Transcendental Logarithmic Utility Functions Author(S): Laurits R

American Economic Association

Transcendental Logarithmic Utility Functions Author(s): Laurits R. Christensen, Dale W. Jorgenson and Lawrence J. Lau Source: The American Economic Review, Vol. 65, No. 3 (Jun., 1975), pp. 367-383 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/1804840 . Accessed: 09/08/2013 14:37

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By LAURITS R. CHRISTENSEN, DALE W. JORGENSON, AND LAWRENCE J. LAU*

The traditional starting point for econo- ture proportions are constant, and elas- metric studies of consumer demand is a ticities of substitution among all pairs of system of demand functions giving the commodities are constant and equal to quantity consumed of each commodity as unity. a function of total expenditure and the Hendrik Houthakker and Stone have de- prices of all commodities. Tests of the veloped alternative approaches to demand theory of demand are formulated by re- analysis that retain the assumption of ad- quiring that the demand functions be ditivity while dropping the assumption of consistent with utility maximization. Ad- homotheticity.4 Stone has employed a lin- ditive and homothetic utility functions ear expenditure system, based on a utility have played an important role in formulat- function that is linear in the logarithms of ing tests of the theory of demand. If the quantity consumed less a constant for each utility function is homothetic, expenditure commodity. The constants are interpreted proportions are independent of total ex- as initial commitments; incremental expenditure. If the utility function is addi- penditure proportions, derived from quan- tive and homothetic, elasticities of sub- tities consumed in excess of the initial stitution among all pairs of commodities commitments, are constant for all varia- are constant and equal.' tions in total expenditure and prices. If all An example of the traditional approach initial commitments are zero, the utility to demand analysis is the system of double function is linear logarithmicin form. Non- logarithmic demand functions employed zero commitments permit expenditure pro- in the pioneering studies of consumer de- portions to vary with total expenditure. mand by Henry Schultz, Richard Stone, Houthakker has employed a direct addi- and Herman Wold. If the theory of de- log system, based on a utility function mand is valid and demand functions are that is additive in functions that are double logarithmic, the utility function is homogeneous in the quantity consumed linear logarithmic.2Similarly, the Rotter- for each commodity. The degree of homo- dam system of demand functions em- geneity may differ from commodity to ployed by A. P. Barten and Henri Theil is commodity, permitting expenditure pro- consistent with utility maximization only portions to vary with total expenditure. If if the utility function is linear logarithmic.3 the degree of homogeneity is the same for A linear logarithmic utility function is all commodities, the addilog utility func- both additive and homothetic; all expendi- tion is additive and homothetic. Robert Basmann, Leif Johansen, and Kazuo Sato * University of Wisconsin-Madison, Harvard Uni- have combinedthe approachesof Houthak- versity, and Stanford University, respectively. ' The class of additive and homothetic utility func- ker and Stone, defining each of the homo- tions was first characterized by Abram Bergson. geneous functions in the direct addilog 2 See Schultz, Stone (1954a), Wold, and Robert Bas- mann, R. C. Battalio, and J. H. Kagel. I See Houthakker (1960) and Stone (1954b). The 3 See Barten (1964, 1967, 1969), Daniel McFadden, linear expenditure system was originally proposed by and Theil (1965, 1967, 1971). Lawrence Klein and Herman Rubin. 367

