Munich Personal RePEc Archive
Building an input-output Model for Buenos Aires City
Chisari, Omar Osvaldo and Mastronardi, Leonardo Javier and Romero, Carlos Adrián
Instituto de Economía UADE, Instituto de Economía UADE and CONICET
28 February 2012
Online at https://mpra.ub.uni-muenchen.de/40028/
MPRA Paper No. 40028, posted 13 Jul 2012 14:12 UTC
B UILDI NG AN I NPUT OUTPUT MODEL FOR B UENOS A I RES C I TY
Mastr onar di , Leonar do J.
Insti tuto de Economía UADE and CONICET
Romer o, Car los A.
I nstituto de Economía UADE Chisar i, Omar O.
Insti tuto de Economía UADE and CONICET
Abstr act:
Buenos Air es City (BAC) is the Ar genti na’s biggest cit y and the second lar gest met r opolitan ar ea i n South Amer ica after Sao Paulo (Br azil ). Assessi ng r egional effects might be useful t o take politi cal or / and economic deci sions, consider i ng the di mension and t he economic impor tance of Buenos Ai res Cit y. Taking into consider ation the latter backgr ound i nfor mation, t he aim of this paper is to quantify the BAC’s i nt er r egional flows, evaluat ing dir ect and indir ect r egi onal effects wit h other r egi ons of Ar gentina. At this r egar d, differ ent l evels of i ntegr ation and dependence bet ween BAC and the other r egi ons countr y can be esti mat ed applying and Inter r egional Input Output model.
This i s the fir st time a input-output mat r ix is constr ucted for Buenos Air es, whi ch does not have a Regional Account s Syst em available. To tackl e this pr oblem, our model uses non-sur vey and calibr ation techni ques.
The paper focuses on the building pr ocess of that Input–Output Model and pr esents the estimations for intr ar egi onal and inter r egional tables. I n par ticular , Ar gent ina is separ ated in two r egions, BAC and the r est of the countr y. The esti mati ons to measur e t he Intr ar egional coefficients for each r egi on ar e based on non-sur vey techniques, usi ng Locati on Quotient s (Si mpl e Location Quoti ent, Cr oss Industr y, Flegg’s Location Quotient and Augment ed Flegg’s Location Quoti ent). Two common alter nati ve ways to balance t hese mat r ices, the RAS and cr oss entr opy methods ar e adapted to estimate the inter r egi onal coeffi cients.
JEL: C67 – D57 – R15 – R58
1. I
NT RODUCT I ONThis paper focuses on the building pr ocess of a r egional i nput-output tabl e for Buenos Air es Cit y (BAC), the capi tal of Ar gentina and the second biggest city in South Amer i ca. Our aim is to estimate tr ansaction matr i ces for BAC and the Rest of Ar gentina (ROC), using r egional input-output methodology. Thi s paper i s par t of a broader objective:
the constr uction of a CGE model of Ar genti na with two r egions that tr ade among them and with the r est of the wor ld. Par ticular l y, our wor k is a fir st step to build an Inter r egi onal Social Accounting Matr ix for BAC.1
At thi s r egar d, Ar gentina is separ ated in two r egions to cr eate the input out put tables, BAC and the ROC. An estimati on of i nter regional and intr ar egional flows for ten pr incipal sector s in each r egion will be pr ovided in this paper. The key of the estimation i s the infor mation availabi lity. Unfor tunately, ther e is not a census or other r egional stat s (sur vey methods) that can be used to compar e wit h nat ional data. Accor di ngly, hybrid and non-sur vey methods wer e used to build the tables in this study. Ther efor e t o measure an int rar r egional coefficient for each region we based our esti mations on non-sur vey techniques such as Location Quotients (Simple Location Quoti ent, Cr oss Industr y, Flegg’s Locati on Quotient and Augmented Flegg’s Location Quotient ). Two common alter native ways to bal ance these matr ices, the RAS and the Cr oss Entr opy Method, have been adapted to esti mate inter r egional coeffi ci ents.
The paper is or ganized as follows. In section 2, the paper pr esents methods based on backgr ound liter atur e as Jensen et al . (1978) and Flegg et. al (1995, 1997, 2000). They will be used to estimate the intr ar egional fl ows usi ng t he national technical coefficients. The idea is “to r egionalize” the nati onal input output coeffi ci ents using a l ocation quotient (it depends on the r el ati onships between the r egi on and the nati onal data) that assigns a value for the r egional technical coefficient. In secti on 3, we pr esent calibr ati on methods that have been applied in the liter atur e, based on Robinson, Cattaneo and El Said (2001) and Romero (2009). In thi s section, the Bipr oportional Adjustment (her eafter RAS) and Regi onal Cr oss Entr opy wi ll be used to estimate the final tables. Compar ative per for mance indicator s ar e used for these estimates all owing to choose a method in the sect ion 4.
Fi nally, in the section 5 we present conclusions based on the estimated matr ix.
Socio-Economic char acter istics of Buenos Air es
In 1994, BAC has become an autonomous city of Ar genti na, changing its institutional status. It has an appr oxi mated ar ea of 202 squar e kilometer s and thr ee million inhabitants that r epr esents the 7.5% of the Ar gentina population. It is the thirti eth urban area with r espect to the mar ket size and the best city of Lat in Amer ica in ter ms of life quality2. The r egional Gr oss Domestic Pr oduct (hereafter GDP) of BAC is about 60 billions of dollar s and it r epr esented about 28% of Ar gentina’s GDP i n 2006. Mor eover , Buenos Air es i s the
1 This i s the fir st appr oach to estimate r egi onal input out put tables for BAC and ROC. Mastr onar di (2010) pr esents an intr ar egi onal input-output table for BAC and Mastr onardi and Romer o ( 2012) show a methodologi cal appr oach to build a regi onal input-output model.
2 See Minister io de Desarr ollo Económico (2009).
r ichest region of the countr y wit h a GDP per capita of U$20,000, when the aver age of Ar gentina i s about U$6,500.
In r elation to the r egional pr oduct, Table 1 shows that BAC is specialized in the ser vice sector , especially in fi nancial, r eal estate and tour ism.
