Optimal growth with renewable resources and pollution

European Economic Review 35 (1991) 650-661. North-Holland

Optimal growth with renewable resources and pollution Olli Tahvonen Helsinki Schcol

0; Economics, SF-00100 Helsinki, Finland

Jari Kuuluvainen Finnish Forest Research Institute, SF-00381 Helsinki, Finland

This paper presents three models which combine the optimal dynamics of economic growth, renewable resource harvesting and pollution. We investigate growth, pollution and renewable resources as a three state variable problem. The economy has three inputs, man made capital, renewable resources and emissions. Endogenous pollution affects both the growth of the renewable resource stock and the social welfare. It is shown that the existence of the steady state depends, in a complicated manner, and in addition to preferences,on the production technology, growth of the renewable resource and the rate of decay of pollution. The suflicient conditions for the existence of a global saddle point are given. Conditions under which the market solution equals the optimal solution of the social planner, and the consequences of a too low externality tax on emissions or on common property resource harvesting are discussed.

1. introduction

This paper examines an economy which produces commodities using man-made capital, renewable raw material and emissions as necessary factors of production. It will be assumed that emissions accumulate as a slowly decaying stock in the environment which, in addition to direct welfare effects, affects the growth of biologically regenerating resources. Our framework has well established roots in resource and environmental economics. Nevertheless, the research on economic growth and natural resources has focused on nonrenewable resources. This has also affected the manner in which the problems of pollution have been dealt with. Until recently, the consequences of pollution that have been considered important have been the direct disutility to consumers and abatement costs to producers [Keeler et al. (1972)]. In our opinion, the core of the problem is changing. As a flow variable, pollutants often have a positive marginal effect on production (pesticides in agriculture and forestry, emission generation in production). However, the accumulating stock of pollutants has direct 0014-2921/91/$03.50 ((*)1991-- Elsevier Scienc- Publishers B.V. (North-Holland)

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651

negative marginal effects on human welfare but, at least as importantly, affects the reproductive capacity and growth of renewable populations. This leads to another important feature, that is the ease of substitution between emission generation and other inputs in production (in industry, agriculture, forestry, fisheries management etc). Although technological development and accumulating knowledge [cf. Romer (1986)] may weaken the link between production and natural resources, it is difficult to imagine an economy without some degree of dependence on natural resources. Therefore the optimal management of renewable natural resources, the optimal level of emissions and the rate of capital accumulation must be decided simultaneously. The paper is organized as follows. Section 2 presents a neoclassical model of economic growth with renewable resources and stock pollution. First, we prove the most essential mathematical properties of this model, i.e. the existence and the dynamic properties of the steady states. It turns out that, in contrast to the renewable resource models without capital accumulation [cf. Clark (1976)], the steady state may fail to exist although the growth potential of the renewable resource stock exceeds the rate of discount. In section 3 we consider whether the socially optimal solution can be realized in a market economy as well as the general equilibrium implications of nonoptimal environmental policies. Section 4 concludes. 2. Socially optimal solution The problem of the social planner is to choose time paths for consumption, resource harvesting and emissions in an economy where production is based on renewable resources which are sensitive to stock pollution. Formally: max

(1)

W= j U(c,$e-“‘dr,

s.t.

Ii = Q( k, h,e) .t =

(2)

c,

(3

F(x, z) - h,

(4)

i=e-az,

k(0) = l&-J > 0,

x(0) =

lim k(r)zO,

lim x(t)zO,

x0 >

0,

&

z(0) = zo 2 0,

(5)

Z(t)zO,

(6)

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0. Tahvonen and J. Kuuluoainen, Optimal growth with renewable resources

where U gives the instantaneous utility, Q is the production function, F denotes the growth of the renewable resource, x, c is consumption, e is emissions, k is the stock of capital, z is the stock of pollution, h is the rate of harvest, S( 20) is the rate of discount, and ac(>O) is the rate of decay. We assume that U(c,z) is w2, strictly concave and increasing with consumption and decreasing with pollution. We also suppose that lim U,(c, 2) = Go, UC=s 0,

lim W&c,2) = 0.

