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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review. http://www.jstor.org 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 TranscendentalLogarithmic Utility Functions 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 ex- penditure. 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 func- tions 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.