I a t e r n a t i o n a l Bank f o r Feconstruction acd Development Development Research Center Discussion Papers No. 1 APPLICATIONS OF CONTXOL THEORY TO A LEONTIEF- TYPE FL'JTNING MOD= by Charles R. B l i t z e r and Han K. Kim May, 1973 - SOTE: Discussion Papers a r e p r e l j . n i n a 3 m a t e r i a l s circulated t o s t i m u i a t e discussion and c r i t i c a l comment. Seferences i n publication t o Discuss:on Papers should be cleared with t h e author(s) t o p r o t e c t the t e n t a t i v e character of t h e s e papers. Q - n m i c multi- sector models a r e by now a f = i l i a r tool of the development p l .nner . / Since s f f i c i e n t general rnatheaatical programing algorithms a r e not y e t available, planning models have normally been .;olved using l i n e a r programing techniques. Because i t is comput&tionalLy d i f f i c u l t and expensive to l i n e a r i z e non- linear r e l a t i o n s , th2v have been e i t h e r simplified substantially o r avoided e n t i r e l y . I n p a r t i c u l a r , while i t may be s t a t i s t i c a l l y reasonable c?r necessary t o assume l i n e a r I production functions, the wide-spread use of l i n e a r maximands (often implying constant marginal u t i l i t y of consumption) is b a s i c a l l y a computational canvenience. Secondly, adding additional time periods f o r dynanic models. very rapidly presses on the capacity l i m i t a t i o n s of even very large- scale computers, once again pushing costs beyond reasonable l i m i t s . In t h i s paper, the application of zontrol theory t o dynamic planning models is discussed. Optimal control theory o f f e r s i t s e l f a s * a possible a l t e r n a t i v e t o l i n e a r programing i n t h a t , computationally, it easily handles dynamic relationships and various non- linearities. Most economic work i n control theory has been concerned with t h e o r e t i c a l * aspects -- a n a l y t i c solutions of abstract (usually one-se-tor) growth models, such a s those of Arrow (1968), Shell e t . a l . (1967) , o r Dobell - and Ho (1967). .? - I 6 * rl, -1/ Ve have benefited from comments and suggestions made by Hans Sergendorf and Alan Xanne on an e a r l i e r d r a f t of t h i s paper. -21 See Bruno e t . a l . (1969), B l i t z e r (1971), Eckaus and Parikh (1968), o r Mznne (1972). - The f i r s t 5ur.erFczl a?plica:ions a l s o d e a l t with r a t h e r simple an2 snall-scale n2dels. Gooarcin (1961) and. S t o l e r u (1963) derived i l l u s - t r a t i v e paths q h c p t i m a l savizgs f o r deveioping countries. Chakravartv (1962, 1969: a?c Sa-r and Yanne (19F7) have a l s o coaputed t h e t~lrnerical properties nf o p t i n a l growth ~ a r h sf o r nodels ~ i t hnon- linear welfare f-2ctions iv.2 l i n e a r technologies. C3r'mal control versions of multi- sector models have been more recent. Kezdrick and Taylor (1970) discussed a small non- linear growth model f o r Korea which focused on investment and t r a d e choices. Subsequent I work by Pindyck (19711, Norman (1971), end Kim e t . a l . (1372) s t u d i e d s t a b i l i z a t i o n p o l i c i e s . However, nuicerical applications have been limited p a r t l y by t h e awkwardness of control theory i n handling i n e q u a l i t y c o n s t r a i n t s and t h e high c o s t s of solving systems with many s t a t e v a r i a b l e s , as world be t y p i c a l with a multi- sector planning model. The work of Kendrick and Taylor represents t h e f i r s t s t e p i n the a p p l i c a t i o n of control theory t o economy-wide development planning. The next s t e p is t o solve a l a r g e and e r e a l i s t i c model a t l e s s c o s t + h a a l t e r n a t i v e l i n e a r progrannniiag methods. I n t h i s Faper, we attempt t o g:, p a r t of the way, by a formulation which w i l l reduce t h e n u ~ b e rof