Social Accounting Matrix Balanced Based on Mathematical optimization Method and General Algebraic Modeling System

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1 Social Accoutig Matrix Balaced Based o Mathematical optimizatio Method ad Geeral Algebraic Modelig System Mig-Chag Lee, Li-Er Su 2 Natioal Kaohsiug Uiversity of Applied Scieces, Taiwa, No. 45 Chie Kug Rd. Kaohsiug Taiwa. 2 Shih Chie Uiversity, Kaohsiug Campus, Kaohsiug Taiwa. Mig-Chag Lee The correspodet author: Mig-Chag Lee; mig_li@mail2000.com.tw Natioal Kaohsiug Uiversity of Applied Scieces (Taiwa) 45 Chie Kug Road, Kaohsiug, Taiwa ABSTRACT Aims: Social accoutig matrix (SAM) is a kid of importat form i the atioal accouts calculatio. a umber of differet sources. The requisite data are typically collected from Therefore, much of data used will ot be immediately cosistet. The aim of this paper preset the various methods of balacig applied to social accoutig matrix. Study desig: This paper solves the social accoutig matrix balaced problem usig iterative method (i RAS approach) ad usig the computer programmer about Geeral Algebraic Modelig system (GAMS) (i other approach). The sum square error of the estimated table from the iitial table is used as the mai istrumet to measure deviatio of the updated matrices from the matrix. Methodology: These methods (RAS, Cross-etropy (CE), least square, liear programmig with L, ad liear programmig with L for develop ad keep ad time cosistet multi-sector datasets. method) are applied widely Results: It is foud that CE method is more accurate tha the other methods. Keywords: Social accoutig matrix (SAM); System of Natioal Accouts (SNA); Cross etropy (CE), Least square method, Liear programmig orm L, liear programmig orm L. INTRODUCTION SAM is simply defied as a sigle etry accoutig system whereby each macroecoomic accout is represeted by a colum for outgoig ad a row for icomigs (Roud, 98). SAM is a approach for data orgaizatio, recociliatio, ad

2 descriptive aalysis of the structure of ecoomy. It provides a cosistet ad coveiet approach to orgaizatioal ecoomic data for a coutry ad it ca provide a basis for descriptive aalysis ad ecoomic modelig i order to aswer various ecoomic policy questios (Pleskovic ad Travio, 985; Reiert ad Rolad-Hoist, 997). SAM is a way of logical arragemet of statistical iformatio, cocerig icome flows i a coutry s ecoomy withi a particular period (usually a year). It ca provide a coceptual basis to aalysis both distributio ad growth issues withi a sigle framework. For example, SAM o the coutry the atioal accouts attempt to classify various istitutios to their socio-ecoomic backgrouds istead of their ecoomic or fuctioal activities Chowdhury ad Kirkapatrick (994). Bellu (202) offers a methodological itroductio to SAM for aalyzig socio-ecoomic impact of agricultural ad rural developmet policies. SAM eeds to be supported by a large umber of statistical data, but these statistics are ofte differet sources, such as atioal accouts, trade data, iput-output tables, the data are ot cosistet. How these differet data sources through a meas to the cosistecy of the data is a very importat ad difficult task. SAM imbalace usually appears: i exact data sources, the sum of accout rows ad colums are ot equal which destructio the SAM table raks of the accout the priciple of equal; or ofte abormal data exist i the table. Table is show as a example of Macro Social Accoutig Matrices. Table : A example of the Macro Social Accoutig Matrices Receipts Total. Suppliers C G I X Demad 2. Households Y Icome 3. Govermet T Receipts 4. Capital Acc. S h S g S f Savigs 5. Rest of world Z Imports Total Supply Expediture Expediture Ivestmet Row Note: S h =Private Savigs, S g = Govermet Savigs, S f = Foreig Savigs, T = Tax paymets, G = Govermet Spedig We have: Y + Z = C+ G+ I+ X (GNP) C + T+ S h = Y (Icome) G + S g = T (Govermet, Budget) I = S h + S g + S f (Savigs-Ivestmet) X + S f = Z (Trade Balace) Estimatig from primary or disparate secodary source are ofte icosistet ad several alterative matrix balacig methods are available to adjust the iitial estimates for