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions 368 THE AMERICAN ECONOMIC REVIEW JUNE 1975 utility function on the quantity consumed direct demand functions is dual to the sys- less a constant for each commodity. The tem consisting of indirect utility function resulting utility function is additive but and direct demand functions.7 not homothetic.5 We represent the indirect utility func- Our first objective is to develop tests of tion by functions that are quadratic in the the theory of demand that do not employ logarithms of ratios of prices to total ex- additivity or homotheticity as part of the penditure, paralleling our treatment of the maintained hypothesis. For this purpose direct utility function. The resulting in- we introduce new representations of the direct utility functions provide a local utility function in Section I. Our approach second-order approximation to any in- is to represent the utility function by func- direct utility function. These indirect util- tions that are quadratic in the logarithms ity functions are not required to be addi- of the quantities consumed. The resulting tive or homothetic. The duality between utility functions provide a local second- direct and indirect utility functions has order approximation to any utility func- been used extensively in Houthakker's tion. These utility functions allow expendi- pathbreaking studies of consumer demand. ture shares to vary with the level of total Paralleling the direct addilog demand sys- expenditure and permit a greatervariety tem, Houthakker has employed an indirect of substitution patterns among commod- addilog system, based on an indirect utility ities than functions based on constant and function that is additive in ratios of prices equal elasticities of substitution among all to total expenditure.8 pairs of commodities. We refer to our representation of the Our second objective is to exploit the direct utility function as the direct trans- duality between prices and quantities in cendental logarithmic utility function, or the theory of demand. A complete model more simply, the direct translog utility of consumer demand implies the existence function. The utility function is a trans- of an indirect utility function, defined on cendental function of the logarithms of total expenditure and the prices of all com- quantities consumed. Similarly we refer to modities.6 The indirect utility function is our representation of the indirect utility homogeneous of degree zero and can be function as the indirect transcendental expressed as a function of the ratios of logarithmic utility function, or, more prices of all commodities to total expendi- simply, the indirect translog utility function. ture. The indirect utility function is useful Earlier, we introduced transcendental log- in characterizing systems of direct demand arithmic functions into the study of pro- functions, giving quantities consumed as duction.9 The duality between direct and functions of the ratios of prices to total indirect translog utility functions is anal- expenditure. The direct utility function is ogous to the duality between translog pro- useful in characterizing systems of indirect demand functions, giving the ratios of I Indirect demand functions were introduced by prices to total expenditure as functions of Antonelli The duality between direct and indirect the quantities consumed. The system con- utility functions is discussed by Lau (1969a); the dual- sisting of direct utility function and in- ity between systems of direct and indirect demand functions is discussed by Wold, John Chipman, and Leonid Hurwicz 5 A recent survey of econometric studies of consumer 8 See Houthakker (1960). This demand system was (lemand is given b-y Alan Brown and Angus Deaton. originally proposed by Conrad Leser and has also been 6 The indirect utility function was introduced by employed by W. H. Somermeyer, J. G. M. Hilhorst, Giovanni Antonelli, and independently by Harold and J. W. W. A. Wit. Hotelling. 9 See the authors (1971, 1973).

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 369 duction and price frontiers employed in of demand based on time-series data for our study of production. the United States for 1929-72 in Section For an additive direct utility function III. The data include prices and quantities ratios of indirect demand functions, giving consumed of the services of consumers' the ratios of prices, depend only on the durables, nondurable goods, and other quantitiesof the two commodities involved. services. For these data we present direct The direct addilog and linear expenditure tests of the theory of demand based on the systems, together with the system em- direct translog utility function, and in- ployed by Basmann, Johansen, and Sato, direct tests of the theory based on the in- have this property. Similarly, for an addi- direct translog utility function. For both tive indirect utility function, ratios of di- direct and indirect tests we first test the rect demand functions giving the ratios extensive set of restrictions implied by the of quantities depend only on the prices of theory of demand. Proceeding condition- the two commodities involved. The in- ally on the validity of the theory, we test direct addilog system has this property. restrictions on the form of the direct and For an additive and homothetic direct indirect utility functions implied by the as- utility function the ratios of indirect de- sumptionsof additivity and homotheticity. on the ratios mand functions depend only I. TranscendentalLogarithmic of quantities. Furthermore, the indirect Utility Functions utility function is also additive and homothetic, so that ratios of direct demand A. Direct Translog Utility Function functions depend only on ratios of prices. The directutility function U can be repre- The use of direct and indirect translog sented in the form: utility functions permits us to test these (1) In U = In U(X1, X2, * * , Xm) restrictions on direct and indirect demand functions. We do not impose the restric- where Xi is the quantity consumed of the tions as part of the maintained hypothesis. ith commodity. The consumer maximizes We present statistical tests of the theory utility subject to the budget constraint: of demand in Section II. These tests can = M be divided into two groups. First, we test (2) E piXi restrictionson the parametersof the direct where pi is the price of the ith commodity translog utility function implied by the and M is the value of total expenditure. theory of demand. We test these restric- The first-order conditions for a maxi- tions without imposing the assumptions of mum of utility can be written: test additivity and homotheticity. We pre- 0 liz U p1X1 cisely analogous restrictions on the parameters of the indirect translog utility func- (3) Inx, Uu (j= 1,2,...,m) tion. Second, we test restrictions on the where,u is the marginal utility of income. direct translog utility function correspond- From the budget constraint we obtain: ing to restrictions on the form of the utility function. In particular, we test re- ( ~ 1 all,u (4) = E - strictions correspondingto additivity and U M c In Xi homotheticity of the direct translog utility so that: function. Again, we test precisely analo- 9 In U pJXj a In U gous restrictions on the indirect translog _ (5) - M utility function. C,in Xj M a In Xi = We present empirical tests of the theory (j 11 21 M

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To preserve symmetry with our treat- B. Indirect Translog Utility Function ment of the indirect utility function given The indirect utility function V can be below, we approximate the negative of the represented in the form: logarithm of the direct utility function by a function quadratic in the logarithms of (13) In V = In V (- . the quantities consumed: ....

