Siebe , Ho s
A icle
En i onmen al Policy, Alloca ion o Resou ces, Sec o
S uc u e and Compa a i e P ice Ad an age
Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de
Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik
P o ided in Coope a ion wi h:
Duncke & Humblo , Be lin
Sugges ed Ci a ion: Siebe , Ho s (1978) : En i onmen al Policy, Alloca ion o Resou ces, Sec o
S uc u e and Compa a i e P ice Ad an age, Zei sch i ü Wi scha s- und Sozialwissenscha en
(ZWS) - Vie eljah essch i de Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü
Socialpoli ik, ISSN 0342-1783, Duncke & Humblo , Be lin, Vol. 98, Iss. 3, pp. 281-293,
h ps://doi.o g/10.3790/schm.98.3.281
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/291410
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En i onmen al Policy, Alloca ion o Resou ces, Sec o
S uc u e and Compa a i e P ice Ad an age
Von Ho s Siebe *
The pape analyzes how en i onmen al policy a ec s sec o s uc u e, he
alloca ion o esou ces and compa a i e p ice ad an age in a wo-sec o
equilib ium model.
En i onmen al quali y can be conside ed as a public good being in-
luenced by emissions om consump ion and p oduc ion p ocesses.
Since he assimila i e capaci y o he en i onmen has been used so
a as a ee p oduc ion inpu , he common-p ope y- esou ce has been
hea ily o e used. En i onmen al policy a emp s o decide which o
he compe ing uses o he en i onmen should ha e p io i y and by
which policy ins umen s en i onmen al sca ci y can be in e nalized
in o he decisions o he subsys ems o he economy.
In his pape we analyze o wha ex en en i onmen al policy will
a ec sec o s uc u e, he alloca ion o esou ces and ela i e p ice.
Since ela i e p ice in a closed economy de e mines compa a i e ad-
an age ( ela i e o he o eign coun y), he pape also answe s he
ques ion o wha ex en en i onmen al policy will a ec he com-
pa a i e p ice ad an age o a coun y. In o me models (Siebe 1974,
1976) only pa ial equilib ium models we e de eloped neglec ing he
demand side o no closing he model wi h espec o eceip s om he
emission ax. In his pape , he model is closed wi h espec o he
demand side, and demand condi ions a e explici ly aken in o con-
side a ion.
The ame o e e ence is a wo-sec o -model in which p oduc ion
gene a es pollu an s as a join p oduc . The go e nmen le ies an
emission ax pe uni o pollu an emi ed o he en i onmen . We
ea he en i onmen as a na ional public good and igno e in e -
na ional, ansna ional and egional en i onmen al sys ems1. Sec ion I
p esen s he assump ions, Sec ion II de elops he model, and in Sec ion
III he implica ions o he model a e discussed. Sec ion IV poin s ou
possible ex ensions.
* Uni e si ä Mannheim (WH), D-6800 Mannheim A 5.
1 On hese p oblems compa e Wal e (1975).
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282 Ho s Siebe
I. Assump ions
A 1. In o de o keep he model as simple as possible we assume only
one ype o esou ce, R, and a p oduc ion unc ion
A 2. The p oduc ion o commodi ies i
—
1,2 gene a es pollu an s
as a join p oduc . Fo simpli ying pu poses he e is only one ype o
pollu an s. I is assumed ha pollu an s emi ed ise p opo ionally
o p og essi ely wi h esou ces used, i. e.
(2)
S i
=
Hi
(Q;) = H{
[F{ (Ri)]
=
Zi (Ri)
wi h Z; >0, Z'/ ^ 0; H; > 0
The con ex emission unc ion Zi is sugges ed in ui i ely by enginee -
ing p oduc ion unc ions2. Wi h he ac i i y le el o an engine eaching
o exceeding capaci y, i is ealis ic o assume ha inpu s ha e o be
inc eased p og essi ely o an addi ional uni o ou pu . This sugges s
ha emissions ise p og essi ely. A mo e p ecise explana ion o
he con exi y o he emission unc ion ollows om he applica ion
o he mass balance concep o p oduc ion unc ions. In e ms o
weigh a mass balance exis s be ween inpu and ou pu . Since mass
canno be los in a p oduc ion p ocess, egula ou pu and emissions
mus be iden ical o inpu s, in weigh e ms. De ine he p oduc ion
unc ion in weigh e ms. Fo a gi en echnology his unc ion mus
be conca e, i he egula p oduc ion unc ion is conca e. Then i ol-
lows ha he emission unc ion Zi mus be con ex.3
A3. Resou ces may also be used o aba emen pu poses. Le S?
indica e he quan i y o pollu an s educed in Sec o i. The aba emen
unc ion is gi en by
The aba emen unc ion desc ibes a echnology ha p e en s pol-
lu an s om en e ing he en i onmen . Addi ionally i could be as-
sumed ha a echnology exis s o educe pollu an s ambien in he
en i onmen (wa e ea men ).