(7)

c+m

C-@O

The production function Q is 99’ and increasing with all arguments. All inputs are necessary and Q is strictly concave. In addition Qke, Qkk, Qbe>O and

Vk, Vh, 3

lim Q,( k, h, e) = 0,

lim Qk(k, h, e) = 0.

444

k-m

(8)

The growth function of the renewable resource stock is F(x, z) M2, concave, and, F,cO,

F&O,

Vz>O,

F(O,Z)=F(_~~,Z)=O,

5.~0,

strictly

lim FE(x,z)=O;

:+o

F,,~:O

for s>O,

F,,
(9

Note that with all pollution levels there are two stationary states of the population: x = 0 is unstable and x =& is stable. .& is the pollution-intensive carrying capacity of the population. Write the current value Hamiltonian H = U(c, z) + L[Q( k, e, h) -cl + N-F(x, 2) - hl + +(e- az) and the necessary conditions for optimal infinite time solutions: (?H/Sc = U,( c, 2) - E,= 0,

(10)

?H/i%=i.Q,,(k,h,e)-$50,

h?H/?h=O,

hz0,

(11)

(7H/?e = i.Q,( k, h, e) + $ s 0,

e ?H/c7e = 0,

e 2 0,

(12)

I=E.[S-Q,(k,h,e)],

(13)

4 = 40

(14

- U-L a],

0. Tahvonen and J. Kuuluvainen, Optimal growth with renewable resources

3/= - w,

4 - q&(x, z) + $46+ Gt),

653

(1%

and (2)-(6). The first two propositions give the basic mathematical properties of the model. Note that the Hamiltonian of the model is strictly concave in (c, e, h, k, x, z). This means by Theorem 3.13 in Seierstad and Sydsaeter (1987) that if one finds a path converging towards the steady state one has found the optimal solution. We start by considering the dynamic properties of the steady states. Proposition I. lf the rate of discount is small enough all bounded solutions converge to a unique steady state, given it exists. Proof.

See Appendix 1.

Bounded solutions satisfy the necessary and sufficient conditions for optimality. Thus the above proposition gives a sufficient condition for the property that, with any initial levels of capital, resource and pollution stocks, the optimal path converges toward a unique steady state. However, this result is not very useful until we have established that, at least with low rates of discount, an optimal steady state exists. To show this assume further that Q satisfies lim Q,Jk,h,e)>&

Vh>O,

Ve>O.

(16)

k+O

Next define 2 as the largest steady state emission level which does not reduce the growth potential of the renewable resources stock below the rate of discount, i.e. lim,,, F,(x, $a) = 6. Recall that lim,,& Qe(k, h, e) = 0. 2 Proposition 2. ProoJ

Given (16) and C< 6, at least one steady state exists.

See Appendix 2.

TO interpret the above result note that j is the higher the higher is the rate of decay of pollution, the lower is the rate of discount and the lower is the effect of the pollution stock on the marginal productivity of the renewable resource stock. But when does the marginal productivity of emissions decline toward zero even at low levels of emissions? This may occur e.g. if the recycling technique is so well developed that ‘low’ levels of emissions mean waste of useful raw material. Thus the steady state exists at least if the rate of discount is not too high and if the productive activity does not require too large an emission stream compared to the rate af decay and th2 negative effects of pollutants on the growth of the resource stock. Together Propositions 1 and 2 imply that given any strictly positive initial

