k'orld Bank Development Research Center Discussion Papers No. 23 A MIJLTI-SECTORAL NODEL vJITH ENDOGENOUS TEN.1INAL CONDITIONS Richard Inman, K i m Yoon Hyung, a d Roger Korton I I I I * I I r I I ?;OTT: Discussion Pnpcrs a r e yveliminary materials i i r c n l a t e d t o s t i m u j a t e s* d i s c u s s i o n and c r i t i c a l comment. R e f a e n c c s i n p u b l i c a t i o n t o - Discussion Papers sllould be cleared with the a u t h o r ( s ) t o proteqt the t e n t a t i v e c h a r a c t e r of these p a p e g . The papers express t h e .I v i e c s of t h e author and should not be s n t e r p r e t e d t o r e f l e c t those of t h e World Bank. ~ Richard IIL'QiJ, .YJ!.I Yoor. Fipng, and Joger I!ORTO!i 1. Introduction and Summary 2. The Strucrure of tho, Korean hIodel 3. Iteration Rules 4. Outline of a Convergence Proof 5. Numerical Results 6. Concluding Rerarks - I ~ rC Richard _Tl?A!J, K_.LY YOOX ,??xn?, and Roger lGL?TOIJ ~ ~ 1. Introductio~,ard ,Si.-?maq I F I n c e r t a i n c l a s s e s of planning rncdels, t h e numerical proje' t i o n s a r e influenced by t h e nanner i n which t h e time horizon is truncated.1 3is- F t6;tions can occur simply because t h e planning period is f i n i t e . A t empts t o overcome these problems have l e d t o various a l g e b r a i c s p e c i f i c a t i b n s of " terminal conditions" . I n t h e s i n p l e s t case, terminal conditions I r e q u i r e t h a t a given l e v e l of c a p i t a l s t o c k be accumulated a t the en# of I the p l a n i n g period; i n other cases, they p o s t u l a t e s p e c i f i c post-tLrrnina1 I growth r a t e s f o r one o r more periods, as an inducement t o investment ( u u r i n g I the planning period i t s e l f . As these f o r o u l a t i o n s suggest, t h e problem a r i s e s because,)with- l out t e r n i n a l conditions, t h e model is s t r u c t u r e d so t h a t t h e r e is no1 L i n c e n t i v e t o accumulate c a p i t a l at t h e end of t h e horizon. This typ occurs when t h e model maximizes consumption, e i t h e r i n t h e terminal. j e a r I A rnman and Norton a r e a t the I.:orld !dank'$ ~evelopmentResearch Center, Washington, D.C., and Kin is a t t h e Korea Devel-opment I I n s t i t u t e , Seoul. The authors a r e g r a t e f u l f o r h e l p f u l corrzents from the following perscns: l:ontek Ahluwalia, Bela Balassa, Charles B l i t z e r , Syamaprasad Gupta, 1,i:l 1'~?,nc , ? :,i.rcs'i l e ~ i u l k n r ,Zezn !'ael brcec'.: . ;.--tf ] ,.I r y \lc.:t ~ i l : ~ ? or over t h e planning period. (Son-optimi.zing nodels and r e c u r s i v e optimi- zing ~ c d e l sdo not encounter the terminal conditions problem.) I I The usual cure f o r t h e problem r e q u i r e s t h e introduction bf P a d d i t i o n a l (and perhaps undesirable) a s s u n p t i ~ n s ,about terminal ca i t a l s t o c k s , o r about post- terminal growth paths. It is recognized that)there I is a logical inconsistency i n making assumptions about terminal or post- I t e r m i c z l behavior before t h e planning period behavior is projected. Nanne has at teopted t o handle t h e problem by designing terminal condition$ t h a t, I I with s o c e confidence, can be assumed t o be approximately c o n s i s t e n t w i c h I t h e ei~dogenousgrowth path which w i l l emerge from t h e model [3, pp. 270-711. I f I n t h i s paper, a procedure is offered which makes t h e t e r i n a l I c o n d i t i c n s f u l l y c o n s i s t e n t with t h e endogenous behavior, i n t h e c o j t e x t of a uiulti-sectoral optimization model. The procedure involves stodk-f low f a c t o r s and an i t e r a t i v e sequence of s o l u t i o n s of t h e optimization dodel. Evidence is given on its numerical r a t e of convergence. I The model used i n t h i s paper is a s t a t i c model f o r the t e + i n a l I i year of a planning period, with socc c o n s t r a i n t s defined on cumulat've v a l u e s over t h e e n t i r e planning period. S p e c i f i c a l l y , t h e equilibrdlum . t ~ c o n d i t i c n s on savings and investment a r e s t a t e d i n terms of cunulati/ve - I plan period values of t h e r e l e v a n t v a r i a b l e s . - Also, the capacity c o n s t r a i n t s on s t - 1 production levels a r c stated on the basis of (depreciated) curnulntivg investnent i n each s e c t o r . The stock- flow f a c t o r s convert terminal- year s o l u t i o n valdcs i n t o cuculat i v e v a l u e s , as required. I n i t i a l numbers f o r these f a c t o r s a r e taken from 0t5cr Korean s t u d i e s such as [ 3 ] ; of