Table 1 – BAC and Argentina’s GDP and relative shares ( In millions of Argentine Pesos and percentage)
N° Sectors BAC’s GDP ( 1) Argenti na’s GDP ( 2) Rel ative share
( ( 1) / ( 2) ) *100
1 Agr icultur e, f or estr y and hunt ing 807 41962 2%
2 Fishi ng 45 1707 3%
3 Mi ning and quar r yi ng 3534 33455 11%
4 Indust r y 26454 108366 24%
5 Water , Elect r ici ty and gas 1939 8883 22%
6 Constr uct ion 7480 31822 24%
7 Comm er ce 16074 65732 24%
8 Hotels and r estaur ants 7209 15377 47%
9 Tr anspor t and com municat ion 18458 47441 39%
10 Financi al inter mediat ion 14714 26432 56%
11 Real est at e, r enti ng and busi ness 31773 61993 51%
12 Publi c admi nist r ati on 7834 32407 24%
13 Educati on, healt h and soci al ser vices 10,927 45192 24%
14 Other ser vices 6,695 23592 28%
Total 153943 544361 28%
Sour ce: Inst it ut o Nacional de Est adísticas y Censos and Di r ecci ón Gener al de Est adíst ica y Censos (Mi nist er io de Hacienda GCBA) .
Regar di ng to the job mar ket, BAC has many commut er s fr om Gr eater Buenos Air es (her eafter GBA). GBA is the name to call the suburbs of BAC (See Figur e 1).It has appr oxi mately ten (10) million inhabitants (25% of Ar gentina’s population) and is part of the l ar gest pr ovince of Ar gentina (in ter ms of population and GDP).
Figur e 1. BAC and GBA
BAC GBA
Argentina
Metropolitan Area BAC
GBA BAC
GBA BAC
GBA
Argentina
Metropolitan Area
The migr ation fl ow between BAC and the r est of the region i s an impor tant pr oblem for the economic modeling because it must be differ entiated wher e the people wor k,
wher e the people live and which i s the pr opor tion of that people that consume and invest in their or iginal r egions or i n another r egi on. At this r egard, Table 2 pr esents statistics of occupied peopl e i n the metr opol itan ar ea (BAC and GBA). It differ enti ates wher e people wor k and wher e people l ive.
Table 2 – The occupied people in BAC and GBA
People wor king at
BAC GBA Both
BAC 1,210,089 178,787 65,023
People livi ng
at GBA 908,808 2,939,740 177,411
Sour ce: Encuest a Per m anent e de Hogar es ( INDEC)
Table 2 has shown that commut er s r epr esent a r elevant per centage (24.2%) of people. Additionally, about 4.5 million people wor k in the r est of the countr y (excluding GBA).
2. I
NT RAREGI ONAL I NPUT-
OUTPUT: T
HE USE OF LOCAT I ON QUOT IENT S The nati onal input-out put table has been used to show the fl ows between sector s within a countr y. Each industr y has pr oduced a single output, usi ng the pr oducts pr oduced by other industri es as inputs. These tabl es have not descr ibed the specifi c location of the industr y within the countr y.However , a national input-output tabl e can be disaggr egated in r egi onal tabl es, taking into account separ ately intr ar egi onal and inter r egi onal tr ansacti ons (Fuentes Fl or es, 2002).
Two pr incipal methodologies to regionalize a national input output table can be found in the liter atur e. The key to choose bet ween them is the data availability. On one hand, sur vey techniques ar e based on parti cular data or samples, but it pr esents the disadvantage of a str ongly, costly and slowl y pr ocess. On the other hand, the non-sur vey techniques do not need samples or parti cular census, because they use available annual data and economic census.
Statistics techniques have been used to der ive r egi onal i nput-output tables fr om a National Input-Output t able. Gener ally, these techniques have been employed to adjust a national technical coeffi ci ent to r efl ect the str uctur e of r egi onal pr oduction and thei r r el ationships with al l the sector s of the economy.
In r espect to technology, the national input-output table r epr esents the national aver age r equir ements of inputs to pr oduce the outputs. Those r equir ements ar e obtained fr om the sum of the compani es of the r egions. Instead, if a r egion is specialized in some acti viti es, it could have a di ffer ent technol ogy compar ed with other r egions. Another differ ence between the national and r egional tables is that the r egi onal tables contai n the r egional commer ce. Additionally, the r egional impor ts ar e defi ned by the goods and ser vices that come fr om another r egion. They ar e fundamental to the analysis, because the r egional inter mediate consumption is consider ed as a r egional impor t and r egi onal inter mediate sal es ar e tr eated like a r egi onal expor t, r especti vely.
The annex I pr esents the nati onal input-output table for Ar gentina dated in 2006 and based on Chisar i et al. (2010). This tabl e was the star ting point to apply the methods listed bel ow and to build the intr ar egional techni cal coefficients. Calibr ation techniques wer e applied to tr ansfor m this coefficient into r egi onal input-output tables for 2006.
The pr imar y aim of this study is t o separ ate Argentina in two r egions, BAC and the ROC. Ther efor e, the national input-output tabl e i s br oken down into four regional tabl es, which r epr esent intr aregional and inter r egional (expor ts and impor ts fr om/ to other r egion) commer ce between r egions. Table 3 show s a scheme for N sector s of the economy in each r egion to descr i be the tables.
Table 3 – An example of Regional I nput-Output Table for N sectors.
BAC acti vity sector s ROC activi ty sector s
S1 ... Sn S1 … Sn
S1 BAC activity …
sectors
Sn
BAC I nput-Out put BAC Expor ts – ROC I mpor ts
S1 ROC activity …
sectors
Sn
ROC Expor ts – BAC I mpor ts ROC Input-Output
Sour ce: Ow n el abor ation
Non-sur vey techniques wer e used to build the intr ar egional input-output t ables. In par ticul ar , the Flegg and Webber’s (1995, 1997, and 2000) methodology of locati on quoti ents (her eafter LQ) was used to model the r egional commer ce. Ther e ar e differ ent LQ’s and these techni ques have become mor e complex over time. In this paper each one i s mentioned, but the most recent LQ i s used to built the r egional input-output tables.
This methodology has assumed that the intr ar egional coeffi ci ents (rij) di ffer fr om the national coeffi ci ents (aij) only by a shar e, which has explained the r egional tr ade (lqij) (Jensen et . al, 1979)):
[ 1] rijlqijaij
The subscr ipts j and i r efer to the pur chasing and supplying sector s r espectivel y. The rij coeffi ci ent r epr esents an i ntr aregional quantity of input i that is needed by the sector t o pr oduce a unity of j pr oduct. It has been called “r egi onal pur chasi ng coeffi ci ent” (Fuentes Fl or es, 2002).