s t a t e variables, and by suggesting ah appro- p r i a t e and e 5 f i c i e n t s o l u t i o n algorithm. Thrgughout, w e w i l l d e a l only with l i n e a r system3 of production and trade, r o t because they a r e methodo- l o g i c a l l y Eore appealing, 5 u t because of t h e well-known econometric d i f f f c u l t i e s of estimating non- linear production o r foreign t r a d e functions. S e c t i o n 1 is levcte? LO a d e t a i l e d description of t h e dynamic p r o p e r t i e s of t h e open Leontief system without f o r e i g n t r a d e o r primarv f a c t o r c o n s t r a i n t s . A f t e r d e s c r i b i n g zn a p p r o p r i a t e c o n t r o l theory transformation of t h i s simple modei, we show how realism can be added - i n Section 2 . Here f o r e i g n t r a d e a c t i v i t i e s , human c a p i t a l formation, and primary f a c c o r s a r e introduced. I n S e c t i o n 3 , w e propose an algorithm f o r s o l v i n g t h e model both w i t h and w i t h o u t i n e q u a l i t y con- straints. 1- 5 l m i c Leontief Systems w i t h Non- linear O b j e c t i v e Functions The open dynamic Leontief Model f o r a c l o s e d econony can be s t a t e d a s follows: The Basic Demand - _Supply Balance Equatior; Xt = A x t + 1 C t (1.1) Jt + e . .-- m where X = o u t p u t l e v e l s (n- vector), , * A = Leontief m a t r i x Eor i n t e r m e d i a t e i n p u t s (n x n m a t r i x ) , -- C = consumption demand (n- vector), '2 J = investment demand by sector of origin (n- vector), I w-re t h e volume i n d i c e s of q u a n t i t i e s and s a l e s a r e i n b a s e y e a r p r i c e s . e . m The underlying assumptions of o u r demnd-sdpply e q u a l i t y (1.1) are that there are no j o i n t p r o d u c t s ; t h a t p r o d u c t i o n f u n c t i o n s a r e of f i x e d c o e f f i c i e n t s form; t h a t output is n o t thrown away; t h a t input- I outpct c o e f f i c i e n t s re?nain constant over Znvestxaent ?)errands DFstribution Relationship where 3 = a matrix of sector- of- origin r a t i o s for investment demands, (n x n matrix) I = aggregate investment demands by s e c t o r of use (n vector) An element of B = ( 5 . . ) represents the proportion of c a p i t a l 13 goods originating i n s e c t o r i per unit of aggregate investment in sector j . Since few s e c t o r s produce f o r investment goods, many rows of B w i l l be equal t o zero, thereby causing B t o be singular. Once again, f o r s i m p l i c i t y , we assume the cbnstancy of elements i n t h e B matrix. Accelerator Relationship S I = X t - + l - X t 9 (1.3) t . L I o r -1 - I (1.3)' t = G- (X,+l - Xt) , .* .% where G is a diagonal matrix gf order n of aggregate incremental e 0 output- capital r a t i o s . T h i s expression assimes there is no excess capacity i n any s e c t o r ~f the econorny and s t a t e s that increases i n capacity should be equal co increases i n output. A s usual, the incremental cutput - capital 1 r a t i o s a r e assumed t o be constant. Growth paths e x i b i t i n g ihese properties a r e usually referred t o a s Leontief t r a j e c t o r i e s . There a r e a t l e a s t th:xe vays i n which l i n e a r dynamic system can be deduced from t h e Leontief dynamic midel (1.1) - (1.3). 2. a Direct Deduction Substituting (1.2) and (1.3) i n t o (1.1) and arranging the terms, we obtain a whence, l e t t i n g = 9 6 l and H = I - A -ri, we have .. = H X t - C t (1.4)' BXt+l It should be pointed out that 1 . 