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A Complete Characterization of the Implications of Slutzky Symmetry for the Linear, LogLinear, and Semi-Log Incomplete Demand System Models

Roger H. von Haefen, U.S. Bureau of Labor Statistics

Working Paper 332

September 2000

 

The views expressed are those of the author and do not reflect the policies of the U.S. Bureau of Labor Statistics or the

views of other staff members.
A Complete Characterization of the Implications of Slutsky Symmetry

for the Linear, Log-Linear, and Semi-Log

Incomplete Demand System Models

Roger H. von Haefen✝

Research Economist

Division of Price and Index Number Research

U.S. Bureau of Labor Statistics

Room 3105

2 Massachusetts Ave, NE

Washington, DC 20212

Phone – (202) 691-6593

Fax – (202) 691-6583

Von_Haefen_R@bls.gov

First Draft – March 17, 2000

Revised – July 20, 2000

✝ I would like to thank V. Kerry Smith and J. Scott Shonkwiler for helpful comments on earlier drafts of

this paper. All remaining errors are my own responsibility.
1

Abstract

This paper extends previous research by LaFrance [1985, 1986, 1987] by deriving the

necessary parameter restrictions for two additional classes of incomplete demand system models

to have symmetric Slutsky substitution matrices in a local neighborhood of price and income

values. In contrast to LaFrance’s previous research, this paper considers models that treat

expenditures and expenditure shares as the dependent variables in the specified incomplete

demand systems. Along with the eight models considered by LaFrance, the sixteen alternative

specifications considered here represent a complete characterization of the implications of

Slutsky symmetry for the linear, log-linear, and semi-log incomplete demand system models.
2

I. Introduction

In a series of papers, LaFrance [1985, 1986, 1990] derives the necessary restrictions for

the linear, log-linear, and semi-log incomplete demand system specifications to have symmetric

Slutsky matrices in an open neighborhood of relevant price and income levels. These models

employ the quantities demanded by the individual as the dependent variables in the specified

demand equations, share a common linear-in-parameters and additive structure, and have been

extensively used in applied work (e.g., Englin, Boxall, and Watson [1998], Shonkwiler [1999]).

This paper extends LaFrance’s results by examining the implications of Slutsky symmetry for

sixteen additional incomplete demand system models. In contrast to the specifications considered

by LaFrance, these structures treat the individual’s expenditures and expenditure shares on the

goods of interest as the dependent variables. Although complete expenditure and expenditure

share demand systems models are frequently employed in empirical work, incomplete

expenditure and expenditure share systems have received far less use. This paper’s objective is to

introduce to applied researchers a menu of theoretically consistent incomplete demand system

specifications that employ expenditures and expenditure shares as the dependent variables. The

necessary parameter restrictions for the linear, log-linear, and six variations of the semi-log

expenditure and expenditure share models are derived. When closed form solutions exist, the

quasi-indirect utility functions for the restricted demand models are also derived. Along with

LaFrance’s earlier work, the new results presented in this paper give a complete characterization

of the implications of Slutsky symmetry for the linear, log-linear, and semi-log incomplete

demand system models.

The paper is organized as follows. The next section reviews the theory of incomplete

demand systems. Section III presents the necessary parameter restrictions, the implied structure

of the expenditure and expenditure share models, and the structures of the quasi-indirect utility

functions when closed form solutions exist. For completeness, results for the eight models
3

considered by LaFrance are also presented. Because the necessary restrictions for Slutsky

symmetry to hold in an open neighborhood of price and income levels are generally found to be

quite restrictive, the paper concludes with a discussion of the relative merits of imposing Slutsky

symmetry in an open neighborhood versus at a single set of price and income values.

II. Incomplete Demand Systems

Applied researchers are often interested in modeling the demand for a subset of goods

entering an individual’s preference ordering. To consistently model consumption for these goods

within a demand systems framework, the analyst may employ one of three sets of assumptions.

One approach assumes the goods of interest enter consumer preferences through a weakly

separable subfunction. In this case, the analyst models consumption for the goods of interest

conditional on total expenditures allocated to them. Alternatively, the analyst may assume that

the other goods’ prices vary proportionately across individuals and/or time. In this situation, the

other goods can be aggregated into a single Hicksian composite good, and the analyst models the

demand for the goods of interest as functions of their prices, total income, and the composite

good’s price index. A third approach involves the specification of a demand system for the goods

of interest as functions of their own prices, total income, and the other goods’ prices that are

assumed quasi-fixed. This latter strategy falls under the rubric of incomplete demand system

approaches and has been systematically investigated by Epstein [1982] and LaFrance and

Hanemann [1989].

The incomplete demand system framework assumes that consumer demand for a set of n

goods can be represented by the following system of Marshallian demand functions:

x x y i n i i = ( p,q, , β), = 1,..., (1)
4

where i x is the Marshallian consumer’s demand for good i, p is a vector of prices for the n

goods in (1), q is a vector of prices for m other goods whose demands are not explicitly

specified, y is the consumer’s income, and β is a vector of structural parameters. Following

LaFrance [1985, 1986, 1990], p, q, and y are all normalized by π (q) , a homogenous of degree

one price index for the m other goods, to insure the demand equations are homogeneous of degree

zero in prices and income. It is because the analyst models the demand for the n goods in x as

functions of all n + m prices and income that the demand specification in (1) is incomplete.

In principle, the analyst can generate (1) by either: 1) specifying an indirect utility

function and using Roy’s Identity; or (2) specifying the system of incomplete demand equations

directly. With either approach, a significant question for analysts attempting to use (1) to

generate consistent Hicksian welfare measures for a set of price changes is whether the system is

consistent with a rational individual maximizing her utility subject to a linear budget constraint.

This is the classic integrability problem. As noted by LaFrance and Hanemann [1989], there are

at least three distinct concepts of integrability in the incomplete demand systems framework.

This paper employs LaFrance and Hanemann’s concept of weak integrability. This concept

implies that within a local neighborhood of price and income values, there exists a continuous and

increasing preference ordering that both gives rise to and is quasiconcave in x and s, where s is

defined as total expenditures on the m other goods, i.e., = − ∑ =

n

i i i s y 1 p x . Compared to other

concepts of integrability in an incomplete demand systems framework, weak integrability

represents the minimal set of assumptions that allow the analyst to construct exact welfare

measures for changes in p conditional on quasifixed values of q.

Theorem 2 in LaFrance and Hanemann states that an incomplete demand system is

weakly integrable if the following four conditions are satisfied: 1) x is homogenous of degree

zero in prices and income; 2) x is nonnegative, i.e., x ≥ 0 ; 3) expenditures on the n goods
5

included in the incomplete demand system are strictly less than income, i.e., p x y n ∑i=1 i i < ; and

4) the Slutsky substitution matrix, i.e., the n × n matrix whose elements consist of:

x i j n

y

x

p

x

s j

i

j

i

ij , , ∈1,..., ∂

∂

+

∂

∂ = (2)

where i p j ∂x / ∂ and x y ∂ i / ∂ are partial derivatives of the Marshallian demand functions with

respect to price and income, respectively, is symmetric and negative semidefinite. Symmetry is

accomplished if for each good i, j ∈ n , i ≠ j , ij ji s = s , whereas negative semidefiniteness

requires that the Slutsky matrix’s eigenvalues are nonpositive. The normalization of prices and

income by the price index, π (q) , implies that the first condition is satisfied, and the second and

third conditions are innocuous in many applied situations and assumed to hold in an open

neighborhood of prices and income. Thus, the necessary conditions for weak integrability that

imply added structure for (1) are the symmetry and negative semidefiniteness of the Slutsky

matrix.

In a series of papers, LaFrance derives the necessary parameter restrictions for the

Slutsky matrix to be symmetric for eight incomplete demand system specifications – the linear

model [1985], the log-linear or constant elasticity model [1986], and six alternative semi-log

models [1990]. These models or their logarithmic transformations share a common linear-inparameters structure and are additive in their arguments. Table 1 lists the eight demand

specifications LaFrance considers. This paper extends his results by deriving the implications of

Slutsky symmetry for two additional classes of incomplete demand system models. Sixteen

additional specifications are considered that treat either expenditures ( = , > 0 i i i i e p x e ),

expenditure shares ( = / , 0 < < 1 i i i i s p x y s ), or their logarithmic transformations as the

dependent variables. Tables 2 and 3 list the expenditure and expenditure share specifications

considered in this paper.
6

In addition to expanding the menu of specifications from which analysts can choose,

these models may be of interest to applied researchers for at least two additional reasons. Since

Stone’s [1954a, 1954b] pioneering work, it has been common in applied demand analysis for

expenditures, expenditure shares, or transformations of them to be specified as the dependent

variables in the estimated system of equations. Three of the most widely used empirical

specifications, the linear expenditure system (Klein and Rubin [1947-1948]), the Indirect

Translog (Christensen, Jorgenson, and Lau [1975]), and the Almost Ideal Demand System

(Deaton and Muellbauer [1980]), treat expenditures or expenditure shares as the system’s

dependent variables. Furthermore, many of the expenditure and expenditure share structures

explored here may be of interest to researchers estimating dual specifications of generalized

corner solution models (Bockstael, Hanemann, and Strand [1986]), i.e., demand systems that

consistently account for the presence of both interior and corner solutions. At present, empirical

specification of these models proposed by Lee and Pitt [1986] and recently implemented by

Phaneuf [1999] only consider complete or weakly separable demand systems, but these models

can also be estimated within an incomplete demand system framework.1

 Moreover, the

implementation of the dual models proposed by Lee and Pitt depend critically on the existence of

closed form solutions for the implied virtual price functions, i.e., the prices that would drive the

consumer’s demand for the nonconsumed goods to zero (Neary and Roberts [1980]). Because

LaFrance’s (x1)-(x4) specifications and the proposed (e1)-(e4) and (s1)-(s4) structures allow for

corner solutions and have p entering linearly or log-linearly, they can in principle be inverted to

solve for the implied virtual price functions.

For any pair of goods i, j ∈i,..., n;i ≠ j , the Slutsky symmetry restrictions require that

in an open neighborhood of prices and income, the following conditions must hold for the

demand, expenditure, and expenditure share equations, respectively:

 

1

 See von Haefen [2000] for a discussion of some practical issues associated with estimating these models.
7

j

i

j

i

i

j

i

j x

y

x

p

x

x

y

x

p

x

∂

∂

+

∂

∂ = ∂

∂

+

∂

∂

(3)

















∂

∂

+

∂

∂ =

















∂

∂

+

∂

∂

j

i

j

j

i

i j

i

j

i

i

j

i j

e

y

e

p

p

e

p p

e

y

e

p

p

e

p p

1 1

(4)





























∂

∂

+ +

∂

∂ =

































∂

∂

+ +

∂

∂

j

i i

j

j

i

i j

i

j j

i

i

j

i j

ys

y

s

y

s

p

p

s

p p

y ys

y

s

y

s

p

p

s

p p

y (5)

where all derivatives are with respect to the Marshallian demands, expenditures, and expenditure

shares, respectively. Table 4 reports the structure of these restrictions for the eight demand

specifications explored by LaFrance, and Tables 5 and 6 report the restrictions for the sixteen

expenditure and share models from Tables 2 and 3, respectively.

In addition to ascertaining the necessary parameter restrictions implied by Slutsky

symmetry, determining whether the restricted demand systems can be linked to closed form

representations of preferences may be of interest to applied researchers. For example, virtually

every recently proposed method for linking intensive and extensive margins of consumer choice

in a behaviorally consistent framework (e.g., Cameron [1992], Eom and Smith [1994]) assumes

that consumer preferences can be represented by a utility function with a closed form solution.

