The Barten-Gorman Model in the Aids Frameword

No. 347 July 1992 'I'D BARTZN-GOIQC!Uf N:>DZL IN 'I'D AIDS l'RAMEWORlt By M. Luisa Ferreira and C. Federico Perali University of Wisconsin-Madison :1 Abstract The Barten-Gorman model has been difficult to estimate due to the problems in identifying all demographi c parameters and the highly nonlinear nature of the specification. Restrictions on the demographic parameters that allow the identification of the model and make it estimable are derived. An empirical application shows that the Barten-Gorman model behaves well and preferred to the nested Translating and Scaling specifications. Key Words: Barten-Gorman Model, Identification , Concavity and Demographics. */ Both authors are graduate students , Department of Agricultural Economics , University of Wisconsin-Madison. The authors would like to thank, without implicating , Rueben C. Buse, and Thomas L. Cox for their comments. 1 . Introduction Incorporating demographic or other factors into a demand system is gaining renewed interest due to the results derived by Lewbel (1985). Using a somewhat cryptic notation, Lewbel shows how to modify complete demand systems with the introduction of demographic characteristics that interact with prices and income in an almost unlimited variety of functional specifi cations . Such modifications include the popular Barten (1964) scaling and translating specifications, first introduced by Pollak and Wales (1981), as special cases . Lewbel describes the properties that the modifying functions must maintain to ensure integrability of the modified demand system. This guarantees that the original preference structure is preserved and that the modified model is still theoretically plausible. These conditions are necessary to compute the true equivalence scales and to derive correct measures of welfare. This study concentrates on the Barten- Gorrnan modifying specification which combines both translating and scaling. Deaton and Muellbauer (1986) shows that this model is metrically superior to competing specifications for generating equivalence scales. Similar to the Slutsky decomposition of income and substitution effects, the Barten-Gorrnan specification translates the budget line through the fixed cost element (translating) and rotates the budget constraint by modifying the effective prices with the related substitution effec ts via demographic characteristics (scaling ). The Barten construction is motivated by German's (1976) observation that "when you have a wife and a baby, a penny bun costs threepence." However, some characteristics may affect consumption only through income effects , or may show both a translating and a scaling effect. Only a model that nests both could resolve this empiri cal question. In the past, the Barten-Gorrnan model was not very popular because its estimation is highly nonlinear and the model was thought to have serious identification problems. Moreover, Lewbel states that there is no guaranteee that all parameters can be identified, just as there is no guarantee that all t he parameters in any demand system can be identified (1985 : 5) . Deaton and 1 Muellbauer assert that in practical applications it will always be extremely difficult to estimate the parameters of the Gorrnan- Barten model (1986 : 740) . Chung (1987 ) circumvents the problem by arbitrarily introducing into the translating function demographic variables different from the ones incorporated into the scaling function. Davidson and McKinnon (1981) identify the artificial par ameters used to compound alternative specification hypotheses into a nesting model by estimating the artificial parameters conditional on the knowledge of the estimated parameters of the single alternatives (Atkinson 197 0) . In the present study we specify the Barten-Gorrnan model in a theoretically plausible way . We use the property of homogeneity of degree one in p of the cost function to derive the identifying restrictions that make the model