„- '~~R Discussionfor a er8414 ~omic Research1993

76

iiumiu~ im~uii piigiii~iini~m ~ in~

Centerfor

Economic Research

No. 9376POLLUTION CONTROL AND THE

DYNAMICS OF THE FIRM: THE EFFECTSOF MARKET BASED INSTRUMENTS ONOPTIMAL FIRM INVESTMENTS

by Peter M. Kort

November 1993

ISSN 0924-7815

Pollution Control and the Dynamics of the Firm:the Effects of Market Based Instruments on

Optimal Firm Investments

Peter M. ICort

Econometrics Department and CentER, Tilburg University, 5000 LE Tilburg, The

Netherlands

E-Mail: Kort.CdKIJB.NL

This contribution belongs to a category of papers that attempts to determine the ef-fccts of environmental regulation on the growth of an individual firm. It extends theexisting literature in at least two ways. First, our pollution function explicitly deals

with t.he facl. Lhal, it is more di(ficult to reduce pollution by abatement activities when

pollution is already low. Second, accotding to our knowledge it is for the first time that

marketable pollution permits are incorporated in a dynamic model of the firm.

In the paper we establish the effects of a pollution tax and marketable permits on the

behavior of tt~e firm. For the tax model as well as the marketable permits model we

prove that the equilibrium is stable and approached monotonically, and we derive for-

tnul.~.ti for opl.imal invc~stmcnt policics. I~urthcrmorc~, we carry out a phase plane analysis

for Lhe pollntion tax case, and, linally, a condition is obtained under which long run firm

behavior is the same when either a tax or marketable perr}tits are imposed.

~ ~~j~K.t~.t~.„ ~iE~Li~iTF-lEcK

T~~B!!~G

i

1 Introduction

One of the most important problems of firm behavior in practice these days is how toreact on environmental regulation. The government has several policy instruments at its

disposal to give the firm an incentive to reduce pollution.

In case environmental problems develop smoothly and gradually it is well known thatmarket-based approaches, like taxes and marketablo permits, have important efficiencyadvantages over pollution standards, thus restricting pollution emissions directly (seee.g. Baumol and Oates (1971)). But, as argued by Hahn and Stavins (1991), in practicepolitical and technological constraints can occur that lead to a poor performance of thesemarket-basecl instruments. Therefore, it is important to recognize that the nature of in-dividnal cnvirunn,rntal prol,l~~niti c~;rn ,Irainatic~ally a(firt thc~ choicc uf prcfcrrccl policy

instruments. 7'hus, for exarnple, for highly localized pollution problems with threshold

damage functions (e.g. Dasgupta (1982), Figure 8.3), source-specific standards may be

appropriate, whereas for pollution problems characterized by more uniform mixing over

larger geographical areas, market-based approaches may be particularly desirable. Thisleads to the conclusion that the ideal policy package contains a mixture of instruments,

with taxes, marketable pcrmits and standards each used in certain circumstances to reg-

ulatc thc suurccs of cnvironmcntal damagc (Baumol and Oates (1988), p. 190).

Ilence, in detennining a theoretical [rarnework (or establishing the effects of environ-

mental regulation on the firm's investment policy, each of the above mentioned policy

instruments should be addressed. Contributions in this area include Helfand (1991,

standards), Kort, Van Loon and Luptacik (1991, tax) and Xepapadeas (1992, tax f

standards).

1'he aim of this paper is to establish the optimal dynamic behavior of a firm facing either

imposition of a pollution tax or a market for pollution permits. Here the specification

of the pollution function is improved compared to models in, e.g., Kort, Van Loon and

Luptacik (1991) and Xepapadeas (1992). In these contributions a given abatement ex-

pcnditnrc I~~ads to a givcn pollntion rcduction, which is thus irrespective of the amount of

pullutiuu ~'ausecl Ly the prodnctiuu process. '1'his suggests that pollution can be driven

to zero or even become negative, and, indeed, in these models a constraint occurs to

prevent that pollution becomes negative. In our formulation we adopt the more realistic

assumption that marginal costs of abatement increase sharply as the level of pollution

shrinks, which implies that driving pollution to zero is very expensive if not impossible.

A dynamic rnodel of the firm in which the firm faces a pollution tax is developed in

3

Sc~ct,ion 2. lu Sc~ct,iun :3 wc solvc~ Lhis rnodcl by performing a stability analysis, deriving

neL presenL valuc forrnulas and dcvelopiug dynarnic adjustment paths. In Section 4 we

consider a firm that has to buy marketable pollution permits in order to be allowed to

pollute the environment, and we compare the results with the tax case. The paper is

concluded in Section 5.

2 Pollution tax model

Consider a firm that has the possibility to invest in two different sorts of capital goods.

One is productive but also causes pollution as an inevitable byproduct. The other one is

non-productive but cleans pollution. The firm's pollution function is given by E(Kr, K2)

which satisfics:

E(lí r, KZ) ~ 0 for all Kr ~ 0 and Kz ~ 0; E(O, ICz) - 0 (1.1)

E,,.,(Ií~,líz) ~ o; E~..,h~,(Kr,K~) ~ o (l.a)

hh,(líi,líl)Gllfurall lii~U; l.hzhz(li~,h~)~llforalllír~0 (1.3)

EK,K~(Kr, líz) - Eh,F,(Iír,Iíz) G 0 (1.4)

Ex~ti~Ílír,lís)Eti-zKz(Ií~,líz) ~ ~En,x2(lít,líz)~~ (1.5)

in which:E(lí ~, Ií2) :(low of pollution generated by the firrn,

being a function of Kr and líZr

Kr - Ií~(t) : stock of productive capital goods at time t

Kz - Ií2(l) : stock of abatcrnent capital goods aL tirne t

(1.1) implies that pollution output is positive as long as the productive capital stock is

positive. Thcrefore, pollution is never non-existent when the firm produces.

