Slutsky Matrix Norms: The Size, Classi cation, and ...Slutsky Matrix Norms: The Size, Classi cation,...

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Slutsky Matrix Norms: The Size, Classification, and Comparative Statics of Bounded Rationality Victor H. Aguiar and Roberto Serrano This version: March 2016 Abstract Given any observed demand behavior by means of a demand function, we quantify by how much it departs from rationality. The measure of the gap is the smallest Frobenius norm of the correcting matrix function that would yield a Slutsky matrix with its stan- dard rationality properties (symmetry, singularity, and negative semidefiniteness). A useful classification of departures from rationality is suggested as a result. Errors in comparative statics predictions from assuming rationality are decomposed as the sum of a behavioral error (due to the agent) and a mis-specification error (due to the modeller). Comparative statics for the boundedly rational consumer is expressed as path dependence of wealth com- pensations in price-equivalent paths, or as violations of the compensated law of demand. Illustrations are provided using several bounded rationality models. JEL classification numbers: C60, D10. Keywords: consumer theory; rationality; Slutsky matrix function; bounded rationality; comparative statics. 1 Introduction The rational consumer model has been at the heart of most theoretical and applied work in economics. In the standard theory of the consumer, this model has a unique prediction in the form of a symmetric, singular, and negative semidefinite Slutsky matrix. In fact, any demand system that has a Slutsky matrix with these properties can be viewed as being generated as the result of a process of maximization of some rational preference relation. Nevertheless, empirical evidence often derives demand systems that conflict with the rationality paradigm. In such cases, those hypotheses (e.g., symmetry of the Slutsky matrix) are rejected. These important findings have given rise to a growing literature of behavioral models that attempt to better fit the data. This paper aims to unify and systematize the implications of many of these models, and in doing so, it uses a well-known tool in microeconomic theory, namely, the Slutsky matrix. At this juncture three related questions can be posed in this setting: This paper subsumes Aguiar and Serrano (2014). We are especially indebted to Xavier Gabaix, Michael Jerison, and Joel Sobel for suggesting many specific improvements to that earlier version of the paper. We also thank Bob Anderson, Francis Bloch, Mark Dean, Federico Echenique, Drew Fudenberg, Peter Hammond, Susanne Schennach, Larry Selden, Jesse Shapiro and the participants at several conferences and seminars for helpful comments and encouragement. Department of Economics, Brown University, victor [email protected] Department of Economics, Brown University, roberto [email protected] 1

Transcript of Slutsky Matrix Norms: The Size, Classi cation, and ...Slutsky Matrix Norms: The Size, Classi cation,...

Page 1: Slutsky Matrix Norms: The Size, Classi cation, and ...Slutsky Matrix Norms: The Size, Classi cation, and Comparative Statics of Bounded Rationality* Victor H. Aguiar and Roberto Serrano

Slutsky Matrix Norms: The Size, Classification, and

Comparative Statics of Bounded Rationality*

Victor H. Aguiar�and Roberto Serrano�

This version: March 2016

Abstract

Given any observed demand behavior by means of a demand function, we quantify by

how much it departs from rationality. The measure of the gap is the smallest Frobenius

norm of the correcting matrix function that would yield a Slutsky matrix with its stan-

dard rationality properties (symmetry, singularity, and negative semidefiniteness). A useful

classification of departures from rationality is suggested as a result. Errors in comparative

statics predictions from assuming rationality are decomposed as the sum of a behavioral

error (due to the agent) and a mis-specification error (due to the modeller). Comparative

statics for the boundedly rational consumer is expressed as path dependence of wealth com-

pensations in price-equivalent paths, or as violations of the compensated law of demand.

Illustrations are provided using several bounded rationality models.

JEL classification numbers: C60, D10.

Keywords: consumer theory; rationality; Slutsky matrix function; bounded rationality;

comparative statics.

1 Introduction

The rational consumer model has been at the heart of most theoretical and applied work in

economics. In the standard theory of the consumer, this model has a unique prediction in the

form of a symmetric, singular, and negative semidefinite Slutsky matrix. In fact, any demand

system that has a Slutsky matrix with these properties can be viewed as being generated as the

result of a process of maximization of some rational preference relation. Nevertheless, empirical

evidence often derives demand systems that conflict with the rationality paradigm. In such

cases, those hypotheses (e.g., symmetry of the Slutsky matrix) are rejected. These important

findings have given rise to a growing literature of behavioral models that attempt to better fit

the data. This paper aims to unify and systematize the implications of many of these models,

and in doing so, it uses a well-known tool in microeconomic theory, namely, the Slutsky matrix.

At this juncture three related questions can be posed in this setting:

*This paper subsumes Aguiar and Serrano (2014). We are especially indebted to Xavier Gabaix, MichaelJerison, and Joel Sobel for suggesting many specific improvements to that earlier version of the paper. Wealso thank Bob Anderson, Francis Bloch, Mark Dean, Federico Echenique, Drew Fudenberg, Peter Hammond,Susanne Schennach, Larry Selden, Jesse Shapiro and the participants at several conferences and seminars forhelpful comments and encouragement.

�Department of Economics, Brown University, victor [email protected]�Department of Economics, Brown University, roberto [email protected]

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� (i) How can one measure the distance of an observed demand behavior –demand function–

from rationality?

� (ii) How can one compare and classify two behavioral models as departures from a closest

rational approximation?

� (iii) Given an observed demand function, what is the best rational approximation model?

The aim of this paper is to provide a tool to answer these three questions in the form of a Slutsky

matrix function norm, which allows to measure departures from rationality in either observed

Slutsky matrices or demand functions. The answer, provided for the class of demand functions

that are continuously differentiable, sheds light on the size and type of bounded rationality that

each observed behavior exhibits.

Our primitive is an observed demand function. To measure the gap between that demand

function and the set of rational behaviors, one can use the “least” distance and try to identify

the closest rational demand function. This approach presents serious difficulties, though. Leav-

ing aside compactness issues, which can be addressed under some regularity assumptions, the

solution would require solving a challenging system of partial differential equations. Lacking

symmetry of this system, an exact solution may not exist, and one needs to resort to approxi-

mation or computational techniques, but those are still quite demanding.

We take an alternative approach, based on the calculation of the Slutsky matrix function of

the observed demand. We pose a matrix nearness problem in a convex optimization framework,

which permits both a better computational implementability and the derivation of extremum

solutions. Indeed, we attempt to find the smallest correcting additive perturbation to the ob-

served Slutsky matrix function that will yield a matrix function with all the rational properties

(symmetry, singularity with the price vector on its null space,1 and negative semidefiniteness).

We use the Frobenius norm to measure the size of this additive factor, interpreting it as the

“size” of the observed departure from rationality.

We provide a closed-form solution to the matrix nearness problem just described. Interest-

ingly, the solution can be decomposed into three separate terms, whose intuition we provide

next. Given an observed Slutsky matrix function,

� (a) the norm of its anti-symmetric or skew symmetric part measures the “size” of the

violation of symmetry;

� (b) the norm of the smallest additive matrix that will make the symmetric part of the

Slutsky matrix singular measures the “size” of the violation of singularity; and

� (c) the norm of the positive semidefinite part of the resulting corrected matrix measures

the “size” of the violation of negative semidefiniteness.

Our main result shows that the “size” of bounded rationality, measured by the Slutsky matrix

squared norm, is simply the sum of the squares of these three effects. In particular, following any

observed behavior, we can classify the instances of bounded rationality as violations of the Ville

axiom of revealed preference –VARP– if only symmetry fails, violations of homogeneity of degree

1We abuse terminology slightly here. By “singularity with the price vector on its null space” or “singularityin p,” we mean that p is a right eigenvector of the Slutsky matrix associated with a zero eigenvalue, since Walras’law (assumed throughout the paper) implies that p is a left eigenvector of the matrix.

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zero or other money illusion phenomena if only singularity fails, violations of the weak axiom of

revealed preference –WARP– by a symmetric consumer if only negative semidefiniteness fails,

or combinations thereof in more complex failures, by adding up the nonzero components of the

norm. On the other hand, the decomposition offers new interesting insights: there are cases in

which, although all regularity properties fail, it suffices to “treat” only some of these failures in

order to end up with rational behavior.

The size of bounded rationality provided by the Slutsky norm depends on the units in which

the consumption goods are expressed. It is therefore desirable to provide unit-independent

measures, and we do so following the approach of modifying the Slutsky matrix by a weighting

matrix. For example, one can translate the first norm into dollars, hence providing a monetary

measure, or one can instead use a budget shares version or a normalized goodness of fit R square

version, which are unit-free.

The Slutsky matrix function is the key object in comparative statics analysis in consumer

theory. It encodes all the information of local demand changes with respect to small Slutsky

compensated price changes. We obtain comparative statics results for a boundedly rational

consumer. Importantly, one can decompose the error in comparative statics from assuming a

given form of rationality as the sum of two independent terms. The first is the behavioral error

(measured by our Slutsky norm and its decomposition), and the second is a mis-specification error

given the assumed parameterized rationality model. Further, unlike her rational counterpart, a

boundedly rational consumer may exhibit path dependence of wealth compensations in price-

equivalent paths or may violate the compensated law of demand, and we relate such phenomena

to the different nonzero terms in our decomposition of the norm.

The closest precurssor to our work is the approximately rational consumer demand proposed

by Jerison and Jerison (1992; 1993). They relate the local violations of negative semidefiniteness

and symmetry of the Slutsky matrix to the smallest distance between an observe smooth demand

system and a rational demand. Russell (1997) proposes a notion of quasi-rationality by linking

the Slutsky matrix anti-symmetry part with the lack of integrability of a demand system. Our

work is global, not necessarily tied to a small neighborhood of a price-wealth combination, and

it allows for a generalization that treats the three kinds of violations of the Slutsky conditions

simultaneously.

The rest of this paper is organized as follows. Section 2 presents the model. Section 3 deals

with the matrix nearness problem, and finds its solution. Section 4 emphasizes its additive

decomposition, and provides interpretations of the matrix nearness problem in terms of the

axioms behind revealed preference. Section 5 presents weighted Slutsky norms. Section 6 goes

over comparative statics. Section 7 presents several examples and applications of the result,

including hyperbolic discounting and the sparse consumer model. Section 8 is a brief, but

expanded, review of the literature, and Section 9 concludes. All the proofs are collected in an

appendix.

2 The Model

Consider a demand function x : Z 7→ X, where Z ≡ P ×W is the compact space of price-wealth

pairs (p, w), P ⊆ RL++, W ⊆ R++, and X ≡ RL is the consumption set. This demand system is

a generic function that maps price and wealth to consumption bundles.

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Assume that x(p, w) is continuously differentiable and satisfies Walras’ law: p′x(p, w) = w for

all (p, w) ∈ Z. Let X ⊂ C1 denote a set of functions that satisfy these characteristics. To be more

explicit noting the dependence on the domain, define also X (Z) ⊂ C1(Z), with C1(Z) denoting

the complete metric space of vector-valued functions f : Z 7→ RL, continuously differentiable,

uniformly bounded with compact domain Z ⊂ RL+1++ , equipped with a norm.2

Our attempt in this paper is to shed light on the size and types of bounded rationality.

To that end, let R ⊂ X be the set of rational demand functions (i.e., xr ∈ R is the solution

to maximizing a complete, locally nonsatiated and transitive preference over a linear budget

constraint). We define analogously R(Z) ⊂ X (Z) to be the set of allowable rational demand

functions.

Definition 1. Define the distance of x ∈ X to the set of rational demands R by the “least”

distance from an element to a set: d(x,R) = inf{dX (x, xr)|xr ∈ R}.

We shall refer to this problem of trying to find the closest rational demand to a given demand

as the “behavioral nearness” problem (Varian, 1990). Observe that the behavioral nearness

problem at this level of generality presents several difficulties. First, the constraint set R(Z),

i.e., the set of rational demand functions, is not convex. In addition, the Lagrangian depends not

only upon xr but also on its partial derivatives. The typical curse of dimensionality of calculus

of variations applies here with full force, in the case of a large number of commodities. Indeed,

the Euler-Lagrange equations in this case do not offer much information about the problem and

give rise to a large partial differential equations system. Finally, to calculate analytically the

solution to this program is computationally challenging.3

Thus, instead of solving directly the “behavioral nearness” problem, our approach will rely on

matrix spaces. Our next goal is to talk about Slutsky norms. LetM(Z) be the complete metric

space of matrix-valued functions, F : Z 7→ RL×L, equipped with the inner product 〈F,G〉 =´z∈Z Tr(F (z)′G(z))dz. This vector space has a Frobenius norm ||F ||2 =

´z∈Z Tr(F (z)′F (z))dz.4

Let us define the Slutsky substitution matrix function.

Definition 2. Let Z ⊂ P ×W be given, and denote by z = (p, w) an arbitrary price-wealth

pair in Z.5 Then the Slutsky matrix function S ∈M(Z) is defined pointwise: S(z) = Dpx(z) +

Dwx(z)x(z)′ ∈ RL×L, with entry sl,k(p, w) = ∂xl(p,w)∂pk

+ ∂xl(p,w)∂w xk(p, w).

The Slutsky matrix function is well defined for all x ∈ C1(Z). Restricted to the set of rational

behaviors, the Slutsky matrix satisfies a number of regularity conditions. Specifically, when a

matrix function S ∈ M(Z) is symmetric, negative semidefinite (NSD), and singular with p in

its null space for all z ∈ Z, we shall say that the matrix satisfies property R, for short.

Definition 3. For any Slutsky matrix function S ∈ M(Z), let its Slutsky norm be defined as

follows: d(S) = min{||E|| : S − E ∈M(Z) having property R}.2For instance, the norm ||f ||C = max({||fl||C,1}l=1,...,L), with ||fl||C,1 = max(||fl||∞,1) where f(z) =

[f1(z) · · · fL(z)]′ ∈ RL). We use also the notation for the norm || · ||∞,m = supz∈Z |g(z)| for g : Z 7→ Rm, forfinite m ≥ 1 and | · | the absolute value. By the set C1(Z) we mean the set of functions that have C1 extensionsto an open set containing the compact domain Z.

3 We refer the reader to our expanded discussion in Aguiar and Serrano (2014), which includes the use of the“almost implies near” approach from (1986), central in the inception of our ideas in this work.

