jorgenson and lau traslog

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American Economic Association

Transcendental Logarithmic Utility FunctionsAuthor(s): Laurits R. Christensen, Dale W. Jorgenson and Lawrence J. LauReviewed work(s):Source: The American Economic Review, Vol. 65, No. 3 (Jun., 1975), pp. 367-383Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1804840 .

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TranscendentalLogarithmic U t i l i t yFunctions

By LAURITS R. CHRISTENSEN, DALE W. JORGENSON, AND LAWRENCE J. LAU*The traditionalstarting point for econo-metric studies of consumer demand is asystem of demand functions giving thequantity consumed of each commodity asa function of total expenditure and theprices of all commodities. Tests of thetheory of demand are formulated by re-quiring that the demand functions be

consistent with utility maximization. Ad-ditive and homothetic utility functionshave played an important role in formulat-ing tests of the theory of demand. If theutility function is homothetic, expenditureproportions are independent of total ex-penditure. If the utility function is addi-tive and homothetic, elasticities of sub-stitution among all pairs of commoditiesare constant and equal.'An example of the traditional approachto demandanalysis is the system of doublelogarithmic demand functions employedin the pioneering studies of consumer de-mand by Henry Schultz, Richard Stone,and Herman Wold. If the theory of de-mand is valid and demand functions aredouble logarithmic, the utility function islinear logarithmic.2Similarly, the Rotter-dam system of demand functions em-ployed by A. P. Barten and Henri Theil isconsistent with utility maximization onlyif the utility function is linearlogarithmic.3A linear logarithmic utility function isboth additive and homothetic; all expendi-

ture proportions are constant, and elas-ticities of substitution among all pairs ofcommodities are constant and equal tounity.Hendrik Houthakkerand Stone have de-veloped alternative approachesto demandanalysis that retain the assumption of ad-ditivity while dropping the assumption ofhomotheticity.4 Stone has employed a lin-ear expenditure system, based on a utilityfunction that is linear in the logarithms ofquantity consumed ess a constant for eachcommodity. The constants are interpretedas initial commitments; incremental ex-penditure proportions,derivedfrom quan-tities consumed in excess of the initialcommitments, are constant for all varia-tions in total expenditureand prices. If allinitial commitments are zero, the utilityfunction is linearlogarithmic n form.Non-zerocommitmentspermit expenditurepro-portionsto vary with total expenditure.Houthakker has employeda direct addi-log system, based on a utility functionthat is additive in functions that arehomogeneous in the quantity consumedfor each commodity. The degree of homo-geneity may differ from commodity tocommodity, permitting expenditure pro-portions to vary with total expenditure.Ifthe degree of homogeneity is the same forall commodities, the addilog utility func-tion is additive and homothetic. RobertBasmann, Leif Johansen, and Kazuo Satohave combined he approachesof Houthak-ker and Stone, definingeach of the homo-geneous functions in the direct addilog

* University of Wisconsin-Madison, Harvard Uni-versity, and Stanford University, respectively.' The class of additive and homothetic utility func-tions was first characterized by Abram Bergson.2 See Schultz, Stone (1954a), Wold, and Robert Bas-mann, R. C. Battalio, and J. H. Kagel.3 See Barten (1964, 1967, 1969), Daniel McFadden,and Theil (1965, 1967, 1971).

I See Houthakker (1960) and Stone (1954b). Thelinear expenditure system was originally proposed byLawrence Klein and Herman Rubin.367


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368 THE AMERICANECONOMIC REVIEW JUNE 1975utility function on the quantity consumedless a constant for each commodity. Theresulting utility function is additive butnot homothetic.5Our first objective is to develop tests ofthe theory of demand that do not employadditivity or homotheticity as part of themaintained hypothesis. For this purposewe introduce new representations of theutility function in Section I. Our approachis to represent the utility function by func-tions that are quadratic in the logarithmsof the quantities consumed. The resultingutility functions provide a local second-order approximation to any utility func-tion. These utility functions allow expendi-ture shares to vary with the level of totalexpenditure and permit a greatervarietyof substitution patterns among commod-ities than functions based on constant andequal elasticities of substitution among allpairs of commodities.Our second objective is to exploit theduality between prices and quantities inthe theory of demand. A complete modelof consumer demand implies the existenceof an indirect utility function, defined ontotal expenditure and the prices of all com-modities.6 The indirect utility function ishomogeneous of degree zero and can beexpressed as a function of the ratios ofprices of all commodities to total expendi-ture. The indirect utility function is usefulin characterizing systems of direct demandfunctions, giving quantities consumed asfunctions of the ratios of prices to totalexpenditure. The direct utility function isuseful in characterizing systems of indirectdemand functions, giving the ratios ofprices to total expenditure as functions ofthe quantities consumed. The system con-sisting of direct utility function and in-

direct demand functions is dual to the sys-tem consisting of indirect utility functionand direct demand functions.7We represent the indirect utility func-tion by functions that are quadratic in thelogarithms of ratios of prices to total ex-penditure, paralleling our treatment of thedirect utility function. The resulting in-direct utility functions provide a localsecond-order approximation to any in-direct utility function. These indirect util-ity functions are not required to be addi-tive or homothetic. The duality betweendirect and indirect utility functions has

been used extensively in Houthakker'spathbreaking studies of consumer demand.Paralleling the direct addilog demand sys-tem, Houthakker has employed an indirectaddilog system, based on an indirect utilityfunction that is additive in ratios of pricesto total expenditure.8We refer to our representation of thedirect utility function as the direct trans-cendental logarithmic utility function, ormore simply, the direct translog utilityfunction. The utility function is a trans-cendental function of the logarithms ofquantities consumed. Similarly we refer toour representation of the indirect utilityfunction as the indirect transcendentallogarithmic utility function, or, moresimply, the indirect translog utility function.Earlier, we introduced transcendental log-arithmic functions into the study of pro-duction.9 The duality between direct andindirect translog utility functions is anal-ogous to the duality between translog pro-

5 A recent survey of econometric studies of consumer(lemand is given b-y Alan Brown and Angus Deaton.6 The indirect utility function was introduced byGiovanni Antonelli, and independently by HaroldHotelling.

