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Untestability of Average Slutsky Symmetry
Rationality is a central concept in economic theory, serving as a fundamental assumption in the analysis of consumer behavior. It assumes that consumers make decisions that are consistent with utility maximization, a principle that supports many economic models and empirical research. Testing this assumption is crucial for validating theoretical models and understanding consumer decision-making.
When individual demand functions are available, the Hurwicz-Uzawa theorem (hurwicz1971integrability) provides a complete characterization of rationality. To state the theorem, let $p$ be a $d$-dimensional price vector and $y$ be an income, both of which are relative to the price of the numeraire. Let $q (p, y)$ be the $d$-dimensional vector of quantities demanded when the price is $p$ and the income is $y.$ We say that a demand function $q$ is rational if it maximizes a utility function subject to the budget constraint. According to the Hurwicz-Uzawa theorem, a demand function $q$ is rational if and only if its Slutsky matrix, which is defined as a $d \times d$ matrix $$ S_q (p, y) \coloneqq D_p q (p, y) + D_y q (p, y) q (p, y)^\prime , $$ is both symmetric and negative semidefinite.
In many real-world applications, individual demand functions are not observed and only cross-sectional data is available. To investigate the population rationality, researchers have explored the rationalizability of the average demand function conditional on observable characteristics, such as price and income, treating it as if it is generated by an individual representing the population. See, for example, lewbel1995consistent, lewbel2001demand, haag2009testing, and hoderlein2011many.
However, this approach involves some disadvantages. For instance, as Theorem 1 of lewbel2001demand demonstrates, the rationality of the average demand function is irrelevant to the rationality of individual demand functions: it is possible that all individuals in a population are rational while the average demand is not, and vice versa. Moreover, it is often reported that cross-sectional mean regressions fail to explain the variation of demand adequately because of the unobserved preference heterogeneity. (hoderlein2011many, hausman2016individual). These facts indicate the need to investigate all the information about the heterogeneity contained in the data.
Compared with testing the rationality of an individual demand function, testing whether a cross-sectional demand data is consistent with a rational population is a harder problem because individual Slutsky matrices are not observed. This difficulty raises a fundamental question: can we infer the rationality of a population from cross-sectional demand data? More specifically, is it possible to construct a statistical test with nontrivial power to detect consumer irrationality only from demand distributions?
Recent research has made progress in addressing this question. hausman2016individual provide a necessary and sufficient condition for cross-sectional demand distributions to be rationalizable when only two goods are present. For cases involving more than two goods, dette2016testing provide the empirical content of the negative semidefiniteness of the Slutsky matrix. Specifically, they show that the quadratic form of the average Slutsky matrix $\bE [S_{Q^\ast} (p, y)]$ conditional on price and income, where $Q^\ast$ is the random demand function representing the population, is identified. If an estimate of the quadratic form is significantly positive, then it means that the observed data is likely to be inconsistent with a population having an negative semidefinite Slutsky matrix---let alone rationality. These studies demonstrate the feasibility of testing rationality by means of the negative semidefiniteness of the Slutsky matrix.
However, these methods neglect the symmetry condition, leading to statistical tests that are overly conservative and fail to fully leverage the implications of rationality. The symmetry condition can be more critical to the rationality than the negative semidefiniteness, given that the former is highly sensitive to small perturbations while the latter is not. If Slutsky symmetry is testable, one should be able to construct a much more powerful test for rationality by testing both conditions.
Despite its theoretical importance, the empirical content of the Slutsky symmetry condition remains poorly understood. This paper provides a first step toward elucidating the implications of individual Slutsky symmetry for cross-sectional demand distributions. The main result of this paper claims that the average Slutsky matrix $\bE [S_{Q^\ast} (p, y)]$ is not identified, even though its quadratic form is identified, as shown in dette2016testing. Moreover, we show that the identified set of the average Slutsky matrix always contains a symmetric matrix.
An immediate consequence of our results is the fundamental untestability of individual Slutsky symmetry via the average Slutsky matrix. This fact indicates that the Slutsky symmetric condition is totally distinct to the negative semidefiniteness condition, which is testable based on the quadratic form of the average Slutsky matrix (dette2016testing).
