Supply behavior with production quotas and quasi-fixed factors

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Supply behavior with production quotas and quasi-fixed factors

L.P. Mahé, Herve Guyomard, . Cresep. Centre de Recherche Sur l’Emploi Et La Production

To cite this version:

L.P. Mahé, Herve Guyomard, . Cresep. Centre de Recherche Sur l’Emploi Et La Production. Supply behavior with production quotas and quasi-fixed factors. 6. Journees de microeconomie appliquee, Jun 1989, Orleans, France. 36 p., 1989. �hal-02857316�

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ô (

IN$TITUT NATIONAL DE LA RECHERCI.IE AGRCINOI'1TOUE

Sbutlon d' Economle Runalc

65 rue de $alnt tsrleuc 350ô2 RENNES CEDEX

SUPPLY BEHAVIOR }TITH PRODUCTION OUOTAS AND OUASI-FIXED FACTORS

Louis Hervé

I"IAHE GUYOI"IARD

À

I July ^19Açl

1,

eocT less

OMIE RUNI\LE

.NEtrilE$

(

1, Introductlon

In the reçent rreriocl ^pc'1. icy nrækens hæve b'econre nelucbænt to cut ç,rice support :i.n or'der to re,Jutce tl-re. widespnen,l erxcessi.ve

capacity of tl-re farm secton indevel,:rred,:c.runtr'ies" Inter'vention

$chemes which,lr:r r:ot disturb the ^r,,esb*d i.nterests br-rt ^prevent buclget cc,sts f rom j.ncreasinÇ f urther lrmve become môre p,:puli*n..

As ma[.r.inç tl"re trængf*r æs ôhrsc,çrr* i3s trûss:ih,].e i.s i*Is;,: ^æ banget of farm r:'ress;ure,.?roups, it,loes nÔt conte *s a.surprj.se thæt ^product ion qurotns. lan,J set msicle, f ert i l izer rat ioning mnd

al I socæl led ^supË,.1 y rfianagernent pol i.cies Frlay æn j.ncreasing nole in

f arm Ê,r^ograms "

There i.s â çrowi.ng f itenatr:r* ûn r:roducbion ^quçtæs, ænrJ s;ur>ply

r:orrtr*ol policies. c,anticularly in ^Cænacla where ^'bhese type.s of ins,trum*nts hæve beerr implemenbed for sûrne time lÀrcus. 1"978 i

Bar'icl-relIo, 1"981^ , 1"98/+ î ^{Veeman, 1.gt:rz} ; $chmi.tz., 1,t83 I G,ruin"

J-988 ) , Tn Hurc,rre " the ,Jairy ^quotæ has attnactecl frn *:xtensive ^w,rr*l.r of ænalysiE IHanvey and Hubbard , 1984 ; ^{Perr'æud ed} ,," l-v68)"

Mosb stt*tclies deal witl-r bhe er/ôluætj.,:n c'f the r*êrrt or of tlre

brærrsf ens ænd welf are .l.ass in 'ù r,ar'tiæl *quilibr^iurn f i^amewonk

apprôach in the context of mar*kets far qr-rotæs" Br,;t ^qur:tæs hæve

ælso become & challenge to model build*r's, since rnodels estimastsd and desiçned in an environment çf c,nice supË'ort pr:.l.icy inetruments have to be usecl ncuJ in the context ctf a sLlË'p1y mmnægement scheme.

The natunal response af the ^econ,rnr iet in tl^rêl cc'ntext iç, ta fin'J ,:ut the ^" pr^ice cut equivalent to tli* ^rruatæ c,:ns,tratrrt ^" " ^f iris ^l-ressi been done by $ome aubhon$ ^[ Tonnicl is, L981. ; Bingi.ev" Buntorr"

$tnmk, 1985) " But a real en,loçeneiaation of the shædor"r pnice equivalent bo bhe quota, which braces the'cros$ effects in the modelo is rarely done. This is l'rouever i.mporbant in multioutput multllnput commcrdity rnodels. l'lunk ⁽ t9A5) hra$ echieved an actuerl

