The Structure of Consumer Preferences · 2020. 3. 20. · consumer preferences and changes in preferences over time in Section 4. Our tests are based on time series data for U.S

This PDF is a selection from an out-of-print volume from the NationalBureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume4, number 1

Volume Author/Editor: Sanford V. Berg, editor

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm75-1

Publication Date: 1975

Chapter Title: The Structure of Consumer Preferences

Chapter Author: Dale W. Jorgenson, Lawrence J. Lau

Chapter URL: http://www.nber.org/chapters/c10218

Chapter pages in book: (p. 49 - 101)

Annals of Economic and Social .1feocuremenl, 4 I 1975

THE STRUCTURE OF CONSUMER PREFERENCES

13Y

DALE W. JORGENSON

AND

LAWRENCE J. LAU.

The purpose of this paper is to present an econometric methodology for select tug among alternativespecificutions of the structure of consumer pref't-ences in statistical demand eauils:s. Wi' first derire para-metric restrictionS for direct and indirect transcendental logarithmic utilltv Iiinctions corresponding torestrictions on the /ir,n of consumer preferences and on changes iii prefrreiues ot'cr rune. We considerrestrictions corresponding to graupwise separability in goods and in time, groupwise homotheticity.groupwi.se linear logarithmic utility, and groupwise equality of rates of commodity augmentation. Second.we formulate statistical tests of these restrictions based on the likelihood ratio principle. Finally, we presentempirical tests of each set of restrictions for U.S. time series data on personal consumption expendituresfor the period 1947-Ic7l.

1. INTRODUCTION

The purpose of this paper is to present an econometric methodology for charac-terizing the structure of consumer preferences and changes in preferences overtime.' For this purpose we introduce new representations of consumer preferences.Our approach is to represent the underlying utility function by functions that arequadratic in the logarithms of the quantities consumed and time. Similarly, werepresent the underlying indirect utility function by functions that are quadraticin the logarithms of ratios of prices to total expenditure and time. These representa-tions of consumer preferences do not require the assumptions of additivity,homotheticity, and stationarity of preferences implicit in the traditional approachto statistical demand analysis.

We refer to our representation of the utility function as the direct transcen-dental logarithmic utility function with time-varying prelerences, or more simply,the direct translog utility function. The utility function is a transcendentalfunction of the logarithms of the quantities consumed and of time.2 Similarly, we

refer to our representation of the indirect utility function as the indirect transcen-dental logarithmic utility function with time-varying preferences or, more simply,the indirect translog utility function. Direct and indirect translog utility functions

* Harvard University and Data Resources. Incorporated, and Stanford University, respectively.The research of Dale W. Jorgenson was supported by The National Science Foundation under GrantGI-43097. The research of Lawrence J. Lau was supported by the John Simon Guggenheim MemorialFoundation and by the National Science Foundation through Grant GS-40104 to the Institute forMathematical Studies in the Social Sciences, Stanford U niversisy. We are grateful to Christophe Cham-

Icy arid Paul Swaim for expert research assistance and to Laurits Christensen for helpful advice.Responsibility for any remaining deficiencies rests entirely with the authors.

Direct and indirect utility functions with time-varying preferences are discussed by Lau [1969a].

A function U = F(X) is an algebraic function if U can be defined implicitly by an equationG(U, X) = 0, where G is a polynomial in U and X. All functions which are not algebraic aretranscendental. See Courant (1936), p. 119.

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without jrne-varYing preferences were introduced by Christensen. Jorgenson and

Lau and used by them to test the theory ofdeniand and to characteriie substutjo

patterns among commoditY groups.3 Lau and Mitchell arid Christensen and

Mauser have employed hornOthetic indirect translog utility functions to charac-

terize substitution patterns.4As an illustration of the traditional approach to demand analysis, we Call

consider the double logarithmic demand functions employed in the Pioneering

studies of consumer demand by Schultz. Stone, and Wold.5 If the theory of demand

is valid and demand functions are double logarithmic with time trends, the utility

function is neutral linear logarithmic. A neutral linear logarithmic utility function

is additive, homothetic, and stationarY. Elasticities of substitution among all pairs

of commodities are constant and equal to unity. All expenditure proportions are

constant for all values of prices, total expenditure, and time. Similarly, the Rotter.

dam system of demand functions with time trends employed by Barten and Theil

is consistent with utility maximization only if the utility function is neutral linear

logarithmic.6 We conclude that the double logarithmic and Rotterdam demand

systems implicitly maintain the hypotheses of additivity. hornotheticity, tnd

stationarity.Houthakker and Stone have developed alternative approaches to demand

analysis that retain the assumption of additivity while dropping the assumption of

homotheticitY.7 Stone has employed a linear expenditure system, based on a

utility function that is linear in the logarithm of quantities consumed less a con-

stant for each commodity, representing initial commitments of expenditure. Non-

zero commitments permit expenditure proportions to vary with total expenditure.Houthakker has employed a direct addilog system. based on a utility function that

is additive in functions that are homogeneous in the quantity consumed for eachcommodity. The degree of homogeneity may vary from commodity to commodity.

again permitting expenditure proportions to vary with total expenditure. Parallel!-

ing the drect addilog demand system, Houthakker has also employed an indirectaddilog system, based on an indirect utility function that is additive in the ratiosof prices to total expenditure.

Basmann, Johansen, and Sato have combined the approaches of flouthakkerand Stone. defining each of the homogeneous functions in the direct addilogutility function on the quantity consumed less a constant for each commodity.8The resulting utility function is additive but not homothetic. We conclude that

See Christensen, Jorgenson and Lau [1975]. Earlier Christensen. Jorgenson and Lau 11971.

19731 introduced transcendental logarithmic functions Into the study of production.See I.au and Mitchell [1971] and Christensen and Manser [1974a. 1974h].See Schultz [1938]. Stone [1954a]. and Wold [1953]. For a proof that an iniegrabie system ol

double logarithmic demand functions with time trends implies neutral linear logarithmic utility. 'Jorgenson and Lau [1974].

See Barten [1964, 1967, 1969], McFadden [1964], and Theil [1965. 1967, 1971]. For a proof that

an integrable Rotterdam system with time intercepts implies explicit neutral linear logarithmic utility.see Jorgenson and Lau [1974].

'See Houthakker [1960] and Stone [l954h]. The linear expenditure sstem Was originally )f

posed by Klein and Robin [1947-1948].See Basmann [1969]. Johansen [1969] and Sato [1972j. For an empirical application. see Broan

and Heren [ 1972]. A recent survey of econometric studies of consumer demand is gisen by Brown and

Deaton [19721.

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t1

t

I-

t

the linear expenditure system, the direct and indirect addilog systems, and thecombined systems introduced by Basmann, Johansen, and Sato maintain thehypotheses of direct or indirect additivity. By employing direct and indirecttranslog utility functions with time-varying preferences we can test additivity,honiotheticity, and stationaritv restrictions rather than maintaining these re-restrictions on preferences as part of our econometric model.

In the following section we introduce direct and indirect translog utilityfunctions with time-varying preferences and the corresponding systems of indirectand direct demand functions. We consider restrictions on the demand functionsimplied by utility maximization. We impose these restrictions as part of ourmaintained hypothesis. In Section 3 we consider demand systems associated withrestrictions on the structure of consumer preferences and changes in preferences

over time. We begin with groupwise separability and groupwise homotheticityof preferences. For each set of restrictions on preferences. we derive parametricrestrictions on the corresponding system of demand functions. These parametricrestrictions provide the basis for statistical test of alternative hypotheses about

the structure of consumer preferences.We consider two alternative sets of restrictions on the variation of consumer

preferences over time. The first. set corresponds to separability of goods and time;

a commodity group is separable from time if the ratios of any pair of demandfunctions for all commodities within the group are independent of time. Analternative set of restrictions on changes in preferences is associated with com-modity augmentation; commodity augmentation by itself is not a testable hypoth-

esis since any change in preferences over time can be regarded as commodityaugmenLing or commodity diminishing. We impose restrictions on the variationof preferences with time by imposing restrictions on rates of augmentation ofcommodities within a given group: in particular, we formulate tests of equality of

rates of commodity augmentation within a group. Groupwise separability fromtime and groupwise equality of rates of commodity augmentation are not mutually

exclusive; however, they coincide only under additional restrictions such as neutral

linear logarithmic utility.We present empirical results of tests of alternative sets of restrictions on

consumer preferences and changes in preferences over time in Section 4. Our tests

are based on time series data for U.S. personal consumption expenditures of threecommodity groups---durables. non-durables, and energyfor the period 1947-1971. Our concept of personal consumption expenditures differs from the cor-responding concept in the U.S. national income and product accounts in the

treatment of consumers' durables.9 We treat expenditure on consumers'durables as part of gross private domestic investment rather than personalconsumption expenditures. We add an imputed flow of services from consumers'durables to personal consumption expenditures, so that our concept of durables

services is perfectly analogous to the national accounting concept of housing

services.

A detailed reconciliation of our concept of personal consumption expenditures and the nationalaccounting concept is given by Christensen and Jorgenson [1973]. pp. 33 1-348.

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1

:1

2.1. The direct trWL!Vg utility Ju?UtWfl

A direct utIlity fimciiofl U with time-Varying JefeRuces 'aii be written in

the form:

(2.1) In U F(X1.X,

where X(i = 1, 2, 3) is the quantity consumed of the ith commodity and I is time.

