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TRANSCRIPT
Development R e s e a r c h C e n t e r
D i s c u s s i o n P a p e r s ,
No. 24
I 1
A COMPUTABLE CLASS OF GENERAL EQUILIBRIUM MODELS I 1
b Y I
Roger D. Nor ton and P a s q u a l e L. S c a n d i z z o I
I
I I
May 1 9 7 7 I
I I
1 I I
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- I D i s c u s s i o n P a p e r s a r c p r e l i m i n a r y m a t e r i a l s c i r c u 1 a t e . a t o s t i r n u i a t e 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 n t o D i s c u s s i o n P a p e r s s h o u l d b e c l e a r e d w i 3 t h e 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 o f t h ~ & papefs . T h e p a p e r s e : ->ress t!le v i e w s o f t h e n u t h o r s a d s h o u l d n o t b e interpreted t o r c f l c c t ' t h o s e o f t h e ~ o ; l c { ]jack
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Table of Conten ts
1. I n t r o d u c t i o n
2 . A S t a t i c Genera l Equi l ib r ium Model
3. A Quadra t i c Programming Formulat ion
4. A L inea r Prograrmnirig Approximat i o n
5 . Numerical R e s u l t s
Appendices:
A. Compensated Changes and I n t e g r a b i l i t y Condi t ions
B. A Recu r s ive Dynamic Version
C . Equat ions of PROLOG 1 and 2
D. Data f o r PROLOG 2
T h i s p a p e r d e s c r i b e s n s in ip le anrl power fu l p r o c e d u r e foC o b t a i n i n g
g e n e r a l 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 p r o c e u r e is d c a p a b l e of r s t e n s i o n t o c a s e s where m a r k e t s a r e s u b j e c t t o m u l t i p 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 i s u s a b l e i n t h e c o n t e x t o f p r i e- l endogenous models f o r development p l a n n i n g .
I n v e r y r e c e n t y e a r s , t h e r e h a s been a r e m a r k a b l e amounk of pro-
g r e s s r e g a r d i n g methods f o r s o l v i n g n u m e r i c a l 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 ~ : b s e q u c n t l y have a t t e m p t e d t o improve on h i s a l g o r i t h m [ l , 31. " P
t i le a l g o r i t h c x r e n n i n s o n w h a t e x p e n s i v e and d i f f i c o l t co u s e , and t h e r e f o r e
:iley c a n r o t g e n e r a l l y b e a p p l i e d t o l a r g e- s c a l e n o n l i n e a r models.
- L h e procedure p r e s e n t e t l h e r e a t t e m p t s t o c o n t r i b u t e i n h e [allow- I i n s t h r e e x a p s : a ) t o p r o v i d c g r e n t c r c o s t - e f f i c i e n c y f o r l a r g e - c a l e I r . o d e l s , b ) t o p r o v i d e 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
7 I i
not have t o be r e - p r o g r a m e d f o r d i f t c r e n t models, and c ) a s men i o n e d , t o i; -
a ) :'he ; l u ~ : ~ o r s Arc merr.!~cr:; o l t i ~ c I!orld r,ani\ 's R e v e l o ; ~ ~ ~ e n t Kcsr.. c'i 5
C c ~ ~ t e r . ''!icy wish to tliank tile l z r g c number o f p e o p l e wl;o ha e .c
rri,ltlt: i ; e : p f ~ ~ l cori1n:ents ,it ~ : < i r i o ~ i s s t ( i g e s i n t h e development u f tlri:;
I
I ) r : .1 , i3i s s c h o ! ~ , :i. C ; ~ n d l c r , . I . I lc i lo)~, P . H a z c l l , K . T n n ~ \ l , ,
(:. ~.l::cti, is. I v s y , 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 ach ieved by c a s t i r l g t h e g e n e r a l
j r i u a p r c b l e r , i n l i c e a r ~ r o g r z c i ~ i n g E o r c a t , s o t h a t t h e c o ~ p u t a t l o r ~ a l
.zovcr of t h e s i z p i e x s o l ! : t i m a i ~ c r i t h ~ a (and more r e c e n t l i n e a r p r grarming n c o d e s ) c a n b e e q l o i t e d . ?:like t h e Ginsburgh- '{aelbroeck arid 3 i x o
11, 31 , which u t i l l z e i t e r a t i - . , e s e q u e r ~ c e s of i i i l e a r programming s o
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 1 r i z a t i o n
1 / t e c h n i q u e s a r e enFioyed f c r r . r :n l inear i t f es . -
The l j n e a r p rogrs r i z ing framework a l s o i s p a r t i c u l a r l y a m
t o e x t e n s i o n s t o t h e non- genera l e q u i l i b r i u m c a s e s . 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 unequa l n u n b e r i of
v a r i a b l e s and c o n s t r a i n t s i s h e l p f u l i n - t h i s r e s p e c t . I t i s f r e q u n t l y 4 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 b e h a v i o r a l c o n s t r a i n t s c o - d x i s t
cx a n t e , i n o v e r l a p p i n g ways, w i t h o u t i t b e i n g known beforehand r~;7 ch ones i w i l l b e e f f e c t i v e .
Xost of t h e remainder o f t h i s p a p e r i s dcvo tcd Lo
t o r e p r e s e n t t h e r e l ~ t i o n s h i p s of a g e n e r a l equjpl ibr iurn *
n a t i c a l p r o g r a m i n g , a l o n g w i ~ h p s o o i s of t h c d e s i r r . d
models arc. r i i ~ c u s s c c ? I n s c c t i e n s g - 4 , 2nd exLensioiis i o ;: ~ - c c ~ ~ r : ; i v e ' . i ---____--- _ _
? / Some o f t h c newer q u a d r a t i c p m g r n n ~ l i n g r:cdcas also ;!]:pear t o ! , e - p o w e r f u l ; i f t!le r ~ o d c l wcrc m t too I n r y ; ~ , ;!ncl i F t i l e cocic?
C
dyir;:r::ic
c l u i ~ c i ! c r ~ - ~ i t t ~ ~ d
c;untir;tt i rorr.s in t l : ~ cj2.y t-,iinL se t , tl!cx ;i;~prc~.?cl: !'f t h i s ps:-Icr zoul t i [)P in~,]i2:;pnte<! d i j -prt ly v l t i i r i ~ r r i t i c r ~ r 7 ~ : ; r e . . d C J ~ 1 l i n c n r i z ing fo t - I.P. I ~
v e r s i o n d r e g i v e n i n ail appendix . Nulnerical examples of how t h e p r o c e d u r e
I w < ) r k s a r c p r e s e n t e d i n s e c t i o n 5. i ~
2. A S5ctic General ,?pi ZibK-m ?.bdeZ
I n t h i s s e c t i o n i t i s shown, i n t h e c o n t e x t of a s i m p l e -
model , t h a t t h e r e e x i s t s a maximiza t ion problem whose s o l u t i o n i s t/he
g e n e r a l e q u i l i b r i u m s o l u t i o n i n p r i c e s , q u a n t i t i e s , and incomes. of c o u r s e ,
i t may w e l l b e p o s s i b l e t o a t t a i n t h e same s o l u t i o n v i a non- optimi
methods. But , i n p r i n c i p l e , an o p t i m i z a t i o n f o r m u l a t i o n a l l o w s t h
t o b e e x t e n d e d w i t h s p e c i f i c a t i o n s which c a n n o t b e hand led i n a s i u l t a n e o u s -
e q u a t i o n framework, a s n o t e d above. Somewhat more s p e c i f i c a l l y : 1 ) a d d i t i o n a l c o n s t r a i n t s c a n b e added t o r e p r e s e n t , s a y , a s i t u a t i o n
m u l t i p l e p r i c e f l o o r s (and c e i l i n g s ) i n which i t i s n o t known e x a
of t h e c o n s t r a i n t s w i l l b e b i n d i n g ; i i ) e x p l i c i t a l t e r n a t i v e t e c h
( i m p l i c i t s u p p l y r e s p o n s e s t r u c t u r e s ) c a n b e c o n s i d e r e d , i n t h e a c i v i t y i a n a l y s i s manner; and i i i ) m o a l f i c a t i o n s t o p rof i t- maximiz ing beh v i o r 1 r u l e s , s u c h a s r i s k a v e r s i o n 141, c a n b e i n t r o d u c e d r e a d i l y .