4While one could use a different norm, we justify our choice of the Frobenius norm in the sequel.5Since Z is closed, we use the definition of differential of Graves (1956) that is defined not only in the interior

but also on the accumulation points of Z.

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The use of the minimum operator is justified. Indeed, it will be proved that the set of Slutsky

matrix functions satisfying R is a closed and convex set. Then, under the metric induced by the

Frobenius norm, the minimum will be attained in M(Z). We shall refer to the minimization

problem implied in the Slutsky norm as the “matrix nearness” problem.

3 The Matrix Nearness Problem: Its Solution

In this section we provide the exact solution to the matrix nearness problem, which allows us

to quantify the distance from rationality by measuring the size of the violations of the Slutsky

matrix conditions.

3.1 Preliminaries on Matrices

We begin by reviewing some definitions.

It will be useful to denote the three regularity conditions of any Slutsky matrix function with

shorthands. We shall use σ for symmetry, π for singularity with p in its null space (p−singularity),

and ν for negative semidefiniteness.

Given any square matrix-valued function, S ∈M(Z), let Ssym = 12 [S + S′] be its symmetric

part. With our new notation, if S is a Slutsky matrix function, Sσ = Ssym. Equivalently, Sσ is

the projection of the function S on the closed subspace of symmetric matrix-valued functions,

using the inner product defined for M(Z).

Every square matrix function S ∈ M(Z) can be written as S(z) = Ssym(z) + Sskew(z)

for z ∈ Z, also written as S = Sσ + Eσ, where Sσ = Ssym is its symmetric part and Eσ =

Sskew = 12 [S − S′] is its anti-symmetric or skew-symmetric part (the orthogonal complement of

the symmetric part).

Every symmetric matrix function Sσ ∈M(Z), in particular the symmetric part of a Slutsky

matrix function S, can be decomposed into the sum of a singular and not singular parts (with

prices in the null space): see Claim 3.6 With our notation, the matrix function part that is

singular with p in its null space will be denoted Sσ,π, that is, Sσ,π(z)p = 0. Then Sσ = Sσ,π+Eπ

where Sσ,π = PSσP and Eπ = Sσ − PSσP is its orthogonal complement with P = I − pp′

p′p a

projection matrix.7

Any symmetric matrix-valued function Sσ ∈ M(Z) and in particular any matrix function

that is the singular in prices part Sσ,π ∈M(Z) of a Slutsky matrix can be pointwise decomposed

into the sum of its positive semidefinite and negative semidefinite parts. Indeed, we can always

write Sσ,π(z) = Sσ,π(z)+ + Sσ,π(z)− , with Sσ,π(z)+Sσ,π(z)− = 0 for all z ∈ Z. Moreover, for

any square matrix-valued function S ∈M(Z), its projection on the cone of NSD matrix-valued

functions under the Frobenius norm is Sσ,π− .

In general, a square matrix function may not admit diagonalization. However, we know

thanks to Kadison (1984) that every symmetric matrix-valued function in the set M(Z) is

diagonalizable. In particular, Sσ,π can be diagonalized: Sσ,π(z) = Q(z)Λ(z)Q(z)′. Here, Λ(z) =

Diag[{λl(z)}l=1,...,L], where Λ(z) ∈ M(Z), with λl : Z 7→ R a real-valued function with norm

6This means that we consider the space of symmetric matrix functions Sσ that satisfy p′Sσ(z)p = 0. This isa consequence of Walras’ law because p′S(z) = 0, although not necessarily S(z)p = 0.

7Because of Walras’ law when S is the Slutsky matrix of some demand function x ∈ X (Z), p′S(z) = 0 andSσ,π = Sσ − Eπ and Eπ(z) = 1

p′p [Sσ(z)pp′ + pp′Sσ(z)].

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|| · ||s (a norm in C1(R)), and λ1 ≤ λ2 ≤ . . . ≤ λL with the order derived from the metric induced

by the || · ||s norm.8 Let Q(z) = [q1(z) . . . qL(z)], where Q ∈ M(Z) and its columns ql ∈ C1(Z)

are the eigenvector functions such that for l = 1, . . . , L: Sσ,π(z)ql(z) = λl(z)ql(z) for z ∈ Z.

Any real-valued function can be written as λ(z) = λ(z)++λ(z)−, with λ(z)+ = max{λ(z), 0}and λ(z)− = min{λ(z), 0}. This decomposition allows us to write Sσ,π,ν(z) = Sσ,π(z)− =

Q(z)Λ(z)−Q(z)′ for Λ(z)− = Diag[{λl(z)−}l=1,···L] with λl(z)− the negative part function for

the λl(z) function. We can write also Eν(z) = Sσ,π(z)+ = Sσ,π(z) − Sσ,π(z)− for z ∈ Z,

or Sσ,π(z)+ = Q(z)Λ(z)+Q(z)′ with Λ(z)+ defined analogously to Λ(z)− as the orthogonal

complement of Sσ,π,ν .

3.2 The Result

We are ready to state the solution to the matrix nearness problem:

Theorem 1. Given a Slutsky matrix S, it has the unique decomposition S = Sr + E, where

Sr = Sσ,π,ν is the solution to the matrix nearness problem and E is the sum of the orthogonal

complements of the symmetric, singularity in p and NSD parts of S: E = Eσ + Eπ + Eν .

Furthermore, the Slutsky norm of the solution to the matrix nearness problem can be decomposed

as follows:

||E||2 = ||Eσ||2 + ||Eπ||2 + ||Eν ||2.

We elaborate at length on the different components of this solution in the next section, right

after outlining the proof of the theorem.

Proof. We first establish that the matrix nearness problem has a solution, and that it is unique.

This is done in Claim 1.

Claim 1. The solution to the matrix nearness problem exists, and it is unique.

The rest of the proof of the theorem is done in two parts. Lemma 1 gives the solution

imposing only the singularity with p in its null space and symmetry restrictions.

Lemma 1. The solution to minA||S − A|| subject to A(z)p = 0, A(z) = A(z)′ for all z ∈ Z is

Sσ,π.

If Sσ,π were NSD, then we would be done, since it would have property R and minimize

||E||2. Otherwise, the general solution is provided after the following lemma, which rewrites the

problem slightly.

Lemma 2. The matrix nearness problem can be rewritten as minA||Sσ,π − A||2 subject to A ∈M(Z) having property R.

The orthogonalities of the relevant subspaces of matrix functions, as made evident in previous

steps of the proof, are responsible for the additive decomposition of the norm in the second part

of the statement. The Frobenius norm and its weighted variants are essentially the only ones

that correspond to an inner product on the space of matrix-valued functions. For this reason,

8The order of the eigen-values is inessential to the results but it is convenient for the proofs. Observe that wecan have many diagonal decompositions all of which will work for our purposes.

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they are the only ones that lead to the additive decomposition of the three components of the

residual matrix E.

Furthermore, we close this section by stating that the solution Sr to our problem is a con-

tinuous matrix-valued function:

Claim 2. Sr is continuous.

The proof is omitted because we note that Sr is the result of three projections on closed

subspaces applied to the convex set of constraints. Such projections are continuous mappings

under the conditions that we have imposed, and then Sr is continuous by construction in all

z ∈ Z.

Finally, we establish the pointwise orthogonality of Eπ and Eν :

Claim 3. The matrix Eπ(z) is pointwise orthogonal to Eν(z). That is Tr(Eπ(z)′Eν(z)) = 0 for

all z ∈ Z.

4 Decomposition of the Matrix Nearness Solution

The importance of Theorem (1) is to provide a precise quantification of the size of the departures

from rationality by a given behavior, as well as a revealing decomposition thereof.9

We should think of the three terms in the decomposition of ||E||2 as the size of the violation

of symmetry, the size of the violation of singularity, and the size of the violation of negative

semidefiniteness of a given Slutsky matrix, respectively (this idea is illustrated in figure 1). The

three terms are the antisymmetric part of the Slutsky matrix function, the correcting matrix

function needed to make the symmetric part of the Slutsky matrix function p-singular, and

the PSD part of the resulting corrected matrix function. Note that if one is considering a

rational consumer, the three terms are zero. Indeed, if S(z) satisfies property R for all z ∈ Z,

S(z) = Sσ(z) and Eσ(z) = 0, Sσ,π(z) = Sσ(z) and Eπ(z) = 0, and Sr(z) = Sσ,π,ν(z) = Sσ,π(z)

and Eν(z) = 0. If exactly two out of the three terms are zero, the nonzero term allows us

roughly to quantify violations of the Ville axiom of revealed preference –VARP–, violations of

homogeneity of degree 0, or violations of the compensated law of demand (the latter being

equivalent to the weak axiom of revealed preference –WARP–), respectively. We elaborate on

these connections with the axioms of consumer theory in Subsection 4.1 below.

The violations of the property R have traditionally been treated separately. For instance,

Russell (1997), using a different approach (outer calculus), deals with violations of the symmetry

condition only. In this case, ||E||2 = ||Eσ||2.

Another application of our result connects with Jerison and Jerison (1993), who study the

case of violations of symmetry and negative semidefiniteness independently. They prove that

the maximum eigenvalue of Sσ(z) can be used to bound the distance of any demand to the set of

rational demands d(x,R) locally when NSD is violated and Eσ(z) can be used to bound d(x,R)

when symmetry is violated (under the sup norm in the space of continuous differentiable demand

functions). Indeed, this is consistent with our solution. In this case ||E||2 = ||Eσ||2 + ||Sσ+||2,

where max({λl(z)+}l=1,...,L) ≤ ||Sσ+||2 ≤ d ·max({λl(z)+}l=1,··· ,L), with d = Rank(Sσ(z)+)

(by the norm equivalence of the maximum eigenvalue and the Frobenius norm).

9The decomposition, a consequence of orthogonalities, can be seen as a generalized Pythagorean theorem.

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ÈÈ EΣ

ÈÈ2

ÈÈ EÈÈ2 =ÈÈ E Σ

ÈÈ2 +ÈÈ E Π

ÈÈ2 +ÈÈ E Ν

ÈÈ2

ÈÈ EΠ

ÈÈ2

ÈÈ EΝ

ÈÈ2

S

SΣ,Π

Sr

Figure 1: Decomposition of the Slutsky matrix norm. The size of the violations of rationality isthe sum of the violations of symmetry, singularity in prices and negative semidefiniteness.

4.1 Connecting with Axioms of Revealed Preference

Since Hurwicz and Uzawa (1971) it is known that for the class of continuous differentiable

functions a demand that satisfies Walras´ law can be rationalized if and only if its Slutsky

matrix function is symmetric (σ) and NSD (ν). A corollary of this result is that its Slutsky matrix

function is singular in prices (π) (John, 1995). These Slutsky regularity conditions (σ, π, ν) are

linked to behavioral demand axioms. In fact, Houthakker (1950) proved that, for the class of

continuous demand functions, the strong axiom of revealed preference –SARP– implies that a

demand can be rationalized. In this sense, SARP is indeed strong. A weaker axiom implies

only the Slutsky matrix function symmetry condition: the Ville axiom of revealed preference

–VARP– is equivalent to the symmetry condition and therefore to integrability of the demand

system (Hurwicz & Richter, 1979). VARP postulates the nonexistence of a Ville cycle in the

income path of a demand function. The weak axiom of revealed preference –WARP– implies

that the Slutsky matrix function is NSD and singular in prices. Furthermore, the NSD and

singularity in prices are equivalent to a weak version of WARP (Kihlstrom et al., 1976). VARP

is a differential axiom and does not imply SARP or WARP (Hurwicz & Richter, 1979). It is also

known that WARP does not imply VARP or SARP for dimensions greater than two.

A continuously differentiable demand function is said to be rationalizable if and only if

it fulfills SARP. However, we can also combine the other weaker axioms to have the same

result while drawing connections to the additive components of our Slutsky norm. For example,

VARP and WARP imply that a demand function is rationalizable, and so do the Wald Axiom,

homogeneity of degree zero and VARP.

Before presenting the main remark of this subsection, for completeness, it is useful to posit

the axioms that we employ.

Assumed throughout, the first basic condition (price is a Slutsky matrix left eigenvector) is

given by Walras’ law.

Axiom 1. (Walras’ law) p′x(p, w) = w.

We have that if x ∈ X (Z) satisfies Walras’ law then its Slutsky matrix function S ∈ M(Z)

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has the following property: p′S(z) = 0 for z ∈ Z. The converse holds when Z is path connected.

The condition that the price p is the Slutsky matrix right eigenvector or singularity in prices

is given by “no money illusion”.

Axiom 2. (Homogeneity of degree zero -HD0-) x(αz) = x(z) for all z ∈ Z and α > 0.

It is easy to prove that if x ∈ X (Z) satisfies Walras’ law and HD0 then S(z)p = 0 for z ∈ Z.

More generally, p is an eigenvector associated with the null eigenvalue of S if and only if x

satisfies Walras’ law and HD0.

The symmetry of the Slutsky matrix function is given by VARP.10 To state this axiom we

need to define an income path as w : [0, b] 7→W and a price path p : [0, b] 7→ P . Let (w(t), p(t))

be a piecewise continuously differentiable path in Z. Jerison and Jerison (1992) define a rising

real income situation whenever (∂w∂t (t), ∂p∂t (t)) exists, leading to ∂w∂t (t) > ∂p

∂t (t)′x(p(t), w(t)). A

Ville cycle is a path such that: (i) (w(0), p(0)) = (w(b), p(b)); and (ii) ∂w∂t (t) > ∂p

∂t (t)′x(p(t), w(t))

for t ∈ [0, b].

Axiom 3. (Ville axiom of revealed preference -VARP-) There are no Ville cycles.

Hurwicz and Richter (1979) proved that x ∈ X (Z) satisfies VARP if and only if S is symmetric

and x satisfies Walras’ law.

The negative semidefiniteness condition of the Slutsky matrix is given by the Wald axiom.

The Wald axiom itself is imposed on the conditional demand for a given level of wealth. Following

John (1995) a demand function that satisfies Walras’ law is said to fulfill the Wald axiom

when so do the whole parametrized family (for w ∈ W ) of conditional demands. Formally, a

demand function can be expressed as the parametrized family of conditional demands. That is:

x(p, w) = {xw(p)}w∈W where xw : P 7→ X such that p′xw(p) = w for all p ∈ P .

Axiom 4. (Wald axiom) x ∈ X (Z) is such that for every w ∈W and for all p and p, p′xw(p) ≤ w=⇒ p′xw(p) ≥ w.