At each time the consumer maximizes utility, subject to the budget constraint,

(2.2) = M,

where p(i = 1, 2, 3) is the price of the i-th commodity and M is the value of total

expenditure.Maximizing utility, subject to the budget constraint, we obtain the identjt

lnU p.X .,iflnU(2.3) =

This identity gives the ratios of prices to total expenditure as functions of the

quantities consumed:

(2.4)

2. TRANscFM)NTAt LooRlT}IMiC !JT1LITY FL'NciioNs viiii TIMi-VAkyIN(;PR1FFRENCES

In U

p_ ?lnXlnU'

X

52

(1= 1,2,3).

(1= 1,2,3).

(1= 1,2,3),

We refer to these functions as indirect demand functions.Utility is nondecreasing in the quantities consumed, so that the negative

of the logarithm of utility is nonincreasing in the logarithms of the quantitiesconsumed. A necessary and sufficient condition for monotonicity of the negativeof the logarithm of the utility function at a particular point is that the budgetshares are non-negative at that point. The utility function is quasiconcave, so thatthe negative of the logarithm of the utility function is quasiconvex. Monotonicityand quasiconvexity of the negative of the logarithm of the utility function are thebasic assumptions of the theory of demand.

We approximate the negative of the logarithm of the utility function by a

function quadratic in the logarithms of the quantities consumed and t:

(2.5) In U = + ; In X, + . r + fl In X. In X

+ E/3111nX1.t

Using this form of the utility function we obtain:

+ flIn L + /3. .t = PJXJV(2+ flkIln X + I), (j = 1,2.3).

To simplify this notation we write:

(2.7) M , II(1=>.:#kl, I3Mt

so that

(8 ( l )

M - z + In X1 + f, r' -n We note that the paiaiueters z, and fl, have no effect on the utility_maximi/itig

quantities consumed. These two parameters cannot he identified from data on

prices and quantities.The budget constraint implies that:

(2,9) = I.

so that, given the parameters of any two equations For the budget shares, p1XM

(j = 1,2,3), the parameters of the third equation can be determined from the

definitions of CM, (j = 1, 2,3), and fl.Since the equations for the budget shares are homogeneous of degree zero in

the parameters. normalization of the parameters is required for estimation. A

convenient normalization for the direct translog utility function is:

(2.10) = = 1We estimate only two of the equations for the budget shares, subject to

normalization of the parameter CM appearing in each equation at minus unity.

Unrestricted, there are eighteen unknown parameters to be estimated from the two

equations. If the equations are generated by utility maximization, the parameters

f3.1(j = 1.2, 3) and J3 appearing in each equation must be the same. This results

in a set of restrictions relating the four parameters appearing in each of the two

equations, a total of four restrictions. We refer to these as equa lily restrictions.

C The negative of the logarithm of the direct translog utility function is twice

s differentiable in the logarithms of the quantities consumed, so that the Hessian

e of this function is symmetric. This gives rise to a set of restrictions relating the

t parameters of the cross-partial derivatives:

(2.11) fl = f3. (i = 1,2.3).

y There is one restriction of this type among the parameters of the two equations

we estimate directly and two such restrictions among the parameters of the equa-

a tion we estimate indirectly from the budget constraint. We refer to these as symmetry

restrictions. The total number of symmetry restrictions is three.

If equations for the budget shares are generated by maximization of a direct

translog utility function, the parameters satisfy equality and symmetry restrictions.

There are seven such restrictions. Given the seven equality and symmetry restric-

tions, eleven unknown parameters remain to be estimated. Our approach to the

analysis of consumer demand takes as assumptions the restrictions on expenditure

allocation implied by utility maximization and the existence of the three commodity

groupsdurables, non-durables, and energyas well-defined economic aggre-

gates. Given these assumptions, we estimate the unknown parameters of our

complete demand system simultaneously.Given the hypothesis of consistency between our system of indirect demand

functions and the maximization of utility and the grouping of commodities into

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three aggregates. we could proceed to impose further constraints ()fl the allocat ion

of personal consumption expenditures. such IS COflStaflt price and inCome elastici-

ties of demand or constant elasticities of substitut on among COniniodity groupsHowever, such an approach would frustrate our primary research objective ofcharacterizing the pattern of consumer demand empirically. I his approach

Wouldconvert hypotheses about budget allocation and patterns of substitution intoassumptions rather than hypotheses to be tested. Instead we propose to test allfurther restrictions on the structure of the direct utility function.

2.2. The indireci trans/ag uti!tV /iinction.

An indirect utility finction V with time-varying preferences can be writtenin the form:

F' 1' PlnV=G M M Mwhere V is the maximum level of utility corresponding to the prices p1(i

1,2,31and the level of total expenditure M.

We can determine the budget share from the J-th commodity from theidentity

olflVpjXjv.C1flV 12 )in p/M M ' In p1/M'

This identity gives the quantities consumed as functions of the ratios of prices tototal expenditures:

1n V

i) lnp.IM(2.14) ' , ( = 1,2, 3).

"ic' n

ML ' In p1/M

We refer to these functions as direct demand lunct ions.Utility is nonincreasing in the prices, so that the logarithm of utility is non-

increasing in the logarithms of the prices. A necessary and sufficient condition formonotonicity of the logarithm of the indirect utility function at a particular pointis that the budget shares are non-negative at that point. The indirect utility functionis quasiconvex, so that the logarithm of this function is quasiconvex.

The system consisting of the negative of the logarithm of the direct utilityfunction and the indirect demand functions is dual to the system consisting of thelogarithm of the indirect utility function and the direct demand functions. Onesystem can be obtained from the other by simply interchanging the quantitiesconsumed X (1 = 1,2,3) with the ratios of prices to total expenditure p'M(i1, 2, 3). All the properties of one system carry over to the other system with therole of these two sets of variables interchanged.

!O Systems of direct and indirect demand funtion with these properties arc discussed by Christen-sen, Jorgenson and Lau [1975].

'This is the logarithmic form of Ro's Identity. See Roy [1943]

54

(2.12)

(2.13)

We approximate the logarithm of the indirect utility function by a functionquadratic in the logarithms of the ratios of prices to the value of total expenditureand t:

(2.15) In V + s1ln + +

+ fi1,ln-t -f

Using this form of the indirect utility function we obtain:

(2.16) j+>I3JiIfl+[1il.t_(k+>.f1ki1flçj+flkro).

(/= 1,2,3).

As before, we simplify notation by writing:

(2.17) M = ' fiMi = Iki' flM (i= 1,2,3),

so that:

(218)+ /3flInp'M + flt - 1 2 3)

M - jq + flMIlnP/M + [11t' -The parameters ; and f3, cannot be identified.

The budget constraint implies that, given the parameters of any two equationsfor the budget shares, the parameters of the third equation can be determined fromthe definitions of cLM,I3MJ(j = 1,2,3), and f3M. As before, we normalize the para-meters of the indirect translog utility function so that:

(2.19) aM =

As in the case of the direct transiog utility function with timevarying preferences,we estimate only two of the equations for the budget shares, subject to normaliza-tion of the parameter aM appearing in each equation at minus unity. We alsomaintain the assumptions of utility maximization and the existence of the threeaggregates. The equality and symmetry restrictions resulting from these assump-tions are strictly analogous to those for the direct translog utility function withtime-varying preferences.

2.3. Stochastic specfication

The first step in implementing an econometric model of demand based onthe direct translog utility function with time-varying preferences is to add astochastic specification to the theoretical model based on equations for thebudget shares p3X/M( / = 1,2,3). Given the disturbances in any two equations,the disturbance in the remaining equation can be determined from the budget

55

constraint. Only two equations are required for a complete econometric model of

demand. We assume that the noncuntenlporafleOUS disturbances, whether from

the same or different equations. have /eiO covartance. No additional restrictioii5

are placed on the disturbances, other than the IccItlirenlent that (llsttirhances from

the three equations must add up to zero. VVe also assume that the right hand side

variables of the equationS for the budget shares are uncorrelated with the sto-chastic disturbances. l'his latter assumption facilitates the use of the method ofmaximum likelihood in estimation of the parameters.

In implementing an econometric model of demand based on the indirectutility function with time-varyilig prefereiices the first step. as before, is to add a

stochastic specification to the theoretical model based on equations for the budget

shares p3X,/M(j = I, 2, 3). Only two equations are required for a complete modelThe assumptions that we make here are strictly analogous to those for the directtranslog utility function with time-varying preferences. We note, however, that theimplications of the stochastic specification are different for the direct and indirectmodels and hence the results for the two models are not directly comparable

To summarize: We have derived models for the allocation of personal con-sumption expenditures from direct and indirect translog utility functions withtime-varying preferences. We take the hypothesis of utility maximization to be anassumption r2ither than a hypothesis to be tested. Utility maximization impliesthat the parameters of equations for the budget shares in each model satisfy sevenequality and symmetry restrictions that enable us to reduce the number of unknownparameters from eighteen to eleven. These parameters are further constrained bycertain inequalities that embody monotonicity and quasiconvexity restrictions othe negative of the logarithm of the direct utility function and the logarithm of theindirect utility function. We estimate the parameters of our models of consumptionsubject to the equality and symmetry restrictions; at a later stage we incorporatethe monotonicity and quasiconvexity restrictions.'2

3. PREFJRENCE STRUC-TURI

3.1. Approximation

The primary objective of our research is to ascertain and characterize thestructure of consumer preferences empirically, without maintaining restrictiveassumptions on the specific form of the utility function other than monotonicilyand quasiconvexity. We wish, first, to determine the effects of changes in totalexpenditures and changes in preferences over time on the allocation of the con-sumer budget among commodity groups and, second, to determine the effects ofchanges in relative prices on the allocation of the consumer budget, that is, tocharacterize the patterns of substitution among commodities.