estimable. We complete the description of the Barten-Gorrnan model by providing the expres sion to analyze the concavity properties. We also derive the expressions for the elasticities . The paper first lays out the notation and definitions . section specifies the model . The third The fourth section demonstrates how all the demographic parameters can be identified . The next section derives the conditions to test concavity in the Barten-Gorrnan model . The sixth section applies the model to the estimation of the demand for food at home and food away from home in the United States during the period 1953-1988. The conclusions complete the discussion . 2 . Notation and D•~ i nit i ons Set the basic notation according to the following definitions: h (h 1 , .. ,hH)El+H (I the real numbers, I+ the non negative reals ) is the index vector for demographic profile s at year h=l, . . ,H: h=O designates the reference profile , h=l designates the profile chosen for comparison; Pj P the price of commodity i =l , .. ,n assumed constant across profiles; (P11··1Pn ) El+n; 2 qih the quantity of the ith commodity consumed by the h th demographic profile; q = ( ql, . . , qn) EI+ n; wih = the budget share of the i th cormnodity by the h th profile; yh the total expenditures of the h th demographic profile (income in short ); dr the r th demographic characteristic; d (d1, .. ,dR) El+R; ai r the scaling demographic parameter for the ith cormnodity and the r th characteristic; Oi = (61 1 • • ,0R) EIR the R vector of scaling parameters for the ith cormnodity; mi (d; Oi) : l +R- 1 =the scaling function specific to the i th commodity; = the translating demographic parameter for the ith commodity and the r th ~ ir characteristic; ~ i = ( ~ 1 , .. , ~ R) EI R the R vector of translating parameters for the ith commodity; t i( d; ~ i): R-1 I =the translating function specific to the ith commodity; C(u,p) = the minimum cost of attaining utility level u at prices p. By definition, yh=C(u , p) . This cost function is assumed to be twice continuously differentiable and theoretically plausible . V(y,p) = the indirect utility function at income y and prices p . t (u) = the level o f utility of the reference demographic profile . 3. Spec i ~ ic a ti on o~ th• Bar t e n-Gorman Model Following Lewbel (1985) , consider the relation: y::: C ( u,p,d) ::: f{c•[u,h(p,d ) ] , z (p , d ) , d) where c*( u , p*) is a well- behaved expenditure function, y* = c*[ u , h(p,d)J = c*(u,p* ) is the minimum expenditure necessary to attain utility level u at s ome scaled prices Pi*= hi( p,d) and translated prices PiT=zi(p ,d ) f or some vector valued functions h and z. 3 (1) Using the facts y=C=f and y*=c*, the Barten-Gorman specification 1 is obtained from equation (1) using the following f (y,p, d) modifying function: y = C(u,p, d) .a f(y•,z (p,d),d) • y • p T with pT • TI (z1 (P 1 1 d) )t1( d ) • l This expression corresponds to the Barten (1964) specification with the addition of fixed overheads pT for "necessary" or "subsistence" quantities (Gorman 1976) . Several different demographic specifications can be derived by making explicit assumptions about the functions h(p,d) and z (p,d). The specifications are: (a) h(p ,d)=z(p ,d)=p (b) h(p,d)=z(p,d)=pm=p * (c) h(p,d)=p * and z(p,d)=l (d) h(p,d)=p * and z(p,d )=p - budget share Translating budget share Reverse Gorman budget share Scaling b•Jdget share Gorman. These definitions comply with Pollak and Wales (1981) terminology. empirical convenience 2 the translating demographic functions ti(d) are specified as ~(dl=ri a ir, ti(d)= For and the scaling demographic functions mi (d ) as E r ~ irln(d for r=l, .. , n. Assume quasi-homothetic preferences as described by the demographically modified Gorman Polar cost function: C(u , p , d) = (A(p,d) ( ~ (u) (2) )B(p,dl) pT. The linear in logarithm analog is: 1 The Barten-Gorman demographic specification that we refer to in this paper is the Reverse Gorman modifying structure. 