Without the closed form, these strategies would not be econometrically viable.

As noted by LaFrance and Hanemann [1989], a difficulty with the incomplete demand

system framework is that one cannot recover the complete structure of preferences with respect to

all n + m goods from an n-good demand system that satisfies the conditions for weak

integrability. However, one can recover what Hausman [1981] has called the quasi-indirect

utility function by solving a series of partial differential equations. For the demand, expenditure,

and expenditure share models, this can be accomplished sequentially by first solving one of the

following partial differential equations:
8

( , , ( ), ) ( )

1

1

= ⋅ β ∂

∂ ⋅ x E

p

E

p q (6)

( , , ( ), ) ln

( )

1

1

= ⋅ β ∂

∂ ⋅ e E

p

E

p q (7)

( , , ( ), ) ln

ln ( )

1

1

= ⋅ β ∂

∂ ⋅ s E

p

E

p q (8)

where E(⋅) is the expenditure function evaluated at the baseline utility, U , and good 1 is chosen

arbitrarily with no loss in generality. In some cases, the techniques of differential calculus can be

used to derive closed form solutions for E(⋅) (or ln E(⋅) ) up to a constant of integration,

( , , ) 1

1 p q − K U , where −1 p is the price vector for the n – 1 remaining goods in the specified

incomplete demand system. Because the constant of integration depends on the n – 1 other

prices, one can recover additional information about the structure of the expenditure function by

sequentially solving the following differential equations for i = 2,..., n:

( , , ( ), ) ( ) ( ) ~ 1 = ⋅ β ∂

∂ ⋅ +

∂

∂ ⋅ − x E

p

K

p

E

i

i

i

i

p q (9)

( , , ( ), ) ln

( )

ln

( ) ~ 1 = ⋅ β ∂

∂ ⋅ +

∂

∂ ⋅ − e E

p

K

p

E

i

i

i

i

p q (10)

( , , ( ), ) ln

( )

ln

( ) ~ ln 1 = ⋅ β ∂

∂ ⋅ +

∂

∂ ⋅ − s E

p

K

p

E

i

i

i

i

p q (11)

where ( ) 1 ⋅ Ki− is the constant of integration arising from the evaluation of the first i – 1 partial

differential equations and ( ) ~

E ⋅ is the identified component of the individual’s expenditure

function (i.e., that portion of the expenditure function excluding the constant of integration).

When the analyst has solved all n differential equations, the individual’s expenditure

function is identified up to the constant of integration, K (U ,q) n , which is independent of p.

The fact that the constant of integration is a function of the baseline utility as well as the other m
9

goods’ prices suggests that the analyst cannot identify the full structure of the expenditure

function with respect to all n + m goods from an incomplete demand system. However, one can

obtain the quasi-indirect utility function by treating K (U ,q) n as the quasi-baseline utility and

inverting, i.e.:

( , ) ( , , , ) ~U = Kn U q = φ p q y β (12)

LaFrance and Hanemann formally demonstrate that φ( p,q, y, β ) can be used to consistently

evaluate the welfare implications of one or several price changes for the n goods.

II. Necessary Parameter Restrictions, the Structure of the Restricted Demand Systems, and

the Quasi-Indirect Utility Functions

Tables 7, 8, and 9 reports all possible combinations of parameter restrictions that satisfy

Slutsky symmetry for the demand, expenditure, and share specifications reported in Tables 1, 2,

and 3, respectively.2

 The results in Table 7 were reported originally in LaFrance [1985, 1986,

1990] and are presented here mainly for completeness.3

 For expositional purposes, these tables

employ some simplifying notation used by LaFrance. Let J, K, and N denote index sets satisfying

0/ ⊂ J ⊂ K ⊂ N ≡ {1,2,...,n}, and let ~ denote set differences, e.g., N ~ J ≡{ i ∈ N;i ∉ J }.

Further assume that if J ≠ 0/ , 1∈ J , or if K ≠ 0/ , 1∈ K .

The derivation of these results follows the logic laid out in LaFrance [1985, 1986]. For

each specification, three mutually exclusive and exhaustive types of income effects for goods i

and j are considered: 1) no income effects, i.e., = = 0 i j γ γ ; 2) both goods having income

 

2

 A technical appendix with the derivations for the expenditure and expenditure share parameter restrictions

can be obtained from the author upon request. The appendix also contains derivations for LaFrance’s (x2)-

(x7) models, and the interested reader can consult LaFrance [1985, 1986] for the derivations of the (x1) and

(x8) specifications.
10

effects, i.e., ≠ 0; ≠ 0 i j γ γ ; 2) only one good having income effects, ( ≠ 0; = 0 i j γ γ ). For each

of these possibilities, the necessary parameter restrictions for Slutsky symmetry to hold in an

open neighborhood of relevant prices and income were derived. The derivative properties of the

Slutsky symmetry conditions were used extensively to identify these parameter restrictions.

Because equations (4), (5), and (6) are assumed to hold over a range of price and income values,

they are identities that can be differentiated to generate additional restrictions. Theorem 2 in

LaFrance and Hanemann [1989]) identifies the following two equalities:

i j k n i j

p

s

p

s

k

ji

k

ij ∈ ≠

∂

∂ = ∂∂ , , , 1,..., ; (13)

i j n i j

y

s

y

sij ji ∈ ≠

∂

∂ = ∂

∂ , , 1,..., ; (14)

One should note, however, that these equalities are only a subset of the restrictions that can be

generated by differentiating the Slutsky symmetry identities. In principle, one can multiply

and/or add the same functions of market prices and income to both sides of the Slutsky symmetry

conditions and still preserve the identity relationship. One can then differentiate these modified

Slutsky identities to generate additional equalities that may help to identify the necessary

parameter restrictions. Once the parameter restrictions were identified for the three distinct

income relationships, consistent combinations of the three sets of parameter restrictions were then

determined, and the results are reported in Tables 7, 8, and 9.

 To help clarify the implications of the parameter restrictions reported in Tables 7, 8, and

9, Tables 10, 11, and 12 present the structure of the restricted incomplete demand systems. Not

all cross equation restrictions within sets of good can be represented in the restricted demand

specifications, so these tables should only be interpreted as suggestive of the general structure.

Tables 13, 14, and 15 also present the structure of the quasi-indirect utility functions for all

 3

 A review of the results reported in LaFrance [1990] uncovered minor extensions for the (x5) and (x6)

specifications as well as a few typographical errors for the remaining specifications. The results reported in
11

restricted models with closed form solutions. These tables suggest that roughly one-half of the

restricted models can be linked to closed form representation of consumer preferences.

Collectively, the results reported in Tables 7 through 15 imply that none of the twentyfour structures considered in this paper allow for both flexible income and Marshallian crossprice effects, and some do not allow for either. Perhaps the most general specifications are the

(s3-1) and (s4-1) models that allow for general cross price effects but restrictively assume that all

consumer demand equations are homothetic in income. Collectively, these findings suggest that

strong and in many cases implausible assumptions about the structure of consumer preferences

are required for analysts employing linear, semi-log, and log-linear incomplete demand system

models.

III. Discussion

This paper has extended LaFrance’s earlier research by identifying the necessary

parameter restrictions for systems of linear-in-parameters incomplete expenditure and

expenditure share equations to satisfy the integrability condition of Slutsky symmetry. Although

Slutsky symmetry is a necessary condition for the existence of a rational underlying preference

ordering, it is not sufficient. As noted in Section II, integrability also requires that the Slutsky

matrix must be negative semidefinite, i.e., the matrix’s eigenvalues must be nonpositive.

Imposing this later condition is confounded because the Slutsky matrix’s elements are in general

nonlinear functions of prices, income, and the demand system’s structural parameters. As a

result, the Slutsky matrix may not be negative semidefinite over the full range of relevant price

and income values for the welfare scenarios under consideration.

Existing approaches to imposing curvature restrictions on systems of equations can be

grouped into two broad categories: 1) those that impose negative semidefiniteness of the Slutsky

 

Tables 7, 10, and 13 incorporate these extensions and correct for the errors.
12

matrix at a single point (such as the each individual’s observed prices and income or the sample

average of these values); and 2) those that impose negative semidefiniteness globally over the full

range of relevant price and income values through parameter restrictions (see Pitt and Millimet

[2000] and Diewert and Wales [1987] for discussions of existing approaches). Although the

latter approach is similar in spirit to the strategy for insuring Slutsky symmetry described in this

paper, the former suggests a conceptually different strategy. In principle, the analyst could use

the conditions in Tables 4, 5, and 6 as binding nonlinear constraints evaluated at the observed

market price and income values when estimating the structural parameters of the demand

equations. Although estimation of a system of equations subject to side constraints can be

computationally burdensome, the approach has some precedence in the existing literature

(LaFrance [1991]) and has both advantages and drawbacks. On the one hand, the results

presented in the previous sections strongly suggest that imposing Slutsky symmetry on linear-inparameters demand, expenditure, and expenditure share systems greatly limits the analyst’s

ability to allow for flexible income and Marshallian cross price effects. Imposing symmetry on

the Slutsky matrix at a single point allows the analyst to incorporate these effects while

preserving some degree of theoretical consistency. On the other hand, economists interested in

using the estimated system of equations to evaluate the welfare implications of nonmarginal price

changes may find it troubling that the model is capable of generating only approximate Hicksian

values. Moreover, because symmetry of the Slutsky matrix is not preserved over the entire range

of the relevant price changes, the approximate welfare measures are not independent of the

ordering of the price changes.4

 Although these factors suggest that imposing Slutsky symmetry at

a single point does not strictly dominate the approach pursued in this paper, it may be preferable

with some applications.

 

4

 See LaFrance [1991] for a possible resolution to this problem.
13

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Conditions,” Econometrica, 55(1): 43-68.

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LaFrance, Jeffrey [1985]: “Linear Demand Functions in Theory and Practice,” Journal of

Economic Theory, 37(1): 147-166.

LaFrance, Jeffrey [1986]: “The Structure of Constant Elasticity Demand Models,” American

Journal of Agricultural Economics, 68(3): 543-552.

LaFrance, Jeffrey [1990]: “Incomplete Demand Systems and Semilogarithmic Demand Models,”

Australian Journal of Agricultural Economics, 34(2): 118-131.

LaFrance, Jeffrey [1991]: “Consumer’s Surplus versus Compensating Variation Revisited,”

American Journal of Agricultural Economics, 73(5): 1496-1507.