The possibi lity to quantify the shar e of r egional requir ements for a sector i n a specific r egi on has been ar gued to be the mai n advantage of the LQ. The rule presented on equation [2] has been consi der ed the fundamental constr ai nt of the LQ’s (Jensen, 1979).
The l atter r efer r ed constr aint implies that i f the r egion sector is self-sufficient or a net exporter , the LQ is higher than one (l qi j ≥1) and the regional coefficient (ri j) is exactly the nati onal techni cal coefficient (ai j). Instead, if the r egion sector is a net impor ter , the LQ is smal ler than one and the regional coefficient wi ll be a shar e of nati onal coefficient.
[ 2]
if 1
if 1
ij ij ij ij
ij ij ij
lq a lq
r
a lq r
In the next subsecti ons, sever al di ffer ent LQ’s to constr uct the r egional input-output tables will be presented. Fi nally, an augment ed Fl egg Locati on Quotient (her eafter AFLQ) and its estimat ion for intr ar egional tabl es wi ll be offered.
Simple location Quotient ( SLQi)
The Si mple Location Quotient (her eafter SLQ) compares a r egional sector shar e in r el ation to the r egional pr oduction wi th the national shar e with r efer ence to the nati onal pr oducti on.
[ 3] ,
, Si Ru
u i
Si TC
PV SLQ RPV
PV NPV
Wher e PVSi,Ru i s the pr oduction value of the sector i in the uth r egion, RPVu is the pr oducti on value of the uth region, PVSi,TC is the pr oduction value of the sector i in tot al countr y and NPV is the total pr oducti on of the country. As it was mentioned, the sector in the r egion is a net r egional expor ter if the SLQ is gr eater than one and a net r egi onal impor ter if SLQ is less than one.
A major cri ticism to this type of quoti ent is t hat its resul ts over estimate the r egi onal pr oducti on of many industr ies, i.e. it usually overestimates the i ndustr ies self-sufficient (Fl egg and Webber, 1997 and Fuentes Flor es, 2002). For thi s r eason, it has been suggested that other LQ’s have a gr eater pr ecision like Flegg’s Location Quotient (hereafter FLQ) or AFLQ, but calculati ons have appear ed to be mor e compl ex.
The annex II shows the pr oduction value in each region and the cor r esponding SLQ, using national data and another calculus based on Chisari et. al (2010). It has been affir med befor e in this paper that, i f the LQ is higher than one, the r egional technical coefficient is exactl y the national value.
Cr oss-industry location quotient (CILQi j)
The Cr oss-Industr y Location Quotient (her eafter CILQ) measur es the relative impor tance of the supplying industr y i with r espect to the pur chasing industr y j, i n a specific r egion:
[ 4]
, ,
, , Si Ru
Si TC i
ij
Sj Ru
j Sj TC
PV PV SLQ CILQ
PV SLQ
PV
Wher e PVSi,Ru i s the production value of the suppl ying sector i in the uth r egion, PVSi,TC
is the pr oducti on val ue of the supplying sector i i n t he countr y, and PVSj,Ru is the pr oduction value of the pur chasing sector j in the uth r egion, PVSj,TC is the production value of the pur chasi ng sector j i n total countr y. The latter for mul a is similar to the r atio between supplying and purchasing SLQi. (Flegg and Webber, 1997).
On one hand, if the r egional pr oduction of the supplying indust ry i (in ter ms of i ts national pr oduction) i s gr eater than the r egi onal pr oducti on of the pur chasing industry j (in terms of its nati onal pr oduction), the CILQij i s greater than one and the input r equir ements of j sector could be satisfied within the r egi on (Fuentes Flor es, 2002). On the other hand, if CILQijis l ower than one, the inputs needed by the purchasi ng indust ry might not be pr oduced by the supplying sector and, consequently, they would need to impor t the inputs fr om another r egion.
The just descr ibed method all ows to make r egi onal estimations without extensive sector ial data. It only r equer i res pr oduction data from the r egions. The main disadvantage of this method is that it r educes the industr y technical coefficient and magnify the impor tant sector s of the r egi on (Flegg and Webber, 2000). For this reason, it has been consi der ed that i t under esti mates the r egional impor t pr opensity and gener ates a higher sel f-suffi ci ent, l ike the SLQ. Annex III shows the cr oss-indust ry locat ion quoti ent for each r egion.
The FLQ ij for mula
The Fl egg Location Quotient (FLQ) attempts to solve the over esti mation of the industr y sector ’s self-suffi ci ency pr oblem, ascr ibed to CILQ and SLQ. This appr oach includes a cor rection to the CILQ method, which i s a measur e of the si ze of the r egion. The ai m of the cor r ection is to weight the importance of each r egion compari ng the r egi onal pr oducti on value with the national pr oduction value.
[ 5]
FLQ
ij CILQ
ij
[ 6] log2 1 RPVu , with 0 <1
NPV
Where λ* is the size factor that weight the r egional relative i mpor tance for the country. A crucial parameter for this quotient is δ (constant across the sectors), which is a measur e of the r egional impor ts.3 On one hand, if the par ameter i s close to one, the r egional impor ts will be higher . On the other hand, if the parameter is exactly zer o, the FLQ i s equi valent to the CILQ (Flegg and Webber , 1997)). Finall y, the ter m that has ri sen t o the power in questi on, i s a logar ithm of base two. It measur es the size of the r egi on using the resul ting share over the total pr oduction in the region (RPVu) and the nati onal pr oducti on (NPV).
3 A r ecent study of Faye, Romer o and Mastr onar di (2012) for t he Ar genti nean pr ovince of Cór doba have found that it was preferred a sectorial δ because it reduces the sectorial bias in terms of intermediate consumption and repr esents a better cost st ructure.
Empiri cal r esults in Fl egg and Webber (1996a and 1996b) have proved that thi s method is better than SLQ and CILQ because it r educes the standard err or on the non- sur vey estimation. However , this part icular LQ has dr awed some cr iticisms t hat the formula explained in the foll owing section will tr y to solve.
The cor r ection by a specialization coefficient: the AFLQi j for mula
McCann and Dewhur st (1998) have cr iti ci zed the FLQ for mula because it has not al lowed a regional technical coefficient of some par ticular industr y to be gr eater than the national techni cal coefficient of that. Flegg and Webber (2000) have offered a new LQ methodology called the augmented FLQ for mul a (AFLQ). It new method has included a special izati on effect of each indust ry.