2 It should be emphasized that the choice of the functi ona l form of the demographic functions is not restricted to any particular form partly because only the relative magnitudes of the estimated demographic functions have a meaningful interpretation. The researcher, howeve r, can specify a more complex form such as a translog if interested in modelling economies of scale. 4 ln C( u,p, d) "' (ln A(p , d) + B(p, d) ln ~ ( u)) + ln pT where: ln A (p, d) = cxo r + ~ i ex · ln p •· l l + • 5 ~ r~ r i j • yijln p i• ln P j• n B(p , d) "' Po II (p ;) P1 • i •l The corresponding Barten-Gorrnan AIDS indirect utility function is given by: ln y• - («o ln V• ~ + ~ l;: l «i ln pj + .5 l;: l;: l ~,- Po II J Yi j ln pj ln pjl ~ (3) ~- (p;) P1 i where Yi j=Yij*+Yji* , V= ~ (u) and ln y*=ln y - t iti(d)ln p* from equation (2). This extension pe rmits distinguishing between the intercept shifting function and the transla ting function . income elasticities . Moreover, it allows deriving profile specific Roy ' s identity yields the Barten- Gorman AIDS budget shares: 3 (4) Lewbel shows that a theoretically plausible specification of a modified Marshallian share demand system can be obtained from equation (1) by applying the following trans formation (1985 , Theorem 4): where t iti (d)=O due to the homogeneity restrictions. It is important to note that the system represented in equation (4) is not a unique specification of the Barten-Gorman. Many other specifications can be obtained by applying Lewbel ' s technique. Consider, for example, the 3 The term ln A' (p,d ) is the same as ln A(p,d) with Yij in place of y · · ' . Henceforth, to simplify notation ln A(p,d) will be used in lieu of 1:1n A' (p,d ) . 5 af(y • ,p, d) ay • y• ~ - '" (1 - l: t i (d)) i ~ y Pi •( • • af(y* ,p, d l P i wi y ,p 1 + - - - - - ap i y P j• ah i (p, dl J· apl · wj (y*,p * ) '"'wj + t 1 ( d) + t i (d) following exponential specification of the h (p , d) function hi (p , d) * exp(pimi(d) ) =exp(pi). The derived Barten-Gorrnan shares are : (5) This specification is interesting because the translation term l ooks much like the committed quantity term of the linear expenditure system. However, the The supernumerary quantities increase as the ratio overhead is not fixed. . . p *1 y * a l so increases Hence the degree to which a good is perceived as a necessity is subjective and varies from individual to individual (Lewbel 1985). The Barten-Gorrnan model in equation (4) nests the following demographic specifications: (a) Scali ng w · == l al · + l: j y l·]· ln P j• + A • ... 1 ln ( ( Y• (6) ) A p , d) where h (p , d)=p*=pm, and ln y*=ln y ; (b) Trans lating w 1· == a 1· + t 1• ( d) + l: y · · ln p · + j lJ J A· ,..1 ln ( AY(p) • ) (7) where h (p , d) =p , and ln y*=ln y-ln P1 . Integrability r equires that these specifications be estimated including y * and without linearization of the deflating index A(. ) . This guarantees 6 recovering exactly the underlying modified cost and indirect utility function which can then be used to derive equivalence scales and to make welfare comparisons that are fully cardinal. 4. Identi~ i cation From the previous section , recall the Barten- Gorrnan specification of f(y*,p*,d) where h(p,d)=z(p , d)=pm=p*: (8 ) that nests both the scaling and translating specifications. A straightforward application of Lewbel's Theorem 1 (1985) leads to the following proposition . Proposi t i o n 1. Let c* (u,p*) be a cost function homogeneous of degree one in prices. C(u,p,d) will maintain the same property if and only if (iff): n ohi (p, d) since _____ ap j E j •l n 1 - :E n of(y• ,p, dl Pj opj f(y* , p, d) j •l of(y• ,p, dl y • ay • y • 1 - n 1, iff E t i ( d) i •l E t 1 !dl =0 V j ,,.i , - ; 0 .I i •l Using Euler's law, the legitimate cost function C(u,p,d) can be rewritten as: 7 Then C( u,p, d) oc• op; ·Ei - - - Pi Ei -opi opi (9) Given the Barten-Gorman specification of C(u,p,d ) in equation (8), the Euler relation in (9) becomes: f(y •,p, d) ~ oC i opi - LJ - - Pi .. p T ~ oc• • c LJ ~ ti i opl i LJ--Pi + (d) I (10) where: of Equation (10) gives the fundamental economic relationship that is used to derive the econometric restrictions at the share demand level. The homogeneity, or invariance, property implicit in the budget constraint modified by the Barten-Gorman specification is a sufficient condition to identify all demographic parameters. It makes the derived share demand models econometrically tractable. By applying Shephard's lemma to (10) and dividing both sides by C(u,p,d)=c*( u,p*)pT, we obtain: E i wi .. 