course, both the terninnl 3. y e a r s o l u t i o n values and t h e stock- flow f a c t o r s imply c e r t a i n growth r a t e s over t h e planning period. Even though t h e f a c t o r s a r e c a r e f u l l y s 4 l e c t e d I I beforehand, they i n e v i t a b l y t u r n out t o b e i n c o n s i s t e n t w i t h t h e I I I endogenous termical values, i n t e r n s of t h e implied growth r a t e s . !The I I i t e r a t i v e procedure r e v i s e s t h e stock- flow f a c t o r s u n t i l c 0 n s i s t e n 4 ~is I achieved i n t h i s r e s p e c t . I I Hence, when convergence is a t t a i n e d t h e model's s o l u t i o n r e p r e s e n t t h e e n t i r e planning period, i n s t e a d of j u s t t h e terminal The assunption which permits t h i s is t h e assumption of smooth growth paths I (conscant conpound growth r a t e s ) f o r investment i n each s e c t o r . 1 I I n a l i m i t e d sense, t h e model is a kind of multi- period m d e l , P I- and t h e choice of year t o t r u n c a t e i t ( t h e terminal year) does not f f e c t t h e growth paths. Kor does t h e model tend t o consume excessively i b t h e terminal year; i.e., the " terminal conditions problem" has vanishe !. I After t h e model and its p r i n c i p a l r e s u l t s a r e presented, b ome remarks a r e offered on how these procedures might be used t o make t e 1 t terminal conditions endogenous i n more conventional multi- period mo e l s . t Xevertheless, we do not claim much g e n e r a l i t y f o r t h e i t e r a t i v e met od i n a , I i t s present form, precisely because i t has been tested just for a sdecific kind of model. A t a g e n e r a l l e v e l , t h e only conclusions we can dra4 is - ** I - rkivt t h e b a s i c approach of seeking: i t e r a t i v e l y compatibility betwein t h e I * .. terminal c o ~ dtiion values and e n d o l p ~ o u ss o l u t i o n values appears pr{n'l s i n g , I s o perhaps ochers w i l l be stimulated t o modify the method t o s u i t t d e i r It also appears t c be important to carry out a procedure like I - t h i s , o r something e q u i v a l e n t , i n L ~ I : ~ ; ; uf the e f f e c t s on t h e ntmeri/=al I r e s u l t s . I n comparing t h e f i r s t - i t e r a t i o n s o l u t i o n w i t h t h e s o l u t i o b s I I " at convergence, I' a measure of t h e importance of making termink1 conbi- I t i o n s ecdogenous i s gained. The r e s u l t s suggest t h a t exogenous term nal. i I I I c o n d i t i o n s , o r s p e c i f i c a t i o n s l i k e t h e Hanne approximation, may w e l l b e I acceptable i n t e r n s of macroeconomic v a r i a b l e s , b u t perhaps unaccept b l e t. t a t t h e l e v e l of i n d i v i d u a l s e c t o r s . For sone s e c t o r s , cumulative i n e s t - i nent l e v e l s show l i t t l e change a s i t e r a t i v e convergence is approache , b u t f o r many o t h e r s e c t o r s the change is q u i t e s u b s t a n t i a l , exceedin4 50% I i n some instances. I n summary, t h e r e a r e two broad obiectives of t h i s psper: t o I I r e p o r t a n experience w i t h a s p e c i f i c terminal- year economy-wide mode4 i which incorporates some multi- period considerations, and t o r e p o r t a i t e r a t i v e procedure f o r reducing t h e approximation e r r o r induced by i n p o s i t i o n of terminal conditions. i The remainder of t h i s paper is divided i n t o f i v e s e c t i o n s , a s I follows: I L I 9 1 - Section 2 presents the Korean linear programing model I used f o r thcsc experiments; -- '2 - Section 3 discusses the method of endogenizing he - * . W terminal conditions: t h e stock- flow f a c t o r s a n 8 t h e i t e r a t i o n r u l e s ; I ccnverge; - Section 5 p r e s e n t s numerical r e s u l t s ; and - Section 6 o f f e r s some concluding remarks. ~ 1 2. 2r.e Stmzcture of the ZOT~CLE !'.AOdeeZ 4 I I I b 1/ The Korean nodel, YULGOK,- is a 53-sector input- output b sed l i n e a r p r o g r a m i n g oodel f o r t h e year 1981. The base y e a r , o r refekence 1 y e a r , f o r p r o j e c t i n g t h e exogenous v a r i a b l e s was taken t o be 1974. The major p o l i c y concerns addressed w i t h t h e model a r e (2) t h e i n t e r s e k t o r a l 1 a l l o c a t i o n of cumulative 1974-81 investment and (b) t h e i n t e r s e c t o - a 1 I p a t t e r n s of foreign t r a d e i n 1981. I Aggregate consumpt ion i n 1981 is t h e maximand, and t h e r e f j r e some conditions a r e required s o t h a t t h e model has an i n c e n t i v e t o 4 a t r a t e s c o n s i s t e n t w i t h