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Slutsky equation

The Slutsky equation is a fundamental principle in microeconomic consumer theory that decomposes the total effect of a change in the price of a good on the consumer's demand for that good (or another good) into two components: the substitution effect, which arises from the relative change in prices while holding purchasing power constant, and the income effect, which stems from the change in the consumer's real income due to the price change.^[1]^[2] Named after Russian economist and mathematician Eugen Slutsky, who derived it in his 1915 paper "Sulla teoria del bilancio del consumatore," the equation provides a precise mathematical framework for understanding consumer behavior under varying prices and income levels.^[1]^[3] In formal terms, the Slutsky equation relates the Marshallian demand function x_i(p, m), which gives the quantity demanded of good i at prices p and income m, to the Hicksian demand function h_i(p, u), which minimizes expenditure to achieve utility level u at prices p.^[4] The core relation is \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}, where the first term on the right captures the substitution effect (always non-positive for own-price changes due to the concavity of the utility function) and the second term represents the income effect, whose sign depends on whether the good is normal (positive demand response to income) or inferior (negative).^[4]^[2] This decomposition highlights that the substitution effect reinforces the law of demand, while the income effect can oppose it in the case of inferior goods, potentially leading to Giffen goods where demand rises with price.^[2] The equation's derivation stems from the identity linking Marshallian and Hicksian demands through the expenditure function e(p, u), via Shephard's lemma, which equates the partial derivative of expenditure with respect to price j to the Hicksian demand for good j.^[4] Extending to multiple goods, the Slutsky matrix—comprising all cross-price substitution effects—is symmetric and negative semi-definite, ensuring consistency with utility maximization and providing testable restrictions on empirical demand data.^[4] Historically overlooked until rediscovered by John Hicks and Roy Allen in 1934, Slutsky's work brought mathematical rigor to demand analysis and remains central to modern applications in welfare economics, labor supply modeling (e.g., decomposing wage effects on hours worked), and policy evaluation, such as assessing tax incidence or subsidy impacts on consumption.^[1]^[5]

Introduction

Definition and Purpose

The Slutsky equation establishes a core relationship in microeconomic consumer theory between the Marshallian demand function, which reflects uncompensated consumer choices given market prices and income, and the Hicksian demand function, which represents compensated choices holding utility constant. Formally, for the demand of good i, it expresses the partial derivative of the Marshallian demand x_i with respect to the price p_j of good j as equal to the corresponding partial derivative of the Hicksian demand h_i minus the quantity demanded of good j (x_j) multiplied by the partial derivative of the Marshallian demand x_i with respect to income m:

\frac{\partial x_i}{\partial p_j} = \frac{\partial h_i}{\partial p_j} - x_j \frac{\partial x_i}{\partial m}

This identity, derived from the budget constraint and utility maximization principles, links observable market behavior to underlying preferences.^[3] The primary purpose of the Slutsky equation is to decompose the total effect of a price change on consumer demand into two distinct components: the substitution effect, measured by \frac{\partial h_i}{\partial p_j}, which isolates how consumers adjust their consumption while maintaining the same utility level; and the income effect, captured by -x_j \frac{\partial x_i}{\partial m}, which reflects the change in real income due to the price shift and its impact on purchasing power. By separating these effects, the equation allows economists to predict and interpret how price variations influence consumer choices, such as whether demand for a good is more responsive to relative price signals or overall affordability. This analytical tool is particularly valuable in assessing elasticities and behavioral responses in markets with multiple goods. In broader economic analysis, the Slutsky equation underpins applications in revealed preference theory, where it imposes empirical restrictions on demand data to test for consistency with rational utility maximization, and in welfare economics, where it aids in calculating measures like compensating variation to evaluate the distributional impacts of policy-induced price changes without requiring direct utility observations.^[6]