1 . Hence, homogeneity of degree one in prices of the cost function or of degree zero in prices and income of the share demands require: In the Barten-Gorman framework the number of "free" demographic parameters doubles from r (n-1), as in scaling or translating, to 2xr(n-l). The procedure of specifying the same demographic functions for scaling and translating gives rise to problems of perfect correlation between the 8 translating and scaling set of parameters. Moreover, this procedure would imply the behavioral assumption that the translating effect is the same as the scaling effect. Thus, this approach would result in a loss of economic information. Instead, the hypothesis that the translating demographic effects differ statistically from the scaling effects is interesting and can be tested by using the nested procedure suggested by Pollak and Wales (1981). In line with previous work (Atkinson 1970; Davidson and McKinnon 1981; and Pesaran and Deaton 1981), Pollak and Wales (1981) propose the following parsimonius solution to estimate the compound Barten-Gorrnan model: t1 a (l-v1) ln mj for i=l , . . ,n Since the m1 and ti functions include the same parameters and have the same functional form, the approach generates only r(n-1) "free" demographic slopes and n artificial parameters vi. An additional benefit of this structure is that it nests the translating and scaling hypothesis as special cases. If vi =l, the model collapses to scaling; if vi =O, it collapses to translating. These restrictions are amenable to standard null hypothesis tests against the more general compound specifi cation. To evaluate the implications of (10) on this specification, it is insightful to reconsider the share equations of the Barten-Gorman AIDS model: wi = « i + (l-v1)ln mj + l: j Yij (ln Pj + Vj ln mjl where: 9 +Pi ln ( y• ) . A(p,d) (11) ln y• • ln - 1;: [ (1-vj )ln mj(ln C(u,p,d) Pj + Vj ln mj l .l t1],. Let us examine the necessary conditions for an AIDS Bar ten- Gorrnan system to generate a cost function homogeneous of degree l in prices. t his is obtained by ens uring that the shares add to one. recognize the restrictions on the vi's and A:3 seen before, In order to parameters when adding up is ~ j r's applied to equation (11) , it is convenient to isol ate three elements GT ( . ) , GS (.) and GCI(.) as follows: GT(d;v ,~ ) · 1;: [(1-vi )lnmi]-o, .l GS ( d ;y ,v, tl ) GCI(d,p; y ,v, "'r[r Yij Vj ln mJ],. 0, ~ ) a l;: [Pi ln y•] .. ln y• ~ .l l Pi .. o. Each of these terms is examined sequentially. (1) The GT ( . ) term E ( (1-vi) ln mi"'~ i ln l mi - ~v i ln l mi,. O, hence, homogeneity of C(u,p,d) in p and adding up imply : 1;: ln 1 mi "' EE ~ ir i r ln dr 0 - for each r, - ln dr E i for each r, () i r ln O, and d r l;:vi ~ ir .l which implies: 10 "' 0, E v1 i t>ir = 0, and ( 12) (2) Th• GS( . ) term l;: [l;: l J Yij v 1 E r t>1r ln dr] • O, hence, for each r : where x is a finite constant. Note that the restriction on the function GT(d;v, t> ) requires x be equal to 0. Hence, as before: (13 ) (3) The GCI ( . ) term It can be easily shown that the function GCI (d , p; y ,v, t> ) does not generate further restrictions . In matrix notation, the identifying restrictions of the Barten- Gorrnan model can be written as A\ =O , and AT=O , where A is a rxn matrix of demogr aphic parameters with n be ing the number of equations , \ is a nxl vector of ones , T is a n column vector of vi parameters and 0 is a rxl vector of zeros . It is important to note that, given