ccntinuing long- term growth. Follcwing Ch nery and Bruno [ I ] and Manne [ 3 ] , stock- flow r e l a t i o n s , together with t h E 4 asssm,ption of smooth exponential growth of a l l q u a n t i t i e s during 1 74-81, 4 a r e used t o i n s u r e t h a t s u f f i c i e n t investment takes placc t o s a t i s f post- I 1981 growth requirements. E f f e c t i v e l y , t h e 1981 investment l e v e l i d I I transf ormed i n t o a r e p r e s e n t a t i o n of 1974-80 investment l e v e l s a l s o ,i and * these a r e rhquired t o s u s t a i n production capacity (and t h e r e f o r e cor/surnp- I Lion) a t desired l e v e l s i n 1981. i5xport and import l e v e l s a r e d e t c mined I cndogr:nously within a f i x e d range of v a r i a t i c n (which is f a i r l y wid ; .I o f t e n exceeding 1002). ldhen the upper i n d lower bounds on t h e trad$ . ~ I I -I/ Yi Yul-gok was a 16th century Korean philosopher and cabinct m i n i s t e r who made s o w accurate (and unpopular) f o ~ c c a s t sof , n a t i o n a l policy needs a t t h a r tine. v a r i a b l e s a r e binding, t h e d u a l values of those bounds provide a b a s i s f o r making conparative advantage calculation^, a s i l l u s t r a t e d i n thd s e c t i o n on nur-erical r e s u l t s . I 1 I Table 1 lists t h e p r i n c i p a l v a r i a b l e s i n YGLGOK, and t cquations are presented below. I n terns of s i z e , YULGOK has i i n d i v i d u a l e q m t i o n s and 5000 nonzero c o e f f i c i e n t s i n t h e LP mat ix. Table 1: CLASSIFICATIGN OF YULGOK VARIABLES AND PARAMETERS I ( f o r t h e year 1981 except where s p e c i f t e d o t h e w i s e ) l I I 1. Exogenous variables Government consumption and its s e c t o r a l cornposition 1~ Exchange r a t e Net f a c t o r income from abroad Foreign savings 1 2. Endogenous v a r i a b l e s P r i v a t e consumption and its s e c t o r a l composition * Gross flxed i n ~ ~ e s t m e nand its s e c t o r a l cornpositio~l t Inventory investment and its s e c t o r a l composition* GDP and GhT Donestic savings* 7 ,Exports by s e c t o r endogenous w i t h i n a f a i r l y wide range Imports by s e c t o r of bounded v a r i a t i o n Emp10,ment by s e c to r I I n i t i a l condition parameters Rase year (1974) production l e v e l s by s t c t o r b Other key p a r a e t e r s Xarginal savings r a t e (1974-91) i Capital- output r a t i o s by s e c t o r ( 1 9 7 4 - 6 1 ) Cpper bound on f o r e i g n c a p i t a l inflow I I m For 1981 and a l s o cumulative 1974-81. -.".. - ?.ia.;ly of t h e equations a r e f i r s t F'resented i n t h e most recogniz- a b l e fashion, and then transforned i n a s e r i e s of s t e p s t o t h e b e r s i o n II most s u i t a b l e f o r d i r e c t use i n tlie model. The model v e r s i o n i always denoted by t h e s y ~ ~ b oMl i n t h e equation number. I I P r i v a t e Consunption Denand 1 1 The consumption equations a r e not e x p l i c i t but r a t h e r are incor- porated i n t h e m a t e r i a l balances. S e c t o r a l consumption l e v e l s r e obtained by assuming a constant expenditure ~ l a s t i c i t yf o r each good d u r i c g t h e plannir-g period: where N = population, ni = a constant, E = t h e Engel e l a s t i c i t y , i C = aggregate household consumption expenditur b. I F Since t h e s e c t o r a l consumption l e v e l s r e s u l t i n g f om (1.1) i u s u a l l y w i l l n o t add up t o C ,t h e consumption function is 'lineatized I/ around t h e base year 1974 consumption p a t tern:- - 'f -- where and p is a normalization such that Letting and (1.2) can be rewritten as Private consumption expencitures in each sector consists (ofout- lays on two categories of gccds, domestic goods plus competitive i+ports, I l and noncompetitive imports. Thus, -- z ~ ++C; ~ . e (1.4) 4 Si i I - Using estimated coefficients for the allocation of consudption 1 I -@ over these two-categories, we have - - i I C - .I - d+c Jr 'i - zipi + aiqic 11 (1.5) l and I C: = (1 - ailpi + (1 - oi)qic (1.6) I I ;-lateri a l Balance Equations I where t h e following a d d i t i o n a l syxxbols a r e used: I = t h e grr ss o ~ t p u tl e v e l from s e c t o r i, 'i I a = t h e input- output coefficients (current ij I account i n t e r i n d u s t r y demands on item i per unic output of process j), = investment demand f o r s e c t o r i induced i by output i n c r e a s e s , = inventory accumulation demand f o r sector Hi = exports from s e c t o r i, Ei Mi = imports i n t o s e c t o r i, G = government expenditures f o r products of i s e c t o r i. , I C