(

'ï,ndôgc'n,*isation ,:lf the ^rni. 1k arnd suçffi1^ shmclow pri.ces i.n m mC. E,r;ppl.y

model cl.osely based on moçlern r:nocluct ion bheory" M"rl-r6,, Tavérm,

Trochet ⁽ f çeA) nave elso f ul.Iy endoçleneised the dæi.ry mncJ suÇeâr

shadow prices in ân intennational. msriculturel tnmde mo'Cel

f ocussing on ^EC antl ^U$"

Lau 1,.L976) has charactenized the cc,n'litions unclen which

eupplv ænd demend rraræm*ter s withcut quanbity constrmints can ^be d*ducecl from the paræmebers *etimate.,l urnden s']m* input 'rl- cutput fixitv ^{ancJ l'rês} showtt l-raw Le t)hmtelier's Enj.ncirrj^e mpç, lieg" Brown æncJ Chrietens*n ^{( 1^9SJ-} ) ; Kul.mtilekæ ^{ 1"9S5) ; $quines ^{ 1"9871 ;

Guyomærd mnd Vermersch fL9S9) hæve irnplemenLed practicælIy _tl-re procedure proposed by Læu in order bo derive long*run inrrut ^denrand eIæsticities from short*run parameber-s estimated under input

ancl ,/c,r output fixity"

Recerrtly Moschini {1-988) l-rms, fôr thre first time mpp*r'*ntIyn tisecl the duæI mpproech to estimæte æ multioutput multi.inpr-rL sur:F:lrz model in æ context of supply management pol.ici.es" _{He ræises} the issue of ænalysinç thc comparative statics of proclucrr's respôn$e under supply management Énd, after'stucjyinçi turo sources ,rf .jolntness in input ^quant it ies ^{I Koht} i , ^1"9ô1 | $humr,.ray, Porre,

Nas:,h " ^{l-984 )} , he states that without â gen*raJ sluj</el jne'., _arfrefher^

ar' not suË,rly r*6.$fr^j,:f inçt ^p'c,.I ir:Îes ^w.i I J tlrrc,^ease ûr cle,cr*æçe fh{}'

sup,ç,Jv of ^a/,?r*a.s tr'icted autp,ut and the demancl of vari*L'l.e i,*iËtL,â.$

r^tilJ t'e ^p,ursued at the emp,i,^icaJ .].etr*L.

The purpo$e af the p,r*sent r>aper is to expl.ore vær'iorJs cases

under ulhi.cl^t the cornparaLive statics c"tf supply [:ehævior urrclen input ancl output rat ioningj can bcl characLerized in mûnô rlebai 1s. " ^{l^le} treæt jointly the casss of a quota and a factor fixity a.$ they

reæ1. ly are mirror imageg c,f the same r:roblem. ^Tn order tçt cjrr that ⁿ the theoretical basis of ll-re approach is laid down in section 2

whe're the use of shadow pri.ces and disequilibriurn geps is i.ntnoduced as a simple wây to car"ry the analysis" Irr section 3o we

?

I

clevelop the essenLimls of the conrpanmtive stmtics of supply behavior with respecb to tl-re level of the quote and the ^{f i xed} fmctor, Ëmphasis ls put on the compmnison of the supply behevior af unconstrmined quantitles while some other lnput.s ôr oubputs mre

netloned, with respect to the ^$ame behævior without rationins"

Attenbion i$ centered on cross price ^{ef f} ecbs. Section ^/* deals wittr æ dynamic mpproacl-r of the sâme problemo ænd stmple ænmlyticml relations ^betroeen short-run ond long*nun re$pon$es are deriv*qJ"

They I.emd to ûn estimable model nf the ^demmnd for quasi*fixed inputs which i.s J.ess sË.nerml but simpler to specify tlrfrrr the ^orre of Hpstein (19S1), A numerical lltustrmtion of bhe snadual evolution of supply and derlved demand elæsticitiee ûven the ^bime Iags afber the inibial prlce _chanÇesn i$ finallv ^provlded,

&

2. A theonetical framework Lo analyse supply behavion under ratlonlng

In oncler to chanacterize the behavi.or unqJer constraints, ^w€!