I Indirect demand functions were introduced byAntonelli The duality between direct and indirectutility functions is discussed by Lau (1969a); the dual-ity between systems of direct and indirect demand func-tions is discussed by Wold, John Chipman, and LeonidHurwicz8 See Houthakker (1960). This demand system wasoriginally proposed by Conrad Leser and has also beenemployed by W. H. Somermeyer, J. G. M. Hilhorst,and J. W. W. A. Wit.9 See the authors (1971, 1973).


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 369duction and price frontiers employed inour study of production.For an additive direct utility functionratios of indirect demand functions, givingthe ratios of prices, depend only on thequantitiesof the two commodities nvolved.The direct addilog and linear expendituresystems, together with the system em-ployed by Basmann, Johansen, and Sato,have this property. Similarly,for an addi-tive indirect utility function, ratios of di-rect demand functions giving the ratiosof quantities depend only on the pricesofthe two commodities involved. The in-direct addilog system has this property.For an additive and homothetic directutility function the ratios of indirect de-mand functions depend only on the ratiosof quantities. Furthermore, the indirectutility function is also additive and homo-thetic, so that ratios of direct demandfunctions depend only on ratios of prices.The use of direct and indirect translogutility functions permits us to test theserestrictionson direct and indirect demandfunctions. We do not impose the restric-tions as part of the maintainedhypothesis.We presentstatistical tests of the theoryof demand in Section II. These tests canbe divided into two groups. First, we testrestrictionson the parametersof the directtranslog utility function implied by thetheory of demand. We test these restric-tions without imposing the assumptionsofadditivity and homotheticity. We test pre-cisely analogousrestrictionson the param-eters of the indirect translog utility func-tion. Second, we test restrictions on thedirecttranslogutility function correspond-ing to restrictions on the form of theutility function. In particular, we test re-strictions corresponding o additivity andhomotheticity of the direct translogutilityfunction. Again, we test precisely analo-gous restrictions on the indirect translogutility function.We present empiricaltests of the theory

of demand based on time-series data forthe United States for 1929-72 in SectionIII. The data include pricesand quantitiesconsumed of the services of consumers'durables, nondurable goods, and otherservices. For these data we present directtests of the theory of demandbased on thedirect translog utility function, and in-direct tests of the theory based on the in-direct translog utility function. For bothdirect and indirect tests we first test theextensive set of restrictionsimplied by thetheory of demand. Proceeding condition-ally on the validity of the theory, we testrestrictions on the form of the direct andindirectutility functions impliedby the as-sumptionsof additivity and homotheticity.

I. TranscendentalLogarithmicUtility FunctionsA. Direct TranslogUtility Function

The directutility unction U can be repre-sented in the form:(1) In U = In U(X1, X2, * * , Xm)where Xi is the quantity consumed of theith commodity. The consumer maximizesutility subject to the budget constraint:(2) E piXi = Mwhere pi is the price of the ith commodityand M is the value of total expenditure.The first-order conditions for a maxi-mum of utility can be written:

0 liz U p1X1(3) Inx, u (j= 1,2,...,m)where,u is the marginal utility of income.From the budget constraint we obtain:

( ~ 1 all,u(4) = E -U M c In Xiso that:9 In U pJXj a In U(5) - M _C,inXj M a InXi

(j = 11 21 M


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370 THE AMERICAN ECONOMIC REVIEW JUNE 1975To preserve symmetry with our treat-ment of the indirect utility function givenbelow, we approximate the negative of the

logarithm of the direct utility function bya function quadratic in the logarithms ofthe quantities consumed:(6) -In U = ao + Z ai In Xi

+ 2 ZZ3ijlnXlnXjUsing this form for the utility function weobtain:(7) ahj+ Z3 ji In Xi

Mj (aXk 1: ki InXi)(j = 1, 2, . ,m)

To simplify notation we write:(8) aM = E ak(9) 3Mi= Zki (i = 12, ..., m)so that:

pjXj aj + , fji In Xi(10) M= MZ3ifXM ahm+ E mi InXi(j = 1, 2, ..., m)

The budget constraint implies that:(11) E pixi

so that, given the parameters of any m- 1equations for the budget shares pjXj/M(j= 1, 2, . .. , m), the parameters of themthequation,a,mandmj (j = 1 ,2, . . .., m),can be determined from the definitions ofaxM and 3Mj (j= 1 2, . . , m).Since the equations for the budget sharesare homogeneous of degree zero in theparameters, a normalization of the param-eters is required for estimation. A con-venient normalization for the parametersof the direct translog utility function is:(12) aM = Zai = - 1

B. Indirect Translog Utility FunctionThe indirect utility function V can berepresented in the form:

(13) In V = In V (- .....We determine the budget share for thejth commodity from the identity:10

(14) Pj jX _ tnVI nVM tinpj aInM

(j= 1,2,... ,m)Preserving symmetry with the directutility function, we approximate the loga-rithm of the indirect utility function by afunction quadratic in the logarithms of theratios of prices to the value of total ex-penditure:

(15) In V = ao + ai In-M

+ OijIn Pln- n-Using this form for the utility function weobtain:(16) in + Ina In pi M

(j=1,2,...,m)0On V PA(17) - ak /+ OkiIn-)a In M

As before, we simplify notation by writing:(18) am= ak(19) OMi Oki (i = l, 2,.., m)so that:

ai + ji InPi(20) M M

Om + E mi In Pi(j = 1, 2, ... , m)

'0 This is the logarithmic form of Rene Roy's Identity.