Historical Development

The Slutsky equation originated with the work of Russian economist and statistician Eugen Slutsky, who introduced it in his 1915 paper "Sulla teoria del bilancio del consumatore," published in the Giornale degli Economisti. Slutsky developed the equation as a means to decompose the effects of price changes on consumer demand, building on foundational ideas from Vilfredo Pareto's partial derivation in his 1906 Manual of Political Economy and Francis Ysidro Edgeworth's earlier explorations of ordinal utility and indifference curves in the 1880s and 1890s.^[3]^[6] Slutsky's formulation advanced the understanding of compensating variation, allowing for a precise separation of substitution and income effects while maintaining the consumer's original utility level.^[3] Nearly two decades later, British economists John R. Hicks and Roy G. D. Allen independently derived the equation in their 1934 article "A Reconsideration of the Theory of Value," published in Economica. Their work integrated the decomposition into the emerging framework of ordinal utility theory, emphasizing measurable demand responses without reliance on cardinal utility assumptions, and it played a pivotal role in shifting economic analysis toward observable behavior.^[7]^[8] This rediscovery of Slutsky's earlier contribution, facilitated by Hicks in the 1930s, bridged pre-war Italian and Russian economic thought with Anglo-American developments.^[6] Following World War II, the Slutsky equation gained widespread adoption in modern general equilibrium theory during the 1940s and 1950s, as exemplified in the axiomatic models of Kenneth Arrow and Gérard Debreu, where it underpinned the symmetry and negative semi-definiteness properties essential for equilibrium existence. Slutsky's ideas also influenced Paul Samuelson's 1948 paper "Consumption Theory in Terms of Revealed Preference" in Economica, which incorporated the equation to test consistency in consumer choices through observable data, further solidifying its role in empirical and axiomatic economics.^[9]

Theoretical Foundations

Marshallian and Hicksian Demands

The Marshallian demand function, also known as the uncompensated demand, denoted x_i(\mathbf{p}, m), specifies the quantity of good i that a consumer chooses to maximize utility subject to a budget constraint \sum_j p_j x_j \leq m, where \mathbf{p} is the vector of all prices and m is the consumer's income. This function arises from the primal problem of utility maximization and reflects how demand responds to changes in prices and income while holding the budget fixed. As a result, shifts in relative prices under Marshallian demand capture both the direct substitution between goods and an indirect income effect due to altered purchasing power.^[10]^[11] In contrast, the Hicksian demand function, or compensated demand, denoted h_i(\mathbf{p}, u), indicates the quantity of good i required to achieve a target utility level u at minimum cost, solving the dual expenditure minimization problem: minimize \sum_j p_j x_j subject to u(\mathbf{x}) \geq u. Derived from this setup, Hicksian demand adjusts income implicitly to maintain constant utility as prices change, thereby excluding income effects and focusing solely on substitution responses. This compensation mechanism ensures that Hicksian demands are more suitable for welfare analysis and decomposition of price effects.^[12]^[13] The primary difference between Marshallian and Hicksian demands lies in their treatment of income adjustments: Marshallian demands hold nominal income constant, incorporating the full impact of price changes on real income and thus blending substitution and income effects, whereas Hicksian demands hold utility constant through hypothetical compensation, isolating the substitution effect alone. This distinction is crucial for understanding behavioral responses, as Marshallian demands align with observable market data under fixed budgets, while Hicksian demands provide a theoretical benchmark for pure price responsiveness.^[14]^[13] These demand functions rely on standard assumptions about consumer preferences to ensure well-defined solutions and desirable properties. Convex preferences guarantee that the feasible set is convex, leading to unique or single-valued demand under strict convexity. Local non-satiation implies that more is always preferred locally, ensuring the budget constraint binds with full income expenditure. Furthermore, assuming twice-differentiable utility functions facilitates the analysis of demand responses through first- and second-order derivatives, enabling precise measurement of elasticities.^[15]^[15]^[16]

Substitution and Income Effects

The substitution effect captures the change in a consumer's demand for a good resulting from a shift in relative prices, while maintaining the consumer's utility level constant. This effect reflects the consumer's tendency to substitute toward the now-relatively cheaper good, moving along the same indifference curve to reoptimize consumption.^[2] In contrast, the income effect arises from the change in a consumer's purchasing power due to the price variation, which effectively alters their real income and allows access to a different indifference curve. For a price decrease, this typically enables the consumer to afford more goods overall, shifting the budget line outward in a parallel manner to the original slope adjusted for the new prices.^[2] Graphically, consider a price decrease for good X: the initial budget line pivots outward from the Y-intercept, tangent to the original indifference curve at point A. The substitution effect traces the movement along this curve to a new tangency point C with a hypothetical budget line parallel to the new one but tangent to the original curve, illustrating the pure relative price response. The income effect then shifts from C to the final point B on a higher indifference curve, capturing the purchasing power gain.^[2] The direction of the income effect depends on whether the good is normal or inferior. For normal goods, where demand increases with income, the income effect reinforces the substitution effect, amplifying the total increase in demand following a price fall. For inferior goods, where demand decreases as income rises, the income effect opposes the substitution effect, potentially leading to ambiguous total effects.^[2] Eugen Slutsky formalized these effects as additive components of the total price change impact on demand in his 1915 paper, providing the foundational decomposition that separates relative price influences from income adjustments.^[17]