~Stock characteristics of environmental pollution are not considered here. This is because according

to Xepapadeas (1992) stock effects are particularly important in a model where the objective is tomaximize some welfare indicator and not in a model where private profits are maximized. The pollution

(low generated by the finn contributes to the total pollution stock of the whole economy.

4

(1.2) states that pollution increases in a convex way with increasing productive capitalstock, for a givcn Icvcl of abatrnicnt capital stock.

(1.3) means that pollution output is smaller the larger the current level of abatementcapital stock, for a given positive stock of productive capital. (1.3) also implies that wehave diminishing returns to abatement capital stock.

(1.4) states that the increase of pollution, due to an extra unit of productive capitalstock, is smaller the larger the level of abatement capital stock. Alternatively, the de-crease in pollution output, due to an extra unit of abatement capital stock, is larger thelarger the current productive capital stock.

Because Eh~,r,., G 0 and EF,h., ~ 0, our formulation has the realistic property that moreabatement investments are required to reduce pollution with some fixed amount as thelevel of pollution shrinks.

In Xepapadeas (19J`l) ancl in Kort, Van Loon and Luptacik (1991) the pollution func-tion is separablc in lí~ and líz. '1'his implies that a given amount of abatement capitalstock leads to a given reduction of pollution, irrespective of the current level oí pollution,and also that pollution can be zero. In our formulation we got rid of these unrealisticfeatures.

Finally, (1.2), (1.3) and (1.5) imply that function E is strictly convex in (Kr,Kz).

According to Jorgenson and Wilcoxen (1990, p. 330) Lhe most important response of afirm to environmental regulations is investment in costly new equipment for pollution

abatcment. Therefore, by introducing abatement capital stock as a state variable we as-

sume that the firm can build up abatement capital. Both capital goods evolve according

to the standard capital accumulation dynamics:

Kr - h - ac Ií r, Kr(0) ~ 0 (2)

líz - Iz - azlíz, líz(0) ~ 0

in which:Ir - li(l) : ratc of invcstmcnt in productivc capital goods at tirnc t

Iz - Iz(t) : ratc of investment in abatement capital goods at time t

ai : clepnr.ial.iuu ratc- of thc prodn~ tivc~ capital guods

(ar 1 0 and constant)az : depreciation rate of the abatement capital goods

(az ~ 0 and constant)

(3)

5

Gross earnings of the firm are given by the instantancous revenue function S- S(Kl).

Assurne that S is twice continuously differentiable, S(Kr) ~ 0 for Kr ~ 0, S'(Kl) ~ 0,

S"(Kr) G 0, S(0) - 0. [Function S(lír) is defined as revenue after maximization with

respect to variable inputs, e.g. labor].

Investment is costly. Let, for i- 1,2, C;(I;) be the cost of investment with C; a convexand increasing function, C;(!;) 1 0, C;'(I;) ~ 0, C(0) - 0.The objective of the firm is to maximize the net cash flow stream:

maximize: J~[S(ICi) - Cr(Ir) - C~(IZ) - rE(lít, líZ)]exp(-rt)d1 (4)

in which:r: discount rate (r ~ 0 and constant)r: pollution tax rate (r ~ 0 and constant)

As argued by Pindyck (1991) investment expenditures are largely irreversible; that is,

they are mostly sunk costs that cannot be recovered. This comes from the fact that

usually capital is firm or industry specific, that is, it cannot be used productively by a

different firm or in a different industry. To include irreversibility of investment in our

model we add the following two non-negativity restrictions:

1~ ~ 0 (5)

h 1 0 (6)

'I'hr di~cisi~m prublrin uf Lho linu is tu cl~~t~,rinin~~ ~ui invrsl.nirnl. pal,h. {li(l), ~l(f.)} ovcr

an infinite planning period [0, oo), such that the objective functional in (4) is maximal,

subject to the constraints (2), (3), (5) and (6).

3 Solution

"Ib obtain the optimality conditions for the optitnal coutrol problern described at the end

of the previous section we use Pontryagin's maximum principle (see e.g. Feichtinger and

Ilartl (1986)). The current value Hamiltonian and Lagrangian for this problem are:

6

ll - S(ltit) - Ct(It) - G`x(Ix) - rF.(lí~, líx) -~ at(Jt - atlít) f~z(fx - azKz) (7)

L-llfvtltfnxlx

in which:a; -~;(t) : co-state variable belonging to lí; at time t; i- 1,2

p; - rt;(t) : dynamic Lagrange multiplier belonging to the constraint

I;~Oattimet;i-1,2

The necessary optimality conditions are:

-C.;(I~) t a~ f n~ - ~

-C~(Iz)f~xfrtx-~

ai -(r f at)~~ - S'(Kt) f rEh-,(lít,líx)

~z - (r t ax)~z f rEh.z(Itt, !íx)

7t ? ~, ntli - ~

riz?0, rlxlx-p

If furthermore the transversality conditions

ihm exp(-rt)a;(t)[K;(t) -!í;(t)] 1 0; i- 1,2

(8)

(9)

(10)

(12)

(13)

(14)

(15)

hold for evc,ry (casible solution (lí~, lí1), then (9)-(14) are also sufficient for optimality

sincc thc maxiniircd llamiltonian is concave in (lí~, Kz).