3 cosistecy (Robiso ad EI-Sald, 200; Roud, 2003). Therefore, the aim of this paper is usig the mathematical optimizatio method ad GAMS software to balace the multi-sector datasets to estimate a SAM. 2. SOCIAL ACCOUNTING MATRIX AND SYSTEM NATIONAL ACCOUNTS 2. Costructig a SAM The mai features of a SAM are threefold. First, the accouts are represeted as a square matrix; where the icomigs ad outgoigs for each accout are show as a correspodig row ad colum of the matrix. The trasactios are show i the cells, so the matrix displays the itercoectios betwee agets i a explicit way. Secod, it is comprehesive, i the sese that it portrays all the ecoomic activities of the system (cosumptio, productio, accumulatio ad distributio), although ot ecessarily i equivalet detail. Thirdly, the SAM is flexible, i that, although it is usually set up i a stadard, basic framework there is a large measure of flexibility both i the degree of desegregatio ad i the emphasis placed o differet parts of the ecoomic system. As it is a accoutig framework ot oly is the SAM square but also the correspodig row ad colum totals must be equal. Clearly, at oe extreme, ay set of macroecoomic aggregates ca be set out i a matrix format. Therefore, Keuig (99) suggested that SAM is a basis of ew system atioal accouts (SNA). I a ope macroecoomic system, ay of the implemetatio of the policy will have a impact o ecoomic activity at all levels. Social Accoutig Matrix has become a stadard data structure of the computable geeral equilibrium (CGE) model. The costructio of a SAM with ay sigificat degree of disaggregatio of the priciple accouts icludes: () Supply ad use tables (iput-out tables), or the ecessary priciple survey data to compile them. (2) Household survey icorporatig a labor force survey (a multi-purpose, itegrated household survey). (3) Govermet budget accouts, trade statistics ad balace of paymets statistics. (4) Natioal accouts. May compilers begi by assemblig a macro SAM from the atioal accouts. This defies a set of cotrol totals for the subsequet disaggregatio ad meas that the SAM is cosistet with ay macro aalysis. Ofte macro SAMs available for a more recet year tha the detailed datasets are iput-output tables, household survey. Gallardo ad Mardoes (203) uses iformatio from I-O table 2008, atioal accout household survey, ad evirometal pollutat emissios to elaborate a evirometally exteded SAM for chile. Payatt ad Roud (984) have poited out that compilig detailed SAMs ca be part of a process to improve the atioal accouts estimates. May coutries ow re-base their ratioal accouts periodically i accordace with a set of commodity balaces

4 (iput-output table such as a Ugada social accoutig matrix (Thurlow, 2008). Otherwise, household survey data is ot always fully i estimatig the atioal accouts (e.g. cosumer expediture is obtaied as a residual i the commodity balaces), so there might be a case of adjustig the macro SAM i some circumstaces. 2.2 Basic Social Accoutig Matrix Structure A simple, stylized SAM framework is show i Table 2. It is a square matrix that represets the trasactios takig place i ecoomy durig a accoutig, usually oe year. Table 2 shows clearly the three basic forms of ecoomic activity, productio (accouts,, 2, ad 3), cosumptio (accouts 4, 5, ad 6), accumulatio (accout 7), ad the trasactios with the rest of the world (accout 8). Table 2 : A Basic Social Accoutig Matrix (SAM) Accout () (2) (3) (4) productio commodities () Itermediate cosumptio Household cosumptio Activities (2) Domestic sale Factors of productio (3) Gross value added paymets to factors Istitutio Households (4) Labor ad mixed icome Iter- household trasfers Corporate eterprise (5) Operatig surplus Govermet (6) Net taxes Direct Taxes o product Combied capital accouts (7) Eterprise savigs Rest of World (RoW) (8) Imports Total Supply of products Cost of productio activities Factor icome paymets Curret household outlays Table 2: A Basic Social Accoutig Matrix (SAM) (cotiuous) Accout (5) (6) (7) (8) Total productio commodities () Goverm et Fixed capital Exports Demad for products cosump tio formatio ad chage i stocks Activities (2) Sales of commodities Factors of productio (3) Net factor icomes from RoW Istitutio Households (4) Distribute d profits to househol d Corporate eterprise Curret trasfers to househol d (5) Curret trasfers to Labor ad mixed icome Operati g surplus Net curret trasfers from RoW Net curret trasfers Factor icome receipts Curret household receipts Curret eterprise receipts