Here t h e s u p e r s c r i p t s d+c and n (on Ci, Ji, Hi, )Ii, and domestic d e l i v e r i e s p l u s competitive i n p o r t s and - r e s p e c t i v e l y . - m t .) From (2. I), t h e following transformat i o n s were made: I I I I I I I I I \ahere t h e a d d i t i o n a l symbols a r e hs follows: = t h e proportion of s e c t o r i i n v e n t o r i e s which i comes from domestic production and competitive i n p o r t s ( ( 1-8 . ) is the p r ~ p o r t i o nsupplied by 1 i noncompetitive imports), = inventories- to- output r a t i o , sector i, j = co~poundannual growth r-ate of s e c t o r i output, Cx 1974-81, WI = t h e proporLion of s e c t o r i i n v e s t ~ c n tgoods which 'i cones from domestic product< and competitive i x p o r t s c a p i t a l goods input- output c o e f f i c i e n t s , f o r f ixed investment I = f i x e d in-~estmentby s e c t o r of d e s t i n a t i o n j j a = s h a r e of s e c t o r i conscmption met by domestic i 1 s u p p l i e s and c o a 2 e t i t i v e imports (as opposed t o noncompetitive imports) = s h a r e of gover~mentconsumption spent on i s e c t o r i produc.ts pi = s h a r e of government conssaption of s e c t o r i products which is s a t i s f i e d by domestic s u p p l i e s p l u s ronpetii:ive imports. '* - r r - ai q i i = -I- = vector { a p + RigiG 1 I I 1 i i 'I . L !fonconpetftive Import Balance Equations * E I I n + nn + n J: + c2 - M; -< -G i i (3.4) ~ I where t h e s p b o l s a r e as 1..iPned above and the s u p e r s c r i p t n incricates I nonconpeti:ive imports. Therefore I I where = the vector 1(1- nilpi + ( 1 - Gi)giGj - r 2 , I L Capacity Constraints -.. 1 Assuming a ccnstnnt depreEiation r a t e p f o r s e c t o r j c a p i t a l j I s t o c k , t h e c a p i t a l accumulation equation is e m L e t t i n g k be t h e s e c t o r j capital- output r a t i o and u be t h e j j ,m n a x i ~ u ncapacity u t i l i z a t i o n r a t e , r allowing f o r a one-year g e s t a t i o n l a g , The base year 0 = 1974, a d v a r i a b l e s without a t i ~ seu b s c r i p t r e f e r t o t h e terminal s o l u t i o n 1981. Then where 4 ; is the investrcent stock- flow factor over whicl- iterations take , t d C place (discussed i n s e c t i o n 3 below). I - Savings- Investment I d e n t i t y If, savings behavior is represented a s I I S - S I (5.1) t 0 -< u(Yt - Yo) I I where Y = G?;P I I C = t h e marginal p r o p e n s i t y t o save I I I i I then where YD = g r o s s domestic product - NFIt = n e t f a c t o r income from abroad, year t 81 AFIT = 1 NFIt t=75 a = another s t o c k flow f a c t o r : = t h e CDP growth r a t e &Y i L e t t i n g t o t a l f i x e d investment t h e savings inves ment J cons t r a i .t is - where F = n e t f o r e i g n c a p i t a l inflows - But E I 81 81 j: ( T I ~ + ~ )I ~= T I t + l h (X - X ) t = 7 5 j j ,t t = 7 5 j j j,gl J , 7 4 where +* = stock flow factor for total investment: 1 and IH = cumulative total inventory investment, sector j j where the inventory investment stock-flow factor is (see section 3 below). - where VG = value added by the government. GDP Definition Y3 .- Y - NFI Balance of Payments Constraint L M +~ - - - l E i F + NFI i i i i Objective Funczion C -+ max Equation (0.M) completes t h e d e s c r i p t i o n of t h e model. For purposes of s o l u t l o n design, bounds were added f o r exports and I 1 . competitive imports, a s discussed subsequently. 3. Iteration Rules I To r e c a p i t u l a t e , any s o l u t i o n of YULGOK r e q u i r e s given of t h e following s t o c k flow Factors: c a p a c i t y total investment - - * .b * s a v i n g s 0 81 t 1 1( --->1 cy =7'e 9 =--t=75 - t=o "3 21 YPr < . 6 -- -t 1 g~ t=o For n s e c t o r s , t h e r e are 2ni-2 stock- flow f a c t o r s i n a l l . I n t e c a s e h of YULGOK, 2ni-2 = 108. 1 l h e s a f a c t o r s convzrt 1981 s o l u t i o n values i n t o c u m u l a t i e 1975-81 v a l u e s (1974-81 v a l u e s i n the case of capacity increments, where ilnvest- Kent f u new capacity has a onc-year g e s t a t i o n l a g ) . Hence they pla k i n e s t a b l i s h i n g t h e c o n s t r a i n t s which a r e defined w i t h r e s p e c t t o c mula- F Live plan period behavior: t h e s e c t o r a l capaCity c o n s t r a i n t s and t h savings- inves t ~ e n tc o n s t r a i n t . L - - '3 The above expressions f o r t h e stock- flow f a c t o r s make i t l e a r `% - *-* t h a t they embody a s s ~ m ~ t i o n s ~ a l i ot huet compound annual growth r a t e I s e c t o r a l investmer~t(g ). t o t n l i n v c s t n ~ n