We s t a r t w i t h a f o r m u l a t i o n which is n o t e a s i l y computab 1 e b u t is
p e r h a p s t h e s i m p l e s t i n c o n c e p t u a l t e r m s , and then we move i n s t a g i!
f o r m u l a t i o n s which a r e more p r a c t i c a l f o r c o m p u t a t i o n a l p u r p o s e s .
Con:;idcr n marke t w i t h t h e f o l l o w i n g s t r u c t u r e z .r
( 3 ) - t h e i t h consrlner behaves i n a c c o r d a n c e w i t h a n j l g g r e g a t e demand C
i u n c t i o n f o f t h e t y p e :
A
C
C
.r(
U E" 3
V]
V]
K!
h
U
*E! n
a, z 0
U
al
I-i
P
K!
U
4
3
b? a,
ki
'd
3
0
a,
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r-l
C:
0 3
0
ri
al P
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0
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Ll a
a, s
k
Theorem 1: S o l u t i o n o f r h e problem (4 ) - (8 ) y i e l d s a c o m p e t i t i e market V e q u i l i b r i u m . 1
Proof : 31
Form t h e Lagrangean from (f+) - ( 8 ) : -
Max L := l " I f i (P , - AIIb i + P'I P y y i , A , Q , ~ , 5 , ~ S T i
y i ) i i
v e c t o r y h a s r e s o u r c e dimension 1 , . r. I n a p u r e e x c h s ge economy,
n = r.
The Kuhn--Tucker c o n d i t i o n s f o r t h e s o l u t i o n of (9 ) a r e : " where t h e v e c t o r s 6 and p have c o m o d i t y d imension 1 , .
3 / I n forming t h e Lagrangean, we adopt: t h e conver j t ion o f g i v i n - ( n e g a t i v e s i g n s ) t o t h e i n e q u a l i t y ( 2 ) const r a i n t s r e f e r r p o s i t i v e ( n e g a t i v e ) v a r i a b l e s i n t h e maximand.
, n and t h e
By t h e complementary s l a c k n e s s and n o n- n e g a t i v i t y cond t i o n s 4 I
[ e q r a t i o n s (15 ) and ( 1 9 ) ] , a l l i n e q u a l i t i e s i n (10) - (17 ) h o l d a s
e q u a l i t i e s f o r non- ze ro a c t i v i t i e s . Abus ing t h e n o t a t i o n , l e t tde same 8 t
symbo l s now i n d i c a t e t h e a c t i v i t i e s , and t h e i r c o r r e s p o n d i n g mat
and v e c t o r s , i n t h e o p t i m a l b a s i s . Yrom a b a s i c P r o p e r t y o f anGe - 4 /: 1
m u l t i p l i e r s ? - t h e f o l l o w i n g r e s u l t s c a n b e i m m e d i a t e l y s t a r e d f o r primal
au . * = - a u a n d , by d e f i n i t i o n , A = - -- === .I* = -),
a PJi ( 2 0 ) a l b i i i
41 S e e L a n c a s t e r [ 5 1, p . 50. -
- and d u a l v n r a b l e s a t t h e optimum.
- - - E .
- - 3 (D' A )
n o t c l e a r from t h e c o n t e x t s .
With t h e above r e s u l t s , w e c a n immediate ly show t h a t n g e l and E '>
Cournot a g g r e g a t i o n c o n d i t i o n s h o l d , f o r t h e o p t i m a l a g g r e g a t e 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 t h d demand
funct ions ,? ' and r e g a r d l e s s of t h e u n d e r l y i n g i n d i v i d u a l u t i l i t J maximiea-
t i o n p r o c e s s . From ( 2 1 ) , (11) and ( 2 3 ) , i n f a c t , w e o b t a i n :
a f i
a f i p -- = - = 1 f o r a l l yi > 0
3yi ayi
and f rom ( 2 1 ) , ( l o ) , ( 2 2 ) and (23)
s
Engel a g g r e g a t i o n
3f a f i If 1 . ( p , y i ) + F ' = p'! ap + C = O i
1 (25) 1'.
'5 5
f o r a l l P . > 0 J - - - Cournor a g g r e g a t i o n
B * C -
- - 1 / S u i t a b l e convexity assumptions, liowever , w i l l liave to b e met .-
t o ensure c o n v e x i t y o f the constraint s e t .
l
i t i s now easy t o .i:~o.: ~'!~^lt tile j\.xi~n-Tucker c o n d i t i o n s j z p l y 3
I c o ~ ~ p c t i t l r s m r k e t e q u i l i b r i : ~ . Three p r i n c i p a l c h a r a c t e r F s t Lcs
c o n p c t i t i v e e q u i i i b r i u r n r'oLlow i i r c c t l y f r c x 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 l b a s i s as f o l l o w s :
o r t o t a l o u t p u t = t o t a l deaand = d u a l v a l u e o f z e r o e x c e s s p r o f i t s
o r p r i c e = o p p o r t u n i t y c o s t = - d u a l v a l u e of commodity b a l a n c e co s t r a i n t ; I A = - Y
o r r e s o u r c e c o s t = - d u a l v a l u e of r e s o u r c e c o n s t r a i n t .
T h i s conc ludes t h e p r o o f . Equa t ion (26) is t h e market c . ."
c o q d i t i o n which i s r e q u i r e d of e q u i l i b r i u m s o l u t i o n s . Equa t ion (2
t h a t p r i c e e q u a l s m a r g i n a l c o s t , a n o t h e r 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 . Toge ther they i n p l y t l l a ~ t 3 t a 1 f a c t o r ren lunera t i an e q u a l s t h e
T\;ese c o r i d i t i o n s a r e the. ones normal ly invoked when ;i co r c t i t i v e .P
v a l ~ e o f ( f i n a l ) o u t p u t . Equa t ion (27) a l s o i m p l i e s t h a t a g iven L
pw p r ~ c e i s t h e s a n e f o r a l l groLps of consumers.
e q u i l i b r i u m 1s b e i n g c l . a r n c t e r i 7 c d . We have chosen t o show t h a L ($) and -
J - i o d u c t ' s . I
(25 ) a r e a l s o s n t i m f i c d , i . e . , t i l i l ~ i n t h e a g g r e g a t e ro;lsurncrs i n +is rn
model b e h a v e .iccording t o tenet: ; o r deaond t h e o r y . 11 - ; i l l brc~i :e J.lypnrc.nt.
s u b s c q u c n t l y t h a t t h i s l a s t propclr ty i s ilnyor' ?:.t t o ti7rb c lcf in i t i r l i c : t i i r x
more c o n p u t n b l c v e r s i o n s of :he model. I
r - - . - - . - . . . 7 '
, ! ;\7:1.12 ' farcn' ; fLrxula t j .c . r , :lac! irLccr3e p redr - t e 5 , I ~~~r.~ y : : y c r j
+-. . Lr,e i-.; t ~ a l per-~d (;.22 : l :~ :eE~re exogenous i n the- s t a t i c f ~ r x u l
o u r 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 I- t h e s z t of endcgenous v a r i a b l e s . Second, t h e problem f o r m u l a t i o i s o f n t h e p r i n a l - d u a l t y p e , s l n c e p r i c e and q u a n c i t y v a r i a b l e s a p p e a r imul t , ln- I e o u s l y a s p r i m a l and d u a l v a r i a b l e s . T h i r d , w i t h o u t c o n s t r a i n t 6 ) , t h e k s o l u t i o n would c o r r e s p o n d t o t h e m o n o p o l i s t s ' e q u i l i b r i u m . Four
o p t i m a l s o l u t i o n i s a s s o c i a t e d w i t h a z e r o v a l u e o f t h e o b j e c t i v
The model p r e s e n t e d i n s e c t 2 o n 2 s u g g e s t s t h a t t h e Co r n o t and i
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
demand f u n c t i o n s i s c o t r e q u i r e d .