this set of restrictions, the artificial paramete r s vi of the Barten-Gorrnan model a r e overidentified. To clarify, note that the condition A\ =O is derived from the homogeneity condition of degree one in prices of the cost function. This implies that A is o f rank n-1. Hence , the parameter v 0 can take infinitely many solutions and 11 there is not a unique way to reconcile the values of vn. Remarkably, it is neither necessary nor interesting to recover the value of vn uniquely from the product vn6nr• Due to the homogeneity of degree zero in prices of the demand system only n-1 of the artificial parameters, vn, have to be uniquely identified to fully compound the translating and scaling effects in the Barten-Gorman framework. The existence of at least one solution is ensured by the fact that the rank of 4 is Observe that the system would be otherwise consistent if all ~r. the vi are equal . This option is not satisfactory since v would act as a normalization that could take any value. Finally, note that neither equation (12) nor equation (13) imply linear or nonlinear restrictions on the parameters vi's per se when the vi's are equal. Equations (12) and (13) are the restrictions needed to identify and estimate the Barten-Gorman AIDS model. They are normalizations derived from the homogeneity requirement of the cost function. Therefore, these restrictions do not alter the economic content of the model. 5 . Concavity Following Lewbel's proof of Theorem 3, let c * (u,h(p , d))=C*(u , p*)=y* be a modified cost function and let C(u,p , d)=f(C* (u,h(p , d)),p*,d)]. For all arbitrary n vectors v, C(u,p,d) is concave in p if and only if 4 V / a2 c(_ u,p, d) v __ _ __ ~ 0, apap' where: As in Lewbel (1985), all gradient vectors, Jacobian and Hessian are represented as derivatives or by vectors. 12 v' o 2c(u,p,d) v opop 1 o 2 f(y*,p,dl v] ______ op op 1 + [ of v , oh o2c• oh ] ) oy• op op• op•' v o/ Assume that f[C*(u ,h (p,d)) ,p*,d] describe 8arten-Gorrnan preferences as: Note that 8 1=0 since and 8 2=0 because t i ti=l. In fact , Thus, C(u ,p,d ) is concave in p if B 3 ~o and c* in 8 4 is also concave given that y* is only differentiable once and oh/opi>O because p lies in the positive orthant and m>O is an exponential function. 5 Thus, in a BartenGorman context a sufficient test for concavity reduces to testing the concavity in p of both f and c*. 5 Observe that for more general forms of f(y*,p* , d) where, for example , y* is twice differentiable due to the introduction of a function of demographics that also shifts the parameters , then the conditions for C(u , p,d ) to be concave in p are more restrictive (See Lewbel Theorem 3) . 13 Assume again a Gorman Polar form for the cost function whose el~nts are specified by the AIDS model. Straightforward differentiation shows that a 1n f(y •, p,d) a1n P i gives the Barten-Gorman specification of the AIDS model presented in equation ( 11) and that a1n f 2 (y •,p, d) aln P i aln pj gives the compensated component s ij of the Slutsky equation. As suggested by Deaton and Muellbauer (1980), it is possible to obtain a simpler algebraic expression to test for concavity by adopting the following transformation : and, where Eij is the compensated price elasticity and Aij =l for i=j and 0 otherwise is the Kr onecker operator. The concavity of c* implies the concavity of f since f is continuous, monotonic and separable in c* and pT as shown above. This complete s the proof. 6 . App1ication to Food at Som. and Food Away from Hom. in th• USA During th• P•riod 1954-1990 6 . 