5 Govermet (6) Direct Taxes Combied capital accouts (7) Eterpris e savigs eterpris e Goverm et savigs Capital trasfers Rest of World (RoW) (8) Curret exteral balaced Total Source: Roud (2003) Curret eterpris e outlays Curret goverm et outlays Capital outlays from RoW Net curret trasfers from RoW Net capital trasfers from RoW Aggregat e outlays to RoW Curret govermet receipts Capital receipts Aggregate outlays from RoW 2.3 System of Natioal Accouts (SNA) The System of Natioal Accouts (SNA) cosists of a set of accouts that cover all ecoomic aspects of the workig of the ecoomy, icludig its dealigs with the rest of the world. It is a importat tool for the aalysis ad evaluatio of ecoomic performace withi a accoutig time period, usually a oe-year period ad, i more sophisticated situatios, o a quarterly basis. SNA ca aalysis: (SNA, 993; Wilso, 2004). (). The System of Natioal Accouts (SNA) helps ecoomists to measure the level of ecoomic developmet ad the rate of ecoomic growth, the chage i cosumptio, savig, ivestmet, debts ad wealth (or et worth) for ot oly the total ecoomy but also each of its istitutioal sectors (such as govermet, public ad private corporatios, households ad o-profit istitutios servig households); (2). with data from SNA, ecoomists ca either forecast the future growth of the ecoomy or study impacts o the ecoomy ad its sectors of alterative govermet policies. (3). SNA promotes the itegratio of ecoomic ad related statistics i a system that is based o cosistet ecoomic ad statistical cocepts ad methods. As such, it allows domestic ad iteratioal comparative aalysis. (4). SNA cosists of a coheret, cosistet ad itegrated set of macroecoomic accouts; balace sheets ad tables based o a set of iteratioally agreed cocepts, defiitios, classificatios ad accoutig rules. (5). It provides a comprehesive ad detailed record of the complex ecoomic activities takig place withi a ecoomy ad the iteractio betwee differet ecoomic agets

6 ad groups of agets that takes place i markets or elsewhere. The various accouts were related to specific types of aalysis related to research or policy maagemet as illustrated table below (Table 3) Table 3: Aalysis uderpiigs of the system of accout Supply Use Operatig assets Productio Cosumptio Fiacial claims 2. Net assets 3. Commercial productio 4 No Commercial Iput-output aalysis; aalysis of productivity; Cosumer s demad aalysis, studies of govermet spedig productio busiess cycle studies 5.Fial Cosumptio 6.Icome ad outlay Productio fuctios; aalysis of productivity; aalysis of factor aalysis Distributio ad redistributio of icome, fiscal policy 7. Chage i stocks 8. Fixed ivestmet 9.Fiacial claims 0. Capital fiace Studies of et worth Aalysis of savig behavior.curret Import demad Short-term foreig ad policy trasactios 2. Capital trasactio aalysis 3. Fiacial claims 4.Net assets 5. Fiacial claims 6. Net assets Table 3: Aalysis uderpiigs of the system of accout (cotiuous) Supply Use Accumulatio Rest of the world Revaluatios Closig assets Fiacial claims Studies of atioal wealth; 2. Net assets aalysis of productivity 3. Commercial productio 4 No Commercial productio Models of sock buildig ad fixed capital formatio; ivestmet policy Expert demad aalysis; studies of the globalizatio pheomeo

