Optimization model in Input output analyi and computable general equilibrium by uing multiple criteria non-linear programming Jing He * Intitute of ytem cience, cademy of Mathematic and ytem cience Chinee cademy of cience, Beijing 100080, China btract. Thi paper develop an optimization theory on aggregation of input output (IO) analyi and computable general equilibrium (CGE) theory, in contrat to the ituation extant, where it i only partially decribed.n overall general equilibrium on IOCGE i therefore developed and preented, baed on a model uing multiple criteria non-linear programming (MC). Here, demand and upply model are conidered a two part of a ingle, compoite flow IO model. Thi undertanding implie that each IO model ha dual counterpart in the demand and upply model. Through conitent ue of thi counterpart principle ignificant CGE model are developed for both model type; and a balance of the price between demand and upply can be tated in a impler way. pecifically, multiple criteria non-linear programming for perfect aggregation of the balance between the output, a well a the price of demand and upply.in thi context, equilibrium i obtained when there i a bet compromie olution among the MC. Thi proce enable u to etablih feedback not only between upply and demand but alo between theory and application. It i a great advantage and an important tool for planner and deciion-maker. Keyword: Computable general equilibrium; Input output analyi; Multiple criteria non-linear programming; Weighting method; Bet compromie olution JEL claification: C50; C62; C67; C68; 57; 58 1.Introduction In the year ince Leontief firt propoed hi claic Input output (IO) model, IO i hown a a general equilibrium ytem (Leontief, 1953). general equilibrium theory ha to conider the quantitie and the price of both demand and upply at the ame time (rrow & Hahn, 1971); the IO Tel: +86-10-82683600;fax: +86-10-82683600 E-mail addre: hejing2005@hotmail.com (J.He) 1
theory would have to be the ame. But o far the ue of IO within the framework of the general equilibrium model ha limited to one ide for both apect. From the very beginning, IO technique do not wholly decribe a general equilibrium ytem (Chrit, 1955). Optimization model ha been ued in CGE analyi ince 1971. pecifically, the purpoe of uing the optimization method for CGE i to find the general equilibrium point of the overall economic ytem along with attaining the objection from upply and demand. There are three theorem can often ued in the optimization model of CGE Model: Fermat Theorem (Fermat 1629), Lagrange Multiplier ule (Lagrange 1788), Kuhn-Tucker Theorem (Kuhn & Tucker 1951). The proce of the Lagrange multiplier method, which i the mot familiar way to olve the optimization in CGE i divided into three part: optimization in the model of demand, optimization in the model of upply, equilibrium in demand and upply. The firt two part ha no interior relationhip with each other and alway regard ome function, uch a price, interet rate, wage of unit labor a fixed rate. Optimization model in IOCGE by uing multiple criteria nonlinear programming i derived from well known linear model. When we deal with the conditional extremum problem with the inequality in the contraint, the multiple criteria nonlinear programming can be applied with Kuhn-Tucker Theorem. In thi paper we aggregate the dual counterpart and the equilibrium within a overall mathematic programming. ince the objective function of the propoed method i nonlinear, we hall call it multiple criteria nonlinear programming. Thi paper will proceed a follow: ection 2 will contitute the baic concept of general equilibrium tructure in IO analyi. ection 3 will elaborate the definition and algorithm of the propoed MC model for the IOCGE.The concluion will be outlined in ection 4. 2.The general equilibrium model tructure of IOCGE The general equilibrium model commonly include: model of demand, model of upply, model of price and model of equilibrium. While the framework of IO model ha been limited to upply ide for both apect. In order to enable the IOCGE model to earch the equilibrium point it i neceary to conit the dual counterpart of IOCGE tructure (Qiyun Liu, 1993). 2