make use the knowledçe of supply response without on before ^the implementmtion of the rationi.ng" Tl-ri.s is made eâsy urnden fairl.y general. ccrndj.tions ms the ûomrranettive statics af unccrnstreli.ned

sug'ply behævi.on i.s b'etter known (e" Ç, $ækmi, ^Lq7(+) " As thene is an evident symmetry betwee,n introclucinç consbrnints on the ^one hand and relaxing bhem on the othen, tt*re comparison with the problem of movi.r-rg from shont*run to long*run nespon.se is of ^some interest "

2^ 1." Ths ef fect af r'atianinç .in pr-odLtcer's t,ehattior lalhen aI l pni.ces ôre

immeclj.mtely, bh* familian

çiven and ^producens âre free t,) mdjusb

proclucer pnoblerm _i.s n

(2.1") Max { voq ; q € T ^}

q

n' ^(v)

where q is the vector of {n+m) n*tput quantitiesï v' the itranspcsed) vector of corresponqling r:r'ices and ntt(v) ^the (uncenstrmined) profit function" The feasible seb T is assumecl

strictty cionvex so that optimal quanbi.ties ar'e uniquely deternr i.ned

and well beheved fr:nction af ririces, Tl-re vector^ q i.s Ë,antibionned j.nto two su.rbvectors of quantiti.es a 1 always variæble ^anci quantities ao susceptible of beinç constræined- A similar subdivision applies to tl're vector of pricee v. Problem (.2"t\ ^can bhen be wri.bten 6$.

5

(2"2) l'1a x u

Itr 9O

( o., oo) €T ) n

{"i e1 * v', ao i Iv to)

$upply ^f unctions of netputs are c'btainerj ^{f nqm} HolelI{ng's ^Lemmd,

(a"s) a) ^{n u}₇ ^(vrn

(a".3) t)) T'v ⁽ v ,,

o

where ilu ^mncl ntt are Lhe çracjients of the profit function with nespect tc t1 and vO, ^tJhen quantitites âne ^peSSed mt $æv AO bv

p,:l.icy instrumenbs (truotae, set aside.., ) Çr ^mmrkeb risidi.ties ^ænd ædjr-tstment cost ^(f actor ^f ixity in short*run), vmriah,le qumntities

û t do not ^behæve in the $ame way wibh respect bo ^{*xogenous P'r'ices}

V to since they ane also a function of ;o, Fotlowing ^Rarbærth (1q/u1), rlefine ,o the vector of virtual pnices which eljnsure tl-rmt

thel unconsbrained quarrtities oi es funcLions of prices wiIl. ^stav at level ^q_oo

(2.6) ao^u (vro ,o) x ^q_oo

Tl^ren f rom the knOwledge Ôf (e-.:,J we cmn inf er ^{hoi^,} sur:'ply nrârlffiÇÛd

gçods witt modify the behavior of the other suppl,ied ^{ænd demænded} goods and factor's. Followins Sakai's (.L976) decÛrnposition nulq:.

ancj if (A"6) ^Çr,rt be solved for ^tl-re virtual prices ,o es function nf ^V 1 and eoo ue cen ^def ine the retatlon ^between bhe nestnicted behavional funcblons .rf ^anU the unconstnain*d functions defitred ^ag

in [2.3),

7

oo)

v q

vl_'o-

Qou

o

1

by:

6

?r

[2.5) ^qR (vrn _oo)

1

q₇ Irr" ,o Ivr, ^cro)l

l-les.sian of the urnrestricted ^pnc,f it f unct ion ,lifferentimbi.Iity i.* æssumed.

with ,o B.s defined by (2"/*)" Res'Lr-icted supplv and demand

functions ol Çan then be R ænæJ"ysed, ê$ thev mrs equivmlent bo the scrlution of tl-re system ^(e" S m) ænd ^{(.2" {,")} " In par*ticular the comË'arative. stmtics af Q.S) cæn be easily deni.ved from the nt( v r, ^,oJ , ^{when t-wice}