4 ) does not define a + . dynamic system because B is singular i n general. Therefore, Xt+l t cannot be uniquely determined from t h e s t a t e s Xt and controls Ct i n each time period. Following endr rick's (1972) suggestion, t h i s d i f f i c u l t y - + i can be circumvented by rearranging t h e B matrix so t ~ g tthe f i r s t m - rows consist entzrely of zeroes and the remaining (n - rnF rows, which rep- s * 1, resent t h e c a p i t a l producing s e c t o r s , contain noil-zero olenents. =(' We can 0 then w r i t e the partittoned form of as where iZ2 B21 22 is a square matrix, which we shall assme to be non-singular. P a r t i t i o n i n g t h e H m a t r i x s i m i l a r l y and a r r a n g i n g Ct and X a c c o r s i n g l y i n t o , Xlt and X2t , re can rewrite t Clt ' C2t ( 4 as (1.5) and (1.6). - -. - Xl,t+l B 2 '2, t+l - H21 ~ Xlt + H22 x2t - C2t (1.6) + Assuming t h a t Hll is non- singular, we obtain from (1.5) whence, s u b s t i t u t i n g i n t o (1.6) , we derixre as a l i n e a r function X2,t+1 -1 where (B22 - 32L HI1 H12) is assumed nonsingular. So f a r , t h i s f o m l a t i o n of t h e ?roblea closely resembles t h e dynamic inverse problem f i r s t posed by Leontief (1969) i n which,for computati c n a l reasons, t h e dynamic node1 is solved through a backward integration schene starting wi'h values f o r a l l and t h e l e v e l of ter- Ct rainal y e a r ' s o u t p u t - I n our formulation, the same decomposition is done through forward i n t e g r a t i o n , thus avoiding the prablem of inconsistent l e v e l s of i n i t i a l year's output Xo . I n other words, given Xo and a l l C t , the model can be solved dynamically without inverting the entire 11 matrix f o r a l l time periods.- We should point out t h a t the control variables C appear 1,t+l i n t h e system equation (1.8) simultaneously with the s t a t e variables X 2,t+l The usual way of t r e a t i n g t h i s simultaneity problem is t o define a new s t a t e variable for t h e leading control variable, i . e . , t o define Then, from (1.8) and (1.9) , we f i n a l l y have the following l i n e a r dynanic system: ' ' where -1/ For d e t a i l s , s e e Kendrick (1972). I n t h i s Gynamic sysrem, a r e s t a t e variables Clt and X2t and AClt a i ~ d C2t a r e c s n t r o l variables. Thus, w e hzve n s t a t e v a r i a b l e s and n control variables f o r each period of t i m e . The f u t u r e paths of the economy a r e constrained over t i m e by - equations (1.10) , s t a r t i n g from h i s t o r i c a l l y given i n i t i a l values c - 1,o Any ~ a t hof t h e economy which s a t i s f i e s t h i s equation w- *290. - - -- - - -- - . - -.nd the icitial conditg-ons is c a l l e d a f e a s i b l e path. The aim of t h e planning model is t o choose t h e b e s t among a l l f e a s i b l e paths. H - m r e ue thus specffy a c r i t e r i o n by which optimality can be judged. Sollo-ding Arrow (1S5S), we take, as thO c r i t e r i o n of optimality, the maxi- '2 o l z a t i o n of the s m ef discoun?ed u t i l i t i e s of per c a p i t a consumption f o r - c B all members of the soclety over a plar,@Lng horizon: .. where p = t h e u t i l i t y discount r a t e , P = t h e s i z e of t h e population, U = t h e r a t e of s o c i a l u t i l i t y. To Sum Up : The optimality problem is posed a s one of choosing t h e f e a s i b l e path of AClt and C2t f o r each time t so as to maximize subject t o constraints n > 0 are given. W e assume t h z t t h e per c a p i t a u t i l i t y function is twice differen- ,,a51e, increcslng, and s t r i c t l y convex. I f we assune f u r t h e r t h a t L2 equation (1.15) implies t h a t t h e per capita conswnption a t any mcment of t t m e t vl11 necessarily be positive, so that the inequality constraint (1.16; w i l l 5e i n e f f e c t i v e along an o p t i m l path. Although wc have derived a f u l l y dynamic optimal control system, t h e r e s t i l l remain sever;?