LaFrance, Jeffrey, and W. Michael Hanemann [1989]: “The Dual Structure of Incomplete

Demand Systems,” American Journal of Agricultural Economics, 71(2): 262-274.
14

Lee, Lung-Fei, and Mark Pitt [1986]: “Microeconomic Demand Systems with Binding

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Measurement,” Working Paper, Division of Price and Index Number Research, Bureau of Labor

Statistics.
Table 1

Incomplete Demand System Models1

(x1) x p yi

n

k

i i ik k = α +∑β + γ =1

(q)

(x2) x p y i

n

k

i i ik k ( ) ln

1

= α +∑β + γ =

q

(x3) x p yi

n

k

i i ik k = α +∑β + γ =1

(q) ln

(x4) x p y i

n

k

i i ik k ( ) ln ln

1

= α +∑β + γ =

q

(x5) 









 = ∑ + =

x p yi

n

k

i i ik k α β γ

1

(q) exp

(x6) i x p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp

(x7) ( ) exp( )

1

x p yi

n

k

i i k

ik α γ β

∏=

= q

(x8) ik i x p y

n

k

i i k

β γ α ∏=

= 1

(q)

1 The (x1) model is considered by LaFrance [1985], the (x2)-(x7) models are considered by LaFrance

[1990] and coincide with models (m3), (m1), (m2), (m4), (m5), and (m6), respectively, and the (x8) model

is studied in LaFrance [1985].
Table 2

Incomplete Expenditure System Models

(e1) e p yi

n

k

i i ik k = α +∑β + γ =1

(q)

(e2) e p y i

n

k

i i ik k ( ) ln

1

= α +∑β + γ =

q

(e3) e p yi

n

k

i i ik k = α +∑β + γ =1

(q) ln

(e4) e p y i

n

k

i i ik k ( ) ln ln

1

= α +∑β + γ =

q

(e5) 









 = ∑ + =

e p yi

n

k

i i ik k α β γ

1

(q) exp

(e6) i e p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp

(e7) ( ) exp( )

1

e p yi

n

k

i i k

ik α γ β

∏=

= q

(e8) ik i e p y

n

k

i i k

β γ α ∏=

= 1

(q)
Table 3

Incomplete Expenditure Share System Models

(s1) s p yi

n

k

i i ik k = α +∑β + γ =1

(q)

(s2) s p y i

n

k

i i ik k ( ) ln

1

= α +∑β + γ =

q

(s3) s p yi

n

k

i i ik k = α +∑β + γ =1

(q) ln

(s4) s p y i

n

k

i i ik k ( ) ln ln

1

= α +∑β + γ =

q

(s5) 









 = ∑ + =

s p yi

n

k

i i ik k α β γ

1

(q) exp

(s6) i s p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp

(s7) ( ) exp( )

1

s p yi

n

k

i i k

ik α γ β

∏=

= q

(s8) ik i s p y

n

k

i i k

β γ α ∏=

= 1

(q)
Table 4

Slutsky Symmetry Conditions for Incomplete Demand System Models

(x1) ji j i ij i j β + γ x = β + γ x

(x2) j

i

i ij

j

ji x

y

x

y

γ β γ β + = +

(x3) i j

j

ij

j i

i

ji x

p

x

p

γ

β

γ

β

+ = +

(x4) j

i

j

ij

i

j

i

ji x

p y

x

p y

β γ β γ + = +

(x5) ji j j i j ij i i i j β x + γ x x = β x + γ x x

(x6) i j

i

i j ij i

j

ji j x x

y

x x x

y

x γ β γ β + = +

(x7) i i i j

j

ij

j j i j

i

ji x x x

p

x x x

p

γ

β

γ

β

+ = +

(x8) i j

i

i

j

ij

i j

j

j

i

ji x x

y

x

p

x x

y

x

p

β γ β γ + = +
Table 5

Slutsky Symmetry Conditions for Incomplete Expenditure System Models

(e1) { } { ij j i j}

i j

ji i j i

i j

p e

p p

p e

p p

β + γ = β + γ

1 1

(e2) 









 = +













+ j

i

ij j

i j

i

j

ji i

i j

e

y

p

p p

e

y

p

p p

γ β γ β 1 1

(e3) { } { } ij i j

i j

ji j i

i j

e

p p

e

p p

β + γ = β + γ

1 1

(e4) 









 = +













+ j

i

ij

i j

i

j

ji

i j

e

p p y

e

p p y

γ β γ β 1 1

(e5) { } { } ij j i i i j

i j

ji i j j i j

i j

p e e e

p p

p e e e

p p

β + γ = β + γ

1 1

(e6) 









 = +













+ i j

i

ij j i

i j

i j

j

ji i j

i j

e e

y

p e

p p

e e

y

p e

p p

γ β γ β 1 1

(e7) { } { ij i i i j}

i j

ji j j i j

i j

e e e

p p

e e e

p p

β + γ = β + γ

1 1

(e8) 









 = +













+ i j

i

ij i

i j

i j

j

ji j

i j

e e

y

e

p p

e e

y

e

p p

γ β γ β 1 1
Table 6

Slutsky Symmetry Conditions for Incomplete Expenditure Share Models

(s1) { } { } ij j i i j

i j

ji i j j i

i j

p s y s

p p

y p s y s

p p

y β + ( + γ ) = β + ( + γ )

(s2) { } { ij j i i j}

i j

ji i j j i

i j

p s s

p p

y p s s

p p

y β + ( + γ ) = β + ( + γ )

(s3) { } { ij i i j}

i j

ji j j i

i j

s y s

p p

y s y s

p p

y β + ( + γ ) = β + ( + γ )

(s4) { } { } ij i i j

i j

ji j j i

i j

s s

p p

y s s

p p

y β + ( + γ ) = β + ( + γ )

(s5) { } { ij j i i i j}

i j

ji i j j i j

i j

p s y s s

p p

y p s y s s

p p

y β + (1+ γ ) = β + (1+ γ )

(s6) { } { } ij j i i i j

i j

ji i j j i j

i j

p s s s

p p

y p s s s

p p

y β + (1+ γ ) = β + (1+ γ )

(s7) { } { ij i i i j}

i j

ji j j i j

i j

s y s s

p p

y s y s s

p p

y β + (1+ γ ) = β + (1+ γ )

(s8) { } { ij i i i j}

i j

ji j j i j

i j

s s s

p p

y s s s

p p

y β + (1+ γ ) = β + (1+ γ )
Table 7

Slutsky Symmetry Restrictions for Incomplete Demand System Models

(x1)1

i N

i j N

i

ij ji

= ∈

= ∈

1b. 0,

1a. , ,

γ

β β

i J

i

i i

i ∈











 2a. ( ) = ( ) + − , 1

1

11

1

1 γ

β

γ

β α

γ

γ α q q

2b. β ij = (γ i / γ 1 )β1 j , i ∈ J , j ∈ N

i N J

i N J

i N J j N

i J

i i

i

ij

i

2f. ( ) / 0, ~

2e. 0, ~

2d. 0, ~ ,

2c. sgn( ) sgn( ) 0,

1 1

1

= − > ∈

= ∈

= ∈ ∈

= ≠ ∈

α β γ

γ

β

γ γ

q

(x2)

i N

ij ji i j N

= ∈

= ∈

1b. 0,

1a. , ,

i γ

β β

i N

i j N

i N

i

ij i j

i i

= ≠ ∈

= ∈

= ∈

2c. sgn( ) sgn( ) 0,

2b. ( / ) , ,

2a. ( ) ( / ) ( ),

1

11

2

1

1 1

γ γ

β γ γ γ β

α q γ γ α q

(x3)

 &

(x4) i N

i j N i j

i

ij

= ∈

= ∈ ≠

1b. 0,

1a. 0, , ,

γ

β

i N

i N

i j N

i

i i

ij

= ≠ ∈

= ∈

= ∈

2c. sgn( ) sgn( ) 0,

2b. ( ) ( / ) ( ),

2a. 0, ,

1

1 1

γ γ

α γ γ α

β

q q

(x5)2,3 i J 1a. αi (q) = (βii / β11 )α1 (q) > 0, ∈

1b. γ i = γ 1 , i ∈ K

i j J 1c. β ij = β jj , , ∈

i N K

i N K

j K i j i N K j N

i J j K J i K J

i i

i

ij

1f. ( ) / 0, ~

1e. 0, ~

 , ; ~ ,

1d. 0, , ~ ; ~ ,

= − 1 1 > ∈

= ∈

∈ ≠ ∈ ∈

= ∈ ∈ ∈

α β γ

γ

β

q

(x6)3

i N

i J

i

i ii

= ∈

= > ∈

1b. ,

1a. ( ) ( / ) ( ) 0,

1

11 1

γ γ

α q β β α q

 ~ , , 

1d. 0, , ~ ;

1c. , ,

i N J j N i j

i J j N J

i j J

ij

ij jj

∈ ∈ ≠

= ∈ ∈

= ∈

β

β β

i J 2a. αi (q) = (βii / β11 )α1 (q) > 0, ∈

2b. γ i = γ 1 , i ∈ K

i N K

i N K

j K i j i N K j N

i J j K J i K J

i j J

i i

i

ij

ij jj

2f. ( ) 0, ~

2e. 1, ~

 , ; ~ ,

2d. 0, , ~ ; ~ ,

2c. , ,

= 1 > ∈

= ∈

∈ ≠ ∈ ∈

= ∈ ∈ ∈

= ∈

α β

γ

β

β β

q

(x7)2,4

i K j N K

i j J i j

i J

ij j

ij jj

ii

i

1c. , , ~

1b. 1 , , ,

0, 1

1 1a. ( ) ( )

1

11

1

= ∈ ∈

= + ∈ ≠

> ∈













+

+ =

β β

β β

β

β α q α q

1d. γ i = γ 1 , i ∈ K

j K i j i N K j N i j

i J j K J i K J ij

∈ ≠ ∈ ∈ ≠

= ∈ ∈ ∈

 , ; ~ , ,

1e. β 0, , ~ ; ~ ,

i N K

i N K

ii

i

1g. 1, ~

1f. 0, ~

= − ∈

= ∈

β

γ

1h. αi(q) = −β1i / γ 1 > 0, i ∈ N ~ K

1

 LaFrance [1985] notes that an additional restriction arising from the negative semi-definiteness of the

Slutsky matrix assumption is 0 11 1 1 β + γ x ≤ for the J subset.

2

 Note that the N~K subset is empty if 0 1 γ = . 3

 For the (x5-1) and (x6-2) restricted models, LaFrance [1990] further decomposes the K~J subset into one

set with 0 = kk β and another with 0 ≠ kk β . Similarly for the (x6-1) restricted model, LaFrance

decomposes the N~J subset into one set with 0 = kk β and another with 0 ≠ kk β .
Table 7 (cont.)