t(gTI),r;l,y(gy), and sec I . Applying t h o s e growth r a t e s t o t h e 1974 base- year values, t h e 1981 values implied by t h e stock-flcw f a c t o r s can be derived. Th~e1981 I s o l u t i o n t o PULGOK, on t h e o t h e r hand, is not l i k e l y t o agree w i t h those values. Thus i t e r a t i o n r u l e s were designed i n order t o b r i n g t h e two s e t s ~f 1981 values i n t o accord, over t h e course of a number of ( i . e . , t o bring t h e stock- flow f a c t o r s i n t o conformity with t h e s o l u t i o n values). The i t e r a t i o n r u l e s specify procedures f o r updating t h e stock- flow f a c t o r s on t h e b a s i s of s o l u t i o n experience. W e use t h e capbcity 1ere stock- flow f a c t o r s i n t h i s d i s c u s s i o n ; p a r a l l e l procedures lJ J a p p l i e d t o t h e o t h e r f a c t o r s . The s i m p l e s t updating procedure wo I d b e t o b t h t a k e t h e (1981) investment l e v e l s from t h e n i t e r a t i o n s o l u t i n, c a l c u l a t e t h e implied growth r a t e s , and c a l c u l a t e ' f o r use In t h e j (n+lls i t e r a t i o n from equation (10). How--~er,t h i s r u l e leads t4 d. u n s t a b l e behavior over i t e r a t i o n s s o a damping mechanism is neede The most n a t u r a l one is t o use an average of i ~ v e s t m e n tgrowth r a t e s fn t h e l a s t s e v e r a l i t e r a t i o n s , n-s through n , i n computing expected growth of investment a t i t e r a t i o n n+l can be defined a$ = w 1" + ( 1 - w) E I ~ (14) j J E ~ >!ow w is the parameter vhich controls the amount of-damping; tile smaller w, t h e g r e a t e r t h e damping e f f e c t . I 20. S t a b i l i t y can be c o n t r o l l e d i n another dipension a l s o . I n t h e c a p a c i t y c o n s t r a i n t s (4.?.1), t h e stock- flow f a c t o r s a r e multiplieq! by t h e + I endogenous investment t o y i e l d t h e plan- period capacity incremen.. 4 A l t e r n a t i v e l y , t h e r o l e of c u r r e n t endogenous investment could b elimin- a t e d e n t i r e l y by using expected investment a s t h e determinant of t h e capacity increment. Current investment would then c o n t r i b u t e t o revised capacity c a l c u l a t i o ~ i sonly i n f u t u r e i t e r a t i o n s , through its con r i b u t i o n i t o f u t u r e values of expected investment. This a l t e r n a t i v e gives an excessive damping e f f e c t , s o a compronise w a s developed: equation (4.~!) is r e w r i t t e n I a s : \*,'hen v = 9;n , a l l of endogenous investment works t o c r e a t e capa J through t h e stock- flow f a c t o r ; when v = 0 , endogenous investnen P e f f e c t and expected investment does a l l t h e work of increasing ca a c i t y . At v a l u e s 0 < v < $n , some weight is given to each of the extre j i; ( I f v is s e t uniformly f o r a l l sectors, with varying by s e c o r it 'j sometimes happens than .. v a t - t o causczany problems w i t h C f i 1 n p r a c t i c e , it was found t h a t a value of v = I was t h e " typical" value over s e c t o r s j of 4 seemed t o give a f a r l y r a p i d .i' i I r a t e of convergence. + 21. I n smmary, t h e i t e r a t i o n r u l e s were implemented by e n t e r i n g (15) i n place of (4.3) and applyicg (14) t o c a l c u l e t e t h e E I ~i n a j 11between- iterations" subroutine. These equations enboc'y two new p I w and v, which must b e assigned values. Choice of a p p r o p r i a t e v a l i e s of w and v is an empirical question; we only can o f f e r suggestions ob t h e i r ~ v a l u e s , based on l i n i t e d numerical experience. It seens Ispartant that I C, w s 0.5. - Our " typicalt1 $ (which depends on t h e length of t h e planbing I horizon) was about 6.0, and s o v = 3.0 gave s a t i s f a c t o r y r e s u l t s -(much b e t t e r than, say, v < 1 . 0 o r v > 6.0. - - A s e t of t a b l e s w i t h numeri r e s u l t s by s e c t o r is given i n t h e s e c t i o n 5. 4. GatLine 0.7 a Comergence Proof i I I The numerical r e s u l t s show a f a i r l y rapid tendency of t h '7 algorithm t o converge towards l i m i t i n g values i n each s e c t o r . The con- I vergence is due t o two c h a r a c t e r i s t i c s of t h e algorithm, which can ibe o u t l i n e d i n a f e w words. F i r s t , f o r any s e c t o r j t h e sequence of I n n n f l n+2 n+2 1 v a l u e s $ 1 ... 