In the remainder of this section, we develop tests of a series of possible re-strictions on the underlying structure of consumer preferences. First, we considergroupwise separability of preferences in commodities and in time. Second, weconsider overall hornotheticity and groupwise honiotheticity of preferences.

2 Monotorncity and quasiconvexily restrictions are discuscd h) Lau [1)74j. See also Jorgensonand Lau [1974].

56

3

Third, we consider groupwise linear logarithmic utility as a possible restriction onpreferences. Finally, we consider groupwise equal rates ofconimodity augmentationas a possible restriction on changes in the structure of preferences over time.

The transcendental logarithmic utility function with time-varying preferencescan be interpreted as a local second-order Taylor's series approximation of anarbitrary utility function with time-varying preferences that is differentiable atleast up to the third order. In practical applications the latter condition is hardlyany restriction as any utility function can be approximated arbitrarily closely byan infinitely differentiable function. Using this local approximation property,the translog utility function can be used to test specific hypotheses on the structureof the underlying utility function.

The parameters of the translog utility function can be identified with thecoefficients in a Taylor's series expansion to the underlying utility function. Theytake the values of the IIrst and second partial logarithmic derivatives of the nega-tive of the logarithm of the underlying utility function at the point of expansion.Specific hypotheses on the structure of preferences imply restrictions on the Hessianof the negative of the logarithm of the utility function and can be tested by imposingthese restrictions on the parameters of the translog utility function.

Restrictions on the structure of preferences do not necessarily imply thecorresponding restrictions on the translog utility function itself. Properties of theunderlying utility function and its translog approximation agree up to and includ- I

ing second-order derivatives at the point ofapproximation. We distinguish betweensituations where the translog utility function provides an approximation to anunderlying utility function with a certain property and situations where thetranslog utility function also possesses that property. In the latter case, we saythat the translog utility function possesses the property intrinsically.

3.2. Groupit'ise separability

The first set of restrictions on consumer preferences that we propose to test aregroupwise separability restrictions. A direct utility function U with time-varyingpreferences that is separable in X1 and X2 from X3 can be written in the form:

(3.1) In U = F(. In U'(X,X2,t),X3,r),

where the function - In W depends only on X1, X2 and time and is nonincreasingand quasiconvex in X1 and X2. A necessary and sufficient condition for groupwiseseparability of the direct utility function in X1 and X2 from X3 is that the ratio ofthe indirer demand functions for X1 and X, is independent of the quantity of X3.A direct utility function that is groupwise separable in X1 and X2 from time canbe writt:n in the form:

(3.2 In U F(ln U'(X1,X2,X3),X3,t),

which is analogous to equation (3.1) with the roles of X3 and t interchanged. Anecessary and sufficient condition for groupwise separability of X1 and X2 fromtime is that the ratio of the indirect demand functions of X1 and X2 is independentof time. Groupwise separability in time is also referred to as groupwise neutrality.

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Partiall differentiating equation (3.1) fIrst With respect to In V3 and thcwith respect to In X and In X 2 separately, we Obtain:

- In U (2f. - In U'nX,InX3rInt.'(lnX3 In.V,

- In 1.! (2 j.' -- In U'iIn "2 lnX1 e In In ( In \

By observing that:

1' In U jj;' In U'In X, - - In U' In .V,

- Iii L [F (' In UIn X2 In f hi A

2

we can rewrite:

'2--InL! 2F(!nL''(lnV3)c In X, In \3 iF( - In U') j In ,Y,

so that:

(3.4)

U 2F'( - In U' 1 In X3) - In U(35)In X, In X eFIfr - In Li') c In ,y2

Given groupwise separability, equations (3.5 must hold everywhere inparticular, they must hold at the point of approximation in this case, In A' == 1,2, 3), t = 0, where we can identify the first and second partial derivativeswith the parameters of the direct translog utility function with time.varyingpreferences:

(2 - In U ('2 - In UInX' = /113, 1I\. ii .k= /123.

lnU ilnt' (lflX; 2

Thus, given groupwise separability of X1 and A2 from A3. the parameters of thedirect translog utility function must satisfy the restrictions:(3.6)

P13

where p3 is a constant given by:

- In Li' j' In A3)f) =- F/( - In U')

at the point of approximation.Similarly, in a manner strictly analogous to the derivation of equation (3.6).it can be shown that given groupwise separability of A' and A2 from time. theparameters of the direct translog utility function must satisfy the restrictions:

(3.7)

= p32,

fill = , [12, =

58

We note that there are no analogous restrictions on the direct translog parametersfor groupwise separability of the type X and time from X2 because the parameterz cannot be identified.

We distinguish among three commodity groups. Each pair of commodities,such as X1 and X2, can be separable from the remaining commodity, X3 in thisinstance, and time. Corresponding to the three possible pairs of commodities,

there are six possible sets of groupwise separability restrictions analogous toequation (3.6) or equation (3.7). Each set of two restrictions involves the ntroduc-tion of one new parameter p3 and p in the examples given above. Under each

set of such restrictions, maintaining the symmetry and equality restrictions, tenunknown parameters remain to be estimated.

The translog approximation to a groupwise separable utility function is not

necessarily groupwise separable. For a direct translog utility function to be

groupwise separable in X1 and X2 from X3, the ratio of the indirect demandfunctions generated by the direct translog utility function must be independent of

X3. We refer to a direct translog utility function as intrinsically groupivise separable

if it is groupwise separable. Two alternative sets of restrictions on the parametersof the direct translog utility function are jointly necessary and sufficient for intrinsic

groupwise separability of the direct translog utility function. The first set consists

of the restrictions given in equation (3.6) and the additional restriction:

(3.8) P3 = 0.

This restriction implies that the cross partial derivatives of the direct translogutility function with respect to X1 and X3 and X2 and K3. respectively, areidentically zero at the point of approximation. Thus the indirect demands of X1

and X2 do not depend on X3. We refer to this set ofrestrictions as explicit groupwise

separability restrictions.A second set of restrictions that implies intrinsic groupwise separability of the

direct translog utility function is that p is different from zero, but that the ratioof the budget shares of X1 and K2 is constant for all prices, total expenditure and

time. This means that the parameters of the direct translog utility function must

satisfy the restrictions:

(3.9)1fl12 = 2flhi' z1fl22 = 2fli2' i123 = 2fll3' ziP2, 2flui.

that is, the second order trans log parameters corresponding to the first and second

commodities must be in the same proportion as the first order translog parameters.

If the ratio of the optimal budget shares of X and X2 is constant, the direct utility

function takes the form:

In U = F(i1lnX1 + ö2lnX2,X3,t),

where 5 and are constants. This utility function is both groupwi.se linear

logarithmic in X1 and X2 and groupwise separable in X1 and X2 from time. We

say that such a utility function is groupwise neutral linear logarithmic. This con-

dition is much more restrictive than groupwise separability or explicit groupwise

separability; we will discuss it in more detail in Section 3.4 below.Similarly, two alternative sets of restrictions on the parameters of the direct

translog utility function are jointly necessary and sufficient for intrinsic groupwise

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separability of X and X2 from time. The first set consists of the restrictions givenin equation (3.7) above and the additional restriction(3.10) = 0.that is, the direct translog utility hi nctoii is explieitl graupise .ceparahk' inX and X2 from lime. A second set of restrictions that also implies intrinsicgroupwise .sc'poral'ilitv of X1 and X, from time are the restrictions of groupwiseneutral linear logarithmic utility.

We can show that restrictions analogous to equations (3.) and (3.10) musthold for any one of the six possible types of explicit groupwise separability, givengroupwjse separability. Under each set of explicit groupwise separability restric-tions, nine unknown parameters remain to be estimated.

A direct utility function with time-varying preferences is additive in A'1 . X2and X3 if it can be written in the form:(3.11) In U F((In U'(Xt) + In U2(X2,,) + In U3(X3,t)),i)A necessary and sufficient condition for additivity in commodities is that the directutility function is groupwise separable in any pair of commodities from theremaining commodity In particular, since there are only three commoditiesgroupwise separability of any two pairs of commodities from the third is sufficientfor additivity. A direct transiog utility function with time-varying preferences isexplicit/t' additizc' if it can be written in the form

(3.12) - in U = - In U (X1 . t) - In U2(X2. 1) - in U3(X3, t).where each function - In tJ'(i = 1, 2, 3) is nonincreasing and Convex. The translogapproximation to an explicitly additive utility function is necessarily explicitlyadditive. A necessary and sufficient condition for explicit additivity in commoditiesis that the direct translog utility function is explicitly groupwise separable in anypair of commodities from the remaining commodity. Since there are only threecommodities explicit groupwise separability for any two pairs of commoditiesfrom a third commodity is sufficient for explicit additivity.

A direct utility function with time-varying preferences is neutral if it can bewritten in the form:

---In U = F(in U'(X1 X2, X3), t),where In U' is independent of time. A necessary and sufficient condition forneutrality is that the direct utility function is groupwise separable in any pair ofcommodities from time. in particular, since there are only three commoditiesgroupwise separability of any two pairs of commodities from time is sufficient forneutrality. A direct utility function with time-varying preferences is expljjt Ijneutral if it can be written in the form

(3.13) --In U = In U1(X, , X2, X3) + F(t).The translog approximation to an explicitly neutral utility function is necessarilyexplicitiy neutral. A necessary and sufficient condition for explicit neutrality is thatthe direct translog utility function is explicitly groupwise separable in any pair ofcommodities from time. In particular, since there are only three commodities

60

a

exphcit groupwise separability of any two pairs of commodities from time issufficient for explicit neutrality.