The untestability of the average Slutsky symmetry does not immediately rule out the possibility of testing Slutsky symmetry from cross-sectional distributional data. After the first draft of this paper was made public, gunsilius2025nonparametric derive the empirical content of individual Slutsky symmetry without relying on the average Slutsky matrix. They show that individual Slutsky symmetry implies a nonparametric conditional quantile restriction, which is testable.
Although the average Slutsky symmetry is untestable in general, it can become testable under additional assumptions. In particular, we consider imposing bounds on the income elasticity of demand. By restricting the patterns of demand substitution in this way, Slutsky symmetry leads to functional inequality conditions that are, in principle, testable. Constructing a formal statistical test based on these conditions is beyond the scope of this paper, but existing methods for testing functional inequalities could be applied.
A key theoretical contribution of this paper is the development of a constructive method to generate a random demand function that satisfies prescribed marginal distributions while remaining consistent with the Slutsky conditions. In general, both the levels and the derivatives of individual demand functions vary across consumers, reflecting heterogeneous preferences. The construction in this paper shows, however, that even if one fixes a deterministic rule describing how demand changes with prices and income, it is still possible to produce random demand distributions that are consistent with observed marginals and with rational behavior. The key insight is that once the local response of demand to prices and income is fixed, the distribution of demand at all other price-income pairs can be obtained from the distribution at a single reference point by following the trajectories implied by this response rule. Mathematically, this propagation can be interpreted as the solution to a system of differential equations that transports the reference distribution along price-income changes.
Related literature. hurwicz1971integrability investigate the integrability of individual demand functions and give a necessary and sufficient condition for rationality based on the Slutsky matrix. lewbel2001demand considers a population that is heterogeneous in preference and provides conditions for the average demand function to be rational. haag2009testing and hoderlein2011many among others consider the estimation and inference on the average demand function under the rationality shape restriction.
For cross-sectional demand distributions, hausman2016individual give a necessary and sufficient condition for observed datasets to be consistent with a rational demand system for the cases when there are only two goods. When more than two goods are present, the characterization of rationalizability is largely open. dette2016testing provide a necessary condition that the data needs to satisfy for it to be rationalizable focusing on the negative semidefiniteness of the Slutsky matrix and propose a statistical testing for rationality. maes2024beyond construct a testing procedure of rationality based on higher order moments of demand distributions. They also observe that the average Slutsky matrix is not identified from those moments. More recently, gunsilius2025nonparametric derive the empirical content of individual Slutsky symmetry without relying on the average Slutsky matrix.
This paper is also related to the literature on random utility models. mcfadden1990stochastic and mcfadden2005revealed show that the axiom of revealed stochastic preference characterizes rationalizability of stochastic choice functions defined on a finite number of choice sets. In a similar setup, kitamura2018nonparametric constructed a statistical test for the axiom. In the context of discrete choice, bhattacharya2025integrability recently gives the complete characterization of the rationalizability of demand distributions.
For $d \geq 2,$ we consider an economy with $d + 1$ goods. Relative to the first good, their prices are encoded into a price vector $p \in \cP \subset \bR_+^d$ where $\bR_+ \coloneqq (0, \infty).$ Let $y \in \cY \subset \bR_+$ be the relative income. For given price $p$ and income $y,$ a consumer demands $q (p, y) \in \bR_+^d.$ Notice that we assume the homogeneity of demand functions at this point. We also assume Warlas' law, i.e., the demand for the numeraire is $y - p^\prime q (p, y),$ and consequently, $p^\prime q (p, y) < y$ is assumed.