7

3.1 Stability analysis

To get insight into the dynamics oí the optimal solution of our model, a stability analysis

has to be carried out. By linearizing the non-linear canonical system around the steady

state ( Ií ~, !í2i a~, aZ, h, I2 ) we obtain the Jacobi matrix evaluated at the equilibrium.

An optimal steady state is a solution of:

Il-a~K1-0

Ia - azKz - 0

(r f ai)a~ - S'(líl) f rEh-,(líi, líz) - ~

(r f az)~z f rEk'z(Ki, I{z) - ~

-C,(h) f ~~ - 6

-C2(Iz) f ~z - 6

Then the Jacobian of the canonical system evaluated at the equilibrium has the following

furni:

J-

-a~ 0 1~Cï 0

0 -az 0 1~C2-4" -F rEh-,K, rEh,F., r f a~ 0

rE~.,h, rGh,F-, 0 r-} az

(16)

'I'hc~ dctcnninant of Lhc~ .lacobian ~waluati~d at the stc~ady st.atc is given by the cxpression:

detJ - a~az(rtai)(r~az)f

}a~(r t ai)rExzF'~ ~ az(r f az)(rEF,F~ - S")- rS„EK'K' fC2 C',' C;C2

T 1-}C'C2 (Is1h'~h'i Ish~zh-, - Lh.~h,) ~ Q (17)

According to reichtinger and Hartl (1956, Theorern 5.4) we have saddle-point stabilitywith monotonic rnotions if the Collowing conditions are satisfied:

ti ~0

0 G det J G líZ~4

in which:

IC -

alí i rllí ~

alf, aa,

a~, a~,ai~, áa,

i)lí.t illí~

a1í2 aaz

aa2 aa,arí2 áa,

Calculating lí Icads to:

-F 2

(18)

(19)

lllí i illí ialí2 aa2

aa, aa,ai, 2 áa2

TEh'x~,', l TEF,K~ ~ S~~ jIti -- C,~ - a2(r f az) J f~- Cl - a,(T' f a,)J G 0 (20)

To determine the sign of IC~~4 - det J we rewrite det J into (cf. Xepapadeas (1992)):

rl;h' h' r TGh ti- f S" l TzE2det J- - C ,~' x- az(r f az) I- C,~; - ar(r f ar)J - C,1C2 ' (21)

Due to (20) and (21) we obtain:

If2 - 4 det J-~L-TCkxFx - az(r ~ az) 1 ~-

z

TEh'~h'~ -} s~~ l l ~ 4T2i'iK~Ax-[- c~~ - ar(r f ar )J 1 f G,,,C~~ ~ 0 (22)

~ i z

We conclude that conditions (18) and (19) are satisfied so that the equilibrium is a

saddle-point which is approached in a monotonic way.

3.2 Optimal investment policies

The firm's equilibrium values of the productive and abatement capital stock satisfy:

S~(tii ) - (r -f ar)Ci(arlíi ) -f rEx~(hi ~ hx) (23)

-rER'~(I~i, Iís) - (r f ax)Cs(asKz) (24)

For both capital stocks it holds that in equilibrium marginal revenue equals marginal

cost. Cornparcd to thc standard invcstment modcls with convcx investment costs (e.g.

Takayama (1985), pp. 698-699), here marginal cost of productive investment has in-

creased with rEh-,. This is because owning an additional unit of productive capital

stock increases pollution with E~, so that additional tax must be paid at the expense

of rEF,. We conclude that introducing a tax on pollution results in a lower equilibrium

value of productíve capital stock, and thus in a lower level of production and pollution.

The marginal revenue of abatement capital stock consists of a decrease in taxation ex-

penses due to thc fact that the extra unit of abatement capital stock reduces pollution.

Comparative static analysis shows that, surprisingly enough, the level of pollution in

equilibrium does not necessarily decrease when the pollution tax rate increases. But,

from comparative dynamic analysis we can conclude that cumulative discounted pollu-

tion certainly decreases with increasing pollution tax rate (cf. Xepapadeas (1992), pp.

264-266).

Let us assume for the moment that the productive investment rate is positive. Then,

after solving the differential equation (11), substituting (9) (with rfr - 0) into this rela-

tion, and using (23) as a fixcd point, we derive that at each moment of time the level of

productive investment must satisfy

1,x,{S"(Ki(s)) - rl;ti-~(Iíi(s), h2(s))}

exp(-(a~ t r)(s - t))ds - C~(lr(t)) - 0 (25)

whcrc thc Icfl.-haud sidc is thc "net prescnt valuc of marginal investmcnt". For an intcr-

pretation cousider the acyuisitiou of an extra w~it of capital at time l. 'fhe firm incurs an

extra expense at timc t in arnount of marginal investment cost C~. On the other hand,

the rnarginal unit of productive capital generates - as of time t- a stream of cash flows

consisting of revenue from selling products (S') minus pollution tax payments (rEK,).