1 Estimation The application is carried out estimating a complete demand system over the period 1953-1988 whose separable components are food- at-home, food-away- 14 from-home, and non-food. In recent years the empirical examinati on of the food-at-home/food-away-from-home issue has received increasing attention. From the point of view of welfare measurement, the decomposition between food and non-food is interesting because it is ethically in line with the Engel way of associating utilities with well-being. In the data set, personal consumption expenditure represents income. Expenditure information was obtained from the National Income and Product Accounts of the United States as published by the United States Department of Commerce. Price indices were derived from the annual city averages of consumer price indices from the regular urban National Statistical Accounts with base years 1983-84. The demographic variables included in the model are the percentage of the U. S . population falling in the 0-15 age (01) category and the percentage of U.S . population enrolled in schools in each year (02). Demographic information was drawn from Current Population Reports o f the U.S . Bureau of the Census. Descr.iptive statistics of the data used in the analysis are presented in the Data Appendix. The stochastic Barten-Gorman model is given by: E(wi) = «i + t i( d) + J:lYi j lnpj J +Pi ln ( y• A (p,d) (14) ) . We assume that the errors across equations (Ei) are no rmally distributed, with a constant covariance matrix C. They are uncorrel ated over time, but correlated in each period: E( Efr Ejsl • l Oi j for r = s 0 f or r,,. s Moreover, all variables affecting demand are ass umed as exogenous. The system of equations (14) formed by food-at-home (FH) , food-awayfrom-home (FAH) and non-food (NF) was estimated jointly using maximum likelihood (ML) estimation. Because the adding up restrictions were i mposed to identify the parameters in the model, \ 'wi =l and \ 1 Ei =O, 15 the cova riance matrix is singula r and the system is estimated by invariantly dropping the non-food equation. Following Atkinson (1970), Pollak and Wales (198 1 ) , and Davidson and McKinnon (1981) , the Barten Gorman model can be compounded in a parsimonious fashion that allows testing the specification of both the nested and nonnested models. This is accomodated by defining the demographic functions as follows: ti (d) a (1 - vi) ~ ~ ir ln dr • ~ .l ~ ir ln dr for ~ ir • (1 - vi) ~ ir , and .l for s ome constant vi. The models were estimated with the maintained hypothesis of homogeneity and symmetry. Adding up was explicitly imposed since the model is non- linear. As shown in section 3, the artificial parameters vi of the Barten-Gorman model are overidentified , but all demographic parameters can be uniquely identified. The derivation of the income , price and demographic elasticities is presented in Appendix A. 6 . 2 Resu1ts Three demographic specifications were estimated . The parameter estimates for the Translating (T), Scaling (S) , and Barten-Gorman (BG) specification can be found in Appendix B. To test for the econometric superiority of one demographic specification over the other we rely upon the Likelihood Ratio . The statistical difference of the v terms from either 1 or 0 is non-informative since the v 's are not exactly identified. The likelihood ratio tests of translating against Barten-Gorman, and Barten-Scaling against Barten-Gorman, are shown in Table 1 . The values of the test fail to reject the null hypothesis that scaling and translating are as good as the Barten- Gorman. Nevertheless, according to the likelihood 16 dominance criterion introduced by Pollak and Wales (1991), the Barten- Gorrnan model is econometrica lly superior to both . The values of the likelihood functions are presented in Appendix B, Table B. l. Tabl• l. Likelihood Ratio Test for demographic specification D.moqraphic Sp9ci:fi cati on LR-2 (L•-L) x2 (.0 l;d.f.l x2 ( . 05; d. fl BG vs BS d . f. - 2 1.12 9 . 21 5.99 BG vs T d . f. - 2 1.23 9 . 21 5.99 Note: L• is the unrestricted log- likelihood value . x 2 (s;d .f. ) wheres- significance level and d .f. -number o f restrict ions. If the model is to be used for estimating equivalence scales and money metrics for u tility, Blackerby and Donaldson (1988) and Lewbel (1989) point out that the money metric representations should be concave at all price levels. This requires that the Slutsky matrix must be negative definite at all prices. Concavity ensures that social judgements do not contradict distributional judgments derived from a social welfare function which is quasi-concave when each of its arguments is concave. We test for "single-peaked" preferences, by computing the eigenvalues of the Slutsky matrix incorporating demographic factors . 