Cross-Section Application of Linear Expenditure Systems: Responses to Sociodemographic Effects Howard Howe American Journal of Agricultural Economics, Vol. 59, No. 1. (Feb., 1977), pp. 141-148. Stable URL: http://links.jstor.org/sici?sici=0002-9092%28197702%2959%3A1%3C141%3ACAOLES%3E2.0.CO%3B2-9 American Journal of Agricultural Economics is currently published by American Agricultural Economics Association.
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Research on the Economics of Household Consum tion Behavior (Rueben Buse, University of Wisconsin, Chairman
Cross-Section Application of Linear
Expenditure Systems: Responses to
Howard Howe Analysis of consumption allocation with expenditure systems is well known in aggregate, time-series applications. Yet, until recently, studies of individual household behavior have not generally brought available demand theory to bear on the empirical estimates. Pollak and Wales cited two advantages of using complete demand systems for the study of demographic effects on consumption. Demand systems incorporate the budget constraint into the analysis and permit the separation of demographic effects from own- and cross-price effects as well as from income effects. Unless these effects are separated, the demographic effects estimated with one set of prices cannot be presumed to apply in a different price situation. This paper discusses the interpretation of linear Engel curves as reduced forms of the linear expenditure system (LES) and the extended linear expenditure system (ELES). Parameters of these systems can be made functions of the sociodemographic characteristics of the household. With constant prices, the marginal budget shares in the LES are identified, but the subsistence expenditures are not. The ELES, which uses income in place of total expenditure, is exactly identified; all its parameters can be estimated by indirect least squares. Subject to the assumption of additive separability underlying the LES and the savings behavior implicit in the Howard Howe is an economist in the International Finance Division, Board of Governors of the Federal Reserve System. The views expressed herein are solely those of the author and do not necessarily represent the views of the Federal Reserve System. Richard Berner and Philip Musgrove made valuable comments on an earlier draft of this paper.
ELES, price effects can be estimated with data from a single cross section. Linear Expenditure System Properties The demand functions of the LES can be written
for i
=
1 , . . . , n,
where xi and pi are the quantity and price, respectively, of good i, and yh is the total expenditure of household h. Klein and Rubin developed the LES as the most general linear formulation in prices and income satisfying homogeneity, the budget constraint, and Slutsky symmetry. These demand functions represent the utility function
The ats are marginal budget shares, and the b{s have been interpreted as a "necessary" or "committed" set of goods. Pollak cautioned, however, that strict interpretation of bi as a necessary level of consumption could be misleading because b; can take on negative values. The utility function is defined wherever x ; - bi is positive. The admission of b; < 0 permits price-elastic goods. The Stone-Geary utility function, as equa-
142
Amer. J . Agr. Econ.
February 1977
Expenditure Equations and Engel Curves
tion (2) is known, underlying the LES is additive and nonhomothetic. The additivity permits price responses to be expressed in terms of the subsistence quantities (or expenditures). The nonhomotheticity introduced by the subsistence quantities allows the LES to exhibit nonunitary expenditure elasticities.
In a single cross section, it is assumed that all households face identical prices. The demand functions, equation (I), can be written in expenditure form
Sociodemographic EfSects
(4) pix,
In cross-section studies, the sample is often subclassified by sociodemographic characteristics of the household to reduce unexplained variation in expenditure behavior, and Engel curves are estimated separately for these subsamples. Another means of taking sociodemographic effects into account is to postulate that parameters of the expenditure system are functions of household characteristics.' Pollak and Wales have described this procedure as "translating" the demand system. The "subsistence quantity" of the LES (bi) can be made an explicit function of the sociodemographic characteristics of the household such as age, education, occupation, number of members, etc. For simplicity, bi is postulated to be a linear combination of these effects. Let
n
eih =
pibih + ~ i ( l ~. h 2 bkh~k) k= 1
where the subscript h on bi indicates that subsistence quantities now vary across households. Introducing equation (3) and defining yig = picig, the expenditure equation becomes
where yig is the value (at sample prices) of the gth characteristic's contribution to the subsistence expenditure of good i. The reduced form of equation (5) is a linear Engel curve
where cig is,the effect of the gth characteristic where, by comparison with equation (5), 6ig is on the subsistence quantity for good i and zgh implicitly defined as is either a dummy variable or a quantitative variable. In principle, the marginal budget shares (ai) could also be made functions of house- As evident in equation (6), the marginal m hold characteristics: aih = 2 crihzgh. This budget shares are all identified. The subsisP= 1 tence expenditures yi cannot be recovered generalization complicates ?he formulation from the Sip because tbe system is underidenbecause the LES requires 2 ak = 1. The tified by degree 1. Additional information or marginal budget shares would have to be restrictions are required to permit identifican m tion of the yig and thereby the inference of divided by akgzgh because there is price effects. k = l g=l no single scale factor that could be applied across all observations. In the studies report- Extended Linear Expenditure ed here, ai was treated as constant across System households; only bi was made a function of household size. Lluch developed the ELES from an intertemporal maximization of the Stone-Geary utiliI Barten incorporated household composition effects using ty function, equation (2) (1973). The ELES specific adult equivalent scales. Muellbauer expanded this employs (a concept of permanent) income as approach using the indirect utility function. When considering the explanatory variable and incorporates an the LES, Muellbauer obtained the same result as equation (5).