The Wald axiom implies that S ≤ 0 (John, 1995) for demands that satisfy Walras’ law.

The Slutsky singularity in prices and the NSD conditions are equivalent to the following

version of WARP.

Axiom 5. (Weak axiom of revealed preference -WARP-) If for any z = (p, w) z = (p, w):

p′x(p, w) ≤ w =⇒ p′x(p, w) ≥ w.

This is the weak version of WARP, as in Kihlstrom et al. (1976). We follow John (1995),

who proves that for continuously differentiable demands (that satisfy Walras’ law) WARP is

equivalent to the Wald Axiom and HD0. Kihlstrom et al. (1976) and John (1995) prove that

x ∈ X (Z) satisfies WARP if and only if S is NSD and S(z)p = 0.

We are ready to summarize the main point of this subsection in the following remarks.

Remark 1. The Slutsky matrix nearness norm ||E||2 will be equal to zero if and only if x satisfies

VARP, homogeneity of degree zero, and the Wald Axiom (in addition to Walras’ law).

Moreover

10We present the axiomatization due to Ville as reinterpreted by Hurwicz and Richter (1979) and Jerison andJerison (1992). There are alternative discrete axioms due to Jerison and Jerison (1996), that also do the job andare potentially testable.

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(i) ||E||2 = ||Eσ||2 > 0 only if VARP fails. In this case, either HD0 and the Wald Axiom

hold, or Warp holds.

(ii) ||E||2 = ||Eπ||2 > 0 only if Walras’ law fails. In addition, HD0 may also fail; VARP and

the Wald axiom hold.

(iii) ||E||2 = ||Eν ||2 > 0 only if the Wald axiom fails. In this case, VARP holds, and either

WARP fails or HD0 holds.

In addition, under additional technical regularity conditions on the domain and on the de-

mand (that are not necessary but sufficient and capture the current state of the literature), we

can state even stronger connections. Also note that our results go through with an open price

wealth region Z, provided it is Lipchitz continuous.

Remark 2. Suppose Z is open, convex, path connected and the demand x ∈ X (Z) is smooth,

bounded and Lipchitz continuous (at least with respect to wealth) and satisfies Walras’ law .

(i) Weak WARP holds if and only if ||Eπ|| = 0 and ||Eν || = 0 (John, 1995; Kihlstrom et al.,

1976).

(ii) The Ville Axiom holds if and only if ||Eσ|| = 0 (Hurwicz & Richter, 1979; Hurwicz &

Uzawa, 1971).

(iii) No money illusion (homogeneity of degree zero) holds if and only if ||Eπ|| = 0.11

(iv) The Wald axiom holds if and only if ||Eν || = 0 (John, 2000, 1995).12

We do not impose the additional sufficient conditions on x ∈ X (Z) described in the previous

remark for the general results in the rest of the paper because the necessity part of the above

statements hold without them. Such necessity parts of the statements suffice to implement an

econometric test of the revealed preference axioms due to the fact that if any of the parts of the

Slutsky matrix norm are non zero the corresponding axiom cannot hold for the given demand.

5 Weighted Slutsky Norms

The norm of bounded rationality that we have built so far is an absolute measure. Therefore,

for a specific consumer, this distance quantifies by how far that individual’s behavior is from

being rational. Furthermore, one also can compute how far two or more consumers within a

certain class are from rationality, and induce an order of who is closer in behavior to a rational

consumer. However, we are limited to the case where the setting of the decision making process

is fixed in the sense that the decision problem facing each of the individuals is presented in the

same way. This implies that the measure is unit dependent, being stated in the same units

(the units in which the consumption goods are expressed). A second undesirable feature, is that

the Frobenius norm depends on the area of the arbitrary compact region Z that has positive

Lebesgue measure. This section addresses these issues.

Let W , U be square and PSD (i.e., with symmetric part also PSD) matrix functions of

weights such that it is non-null and W (z), U(z) ∈ RL×L. This matrix pair is meant to encode

all suitable normalizations and priorities of the modeller. Let Z ⊆ P ×W be sigma-measurable

and have positive measure µ with µ continuous, strictly positive function, such that S ∈ M(Z)

11This result is trivial under path connectedness because Eπ = 0 means that Dpx(p, w)p+Dwx(p, w)w = 0.12This result is trivial to establish by noticing that the related Slutsky matrix function S is nowhere zero due

to smoothness, and the fact that Eν = 0 guarantees that Dpx(p, w) is NSD for any w.

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is square integrable. Let || · ||M be the Frobenius matrix norm where M = RL×L. The weighted

semi-norm ||S||WU,µ is defined as follows:

||S||2WU,µ =

ˆz∈Z||W (z)S(z)U(z)||2Mµ(z)dz <∞,

when µ is continuous and strictly positive, let B(Z) denote the Borel set on Z, then the spaces

L2(Z,B(Z);M(Z)), and the weighted L2(Z,B(Z);M(Z);µ) are unitary equivalent or unitary

isomorphic.13 This means that our orthogonality relations under the inner products defined by

the Lebesgue measure are preserved by inner product weighted by µ. Hence, suitable variants

of Theorem 1 can be obtained for these weighted norms.

Remark 3. With a W = U that is positive definite and symmetric, the solution to the matrix

nearness problem, S∗r = argminA∈M(Z)||S − A||W,µ subject to A having the property R, is

S∗r = W−1(WSσ,πW )−W−1 and the quadratic norm is given by:

||E∗||2W,µ = ||E∗σ||2W,µ + ||E∗π||W,µ + ||E∗ν ||2W,µ,

where E∗σ = Eσ, E∗π = Eπ, and E∗ν = W−1(WSσ,πW )−W−1.

Observe that only the part concerning the NSD matrix function of violations (ν property)

changes due to the introduction of the weighting matrix function W .

Remark 4. Using the identity matrix as W = U and the multivariate uniform as µ, one obtains

the closest results to Theorem 1. However, the resulting Slutsky norm is independent of the area

of the domain Z.

Remark 5. If W = Diag(p) and U = W , the weighted Slutsky matrix is expressed in (square)

dollar amounts. The entries of this normalized matrix express changes in expenditure on each

good following a percent change in each price.

Remark 6. If W = 1√wDiag(p) and U = W , the weighted Slutsky matrix is unit-free. The

entries of this normalized matrix express changes in expenditure shares on each good following

a percent change in each price.

Remark 7. When we have a degenerate µ such that µ(z) = 1 (say an equilibrium price and a

given exogenous wealth) and W,U are the identity, ||S||µ = ||S(z)||M and we can apply the local

results of Jerison and Jerison (1992).

The standard Frobenius norm we use is chosen when one gives equal weight to each entry of

the Slutsky matrix, thus avoiding that a specific price change on a specific commodity receive

greater weight, but the modeller may adopt such choices at will and use arbitrary weighting

matrices as a function of her interests. Finally, the integral in the norm embodies an expectation

operator, which can be justified with Savage-type ideas of independence across consumption data

points.

13Two Hilbert spaces are unitary isomorphic if there is an isometry between them, i.e., there exists a mappingT that is linear, bijective and preserves the inner products, for all x, y ∈ H with H the Hilbert space 〈x, y〉 =〈Tx, Ty〉.

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6 The Comparative Statics of Bounded Rationality

An approach based on the Slutsky matrix cannot be silent on comparative statics analysis.

This section goes over a number of results that shed light on the meaning of the decomposition

found in Theorem 1 and that connect with the behavioral nearness problem in the space of

demand functions. The first subsection below focuses on mis-specification errors, offering another

result based on orthogonalities. In the second subsection we offer a related goodness of fit

measure of bounded rationality, a normalization of our main measure. The results in the last

two subsections relate errors in comparative statics predictions to the error terms in our Slutsky

norm decomposition.

6.1 Connecting with Demand: Mis-Specification Errors

We have solved the matrix nearness problem on the basis of the Slutsky regularity conditions.

Now, in order to connect with demand, the exercise is one of integrating from the first-order

derivatives of the Slutsky matrix terms. In such an integration step, a constant of integration

shows up, which we would like to envision as a “residual” or “mis-specification error.” That

is, starting from our observed Slutsky matrix function S(x), we find thanks to Theorem 1 the

nearest matrix function Sr satisfying all the regularity properties. Now suppose that we specify

x1 and x2 as two demand functions with which we wish to explain the consumer’s behavior

(for example, x1 might be a Cobb-Douglas demand properly estimated, whereas x2 might be a

different estimate, perhaps allowing the entire class of CES demands).

As will be shown, the error in comparative statics analysis from using each of those spec-

ifications is, respectively, ||S(x) − S(x1)||2 = ||Sr − S(x1)||2 + 2〈Eν , S(x1) − Sr〉 + ||E||2 and

||S(x) − S(x2)||2 = ||Sr − S(x2)||2 + 2〈Eν , S(x2) − Sr〉 + ||E||2. This means that ||E||2 =

||S(x) − Sr||2 found in Theorem 1 –the behavioral error– does not change with x1 or x2, this

term is invariant to the choice of the parametrized rational approximation. The other term is

the mis-specification error, as a function of the parametrized class considered and a positive

correlation term that depends only on the failures of the ν propery and the miss-specification

error. This is a useful separation. We establish it in our next result.

Let Rr(Z) ⊂ R(Z) be an arbitrary closed and bounded subset of rational demand functions

in X (Z).14 In particular, we are interested in a symmetric error, where we penalize any mistake

symmetrically which corresponds to the Frobenius matrix norm:

minxr∈Rr(Z)||S(x)− S(xr)||2.

Our next result follows:

Proposition 1. For any solution x∗r ∈ argminxr∈Rr(Z)||S(x)−S(xr)||2, it follows that ||S(x)−S(x∗r)||2 = ||Sr − S(x∗r)||2 + 2〈Eν , Sr − S(x∗r)〉 + ||E||2, where 0 ≤ 〈Eν , Sr − S(x∗r)〉 ≤||Sr − S(x∗r)|| · ||Eν ||. Here, ||E||2 is the Slutsky matrix norm –behavioral error term–, and

the other two terms are a generalized mis-specification error, consisting of ||Sr − S(x∗r)||2 –the

proper miss-specification error– and the last term is a nonnegative and bounded correlation term

(penalty) that relates the lack of ν (NSD) and the proper miss-specification error.

14In fact, R(Z) is not closed (nor is it open), but we can always use a closed and bounded ball (under somesuitable metric) R

r(Z, ε) with radius ε ≤ M for an arbitrarily large positive constant M , approximating R(Z)

arbitrarily closely.

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Remark: The correlation term shows up because the error capturing the failure of NSD is not

orthogonal to the mis-specification error, while the errors associated with the other failures of

the Slutsky properties are, as the proof of the proposition reveals.

In the same spirit, we close this subsection with a local result. Around a given reference price

p, if one searches for a minimizer of error in comparative statics with respect to Sr among all

twice continuously differentiable functions with bounded Hessians, satisfying Walras’ law and

defined on an open ball around p, the result is a rational demand function. Note that the space

of demands that satisfies Walras’ law is convex.

Define the local problem. Fix a reference price p. Define the open set Z(p, ε) ≡ B(p, ε)× Wwhere B(p, ε) = {p ∈ P : ||p − p||RL < ε} is an open ball of prices around p for a small

ε > 0 and W is an open set of wealth levels. In this subsection we refer to the norm ||S||2 =´(p,w)∈Z(p,ε)

||S(p, w)||2RL×Ld(p, w) abusing notation, that is defined only for an open ball of prices

(the inner product is〈S,A〉 =´(p,w)∈Z(p,ε)

Tr(S′(p, w)A(p, w))d(p, w)).

We can compute the Slutsky matrix of x at p or any p ∈ B(p, ε) for small ε > 0 S(p, w). We

can compute Sr(p, w) using the same techniques developed for the general case point-wise.

Abusing notation we keep referring to the restriction of the matrix function S(·, ·) to Z(p, ε)

as S and the analogous restriction of the matrix function Sr(·, ·) to the same region as Sr. Let

X (p) be a closed and bounded set of demand functions that have X extensions to an open set

containing the domain Z(p, ε); we also assume that the demand functions in X (p) are twice

continuously differentiable, with a bounded Hessian. Let R(p) ⊆ X (p) be the subset of rational

demands with the same properties.

We also define a notion of Slutsky matrix function integrability:

Definition 4. [Slutsky integrability] We say a matrix function S ∈M(Z) is Slutsky integrable

if there exists some x ∈ X (Z) such that S(x) = S.

We now show that even when globally Sr may not necessarily be the Slutsky matrix of some

rational demand function xr ∈ R(Z) (i.e., or it is not Slutsky integrable), locally it approximately

is. Solving the local inverse problem:

miny∈X (p)||S(y)− Sr||2

results in finding a rational demand solution (i.e., it is locally Slutsky integrable), and the

rational demand solution has a Slutsky matrix function that at p coincides exactly with Sr.

Proposition 2. If y∗ ∈ argminy∈X (p)||S(y)− Sr||2, then y∗ ∈ R(p).

Proof. The proof is organized in the following series of lemmata. We start with a definition due

to Jerison and Jerison (1992; 1993).

Let (p, w) ∈ RL+1++ , and let a price path be defined p(t) = tp+(1−t)p, where p ∈ RL++ is fixed.

Then we study the Slutsky wealth compensation problem: ∂w(p,w,p,t)∂t = ∂p(t)

∂t x(p(t), w(p, w, p, t))

with initial condition w(p, w, p, 0) = w. The unique solution to this problem is wx,∗(p, w, p, ·).The following definition of approximate local indirect utility is borrowed from Jerison and

Jerison (1992; 1993). The indirect local utility at (p, w), where p is close enough to p ( is defined

by vx,σ(p, w) = wx,∗(p, w, p, 1). Jerison and Jerison (1993) prove that wx,∗ is continuously

differentiable when x is continuously differentiable, and it is known that continuous functions

can be approximated arbitrarily closely (in the sup norm) by smooth functions (Azagra & Boiso,

2004), thus we assume that w∗ is smooth.

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Lemma 3. [Jerison and Jerison (1993)] For every demand x ∈ X (p) that is twice continuously

differentiable there exists xσ ∈ X (p) such that S(xσ)(p, w) = Sσ(p, w). Moreover, xσ(p, w) =

−∇pvx,σ(p,w)

∇wvx,σ(p,w) .

Proof. Remark 1 of [Jerison and Jerison (1993)].

