Development Research Center Discussion Papers , No. 24 I 1 A COMPUTABLE CLASS OF GENERAL EQUILIBRIUM MODELS I1 bY I I Roger D. Norton and Pasquale L. Scandizzo I I I May 1977 I I I 1 I I I I I L I - I Discussion Papers a r c preliminary m a t e r i a l s circu1ate.a t o stirnuiate d i s c u s s i o n and c r i t i c a l comment. ~ e f e r e n c e a; i n I p u b l ~ c a i i o nt o Discussion Papers should be c l e a r e d w i 3 the m t h o r s t o p r o t e c t t h e t e n t a t i v e c h a r a c t e r of t h ~ & papefs. T h e papers e:->ress t!le views o f t h e n u t h o r s a d s h o u l d not be interpreted t o rcflcct' those of t h e ~ o ; l c { ]jack Table of Contents 1. Introduction 2. A S t a t i c General Equilibrium Model 3. A Quadratic Programming Formulation 4. A Linear Prograrmnirig Approximat ion 5. Numerical Results Appendices: A. Compensated Changes and I n t e g r a b i l i t y Conditions B. A Recursive Dynamic Version C . Equations of PROLOG 1 and 2 D. Data f o r PROLOG 2 This paper d e s c r i b e s n siniple anrl powerful procedure foC o b t a i n i n g general e q c i l i b r i u m s o l u t i o n s f o r economy-wide rcodels. The proce ure isd l capable of r s t e n s i o n t o c a s e s where markets a r e s u b j e c t t o multip e c o n t r o l s and d i s t o r t i o n s , and t h e r e f o r e i t is u s a b l e i n t h e context of p r i e- endogenous models f o r development planning. " I n very r e c e n t y e a r s , t h e r e h a s been a remarkable amounk of pro- g r e s s regarding methods f o r s o l v i n g numerical g e n e r a l e q u i l i b r i u m problems. S c a r f ' s 1973 book [81 s t i n u l a t e d renewed i n t e r e s t i n t h i s a r e a , a d o t h e r s s~:bsequcntlyhave attempted t o improve on h i s algorithm [ l , 31. "P tile a l g o r i t h c x rennin s o n w h a t expensive and d i f f i c o l t co u s e , and t h e r e f o r e I :iley c a n r o t generally b e applied t o large- scale nonlinear models. -Lhe procedure presentetl here attempts t o contribute i n he [allow- I i n s t h r e e xaps: a ) t o providc g r e n t c r cost- efficiency f o r large- c a l e r.odels, b ) t o provide s u f f i c i e n t f l c l x i b i l i t y s o t h a t t h e a l g o r i t i 7 I i; not have t o be re-programed f o r d i f t c r e n t models, and c ) a s men ioned, t o - I, :'he ; l u ~ : ~ o rArc merr.!~cr:; o l t i ~ cI!orld s r,ani\'s R e v e l o ; ~ ~ ~ eKcsr.. c'i n t a) 5 C c ~ ~ t e r ''!icy . wish to tliank tile l z r g c number of people wl;o ha e .c rri,ltlt: i;e:pf~~lcori1n:ents ,it ~ : < i r i o ~st(iges i n t h e development u f tlri:; i s ) r : .1 , i3i sscho!~,:i.C;~ndlcr,.I. Ilcilo)~, P . H a z c l l , K. T n n ~ \ l , I (:. ~.l::cti, is. I vsy, 11. ::eernu!;, (;. ~j'v'rircl, Y . 1'1 c s s n c r , R . i'tltb;?r,r, d u a l i s z , p r i c e c o r ~ t r o l z ,e t ~ . Tk.e f i r s t two a i z s '3rc achieved by castirlg t h e general n j r i u a prcbler, i n l i c e a r ~ r o g r z c i ~ i nEorcat, so t h a t t h e c o ~ p u t a t l o r ~ a l g .zovcr of t h e s i z p i e x sol!:tim a i ~ c r i t h ~(and more r e c e n t l i n e a r pr grarming a codes) can be e q l o i t e d . ?:like t h e Ginsburgh-'{aelbroeck arid 3ixo 11, 31, which u t i l l z e iterati-.,e sequer~cesof iiilear programming so 1 t h i s a l g o r i t h n r c q i l i r e s cnl;- a s i n g l e LP s o l u t i o n . E f f i c i e n t l i n e r i z a t i o n 1/ techniques a r e enFioyed f c r r.r:nlinearitfes.- The l j n e a r progrsrizing framework a l s o is p a r t i c u l a r l y a m t o extensions t o t h e non- general equilibrium cases. The c o n s t r a i n t s a r e a p p r o p r i a t e f o r t h e r e p r e s e n t a t i o n of many i n s t i t u i o n a l t! i n f l u e n c e s on t h e econo.:y. And t h e freedom t o have unequal nunberi of 4 v a r i a b l e s and c o n s t r a i n t s is h e l p f u l i n - t h i s respect. I t i s frequ n t l y r h e c a s e t h n t s e v e r a l i n s t i t u t i o n a l and behavioral c o n s t r a i n t s co-dxist i cx a n t e , i n overlapping ways, without i t being known beforehand r~;7ch ones w i l l be e f f e c t i v e . Xost of t h e remainder of t h i s paper i s dcvotcd Lo t o represent t h e r e l ~ t i o n s h i p sof a general equjplibriurn * . n a t i c a l p r o g r a m i n g , along w i ~ hpsoois of t h c d e s i r r . d models arc. rii~cusscc?In s c c t i e n s g-4, 2nd exLensioiis i o ;: ~ - c c ~ ~ r : ; idyir;:r::ic v e ' ---____--- _ _ i -?/ Some of thc newer q u a d r a t i c pmgrnn~lingr:cdcas also ;!]:pear to !,e c l u i ~ c powerful; if t!le r~odclwcrc mt too I n r y ; ~ , ;!ncl i F t i l e cocic? i ! c r ~ - ~ i t t ~ ~ d C c;untir;tt i rorr.s inl:~cj2.yt-,iinL set, tl!cx ;i;~prc~.?cl: t !'f t h i s ps:-Icr zoulti [)P in~,]i2:;pnte Cournot aggregation c o n d i t i o n s hold, f o r t h e optimal aggregate n/arket s o l u t i o n of t h e above model, r e g a r d l e s s of t h e p r o p e r t i e s of thd demand J functions,?' and r e g a r d l e s s of t h e underlying i n d i v i d u a l u t i l i t maximiea- t i o n process. From (21), (11) and ( 2 3 ) , i n f a c t , w e o b t a i n : a f a f p -- =i -= i 1 f o r a l l yi > 0 3yi ayi Engel aggregation and from ( 2 1 ) , ( l o ) , (22) and (23) s 3f If. a f i 1 1 ( p , y i ) + F' = p'! ap + C = O (25) i 1'. '5 f o r a l l P . > 0 - 5 J - Cournor aggregat i o n B *- 1 - C - / S u i t a b l e convexity assumptions, liowever , w i l l liave to b e met .