2 2
Household Consumption Behavior 143
Howe
aggregate consumption function. The typical ELES expenditure equation is written
for i
=
1 , . . . , n,
then obtained from ai = qi/m. The y's are identified from the Aig and qi. The yig identified in this way are subsistence expenditures at sample prices rather than the subsistence quantities themselves of equation (3). Strictly speaking, the cig that would appear in the structural ELES equation (9) are not identified. As Powell pointed out, measurements on pi at the time of the sample are sufficient to identify the quantities. Price elasticities are readily expressed in terms of the subsistence expenditure; quantities are not required to evaluate price responses.3
where m is the marginal propensity to consume and { is permanent income. The ELES is a particularly useful generalization of the LES in cross-section applications in that the independence of income from the expenditure error terms overcomes the least-squares bias encountered using total expenditure and that the ELES permits indirect least-squares estimation of all parame- Cross-section Estimates of Linear ters since it is exactly identified.2 The ELES Systems can be developed from an atemporal utility maximization by treating saving as the (n Prior to the introduction of the ELES, Betan+ l)th commodity, with bv+l = 0 (Howe court applied the labor-leisure choice and 1975). This a priori specification of one of the wage rate information to the LES to estimate b's identifies the remaining subsistence total subsistence expenditure and thereby expenditures in the absence of price varia- identify all parameters (1971). Belandria emtion. ployed the LES and the ELES to analyze The alternative development of the ELES cross-section household consumption data can be repeated using current income ('y) in for three Venezuelan cities. He showed that the budget constraint instead of permanent the LES is underidentified where prices are income ({). In this case saving is out of constant over the sample and that the ELES current income, and m becomes the marginal is identified under the same conditions. Bepropensity to consume current income. Us- landria treated the parameters of the expening the same formulation for modifying the diture systems as constant across households. subsistence expenditures as with the LES, Subsequently, Powell systematized Belanequation (3) can be substituted into equation dria's results on cross-section application and (8) to obtain ELES functions reflecting the integrated them with new results on large sociodemographic characteristics of the sample biases in LES estimates. household. The current income version is In the following review of LES and ELES cross-section applications, the first two sets of studies translate the demand system with respect to household size or composition. The third group of studies follows the more conventional procedure of estimating the systems separately for demographically similar The reduced form of equation (9) is an Engel subsamples. curve based on income Estimates of Household Composition Efects The marginal propensity to consume (m) equals I:vk. The marginal budget shares are 2 Transitory components of current income could be correlated with the error terms of certain expenditure items. The use of normal income as a function of socioeconomic variables amounts to an instrumental variables procedure and results in unbiased estimates.
In an application of LES and ELES to Colombian cross-section data for 1967-68, Howe emphasized the effects of household composition on expenditure (1974). Following equation (3), the subsistence quantities
' The own-price elasticity can be expressed as e, = @,b,/p,x,) (1 - ai)-' and the cross-price elasticity can be written in the form cv = -ai(pjb,/pixi).