10

This cash flow strearn is corrected for depreciation by multiplication by exp(-ar(s - t))

and is discounted to time t by multiplication by exp(-r(s - l)). Condition (25) states

that the nct prescnt valuc of marginal productive invcstmenL equals zero. Hence, the

optimal level of productive investment satisfies the fundamental economic principle of

balancing marginal revenue with marginal expenses.

If the productive investment rate equals zero (25) changes into:

-rlr - f W{S'(!í r(s)) - rEk-,(Iír(s), K2(s))} exp(-(ar ~ r)(s - t))ds - Ci(0) (26)

This expression shows that the firm does not invest when the net present value of

rnarginal invcstrncnt is negative. 1'his makes sense, because now rnarginal expenses

exceed marginal revenue. (26) shows that it will be optimal to have a zero productive

investment rate when the productive capital stock is large and the abatement capital

stock is low.

From (10), (12) and (24) we obtain that a positive abatement investment rate always

satisfies the following equation:

1~ -rEh~z(lír(s), Kx(s)) exp(-(as f r)(s - t))ds - Cz(12(t)) - 0 (27)r

(27) implies that thre level of abatement investments is such that marginal abatement

investment expenses (C2) balance the discounted decrease in pollution tax payments over

the whole planning period, caused by an extra unit of abatement investment carried out

aL Lime t. We conclnde that the net present value of marginal abatement investment

eyuals zero. Aualogous Lo the case of productive invcstment it holds that abatement

investment will be zero whenever this net present value is negative.

3.3 Phase plane analysis

Linearization of (2), (3), (II) and (12) around the steady state yields the following

system

i - J(z - z'), (28)

where z -(lír,lti2,ar,az)T E R', z' is the stationary state and J is the Jacobian

presented in (16). A general solution is of the form

11

4z(t) - ~ c;z; exp(~;t) ~- z', (29)

~-i

where ~~, ~zi ~3i ~q are the eigenvalues of J and z~, zz, z3, zq are the corresponding eigen-

vectors, as far as all ~; have multiplicity 1. For this model the latter is assured by

inequality (22) (cf. eqn. (5.36) of Feichtinger and Hartl (1986)).

Because conditions (18) and (19) are satisfied, we know from Theorem 5.4 of Feichtinger

and Hartl (1986) that all eigenvalues are real, with t,,z G 0 and 13,q ~ 0. For all solutions

in Lhc stablc, rnanifol~l iL holds t.hat cs - cq - 0 in (29). Othc~rwisc thc solution would

explode, because ~3,q ~ 0.

If we only consider the first two coordinates of the linearized system, and we set the

second coordinates of z~ and zz equal to unity (this can be done without any objection

since the eigenvectors are unique only up to a scalar multiple), we obtain:

K~ r ~i~ "1~ ~í~~ K ~- c~ exP(~it) I 1~ f cz exP(~zt) ~1 f K,

z ` z

After differentiation of (30) w.r.t. time we get:

r K~ r z~i zz~I Kz ~-~~ci exP(~~t) I 1~ f~zczexP(~zt) 1

Next, (30) may be solved for cl exp(~lt), cz exp(~zt),

IC~ - Iíi - zx~(Kz - Kz)c~ exp({~t) - ,

zii - zzi

zii(~íz - Kz) -(K~ - líi)czexP(~zt) - ,z~l - zn

and substituting from (32), we may write (31) as:

r tíi ~ - ~ - ( fi-ii -fz-xi -ii-ziÍfz -fi) ~ ( Ki - tí'

~Kz zj~ - z21 ~~ - fz fzz~i -~izzi Kz - KZ

(30)

(31)

(32)

(33)

To draw a phase diagram we are interested in the slope of the isoclines KI - 0 and

!CZ - 0:

12

dIí ~d IC,

-aK,~aK, ~zZZ1- ~,Z„,;',-o - aK,~aK, - z~~z2~(~z - f~)

dlí2 -alíZ~aK, i;z - ~,dK, Kl-o - a1Cz~aK2 - fzzii - ~izzi

(34)

(35)

To calculate these slopes we have to restrict our attention to the case where r- 0.Furthcrrnorc, Lu simplify thc Lcvlious ca~c'ulxtiuns wc imposc Lhat a, - az - n. By

following the method employed in Turnovsky (1981) we can show that the slopes of the

two lines are respectively (the proof is relegated to the Appendix):

dK,dK,

d lí~dlf,

h', -u

Kz-O

- -C~ {a2 -~ (rEF-,r;, - S")~C~ f detJ} ) 0Tl.h~h"i

rEx,h,) 0

-CZ {a~ -f rEF,~,~Cz -F det J}

A straightfurward exrrcisc shows that

dK~dlí,

dK2

ti.,-o' dlí, Fz-O ~

(36)

(37)

(38)

so that the lí, - 0 isocline is steeper than the Iíz - 0 isocline.

Now, we arc ready to draw the phase diagram, which is done in Figure 1. The line

K, - 0 traccs out thc combinations of K, and lí2 for which lí, is stationary. Likewise

lí2 - 0 is tlie locus of combinations uf lí, and Iíz corresponding to stationary values

of Ií2. Tt~ese two lincs divide the (lí,, Kz)-space into four quadrants. Above the line

IC, - 0, K, ~ 0, while below it K, C 0. Similarly above the line KZ - 0, K2 G 0, while

below it KZ 1 0. Accordingly, the overall adjustment of the system in the four regions

is in the directions indicated by the arrows.