We assume $$ \cP = \prod_{i = 1}^d \left[\underline p_i, \overline p_i\right] \text{ and } \cY = [\underline y, \overline y] $$ for $0 < \underline p_i < \overline p_i$ and $0 < \underline y < \overline y.$ Let $\cX \coloneqq \cP \times \cY \subset \bR_+^{d + 1}.$ We restrict ourselves to demand functions that are in the space $$ \cQ \coloneqq \left\{ q : \cX \to \bR_+^d \ \ \Big | \
\right\} . $$ \footnotetext{This is a weaker condition than the demand function being $C^1$ because the partial derivative with respect to one variable need not be continuous in other variables.}
In the real world, consumer's preference heterogeneity is present. In this sense, consumer's demand is stochastic from the perspective of an econometrician. Let $Q^\ast$ be a random individual demand function drawn from a probability distribution on $\cQ.$ The econometrician is assumed to observe cross-sectional demand distributions, that is, (s)he observes the distribution $\mu_x$ of $Q^\ast (x)$ for each $x \in \cX,$ but no joint distribution of demands across different price-income levels is available. Regarding $Q^\ast$ as a stochastic process indexed by $\cX,$ we often call $\mu_x$ the (one-dimensional) marginal distribution of $Q^\ast$ at $x.$ Note that although $(\mu_x)_{x \in \cX}$ is a population object, we assume that it is available since we are interested in identification. Assume that the interior of the support of $\mu_x,$ denoted by $\Omega_x,$ is not empty.
Recall that an individual demand function $q \in \cQ$ is said rational if it is induced by utility maximization. The Hurwicz-Uzawa theorem states that $q$ is rational if and only if its Slutsky matrix
is symmetric and negative semidefinite for each $x = (p, y).$
As we discussed in the previous section, dette2016testing propose a method to test individual rationality based on the negative semidefiniteness. Their testing procedure is roughly as follows. In their Theorem 1, they show that the quadratic form of the average Slutsky matrix is identified, that is, $v^\prime \bE [S_{Q^\ast} (x)] v$ is identified for each $v \in \bR^d$ and $x \in \cX.$ If the population consists of rational individuals, then $S_{Q^\ast} (x)$ is negative semidefinite almost surely, and hence, $v^\prime \bE [S_{Q^\ast} (x)] v \leq 0$ should hold. One can reject the null hypothesis that the population is rational if an estimate of $v^\prime \bE [S_{Q^\ast} (x)] v$ is significantly positive for some $x$ and $v.$\footnote{To be more precise, dette2016testing show a stronger result that the quadratic form of the average Slutsky matrix conditional on the value of $Q^\ast.$ Thus, their test is more powerful than what is described here.}
Although this method has nontrivial power, it is likely to be overly conservative since it completely neglects the other critical component of rationality, Slutsky symmetry. We shall investigate its testability based on the average Slutsky matrix. This question is important since the symmetry property is not robust to small perturbations while the negative semidefiniteness is. If Slutsky symmetry is testable, one should be able to construct a much more powerful test for rationality by testing both conditions.
If individual Slutsky matrix $S_{Q^\ast} (x)$ is symmetric (almost surely), so is its average. In what follows, we address the question of what we can say about the average Slutsky matrix $\bE [S_{Q^\ast} (x)]$ from cross-sectional demand distributions $\mu_x.$
First, we observe that the average Slutsky matrix is not identified in an explicit way because of the second term of ((ref)). The expectation of the second term is written as
but it is unclear how to identify $\bE [Q^\ast (p, y + \Delta y) Q^\ast (p, y)^\prime]$ because it involves the joint distribution of demand at different income levels, and it is indeed not identifiable as Theorem (ref) implies below.
The nonidentifiability of the average Slutsky matrix does not immediately imply that we cannot say anything about its symmetry. The individual symmetry could have some empirical implications for the observable demand distributions. Unfortunately, however, there is nothing we can say about the symmetry of the average Slutsky matrix from the cross-sectional demand data. The following theorem formalizes the argument so far. The proof is given in Appendix.