Amer. J . Agr. Econ.
were postulated to be linear combinations of the number of persons in each of three age classes: zero to seven years, eight to seventeen years, and over seventeen years, and zl was set to 1 to represent the head of the household. Estimates of the ELES parameters of equation (9) were obtained by indirect least squares and compared for four cities. Table 1 presents ELES estimates (and their standard errors) for Bogota. The estimates of the marginal budget shares and the marginal propensity to consume out of current income are reasonable. The negative values for fig are not easily interpreted in terms of subsistence expenditure, but none of the negative values are significantly different from zero. Also, the income effects are estimated with much greater precision than the subsistence expenditures. Interestingly, the significant subsistence expenditures are all for persons aged eight to seventeen years and over seventeen years. It could well be that, at this level of aggregation, children under seven years of age simply do not affect consumption allocation greatly or consistently enough across income levels to yield significant y's. In interpreting the subsistence expenditure of the head of household, it is useful to recall its role as an intercept term; its insignificance could reflect the lack of a constant term (or nonlinearity near the origin). In retrospect, a better formulation for subsistence quantity could have been Table 1. ELES System Parameters, Bogoth Subsistence Expenditure 2 Food
0.3742 (0.0077) 0.2720 Shelter (0.0058) Clothing 0.1020 (0.0021) Medical 0.0350 (0.0007) Durables 0.0350 (0.0007) Transportation 0.0316 (0.0006) Miscellaneous 0.1402 (0.0028) 0.8482 MPC(m) (0.0166)
a
PO-7
P8-17
85 (75) -49 (54) -8 (20) 0 (8) -15 (12) 3 (5) -35 (27)
384 (68) 217 (49) 76 (18) 21 (3) 35 (1 1) 29 (6) 153 (24)
P17+
Head
809 130 (86) (239) 541 81 (62) (172) 170 -10 (27) (65) 49 5 (10) (28) 55 -3 (35) (39) 61 0 (7) (21) 268 -103 (31) (86)
Colombian pesos per quarter at 1967-68 prices
where the head would be counted as a member of the over seventeen years class, and the intercept could be tested explicitly.
Nutritional Information Just as the a priori specification of b,+l = 0 for saving permits identification of the ELES in the cross section, independent information on any other subsistence expenditure identifies the remaining y's of the LES. The use of nutritional requirements to estimate subsistence expenditures on food was inspired by Orshansky's work on defining the poverty line in the United state^.^ For the Colombian study, independent measures of the minimum cost diets necessary to maintain adequate nutrition for persons of various ages were obtained. tary values of the minimum cost diets at prices prevailing during the sample period were used as extraneous estimates of the subsistence expenditure for The resulting LES estimates are presented in table 2. The estimated marginal budget shares of the two systems are very close. The subsistence expenditures for ages eight to seventeen years and over seventeen groups are within two standard deviations of the f's estimated with the ELES. The T's for the head of household are very different, a result of the intercept problem discussed above. The f's for children under seven years old are much larger than those in table 1. The poor estimates of y for children obtained with ELES are probably a result of very different expenditure responses to children across income strata. Use of dummies for socioeconomic stratum in equation (3) for intercept and interactive effects would be the next step in improving the estimates of y with the ELES.
one-
In an attempt to count the number of persons living in poverty in the United States, Orshansky observed that "there is no generally accepted standard of adequacy for essentials of living except food" (p. 5). She then proceeded to use low-cost food plans of the U.S. Department of Agriculture as the minimum outlays necessary to maintain adequate nutrition. The food plans are tailored to the age-sex composition of the household. A poverty budget approach was also tested in the Colombian study. The poverty budget for the individual household was substituted for the "subsistence expenditure" 2 I:ykgzgh in equation (5), thus identifying the individual yi However, the estimates of yig for food were not constraines to equal the values of the minimum cost diets.
Household Consumption Behavior 145
Table 2. LES System Parameters Identified with ~xtraneou; Nutritional Information, Bogota Subsistence Expenditure
cii Food
P 0-7
P 8-17 P 17+
0.3965 [291Ib [480] (0.0070) Shelter 0.2797 99 298 (0.0074) Clothing 0.0994 47 100 (0.0038) Medical 0.0294 13 29 (0.0027) Durables 0.0342 15 31 (0.0032) Transportation 0.0319 18 39 (0.0025) Miscellaneous 0.1290 33 176 (0.0040) a
(ri.)"