Figure 1 depict the general equilibrium tructure of IOCGE. Each formulation in thi figure i given in money term and i determined a follow: The model of upply can be outlined a: = ( I ) (1) i the vector of gro input (total production), in order to ue primary input (upply). i the vector of final demand, i the direct input coefficient. The model of demand i more widepread amongt the model of input-output, in theory a well a in practice. By mean of thi model it i poible to determine gro input (total production) of upply when the final ue (demand) are given. In thi cae, upply adapt itelf to demand. In other word, demand become the major factor and upply ha a minor function. Thi i characteritic from the demand ide of quantitie in general equilibrium. Thu the upply price are identified with them. The model of demand can be outlined a: = ( I ) 1 (2) i the vector of gro output (total production), in order to meet final ue (demand). i the vector of primary input, i the direct ditribution coefficient. efine 1.irect input coefficient ij = ij j (3) efine 2.irect ditribution coefficient ij = ij i ij (4) i the quare matrix ( n n) of the inter-branch flow of intermediate ue. Each element of thi matrix i the output of the branch i earmarked for branch j, and i the input of the branch j to branch i, j i the row vector (1 n ) of gro output (total production), j i the column vector ( n 1) of gro output (total production). 3
The model i no le wide pread and i ued more in theory than in practice. By mean of thi model it i poible to determine the gro output of demand when the primary input are given. In thi cae, demand adapt itelf to upply. In other word, upply become the major factor and demand ha a minor function. Thi i characteritic from the upply ide of quantitie in general equilibrium. Thu the demand price are identified with them. The model of price i determined a follow: The model of upply price: = ( + + )( I ) g v M g + v + M = 1 (5) i the price of upply, g i the direct input coefficient of fixed aet depreciation, v i the direct input coefficient of labor income & welfare, M i the direct input coefficient of ocial profit & Taxe. i the added value per output. The model of demand price: = ( ) ( z + w + k + f ) I + + + = z w k f (6) i the price of demand, z i the direct ditribution coefficient of fixed capital formation, i the direct ditribution coefficient of conumption, i the direct ditribution w coefficient of accumulation, ditribution coefficient. The model of general equilibrium: f i the direct ditribution coefficient of export, i the final k = (7) = (8) If we want to get the general equilibrium in the model, we mut atify the following condition: 4
( ) T T T ( ) = [( I ) ] = ( I ) T = T T H = ( I )( I ) = H i i J.He /Optimization model in IOCGE (9) i the price of upply, i the price of demand, i the direct coefficient, i the ditribution coefficient, i the added value per output, i the final ditribution coefficient. 3.Input-output computable general equilibrium by uing multiple criteria non-linear programming 3.1 Baic tructure of IOCGE There are many optimization model we can contitute to attain the equilibrium olution of the economic ytem, uch a operation reearch & mathematical programming, Mathematical analyi, control theory (Qiyun liu, 1993). While Lagrange multiplier method and the linear & nonlinear equation are familiar in calculation of CGE, the multiple criteria programming model in the IOCGE i a largely unexplored ubject. The MC model of IOCGE i tated a : max : L = f (, ) max : L = f (, ) ( I ) 0 ( I ) 0 B B F(, ) 0 = 0, 0 0, 0 (10) i the vector of gro input (total production), in order to ue primary input(upply). i the vector of final demand, i the direct input coefficient. i the vector of gro output (total production), in order to meet final ue(demand), i the vector of primary input, 5
i the direct ditribution coefficient, B i the reource conuming coefficient, B i the gro reource available. f () i i the revenue function from upply ide, f () i i the total utility function from the demand ide, B B i the contraint of reource. There are three reaon for thi model:(1) CGE and IO are prepared and ued in certain function, not the fuzzy or random function.ll thoe can be combined with mathematical programming. Thi allow u to link CGE and IO with MC.(2) Even if CGE&IO model exit in phyical term, it i not poible to ue the model to equate unitility with money term. When we ue the weighting method to olve MC, the problem can be olved eaily.(3) MC can be eaily rewritten to dynamic programming, which can be ued a dynamic CGE model.(4)we can get the bet compromie olution from the MC model.the olution i the global olution to the demand and upply ide at the ame time, which can meet the baic aumption of CGE in economic ytem. 3.2Exitance of the bet compromie olution of IOCGE When we ue the computer-baed algorithm to olve IOCGE, we firtly ue the weighting method to change the multiple criteria programming to the ingle criteria programming. The original MC model ban be rewritten a : max : ω f (, ) + ω f (, ) 1 2 ( I ) ( I ) B B F(, ) 0 = 0, 0 0, 0 0 0 (11) Then MC can be olved a ingle objective nonlinear programming. Becaue the elected weighting i poitive(we often elect ω = 0.5, ω = 0.5 1 2 ),the optimal olution in model (11) i equal to the bet compromie olution in model (10)(ee proof 1).The objective function in model(11) i differentiable and continuou with the feaible region i bounded. o the exitence of 6