- t o ensure convexity of the constraint s e t . l i t i s now easy t o .i:~o.: ~'!~^lt tile j\.xi~n-Tucker conditions jzply 3 I c o ~ ~ p c t i t l msr k et e q u i l i b r i : ~ . Three p r i n c i p a l characterFst Lcs r c o n p c t i t i v e equiiibriurn r'oLlow i i r c c t l y frcx t h e above; thcy can f o r a l l a c t i v i t i e s i n t h e o p t i ~ a lb a s i s as follows: or t o t a l output = t o t a l deaand = d u a l value of zero excess p r o f i t s I or p r i c e = opportunity c o s t = - dual v a l u e of commodity balance co s t r a i n t ; A = -Y o r resource c o s t = - d u a l value of resource c o n s t r a i n t . This concludes t h e proof. Equation (26) is t h e market c . ." coqdition which is required of equilibrium s o l u t i o n s . Equation (2 t h a t p r i c e equals marginal c o s t , another c h a r a c t e r i s t i c of t h c corn e t i t i v e 1 e q u i l i b r i r m . Together they inply t l l a ~t 3 t a 1 f a c t o r renluneratian equals the v a l ~ eof ( f i n a l ) output. Equation (27) a l s o implies t h a t a given J-ioduct's . I L p r ~ c eis t h e sane f o r a l l groLps of consumers. pw .P T\;ese coriditions are the. ones normally invoked when ;i cor c t i t i v e - equilibrium 1s being cl.arncteri7cd. We have chosen t o show thaL ($) and (25) a r e a l s o s n t i m f i c d , i . e . , t i l i l ~in t h e aggregate ro;lsurncrs i n +is rn model behave .iccording t o tenet:; or deaond theory. 11 -;ill brc~i:e J.lypnrc.nt. subscqucntly t h a t t h i s l a s t propclrty is ilnyor' ?:.t t o ti7rb clcfini t i r l i c: tiirx more conputnblc v e r s i o n s of :he model. I , I - - - . - . . . 7 ' r - ~~~r.~. y::ycrj 5 ,! ;\7:1.12 'farcn'; fLrxulatj.c.r, :lac!irLccr3e predr-te +-. . Lr,e (;.22 :l:~:eE~re exogenous i n the- s t a t i c f ~ r x u l i-.; t ~ a per-~d l nI- our f c m c l a t i o r : i n c l u d e s incoc;r (2nd t h e income-expendlcure cons ri--:lt) i n the s z t of endcgenous v a r i a b l e s . Second, t h e problem formulatio is of I t h e prinal- dual type, s l n c e p r i c e and quancity v a r i a b l e s appear imult,ln- eously a s primal and d u a l v a r i a b l e s . k Third, without c o n s t r a i n t 6 ) , t h e s o l u t i o n would correspond t o t h e monopolists' equilibrium. Four optimal s o l u t i o n is a s s o c i a t e d w i t h a z e r o value of t h e o b j e c t i v And f l f t h , when t h e problem is s t a t e d i n t h i s way, i n t e g r a b i l i t y of t h e demand f u n c t i o n s is c o t required. The model presented i n sect2on 2 suggests t h a t t h e Co r n o t and i Engel aggregation c o n d i t i o n s can be exploited t o make endogenou process o r income formation i n t h c corllputation of competitive e through mathematical programming models. Although t h e r , I of t h e model presented is q u i t e g e n e r a l , i t s p n r t i c u l d r f u n c t i o n a l forms of t h e i n d i v i d u a l 5 wc now considor some e p p l i c n t i o n s t o t h e c a s e of l i n e a r and con t a n t e l a s t i c i t i e s dcmnd f u n c t i o n s . To f n c i l i t t e computations, t h e income 3 I I 7 ~ ~ 1 r i a bno longer e n t e r s t h r nnximancl e x p l i c i t l y i n t!ie follow ng v e r s i o n s l e o f t h e model. C o n ~ i a e sthe ioiloi;ing n.odifl-:aticn of t h e rnarkst s t r u c t u r e defined i n (aj - (c): (a') Conscsers behave i n accordance w i ~ harl aggregate i v e r s e b deciand function of t h e type: where A > 0 and B is a ncn- singular symmetric a t r i x of demand c o e f f i c i e n t s . (b') Producers maximize p r o f i t s s u b j e c t t o an aggregate resource c o n s t r a i n t : C ( c ' ) The q u a n t i t i e s demanded cannot exceed t h e q u a n t i t i s produced: and t h e r e is f r e e d i s p o s a l of ovcrpro2uction. Theorem 2 : Solution of t h e following aggregate maximizing y i e l d s t r L a s t a t i c competitive market equilibrium: Max X' (A - 0.5 BX) - O ' Q x,Y - where the C' ere t h e c o s t s of a l l primary factairs i l i i c h ;Ire I - a v q i l a b l e i n jnf I n i t e l y e l a s t i c supply, I scilject- to X ' O - 1 (34) wllich is Engel aggregation, obtaiqed by d i f f e r e n t i a t i n g (30) t 3 incone (and assuming that consumers are on their budget lineb) ; and I s u b j e c t a l s o to: DQ - b - .: 0 Resource c o n s t r a i n t s (35) x - ( 2 -< 0 Commodity balances (36) where t h e maximand (33) can be i n t e r p r e t e d a s t h e sum of consum r and pro- e ducer surplus over a l l ;roduct markets. The rcotivatioll behind t h e intro- Y dllctj.on of (34) is not t h e assumptior of u t i l i t y maximization b individual t consumers, but r a t h e r t h e r e s u l t of Theorem 1 which s t a t e s t h a t Engel aggregation must hold i n t h e aggregate f o r t h e equilibrium solu ion. Since we a r e dealing with aggregate markets, t h e p a r t i a l d e r i v a t i v e - . defined. Proof: Form the Lagrangean from (33) - (36) : hIax L = X' (A - 0.513X) - C'Q + p(+'X - 1 ) ~ , Q , P , P , ~ 0 t + p'(X - Q) - Xt(DQ - b) The Kuhn-Tucker c o a i t i o n s f o r t h e solution of (37) ale: -2L = X - Q 80 From (38) and (39) and t h e ccmplementarity s l a c k n e s s c o n d i t i o n s (16) i t is c l e a r t h a t f o r t h e