[ 7]
AFLQ
ij CILQ
ij
log 1
2 SLQ
j FLQ
ij log 1
2 SLQ
j
[ 8] log2 1 RPVu , with 0 <1
NPV
The cor r ecti on of the equati on [7] (with r espect to the equati on [5]) will be oper ative if and only if the industr y is self-suffici ent, which cor responds wi th a SLQ gr eater than one. If that occur s, the specialization ter m wi ll r aise the FLQ formula and, consequently, the r egional impor t wi ll decr ease.
It has been affirmed that the parameter δ is important to make the estimation. Flegg and Webber (2000) have said that a r easonable value could be 0.3. In addition, they have al so advised a small er value i f the r egion is smal ler and vi ce ver sa.
For the cur r ent study, it has been decided to work with a parameter δ close to 0.4, because this specific case is about two lar ge r egions. It must be remar ked that non-sur vey methods use only pr oduction figur es. In our case, we also have i nfor mation on inter mediate consumption and value added. Hence, we have a mor e precise notion about the existent technology at the sector ial l evel4. These are i ncluded as additional constr aints that our estimati on of the r egional input output tabl es has to enfor ce. The next sect ions will show cal ibr ation techniques to deal wi th these constr aints.
The AFLQ coefficients and the intr ar egi onal input-output t ables ar e pr esented in the annex IV and in the annex V, r espectively. These tables change when the inter r egi onal commer ce i s incor por ated. It is important t o know that ever y LQ const raint must be enforced when the CILQ has been put in the equation [7], i.e. i f CILQi j is greater than one, the CILQij on the equati on [7] is one.
Once the AFLQ is computed, the r egional technical coefficients ar e obtained. These coefficients ar e used to multiply the r egi onal pr oduction value and to esti mate the intr ar egional input-output table. Wi th r espect to the interr egional tables, it has been assumed that a r egion is a r egional net-expor ter if and only i f the SLQ is gr eater than one (self-suffi ci ency). For thi s r eason, i t might be consider ed that BAC is net expor ter of ser vices, because it is mor e specialized i n that sector (the SLQ can be checked). In the next
4 The ratio betw een r egional intermediate consumption and r egional pr oducti on value obtained fr om a parameter δ of 0.37 for BAC and of 0.4 for the ROC has been close to the observed data in each region.
section, the inter r egional i nput-output tabl es for Ar genti na with calibration techniques will be estimated.
3. I
NT ERREGI ONAL I NPUT-
OUT PUT T ABLES: C
ALI BRAT I ON T ECHNI QUES Additional constr aint s must be added to br ing consi stency to r egi onal input-output tables. Fir st, special attenti on to the national accounts must be pai d, because the sum of tr ansactions car r ied out by sector must r epr oduce the national sector in r el ati on to the inter mediate consumption and the i nter mediate sales. Mor eover , the sum of r egional ij’s tr ansactions for a par ticular sector must r epr oduce the ij national tr ansaction for that sector . This constr aint implies to enfor ce the national technical coeffici ents and it coul d be summar izing by the equation [9]:[ 9] 1 1
P S
n ps
ij ij
p s
t t
Wher e rijpsi s the r egi onal ij tr ansaction fr om the pur chasi ng r egion “p” and the supplying region “s”, and anij is the national ij tr ansaction.
It can be argued that ther e ar e many probl ems in connection wi th the consistency of the intr ar egional tables. It has been menti oned that t he LQ theory needs only pr oduction data. At the local level, inter mediate consumption data ar e available, so ther e ar e additional constr aints to enforce. At this r egar d, si nce the quotient between the inter mediate consumption and the pr oducti on value i s differ ent across the r egions, the technology of each sector coul d be similar but no identical.
Taki ng into consider ation the probl ems descr ibed i n the l atter par agr aph, the inter r egional tables have been buil t using calibr ation techniques to enforce the nati onal table, to r eply it aft er the adjustment.
Bipr opor tional Adjustment (Stone, 1962 and Bachar ach, 1970) and Cr oss-Entr opy (Kul lback and Leibler , 1951) wer e the techniques used to solve those pr oblems. It has been affi r med by Mc Dougall (1999) that RAS is an entr opy optimizati on method, concl uding that entr opy optimization method is prefer r ed when a matr ix-filli ng probl em i s pr esent. However , i t also has been suggested that RAS i s prefer r ed for the balancing matr ix pr oblem.
An ini tial table was used by these techniques to build the final tables (see Table 3).
For this pur pose, the initial table was cal ibr ated taki ng additional assumptions. Fir st, it was put the i ntr ar egional tables which ones wer e calculated by LQ on the diagonal.
Second, the ini tial commer ce between r egions was needed. Subsequently, assumpti ons based on the theor y of LQ w er e used t o build tables for the two regions, as foll ow5:
a. A r egional sector is an exporter if and only if its SLQi is gr eater than one.
Then, the sector s that have br oken this rule only supply to the intr ar egi onal commer ce.
5 I t w ould be impor tant to poi nt out t hat if the sector can be disaggregated into smaller specific sector s, these techniques offer a mor e accur ate measurement . Unfor tunately, the data collected allow ed the disaggr egat ion into only fourteen sector s, given t he few informati on at l ocal level.
b. Subsequentl y, the equation [9] must be enfor ced. Using (a), if a sector j of BAC expor ts, the sector j of ROC does not expor t. Thus, it should be under stood that or
r
ijBAC ROC, 0
or rijROC BAC, 0. The latter sentence i s summar ized by the equation [10]:[ 10] ijps
0
p s
r
Although an initial table that enfor ces the inter mediate sal es w as obtai ned, it gener ally does not enfor ce the int er mediate consumption at regional level. A table wher e the total sales and consumption (r ows and columns) conver ge, di fferent fr om initi al table, is needed to solve the pr oblem. The availabl e data wer e: the production value, the inter mediate consumption and intermediate sales (for national data) and the r egi onal pr oducti on value (for intr ar egional tables), r egional i nter mediate consumption (total columns) and i nter mediate sales6 (total r ow s). Calibration techniques can be appli ed to solve the latter pr oblem.
It was deci ded to take addi tional assumptions to appl y the calibr ation techniques.