- problems - ~ h i c hwould make a numerical s o l u t i o n c o r n ~ u t a t i o n d l yd i f f i c u l t . I n the f i r s t place, we have a s many s t a t e variables as s e c t o r s , which would be most costly i n a l a r g e system. Seco~ldly,i n t h e mcdel as s e t out i n equations (1.12) - (1.17), t h e Hamil- tonian is l i n e a r i n t h e control variable I AClt . I n control theory, t h i s is referred to as a singular arc problem,y f o r which optimal solutions a r e d'fficult t o conpute. This problem can most e a s i l y be avoided by deducing a d i f f e r e n t , b u t equivalent, dynamic c o n t r o l problem using sec- t o r a l c a p i t a l stocks r a t h e r than s e c t o r a l output a s s t a t e variables.- 2/ - -2 - -1/ - For d e f i n i t i o n s , see 3ryson an8 Ho (1969). * w 136 2 1 Alternatively, t h e singular a r c problem can b e overccire by respecify- iag t h e u t i l i t y functional i n terms of increments r a t h e r than absolute l e v e l s of p e r c a p l t a conscmption. Such a formulation is disciissed i n Chakravarty and Xanne (1968) and l i e s behind the use of " gradualist" con- sunption paths i n iixnerical ap>lications, e.g. Manne (1972). ~ h , functional acd :he dynamic systems can then be r e w r i t t e n zs (1.19) and !1.20), respectively. {coxfd.) ~ 1.b Capital Stock by Sector of Use. I f the matrix G is assumed constant over tine, the solurion f o r t h e difference equation (1.3) ' can be w r i t t e n as follows: We define s e c t o r a l c a p i t a l stocks Kt as (1.22) : whence, from (1.21) , we obtain the following c a p i t a l accumulations re- l a t i o n s h i p s - 2/ (con'd.) where a l l v a r i a b l e s a r e defined as before. Note t h a t i n t h i s new system (1.20) , Clt , CzL and XZt are I s t a t e v a r i a b l e s and PClt and a r e c o n t r o l variablLs. Compared ~ i ~ the previous system (1.13) , the number of s r a t e variables has b e e n Zn- creased from n t o (2n - m) with t h e same number of controls. Substituting (1.22) i n t o the balance equation (1.1) and solving f o r C t , we have Retaining t h e sane objective function a s before, we can s t a t e the optimal control planning model a s : TO maximize N- 1 N- 1 (I-A) G Kt-BIt 1 (1 + Pt U (Ct ! Pt) = L (1 + p F t Pt U[ I , t=o t=o Pt subject to the constraints I n t h i s model, we have n s t a t e variables, K t ( c a p i t a l s t o c k by s e c t o r of use), and n c o n t r o l variables, It (gross investment by s e c t o r of use). 1.c C a p i t a l Stock by Sector of Origin As 39:CY above, there is a major colnputational problem i n formulating the optimal control problem a s i n e i t h e r (1.a) o r (1.b) - For optimal control a1goriCh.m the number of s t a t e var+-bles is the mose Cmpostant determinant of conputation time. In order t o take advantage of the decomposition properties of control theory, it is important t o re- duce the number of s t a t e s as much as possible. In t h i s section, we w i l l s t a r t with our basic open dpoainic Leontief model and put it i n t o control theory format while reducing the number of s t a t e s t o the number of capiezxl producing sectors. To begin with, define R = I - A . (E-29) - Partitioning R i n the same way as 9 was partitioned and partitioning Jt accordingly into Jlt = 0 and JZt, w e mewrite (1-1) i n the following form: 9 e Substituting 1 - 3 1 into (1.2) , we have - - (1.32) B21 (Xl,t+l - X1t) + *22 (X2,t+1 X2t) - = J2t The s o l u t i o n f o r (1.32) is LettFng t h e (n- m) - component vector of c a p i t a l stocks by s e c t o r of o r i g i n a t t i n e t be defined as - - K t = B21 Xlt + B22 X2t (1.34) the c a p i t a l accunulation equation can be w r i t t e n a s (1.35) Kt+l = Kt J2t y + - - - - - %here - *21 5 0 '22 X20 - + - A s s d n g t h a t is nonsingular, we derive from (1.34) B22 L , .) - - '-1 - X2t = .B22 Kt - --I B22 B21 Xlt (1.36) .3 - -L whence, s a s t i t u t i n g i n t o (1.30) , we obtain * 6 I m - - - - - - I -1 --I '-1 (1-37) Xlt - !R1l 5 2 B22 8- - ~1 Clt - (Rll - R12 B22 B21)-1 R12 RZ2 Kt , acd, I I I ~ I