Lemma 4. If y ∈ argminy∈X (p)||S(y) − Sr||2 then S(y)(p, w) is symmetric for all (p, w) ∈Z(p, ε). Moreover, if Walras’ law holds and x satisfies Weak WARP then there is a y∗ ∈argminy∈X (p)||S(y)− Sr||2 such that y∗ = xσ.

Lemma 5. [CCD] Convex concave decomposition. Any scalar twice continuously differentiable

function with a bounded Hessian can be decomposed in the sum of a concave and convex function

parts.

Proof. Yuille & Rangarajan (2003)

Proof. Our task now is to define a local expenditure function. For a fixed w, and a given

x ∈ X (p), let ex,σ(p, u) be defined implicitly as vx,σ(p, ex,σ(p, u)) = u. The ex,σ is not necessarily

concave in p. The function ex,σ is unique and continuous when the indirect utility is monotone

in wealth.15

Now notice that, by construction, the integrable demand xσ can be written as xσ(p, e(p, u)) =

∇pex,σ(p, u) because by differentiating vx,σ(p, ex,σ(p, u)) = u with respect to prices we obtain:

∇pex,σ(p, u) = −∇pvx,σ(p,ex,σ(p,u))

∇wvx,σ(p,ex,σ(p,u)) . In turn, the Slutsky matrix of a “integrable” demand xσ

can be written as: S(xσ)(p, ex(p, u)) = Hex,σ(p, u), where H stands for the Hessian operator.

Observe that by definition ex,σ(p, vx(p, w)) = w.

Also any expenditure can be decomposed into a concave part and a convex part ex,σ =

ex,σcave + ex,σvex such that 〈Hex,σcave,Hex,σvex〉 = 0, this is always possible since we are dealing with a

local problem and we can limit ourselves without loss of generality to quadratic quasi-expenditure

functions.

Definition 5. [Concave part local indirect utility] We define the “concave part” of the indirect

utility of x ∈ X (p) vxcave implicitly ex,σcave(p, vxcave(p, w)) = w.

Lemma 6. For any demand y ∈ X (p) that is twice continuously differentiable such that it has

a symmetric and bounded Slutsky Matrix S(y)(p, w) for all (p, w) ∈ Z(p, ε):

(i) There is an scalar function vy : Z(p, ε) 7→ R such that y(p, w) = −∇pvy(p,w)

∇wvy(p,w) . Moreover:

y(p, w) = yσ(p, w).

(ii) There exists a function yr ∈ R(p) such that yr(p, w) = −∇pvycave(p,w)

∇wvycave(p,w)with vycave the

concave part local indirect utility function of vy.

Lemma 7. If y ∈ argminy∈X (p)||S(y)− Sr||2, then S(y) is NSD.

15If it were not monotone, in particular if existence or uniqueness are not guaranteed (i.e., the problem isill-posed) then we could use a regularization procedure to obtain a unique continuous expenditure (Tichonovregularization). However, the result goes through also if there exists at least one continuous and differentiablesolution eσ to the equation above so we do not need uniqueness. Thus without loss of generality we assume thatvσ(p, w) is monotone in wealth.

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Lemma 8. If y∗ ∈ argminy∈X (p)||S(y)−Sr||2, then S(y∗)(p, w)p = 0 ( singular in p). Moreover,

y∗ is HD0.

Now we turn to study the relation between the problem of minimizing its comparative statics

error locally and the Slutsky matrix norm. In particular we find a local solution to the closest

demand that minimizes its comparative statics error problem that will be correct to the first

order of approximation of prices o(||p− p||). Before proceeding we need some preliminaries.

Fix a demand x ∈ X (p) and we obtain the quasi-indirect utility vx,σ with associated quasi-

expenditure ex,σ. For any w, we have u = vx,σ(p, w).

The quasi-expenditure function taylor expansion at p for a given w is:

ex,σ(p, u) = ex,σ(p, u) + [p− p]′∇pex,σ(p, u) + 12 [p− p]′Hex,σ(p, u)[p− p] + o(||p− p||3), with

vx,σ satisfying ex,σ(p, vx,σ(p, w)) = w for any w.

We define a quasi-expenditure function ex,σ,π that has the property that its hessian is singular

at prices p, thus making it behave locally as an homogeneous of degree 1 in prices expenditure.

ex,σ,π(p, u) = ex,σ(p, u) + [p− p]′∇pex,σ(p, u) + 12 [p− p]′PHex,σ(p, u)P [p− p] + o(||p− p||3).

With Hessian that is singular in p, PHex,σ(p, u)P . The matrix P = I − pp′

p′p is the analogous to

P = I − pp′

p′p .

Finally we define the concave part of expenditure eσ,π, such that it has a Hessian that is

equal to the NSD part of PHeσ(p, u)P .

ex,r(p, u) = ex,σ,π,ν(p, u) = ex,σ(p, u) + [p − p]′∇pex,σ(p, u) + 12 [p − p]′[PHex,σ(p, u)P ]−[p −

p] + o(||p− p||3), this in fact is the concave part of eσ,π (i.e., ex,σ,π,ν(p, u) = ex,σ,πcave (p, u)).

We can define xσ,π,ν(p, w) = ∇pex,σ(p, vx,σ(p, w)) + [PHex,σ(p, vx,σ(p, w))P ]−[p − p], alter-

natively we have vx,σ,π,ν = vx,r defined implicitly as ex,r(p, vx,r(p, w)) = w.

We have xr(p, w) = −∇pvx,r(p,w)

∇wvx,r(p,w) .

That has Slutsky matrix:

S(xr)(p, w) = [PHeσ(p, vx,r(p, w))P ]− with properties σ, π, ν.

Observe that S(xr)(p, w) = [PHeσ(p, vx,r(p, w))P ]− and it is such that S(xr)(p, w) =

[PHeσ(p, vx,σ(p, w))P ]− with vx,r(p, w) = vx,σ(p, w) by construction for any w because ex,r(p, u) =

ex,σ(p, u) = w.

Proposition 3. The closest rational demand to x ∈ X (p) in the sense of minimizing its com-

parative statics error at least to first order xr ∈ R(p) defined locally as xr(p, w) = −∇pvx,r(p,w)

∇wvx,r(p,w) .

Moreover when ε = ||p − p|| is sufficiently small its value function can be decomposed in four

components miny∈R(p)||S(y) − S||2 = ||Eι||2 + 2〈Eν , Eι〉 + ||Eσ||2 + ||Eπ||2 + ||Eν ||2, where

||Eι||2 = miny∈X (p)||S(y) − Sr||2 is the mis-specification error (distance of Sr from being the

Slutsky matrix of some demand in X (p), i.e., Slutsky integrability). Moreover ||S(xr)−S(x)||2 =

||Eσ||2 + ||Eπ||2 + ||Eν ||2 + o(||P − P ||) where P → P as p→ p.

6.2 A Slutsky Goodness of Fit Measure of Bounded Rationality

Our bounded rationality measure ||E|| = ||S − Sr|| has cardinal meaning. In fact, the main

decomposition result shows that it is unique up to an affine transformation. But, importantly,

we can interpret our matrix nearness problem locally in terms of goodness of fit. Formally, the

goodness of fit is related to the minimum error in doing comparative statics analysis under the

rationality hypothesis.

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Definition 6. The Local Slutsky Goodness of Fit Measure of Bounded Rationality is a mapping

ρ : X (p) 7→ [0, 1] given by ρ(x) = ||Sx−S(x∗r)||2||s(x)||2 , with Sx the Slutsky matrix of x ∈ X (p) and

x∗r ∈ argminy∈R(p)||S(y)− S||2when Sx 6= 0 and ρ(x)=0 otherwise.

We have the following claim (it follows directly from Proposition 3 and Theorem 1.

Claim 4. The Local Slutsky Goodness of Fit Measure of Bounded Rationality of any given

x ∈ X (p) is given by ρ(x) = ||Ex||2+o(||P−P ||)||Sx||2 ' ||Ex||2

||Sx||2 . Moreover, it can be approximately

decomposed into the sum of three terms ρ(x) = ρσ(x) + ρπ(x) + ρν(x) + o(||P − P ||), with

ρi(x) = ||Ex,i||2||Sx||2 with i ∈ {σ, π, ν}.

This goodness of fit measure is advantageous because it provides a normalized decomposition

of the divergence between the local behavior of the demand function x around a price reference

p with respect to its closest rational approximation (in a restricted space of rational demands

R(p)). Because each of these terms is connected with violations of revealed demand axioms (i.e.,

corresponding to violations of the Ville axiom (path independence), homogeneity of degree zero

and the Wald axiom respectively), this index allows us to account for the relative importance

of each type of inconsistency of behavior with the failures in doing correct comparative statics

analysis under the rationality hypothesis. Of course, ρ(x) tends to 0 whenever the rational

demand approximation x∗r has a Slutsky matrix S(x∗r) that behaves approximately the same

as the observed Slutsky matrix Sx corresponding to demand function x.

Remark 8. A heuristic but useful interpretation is that ρ(x) represents how much of the lo-

cal behavior of x around p, as captured by Sx, is not “explained” by the rational hypothesis

(restricted to R(p)).

6.3 Path Independence of Wealth Compensations

In what follows we study the Slutsky Wealth compensation problem and its relation with the

Slutsky Error Eσ. In this and the next subsection we assume that the price-wealth region Z is

path connected. We need some preliminary definitions.

Definition 7. A price path is p : [0, 1] 7→ RL++ with p(0) = p0 and p(1) = p1, for any arbitrary

pair of prices p0, p1, is a continuous function that connects the two points in a convex subset of

RL++.

The Slutsky wealth compensation problem is that of compensating a given price change

through a change in wealth in order to keep the initial demand quantities affordable after the

change in prices. We ask what is the local wealth compensation needed to offset a small price

change at any point in the price path. We want to compute the total Slutsky wealth compensation

that is required to move from price situation p0 to p1. This means that along the path the

consumer is always compensated in the sense of Slutsky for small price changes.

The local Slutsky compensation quantity change is related to the actual amount or level of

wealth compensation by the following relation : ∂w(t)∂t = ∂p(t)

∂t

′x(p(t), w(t)) with Walras’ law

holding along the path w(t) = p(t)′x(t), in particular we fix w(0) = w0 at some level. Hence, the

Slutsky compensation level at some t is given by w(t) = w(0) +´ t0∂p(τ)∂τ

′x(p(τ), w(τ))dτ along

a price path p. Of course, the sum total wealth compensation change from (p0, w0) to p1 is:

w(1)− w(0) =´ 10∂p(τ)∂τ

′x(p(τ), w(τ))dτ .

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We next show that Eσ is related to a notion of path independence in this wealth compensa-

tion. To introduce the path independence definition, we define a composite price path.

Definition 8. (Composite price path) Let p and p∗ be price paths such that p(0) = p∗(0) and

p(1) = p∗(1). A composite price path is defined as p(s, t) = ps(t) = sp(t) + (1 − s)p∗(t) for all

s ∈ [0, 1] and all t ∈ [0, 1].

To define path independence, we first need to consider the Slutsky wealth compensation prob-

lem for a fixed s: ∂ws(t)∂t = ∂ps(t)

∂t

′x(ps(t), ws(t)) with solution ws(t) = ws(0)+

´ t0∂ps(τ)∂τ x(ps(τ), ws(τ))dτ .

Next, we need to allow s to vary in order to capture the sensitivity to the paths. This leads to

the following definition:

Definition 9. (Slutsky Wealth Compensation Across -Paths Gap) The Slutsky Wealth Com-

pensation across-Path gap is w0(1)− w1(1).

In this exercise, we have allowed τ first to trace the interval [0, 1], followed by a similar move

in the parameter s.

Alternatively, we can allow first s to move, followed by a move in τ in determining the wealth

compensations. To study first the relation of the dependence of the wealth compensation on the

path (parameter s), we propose the following quantity (whose exact expression stems from the

integral calculations in the ensuing proof). This is well defined for any composite price and

wealth paths ps and ws.

Definition 10. (Uncompensated across-path demand correction term) The uncompensated

across-path demand correction term c = 12

´ 10

´ 10

[∂ps(τ)∂s

′Dwx(ps(τ), ws(τ))][ws(τ)−∂ps(τ)∂s

′xs(τ)]dsdτ

We refer to c as uncompensated across-path demand correction term because in general

changes in wealth are not required to be compensated with respect to the parameter s, which

is first allowed to move, and c captures this divergence (i.e, the weighted sum of the pointwise

uncompensated gap c = 12

´ 10

´ 10

∑Ll=1 λl(τ, s)[ws(τ)− ∂ps(τ)

∂s

′xs(τ)]dsdτ with weights λl(τ, s) =

∂ps.l(τ)∂s

′ ∂xl(ps(τ),ws(τ))∂w capturing the interaction of the composite price sensitivity to the mixing

parameter and the wealth effect).

When the ps and ws are not Slutsky compensated paths with respect to s, the c correction

term allows us to obtain an equivalence between a quadratic form of Eσ and a monetary mea-

sure of the possible departures from rationality. We remark that the extra term beyond the

quadratic form in our next result also vanishes when Eσ = 0, and hence, the entire expres-

sion is “controlled” by the failure of the Slutsky symmetry property. We propose the following

definition:

Definition 11. (Slutsky wealth compensation path independence) Slutsky wealth compensation

path independence holds whenever the Slutsky Wealth Compensation across-Path gap vanishes:

[w0(1)− w1(1)] = 0.

Our next result follows:

Proposition 4. The Slutsky Wealth Compensation is path independent if and only if Eσ = 0.

Moreover, for any price ps connecting any two price vectors p0, p1, and wealth ws paths, the path

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dependence wealth gap is

1

2[w1(1)− w0(1)] =

ˆ 1

0

ˆ 1

0

∂ps(τ)

∂s

′Eσ(ps(τ), ws(τ))

∂ps(τ)

∂τdτds+ c.

Next, we establish the connection with the Slutsky matrix norm. The local change of wealth

compensation with respect to the change of the corrected path is measured by∂2ws(t)∂t∂s −cs(t), and

this term is controlled globally by a weighted norm of Eσ –suitable marginal quantity changes

correcting for possible path dependence– properly weighted by the changes in prices. Specifically:

Proposition 5. Given a composite price path, for any Slutsky wealth compensated path such that

ws(τ) = ps(τ)′x(τ), it is the case that [∂2ws(t)∂t∂s −cs(t)]

2 = ||Eσ(s, t)||2WU with ||·||WU the weighted

Frobenius norm with weighting Ws(t) =√

2∂ps(t)∂s∂ps(t)∂t

′when W is PSD pointwise and U is the

identity. Moreover, the path dependence wealth gap from any two paths is bounded above by the

average weighted Frobenius norm: [ 12 [w1(1)−w0(1)]− c]2 ≤ ||Eσ||2WU , with [w1(1)−w0(1)] = 0

and c = 0 when Eσ = 0.

