, N. H. (2008). "Giffen Behavior and Subsistence Consumption." The American Economic Review, 98(4), 1553–1577. https://doi.org/10.1257/aer.98.4.1553