Again, substituting (1.37) and (1.38) i n t o (1.31) , we derive J2t as a linear function of Kt ,Clt and C2t : Finally, substituting from (1.39) i n t o (1.35) and J2t --I - --I - l e t t i n g F = (R22 B22 BZ1 - RZ1) (Rll -R12 B22 B 21 , we have the following (n-m) linear dynamic systems, - . The optimality prrblem c a be posed a s one of choosing t h e feasible path of C and C2t f o r a l l t so as to maximize It s ~ b j e c t:o the equality c o n s t r a i n t (1.50) and t h e i n i t i a l and terminal I n t h i s system, we have (n - n> s t a t e v a r i a b l e s , Kt ( c a p i t a l stocks by s e c t o r of o r i g i n ) , and n control variables, C,~t and C Z t f o r each t l m e t . Once again, the per c a p i t a ccnsumption w i l l neczssarily be p o s i t i v e along an optimal path. As illustrated by Bruno e t al. (1969) a ~ Kendricic (1972), the d nuaber of c a ? i t a l goods producing i n d u s t r i e s is approxinntely equal t o one tenth of t h e t o t a l number of sectors. Thus, i f n = 100 , then t h e number of s t a t e variables can be r e h c e d from 100 t o 10 i n t h i s system compared with previous models (1. a) and (1.b). Siace computation time is roughly a i i n e a r funcLion of the number of s t a t e variables and a l e s s than l i n e a r function of t h e number of control variables, we m2y be able t o solve a r e l a t i v e l y l a r g e aodel (100 s e c t o r s ) i n reasonable conputation time by 0 r using model ( 1 . . Purther gains still i n compuational e f f i c i e n c y - - could he ma& by using the Taylor s e r i e s approximations t o make a linear- '9 * quadratic problem, as w i l l be shown i n Section 3 . - *** w -l/ \v?1i1eitnayseenarbitraryatfirstglancetochoosete'minal stocks of the c a p i t c l goods, 2 t is easy t o s e e t h a t a l t e r n a t i v e ternlinal canditicns aze f u l l y cn~:patab'.ewith t h i s formulation. In p a r t i c u l a r , . valuation Lor tcl-mizal caFIL,j?_can be s u b s t i t u t e d without l o s s of genera- l i t y . S u c i ~a ~t-oceciarew i l l be u t i l i z e d i n Section 3 when discussing algorit!~:-.sf o r s o l - ~ i n gthe optimal control problem. 2. More General Formulations The discussion i n Section 1 centered around an open Leontief model without foreign trade o r primary f a c t o r constraints. Although simple, t h i s f orm;;i.ation provided us with considerable insight i n t o the dynamic properties of input- output systems. Real world planning models, however, a r e more complex and we must now demonstrzte how the model of analysis used previously would apply to i n more r e a l i s t i c situations. 2. a Foreign Trade Activi--ties Past multi- sector dynamic planning models have focused con- 1/ siderable attention on deriving optimal trade patterns over time:- Here we w i l l expand our b a s i c Leontief model t o include exports, c~mpeti- t i v e and non-competitive imports, and foreign debt accumulation. Defining Et and Mt a s n-vectors representing the levels of export deliveries from sectors and competitive imports t o sectors a t time t , w e re- write the balance equation (1.1) as (I-A) Xt = Ct + Jt + Et - (2.1) Mt 9 - where a l l o t h e p variables z r e defined as i n Section 1. I n addition t o -.. supply and d e h d balances f o r each s e c t o r , we now a l s o require a w balance equation for foreign exchange. Letting CMt , JMt stand f o r levels of non-competitive imports of consumption goods and inlSca'--tment . . . - 1/ For example, see Bruno et a 1 (1969) o r Manne (1.972) ~ o o d srespecSivaly, ar'd Ft a7.d NF represent l e v e l s of foreign debt t 2-2 deSt accurnulat~on,we define ;he balance equation i n terms of - fareigz e x c h ~ ~ g e : where, r is the average interest rate on foreign de5t, a Is a rcw vector of foreign exchange costs of intermediate non-con.pet it i v e imgor ts, m is