s o l u t i c ~ n : where a g a i n w i t h some abuse of t h e n o t a t i o n w e have used t h e same synlbols t o i n d i c a t e t h e nonzero q u a n t i t i e s of t h e a c t i v i t y l e v e l s i n t h e optimal s o l u t i o n ( a n d t h e i r corresponding parameter m a t r i c e s ) . Equation (45) i s t h e market e q u i l i b r i u m c o n d i t i o n t h a t p r i c e s = marginal c o s t s i f f p = t o t a l income. I n o r d e r t o show t h i s we prove now the following: Corollary: Under t h e c o n d i t i o n s (38) - (43) p is equal to t o t a l remuneration of production f a c t o r s . Proof: Because of complementary s l a c k n e s s and equacion (41) i n t h e s o l u t i o n , X = Q; and thus p r e m u l t i ~ l y i n gby . L - we o b t a i n : 7 .! A .. ~ . >cCc-sz of t5e d?mr:.r,< ~ t eq~;iio:is in (23) 222 t h e Engel ::~;C~CF;F.~:CF. ~ C Z L L- I - X' (A - BX) = 0 , s o t h a t ( 4 6 ) becomes ( 4 9 ) p = X'D'X + X'C n + I t h a t is, p is equal t o t o t a l cost (X'C) t o t a l imputed valu s of sc?rce f a c t o r s , o r t h e t o t a l remuneration of productive f a c t o r s . That t is magnitude has t o b e equal t o t o t a l expenditure i n e q u i l i b r i a is e a s i l y s h o i n a s follows. Assume p r i c e s a r e e q u a l t o marginal c o s t s : A - BX + $y = D'X + C ( 5 0 ) I Premultiplying ( 5 0 ) by X and applying Engel a g g r e g a t i s y = x'D'X + X ' C Hence t h e d u a l analogue of ( 3 4 ) is the constraint that ~ x p e n d i t u r cannot * I-~ exceed income; i n equilibrium t h i s r e l a t i o n becomes an e q u a l i t y . .;. - cl Linear Progrcurmin!] /i?proxima tion In order t o w r i t e th! f o r ~ g o i n gproblem i n terms of l i n e a r pro- grarnming, s e v e r a l transformations a r e required. The f i r s t s t e p i l i n e a r i z e t h e q u a d r a t i c o b j e c t i v e ?unction. Following Duloy and N o r ~ o n[ 2 ] , w e r e w r i t e t h e p r o g r a m i n g p r o b l e n i n (33) - (36) i n terns of a s t e p w i s e l i n e a r f u n c t i o n which p e r n i t s d i r e c t measurement of t h e a r e a under t h e demand f u n c t i o n : I where 12 I r e p r e s e n t s t h e a r e a under t h e denand f u n c t i o n o r i t h e i t h good; I t a r e t h e a c t i v i t y l e v e l s f o r t h e segments of t h f u n c t i o n wi ; 1 a r e t h e v a l u e s of Wi corresponding t o D i,s a r e , a s b e f o r e , parameters of a l i n e a r (or I ai,bi Qi 9 l i n e a r i z e d ) i n v e r s e demand f u n c t i o n of t h e typ i C f o r s i m p l i c i t y , w e a s s e t h a t a l l c r o s s - e l a s t i c i t i e s are z e r o ; Y is an i n i t i a l value of income. 0 Xote t h a t t h e a r e a under demand f u n c t i o n W can be combutcd f o r . . i L P o n l y f l ~ n c t i o n a ls p e c i f i c a t i o n w i t h a n a r b i t r a r y degree of approximation -- depending o n l y on t h e number of s t e p s . The l i n e a r f u n c t i o n is uscd i n t h c 'r - e x p o s i t i o n o n l y t o simpl.ify t h e argument. I B u i l i ing on t h e approach in [ Z ] , w e can now s p e c l t y I .neor?n 3: Solution of t h e foilowing LP node1 y i e l d s a competitive P-- ~ I market equilibrium: subject t o ... 1 -< 0, i m1, ,n 'i,s Di,s - Qi s Commodity balan n c o n s t r a i ts . ,b 1 - 1 i = 1, ... ( 5 8 ) Di,s s Ei 'Convex combinat on c o n s t r a i n t where t h e 8 represent t h e q u a n t i t i e s sold a t the l i m i t of eac . i,s Y - Y o of t h e function Wi, E is Engel elasticity and y = i Yo index of income change. ~ Proof: Forming the Lagrangean f o r t h e problem above, we obtain: t h e n be a n a l y z e d a s follows. For t h e i,s a c t i v i t y D i,s e i t h e r o r and Also, f o r t h e i t h a c t i v i t y Qi, - e i t h e r Qi - and b Analogous statements hold f o r t h e shadow p r i c e s and t h e c n s t r a i n t s . I n p a r t i c u l a r , e i t h e r = 0 Pi o r and i . e . demand e q u a l s supply i n t h e i t t market and t h e shadow p r i c e of t h e it,h , * commodity balance c o n s t r a i n t equals t h e opport~..:ity c o s t of tAe "i I f a c t o r s employed i n t h e production of t h e i t h commodity. It fs c1e.r by '5 i - - now t h a t t h e m u l t i p l i e r i n (60) must equa1,at e q u i l i b r i u m*t o t a l income , change y - 9= y yo . I n f a c t , assuming t h a t Y m Y )I= E u f f i iency) 0 and multiplying (60) and summing over s, w e can write: f o r a l l yi > 0 where w e have used t h e r e s u l t i n (61). Equation (63) then characte a s a measure of consumer s u r p l u s , f o r a given-income l e v e l y, i n market . / Clearly, t h i s implies competitive equilibrium, since, s u t h i s r e s u l t i n t o (60) again, we can d e r i v e t h e following expression revenue f o r t h e i,s consumer surplus i n t h e a r e a under t h e demand a c t i v i t y i t h market function f o r t h e i,s a c t i v i t y o r , by dividing both s i d e s by 8 i,s supply p r i c e demand p r i c e I n order t o prove necessity, on the other hand, assume that (64) holds with p i n l i e u of y yo : ~ - 6 / Note t h a t yi equals consumer surplus CSiat initial incomt? yo . Indicating sgch a value of t h e surplus a s CSio ; i n f a c t , w .? -- i write (63) as: "io s . ( 1+ E ~ $ ) c s= ~cSi ' = .