A4. Ne emissions o pollu an s ambien in he en i onmen a e
de ined as emissions p oduced minus emissions aba ed. A di usion
unc ion is no explici ly in oduced.
(1) Qi = F
(Ri)
wi h
F'i
> 0,
F'l
< 0
(3) s . =
Fj:
( î) wi h
F1.'
> 0,
F i
< 0
(4) Si = S -Sl
2 Gu enbe g (1972), S. 326.
3 Compa e Son heime (1975).
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En i onmen al Policy, Alloca ion o Resou ces 283
A 5. Fi ms maximize p o i s and ega d commodi y p ices, ac o
p ices and he emission ax as gi en.
A 6. The esou ce can be used o p oduc ion and aba emen and
is gi en
(5) + H2 + R[ + R
2 = R
A 7. Commodi y demand is gi en by
(6) Q? =Di(p, Y)
wi h p = pi/p2 whe e pi indica e nominal p ices and p is he ela i e
p ice.
A 8. Income Y is de ined om he p oduc ion side. The e a e no
sa ings. In o de o close he model, we assume ha he go e nmen
spends he ax income ecei ed in o m o ans e s o he households.
Consequen ly disposable income o he households is iden ical o ne
na ional income a ma ke p ices and is de ined as
(7) Y = pQi + Q2
Obse e ha Y includes ans e s no explici ly shown and ha p
is consume s p ice and no p oduce s p ice. I Y would be de ined
wi h espec o p oduce s p ice p*, emissions axes (and ans e s)
would appea explici ly on he igh side o (7).
A 9. Commodi y ma ke s mus be in equilib ium so ha
(8) Qi = Q
A 10. The go e nmen le ies an emission ax z (in nominal e ms)
on ne emissions Si, wi h z being changed pa ame ically4.
II. The model
1. Fac o demand by he p o i maximizing i m is gi en by maxi-
mizing
U =
Pi
Qi - (Ri + R]) — zSi
s. . Qi-FilRj)^ 0
4 Addi ionally, a damage unc ion may be in oduced indica ing ha en i-
onmen al quali y is a ec ed by ne emissions
U = G (S) wi h G' < 0, G" < 0.
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284 Ho s Siebe
H{ (Qi)-S?< 0
^ - F (R ) < 0
- Si + SV - sj < 0
whe e deno es he esou ce p ice in nominal e ms. Assuming ha
p oduc ion akes place in bo h sec o s, i. e. ha we ha e a meaning ul
p oblem, and de ining = ?/p2 and z = z/p2 he condi ions o p o i
maximizing ac o demand a e gi en as
R = (P- ZHJ F[ (Rx)
(9) = (1 - ZH2) F2 (Rg)
= zFl' (Rj)
2. The sys em o equa ions (1) - (9) has he 17 a iables S?, Si, S {,
Qi, Ri, R i Q , p, Y and and 18 equa ions. The de ini ion o Y in equa-
ion 7 s a es ha o al demand is equal o income, so ha in a wo-
sec o model he equilib ium condi ion o one o he p oduc ma ke s
is edundan (Wal as Law) and should be omi ed. By subs i u ion he
sys em can be simpli ied o
F1 (Ri) = D1
(p,
pFi (Ri) + F2 (R2)) (i)
= zF l (Rj) (ii-iii)
= (p-zH1) F[ (R{) (i )
= (1 -ZH2) F2 (Rg) (V)
R = Rj -j- Rg Rj R*2 (Vi)
To al di e en ia ion and subs i u ion o (10 i) in o (i) - ( ) yields
al 0 0 1 dR1 — H j Fj dz
0 a2 0 1 0 dR2 -H2F'2dz
(11) U 0 1 0 dR[ = F i dz
zF 2 zF 2 zF 2 1 0 d F 2dz
~D1YF2 0 0 -b2 dp 0
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En i onmen al Policy, Alloca ion o Resou ces 285
The coe icien s a e de ined as ollows5
(12)
o<
= zH'l F? - (Pi - zH ) F'l >
0
(i)
b1 = l-DlY == D2Y (ii)
b2 = Dlp + D1Y Fx<
0
(iii)
In
(12 i)
we ha e
Pi
= p o i = 1 and
p*
=
1
o i = 2.