Taki ng into account that the wor k is based on the BAC, it was decided to fix the intr ar egional tables for this r egion. The latter assumption has impli ed that the LQ appr oach’s have a val id theor y as sour ce. Moreover , the calibr ation techniques wer e appl ied i n t he intr ar egional tabl es for ROC and the inter regional tables.
A cr ucial aspect for the calibr ation techniques is the star ting poi nt for the inter r egional tables i n the beginni ng of that pr ocedur e. A gener al appr oach to build the ini tial tables was not found in the liter atur e. For thi s purpose, two star ti ng point s wer e included based on supplyi ng and pur chasi ng assumptions. It has been pointed out befor e in thi s study that a BAC’s sector expor ts to a ROC’s sector if and onl y if their SLQ is gr eater than one.
With r espect to the sales theor y, it has been assumed that the supplying sectors sell their products in the same pr opor tion in each r egi on, i.e the sector one fr om ROC has a SLQ gr eater than one, so i nitiall y sell to BAC’s sect or in the same pr opor ti on as i t sel l to ROC. Certainl y, this shar e changes when the i terations to enfor ce the r estr ictions for inter mediate sal es and consumption ar e applied.
The other star ting point has a pur chasing assumption but differ s in each r egion. As the objective of this wor k is to estimate pr incipall y BAC tables to analyze their str uctur e, it has been taken the cost str uctur e fr om LQ techniques as well. To that end, it has been modeled the star ting BAC’s i mpor ts usi ng the inter mediate consumption pr opor tion of BAC intr ar egional tabl es. For the BAC’s expor ts, i t has been taken a tr ansacti onal appr oach7. The ROC’s imports have been distr ibuted i n tr ansactional propor tions of ROC’s
6 In fact, intermediate sales w ere not considered l ocal data. The total s come fr om t he assumption t hat if a r egion is self-sufficient , it can export. If it is not an export er region, t he total intermediate sales was given by the method of regionali zation of I-O tables. I f it i s an expor ter region, t he total intermediate sales w as or iginated fr om the di fference betw een nati onal i nt er mediate sal es and the sales in the other regi on.
7 The same appr oach could not be used because the LQ method overestim at es the ROC intr aregional tables for many sector s. The method has estimated an i nt er mediate consumption gr eater than the regional account s onl y for the intrar egional t ransactions. It has l ed t o the resul t that the LQ theor y does not need intermediat e consumpti on dat a to make the int rar egi onal tables.
intr ar egional tables. Accor dingly, it has been separ ated the BAC’s sectors that sell to ROC’s sector s and a shar e of ROC’s intrar egi onal tables have been computed as wel l. It is a pur chasi ng/ supplyi ng appr oach because i t has enfor ced the cost structur e and the impor tance of the sector at the inter medi ate l evel.
The RAS method
Bipr opor tional Adjustment, usually called RAS method (Stone, 1962 and Bachar ach, 1970) is the fir st calibr ation technique that to be explained i n this paper . Basically, the technique t akes an initial matrix (in the pr esent case the inter r egional input output tables) and a set of r ow and column vector s as benchmar k to enfor ce. After sever al iter ati ons, the method offers a new table with tr ansactions that has similar str ucture to the initial matr ix but it enfor ces the constraints (at r ows and columns level )8.
The logic of the iter ative pr ocedur e is to fi nd rj and sj vector s such that:
[ 11]
a
*ij r a s
i ij jWher e ri is the total of i column (intermediate consumpti on), aij is t he initial matr ix coefficient of consumption (not the tr ansaction), sj i s the total of the j column (inter mediate sal es) and aij* i s the fi nal matr ix of coeffi ci ents. The pr ocedure i s an i ter ative al gor ithm that is enfor ced in each iteration with the r ow or column total thr ough the change of the initial aij.
RAS has been fr equently used to calibr ate tables in the soci al accounti ng matr ices (see Chisar i et al , 2009 and 2010), l ike nat ional input -output table and pr ivate consumption tables. It has been suggested that t he disadvantage of this method is that r equir es r ow and column tot als and an ini tial mat rix to begin the pr ocedur e. Mor eover , it has been consi der ed not flexible for the matr ix additional constr aints (lineal or not lineal).
The regional input output tables are shown as an example because under that method the national tabl es cannot be r eplicated in the pr ocess to calibr ate the r egional tables (they can be si mi lar but not equi valent).
Once the final r egi onal tables ar e obtained, the national table could be r emade. As was menti oned befor e, it might be difficult to r each the or iginal national tabl e. In addition, many tr ansactions should be fixed for the BAC, so if one tr ansaction for this r egi on i s gr eater t han for the nation (it could happens appl ying LQ methods), it may be impossible to ar r ive to the or iginal table.
Regional Cross-Entropy: additional constr aints for the r egional pr oblem
It has been ar gued that the tr adit ional cr oss-entr opy appr oach is an infer ence stati st ic application based on infor mation theor y.9
To i llust rate the pr oblem in an intuiti vely way, the Figur e 1 shows the method.
Fi rstl y, a set of events (E1, .., En) wer e assumed t hat initially have qi pr obabi lity to occur.
8 I t has been show n by Bacharach (1971) that RAS converges under some necessar y and suffi cient condit i ons.
9 Technical beari ngs and different applications coul d be seen in Jaynes (1982) and Golan, Judge y Miller (1996).
Secondl y, it has been supposed that a message implies a change of those pr obabil ities and they are t ransfor med in pi. The pr ocedur e impli es to minimize a cr oss-entr opy measur e of distance (Kull back-Lei bler,1951) between the i niti al and the new pr obabi lities.
F
Fiigugurree 11:: TThhee CCrrososss--EEnnttrropopy y mmeetthohodd
Sour ce: Ow n el abor ati on
Thir dly, it has been assumed that it is focused on some par ti cular event Ej. The r eceived infor mation fr om the message has been –ln pj10, but the additi onal infor mation has been defined as follow: (–ln pj – ln qj) = – ln (pj /qj). Subsequentl y, expect factor has been applied separ ately over t he infor mat ive values of each event, the expected infor mati ve value has been found fr om the message (Robinson, Cattaneo and El-Sai d, 2001):
[ 12]
:
n iln ii i
I p q p p
q
Once the pr ocedur e t o estimat e i nter r egional input-output tables had been applied, the pr oblem has become to fi nd a new matr ix close to t he alr eady existing matr ix11, minimizing the cr oss-entr opy distance but enfor ci ng the constr ai nts. It could be consi der ed that thi s method as mor e flexible than RAS because i t allows updating the tables star ting fr om inconsistent data. Mor eover , it allows incl uding additional constr aints like non-linear constr aints of information on each tr ansaction or a set of them (not necessar y total row or col umn).