(6) -In U = ao + Z ai In Xi We determine the budget share for the jth commodity from the identity:10 + 2 ZZ3ijlnXlnXj tnVaI n V Using this form for the utility function we (14) Pj=X j _ obtain: M tinpj aInM (j= 1,2,... ,m) (7) ahj+ Z3 ji In Xi Preserving symmetry with the direct MEj (aXk+ 1: ki In Xi) utility function, we approximate the logarithm of the indirect utility function by a (j = 1, 2, . ,m) function quadratic in the logarithms of the ratios of prices to the value of total ex- To simplify notation we write: penditure: (8) aM = E ak (15) In V = ao + ai In-M (9) 3Mi = Zki (i = 12, ..., m) so that: P + Oij Inln- n- pjXj aj + , fji In Xi (10) M= MZ3ifX Using this form for the utility function we M In ahm+ E mi Xi obtain: (j = 1, 2, ..., m) (16) in + In The budget constraint implies that: a In pi M (11) E pixi (j=1,2,...,m) 0OIn V PA (17) - ak /+ OkiIn-) a In M so that, given the parameters of any m- 1 equations for the budget shares pjXj/M As before, we simplify notation by writing: (j= 1, 2, . .. , m), the parameters of the (18) am= ak mth equation,a,m and jmj (j = 1 ,2, . . .., m), can be determined from the definitions of (19) OMi O ki (i = l, 2,.., m) axM and 3Mj (j= 1 2, . . , m). so that: Since the equations for the budget shares Pi are homogeneous of degree zero in the ai + ji In M parameters, a normalization of the param- (20) M eters is required for estimation. A con- Pi Oam + E mi In venient normalization for the parameters of the direct translog utility function is: (j = 1, 2, ... , m) (12) aM = Zai = - 1 '0 This is the logarithmic form of Rene Roy's Identity.

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The budget constraint implies that given There are (1/2)(m-2)(m-1) restrictions the parameters of any mr-I equations for of this type among the parameters of the the budget shares, the parameters of the m-1 equations we estimate directly and mnthequation, am and I3mj (j= 1, 2, ... . m), rn-1 such restrictions among the param- can be determined from the definitions of eters of the equations we estimate in- atM and f3j (j= 1, 2, . . . , m). As before, directly from the budget constraint. We we can normalize the parameters so that: refer to these as symmetry restrictions. The total number of symmetry restrictions is (21) afm = ai = - 1 E (1/2)m(m- 1). If equations for the budget shares are II. Testing the Theory of Demand generated by maximization of a direct A. StochasticSpecification translog utility function, the parameters The first step in implementing an econo- satisfy equality and symmetry restrictions. metric model of demand based on the There are (1/2) m(3m-5) such restrictions. direct translog utility function is to add a stochastic specification to the theoretical C. A dditivity model based on equations for the budget If the direct utility function U is addi- shares pjXj/M (j= 1, 2, . . . , m). Given tive, we can write the disturbances in any mr-I equations, the disturbance of the remaining equation (23) In U = F(Z In Ui(Xi)) can be determined from the budget con- where each of the functions Ui depends on straint. Only m-1 equations are required only one of the commodities demanded Xi, for a complete econometric model of de- and F is a real-valued function of one vari- mand. able. The parameters of the translog approxi- B. Equality and Symmetry mation to an additive direct utility func- We estimate m- I equations for the tion can be written: budget shares, subject to normalization of a In U a In Ui the parameter am appearing in each equa- (24) - =-F' = tion at minus unity. If the equations are aInXi aInXi generated by utility maximization, the (i = 1, 2, . .. m) parameters lmj (j= 1, 2, . . . , m) appear- in each equation must be the same. a2lIn U ing (25) - x X This results in a set of restrictions relating a In Xio In Xi the m parameters appearing in each of the a In Ui a In Ui m-1 equations, a total of m(m-2) re- -F" - strictions. We refer to these as equality a lit Xi a In X- restrictions. (k $ j; i, j = 1, 2, ... , m) The logarithm of the direct translog utility function is twice differentiable in where the logarithmic derivatives, the logarithms of the quantities consumed, a In U so that the Hessian of this function is (26) F' = (i=1,2, ..., n) symmetric. This gives rise to a set of re- aIln Ui strictions relating the parameters of the a2In U (27) F" = cross-partialderivatives: a In Uia In Uj

(22) j=l3ji (i $ j; i, j = 1, 2, . . , m) (i. j 1. 27 ...... m)