a row vector of foreign exchange costs of competitive i n p o r t s , and e is a row vector of foreign exchange earnings from exports. Note, t h a t w e a r e assuming constant n a r g i n a l revenue and narginal c o s t f o r t h e export and Fmport a c t i v i t i e s . As long as these assumptions a r e made, it L s c l e a r t h a t an optimal path would opt f o r complete specializa- t i o n i n production. To avoid these problems, which a r e e s s e n t i a l l y caused - . . -. elw P. r C . by our l i n e a r formulation, i t is necessary t o impose upper bounds on Et 1/ and NF . --r t 2 - - . 8 Since w e now have zn a c t i v i t y , JM Q, which represents imports- t; t i o n of non-competltiv~ c a p i t a l goods, our a c c z l e r a t o r relationship (1.3)' must be expended ~c i n d u d e G,is a d d i t i o n a l c a p i t a l constraifit. This Lnvolves : 0 ,we have p = 0 and for g = O , we have u 2 0 , along the optimal trajectory. The optimal control Ct is then obtained by solving (3.1) simultaneously with the Euler-Lagrange equations, - whence, a f t e r differentiation, we have; where u t = O for g, (Kt , C,) > 0 and > 0 for g - vt- (Kt , Ct) = Q i+1 A Letting C = C - C; , equation (3.16) can be rewritten t t (3. IT) N-1 (+TIN- (t+l) Substituting A - a + ( 0 1'-('+l) nT y T from (3.15) t+l T=t+l h i n t o (3.17) , we f i n a l l y derive an improved control C t a s a l i n e a r func- t i o n of unknown shadow prices, pt , ut+l , ..., y ~ -:l Since equation (3.18) holds f o r a l l t = 0 1 . - 1 it can be rewrit ten as: . e where .. C = (n x N) x 1 column vector - D = (n x N) x (n x N) d i a g w a l matrix S (n x N) x (k x N) matrix R Ar nmr . . . A +N-2 r 31 u = (k x N) x 1 column vector of shadow p r i c e s f o r inequality constraints g = (n x N ) x 1 column vector of given constants t-1 t Next, s u k s t i t u t i n g ( KO ,+ 1 pT - r ci+l T=O T t-1 t-1-T t-1 i [ f t KO + 1 $J r cT] + 1 (t-l-T r CT from (3.1) and A i T=O T=O *3 - L A fro? the d e f i n i t i o n Ct = Cti+l C: - = i n t o the inequality constraints rn A A A C ( 3 . 4 ) , we derive t h e constraints as l i n e a r function of . Co, ClY - a - Y Ct : - Since the inequality constratnts (3.23) hoid f o r 211 t = o , , ..., 1 d = (k x N) x 1 colmn vector of exogenously given constants m.d 5 anl C a r e defined a s before. .. Finally, s u b s t i t u t i n g improved control C from (3.19) i n t o the inequality constraints (3.21) , it immediately follows t h a t T where A = - S D S , r 3 = - s T g + d . , f !-" A O sum up, t h e constrained (linearly) optimization problem f o r - l i n e a r dynamics and quadratic c q t e r i o n is reduced t o solving for the 5 non-negative d u a l vsriables p ?-hich s a t i s f y (3.22) and the following * conplimentary s l a c k ~ e s sc o n d i f i a : pT + . (AL.,! 5) = 0 (3.23) From fhe definition of A = -sT D S and the diagonal nature of D , w e c o t e A is a syrmetric matrix; therefore, (3.22) and (3.23), together with the non-negatively constraints on p , can be viewed a s the Kuhn- Tucker conditions of the following quadratic programing problem: T I f A =- S D S is positive definite, it is then clear that * there e x i s t s a unique minimum s o l u t i o n p f o r t h e problem (3.24). From -1 the neoclassical properties cf t h e u t i l i t y function, Ucc (Ct) is negztive definj.'ie, and it s u f f i c e s t o prove t h a t S has f u l l rank, i n order t o show the p o s i t i v e definiteness of A. It is intuitively clear, and can be shown, t h a t the rank of S depends on the constraints (3.11). I f none a r e redundant, then S w i l i be of f u l l rank; i f n o t , then the constraint set should be reduced accordingly. With t h i s procedure, we have reduced t h e dimensionality of the csncrol problem t o t h a t of the boast r a i n t set. A The solutipn p for the quadratic problem (3.24) can theh be substituted i n t o (3.18) to obtain improved nominal controls and A i t e r a t i o n continues as i n section (3.a) u n t i l Ct i s Z a r b i t r a r i l y near d- 0 - zero. Since * * E r - r (O I T i T T N-(t+l) T .C lin: Uc K t ) = 1 a + n ut i-w - 1/ We are deeply indebted to our colleague Hans Bergendorf for this suggestion and f o r h i s help i n proving the following properties of t h e matrix A. * .. C_ is the opti-21 valce of the controls for these values of p . * L As I n s e c t l o n (3. a) , can be deduced through i t e r a t i o n o r Ct d i r e c t l y a s the s o l u t t o n of the non- linear equations (3.25) . The choice of ~ r o c e d u r ewould seem t o depend on the nature of u t i l i t y fqnc- t l o n U (C \ t t" * The values f o r Ct cb? then be used t o s t a r t ariother i t e r a t i o n through recom?utation of A ar,d b f o r the next quadratic programming problem. Using the index "n" t o denote i t e r a t i o n s involving values * of Ct , the ~ r o c e d u r eis continued u n t i l f o r some n , I L q f o r a l l t , where rl is any desired degree of approximation. It should be remarked that there are two separate iterations going on i n our algorithms. The main i t e r a t i o n is of successive approxi- mations t o a nonlinear u t i l i t y function by a quadratic equation. Fot each I main i t e r a t i o n , t h e r e is a sub- iteration which produces control functions t h a t successLvely improve the approximation t o the constrained optimal R. l *A. I. c o h t r o l functions. - It should also be poFnted out that our solution pracedure can i be extended r e a d i l y t o deal with inequality constraints which a r e quadra- -z - t i c functions of the s t a t e and c o n t r o l variables. For inequality c o n s t r a i n t s I B r which a r e neither l i n e a r nor quadratic, the augmented Hamiltonian would It become a non- quadratic function of controls and s t a t e s . Once again, we could use Taylor s e r i e s approximations t o make a quadratic inequality con- s t r a i n t , but this would involve an additional sub- iteration which might prove costly. Finally, an i t e r a t i v e LQP solution procedcre can a l s o be used f o r solving multi- sectoral planning models with non- linear welfare and production relations. However, we have a strong feeling t h a t successive approximations of complicated non- linear production functions by l i n e a r equations would make t h e convergence intolerably slow. A p r i o r i it appears more e f f i c i e n t computationally t o use first- order d i f f e r e n t i a l dynamic programing algorithmsl/ t o g e t "close" t o the optimal solution, with a truely non- linear model, but more e f f i c i e n t t o use our solution i - 1 algorithm ( a i~t e r a t i v e LQP sollltion procedure) t o 50 the remaining distance t o optimality . 4. Conclusions I n Sections 1 and 2 w e d e v n s t r a t e d how it is possible t o transpose even very l a r g e and complex Leontief- type planning models i n t o a format suitable f o r cnntrol theory solution techniques. Since these techniques a r e severly limited, computatioilally, by the number of system a equations o r s t a t e variables i n the problem, w e showed how the number of s t a t e variables could be reduced t o the number of c a p i t a l goods which a r e required i n production, including human capital. A solution algorithm, i t e r a t i v e LQP , was zuggested i n Section 8 I 3 a s an e f f i c i e n t method f o r solving the planning model t o any degree of approximation required f o r a non- linear objective function. I n e q u a l i t y constraints, up t o quadratic functions, can be included i n the procedure, -1/ Kim, Goreux and Kendrick (1972) have developed new solution algarithms based on Jacobson's and Mayne's (1970) first- order DDP which allow