Head
[750]
[750]
463
576
148
138
37
48
38
54
52
59
216
125
Colombian pesos per quarter at 1967-63 prices. Values of minimum cost diets.
Economies of Scale The linearity assumption of equation (3) was relaxed by redefining z as a dummy variable for each size of Lousehold from two members to eleven or more. Although this modification lost the distinction between children and adults, it allowed subsistence expenditure to vary nonlinearly with household size. The profiles of the T's as a function of size exhibited puzzling declines at five, eight, and eleven or more members. For the subsistence interpretation of the y's to hold, subsistence expenditure must at least be increasing with respect to size of household. These anomolous results are probably due to the same lack of consideration of socioeconomic stratum that caused problems with the household composition effects. With further refinement, this procedure could be useful in studying economies of scale in household expenditure. Ferber and Musgrove plotted the general trend of these estimates of food and total subsistence expenditure by household size. They found elasticities of subsistence expenditure with respect to size close to 1.0 for the range three to eleven or more persons. Other studies investigated household size effects in translated linear expenditure systems without distinguishing household composition. Howe and Musgrove applied the ELES to four Latin American cities-Bogota, Caracas, Guayaquil, and Lima-making only the distinction between large (five or
more) and small households. The systems were also estimated separately by (ex ante) socioeconomic stratum (three strata approximating permanent income levels) and age of head (under forty-five years and over fortyfive). Committed expenditures (y) tended to be larger for higher stratum households, showing that the y more nearly represent subsistence needs at low incomes, while habits and expectations become more important as income rises. Younger households tended to have lower mean expenditures but higher committed spending. They behaved as if their notion of a minimally acceptable standard of living, or their expectations about future income, was higher than that of older households. The difference between yi (large) and y, (small) was greater for young than for old households. Estimates of the marginal propensity to consume were extremely varied and often too low. They did not always decline as socioeconomic level went up, and they were often low for very low income households. Separate estimation by sthtum overcomes a major criticism of linear systems-the constancy of marginal budget shares across income levels, but it may complicate the study of household saving. If the share of income variation due to permanent income is not constant across strata, the bias in estimating the marginal propensity to consume due to transitory elements may be made worse when the sample is stratified (Howe and Musgrove). This problem was more serious with young than with old households. Betancourt applied current income (1976) and permanent income (1973) versions of the ELES to Chilean data for 1964. He reformulated the ELES to use discrete time and a finite horizon and allowed subsistence expenditure to vary as a function of household size with an economies-of-scale effect. The ELES was then estimated separately by twelve subsamples (six age classes and urban-rural location). Shift dummy variables for occupation (also interpreted as socioeconomic class) were applied to income and household size. Betancourt found that, although failure to control for socioeconomic stratum led to differences of less than 10% in the incomerelated estimates of ELES, differences of substantially greater than 10% resulted in the price-related ELES estimates (1976). Esti-
146
Februuty 1977
mates of the economies-of-scale parameter varied widely. Pollak and Wales applied two cross-section data sets to obtain price variation for estimating the LES and a quadratic expenditure system. They used family expenditure data from the United Kingdom for 1966 and 1972. Mean consumption expenditures on three categories for twenty-nine cells of various income and family size classifications were combined with price indexes for the two periods. The subsistence quantities were translated as functions of the number of children in the household. The estimation employed the same methods used for timeseries data with price variation. Although not the first to pool cross-section data, Pollak and Wales demonstrated that sufficient price variation existed in only two cross sections to permit identification of the parametersS6 Estimation by Subsample
In addition to the studies presented above that also incorporated variable subsistence expenditures, three recent cross-section studies investigated the influences of sociodemographic characteristics with separate estimates by subsample using constant subsistence expenditures. Lluch, Powell, and Williams (chaps. 5, 6, and 9) estimated the ELES for Korea, Mexico, and Yugoslavia. For Korea, urban-rural differences in demand behavior were explored by estimating the ELES separately for urban and farm households. Per capita income and expenditure data from annual sample surveys were combined with consumer price indexes for each of these areas. Then urban cross-section data for 1971 were separated into six subsamples (salary and wage earners, young and old head of household, small and large household), and separate ELES estimates were obtained for each type of household. The authors show that for expenditures on nondurable~,the cross-section data yielded disaggregated demand parameters and elasticities that, when averaged, were broadly consistent with time-series estimates. For Mexico, separate ELES estimates were obtained for thirty-two subsamples (ruralurban location, socioeconomic class (3), large and small family, young and old head of 6 Lluch used price variations among regions to estimate an LES with cross-section data from Spain (1971).