R Ivro ^{PoJ" (ôuo/ôv,)}

(2"7) *À (v1, oo)/ôv,

t2"6) *ri ^(vr, ôo)/au, = ^{âaiirrr, uo) /ôv +} (ati (v

!' p \/o" ^fu o) " ^{( ôpo// ôv} _lJ

1

Of.r

It + il^u

l)o

7 7

1 7 1 o

The grædient of vj.rtuml price ^vecton

f rom dif f erentiatins (2"/+), takins ^f 2..31 bl

ur, r, t, v 1 is derived into account,

(2. ₈₎ I

Arr

dr, and resu lt

u u

'o ₁ ¹ toVo

explicit solution nf 12"7)

doo around the point (vr, can be obtained by total

lvr, ,o) ^dPo * ^qlq

o

mnd ^(2" s) for do,^R ) can be clerived "

( + n

7

pl ,J

o'

,Jifferentiation o af

irr terms ^r:f Tl-r* same

(2" 5) ^r"rith

l)

respect to r1 and ^tro wfrerÉ. dro and doo are relatecl b'y ^equabion (2,,8) which clef ines ,o implicitelv. The complete svstem of supÊ,Iv

re:$r)onse can then h:e writben in terms of the Jacobian Of ^the unconstrairre'J prof it functi,:n.

7

HJ xlù ^{u o(v1o}

nu (v

7' ]HJ

v n l)

u)_o'

l, I o'

1

o o

e.e) ^{1 1} 7

(v ro VoVo

otl

Tt shouLd be emphæsized t nt arq evalualte.d ab thq point ,Jan be sç:Ivecl for the ^acLumL

r^r i. t l-r resrrect t ci t Ire rrew set

l^rat the cross r:urtiæI cleri.vertives of (vro oo) j.,E":" (vr, ,o)- Tl^ren (2"9) encl*genous vanimbtes i"*" (.clcr, _clpoi

of exo(Jenous ene.Ë wl'rich $re _{tclv r,}

O" ) , thæt is, ^clropping the srguments in the ^prof i.t f unct icn,

o

[,j ^{L,;; lu-1 (r" 'o oI -1 nt'o 1 n' VVV\,/10 (II" 00 )-1l[;j}

{2.ln)

ot lto

nt

ot'o

7

) (tl" ₎ ^-1

1 Vot'o

The slnict convexity of tl^re prociuction set ensunes thæt ^nt_toVo is r:ositive def inite and inr,tertible (1)" Systern ^(2" 1n) ællov,rs to Çompletely':hæracterize tire ccmpar'aLive ebatics unc]er constnaints, ûn tl^re basis of ^{tl^re.} inf ^ormat ion of the uncc'nstrni.rred suË,ply anci

rlenived c.lemand responses, evaluated at lhe relevmnb Çontr'ained equli.libr'ium ^(clo, 'ol ^"

There i.s â simitariby ^-between the system of equætioris (2,1"Û)

which allows to describe the consbraineci behavion on the basie of the unconstræined one, anci the problem of derivins long run behævior f r om the shor^t*r'un re$ponses" T+ be ntôr'e specif ic" the

,same, infonmætiorr ae l.n ^(2,, 1.û) cc'utd Yte ob'tainecl directtv i.'f the Jæcobian of tl-re nestr'icted Krrofit function 1R{tr, ïo) is ^knr;wn.

nR(vr, âo) ls def ine<J by the pnoducer's problem ^:

I

t.

(2 " 11) r,te

q

,AppIying functions of constrained _at shadow rrrice _,o

(2.12) ar) nR

(2" x?) b)

An _equation ,Jiffenentietion

of

f"il

,tu,' ^{oo) x} .i {.,,, À"ol

X 7

Fr q^{I (q} 7'

q

similan

systern

io)tt]ËnR(ur,io)

the _enveloppe

rhe unresrni::::".:"ï.'z'tx') p.ovi er when

level;'o" ïhe _{smme pnocedune}

al"scr æs a function crf v, _{and ;o,}

des the

goods ^{supp I y}êre the

Ço

prov i des

o

nJo '' "

o)

'o ^{{v 1o oo)} to ^(.2 _{" 1.û)}

(2,, t^2),

TP q_o) nR

reo {vr, _îo)