Engel a g g r e g a t i o n c o n d i t i o n s can b e e x p l o i t e d t o make endogenou
p r o c e s s o r income f o r m a t i o n i n t h c corl lputat ion o f c o m p e t i t i v e e
o f t h e
th rough m a t h e m a t i c a l programming models. Although t h e , I r
o f t h e model p r e s e n t e d 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 o f t h e i n d i v i d u a l
wc now c o n s i d o r some e p p l i c n t i o n s t o t h e c a s e o f l i n e a r and con t a n t 5 e l a s t i c i t i e s d c m n d f u n c t i o n s . To f n c i l i t t e c o m p u t a t i o n s , t h e income 3
I
7 ~ ~ 1 r i a b l e no l o n g e r e n t e r s t h r nnximancl e x p l i c i t l y i n t!ie f o l l o w ng v e r s i o n s
o f t h e model. I
C o n ~ i a e s the i o i l o i ; i n g n .od i f l - : a t i cn of t h e rnarkst s t r u c t u r e
d e f i n e d i n ( a j - ( c ) :
(a') Conscse rs behave i n a c c o r d a n c e w i ~ h arl a g g r e g a t e i v e r s e b deciand f u n c t i o n o f t h e t y p e :
where A > 0 a n d B is a n c n- s i n g u l a r symmetric
of demand c o e f f i c i e n t s .
a t r i x
( b ' ) P r o d u c e r s maximize p r o f i t s s u b j e c t t o a n a g g r e g a t e r e s o u r c e
c o n s t r a i n t :
( 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 C produced:
and t h e r e is f r e e d i s p o s a l of o v c r p r o 2 u c t i o n .
Theorem 2 : S o l u t i o n of t h e f o l l o w i n g a g g r e g a t e maximizing y i e l d s
r L t
a s t a t i c c o m p e t i t i v e marke t e q u i l i b r i u m :
Max X' (A - 0.5 BX) - O ' Q
x, Y
where the C' e r e t h e c o s t s of a l l pr imary fac ta i r s 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 s u p p l y , I
scilj ect- to X ' O - 1 (34)
wllich is Engel a g g r e g a t i o n , ob t a iqed by d i f f e r e n t i a t i n g (30)
t 3 incone (and assuming t h a t consumers a r e on t h e i r budget l i n e b ) ; and I
s u b j e c t a l s o t o :
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
DQ - b .: 0 - Resource c o n s t r a i n t s
x - ( 2 - < 0 Commodity ba l ances
ducer s u r p l u s over a l l ; r o d u c t markets . The rcotivatioll behind h e i n t r o - t
(35)
(36)
dllctj.on of (34) is n o t t h e assumptior of u t i l i t y maximization b i n d i v i d u a l Y consumers, bu t 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
agg rega t ion must hold i n t h e agg rega t e f o r t h e e q u i l i b r i u m s o l u ion. S ince
- . t
we a r e d e a l i n g w i t h agg rega t e marke ts , t h e p a r t i a l d e r i v a t i v e
de f ined .
Proof : Form t h e Lagrangean from (33) - (36) :
hIa x 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 s o l u t i o n of (37) a l e :
2 L - = X - Q 80
From (38) and (39) and t h e c c m p l e m e n t a r i t y s l a c k n e s s c o n d i t i o n s
(16) i t i s 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 a b u s e 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 n o n z e r o q u a n t i t i e s o f t h e a c t i v i t y l e v e l s i n t h e o p t i m a l
s o l u t i o n ( a n d t h e i r c o r r e s p o n d i n g p a r a m e t e r m a t r i c e s ) . E q u a t i o n (45) i s t h e
marke t 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 = m a r g i n a l 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 p r o v e now t h e f o l l o w i n g :
C o r o l l a r y : Under t h e c o n d i t i o n s (38) - (43) p is e q u a l t o t o t a l
r emunera t ion of p r o d u c t i o n f a c t o r s .
P r o o f : Because o f complementary s l a c k n e s s and e q u a c i o n (41)
i n t h e s o l u t i o n , X = Q ; and t h u s p r e m u l t i ~ l y i n g by . L
7 . ! A we o b t a i n : -
. . ~ . ~ t >cCc-sz of t5e d?mr:.r,< e q ~ ; i i o : i s in ( 2 3 ) 2 2 2 t h e Engel ::~;C~CF;F.~:CF. ~ C Z L L-
I I
p = X ' D ' X + X ' C
-
X' (A - BX) = 0 , s o t h a t ( 4 6 ) becomes
I t h a t is, p is e q u a l t o t o t a l c o s t (X'C) + t o t a l imputed v a l u s o f s c ? r c e
f a c t o r s , o r t h e t o t a l r e m u n e r a t i o n o f p r o d u c t i v e f a c t o r s . Tha t t is magni tude n ( 4 9 )
h a s t o b e e q u a l t o t o t a l e x p e n d i t u r e 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 f o l l o w s .
Assume p r i c e s a r e e q u a l t o m a r g i n a l c o s t s :
s y = x ' D ' X + X ' C
A - BX + $y = D'X + C I
Hence t h e d u a l a n a l o g u e o f ( 3 4 ) i s t h e c o n s t r a i n t t h a t ~ x p e n d i t u r c a n n o t * I-
( 5 0 )
exceed income ; i n e q u i l i b r i u m t h i s r e l a t i o n becomes a n e q u a l i t y . ~
P r e m u l t i p l y i n g ( 5 0 ) by X and a p p l y i n g Engel a g g r e g a t i
grarnming, s e v e r a l t r a n s f o r m a t i o n s
.;. c l Linear Progrcurmin!] /i?proxima t i o n - I n o r d e r t o w r i t e th! f o r ~ g o i n g problem i n t e r m s of l i n e a r
a r e r e q u i r e d . The f i r s t s t e p i
pro-
demand f u n c t i o n : 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 ? u n c t i o n . F o l l o w i n g Duloy a n d N o r ~ o n [ 2 ] ,
where 12 i
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) - ( 3 6 ) i n terns o f a
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 o f t h e a r e a u n d e r
r e p r e s e n t s t h e a r e a u n d e r t h e denand f u n c t i o n o r I
s t e p w i s e
t h e
t h e i t h good; I 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 segmen t s of t h t f u n c t i o n wi ;
a r e t h e v a l u e s o f Wi c o r r e s p o n d i n g t o D i, s 1
a i , b i Qi a r e , a s b e f o r e , p a r a m e t e r s of a l i n e a r ( o r
9 I 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 o f t h e t y p i
f o r s i m p l i c i t y , w e a s s e C 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 i s a n i n i t i a l v a l u e of income. 0
Xote t h a t t h e a r e a u n d e r demand f u n c t i o n W c a n be combutcd f o r . . P
i L
o n l y f l ~ n c t i o n a l s 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 d e g r e e of a p p r o x i m a t i o n
' r - e x p o s i t i o n o n l y t o s i m p l . i f y t h e argument . I depend ing o n l y on t h e number o f s t e p s . The l i n e a r f u n c t i o n i s uscd - -
B u i l i ing on t h e a p p r o a c h in [ Z ] , w e c a n now s p e c l t y I
i n t h c
P-- .neor?n 3: So lu t ion of t h e fo i l owing LP node1 y i e l d s a compe t i t i ve I
market equ i l i b r ium: ~
subject t o
D 1 'i,s i ,s - Qi < 0, i m 1, ... , n -
s Commodity ba l an
c o n s t r a i ts n 1 Di,s
. - 1 i = 1, ... ,b ( 5 8 ) s Ei 'Convex combinat on
c o n s t r a i n t
where t h e 8 r e p r e s e n t t h e q u a n t i t i e s s o l d a t t h e l i m i t of eac i , s . Y - Y o
of t h e f u n c t i o n Wi, E is Engel e l a s t i c i t y and y = i
Yo
index o f income change. ~ Proof: Forming t h e Lagrangean f o r t h e problem above, w e ob t a in :
t h e n b e a n a l y z e d a s f o l l o w s . 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
A l s o , 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
Analogous s t a t e m e n t s h o l d 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 . b I n p a r t i c u l a r ,
e i t h e r Pi = 0
o r
and
commodity b a l a n c e c o n s t r a i n t "i
e q u a l s t h e oppor t~. . : i ty c o s t of tAe
f a c t o r s employed i n t h e p r o d u c t i o n o f t h e i t h commodity. I t fs c 1 e . 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 e q u a 1 , a t 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 )I = Y m Y E u f f i i e n c y )
0 I i . e . demand e q u a l s s u p p l y i n t h e i t t market and t h e shadow p r i c e of
*
and m u l t i p l y i n g (60)
t h e i t , h ,
and summing o v e r s , w e can wr i te :
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) . Equat ion (63) t h e n c h a r a c t e
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 . / C l e a r l y , t h i s i m p l i e s compe t i t i ve equ i l i b r ium, s i n c e , s u
t h i s r e s u l t i n t o (60) aga in , we can d e r i v e t h e fo l lowing exp re s s ion
o r , by d i v i d i n g bo th s i d e s by 8 i , s
revenue f o r t h e i , s consumer s u r p l u s i n t h e a r e a under t h e a c t i v i t y i t h market f u n c t i o n f o r
a c t i v i t y
supply p r i c e demand p r i c e
demand t h e i , s
I n o r d e r t o prove n e c e s s i t y , on t h e o t h e r hand, assume that (64)
ho lds w i t h p i n l i e u of y yo : ~ 6 / Note t h a t yi equa l s consumer s u r p l u s - CSi a t i n i t i a l incomt? yo .