It has been suggested by Golan, Judge and Robi nson (1994) that differ ent techniques to solve the estimati on -pr eviously mentioned- have focused on the national input-output table.
The pr oblem to minimize the cr oss-entr opy measur e consi sts in finding a new set of coefficients (A) that minimize the measur e between the ini tial coefficient and the esti mated one.
[ 13]
i j
j i j i j
i a a
a, ln , *,
min Such as:
10 An experiment wit h n possible r esults is considered. A measure of uncertai nt y S(n) t hat has three pr oper ties has been searched: (i) S ≥ 0, (ii) S(1) = 0 y (iii) S(mn) = S(m) + S(n). It could be demonstrated that the logar ithm enforces t hese pr operties. So S(n) = k ln n, w her e k is a scale factor that normali zes to one the measur e
11 It should be remembered the impor tance of i ni tial tables on the previous sections.
E
j1
,...,
nq q p
1,..., p
nln
jj
p
q
Data 1
,...,
nE E
Additional information of each event
[ 14]
, 1i j
ai , j j
i j
i y y
a
, wi th 0ai,j 1The solution can be obtai ned solving a Lagr angian that includes equati ons [13] and [14]. The results combi ne information of the new matr ix and the initial one:
[ 15]
* ,
, * *
, ,
exp( )
exp( )
i j i j
i j
i j i j
i j
a y
a a y
Wher e i ar e the Lagr ange mul tipl ier s associated by the r ow -columns sum and the denominator is the nor mali zation factor. This methodology is used to update soci al accounting matr ices.12
It might be ar gued that Cr oss-Entr opy i s a more gener al technique than RAS because:
i. It does not need al l the new total s of row s or columns (although the pr edi ction will be less accur ate).
ii. It does not need a balanced initial matr ix (the sum of r ows could be mor e/ l ess than the sum of columns).
i ii. New r ims could contain an er r or ter m.
iv. New r ims can be non-fixed par ameter s.
v. Many values on the final matr ix could be fi xed (not necessar ily a par ameter , which will be explained further on this wor k).
vi. It al lows non-linear constr aints.
It has been obser ved that the initial constr ai nts ar e the same as the nati onal input- output pr obl em when the latter techniques on the r egional appr oach have been appl ied.
This paper intr oduces additional constr aint that allow s a better adjustment to r emake the national t able.
The same star ting point than RAS has been used under pur chasing assumption because it has better r esults for the measur e of the er r or . It all ows to compar e the per for mance of the methods. In the case of cr oss-entropy, it has been establ ished that additional constr aints usually take into account the objecti ve to have a l ower er r or mor e than RAS. The constr aints have specified by the tr ansactional equation [16]:
[ 16] ij ijp s,
p s
t t
Wher e p and s ar e the purchasing and supplying r egion and i j ar e t he specific sector s.
The latter constr aint (equation [16]) cannot be applied for t he entir e matrix because the BAC int rar egional tables have been fixed, being the loss of degr ees of freedom the main pr oblem. Instead, the equation [10] was enfor ced for each i nter r egional tr ansaction.
12 A methodological appr oach has been show n by Chi sar i et . al ( 2010) and Romer o (2009). In addition, it coul d also be seen in Ar ndt, Robi nson and Tar p (2002) to view applicat ion focuses on computable gener al equilibri um models.
To that end, it has been deci ded to r un the cross-entr opy pr ogr am with differ ent quantities of r estri ctions at the sector ial l evel, having in mind the objecti ve to analyze the r esults in ter ms of the estimated nati onal table and the ori gi nal i nput-output table.
Fir stly, the progr am without these constr ai nts was r un. Secondly, the fir st pri ncipal pur chasi ng tr ansaction for each sector at national level was fixed, applying the equation [16]. Finally, the second pur chasi ng tr ansaction was computed. This pr ocedur e was followed until the eighth pur chasing tr ansaction.
In the next secti on, statistics will be presented t o deci de what assumption coul d be better in ter ms of measur e the er r or between the estimated table and the or iginal one.
4. P
ERFORMANCE I NDI CAT ORS AND RESULT ANALYSI SThe objective of this section is to select the inter r egi onal tabl es that ar e mor e
“accur ate”. For thi s pur pose, it was deci ded to contr ast the esti mated national input output table with the or i ginal ones.
Stats for eleven estimations mentioned in pr evious subsect ions ar e offer ed: two for RAS esti mation (differ entiating the assumption around the initial matr ix) and nine for cross-entr opy technique (differ enti ati ng the quantity of fix sector ial tr ansactions in the pr oblem).
Fir st, it could be obser ved the absolute aggr egate bias, measur ed as equation [17].
[ 17]
^
ij ij
ij ij ij
t t
A B t
This indicator is the r esult of compar ing the tr ansactions in the final aggregated matr ix (t^ij ) and the star ting one (tij ). The indicator i s presented on Tabl e 4.
Table 4 – Aggregate bias by calibration method
Method AB
Supplyi ng RAS 8.9%
Pur chasi ng RAS 7.0%
Entr opy 0 t r ansacti on 12.2%
Entr opy 1 t r ansacti on 7.7%
Entr opy 2 t r ansacti ons 5.6%
Entr opy 3 t r ansacti ons 4.0%
Entr opy 4 t r ansacti ons 3.3%
Entr opy 5 t r ansacti ons 1.8%
Entr opy 6 t r ansacti ons 1.4%
Entr opy 7 t r ansacti ons 1.2%
Entr opy 8 t r ansacti ons 0.8%
Sour ce: Ow n el abor ation
It can be obser ved that RAS method i s pr eferr ed than Entr opy method i f and only if any constr ai nt or one constr ai nt ar e enfor ced. Compar ing the star ting point on RAS method, pur chasing method is pr efer r ed than supplying matr ix because the aggr egate bias ar e l ower .
Using the equati on [18], sector ial bias in terms of sales ar e comput ed. Unfortunately, a tr ade-off between add constr ai nts and the absol ute sector ial er r or w as found.
