Chapter 20. Aggregation and calibration. Betina Dimaranan, Thomas Hertel, Robert McDougall

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1 Chapter 20 Aggregaton and calbraton Betna Dmaranan, Thomas Hertel, Robert McDougall In the prevous chapter we dscussed how the fnal verson 3 GTAP data base was assembled. Ths data base s extremely large. For example, just one of the arrays assocated wth ntermedate usage of commodtes s of dmenson: 37 x 37 x 30 = 41,070 elements. For ths reason, the resultng model s also extremely large. Indeed for most researchers, a model based on the fully dsaggegated data base s smply too demandng of computer memory and ds space. As a result, most GTAP users wor wth aggregatons of the data base. Procedures for aggregaton are dscussed n secton Once the data base and elastctes have been aggregated, there remans the problem of calbraton. Fortunately, the extent of ths problem s mnmzed by solvng the non-lnear GTAP model va ts lnearzed representaton. Ths approach has the advantage of not requrng calbraton of the so-called shft parameters n the producton and utlty functons, as they are embedded n the cost and expendture shares whch appear n the dfferental form of the model. In the case of the CES functonal form, where the shares and the constant substtuton elastcty parameter are all that appear, no further calbraton s requred. However, n the case of the CDE functonal form used to represent prvate household demands, a further calbraton step s requred. Ths s dscussed n secton 20.2.

2 Aggregaton Procedures Aggregaton of value flows n the benchmar data s nvolves the smple summaton of approprate component elements. For example, dsaggregate consumer expendtures for agrcultural and processed food products are summed to obtan total consumer expendtures for the aggregate category, food. Smlarly, when aggregatng regons, consumer expendtures for each commodty are summed across dsaggregate regons. For varables that are source-specfc (e.g., trade flows), the regonal aggregaton also aggregates across these sources. Aggregaton of behavoral parameters of the model s more nvolved than aggregaton of value flows. The relevant behavoral parameters are the uncompensated own-prce and ncome elastctes of demand, the elastctes of substtuton n value-added, and the elastctes of substtuton among foregn goods and between domestc and foregn goods. The own-prce and ncome elastctes of demand vary by commodty and regon, whereas the substtuton elastctes vary only by commodty. Rather than smply computng arthmetc means of the dsaggregate parameters, we compute ther expendture share-weghted means. In the case of consumer demand elastctes, we proceed as follows: We ensure that the Engel aggregaton condton holds for the dsaggregate data and parameters by proportonally adjustng all ncome elastctes as requred. We apply the Slutsy condton to derve compensated own-prce elastctes based on the GTAP data for uncompensated own-prce elastctes and the ncome elastctes from the prevous step. We obtan expendture share-weghted means of the dsaggregate ncome and compensated own-prce elastctes for the aggregaton n queston. We agan mae sure that the Engel aggregaton condton holds for the aggregated data and ncome elastctes by proportonally adjustng all ncome elastctes. Snce the elastctes of substtuton are not regon-specfc, only aggregaton across commodtes s necessary. Hence, the share weghts used n commodty aggregaton are global values. For the elastctes of substtuton n value-added, sectoral payments to land, labor, and captal are summed to obtan total value-added by sector, whch s the weght used n the aggregaton. For the elastctes of substtuton among foregn goods and between domestc and foregn goods, total expendtures for mports and total expendtures for mported and domestc commodtes are calculated. As above, share weghts are then calculated for the dsaggregate members of each aggregate commodty and used to weght the dsaggregate substtuton elastctes pror to summaton. Table 20.1 reports the parameter fle used n the 3x3 aggregaton taen from Hertel et al. It begns wth the calbrated CDE parameters, SUBPAR and INCPAR, whch are commodty- and

3 20-3 regon-specfc (see next secton for more detals). These are followed by the regon-generc e l a s t c t e s o f

4 20-4 Table 20.1 Parameters for the 3x3 Aggregaton The matrx SUBPAR(%1,%2) wth %1 n IND, %2 n REG. %2= USA EU ROW %1=food %1=mnfcs E E E-03 %1=svces The matrx INCPAR(%1,%2) wth %1 n IND, %2 n REG. %2= USA EU ROW %1=food %1=mnfcs %1=svces Values of ESUBD(IND) - an array of sze food mnfcs svces Values of ESUBM(IND) - an array of sze food mnfcs svces Values of ESUBVA(IND) - an array of sze food mnfcs svces CGDS Values of ETRAE (a sngle number) Values of RORFLEX(REG) - an array of sze USA EU ROW Values of RORDELTA (a sngle number) Source: Global Trade Analyss: Modelng and Applcatons, Thomas W. Hertel (ed) Chapter 5, p. 156.