654

0. Tahronun and J. Kuulurainun, Optimal growth with renewable resow-es

levels of the three stocks the optimal path converges toward a steady state which exists and is unique if the rate of discount is not too high. We next come to one of our main results. If, without capital accumulation and pollution, the growth potential’ of the resource stock exceeds the rate of discount the existence of the optima1 steady state and the optimal sustainable harvesting policy are guaranteed [Clark (1976)]. Above we have shown that the existence of the steady states depends on the time preference, on the properties of the production technology, on the rate of decay of pollution and on the effects of pollution on the renewable resource stock. Next we show that the traditional rule for the existence of the steady state harvest is only necessary but not suf?‘icient in this general equilibrium framework. For the existence of the steady state. it is necessary that with some strictly positive level of pollution, the growth potential of the resource stock exceeds the rate of discount i.e. 3z >O 1lim,,, C;,(.u,Z)> S. If this requirement is satisfied, (13), together with (3) and (4), gives the steady state harvest as a declining function of the steady state emission level, given e c;. Further, it is necessary that with some OS. However, if the marginal productivity of capital is bounded this condition can be satisfied only if the levels of harvest and emissions are high enough. Thus, a low rate of decay of pollution and low absolute growth rate of the resource stock may imply that there do not exist any combinations of [h(e),e] that satisfy the modified golden rule of capital accumulation. Furthermore. note that some level of e which satisfies the above two conditions must also simultaneously satisfy the modified golden rule condition for the pollution stock. We do not study this condition here because the above arguments are enough to establish the claim that the traditional necessary-and-sufficient condition is only necessary here.2 This means that in the general equilibrium framework with many capital stocks in interaction the conditions for an optimal sustainable resource harvesting policy may be more restrictive than the ordinary partial analysis of renewable resource harvesting suggests. Next we consider under what conditions the socially optimal solution can be realized in a market economy. et solution with externalities

Assume an economy

which consists

of a representative

consumer,

a

‘We defme growth potential as the marginal growth rate of the resource stock when the stc;k size approaches zero. ‘The interesting question here is what are the properties of the optimal solution if no steady states exist. Because of the inada conditions in the utility function consumption can approach zero only t + x . Without them, the optimal planning horizon may be finite and both of the optimal terminal stc.rLksare zero.

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655

representative firm, the government and the common property environment [cf. Brock (1977)]. The utility of the consumer increases with consumption and decreases with pollution. He sells a fixed labor input (1) at a given wage (w) and rents capital at the market interest rate (r) to a firm which maximizes the instantaneous profits (II) (received by the consumer). The representative firm produces a composite commodity using four factors of production: labor, capital (composite commodity), a natural resource and emissions. Perfect information and perfect foresight and similar functional forms as in Section 2 are assumed. The problem of the representative consumer is, therefore, to

max W= 3 U(c,z)e-“dt :csoI 0

(17)

s.t.

C;=n+rk+wl+T+f

-c,

(18)

where T and f are the redistributed externality taxes on harvest flow and emissions. The problem of the representative firm is to, max

n=Q(k,k,e)-rk-qlt-be--1.

(20)

&.h.I.e2$0

The government collects the externality taxes, f=re

and

T=qh.

(21)

Finally, nature produces raw material which is harvested by the firm. The emission stream accumulates as a slowly decaying stock which affects the growth of the resource stock. i =

2=e-a:,

F( x, z) - h,

X(O)= so, and lim s(t) 2 0, l-+X

(22)

and _iim s(t) 2 0. I’X

(23)

z(0) =Zo,

Assuming interior solutions, the first order conditions ing firm require that

of the profit maximiz-

0. Tahrcwn

656

and J K uuluvuinen,

Q,,=q,

Q k =r,

I

.

Q e = T,

Optimal

growth

with renewable

resources

(24)

which are the implicit instantaneous demand functions for capital, the renewable resource and pollution. The consumer solves the problem ( 17)-( 18) in or&r to determine optimal consumption, additions to capital, and the stock supply of capital. The present value Hamiltonian for the consumer’s problem is ff = U(C,2) + p( II + rk + M’I+ T + f - c), and the necessary conditions are H,=U,--p=O,

(25)