Slutsky Symmetry Restrictions for Incomplete Demand System Models

(x8)4

i N J j N i j

i J j N J

i N

i j J i j

i J

ij

i

ij jj

ii

i

∈ ∈ ≠

= ∈ ∈

= ∈

= + ∈ ≠

> ∈













+

+ =

 ~ , ,

1d. 0, , ~ ;

1c. ,

1b. 1 , , ,

0, 1

1 1a. ( ) ( )

1

11

1

β

γ γ

β β

β

β α q α q

i J ii

i > ∈













+

+ = 0, 1

1 2a. ( ) ( )

11

1 β

β α q α q

i j J i j 2b. β ij = 1+ β jj , , ∈ , ≠

2c. αi (q) = β1i > 0, i ∈ N ~ K

2d. γ i = 0, i ∈ K

2e. γ i = 1,i ∈ N ~ K

i N K j N i j

i K J j K i j

i J j K J

i N K

i K j N K

ij

ii

ij j

∈ ∈ ≠

∈ ∈ ≠

= ∈ ∈

= − ∈

= ∈ ∈

 ~ , ,

 ~ , , ;

2h. 0, , ~ ;

2g. 1, ~

2f. 1 , , ~

β

β

β β

4

 For the (x7-1) and (x8-2) restricted models, LaFrance [1990] further decomposes the K~J subset into one

set with 1 = − kk β and another with 1 ≠ − kk β . Similarly for the (x8-1) restricted model, LaFrance

decomposes the N~J subset into one set with 1 = − kk β and another with 1 ≠ − kk β .
Table 8

Slutsky Symmetry Restrictions for Incomplete Expenditure System Models

(e1) i j N i j 1a. βij = 0, , ∈ , ≠

1b. γ i = 0, i ∈ N

i J 2a. γ i = 1, ∈

i J 2b. β ii = 0, ∈

2c. αi (q) = (γ i / γ 1 )α1 (q), i ∈ K

i K J 2d. γ i ≠ 0,1, ∈ ~

i K j K J 2e. β ij = γ i β jj /(γ j −1), ∈ , ∈ ~

k i j

ik i j jk i j K k N

 ,

2f. ( / ) , , , ,

≠

β = γ γ β ∈ ∈

i N K j N i j 2g. β ij = 0, ∈ ~ , ∈ , ≠

2h. γ i = 0, i ∈ N ~ K

2i. αi(q) = 0, i ∈ N ~ K

2j. β ii = −β1i / γ 1 > 0, i ∈ N ~ K

3a. γ 1 = 1

3b. = 0, i ∈ N,i ≠ 1 i γ

3c. 0, i, j N,i 1, j i,1 βij = ∈ ≠ ≠

3d. = − 1 , i ∈ N,i ≠ 1 β ii β i

3e. ( ) = 0, i ∈ N,i ≠ 1 i α q

(e2) i j N i j 1a. βij = 0, , ∈ , ≠

1b. γ i = 0, i ∈ N

2a. β ij = 0, i, j ∈ N

2b. αi(q) = (γ i / γ 1 )α1 (q), i ∈ N

2c. sgn(γ i) = sgn(γ 1 ) ≠ 0, i ∈ N

(e3) 1a. β ij = β ji , i, j ∈ N

1b. γ i = 0, i ∈ N

i j J 2a. β ij = (γ i / γ 1 )β1 j , , ∈

2b. β ij = 0, i ∈ N ~ J , j ∈ N

i J 2c. sgn(γ i) ≠ 0, ∈

i N J 2d. γ i = 0, ∈ ~

i J

i

i

i

i i

i ∈











 2e. ( ) = ( ) − + , 1 1

1

1 γ

β

γ

β α

γ

γ α q q

i N J 2f. αi (q) = −β1i / γ 1 > 0, ∈ ~

(e4) 1a. βij = β ji , i, j ∈ N

1b. γ i = 0, i ∈ N

2a. αi (q) = (γ i / γ 1 )α1 (q), i ∈ N

2b. ij = ( i j / ) 11 , i, j ∈ N 2 β γ γ γ 1 β

2c. sgn(γ i ) = sgn(γ 1 ) ≠ 0, i ∈ N

(e5) i j N i j 1a. βij = 0, , ∈ , ≠ 1b. γ i = γ 1 , i ∈ N

(e6) i j N i j 1a. βij = 0, , ∈ , ≠ 1b. γ i = γ 1 , i ∈ N

(e7)1

i K j N K

i j J

i J

ij j

ij jj

i ii

1c. , , ~

1b. , ,

1a. ( ) ( / ) ( ) 0,

1

11 1

= ∈ ∈

= ∈

= > ∈

β β

β β

α q β β α q

1d. γ i = γ 1 , i ∈ K

j K i j i N K j N

i J j K J i K J ij

∈ ≠ ∈ ∈

= ∈ ∈ ∈

 , ; ~ ,

1e. β 0, , ~ ; ~ ,

i N K

i N K

i i

i

1g. ( ) / 0, ~

1f. 0, ~

= − 1 1 > ∈

= ∈

α β γ

γ

q

1 Note that the N~K subset is empty if 0 1 γ = .
Table 8 (cont.)

Slutsky Symmetry Restrictions for Incomplete Expenditure System Models

(e8)

i N J j N i j

i J j N J

i N

i j J

i J

ij

i

ij jj

i ii

∈ ∈ ≠

= ∈ ∈

= ∈

= ∈

= > ∈

 ~ , ,

1d. 0, , ~ ;

1c. ,

1b. , ,

1a. ( ) ( / ) ( ) 0,

1

11 1

β

γ γ

β β

α q β β α q

i J 2a. αi (q) = (βii / β11 )α1 (q) > 0, ∈

i j J 2b. β ij = β jj , , ∈

2c. αi (q) = β1i > 0, i ∈ N ~ K

2d. γ i = 0,i ∈ K

2e. γ i = 1,i ∈ N ~ K

i N K j N

i K J j K i j

i J j K J

i K j N K

ij

ij j

∈ ∈

∈ ∈ ≠

= ∈ ∈

= ∈ ∈

 ~ ,

 ~ , , ;

2g. 0, , ~ ;

2f. 1 , , ~

β

β β
Table 9

Slutsky Symmetry Restrictions for Incomplete Expenditure Share System Models

(s1) 1a. γ i = 0, i ∈ N

i j N i j 1b. β ij = 0, , ∈ , ≠

2a. αi (q) = (γ i / γ 1 )α1 (q), i ∈ N

2b. β ij = 0, i, j ∈ N

2c. sgn(γ i ) = sgn(γ 1 ) ≠ 0, i ∈ N

(s2) 1a. γ i = 0, i ∈ N

i j N i j 1b. β ij = 0, , ∈ , ≠

i J 2a. γ i = 1, ∈

i N K

i K J

i

i

2c. 0, ~

2b. 0,1, ~

= ∈

≠ ∈

γ

γ

2d. αi(q) = (γ i / γ 1 )α1 (q), i ∈ K

i J 2e. β ii = 0, ∈

i K j K J i j

j

i jj

ij ∈ ∈ ≠ − = , , ~ , 1

2f. γ

γ β β

k i j

ik i j jk i j K k N

 ,

2g. ( / ) , , , ,

≠

β = γ γ β ∈ ∈

i N K j N i j 2h. β ij = 0, ∈ ~ , ∈ , ≠

2i. αi(q) = 0, i ∈ N ~ K

2j. β ii = −β1i / γ 1 > 0, i ∈ N ~ K

3a. γ 1 = 1

3b. = 0, i ∈ N,i ≠ 1 i γ

3c. 0, i, j N,i 1, j i,1 βij = ∈ ≠ ≠

3d. = − 1 , i ∈ N,i ≠ 1 β ii β i

3e. ( ) = 0, i ∈ N,i ≠ 1 i α q

(s3)

i N

i j N

i

ij ji

= ∈

= ∈

1b. 0,

1a. , ,

γ

β β

2a. αi (q) = (γ i / γ 1 )α1 (q), i ∈ N

2b. ij = ( i j / ) 11 , i, j ∈ N 2 β γ γ γ 1 β

2c. sgn(γ i ) = sgn(γ 1 ) ≠ 0, i ∈ N

(s4)

i N

i j N

i

ij ji

= ∈

= ∈

1b. 0,

1a. , ,

γ

β β

i J

i

i

i

i i

i ∈











 2a. ( ) = − + ( ) , 1

1 1

1

q α q

γ

β

γ

β

γ

γ α

2b. β ij = (γ i / γ 1 )β1 j , i ∈ J , j ∈ N

i N J

i N J j N

i N J

i

ij

i i

2e. 0, ~

2d. 0, ~ ,

2c. ( ) 1 / 1 0, ~

= ∈

= ∈ ∈

= − > ∈

γ

β

α q β γ

i J 2f. sgn(γ i) ≠ 0, ∈

(s5) i j N i j 1a. βij = 0, , ∈ , ≠ 1b. γ i = γ 1 , i ∈ N

(s6) i j N i j 1a. βij = 0, , ∈ , ≠ 1b. γ i = γ 1 , i ∈ N

(s7)

i N

i j J

i J

i

ij jj

i ii

= ∈

= ∈

= > ∈

1c. ,

1b. , ,

1a. ( ) ( / ) ( ) 0,

1

11 1

γ γ

β β

α q β β α q

i N J j N i j

i J j N J ij

∈ ∈ ≠

= ∈ ∈

 ~ , ,

1d. β 0, , ~ ;
Table 9 (cont.)

Slutsky Symmetry Restrictions for Incomplete Expenditure Share System Models

(s8)

i N

i j J

i J

i

ij jj

i ii

= ∈

= ∈

= > ∈

1c. ,

1b. , ,

1a. ( ) ( / ) ( ) 0,

1

11 1

γ γ

β β

α q β β α q

i N J j N i j

i J j N J ij

∈ ∈ ≠

= ∈ ∈

 ~ , ,

1d. β 0, , ~ ;

i j J

i J

ij jj

i ii

= ∈

= > ∈

2b. , ,

2a. ( ) ( / ) ( ) 0, 11 1

β β

α q β β α q

2c. αi (q) = β1i > 0, i ∈ N ~ K

2d. γ i = −1, i ∈ K

2e. γ i = 0, i ∈ N ~ K

2f. β ij = β1 j , i ∈ K, j ∈ N ~ K

i N K j N

i K J j K i j

i J j K J ij

∈ ∈

∈ ∈ ≠

= ∈ ∈

 ~ ,

 ~ , , ;

2g. β 0, , ~ ;
Table 10

Restricted Incomplete Demand System Models

(x1) 1. x p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ,

2.

x i N J

x p y i J

i i

k N

k k

i

i i

i

/ , ~

( ) ,

1 1

1 1

1

1

11

1

1

= − ∈

∈











 = + − + ∑ + ∈

β γ

β γ

γ

β

γ

β α

γ

γ q

(x2) 1. x p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ,

2.

x p y i N

k N

k k

i

i ∈











 = + ∑ + ∈

( ) ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

(x3) 1. xi = αi (q) + β ii ln pi , i ∈ N

2. xi = (γ i / γ 1 )(α1 (q) + γ 1 y), i ∈ N

(x4) 1. xi = αi (q) + β ii ln pi , i ∈ N

2. xi = (γ i / γ 1 )(α1 (q) + γ 1 ln y), i ∈ N

(x5) 1. 1

x i N K

x p p y i K J

x p p y i J

i i

k N K

i i ii i k k

k N K

k k

k J

kk k

ii

i

/ , ~

( ) exp , ~

( ) exp ,

1 1

1

~

1

1

~

1 1

11

= − ∈

∈











 = + +

∈











 = + +

∑

∑ ∑

∈

∈ ∈

β γ

α β β γ

α β β γ β

β

q

q

(x6) 1.

x p y i N J

x p y i J

i i ii i

k J

kk k

ii

i

( ) exp( ) , ~

( ) exp ,

1

1

1

11

= ∈

∈











 = ∑∈

γ

γ

α β

α β β

β

q

q

2.

x y i N K

x p p i K J

x p p i J

i i

k N K

i i ii i k k

k N K

k k

k J

kk k

ii

i

, ~

( ) exp , ~

( ) exp ,

1

~

1

~

1 1

11

= ∈

∈











 = +

∈











 = +

∑

∑ ∑

∈

∈ ∈

β

α β β

α β β β

β

q

q

(x7) 1.1

x p i N K

x p p y i K J

x p p p y i J

i i i

k N K

i i i k

k N K

k

k J

i k

ii

i

ii k

kk k

( / ) , ~

( ) exp( ), ~

exp( ), 1

1 ( )

1

1 1

1

~

1

~

1 1

11

1

1

1

= − ∈

= ∈

∈













+

+ =

−

∈

∈ ∈

− +

∏

∏ ∏

β γ

α γ

γ β

β α

β β

β β

q

q

1 Note that the N~K subset is empty if 0 1 γ = .
Table 10 (cont.)

Restricted Incomplete Demand System Models

(x8) 1.

x p y i N J

x p p y i J

ii

kk

i i i

k J

i k

ii

i

( ) , ~

, 1

1 ( )

1

1 1 1

11

1

= ∈

∈













+

+ = ∏∈

− +

β γ

β γ

α

β

β α

q

q

2.

x p y i N K

x p p i K J

x p p p i J

i i i

k N K

i i i k

k N K

k

k J

i k

ii

i

ii k

kk k

, ~

( ) , ~

, 1

1 ( )

1

1

~

~

1 1

11

1

1

1

= ∈

= ∈

∈













+

+ =

−

∈

∈ ∈

− +

∏

∏ ∏

β

α

β

β α

β β

β β

q

q
Table 11

Restricted Incomplete Expenditure System Models

(e1) 1. ei = αi(q) + β ii pi , i ∈ N

2.

e p i N K

e p p y i K J

e p p y i J

i i i

i

k N K

k k

i

k K

ik k

i

i

k N K

k k

k K i k

i ik k

( / ) , ~

( ) , ~

( ) ,

1 1

~

1

1

1

1

~

1

,

1

= − ∈

= + + + ∈

= + + + ∈

∑ ∑

∑ ∑

∈ ∈

∈ ≠ ∈

β γ

β γ

γ

γ α β

γ

γ

α β β

q

q

3.