5 n 20-5 substtuton between domestc goods and mports (ESUBD), between mports from dfferent sources (ESUBM), and among the components of value-added (ESUBVA). The next entry, ETRAE, specfes the elastcty of transformaton between alternatve uses of the sluggsh endowment commodtes ths case, farmland. The fnal sets of parameters are relevant to the nterregonal allocaton of nvestment. RORFLEX, the flexblty of nvestment allocaton, s permtted to vary by regon. It s operatonal only when RORDELTA s equal to 1. In ths parameter fle RORDELTA = 0, so that the global banng sector's regonal nvestment portfolo s fxed The CDE Functonal Form For GTAP, we choose the CDE functonal form for the prvate household s expendture functon. The CDE expendture functon was ntroduced by Hanoch (1975), who dscussed models that were more general than the CES but less general than a flexble functonal form, for example, the translog. 3 The CDE s based on the assumpton of mplct addtvty. whch, n the case of N commodtes, constrans the symmetrc N N matrx of elastctes of substtuton to depend on only N parameters. The CDE also allows for a rcher representaton of ncome effects n the demand system. In partcular, margnal budget shares may vary wth expendture levels. Secton gves a bref overvew of the CDE form. Secton dscussed the calbraton problem n detal Theoretcal Development of the CDE In general, an expendture functon can be represented n the followng manner: E G(p,u) {mn p x:f(x)u}, (20.1) where p and x are N-dmensonal vectors of prces and demands, u s utlty, and E s mnmum expendture. The functon f() represents utlty, and G() s the mnmum expendture functon. Functon G() s homogeneous of degree 1 n prces, allowng the followng normalzaton of prces and expendture by mnmum expendture: G(E 1 p,u) G(z,u) 1, (20.2) where the z's are the normalzed prces. To obtan the CDE expendture functon, Hanoch (1975) restrcts the number of substtuton effects to N by mposng addtvty n the normalzed prces. The mplct functon proposed taes the form:

6 20-6 G(z,u) N 1 B u e b z b 1, (20.3) where the b 's are the N parameters, whch determne substtuton possbltes among commodtes n consumpton; the e 's are N expanson parameters, whch appear owng to nonhomothetcty n consumpton; and the B 's are scale parameters necessary to specfy the functon. It s requred that B > 0 and e > 0, and b < 1, wth ether 0 < b < 1 or b < 0 for all (Hertel et al. 1991). If the substtuton parameters are rewrtten as = 1 -b, the Allen partal elastctes of substtuton can be expressed as: j j s j s, (20.4) where = 1, and j = 0 for j, and the s 's are expendture shares. The name constant dfference n elastctes arses due to the fact that the dfference between the elastctes of substtuton j and h s nvarant to ndex (Hertel et al. 1991) ( j - h ) = ( j - h ) (20.5) The expressons for ncome elastctes of demand are: s e 1 e (1 ) s e s, (20.6) and the expressons for compensated own-prce elastctes of demand are: v j s 2 s /s. (20.7) Calbraton of the CDE The ncome and uncompensated own-prce elastctes of prvate household demand used n GTAP derve from a varety of sources, none of whch nvolve econometrc estmaton usng the CDE form. Ths s a common dffculty when choosng elastctes for AGE models. That s, the functonal forms used n the econometrc studes from whch estmates are drawn do not correspond to the functonal forms specfed n the model. Furthermore, the sample perod used n estmaton may not nclude the benchmar equlbrum year. Fnally, econometrc studes are based on consumer goods prces, whereas AGE models typcally evaluate consumpton at producer prces. These dvergences cause