~ 3 E y = i c s i o 4 a r c e l a s t i c i t y of cons-mp,r s!:rplus with mateiy t h e same a s t h e Engel e i a s t i c i t y . ( I t is exactly the s a constant e l a s t i c i t y denland functions. ) where i n d i c a t e s t h a t y i ( l + E ~ Y )is now assumed t o equal c o .sumer 11 CSi s u r p l u s i n t h e i t h market. H u l t i p l y i n g both s i d e s of (66) by D i,s and summing o v e r i,s yields: t D = 1 by Engel aggregation and t h e sum of cons er But + i 'i,s i,s i,s s u r p l u s e s i n a l l markets e q u a l s I Wi,sDi,s + (Y - Yo) 5, s This c l a s s of p r o g r a m i n g models is c a l l e d t h e PROLOG models, f o r mathematical PROgramming with LOG- derivative number of simple PROLOG v a r i a n t s have been worked our numerically, beginning w i t h t h e s i m p l e s t v e r s i o n given i n appendix c . A s an e p i r i c a l ~ 7 t e s t , i n each c a s e t h e e q u i l i b r i u m s o l u t i o n is found beforehand w i t 4 a set of simultnneous equations and t h e LP s o l u t i o n is checked a g a i n s t it. Eventually, t h e r e w i l l b e v e r s i o n s which cannot b e expressed i n simdrtaneous L equatioris,' but f o r t h e b a s i c formulktions t h i s check is useful. -- I n t h e r e s u l t s reported below, t h e values of t h e rate-of-c .* of I ; variables areexpressedwithreference to thesimultaneous solution - t h c * - 1 * I * -1 7 Some indicntdon of t h e wider s e t of formulations which a r e p o s s i b l e i n t h e PROLOG framework is given i n i5]. s y s t e c , s o t h e performance of t h e PROLOG model can be r e a d i l y nea ured by s tile d e v i a t i o n from zero of t h e solution's rate- of- change v a r i a b l e b. 5. !PmericaZ Results To i l l u s t r a t e t h e numerical behavior of t h e model, two v e l J i o n s of PROLOG air. presented. PROLOG 1 c o n t a i n s t h r e e s e c t o r s and onc: primary f a c t o r ; PROLOG 2 c o n t a i n s r h e same input- output s t r u c t u r e f o r intermediate goods but includes two primary f a c t o r s of production and hence t w o income I groups, w i t 5 corresponding s e t s of demand functions. It also includes savings and is recursive- dynamic. The extensions of t h e foregoing proofs t o t h e r e c u r s i v e c a s e a r e given i n appendix B, and t h e set of PR0,LOG 1 and 2 equations i n appendix C. For each case, two kinds of s o l u t i o n s have been made. The i n i t i a l s o l u t i o n is a t e s t t o s e e i f PROLOG w i l l -sproduce a s e t of general e q u i l i - I brium i n i t i a l conditions. Subsequent s o l u t i o n s demonstrate t h e 'mpact of i exogenous changes o r changes over t i m e . F3r PROLOG 1 t h e exogenous changes a r e a ) an i n c r e a s e i n t h e given endowment of t h e ( s i n g l e ) resou b) technological change i n t h e production of one of t h e goods. I n t h e c a s e of PROLOG 2', r e c u r s i v e dynamic s o l u t i o n s a e conducted d f o r ten periods. TSere a r e two v a r i a n t s : one without technologjcal change - and one with technGlogica1 change. - i For PROLOG 1 t h e input d a t a r e q ~ i r e da r e a s follows (p rameter !e 4 I values and i n i t i a l conditions' : C n t a for PROLOG1 In Table 1, t h e f i r s t column gives the solution t o PRO t h e foregoing data set. The second cciiunn shows the consequence increased c a p j t a l stock (K = 3300). The third column shows the t with higher productivity i n the production of good C (aCC cha ged from .08333 t o .03333), and the fourth column shows a solution with b increase capital and the technological change. Table 1: NUMERICAL RESULTS OF PROLOG 1 Solution Number ~ Variable 1.1 1.2 1.3 1.4 -.001 QA ,084 .001 .088 ~i 0 .lo4 0 ,103 ~b .001 .I21 .055 .I15 p i .003 .003 .030 .030 p i -003 .003 .027 .027 ~6 .003 .003 -.021 -.G21 ? ,008 .lo8 .044 .I48 XA .001 .081 .004 .087 4 XC .002 .I32 .071 .205 QA 315.954 342.713 316.534 344. L14 - Q B 76.859 84.846 76.823 84.759 QC E 141.542 158.508 148.901 157.694 I Note: A zero entry means zero a t three right- hand dec places of accuracy. The f i r s t i s s u e t o be disccssed i n connection with ~ q b l / e1 is t h e sense i n which t h e LP mole1 is an approximation. There a r e ~ w o a p p r ~ x i n a t i o n si m p l i c i t i n equations (C.4) - (C. 13). One is t h e approximatinn i n h e r e n t i n t h e stepped demand functions. The o t h e r is t h e appro imation x i' introduced by expressing sane r e l a t i o n s i n log- linear form i n s t e a of level. fcm; equations (C. 9) and (C.13) a r e examples. When t h e icg-linear form is invoked, change is defined by a Laspeyres i n a e x ( C . 3 ) , and t h i s t o i n t r o - b I duces a degree of Inexactitude. I The f i r s t s o u r c e of e r r o r , t h a t of t h e denand segmentat on, t u r n s o u t t o be n e g l i g i j l e . For c q l u t i o n s 1.1 t o 1.4, f i f t y segments trkre used f o r each demand curve, and expanding tllose t o 200 segments (over range) produced imperceptible changes i n t h e numerical r e s u l t s . 1 source of e r r o r is s u f f i c i e n t l y important, however, t h a t it is no p o s s i b l e t o expect a l l t h e r e l a t i o n s h i p s i n the model t o hold exactly. A fu l l y exact s p e c i f i c a t i o n produces a n i n f e a s i b l e solution. We a r e still exploring a l t e r n a t i v e treatments of t h e ap t i o n problem, b u t a s of t h i s w r i t i n g t h e s i m p l e s t answer seems t o a/ allow some c o n s t r a i n t s t o h31d within a c e r t a i n e r r o r t o l e r a n c e The Table 1 s o l u t i o n s were conducted w i t h a t o l e r a n c e of .002 ( . 