, , 1

( )

1 1 1

1 1 1

= − ∈ ≠

= + ∑ + ∈

e p p i N i

e p y

i i i i

k N

k k

β β

α q β

(e2) 1. ei = αi(q) + β ii pi , i ∈ N

2. ei = (γ i / γ 1 )(α1 (q) + γ 1 ln y), i ∈ N

(e3) 1. e p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2.

e i N J

e p y i J

i i

k N

k k

i

i

i

i i

i

/ , ~

( ) ln ,

1 1

1 1

1 1

1

1

= − ∈

∈











 = − + + ∑ + ∈

β γ

β γ

γ

β

γ

β α

γ

γ q

(e4) 1. e p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2.

e p y i N

k N

k k

i

i ∈











 = + ∑ + ∈

( ) ln ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

(e5) 1. ei = αi(q) exp(βii pi + γ 1 y), i ∈ N

(e6) 1. ei = i( ) exp( ii pi ) y 1 , i ∈ N γ α q β

(e7)1 1.

e i N K

e p p y i K J

e p p y i J

i i

k N K

i i i k

k N K

k

k J

i ii k

ii k

kk k

( / ), ~

( ) exp( ), ~

( / ) ( ) exp( ),

1 1

1

~

1

~

11 1

1

1

= − ∈

= ∈

= ∈

∏

∏ ∏

∈

∈ ∈

β γ

α γ

β β α γ

β β

β β

q

q

(e8) 1.

e p y i N J

e p y i J

ii

kk

i i i

k J

i ii k

( ) , ~

( / ) ( ) ,

1

1

11 1

= ∈

= ∏ ∈ ∈

β γ

β γ

α

β β α

q

q

2.

e y i N K

e p p i K J

e p p i J

i i

k N K

i i i k

k N K

k

k J

i ii k

ii k

kk k

, ~

( ) , ~

( / ) ( ) ,

1

~

~

11 1

1

1

= ∈

= ∈

= ∈

∏

∏ ∏

∈

∈ ∈

β

α

β β α

β β

β β

q

q

1

 Note that the N~K subset is empty if 0 1 γ = .
Table 12

Restricted Incomplete Expenditure Share System Models

(s1) 1. si = αi(q) + β ii pi , i ∈ N

2. si = (γ i / γ 1 )(α1 (q) + γ 1 y), i ∈ N

(s2) 1. si = αi(q) + β ii pi , i ∈ N

2.

s p i N K

s p p y i K J

s p p y i J

i i i

i

k N K

k k

i

k K

ik k

i

i

k N K

k k

k K i k

i ik k

( / ) , ~

( ) ln , ~

( ) ln ,

1 1

~

1

1

1

1

~

1

,

1

= − ∈

= + + + ∈

= + + + ∈

∑ ∑

∑ ∑

∈ ∈

∈ ≠ ∈

β γ

β γ

γ

γ α β

γ

γ

α β β

q

q

3.

, , 1

( ) ln

1 1 1

1 1 1

= − ∈ ≠

= + ∑ + ∈

s p p i N i

s p y

i i i i

k N

k k

β β

α q β

(s3) 1. s p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2.

s p y i N

k N

K k

i

i ∈











 = + ∑ + ∈

( ) ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

(s4) 1. s p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2.

s p y i J

k N

k k

i

i

i

i i

i ∈











 = − + + ∑ + ∈

( ) ln ln , 1 1 1

1 1

1

α β γ

γ

β

γ

β

γ

γ q

s i N J i = −β1i / γ 1 , ∈ ~

(s5) 1. si = αi(q) exp(βii pi + γ 1 y), i ∈ N

(s6) 1. si = i( ) exp( ii pi ) y 1 , i ∈ N γ α q β

(s7) 1.

s p y i N J

s p y i J

ii

kk

i i i

k J

i ii k

( ) exp( ), ~

( / ) ( ) exp( ),

1

11 1 1

= ∈

= ∏ ∈ ∈

α γ

β β α γ

β

β

q

q

(s8) 1.

s p y i N J

s p y i J

ii

kk

i i i

k J

i ii k

( ) , ~

( / ) ( ) ,

1

1

11 1

= ∈

= ∏ ∈ ∈

β γ

β γ

α

β β α

q

q

2.

s i N K

s p p y i K J

s p p y i J

i i

k N K

i i i k

k N K

k

k J

i ii k

ii k

kk k

, ~

( ) , ~

( / ) ( ) ,

1

1

~

1

~

11 1

1

1

= ∈

= ∈

= ∈

−

∈

−

∈ ∈

∏

∏ ∏

β

α

β β α

β β

β β

q

q
Table 13

Quasi-Indirect Utility Functions for Incomplete Demand System Models1

Model Restrictions Quasi-Indirect Utility Function(x1)-1 &(x2)-1

∑ ∑∑

∈ ∈∈

=

−

−

k N N k N j

k k kj k j φ

y

y

α

p

β

p

p 2

1 ( p,q, ) (q)

(x1)-2 





−





 

 =

+

∑

+

+

∑ ∈ k∈J

k k

k N

y

y k pk γ

p

γ

β

β

α

γ

φ ( ) exp 1 ( , , )

1

11

1 1

1

p

q

q

(x3)-1 &(x4)-1

∑ ∑

∈ ∈

=

−

−

−

k N N k

k k kk k k φ( p,q, y) y

α (q) p

β

p (ln

p 1)

(x3)-2 





−



 =

+

∑k∈N

k k y

y

γ

p

γ

α

φ exp ( ) ( , , )

1

1 q

p

q

(x5)-1

γ 1 =

0

N

~

K

=

0/ ∑

∑

∑

= ∈ ≠ ∈ ∈

−

− 



 =

−

0

~

0

11 ~

1 exp( ) ( ) ( ) exp ( ) ( , , )

kk kk

k K J

k k

k K J

kk k

kk

k

k J

kk k y

y

p

p

p

β

β

β

α

β

α

β

β

α

φ

q

q

q

p

q

 (x5)-1

γ 1 ≠

0

∑

∑

∑

≠

∈ ∈ ∈

−



 −





−

−

−

=

0

11 ~

1

~

1

1

1 exp( 

( ) exp ( ) exp exp( ) ( , , )

kk 

k K J

k

kk

k

k J

kk k

k N

K

k k p

p

y

y

β

β

β

α

β

β

α

β

γ

γ

φ

q

q

p

q

(x6)-1

γ 1 =

0

∑

∑

∑

=

∈

≠

∈ ∈

−

− 



 =

−

0

~ 0 11 ~

1 exp( ) ( ) ( ) exp ( ) ( , , )

kk kk

k N J

k k

k N J

kk k

kk

k

k J

kk k y

y

p

p

p

β

β

β

α

β

α

β

β

α

φ

q

q

q

p

q

(x6)-1

γ 1 = 1 ∑

∑

∑

=

∈

≠

∈ ∈

−

− 



 =

−

0

~

0

11 ~

1 exp( ) ( ) ( ) exp ( ) ( , , ) ln

kk kk

k N J

k k

k N J

kk k

kk

k

k J

kk k y

y

p

p

p

β

β

β

α

β

α

β

β

α

φ

q

q

q

p

q
Table 13 (cont.)

Quasi-Indirect Utility Functions for Incomplete Demand System Models

Model Restrictions Quasi-Indirect Utility Function(x6)-1 γ 1 ≠ 1,0 ∑ ∑ ∑

=

∈

≠

∈ ∈

−

− −

  − − =

0

~

0

11 ~

1

1

1

exp( ) ( ) ( ) exp ( ) 1 ( , , ) 1

kk kk

k N J

k k

k N J

kk k

kk

k

k J

kk k p p p y y

β β

γ

β α β α β β α γ φ q q q p q

(x6)-2 ∑ ∑ ∑ ∑

=

∈

≠

∈ ∈ ∈

− −

  −  = −

0

~

0

11 ~

1

~

1 exp( ) 

( ) exp ( ) ( , , ) exp

kk kk

k K 

k K J

kk k

kk

k

k J

kk k

k N K

k k y y p p p 

β β

β 

β α β β α φ β 

q q p q

(x7)-1 γ 1 = 0 & N ~ K = 0/ ∏ ∑ ∑ ∈ ∈ + ∈ + − + − + = − k K J k k N K k k kk k k J k y y p p p kk kk ~ ~ 1 1 11 1 ( )ln 1 ( ) 1 ( ) ( , , ) q q q p q α β α β α φ β β

(x7)-1 γ 1 ≠ 0 

k K J k k K J k kk

k

k J

k

k N K

k p p p 

y

y

kk kk

k kk kk ( )ln 1 ( ) 1 exp( ) ( ) ( , , )

1

~

1

~

1 1

11

1

1 ~

1 1 ∏ ∏ ∑ ∑

=−

∈

≠−

∈

+

∈

+

∈

−

−

+

−

+

−

− −

=

β β

β β β α

β

α

β

α

γ

γ φ q q q p q

(x8)-1 γ 1 = 1 ∏ ∑ ∑

≠−

∈

=−

∈

+

∈

+

−

+

−

+

= −

1

~

1

~

1 1

11

1 ( )ln 1 ( ) 1 ( ) ( , , ) ln

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k y y p p p

β β

β β α

β

α

β

α φ q q q p q

(x8)-1 γ 1 ≠ 1 ∏ ∑ ∑

≠−

∈

=−

∈

+

∈

+

−

−

+

−

+

−

−

=

1

~

1

~

1 1

11

1

1

1

( )ln 1 ( ) 1 ( ) 1 ( , , ) 1

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k p p p y y

β β

β β

γ

α

β

α

β

α

γ

φ q q q p q

(x8)-2 ∏ ∏ ∑ ∑

≠−

∈

=−

∈

+

∈

+

∈

−

−

+

−

+

= −

1

~

1

~

1 1

11

1

~

( )ln 1 ( ) 1 ( ) ( , , ) 1

kk kk

k kk kk

k K J

k

k K J

k k

kk

k

k J

k

k N K

k y y p p p p

β β

β β β α

β

α

β

α φ q q q p q

1 The results reported here correct for typographical errors found in LaFrance [1990].
Table 14

Quasi-Indirect Utility Functions for Restricted Incomplete Expenditure System Models