the optimal olution in model(11) can be proven, a well a the bet promie olution in model(10)(ee proof 2). roof 1. Thi i an application of Kuhn-Tucker Condition reult to the noninferior olution in multiple criteria programming. roof 2. Thi i an application of Kuhn-Tucker Condition reult to the optimal olution in Linear programming. 3.3Computer-Baed lgorithm of IOCGE Figure 2 depict the framework of the algorithm. heuritic IOCGE algorithm can be outlined a: tep 1: Get the baic parameter,uch a the direct input coefficient,the direct ditributing coefficient,the function of the revenue and the utility etc. The proce preent requirement for Input-output table. The production part in the input-output table of today i preented in detail. In contrat to thi,the demand and factor part are highly aggregated.we cannot expect to gain very good reult from thi proce if change are not made to give correct proportion between thee part(ezra davar,1989). tep 2: Contitute the MC model. tep 3: Ue the oftware Lingo to olve the IOCGE.uring calculation of IOCGE, we find the cale of calculation i o enormou that we have to try the different toolbox to find the bet, uch a Matlab, MIO, LOQO and o on. Finally,we ue the Lingo oftware (ee http://www.lindo.com) to do and get the fitful reult. rojection Method, one kind of implex Method,i the kernel algorithm in Lingo which can deal with the variable beyond 8 1 10 unit and i the bet algorithm for our IOCGE. ow Lingo (5.0 verion) i unlimited for intallation on contraint,variable, nonlinear variable and integer variable along with the limit of generator memory i 1031.5 M. tep 4: enitivity analyi for the IOCGE. nother advantage of uing the Lingo i that it ha a pecial package to calculate the enitivity analyi reult. 7
4.Concluion The dicovery of new dual counterpart characteritic of IO model enable u to contruct the IOCGE model by uing MC. Thee model,along with exiting one, generate general tructure of model which enable u to etablih linkage and feedback between demand and upply, with repect to both quantitie and price. The IOCGE model uggeted in thi paper, baed on the above-mentioned model enable u to find the general equilibrium at the ame time in one model a to etablih feed back between upply and demand. We can ue the oftware for the large-cale computation in order to attain the bet compromie olution. Thi i an important tool for planner and deciion-maker. 8
Figure 1. General tructure of IOCGE ( I ) 1 f * Λ Λ = max I O C G E = * ( I ) 1 f max Figure 2 Calculation proce of IOCGE 9
cknowledgement Thi reearch ha been partially upported by grant a ational ature Foundation of China (o.70131002). ppendix Lit of variable ame ecription Gro input of upply Final demand irect input coefficient Gro output of demand B B g v M rimary input of upply irect ditribution coefficient eource conuming coefficient Gro reource available rice of upply irect input coefficient of fixed aet depreciation irect input coefficient of labor income & welfare irect input coefficient of ocial profit & Taxe dded value per output z w k f rice of demand irect ditribution coefficient of fixed capital formation irect ditribution coefficient of conumption irect ditribution coefficient of accumulation irect ditribution coefficient of export Final ditribution coefficient 10
eference Waily W. Leontief, 1953.tudie in the tructure of the merican Economy. 1-5.Oxford Univerity re. rrow, K., 1971.General Competitive nalyi.2-8.an Francico. C. Holden-ay. Chrit,C.,1955. review of input-output analyi, an ppraial tudie in Income and Wealth. 16-22.rinceton Univerity re. Qiyuan Liu, 1993. Model and Method on economic ytem. 17-36.enMin re. Jinhui Zhang, 2000.on-linear dynamic input-output model and the dynamic CGE model. 77-102.Tinghua Univerity re. Ezra davar, 1989. Input-output and general equilibrium. Economic ytem eearch.331-343. 11