household) of the population surveyed in 1968. The authors found that if the data were to be pooled for estimation, the model should allow for family size effects, the subsistence parameters should shift by socioeconomic class, and marginal budget shares should be allowed to vary with income. The first two modifications could be accomplished in a translated ELES using the form of equation (3). . , Separate ELES estimates were applied to twelve subcells of Yugoslavian farm households separated by region and type of farm. Income levels varied substantially across these groups; mean income more than doubled from poorest to richest. Marginal budget shares showed little variation by income level. Although the farm households were reasonably homogeneous with respect to demand behavior, savings behavior was influenced by income level. Income Concepts for the ELES
All of the results reported thus far proceeded from an ELES using current income. As developed theoretically, the ELES employs a concept of permanent income. In an attempt to develop a measure of permanent income, Belandria estimated relationships between current income and age, education, and occupation. The estimated coefficients were then used to impute expected or normal income for each household. Howe employed the same procedure to generate a measure of normal income except that the log of income was used as the dependent variable (1974). Betancourt applied a method whereby the household in question is assumed to focus upon a similar household in the next highest age class to form an estimate of its permanent income (1973). Howe found that although the normal income ELES provided superior estimates of the marginal propensity to consume for two of the four cities, the number of observations for which subsistence expenditure exceeded actual expenditure (a violation of the LES property, equation (2)) was much greater than with current income (1974, pp.201-13). Belandria and Betancourt also found that normal or permanent income produced significantly higher estimates of the y's. Because the normal income procedures underestimate the variance of permanent income,
Howe
Household Consumption Behavior
147
the other sociodemographic characteristics known to affect importantly the parameters of the system. In this way the restrictions of demand theory are imposed on the effects under study, while unexplained variation in Conclusions expenditure behavior resulting from other Modest success has been achieved in estimat- sociodemographic differences is reduced. ing price effects along with income and socio- However, as found in Howe and Musgrove, demographic effects from cross-section data. stratification of the population may improve Although such estimates are obtained at the the expenditure functions but lead to unrelicost of very restrictive assumptions, the re- able descriptions of savings behavior. Anothsults have often been reasonable enough to er aspect of such a procedure requiring careencourage further application to cross-sec- ful judgment is that the number of subclasses tion data. The ELES could prove particularly could proliferate to the point of making useful in developing economies where suffi- sample sizes too small and simulation of the ciently long time-series data on the compo- full system unwieldly. nents of consumption are lacking. Although the LES and ELES provide wide scope for analyzing a single cross-section of References household data, some caveats are in order. Aside from the restrictiveness of the LES Barten, A. P. "Family Composition, Prices and Expendititself, the complications arising from transiure Patterns." Econometric Analysis for National Ecotory behavior in the cross section are considnomic Planning, eds. P. E. Hart, G. Mills, and J. K. erable. Because of the severity of the assumpWhitaker, pp. 277-92. London: Butterworths, 1964. tions, the ELES parameter estimates could be Belandria, Francisco. "An Empirical Study of Consumer Expenditure Patterns in Venzuelan Cities." Ph.D. thevalid only locally. Care should be exercised sis, Northwestern University, 1971. in extrapolating the system far from the Betancourt, Roger R. "The Estimation of Price Elasticities sample prices and incomes. from Cross-Sectional Data under Additive PreferBoth translated expenditure systems and ences." Znt. Econ. Rev. 12 (1971):283-92. separate systems by subsample are applicable -. "Household Behavior in Chile: An Analysis of to policy-oriented models. Since the translatCross-Section