(Insert Figure 1 about here]

13

For reasons of surveyability we have only drawn those trajectories where abatement

capital stock increases all the way and where productive capital stock is not very large

in the beginning of the trajectory. We see that then both productive and abatement

capital stock increase on the final path that ends at the equilibrium. This policy can

be preceeded by a phase where !~r decreases (productive investment can even be zero

(cf. eqn. (26))) when ICr is large, initially (trajectory (a)). Trajectory (a) could be the

final phase of a solution where, at time zero, the firrn is at its unregulated equilibrium

at the moment that thc government imposes a pollution tax (the unregulated steady

statc satisfies eqn. (23) without the term rEk.,. This makes that ICr is larger in the

unregulated equilibrium compared to the regulated one). Remember that the phase

diagrarn only holds locally around the steady state.

4 Marketable permits model

The implementation of marketable permits involves several steps (cf. Hahn (1989)).

First, a target level of environmental quality is established. Next this level of environ-

mental quality is defined in terms of total allowable pollution. Permits are then allocated

to firms, with each permit enabling the owner to pollute a specified amount. Firms are

allowed to trade Lhese permits among themselves.

In the USA t}rere has been some limited experience with programs of marketable permits

for the regulation of air and water quality, while in Europe there is hardly no experi-

ence with rnarketable permits. The major program of imposing marketable permits as

a mechanism for providing economic incentives for pollution control in the USA is the

Environmental Protection Agency's Emissions Trading Program for the regulation of air

quality (see Tictenberg (1985)).

I:xisl,ing systi~ins uf iua.rkctahlr prnnits in Lhc Ilnitrd SLatr~ r~rnbudy a kiud of "grandfa-

theringr scheme involving an initial distribution of pollution permits or "rights" among

polluters based on historical pollution. According to Cropper and Oates (1992) a draw-

back of this system is that heavy polluters are rewarded by receiving a lot of permits,

which they can then use either to validate their own pollution output or sell to other

firms. In this way the "Polluter Pays Principle" is violated.

Comparcd to a pollution tax system, a major advantage of the marketable permit ap-

proach is that it gives the government direct control over the level of pollution. Under

the taxes approach, the government must set a tax, and if, for example, the tax turns

14

ouL tu bc luw, pullul.iun still cxccc,d pcrniissiblc Icvols. 'fhc govcrnmcnL will find itself inthe uncomfortable position of óaving to adjust and readjust the tax to ensure that theenvironment is noL severely damaged (Cropper and Oates (1992)).In this section we consider a firm that has to buy permits in order to be allowed topollute the environment. According to Siebert (1992, p. 142) marketable permits maybe defined on a temporary basis or without a time limit. We will assume here that oncea pcrmit is boughL it rcmains valid forcvcr (conl.rary Lo e.g. Lhe Wisconsin Fox R.ivcrWater Permits which are only valid for five years (1[ahn (1989))).If it has good growth prospects the firm will increase production and, after assuming forthe moment that abatement capital is too costly, this will also increase pollution whichimplies that the firm needs to buy extra permits. These permits can be sold to otherfirms aL thc momcnt. that thc firm reduces pollution by either a sufficient increase ofabaLcrncnt ra{,ilal sLuc k ur a rc~ductivn uf pruduc l.iun. If Lhc pricc uf a pcrtnit cyualsp and the firm needs one permit per unit of pollution, then the firm's expenses on thepermiL markct aL time t equal

PF - P{Eh-,lír f F.h-,líz} - P{EF',(Ir -arK,) ~ En~z(Iz-az1íZ)}. (39)

Nuticc Lhal, sprndings turu inLu rccciviugs as suun .GS pullutiun dccrcasc~s ovcr tirnc.Whether Lhc price of a permit, will go up or down depends on the behavior of all com-petitors in the market. Leaving abatement activities aside for the moment, if all firmswant to produce more they implicitly want to increase pollution. Therefore, the demand

for pollution permits goes up and the price of the permits increases. Notice in this re-spcct Lhal, Lhc :unuunt uf pcrn,it, on thc market is fixcd, which in turn leads to a fixedIevel of pulluLion generaLed by thc whole sector.

Thc above dcscription refcrs to a market for pollution pcrmits Lhat provides great flexi-

bility due Lo the absence of transactions costs and other obstacles to trading. However,

in practice the rules of the marketable permits can be so restrictive that the flexibility

they offer is more imaginary than real (see Cropper and Oates (1992), Hahn (1989)).

Nevertheless, in this paper we assume that trading barriers are absent on the permit

market.