9 , Ij , ) is either monotonically convergent o r o s c i l l a t o r y around t h e l i m i t i n g values (40 1:). Thils can j' J I I be shown simply a s follows. I f 4 3 < 4; , then, ,by t h e d u a l forbula- I L . n t i o n , t h e shadow p r i c e of capacity 'i > n o ,and hence > 1' j j j j Therefore, b y ( 1 0 ) , $3" > $jn T h i s z w i l l continue i n t h e pat t e r n '? > $ , etc., until (unless) a point is reached a t which - $ j n+r ,It that point, tfle rcasoning is reve$ed, and ' . @ j I f / h a t I I point of r e v e r s a l is never reached, t h e procedure is monotonic, w i t h n getLing even c l o s e r t o 4; . I f i t is reachcd, t h e procedure o s c i l l a t e s j around I f it is monotonic, t h e r e is no problem. If, however, it oscill?tes f o r a t l e a s t one s e c t o r , then t h e second important characteristilc of the I procedure is needed: t h e damping given by t h e c o e f f i c i e n t s w an k v i n (13) and (14). Both of t h e s e coefficients o p e r a t e i n a way t o reduce t h e a c p t i t u d e of t h e - o s c i l l a t i o n s . I f i n (13) the suvmation is a l l o t e d t o run from i=lto i = n, then expected investment EI" always is an j of & p r e v i o u s i t e r a t i o n s values; provided t h a t t h e undamped 0 1 c i l l a t i o n s do n o t continuously i n c r e a s e i n amplitude, then E I ~-+ EIO = I? , a s I n j J increases. F i n a l l y , it can be shown t h a t t h e undamped o s c i l l a t i n s cannot continuously i n c r e a s e i n amplitude. Given resource c o n s t r a i n t s , t h e r e - e x i s t s an upper l i m i t t o investment i n each s e c t o r : I I = I j - j a t most, ,.I1 t h e investment i n t h e economy could flow i n t o one e j c t o r . I I The lower l i m i t is of course zero. Hence f o r each s e c t o r j, 0 I- 1" < I, 1 j - so :here is an upper bound t o the amplitude of t h e o s c i l l a t i o n s . By (9) t h i s impiies a bound t o t h e variance i n t h e values of also. 0 0 Therefore t h e algorithm must converge t o ($ I . ) i n t j' J investnent values. Given the nature of t h e input- output ~ r o d utim c functions, t h i s means collvergence i n t k e X? and 3" , and therefo e J j 1X? and t h e en. n It also means convergence in the 1I and t h e r f o r e i n , J I j j j the +*n. - * krhnt i f the o s c i l l a t i c n s are asymmetric i n the sense i t e r a t i o n s , the sun of p o s i t i v e deviations around 1' than (er l e s s t!lzn) the s u of negative deviations? ~ by l e t t i n g v -> 0 i n (15) tends toward complete of I? . E I T If the sum in (14) runs from i=lto i=n, - contains a complete " history" of a l l previous i t e r a t i o n s , andl asyrrmetry would be r e f l e c t e d i n E I ? # EI? . Using t h i s , we 1 13 car, re- write t h e f i r s t paragrzph ofJti:;r: J s e c t i o n , replacing with ET? . Thus S t can be seen t h a t tile o s c i l l a t i o ~ s ,if J I p r e s e n t , would tend toward s?.rnrnetry i f v wel-e c l o s e t o zero. Convergence is u n l i k e l y t o be absolute i f the model is reasonably l ~ r g e ,given c c n s t r a i n t s on computing resources. I n t h e exper 'nents t reported here, t h e following c r i t e r i o n was adopted: I I n p r a c t i c e , the r a t e of convergence was n o t much of la problem, I b u t t h e r e were c a s e s i n which t h e model became i t e r a t i v e l y i n f e a s i b l e . Ey t h i s w e mean t h a t t h e model was f e a s i b l e i n t h e f i r s t i t e r a t i o n b u t , through t h e algorithm's reassigning of values t o the stock- flow i t redefined the feasible sgace to a point where it became a nu T h i s cannot occur i f zero l e v e l s of production, consumpticn, infestment , e t c . , a r e included i n the f e a s i b l e s e t , but by imposing lowez b s e c t o r a l export l e v e l s i n YULGOK, we had bounded t h e i e a s i b l e s from t h e o r i g i n . The s o l u t i o n t o t h i s problem was found i n cha values of v and w. The process i e f i n c d above con-~ergest o nn cquilihrium i n t h e s e n s e t h a t f u r t h e r i t e r a t i o n s w i l l n o t a l t e r t h e v a l u e s of the ~ e c t o r a l L investlnent l e v e l s (or t h e stock- flow f a c t o r s ) . It remains to show that t h i s equilibrium is unique. Assume t h a t it is not, unique. ~ h e tih e r e - a r e a t l e a s t two e q u i l i b r i a such t h a t and I - L e t tsio d i f f e r i n g e q u i l i b r i a b e denoted by t h e s i b s c r i p t s 1 and 2 , and f u r t h e m o r e , l e t (J) denote t h e non-empty set such I I ha t l and (J*) t h e set such t h a t k ( t ) I f J is non-enpty, t h e d u a l p r i c e s TI i~ (t) on t h e c a l z c i t y con- j , l Y j , 2 s t r a i n t s (4 .?