3.3. Groupwise hwnotjiencitv and homogeneity

The second set of functional restrictions on consumer preferences that wepropose to test are 1:mnotJietic,ty restrictions. First, we consider overall homo-theticity of preferences. A direct utility function with time-varying preferencesthat is Iwmotlietic can be written in the form

(3.14) In U F(ln H(X1, X2, K, t), 1),

where H is homogeneous of degree one in the quantities X1, X2, and X3. Underhomotheticity, the optimal budget shares for all three commodities depend onlyon prices and time and are independent of total expenditure. An equivalentcharacterization of homotheticity is that the ratios of indirect demand functionsare all homogeneous of degree zero in X1, X2 and X3.

Paitially differentiating equation (3.14) with respect to In X)(/ = 1.2, 3), weobtain:

Second, differentiating again with respect to In Xk (k = 1.2. 3). we obtain:

- In (,T r In H In H(3.16)

and:

ilnxkiflnX3 InH iInXk flnXJ

F "2InH+

lnH InXklnXJ'(j.k = 1,2,3).

Finally, summing over k and using homogeneity of degree one of the function H,we can write:

2InU 12F ?InH(3.17) = (j = 1.2,3).

clnX&?lnX1 InHiInX1

Given homotheticity, equations (3.17) must hold everywhere; in particular,they must hold at the point of approximation, where we can identify the first andsecond partial derivatives with the parameters of the direct translog utility functionwith time-varying preferences:

InX nX = flkj = fiMJ.

- In U

alnx =

61

(j= 1,2.3).

(1= 1.2.3).

- Intl i?F ?!n If(3.15)

?InX5 = ?lnH ?lnX1' (.i= 1,2.3).

S

Given homotheticit', the parameters of the direct translog utility function mustsatisfy the restrictions

(3.18) IJ,tI I = I' I.f2 = /1's13

where a is a constant given by:

i2F1'(? In H)

We introduce one new parameter, ;, so that these restrictions reduce the numberof parameters by two, leaving nine unknown parameters to he estimated.

The translog approximation to a liomothetic direct utility ftinction is notnecessarily homothetic, even though it must satisfy the restrictions given inequation (3.18; above. For a direct translog utility function to be homothetic, theratios of the indirect demand functions generated by the direct translog utilityfunction must he homogeneous of degree zero in the quantities consumed. Werefer to a direct translog utility function as intrinsically lunnotiwlic if it is itselfhomothetic. Two alternative sets of restrictions on the parameters of the directtranslog utility function are jointly necessary and sufficient for intrinsic homo-theticity of the direct translog utility function. The first set consists of the restric-tions given in equation (3.1$) above and the additional restriction:(3J9 o=0.We refer to this set of restrictions as explicit homot/wticitv restrictions. Under theexplicit homotheticity restrictions, only eight unknown parameters remain to beestimated.

A second set of restrictions that implies intrinsic hornotheticity of the directtranslog utility function is that o is different from zero, but that the ratios of allpairs of optimal budget shares are constant for all prices, total expenditure andtime. This means that the parameters of' the direct translog utility function mustsatisfy:

(3.20) iI'I2 21iII. iII3 = 3llui. 12/113 3flI2,

II22 = :2IJl2, i3fl12. == 2flI3' I/33 - 3fll3, 12/133 13/123,

2Il,' iI13, = 2fl3r 13112,

not all of which are independent. In other words, the second order parameters ofeach commodity must be in the same proportion as the first order parameters.If the ratios of all pairs of optimal budget shares are constant, the direct utilityfunction takes the form:

(3.21) In U = F(ö1 lnX1 + ô2lnX2 + lnX3,t)where ö2. and Ô3 are constants. We refer to such a utility function as neutrallinear logaritlinije. This condition is much more restrictive than homotheticjt orexplicit homotheticity and we will discuss it in more detail in Section 3.4.

62

A direct utility function with time-varying preferences is Iumiogeneoii ii it

can he written in the form:

(3.22) In ' !11'i."2, 3.t).

where H is a homogeneous functron of degree one in X X2 and K . Homogeneityis, of course. a specialization olhomotheticity. Under homogeneity the parametersof the direct translog utility function must satisfy the explicit hornoiheticitvrestrictions given in equation (3.19) above and the additional restriction:

(3.23) = 0.

We refer to this set of restrictions as Iio,noeeueitr restrictions. Under these re-strictions only seven unknown parameters remain to be estimated. We note thatthe translog approximation to a homogeneous direct utility function is necessarilyhomogeneous.

An alternative form of homotheticity of preferences is groupwise houw-theiicitv. A direct utility function with time-varying preferences that is groupwisehomothetic in X1 and X2 can he written in the form

(3.24) lnU=F(lnH(X1,X2.X3M,X3.t)

where H is homogeneous of degree one in the quantities X1 and X2. Undergroupwise homotheticity in X and X 2 the ratio of the indirect demand functionsof K1 and K2 is homogeneous of degree zero in K1 and X2. In other words, theratio of the indirect demands remains invariant under proportional changes in thequantities consumed ofX1 and X2. Under groupwise homotheticity the parametersof the direct translog utility function must satisfy the restrictions:

(3.25) fin + 1112 a2I, Ii2 + 1122 =

This set of two restrictions involves the introduction of one new parameter,a12 so that only ten unknown parameters remain to be estimated. Correspondingto the three possible pairs of commodities, there are three possible sets of group-wise homotheticity restrictions. Restrictions analogous to those given iii equations(3.25) above must hold for any one of the three possible sets of groupwise homo-theticity restrictions.

The translog approximation to a groupwise homothetic direct utility functionis not necessarily groupwise homothetic. For a direct translog utility function tobe groupwise homothetic, the ratio of the indirect demand functions of X1 and X2generated by the direct translog utility function must be homogeneous of degreezero in K1 and X2. We shall refer to a direct translog utility function as intrin.sica Hr

groupwis' homotheoc if it is itself groupwise homothetic. Two alternative sets ofrestrictions on the parameters of the direct translog utility function are jointlynecessary and sufficient for intrinsic groupwise homotheticity of the direct translogutility function. The first set consists of the restrictions given in equations (3.25)above and the additional restriction:

(3.26) C12 = 0.

We refer to this set of restrictions as explicit groupwise Iwnioiheiicuv restrictions.

63

Under the explicit grotipwise hornotheticity restrictions, only nine unknownparameters remain to be estimated.

A second set of restrictions that implies intrinsic groupwise liolnotheticityof the direct translog utility function is that a12 is different from zero. hut that [heratio of the optimal budget shares of X and X2 is constant for all prices, totalexpenditure and time. This is precisely the case of groupwise neutral linear loga-rithmic utility discussed in Section 3.2 above with the restrictions given in equation(3.9). Corresponding to the three possible pairs of commodities, there are threepossible sets of explicit groupwise homotheticity restrictions. Restrictions analo-gous to those given in equation (3.26) above must hold for any one of the threepossible sets of explicit groupwise homothet icily restrictions.

A direct utility function with time-varying preferences is incIusirel group1vis''unnotlu'jje in X and \3 if it can be written in the form(3.27) -in U = F(ln H(X1, X,, X3, t). t),

where H is homogeneous of degree one in the quantities X1 and X2. Givengroupwise homotheticity, this condition implies in addition that the ratios ol allthe indirect demand functions are homogeneous of degree zero in the quantitiesX1 and A',. Under inelusire groupwisc' Iw,noil,etici,1 in X and X7 the parametersof the direct Iranslog utility function must satisfy the groupwise homotheticityrestrictions given in equation (3.25) above and the additional restriction:(3.28)

1)13 + fl23

tJnder the inclusive groupwise homotheticity restrictions, only nine unknownparameters remain to be estimated. Again, there are three possible sets of inclusivegroupwise homotheticity restrictions corresponding to the three possible sets ofgroupwise hornotheticity restrictions. Restrictions analogous to those given iiiequation (3.28) must hold for any one of the three possible sets of grolipwisehomothetjcit restrictions.

The translog approximation to a inclusively groupwisc homothetic directutility function is not necessarily inclusively groupwise homothetic. For a directranslog utility function to be inclusively groupwise homothetic, the ratios of allpairs of indirect demand functions generated by the direct translog utility functionmust be homogeneous of degree zero in X1 and X2. As before, two alternative setsof restrictions on the parameters of the direct translog utility function are jointlynecessary and sufficient for inclusive groupwise homotheticity of the directtranslog utility lunction. The first set Consists of the restrictions given in equations(328) above and the additional restriction:(3.29)

a12 -0.We refer to this set of restrictions as explicit iflcluii groupwfsc' I1o,notI:etiirestrictions Under this set of restrictiois only eight unknown parameters remainto be estimated.

A second set of restrictions that implies intrinsic inclusive groupwise homo-theticity of the direct translog utility function is that a12 is different from zero butthat the direct utility function is groupwise neutral linear logarithmic. Correspond-ing to the three possible pairs of commodities there are three possible sets of64

explicit inclusive groupwise homotheticity restrictions, Restrictions analogousto those given in equation (3.9) above must hold for any one of the three possiblesets of explicit inclusive groupwise hc,niot heticity restrictions.