Model Restrictions Quasi-Indirect Utility Function(e1)-1 &(e2)-1

∑ ∑

∈ ∈

= − −

k N

kk k

k N

k k φ( p,q, y) y α (q)ln p β p

(e3)-1 &(e4)-1

∑ ∑∑

∈ ∈ ∈

= − −

k N N j

kj k j

k N

k k ( , , y) y ( )ln p ln p ln p 21 φ p q α q β

(e7)-1 γ 1 = 0 N ~ K = 0/ ∏ ∑ ∑

≠

∈

=

∈ ∈

= − − −

0

~

0

11 ~

1 ( )ln ( ) ( ) ( , , )

kk kk

kk kk

k K J

k

k K J

k k

kk

k

k J

k y y p p p

β β

β β α

β

α

β

α φ q q q p q

(e7)-1 γ 1 ≠ 0 k k K J k k K J k kk

k

k J

k

k N K

k p p p p

y

y

kk kk

k kk kk ( )ln exp( ) ( ) ( ) ( , , )

0

~

0

11 ~

1

1 ~

1 1 ∏ ∏ ∑ ∑

=

∈

≠

∈ ∈ ∈

−

− − −

− −

=

β β

β β β α

β

α

β

α

γ

γ φ q q q p q

(e8)-1 γ 1 = 1 ∏ ∑ ∑

≠

∈

=

∈ ∈

= − − −

0

~

0

11 ~

1 ( )ln ( ) ( ) ( , , ) ln

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k y y p p p

β β

β β α

β

α

β

α φ q q q p q

(e8)-1 γ 1 ≠ 1 ∏ ∑ ∑

≠

∈

=

∈ ∈

−

− − −

−

=

0

~

0

11 ~

1

1

1

( )ln ( ) ( ) 1 ( , , ) 1

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k p p p y y

β β

β β

γ

α

β

α

β

α

γ

φ q q q p q

(e8)-2 ∏ ∏ ∑ ∑

≠

∈

=

∈ ∈ ∈

−

= − − −

0

~

0

11 ~

1

~

( )ln ( ) ( ) ( , , ) 1

kk kk

k kk kk

k K J

k

k K J

k k

kk

k

k J

k

k N K

k y y p p p p

β β

β β β α

β

α

β

α φ q q q p q
Table 15

Quasi-Indirect Utility Functions for Restricted Incomplete Expenditure System Models

Model Restrictions Quasi-Indirect Utility Function(s1)-1 &(s2)-1   = ∏ − ∑ ∈ ∈ − k N kk k k N k y y p p k φ β α ( , , ) exp (q) p q

(s3)-1 &(s4)-1

∑ ∑∑

∈ ∈ ∈

= − −

k N N j

kj k j

k N

k k ( , , y) ln y ( )ln p ln p ln p 21 φ p q α q β

(s8)-1 γ 1 = 0 ∏ ∑ ∑

≠

∈

=

∈ ∈

= − − −

0

~

0

11 ~

1 ( )ln ( ) ( ) ( , , ) ln

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k y y p p p

β β

β β α

β

α

β

α φ q q q p q

(s8)-1 γ 1 ≠ 0 ∏ ∑ ∑

≠

∈

=

∈ ∈

−

− − −

−

=

0

~

0

11 ~

1

1

( )ln ( ) ( ) ( , , ) 1

kk kk

kk kk

k N J

k

k N J

k k

kk

k

k J

k p p p y y

β β

β β

γ

α

β

α

β

α

γ

φ q q q p q

(s8)-2 ∏ ∏ ∑ ∑

≠

∈

=

∈ ∈ ∈

−

= − − −

0

~

0

11 ~

1

~

( )ln ( ) ( ) ( , , ) 1

kk kk

k kk kk

k K J

k

k K J

k k

kk

k

k J

k

k N K

k y y p p p p

β β

β β β α

β

α

β

α φ q q q p q
Technical Appendix

This appendix derives the necessary parameter restrictions for Slutsky symmetry to hold in an

open neighborhood around observed prices and income. The approach employed is similar to LaFrance

[1985, 1986]. For each of the 24 models, three mutually exclusive and exhaustive cases with alternative

income effects for goods i and j (i ≠ j ) are considered: 1) no income effects, i.e., = = 0 i j γ γ ; 2) both

goods having income effects, i.e., ≠ 0; ≠ 0 i j γ γ ; 2) only one good having income effects,

( ≠ 0; = 0 i j γ γ ). For each of these possibilities, the necessary parameter restrictions for Slutsky

symmetry to hold regardless of prices and income were derived. Restrictions implied by the derivative

properties of the Slutsky symmetry conditions were used extensively for this task. Once the parameter

restrictions were identified for the three distinct income relationships, consistent combinations of the three

sets of parameter restrictions were then determined.

1. The (x1) Model

Consider the (x1) unrestricted model specification:

x p yi

n

k

i i ik k = α +∑β + γ =1

(q) (x1)

The implied Slutsky symmetry conditions are:

ji j i ij i j β + γ x = β + γ x (1)

See LaFrance [1985] for the derivation of the necessary parameter restrictions.
2

2. The (x2) Model

Consider the (x2) unrestricted model specification:

x p y i

n

k

i i ik k ( ) ln

1

= α +∑β + γ =

q (x2)

The implied Slutsky symmetry conditions are:

j

i

i ij

j

ji x

y

x

y

γ β γ β + = + (1)

The derivative of (1) with respect to k p , k = 1,..., N, implies the following restriction:

j ik i β jk γ β = γ (2)

The derivative of (1) with respect to y implies the following restriction:

j i i j γ x = γ x (3)

Case I. = = 0 i j γ γ

• (1) implies:

β ji = βij (4)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies:

jk j i βik β = (γ / γ ) , ∀ k (5)

• (3) and (5) together imply:

(q) ( / ) (q) j j i αi α = γ γ (6)

• Plugging (5) and (6) into (1) and simplifying implies:

β ij = β ji (7)
3

• One can combine (5) and (7) as:

ij i j k β kk β (γ γ / γ )

2 = , ∀ k (8)

• (6) and (8) jointly imply that:

sgn( ) = sgn( ) ≠ 0 i j γ γ (9)

• Thus, (6), (8), and (9) are the necessary parameter restrictions.

Case III. ≠ 0; = 0 i j γ γ

• (3) implies this case is only possible if = 0 i γ , a contradiction.

The restricted model specification takes the form:

1. x p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ,

2. x p y i N

k N

k k

i

i ∈











 = + ∑ + ∈

( ) ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

3. The (x3) Model

Consider the (x3) unrestricted model specification:

x p yi

n

k

i i ik k = α +∑β + γ =1

(q) ln (x3)

The implied Slutsky symmetry conditions are:

i j

j

ij

j i

i

ji x

p

x

p

γ

β

γ

β

+ = + (1)

The derivative of (1) with respect to j p implies:

ij j i jj j ij − β / p + γ β = γ β (2)
4

Case I. = = 0 i j γ γ

• (1) and (2) are only satisfied if:

= = 0 β ij β ji (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) holds in general only if:

k β ik = β jk = 0, ∀ (4)

• (4) and (1) imply:

(q) ( / ) (q) j j i αi α = γ γ (5)

which further implies:

sgn( ) = sgn( ) ≠ 0 i j γ γ (6)

Case III. ≠ 0; = 0 i j γ γ

• (2) implies the restriction in (4), which with (1) implies = 0 i γ , a contradiction.

The restricted model specification takes the form:

1. xi = αi (q) + β ii ln pi , i ∈ N

2. xi = (γ i / γ 1 )(α1 (q) + γ 1 y), i ∈ N

4. The (x4) Model

Consider the (x4) unrestricted model specification:
5

x p y i

n

k

i i ik k ( ) ln ln

1

= α +∑β + γ =

q (x4)

The implied Slutsky symmetry conditions are:

j

i

j

ij

i

j

i

ji x

p y

x

p y

β γ β γ + = + (1)

The derivative of (1) with respect to y implies:

j i i j γ x = γ x (2)

Case I. = = 0 i j γ γ

• (1) is satisfied only if:

= = 0 β ij β ji (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• Plugging j j i i x = (γ / γ )x from (2) into (1) implies (3). (3) and (2) along with the structure

of (x4) imply the following three restrictions:

k β ik = β jk = 0, ∀ (4)

(q) ( / ) (q) j j i αi α = γ γ (5)

sgn( ) = sgn( ) ≠ 0 i j γ γ (6)

Case III. ≠ 0; = 0 i j γ γ

• (2) implies = 0 i γ , a contradiction.

The restricted model specification takes the form:

1. xi = αi (q) + β ii ln pi , i ∈ N

2. xi = (γ i / γ 1 )(α1 (q) + γ 1 ln y), i ∈ N
6

5. The (x5) Model

Consider the (x5) unrestricted model specification:











 = ∑ + =

x p yi

n

k

i i ik k α β γ

1

(q) exp (x5)

The implied Slutsky symmetry conditions are:

ji j j i j ij i i i j β x + γ x x = β x + γ x x (1)

The derivative of (1) with respect to y implies:

i ij j ji γ s = γ s (2)

The derivative of (1) with respect to k p , k=1,...,N, implies:

( ) ( ) ik ij j i j jk ji i i j β s − γ x x = β s −γ x x (3)

Case I. = = 0 i j γ γ

• (1) implies ji j ij i β x = β x which is satisfied only if:

= = 0 β ij β ji (4)

or:

β ik = β jk = β kk , ∀ k (5)

(q) ( / ) (q) αi = βii β jj α j (6)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies i j γ = γ and this case collapses into Case I above.

Case III. ≠ 0; = 0 i j γ γ
7

• (2) implies = 0 ij s . (1) implies that:

j ij i x = −β / γ (8)

The restricted model specification takes the form:

1.

x i N K

x p p y i K J

x p p y i J

i i

k N K

i i ii i k k

k N K

k k

k J

kk k

ii

i

/ , ~

( ) exp , ~

( ) exp ,

1 1

1

~

1

1

~

1 1

11

= − ∈

∈











 = + +

∈











 = + +

∑

∑ ∑

∈

∈ ∈

β γ

α β β γ

α β β γ β

β

q

q

Note that the subset N ~ K is empty if 1 γ = 0.

6. The (x6) Model

Consider the (x6) unrestricted model specification:

i x p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp (x6)

The implied Slutsky symmetry conditions are:

i j

i

i j ij i

j

ji j x x

y

x x x

y

x γ β γ β + = + (1)

The derivative of (1) with respect to y implies:

(s x x / y) (s x x / y) i ij − i j = j ji − i j γ γ (2)

Case I. = = 0 i j γ γ

• (1) simplifies to ji j ij i β x = β x . As with the (x5) model, this condition is satisfied only if:
8

= = 0 β ij β ji (3)

• or:

β ik = β jk = β kk , ∀ k (4)

(q) ( / ) (q) αi = βii β jj α j (5)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies i j γ = γ or s s x x y ij ji i j = = / , but the later condition is only satisfied if

= = 1 i j γ γ and = = 0 β ij β ji . Thus, the following condition must hold:

i j γ = γ (6)

• (6) implies ji j ij i β x = β x , and thus either the conditions in (3) or (4) and (5) must be

satisfied.