I n d i c a t i n g sgch a v a l u e of t h e s u r p l u s a s CSio ; i n f a c t , w .? i
' "io = s
. -
write (63) a 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 w i t h
mateiy t h e same a s t h e Engel e i a s t i c i t y . ( I t i s e x a c t l y t h e s a
c o n s t a n t e l a s t i c i t y denland func t ions . )
where CS i
i n d i c a t e s t h a t y i ( l + E ~ Y ) is now assumed t o e q u a l c o .sumer
s u r p l u s i n t h e i t h marke t . H u l t i p l y i n g b o t h s i d e s o f (66) by D
11 i , s and
summing o v e r i , s y i e l d s :
D But + i 'i,s i,s
= 1 by Engel a g g r e g a t i o n and t h e sum o f cons er i , s t
s u r p l u s e s i n a l l m a r k e t s e q u a l s I Wi,s D i , s + (Y - Yo) 5, s
T h i s c l a s s o f p r o g r a m i n g models i s c a l l e d t h e PROLOG
number o f s i m p l e PROLOG v a r i a n t s have been worked our n u m e r i c a l l y ,
models , f o r ma themat ica l PROgramming w i t h LOG-derivative
b e g i n n i n g w i t h t h e s i m p l e s t v e r s i o n g i v e n i n append ix 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 i s found be forehand w i t 4 a set
o f s i m u l t n n e o u s e q u a t i o n s and t h e LP s o l u t i o n i s checked a g a i n s t i t .
E v e n t u a l l y , t h e r e w i l l b e v e r s i o n s which c a n n o t b e e x p r e s s e d i n s imdr taneous
.* I ; v a r i a b l e s a r e e x p r e s s e d w i t h r e f e r e n c e t o t h e s i m u l t a n e o u s s o l u t i o n of t h c
equat ior is , ' b u t f o r t h e b a s i c f o r m u l k t i o n s t h i s check is u s e f u l .
- * - 1 * I
* - 7 1 Some i n d i c n t d o n o f t h e w i d e r s e t o f f o r m u l a t i o n s which a r e p o s s i b l e i n t h e PROLOG framework i s g i v e n i n i5].
L
I n t h e r e s u l t s r e p o r t e d below, t h e v a l u e s of t h e r a t e - o f - c - -
s y s t e c , s o t h e performance of t h e PROLOG model c an be r e a d i l y nea u red by s t i l e d e v i a t i o n from z e r o o f t h e solution's rate- of- change v a r i a b l e . b 5. !PmericaZ Results
f a c t o r ; PROLOG 2 c o n t a i n s r h e same inpu t- ou tpu t s t r u c t u r e f o r i n t e r m e d i a t e
To i l l u s t r a t e t h e numer i ca l behav io r of t h e model, two e l J i o n s o f v
goods b u t i n c l u d e s two pr imary f a c t o r s of p roduc t i on and hence t w o income I
PROLOG a i r . p r e sen t ed . PROLOG 1 c o n t a i n s t h r e e s e c t o r s and onc:
g r o u p s , w i t 5 co r r e spond ing s e t s of demand f u n c t i o n s . It a l s o i n c l u d e s
pr imary
2 e q u a t i o n s i n appendix C.
s a v i n g s and is recurs ive- dynamic. The e x t e n s i o n s of t h e fo r ego ing
For each c a s e , two k i n d s of s o l u t i o n s have been made. The i n i t i a l
p r o o f s
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 g e n e r a l e q u i l i - I
t o t h e r e c u r s i v e c a s e a r e g iven i n appendix B, and t h e set of PR0,LOG 1 and
br ium i n i t i a l c o n d i t i o n s . Subsequent s o l u t i o n s demons t r a t e 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 g iven endowment of t h e ( s i n g l e ) resou
b ) t e c h n o l o g i c a l change i n t h e p r o d u c t i o n 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 d e conducted
f o r t e n pe r i ods . TSere a r e two v a r i a n t s : one w i thou t t e c h n o l o g j c a l change -
and one w i t h t echnGlogica1 change. i -
For PROLOG 1 t h e i n p u t d a t a r e q ~ i r e d a r e a s f o l l ows (p r ame te r !e 4 I
v a l u e s and i n i t i a l c o n d i t i o n s ' :
C n t a f o r PROLOG 1
In Table 1, t h e f i r s t column gives the s o l u t i o n t o PRO
t h e foregoing d a t a s e t . The second cciiunn shows the consequence
increased c a p j t a l s t o c k (K = 3300). The t h i r d column shows t h e
with higher p roduc t iv i ty i n the production of good C (aCC cha ged from t .08333 t o .03333), and t h e four th column shows a s o l u t i o n with b
i n c r e a s e c a p i t a l and t h e technological change.
Table 1: NUMERICAL RESULTS OF PROLOG 1
Solut ion Number ~
Note: A zero en t ry means zero a t three right-hand dec p laces of accuracy.
Q B 76.859 84.846 76.823 84.759 - QC E 141.542 158.508 148.901 157.694
I
1.2
,084
. l o 4
. I 2 1
.003
.003
.003
. l o 8
.081 4
. I32
342.713
Var iable
Q A
~i ~b
p i
p i
~6
?
XA
XC
Q A
1 . 3
.001
0
.055
.030
.027
-. 021
.044
.004
.071
316.534
1.1
-. 001
0
.001
.003
-003
.003
,008
. 001
.002
315.954
1.4
.088
,103
. I15
.030
.027
-.G21
. I48
.087
.205
344. L14
The f i r s t i s s u e t o b e d i s c c s s e d i n c o n n e c t i o n w i t h ~ q b l / e 1 i s t h e
s e n s e i n which t h e LP mole1 is an approx imat ion . There a r e ~ w o
a p p r ~ x i n a t i o n s i m p l i c i t i n e q u a t i o n s (C.4) - (C. 1 3 ) . One i s t h e a p p r o x i m a t i n n
i n h e r e n t i n t h e s t e p p e d demand f u n c t i o n s . The o t h e r is t h e a p p r o i m a t i o n x i n t r o d u c e d by e x p r e s s i n g s a n e r e l a t i o n s i n l o g- l i n e a r form i n s t e a of level .
f c m ; e q u a t i o n s (C. 9) and (C.13) a r e examples. When t h e i c g - l i n e a r i' form is
invoked , change i s d e f i n e d by a L a s p e y r e s i n a e x ( C . 3 ) , and t h i s t o i n t r o - b duces a d e g r e e o f I n e x a c t i t u d e .
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 I o u t t o b e 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 e a c h demand c u r v e , and expanding t l l o s e t o 200 segments (over
range) produced i m p e r c e p t i b l e changes i n t h e n u m e r i c a l r e s u l t s .
s o u r c e of e r r o r i s s u f f i c i e n t l y i m p o r t a n t , however, t h a t i t is no p o s s i b l e 1 t o e x p e c t a l l t h e r e l a t i o n s h i p s i n t h e model t o ho ld e x a c t l y . A u l l y f e x a c t s p e c i f i c a t i o n p r o d u c e s a n i n f e a s i b l e s o l u t i o n .
We a r e s t i l l e x p l o r i n g a l t e r n a t i v e t r e a t m e n t s o f t h e a p
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 / a l l o w some c o n s t r a i n t s t o h31d w i t h i n a c e r t a i n e r r o r t o l e r a n c e
The T a b l e 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
L a s p e y r e s change i n d e x G u a t i o n s . T h i s a l lowed t h e r e s u l t s t o de i a t e from V * t h e b a s e v a l u e s by t h e amounts i n d i c a t e d by t h e ra te- of- change va i a h l e s 7 i n column 1 of T a b l e 1.
8/ T h i s p r o c e d u r e c a n b e 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" d e v i c e i n s p e c i f y i ~ g t h e e q u a t i o n s .