[) w i l l behave s o t h a t - - From (19), we know t h a t '9 - C * I-I ( t ) m ( ' I ; j J : ! - j,1 [ ~ j , l < and hence, by ( l o ) , But 4 (t+l) a t equilibrium, and s o (21) is i n c o n s i s t e n t with (17) I and (18). Therefore t h e r e cannot e x i s t two d i f f e r i n g e q u i l i b r i a . ' ~ I I Suppose (J*) is empty. Then I I I Then t h e r e a r e a t l e a s t sone s e c t o r s j f o r which and hence some s e c t o r s f o r whish - Rut (24) c o n t r a d i c t s G h e equilibrium assumption and (22). Hence, once again t h e r e cannot e x i s t two e q u i l i b r i a which d i f f e r numerically. Tr~os e t s of n u n e r i c a l r e s u l t s a r e presented: evidence dn t h e i I rat-e of convergence of t h e procedure, and evidence on t h e numeri.cql be- I I havior of t h e no?el l t s e l f "at convergence." 1 Tables 2, L, ~ ~ 5l cdo n t a i n t h e basic r e s u l t s by i t e r a t i d n num- I I h e r . Table 2 gives t h e m c r o r e s u l t s , and t a b l e s 4 aod 5 give thd d i s - I aggregated outcopes f o r investment. Several conclusions a r e apparlent. I F i r s t , t h e i t e r a t i v e proced.~res e t t l e s down q u i t e f il-mly by t h e 301+hround. I- I n f z c t , most s e c t o r s ' investment l e v e l s appear t o have converged l o w i t h i n l one percent of t h e i r asymptotic values by t h e 20th i t e r a t i o n , althbugh t h e r e a r e n few n o t a b l e exceptions. Second, i f ve take the 30th i l e r a t i o n I as representing an acceptable degree of convergence, then a l l the bacro , I v a r i a b l e s a r e w i t h i n 0.5Z of those l i m i t i n g values by t h e 10th i t e i a t i o n l and w i t h i n 2.0% by t h e 5 t h i t e r a t i o n . Except f o r aggregate cunula i v e i i n v e s t c e n t , a l l v a r i a b l e s a r e w i t h i n 2.0Y of t h e l i m i t i n g values o / t h e f i r s t i t e r a t i o n . . t I I Q Table 3. 1-alues of !kcroeconorr.ic Variables (1361) ~ I over S e l e c t e J Tterations '-of YUI.COI< I I ( b i l l i o n 1974 1'0'8) I W I t e r a t i o n h ' u ~ h e E - i I .. - : 31Y I vriri,iblv -, I:I --- 1 - A 2 ------- 1- - 25- r - i ; s ~ ! ~ - ~ t i ~ n 9551 85?7 8417 F [+;1s ' I a 8 4 5 3 PA20 9422 , I ,--- l??-c; 13377 1'3366 13235 13233 ?3?17 1S?<13 -:norts [,'-:? A435 4PCfi L G q l 4F57 / z rPI 4 i ; C h i ".:-ulat i v e I Investrzer,t - (.,-- 1 5 L 1 5 15251 15201, 151?lr 15I:iQ 15156 I I I l I S e c t o r ';u~.ber Def in Ltion 1 I I t A g r i c u l t u r e acd F o r e s t r y F i s h e r y Coal ? : e t a l l i c Ores 7 Geverage and Tobacco 8 F i b e r Spinning - 9 F a b r i c s 1 0 F i n i s h e d T e x t i l e s 11 L e a t h e r and Lest!ler E'roducts i2 L u i b e r tc< P1;~:ood 1 3 L'ood P r o l u c t s and F n r n i t u r e 14 P u l p , P a p r ncd Pa;er P r o d u c t s 1 5 P r i n t i n g and P----l i s h i-c ~ .- c h 16 I n o r g a n i c Chenicals 1 7 Organic f i e r - i c a l s 1 8 C h e r i c a l F e r t i l i z e r s 1 9 S y n t h e t i c Resin and C h e n i c a l F i b e r s 2G Other C h e a i c a l s 21 ~ z r o l e m .products 22. Coal P r o d u c t s 23 ., Rubber P r o d u c t s 24 Cemcnt 25 C l a s s , Clay 2nd Stone P r o d u c t s 26 Jron and S t e e l 27 Rolled S t e e l 28 S t e e l P i p e s and P l a t e d S t a e l ~: ~ I 29 C a s t and Forged S t e e l 30 Eon- Ferrous ::etols 3i b : e t a l l i c Products i 32 - i R o n - E l e c t r i c a l I.nchincry 33 I n d u s t r i a l E l e c t r i c a l Machinery 34 E l r c t r o n i c s f -- i 35 ---- -- ----- -- ---- -- -- Ilouschold CIec t r i c n l ?Inchincry .b q S h i p b u i l d i n g 2nd R e p a i r i n g 37 R a i l r o a d T r a n s p c r t I 36 Kotor V e h i c l e s i ' 39 P r c c : s i o n and O p t i c a l P r o d u c t s i 40 -- 0 t h-~- :-.- -- !!jntrfncturirti: --- ---- ----------- - i - i 41 1:csj ~lcrrcen ~ ~1;uidl(!ini; 4 2 I'ubl c r.nd Cthcar C o r ~ s t r u c t i o n 43 Clcc: r i c i t y *- 4 4 Water i n d S a n i t a r y ~ r r v i c c ! ---- -------- 45 Bn:ik- n q 2nd Irisurnnce ---. ---. --- .r ~ i 6 4 6 - Ilous. r.2 4 7 C o r a ~ ~ n i c a t i o n I *= - 48 .· 'Srcn!:port znd S t o r a g e ~ t 4 9 Cor.~?~~rcc i; -- - -5-0- - -----I'f!ll~orts. Tn bcth cases, t h e cnlns r ---- i n c r e ~ c n t n lunit:; cr trade nbnve and beyond t11or;c Jcvel:: rcpnrt - next- to- last c011::-n. textile:; (;!lo) .'