and eqs. ( 18)--(19). It is obvious from (25) and ( 10) that i, = p, while (26) is identical to (13). The government has to find values for the externality taxes, q and r, which maximize s&al welfare. It can do no better than the social planner in the previous section and if it does worse, it has not found the social optimum . By (11) and (24) it must hold that qy +,G. In order to find the optimal time path for the tax on harvest flow the government has to actually find a solution to the resource owner’s problem: maximize the present value of the resource subject to (22), which yields a necessary condition identical to ( 14) determining optimal q/q. Using (12) and (24) the emissions are controled optimally by setting the externality tax f = -- e/i. Thus the price of emissions in the profitmaximizing firm is the social cost of pollution. The evolution of r is obtained using (15), only then the necessary conditions for the infinite time market solution are exactly the same as those in the central planner’s problem. Thus, by Proposition 1 we know that the steady sta?e, if it exists, is a saddle point. We have actually established the following proposition: Proposit ion 3. Assume a society with a representatice utility-maximizing consumer, representative profit-maximizing firm. the government and a common property erwironment producing a renewable resource and assimilating pollution. If the consumer and the gowrnment hare perfect knowGdge and perfect foresight and the karwst flow and emissions are taxed optimally, and the assumptions made on utility and production functions hold, the perfect market solutir with market clearing equals the socially optimal solution.

The

above

information

discussion indicates that, even in the market solution;, the requirements (from the empirical point of view) are strin

0. Tahvonen and .I. Kuuluoainen,

Optimal growth

with renewable resources

657

indeed, as stringent as in the social planner’s problem. In fact, the government has to find the solution to the social planner’s problem of the previous section in order to be able to tax emissions optimally. But what happens if the tax on emissions and common property harvesting are set nonoptimally? We conclude by examining the consequences of nonoptimal environmental policy. It is possible to solve c, h, and e as functions of prices and stocks [cf. Arrow and Kurz (1970)]. In particular, at any point in time, harvest and emission generation are both decreasing functions of 4 and T. If the government sets both 4 and r, they can be considered instruments and before analyzing the steady state we give the following proposition. Proposition 4. At given levels of capital, the renewable resource and pollution, if either q or T are set too low compared to the social planner’s optimal values, the rate of harvest, emissions and the interest rate are higher than on the social planner’s optimal path.

This result follows immediately by comparing the marginal productivities of capital when taxes are set optimally and when they are not (see proof in Appendix 3). It means that in an economy in which the price of the harvest or emissions are set too low, the environment, as a raw material source and as a sink for waste, is in danger of becoming over-exploited. Further, the use of the market interest rate in cost-benefit calculations, as is often suggested, may lead to biased results [cf. Brock ( 1977)]. Fcr the steady state we can give the following proposition. Proposition 5. (i) If the emission 1~1.xis lower than the social optimum, the steady state level of the pollution stock will be higher and stock of the resource will be lower than in the social planner’s optimal solution but the effect on the capital stock and consumption remains ambiguous. (ii) lf the tax on the harvest flow is lower than the social optimum, the stock of-the natural resource will be higher and the stock of pollution lower, but the erect on the stock of capital . and the steady state rate of consumption is again ambiguous.

The proof (Appendix 4) uses the fact that the equilibrium conditions for harvest and production must be met simtiltaneously. The possibility that pollution control may increase steady state consumption and capital levels in (5) should be compared to the result which states that when renewable resources are neglected, pc>llution control always reduces steady state capital and and consumption levels (but increases welfare) (see Tahvonen Kuuluvainen ( 1990) corollaries I and 1’). The result in (ii) showing that a nonoptimally low tax on resource harvesting increases the ievel of the renewable resource stock may seem counterintuitive. The explanation is that

658

0. Tahronen and J. Kuulurainen. Optimal growth with renewable resources

a nonoptimally low tax on harvest flow causes distortions in pollution control activity in the form of too low levels of emissions and the pollution stock [for further discussion see Tahvonen (1989)]. Note that these last consequences hold only if the distortions do not cause the nonexistence of the steady state. If this is the case the perfect foresight market solution path exhausts and po’llutes the raw material bases of the economy.

4. Conclusions We have studied the optimal growth of an economy with a resource base that is affected by endogenous pollution. The important question considered is whether an optimal and sustainable consumption and resource harvesting policy exists. It turns out that, opposite to the case of partial renewable resource models, an optimal sustainable path may fail to exist although the growth potential of the renewable resource stock exceeds the rate of discount. The absolute growth rate may be too low to satisfy the modified golden rule of capital accumulation. The core of the problem is, whether, with the given production technology, preferences, growth function of the resource stock and the rate of decay of pollution, the equilibrium conditions for three stocks can be met simultaneously. We further show that uncontrolled pollution may decrease steady state consumption and capital levels, opposite to the case where the effects of pollution on the resource stock are ignored. Accordingly, free access harvesting causes distortions in pollution control activity.