Assumption (ref) imposes regularity on the family $(\mu_x)_{x \in \cX}$ of demand distributions. Specifically, it requires that $\mu_x$ has a smooth density, and that both the density and support vary smoothly with $x.$
Testability of Slutsky symmetry. Theorem (ref) has several useful implications. First, as stated in the theorem, the average Slutsky matrix is not identified, and the identified set $\cS_d$ is unbounded in the sense that $S_{i, j} (x)$ can be arbitrarily large by $S_{j, i} (x)$ is small. Moreover, Theorem (ref) implies that no matter what cross-sectional demand distributions satisfying the regularity condition we observe, there always exists a stochastic demand system such that it is observationally equivalent to the true demand system and its average Slutsky matrix is symmetric. Consequently, there is no way to infer whether $\bE [S_{Q^\ast} (x)]$ is symmetric or not from $(\mu_x)_{x \in \cX}.$
This negative result suggests that researchers collect additional demand data in order to identify the average Slutsky matrix and test Slutsky symmetry. For example, let us assume that the joint distribution $\mu_{x, \tilde x}$ of $(Q^\ast (x), Q^\ast (\tilde x))$ is available for all $(x, \tilde x) \in \cX^2,$ rather than the marginal distribution $\mu_x.$ This setup corresponds to the situation where analysts can observe each individual's choice twice. In this setup, the average Slutsky matrix is identified because the RHS of ((ref)) is identified, and hence, it is possible to test the Slutsky symmetry by testing whether $\bE [S_{Q^\ast} (x)]$ is symmetric or not.
It is also worth mentioning that even though Theorem (ref) does show that Slutsky symmetry is not testable based on the average Slutsky matrix, it does not immediately rule out the possibility of testing Slutsky symmetry from cross-sectional distributional data. Indeed, individual symmetry could have observable implications through nonlinear statistics rather than the average, but we leave this for future work.
Adding income elasticity bounds. Now, we shall see that the average Slutsky symmetry can be testable by imposing bounds on the income elasticity of demand. Let $$ \varepsilon_i (x) \coloneqq D_y q_i (x) \cdot \frac{y}{q_i (x)} $$ be the income elasticity of demand for $i$th good for demand system $q.$ Then, we have $$ D_y q_i (x) \cdot q_j (x) = \frac{1}{y} \varepsilon_i (x) q_i (x) q_j (x) . $$ Let $\varepsilon^\ast$ be the income elasticity of demand $Q^\ast.$ It is easy to check $$ (E [S_{Q^\ast} (x)])_{i, j} = D_{p_j} \int q_i d \mu_x + \frac{1}{y} \bE [ \varepsilon_i^\ast (x) Q_i^\ast (x) Q_j^\ast (x) ] . $$
It is often reasonable to put bounds on the income elasticity of demand $\varepsilon_i^\ast.$ Assume that there are functions $\ell, u : \cX \to \bR$ such that $\ell (x) \leq \varepsilon_i^\ast (x) \leq u (x)$ for all $x$ and $i.$ Then we have $$ D_{p_j} \int q_i d \mu_x + \frac{\ell (x)}{y} \int q_i q_j d \mu_x \leq (E [S_{Q^\ast} (x)])_{i, j} \leq D_{p_j} \int q_i d \mu_x + \frac{u (x)}{y} \int q_i q_j d \mu_x . $$ This inequality leads to a bound on the difference $(E [S_{Q^\ast} (x)])_{i, j} - (E [S_{Q^\ast} (x)])_{j, i},$ for $i < j:$ it must be lie in the interval $$ I_{i, j} (x) \coloneqq \left[ \left( D_{p_j} \int q_i d \mu_x - D_{p_i} \int q_j d \mu_x \right) \pm \left( \frac{u (x) - \ell (x)}{y} \int q_i q_j d \mu_x \right) \right] . $$ Since $I_{i, j} (x)$ is identified and estimable, the average Slutsky symmetry---$(E [S_{Q^\ast} (x)])_{i, j} = (E [S_{Q^\ast} (x)])_{j, i}$ for all $i < j$ and $x \in cX$---can be tested by checking whether the null hypothesis $$ H_0 : 0 \in I_{i, j} (x) \text{ for all } i < j, x \in \cX . $$ While constructing a statistical test for this hypothesis is beyond the scope of this paper, the literature on testing functional inequalities, such as lee2013testing, lee2018testing, and li2025general, will work.
Let $S \in \cS_d$ be an element of the identified set of the average Slutsky matrix. To prove Theorem (ref), it is sufficient to construct a $\cQ$-valued random demand function $Q$ such that
for $x \in \cX.$ The goal of this section is to describe the construction of such a random demand function.