]H

cæn be obtai.ned h:y tobal

(2" x3) fnJ,", ,u,, [nJr,

, ,, _{,, (}

; o) ÇoQo 1

ïhe equivæIence between _(a" fS) and _tA" 1n) _c

exprored bv using e procec.rure simi ran to _a*o *n be _{f urbher}

ï:"":::ï:.:ï:;"".:;.::î.'_uï ";#;.:lr,

;: ... ;: Jï;"*nji

negntive of the expansi*n ^" 1'o ^v

ef f ect of *noto '!ot' is iust _the e r:roblem _of

short*run b. _ê$ rons-run equiribrium _{Mahén

Guv*merd ,.:;:î: ^â

Ï n ^pent _{i cu r} an i t ôpps;rn c. ^"î h

derived _{demond r-esponse}

.**"* in equation _(2" 1tl th vector _v - ,.,o"1.,^1,.ï:.ï of _{sûocJs a}

r ^tnritt-r respecr

etL sur:r:rv _and ' t *s ^now modif ied" mo"*o,r*", ;.;.;,_::""^*_"t ^. *" il're price chatetien,s principre îs ;".;::::"i, .another application of Le

matnix IIu (tl-., -j*

r-*n,*oo*"*t't â'$ the diasonal tenms çf _the posltive

defi.nibe mabrix (IIu )-1 tç;r'tII$ anound _the

"!oro' Thenefone output _supply I

a

responses of Llnconstneined _qLtont

it ies to thein _ûwrl

;::ii:: ;j'iJ";:ïîï:: :;";::_-:ïffi.:.:,,":j ;lï.iïj;

iil"ll;-]i:;.:."::;,;.;;.ï:;:."".". ;; Lau,s nesu,r derived _rnom As noted bv t40schini (rçee) whab _happens bo crô$s price effects is much ress obvious pônticuranly

when one _{LJ$es bhe} r^estricted proff,t function, _{whene cr.e$s}

effecbs _betwe

of the ty'e a R ^L'çLwssn quentitie*

siruæbirn ,. "rt;:l"t."lrtl"' 9o,r' ^{mne nob e&sv ro} inrerr:rer" _ïhe

when ûne sbants with _IIuIvr, uo) _since

il;;r.ï:::ï.ff_" ^{rusb cnos$} prlce erasriciries _encJ êre _more

e" 4-, 1. tne notiôn of ,,fr4?j.r+,,jse

simi lrrri ty ^,, ïn order to chanaeteniue

convenient to introduce the de

is poir of fi

nespecr ro . i:::ï"J;",*"* _ao ^e"_if

and ol witn _nespecb to _{vo have} notmtions if ^I

the behavion under nmbioning, it is finltion rf ,,pæirwise

similænity,, _: will _bhe be $eid tô be similar with cross*r:rice elasticities af q, the $eme $ign, bhat i$o with ,"r"i

( 2. r6) ^€u

u u

Êq

g o rvolcf . ^a ,aitu ro). (volol

0)

r o _ôq /Av >0

{i, h} ; s * {J, k} i i, _{r ll;h}

a

k x 1r

J x I,

Lrl

m

F'rr' ,'lll'pc'çte r-rf cl.atnity, outrrults ui r at ; i. = l, ., a t n ;

with F:nices Ë,i anci inputs *h x *" qh î Fr = L, rin, m ; wi.th _Frrices 'h âre noh, identified. rnequai.ity (z,ta) _mi:y ,.ren be wni.tten _i.n tenms of the second pæntiar.erivebives.f

the pnoflt function II"(v) and in terms af the cross! r:,nice derivatives ûf the facton

demand or outr:'ut suppl'v e.'uati.ons (z)" _{Tæb1.e 2" 1.} $unrmanizes these different exrrnes*ions bv disti.r-rguisr-r ing the r^rætune, output _ôn input, of ther netputs taken into account"

Iinsert tæh,le Z.I]

As frn example, two inputs *h _æncr *k ans sai.d to be similar witr-r respecb to inpub x o if ^b,<lth âre substitutes _{f or-,}