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 371The budget constraint mplies that giventhe parametersof any mr-I equations forthe budget shares, the parameters of the

mnthquation, am and I3mj (j= 1, 2, ... . m),can be determinedfrom the definitions ofatM and f3j (j= 1, 2, . . . , m). As before,we can normalizethe parametersso that:(21) afm= E ai = - 1

II. Testing the Theory of DemandA. StochasticSpecification

The first step in implementingan econo-metric model of demand based on thedirect translog utility function is to add astochastic specification to the theoreticalmodel based on equations for the budgetshares pjXj/M (j= 1, 2, . . . , m). Giventhe disturbances in any mr-I equations,the disturbanceof the remaining equationcan be determined from the budget con-straint. Only m-1 equations are requiredfor a complete econometric model of de-mand.B. Equality and Symmetry

We estimate m- I equations for thebudget shares, subject to normalizationofthe parameteram appearing n each equa-tion at minus unity. If the equations aregenerated by utility maximization, theparameters mj (j= 1, 2, . . . , m) appear-ing in each equation must be the same.This results in a set of restrictionsrelatingthe m parametersappearingin each of them-1 equations, a total of m(m-2) re-strictions. We refer to these as equalityrestrictions.The logarithm of the direct translogutility function is twice differentiable inthe logarithmsof the quantities consumed,so that the Hessian of this function issymmetric. This gives rise to a set of re-strictions relating the parameters of thecross-partialderivatives:(22) j=l3ji (i $ j; i, j = 1, 2, . . , m)

There are (1/2)(m-2)(m-1) restrictionsof this type among the parametersof them-1 equations we estimate directly andrn-1 such restrictions among the param-eters of the equations we estimate in-directly from the budget constraint. Werefer to these as symmetry restrictions. Thetotal number of symmetry restrictions is(1/2)m(m- 1).If equations for the budget shares aregenerated by maximization of a directtranslog utility function, the parameterssatisfy equality and symmetry restrictions.There are (1/2) m(3m-5) such restrictions.

C. AdditivityIf the direct utility function U is addi-tive,we can write

(23) In U = F(Z In Ui(Xi))where each of the functions Ui dependsononly one of the commoditiesdemandedXi,and F is a real-valued function of one vari-able.

The parametersof the translog approxi-mation to an additive direct utility func-tion can be written:a In U a InUi(24) - =-F' =aInXi aInXi

(i = 1, 2, . .. m)a2ln U(25) - x Xa InXio InXi a In Ui aIn Ui-F" -a lit Xi a In X-

(k $ j; i, j = 1, 2, ... , m)where the logarithmicderivatives,

aIn U(26) F' = (i=1,2, ..., n)aIln Uia2 n U(27) F" = aIn Uia In Uj

(i. j 1. 27 ...... m)


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372 THE AMERICAN ECONOMIC REVIEW JUNE 1975are independent of i and j.Under additivity the parameters of thetranslog utility function satisfy the restric-tions:(28) d3ij = Oaciaj

(i j; i, j = 1, 2, . . .,y m)where

F"(29) F=- -(F')2We refer to these as additivity restrictions.The total number of such restrictions is(1/2) (m-2) (m-1) .The translog approximation to an addi-tive utility function is not necessarily addi-tive. The direct translog utility function isadditive if and only if In U can be writtenas the sum of m functions, each dependingon only one of the quantities demanded.We refer to such a function as explicitlyadditive. Explicit additivity of the translogutility function implies the additivity re-strictions given above and the additionalrestriction:(30) 0= 0We refer to this restriction as the explicitadditivity restriction. XVe note that thetranslog approximation to an explicitlyadditive function is explicitly additive.

D. HomotheticityIf the direct utility function is homo-

thetic, we can write:(31) In U = F(ln H(X1, X2, . . . Xm))where the function II is homogeneous ofdegree one.The parameters of the translog approxi-mation to a homothetic direct utility func-tion can be written:

a In U 9F a InH(32) - -- = - - = aid In Xi &InH OInXi(i = 112 . . .m)

&2 In U(33) -( a In X8a In XjaF a2 In H_a zH a n Xja In Xj

Ca2F daInH aInH+- _ __=- IaIn H2 a In X, a In Xj_

(i,j = 1 2, ... ,m)Homogeneity of degree one of the functionH implies that:

a In H(34) E -X=m 2In H(35) E -- = Oj=a InXia InXj

(i = 1, 2, ... ., m)Under homotheticity the parameters ofthe translog utility function satisfy the re-strictions:

(36) f3i =oas (i = 1, 2,.. ,m)where

a2F(37) a n 2d9 n H2and we have used the normalization(38) ,a ai =-1We refer to these as homotheticity restric-tions. There are n- 1 such restrictions.The translog approximation to a homo-thetic utility function is not necessarilyhomothetic. A necessary and sufficient con-dition for homotheticity of the translogutility function is that it is homogeneous,so that:(39) of = OWe refer to this restriction as the homo-geneity restriction. We note that the trans-log approximation to a homogeneous func-tion is homogeneous.


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 373E. A dditivity and Homotheticity

The class of additive and homothetic di-rect utility functions coincides with theclass of utility functions with constantelasticities of substitution among all com-modities. If the direct utility function Uis additive and homothetic, we can write:(40) In U = F( 8iX)or(41) In U = F(ZE 8i in Xi)The second form is a limiting case of thefirst corresponding to unitary elasticitiesof substitution among all commodities.The parameters of the translog approxi-mation to an additive and homothetic util-ity function satisfy the additivity andhomotheticity restrictions given above, sothat:(42) 3ijO=caiaj (i ?j;i,j= 1,2,...,m)(43) /3ti = (a + 6)a-i + 2cxt

(i = 1, 2, . .., m)where we have used the normalization(44) Zai-1

The translog approximation to an addi-tive and homothetic utility function is notnecessarily additive and homothetic. Wecan, however, identify the parameters of atranslog approximation with the param-eters of the additive and homothetic utilityfunction given above, as follows:(45) a + 0 =p(46) avi 5i (i = 1, 2,...,m)As before, the parameter 0 corresponds to:

F"(47) (-F)2(Ft)The translog approximation to a utilityfunction characterized by unitary elas-

ticities of substitution among all com-

modities satisfies the restriction:(48) r+ 0 = p = OUnder this restriction the expendituresshares are constant:

pjXj, j + OajE ai In X- - -I =-02: a n X iThe value of the parameter 0 is arbitraryso that we can let afand 0 be equal to zero.The translog approximation to a utilityfunction with unitary elasticities of sub-stitution has the same empirical implica-tions as a linear logarithmic utility func-tion, which is explicitly additive andhomogeneous.