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 s a r e p o s s i b l e . T a b l e 2 s t svs t h e
e i f e c t s 9f s o l v i n g v e r s i o n 1.i r e p e a t e d l y 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 . I t i s e v i d e n t t h a t t h e d e g r e e of a p p r o x i n a t i o n n e c e s s a y w i l l
m a t i o n s a r e a t t a i n a b l e .
i depend on t h e n a t u r e o f t h e p a r t i c u l a r model, b u t t h a t v e r y c l o e a p p r o x i - 5
T a b l e 2: SOLUTIONS OF PROLOG 1.1 UNDER TOLENCES ON THE INDEX OF CHANGE
T o l e r a n c e Va lue
V a r i a b l e .002 ,001 .0005 . ~ 0 0 0 1
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 va1 iu ; l t s
1 . 2 t o 1-4 p roduce t h e e x p e c t e d k i n d s of movements i n t h e numeri a 1 o u t- - *4
c comes. Vhder a t e n p e r c e n t 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 t e n p e r c e n t ( w i t h 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 1 remain unchanged, and c o n s u n p t i o n demands i n c r e a s e by 1 3 % f o r t h e good
w i t h a n Engel e l a s t i c i t y 1 . 2 9 and by 8;: f o r t h e good w i t h a n Eng 1 r
e l a s t i c i t y cf G . 3 0 . O u t ~ u t l e v e l s c h a ~ g e c o r r e s p o n d i n g l y , t a k i n g i n t c I
a c c o u n t t h e n e e d s f o r i n r e r - . i n ? u s t r y d e l i v e r i e s .
T e c h i l o l o g i c a l change w s s i n t r o d u c e d by means of a2 lcwing 2 5.5%
g r e a t e r o a t p u t l e v e l , w i t h t h e s a n e i n p u t s , i n t h e p r o d u c r i o o f g o ~ d C. n
is assurced t o b e a v a i l a b l e -: 1 i n l i n i t e l y e l a s r i c s u p p l y , a t f i x e d zloney 4
Tk,e cocseqcences (1.3) a r e a lower r e l a t i v e p r i c e of gccd C , h i g h e r o u t s a t
an? consumption l e v e l s of good C , znd a somewhat h i g h e r i n c o h e l e v e l .
S o l u i i o n 1.1; combices 1 .2 end 1 . 3 i n a n a lmos t a d d i t t v e f a s h ' o n . 1 PROLOG 2 i s a n o s e s u b s t a n t i a l qlodel, and hence i t r e s a l t s are C
wage, w h i l e t h e o t h e r f a c t o r ' s s u p p l y ( c a p i t a l ) i s in:l.astic a t any
c o r e i n t e r e s t i n g . i t i s e i f e c t i v e l y a d u a l i s t i c model: one f a c t o r ( l a b o r )
b e summarized n o r e s y s t e m a t i c a l l y a s f o l l o w s ( s e e t h e e q u a t i ?s i n i
I monent, and i t s p r i c e i s de te rmined by t n e economic r e n t s i t
The ways i n which =ode1 2 d i f f e r s from model 1 nay
a p p e n d i x C and t h e d a t a i n a p p e n d i x D ) : ~
g e n e r a t e s .
a ) There a r e two f a c t o r income groups and h e n c e t h e l j c o m r
group s u b s c r i p t g is i n t r o d u c e d f o r many of t h e I I v a r i a b l e s . I
a r e s o u r c e from o u t s i d e t h e sys tem (as lmported goods I 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
would be i n a n open-&onomy mode l ) ; hence i t s wage c o s t 'S
e f f e c t i v e l y a
* is t o r e n a i n a nierisurm of consumer and p r o d c c c r s t l r p l u s . I U l t i m a t e l y , a l l pr imary f a c t o r c o s t s t n g e t t c i c l i ~ i d e t h e
i
must be s u b t r a c t e d f ~ 0 i 2 t h e o b j e c t < v e f u n c t i o n i f -
i m p l i c i t s u p p l y ! u n c t i o n s ; those f a L i o r s ~i l !ose :!vaiiat i l i r i c . : I a r e l i m i t e d ( i n th(! s h o r t- r u n ) a r e zccounted f o r i n t h c d::.:~,
t h e l a t t e r
I
~ 3 - e Ruhn-Tucker : ~ n Z i ~ i o n s . ? t h e r f a c t o r s ' c o s t s must
I 1 ? e x p l i c i c i n t h e p r i n i a l .
c ) S a v i n g s r a t e s a r e n c n z e r o , and t h u s t h e r e i s c a p i t a l
a c c u n u l a t i o n ; s a v i n g s i s e q u a t e d t o i n v e s t m e n t
( e q u a t i o n (c .27) s u b j e c t t o d e f l a t i o n by t h e p i i c e
o f a n " a g g r e g a t e c a p i t a l good" ( e q u a t i o n (C. 28) .
2.) Asset a c c u m u l a t i o n o c c u r s f o r e a c h income group. I n ~ a model w i t h no f i n a n c i a l i n t e r m e d i a t i o n , t h e
endogenous ra te of r e t u r n t o c a p i t a l i s a p p l i e d t o
t h e c u m u l a t i v e s a v i n g s o f b o t h g roups i n o r d e r t o
g e n e r a t e f l o w s o f i n t e r e s t income. See , f o r example,
t h e second term i n e q u a t i o n (C. 23).
The r e s u l t s o f E.'<,lLOG 2 are r e p o r t e d i n T a b l e s 3 , and 4 .
A d i f f e r e n t s t r a t e g y was pursued w i t h r e s p e c t t o t h e
L a s p e y r e s i n d e x e q u a t i o n s , t h e t o l e r a n c e s were s h i f t e d t o t h e equa i o n s t I
e r r o r t o l e r a n c e s t h a n i n PROLOG 1; i n s t e a d of a l l o w i n g t o l e r a n c e s
where t h e c u m u l a t i v e a p p r o x i m a t i o n e r r o r might b e expec ted t o caus e
on t h e
problems: e q u a t i o n s (C.15) and (C.20). A s i n t h e c a s e o f PROLOG , t h e I main s e t s o f r e s u l t s w e r e o b t a i n e d under t o l e r a n c e v a l u e s which we
L I I
what l o o s e r t h a n n e c e s s a r y .
T a b l e s 3 and 4 p r e s e n t some b a s i c m u l t i- p e r i o d r e s d t s w en t h e - L t
o n l y i n t e r - t a p o r a l c o n n e c t i o n s a r e s a v i n g s , c s p i t a l
w a s s e t accum~ll ra t ion. I n a l l o f t h e n u l t i - p e r i o d
s u b r o u t i n e i n j u r e s th'7.t a l l l agged v a l u e s a r e
p r c c e e d i n g p e r i o d .
-- r - 7 - -,, .~;-lc ? r e s e n t i l l g s t r a t i o n s , a s i a p l e r u l e h&s kezn a d o p t e d
f o r z l l o c ~ r i c g Inves t r ren: c2t.r s e c t o y s : i n v e s t m e n t i s a l l o c ~ t e d i n
;:rrJpcrric;n L O e x i s t i n g c 3 p l t a l s t o c k .Levi.ls ti1 e a c h s e c t o r . ( V e r s i o n s w i t h
a l t e r n a t i v e i n v e s t n e n t a l l s c a t i s n r u l e s , srLd Cobb-Douglas t e c h n o l o g i e s ,
have been s o l v e d and w i l l b e r e p o r r e d c n s e p a r a t e l y . ]
As T a b l e 3 shoigs, t h e s y s t e x f o l l o w s a s t e a d y g ro r j th p a t h o v e r t h e
~2:: p e r i o d s . (The d e v i a t i o n i roc : z s y o o f t h e d o t v a r i a b l e s i n p e r i o d z e r o
i s a n Indicator of t h e d - ~ r e . ~ c l a p p r o x i n a t i o n i m p l i e d by t h e l i c e a r i z n t i o c . )
Or: t h i s s t e a d y - s t a t e pat11, wage i n c o z e s grow s l i g h t l y n o r e r a p i d l y t h z n
c a p i t a l i r -cozes and s e c t o r C 5 o a t p u t becomes p r o g r e s s i v e l y l a r g e r r e l a t i - ~ e
t o t h e o u t ~ u t s o f s e c t o r s A and R. T h i s e f f e c t i s c a u s e d by t h e h i g h e r Enge l
e l a s t i c l t l . 3 ~ f o r t h e o u c p u t o f s e c t o r C , a n d , g i v n n t h a t i t i s t h e ~ o s t l a b o r
i n t e n s i v e s e c t o r i n p r o d u c t i o n ( a p p e n d i x C) , i t s r e l a t i v e e x p a n s i o n i n c r e a s e s
t h e s h a r e of wage income i n t o t a l income.