Data." Household Demand and Saving in ed models discussed here all varied subsisEconomic Development: Applications of Linear Demand tence expenditures while holding the MBS System, eds. Constantino Lluch, Alan Powell, and Ross and MPC constant, estimation by subsample Williams, chap. 8. Washington, D.C.: Development could be a useful first step in testing the Research Center, International Bank for Reconstrucvalidity of that assumption. A translated tion and Development, forthcoming. Behavior in a Less Developed Counsystem is superior to a set of subsystems in -."Household try: An Econometric Analysis of Chilean Cross-Secthat it imposes the budget constraint on the tion Data." Mimeographed. College Park: Dep. Econ., analysis of sociodemographic responses. A University of Maryland, 1973. translated system requires that the changes in Ferber, Robert, and Philip Musgrove. "Finding the Poor: expenditure components in response to a On the Identification of Poverty Households in Urban change in household composition sum to Latin America." Mimeographed. Washington, D.C.: zero if total expenditure remains constant. The Brookings Institution, 1976. In all of the studies reported here, separa- Howe, Howard. "Development of the Extended Linear tion into subsamples was employed to some Expenditure System from Simple Saving Assumptions." Eur. Econ. Rev. 6 (1975):305-10. extent. Even in the Colombian study, with "Estimation of the Linear and Quadratic Expenditthe most extensively pooled model, estima- -. ure Systems: A Cross-Section Case for Colombia." tion was separated by city. In most cases, all Ph.D. thesis, University of Pennsylvania, 1974. parameters of the system differed signifHowe, Howard, and Philip Musgrove. "An Analysis of icantly by subsample. ECIEL Household Budget Data for Bogota, Caracas, A rule of thumb for applying demand Guayaguil, and Lima." Household Demand and Saving systems to policy problems would be to inin Economic Development: Applications of Linear Decorporate the sociodemographic effects to be mand System, eds. Constantino Lluch, Alan Powell, analyzed (age distribution, household size or and Ross Williams, chap. 7. Washington, D.C.: Develcomposition, etc.) in the translation of the opment Research Center, International Bank for Resystem and then to subdivide the sample by construction and Development, forthcoming.
current income has been preferred for estimation of the ELES.
148 February 1977 Klein, L. R., and H. Rubin. "A Constant Utility Index of the Cost of Living." Rev. Econ. Stud. 15 (1947-48):8487. Lluch, Constantino. "Consumer Demand Functions, Spain, 1958-1964." Eur. Econ. Rev. 2 (197 1):277-302. "The Extended Linear Expenditure System." Eur.
-. Econ. Rev. 4 (1973):21-32. Lluch, Constantino, Alan Powell, and Ross Williams, eds. Household Demand and Sauing in Economic Deuelopment: Applications of Linear Demand System. Washington, D.C.: Development Research Center, International Bank for Reconstruction and Development, forthcoming.
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Muellbauer, John. "Household Composition, Engel Curves and Welfare Comparisons Between Households: A Duality Approach." Eur. Econ. Rev. 5 (1975): 103-22. Orshansky, Mollie. "Counting the Poor: Another Look at the Poverty Profile." Soc. Secur. BUN. 28 (1965):3-24. Pollak, Robert A. "Additive Utility Functions and Linear Engel Curves." Rev. Econ. Stud. 38 (197 1):401-14. Pollak, Robert A., and T. J. Wales. "Estimation of Complete Demand Systems from Household Budget Data." Dep. Econ. Disc. Pap. No. 345, University of Pennsylvania, 1976. Powell, Alan A. Empirical Analytics of Demand Systems. Lexington, Mass.: D.C. Heath Co., 1974.
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References The Estimation of Price Elasticities from Cross-Section Data Under Additive Preferences Roger R. Betancourt International Economic Review, Vol. 12, No. 2. (Jun., 1971), pp. 283-292. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28197106%2912%3A2%3C283%3ATEOPEF%3E2.0.CO%3B2-I
A Constant-Utility Index of the Cost of Living L. R. Klein; H. Rubin The Review of Economic Studies, Vol. 15, No. 2. (1947 - 1948), pp. 84-87. Stable URL: http://links.jstor.org/sici?sici=0034-6527%281947%2F1948%2915%3A2%3C84%3AACIOTC%3E2.0.CO%3B2-B
Additive Utility Functions and Linear Engel Curves Robert A. Pollak The Review of Economic Studies, Vol. 38, No. 4. (Oct., 1971), pp. 401-414. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28197110%2938%3A4%3C401%3AAUFALE%3E2.0.CO%3B2-1