7 20-7 dffculty, as not all sets of ncome and own-prce elastctes wll be representable va the CDE functonal form, evaluated at 1992 producer prces. The problem can be seen clearly for the consumpton budget shares and own-prce elastctes shown at the top of Table These are derved from a three-regon, three-commodty aggregaton of GTAP (Huff et al). The parameters derved by solvng equaton (20.7) for each regon are also gven n Table Planly, none of these sets of values satsfes the requrement that 0 < < 1 for all. Therefore, a procedure must be formulated that, for any set of ncome and own-prce elastctes, determnes values of the CDE parameters that yeld model ncome and own-prce elastctes suffcently close to the ncome and own-prce elastctes. Ths secton descrbes such a calbraton procedure for determnng CDE parameters. It s evdent from the outset that certan aspects of ths calbraton procedure, such as the choce of a measure of "closeness," wll be, to some extent, arbtrary. Another element of choce s the weght to be gven to devatons of own-prce elastctes from values as compared wth ncome elastctes. In the case of the CDE functonal form, a "natural" choce n ths regard s suggested by equatons (20.6) and (20.7), relatng elastctes to CDE parameters. The own-prce elastctes are determned by the parameters, whle the ncome elastctes depend on both and e parameters. Thus the procedure adopted was to determne 's to ensure closeness of the own prce elastctes mpled by equaton (20.7) wth elastctes, and to use these 's n equaton (20.6) when determnng an approprate set of e parameters. The calbraton procedure for CDE parameters entals solvng two constraned mnmzaton problems. The frst determnes the CDE parameters by seeng a good ft to the own-prce elastctes. The second, tang the values from the frst as gven, determnes the e parameters by seeng a good ft to the ncome elastctes. The constrants mposed are of two types. Frst are those on CDE parameters requred for the regularty of the CDE form. Second s the requrement that ncome elastctes are greater than (less than) or equal to 1 f the correspondng s are greater than (less than) 1. Ths latter constrant ensures that goods that were superor (nferor) as represented by the elastctes reman so after calbraton. The constraned mnmzaton problem for determnng the CDE parameters can be represented formally as: gven, choose, and to mnmze: [log e ( / ) 1] (20.8) subject to , and equaton (20.7), where = -compensated own-prce elastctes; = compensated own-prce elastctes arsng from calbraton; = CDE substtuton parameters arsng from calbraton. The regularty requrement that 0 < < 1 has been replaced by the nequalty constrant shown. Otherwse the problem may not have a soluton f the objectve functon attaned ts

8 20-8 mnmum on the boundary of the regon { : 0 < < 1 for all }. The objectve functon s strctly convex, and the feasble regon convex, so a unque mnmum exsts. The constraned mnmzaton for determnng the CDE e parameters can be represented formally as: gven, choose, and e to mnmze: ( ) 2 (20.9) subject to equaton (20.6), 0.01 e, s 1, and ( -1)*( -1) 0, where = ncome elastctes; = ncome elastctes resultng from calbraton; and e = CDE expanson parameters resultng from calbraton. The regularty requrement that e > 0 has been replaced by the nequalty constrant shown, for the same reason that the regularty condton on the was altered n the frst mnmzaton. It should be noted that the second constrant ensures that the ncome elastctes derved as the soluton to ths mnmzaton wll satsfy the Engel condton. The thrd constrant ensures that the calbrated ncome elastctes le on the same sde of one as the elastctes. Although many software pacages exst n whch these constraned mnmzatons could be formulated and solved wth ease, they have been mplemented n GEMPACK. Ths has been done to ensure that GTAP users do not requre access to any other software n order to perform the entre range of operatons, from data aggregaton to prntng of smulaton results. Fnally, a mportant warnng. It s ey for users to note that the calbraton process nevtably results n elastctes whch depart from ther values. The degree to whch ths occurs wll vary by aggregaton. In many cases the and ftted ncome elastctes of demand are qute close. However, users have also reported some nstances where they depart sgnfcantly. If the prvate household s ncome and prce responsveness s mportant to the problem at hand, the user would be well-advsed to spend some tme scrutnzng the ftted values before proceedng.

9 20-9 Table 20.2 A 3X3 Example of the CDE USA EU ROW Consumpton Shares Foods Manufactures Servces Own-Prce Elastctes Foods Manufactures Servces Parameters Derved by Solvng Equaton (20.7) Foods Manufactures Servces

10 20-10 References Hanoch, G. (1975) "Producton and Demand Models wth Drect or Indrect Implct Addtvty," Econometrca 43: Hertel, T. W., E. B. Peterson, P. V. Precel, Y. Surry, and M. E. Tsgas (1991) "Implct Addtvty as a Strategy for Restrctng the Parameter Space n CGE Models," Economc and Fnancal Computng 1(1): Huff, K., K. Hanslow, T.W. Hertel, and M.E. Tsgas (1997) GTAP Behavoral Parameters. In Thomas W. Hertel (ed), Global Trade Analyss: Modelng and Applcatons, Chp. 4, pages