Case III. ≠ 0; = 0 i j γ γ

• (2) implies that s x x y ij i j = / , which when plugged back into (1) implies:

x y i = β ji (7)

The restricted model specification takes the form:

1.

x p y i N J

x p y i J

i i ii i

k J

kk k

ii

i

( ) exp( ) , ~

( ) exp ,

1

1

1

11

= ∈

∈











 = ∑∈

γ

γ

α β

α β β

β

q

q

2.

x y i N K

x p p i K J

x p p i J

i i

k N K

i i ii i k k

k N K

k k

k J

kk k

ii

i

, ~

( ) exp , ~

( ) exp ,

1

~

1

~

1 1

11

= ∈

∈











 = +

∈











 = +

∑

∑ ∑

∈

∈ ∈

β

α β β

α β β β

β

q

q
9

7. The (x7) Model

Consider the (x7) unrestricted model specification:

( ) exp( )

1

x p yi

n

k

i i k

ik α γ β

∏=

= q (x7)

The Slutsky symmetry conditions are:

i i i j

j

ij

j j i j

i

ji x x x

p

x x x

p

γ

β

γ

β

+ = + (1)

The derivative of (1) with respect to y implies:

j ij i ji γ s = γ s (2)

The derivative of (1) with respect to p k i j k , ≠ , , implies:

( ) ( ) jk ji i i j ik ij j i j β s − γ x x = β s − γ x x (3)

Case I. = = 0 i j γ γ

• (1) simplifies to ji j j ij i i β p x = β p x , which is satisfied if:

= = 0 β ij β ji (4)

• It can also be shown that (1) is satisfied if:

k i j l i j k l β kl = 1+ β ll , = , ; = , ; ≠ (5)

β jk = βik , ∀ k; k ≠ i, j (6)

( ) 1

1 (q) j q

jj

ii

i α

β

β α

+

+ = (7)

Case II. ≠ 0; ≠ 0 i j γ γ
10

• (2) implies that i j γ = γ , and with this restriction the case collapses into Case I above.

Case III. ≠ 0; = 0 i j γ γ

• (2) implies that = 0 ji s , which is satisfied only if:

j ij i j x = −(β / γ )/ p (8)

The restricted model specification takes the form:

1.

x p i N K

x p p y i K J

x p p p y i J

i i i

k N K

i i i k

k N K

k

k J

i k

ii

i

ii k

kk k

( / ) , ~

( ) exp( ), ~

exp( ), 1

1 ( )

1

1 1

1

~

1

~

1 1

11

1

1

1

= − ∈

= ∈

∈













+

+ =

−

∈

∈ ∈

− +

∏

∏ ∏

β γ

α γ

γ β

β α

β β

β β

q

q

Note that the N ~ K set must be empty if γ 1 = 0 .

8. The (x8) Model

Consider the (x8) unrestricted model specification:

ik i x p y

n

k

i i k

β γ α ∏=

= 1

(q) (x8)

The Slutsky symmetry conditions are:

i j

i

i

j

ij

i j

j

j

i

ji x x

y

x

p

x x

y

x

p

β γ β γ + = + (1)

See LaFrance [1986] for the derivation of the necessary parameter restrictions.
11

9. The (e1) Model

Consider the (e1) unrestricted model specification:

∑=

= + +

N

k

i i ik k i e p y

1

α (q) β γ (e1)

The implied Slutsky symmetry conditions are:

{ } { ij j i j}

i j

ji i j i

i j

p e

p p

p e

p p

β + γ = β + γ

1 1

(1)

The derivative of (1) with respect to i p implies:

i ji ji j βii γ β = β + γ (2)

The derivative of (1) with respect to k p , k ≠ i, j , implies:

j ik i β jk γ β = γ (3)

Case I. = = 0 i j γ γ

• (1) and (2) imply:

= = 0 β ij β ji (4)

Case II. ≠ 0; ≠ 0 i j γ γ

• (3) implies:

ik i j β jk β = (γ / γ ) , ∀ k; k ≠ i, j (5)

• For (2) to hold it must be the case that:

= 0 β ii if = 1 i γ (6)

−1 =

i

j ii

ji γ

γ β β if ≠ 1 i γ (7)

Thus, (5), (6), and (7) are the necessary parameter restrictions.
12

Case III. ≠ 0; = 0 i j γ γ

• (1) implies j ij i j ji i i e = −(β / γ ) p + (β / γ ) p . Two possibilities are implied by this structure:

= 1 i γ & β jj = −β ij ⇒ j ij j ji i e = −β p + β p (9)

≠ 1 i γ & = 0 β ji ⇒ j ij i j e = −(β / γ ) p (10)

The restricted model specification takes the form:

1. ei = αi(q) + β ii pi , i ∈ N

2.

e p i N K

e p p y i K J

e p p y i J

i i i

i

k N K

k k

i

k K

ik k

i

i

k N K

k k

k K i k

i ik k

( / ) , ~

( ) , ~

( ) ,

1 1

~

1

1

1

1

~

1

,

1

= − ∈

= + + + ∈

= + + + ∈

∑ ∑

∑ ∑

∈ ∈

∈ ≠ ∈

β γ

β γ

γ

γ α β

γ

γ

α β β

q

q

3.

, , 1

( )

1 1 1

1 1 1

= − ∈ ≠

= + ∑ + ∈

e p p i N i

e p y

i i i i

k N

k k

β β

α q β

10. The (e2) Model

Consider the (e2) unrestricted model specification:

∑=

= + +

N

k

i i ik k i e p y

1

α (q) β γ ln (e2)

The implied Slutsky symmetry conditions are:











 = +













+ j

i

ij j

i j

i

j

ji i

i j

e

y

p

p p

e

y

p

p p

γ β γ β 1 1

(1)

The derivative of (1) with respect to j p implies:
13

y y j ij ij i jj γ β / = β + γ β / (2)

The derivative of (1) with respect to k p , k ≠ i, j implies:

j ik i β jk γ β = γ (3)

Case I. = = 0 i j γ γ

• (2) implies:

= = 0 β ij β ji (4)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies (4) must hold. (1), (2), and (4) imply:

(q) ( / ) (q) i i j α j α = γ γ (5)

0, k β ik = β jk = ∀ (6)

sgn( ) = sgn( ) ≠ 0 i j γ γ (7)

Case III. ≠ 0; = 0 i j γ γ

• (1) implies ( ) ij j ji i

j

i p p y e β β

γ = − , which is inconsistent with (e2).

The restricted model specification implies:

1. ei = αi(q) + β ii pi , i ∈ N

2. ei = (γ i / γ 1 )(α1 (q) + γ 1 ln y), i ∈ N

11. The (e3) Model

Consider the (e3) unrestricted model specification:
14

∑=

= + +

N

j

j j jk k j e p y

1

α (q) β ln γ (e3)

The implied Slutsky symmetry conditions are:

{ } { } ij i j

i j

ji j i

i j

e

p p

e

p p

β + γ = β + γ

1 1

(1)

The derivative of (1) with respect to k p , k=1,..., N, implies:

j ik i β jk γ β = γ , ∀ i, j, k (2)

Case I. = = 0 i j γ γ

• (1) implies:

β ji = βij (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies:

jk

j

i

ik β

γ

γ β = , ∀ k (4)

• Plugging (4) back into (1) implies the following restriction:

− + (q) + (q) = 0 ji ij j i i

α j β β γ α γ (5)

Case III. ≠ 0; = 0 i j γ γ

• (1) simplifies to i ij j ji j e = β / γ − β / γ . To be consist with (e3), it must be the case that:

= 0 β ik , ∀k (6)

i ji j α (q) = −β / γ (7)

The restricted model specification takes the form:

1. e p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,
15

2.

e i N J

e p y i J

i i

k N

k k

i

i

i

i i

i

/ , ~

( ) ln ,

1 1

1 1

1 1

1

1

= − ∈

∈











 = − + + ∑ + ∈

β γ

β γ

γ

β

γ

β α

γ

γ q

12. The (e4) Model

Consider the (e4) unrestricted model specification:

∑=

= + +

N

k

i i ik k i e p y

1

α (q) β ln γ ln (e4)

The implied Slutsky symmetry conditions are:











 = +













+ j

i

ij

i j

i

j

ji

i j

e

p p y

e

p p y

γ β γ β 1 1

(1)

The derivative of (1) with respect to y implies:

i j j i γ e = γ e (2)

The derivative of (1) with respect to k p , k=1,..., N, implies:

j ik i β jk γ β = γ , ∀ k (3)

Case I. = = 0 i j γ γ

• (1) implies:

β ji = βij (4)

Case II. ≠ 0; ≠ 0 i j γ γ

• Plugging (2) into (1) and simplifying implies (4). (4) and (3) together imply:

kk

k

i j

ij β

γ

γ γ β 2 = , ∀ k (5)
16

sgn( ) = sgn( ) ≠ 0 i j γ γ (6)

• Plugging (5) back into (2) then implies:

(q) (q) j

j

i

i α

γ

γ α = (7)

Case III. ≠ 0; = 0 i j γ γ

• (2) implies this case is not possible.

The restricted model specification implies:

1. e p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2. e p y i N

k N

k k

i

i ∈











 = + ∑ + ∈

( ) ln ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

13. The (e5) Model

Consider the (e5) unrestricted model specification:











 = ∑ + =

e p yi

n

k

i i ik k α β γ

1

(q) exp (e5)

The Slutsky symmetry conditions are:

{ } { } ij j i i i j

i j

ji i j j i j

i j

p e e e

p p

p e e e

p p

β + γ = β + γ

1 1

(1)

The derivative of (1) with respect to y implies:

j ji i ij γ s = γ s (2)
17

Case I. = = 0 i j γ γ

• (1) simplifies to ji i j ij j i β p e = β p e , which is not in general satisfied unless:

β ij = β ji (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies:

i j γ = γ (4)

• (4) implies that (1) simplifies to ji i j ij j i β p e = β p e , and thus (3) must also be satisfied.

Case III. ≠ 0; = 0 i j γ γ

• (2) implies this case is not possible.

The restricted model specification takes the form:

1. ei = αi(q) exp(βii pi + γ 1 y), i ∈ N

14. The (e6) Model

Consider the (e6) unrestricted model specification:

i e p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp (e6)

The Slutsky symmetry conditions are:











 = +













+ i j

i

ij j i

i j

i j

j

ji i j

i j

e e

y

p e

p p

e e

y

p e

p p

γ β γ β 1 1

(1)

The derivative of (1) with respect to y implies:
18















 = − 













 − p p y

e e

s

p p y

e e

s

j i

j i

i ij

j i

j i

j ji γ γ (2)

Case I. = = 0 i j γ γ

• (1) simplifies to ji i j ij j i β p e = β p e , which is not in general satisfied unless:

β ji = βij (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (1) is not in general satisfied unless (3) and the following condition are satisfied:

i j γ = γ (4)

Case III. ≠ 0; = 0 i j γ γ

• (2) implies that s e e /( p p y) ij = j i j i . This restriction along with (1) implies e p y i = β ji i ,

which is inconsistent with (e6).

The restricted model specification implies:

1. ei = i( ) exp( ii pi ) y 1 , i ∈ N γ α q β

15. The (e7) Model

Consider the (e7) unrestricted model specification:

( ) exp( )

1

e p yi

n

k

i i k

ik α γ β

∏=

= q (e7)

The Slutsky symmetry conditions are:

{ } { ij i i i j}

i j

ji j j i j

i j

e e e

p p

e e e

p p

β + γ = β + γ

1 1

(1)
19

The derivative of (1) with respect to y implies:

j ji i ij γ s = γ s (2)

Case I. = = 0 i j γ γ

• In this case, (1) simplifies to ji j ij i β e = β e , which is satisfied only if:

= = 0 β ij β ji (3)

or:

k β jk = βik , ∀ (4)

(q) (q) j

jj

ii

i α

β

β α = (5)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies:

i j γ = γ (6)

• Using (6), (1) simplifies to ji j ij i β e = β e , which is satisfied only if (3) or (4) and (5) are

satisfied.