7 5.Fial Cosumptio 6.Icome ad outlay 7.Chage i stocks 8.Fixed ivestmet 9.Fiacial claims 0. Capital fiace.curret trasactios 2. Capital trasactio 3.Fiacial claims 4.Net assets Depreciatio aalysis; ivestmet allowaces Moetary policy ad liquidity Iteratioal fiace ad liquidity, log term foreig ad policy Capital revaluatio Studies of the retur o foreig ivestmet; double taxatio policy Iteratioal fiace ad liquidity, log term foreig ad policy Aalysis of busiess of trade paymets Studies of capital gais ad losses; capital gais tax policy Studies of the wealth effort o cosumptio Studies of et worth 5.Fiacia claims 6. Net assets Studies of atioal wealth; aalysis productivity Source: Wilso (2004) 3. VARIOUS METHODS OF BALACING 3. The RAS method (Fofaa et al. 2005) RAS is a widely used methodology to balace SAMs. The RAS method is a iterative method of biproportioal adjustmet of rows ad colum that has bee idepedetly developed by various researches. t Each elemet i matrix T is idicated ij, where i =, 2, 3, ; is the row idex, j =, 2, 3, ; is the colum idex. Colum sums y j = t ij i= are the colum totals. Row sums xi = tij j= are the row totals. As already metioed, for a give k accout, expediture is equal to receipts ad is show by the fact the sum of row is equal to the sum of the colum. t ij = x i j= tij i= = y j () A = ( a ij ) is a SAM matrix, tij a ij =, (2) y j

8 a ij = (3) i ad Y = AY, that is aij j= y j = yi (4) A classic approach to solve this problem is to geerate a ew matrix matrix A by meas of biproportioal row ad colum operatios: a ij = ri aij s j i matrix otatio : Where (~) idicate a diagoal matrix of elemets ri ad s j ~ ~ R = diag [ r, r2,..., r ], S = diag[ s, s2,..., s ], A, from the old ~ ~ A = R A S (5) a ij = ri aij s j (6) The iterative process is as follows, ad ca be see below (Ahmed ad Preckel, 2007). The origial matrix of coefficiets is multiplied by the row of colum total y j to obtai the matrix f ij f ij = a y ij j (7) The row totals of this matrix are represeted i the vector x i. Let the ratio of be the multiplier r i. Multiplyig i r ad f ij x i to x i, we obtai a ew f ij. Row vector y j of colum totals is obtaied ad used to calculate the multiplier s j. f ij ad s j are the multiplied. The etire sequece of operatios ca be see i EQ. 8. x i = fij ri = xi xi j y j = f ij s j = y j y j i f ij= r i f (8) ij f ij= s j f ij The iterative process i equatio (8) the cotiuous util the coditios x i = x i ad y j = y j are met. The RAS method has advatage of beig simple to apply iteratio steps. But, this simplicity has may disadvatages: () a lack of ecoomic foudatios, (2) iability to accommodate other source of data tha those o row ad colum totals, for example we caot fix ew cell values that we suppose are accurately measured (Robiso et al. 200; Fofaa et al. 2005). I the followig sectio, we itroduce the cross-etropy