Derivation

For a Single Price Change

The Slutsky equation addresses the impact of a change in the price of one good, p_j, on the demand for another good, x_i, distinguishing between the substitution effect, which holds utility constant, and the income effect, which arises from the change in purchasing power. This decomposition is derived under the framework of utility maximization subject to a budget constraint \mathbf{p} \cdot \mathbf{x} = m, where \mathbf{p} is the price vector, \mathbf{x} is the consumption bundle, and m is income. The Marshallian (uncompensated) demand is x_i(\mathbf{p}, m), obtained from maximizing utility u(\mathbf{x}), while the Hicksian (compensated) demand is h_i(\mathbf{p}, u), from minimizing expenditure subject to achieving utility level u.^[18] The derivation begins with the expenditure function e(\mathbf{p}, u) = \min_{\mathbf{x}} \mathbf{p} \cdot \mathbf{x} subject to u(\mathbf{x}) \geq u, whose solution yields the Hicksian demand. By Shephard's lemma, which follows from the envelope theorem applied to the Lagrangian of the expenditure minimization problem, the partial derivative of the expenditure function with respect to p_j equals the Hicksian demand for good j:

\frac{\partial e(\mathbf{p}, u)}{\partial p_j} = h_j(\mathbf{p}, u).

This holds under assumptions of continuity and differentiability of the utility and expenditure functions, ensuring the demands are well-defined and smooth. The duality between Marshallian and Hicksian demands implies h_i(\mathbf{p}, u) = x_i(\mathbf{p}, e(\mathbf{p}, u)), as the income level that achieves utility u at prices \mathbf{p} is exactly e(\mathbf{p}, u).^[18]^[4] To derive the Slutsky equation, differentiate the identity h_i(\mathbf{p}, u) = x_i(\mathbf{p}, e(\mathbf{p}, u)) with respect to p_j, applying the chain rule:

\frac{\partial h_i(\mathbf{p}, u)}{\partial p_j} = \frac{\partial x_i(\mathbf{p}, e(\mathbf{p}, u))}{\partial p_j} + \frac{\partial x_i(\mathbf{p}, e(\mathbf{p}, u))}{\partial m} \cdot \frac{\partial e(\mathbf{p}, u)}{\partial p_j}.

Substituting Shephard's lemma \frac{\partial e}{\partial p_j} = h_j(\mathbf{p}, u) = x_j(\mathbf{p}, e(\mathbf{p}, u)) and rearranging yields the Slutsky equation for a single price change:

\frac{\partial x_i(\mathbf{p}, m)}{\partial p_j} = \frac{\partial h_i(\mathbf{p}, u)}{\partial p_j} - x_j(\mathbf{p}, m) \frac{\partial x_i(\mathbf{p}, m)}{\partial m}.

Here, \frac{\partial h_i}{\partial p_j} captures the substitution effect, and -x_j \frac{\partial x_i}{\partial m} captures the income effect. This form assumes no direct income effects on prices and relies on the differentiability of demands, with local nonsatiation ensuring the budget constraint binds.^[18]^[4] In the own-price case where i = j, the substitution effect \frac{\partial h_i}{\partial p_i} is negative, reflecting the consumer's tendency to reduce consumption of good i when its relative price rises, holding utility constant; this follows from the concavity of the expenditure function. The income effect -x_i \frac{\partial x_i}{\partial m} is negative if good i is normal (\frac{\partial x_i}{\partial m} > 0), reinforcing the downward-sloping demand, but positive if inferior, potentially offsetting the substitution effect. For the cross-price case where i \neq j, the substitution effect \frac{\partial h_i}{\partial p_j} is positive if goods i and j are substitutes (consumption of i increases as j's price rises) and negative if complements, determining the overall sign of the total effect alongside the income term.^[18]^[4]

General Matrix Form

The general matrix form of the Slutsky equation provides a compact representation for analyzing how Marshallian demands respond to simultaneous changes in all prices across multiple goods, building on the scalar case for individual price changes.^[19] The Slutsky matrix S(p, u), an n \times n matrix where n is the number of goods, captures the substitution effects and is defined element-wise as

S_{ij}(p, u) = \frac{\partial x_i(p, m)}{\partial p_j} + x_j(p, m) \frac{\partial x_i(p, m)}{\partial m},

where x_i(p, m) denotes the Marshallian demand for good i, p_j is the price of good j, and m is income (with u = v(p, m) the indirect utility). Equivalently, S_{ij}(p, u) = \frac{\partial h_i(p, u)}{\partial p_j}, where h_i(p, u) is the Hicksian demand for good i.^[20]^[19] In matrix notation, the full Slutsky equation expresses the Jacobian of the Marshallian demand with respect to prices as