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are independent of i and j. &2 In U (33) - Under additivity the parameters of the ( a In X8a In Xj translog utility function satisfy the restrictions: L aF a2 In H _alz H aIn Xja In Xj (28) d3ij = Oaciaj Ca2F daInH aIn H i, = 1, 2, . . (i j; j .,y m) +- _ __=- I a In H2 a In X, a In Xj_ where (i,j = 1 2, ... ,m) F" (29) - F=- (F') 2 Homogeneity of degree one of the function H implies that: We refer to these as additivity restrictions. The total number of such restrictions is a In H (34) E -X= (1 /2) (m-2) (m-1) . to an addi- The translog approximation m 2In H tive utility function is not necessarily addi- (35) E -- = O tive. The direct translog utility function is j=a In Xia In Xj additive if and only if In U can be written (i = 1, 2, ... ., m) as the sum of m functions, each depending on only one of the quantities demanded. Under homotheticity the parameters of We refer to such a function as explicitly the translog utility function satisfy the re- additive. Explicit additivity of the translog strictions: utility function implies the additivity re- (36) f3i =oas (i = 1, 2,.. ,m) strictions given above and the additional restriction: where

(30) 0= 0 a2F (37) a in 2 We refer to this restriction as the explicit d9In H2 additivity restriction. XVe note that the and we have used the normalization translog approximation to an explicitly additive function is explicitly additive. (38) ,a ai =-1 D. Homotheticity We refer to these as homotheticity restric- If the direct utility function is homo- tions. There are n- 1 such restrictions. thetic, we can write: The translog approximation to a homothetic utility function is not necessarily (31) In U = F(ln H(X1, X2, . . . Xm)) homothetic. A necessary and sufficient con- where the function II is homogeneous of dition for homotheticity of the translog degree one. utility function is that it is homogeneous, The parameters of the translog approxi- so that: mation to a homothetic direct utility func- (39) of = O tion can be written: refer to this restriction as the homo- a In U 9F a In H We (32) - -- = - - = ai geneity restriction. We note that the trans- d In Xi &InH OIn Xi log approximation to a homogeneous func- (i = 11 2 . . .m) tion is homogeneous.

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E. A dditivity and Homotheticity modities satisfies the restriction: The class of additive and homothetic di- (48) r+ 0 = p = O rect utility functions coincides with the this restriction the expenditures class of utility functions with constant Under shares are constant: elasticities of substitution among all commodities. If the direct utility function U pjXj3, j + OajE ai In X - - = is additive and homothetic, we can write: -I -- 02: a n X i

(40) In U = F( 8iX) The value of the parameter 0 is arbitrary or so that we can let af and 0 be equal to zero. The translog approximation to a utility (41) In U = F(ZE 8i in Xi) function with unitary elasticities of sub- The second form is a limiting case of the stitution has the same empirical implica- first corresponding to unitary elasticities tions as a linear logarithmic utility func- of substitution among all commodities. tion, which is explicitly additive and The parameters of the translog approxi- homogeneous. mation to an additive and homothetic util- F. Duality ity function satisfy the additivity and homotheticity restrictions given above, so In implementing an econometric model that: of demand based on the indirect translog utility function the first step as before is to (42) (i ?j;i,j= 1,2,...,m) 3ijO=caiaj add a stochastic specification to the theo- for the (43) /3ti = (a + 6)a-i + 2cxt retical model based on equations (j= 1, 2, . . , mi). (i = 1, 2, . .., m) budget shares pjXj/M Only m-i equations are required for a where we have used the normalization complete model. If equations for the budget shares are generated from the indirect (44) Zai-1 translog utility function, the parameters The translog approximation to an addi- satisfy equality and symmetry restrictions tive and homothetic utility function is not that are strictly analogous to the corre- necessarily additive and homothetic. We sponding restrictions for the direct trans- can, however, identify the parameters of a log utility function. translog approximation with the param- Additivity and homotheticity restric- eters of the additive and homothetic utility tions for the indirect translog utility func- function given above, as follows: tion are analogous to the corresponding restrictions for the direct translog function. (45) a + 0 =p The direct utility function is homothetic if (46) avi 5i (i = 1, 2,...,m) and only if the indirect utility function is homothetic." In general, an additive di- As before, the parameter 0 corresponds to: rect utility function does not correspond to F" an additive indirect utility function.'2 the direct utility function is (47) (-F)2 However, if (Ft) additive and homothetic, the indirect util- The translog approximation to a utility 11 See, for instance, Paul Samuelson (1965) and Lau function characterized by unitary elas- (1969b). ticities of substitution among all com- 12 See Lau (1969b).