[ 18]
^
ij i j
i i
i ij
t t
A S S B
t
Table 5 shows the sector ial bi as in ter ms of intermediate sales. It can be obser ved that it is possible t o add constr aint s but these are worse in ter ms of r elative pri ces. It happened because when other tr ansactions ar e enforced, the er r or is put in some sector s that ar e less i mpor tant in ter ms of sal es. It is wor se because the str uctur e of sales of thi s sector at national level changes. When additional constr aints wer e intr oduced, the sector most affected w as the publi c admini str ation (S12).
Table 5 – Sectorial intermediate sales bias
S1 S2 S3 S4 S5 S6 S7 S8 S9 S1 0 S1 1 S1 2 S1 3 S1 4
Supplyi ng RAS 3.5% 9.3% 4.1% 7.0% 33.3% 38.7% 5.6% 8.9% 6.9% 13.7% 5.7% 52.8% 13.4% 16.7%
Pur chasi ng RAS 2.9% 2.7% 4.2% 5.1% 13.3% 45.8% 5.3% 9.0% 6.3% 8.1% 6.8% 11.4% 15.8% 12.5%
Entr opy 0 tr ans. 3.1% 3.3% 2.6% 8.7% 24.7% 59.6% 4.2% 12.3% 6.5% 22.5% 31.6% 19.9% 21.9% 14.2%
Entr opy 1 tr ans. 2.8% 4.1% 7.2% 2.2% 24.1% 8.1% 5.3% 10.9% 5.9% 12.4% 26.8% 33.6% 19.5% 12.8%
Entr opy 2 tr ans. 0.1% 5.0% 0.0% 0.5% 26.9% 5.3% 6.9% 9.9% 6.1% 9.7% 17.5% 30.6% 24.3% 13.1%
Entr opy 3 tr ans. 0.1% 2.9% 0.0% 0.2% 23.3% 4.1% 11.0% 12.3% 0.4% 8.0% 11.5% 29.6% 21.3% 13.9%
Entr opy 4 tr ans. 0.1% 0.0% 0.0% 0.0% 21.1% 5.8% 8.8% 12.8% 0.0% 2.5% 10.2% 23.2% 19.0% 16.7%
Entr opy 5 tr ans. 0.1% 0.0% 0.1% 0.0% 10.1% 7.5% 2.2% 14.9% 0.0% 1.8% 0.0% 15.2% 14.8% 18.7%
Entr opy 6 tr ans. 0.1% 0.0% 0.0% 0.0% 2.4% 4.4% 2.0% 13.5% 0.0% 1.2% 0.0% 27.6% 13.0% 14.8%
Entr opy 7 tr ans. 0.1% 0.0% 0.0% 0.0% 11.2% 2.2% 2.0% 10.3% 0.0% 0.3% 0.0% 47.1% 1.7% 14.8%
Entr opy 8 tr ans. 0.2% 0.1% 0.1% 0.0% 0.0% 1.1% 2.0% 3.9% 0.0% 0.0% 0.0% 57.6% 1.4% 15.7%
Sour ce: Ow n esti mati ons.
Using the equati on [ 19], the sector ial bias in ter ms of pur chases is computed. A tr ade-off between add constraints and the absolute sector ial pur chasing bias was found as wel l.
[ 19]
^
ij ij
j j
j ij
t t
A S P B
t
Table 6 shows the sector ial bias in ter ms of inter mediate pur chases. This bias is the impor tant one because the inputs r equir ement s affect dir ectly on the pr oduction function.
It can be obser ved that it is possible to add constr aints but these ar e wor se in ter ms of r el ative pri ces. When additional constr aints ar e intr oduced, the sector most affected is the r eal estate, r enti ng and business (S11).
Table 6 – Sectorial intermediate purchases bias
S1 S2 S3 S4 S5 S6 S7 S8 S9 S1 0 S1 1 S1 2 S1 3 S1 4
Supplying RAS 11.3% 15.2% 24.5% 4.0% 16.4% 5.3% 15.7% 13.7% 15.6% 23.7% 39.5% 5.4% 3.9% 9.9%
Pur chasing RAS 12.0% 4.6% 17.0% 2.8% 5.0% 3.6% 16.3% 6.9% 7.8% 20.0% 62.2% 5.2% 3.5% 6.6%
Ent r opy 0 tr ans. 9.7% 10.1% 12.3% 11.8% 6.2% 6.1% 21.8% 4.9% 8.2% 11.9% 67.8% 13.2% 4.6% 14.8%
Ent r opy 1 tr ans. 6.7% 8.8% 9.1% 6.1% 4.1% 6.3% 14.5% 4.3% 7.4% 6.1% 33.2% 7.0% 5.2% 12.3%
Ent r opy 2 tr ans. 3.0% 7.6% 8.2% 5.2% 3.7% 5.7% 9.0% 1.1% 2.5% 5.4% 31.3% 8.4% 2.6% 8.6%
Ent r opy 3 tr ans. 2.2% 3.8% 5.5% 4.3% 3.3% 2.7% 5.2% 0.7% 2.6% 2.6% 25.2% 5.9% 2.1% 4.8%
Ent r opy 4 tr ans. 2.6% 2.8% 3.9% 3.7% 3.0% 2.3% 4.8% 0.5% 0.9% 1.0% 19.9% 5.9% 2.1% 2.9%
Ent r opy 5 tr ans. 0.7% 1.4% 2.3% 2.5% 1.7% 0.2% 2.2% 0.3% 0.7% 0.2% 16.5% 2.5% 0.9% 2.1%
Ent r opy 6 tr ans. 0.4% 0.3% 0.9% 2.0% 1.4% 0.1% 1.3% 0.3% 1.1% 0.2% 16.5% 0.8% 0.9% 1.0%
Ent r opy 7 tr ans. 0.1% 0.2% 2.6% 1.5% 0.1% 0.0% 0.8% 0.2% 2.0% 0.1% 15.5% 3.2% 0.6% 0.7%
Ent r opy 8 tr ans. 0.1% 0.3% 0.0% 1.4% 2.2% 0.1% 0.0% 1.7% 0.3% 1.5% 11.7% 0.0% 0.0% 0.0%
Sour ce: Ow n esti mati ons.
It could be appr eciated on Figur e 2 the absolute aggr egate bias and the maxi mum absol ute sales bias. As it was said before on the Table 4, pur chasing RAS has a low er er r or in aggr egate terms than supplyi ng RAS. In terms of Cr oss-Entr opy method, it could be appr eciated that if it is not possible to enfor ce transactional constraints, RAS is better . However , when the tr ansacti onal constr aints are increased, the bias falls to 0.8%. The cri ter ia to choose the final matr ix was based on the last tables and the next figur e.