F. DualityIn implementing an econometric modelof demand based on the indirect translogutility function the first step as before is toadd a stochastic specification to the theo-retical model based on equations for thebudget shares pjXj/M (j= 1, 2, . . , mi).Only m-i equations are required for acomplete model. If equations for the bud-get shares are generated from the indirecttranslog utility function, the parameterssatisfy equality and symmetry restrictionsthat are strictly analogous to the corre-sponding restrictions for the direct trans-log utility function.Additivity and homotheticity restric-tions for the indirect translog utility func-

tion are analogous to the correspondingrestrictions for the direct translog function.The direct utility function is homothetic ifand only if the indirect utility function ishomothetic." In general, an additive di-rect utility function does not correspond toan additive indirect utility function.'2However, if the direct utility function isadditive and homothetic, the indirect util-11See, for instance, Paul Samuelson (1965) and Lau(1969b).12 See Lau (1969b).


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374 THE AMERICANECONOMIC REVIEW JUNE 1975ity function is additive and homothetic.Direct and indirect utility functions corre-sponding to the same preferences are addi-tive only if the direct utility function ishomothetic'3 or the direct utility functionis linear logarithmic in all but one of thecommodities.14 In this relationship theroles of direct and indirect utility functionscan be interchanged.A direct utility function is self-dual if thecorresponding indirect utility function hasthe same form.15 The only direct translogutility function which is self-dual is thelinear logarithmic utility function. Linearlogarithmic utility functions are the onlyadditive and homothetic direct or indirecttranslog utility functions. Translog directand indirect utility functions represent thesame preferences if and only if they areself-dual; that is, if and only if they arelinear logarithmic. Unless this stringentcondition is met, the direct and indirecttranslog approximations to a given pair ofdirect and indirect utility functions repre-sent different preferences.

III. Empirical ResultsA. Summary of TestsWe have developed statistical tests ofthe theory of demand that do not employthe assumptions of additivity and homo-theticity for individual commodities or forcommodity groups. At this point we out-line restrictions on equations for the bud-get shares corresponding to direct and in-

direct translog utility functions. First, wepresent restrictions derived from the theoryof demand; that is, from the basic hy-pothesis of utility maximization. Second,

we present restrictions derived from hy-potheses about functional form such asadditivity and homotheticity.Our empirical tests are based on data forthree commodity groups-services of con-sumers' durables, nondurable goods, andother services-so that we specialize ourpresentation of statistical tests of thetheory of demand to the three-commoditycase, m= 3. A complete econometric modelfor either direct or indirect translog utilityfunctions is provided by any pair of equa-tions for the budget shares. We considerthe system of two equations,

(50) p1 =M

a+1+il In X1+i12 In X2+313 In X3- 1+OM1 In Xl+fM2 In X2+1M3 In X3

p2X2M

a2+2l1 In X1+:22 In X2+0323 In X3- 1+OM1 In Xl+,BM2 In X2+OM3 In X3corresponding to the direct translog utilityfunction and the system of equations,

(52) -M-=

ai+il In'MPi012 In-M+ P13n-MaI+OM, In -+/M2 In M+OM3 In -

P22 P531 M

PinP P2 P3a2+/21 In-+122 In -+123 In -m _Pi P2 P3-1 +OM1 In- +fM2 In -+OM3 In -corresponding to the indirect translog util-ity function. Restrictions for the two sys-tems are perfectly analogous so that the

13See Houthakker (1960), Samuelson (1965), andLau (1969b).14 This is the special case introduced by John Hicks.See also Samuelson (1969).1"See Samuelson (1965) and Houthakker (1965). Wemay also mention the "self-dual addilog system" in-troduced by Houthakker (1965). This system is not

generated by additive utility functions except forspecial cases.


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 375following outline may be applied to eithersystem. We emphasize the fact that thetwo systems representthe same preferencesif and only if(54) /ij = 0 (i, j = 1, 2, 3)1. Equality Restrictions. The parametersI M1, 3M2, OM3} occur in both equationsand must take the same values. There arethree equality restrictions.2. SymmetryRestrictions.One restrictionof this type is explicit in the two equationswe estimate directly, namely,(55) 012 = 021In addition, we estimate the parameters/31 and 32 from the equations:(56) /31 = /M1 - Oil - /21(57) /32 = OM2 - /12 - /22so that the two additional symmetry re-strictions are implicit in the two equationswe estimate. We write these restrictionsinthe form:(58) /13 = /M1 - /3l - /21(59) /23 = /M2 - /12 - /22There are three symmetry restrictions al-together.We can identify tests of the six equalityand symmetryrestrictionswith tests of thetheory of demand. We next consider testsof hypotheses about functional form.3. AdditivityRestrictions.Given the equal-ity and symmetry restrictions,the additiv-ity restrictionstake the form:(60) /12 = 0a1a2(61) /13 = Oal(-1 - a, - a2)(62) /23 = 0a2(-1 - - a2)We introduce the additional parameter0,so that there are two independent restric-tions of this type.

4. HomotheticityRestrictions. Given theequality and symmetry restrictions, thehomotheticity restrictionstake the form:(63) 1M1 = ai(64) 3M2 = 0a2(65) 3M3 = 0(- - a - a2)We introduce the additional parametero-,so that there are two independent restric-tions of this type.5. Explicit Additivity Restrictions. Giventhe equality, symmetry, and additivityrestrictions, a translog utility function isexplicitly additive under the further re-striction:(66) 0 = 06. Homogeneity Restrictions. Given theequality, symmetry, and homotheticityrestrictions, the homogeneity restrictiontakes the form:(67) a= O7. Linear LogarithmicUtility Restrictions.Given the equality, symmetry, additivity,and homotheticity restrictions, the trans-log utility function reduces to linear loga-rithmic form under the additional re-strictions:(68) = = 0

Tests of hypotheses about the form ofthe utility function can be carriedout in anumberof differentsequences.We proposeto test additivity and homotheticity inparallel. Each of these hypotheses consistsof two independent restrictions. Our testprocedure is presented in diagrammaticform in Figure 1. We also present alterna-tive test procedures.Forexample,we couldfirst test additivity and then test homo-theticity only if additivity is accepted. Ourtest procedure is indicated with doublelines while possible alternative proceduresare indicated with single lines.