ârs compremenrs ro *o" îwo {iLrrpurs ;-";;'ï 'll. *o t't' ir ^both Eimirmr wi.th re.q. Éà.* ,-T ,. -:

-

*:'l-.t"* Yi æncl

" j &rs sæid t. be

in bhe o""o,*ilï.:: ::_;""::":::: ;:"::: ::,:ï:;1":"::,J"J:"ï;

'nput *h êncJ sæi,J to be similær wibh nespect r* ,,.or*-îl _,," t:

-"1

to.rr^ is non negative, thæb is if inplrt x ic -rr.o u {€ rr rnpuf *o is, sup}*rj.ûn [i.nferj.,:n]^ê,,Frâ_h in the pnoduction of ^y . and i.nputs _*

h and ^* o âns com'lement:

(subsbituable).

Simi.tar ir't*"o"u.tations of i.neqr-.renl.f.ty (;:":s) _can be exten,Jed to at L the possible _ca$e.s

"

A ^pæirutise simitæri.by of two netpr*rts r^rittr re.spect tc, _{m thin,J} netput is def ined at a given e,ruilib:rium p+int. As a:n exêmtr,.1.e.0

gi'ven pnice vector v' simf.rani.tv is ciefined _crn the bæsis c,f u^re

unconstrained pnofit function" Af ùourse srmilmnj.ty

wirh _resrrecr ro Qo ar this poinr _does ror impl.y simiil,-i;" -iJ

"l;

othen _{equi l} ibrium poinb, anc, rfiore generar r^y, _cr,]e.s nût i.mpl.v

simil'arity slobal'ty" As a conseqLrence, bw' netputs 9, .*r-rcJ e" carl

be eimilar uf.th _nespect

II"(r') ano noL at the pciint E*'conr^esponding to t'(v*)" ït cen be

nemdilv verified thet a technorogv which is _normæt in the _{$en.$e af} sakai (r974) has alr pairs of inc,uts and outputs simiror" rt mây

be called strongly similar. This condition ûf similanrityn _r,rhj.ch

I1"

cloers not seem ovenly nestnictive _î.ûr deflnition of goods, v,,iIl ellow to cro$.$ pniÇe effects bebween unconstnained constnelning _some quantities"

S, The modlfled _cross

nat lonlng

e fmirly _eggresâted chanmcteniee how _the goods êne eltened _by

effects of the compÊnâtive statlcs _unden

The main nesulbs of componabive sbabics uill be carried basj.cally ôn system (Z.tû) which _assumes (2,ç) _known, but in order b.' make the resurbs _more tnansparent _ând easier to

'ntenpnet (i) inputs end outputs are sepôrfrted agâinn anql ⁽ ii ) , t' keep

notmtions hopefulry simple _mnd to rrnovide conveni.enb _formur^ee for empirîcar sss: the basic equation

corrss,,onding to (2.g) is written

'n berms çf price eræsticities, hrith a serf _evidenb pentition of the matnix _ône gets _;

1

I 7

I rtl

Y

(n+rn)xl

o 11 10

F F"

&

i9

b

7

o

7

(ô. _{l" )}

p

^o

*o1 p

*00 r

e

t/l/

(t) 7

o

7

o Y

X

;

o1 oo

oo

E

G

F

77 77 10

H o1

H

+

G

H

Hol

^ ¹⁰

^00

(t o

b (n+m)x1

(n+rn)xl (n+rn) x ⁽ⁿ⁺ⁿ⁾

t2

h,hene

"

t, . i r

nrs the vectons crf pencenteÇe changes in unconsbnæined output and

^ tnput quênt{bies _wlbh

vecrors or pnice _chanees i r end , t _*,..,o

ti* ;;j. ;ïriïï"i;:

technical pnogness

biases expressed in percentege terms ûs wett, ïo keep the analytics _.sirnple

only one outpub guoto end one _{f i xed} fæcton are maintained. io^:* the pencentage

chanse of the quotæ

with _shadow price _po end * o,

"o *.* ** equivarenb notabions for fixed inputs, *o ancJ io ane the rêbe of technical _chançe bias on

fixed quæntities" t3"s) has to Lre sorved for the vect.r t;r, oo,

"7 ^O-

x ' ^o J in berms of rhe vector t;t,

"o, "r, -*.;