6.4 The Compensated Law of Demand

So far we have related our error Slutsky matrix Eσ to a path independence notion in wealth

compensated paths. Now we move on to relate Eπ and Eν to WARP and the Compensated law

of demand.

The compensated law of demand states that prices and quantities “move in opposite direc-

tion,” once the consumer’s wealth is compensated so that she can afford previous bundles. The

compensated law of demand can be written in terms of price paths as ∂p(t)∂t

∂x(t)∂t ≤ 0 for all

t ∈ [0, 1], for every Slutsky Wealth Compensated path. A Slutsky Wealth Compensation path

(p(t), w(t))t∈[0,1] is such that:∂w(t)∂t = x(p(t), w(t))′ ∂p(t)∂t for all t ∈ [0, 1] .

Here we are interested in the sign and magnitude of the term ∂p(t)∂t

′ ∂x(p(t),w(t))∂t for Slutsky

Wealth Compensated paths ∂w(t)∂t = x(p(t), w(t))′ ∂p(t)∂t , which we call locally compensated de-

mand paths. The consumer that does not satisfy the compensated law of demand will have∂p(t)∂t

′ ∂x(t)∂t > 0 for some paths.

It is well known that the locally compensated demand is related to a quadratic form of

the Slutsky matrix function: ∂p(t)∂t

′ ∂x(t)∂t = ∂p(t)

∂t

′S(p(t), w(t))∂p(t)∂t . We note that the locally

compensated demand path is measured in money.

Our next result relates the locally compensated law of demand to Eπ and Eν .

Proposition 6. Consider any locally compensated path. The locally compensated demand path

satisfies the compensated law of demand if and only if Eπ = 0 and Eν = 0. Moreover,∂p(t)∂t

′ ∂x(t)∂(t) = ∂p(t)

∂t

′[Sr + Eπ + Eν ]∂p(t)∂t

The dot product of price and quantity changes defined above is important. Its discretized

version (pt− pt−1)′(xt− xt−1) ≤ 0 corresponds to the Compensated law of demand for compen-

sated price changes pt′x(pt−1, wt−1) = wt. The result above quantitatively connects violations

of this necessary condition for rationality with the error Slutsky matrix functions Eπ and Eν .

Next, we connect this to the Slutsky matrix norm.

Proposition 7. Consider any locally compensated demand path. For two demand functions x

and y such that their nearest rational Slutsky matrix function is the same Sx,r(p, w) = Sy,r(p, w),

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it holds that for the weighting matrix W (t) = ∂p(t)∂t

∂p(t)∂t

′and U the identity, one has that

||Ex,π||2WU + ||Ex,ν ||2WU ≥ ||Ey,π||2WU + ||Ey,ν ||2WU if and only if ∂p(t)∂t

∂x(t)∂t ≥

∂p(t)∂t

∂y(t)∂t .

7 Examples and Applications

The rationality assumption has long been seen as an approximation of actual consumer behavior.

Nonetheless, to judge whether this approximation is reasonable, one should be able to compare

any alternative behavior with its best rational approximation. Our results may be helpful in

this regard, as the next examples illustrate.

7.1 The Sparse-Max Consumer Model of Gabaix (2014)

This model generates analytically tractable behavioral demand functions and Slutsky matrices.

In this example, we compare the matrix nearness distance to the “underlying rational” Slutsky

matrix function proposed by Gabaix and compare it to the one proposed here. This example

shows that there exists a rational demand function that is behaviorally closer to the sparse max

consumer demand proposed by Gabaix than the “underlying rational” model of his framework.

Consider a Cobb-Douglas model xCD(p, w) such that:

xCDi = αiwpi

for i = 1, 2.

xCDi,pi = −αiwp2i

xCDi,w = αipi

sCDi,i = −αiwp2i

+ αipiαiwpi

= −αi(1−αi)wp2i

sCDi,j = αipi

αjwpj

.

The Slutsky matrix function is:

SCD(p, w) =

[−α1α2w

p21

α1

p1α2wp2

α1

p1α2wp2

−α1α2wp22

].

Let us denote Gabaix’s theory of behavior of the Sparse-max consumer by G. Then the

demand system under G is:

xGi = αipGi

w∑j αj

pj

pGj

for i = 1, 2.

This demand system fulfills Walras’ law. This function has an additional parameter with

respect to xCD(p, w), the perceived price pGi (m) = mipi + (1 −mi)pdi . The vector of attention

to price changes m, weighs the actual price pi and the default price pdi .

Consider the following matrix of attention for the sparse-max consumer:

M =

[1 0

0 0

]That is, the consumer does not pay any attention to price changes in p2, but perceives price

changes perfectly for p1. One of Gabaix’s elegant results relates the Slutsky matrix function of

xG, to the Cobb-Douglas benchmark. The behavioral Slutsky matrix evaluated at default prices

(in all this example p = pd) is:

SG(p, w) = SCD(p, w)M

SG(p, w) =

[−α1α2w

p210

α1

p1α2wp2

0

].

This matrix is not symmetric, not NSD, nor singular with p in its null space. Applying

Theorem 1, the nearest Slutsky matrix when p = pd is:

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Sr(p, w) =p22

p21+p22

[−α1α2w

p21

α1

p1α2wp2

α1

p1α2wp2

−α1α2wp22

]Also, one has

E(p, w) = S(p, w)− Sr(p, w)

E(p, w) = −

[[1− b(p)][α1α2w

p21] b(p)(α1

p1α2wp2

)

[1− b(p)][α1

p1α2wp2

] b(p)(−α1α2wp22

)

]with

b(p) =p22

p21+p22.

Now, we compute a useful quantity:

Tr(E′E) =w2α2

2α21

p21p22

.

It is convenient to compute the contributions of the violations of symmetry and singularity

in p separately.

Tr(E′E) = Tr(Eσ′Eσ) + Tr(Eπ

′Eπ) = 1

2w2α2

2α21

p21p22

+ 12w2α2

2α21

p21p22

. In this case, regardless of

the values that w takes, the contribution of each kind of violation is equal and amounts to

exactly half of the total distance. In fact, we have: ||E||2 = 12

´ wwTr(Eσ(w)′Eσ(w))dw +

12

´ wwTr(Eπ(w)′Eπ(w))dw =

(w3−w3

3

)α2

2α21

p21p22

, with p = pd.

Note, however, that in this example the third component of the violations, the one stemming

from NSD, is zero when the prices are evaluated at the default. Since the behavioral model

proposed by Gabaix does not satisfy WARP and its Slutsky matrix function violates NSD, we

conclude that the violation of the WARP is not massive, in fact, it affects the size of the Slutsky

norm only through its interactions with homogeneity of degree zero or “money illusion”. Our

approach can also be used in the general case. We compute the previous quantities at any p,

and any pd with m = [1, 0]′.

Tr(E′E) =w2α2

2α21

p21

[pd2 ]2

[p2+[pd2−p2]α1]4, with Tr(Eσ

′Eσ) = Tr(Eπ

′Eπ) and Eν = 0.

The expression above has a positive derivative with respect to pd2 for α1+α2 = 1, this indicates

that ∂∂pd2

δ(·) > 0 for any p. A sparse consumer that does not pay attention to p2 will be further

from rationality when the default price pd2 is high. Furthermore, the power of our approach lies

in the decomposition of ||E||2 = ||Eσ||2 + ||Eπ||2. In this case, the decomposition suggests that

the violation of WARP can be seen as a byproduct of the violations of symmetry and singularity

stemming from the “lack of attention” to price changes of good 2 and the “nominal illusion” or

lack of homogeneity of degree zero in prices and wealth in such a demand system.16

To show the tractability of our approach in this example, we will study a very simple re-

gion Z, with the aim of illustrating how one can learn from the effect of a behavioral pa-

rameter such as α1 and pd2. Let Z = {w, p1 = 1, p2 ∈ [1, 2]}, then δ(α1, pd2) = 1

3 (α1 −1)α1

2[pd2]2(

1(α1(pd2−2)+2)3

− 1(α1(pd2−1)+1)3

). One can now visualize this δ in the α, pd2 space,

that is at pd2 ∈ [1, 2] and α1 ∈ [0, 1] (figure 2). We can observe that α1 has a non-linear effect on

δ, and the distance toward the rational matrix goes to zero when either α1 → 0 or α1 → 1 for

all pd2 ∈ [1, 2].

To finish this example we study the errors in comparative statics analysis. This is done

locally.17 Fix (p, w) = (pd, w) as the reference point. First, we compare the distance between SG

and SCD, in terms of errors in comparative statics analysis. If we use xCD as an approximation

16This observation is robust to other specifications (e.g. CES family).17In general, our approach does not require finding the closest xr ∈ R(Z) to xG, nevertheless doing so may

help to complement the understanding of how much this particular bounded rationality model differs from thestandard rational one in practical terms

20

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δ(pd2, α)

1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.2

0.4

0.6

0.8

1.0

0.04

0.08

0.12

0.16

-6-4-20246

Figure 2: Level curves sparse-max consumer example . The figure plots α ∈ [0, 1] on its verticalaxis and pd2 ∈ [1, 2] on its horizontal axis.

of xG at the default prices, we obtain the following local comparative statics error ||SG(pd, w)−SCD(pd, w)||2M =

(pd1)2+(pd2)

2

(pd2)2

w2α22α2

1

(pd1)2(pd2)

2 . By our results in Subsection 6.1, we can decompose

this quantity as the sum of a mis-specification error and ||E(pd, w)||2M. In fact, ||SCD(pd, w) −SG(pd, w)||2M = ||SCD(pd, w)− Sr(pd, w)||2M + ||E(pd, w)||2M, with ||SCD(pd, w)− Sr(pd, w)||2M =w2α2

2α21

(pd2)4 and ||E(pd, w)||2M =

w2α22α2

1

(pd1)2(pd2)

2 .

We know also, by the results in Subsection 6.1, that even if we could improve the mis-

specification error, the Slutsky matrix error norm ||E(pd, w)||2M will not change at all. To illus-

trate this, we find the Cobb-Douglas demand that minimizes the error in comparative statics

analysis at the reference price-wealth pair (pd, w). We can write this problem parametrically,

through characterizing the Cobb-Douglas family by a parameter β ∈ [0, 1]: xCD2(p, w, β) =

(βwp1 ,(1−β)wp2

)′. We solve the problem β∗ ∈ argminβ∈[0,1]||SCD2(pd, w, β)− SG(pd, w)||2M. There

are two solutions β∗ = 12 ±√pd′pd[(pd1)

2+(pd2)2[1−2α1]2]

2pd′pd, which depend on the given parameters

and the fixed default price. The mis-specification error is zero, that is ||SCD2(pd, w, β∗) −Sr(pd, w)||2M = 0. Because of this, in this case, the error in comparative statics reduces to the

behavioral error captured by the Slutsky norm: ||SCD2(pd, w, β)−SG(pd, w)||2M = ||E(pd, w)||2M,

where ||E(pd, w)||2M =w2α2

1α22

(pd1)2(pd2)

2 .

Observe that it does not depend on β∗ and remains unchanged with respect to the case of

SCD. Of course, even if we used the “best” rational model to minimize errors in comparative

statics analysis locally, we would not reduce at all the Slutsky matrix error norm that stems

from lack of rationality as captured by properties σ, π, ν.

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7.2 Hyperbolic Discounting

The literature on self-control and hyperbolic discounting has flourished in macroeconomics and

development economics. In this example, we study a three-period model that allows us to

illustrate the use of our methodology. Our aim is to measure the violations of property R by

naive and sophisticated quasi-hyperbolic discounters.

The optimization problem for a consumer that can pre-commit is: max{xpi }i=1,2,3.u(xp1) +

βθu(xp2) + βθ2u(xp3) subject to the budget constraint∑3i=1 pix

pi = w.

Which gives the demand system: (i) xp1 = [p1 + p2[βθ p1p2 ]1σ + p3[βθ2 p1p3 ]

1σ ]−1w. (ii) xp2 =

[βθ p1p2 ]1σ xp1 (iii) xp3 = [βθ2 p1p3 ]

1σ xp1.

The naive quasi-hyperbolic discounter will have the following demand system:

In the first period, the consumer assumes he will stick to his commitment in the second

period and consumes the same amount as in the pre-commitment case (i.e. xh1 = xp1). However,

when period two arrives, he re-optimizes taking as given the remaining wealth w − p1xh1 : xh2 =w−p1xh1

p2+p3[βθp2p3

]1σ

and xh3 = [βθ p2p3 ]1σ xh2 .

The analytical result for the matrix nearness norm has a nice expression:

Tr(E′E) =

(σ − 1)2w2(p21 + p22 + p23

)((βθ2p1p3

) 1σ −

(βθp1p2

) 1σ(βθp2p3

) 1σ

)2

2σ2

(p3

(βθp2p3

) 1σ

+ p2

)2(p2

(βθp1p2

) 1σ

+ p3

(βθ2p1p3

) 1σ

+ p1

)4 ,

which readily gives us that: (i) when σ = 1 then Tr(E′E) = 0 and δ = 0, that is the demand

is rational, (ii) when β = 1 then Tr(E′E) = 0, (iii) finally when β → 0, θ → 0, then δ → 0.

In these three cases by the previous results ε → 0. In fact, in the limit cases the hyperbolic

demand system is rational. Take for instance case (iii), because the agent consumes everything

in the first period and gives no weight to the other time periods then it is trivially rational, with

Sr → 0 and xh1 → wp1 and xh2 , x

h3 → 0. In case (i), the logarithmic utility case, the hyperbolic

discounter manages to keep his commitment and therefore his consumption is time consistent

and ||E|| = 0.

To illustrate further the use of the tools developed here, we find an explicit value for δ

in terms of the behavioral parameters β, θ, for an arbitrary rectangle Z of prices and wealth.