Like in the tax model also here the objective of the firm is to maximize the net cash flow

stream:

maximize : I~{S(Iír) - Cr(Ir) - CZ(Iz) - pE(Ir,12, lír, If2)} exp(-rt)dt (40)

15

The optimal control problem is to maximiae (40) subject to (2), (3), (5) and (6). Similar

calculations like in the model with a pollution tax lead to the tollowing results. The

upl.imal stoa~ly xtal~~ iti a solut.iun oÍ:

I, - a, Ií1 - 0

Iz-azlfz-0

(r -f ai)~i - S'(~~i) - a~PEr~'~(fíi,1~z) - 0

(r f az)~z - azPEtc~(k~, ~~z) - 0

-CI(fl) - P~-h,(~S i, ~~z) f~i - 0

-C2(~z) - PEKz(~~1, ~t7) ~ ~4 - ~

The Jacobian evaluated at the equilibrium has the following form:

J-

rr;,, ~,~.rf i ~,~i

-cEe, x,

Ca

~-~ ~,~,zC~

-yEk'x' - aCz 2

a (i

Q 7

iC~0

r i- ai f pEh~' x'c;PEK, xz

c;

in which:E zEz~~ P h~, ~.~, P h', xza--S - pF.h,x~ ~,,, f 2a, - ~„~

i z

rPEK,h, PEKstisa--PEa-~h's I C,,, t C,,, f a, -F az

` i z

-pzl;h~~.7 - - ~.~

~pl:h-,h~z 1p~;,,,,,, ~„~ -~ z(l,llz

(41)

Using the computer program Mathematica we were able to calculate the determinant:

I fi

detJ - arax(r f ar)(r f ax) -Far(r f a~;mEh'k'

-bx

ax(r f ax)(PrEx,ti, - S") PrS~~E~zKz fC; - C;CzxTx

C„C„(EK,n, EKsh'i - Eti,Ki ) ~ 0r x

The value of the number K(cf. below expression ( l9)) equals:

(42)

Pr Eh'z h"z pr Eh", h~, fs,r 1

K- - C„ - ax( T } ax) f- C„ - a~(r f ar) J G 0 (43)z r

Comparing ( 4`L) and ( 43) with (17) and ('l0) we see that detJ and lí have the same

value in the tax model and the permits model when r- rp. Therefore, we can easily

conclude that also here conditions ( 18) and ( 19) are satisfied so that we have saddle

point stability with monotonic motions.

The firm's equilibrium value of productive capital stock is given by:

s~(~~ i~) -(r f ar )Ci(ar ~í i~) -F- rPEh', (K~', lí2') (44)

Ilcnce marginal revemw equals marginal costs, while margiual pollution costs equal

rpEh-,. This is because increasing productive capital stock with one unit during one

period raises pollution with Eh-,. Therefore, at the beginning of the period the firm has

to buy additional permits at the expense of pEF,. These permits can be sold at the end

of the period (notice that pollution increases only during one period) so that the firm

needs to borrow pEh-, dollars for one period to finance the permits transaction. This

increases interest costs with rpEh,.

Due to equation (44) we obtain that introducing a marketable permits system leads to

a lower equilibrium value of productive capital stock, which in turn leads to lower levels

of production and pollution. rurther, we see that the effects of marketable permits are

particularly significant when the interest rate is large.

At each rnoment of Lirne thc productive investmenL rate, whenever iL is positive, must

satisfy:

1~{s~(Itr(S)) - n(s)~h", (~r(s), ~x(s), Kr(S), ~íx(s))}exp(-(ar f r)(s - t))ds -C~(Ir(t)) - p(t)EK,(lí,(t),Kx(t)) - 0 (45)

17

Incrcasing I,ho prudu~~t.ive capital stock with onc nnít. results in immediate extra expcnses

in arnounL of margiual cosL U„ plus spendiugs on extra permits pEk-, needed to account

for the additional pollution generated by this extra unit of capital. It also generates -

as of time t - a stream of cash (lows consisting o( revenue from selling products (S') and

changes in future expenses on the permit market (-pE~,). Hence, according to (45)

the firm fixes productive investment such that the discounted cash inflows and outfíows,

which are due to marginal investment, are balanced. To state this differently: the netpres~,nt. valu,~ uf inarginal invi~stin~~nt ,~qnals z~~ru.

1'he firm's equilibriwn value of abatement capital stock is given by

(r i- as)Cx(az~~z ) - - rPEt~s(It~',Ifz")

while, for every time t, positive abatement investments satisfy:

- f~P(3)E~.~:(7i(S),7z(S),~~~(s),Itx(3))eXP(-(asfr)(s-t))def,-Cz(h(t)) - p(t)E~,(K~(t), Kz(t)) - 0

(46)

(47)

(47) implies tl~at marginal abatement investment expenses (C2) balance the changes in

cash flow on the permit market, caused by this marginal abatement investment. The

additional cash flows on the permit market consist of an immediate cash inflow (-pEK„

note that E~, C 0) and revenue~ expenses over the remaining planning period.

After comparing (23) and (24) with (44) and (46) we can conclude that a pollution tax

and marketable permits lead to the same equilibrium if it holds that:

rp-r (48)

To understand this equation consider the situation where pollution is increased with one

unit during one period. Then, in the pollution tax case the firm incurs an extra expense

of r dollars during this period, whereas in the markctable pcrmits case the firm has to

buy a permit in the beginning of this period that can be sold again after the period is

over. This implies that the firm has to borrow an amount of money equal to the petmit

price p during this period, which leads to interest costs of rp dollars. Hence, pollution

costs increase with the same amount under both instruments when (48) holds.

18

5 Conclusions

This contribution belongs to a category of papers that attempts to determine the effects

of cuvironmental regulation on the growtli of an individual firrn. [t extends the analysis

of its predccessors (e.g. Kort, Van Loon and Luptacik (1991), flartl (1992), Xepapadeas

(1992)) in at least two ways. First, we incorporated a more realistic (but also more

complex) pollution function. With more realistic we mean that we explicitly deal with

the fact that for lower levels of pollution more abatement effort is required to reduce pol-

lution with some fixed amount. This is contrary to Kort, Van Loon and Luptacik (1991)

and Xepapadeas (1992) where it is, rather unrealistically, assumed that abatement costs

are independent of the pollution output generated by the production process.