Case III. ≠ 0; = 0 i j γ γ

• (2) implies this case is not possible unless = 0 ij s ,

which holds only if:

j ij i e = −β / γ (7)

The restricted model specification takes the form:
20

1.

e i N K

e p p y i K J

e p p y i J

i i

k N K

i i i k

k N K

k

k J

i ii k

ii k

kk k

( / ), ~

( ) exp( ), ~

( / ) ( ) exp( ),

1 1

1

~

1

~

11 1

1

1

= − ∈

= ∈

= ∈

∏

∏ ∏

∈

∈ ∈

β γ

α γ

β β α γ

β β

β β

q

q

16. The (e8) Model

Consider the (e8) unrestricted model specification:

ik i e p y

n

k

i i k

β γ α ∏=

= 1

(q) (e8)

The Slutsky symmetry conditions are:











 = +













+ i j

i

ij i

i j

i j

j

ji j

i j

e e

y

e

p p

e e

y

e

p p

γ β γ β 1 1

(1)

This model is equivalent to (x8). See LaFrance [1986] for the derivation of the necessary parameter

restrictions.

17. The (s1) Model

Consider the (s1) unrestricted model specification:

s p yi

n

k

i i ik k = α +∑β + γ =1

(q) (s1)

The implied Slutsky symmetry conditions are:

{ } { } ij j i i j

i j

ji i j j i

i j

p s y s

p p

y p s y s

p p

y β + ( + γ ) = β + ( + γ ) (1)
21

Case I. = = 0 i j γ γ

• (1) simplifies in this case to ji i ij j β p = β p , which is satisfied only if:

= = 0 β ij β ji (2)

Case II. ≠ 0; ≠ 0 i j γ γ

• (1) simplifies to ji i j i ij j i j β p + γ ys = β p + γ ys which when differentiated with respect to y

implies j i i j γ s = γ s , and when differentiated with respect to i p implies

ji i j ii β (1− γ y) = −γ yβ . These two conditions hold in general only if:

k i j ik jk β = β = 0, = , (3)

Case III. ≠ 0; = 0 i j γ γ

• (1) in this case simplifies to s ( p p )/( y) i ij j ji i j = β − β γ , which is inconsistent with the

structure of (s1).

The restricted model specification takes the form:

1. si = αi(q) + β ii pi , i ∈ N

2. si = (γ i / γ 1 )(α1 (q) + γ 1 y), i ∈ N

18. The (s2) Model

Consider the (s2) unrestricted model specification:

s p y i

n

k

i i ik k ( ) ln

1

= α +∑β + γ =

q (s2)

The implied Slutsky symmetry conditions are:
22

{ } { ij j i i j}

i j

ji i j j i

i j

p s s

p p

y p s s

p p

y β + ( + γ ) = β + ( + γ ) (1)

Case I. = = 0 i j γ γ

• (1) simplifies in this case to ji i ij j β p = β p , which is satisfied only if:

= = 0 β ij β ji (2)

Case II. ≠ 0; ≠ 0 i j γ γ

• (1) simplifies to ji i j i ij j i j β p + γ s = β p + γ s whose derivative with respect to i p is

ji i j βii β (γ −1) = γ and whose derivative with respect to k p , k ≠ i, j, is j ik i β jk γ β = γ .

For these conditions to hold in general, either:

= 1 i γ (3)

= 0 β ii (4)

ik jk j β = β / γ , ∀ k; k ≠ i, j (5)

or:

≠ 1 i γ (7)

−1 =

i

j ii

ji γ

γ β β (8)

Case III. ≠ 0; = 0 i j γ γ

• (1) in this case simplifies to i ij j ji i j s = (β p − β p )/ γ . To be consistent with (e2), this

condition requires that either:

= 1 j γ (9)

or:
23

= 0 β ij (10)

The restricted model specification takes the form:

1. si = αi(q) + β ii pi , i ∈ N

2.

s p i N K

s p p y i K J

s p p y i J

i i i

i

k N K

k k

i

k K

ik k

i

i

k N K

k k

k K i k

i ik k

( / ) , ~

( ) ln , ~

( ) ln ,

1 1

~

1

1

1

1

~

1

,

1

= − ∈

= + + + ∈

= + + + ∈

∑ ∑

∑ ∑

∈ ∈

∈ ≠ ∈

β γ

β γ

γ

γ α β

γ

γ

α β β

q

q

3.

, , 1

( ) ln

1 1 1

1 1 1

= − ∈ ≠

= + ∑ + ∈

s p p i N i

s p y

i i i i

k N

k k

β β

α q β

19. The (s3) Model

Consider the (s3) unrestricted model specification:

s p yi

n

k

i i ik k = α +∑β + γ =1

(q) ln (s3)

The implied Slutsky symmetry conditions are:

{ } { ij i i j}

i j

ji j j i

i j

s y s

p p

y s y s

p p

y β + ( + γ ) = β + ( + γ ) (1)

Case I. = = 0 i j γ γ

• (1) implies:

β ij = β ji (2)
24

Case II. ≠ 0; ≠ 0 i j γ γ

• The derivative of (1) with respect to k p , k=1,...,N, implies:

jk

j

i

ik β

γ

γ β = (3)

• Plugging (3) into (1) implies:

(q) (q) j

j

i

i α

γ

γ α = (4)

β ij = β ji (5)

• (3) and (5) can be combined as follows:

k kk

k

i j

ij = 2 β , ∀

γ

γ γ β (6)

• Thus, (4) and (6) are the necessary restrictions for this case.

Case III. ≠ 0; = 0 i j γ γ

• (1) simplifies to s y i ji ij j = (−β + β )/ γ , which is inconsistent with (s3).

The restricted model specification takes the form:

1. s p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,

2. s p y i N

k N

k k

i

i ∈











 = + ∑ + ∈

( ) ln , 1

1

11

1

1

γ γ

γ

β α

γ

γ q

20. The (s4) Model
25

Consider the (s4) unrestricted model specification:

s p y i

n

k

i i ik k ( ) ln ln

1

= α +∑β + γ =

q (s4)

The implied Slutsky symmetry conditions are:

{ } { } ij i i j

i j

ji j j i

i j

s s

p p

y s s

p p

y β + ( + γ ) = β + ( + γ ) (1)

Case I. = = 0 i j γ γ

• (1) implies:

β ij = β ji (2)

Case II. ≠ 0; ≠ 0 i j γ γ

• The derivative of (1) with respect to k p , k = 1,...,N, implies:

k jk

j

i

ik = β , ∀

γ

γ β (3)

• Plugging (3) into (1) implies:











 = − +

i

ij

i

ji

j

j

i

i

γ

β

γ

β

α

γ

γ α (q) (q) (4)

Case III. ≠ 0; = 0 i j γ γ

• (1) implies i ij ji j s = (β − β )/ γ , but the structure of (s4) requires that:

= 0 β ij (5)

The restricted model specification takes the form:

1. s p i N

k N

i = i + ∑ ik k ∈ ∈

α (q) β ln ,
26

2. s p y i J

k N

k k

i

i

i

i i

i ∈











 = − + + ∑ + ∈

( ) ln ln , 1 1 1

1 1

1

α β γ

γ

β

γ

β

γ

γ q

s i N J i = −β1i / γ 1 , ∈ ~

21. The (s5) Model

Consider the (s5) unrestricted model specification:











 = ∑ + =

s p yi

n

k

i i ik k α β γ

1

(q) exp (s5)

The implied Slutsky symmetry conditions are:

{ } { ij j i i i j}

i j

ji i j j i j

i j

p s y s s

p p

y p s y s s

p p

y β + (1+ γ ) = β + (1+ γ ) (1)

The derivative of (1) with respect to y implies:

j ji i ij (1/ y + γ )s = (1/ y + γ )s (2)

Case I. = = 0 i j γ γ

• (1) simplifies to ji i j ij j i β p s = β p s , which holds in general only if:

= = 0 β ij β ji (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (1) and (2) imply that:

i j γ = γ (4)

• Given (4), (1) simplifies to ji i j ij j i β p s = β p s . As a result, (3) must also hold.

Case III. ≠ 0; = 0 i j γ γ
27

• (2) requires that = 0 i γ , a contradiction.

The restricted model specification takes the form:

1. si = αi(q) exp(βii pi + γ 1 y), i ∈ N

22. The (s6) Model

Consider the (s6) unrestricted model specification:

i s p y

n

k

i i ik k

γ α β 









 = ∑=1

(q) exp (s6)

The implied Slutsky symmetry conditions are:

{ } { } ij j i i i j

i j

ji i j j i j

i j

p s s s

p p

y p s s s

p p

y β + (1+ γ ) = β + (1+ γ ) (1)

The derivative of (1) with respect to y implies:

( )/ / / ( ) /( ) 0 2 2sji − sij y + γ j β jisj p j −γ iβ ijsi pi + γ j − γ i sisj pi p j = (2)

Case I. = = 0 i j γ γ

• (1) becomes ji i j ij j i β p s = β p s , which holds in general only if :

= = 0 β ij β ji (3)

Case II. ≠ 0; ≠ 0 i j γ γ

• (1) simplifies to ji i i j ij j j i β p /s + γ = β p /s + γ , which along with (2) is generally satisfied

only if (3) and the following condition hold:

i j γ = γ (4)
28

Case III. ≠ 0; = 0 i j γ γ

• There are no parameter restrictions under which (1) and (2) hold unless = 0 i γ , a

contradiction.

The restricted model specification takes the form:

1. si = i( ) exp( ii pi ) y 1 , i ∈ N γ α q β

23. The (s7) Model

Consider the (s7) unrestricted model specification:

( ) exp( )

1

s p yi

n

k

i i k

ik α γ β

∏=

= q (s7)

The implied Slutsky symmetry conditions are:

{ } { ij i i i j}

i j

ji j j i j

i j

s y s s

p p

y s y s s

p p

y β + (1+ γ ) = β + (1+ γ ) (1)

The derivative of (1) with respect to y implies:

j ji i ij (1/ y + γ )s = (1/ y + γ )s (2)

Case I. = = 0 i j γ γ

• (1) simplifies to ji j ij i β s = β s , which in general holds either if:

= = 0 β ij β ji (3)

or:

k β jk = βik , ∀ (4)
29

(q) (q) j

jj

ii

i α

β

β α = (5)

Case II. ≠ 0; ≠ 0 i j γ γ

• (2) implies:

i j γ = γ (6)

• With (6), (1) simplifies to ji j ij i β s = β s , which implies either (3) or (4) and (5) must also

hold.

Case III. ≠ 0; = 0 i j γ γ

• (2) implies = 0 i γ , a contradiction.

The restricted model specification takes the form:

1. 

s p y i N J

s p y i J

ii

kk

i i i

k J

i ii k

( ) exp( ), ~

( / ) ( ) exp( ),

1

11 1 1

= ∈

= ∏ ∈ ∈

α γ

β β α γ

β

β

q

q

24. The (s8) Model

Consider the (s8) unrestricted model specification:

ik i s p y

n

k

i i k

β γ α ∏=

= 1

(q) (s8)

The implied Slutsky symmetry conditions are:

{ } { ij i i i j}

i j

ji j j i j

i j

s s s

p p

y s s s

p p

y β + (1+ γ ) = β + (1+ γ ) (1)
30

This model is identical to the (x8) and (e8) models up to a parametric transformation. See LaFrance

[1986] for a derivation of the necessary parameter restrictions.