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376 THE AMERICAN ECONOMIC REVIEW JUNE 1975

UnrestrictedX

CS x Restriction

L Equality and Symmetry |

t wo Restrictions C Two Restriction )Additivity Four Restrictions Homotheticitv

F.oestrictioAs ETwoestrictionsCTAdditivity and Homotheticity

FIGURE 1. TESTS OF ADDITIVITY AND HOMOTHETICITYThe next step in our test procedure de-pends on the outcome of the tests of addi-tivity and homotheticity. If we acceptboth additivity and homotheticity restric-

tions, we continue by testing explicit addi-tivity and homogeneity in parallel. If weaccept both hypotheses, we conclude that

utility is linear logarithmic. A second pos-sibility is that we accept additivity butreject homotheticity. In this case we con-tinue by testing explicit additivity. Thethird possibility is that we accept homo-theticity but reject additivity; we continueby testing homogeneity. If we reject both

Additivity and HomotheticityOne Restriction On etition

Explicit Additivity wo Resrictions Homogeneity

n s on Oneestrictio on_Linear Logarithmic Utility

Additivity Not Homotheticity Homotheticity Not Additivity_m _ I I _One Restriction One Restriction

Explicit Additivity Not Homotheticity Homogeneity Not AdditivityFIGURE 2. TESTS CONDITIONAL ON ADDITIVITY OR HOMOTHETICITY


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 377additivity and homogeneity, we terminatethe sequence of tests. These tests are pre-sented diagrammatically in Figure 2.

B. EstimationOur empirical results are based on time-series data for U. S. personal consumptionexpenditures for 1929-72. The data in-clude prices and quantities of the servicesof consumers' durables, nondurable goods,and other services. We have fitted theequations for budget shares generated bydirect translog and indirect translog utilityfunctions, employing the stochastic speci-

fication outlined above. Under this specifica-tion only two equations are required for acomplete econometric model of demand.We have fitted equations for the servicesof consumers' durables (durables) and fornondurable goods (nondurables). Thereare forty-four observations for each be-havioral equation, so that the number ofdegrees of freedom available for statisticaltests of the theory of demand is eighty-eight for either direct or indirect specifica-tion.For both direct and indirect specifica-tions the maintained hypothesis consists ofthe unrestricted form of the two behavioralequations for the budget shares. We esti-mate the behavioral equations for durablesand nondurables by the method of maxi-mum likelihood and derive estimates of theparameters of the behavioral equation forservices, using the budget constraint.'6 Themaximum likelihood estimates of the pa-rameters of all three behavioral equationsare invariant with respect to the choice ofthe two equations to be estimated directly.

The unrestricted behavioral equations,normalized so that the parameter aM isminus unity in each equation, involve four-teen unknown parameters or seven un-known parameters for each equation. Un-restricted estimates of these parameters forthe direct translog utility function are pre-sented in the first column of Table 1. Thefirst hypothesis to be tested is that thetheory of demand is valid; to test this hy-pothesis we impose the six equality andsymmetry restrictions. Restricted esti-mates of the fourteen unknown parametersare presented in the second column ofTable 1. Unrestricted and restricted esti-mates of the fourteen unknown parameteisfor the indirect translog utility functionare presented in the first and second col-umns of Table 2.Given the validity of the theory of de-mand, the remaining hypotheses to betested are restrictions on the functionalform. First, we test the hypothesis that thedirect utility function is additive; thishypothesis requires that we impose twoadditional restrictions. The correspondingrestricted estimates of the unknown pa-rameters for the direct translog utilityfunction are given in the third column ofTable 1. Second, we test the hypothesisthat the direct utility function is homo-thetic without imposing the additivity re-strictions; this hypothesis requires tworestrictions in addition to the equality andsymmetry restrictions. The restricted esti-mates of the unknown parameters for thedirect translog utility function are given inthe fourth column of Table 1. Finally, weimpose both additivity and homotheticityrestrictions, obtaining the restricted esti-mates presented in the fifth column ofTable 1. The corresponding restricted esti-mates for the indirect translog utility func-tion are given in the third, fourth, and fifthcolumns of Table 2.The next stage of our test procedure iscontingent on the outcome of our tests of

16 We employ the maximum likelihood estimator dis-cussed, for example, by Edmond Malinvaud, pp. 338-41. In the computations we use the Gauss-Newtonmethod described by Malinvaud, p. 343. For the directseries of tests we assume that the disturbances are inde-pendent of the quantities consumed. For the indirectseries of tests we assume that the disturbances are inde-pendent of the ratios of prices to the value of total ex-penditure.


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378 THE AMERICANECONOMIC REVIEW JUNE 1975TABLE 1-ESTIMATES OF THE PARAMETERS OF THE DIRECT TRANSLOG UTILITY FUNCTION

Explicit Homo- ExplicitEquality Additivity Additivity, geneity, Additivity Additivity LinearUnrestricted and Homo- and Homo- Not Homo- Not and Homo- and Homo- LogarithmicParameters Estimates Symmetry Additivity theticity theticity theticity Additivity theticity geneity UtilityDURABLES

ai -.145 -.138 -.144 -.139 -.146 -.147 -.140 -.147 -.147 -.127(.00302) (.00262) (.00357) (.00257) (.00328) (.00279) (.00220) (.00265) (.00280) (.00378).0490 -.0216 -.137 -.268 -.123 -.102 -.317 -.100 -.121 -(.0888) (.0129) (.00971) (.0140) (.00778) (.00694) (.0122) (.00637)012 .500 .139 .142 .170 .0983 - .147 - .0671 -(.194) (.0277) (.0227) (.0293) (.0219) (.0178) (.00386)p1J -.555 -.0737 .115 -.139 .0790 - .170 - .0543 -(.206)

kM1 .659 .0440 .120 .0407 .0542 -.102 - -.100 - -(.864) (.0356) (.0372)6M2 3.53 .165 .213 .136 .176 -.339 - -.323 - -(1.76) (.113)fMm -3.94 .265 .209 .116 .141 -.275 - -.261 - -(1.96) (.107) (.104) (.0318)