2 $ ) on t h e V Laspeyres change index G u a t i o n s . This allowed t h e r e s u l t s t o de i a t e from * t h e base values by t h e amounts indicated by t h e rate- of- change va i a h l e s 7 i n column 1 of Table 1. - 8/ T h i s procedure can be implemented e a s i l y with cbrr,rnercial T,P packages by e x p l o i t i n g t h e "range" device i n s p e c i f y i ~ gt h e equations. I I n f a c t , c l o s e r a ~ p r o n i o a t i o n sa r e possible. Table 2 stsvs t h e i e i f e c t s 9f s o l v i n g v e r s i o n 1.i repeatedly u r d e r d i f f e r e n t e r r o r t o l e r a n c e v a l u e s . It is evident that the degree of approxination necessa y w i l l 5 depend on t h e n a t u r e of the p a r t i c u l a r model, b u t t h a t very c l o e approxi- mations a r e a t t a i n a b l e . Table 2: SOLUTIONS OF PROLOG 1.1 UNDER TOLENCES ON THE INDEX OF CHANGE Tolerance Value V a r i a b l e .002 ,001 .0005 .~0001 I t I The economics of PROLOG 1 a r e s t r a i g h t f o w a r d . The va1iu;lts c 1.2 t o 1-4 produce t h e expected kinds of movements i n t h e numeri a 1 out- - 1 *4 - comes. Vhder a ten percent i n c r e a s e i n c a p i t a l s t o c k ( i . 2 ) , 4.s~m e * i n c r e a s e g b y ten p e r c e n t (with r e s p e c t t o s o l u t i o r (1.1). R e l a t i e p r i c e s I r remain unchanged, and consunption demands i n c r e a s e by 13% f o r t h e good with an Engel e l a s t i c i t y 1.29 and by 8;: f o r t h e good with an Eng 1 e l a s t i c i t y cf G.30. O u t ~ u tl e v e l s c h a ~ g ecorrespondingly, taking i n t c I account t h e needs f o r inrer-.in?ustry d e l i v e r i e s . Techilological change w s s introduced by means of a2lcwing 2 5.5% n g r e a t e r o a t p u t l e v e l , with t h e sane i n p u t s , i n t h e producrio of g o ~ dC. Tk,e cocseqcences (1.3) a r e a lower r e l a t i v e price of gccd C , higher o u t s a t h an? consumption l e v e l s of good C, znd a somewhat higher inco e l e v e l . 1 S o l u i i o n 1.1; combices 1.2 end 1.3 i n a n almost a d d i t t v e fash'on. PROLOG 2 is a nose s u b s t a n t i a l qlodel, and hence it r e s a l t s are C c o r e i n t e r e s t i n g . it is eifectively a dualistic model: one f a c t o r (labor) is assurced t o be available -:1 inlinitely elasric supply, at 4 f i x e d zloney wage, while t h e o t h e r f a c t o r ' s supply ( c a p i t a l ) is in:l.astic a t any monent, and its p r i c e is determined by t n e economic r e n t s it generates. The ways i n which =ode1 2 d i f f e r s from model 1 nay b e summarized n o r e s y s t e m a t i c a l l y a s follows (see t h e e q u a t i ?s i n i ~ appendix C and t h e d a t a i n appendix D): a ) There a r e two f a c t o r income groups and hence t h e ljcomr group s u b s c r i p t g is introduced for many of the I I I v a r i a b l e s . b) Owing t o t h e d u a l i s t i c s p e c i f i c a t i o n , l n b c r is e f f e c t i v e l y a I a resource from outside t h e system (as lmported goods would be i n a n open-&onomy model); hence its wage c o s t 'S i must be s u b t r a c t e d f ~ 0 i 2the object;'c-ilr:i:-: (,. I T a b l e 3. XLTI-PERIOD SOLUTICIN OF PROLOG 2, UITBOUd TECHNOLOGICAL C W G E I - Period V a r i a b l e 9 1 2 3 4 9 - 0 -- QL .0017 .0607 .0605 .0602 .0601 .0998 ,0592 ~i -.0006 -0779 .0779 .0780 ,0781 .0781 ,0782 Q(5 -.0014 .09C15 .0904 .0903 .0902 .0901 .0894 PA .000 1 .0009 .0008 .0009 .0009 .0009 .0007 .000 ~ f i ,0003 .0008 .0009 8 .0008 .0008 .0006 .000 p i .0003 .0003 ,0007 .000 7 7 .OC08 .0006 ~k .0004 .0771 .0773 .0773 .0773 .0774 .0774 Y ~ W .0006 .0793 .0793 .0794 .0794 .O 95 .0798 X k .0005 .0460 .0452 .0444 .0436 -0429 .0403 'Gw .0036 .0656 .0653 .0650 .0648 .0644 .0640 X ( ~ K -.0006 .lo62 .lo55 .lo46 .lo37 . 1 C 28 .0992 X ~ W -.0039 .0955 .0955 .0956 .0955 .OS55 ,0942 378.0 406.9 437.9 471.4 507.4 547.2 733.6 .0757 .0757 .0758 .0758 .O 59 ,0760 Notes: 1 ) Error t o l e r a n c e set a t .0005. 2) Symbols a r e defined i n appendix C; YRW denote r e a l incomes of owners of and wage e a r n e r s , r e s p e c t i v e l y ; q u a n t i t y consur,~edby good and income .q iney l o occxr i n t t e c a s e of I-n!.le 4 , however, where t t c ' , r ~ i c a i 7.1 7 ..,-?.s z ~ s u c e dto take pl;ce Ln ssctor 1;. h s wou',d b e expected, good p r i c e dec1li;es r e i a ~ i - ~,; t h e p r i c e s o f A and 5, a r d t h e growth of e t i o n of ~ ~ Coisdaccelerate-d. (Consm.ption of good C grows nore r than its c u t p u t 2nd i t s i n f e ~ ~ d i a uses grud aore slowly.) t e 1 :.3,ile numerical ex?zriz.ents cannot be d e f i n i t i v e , these s'mpie cases s u f f i c e t o demonstrate t h e workings of s i m p l i f i e d PROLOG nodels, and i n zilese c a s e s r h e ncdel has performed a s expected. I I Table 4: KJiTI-?E?,IGD SOLUTION OF PROLOG 2, WITH TECHKCLOGICAL CHASGE Period -- - 0 1 I 2 3 4 5 9 I -- QA ' . O O i 7 .0596 .0577 .0584 .0591 .0567 .0633 ~i -.0006 .0844 .0868 .0884 .0900 .0923 .0985 Qi: -.0014 .lo63 .I102 .I121 .I139 .I182 ,1222 PA .0001 ,0148 .0148 .0148 ,0149 .0171 .0170 PB .0003 .0099 .0099 .0100 .0101 .0123 .0125 