III. Implica ions
The p oblem is o de e mine how he alloca ion o esou ces is a -
ec ed by en i onmen al policy. We i s s udy how ac o demand o
i ms changes wi h an emission ax o a gi en ela i e p ice p. We
hen analyze how ela i e p ice changes, and how he alloca ion o
esou ces and na ional income a e a ec ed in his case.
1. Assuming a gi en commodi y p ice and conside ing only he ac o
ma ke condi ions and he esou ce cons ain , i. e. (10 ii - i), he change
in ac o use in esponse o an inc ease in he emission ax is gi en
by he subsys em o equa ion 11 no con aining he i h ow o he
sys em and no con aining he i h column o he ma ix o de i a i es.
F om appendix I we ha e he ollowing esul s
dRi
(13
i) —-i > 0
dz
(13 ii) <0: H^F'^H'zF,
ZdR;
(13
iii) -^<0
H- F- indica es he ma ginal endency o pollu e (pe uni o e-
sou ce). Hx Fx > H2
F'2
speci ies ha Sec o 1 is pollu ion in ensi e6. Fo
5 b2 < 0 ollows om Slu sky's ule. Le D^ deno e he pu e subs i-
u ion e ec . We ha e comp,
Di*> = Di*W ^ o Di*w = Di*> + (Rl) = b2 < 0 ,
since he pu e subs i u ion e ec is always nega i e.
S Sp
« I Z1 (Rx) > Z2 (R2) o Rx = Rg,
H^ F^
>
H'2 F'2
can be speci ied as~ > ^
o R{ > R2. This ans o ma ion also holds o R < R2 i ei he Sec o
1
is no
" e y" small compa ed o Sec o 2 o i Z2 has a su icien ly weake cu -
a u e han Zx. Obse e ha
H'x
F^ >
H'2 F'2
demands ha he Z unc ion
should no c oss, i. e. emission in ensi ies should no be e e sed.
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286 Ho s Siebe
gi en commodi y p ices, an emission ax will educe he ou pu o he
pollu ion-in ensi ely p oduced commodi y. Resou ce use in he pollu-
ion aba emen ac i i y o he pollu ion-in ensi ely p oducing sec o
will inc ease. Also we know ha esou ces will be wi hd awn om
p oduc ion and will be used in aba emen ac i i ies. This e ec does
no depend on di e ences in he p oduc ion in ensi y o he wo sec o s.
2. Allowing he commodi y p ice o a y we ha e he ollowing
esul s om appendix II (as su icien condi ions):
d' y > 0
(14i) A>0: { ?F T , ,
I D^ + D^F^O
dR
(14 ii)
--1 > 0: D1YF2 + D^yFj > 0
Assuming (14
i)
is gi en we ha e
2 dR{ ^
(14 iii)
——1 < 0
(14
i )
dz
dRL<0. J D1Y> 0
dz ' H; F[ > H2 F2
dp ^ ( h > H' F'
(14 ) - ->0: > >22 < '
dz y a2 D2Y FJ p > ax DlY F2
dY ^
(14
i)
- -<0:
a z
H1F1> H2F2
F2
1 - —2-
— )lp >
<x
> 1 wi h a =
-
1 -
PFi
In (14
i)
jip deno es he di ec p ice elas ici y o demand.
Diag am 1 summe izes he su icien condi ions o equa ion 14.
We ha e he ollowing esul s:
3. Assume bo h commodi ies a e no in e io , so ha hei ma ginal
p opensi ies o consume (DiY>0) a e nonnega i e. Then he de e -
minan
A
> 0. Assume Sec o 1 is he pollu ion-in ensi ely p oducing
sec o . Then we ha e:
Resou ce use in he pollu ion-in ensi ely p oducing sec o will de-
cline. Resou ce use in each aba emen ac i i y will inc ease; esou ce
in p oduc ion (2 Ri) will be educed.