\nabla_p x(p, m) = S(p, u) - x(p, m) \left( \nabla_m x(p, m) \right)^\top,

where \nabla_p x(p, m) is the n \times n matrix with entries \frac{\partial x_i}{\partial p_j}, x(p, m) is the n \times 1 demand vector, and \nabla_m x(p, m) is the n \times 1 vector of income derivatives (with ^\top denoting transpose). This decomposes the total price effect into a substitution component S(p, u) and an income effect given by the outer product term.^[21]^[19] The derivation follows from the identity x(p, m) = h(p, v(p, m)), or equivalently h(p, u) = x(p, e(p, u)), where e(p, u) is the expenditure function and v(p, m) is the indirect utility. Differentiating the latter with respect to the price vector p using the chain rule yields

\nabla_p h(p, u) = \nabla_p x(p, e(p, u)) + \left( \frac{\partial x(p, e(p, u))}{\partial e} \right) \nabla_p e(p, u)^\top.

By Shephard's lemma, \nabla_p e(p, u) = h(p, u) = x(p, m), and \frac{\partial x}{\partial e} = \nabla_m x(p, m), so rearranging gives the matrix equation above; this aggregates the single-price case via total differentials across all goods.^[21]^[5] This matrix form plays a key role in comparative statics for demand systems, enabling the systematic decomposition of price responses in multi-good models and facilitating empirical estimation of substitution patterns while controlling for income effects.^[19]^[21]

Properties of the Slutsky Matrix

Symmetry and Semi-Definiteness

The symmetry property of the Slutsky matrix S(\mathbf{p}, w) requires that the off-diagonal elements satisfy S_{ij} = S_{ji} for all goods i and j, implying that the compensated cross-price effects are equal in magnitude. This property derives from the identification of the Slutsky matrix as the Hessian matrix of second partial derivatives of the expenditure function e(\mathbf{p}, u) with respect to the price vector \mathbf{p}. Specifically, the elements are given by S_{ij} = \frac{\partial^2 e(\mathbf{p}, u)}{\partial p_i \partial p_j}, and by Young's theorem—which establishes the equality of mixed partial derivatives for twice continuously differentiable functions—these second derivatives commute, yielding \frac{\partial^2 e}{\partial p_i \partial p_j} = \frac{\partial^2 e}{\partial p_j \partial p_i}.^[18] In addition to symmetry, the Slutsky matrix is negative semi-definite on the hyperplane orthogonal to the price vector, such that for any deviation vector \mathbf{z} satisfying \mathbf{z}^\top \mathbf{p} = 0, the quadratic form obeys \mathbf{z}^\top S(\mathbf{p}, w) \mathbf{z} \leq 0. This semi-definiteness property follows directly from the concavity of the expenditure function in prices, as the Hessian of a concave function must be negative semi-definite by definition. The condition S(\mathbf{p}, w) \mathbf{p} = \mathbf{0}, which defines the relevant hyperplane, stems from the homogeneity of degree one of the expenditure function and the corresponding homogeneity of compensated demands.^[18]^[19] A sketch of the proof for negative semi-definiteness relies on the duality between the expenditure minimization problem and utility maximization. By Shephard's lemma, the Hicksian demand functions are the first-order partial derivatives of the expenditure function: h_i(\mathbf{p}, u) = \frac{\partial e(\mathbf{p}, u)}{\partial p_i} for each good i. Differentiating these with respect to prices then produces the Slutsky matrix entries: S_{ij}(\mathbf{p}, u) = \frac{\partial h_i(\mathbf{p}, u)}{\partial p_j} = \frac{\partial^2 e(\mathbf{p}, u)}{\partial p_j \partial p_i}, forming the full Hessian S = D^2_{pp} e(\mathbf{p}, u). The concavity of e(\mathbf{p}, u) in \mathbf{p}—which holds under standard assumptions of convex preferences and local nonsatiation in the underlying utility maximization problem—ensures that this Hessian is negative semi-definite, with the zero eigenvalue associated with the direction of proportional price changes due to homogeneity.^[18]^[19] These mathematical properties carry significant economic implications for consumer behavior. The negative semi-definiteness, in particular, implies that own-price compensated demands are downward-sloping (S_{ii} \leq 0), as the diagonal elements of the matrix must be non-positive, reflecting the substitution effect's tendency to reduce demand for a good when its price rises while holding utility constant. Together with symmetry, these conditions ensure the integrability of observed demands, meaning they can be rationalized by a well-behaved utility function without arbitrage opportunities—such as cycles of price changes that would allow a consumer to achieve higher utility indefinitely (akin to money pump paradoxes in revealed preference theory). This framework thus provides a theoretical foundation for consistency in demand responses under rational choice.^[22]^[18]