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions 374 THE AMERICAN ECONOMIC REVIEW JUNE 1975 ity function is additive and homothetic. we present restrictions derived from hy- Direct and indirect utility functions corre- potheses about functional form such as sponding to the same preferences are addi- additivity and homotheticity. tive only if the direct utility function is Our empirical tests are based on data for homothetic'3 or the direct utility function three commodity groups-services of con- is linear logarithmic in all but one of the sumers' durables, nondurable goods, and commodities.14 In this relationship the other services-so that we specialize our roles of direct and indirect utility functions presentation of statistical tests of the can be interchanged. theory of demand to the three-commodity A direct utility function is self-dual if the case, m= 3. A complete econometric model corresponding indirect utility function has for either direct or indirect translog utility the same form.15 The only direct translog functions is provided by any pair of equa- utility function which is self-dual is the tions for the budget shares. We consider linear logarithmic utility function. Linear the system of two equations, logarithmic utility functions are the only (50) p1 = additive and homothetic direct or indirect translog utility functions. Translog direct M and indirect utility functions represent the a+1+il In X1+i12 In X2+313 In X3 same preferences if and only if they are - In In X2+1M3 In X3 self-dual; that is, if and only if they are 1+OM1 Xl+fM2 linear logarithmic. Unless this stringent p2X2 condition is met, the direct and indirect M translog approximations to a given pair of a2+2l1 In X1+:22 In X2+0323 In X3 direct and indirect utility functions represent different preferences. - 1+OM1 In Xl+,BM2 In X2+OM3 In X3 corresponding to the direct translog utility III. Empirical Results function and the system of equations, A. Summary of Tests We have developed statistical tests of (52) -M-= the theory of demand that do not employ the assumptions of additivity and homotheticity for individual commodities or for ai+il In'MPi+012 In-M+ P13In-M commodity groups. At this point we outline restrictions on equations for the budget shares corresponding to direct and in- aI+OM, In -+/M2 In M+OM3 In - direct translog utility functions. First, we present restrictions derived from the theory P22 P of demand; that is, from the basic hy- 531 M pothesis of utility maximization. Second, PinP P2 P3 a2+/21 In-+122 In -+123 In - m 13 See Houthakker (1960), Samuelson (1965), and _ Lau (1969b). Pi P2 P3 14 case introduced by John Hicks. This is the special -1 +OM1 In- +fM2 In -+OM3 In - See also Samuelson (1969). 1"See Samuelson (1965) and Houthakker (1965). We may also mention the "self-dual addilog system" in- corresponding to the indirect translog util- troduced by Houthakker (1965). This system is not function. Restrictions for the two sys- generated by additive utility functions except for ity special cases. tems are perfectly analogous so that the

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 375 following outline may be applied to either 4. Homotheticity Restrictions. Given the system. We emphasize the fact that the equality and symmetry restrictions, the two systems representthe same preferences homotheticity restrictions take the form: if and only if (63) 1M1 = ai (54) /ij = 0 (i, j = 1, 2, 3) (64) 3M2 = 0a2 1. Equality Restrictions. The parameters (65) 3M3 = 0(- - a - a2) I M1, 3M2, OM3} occur in both equations We introduce the additional parameter o-, are and must take the same values. There so that there are two independent restric- three equality restrictions. tions of this type.

2. Symmetry Restrictions. One restriction 5. Explicit Additivity Restrictions. Given of this type is explicit in the two equations the equality, symmetry, and additivity we estimate directly, namely, restrictions, a translog utility function is under the further re- (55) 012 = 021 explicitly additive striction: In addition, we estimate the parameters 0 = 0 /31 and 32 from the equations: (66)

(56) /31 = /M1 - Oil - /21 6. Homogeneity Restrictions. Given the equality, symmetry, and homotheticity (57) /32 = OM2 - /12 - /22 restrictions, the homogeneity restriction so that the two additional symmetry re- takes the form: strictions are implicit in the two equations we estimate. We write these restrictions in (67) a= O the form: 7. Linear LogarithmicUtility Restrictions. Given the equality, symmetry, additivity, (58) /13 = /M1 - /3l - /21 and homotheticity restrictions, the trans- (59) /23 = /M2 - /12 - /22 log utility function reduces to linear loga- There are three symmetry restrictions al- rithmic form under the additional re- together. strictions: of the six We can identify tests equality (68) = = 0 and symmetry restrictions with tests of the theory of demand. We next consider tests Tests of hypotheses about the form of of hypotheses about functional form. the utility function can be carried out in a number of different sequences. We propose in 3. A dditivityRestrictions. Given the equal- to test additivity and homotheticity consists ity and symmetry restrictions, the additiv- parallel. Each of these hypotheses Our test ity restrictions take the form: of two independent restrictions. procedure is presented in diagrammatic (60) /12 = 0a1a2 form in Figure 1. We also present alterna- For example, we could (61) /13 = Oal(-1 - a, - a2) tive test procedures. first test additivity and then test homo- (62) /23 = 0a2(-1 - - a2) theticity only if additivity is accepted. Our We introduce the additional parameter 0, test procedure is indicated with double so that there are two independent restric- lines while possible alternative procedures tions of this type. are indicated with single lines.