Consider the region Z = {p1, p2, p3 = 1, w ∈ [1, 2]} , with σ = 12 we compute δ(β, θ), which

can be represented graphically in the box β ∈ [0, 1], θ ∈ [0, 1]. The analytical expression for

δ2 =7β4(β2−1)

2θ8

2(β2θ2+1)2(β2(θ4+θ2)+1)4.

The level curves (figure 3) show that the hyperbolic discounter is very close to the rational

consumer, in the matrix nearness sense, for very low values of β, θ and for values of θ ≤ 12 .18 This

makes intuitive sense as a lower θ means heavier discount on the future and lower consumption

of goods of time 2 and 3 that are the ones affected by self-control.

Another observation that we can draw from this example is that for any arbitrary compact

region Z of prices and wealth analyzed ||E||2 = ||Eσ||2. That is, only the asymmetric part

plays a role in the violation of property R. In other words, under this numerical conditions

the hyperbolic discounter violates symmetry but it satisfies singularity in prices and negative

18One can also use the level curves in this example to identify pairs (β, θ) that are “equidistant” from rationality,capturing an interesting tradeoff between the short-run and the long-run discount factors and its effects on theviolations of the Slutsky conditions.

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δ(β, θ)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.02

0.04

0.06

0.08

0.10

0.12

-6-4-20246

Figure 3: Level curves hyperbolic discounter example . The figure plots β ∈ [0, 1] on its horizontalaxis and θ ∈ [0, 1] on its vertical axis.

semidefiniteness. The hyperbolic discounter fulfills WARP.

7.3 Sophisticated Quasi-Hyperbolic Discounting

The sophisticated quasi-hyperbolic discounter is intuitively closer to rationality. However, the

Slutsky norm helps appreciate some of the subtleties and assess which conditions of rationality

are fulfilled by this type of consumer. We build this example as a followup to the naive quasi-

hyperbolic consumer. In this case, the consumer knows that in t = 2 he will not be able to

keep his commitment and therefore will adjust his consumption at t = 1. Then the consumer

maximizes

maxxsh1 u(xsh1 ) + βθu(xh2 ) + βθ2u(xh3 )

where xh2 , xh3 are known to his in t = 1 and depend on period 1 consumption. However, he

can control only how much he consumes in the first period. Then, the first period consumption

under sophisticated hyperbolic discounting is:

xsh1 =

p1βθ + p1βθ

2[βθ p2p3 ]1−σσ[

p2 + p3[βθ p2p3 ]1σ

]1−σ

+ p1

−1

w

The argument in the integral of the expression for δ for a generic Z is given by the quantity:

Tr(E′E) =

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(β − 1)2(σ − 1)2w2θ4/σ(p21 + p22 + p23

)(βp2 + p3

(βθp2p3

) 1σ

) 2σ−2

βp1

p3

(p3(βθp2p3

) 1σ +p2

)2/σ

2σ4

θp1

(p3(βθp2p3

) 1σ +p2

)σ−1(βp2+p3

(βθp2p3

) 1σ

)p2

+ p1

4

As expected, this implies that: (i) when σ = 1, δ = 0 for any Z; (ii) when β = 1, δ = 0;

and (iii) when β = 0, δ = 0. Thus, in all these cases, ε = 0. Also, the decomposition of

||E||2 = ||Eσ||2, which means that only the symmetry property is violated, while the weak

axiom and the homogeneity of degree zero in prices and wealth are preserved.

Finally, we compare this quantity with the case of the naive hyperbolic discounter. We

consider the ratio r = Tr(Esh′Esh)Tr(Eh′Eh)

. When r < 1, the sophisticated hyperbolic consumer has

a lower δ for any Z and any parameter configuration. The first finding is that the ratio r

and the implied values of δ depend crucially on the parameter σ. For σ = 1, both Esh, Eh

are equal to zero: this is a knife-edge case, in which the marginal rates of substitution yield

optimal consumptions equal to the commitment baseline. For σ < 1 (e.g., 1/2), the sophisticated

hyperbolic discounter has a uniformly lower δ and thus r < 1.

However, for σ > 1 (e.g., equal to 2), the naive hyperbolic discounter has a uniformly

lower δ, hence r > 1. Although this may seem counterintuitive, it tells us that the closest

rational type (which need not be the commitment baseline) is closer for the naive than it is for

the sophisticated consumer. To see this, note that the “rational” comparative statics analysis

fails due to two possible sources: (i) The lack of self-control in the consumption of period 2

(overspending); and (ii) in the case of the sophisticated type, an additional effect due to the

strategic change in period 1 consumption (overspending or underspending). So, the “rational”

comparative statics analysis gives a higher failure than in the naive case for σ > 1, because for

an increase in prices for the naive case the change in period 1 consumption equals that in the

rational case and only the second and third period demanded amounts are different. In contrast,

for the sophisticated case, demanded amounts for the three goods all differ, in particular due

to a higher level of wealth remaining at the end of period 1, leading to a higher overspending

in period 2. This observation holds for such CRRA instantaneous utility with any parameter

values of δ,θ ∈ [0, 1].19

8 Literature Review

The canonical treatment of measuring deviations from rational consumer behavior was establish

by Afriat (1973) with its critical cost-efficiency index. Afriat’s index measures the amount by

which budget constraints have to be adjusted so as to eliminate violations of the Generalized

19Furthermore, to enhance the comparison for the case of σ = 12

, we compute explicitly the expression forδ in the same region Z = {p1, p2, p3 = 1, w ∈ [1, 2]} as in the previous example for the naive discounter:

δ2(β, θ) = [14(β − 1)2β6θ8(βθ2 + 1

)2]/[β2θ2

(βθ2

(βθ2 + 2

)+ 2

)+ 1]4. The level curves of this δ expression are

very similar to the naive case, but it is closer to rationality uniformly for all parameter configurations. Evaluatedat the values of β = 0.7 and θ = 0.9, one gets the value δ = 0.0703847, which is slightly lower than the δ ofthe naive hyperbolic case for the same σ = 1

2(δ = 0.074), where the parameters are taken from the empirical

literature.

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Axiom of Revealed Preference (GARP). Varian (1985; 1990) refines Afriat’s measure by focusing

on the minimum adjustment of the budget constraint needed to eliminate violations of GARP.

Houtman and Maks (1985) measure deviations from GARP through identifying the largest subset

of choices that is consistent with maximizing behavior. More recently, Echenique, Lee and Shum

(2011), give a new measure of violations of revealed preference behavior called the “money pump

index” . Also Jerison and Jerison (2012) propose a way to bound Afriat’s index of cost-efficiency

using an index of violations of the symmetry and negative semidefiniteness Slutsky conditions.

It would be interesting to compare our Slutsky matrix norm with these other approaches.

The closest treatment of the problem to our work is the approximately rational consumer

demand proposed by Jerison and Jerison (1992; 1993). These authors are able to relate the vio-

lations of negative semidefiniteness and symmetry of the Slutsky matrix to the smallest distance

between an observe smooth demand system and a rational demand. Russell (1997) proposes a

notion of quasi-rationality. Russell’s argument links the Slutsky matrix anti-symmetry part with

the lack of integrability of a demand system.

Our work takes a different methodological approach to this problem and generalizes the

results to the case of violations of singularity of the Slutsky matrix. More importantly, this new

approach allows to treat the three kinds of violations of the Slutsky conditions simultaneously.

For instance, new behavioral models like the sparse-max consumer Gabaix (2014) suggest the

presence of a money illusion such that prices are not in the null space of the Slutsky matrix. We

also emphasize a positive measure of bounded rationality based on errors in making comparative

statics analysis when one assumes (perhaps incorrectly) that the consumer is rational. The

only other work we are aware of that analyzes comparative statics with behavioral consumers is

Farhi & Gabaix (2015). In their interesting work they propose alternate ideas to the traditional

Slutsky matrix including extensions of the Slutsky matrix for nonlinear budget constraints. It

remains an open task how to extend our methodology to these new concepts.

9 Conclusion

By redefining the problem of finding the closest rational demand to an arbitrary observed be-

havior in terms of matrix nearness, we are able to pose the problem in a convex optimization

framework that permits a better computational implementability and provide a tractable ap-

proach with a closed form solution. We define a metric in the space of smooth demand functions

and finally propose a way to recover the best Slutsky approximation matrix function under a

Frobenius norm. Our approach gives a geometric interpretation in terms of transformations of

the Slutsky matrix or first order behavior of demand functions. As a result, a classification of

the different kinds of violations of rationality is also provided. Comparative statics for a bound-

edly rational consumer is measured with our norm. Our approach is also suited to measure the

mis-specification error from assuming a given form of rationality when the consumer’s behavior

is actually not rational.

References

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method. International economic review, 14(2), 460–472.

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Aguiar, V. H. & Serrano, R. (2014). Slutsky Matrix Norms and the Size of Bounded Rationality.

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Jerison, D. & Jerison, M. (2012). Real income growth and revealed preference inconsistency.

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10 Appendix: Proofs

10.1 Proof of Claim 1

Proof. The problem is minSr ||S − Sr|| subject to Sr(z) ≤ 0, Sr(z) = Sr(z)′, Sr(z)p = 0 for

z ∈ Z.

Under the Frobenius norm, the minimization problem amounts to finding the solution to

minSr´z∈Z Tr([S(z)− Sr(z)]′[S(z)− Sr(z)])dz

subject to the stated constraints.

The objective function is strictly convex, because of the use of the Frobenius norm. This

norm is also a continuous functional.

The constraint set MR(Z) is convex and closed. In fact, the cone of negative semidefinite

matrices is a closed and convex set. Also, the set of symmetric matrices is closed and convex,

and finally the set of matrices with eigenvalue λ = 0 associated with eigenvector p is convex.

To see the last statement, let A(z)p = 0, B(z)p = 0, and let C(z) = αA(z) + (1 − α)B(z) for

α ∈ (0, 1). It follows that C(z)p = 0. Then MR(Z) is the intersection of three convex sets and

is therefore convex itself. It is also useful to note that all three constraint sets are subspaces of

M(Z) and the intersection MR(Z) is itself a subspace of M(Z).

Now we prove that not only the symmetric and the NSD constraints sets are closed but so

is the set of all MR(Z). Any matrix function in the constraint set is a symmetric NSD matrix

with p in its null space. Therefore, every sequence of matrix functions in the constraint set

has the form Dn(z) = Qn(z)Λn(z)Qn(z)′, where Λn(z) = Diag[λni (z)]i∈1,...L with ascending

ordered eigenvalues functions. It follows that the eigenvalue function in position L,L is the

null eigenvalue λL = 0, or the null scalar function. That is, imposing an increasing order

the position 1, 1 is then held by λn1 (z) ≤ λn2 (z) ≤ . . . ≤ 0 where the order is induced by the

distance to the null function using the Euclidean distance for scalar functions defined over Z.20

The matrix function Qn(z) = [qn1 · · · p] is the orthogonal matrix with eigenvectors functions

as columns. For all Dn(z) ∈ MR(Z), λnL = 0 is associated with the price vector qnL = p

always, to guarantee that p is in its null space. The eigenvectors are defined implicitly by the

condition Dn(z)qni (z) = λn(z)qni (z) with pointwise matrix and vector multiplication and qni ⊥ por 〈qni , p〉 = 0 using the inner product for C0(Z) for i = 1, . . . , L− 1 and for all n ∈ N. Take any

sequence of {Dn(z)}n∈N with Dn(z) ∈MR(Z) for each z ∈ Z, with limit limn→∞Dn(z) = D(z).

We want to show that D(z) ∈MR(Z). It should be clear that any Dn(z)→ D(z) converges to

a symmetric matrix function (the symmetric matrix subspace is an orthogonal complement of a

subspace ofM(Z) (the subspace of skew symmetric matrix functions) and therefore, it is always

closed in any metric space). It is also clear that D(z)p = 0 since λnL = 0 for all n and certainly

λnL → 0 with the associated eigenvector qnL = p for all n and qnL → p. This condition, along with

symmetry, guarantees that qni 6=L → q ⊥ p. Finally, the set of negative scalar functions is closed.

Then, λni6=L → λ(z)− with λi 6=L(z)− = min(0, λi6=L(z)). This is a negative scalar function by

construction, since if λi 6=L(z) > 0 then λi 6=L(z)− = 0. Then MR(Z) is closed. Observe that

MR(Z) is closed since it is in the intersection of three closed sets. In conclusion, since the

Frobenius norm in M(Z) is a continuous and strictly convex functional and the constraint set

is closed and convex, the minimum is attained and it is unique.

20These eigenvalue functions can be labeled because the domain Z is simply connected. The order admitscrossing eigenvalue functions.

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Next, we present a claim that is auxiliary to establish Lemma 2, in the proof of Theorem 1.

10.2 Proof of Lemma 1

Proof. The Lagrangian for the subproblem with symmetry and singularity restrictions is: L =´z∈Z Tr([S(z)−A(z)]′[S(z)−A(z)])dz+

´z∈Z λ(z)′A(z)pdz+

´z∈Z vec(U(z))′vec[A(z)−A(z)′],

with λ ∈ RL and U ∈ RL×L. Using that the singularity restriction term is scalar (λ′A(z)p ∈R), as well as the identity Tr(A′B) = vec(A)′vec(B) for all A,B ∈M(Z),21

one can rewrite the Lagrangian as:

L =´z∈Z [Tr([S(z)−A(z)]′[S(z)−A(z)]) + Tr(A(z)pλ(z)′) + Tr(U(z)′[A(z)−A(z)′])]dz

Using the linearity of the trace, and the fact that this calculus of variations problem does

not depend on z or on the derivatives of the solution A, the pointwise first-order necessary and

sufficient conditions in this convex problem (Euler Lagrange Equation) is:

S(z)−A(z) + U(z)− U(z)′ + λ(z)p′ = 0.

Solve for A(z):

A(z) = S(z) + U(z)− U(z)′ + λ(z)p′.

By the symmetry restriction,

2[U(z)− U(z)′] = S(z)′ − S(z) + pλ(z)′ − λ(z)p′.

Replace back in the expression for A(z):

A(z) = S(z) + 12 [S(z)′ − S(z) + pλ(z)′ − λ(z)p′] + λ(z)p′,

We obtain the expression:

A(z) = Sσ(z) + λ(z)p′ + 12 [pλ(z)′ − λ(z)p′].

Now we observe that we can write,

A(z) = Sσ(z) +Wσ(z)with W (z) = λ(z)p′ and Wσ = 12 [pλ(z)′+λ(z)p′], the symmetric part

of W (z).