Interpretation in Terms of Welfare

The Slutsky equation connects Marshallian demands to Hicksian demands, enabling the analysis of consumer welfare through the expenditure function e(\mathbf{p}, u), which represents the minimum cost to achieve utility level u at prices \mathbf{p}. The Hicksian demand h_i(\mathbf{p}, u) is the partial derivative \partial e / \partial p_i, and the Slutsky matrix elements S_{ij} = \partial h_i / \partial p_j capture compensated substitution effects. For a small price change \Delta p_i, the compensating variation (CV), which measures the income adjustment needed to restore the original utility after the price change, can be approximated as CV \approx -h_i \Delta p_i. This approximation arises from the differential form de = \sum_i h_i dp_i, allowing the Slutsky terms to quantify the welfare cost of price changes by holding utility constant.^[23] In cost-of-living indexes (COLI), the Slutsky matrix plays a central role in deriving the true COLI, defined as the ratio e(\mathbf{p}^1, u)/e(\mathbf{p}^0, u) between two price vectors, which reflects the minimum expenditure change to maintain utility. The logarithmic differential d \ln e = \sum_i s_i (dp_i / p_i), where s_i = p_i h_i / e is the expenditure share, uses Slutsky substitution terms to account for consumer responses to relative price changes. This contrasts with fixed-basket approximations like the Laspeyres index, which overstates cost increases by ignoring substitutions, and the Paasche index, which understates them; superlative indexes, such as the Fisher ideal index, better approximate the true COLI by averaging base and current quantities in a manner consistent with Slutsky symmetry and semi-definiteness. The Slutsky equation facilitates empirical estimation of these compensated effects from observed Marshallian demands, improving the accuracy of COLI for inflation measurement and adjustments in social programs.^[23] The Slutsky framework aids policy analysis by enabling the evaluation of price changes—such as taxes, subsidies, or regulations—on consumer surplus without requiring interpersonal utility comparisons, as CV and equivalent variation (EV) provide money-metric welfare measures. By decomposing total demand changes into substitution effects (via the Slutsky matrix) and income effects, policymakers can isolate the distortionary impacts of price policies on efficient allocation, estimating deadweight losses or surplus gains from reforms like tariff reductions. For instance, the compensated demands reveal how resources are reallocated under policy-induced price shifts, supporting ex-ante assessments of welfare neutrality in revenue-neutral tax changes. This approach is particularly valuable in public finance, where Slutsky-derived measures quantify the net benefit to consumers from policy interventions. A key limitation in these welfare interpretations is the reliance on approximations that hold exactly only under homothetic preferences, where the expenditure function scales linearly with utility (e(\mathbf{p}, \alpha u) = \alpha e(\mathbf{p}, u)), making substitution effects independent of the base utility level and higher-order Taylor terms in the CV integral vanish. Without homotheticity, the Slutsky matrix varies across utility levels, leading to path-dependent welfare estimates and biases in COLI approximations that depend on the chosen reference utility. Thus, for non-homothetic preferences—common in empirical settings with varying income elasticities—the Slutsky-based measures require additional assumptions or nonparametric bounds to ensure robustness in policy evaluations.

Applications and Examples

Numerical Example with Two Goods

To illustrate the Slutsky decomposition in a simple two-good economy, consider a consumer with the Cobb-Douglas utility function u(x, y) = x^a y^{1-a}, where x and y are the quantities of goods X and Y, respectively, p_x and p_y are their prices, and m is the consumer's income. The Marshallian (uncompensated) demand functions are x(p_x, p_y, m) = \frac{a m}{p_x} and y(p_x, p_y, m) = \frac{(1-a) m}{p_y}.^[24] The Slutsky equation decomposes the total effect of a change in p_x on the demand for X into a substitution effect and an income effect: \frac{\partial x}{\partial p_x} = s_{xx} - x \frac{\partial x}{\partial m}, where s_{xx} is the substitution term from the Hicksian (compensated) demand. For the Cobb-Douglas case, the total effect is \frac{\partial x}{\partial p_x} = -\frac{a m}{p_x^2}, the substitution effect is s_{xx} = -a(1-a) \frac{m}{p_x^2}, and the income effect is -x \frac{\partial x}{\partial m} = -a^2 \frac{m}{p_x^2}. This verifies the decomposition, as -a(1-a) \frac{m}{p_x^2} - a^2 \frac{m}{p_x^2} = -\frac{a m}{p_x^2}.^[25] For a numerical illustration, assume a = 0.5, m = 100, p_x = 1, and p_y = 1. The initial demands are x = 50 and y = 50, yielding utility u = \sqrt{50 \times 50} = 50. The partial derivatives at these values give a total effect of \frac{\partial x}{\partial p_x} = -50, a substitution effect of -25, and an income effect of -25. Now suppose p_x increases to 2. The new uncompensated demand is x = 25 (a total change of -25), while y = 50. To isolate the substitution effect, compute the Hicksian demand at the new prices holding utility constant at 50; the minimum expenditure required is e = 2 \times 50 \times \sqrt{2 \times 1} = 100\sqrt{2} \approx 141.42, so the compensated demand is h_x = 50 / \sqrt{2} \approx 35.36 (a substitution change of approximately -14.64). The income effect is then the remaining change from h_x \approx 35.36 to the new uncompensated x = 25 (approximately -10.36). These finite changes approximate the derivative-based effects, which become exact for infinitesimal price changes.^[26] In this normal goods case, both the substitution effect (a shift away from the now-relatively more expensive good X) and the income effect (reduced purchasing power leading to lower consumption of the normal good X) decrease demand when p_x rises, with the income effect reinforcing the substitution effect.^[26]