Second, bcsides dcaling with a pollution tax as a form of environmental regulation, we

considcr the effects marketable permits have on the firm's investment policy. As far as

we know this is the first contribution where rnarketable permits are incorporated in a

dynarnic modcl of thc firm.

It turus out that imder both environmental instruments the equilibrium, where marginal

revenue equals marginal cost, is stable and approached monotonically. Imposing a pol-

lution tax as well as a marketable permits system decreases production and pollution in

equilibrium. Effects of marketable permits on dynamic firm behavior are intensified in

case of a high intcrest rate.

Uin'ing t.h~~ whuli~ I~lannin~; prriu~l Lhr ihwrlol~in~~nL uf prudnrl.iw, :4~; w~~ll :r.ti ahaLi,tni~nl.

capital stock is cornpletely dcterrnined by an investment rule that dictates the firm to

stop investing when the rret present value of marginal investment is negative and, when

this is not the case, to fix the investment rate such that the net present value of marginal

investment equals zero. Then the immediate expenses due to acquiring an additional

unit of capital stock exactly balance the discounted future cash flows, such as revenue

from selling products, as far as they result from this unit increase of capital stock.

After comparing the effects on the firm's investment policy of a pollution tax and mar-

ketable permits we concluded that long run firm behavior will be the same when the tax

rate equals the interest rate multiplied by the price of a permit. This rule is economically

interpreted and can be seen as a dynamic extension of the well known static result that

there are similar effects whenever the permit price equals the tax rate (see e.g. Baumol

and Oates (1988), P. 58).

Finally, we mention two intcresting topics for future research. First, it could be inter-

esting to consider a dynamic game of several finns operating on a market for pollution

19

perniits. Such a G:unework cuuld Icad Lo stratcgic intcract.ions likc a firm that buys

mure permits than ncx~ded for covcring its pollutiuu iu order to obstruct the growth of

its competitors.

A second interesting model extension could be to consider permits of limited validity,

as oppositc to the model considered here where permits remain valid forever. Then the

value oí a marketable permit will depreciate over time and the resulting depreciation

costs will certainly influence the above described rule under which the effects of a pollu-

Lion tax and markcl.able permits are cquivalent.

Acknowledgement

The author thanks Jan de Klein for computational assistance and an anonymous referee

for his remarks. This research has been made possible by a fellowship of the Royal

Netherlands Academy of Arts and Sciences.

References

Baumol, W.J., Oates, W.F,., 1971, The use of standards and prices for the protection

of the environment, Swedish Journal of Economics, 73, 42-54.

Raumol, W.J., Oates, W.E., 1988, The Theory of Environmental Policy, second edition,

('ainLii~ly;~. Ilniv~~riil.y I'i~:;:;, ('~uuLri~ly;r.

Cropper, M.L., Oates, W.E., 1992, Environmental economics: a survey, Journal of

Economic Literature, 30, 675-740.

Dasgupta, P., 1982, The Control of Resources, Basil Blackwell, Oxford.

Feichtinger, G., Hartl, R.F., 1986, Optimal Kontrolle Okonomischer Prozesse: An-

wendungen des Maximumprinzips in den Wirtschaftswissenschaften, de Gruyter,

Bcrlin.

liahn, R.W., 1989, Economic prescriptions for environmental problems: how the patient

followed the doctor's orders, Journal oC Economic Perspectives, 3, 95-114.

Ilalin, It.W., 51.avins, II.N., I!191, I?ronomic im~enf,ives for environment.al protection:

intcgratiug thcory aud prac'Licc, CSIA Discussiou Papcr 91-15, licnncdy School of

Govcnmicut, Ilarvard lluivcrsity.

}Iartl, R.F., 1992, Optimal acquisition of pollution control equipment under uncertainty,

Management Science, 38, 609-622.

20

lielfand, G.E., 1991, Standards versus standards: the e(fects of dilferent pollution re-

strictions, Arnc~rican F,conomic Revicw, 81, 622-634.

Jurgenson, U.W., Wilcoxeu, I'..1., 199U, Environmcutal regulation and U.S. economic

growth, Rand Journal of Economics, 21, 314-340.

Kort, P.M., Loon, P.J.J.M. van, Luptacik, M., 1991, Optimal dynamic environmental

policies of a profit maximizing firm, Journal of Economics, 54, 195-225.

Pindyck, R.S., 1991, Irreversibility, uncertainty and investment, Journal of EconomicLiterature, 29, 1110-1148.

Siebert, H., 1992, Economics of the Environment, third edition, Springer, Berlin.

Takayama, A., 1985, Mathematical Economics, second edition, Cambridge University

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Tietenberg, T.H., 1985, Emissions Trading: An Exercise in Reforming Pollution Policy,

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'I'urnovsky, S.J., 1981, The optimal intertemporal choice of inflation and unemployemnt,

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21

Appendix. Derivation of the relations (36) and (37)

In case r- 0 the two negative eigenvalues of J are given by (see Feichtinger and Hartl(1986), p. 134):

~1-- -2t }2Kz-4detJ

ti 1~z-- -2 -2 Kz-4detJ

( A.1)

(A.2)

'I'he eigem~alue ~1 and its corresponding eigenvector zl have the following relationship

~IZI - Jzl, (A.3)

and this equation can be expressed as (remember that here we assume that r- 0 and

al - az - a):

-n 0 1 ~C~ 00 -a 0 1 ~C~-S~~ ~ TEh-~h'i TEKiKz a ~

rEFzh-, rE~~hz 0 a

(A.4)

and solving for zll, we obtain

zll - {CY2(~I -a2)-TFh'ah.a},TEF~zh~,.