NONDURABLESa2 -.467 -.468 -.472 -.464 -.473 -.471 -.461 -.472 -.471 -.495(.00420) (.00353) (.00345) (.00402) (.00385) (.00487) (.00328) (.00440) (.00336) (.00792)621 .272 .139 .142 .170 .0983 - .147 - .0671 -(.299) (.0277) (.0227) (.0293) (.0219) (.0178) (.00386)16= -.536 -.334 -.306 -.168 -.179 -.339 -.216 -.323 -.241 -(.774) (.0563) (.0571) (.0619) (.0448) (.0326)#23 .259 .361 .377 .133 .256 - .0693 - .174 -(.866)15M1 .282 .0440 .120 .0407 .0542 -.102 - -.100 -(.555) (.0356) (.0372)16M2 -.271 .165 .213 .136 .176 -.339 - -.323 -(1.55) (.113)flMa .105 .265 .209 .116 .141 -.275 - -.261 -(1.69) (.107) (.104) (.0318)

additivity and homotheticity. If we acceptadditivity, but not homotheticity, we testexplicit additivity by imposing one furtherrestriction. If we accept homotheticity,but not additivity, we test homogeneity byimposing one further restriction. The re-stricted estimates under these restrictionsare given in the sixth and seventh columnsof Table 1. The corresponding estimatesfor the indirect translog utility functionare given in the sixth and seventh columnsof Table 2.If we accept both additivity and homo-theticity hypotheses, we test the hypothe-sis that the direct utility function is ex-plicitly additive by imposing one furtherrestriction. Similarly, we test homogeneityby imposing one further restriction. Fi-nally, we impose both restrictions underthe hypothesis that utility is linear loga-rithmic. The restricted estimates undereach of these sets of restrictions are given

in the eighth through tenth columns ofTable 1. The corresponding restricted esti-mates for the indirect translog utilityfunction are given in the eighth throughtenth columns of Table 2. The direct andindirect translog utility functions are self-dual if they are linear logarithmic, so thatrestricted estimates for the two alternativeeconometric models given in the final col-umns of the two tables are identical.C. Test Statistics

To test the validity of restrictions im-plied by the theory of demand and restric-tions on the form of the utility function,we employ test statistics based on thelikelihood ratio X, where:max S

(69) X= max Sn


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 379TABLE 2-ESTIMATES OF THE PARAMETERS OF THE INDIRECT TRANSLOG UTILITY FUNCTION

Explicit Homo- ExplicitEquality Additivity Additivity, geneity, Additivity Additivity LinearUnrestricted and Homo- and Homo- Not Homo- Not and Homo- and Homo- LogarithmicParameters Estimates Symmetry Additivity theticity theticity theticity Additivity theticity geneity Utility

DURABLESas -.141 -.125 -.128 -.119 -.119 -.127 -.120 -.120 -.120 -.127(.00215) (.00300) (.00279) (.00390) (.00370) (.00228) (.00392) (.00364) (.00370) (.00378)oil -. 115 -.0970 -.0595 -.131 -.0930 -.0662 -.106 -.105 -.0796 -(.0188) (.0188) (.00972) (.0273) (.00823) (.00684) (.0246) (.00618)012 .695 .0816 .0188 .100 .0282 - .113 - .0454 -(.0447) (.0300) (.0153) (.0403) (.0131) (.0379) (.00238)

s13 -.473 -.0174 .0144 -.0334 .0212 - -.00754 - .0342 -(.0527)#Ml -.801 -.0324 -.0262 -.0644 -.0434 -.0662 - -.105 -(.172) (.0379) (.0216)PM2 4.52 -.489 -.491 -.273 -.183 -.631 - -.440 - -(.419) (.134)jM3 -3.50 -.280 -.274 -.202 -.137 -.393 - -.330 - -(.464) (.114) (.107) (.213)

NONDURABLESa2 -.472 -.499 -.494 -.506 -.503 -.497 -.504 -.503 -.502 -.495(.00472) (.00533) (.00479) (.0576) (.00597) (.00383) (.00612) (.00576) (.00612) (.00792)021 -.872 .0816 .0188 .100 .0282 - .113 - .0454 -(.120) (.0300) (.0153) (.0403) (.0131) (.0379) (.00238)622 1.11 -.668 -.565 -.539 -.300 -.631 -.348 -.440 -.189 -(.247) (.0881) (.0715) (.0810) (.0307) (.0610)02a -.227 .0972 .0560 .166 .0891 - .234 - .143(.164)#Ml -1.63 -.0324 -.0262 -.0644 -.0434 -.0662 - -.105 - -(.212) (.0379) (.0216)OM2 2.98 -.489 -.491 -.273 -.183 -.631 - -.440 - -(.464) (.134),6Ma -1.11 -.280 -.274 -.202 -.137 -.393 - -.330 - -(.334) (.114) (.107) (.213)

The likelihood ratio is the ratio of themaximum value of the likelihood function? for the econometric model of demand wsubject to restriction to the maximumvalue of the likelihood function for themodel Q without restriction. For normallydistributed disturbances the likelihood ra-tio is equal to the ratio of the determinantof the restricted estimator of the variance-covariance matrix of the disturbances tothe determinant of the unrestricted estima-tor, each raised to the power - (n/2).Our test statistic for each set of restric-tions is based on minus twice the logarithmof the likelihood ratio, or:(70) -21nX=n(ln | :, | -In I o |)where 2. is the restricted estimator of thevariance-covariance matrix and 2Q is theunrestricted estimator. Under the null hy-pothesis this test statistic is distributed,

asymptotically, as chi-square with numberof degrees of freedom equal to the numberof restrictions to be tested.To control the overall level of signifi-cance for each series of tests, direct and in-direct, we set the level of significance foreach series at .05. We allocate the overalllevel of significance among the variousstages in each series. We first assign a levelof significance of .01 to the test of equalityand symmetry restrictions implied by thetheory of demand. We then assign a levelof significance of .04 to tests of restrictionson functional form. These two sets of testsare ''nested"'; under the null hypothesisthe sum of levels of significance provides aclose approximation to the level of signifi-cance for both sets of tests simultaneously.We test additivity and homotheticity inparallel, proceeding conditionally on thevalidity of the theory of demand. Thesetests are not nested so that the sum of