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UnrestrictedX

CS x Restriction

L Equality and Symmetry |

t wo Restrictions C Two Restriction )

Additivity Four Restrictions Homotheticitv

1.oTestrictioAsF J ETwoRestrictionsCT

Additivity and Homotheticity

FIGURE 1. TESTS OF ADDITIVITY AND HOMOTHETICITY

The next step in our test procedure de- utility is linear logarithmic. A second pos- pends on the outcome of the tests of addi- sibility is that we accept additivity but tivity and homotheticity. If we accept reject homotheticity. In this case we con- both additivity and homotheticity restric- tinue by testing explicit additivity. The tions, we continue by testing explicit addi- third possibility is that we accept homo- tivity and homogeneity in parallel. If we theticity but reject additivity; we continue accept both hypotheses, we conclude that by testing homogeneity. If we reject both

Additivity and Homotheticity

One Restriction On etition

Explicit Additivity wo Resrictions Homogeneity

n s Restrictioon One on_

Linear Logarithmic Utility

Additivity Not Homotheticity Homotheticity Not Additivity

I _m _ I _ One Restriction One Restriction

Explicit Additivity Not Homotheticity Homogeneity Not Additivity

FIGURE 2. TESTS CONDITIONAL ON ADDITIVITY OR HOMOTHETICITY

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 377 additivity and homogeneity, we terminate The unrestricted behavioral equations, the sequence of tests. These tests are pre- normalized so that the parameter aM is sented diagrammatically in Figure 2. minus unity in each equation, involve fourteen unknown parameters or seven un- B. Estimation known parameters for each equation. Un- Our empirical results are based on time- restricted estimates of these parameters for series data for U. S. personal consumption the direct translog utility function are pre- expenditures for 1929-72. The data in- sented in the first column of Table 1. The clude prices and quantities of the services first hypothesis to be tested is that the of consumers' durables, nondurable goods, theory of demand is valid; to test this hy- and other services. We have fitted the pothesis we impose the six equality and equations for budget shares generated by symmetry restrictions. Restricted esti- direct translog and indirect translog utility mates of the fourteen unknown parameters functions, employing the stochastic speci- are presented in the second column of fication outlined above. Under this specifica- Table 1. Unrestricted and restricted esti- tion only two equations are required for a mates of the fourteen unknown parameteis complete econometric model of demand. for the indirect translog utility function We have fitted equations for the services are presented in the first and second col- of consumers' durables (durables) and for umns of Table 2. nondurable goods (nondurables). There Given the validity of the theory of de- are forty-four observations for each be- mand, the remaining hypotheses to be havioral equation, so that the number of tested are restrictions on the functional degrees of freedom available for statistical form. First, we test the hypothesis that the tests of the theory of demand is eighty- direct utility function is additive; this eight for either direct or indirect specifica- hypothesis requires that we impose two tion. additional restrictions. The corresponding For both direct and indirect specifica- restricted estimates of the unknown pa- tions the maintained hypothesis consists of rameters for the direct translog utility the unrestricted form of the two behavioral function are given in the third column of equations for the budget shares. We esti- Table 1. Second, we test the hypothesis mate the behavioral equations for durables that the direct utility function is homo- and nondurables by the method of maxi- thetic without imposing the additivity re- mum likelihood and derive estimates of the strictions; this hypothesis requires two parameters of the behavioral equation for restrictions in addition to the equality and services, using the budget constraint.'6 The symmetry restrictions. The restricted esti- maximum likelihood estimates of the pa- mates of the unknown parameters for the rameters of all three behavioral equations direct translog utility function are given in are invariant with respect to the choice of the fourth column of Table 1. Finally, we the two equations to be estimated directly. impose both additivity and homotheticity restrictions, obtaining the restricted esti- 16 We employ the maximum likelihood estimator dis- mates presented in the fifth column of cussed, for example, by Edmond Malinvaud, pp. 338- Table 1. The corresponding restricted esti- 41. In the computations we use the Gauss-Newton the indirect translog utility func- method described by Malinvaud, p. 343. For the direct mates for series of tests we assume that the disturbances are inde- tion are given in the third, fourth, and fifth pendent of the quantities consumed. For the indirect columns of Table 2. series of tests we assume that the disturbances are inde- The next stage of our test procedure is pendent of the ratios of prices to the value of total expenditure. contingent on the outcome of our tests of

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TABLE 1-ESTIMATES OF THE PARAMETERS OF THE DIRECT TRANSLOG UTILITY FUNCTION