Similarly, corresponding to the eigenvalue ~z we have

zzl - {Cz(~i - az) - rEti.z~~z}~TEh-zr,,.

(A.5)

(A.6)

Using (20), (21) (where in both equations we put r- 0 and a~ - az - a), (A.1), (A.2),

(A.5) and (A.6) we may now establish the following expressions íor terms appearing in

(34) and (35):

~zzzl - ~IZII - (~z - ~1)C~z Jaz t (TEh.,r,-, - S")~Cï f det J} (A.7)TEn-,F-, l

ziizzi - -CZ~Ci (A.8)

Í~i - ~s)Cï z ~~fszii - ~~zz~ - (a } det J f rEK,K,~Cs ) (A.9)T EKz K~

Nuw, auLtil,iLul.iun uf (A.7)-(A.!)) iul.u (J4) and (:i 5) li,ads Lo (3G) and (37).Q.E.D.

23

i~'igure caption

Figure 1. Phase diagram in case of no discounting and equal

depreciation rates [or both capital stocks.

T K,

~

~,~~lfZ-O

L

Figure 1

Uiscussion Papcr Series, CentER, Tilburg University, The Netherlands:

(For previous papers plcase consult previous discussion papers.)

No. Author(s)

9239 H. Blocmen

9240 F. Drosl andTh. Nijman

9241 R. Gillcs, P. Ruysand J. Shou

9242 I'. Kurt

9243 A.L. Bovenberg andF. van der Ploeg

9244 W.G. Gale andJ.K. Scholz

9245 A. Bera and P. Ng

9246 R.T. Baillic,C.F. Chung andM.A. Tieslau

9247 M.A. Tieslau,P. Schmidtand R.T. Baillie

9248 K. Wámeryd

9249 H. Huizinga

9250 H.G. Bloemen

9251 S. Lijf7ingcr andfi. Schaling

9252 A.L.I3ovenberg andR.A. de Mooíj

9253 A. Lusardi

9254 R. Bcxasma

9301 N. Kahana andS. Nitzan

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9302 W. Giith andS. Nihan

9303 U. Karotkin andS. Nitzan

9304 A. Lusardi

9305 W. Giith

9306 B. Peleg andS. Tijs

9307 G. Imbens andA. Lancaster

9308 T. Ellingsen andK. Wámeryd

9309 H. Bester

9310 T. Callan andA. van Soest

9311 M. Pradhan andA. van Six:sl

9312 T'h. Nijman andE. Sen[ana

9313 K. Wíimeryd

9314 O.P.Attanasio andM. Browning

9315 F. C. Drost andB. J. M. Werker

9316 H. Hamers,P. Borm andS. Tijs

9317 W. Giith

9318 M.1.G. van Eijs

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9320 JJ.G. Lcmmcn andSC. W. I:ijl7ingcr

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9322 K.-E. Wíirneryd

9323 D. Talman,Y. Yamamoto andZ. Yang

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9325 S.C.W. Eijffinger andE. Schaling

9326 T.C. To

9327 J.P.J.F. Scheepens

9328 T.C. To

9329 F. de Jong, T. Nijmanand A. RSell

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9 4 4 I I I I IiuiuiF~,i

9332 V. Feltkamp, A. Koster,A. van den Nouweland,P. Borm and S. Tijs

9333 B. Lautcrbach andll.licn-Iiun

9334 B. Melenberg andA. van Soest

9335 A.I.. Buvcnbcrg andF. van der Ploeg

9336 E. Schaling

9337 G.-J.Otten

9338 M. Gradstein

9339 W. Gíith and H. Kliemt

9340 'i'.C. To

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9346 J.P.C. Blanc andR.D. van der Mei

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9351 F. Vella andM. Verbeek

9352 C. Meghir andG. Weber

9353 V. Feltkamp

9354 RJ. dc Groof andM.A. van Tuijl

9355 Z. Yang

9356 E. van Damme andS. Hurkcns

9357 W. Giith and B. Peleg

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9359 P. Vclla and M. Vcrbcck

9360 W.B. van den Hout andJ.P.C. Blanc

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9361 K. I Icuts and1. dc Klcin

9362 K.-E. Wárneryd

9363 P.J.-1. Herings

9364 P.1.-J. fierings

9365 F. van der Ploeg andA. L. Bovenberg

9366 M. Pradhan

9367 Ii.G. Bloemen andA. Kap~cyn

9368 M.R. Baye, D. Kovenockand C.G. de Vries

9369 T. van de Klundert andS. Smulders

9370 G. van der Laan andD. Talman

9371 S. Muto

9372 S. Muto

9373 S. Smulders andR. Gradus

9374 C Fernandez,J. Osiewalski andM.PJ. Stccl

9375 L.van Danunc

9376 P.M. Kon

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di~ia~~~~~ ~ rarW~~iu