i n r h e a g g r e g a t e , t h e v a i u e added g rowth r n t e ( a t p e r i o d l j Is
7.82 T h i s r a t e e q u a l s t h e Harrcd-Domar w a r r a n t e d g rowth r a t e , ss 3ROLOG
2 i e e f f e c t i ~ e l y a Farrod-Domar model. I n t h i s c o n t e x t , t h e w a r r a n t e d
g rowth r a t e i s w r i t t e n a s r jv /k , w i ~ c r e a i s t h e a g g r e g a t e s e v i n g s r a te ,
v i s t h e v a l u e a d d e d- t o- o u t p u t r a t i o , and k i s t h e c a p i t a l - g u t p u t r a t i o . . e
9/ The i n T-(OLOG 2 , a v / k = 0 . 0 7 6 , wh ich i s I / K a s r e p o r t e d i n T a b l c '3.-
a g g r e g a t e growtl l r n t e of v a l u e a i d e d i s m a r g i n a l l y higher owing t o t h c
c h a n g i n g s e c t o r a l c o m p o s i t i o n o f o u t p u t (and t \ e f a c t t i m t c a p i t a l - o u t p u t
E r a t i o s and v a l u e a d d e d- o u t p u t r a t i o s d i f f e r b y - s c c r c ~ r ) .
-- --up
9 / I n mak ing t h i s cornput.n:ion, cri, e ;nu:; i b r tc?itrn t o cci.\ ~ l r t ~ o i - , i ~ - a l I\.- - v a l u e d c o n c e p t s i n t o r e a i v s l n c i s . See a>;'c-ilr:i:-: (,.
I T a b l e 3. XLTI-PERIOD SOLUTICIN OF PROLOG 2 , UITBOUd
TECHNOLOGICAL C W G E I
Notes : 1 ) E r r o r t o l e r a n c e set a t .0005.
P e r i o d
2) Symbols a r e d e f i n e d i n a p p e n d i x C ; YRW d e n o t e 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,~ed by good and income
-
V a r i a b l e
QL ~i Q (5
PA ~ f i
p i
~k Y ~ W
X k
'Gw X ( ~ K
X ~ W
9
.0017
-. 0006 -. 0014
. 000 1 ,0003
.0003
.0004
.0006
.0005
.0036
- .0006 -.0039
378.0
3 -- .0602
.0780
.0903
.0009
. 000 8
. 000 7
.0773
.0794
.0444
.0650
.lo46
.0956
1
.0607
-0779
.09C15
.0009
.0008
.000 3
.0771
.0793
.0460
.0656
.lo62
.0955
471.4
.0758
2
.0605
.0779
.0904
.0008
.0009
,000 7
.0773
.0793
.0452
.0653
.lo55
.0955
406.9
.0757
4
.0601
,0781
.0902
.0009
.0008
. 000 7
.0773
.0794
.0436
.0648
.lo37
.0955
437.9
.0757
507.4
.0758
9 - 0
.0998
.0781
.0901
.0009
.0008
.OC08
.07
.O
-04
.0644
. 1 C
.OS
547.2
.O 59
733.6
,0760
74
95
29
28
55
,0592
,0782
.0894
.0007
.0006
.0006
.07 74
.0798
.0403
.0640
.0992
,0942
q 7 .1 . 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 .., -? .s z ~ s u c e d t o t a k e pl;ce L n s s c t o r 1;. h s wou',d b e e x p e c t e d , good
p r i c e dec1l i ;es r e i a ~ i - ~ e ,; t h e p r i c e s o f A and 5, a r d t h e growth of
t i o n of ~ ~ o d C is accelerate-d. (Consm.pt ion of good C grows n o r e r
than i t s c u t p u t 2nd i t s i n f e ~ ~ d i a t e uses g r u d aore s l o w l y . )
:.3,ile n u m e r i c a l ex?zr iz .ents cannot be d e f i n i t i v e , t h e s e s ' m p i e 1 c a s e s s u f f i c e t o d e m o n s t r a t e t h e workings of s i m p l i f i e d PROLOG node l s , and
i n zilese c a s e s r h e n c d e l h a s performed a s expec ted . I
I
Table 4: KJiTI-?E?,IGD SOLUTION OF PROLOG 2 , WITH TECHKCLOGICAL CHASGE
Notes: 1 ) Technological change defined t o be 2% higher output of s e c t o r C , from the
Period --- 0 1 I 2 3 4 5 9
I
same inpu t s , each year. - - ' ? .i
2) Symbols have t h e same meaning a s i n - - Table 3. -
Q . -
QA ~i Q i:
PA
PB
PC
YRK
YRW
Xk X ~ W
xh X ~ W
I
I /K
. O O i 7
-. 0006
-. 0014
.0001
.0003
.0003
.0004
. 000 6
.0005
.0036
- .0006
-. 0039
378.0
.0756
.0596 '
.0844
. lo63
,0148
.0099
-.0061
.0990
.0808
.0507
.0606
.I462
. lo35
414.3
.0770
.0584
.0884
.I121
.0148
.0100
-.0061
.lo42
,0836
.0515
.0628
,1503
.lo68
500.9
.0802
.0577
.0868
.I102
.0148
.0099
-.0061
. lo28
.0821
.0519
.0617
.I501
.lo49
455.3
.0786
.0591
.0900
.I139
,0149
.0101
-.0060
.lo56
.0851
.0511
.0638
.I505
. lo86
551.8
.0818
.0567
.0923
.I182
.0171
.0123
-.0041
.I101
.0857
.0513
.0565
.I521
.I182
609.1
.0834
-- .0633
.0985
,1222
.0170
.0125
-.0401
.I164
.0912
.0507
.0611
.I529
.I229
$15.5
.0902
C o ~ p m s a t e d Changes and I n t e p r a b i l i t y
Condi t ions
I f we may b e p e m i t t e d t h e a r t i f i c e o f d i s c u s s i n g a s ingl!e ,
a g g r e g a t e consumer, t h e n o d e l p r e s e n t e d can b e g iven a more genera
i n t e r p r e t a t i o n i n t e rms o f t heo ry o f household behav io r by n o t i n g
t h e Engel a g g r e g a t i o n c o n d i t i o n s o b t a i n e d i n (34) imply corn e n s a t e 4 q u a n t i t y changes . Cons ide r i n f a c t t h e u t i l i t y maximiza t ion problj .
unde r ly ing 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 f u n c t i o n o f t h e r e p r e s e n t a t i v e @onsumer.
Def ine a compensated change i n q u a n t i t i e s a s a change where income i s i
compensated a s t o keep u t i l i t y c o n s t a n t . For t h e " r e p r e s e n t a t i v e onsumer," F we can w r i t e :
a u dU = - (X) dX = U P'dX . ax o
dy > P'dX !+ X'dP -
I
A2pendix A
0
~ 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 t h e r n - g i n a l u t i l i t y
f c r t h e Xggrzgate deaand f u n c ~ i o n : ~
Thus, t h e n c d e l p r e s e n t z d is e q u i v a l e n t t o n a x i m i z i n g t h e s l i m of thk z r e a s
The Enge l a g g r e g a t i o n zond i t_ icn i n ( 3 1 , t h e r e f o r e , i s s v c h t h a t i~
urtder t h e cornpensatad dmsnd f u n c t i a ~ s and i n t h i s s e n s e overcomes f!le
guaran--
u s u z l 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 i t s dependence o t h e t
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 a n a p p r o p r i a t e changes i n p r i l e s .