$scctor sucil ;is finjslle:! -- "/ ::ost s e c t o r s both iriport :mi export jn L ! I ~i.ot?el: "ex:.ortinl:: :;ertnrcqr 2 r c tliose ~ i h i c hoff c r ~ : , i n sfro:: cx?nnsic.i~of e:.:porrs 2nd n p t ii..;~ort.: (and -JS ce-vcrsa f o r "i: p o r t i n ? sectors" ) , I I 1 Tihere a r e sex-era1 i n s t a n c e s i n t h e t a b l e where I:or a ' s cor.?arative 22-;antage i n hig?.ler degrees 05 :.ianufacturing e l a b o r a t i o n . i. , g r e a t e r . i n ? u t s of s k i l l e d l a b o r , is evizent. In t e x t i l e s , f o r exacp e , the deeree of export cocparntive advantage d i r e c t l y follows the degree i f product I e i a b o r a t i o n : f i b r e s , then f a S r i c s , then f i n i s h e d t e s t f l e s . k l e s a c e i s I I t r u e of wood products: pl-ywood and f u r n i t u r e (sectors 1 2 an$ 13) rank above pul:, and Faper (sector 1 4 ) . Yet another i n s t a n c e i s found if chenicals: I "other chemicals" (pha~maceuticals,cosrietics, etc. ) are the only checdcals I t o b e e - q o r t e d i n YLLGCK. The nodel's l a c k of t e c h n i c a l change i n produc i o n is indeed a It s t r o n g caveat t o t h e s e r e s u l t s , but i t is a problem of d a t a dathsr than of ~ ~ d e L s t r u c t u r e . ):ere t h e r e a s u f f i c i e n t l y sound b a s i s f o r p r o j e c i n g t e c h n i c a i 4 chxnge by s e c t o r , then the YULGOI: A-matrix cculd be updated v e r t i n e . In t h e aSsence of such updating, i t probably would he b e t t e r t o s o l v e thri r-odel f o r a t e r n i n a l :.-ear which i s c l o s e r t o t h e base year. I 6. ConeZuding .?smarks r f Some very b r i e f com.ents may be made i n csnclusion. (1) I n a mar; conventional multi- period model, t h e sec -2 i n v e s t n e ~ tgrovth r a t e s i n the f i n a l p e r i o d ( s ) s t i i l must * !e be constrained by t e r n i n a l conditions. lience thos r a t e s can b e subjecte? t - the stock- flow i t e r a t i o n I I I I procedure used here, and i n t h a t s e n s e t:,e procedu& would appear t o Ee capable of g e n e r a l i z a t i c n . (2) I n any event, t h e n u n e r i c a l d i n e n s i o ~ sof t h e terminal I conditions problem a r e perhaps more s e r i o u s than suggesied i n e a r l i e r treatments. ! (3) Souie of t h e main c h a r a c t e r i s t i c s of a multi-pe:lod model I can be captured i n 2 s t a t i c , terminal- year model, by us4 ~ of t h e stock- flow f a c t o r s . (4) However, from t h e a u t h o r s t viewpoint, t h e l a c k of price-1 endogeneity is a s e r i o u s l i m i t a t i o n t o YULGOK and many , I other plancing models. This l i m i t a t i o n places a strong 1 I caveat cn any i n t e r p r e t a t i o n of t h e numerical r e s u l t s . , 1 tr . F t I i ' f . I ; .I : 1 40. I -1 -- Table 10: The Gains from Additional Trade, i n Terns b . of tke I~JTGOKClbjecti-:e Function j : i " f 2 Sector KO. X a ~ e YULSOI: 1981 E x ~ c r t / i n ? o r t ;kt Gains per Lev.-el (billion I?:+ son) Unit oT Incre- - x e n c a l Trade I. a r t i n g Sectors Beverages, Tobacco E l e c t r o c i c s Other s e r v i c e s Wood products Lmber and p l p o o d Leather Oti'er c o ~ l s t r u c t i o n Son-nctallic d n e r a l s Other rranufacturing Other c h e r i c a l s Processed food Finished t e x c i l e s ShipEuildLng RBilroad equipsent Motor v e h i c l e s 11. Inporting S e c t o r s Agriculture, Forestry 630.5 N q t a l l i c o r e s 31.1 Iron and s t e e l 135.9 Glass. Clap, Stonc products 43.5 Petroleum products 65.3 Precision, Optical products 103.1 P r i n t i n g a d publishing 4.7 I n d u s t r i a l e l e c t r i c a l ~ ~ z c h i n e r y 116.6 Coal 0.1 Ken-clcctrical rvcl~inery 511.6 S t e e l pipcs and p l a t e 10.1 Cast and forged s t e e l 8.0 Ilcavy e l c c t r l c o l uachinery 25.2 Fabrics 82.2 Rollcd s t c c l 366.1 )fetal prcdcsts 60.4 Orennic c i ~ c z ~ i c n l s 117.9 Rubber p r o d c t s 4.8 Fibcrs I 45.2 Coal products 12.9 Pulp, I'opcr 113.2 S y n t l ~ e t i c s 183.8 C?lcn~lcalf e r t i l i z e r 8.0 Inorgalri c c l ' c ~ ~ l c a l r ; 31.7 Ron-frrrbus n e t o l s 133.0 F i s h e r i e s 7.1 Ccmnt 0.3 Transport 68.5 [I ] Chenery, I!ollis E. and ?'.ichael Bruno, "Eevelopment Alte4natives I n a n Open Econony ,!' Econon!ic Journal, Vol. 72, pp. 79-403, 1962. I [2] Marine, Alan S., "Key Sectors of t h e Xexican Economy: 19d~-70," i n A. S. YInne and H.S. Yarkowitz, eds., -rudies i n Procqss S Analysis, Wiley, Kew York, 1963. I I [3] K i m ,Yoon Hjrung, "A 53- Sector I n t e r i n d u s t r y Projection piode1 f o r K o ~ e a ,1974-81," mimeo, Korea Development I n s t i t u t e , .July 1975. :4 ] Taylor, Lance, "Ti~eoretica1 Foundations and Technical lmpli- cations," i n C.R. B l i t z e r , P.B. Clark, and L. Taylor, ed~s., -Economy-Wide Kodels and Eevelopment Planning, Oxford L?ni(versity Press, 1975.