Explicit Homo- Explicit Equality Additivity Additivity, geneity, Additivity Additivity Linear Unrestricted and Homo- and Homo- Not Homo- Not and Homo- and Homo- Logarithmic Parameters Estimates Symmetry Additivity theticity theticity theticity Additivity theticity geneity Utility DURABLES 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)

NONDURABLES

a2 -.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 accept in the eighth through tenth columns of additivity, but not homotheticity, we test Table 1. The corresponding restricted esti- explicit additivity by imposing one further mates for the indirect translog utility restriction. If we accept homotheticity, function are given in the eighth through but not additivity, we test homogeneity by tenth columns of Table 2. The direct and imposing one further restriction. The re- indirect translog utility functions are self- stricted estimates under these restrictions dual if they are linear logarithmic, so that are given in the sixth and seventh columns restricted estimates for the two alternative of Table 1. The corresponding estimates econometric models given in the final col- for the indirect translog utility function umns of the two tables are identical. are given in the sixth and seventh columns of Table 2. C. Test Statistics If we accept both additivity and homo- To test the validity of restrictions im- theticity hypotheses, we test the hypothe- plied by the theory of demand and restric- sis that the direct utility function is ex- tions on the form of the utility function, plicitly additive by imposing one further we employ test statistics based on the restriction. Similarly, we test homogeneity likelihood ratio X, where: by imposing one further restriction. Fi- nally, we impose both restrictions under max S the hypothesis that utility is linear loga- (69) X= rithmic. The restricted estimates under max S each of these sets of restrictions are given n

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TABLE 2-ESTIMATES OF THE PARAMETERS OF THE INDIRECT TRANSLOG UTILITY FUNCTION

DURABLES as -.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)

NONDURABLES a2 -.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 the asymptotically, as chi-square with number maximum value of the likelihood function of degrees of freedom equal to the number ? for the econometric model of demand w of restrictions to be tested. subject to restriction to the maximum To control the overall level of signifi- value of the likelihood function for the cance for each series of tests, direct and in- model Q without restriction. For normally direct, we set the level of significance for distributed disturbances the likelihood ra- each series at .05. We allocate the overall tio is equal to the ratio of the determinant level of significance among the various of the restricted estimator of the variance- stages in each series. We first assign a level covariance matrix of the disturbances to of significance of .01 to the test of equality the determinant of the unrestricted estima- and symmetry restrictions implied by the tor, each raised to the power - (n/2). theory of demand. We then assign a level Our test statistic for each set of restric- of significance of .04 to tests of restrictions tions is based on minus twice the logarithm on functional form. These two sets of tests of the likelihood ratio, or: are ''nested"'; under the null hypothesis the sum of levels of significance provides a (70) -21nX=n(ln | :, | -In IIo |) close approximation to the level of significance for both sets of tests simultaneously. where 2. is the restricted estimator of the We test additivity and homotheticity in variance-covariance matrix and 2Q is the parallel, proceeding conditionally on the unrestricted estimator. Under the null hy- validity of the theory of demand. These pothesis this test statistic is distributed, tests are not nested so that the sum of

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions 380 THE AMERICAN ECONOMIC REVIEW JUNE 1975 levels of significance for each of the two of critical values for our test statistics hypotheses is an upper bound for the level given in Table 3 the reader can evaluate of significance of tests of the two hy- the results of our tests for alternative allo- potheses considered simultaneously. There cations of the overall levels of significance are four possible outcomes of our parallel among stages of the test procedure. tests of additivity and homotheticity: Re- Test statistics for both direct and in- ject both, accept both, accept only addi- direct tests of the theory of demand and of tivity, and accept only homotheticity. If restrictions on functional form are presented we reject both hypotheses, our test pro- in Tables 4 and 5. At a level of significance cedure terminates. If we accept either or of .01 we reject the hypothesis that restric- both of these hypotheses, we proceed to tions implied by the theory of demand are test explicit additivity or homogeneity, or valid for either direct or indirect series of both. tests. With this conclusion we can termi- If we accept both additivity and homo- nate the test sequence, since these results theticity, we test explicit additivity and invalidate the theory of demand. homogeneity in parallel, conditional on the One interesting alternative to our test validity of additivity and homotheticity. procedure is to maintain the theory of de- Again, the tests are not nested, so that the mand and to test the validity of restric- sum of levels of significance for the two tions on the form of the utility function. tests provides an upper bound for the tests Proceeding conditionally on the validity of considered simultaneously. If we accept the theory of demand, we could test the only additivity, we proceed to test explicit validity of restrictions on the form of the additivity, conditional on additivity. Simi- utility function. For the direct series of larly, if we accept only homotheticity, we tests we would reject the restrictions im- proceed to test homogeneity. plied by additivity and homotheticity. Since our three alternative procedures This conclusion would hold for our pre- for testing explicit additivity and homo- ferred procedure of testingr these hypothe- geneity are mutually exclusive, we carry ses in parallel, for a test of either hy- out tests of these hypotheses one time at pothesis conditional on the validity of the most. We assign levels of significance to the four hypotheses-additivity, explicit additivity, homotheticity, and homogeneity- TABLE 4-TEST STATISTICS FOR DIRECT AND INDIRECT with the assurance that the sum of levels TESTS OF THE THEORY OF DEMAND AND TESTS OF ADDITIVITY AND HOMOTHETICITY of significance provides an upper bound for the level of significance for all four tests. Degrees of We assign a level of significance of .01 to Hypothesis Freedom Direct Indirect each of the four hypotheses. With the aid Theory of Demand Equality and Symmetry 6 5.64 10.25 TABLE 3-CRITICAL VALUES OF Given the Theory of Demand XI/DEGREES OF FREEDOM Additivity 2 33.63 2.95 Degrees of Level of Significance Homotheticity 2 13.71 22.68 Freedom .10 .05 .025 .01 .005 Additivity and Homo- theticity 4 19.02 16.24 1 2.71 3.84 5.02 6.63 7.88 Additivity Given Homo- 2 2.39 3.00 3.69 4.61 5.30 theticity 2 24.33 9.81 4 1.94 2.37 2.79 3.32 3.72 Homotheticity Given 6 1.77 2.10 2.41 2.80 3.09 Additivity 2 4.42 29.54