9 method. 3.2 Cross-etropy (CE) method Some researcher used cross-etropy approach to SAM balacig, such as (Robiso et al. 998; Robillard ad Robiso, 999; Robiso et al. 200; Robiso ad Ei-Said, 2000). The estimatio procedure is to miimize the Kullback-Leibler (95) cross-etropy measure of the distace betwee the ew ad the prior estimated probability. Gola et al. (994) use cross-etropy method to balace a I-O table. The estimatio procedure is to miimize the Kullback-Leibler (95) cross-etropy measure of the distace betwee the ew ad the prior estimated probability. Applied to A = ( a ) the procedure of updatig a SAM, the problem is to fid a ew SAM ij, close to a existig SAM A = ( a ij ) The formulas are: aij Mi aij log (9) a i, j= ij s. t. = aij aij i= j= From the compariso of these two methods, they foud that if the focus is o colum coefficiets, the the CE method appears to be superior to RAS. However, if the focus is o SAM flows, the the two methods are very similar, with RAS performig slightly better (Ahmad ad Preckel, 2007). 3.3 Method of least squares (LS) We poit out that the priciple of least square is to miimize the sum of the square of deviatios betwee the ew values ad the prior value i absolute or percetage terms. Suppose that the elemets a ij ad a ij are the value of the iitial SAM ad the values of the estimated SAM respectively. The formulas Mi ( aij aij ) 2 i, j=

10 s. t. = aij aij (0) i= j= 3.4 Method of liear programmig by miimizig with orm L (LP- L ) We defie that dij is egative differece betwee. a ij ad a ij, + d ij is positive differece betwee. a ij ad a ij. Where d + ij = max [( aij aij ), 0], d max [ ( ij = a ij aij ), 0] The formulas are: Mi + [( d ij + dij ) / aij ] i, j= s. t. a = a ij ij i= j= () d + ij dij = aij aij 3.4 Method of liear programmig by miimizig with orm L (LP- L ) Mi + ( d ij + d ij ) i, j= s. t. a = a ij ij i= j= (2) d + ij dij = aij aij 4. GAMS SOFTWARE AND PROGRAM FOR BALACING A SAM 4. GAMS for Balacig a SAM Geeral Algebraic Modelig System (GANS) is a high-level modelig system for mathematical programmig ad optimizatio (Kedrick ad Meeraus, 988). GAMS is desiged for modelig ad solvig liear, oliear, ad mixed-iteger optimizatio problems. The system is tailored for complex, large-scale modelig applicatios ad allows the user to build large maitaiable models that ca be adapted to ew situatios

11 There are te steps to balace a ubalaced SAM (Fofaa et al. 2005). Step : Defie matrix accouts Step 2: Iput data Step 3: Normalize iitial matrix cell value Step 4: Treatmet of egative values Step 5: Treatmet of zero values Step 6: Defiitio of variables used i optimizatio process Step 7: Defiitio ad iitializatio of model estimatios Step 8: Iitializatio of Variables Step 9: Model solvig Step 0: Results copied to a ew matrix 4.2 Choose the appropriate method I order to choose the beat optimizatio method, we carry out a approach betwee the four methods (EQ 8~EQ 2). This approach is: ()This approach cosists i calculatig the value D for each method. Sum of squared errors (SSE) is defied as: D = i= j= ( a ij a 2 ij ) (3) (2) We exclude the coditio. The value of elemet cell (i, j) i A (iitial SAM) is ot ull, but the value of elemet cell (i, j) i A (fial SAM). (3) We choose best method accordig to the miimum value of D 5. ILLUSTRATION We use the followig IO tables to compare the accuracy of the RAS, CE, LS, LP- L ad LP- L methods. We use iterative program to solve the social accoutig matrix balaced problem (i RAS approach) ad use the computer programmer about Geeral Algebraic Modelig system (GAMS) to solve the social accoutig matrix balaced problem (i CE, LS, LP- L ad LP- L approach). 5. Defie matrix accouts ad iput data matrix () Defie matrix accouts PRO Productio ACT Activities ALAB Labor value added ACAP Capital value added

12 ENT Eterprise HS Household GOV Govermet GGOV Govermet grats CAP Capital DEP Depreciatio R0W Rest of World TOT Total (2) Iput data matrix Table 4: Iput data matrix PRO ACT ALAB ACAP ENT HS GOV PRO ACT ALAB ACAP ENT HS GOV GGOV CAP INV DEP R0W TOTAL GGOV CAP INV DEP R0W TOTAL PRO ACT ALAB ACAP