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 i s i n s t r n e n t a l C
a s s u m p t i o n 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 a g g r e g a t i o n e n s u r e s
i n o v e r c o n i n g a f u r t h e r p r o b l e n t h a t h a s been p l a g u i n g t h e f a m i l y o n;odels i
a a s i z i -
of t h e t y p e c ~ n s i d e r e d i n t h i s paper : t h e i n t e g r a b i l i t y
c 0 n d i t i o r . s r e q u i r e t h a t t h e m a t r i x 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 i n o u r n o t a t i o n ) b e n o t o n l y q c a s i - n e g a t i v e
10 / symmetric a s w e l l . - Whi le t h e r e a r e some demand s y s t e m s f o r wilich h i s I * c b n d l t i o n i s met , t h e e c o n o n g t r i c 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 i s tile c a s c , tl:zn, di ;! j . r- tnt lc~-
t i o n of t h e f i r s t te rm o f t h e maximnnd X ' ( A - 0.5BX) does n o t vie c! 1 A - BX b u t A - 0 . 5 R X - 0 . 5 B ' X where B # B ' , 30 t h a t h e Kuhn-T: :Xe-
E I t
c o n d i t i ~ n s do n o t y i e l d a marke t e q u i l i b r i u m e q u a t i o n . 1
101 See [lo].
hppendi# A pag,? 3
I
I n t h e c a s e where t h e income e f f e c t i s e x p l i c i t l y cons id r e d ,
hc - eve r , t o t a l d i f f e r e n t i a t i o n of t h e budge t c o n s t r a i n t y i e l d s : e X'dP + P'dX < dy - I ( A . 6 )
and , beca17.se of t h e conipensated change assumed, :-'dP = dy. Equat 'c - ( A . 6 ) 1 y i e l d s t h e n t h e f o l l o w i n g subs ici : a r y r e s t r i c t i on : I
The Lagrangean of t h e maximiza t ion problem can now be fo rmula ted a f o l l o w s : I and t h e r e l e v a n t Kuhn-Tucker c o n d i t i o n s y i e l d :
= A - BX - B ' X + $p + B'5 + p
I t is ea sy t o s e e t h a t , a t e q u i l i b r i m , a nece s sa ry and s u f f i c i e n t
(A* 9)
c o n d i t i o n
f o r e q u i l i b r i u m ( p r i c e = marg ina l c o s t ) Is t h a t 5 = X. I n f a c t ,
t h e n o t a t i o n and w r i t i n g ( A . 9 ) a s an e q u a l i t y f o r t h e op t ima l vari
(A. 10)
(A. 11)
- t i o n ) and i f % ' C I = B ' X = I$'B-' t h en P = -p, ( s u f f i c i e n t cond id ion ) ,
'* If P = -p d e n 8 ' 5 = B ' X = 4 ' ~ - I a s i n t h e c o n s t r a i n t (necessa ry
* condi-
x h 2 r e -D is , by a n o t h e r s e r i e s of Kuhn-Tucker c o n d i t i o n s , equa l .
t o rn.-yginal
Appendix - 3. - d R e c u r s i v e Dynenic V e r s i o z
2% r j inp le e x t e n s i ~ n of t h e f rancwork of s e c t i o n 3 i n t h e t e t can x b e used t o encompass t h e d y c e n i c c a s e of inpu t -ou tpuc t echnology (wi h : L e o n t i e i p r i c i n g ) and Earrod-T)onar growth. (This e x t e n s i o n i s i l l u s r a t e d
. t n u n e r i z a l i y by PROLCG 2 . ) For t h e s a k e o f c l a r i t y , w e d e a l w i t h o.?e t y p e
of c a p i r a l and t h e r e f o r e col:a?se t!le r e s o u r c e v e c t o r t o a s c a l a r .
a g g r e g a t e 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 Enge l a g g r e g a t i o n
11, ~ r e s o u r c e c o n s t r a i n t-
It - S Y t = 0 investment - s a v i n g s i d e n t i t y
- ( I - ?i)Qt + Xt + N I t - l < 0 commodity b a l a n c e -
been i n t r o d u c e d : 1
wh?re b e s i d e s t h e symbols a l r e a d y d e f i n e d t h e f o l l o w i n g new n o t a t i o n
L , t
I = t o t a l i n v e s t m e n t ( i n money t e r n s ) i n p e r i o d t t
has
s = n c o n s t a n t m a r g i n a l p r o p e n s i t y t o s a v e - - M = an nxn m a t r i x of Zgput-output c o e f f i c i e n t s
N = a n nxn d i a g o n a l maFrjx of c a p i t a l goods in : )u t -ou tpu t coef f i c p e n t s .
- 111 I n v e s t m e n t y i e l d s new p r o d u c t i v e c a p a c i t y w i t h n ~ r i c - , ~ o r i n c l -
l a g . The l e n g t h of t h e g e s t a t i o n l a g can h b - v n r i c d i f des i r f23 .
e t s u b s c r i p t d e c o t e s t h e t l c e p e r i o d ar ,d t h e 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
Forn ing t h e Lagrangean from (B. 1 ) - (B. 5 ) , w e o b t a i n : I
A s i d e f rom t h e c o n s t r a i n t s i n (B.3)-(B.5) and n o n- n e g a t i v i t y and komplemen- I
I
t a r i t y , t h e Kuhn-Tucker c o n d i t i o n s c a n now b e s t a t e d a s f o l l o w s : ,
Equa t ion (E.9) is t h e o n l y new f i r s t - o r d e r c o n d i t i o n which is no found i n t h e
s t a t i c model o f s e c t i o n 3 . I n e q u i l i b r i u m P = y f o r t h e r e a s n s e x p l a i n e d t !
b e f o r e and -p e q u a l s 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.
Fur the rmore , because of e q u a t i o n (B .9) , p r i c e s of f i n a l goods w i 1 b e e q u a l 4 I , I
t o m a r g i n a l c o s t s d e f i n e d a s d i r e c t c o s t s C , opportunity c o s t s
khcennedia t e c o s t s -pH ( L e o n t i e f p r i c i n g ) . - - I
Equat ion (B.9) d e f i n e s t h e c o n d i t i o n t h a t has t o bola t e q u i l i - - b b r i u o f o r an e e f i c i e n t inves tment a l l o c a t i o n , where t h e
r e t u r n on s a v i n g s ( y ) e q u a l s clie i n p u t v a l u e of inves tment ( A ) p l u s
t h e o u t p u t v a l u e ( - p x ) .
t h i s i s 2 p u t ty- c lay s 2 e c i i f i c a t i o n . S e c t o r a l investmefit Eunc t icns
r e q u i r e d , and ;he d y n z n i c l in 'kages between p e r i o d s a r e t h o s e i n v e s t e n t re A p p e q d i x 3 page 3
17 e q u a t i o n s ( B . i . 1 - ( B . 5 j , c a p i t a l ss i ~ n o b l l e a c r c s s se lcrors
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 c a p a c t t y . (The i n v s t z e n t E
u i t h i n a p e r i o d and ( i n c r ~ x e n t a i j c s p i t a i i s r c b i l e 'cetwzen p e r i o d s ;
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 o v e r s e k t o r s i n
t h o s e c a s e s where p o l i c y is p r e d o n i n ~ n t . ) S i n c e c a p i t a l i s f i x e d i t h e n
i . e . ,
s h o r t - r u n i t is no l o n g e r 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 e q u a l i z e d a c r o s s s e c t o r s .
Appendix C. Equa t ions of PROLOG 1 and 2
I n our n u m e r i c a l s i m u l a t i o n s , i t i s 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 i n c o n e v a r i e s away f om t \ e
e q u i l i b r i u m v a l u e . For t h i s purpqse , t h e c o n s t a n t e l a s t i c i t y dem nd .i s p e c i f i c a t i o n a p p e a r s t o b e t h e most a p p r o p r i a t e . I n t h i s c a s e , h e s t a t i c ti Engel a g g r e g a t i o n 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 v a r y b u t income d o e s
n o t e n t e r t h e c o n s t r a i n t , i s r e p l a c e d by a v e r s i o n i n which elast c i t i e s are 1 c o n s t a n t and income may v a r y :
where t h e i i , s
a r e p a r a m e t e r s 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 t segment s of t h e demand c u r v e f o r good i, and ci and qi r s p e c t i v e l y , 4, 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 paramete rs $ i , s a r e q e f i n e d e x
a n t e s o t h a t
1 2 1 o r t h e t o t a l v a l u e of consumer e x p e n d i t u r e s on good i.- I t
Another t r a n s f o r m a t i o n f o r t h e n u m e r i c a l v e r s i o n is t o r i t e non w - p o s i t i v e p r o f i t c o n d i t i o n s ( a s i n e q u a t i o n ( 6 ) ) , s o t h a t d u a l v a l e s of
't 5 Y 1 2 1 N a t u r a l l y a l l t h e z q u a t i o n s based c n i i n e a r demand f u n c t i o n can b~
a l s o r e d e f i n e d i n t e r n s of a c o n s t a n t e l a s t i c i t y one. numerical s i m u l a t i o n s p r e s e n t e d , 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 a p p e a r t o make any d i f f e r e n c e .