PC .0003 -.0061 -.0061 -.0061 -.0060 -.0041 -.0401 YRK .0004 .0990 .lo28 .lo42 .lo56 .I101 .I164 YRW .0006 .0808 .0821 ,0836 .0851 .0857 .0912 Xk .0005 .0507 .0519 .0515 .0511 .0513 .0507 X ~ W .0036 .0606 .0617 .0628 .0638 .0565 .0611 xh -.0006 .I462 .I501 ,1503 .I505 .I521 .I529 X ~ W -.0039 .lo35 .lo49 .lo68 .lo86 .I182 .I229 I 378.0 414.3 455.3 500.9 551.8 609.1 $15.5 I/K .0756 .0770 .0786 .0802 .0818 .0834 .0902 Notes: 1) Technological change defined t o be 2% higher output of sector C , from the - same inputs, each year. - ' ? - 2) Symbols have the same meaning as i n -- .i - Table 3. . Q C o ~ p m s a t e dChanges and I n t e p r a b i l i t y Conditions I f we may be p e m i t t e d t h e a r t i f i c e of d i s c u s s i n g a singl!e, aggregate consumer, t h e nodel presented can b e given a more genera i n t e r p r e t a t i o n i n terms o f theory of household behavior by noting t h e Engel aggregation conditions obtained i n (34) imply corn ensate 4 q u a n t i t y changes. Consider i n f a c t t h e u t i l i t y maximization problj. underlying t h e demand f u n c t i o n i n (29) and t h e budget c o n s t r a i n t i{ s u b j e c t t o PIX -< y (A. 2) I where U(X) i n d i c a t e s t h e u t i l i t y function of t h e r e p r e s e n t a t i v e @onsumer. Define a compensated change i n q u a n t i t i e s a s a change where income is i F compensated a s t o keep u t i l i t y constant. For t h e " representative onsumer," we can w r i t e : a u dU = - (X) dX ax = U P'dX . o dy -> P'dX !+ X'dP I ~ A2pendix A v 5 r r e 2 r i a e s i n d i c a t e t r a n s p o s e s as b e f o r e and U is the rn-ginal u t i l i t y 0 ~ f c r t h e Xggrzgate deaand f u n c ~ i o n : The Engel aggregation zondit_icn i n ( 3 1 , t h e r e f o r e , i s svch t h a t i~ guaran-- t e e s t h a t u t i l i c y is he12 c m s t a n t by an a p p r o p r i a t e changes i n p r i l e s . Thus, t h e n c d e l presentzd is equivalent t o naximizing t h e slim of thk z r e a s urtder t h e cornpensatad dmsnd f u n c t i a ~ sand i n t h i s s e n s e overcomes f!le t usuzl l i m i t a t i o n s of c o n s m e r s c r p l u s a n a l y s i s and its dependence o t h e assumption of c o n s t a n t n a r g i n a l u t i l i t y of income. The f a c t t h a t t h e above form of Engel aggregation ensures a a s i z i - z a t i o n of t h e a r e a s under t h e compensated demand f u n c t i o n s is i n s t r n e n t a l C i n overconing a f u r t h e r problen t h a t has been plaguing t h e family o i n;odels of t h e type c ~ n s i d e r e di n t h i s paper: t h e i n t e g r a b i l i t y c0nditior.s r e q u i r e t h a t t h e matrix of f i r s t d e r i v a t i v e s ic t!ie f u n c t i o n (B in our n o t a t i o n ) b e n o t only qcasi-negative I symmetric a s well.- 10/While t h e r e a r e some demand systems f o r wilich h i s * c b n d l t i o n is met, t h e econongtric e s t i m t e s g e n e r a l l y a v a i l a b l e t y ~ d e f l n e a non-symmetric B n a t r i x . When t h i s is tile casc, tl:zn, di;!j.r-tntlc~- 1 t i o n of t h e f i r s t term of t h e maximnnd X ' ( A - 0.5BX) does not vie c! I A - BX b u t A - 0 . 5 R X - 0.5B'X where B # B ' , 30 t h a t h e Kuhn-T: :Xe- E t 1 c o n d i t i ~ n sdo not y i e l d a market equilibrium equation. 101 See [lo]. hppendi# A pag,? 3 e I I n the case where t h e income e f f e c t is e x p l i c i t l y consid r e d , hc-ever, t o t a l d i f f e r e n t i a t i o n of t h e budget c o n s t r a i n t y i e l d s : X'dP + P'dX -< dy I (A.6) and, beca17.se of t h e conipensated change assumed, :-'dP = dy. Equat'c- 1 (A.6) y i e l d s then t h e following subs ici: a r y r e s t r i c tion: I I The Lagrangean of t h e maximization problem can now be formulated a follows: and t h e relevant Kuhn-Tucker conditions y i e l d : = A - BX - B'X + $p + B'5 + p (A*9) It is easy t o see that, a t equilibrim, a necessary and sufficient condition f o r equilibrium ( p r i c e = marginal c o s t ) Is t h a t 5 = X. I n f a c t , t h e n o t a t i o n and w r i t i n g (A.9) a s an e q u a l i t y f o r the optimal vari (A. 10) (A. 11) '* If P = -p d e n 8 ' 5 = B ' X = 4 ' ~ - I as i n t h e c o n s t r a i n t (necessary condi- * - t i o n ) and i f %'C . I = B'X = I$'B-' then P = -p, ( s u f f i c i e n t condidion), x h 2 r e -D is, by another series of Kuhn-Tucker conditions, equal t o rn.-yginal Appendix - - 3. d Recursive Dynenic Versioz 2% rjinple e x t e n s i ~ nof t h e francwork of s e c t i o n 3 i n t h e t e t can x be used to encompass t h e dycenic c a s e of input-outpuc technology (wi h : t L e o n t i e i p r i c i n g ) and Earrod-T)onar growth. (This extension is i l l u s rated . n u n e r i z a l i y by PROLCG 2.) For t h e s a k e of c l a r i t y , w e d e a l w i t h o.?e type of c a p i r a l and t h e r e f o r e col:a?se t!le resource v e c t o r t o a s c a l a r . aggregate n a x i o i z a t i o n ?rob:ex becones now t h e follow!