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380 THE AMERICANECONOMIC REVIEW JUNE 1975levels of significance for each of the twohypotheses is an upper bound for the levelof significance of tests of the two hy-potheses considered simultaneously. Thereare four possible outcomes of our paralleltests of additivity and homotheticity: Re-ject both, accept both, accept only addi-tivity, and accept only homotheticity. Ifwe reject both hypotheses, our test pro-cedure terminates. If we accept either orboth of these hypotheses, we proceed totest explicit additivity or homogeneity, orboth.If we accept both additivity and homo-theticity, we test explicit additivity andhomogeneity in parallel, conditional on thevalidity of additivity and homotheticity.Again, the tests are not nested, so that thesum of levels of significance for the twotests provides an upper bound for the testsconsidered simultaneously. If we acceptonly additivity, we proceed to test explicitadditivity, conditional on additivity. Simi-larly, if we accept only homotheticity, weproceed to test homogeneity.Since our three alternative proceduresfor testing explicit additivity and homo-geneity are mutually exclusive, we carryout tests of these hypotheses one time atmost. We assign levels of significance to thefour hypotheses-additivity, explicit addi-tivity, homotheticity, and homogeneity-with the assurance that the sum of levelsof significance provides an upper boundfor the level of significance for all four tests.We assign a level of significance of .01 toeach of the four hypotheses. With the aid

TABLE 3-CRITICAL VALUES OFXI/DEGREES OF FREEDOM

Degrees of Level of SignificanceFreedom .10 .05 .025 .01 .0051 2.71 3.84 5.02 6.63 7.882 2.39 3.00 3.69 4.61 5.304 1.94 2.37 2.79 3.32 3.726 1.77 2.10 2.41 2.80 3.09

of critical values for our test statisticsgiven in Table 3 the reader can evaluatethe results of our tests for alternative allo-cations of the overall levels of significanceamong stages of the test procedure.Test statistics for both direct and in-direct tests of the theory of demand and ofrestrictions on functional form are presentedin Tables 4 and 5. At a level of significanceof .01 we reject the hypothesis that restric-tions implied by the theory of demand arevalid for either direct or indirect series oftests. With this conclusion we can termi-nate the test sequence, since these resultsinvalidate the theory of demand.One interesting alternative to our testprocedure is to maintain the theory of de-mand and to test the validity of restric-tions on the form of the utility function.Proceeding conditionally on the validity ofthe theory of demand, we could test thevalidity of restrictions on the form of theutility function. For the direct series oftests we would reject the restrictions im-plied by additivity and homotheticity.This conclusion would hold for our pre-ferred procedure of testingr these hypothe-ses in parallel, for a test of either hy-pothesis conditional on the validity of the

TABLE 4-TEST STATISTICS FOR DIRECT AND INDIRECTTESTS OF THE THEORY OF DEMAND AND TESTS

OF ADDITIVITY AND HOMOTHETICITY

Degrees ofHypothesis Freedom Direct Indirect

Theory of DemandEquality and Symmetry 6 5.64 10.25

Given the Theory of DemandAdditivity 2 33.63 2.95Homotheticity 2 13 71 22.68Additivity and Homo-

theticity 4 19.02 16.24Additivity Given Homo-theticity 2 24.33 9.81Homotheticity GivenAdditivity 2 4.42 29.54


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 381other, or for a joint test of the two hy-potheses.For the indirect series of tests we wouldreject the restrictions implied by homo-theticity, but we would accept the restric-tions implied by additivity at a level ofsignificance of .01. For the direct series oftests, we could terminate the test sequenceconditional on the theory of demand atthis point. For the indirect series of tests,we could test the validity of explicit addi-tivity, given additivity but not homo-theticity. We would accept the hypothesisof explicit additivity, given additivity at alevel of significance of .01, conditional onthe validity of the theory of demand.In previous econometric studies of de-mand, the theory of demand has beenmaintained together with the assumptionof additivity. A second alternative to ourtest procedure is to maintain both thetheory of demand and the restrictions im-plied by additivity. We can test homo-

TABLE 5-TEST STATISTICS FOR DIRECT AND INDIRECTTESTS OF RESTRICTIONS ON FUNCTIONAL FORM, GIVEN

ADDITIVITY OR HOMOTHETICITY OR BOTH

Degrees ofHypothesis Freedom Direct IndirectGiven Additivity

Explicit Additivity 1 19.97 1.59Given Homotheticity

Homogeneity 1 1.47 4.73Given Additivity and Homotheticitv

Explicit Additivity 1 11.25 4.02Homogeneity 1 1.74 1.16Linear LogarithmicUtility 2 29.67 48.31Given Explicit Additivity and Homotheticity

Linear LogarithmicUtility 1 48.09 92.61Given Additivity and Homogeneity

Linear LogarithmicUtility 1 57.60 95.46

theticity; for either direct or indirect seriesof tests we reject this hypothesis. Givenadditivity but not homotheticity, wewould reject explicit additivity for the di-rect series of tests and accept explicit addi-tivity for the indirect series of tests. Eitherof these sets of results would rule out linearlogarithmic utility.

IV. Summary and ConclusionOur objective has been to test the theoryof demand without imposing the assump-tions of additivity and homotheticity aspart of the maintained hypothesis. Foreither the direct or the indirect series oftests, we conclude that the theory of de-mand is inconsistent with the evidence.These results confirm the findings of Wold(in association with Lars Jureen) for thedouble logarithmic demand system andBarten for the Rotterdam demand system.'7At the same time our results provide thebasis for more specific conclusions.If the theory of demand were valid, the

double logarithmic form for the system ofdemand functions would imply that theutility function is linear logarithmic. Simi-larly, the validity of the theory of demandand the Rotterdam form for the system ofdemand functions would imply linear loga-rithmic utility. An equally valid interpre-tation of the results of Wold and Barten isthat the theory of demand is valid, butthat utility is not linear logarithmic. Ourresults rule out this alternative interpreta-tion and make possible an unambiguousrejection of the theory of demand.A possible alternative to our test pro-cedure is to maintain the validity of thetheory of demand. Proceeding condition-ally on the validity of demand theory, wewould reject the hypothesis of additivityfor the direct series of tests. Additivity ofthe direct utility function is employed as a

17 See Wold, especially pp. 281-302; Barten (1969). Adetailed review of tests of the theory of demand isgiven by Brown and Deaton, pp. 1188-95.