- r e s o u r c e s ( f a c t o r ~ r i c e s ) a r e d e f i n e d i n t h e p r imal . i?r,d a t h i r d * * m
I . t r a n s f o r m a t i o n
I
Appendix C page 2 I
i s t o d e f i n e l o g- d e r i v a t i v e (percentage-ra te-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 b e used. T h i s
13 I f o r y a l r e a d y , b u t i n general : -
Thus it is a L a s p e y r e s i n d e x of change, based on 3 one- per iod l a g The I above t r a n s f o r n a t i o n s a r e used o n l y f o r purpose of r e t r i e v i n g d u a l a r i a b l e s i. d i r e c t l y i n t h e p r i m a l and d o n o t 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 n o d e l . h Now we a r e i n a p o s i t i o n t o w r i t e t h e f u l l LP model
s i i n p l e s t v e r s i o n which was implemented n u m e r i c a l l y :
1 4 I PROLOG 1 e q u a t i o n s -
Max D 1 i s i , s good i,
i , s segment s
s u b j e c t t o :
[Enge l a g g r e g a t i o n ] ,
[cornrnc?dity b a l a n c e s ] - - i -
131 S e e 161. -
141 Symbols a r e d e f i n e d a f t e r t h e PROLOG 2 eqc a t i o n s . -
(C. 4)
bppcndix C Page 3
[ r e sou rce c o n s t r a i n t ]
[convex combination c o n s t r a i n t ]
[ d e f i n i t i o n of demand p r i c e s ]
[ d e f i n i t i o n of consumption by good]
[non- pos i t ive p r o f i t s ]
[ f a c t o r 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
( p l u s d e f i n i t i o n a l equat ions of t h e Lype ( C . 3 ) f o r t he r a t e - v a r i i b l e s , 6 and ) .
i i
(C. 10)
.;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
s o t h a t i f Y = C j t h en 0 < D < I . - i , s - Each D i,. 1
r e F r s s e n t s a n ( a r b i t r a r i l y s m a l l ) segment s of t h e
denand c u r v e f o r good i. Relevan t ranges of t h e I demand f u n c t i o n s a r e e s t a b l i s h e d a ? r i o r i , and t hen t h
pa r ame te r s w . znd y. a r e computed from t h e 7- , 5 1, s
f u n c t i o n s W and E (marg ina l revenue) . i i
Some o f t h e e q u a t i o n s a r e redundant . For example, (c, 2 )
be s u b s t 3 t u t e d o u t o f t h e model. However, p r e s e n t a t i o
I 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
i n t h i s form h a s t h e advan t ages sf c l a r i t y and e a s e of ~ manipu l a t i on (when many v e r s i o n s a r e b e i n g made).
C ) Equa t ion (12.8) r e q u i r e s t h e s o l u t i o n t o be on t h e demadd
f u n c t i o n . 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 f u n c t i o n r igh tward i n accordance
w i th t h e change i n r e a l income. The paramete r
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
ii a r e
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
e q u a l t o t h e dual. v d r i a b l e s of t h e connodi ty b a l a n c e s , 1 'i b u t t h e y g n u s t b e d e f i n e d i n t h e p r i r l a l problem a l s o
E I I -
i n o r d e r t o e n f o r c e 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 t h e p r i c e of the s i n g l e resource. In
t h i s s imple model, with only one resource, equation
(C. 7) i s c e r t a i n t o hold a s an e q u a l i t y and there-
f o r e (C. 2) can take t h e above form. I n more complex
ve r s ions , where thee may be s l a c k resources , (C.12)
is replaced by a d i f f e r e n t expression.
f ) I n equat ion (C.13), t h e ai a r e budget shares .
I n genera l , t h e l i n e a r programming ve r s ion inc ludes many a r i a b l e s V i i l t h e p r h a l which a l s o appear i n tho dua l , such a s f a c t o r and pro u c t d p r i c e s and incomes. This redundancy i s a necessary f e a t u r e which a
the imposit ion of c e r t a i n primal r e l a t i o n s h i p s which span some of t
v a r i a b l e s .
PROLOG 2 E q u a t i ~ n s
good i segpen t s group g
s u b j e c t t o .I Ei,g r i , s , g D i , s , z - (l-sg)Yg = 0 i , s
[Enge l a g g r e g a ~ l o n f o r e a c h group g ]
[ c o m o d i t y b a l a n c e s ]
[employment d e f i n i t i o n ]
[ c a p i t a l c o n s t r a i n t 1
[convex combina t ion c o n s t r a i n t ]
- * [ d e f i n i t i o n of denand p r i c e s , f o r any o n e income g r o -
[ d e f i n i t i o n o f consumption by goqd j
(C. 1 5 )
(C. 1 6 )
(C. 1 7 )
(C. 1-8)
(C. 20)
(C. 2 1 )
Appendix c p a g e 7
[Labor income definition]
[savings definitions]
[asset accumulation]
[aggregate savings]
[savings-investment identity] ~ [price of investment good]
[income deflatorsli - - * II
yii + P? - ? = 0 g 6 g
[real income definition]
[plus equations defining the log-derivative variables].
(C. 24)
(C. 26)
(C. 2 8 )
(C. 29)
(C. 30)
; Appendix C page 8
I
Syrnbols used i n PROLOG 2
S e t s -
commodi t ies
i n c o x e g roups
s = 1, ... , p segments i n den nd f u n c t i o n s
t , ~ = 1, ... , T tioe p e r i o d s ( a
s u b s c r i p t w l a b o r income g r up ( g=w) C Convent i o n s 1
?) A d o t t e d v a r i a b l e ( ) i n d i c a t e s i t s log- c ange, o r p e r c e n t a g e r a t e o f change. h
2 ) P a r a m e t e r s a r e d e n o t e d by Greek l e t t e r s o r c a s e l e t t e r s .
3) V a r i a b l e s a r e d e n o t e d 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 o v e r t h e ~ u .
D e f i n i t i o n s
w c u m u l a t i v e 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 t h e segments o f t h e i , s , g c u m u l a t i v e demand a r e a f u n c t i o n
s e e e q u a t i o n ( 7 2 ) i n t h e t e x t
- TrJ wage r a t e
N employment
s s a v i n g s r a t e o u t of t o t a l g roup income1 g - - -
w ~ * Y - t o t a l group income -
&
r c u m u l a t i v e area under t h e m a r g i n a l r e v n u e i , s , g f u n c t i o n I
i I - l~pnndix C I page 9
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 C i
g r o s s o u t p u t
bi s h a r e of good i (a s a c a p i t a l good) i n c a p i t a l fo rmat ion
I t o t a l f i x e d investment
6 q u a n t i t y consumed a l o n g t h e demand f u n c t i o u i , s , g
l a b o r input- output c o e f f i c i e n t
k j
c a p i t a l input- output c o e f f i c i e n t
- 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
Pi p r i c e
own-price e l a s t i c i t y
q u a n t i t y 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 l a b o r hcome (Yg , g=w), i . e . , wage i n I t
d iv idends acc ru ing t o l abo r .
S s a v i z g s by income group g
FA a s s e t ho ld ings i n c u r r e n t p r i c e s g , t -
t o t a l s av ings
P I p r i c e of t he agg rega t e c a p i t a l g c ~ d
income
budget
d e f l a t o r
s h a r e s
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
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 l a n d
T u b l i s h i ~ ~ g C'onpany, P a s t e r d a n , 1975. 1 [ 2 1 Duloy, J . K . , and R.D. Xor ton , " P r i c e s and Incomes i n i i n d a r Tro-
[ 3 1 G i r s b u r g h , V i c t o r , and J e s n K a e l b r o e c k , "A G e n e r a l E q u i l Erium Yodel of K o r l d T r a d e , P a r t I , " Cowles Founda t ion 3 i s c u s i o n P a p e r So . 412 , ::r;ven;ber 18, 1975. i g r a r n i ~ g P?odels , I 1 f i inerican J o u r n a l of A g r i c u l t u r a l , v o l . 5 7 , !Joveir.Ser, 1975 , p p . 591-600.
[41 i - Iazel l , P .E.R., and 2 . L . S c a n d i z z o , "Competiti-qe Demand u n d e r R i s k i n A g r i c u l t u r a l L i n e a r P r c g r a m i n g Nod I s , " h e r i c a r . .Jcclrr1=11 or' A g r i c u l t u r a l Ecorcmics , v o l . 6 , 1974, pp. 235- 244.
-- 1
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