Finally, direct utility function with time-varying preferences is groupwisehomogeneous if it can be written in the form:

(3.30) In U = lnH(X,X2,X3,,),where H is homogeneous of degree one in the quantities X1 and X2. Groupwisehomogeneity is, of course, a specialization of inclusive groupwise homotheticitywhich is in turn a specialization of groupwise homotheticity. Under groupwisehomogeneity the parameters of the direct translog utility function must satisfvthe explicit inclusive groupwise homotheticity restrictions given in equation(3.29) above and the additional restriction:

(3.31) fit, + Th, = 0.We refer to this set of restrictions as groupwise homogeneity restrictions. Underthese restrictions only seven unknown parameters remain to be estimated. Wenote that the translog approximation to a groupwise homogeneous direct utilityfunction is not necessarily groupwise homogeneous. Corresponding to the threepossible pairs of commodities, there are three possible sets of groupwise honio-geneity restrictions. Restrictions analogous to those given in equation (3.31) musthold for any one of the three possible sets of groupwise homogeneity restrictions.

We conclude this section by noting that groupwise homotheticity in allpossible groups is neither necessary nor sufficient for homotheticity of the directutility function. Even explicit groupwise homotheticity in all possible groups isnot sufficient for homotheticity of the direct utility function. On the other hand,inclusive groupwise hornotheticity in all possible groups is sufficient, but notnecessary, for homotheticity. Inclusive groupwise homotheticity in all possiblegroups implies linear logarithmic utility. Finally, explicit inclusive groupvisehomotheticity in all possible groups implies explicit linear logarithmic utilityand groupwise homogeneity in all possible groups implies neutral linear logarith-mic utility.

3.4. Groupwise linear logarithmic utility

A direct utility function with time-varying preferences that is groupttisehoniolheticallj separable in X1 and X2 from X3 can be written in the form:

(3.32) - In U = F(ln H(X . X2, 1), X3,

where H is a homogeneous function of degree one and depends only on X1, X2and time. A necessary and sufficient conditions for a direct utility function to begroupwise homothetically separable in X1 and X2 from X3 is that the function isboth groupwise separable and groupwise homothetic in X1 and X2.

Groupwise hornothetic separability implies that the ratio of the indirectdemand functions is independent of X3 and is homogeneous of degree zero in X1and X2. The translog approximation to a groupwise hornothetically separabledirect utility function is not necessarily groupwise homothetically separable.

65

For a direct translog utility function to be itself groupwise homothetically separ-able, the ratio of the indirect demand functions of X and X2 generated from adirect translog utility function must he independent of X and homogeneous ofdegree zero in X1 and A'2 We refer to a direct translog utility function as intrin-sicallv groupwise !ionioihe(ital/i .'eparahlt' if it is groupwise honiotheticallyseparable.

As before, two alternative sets of restrictions on the parameters of the directtranslog utility function are jointly necessary and sufficient for intrinsic groupwisehomothetic separability of the direct translog utility function. The first Consistsof the combination of the explicit groupwise separability, given in equation (3.8)above, and explicit groupwise homotheticity, given in equation (3.26) above.Ve refer to the conjunction of these two sets of restrict ions as the explicit group4t'ise

homorlietic separahilit v restrictions. A second set of restrictions that implies intrinsicgroupwise homothetic separability of the direct translog utility function is that ofgroupwise neutral linear logarithmic utility, given in equation (3.9) above.

A direct utility function U with time-varying preferences is groupwise linearlogarithmic if it can be written in the form

(3.33) - In U = F(51(t) In X1 + 2() In X2 , X3, 1).

where (t) and 62(t) are functions only of time. A necessary and sufficient condi-tion for groupwise linear logarithmic utility in X1 and X2 is that the ratio of theoptimal budget shares ofX1 and X2 is independent of all prices and total expendi-ture and depends only on time. Given groupwise homothetic separability in X1and X from X3, groupwise linear logarithmic utility in X1 and X2 requires theadditional restriction:

(3.34) Ifl12 = 2flI

Under these restrictions only eight unknown parameters remain to be estimated.There are three possible sets of groupwise linear logarithmic utility restrictionsand restrictions analogous to those given in equation (3.34) must hold for any oneof them.

The translog approximation of a groupwise linear logarithmic direct utilityfunction is not necessarily groupwise linear logarithmic. For a direct translogutility function to be itself groupwise linear logarithmic, the ratio of the optimalbudget shares of X1 and X2 generated from a direct translog utility function mustdepend only on time. We shall refer to a direct translog utility function as intrinsi-callv groupwise linear logarithmic if it is itself groupwise linear logarithmic. Asbefore, two alternative sets of restrictions on the parameters of the direct translogutility function are jointly necessary and sufficient for intrinsic groupwise linearlogarithmic utility. The first Consists of the explicit groupwise homothetic separa-bility restrictions and the additional restriction:(3.35) fli2O.Under these restrictions only six unknown parameters remain to be estimated.We refer to these restrictions as explicit groupwise linear logarithmic utility re-strictions. A second set of restrictions that implies intrinsic groupwise linearlogarithmic utility is that of groupwise neutral linear logarithmic utility, given in

66

I

equation (3.9) above. Corresponding to the three possible pairs of commodities.there are three possible sets of explicit group'tise linear karith,nic iitiliti restric-t!ons. Restrictions analogous to those given in equation (3.35) must hold for anyone of them.

A direct utility function with time-varying preferences is liiu'ar logarithmicin X , X2, and X3 if it can be written in the form:

(3.36) -- In U = F(ó (t) In X + 2(t) In X 2 itt) In X3 ,

where 51(t), ö2(t) and 53(t) are functions only of time. A necessary and sufficientcondition for linear logarithmic utility is that the direct utility function is groupwiselinear logarithmic in every pair of the three commodities. In particular, since thereare only three commodities, groupwise linear logarithmic utility for any two pairsof commodities is sufficient for linear logarithmic utility.

A direct utility function U with time-varying preferences is explicitly linearlogarithmic if it can be written in the form:(3.37) - in U = ó1(t) In X1 .f. (t) In X2 + O3(t) In X3 ± F(t).

The transtog approximation to an explicitly linear logarithmic utility function isnecessarily explicitly linear logarithmic. A necessary and sufficient condition forexplicit linear logarithmic utility is that the direct translog utility function isexplicitly groupwise linear logarithmic in every pair of the three commodities. Inparticular, since there are only three commodities, explicit groupwise linearlogarithmic utility for ans' two pairs of commodities is sufficient. Given linearlogarithmic utility, explicit groupwise linear logarithmic utility in any one of thethree possible pairs implies that the direct utility function is explicitly linearlogarithmic. For an explicitly linear logarithmic utility function the budget sharesof all commodities are independent of prices and total expenditure, dependingonly on time.

Finally, a direct utility function U with time-varying preferences is neutrallinear logarithmic if it can be written in the form:

(3.38) - In U = F(ó1 In K1 + 2 In X2 + O3 In X3, I),where ö 2 and are constants. Two alternative sets of conditions are jointlynecessary and sufficient for neutral linear logarithmic utility. First, the directtranslog utility function is both neutral and linear logarithmic arid it is either explic-itly neutral, explicitly linear logarithmic, or both. Alternatively, the direct translogutility function satisfies the restrictions given in equation (3.20), that is, the neutrallinear logarithmic utility restrictions. In either case, the empirical implicationsare identicalthe budget shares of all commodities are constant.

3.5. Groupwise equal raes oJ commodity augmentation

As an alternative point of departure for the analysis of time-varying pref-erences, we suppose that the quantities consumed of X1 , X2 and K3 are augmentedby factors A1(t), A2(t) and A3(z) respectively, where the augmentation factors arefunctions only of time. A direct utility function with commodity-augmenting time-varying preferences can be written in the form:

(3.39) -. In 1] = F(A1(i)X1 , A 2(t)X2, A3(t)X3).

67

Without loss of generality, the augmentation factors can be nornlali2ed so that theyall take the value unity for i = 0. Without further restrictions on the fjtnctjoji Fcommodity augmentation is not a testable hypothesis, since it has no empiricalimplications that can be refuted. Even if one restricts each augmentation factorto be drawn from the family of one-parameter algebraic functions. commodityaugmentation is still not a testable hypothesis since the parameters ; and fl,,are not identified.

A direct utility function with time.varying preferences that is characterizedby groupvise equal rates of coniniod:t augmentation can be written in the form

(3.40) In U = F(A(t)X1 , A(t)X2, A1(i)V3).

The cross partial derivatives of - In U with respect to time and In X In X2 orIn X3 are given by;

12 - In U 12F A 12F A

lnXt1lnXA1nX1lnX A12F /1,

+

(3.41)

(3.42)

3lnX13InX3 /13

a2lnU t2F A (2F A

aInK2tlnX3InX2 AlX2 /1+

I32F 43(flflX2IflX3 43

32 - In U i2F A A3lnX3t 1nX11nX3 A InX23InX3 A

F A

ThnXA3By observing that:

c2F ?2InU(i,j= 1,2,3),ThnXfln X1 ?In XälnX'

and the fact that equation (3.41) must hold everywhere, in particular, at the pointof approximation where t = 0, we can identify the first and second partial deriva-tives of - In U with the parameters of the direct translog utility function with time-varying preferences. Groupwise equal rates of commodity augmentation in X1and X2 implies the following sets of restrictions:(3.43) fl1, = flui + 1112A + fl13A3, fl2, = 11121. + 11221. + 112323,

fl3, = fl132 + 11231. + fl3323,where:

A A32= 1.3=-A /13

are the rates of commodity augmentation at the point of approximation. We note68

that this set of three restrictions involves the introduction of two new parameters.).and in the example given above. Hence under groupwise equal rates ofcommod-ity augnientation ouR Len unknown parameters teinain to be etiiiiaLed. Rctric-Lions analogous to those given in equation (3.43) must hold for the two remainingpossible sets of groupvise equal rates of commodity augmentation restrictions.