Multiple Simultaneous Price Changes

When multiple prices change simultaneously, the Slutsky matrix provides a compact framework for analyzing the resulting changes in consumer demand, capturing both own-price and cross-price substitution effects across all goods. For small price changes \Delta \mathbf{p}, the compensated change in demand (holding utility constant) is approximated by \Delta \mathbf{x}^h \approx S(\mathbf{p}, u) \Delta \mathbf{p}, where S is the Slutsky matrix and \mathbf{x}^h denotes Hicksian demand.^[27] This matrix multiplication allows economists to decompose the total substitution effect into contributions from each price adjustment, ensuring that the overall response respects the negative semi-definiteness of S, which implies that the weighted sum of price and demand changes is non-positive.^[27] For uncompensated demand changes under fixed income (pure price shocks), the total effect is \Delta \mathbf{x} \approx S(\mathbf{p}, m) \Delta \mathbf{p} - \mathbf{x} (\partial \mathbf{x}/\partial m)^\top \Delta \mathbf{p}, where the second term accounts for the income effect induced by the price changes.^[27] In this setting, the uncompensated price derivative matrix \partial \mathbf{x}/\partial \mathbf{p} = S - \mathbf{x} (\partial \mathbf{x}/\partial m)^\top directly yields \nabla_{\mathbf{p}} \mathbf{x} \cdot \Delta \mathbf{p} = S \Delta \mathbf{p} - \mathbf{x} (\partial \mathbf{x}/\partial m)^\top \Delta \mathbf{p}, highlighting how simultaneous price shifts can amplify or offset income effects across goods.^[27] Consider a three-good model where prices of goods 1 and 2 increase simultaneously while the price of good 3 remains fixed. The Slutsky matrix S would feature diagonal elements s_{11} and s_{22} capturing the own-substitution effects (typically negative), and off-diagonal elements s_{12} and s_{21} (equal by symmetry) representing cross-substitution, such as consumers shifting demand toward good 3 if s_{13} and s_{23} are positive.^[27] The net compensated demand change for good 1, for instance, is \Delta x_1^h \approx s_{11} \Delta p_1 + s_{12} \Delta p_2 + s_{13} \Delta p_3, illustrating how the joint price rise might lead to a larger substitution away from goods 1 and 2 than isolated changes would suggest, assuming S remains negative semi-definite.^[27] The properties of the Slutsky matrix, including symmetry (s_{ij} = s_{ji}) and negative semi-definiteness, ensure consistency in multi-price environments, such as uniform inflation where all prices rise proportionally (\Delta \mathbf{p} = \lambda \mathbf{p}).^[27] In this case, homogeneity of degree zero in demand implies S \mathbf{p} = 0, so compensated demand remains unchanged, maintaining budget balance without altering real consumption patterns.^[27] These properties aggregate across consumers, allowing the matrix to model economy-wide responses reliably. In empirical demand system estimation, the Slutsky matrix adjusts for multiple price endogeneities, as seen in the Almost Ideal Demand System (AIDS) proposed by Deaton and Muellbauer.^[28] The AIDS budget share equations incorporate Slutsky terms via parameters \gamma_{ij}, where the compensated cross-price effect on share w_i from a price change in p_j is \gamma_{ij} + \beta_i w_j - \beta_j w_i, enabling estimation of simultaneous price impacts while imposing symmetry (\gamma_{ij} = \gamma_{ji}) for theoretical consistency.^[28] Applied to postwar British consumption data (1954–1974) across eight nondurable goods, AIDS revealed significant cross-effects (e.g., 22 of 64 \gamma_{ij} coefficients statistically significant at 5%), demonstrating how multiple price changes influence shares like food and clothing, though symmetry was rejected in favor of flexible specifications.^[28] This framework has been widely adopted for policy analysis, such as evaluating inflation's differential effects on household demands.^[28]