-ng. ~ s u b j e c t . t o : x i $ t 2 1 compensated Engel aggregation 11, resource c o n s t r a i n t - It - S Y t = 0 investment - savings i d e n t i t y - (I - ?i)Qt + Xt+ N I t - l -< 0 commodity balance wh?re b e s i d e s t h e symbols a l r e a d y defined t h e following new n o t a t i o n has been introduced: 1 L , t I = t o t a l investment ( i n money t e r n s ) i n period t t s = n c o n s t a n t marginal propensity t o save - - M = an nxn matrix of Zgput-output c o e f f i c i e n t s N = an nxn diagonal maFrjx of c a p i t a l goods in:)ut-output coef f icpents. - - 111 Investment yields new productive capacity with n ~ r i c - , ~ o r i n c l l a g . The length of t h e g e s t a t i o n l a g can vnricd i f desirf23. h b - e t subscript decotes the t l c e period ar,d the B n a t r i x is assu3ed sy.:.--etric f o r s i m p l i c i t y . 1 Forning t h e Lagrangean from (B. 1 ) - (B. 5 ), w e obtain: I Aside from t h e c o n s t r a i n t s i n (B.3)-(B.5) and non- negativity and komplemen- I I , t a r i t y , t h e Kuhn-Tucker conditions can now be s t a t e d a s follows: ! Equation (E.9) is the only new first- order condition which is no found i n t h e s t a t i c model of s e c t i o n 3 . I n equilibrium P = y f o r the r e a s ns explained t b e f o r e and -p equals t h e n x l v e c t o r of market p r i c e s f o r thb f i n a l goods. I Furthermore, because of equation (B.9), p r i c e s of f i n a l goods wi 1 be equal 4 , I t o marginal c o s t s defined a s d i r e c t c o s t s C , opportunity costs khcennedia t e c o s t s -pH (Leontief p r i c i n g ) . -- I - Equation (B.9) d e f i n e s the condition t h a t has t o bola b t equili- b r i u o f o r an e e f i c i e n t investment a l l o c a t i o n , where t h e r e t u r n on savings (y) equals clie input v a l u e of investment ( A ) plus the output value ( - p x ) . Appeqdix 3 page 3 17 equations (B.i.1 - (B.5j, capital ss i ~ n o b l l ea c r c s s selcrors u i t h i n a period and ( i n c r ~ x e n t a i jc s p i t a i is r c b i l e 'cetwzen periods; re i . e . , t h i s is 2 put ty- clay s2eciification. S e c t o r a l investmefit Euncticns r e q u i r e d , and ;he dynznic lin'kages between periods a r e those i n v e s t e n t E f u n c t i o n s and t h e g e s t a t i o n l a g e f f e c t s f o r new capactty. (The i n v s t z e n t n f u n c t i o n s m y s i n p l y t a k e t h e f o r n of exogenous a l l o c a t i o n s over s e k t o r s i n those c a s e s where p o l i c y is p r e d o n i n ~ n t . ) Since c a p i t a l is fixed i t h e short- run it is no longer n e c e s s a r i l y t r u e t h a t t h e r e t u r n s t o c a p i a1 a r e t equalized a c r o s s s e c t o r s . Appendix C. Equations of PROLOG 1 and 2 I n our numerical simulations, it is p a r t i c l l l a r l y i n t e r e s t i n g t o d s o l v e t h e node1 under p e r t u r b a t i o n s i n which incone v a r i e s away f om t\e .i equilibrium value. For t h i s purpqse, t h e constant e l a s t i c i t y dem nd s p e c i f i c a t i o n appears t o be t h e most appropriate. I n t h i s case, tihe s t a t i c Engel aggregation c o n d i t i o n (55), i n which e l a s t i c i t i e s vary b u t income does n o t e n t e r t h e c o n s t r a i n t , is replaced by a v e r s i o n i n which elast c i t i e s are 1 constant and income may vary: t where t h e a r e parameters which r e p r e s e n t t h e v a l u e of cons p t i o n on ii,s 4, segment s of the demand curve for good i, and ci and qi r spectively, t h e Engel and t h e p r i c e e l a s t i c i t i e s . The parameters $ i , s a r e qefined e x a n t e s o t h a t 121 o r t h e t o t a l v a l u e of consumer expenditures on good i.- I t Another transformation f o r t h e numerical v e r s i o n is t o w r i t e non Y - p o s i t i v e p r o f i t conditions (as i n equation (6)), s o t h a t d u a l v a l e s of 't 5 - resources ( f a c t o r ~ r i c e s )a r e defined i n t h e primal. i?r,d a t h i r d * transformation * .I m 121 Naturally a l l t h e zquations based cn i i n e a r demand function can b~ a l s o redefined i n t e r n s of a constant e l a s t i c i t y one. numerical simulations presented, however, s h i f t from one £0 mulntioll t o t h e o t h e r d i d n o t appear t o make any d i f f e r e n c e . I Appendix C page 2 I is t o define log- derivative (percentage-rate-qf-change) v a r i 5 b l e s s n u l t i p l i c a t i v e r e l a t i o n s m o n g s r i m a l v a r i a b l e s can be used. This 13I f o r y a l r e a d y , but i n general:- Thus it is a Laspeyres index of change, based on I 3 one- period l a g The above t r a n s f o r n a t i o n s a r e used only f o r purpose of r e t r i e v i n g dual i. a r i a b l e s h d i r e c t l y i n the primal and do not change a t a l l t h e s t r u c t u r e and t e p r o p e r t i e s of t h e nodel. Now we a r e i n a p o s i t i o n t o w r i t e the f u l l LP model siinplest v e r s i o n which was implemented numerically: PROLOG 1 equations - 14 I Max 1 D good i, (C. 4) i s i,s i,s segment s s u b j e c t t o : [Engel aggregation], - [cornrnc?dity balances] - i- - 131 See 161. - 141 Symbols a r e defined a f t e r the PROLOG 2 eqc a t i o n s . bppcndix C Page 3 [resource constraint] [convex combination constraint] [definition of demand prices] (C. 10) [definition of consumption by good] [non- positive p r o f i t s ] [factor income d e f i n i t i o n ] [ r e l a t i o n between nominal income (Y) and LI r e a l income (YR) ] , s (plus d e f i n i t i o n a l equations of the Lype (C.3)- f o r the r a t e v a r i i b l e s , 6 and ). i i .;iczarks 02 t h e LP e ~ c a c L o n s a ) The v a r i a b l e s 3 a r e demand p a r t i t i m i n g variab'es 1,s so t h a t i f