By the restriction of singularity in prices A(z)p = 0 which implies Wσ(z)p = −Sσ(z)p.

To complete the proof we let r(z) = −Sσ(z)p be the feedback error (of non-singularity in

prices).22 Then we can write the matrix function Wσ(z) = r(z)p′+pr(z)′

p′p − [r(z)′p]pp′

(p′p)2 that is the

symmetric part of the outer product of the feedback error and the price vector. Replacing r(z)

by its definition, we get that Wσ = −Eπ:

Eπ(z) = 1p′p [Sσ(z)pp′+ pp′Sσ(z)− 1

p′ppp′Sσ(z)pp′], then Eπ = Sσ−PSσP with P = I − pp′

p′p

and by Walras’ law it simplifies to:

Eπ(z) = 1p′p [Sσ(z)pp′ + pp′Sσ(z)],

this expression along with the implied multipliers λ and U , satisfies all the first-order con-

ditions of the problem. Since we can use arguments identical to those in Claim 1 –only not

imposing NSD–, we know that the solution is unique. Hence, this expression describes the solu-

tion to the posed calculus of variations problem with the symmetry and singularity restrictions.

The proof is complete.

21The vec(A) symbol stands for the vectorization of a matrix A of dimension L × L in a vector a = vec(A)of dimension L2 where the columns of A are stacked to form a. Observe that the symmetry restriction can beexpressed in a sigma notation (entry-wise) but this matrix algebra notation helps us to make more clear the useof the trace operator in the objective function.

22In this step we follow a similar strategy of proof to Dennis & Schnabel (1979); Higham (1989).

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10.3 Proof of Lemma 2

Proof. Recall the matrix nearness problem: minA∈M(Z)||S − A||2 subject to A satisfying sin-

gularity, symmetry, and NSD. This is equivalent, by manipulating the objective function, to:

minA∈M(Z)||Eσ +Eπ +Sσ,π−A||2. Writing out the norm as a function of the traces, and using

the fact that Eσ is skew symmetric, while the rest of the expression is symmetric, we get that this

amounts to writing: minA∈M(Z)||Eσ||2+||Eπ+Sσ,π−A||2 subject to A having property R. This

is in turn equivalent to: minA∈M(Z)||Eσ||2 + ||Eπ||2 +2〈Eπ, [Sσ,π−A]〉+ ||Sσ,π−A||2 subject to

A having property R. Then, exploiting the fact that Eπ and S+ = Sσ,π −A are orthogonal (as

proved in Claim 3), we obtain that the problem is equivalent to minA∈M(Z)||Eσ||2 + ||Eπ||2 +

||Eν ||2 subject to A having property R.

Hence, since the objective function of the matrix nearness problem ||E||2 = ||Eσ||2+||Eπ||2+

||Sσ,π−A||2, solving the program minA∈M(Z)||E||2 subject to A having property R is equivalent

to solving minA∈M(Z)||Sσ,π −A||2 subject to the same constraints.

Now, the best NSD matrix approximation of the symmetric valued function Sσ,π is Sr =

Sσ,π,ν . Then, the candidate solution to our problem is A(z) = Sr(z) for all z ∈ Z. Notice that

Sr(z) is symmetric and singular with p in its null space by construction. Indeed, recall that

Sσ,π(z) = Q(z)Λ(z)Q(z)′ and Sr(z) = Sσ,π,ν(z) = Q(z)Λ(z)−Q(z)′. Then it follows that Sr(z)

is symmetric for z ∈ Z. Moreover, by definition λl(z)− = min(0, λl(z)) for l = 1, . . . , L. Since

Sσ,π(z)p = 0, it follows that λL(z) = 0 is the eigenvalue function associated with the qL(z) = p

eigenvector. Then we have that λL(z)− = 0 is also associated to the eigenvector p, and we can

conclude that Sr(z)p = 0.

As just argued, Sr(z) has property R, i.e., Sr(z) is in the constraint set of the matrix nearness

problem. We conclude that it is its solution.

10.4 Proof of Claim 3

Proof. By definition Sσ,π(z) = Sσ(z) + Eπ(z), with Eπ(z) a symmetric matrix such that

Eπ(z)p 6= 0 when Sσ(p)p 6= 0 and Eπ(z) = 0 when Sσ(p)p = 0. Thus, Eπ(z) is always

singular.

One can then write the direct sum decomposition of the set A(z) of symmetric singular

matrix functions with the property that p′A(z)p = 0 as follows: A(z) = P(z) ⊕ N (z) for all

z ∈ Z, where

P(z) = {Eπ(z) : Tr(Eπ(z)pp′) = 0 and Eπ(z)p 6= 0 for Eπ(z) 6= 0}

and

N (z) = {N(z) : Tr(N(z)pp′) = 0, N(z)p = 0}.

To see that this is a direct sum decomposition, first observe that P(z) ∩ N (z) = {0}, with

0 denoting the zero matrix function, by construction. Furthermore, any A(z) ∈ A(z) can be

written as a sum of A(z) = Eπ(z)+N(z) since A(z)p = 0 or (exclusive) A(z)p 6= 0, for A(z) 6= 0.

Furthermore, p′A(z)p = p′Eπ(z)p+ p′N(z)p = 0 for any Eπ(z), N(z). Then the decomposition

is precisely A(z) = Eπ(z) when A(z)p 6= 0 and A(z) = N(z) when A(z)p = 0. Since every

direct sum decomposition represents the sum of a subspace and its orthogonal complement,

andN (z) is a subspace in the space of symmetric matrix-valued functions, it follows that P(z)

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is its orthogonal complement. In particular, since Eν(z)p = 0 and Tr(Eν(z)pp′) = 0, it follows

that Tr(Eπ(z)′Eν(z)) = 0, for z ∈ Z.

10.5 Proof of Proposition 1

Proof. The proof has three steps: First, we use the decomposition of a matrix into its symmetric

and antisymmetric parts: ||S(x) − S(xr)||2 = ||S − S(xr)||2 = ||Sσ + Eσ − S(xr)||2 = ||Sσ −S(xr)||2 + 2〈Eσ, Sσ − S(xr)〉+ ||Eσ||2,

where Sσ−S(xr) is a symmetric matrix function and Eσ is a skew-symmetric matrix function.

Then 〈Eσ, Sσ − S(xr)〉 = 0. Then ||S(x)− S(xr)||2 = ||Sσ − S(xr)||2 + ||Eσ||2.

Second, we take the p-singularity projection: ||Sσ − S(xr)||2 = ||Sσ,π + Eπ − S(xr)||2 =

||Sσ,π − S(xr)||2 + 2〈Eπ, Sσ − S(xr)〉+ ||Eπ||2. Since Eπ is orthogonal to the matrix functions

that are symmetric and have p as its eigenvector associated with the null eigenvalue (claim 3),

by definition S(xr(p, w))p = 0 and S(xr) is symmetric. Then: 〈Eπ, Sσ,π−S(xr)〉 = 〈Eπ, Sσ,π〉−〈Eπ, S(xr)〉 = 0. Therefore, ||S(x)− S(xr)||2 = ||Sσ,π − S(xr)||2 + ||Eσ||2 + ||Eπ||2.

Finally, we deal with the NSD part: ||Sσ,π − S(xr)||2 = ||Sσ,π,ν +Eν − S(xr)||2 = ||Sσ,π,ν −S(xr)||2 +2〈Eν , Sσ,π,ν−S(xr)〉+ ||Eν ||2. It follows that: 〈Eν , Sσ,π,ν−S(xr)〉 = 〈Eν ,−S(xr)〉 ≤||Sr − S(xr)|| · ||Eν ||. For the lower bound, 〈Eν ,−S(xr)〉 ≥ 0 because both Eν and −S(xr) are

PSD matrices.

This implies that ||S(x∗r)−S(x)||2 = ||S(x∗r)−Sr||2 + 2〈Eν , S(x∗r)−Sr〉+ ||E||2, with the

bounds as asserted.

10.6 Proof of Lemma 4

Proof. Take y∗ ∈ argminy∈X (p)||S(y)−Sr||2 and assume that S(y∗) is not symmetric. Then let

S(y∗) = Sy∗. Now we can compute ||S(y∗)− Sr||2 = ||Sy∗,σ + Ey

∗,σ − Sr||2. Observe that:

||Sy∗,σ +Ey∗,σ − Sr||2 = ||Sy∗,σ − Sr||2 + 2〈Ey∗,σ, Sy∗,σ − Sr〉+ ||Ey∗,σ||2. By the fact that

Sy∗,σ−Sr is symmetric, we have ||Sy∗,σ+Ey

∗,σ−Sr||2 = ||Sy∗,σ−Sr||2+||Ey∗,σ||2. By the lemma

(3) there exists y∗σ ∈ X (p) for the given y∗ such that ||S(y∗σ)−Sr||2 = ||Sy∗,σ−Sr||2. This means

that ||S(y∗)−Sr||2 > ||s(y∗σ)−Sr||2 which is a contradiction to y∗ ∈ argminy∈X (p)||S(y)−Sr||2

because y∗σ improves upon y∗. We conclude that S(y)(p, w) is symmetric for all (p, w) ∈ Z(p, ε).

The moreover part of the statement follows from the existence of xσ such that S(xσ) = Sx,σ

locally. By Walras’ law and weak WARP, we know that x is HD0 Sx,σ(p, w)p = 0 thus xσ is

HD0 around (p, w), this means that locally S(xσ) = Sx,σ,π, finally by Weak WARP Sx,σ,π is

NSD thus S(xσ) is NSD. Thus ||S(xσ)−Sr||2 = 0 when Sx,σ,π = Sr around (p, w), thus making

xσ a solution.

10.7 Proof of Lemma 6

Proof. To prove (i) take y ∈ X (p) we know that S(y) is symmetric in Z(p, ε) then by the

Frobenious Integrability (local version) (Dieudonne (1969)), Theorems 10.9.4, pp. 310–311) we

know that y has a scalar function vy, such that y(p, w) = −∇pvy(p,w)

∇wvy(p,w) . It is trivial to observe

that y(p, w) = yσ(p, w) because vy solves the Slutsky wealth compensation problem for y in the

region Z(p, ε).

To prove (ii) we just notice that by the Concave Convex Decomposition lemma vycave exists

under the boundedness of the Slutsky Matrix S(y) in the region Z(p, ε), in which case it is

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immediate to notice that vycave is continuously differentiable because vy is and yr exists and is

both symmetric and NSD and certainly it satisfies Walras’ law. Then by Hurwicz-Uzawa (1971)

we know yr ∈ R(p).

10.8 Proof of Lemma 7

Proof. By lemma (3) Take y∗ ∈ argminy∈X (p)||S(y) − Sr||2 we know that S(y∗) is symmet-

ric in Z(p, ε) then by lemma (6) we know that y∗ has a scalar function vy∗(p, w), such that

y∗(p, w) = −∇pvy∗ (p,w)

∇wvy∗ (p,w)(that coincides with the quasi-indirect utility of y∗). Also we know

that there is a local quasi-expenditure function ey∗,σ(p, vy

∗(p, w)) = w for a fixed w such that

S(y∗)(p, ey∗,σ(p, u)) = Hey∗,σ(p, u) (not necessarily convex on p). Now take the matrix function

Sr and evaluate it at the same local quasi-expenditure function and obtain the matrix function:

Hx,r(p, u) = Sr(p, ey∗,σ(p, u)).

Next, decompose the quasi-expenditure function in a concave part eσcave = eσ − r with r a

convex function and a convex part eσvex = r. We focus on the local p on quadratic expenditure

functions (this is without loss of generality locally due to the Taylor expansion under twice

continuous differentiability). Then we can ensure that we can find the minimal decomposition in

the sense of Frobenius norm such that Hey∗,σ = Hey∗,σcave+Hey∗,σvex such that 〈Hey∗,σcave,Hey∗,σvex 〉 = 0

and Hey∗,σcave is NSD and Hey∗,σvex is PSD (because locally we can think of the allowable functions

as approximately quadratic forms). We study the equivalent problem ||S(y∗)− Sr||2 = ||Heσ −Hx,r||2.

Assume by contradiction that y∗ is such that S(y∗) is not NSD in Z(p, ε). Observe that ey∗,σ

associated with y∗ provides the value ||Hey∗,σ−Hx,r||2 = ||Hey∗,σcave+Hy∗,σvex −Hx,r||2. This in turn

can be expanded ||Hey∗,σcave +Hey∗,σvex −Hx,r||2 = ||Hey∗,σcave −Hx,r||2 + 2〈Hey∗,σvex ,Hey∗,σcave −Hx,r〉+

||Hey∗,σvex ||2. By construction, 2〈Hey∗,σcave,Hy∗,σvex 〉 = 0. Then ||Hey∗,σcave−Hx,r||2+2〈Hey∗,σvex ,−Hx,r〉+

||Hey∗,σvex ||2 > ||Hey∗,σcave −Hx,r||2 because 〈Hey∗,σvex ,−Hx,r〉 ≥ 0, because both matrices are PSD.

This contradicts that y∗ is not NSD, because by lemma (6) there exists yr(p, w) = −∇pvy∗cave(p,w)

∇wvy∗cave(p,w)

with expenditure ey∗,σcave which improves upon y∗ (which is impossible because y∗ was assumed

to be a solution of the problem). Therefore, y∗ is such that S(y∗) is symmetric and NSD in

Z(p, ε).

10.9 Proof of Lemma 8

Proof. Under Walras’ law and by the results of lemma (7) we apply John (1995) result to conclude

that S(y∗)(p, w)p = 0 so it is singular in p. By Walras’ law again, this leads us to conclude that

Dpy∗(p, w)p+Dwy

∗(p, w)w = 0, this implies that y∗ is HD0.

10.10 Proof of Proposition 3

Proof. First we want to establish that if p → p and if yr ∈ argminy∈R(p)||Sr − S(y)||2 then

yr ∈ argminy∈R(p)||Sr − S(y)||2 + 2〈Eν ,−S(y)〉Locally if S(yr)(p, w) → Sr(p, w) as p → p then Tr(Eν(p, w)′[Sr(p, w) − S(yr)(p, w)]) → 0.

Then the statement we want to prove follows immediately.

Thus we want to establish that S(yr)(p, w)→ Sr(p, w) as p→ p.

First, we compute xσ from x, second and xσ,π from xσ and finally xr = xσ,π,ν .