This content downloaded from 146.186.114.232 on Fri, 9 Aug 2013 14:37:41 PM All use subject to JSTOR Terms and Conditions VOL. 65 NO. 3 CHRISTENSEN, JORGENSON, AND LAU 381 other, or for a joint test of the two hy- theticity; for either direct or indirect series potheses. of tests we reject this hypothesis. Given For the indirect series of tests we would additivity but not homotheticity, we reject the restrictions implied by homo- would reject explicit additivity for the di- theticity, but we would accept the restric- rect series of tests and accept explicit addi- tions implied by additivity at a level of tivity for the indirect series of tests. Either significance of .01. For the direct series of of these sets of results would rule out linear tests, we could terminate the test sequence logarithmic utility. conditional on the theory of demand at this point. For the indirect series of tests, IV. Summary and Conclusion we could test the validity of explicit addi- Our objective has been to test the theory tivity, given additivity but not homo- of demand without imposing the assump- theticity. We would accept the hypothesis tions of additivity and homotheticity as of explicit additivity, given additivity at a part of the maintained hypothesis. For level of significance of .01, conditional on either the direct or the indirect series of the validity of the theory of demand. tests, we conclude that the theory of de- In previous econometric studies of demand is inconsistent with the evidence. mand, the theory of demand has been These results confirm the findings of Wold maintained together with the assumption (in association with Lars Jureen) for the of additivity. A second alternative to our double logarithmic demand system and test procedure is to maintain both the Barten for the Rotterdam demand system.'7 theory of demand and the restrictions im- At the same time our results provide the plied by additivity. We can test homo- basis for more specific conclusions. If the theory of demand were valid, the double logarithmic form for the system of TABLE 5-TEST STATISTICS FOR DIRECT AND INDIRECT demand functions would imply that the TESTS OF RESTRICTIONS ON FUNCTIONAL FORM, GIVEN ADDITIVITY OR HOMOTHETICITY OR BOTH utility function is linear logarithmic. Simi- larly, the validity of the theory of demand Degrees of and the Rotterdam form for the system of Hypothesis Freedom Direct Indirect demand functions would imply linear loga- Given Additivity rithmic utility. An equally valid interpre- Explicit Additivity 1 19.97 1.59 tation of the results of Wold and Barten is that the theory of demand is valid, but Given Homotheticity that utility is not linear logarithmic. Our Homogeneity 1 1.47 4.73 results rule out this alternative interpreta- Given Additivity and Homotheticitv tion and make possible an unambiguous of demand. Explicit Additivity 1 11.25 4.02 rejection of the theory Homogeneity 1 1.74 1.16 A possible alternative to our test pro- Linear Logarithmic cedure is to maintain the validity of the 2 29.67 48.31 Utility theory of demand. Proceeding condition- Given Explicit Additivity and Homotheticity ally on the validity of demand theory, we Linear Logarithmic would reject the hypothesis of additivity Utility 1 48.09 92.61 for the direct series of tests. Additivity of the direct utility function is employed as a Given Additivity and Homogeneity Linear Logarithmic 17 See Wold, especially pp. 281-302; Barten (1969). A Utility 1 57.60 95.46 detailed review of tests of the theory of demand is given by Brown and Deaton, pp. 1188-95.

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