Y = C j then 0 < D - i,s -< I . Each D 1 i,. r e F r s s e n t s an ( a r b i t r a r i l y small) segment s of the denand curve f o r good i. Relevant ranges of t h e I demand functions a r e e s t a b l i s h e d a ? r i o r i , and then t h parameters w .,5 znd y. a r e computed from t h e 7- 1,s functions Wi and Ei (marginal revenue). I Some of t h e equations a r e redundant. For example, (c, 2 ) coxld Se s u b s t i t u t e d i n t o (C. 5); and ((2.9) and (C.lO) could be subst3tuted o u t of t h e model. However, p r e s e n t a t i o ~ i n t h i s form has t h e advantages sf c l a r i t y and ease of manipulation (when many versions a r e being made). C ) Equation (12.8) requires t h e s o l u t i o n t o be on t h e demadd function. The second term on t h e l e f t r o t a t e s t h e I p r i c e - e l a s t i c demand function rightward i n accordance with t h e change i n r e a l income. The parameter t h e Engel e l a s t i c i t y . i d) Eqclation (C. 9) d e f i n e s demand p r i c e s ; t h e e a r e ii own-price e l a s t i c i t i e s of demand. These p r i c e s a r e 1 z 1 equal t o t h e dual. v d r i a b l e s of t h e connodity balances, 'i but theygnust be defined i n t h e prirlal problem a l s o E I - I i n order to enforce t h e -emand r o t a t i o n on t h e b a s i s I of r e a l income changes. e) PK denotes the price of the single resource. In t h i s simple model, with only one resource, equation (C. 7) is certain t o hold a s an equality and there- fore (C. 2)can take the above form. I n more complex versions, where thee may be slack resources, (C.12) is replaced by a different expression. f ) I n equation (C.13), the ai are budget shares. I n general, the linear programming version includes many V ariables i i lthe p r h a l which a l s o appear i n tho dual, such a s factor and pro uct d prices and incomes. This redundancy is a necessary feature which a the imposition of certain primal relationships which span some of t variables . PROLOG 2 E q u a t i ~ n s good i segpent s group g s u b j e c t t o .I Ei,gi,s,gr Di , s , z - (l-sg)Yg = 0 (C. 15) i,s [Engel a g g r e g a ~ l o nf o r each group g] (C. 16) [ c o m o d i t y balances] (C. 1 7 ) [employment d e f i n i t i o n ] (C. 1-8) [ c a p i t a l cons t r a i n t 1 [convex combination c o n s t r a i n t ] (C. 20) - [ d e f i n i t i o n of denand p r i c e s , f o r any one income gro * - (C. 21) [ d e f i n i t i o n of consumption by goqd j Appendix c p a g e 7 [Labor income definition] (C. 24) [savings definitions] [asset accumulation] (C. 26) [aggregate savings] [savings-investment identity] ~ (C. 2 8 ) [price of investment good] (C. 29) [income deflatorsli -- * II yii + P? - ? = 0 (C.30) g 6 g [real income definition] [plus equations defining the log-derivative variables]. ; Appendix C page 8 I Syrnbols used i n PROLOG 2 - S e t s commodities incoxe groups s = 1, ... , p segments i n den nd f u n c t i o n s t , ~= 1, ... , T tioe periods (a C s u b s c r i p t w l a b o r income g r up ( g=w) Convent i o n s 1 h ?) A d o t t e d v a r i a b l e ( ) i n d i c a t e s its log- c ange, o r percentage r a t e of change. 2 ) Parameters a r e denoted by Greek l e t t e r s o r c a s e l e t t e r s . 3) Variables a r e denoted by c a p i t a l l e t t e r s ; v a r i a b l e s have a b a r over the~u. D e f i n i t i o n s w cumulative a r e a under t h e demand f u n c t i o n i , s , g D a c t i v i t y l e v e l s f o r the segments of t h e i , s , g cumulative demand a r e a f u n c t i o n s e e equation (72) i n t h e t e x t - TrJ wage r a t e N employment s ~ -- g -savings r a t e out of t o t a l group income1 w *- I Y - t o t a l group income & r cumulative area under t h e marginal r e v nue i , s , g f u n c t i o n i I-l~pnndixC I page 9 C a input- output c o e f f i c i e n t f o r i n t e r i n d u s t r y s a l s i j gross output i s h a r e of good i (as a capital good) in bi c a p i t a l formation I t o t a l fixed investment 6 q u a n t i t y consumed along t h e demand functiou i , s , g l a b o r input- output c o e f f i c i e n t k c a p i t a l input- output c o e f f i c i e n t j - K i n i t i a l endowment of c a p i t a l s t o c k E Engel e l a s t i c i t y 1 i , g YR r e a l income p r i c e Pi own-price e l a s t i c i t y quantity consumed PK r a t e of r e t u r n t o c a p i t a l YW t o t a l labor hcome (Yg , g=w), i . e . , wage i n I t dividends accruing t o labor. S savizgs by income group g FA a s s e t holdings i n current prices g, t - t o t a l savings P I p r i c e of the aggregate c a p i t a l g c ~ d income d e f l a t o r budget shares Appendix D. Data for PROLOG 2 A. Parameters B. I n i t i a l conditions [Qj -- I p j ,o 1 = Ivj,ol = '7 0 1 i 7 7 L A I Dixon, l e t e r , 7i-.e Theory of J o i n-- t Piaxirnizati.on, Korth--Ha land T u b l i s h i ~ ~C'onpany, P a s t e r d a n , 1975. g [ 21 Duloy, J . K . , and R.D. Xorton, " Prices and Incomes i n i i n d a r Tro- g r a r n i ~ gP?odels, I 1fiinerican J o u r n a l of Agricultural, Economics, v o l . 57, !Joveir.Ser, 1975, pp. 591-600. i [ 31 Girsburgh, V i c t o r , and J e s n Kaelbroeck, "A General Equil Erium Yodel of Korld Trade, P a r t I, " Cowles Foundation 3 i s c u s i o n Paper So. 412, ::r;ven;ber 18, 1975. [41 i-Iazell, P .E.R., and 2 . L . 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