Giffen Goods and Exceptions

A Giffen good is an inferior good characterized by a strong negative income effect that dominates the substitution effect, causing the Marshallian demand to increase with its own price, such that ∂x_i/∂p_i > 0. This anomaly arises specifically from the Slutsky equation, where the condition for Giffen behavior holds only if the magnitude of the income effect exceeds that of the substitution effect: x_i ∂x_i/∂m < ∂h_i/∂p_i (with both terms negative). The historical archetype of a Giffen good is the potato during Ireland's 19th-century famine, where subsistence-level consumption by impoverished households reportedly led to higher potato demand amid rising prices, though subsequent analyses have questioned the empirical validity of this case due to data limitations.^[29] More robust modern evidence comes from field experiments in low-income settings, such as Jensen and Miller's 2008 study in rural China, which identified Giffen behavior for rice among the poorest households; price subsidies reduced rice consumption as households shifted to relatively more expensive proteins, confirming the income effect's dominance.^[30] Similar quasi-Giffen patterns have been observed for staple foods like rice in impoverished Indonesian communities, where nutritional constraints amplify inferiority, with potential for analogous behavior in Indian settings though direct empirical evidence remains limited.^[31] Theoretical constraints on Giffen goods stem from the properties of the Slutsky matrix, particularly its negative semi-definiteness, which ensures that the substitution effects are non-positive and imposes limits on the feasible configuration of income effects; consequently, Giffen behavior cannot manifest for all goods simultaneously, as this would violate the matrix's semi-definiteness and the homogeneity of demand functions—at least one good must display a negative own-price response to maintain consistency.^[32]^[33] Exceptions to the standard Slutsky framework occur when underlying assumptions, such as convex preferences, are violated; non-convex preferences can lead to a non-negative substitution matrix, potentially allowing Giffen-like outcomes without inferiority or even reversing the usual welfare interpretations of price changes.

Labor Supply Application

Beyond consumer goods, the Slutsky equation applies to labor supply, decomposing the effect of a wage change on hours worked. The substitution effect encourages more work as the opportunity cost of leisure rises, while the income effect may reduce labor supply if leisure is a normal good. Empirical studies often find the substitution effect dominates for low-wage workers, leading to upward-sloping supply curves.^[5]