A necessary and sufficient condition for groupwise equal rates of commodityaugmentation of the direct utility function in X1 and X2 is thtt there exist twoscalars and for every t such that:

[aU/X ] (X , X2_ X3, t) [3U/(3X ] (qX1 , qX2, ;;3X3 , 0)

{'U,/X2](X1 , X2, X3, 1) - [i)U1/X2] (qX . qX2, ,

In other words, at every (there exist a proportional scaling of A'1 and X2, and ascaling for K3, so that the ratio of the indirect demands at time zero is the same asthe ratio of the indirect demands at time t. We can verify directly that a translogapproximation to a direct utility function with time-varying preferences charac-terized by groupwise equal rates of commodity augmentation is always character-ized by groupwise equal rates of commodity augmentation.

A direct utility function U with time-varying preferences that is characterizedby groupnise zero rules of con,moditv aug?nenlation can be written in the form

(3.45) - In U = F(X ,X2 A3(t)X3).

The corresponding restrictions on the parameters of the indirect translog utilityfunction with time-varying prelérences can be obtained from equation (3.43) abov3by setting). equal to zero. tJnder groupwise zero rates of commodity augmentationthe parameters must satisfy the restrictions:

(3.46) = fl13'3 f2, = fj23j.3, = fl3i3.

Under these restrictions, only nine unknown parameters remain to be estimatedRestrictions analogous to those given in equation (3.46) must hold for the tworemaining possible sets of groupwise zero rates of commodity augmentation.As before, we can show that the translog approximation to a direct utility functionwith time-varying preferences characterized by groupwise zero rates of commodityaugmentation is always characterized by groupwise zero rates of commodityaugmentation.

A direct utility function with time-varying preferences is characterized byequal rates of commodity augmentation in X , X2 and X3 if it can be written inthe form:

(3.47) In U = G(4(t)X1, A(jX2, ,4(t)X3).

A necessary and sufficient condition for equal rates of commodity augmentationof the direct utility function is that the direct utility function is characterized bygroupwise equal rates of commodity augmentation in every pair of the threecommodities. In particular, since there are only three commedities, groupwiseequal rates of commodity augmentation for any two pairs of commodities issufficient for equal rates of commodity augmentation.

69

(3.44)

Finally, a direct utility function with time-varying preferences is characterj,edby zero rates of cwn,nodjt1 aug?nenlalion if and only if it is characterized by group_wise zero tales of commodity augmentation in every pair of the three cornmodities. lii particular, since there are only three commodities, groupwise zero rates ofcommodity augmentation for any two pairs of commodities is sufficient. In fact,given equal rates of commodity augmentation, zero rates of commodity augmentalion for any pair of commodities implies zero rates of augmentation. In this casethe direct utility function is also explicitly neutral.

3.6. Dualitt'

The implications of separability, honiotheticity, linear logarithmic utilityand equal rates of commodity augmentation for the indirect utility function withtime-varying preferences are strictly analogous to the corresponding propertiesfor the direct utility function with time-varying preferences. They impose restric-tions on the direct demand functions as opposed to the indirect demand functions.Similarly, the parametric restrictions implied by these properties of the indirecttranslog utility functions are strictly analogous to the parametric restrictionsimplied by the corresponding properties of the direct translog function. Theroles of quantities consumed and ratios of prices to total expenditure are, of course,interchanged.

However, a given property of the direct utility function need not imply thesame property of the indirect utility function. For example, a groupwise homotheticdirect utility function does not correspond to a groupwise homothetic indirectutility function. The direct utility function is inclusively groupwise homotheticif and only if the indirect utility function is inclusively groupwise homothetic.Since homotheticity implies group%vise inclusive homotheticity for the groupconsisting of all commodities, direct homothetjcjy is equivalent to indirecthomotheticity. An alternative sufficient condition for groupwise homotheticityof both the direct and indirect utility functions is groupwise separability (eitherdirect or indirect) in the same group of commoditiesSimilarly, a groupwise commodity separable direct utility function does notcorrespond to a groupwise commodity separable indirect utility function. Directand indirect utility functions are groupwise commodity separable in the samegroup of commodities if and only if the utility function (either direct or indirect)is also groupwise homorhetic in the same group of commodities In addition.the direct utility function is groupwise hornothetically commodity separableif and only if the indirect utility function is groupwise homothetically commodityseparable.

In general, a groupwisetime-separable direct utility function does not cor-respond to a groupwjse timeseparable indirect utility function. Two alternativesufficient conditions for groupwise time separability of both the direct and theindirect utility functions in the same group of commodities are, first, inclusivegroupwise homotheticity of the utility function (either direct or indirect) in thesame group of commodities and, second, groupwise homothetic commodityseparability of the utility function (either direct or indirect) in the same group ofcommodities

70

An additive direct utility function does not correspond to an additive indirectutility function. Direct and indirect utility functions are simultaneously additiveonly if the utility function (either direct or indirect) is homothetic or if the utilityfunction (either direct or indirect) is linear logarithmic in all but one of the com-modities.'3 In addition, the direct utility function is additive and homothetic iiand only if the indirect utility function is additive and homothetic. On the otherhand, a neutral direct utility function always corresponds to a neutral indirectutility function. A groupwise linear logarithmic direct utility function alwayscorresponds to a groupwise linear logarithmic indirect utility function. Since agroupwise linear logarithmic utility function is groupwise homothetically corn-inodity separable, a groupwise neutral linear logarithmic direct utility functionalways corresponds to a groupwise neutral linear logarithmic indirect utilityfunction.

Moreover, a direct utility function with time-varying preferences charac-terized by groupwise equal rates of commodity augmentation always correspondsto an indirect utility function with time-varying preferences characterized bygroupwise equal rates of commodity augmentation. Likewise, a direct utilityfunction with time-varying preferences characterized by groupwise zero rates ofcommodity augmentation always corresponds to an indirect utility function withtime-varying preferences characterized by groupwise zero rates of commodityaugmentation. 14

Finally, a utility function is self-dual if both the direct and the indirect utilityfunctions (corresponding to the same preferences) have the same functionalThe only ttanslog utility function which is self-dual is the neutral linear logarithmicutility function. Neutral linear logarithmic utility functions are the only intrinsi-cally additive, homothetic, and stationary direct or indirect translog utilityfunctions. Direct and indirect translog utility functions can represent the samepreferences if and only if they are neutral linear logarithmic. Unless this stringentcondition is met, the direct and indirect translog approximations to a given pairof direct and indirect utility functions correspond to different preferences. so thatthe properties of these approximations are not fully comparable.

4. EMPIRICAL RESULTS4.1. Summary of tests

Tests of the restrictions on preferences we have considered can be carriedout in many sequences. We propose to test restrictions on the structure of pref-erences, given equality and symmetry restrictions, but not monotonicity andquasiconvexity restrictions. Monotonicity and quasiconvexity restrictions takethe form of inequalities rather than equalities, so that these restrictions do notaffect the asymptotic distributions of our statistics for tests of restrictions on thestructure of preferences.' These distributions are the same with or without

See Saniuclson [1965] arid Houthakker [1965]. We may also mention the "self-dual addilogsystem" introduced by Houthakker [l965]. This system is not generated by additive utility functionsexcept for special cases.

This is the special case introduced by Hicks [1969]. See also Samuelson [1969].14 For some of these results on the duality of direct and indirect utility functions, see Houthakker

[1960], Samuelson [1960] and Lau [1969b].See Malinvaud [19701, pp. 366-36&

71

imposing the restrictions assoCiated with fllOflOtOfliCity and cluasiconve\itvAfter the set of acceptable restrictions on the structure of preferences is deter-mined, we can impose the constraints implied by monotonicity and quasiconvexjyof the direct or indirect utility function.

Our proposed test procedure is presented in diagrammatic form in a series offive figures. We propose to test the restrictions derived from groupwise separability.homotheticity, groupwise homotheticity, and commodity augmenting change inpreferences, in parallel. Given groupwise homothetic separability for any group.we proceed to test the additional restrictions implied by groupwise linear logarith-mic utility, conditional on the restrictions implied by groupwise homothetjcseparability. Given the outcome of these tests we can determine the set of accept-able restrictions on the structure of preferences.

Beginning with separability, we recall that, first, groupwise separability fortwo of the three possible groups of two commodities from the third commodityimplies groupwise separability for the third group and additivity of the utilityfunction. Likewise, explicit groupwise separability for two of the three possiblegroups implies explicit groupwise separability for the third and explicit additivityof the utility function. Second, groupwise separability for two of the three possiblegroups of two commodities from time implies groupwise separability of the thirdgroup from time and neutrality ofthe utility function. Likewise, explicit groupwiseseparability kr two of the three possible groups from time implies explicit group-wise separability of the third group from time and explicit neutrality of the utilityfu nct ion.