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En i onmen al Policy, Alloca ion o Resou ces 287
Diag am
1 :
Su icien condi ions
En i onmen al policy will shi he sec o s uc u e o he economy
in a o o he aba emen ac i i ies wi h p oduc ion being a ec ed
nega i ely. We can es ablish ha p oduc ion in he pollu ion-in en-
si ely p oducing sec o will decline7. In he less pollu ion-in ensi ely
p oducing Sec o 2, esou ce use may inc ease o dec ease. The model
allows bo h cases. Thus, in one case Sec o 1 and Sec o 2 lose esou ces
o he aba emen ac i i y, whe eas in he o he case, Sec o 1 loses
esou ces o Sec o 2 and he aba emen ac i i y. Unde he assump ions
made, emissions will be educed and en i onmen al quali y will im-
p o e. This ollows om 2 dRi/dz < 0 and 2 dR {/dz > 0.
4. Condi ion (14 ) o a ise in he ela i e p ice can be spli in o wo
su icien condi ions8.
i) D'2Y > pDlY. Unde he condi ions speci ied below, na ional income
will decline as a consequence o en i onmen al policy. Then D2Y> pD'lY
gua an ies ha demand o he pollu ion-in ensi ely p oduced com-
modi y 1 is educed less han demand o he commodi y 2. This di -
e ence in he income e ec o he wo commodi ies makes su e ha
7 This also holds i Sec o
2
we e he pollu ion-in ensi ely p oducing sec o .
8 No e ha F1P>F2 ollows om H[ FX > H'2F2.
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288 Ho s Siebe
he ela i e p ice o he pollu ion-in ensi ely p oduced commodi y
mus ise.
dR* . dR2
ii) a2 > ai: |—— | > | | speci ies9 ha ( o gi en p and z), Sec o 1
is mo e sensi i e o changes in esou ce p ice han Sec o 2; we may
also say ha Sec o 1 is mo e dependen on he esou ce R. This con-
di ion can be in e p e ed as a udimen a y o m o a ac o -in ensi y
condi ion in a one- ac o model. We can expec ha his condi ion
un olds in o a se o ac o in ensi y condi ions in a mul i- ac o model.
As a esul we ha e:
The ela i e p ice o he pollu ion-in ensi ely p oduced commodi y
will ise i he ma ginal p opensi y o consume o his commodi y
is lowe han he less pollu ion-in ensi ely p oduced commodi y and
i he pollu ion-in ensi ely p oducing sec o depends hea ily on he
esou ce R.
Su icien condi ions o a ise in he ela i e p ice can pa ly sub-
s i u e each o he . Assume Sec o 1 is " e y" pollu ion-in ensi e10. Then
he ela i e p ice o commodi y 1 may ise e en i i has a high income
elas ici y o demand and loses demand quan i ies wi h a decline in
income. O o iden ical pollu ion-in ensi ies o bo h sec o s, he ela-
i e p ice will ise, i Sec o 1 has a su icien ly smalle p opensi y o
consume and hus loses a smalle quan i y in demand equi ing a
highe adjus men in ela i e p ice. Finally, assume Sec o 2 hea ily
depends on esou ce R. Then p can ise ne e heless, i Sec o 1 is su -
icien ly mo e pollu ion-in ensi e o i Sec o 1 has a su icien ly lowe
income elas ici y.
dp/dz > 0 indica es ha en i onmen al policy will unde he con-
di ions indica ed a ec he compa a i e p ice ad an age o a coun y.
Assume a coun y expo s he pollu ion-in ensi ely p oduced commo-
di y 1. Then en i onmen al policy will educe he compa a i e p ice
ad an age o ha coun y11. Conside wo coun ies wi h a di e en
abundance o en i onmen al se ices and assume ha z > z* e lec s
he di e ence in endowmen wi h z* indica ing he emission ax o he
» Di e en ia e he ac o demand condi ions (9) o gi en p and z wi h
espec o
10 Assume o ins ance Sec o 2 does no pollu e a all wi h H2 = 0.
11 I a coun y expo s he less pollu ion-in ensi e commodi y and i i
unde akes en i onmen al policy, i s compa a i e p ice ad an age will be
imp o ed.
dR 1
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