References

[1]
The Mechanics of Demand | Federal Reserve Bank of Minneapolis
The Slutsky equation clarifies the complicated effects of price changes.
[2]
[PDF] Income and Substitution Effects
Answer: Approximate with Marchallisn demand. •. From the Slutsky equation, we know the Hicksian and Marshallian demand functions have approximately the same ...
[3]
ON THE THEORY OF THE BUDGET OF THE CONSUMER - jstor
** Professor Slutsky was at the Institute of Commerce at Kiev when this article was published; he died in Moscow in 1948. Page 2. 174 EUGEN E. SLUTSKY.
[4]
[PDF] Slutsky equation
The budget constraint is p · ˜x = µ, where µ is outside income. The Slutsky equation is derived from the identity. ˜x (p, e (p, ˜u)) = ˜h (p, ˜u). This gives.
[5]
[PDF] A Slutsky derivation - MIT OpenCourseWare
Slutsky for Hours (done in minutes). Josh Angrist. MIT 14.661 (FALL 2017) ... • Re-arrange to get the Slutsky equation for hours: ∂h. ∂hc. ∂h. = + h. ∂w.
[6]
[PDF] Slutsky's 1915 Article: How It Came to be - Department of Economics
In 1915 the Russian statistician and economist Eugen Slutsky sent off from Kiev ... The equality kij = kji (equation [55] in Slutsky's paper) is called the law of.
[7]
A Reconsideration of the Theory of Value. Part I - jstor
A Reconsideration of the Theory of. Value. By J. R. HICKS and R. G. D. ALLEN. Part I. By J. R. HICKS. THE pure theory of exchange value, after a period of ...
[8]
A Reconsideration of the Theory of Value - jstor
A Reconsideration of the Theory of. Value. By J. R. HICKS and R. G. D. ALLEN. Part II.-A Mathematical Theory of Individual. Demand Functions. By R. G. D. ALLEN.
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Consumption Theory in Terms of Revealed Preference - jstor
If the preference field has simple concavity, " indifference " will never explicitly reveal itself to us except as the results of an infinite limiting process.
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[PDF] INCOME AND SUBSTITUTION EFFECTS Two Demand Functions
Marshallian demand xi(p1,…,pn,m) describes how consumption varies with prices and income. – Obtained by maximizing utility subject to the budget constraint. ...
[11]
[PDF] Consumer Theory: The Mathematical Core
Feb 2, 2025 · The function X(p,y) is called this consumer's market or Marshallian demand function. Since this function maximizes utility subject to the ...
[12]
[PDF] Consumer Choice 2
The Hicksian demand function h(p,u) = arg min. x2X Cpi xi subject to u(x). Cu. This is the demand for each good when prices are p and the consumer must achieve ...
[13]
[PDF] Demand III Cross-Price Elasticity of Demand Hicksian Demand ...
Hicksian Demand Functions. • Recall “Marshallian” Demand Functions. – hold income constant. • “Hicksian” or “Utility Constant” or. “Compensated” Demand Function.
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[PDF] Lecture Note 6 – Demand Functions: Income Effects, Substitution ...
The 'Marshallian cross' is the staple tool of blackboard economics. Marshallian demand curves correspond to conventional market-level or consumer-level demand ...
[15]
[PDF] Notes on Microeconomic Theory - Nolan H. Miller
Aug 18, 2006 · the definition of local non-satiation: For any x ∈ X and ε > 0 ... behave, we choose to model consumers as having convex preferences.
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[PDF] Lectures 3—4: Consumer Theory
Approach to comparative statics of Marshallian demand is to relate to Hicksian demand, decompose into income and substitution effects via Slutsky equation.
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Eugene Slutsky - The History of Economic Thought Website
In his 1915 paper, he presented the "Slutsky decomposition" of demand functions into substitution and income effects. His work was ignored and only much later, ...
[18]
[PDF] Hicksian Demand and Expenditure Function Duality, Slutsky Equation
Remember that h (p, v) = x (p, e(p, v)). This is also known as the Slutsky equation: it connects the derivatives of compensated and uncompensated demands. ...
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[PDF] Economics 250a Lecture 1: A very quick overview of consumer ...
(The Slutsky matrix always has maximal rank n 1, where n is the number of commodities. In particular, the original price vector p0 is in the null space of S):.
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[PDF] Chapter 2
The matrix S(p, w) is known as the substitution, or Slutsky matrix. Its elemtns are known as substitution effects. #Explanation of Slutsky matrix (p.34). The ...
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None
### Summary of Slutsky Equation from Lecture 11 PDF
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[PDF] CHAPTER 2 | TOPICS IN CONSUMER THEORY
Negative Semidefiniteness: The associated Slutsky matrix s(p,y) must be negative semidefinite. • Symmetry: s(p,y) must be symmetric. We would like to know ...
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2 Conceptual Foundations for Price and Cost-of-Living Indexes
... price vectors. Note that, because the Slutsky matrix is a negative semidefinite matrix, the quadratic form on the right- hand side of (46) is nonpositive as ...
[24]
[PDF] INDIRECT UTILITY FUNCTION U*(Px,Py,M) = max {U(x, y)
) holding u fixed. Cobb-Douglas example: (Px)1/3 (Py)2/3. PROPERTIES OF ... SLUTSKY EQUATION. Link between Marshallian and Hicksian demands. Equal if u ...Missing: two | Show results with:two
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[PDF] CobbDouglasSlutsky.pdf - Amherst College
Slutsky Equation we have to know the Marshallian demand for good 2 which is. 2. 2. (1 )I x p α. −. = . So the Slutsky components are: 1. 1. 1. 1. 1. 1. 2. 1. 2.
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[PDF] The Slutsky Equation Seminar Handout
Oct 23, 2019 · Slutsky equation holds using the Cobb-Douglas utility function u(x1,x2) = x. 1. 2. 1 x. 1. 2. 2 where x1 denotes the quantity of good 1 and x2 ...
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[PDF] Microeconomic Theory
For semidefiniteness of the symmetric matrix A, we replace the strict inequalities by weak inequalities and require that the weak inequalities hold for all ...
[28]
[PDF] An Almost Ideal Demand System - American Economic Association
Our model, which we call the Almost. Ideal Demand System (AIDS), gives an ar- bitrary first-order approximation to any de- mand system; it satisfies the axioms ...
[29]
Robert Giffen and the Irish Potato - jstor
It is likely that potatoes were not inferior goods for the Irish peasantry during the famine years. Inferior- ity is necessary for a good to be Giffen.
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[PDF] notes for micro i: single person and multiperson decision theory
... symmetric, negative semi-definite Slutsky matrix, then x(p, w) can be rationalized by a preference relation. Summary of Demand Theory in Chapter. Preference ...
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[PDF] Chapter 4: Topics in Consumer Theory - Nolan H. Miller
1. Since there are no wealth effects for apples or bananas, the Walrasian and Hicksian demand curves coincide, and the change in Marshallian consumer surplus is ...<|control11|><|separator|>