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382 THE AMERICANECONOMIC REVIEW JUNE 1975maintained hypothesis by Houthakker inthe direct addilog demand system, byStone in the linear expenditure system, andby Basmann, Johansen, and Sato in thedemand system incorporating features ofboth.Again, proceeding conditionally on thevalidity of the theory of demand, we wouldaccept the hypothesis of additivity for theindirect series of tests. This hypothesis ismaintained by Houthakker in the imple-mentation of the indirect addilog demandsystem. Additivity of the direct and in-direct utility functions is mutually con-sistent if and only if the direct utility func-tion is homothetic, so that the acceptanceof additivity of the indirect utility func-tion and the rejection of homotheticityrule out acceptance of additivity for thedirect utility function.

REFERENCESG. B. Antonelli, "On the MathematicalTheory of Political Economy," Italy 1886,translated by J. S. Chipman and A. P.Kirman, in Chipman et al., eds., Prefer-ences, Utility, and Demand, New York1971.A. P. Barten, "ConsumerDemand Functionsunder Conditions of Almost Additive Pref-erences," Econometrica,Jan./Apr. 1964, 32,1-38.,"Evidence on the Slutsky Conditionsfor Demand Equations,"Rev. Econ. Statist.,Feb. 1967, 49, 77-84.'"MaximumLikelihood Estimation of

a CompleteSystem of Demand Equations,"Eur. Econ. Rev., Fall 1969, 1, 7-73.R. L. Basmann, "Hypothesis Formulation inQuantitative Economics: A ContributiontoDemand Analysis," in J. Quirk and H.Zarley, eds., Papers in Quantitative Eco-nomics, Lawrence 1968, 143-202., R. C. Battalio, and J. H. Kagel,"Commenton R. P. Byron's 'The RestrictedAitken Estimation of Sets of Demand Rela-tions,' " Econometrica,Mar. 1973, 41, 365-70.

A. Bergson, "Real Income, Expenditure Pro-portionality, and Frisch's New Methods,"Rev. Econ. Stud., Oct. 1936, 4, 33-52.J. A. C. Brown and A. S. Deaton, "Models ofConsumerBehaviour: A Survey," Econ. J.,Dec. 1972, 82, 1145-1236.J. S. Chipman, "Introduction to Part II," inChipman et al., eds., Preferences, Utility,and Demand, New York 1971, 321-31.L. R. Christensen,D. W. Jorgenson, and L. J.Lau, "ConjugateDuality and the Transcen-dental Logarithmic Production Function,"Econometrica,July 1971, 39, 255-56.- , ',and , "TranscendentalLogarithmic Production Frontiers," Rev.Econ. Statist., Feb. 1973, 55, 28-45.J. R. Hicks, "Direct and Indirect Additivity,"Econometrica,Apr. 1969, 37, 353-54.H. Hotelling, "Edgeworth'sTaxation Paradoxand the Nature of Demand and SupplyFunctions," J. Polit. Econ., Oct. 1932, 40,577-616.H. S. Houthakker, "Additive Preferences,"Econometrica,Apr. 1960, 28, 244-57."A Note on Self-dual Preferences,"Econometrica,Oct. 1965, 33, 797-801.L. Hurwicz, "On the Problem of Integrabilityof Demand Functions," in J. S. Chipman,et al., eds., Preferenres, Utility, and De-mand, New York 1971, 174-214.L. Johansen, "On the Relationships betweenSomeSystemsof Demand Functions,"Liike-teloudellinenAikakauskirja,no. 1, 1969, 30-41.L. R. Klein and H. Rubin, "A Constant-Utility Index of the Cost of Living," Rev.Econ. Stud., 1948, 15, 84-87.L. J. Lau, (1969a) "Direct and Indirect Util-ity Functions: Theory and Applications,"Inst. Bus. Econ. Res., workingpap. no. 149,Berkeley, Mar. 1969., (1969b) "Duality and the Structureof Utility Functions," J. Econ. Theory,Dec. 1969, 1, 374-96.C. E. V. Leser, "Family Budget Data andPrice Elasticities of Demand," Rev. Econ.Stud., Nov. 1941, 9, 40-57.D. L. McFadden, "Existence Conditions forTheil-Type Preferences," mimeo., Univ.California, Berkeley 1964.


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VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 383E. Malinvaud, Statistical Methods of Econo-metrics, 2d ed., Amsterdam 1970.R. Roy, De l'Utilite':Contribution2 la T1euoriedes Choix, Paris 1943.P. A. Samuelson, "Using Full Duality to Showthat SimultaneouslyAdditive Direct and In-direct Utilities Implies Unitary Price Elas-ticity of Demand,"Econometrica,Oct. 1965,33, 781-96.-, "CorrectedFormulationof Direct andIndirect Additivity," Economnetrica,Apr.1969, 37, 355-59.K. Sato, "Additive Utility Functions withDouble-Log ConsumerDemand Functions,"J. Polit. Econ., Jan./Feb. 1972, 80, 102-24.H. Schultz, The Theory and MeasurementofDemand, Chicago 1938.W. H. Somermeyer, J. G. M. Hilhorst, andJ. W. W. A. Wit, "A Method for Estimat-ing Price and Income Elasticities from Timo

Series and its Application of Consumers'Expenditures in the Netherlands, 1949-1959," Statist. Stud., 1961, 13, 30-53.J. R. N. Stone, (1954a) Measurementof Con-sumers' Expenditures and Behaviour inthe United Kingdom, Vol 1, Cambridge1954.,(1954b) "LinearExpenditureSystemsand Demand Analysis: An Application tothe Pattern of British Demand," Econ. J.,Sept. 1954, 64, 511-27.H. Theil, "The Information Approachto De-mand Analysis," Econometrica, Jan. 1965,33, 67-87., Economics and Information Theory,Amsterdam1967.-, Principles of Econometrics, Amster-dam 1971.H. Wold, Demand Analysis: A Study in Econ-ometrics,New York 1953.

jorgenson and lau traslog

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