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Observe that in general Sx,r(p, w) = Sx,σ,π,ν(p, w) = (PS(xσ)P )−(p, w) for the closest inte-

grable demand xσ.

We now want to prove that S(xr)(p, w) → Sx,r(p, w) as p → p, this automatically will

guarantee that xr is equivalent to yr to the first order and that S(yr)(p, w) → Sr(p, w) as

p→ p.

Notice that (PS(xσ)P )−(p, w) = P [Hex,σ(p, vx,σ(p, w))]P−.

Also (PS(xσ)P )−(p, w) = [P [Hex,σcave(p, vx,σ(p, w))]P + P [Hex,σvex(p, vx,σ(p, w))]P ]−.

Due to John (1995) we have that Hex,σcave(p, u) is singular in prices because ex,σcave is an actual

expenditure function, thus P [Hex,σcave(p, u)]P = [Hex,σcave(p, u)]. If x is HD0 then Hex,σvex(p, u) is

also singular in p.

In general,

Sx,r(p, w) = (PS(xσ)P )−(p, w) = [Hex,σcave(p, vx,σ(p, w)) + P [Hex,σvex(p, vx,σ(p, w))]P ]−.

The solution xr has Slutsky matrix S(xr)(p, w) = [PHeσ(p, vx,σ(p, w))P ]− = [Hex,σcave(p, vx,σ(p, w))+

P [Hex,σvex(p, vx,σ(p, w))]P ]−.

Then outside of p it will differ in proportion to o(||[Heσcave(p, u)+PHeσvex(p, u)P ]−−[Heσcave(p, u)+

PHeσvex(p, u)P ]−||) = o(||P − P ||).Then we have S(xσ,r)(p, w) − Sx,r(p, w) = o(||P − P ||), and P − P → 0 as p → p which

establishes that if yr ∈ argminy∈R(p)||Sr − S(y)||2 then yr ∈ argminy∈R(p)||Sr − S(y)||2 +

2〈Eν ,−S(y)〉.The moreover statement follows from the fact that in an open ball around p: miny∈R(p)||S(y)−

S||2 = miny∈R(p)[||Sr − S(y)||2 + 2〈Eν ,−S(y)〉] + ||Eσ||2 + ||Eπ||2 + ||Eν ||2, by Proposition 2,

we have that:

miny∈R(p)||S(y)−S||2 = ||Sr−S(y∗)||2 +2〈Eν , Sr−S(y∗)〉+ ||Eσ||2 + ||Eπ||2 + ||Eν ||2 when

p→ p, with y∗ ∈ argminy∈X (p)||Sr−S(y)||2. This holds because, by Proposition 2, y∗ ∈ R(p) is

rational then by the first part of the proof it also solves miny∈R(p)[||Sr−S(y)||2 +2〈Eν ,−S(y)〉]when p→ p.

In conclusion, we have the decomposition miny∈R(p)||S(y) − S||2 = ||Eι||2 + 2〈Eν , Eι〉 +

||Eσ||2 + ||Eπ||2 + ||Eν ||2.

Moreover, we notice that ||S(xr)− S(x)||2 = ||Eσ||2 + ||Eπ||2 + ||Eν ||2 + o(||P − P ||)Thus we conclude miny∈R(p)||S(y)−S||2 = ||E||2 +o(||P −P ||). Notice also o(||pp

p′p−pp′

p′p ||) =

o(||p′ppp′−pp′p′pp′pp′p ||) = o(||p− p||).

10.11 Proof of Proposition 4

Proof. The first part of the statement follows from the Frobenius-Stokes theorem of integrability

(Guggenheimer, 1962).

For the “moreover” part of the statement, given s, we consider the function ws(t), the solution

for that s of the Slutsky Wealth Compensation problem: ws(t) = ws(0)+´ t0∂ps(τ)∂τ

′x(ps(τ), ws(τ))dτ .

We compute the derivative with respect to s, ∂ws(t)∂s :23

23This function is differentiable (by the results given by Guggenheimer (1962)).

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∂ws(t)

∂s=

ˆ t

0

∂ps(τ)

∂τ

′Dpx(ps(τ), ws(τ))

∂ps(τ)

∂sdτ +

ˆ t

0

∂ps(τ)

∂τ

′Dwx(ps(τ), ws(τ))

∂ws(τ)

∂sdτ+

ˆ t

0

∂2ps(τ)

∂s∂τ

x(ps(τ), ws(τ))dτ.

Focusing on ws(1) and using integration by parts on the third term:

ˆ 1

0

∂2ps(τ)

∂s∂τ

x(ps(τ), ws(τ))dτ =∂ps(τ)

∂s

′x(τ)|10 − [

ˆ 1

0

∂ps(τ)

∂s

′[Dpx(ps(τ), ws(τ))]

∂ps(τ)

∂τ+

ˆ 1

0

∂ps(τ)

∂s

′[Dwx(ps(τ), ws(τ))]

∂ws(τ)

∂τdτ ].

Notice that ∂ps(τ)∂s

′x(τ)|10 = p∗(τ)′x(τ)|t0 − p(τ)′x(τ)|t0. By Walras’ law this term vanishes.

Thus:

∂ws(1)

∂s=

ˆ 1

0

∂ps(τ)

∂τ

′[Dpx(ps(τ), ws(τ))−Dpx(ps(τ), ws(τ))′]

∂ps(τ)

∂sdτ+

ˆ 1

0

∂ps(τ)

∂τ

′Dwx(ps(τ), ws(τ))

∂ws(τ)

∂sdτ −

ˆ 1

0

∂ps(τ)

∂s

′[Dwx(ps(τ), ws(τ))]

∂ws(τ)

∂τdτ.

By definition of the price path ps and by Walras’ law, we have ws(τ) = ps(τ)′xs(τ). This

implies:∂ws(τ)∂s = xs(τ)′ ∂ps(τ)∂s + ∂xs(τ)

∂s

′ps(τ) and by the Slutsky compensation equation ∂ws(τ)

∂τ =

x(τ)′ ∂ps(τ)∂τ .

Now, plugging this inside the integral equation above:

ˆ 1

0

∂ps(τ)

∂τ

′Dwx(ps(τ), ws(τ))

∂ws(τ)

∂sdτ −

ˆ 1

0

∂ps(τ)

∂s

′[Dwx(ps(τ), ws(τ))]

∂ws(τ)

∂τdτ =

ˆ 1

0

∂ps(τ)

∂τ

′[Dwx(ps(τ), ws(τ))x(τ)′ − x(τ)Dwx(ps(τ), ws(τ))′]

∂ps(τ)

∂sdτ+

ˆ 1

0

∂ps(τ)

∂s

′[Dwx(ps(τ), ws(τ))][

∂xs(τ)

∂s

′ps(τ)]dτ.

Therefore,∂ws(1)∂s = 2

´ 10∂ps(τ)∂τ

′[Eσ(ps(τ), ws(τ))]∂ps(τ)∂s dτ + 2c(s) with

2c(s) =´ 10∂ps(τ)∂s

′[Dwx(ps(τ), ws(τ))][∂xs(τ)∂s

′ps(τ)]dτ, ∂ws(τ)∂s = xs(τ)′ ∂ps(τ)∂s + ∂xs(τ)

∂s

′ps(τ),

and

Eσ(ps(τ), ws(τ)) = 12 [S(ps(τ), ws(τ))− S(ps(τ), ws(τ))′].

Finally, by integration:

w1(1)− w0(1) =´ 10∂ws(1)∂s ds+

´ 10c(s)ds

w1(1)−w0(1)2 =

´ 10

´ 10∂ps(τ)∂τ

′Eσ(ps(τ), ws(τ))∂ps(τ)∂s dτds+ c.

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10.12 Proof of Proposition 5

Proof. Observe that ∂2ws(t)∂t∂s − cs(t) = 2∂ps(τ)∂s

′Eσ(ps(τ), ws(τ))∂ps(τ)∂τ .

Note that we can write [∂2wxs (t)∂t∂s − cs(t)]

2 = Tr([∂2wxs (t)∂t∂s ][

∂2wxs (t)∂t∂s ]), where the left hand side is

a scalar and thus its trace is itself.

Tr([∂2wxs (t)∂t∂s ][

∂2wxs (t)∂t∂s ]) =

2Tr(∂ps(t)∂t

′Eσ(ps(t), ws(t))

∂ps(t)∂s

∂ps(t)∂t

′Eσ(ps(t), ws(t))

∂ps(t)∂t ).

By the properties of the trace,

Tr([∂2wxs (t)∂t∂s − cs(t)][

∂2wxs (t)∂t∂s − cs(t)]) =

2Tr(2Ex,σ(ps(t), ws(t))∂ps(t)∂t

∂ps(t)∂s

′Ex,σ(ps(t), ws(t))

∂ps(t)∂s

∂ps(t)∂t

′).

Then we can establish the first part of the result:

Tr([∂2wxs (t)∂t∂s − cs(t)][

∂2wxs (t)∂t∂s − cs(t)]) = ||Ex,σ(ps(t), ws(t))

√2∂ps(t)∂t

∂ps(t)∂s

′||2M and

||Ex,σ(ps(t), ws(t))√

2∂ps(t)∂t∂ps(t)∂s

′||2M = ||Ws(t)E

x,σ(ps(t), ws(t))||2M.

Finally, for the second part of the statement we apply Holder’s inequality to conclude that:

[ 12 [w1(1)− w0(1)]− c]2 ≤´ 10

´ 10

[∂ps(τ)∂τ Eσ(ps(τ), ws(τ))∂ps(τ)∂s ]2dτds.

By the first part of the statement, we conclude:´ 10

´ 10

[∂ps(τ)∂τ Eσ(ps(τ), ws(τ))∂ps(τ)∂s ]2dτds =´ 10

´ 10||Ws(t)E

σ(ps(t), ws(t))||2Mdτds = ||Eσ||2WU .

10.13 Proof of Proposition 6

Proof. We start by stating the definition ∂p(t)∂t

′ ∂x(t)∂(t) = ∂p(t)

∂t

′S(p(t), w(t))∂p(t)∂t for (p(t), w(t))t∈[0,1]

such that ∂w(t)∂t = x(p(t), w(t))′ ∂p(t)∂t . By virtue of the decomposition in Theorem 1, we can write

S = Sr +Eσ +Eπ +Eν . Also, we notice that ∂p(t)∂t

′Eσ(t)∂p(t)∂t = 0 for all t ∈ [0, 1] and any price

path because Eσ is a skew-symmetric matrix function and it vanishes in a quadratic form.

This establishes the second statement in the proposition. That is, this implies that: ∂p(t)∂t

′ ∂x(t)∂(t) =

∂p(t)∂t

′[Sr(t) + Eπ(t) + Eν(t)]∂p(t)∂t .

Now, we turn to the first statement. If the compensated law of demand holds for any price

path and Slutsky wealth compensated path, it follows that: ∂p(t)∂t

′[Sσ(t)]∂p(t)∂t ≤ 0 which implies

that Sσ(t) is NSD so Sσ,π is NSD and Eν = 0 must be zero by definition. This also implies that

S is NSD because so is its symmetric part, and by John (1995) (theorem 3) this implies that x

is HD0 which in turn implies that Eπ = 0.

Now, if Eπ = 0 and Eν = 0 then we conclude that S = Sr + Eσ. This implies in turn

that ∂p(t)∂t

′ ∂x(t)∂t = ∂p(t)

∂t

′Sr(p(t), w(t))∂p(t)∂t because Eσ is skew-symmetric. By construction,

∂p(t)∂t

′Sr(p(t), w(t))∂p(t)∂t ≤ 0 so the compensated law of demand holds for any locally compensated

demand path.

10.14 Proof of Proposition 7

Proof. If ∂p(t)∂t

∂x(t)∂t ≥ ∂p(t)

∂t∂y(t)∂t under the assumption Sx,r(p, w) = Sy,r(p, w) it holds that

∂p(t)∂t [Ex,π(t) + Ex,ν(t)]∂p(t)∂t ≥

∂p(t)∂t [Ey,π(t) + Ey,ν(t)]∂p(t)∂t .

Observe that Tr(∂p(t)∂t

′[Ex,π(t)+Ex,ν(t)]∂p(t)∂t

∂p(t)∂t

′[Ex,π(t)+Ex,ν(t)]∂p(t)∂t ) = [∂p(t)∂t [Ex,π(t)+

Ex,ν(t)]∂p(t)∂t ]2 because the expression is a quadratic form, a scalar.

By the cyclical properties of the trace:

Tr(∂p(t)∂t

′[Ex,π(t) + Ex,ν(t)]∂p(t)∂t

∂p(t)∂t

′[Ex,π(t) + Ex,ν(t)]∂p(t)∂t ) =

Tr([Ex,π(t) + Ex,ν(t)]∂p(t)∂t∂p(t)∂t

′[Ex,π(t) + Ex,ν(t)]∂p(t)∂t

∂p(t)∂t

′).

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And we recognize that:

Tr([Ex,π(t) +Ex,ν(t)]∂p(t)∂t∂p(t)∂t

′[Ex,π(t) +Ex,ν(t)]∂p(t)∂t

∂p(t)∂t

′) = ||W (t)[Ex,π(t) +Ex,ν(t)]||2M

where W (t) = ∂p(t)∂t

∂p(t)∂t

′.

By an analogous argument to that in Theorem 1 above, ||W (t)[Ex,π(t) + Ex,ν(t)]||2M =

||W (t)[Ex,π(t)]||2M + ||W (t)Ex,ν(t)]||2M because W (t) = ∂p(t)∂t

∂p(t)∂t

′is a Positive Definite matrix

for all t such that ∂p(t)∂t 6= 0 ∈ RL and W (t)Ex,π(t), W (t)Ex,ν(t) are orthogonal.

This implies that ||Ex,π||2WU + ||Ex,ν ||2WU ≥ ||Ey,π||2WU + ||Ey,ν ||2WU .

Conversely, if ||Ex,π||2WU + ||Ex,ν ||2WU ≥ ||Ey,π||2WU + ||Ey,ν ||2WU and Sx,r(p, w) = Sy,r(p, w)

it follows that [∂p(t)∂t∂x(t)∂t ]2 ≥ [∂p(t)∂t

∂y(t)∂t ]2 and this inequality is preserved under monotone

transformations so that ∂p(t)∂t

∂x(t)∂t ≥

∂p(t)∂t

∂y(t)∂t .

36