In overview, the construction proceeds in two steps. First, we construct a preliminary random demand function $\bar Q$ such that ((ref)) holds but not necessarily ((ref)). In the second step, we modify $\bar Q$ to obtain another random demand function $Q$ that satisfies ((ref)) as well as ((ref)).
Step 1. We begin with constructing a preliminary random demand function $\bar Q$ satisfying ((ref)). Let $\underline x \coloneqq (\underline p_1, \dots, \underline p_d, \underline y).$
Before moving on to the next step, we observe that the random demand function $\bar Q (x) = \bar \Phi (x, \omega)$ is characterized by the solution to an ordinary differential equation (ODE). Fix $p \in \cP.$ Let $\bar v_x = \bar v_{p, y} : \bR^d \to \bR^d$ be the income derivative of the demand function of consumer $\omega,$
where $\bar \Phi (x, \omega) = q.$ Notice that this is well-defined since $\bar \Phi (x, \cdot)$ is homeomorphic. The family $(\bar v_x)_x$ of vector fields, combined with an initial condition, pins down the demand function, as the Cauchy-Lipschitz theorem implies that $\bar \Psi (y) = \bar \Phi (p, y, \omega) = \bar \Phi (x, \omega)$ is the unique solution to the ODE
under the global Lipschitz condition on $\bar v_x.$
Step 2. Although $\bar Q$ in Lemma (ref) satisfies the marginal compliance ((ref)), it does not satisfy the average Slutsky symmetry condition $\bE [S_{\bar Q} (x)] = \bE [S_{\bar Q} (x)^\prime]$ in general. Our strategy is to construct another random demand function $Q$ satisfying both conditions by modifying $\bar Q.$ To do so, we modify the ODE ((ref)) so that its solution respects ((ref)). More precisely, we rectify $\bar v_x$ by adding an auxiliary vector field $w_x$ constructed in the following lemma.
For $1 \leq i, j \leq d,$ set $$ a_{i, j} (x) \coloneqq S_{i, j} (x) - D_{p_j} \int q_i d \mu_x - \bE [\bar v_{x, i} (\bar Q (x)) \bar Q_j (x)] , $$ which is a continuous function on $\cX$ under Assumption (ref). Then we have $a_{i, j} (x) + a_{j, i} (x) = 0$ if $i \neq j$ since $S_{i, j} (x) + S_{j, i} (x) = T_{i, j} (x)$ and
where the first equality holds by the law of motion of the ODE ((ref)), and the second equality holds by the fact that $\bar Q$ satisfies ((ref)). Take a vector field $w_x$ from Lemma (ref) for this $(a_{i, j}),$ and let $v_{p, y} \coloneqq \bar v_{p, y} + w_{p, y}.$ Consider a modified ODE
By the Cauchy-Lipchitz theorem, this ODE admits a unique solution $\Psi (y) = \Psi_{p, \omega} (y)$ for each $(p, \omega).$ Define $$ \Phi (x, \omega) = \Phi (p, y, \omega) \coloneqq
. $$ The following lemma shows that this flow induces a random demand function that satisfies the desired requirements.
To sum up, for given demand distributions $(\mu_x)_{x \in \cX}$ and function $S \in \cS_d,$ there exists a random demand function $Q,$ constructed in Lemma (ref), such that $Q (x) \sim \mu_x$ and $\bE [Q (x)] = S (x)$ hold. This is a key component of Theorem (ref).
In this paper, we have shown that the average Slutsky symmetry is not testable using cross-sectional demand data. To establish this, we explicitly derived the identified set of the average Slutsky matrix and demonstrated that it always contains a symmetric matrix. This finding implies that individual Slutsky symmetry cannot be tested through the average Slutsky matrix, although it does not rule out the possibility of testing the symmetry hypothesis without relying on the average. See gunsilius2025nonparametric for recent developments. Furthermore, by imposing bounds on the income elasticity of demand, we showed that average Slutsky symmetry leads to a set of functional inequality constraints that are, in principle, testable. A promising direction for future research is to develop a statistical test for Slutsky symmetry based on these inequalities.