We first test groupwise separability restrictions for each possible group. Ifwe accept groupwise separability for any group, we proceed to test explicit group-wise separabil!ty for that group. If we accept the hypothesis of groupwise separa-bility from the third commodity for any two ofthe three possible groups, we acceptthe hypothesis of additivity. If we accept the hypothesis of explicit groupwiseseparability from the third commodity for any two of the three groups, we acceptthe hypothesis of explicit additivity. If we accept the hypothesis of groupwiseseparability from time for any two of the three possible groups. we accept thehypothesis of neutrality. If we accept the hypothesis of explicit groupwiseseparability from time for any two of the three groups, we accept the hpothesisof explicit neutrality.

Our test procedure for separability is presented diagrammatically in Figure I.There are three sets of tests of this type; the diagram gives only one set of such tests.For each group we test groupwise separability from the third commodity andfrom time. Conditional on the corresponding groupwise separability restrictions,we proceed to test the hypothesis of explicit groupwise separability from the thirdcommodity and from time. Combining results from the tests for each of the threecommodity groups, we can test the hypotheses of additivity, explicit additivity,neutrality, and explicit neutrality.Continuing with homotheticity, we first test groupwise homotheticityrestrictions for each possible group. In parallel we test homotlietjcit' restrictionsfor the group consisting of all three commodities. If we accept homotheticity forall three commodities, we proceed to test explicit homotheticity If we acceptexplicit homotheticity for all three commodities, we proceed to test homogeneity.

72

r1. 2 Explicit Separability

from 3

Elualii aitI SviiiiiiirY1

2} Separability from 3

I

Sepa ahiht fron]

II

Equality Restrion neEuatit Restrfttiofl

I. 2 Explicit Separability

[

Figure I Tests of Separability. (There are three sets of tests of this type; this diagram goes onI one

set of such tests corresponding to the group 1. 2.)

Our test procedure for homotheticity, homotheticity, explicit homotheticity. and

homogeneity is presented diagrammatically in Figure 2.

IF we accept groupwise homotheticity for any group, we proceed to test

explicit groupwise homotheticity and inclusive groupwise homotheticity for that

group in parallel. If we accept both explicit groupwise homoetheticity and inclusive

groupwise homotheticity for any group. we accept the hypothesis of explicit

groupwise inclusive homotheticity. Conditional on explicit groupwise homo-

theticity for any group, we proceed to test groupwise homogeneity for that group.

Our test procedure for explicit and inclusive groupwise homotheticity is presented

diagrammatically in Figure 3. There are three sets of tests of this type; the diagram

gives only one set of such tests.We observe that a utility function with time-varying preferences is charac-

terized by linear logarithmic utility if it is groupwise linear logarithmic in all three

possible groups consisting of two commodities each. Inclusive groupwise homo-

theticity for all three groups implies that the utility function is linear logarithmic;

if we accept inclusive groupwise homotheticity for all three groups, we accept the

hypothesis of linear logarithmic utility. If we accept explicit inclusive groupwisehomotheticity for all three groups, we accept the hypothesis of explicit linear

logarithmic utility. Finally, if we accept groupwise homogeneity for all three groups,

we accept the hypothesis of neutral linear logarithmic utility.We can combine the results of our parallel tests of separability and homo-

theticity in order to draw conclusions about homothetic separability, If we accept

the hypothesis of groupwise separability for a group consisting of two commodities

from the third, and For the same group we accept the hypotheses of groupwise

73

One Equality Restriction One Equality Restriction

(Two EqualityRestrictions

[Homothet icity

One EqualityRestriction

ExplicitFiornothet icitv

One EqualityRestrict On

[moeneitY


ility and Svnsmetrv

One EqualityRestriction I

IT

I1, 2 HomotheticityL

i2, 3} Homotheticity

Figure 2 Tests of Honiotheticity.

homotheticity, explicit groupwise hornotheticity, inclusive groupwise homo-theticity, or groupwise homogeneity, we accept the hypotheses of groupwisehomothetic separability, groupwise explicitly homothetic separability, groupwiseinclusive homothetic separability, or grotipwise homogeneous separability,respectively, for that group. Similarly, if we accept the hypothesis of explicit group-wise separability for a given group, and for the same group we accept the hypothesisof groupwise homotheticity, explicit groupwise homotheticity, inclusive groupwisehomotheticity and groupwise homogeneity, we accept the hypotheses of groupwisehoniothetic explicit separability, explicit groupwise homothetjc separability,groupwise inclusive homothetic explicit separability and explicit groupwisehomogeneous separability, respectively, for that group. Finally, if we accept thehypotheses of additivity and homotheticity, we accept the hypothesis of homo-thetic additivity, If we accept the hypotheses of explicit additivity and eitherexplicit homotheticity or homogeneity, we accept the hypotheses of explicit linearlogarithmic utility and neutral linear logarithmic utility, respectively.

Proceeding under the hypothesis of additivity, if we accept inclusive groupwisehomotheticity of any one of the three possible groups of two commodities each,we accept the hypothesis of groupwise linear logarithmic utility for that group.If we accept inclusive groupwise homotheticity of any two of the three possible

74

l. 3} Homotheticity

One EqualityRest iction


[11.2) Inclusive Two Equality I {I.2} Explicit

Homotheticity Restrictions Hoinotheticity


I, 2) Hornotheticity

I. 2 Explicit

Inclusive Homotheticity

( One Equality\.. Restriction-

11

I. 2) Homogeneity1

Figure 3 Tests of Groupwise Homotheticity (There are three sets of tests of this type; this diagram

gives only one set of such tests corresponding to the group i. 2.)

groups of two commodities each, we accept linear logarithmic utility of the utility

function. 11 we accept explicit inclusive groupwise homotheticity of any one of the

three possible groups of two commodities each, we accept the hypothesis of explicit

groupwise linear logarithmic utility for that group. If we accept explicit inclusivegroupwise homotheticity of any two of the three possible groups of two commodi-

ties each, we accept the hypothesis of explicit linear logarithmic utility of the utility

function.Alternatively, proceeding under the hypothesis of explicit additivity, if we

accept inclusive groupwise homotheticity of any one of the three possible groupsof two commodities each, we also accept the hypothesis of explicit groupwiselinear logarithmic utility for that group. If we accept inclusive groupwise homo-

theticity of any two of the three possible groups of two commodities each, weaccept the hypoihesis of explicit linear logarithmic utility.

If we accept the hypothesis of groupwise homothetic separability for all three

possible groups of two commodities each and, in addition, we accept the hypoth-

esis of inclusive groupwise homotheticity of any one of the three possible groupsof two commodities each, we accept the hypothesis of linear logarithmic utility. If

75

either of these two hypotheses are strengthened to hold expIicitl, We accept thehypothesis of explicit linear logarithmic utility.

If we accept the hypothesis of groupwise honiothctic separability for anygroup of two commodities from the third, we proceed to test the hypothesis ofgroupwise linear logarithmic utility for that group, conditional on groupwjsehomothetic separability. liwe accept the hypothesis ofgroupwise linear logarithmicutility for group consisting of two commodities, and for that group we accept anytwo of the three hypotheses of explicit groupwise separability, explicit groupwisehomotheticity, and inclusive groupwise homotheticity, we accept the hypothesisof explicit linear logarithmic utility for that group. If, in addition, we accept thehypothesis of groupwise homogeneity for that group, we accept the hypothesis ofexplicit neutral linear logarithmic utility for that group. Ifwe accept the hypothesisofgroupwise linear logarithmic utility for any two of the three possible commoditygroups, we accept the hypothesis of linear logarithmic utility. Our test procedurefor groupwise linear logarithmic utility, given groupwise homothetic separabilityrestrictions, is presented digiammatically in Figure 4.

I. 2 HomothctjcSeparabihty

Inc Equalit\. Restoction

r .

Logarithmic tltiiitv

I. 3} Hornothetic

Separability

One Equality

Restriction

i. 3} Linear

Logarithmic Utility

Figure 4 Tests of Linear Logarithmic Utility.

2. 3 Homothetic

Separability

One Equality

Restriction

2.3} Linear

Logarithmic Utility

Finally, we consider tests of restrictions associated with commodity augment-ing changes of preferences over time. First we test the hypothesis of groupwiseequal rates of commodity augmentation for all three possible groups of twocommodities each. If we accept the hypothesis of equal rates of commodityaugmentation for any two of the three groups, we accept the hypothesis of equalrates of augmentation for all three commodities, and hence for all three groups.There is then no need to test zero rates because equal zero rates for all commoditiesis implied by explicit neutrality, which has been tested under separability. If weaccept the hypothesis of equal rates of commodity augmentation for only a singlegroup of two commodities we proceed to test the hypothesis that the rate of aug-mentation for that group is equal to zero. Our test procedure for equal rates ofcommodity augmentation is presented diagrammatically in Figure 5.

76

I


One Equality

Restriction

One EqualityRestrict ion

One Equality

Restriction


I

One Equality

Restriction

Figure S Tests of CommodityAugmefltiflg Change in Preferences.

4.2. Estimation

Our empirical results are based on time series data for prices and quantities

of durables, non-durables, and energy and time. We have fitted the equations for

budget shares generated by direct and indirect translog utility functions with time-

varying preferences, using the stochastic specification outlined above. Under this

specification only two equations are required for a complete econometric model of

demand. We have fitted equations for durables and for energy.i? For both direct

and indirect specifications we impose the hypothesis that the model of demand is

consistent with utility maximization, so that the parameters of this model satisfy

equality and symmetry restrictions. Given these restrictions, and the normaliza-

tion of at minus unity, eleven unknown parameters remain to be estimated in

our econometric model. Estimates of these parameters for the direct translog

utility function with time-varying preferences are given in the first column of

Table I. Estimates of these parameters for the indirect transiog utility function

with time-varying preferences are presented in the first column of Table 2.