Figure 2: Aggregate bias and sectorial supply bias
12.2%
7.7%
5.6%
4.0%
3.3%
1.8%
1.4% 1.2%
0.8%
8.9%
7.0%
60%
34%
31% 30%
23%
19%
28%
47%
58%
53%
46%
0%
10%
20%
30%
40%
50%
60%
70%
Supply i ng RAS
Pur chasi ng RAS
Entr opy 0 tr ans.
Entr opy 1 tr ans.
Entr opy 2 tr ans.
Entr opy 3 tr ans.
Entr opy 4 tr ans.
Entr opy 5 tr ans.
Entr opy 6 tr ans.
Entr opy 7 tr ans.
Entr opy 8 tr ans.
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
Aggr egate Bi as Absolute sector al supply bi as
Sour ce: Ow n est imat ions.
The cr iter ia could change in terms of the objecti ve. For example, the r egional tables ar e needed to constr uct a gener al equilibr ium model. Then, if the technique is taken with the eighth biggest pur chasing tr ansactions, i t is not well when the sector ial r elative pr ices must be computed. However , these concl usions can contr ibute to the fi nal sector i al aggr egation.
For thi s paper ’s aims, i t was pr eferr ed to take for t he final matr ix the cr oss-entr opy technique, which fi xes the fifth principal pur chasing tr ansactions13. The estimated nati onal input-output table could be seen on annex VI and the final r egional matr ix could be seen on annex VII, respectivel y. It can be obser ved that the suppl ying bias are concentr ated pr incipally in sector s 12 and 14. This indicates that these sector s could be aggr egated with the purpose to enhance in ter ms of bias and not compound a distor ted sect or for the model.
Other cr iter ia could observe onl y pur chasi ng t ransactions with the objective to analyze the Leonti ef multiplier s and techni cal coefficients. If thi s wer e this paper ’s aim, pur chasi ng indicator s must be analyzed. These indi cator s suggest that mor e constr aints can be put to have better r esults.
Some i mplications can be obtained from the final matr ix. The inter r egi onal pr opensiti es to impor t, inter r egional pr opensities to expor t and final demand shar e ar e impor tant to be shown after the fi nal interr egional input-output tables ar e bui lt. Table 7 shows these regional shar es in t er ms of pr oducti on value. In addition, the r equir ement of industr y impor ts for BAC ar e presented, because it expl ai ns the 55% of the BAC impor ts. It could be an under / over estimation measur e of the accur acy that have the location quoti ents methods.
Table 7 – Exports ( X_reg) and imports ( M_reg) requirements and Final Demand( FD) . I ndustry imports for BAC (I n terms of production value)
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 14
M_reg BAC 0.15 0.15 0.06 0.11 0.20 0.07 0.15 0.24 0.08 0.02 0.20 0.11 0.22 0.22 I ndustry BAC 0.10 0.13 0.03 0.08 0.02 0.05 0.11 0.22 0.06 0.01 0.05 0.06 0.06 0.16
X_reg BAC - - - - - - 0.02 0.22 0.37 0.56 0.04 - - 0.02
FD BAC 0.03 0.03 0.03 0.68 0.59 0.99 0.90 0.62 0.26 0.00 0.78 0.96 0.86 0.89 M_reg ROC 0.01 0.04 0.01 0.03 0.02 0.05 0.06 0.01 0.05 0.07 0.01 0.06 0.01 0.06
X_reg ROC 0.02 0.01 0.04 0.06 0.17 0.05 - - - - - 0.02 0.09 -
FD ROC 0.24 0.72 0.20 0.47 0.29 0.86 0.78 0.70 0.23 0.00 0.09 0.93 0.75 0.83 Sour ce: Ow n Estim ations
It was observed that BAC has impor tant r egi onal r equir ements when it i s compar ed with ROC As it was pr esented in the second row, the indust ry r egi onal imports explai n the most impor tant pur chase i n ever y sector in ter ms of pr oduction value. An example is the sector 8, which imports twenty four percent (24%) of thei r pr oduction value fr om ROC, but twenty two per cent (22%) of these come fr om ROC indust ry. Those pur chases ar e i mpor tant in ter ms of BAC sector s, because if the industr y r egional export shar e is seen, i t i s 6%, i .e. the industr y sales to BAC onl y six percent (6%) of thei r pr oducti on.
When the r egions w er e compar ed, i t was obser ved the self-sufficiency of ROC that has r egional import s shares and r egional expor t shar es behi nd ten per cent (10%) of thei r pr oducti on value, except for sector 5 that expor ts the seventeen per cent (17%) of thei r pr oducti on to BAC
13 An additi onal perfor mance i ndicat or could be t he value functi on of the entr opy funct ion. How ever , it was not pr esented because if constr aints are added i n the pr oblem, the value to mini mize w ill be greater . It happened because the degr ees of fr eedom are l ost w hen t he constr aints ar e added.
5. C
ONCLUSI ONSThis paper is the fir st appr oach i n Ar gentina to build a r egional input output model for Buenos Air es Ci ty. Regi onal input output tables wer e built wi th the final objective to esti mate a r egional social accounting matr ix, which wil l include Buenos Air es ci ty and the r est of Ar gentina.
The r egi onal tables wer e separ ated i n intr ar egional and interr egional tables. The constr uction methodology of the intr ar egional tabl es was based on Fl egg and Webber (1995, 1997, 2000). Then, the RAS and the Cr oss-Entr opy methodologies wer e i ntr oduced for the calibr ati on of interr egional tabl es.
It was concluded that the entr opy methods per for ms better t han RAS method because i t r eplicates more accur ately the nati onal input output table and it has a l ower sector al biases, so it can be expected that the final distor tion on r el ati ve pr ices will be lower i n a CGE cali br ation. The final r esult is a r egi onal input-output matr ix that has a 2.2 per cent of bias in ter ms of national input output table, and thi s bias is concentr ated on specific sector s, par ticul arly in financial inter mediati on and public administr ation. Thi s sector will be aggr egated in the CGE model following this estimate.
6. R
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Chisar i, O. et al., 2010. “Un modelo de equili br i o gener al comput abl e par a l a Ar genti na 2006”. Seri e
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