6 In all the three demographic specifications the test for the violation of the second order conditions was performed at all data points and at the data means following the procedure explained in section 5. No violations in sign were encountered. All eigenvalues were negative. The elasticities for the Scaling, Translating and Barten-Gorrnan models are presented in Table 2. Comparing the results provides some in~ghts into whether or not the estimated elasticities are sensitive to the demographic specification. 6 Note that non- positive compensated elasticities is a necessary (minimal ) but not sufficient condition for negative semi - definiteness of the Slutsky matrix. 17 Table 2 . Elasticities Estima t es for the Translating, Scaling and Barten- Gorman Models Almost Ideal Model Food at Home (fh) Food \ Home (fah) Others (othl T s BG T s BG T s BG pfh - .556 ( . 095) - .558 ( . 095) -. 569 ( . 106) -. 035 ( . 197) - .032 ( .199) -. 019 ( .196) .116 ( .021) . 116 (. 021) . 119 ( . 022) pf ah -. 012 ( . 069) -. 011 ( . 069) -. 007 ( . 067) - .088 (. 450) - . 0 91 ( . 45 6 ) -. 113 (. 009) . 009 (. 019) . 009 ( . 019) . 009 ( . 018) poth . 569 ( .10 1 ) . 56 9 ( .1 03) .575 ( .108) . 123 ( . 273) .123 ( .277) . 132 ( .258) - . 125 ( . 014) -. 125 (. 014) -. 128 ( . 015) x . 381 (. 078) . 380 ( . 078) .396 ( . 083) .864 ( . 092) • 967 ( . 094) . 87 9 ( . 101) 1.14 ( . 012) 1.14 ( . 012) 1.133 (. 013) Dl -. 330 ( . 126) -. 332 ( . 125) - .378 ( .226) .102 ( . 222) . 106 (. 019) .105 ( . 0 41 ) . 061 ( .020) . 060 ( .230) . 07 ( . 219) 02 . 364 ( . 123) .371 ( .226) .333 ( .227) -.4 5 1 (. 226) -.4 53 ( . 129) - . 464 ( .308) -. 043 (.019) - . 04 4 (.019) - . 036 ( . 059) Note: Asymptotic standa r d e r rors are i n paren t heses . The price elasticities are compensa ted . The results indicate that the statistical and economic differences between the estimated elasti ci ties across demographic specifications are not significant. The estimates are consistent with the theory and conform with estimates from other time-series studies. Though the elasticity estimates do not vary statistically and economically across demographic specifications , it is worth noti ng that the estimated elasti cities are similar for the Translating and Scaling but differ somewhat from the Barten-Gorman results. This is an i ndication t hat the Barten-Gorman model signifi cantly captures the economic information conveyed by either Translating or Scaling. 7 . Conclusions This study provides a theo retical cons i s t ent specifcation of the BartenGorma n mod el , illustrates how i t can be ide ntified a nd estimates it in the AIDS framework. We use the homogeneity property of the modified Gorman cost function t o derive conditi ons that allow identifying all demographic parameters . We also show how to test for concavity in the Barten-Go rman context . 18 With the present data set the Likelihood ratio test did not indicate the Barten-Gorman model as statistically superior. Neverthless , the Barten-Gorman specification can still be deemed as more interesting. In fact, the v's artefact permits singling out the price and income component of the demographic effect and distinguishing whether a variable affects consumption through an income or a price effect . This knowledge is usually not available a priori . On this ground , the Barten-Gorman model is preferred to a less theoretically rich specification such as Translating and Scaling. 19 Appendix A . Derivation of the Income, Price and Demographic Elasticities for the Barten- Gorman model . Recall from equation (11) the estimated version of the Barten-Gorman model: + Pi ln ( ) y• A(p,d) • « i + ( 1-vi ) l n mi +°1?