;""if the b,iases

;;," Ët b'$' (3'x) is firsb solved for rhe virruar price _chanses

l:;1 ;::l

oo

Goo *or

E

f;:1 *

F;:l

oo

H f;;1+f;,1

oo o1 o1

Fo1

,.

p 1

p^o ^o

^o b

+l (,

(3.2)

crn _elf ter tosether _i

H

G

oo oo oo

7

o w

6J

collecting unnætioned

end rationed oubputs and _fæcbors

(3.3)

the matrix

D x _{Hoo H}

f,:l l.

f:"";::l

o1 o1

F r

o1 o1

G H

Fnom bhe sbrict convexity _of the production seb, _{hre know thab} is non singular and j,ts determinant

G F is nesmbive (3).

x.3

The \fulr

companstive statics of endogenous ver.rabres i.e"

shadow pniceer _{QUôta nent}

, frxlbv ross end unconsbnained _suppry nnd demandr

câr. be investlcated by sorving equation _{(3. 1)}

" ^Thrs has been d.ne ersewhene _{(Hahé mnd}

Guvomard, lggg) ano wirt not be

punsued here (4)" The impacb of quobas and fact'n fixity on cnoss

price e'esticities wirl be explcred unden the c.ndition .f painwise similarity introduced _ebove.

ïn

a$sLJme

chanses

onder

NOhJ Y^(} :CX

Lo _oi,nvest _{isæt e} only the

o crôss _{pr i} ce _{ef f ect s} r ^v/ê x O" Then bhe ^percentage be expnessed a$ follows _:

rê*b o ^1 _^.r

cr=l)

in unconstræined guæntibies _cmn

(3,4) ⁷

(3.5)

* _[E ^11*

+ F10

+ _[F ^1l-n + F10

r*Foo _^o1

\k()+

* ¹⁰

L

(*Eoo

(*Hoo Io1*

(*rioo _Fo1*

Ho1

v _F.¹⁰

cot)

HotJ

GotJ Goo Eo1)

rot)

_o1,h.)

rotJ

00 -1

-1

F _D

D

p* ^tJ _p^{^7}

*ool*

$^oo

* o* tl t$/ 1

*oo -7

F 7

X x _{IG11*- Glo} (*uoo Eol*

+ _{Hlo (*roo oo1} + Goo

+ [Hlt* G10 (*noo Fol*

+ Hlo (-Ëoo _F.tot ," coo

D

p* ^1l

t-'tot)

1

*ool-

p

D-7

7-

D J ⁷

propenties

(s) con+erning the ôros$ price unconstneined quanbibies _{cÉn now}

be derived, _under of similaniby of eech pei.r of _sood$ with respect to first _examine

the one quole on one fixeo factor

unembiguous.

effecbs _ômong the _assumpbion y and x" ; On ^rre

ca$e, which is pnopenty 1 : unden the similmr{ty conditlon s blnding constnalnt

T4

on en ôutput or sn

mone substltutable, ^{input makes} the unnestnicted inputs end outputs rrnopenty Z : unden the simi IanJ,by condition, a binding _constnmint on ên output or ân input _makes the unrestnicted _oubpubs negressive and consequently th* unrestnJ.cbed inpubs _more Inferlor.mône

yf only one quoba is implemented, bhe system (J" _t^ ) is wnibten _wibhout

and _{ô. S) simplify _inbo

t"thich _emounts bhe I mst _r*oçr,

to æssuming

e x r:re ss i on.s

bhat

(.b, _{4 )}

(3" _6.a)

(3. _sa)

"t = lrtt * _{'-ro (Eoo)*r *orl} L*J ir

+ [rtt - ^{Elo (Eoo)} *t

FrJ;,

7

X [r" (roo;*r

[*

tt - o'o ^(Eoo) *,

G10 Eol ;

"'J;'

1

*

A typicat element _of between outputs i and i in

the matnix (s, ae) is E

()f _cno$.s

11R

iJ

price elasbicities

(s. o) E11R

rJ 7t

iJ

under the similænity {s positive,

o1

oj

conditionr h,* have E o1 oJ

-oor - ¹

tr) ^* _TF'¹⁰ to *

> û and since H oo 77

tJ

.utpute tend to become more substitutabre _{unden the}

10

to ^Ë

11R TJ

(s.z ) r sr

And the bw<>

output guote.