f o r inequality constraints. usicg a root-ficding method. O f course there i s a cost co pay f o r these constraints i n the £ o m of additional i t e r a t i o n s of t h e algorithm. Alchough not recommecded, nore complex inequality functions could be handled through even more i t e r z t i o n s around t h e i r quadratic approximations. We have takea p a r t of the "next ste?" which was mentioned i n t h e Introduction. To go the whole way i n t e s t i n g t h e usefulness of control theory f o r numerical planning sodels, a c t u a l computer runs must be made f o r a p r a c t i c a l planning mods1 and compared with t h e r e s u l t s from l i n e a r programdng techniques. I f our suppositions a r e correct, t h e computatioaal costs of solving l a r g e dynamic models could be s i g n i f i c a n t l y reduced. REFERENCES [ 11 Arrow, Kenneth (1968), "Applications of Control Theory t o Economic Growth" , Lectures i n Applied Mathematics, Vol. 12. [ 21 Barr, Jmes and Mail Manne (1967) , "Numerical Experiments with a F i n i t e Horizon Planning Model", Indian Econo- rcic Xeview, Vol. 111. [ 31 B l i t z e r , Charles R. (1971), A Perspective P l a c n i n ~Model f o r Turkey: 1969-1984 , Research Center i n Economic Growth, Stanford University, Memorandum No. 114. Bruno, Fdchael, C. Dougherty and M. Fraenkel (1969), "wnamic Input-Output, Trade and Development", i n A. Carter and A. Brody (eds.), Applications of Input-Output Analysis, Amsterdam: North-Holland. Bryson, Arthur and Yu-Chi Ho (1969), Applied Optimal Conqrol, B l a i s d e l l Publishing Company, Waltham, Massachusetcs. Chakravarty , Sukhamoy (1962) , "Optimal Saving with a Finite Planning Horizon", International Economic Review, Vol. 111. (1969), Capital and Development Plan- ning, MIT Press. Chakravarty, S. and A. S. Manne (1968), "Optimal Growth and Rate of Change i n Consumption", The American Econcrmic Review, Vol LVIII. I S Dobell, A. and Yu-Chi Ho (1967), "Optimal Investment Policy: An Example of a Control Problem i n Economic Theory" , I X E Transactions an Automatic Control. - - Eckaus, R. S . and K. S. P a d k h (1968), Plenning f o r Growth, MIT. -3 Goodwin, R. (1961), "Th Growth Path $or an Under- developed Economy Jacobson, David and David Mayne (1970), D i f f e r e n t i a l Dynamic Programming, American Elsevier Publishing Company, Inc. , New York. Kendrick, David (1472), "On the Leontief Dynamic Inverse", Quarterly Journal of Economics, Vol WMVI, No. 4 pp. 693 - 696. [14] Kendrick, David and Lance Taylor (1970), "Numerical Solution of Sonlinear Planning Models", Econornetrica, Vol. 3E. [IS] K i m ,Han, L. Goreux and D. Kendrick (1972), "Feedback Stochastic Decision Rule f o r Commodity Stabilizations: An Application of Control Theory t o the World Cocoa Markets", presented a t t h e 1972 IEEE Conference on Decision and Control, New Orleans, Louisiana. [16] Leontief, Wassily (1969), "The Dynamic Inverse", i n A. Carter and A. Brbdy (eds. ), Contributions t o Input-Output Analysis, Amsterdan : North-Holland. [17] Manne, Alan S. (1972), "DINAMICO, A Dynamic Multi- Sector, Multi- Skill Model", i n L.M. Goreux and A.S. Manne (eds.), Multi- Level Planning : Case Studies i n Mexico, Amster- dam: North-Holland. [18] Norman, A. (1971) , "Optimal Economic Policy and Econometric Model", presented at t h e 1971 IEEE Conference on Decision and Control, Miami Beach, Florida. - -. [I91 Pindyck, R. (1371), "An Application of the Linear-Quadratic Tracking Problem t o Economic S t a b i l i z a t i o n Policy", presented at t h e 1971 IEEE Conference on Decision and Control, Miami Beach, Florida. [201 Shell, K a r l (editor) (1967). Essays on t h e Theory of Optimal Economic Growth, MIT Press. [211 Stoleru, Lionel (1963), A Quantitative Model of Growth of t h e Algerian Economy, Unpublished Ph.D. d i s s e r t a t i o n , Stanford University.