13 ENT HS GOV GGOV CAP INV DEP 54.. R0W TOTAL Results () A balaced SAM results Sice the requisite data are typically collected from a umber of differet sources. So the estimated table may have zero elemets where the iitial SAM has positive value. As the result, whe root mea square error betwee the iitial SAM ad the fial SAM. It is possible that we will get some results that are very iaccurate, especially if the iitial table s values for the correspodig zero cell is very large (Ahmed ad Precket, 2007). GAMS are a high-level modelig system for mathematical programmig ad optimizatio. GAMS program has bee specially treated the iitial table's values for the correspodig zero cell. The matrices of coefficiets are compared. These matrices are basically the full matrices with the cell i a colum divided be the colum total. Table 5 is deoted as the compariso of SSE values from matrices coefficiet for each method Table 5: Compariso of SSE values from matrices coefficiet for each method method Root mea square error (D) zero cell s RAS.46 CE.07 LS Cell (8, 5) LP- L.02 LP- L 0.97 Cell (8,5) ad cell(8, 6) (2) Compariso of SSE value of matrices of coefficiets for each estimatio methods The choice of the best method, we use method of choose the appropriate method

14 (sectio 4.2). We obtai the followig iequality betwee the five values of D. D LS < D LP L < D CE < D LP L < D RAS From the fiial matrix of SAM through the LS ad LP- L, we otice the values of the elemets are ull: Cell (8, 5) i LS method, Cell (8, 5) ad cell (8, 6) i LP- L method. Where as these values is ot ull i iitial matrix of SAM. The LS ad LP- L method do ot reflect the latter. Therefore, the value of D is very low i the method of CE. 6. CONCLUSION A SAM is a sigle accoutig framework with the rows ad colums, arragig icome ad expediture accouts of various ecoomic agets i a coutry. A SAM framework is ot oly a statistical tool but also a framework for macroecoomic aalysis. It provides a framework for the orgaizatio of iformatio about ecoomic ad social structure of a coutry ad a service as dataset foe a model of the ecoomy. Sice estimatig cell value from primary or disparate secodary source are ofte icosistet. There are several alterative matrix balacig methods are available to adjust the iitial estimates for cosistecy. We detail describe five types the matrix balacig methods The RAS method is a iterative method of biproportioal adjustmet of rows ad colum that has bee idepedetly developed by various researches. We preset a GAMS code ad CE, LS, LP- L ad LP- L to balace a ubalaced SAM i te steps. Fially, we preset a compariso of SSE values from matrices coefficiet for each method. REFERENCES Ahmed, S. A. ad Preckel, P. V., A compariso of RSA ad etropy methods i updatig IO tables. America Agricultural Ecoomics Associatio, 2007 Aual Meetig, July 29-August, 2007, Portlad, Orego (accesses 8 august 203). Bellu, L. G., Social accoutig matrix (SAM) for aalyzig agricultural ad rural developmet policies coceptual aspects ad examples, Aalytical tools Easypol module 30, (accesses 5 august 203) Chowdhury, A. ad Kirkapatrick, C., Developmet policy ad plaig: a itroductio to models ad techiques, Lodo ad New York: Routledge, 994. Chug-I Li, J., A 999 social accoutig matrix (SAM) for Thailad, TWD Discussio paper No 95, Iteratioal Food Policy Research Istitute, ad Washigto D.C

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16 SNA, System of Natioal Accouts, Commissio ad Europea Commuities, Iteratioal Moetary Fud, Orgaizatio for Ecoomic Co-operatio ad Developmet, Uited Natios, World Bak / Luxembourg, New York, Paris, Washigto D.C Tames, T., A, 2007 social accoutig matrix for Ugada, Iteratioal Food policy Research Istitute (IFPRI), 2008, -9. Wilso, K., the architecture of the system of atioal accouts: a three-way iteratioal compariso Caada, Australia ad the Uited Kigdom (accesses 8 august 203)