15

ïn

bet _ween

(s" s) ^r-r_hl^11R xH ⁷⁷

hk

In thre cfrse crf

s O, Lhenefone _hi complements _ôn

ê sim j.Inn _{f ashion}

inpubs f.n (J.Sa) is _{given by} ^{a typicat} _: ^nat ioned cro$s _e,tast icity

(E ^{o 10}

ho ok^o1

oo

o 1

G F

H

similanity Glo ^Ê

1lR ,q ^hO

hk = H;;, ^hence rnone substJ.butes-

argument j.t cet-"t be seen that _cro$s

. t " output rrnices _{become smal} tourard inferj.oriby, Ier

By ê simitar

j. nput _{demând$ hr.} r hence the _tendency (s. sl e ^11P

hJ

77 hJ

(s. ro) _riJ^11R 7

r _o1_or.

I ess

10

ko > O, ^ancj inpr"lts h mnd

by symmetny _G10

k tend _{tc: become}ho

(roo)*1 _G

h and output i w" r one fixed fæcton"

*Iasticity _of algebnaiclyo

out put _rJnder

-;1" _$imiran

10 ho ^L

o1

oj'

$imilerity of ^j.nput quota ao impl ies _{G.1: E} ^o1 resurrs _{cæn be} deriJ:";:i"

t . the

11R

hj<

This is as fan ôs similari.ty _cæn bning unambiguous resurts"

As rnentioned before, _when

one stærbs with the ]ong*nun _nor.nrcr.

technology af $akai, where similmrity is venifi.ed, implementing qu*tas on fixed fac'tors

one æt a bime r^,ri.rr induce _m.rs substitution and more inpul inferiority" _Hence.

we get awæy fr.om

the n.r'ma. technorogv and at sorfie poinb there _{wi r} r be Ê:a'rs _of inputs or outputs which _mav

n.' l.nger be simirar" ït would _be convenient bo have _{mone çene.naI} re$ul^ts _when srv*rêl outputs _on inputs âre constrained in the _same time as in exr:ressi.rr _{(3. 1)}

r^rrri'ch reads to thie foll0wins reretion _between

rr]R *n.r -;; r

77 *1

IJ + ^D t-

10

io r tol

i0J _oo

G ^-FE;lLït

1.6

ït r.s convenient

with _known r>ropenties

(3. ea) r_iJ^11P

to r"ewrite _{(g. XO)}

eppeor$,

so thot _{submatrlx n}

Dl)oo

E ^J

J 7

I

p

Y o

o w

X

vo

Y

D 1[

7 +

o û, with

-ô*o / a'

ô", / ^ôpo

*ôvr / ^auto

(s. en) o (Ar /Alo, ôr lô^ro:

ôxo

^o

ot^J

ô*o q)^o

ôvo

^o

olrJ

3j

ôr>

ôy^o

i j

i

ôp _J

As D is nege*ive, if 0 is non negative, then the _tendency toward subsbitution wourd f.r10ç,r. ït is wonth notins first _bhet when i' x i' û is _now û quadrabic form _êround a .$vrnmetric p*sitive def inite _mætrix

" ^hence the Le Chatelier, s ef f ect is verif ied" t^Jhen

i # j, the simileniby condition

can be used bo write e êsà

quesi*def înite _f orm _{e" s. l,lunebæ , 1977, p. _{6?) ,}

?:i ô", ^ôY,

for;"" x";t,with<r)0

CJ

^o

OX

tu

**lôv

ohr^o

J

p*tôy-

^o

Ow wirhp>û

wnitben as ê quddnôtic

form around ê non

ô*o

Then

symmebric û cân be mætri x, (av _.

û * _l* j

[*'

ôy ôvo

o

cr p

p

dW^o

ôxo

^o

oË)

l-odw

{

ôw

^o

Ol,

T7