' We employ the maximum likelihood estimator discussed. for esample. by Malinvaud [1970].

pp. 338-341 For the direct series of tests we assume that the disturbances are independent of the

quantities consumed. For the indirect series of tests we assume that the disturbances are independent

of the ratios 01 prices to the value of total expenditure.

77

Equality and Symmetry

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{2.31 Equal

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{l.3} ZeroRates

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preferences. For each of these hypotheses we impose equality and symmetryrestrictions and the corresponding groupwise separability and groupwise homo-theticity restrictions. The thirty-seventh column of Table I gives restricted esti-

mates for groupwise homothetic separability for the group {!, 2. Correspondingestimates tor groups { 1, 3 and {2, 3} are given in colunins thirty-eight and thirty-nine of Table I. Restricted estimates for groupwise linear logarithmic utility aregiven in columns forty through forty-two. for explicit groupwise linear logarithmicutility in columns forty-three through forty-five, and for groupwise neutral linearlogarithmic utility in columns forty-six through forty-eight. The correspondingrestricted estimates for the indirect transiog utility function are given in columns

thirty-seven through forty-eight of Table 2.The fifth and final set of restrictions on functional form is associated with

restrictions on the form of commodity augmenting change in preferences for thedirect translog utility function with time-varying preferences. We present restrictedestimates corresponding to the hypotheses of groupwise equal rates of commodityaugmentation in columns forty-nine through fifty-one of Table I and restricted

estimates corresponding to the hypotheses of zero rates of commodity augmenta-

tion in columns fifty-two through fifty.four. Corresponding estimates for theindirect translog utility function is given in columns forty-nine through fIfty-foui

of Table 2.

43. Test statistics

To test the validity of equality restrictions implied by the theory of demand

and restrictions on the form of the utility function, we employ test statistics basedon the likelihood ratio A, where:

maxA

max .°

The likelihood ratio is the rati' of the maximum value of the likelihood function

for the econometric model ofdemand without restriction to the maximum value

of the likelihood function for the model w subject to restriction.We have estimated econometric models of demand from data on U.S. personal

consumption expenditures for 1947-1971. There are twenty-five observations for

each behavioral equation, so that the number of degrees of freedom available for

statistical tests of the theory of demand is fifty for either direct or indirect specifica-

tion. For normally distributed disturbances the likelihood ratio is equal to the

ratio of the determinant of the restricted estimator of the variance-covariancematrix of the disturbances to the determinant of the unrestricted estimator, each

raised to the power - (n/2).Our test statistic for each set of restrictions is based on minus twice the

logarithm of the likelihood ratio, or:

2 In A = n(In itj - In

where t is the restricted estimator of the variance-covariance matrix and is

97

the unrestricted estimator. Under the null hypothesis the likelihood ratiotest

statistic is distributed, asymptotically as chi-squared with a number of degrees offreedom equal to the number of restrictions to be tested.

To control the overall level of significance for each series of tests, direct ancindirect, we set the level of significance lôr each series at 0.05. We then allocatethe overall level of significance among the various stages in each series o tests.We test groupwise separability, homotheticity, groupwise homotheticity gfoup.wise linear logarithmic utility, and groupwise equal rates of commodity augmenition proceeding conditionally on the validity of the equality and symmetryrestrictions implied by the theory of demand. These tests are not "nested" so thatthe sum of the levels of significance for each of the five sets of hypotheses is an Upperbound for he level of significance of tests of the sets of hypotheses consideredsimultaneously. We assign a level of significance of 0.01 to each of the five sets ofrestrictions.

There are twelve restrictions associated with groupwise separability andexplicit groupwise separability: we assign a level of significance of 0.0008 to each.There are three restrictions associated with honiotheticity; we assign o.00 toeach. There are twelve restrictions associated with groupwise homotheticity.we assign t10008 to each. There are three restrictions associated with groupwiselinear logarithmic utility; we assign 0.0033 to each of these restrictions. Finally,there are six restrictions associated with groupwise equal rates of commodityaugmentation; we assign a level of significance of 0.0017 to each.

For our econometric models of demand based on the direct and indirecttranslog utility functions with time-varying preferences we have assigned levels ofsignificance to each of our tests of hypotheses about the structure of preferences soas to control the overall level of significance for all tests at 0.05. The probabilityof a false rejection for one test among the collection of all tests we consider isless than or equal to 0.05. With the aid of critical values for our test statistics givenin Table 3, the reader can evaluate the results of our tests for alternative significancelevels or for alternative allocations of the overall level of significance amongstages of our test procedure. Test statistics for each of the hypotheses we haveconsidered about the structure of preferences are given in Table 4.

TABLE 3

CIUTIcAL VALUES OF 72/DEGRELS or FREEDOM

Degrees of Level of significancefreedom 0.10 0.05 0.01 0.005 0.001 0.0005

I 2.71 3.84 6.64 7.88 10.83 12.122 2.30 3.00 4.61 5.30 6.91 7.60

The results of our tests of restrictions on preferences based on the directtranslog utility function, as presented in Table 4, are, first, that the group { I, 2},durables and non-durables, is separable from commodity 2, energy, and that thegroup {2, 3}, non-durables and energy, is separable from commodity 1, durables.These two sets of restrictions imply additivity. Second, the group {l, 3}, durables

98

99

TAIILE4TEST STATSTtCS

Degrees of Critical Test StatisticsHypothesis Freedom Values Direct Indirect

Given equality and .cynimetrvGroupwise separability

l,2}from3 I 11.35 4.40 0.551,3) from 2 1 11.35 27.52 15.142, 3) from 1 I 11.35 1.86 30.35

ll.2}fromt 1 11.35 15.44 3.8311, 3) from t I 11.35 7.08 27.96{2,3}fromt I 11.35 4.11 37.73

Homotheticity 2 5.98 28.24 25.37Groupwise homotheticity

1,2) 1 11.35 1.87 1.08{I,3} I 11.35 1.90 24.68(2.3) I 11.35 3.21 17.65

Groupwise equal rates(1.2) I 1032 11.89 2.13

1,3) 1 10.32 12.45 16.50(2,3) 1 10.32 14.18 30.38

Given groupwise separabilityGroupwise explicit separability

1,2)iroin3 I 11.35 12.61 1.391. 3} from 2 1 11.35 0.88 0.38

(2, 3) from I 1 11.35 11.61 5.271,2) fromi I 11.35 0.00 3.99

(1.3) from t 1 11.35 4.97 0.67{2.3}fromt I 11.35 28.16 15.17

G iren hortwtltetieiryExplicit homotheticity 2 5.98 10.09 1.20

Given groupwise homothelicityGroupwise inclusive homothesicity

1,2) 1 11.35 29.25 13.04I, 3) I 11.35 3.56 21.99

(2,3} 1 11.35 20.77 13.11Groupwise explicit homoetheticity

(I,2} I 11.35 12.70 1.6311,3) 1 11.35 10.77 0.16(2,31 1 11.35 26.20 13.99

Given groupvise equal ratesGroupwise zero rates

(1,2) 1 10.32 0.23 5.12(1,3) 1 10.32 2.21 2.60(2, 3) I 10.32 0.08 4.90

Given explicit homotheticityHomogeneity 1 9.13 3.21 45.50

Given group wise explicit inclusive hornotheticityGroupwise homogeneity

(1,2} I 11.35 3.69 38.89(1,3) I 11.35 32.01 13.12(2, 3) I 11.35 13.82 52.24

Given groupwise homothetic separabilityGroupwise linear logarithmic utility

(1,2) I 9.13 15.72 27.06(1,3) 1 9.13 1.57 20.50(2,3) 1 9.13 16.02 10.35

- k

and energy, and the group 2, 3 }, non-durables and energy, are separable from timeThese two sets of restrictions imply neutrality. Third, all three possible groups o'two commodities each are groupwise hornothetic; hence, each of these groups ishomotheticallY separable. Fourth, the gioup 1, 3}, durahies and energy, isexplicitly inclusive groupwise homothetic, which implies explicit linear logarithn.1jutility. Finally, the group { I, 3} is explicitly separable from time, which

impliesneutral linear logarithmic utility or constant budget shares. This specification isdetermined by only two unknown parameters.

Turning to the results of our tests of restrictions on preferences based on theindirect translog utility function, as presented in Table 4, we find that the group

1, 3}, consisting of durables and non-durables, is explicitly groupwise separablefrom commodity 2, energy, and from time. This group is also explicitly groupwj5ehomothetic and has equal rates of commodity augmentation equal to zero.The form of the system of equations corresponding to the indirect utility functionis as follows:

p1X1 + fit, (In [p1/M] - In {p,/M])Al - - I + fl33 In (p3!Al) + fi3

p2X2 -- fi11 (In [p1/M] - In [p2/M])M - 1 4.1133 In(p3/M) +

p3X3 + fi33 In (p3/M) + [i3'M - I + /?33 In (p3/M) ± /33, . I

This specification is determined by five unknown parameters. We recall that thedirect and indirect utility function represent the same preferences only if they areself-dual. The dual of the neutral linear logarithmic direct utility function is theneutral linear logarithmic indirect utility function. We conclude that the testresults for the two models do not coincide. This is not surprising, since thestochasticspecifications used in the two sets of tests are different.

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The Structure of Consumer Preferences · 2020. 3. 20. · consumer preferences and changes in preferences over time in Section 4. Our tests are based on time series data for U.S