-Yij (lnpj +lnmi) +Pi( (lny-1;: (1-vi) lnmi (lnpi +lnmi) ) -lnA (p, 1 J d)) (A .1) where: ln A(p*) = ln A(p, d) =Cl() + ~ « i ln pj + .5 EE y ~ ·ln p~ i j l l] l ln p~ J and, for i=l, .. , n, n being the number of equations and r=l , . . ,R, R being the number of demographic variables. Let: A nxl vector of parameters , parameters, of demographic parametrs , of p parameters, of ~ v parameters, 7 of the means of the natural log of the price variables, of the means of the natural log of the demographic variables, denoting the mean of the natural log of income, and of shares, of the natural log of quantities, of 1. r = nxn matrix of A = nxR matrix nxl nxR p = nxl D Rxl y lxl w nxl Q = nxl \ = nxl B A Define: M = A*D S = A*D T = M-S P*= P+S 8 vector matrix vector vector scalar vector vector vector Cl x = nxl vecto r valued demographic function , nxl vector valued scaling component of M, nxl vector valued translating component of M, and nxl vec tor valued fun c tion of demographically modified pri ces. Given the overidentification of the Barten- Gorman model, t he e l ements o f the row of the A matrix corresponding to the omitted equation are not uniquely separable into its ~ and v components . 20 In matrix notation, equation (A.l) can be rewritten as: where .* is the element-wise multiplication operator. Using the definition of a share as wi=piqi/y, specify the following relationship in matrix notation: Q"' ln w + Y - P. Then, use the rule of vector differentiation to derive expenditure, price and demographic elasticity. Expenditure Elasticity 'l 'l .. w "' aY"" oln oQ oY \ + oln w aw aw TY UncOD1p9nsated Price Elastici t y Eu oQ CJP oln w aw .. 1 aw oP (r - a (A + (M-sl + 1\7 where I is an nxn identity matrix. Compensated Pri ce Elasti city E • Eu + E 'l W1 using the Slutsky relationship. Demoqraphi c Elasticity oQ aD a1n w aw -aw- aD = 1 . • (('1-Al + (r•Al 1\7 21 1r•P *l )'] - r Appendix B . Tab1• B . 1 . Values of the Likelihood Functions Demographic Specit'ication Barten-Gorman 356.83 (BG) Barten Scaling (BS) 356.27 Translating (T) 356.21 Table B. 2. Parameter Estimates pa ram T s BG a1 0 .44 82 (0.038 4) 0 .4456 (0.0373) 0 . 3883 (0 . 0695) a2 0 . 01645 (0.0212) 0 . 0 1816 (0 . 0213) 0 . 02271 (0 . 02 42) 1'11 - 0 . 01280 (0.0119) - 0 . 0 1291 (0 . 0118) - 0 . 011 4 6 (0.0121) I' 12 - 0 . 09936 (0.0124) - 0.09921 (0 . 0 125) - 0.09737 (0 . 0135) 1'22 0 . 05143 (0 . 0140) 0 . 05170 (0 . 0 136) 0 . 04603 (0. 0139) ~l o. 04717 (0 . 0251) 0 . 0 4733 (0.02 48) 0 . 0 4 607 (0 . 0235) ~2 - 0 . 00739 (0.0052) - 0 . 00756 (0.0051) - 0 . 00673 (0 . 0056) 5u - 1. 605 (1. 76 4 ) - 0 . 052 44 (0 . 0198) -0.9252 (11.57) 512 1. 422 ( 1. 4 97) 0 . 05772 (0 . 0195) - 0.7986 (3 . 653) 521 o. 5119 (0 . 6813) 0 . 005695 (0 . 0 123) 3.085 (3 . 698) 522 - 0 . 7067 (0 . 5806 - 0 . 02506 (0 . 0125) - 0.9219 (2. 721) Vl 1. 042 V2 1.076 (0 . 0765) (0 . 0 911) Note : Standard deviations are in parentheses 22 Appendix c. Swmnary Statistics - Years 1953 to 1988 Unit Minimum Maximum Mean Std Dev Sbil 232 . 6 3235.1 1070 . 61 904 . 2203 sha re(food at home) % 0.1151 0.2056 0 . 1604 0.02 64 share(food away from home) % 0.0529 0 . 0606 0 . 0554 0 .0019 s hare( non food) % o. 7347 0 . 8302 0 . 78 43 0 . 0274 p(food at home) $ 29 . 5 116. 6 50.2916 1. 6450 p(food away from home) $ 21.5 121. 8 44.6150 1.8094 p(non food) $ 23.3238 124 .688 45 . 1786 l. 7558 population 0- 15 of age % 0.2285 0.3306 0.2827 1.1421 pop enrolled in school % 0.2047 0 . 2943 0 . 2584 1. 0905 Variable total PC Expenditur e 23 Atkinson, A. 1970. "A Method for Discriminating between Models." Journal of the Royal Statistical Society . 32 : 323-53 . Barten, A.P. 1964. "Family Composition , Prices and Expenditure Patterns ." In Econometric Analysis for National Economic Planning: 16th Symposium of the Colston Society, ed. by P. Hart, G. Mills, and J.K. Whitaker. London: Butterworth . • Blackorby, c . and D. Donaldson. 1988. 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