Appendix 1

Proof of Proposit ion I

We apply Sorger ( 1989), Corollary 2c. Let us first derive MHDS. Conditions (lO)-( 12) define the optimal levels of consumption, emissions and resource harvesting as functions of the shadow prices and stock levels: c = c(i., z), C~ 0, 11,>O, h, > 0, (or h =0), e =e(J/, 4, i,, k), eJI>O, eA> 0, eti 0, (or e = 0). Using this in (2)-(4), (13)-(15) gives the MHDS: h=?[k,h($,i,,I(/,

k),e($,&j.,

k)] -c(j.,z),

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659

tb=-U,-Cc(i,,z),z]-_F=(x,z)+~(S+a). The equilibrium of this system possesses the saddle point property ‘curvature matrix’

if the

W) I, -H$

1

’

where H* is the maximized Hamiltonian, i = k, x, z and j = i., 4, $, is negative definite. Matrix Q is negative definite with a small rate of discount because matrices HE and Hz are negative definite with minimum eigenvalues below zero. By Sorger (1989), corollary 2c, this implies that the equilibrium, given it exists, is globally stable for bounded solutions. 0

Appendix 2 Proof 0s Proposition 2

-&=;=4=$=0. (14) and (4) now define .u=.Y(u), At the steady state _t= Z-x’(e)<0 when e<6. Using this with (3) gives !r(e)=F[_x(e),e/r], K(e)<0 when e ~6. By ( 13) we now get S - Qk[k, It(e),e] =O. This equation has solutions with some k>O when e
fk e)z~$ = U,(c,e/a)+ &(c, e/a) { QJk(e), + Q,C&), Me), e](6 +-a)} = 0, J+i;&&()

h(e), e]

F&(e), e/a]

H(c, e) = & = Q[k(e), h(e), e] -c = 0. J++&+o

By using the concavity assumptions on U, Q, and F, it can be computed that &(c, e) ~0, &,(c, e) CO and thus f(c, e) =0 defines c as a declining function of e. Because for al1 c>O it holds that Cm,,, f(w)>0 and lim,,,f(c,e)eO,

660

0. Tahronen

and J. Kuuluoainen,

Optimal

growth

with renewable

resources

there exists with all c >O some level of e between 0 and G which satisfies f(c,e)=O. When e-+0 T(c,e)=O requires that c--,00. When e-G from below f(c, e) = 0 requires that c +O. Consider next n(c, e) =O. Because n,(c, e) = - 1 and /7&c, e) $0, function f7(c, e) defines c implicitly as a function of e with an ambiguous sign for the derivative. However, by the necessity assumption e = 0 must imply c =0 for f7(c, e) =0 to hold. Note next that when e = $ it holds that h =O. Thus by the necessity assumption I?(c, e) =0 implies I7(0, i) =O. By (16) U(c, e) =0 requires strict&y positive c between 0 ce &. with at least one (c,e) combination This means that if &d, 0 f(c,e)= U(c,e)=O exists. Appendix 3 Proof of Proposition

4

(i) Let i” and i* denote the market and central planner’s solutions respectively. Assume rm< T*. Since e,, h, ~0, for all k, .Yand z it follows that em> e* and hm > h*, and P ,J k, e*, h*) c Pk( k, em, h”) = r. Analogously if 4” c q*, it follows that em> e* and hm > h* since e,, h,
5

(i) Assume T< P. By (24) ey > e*,, and therefore z: > z*,. By (9), S > F,( x*, , z”). For S = F,( xy, z: ) to hold, xz < x*, . Therefore F( x:, z:) = h”, < h*,,, and S = P,(k”, ,ez, h:) may require a lower or higher level of capital than the social optimum. Thus c’!JSC+, . (ii) Assume 4 c q*. By (4.28) h”, > hS, and S > F,Jx”, , z*, ). Because _Y: is given, the stock of pollution has to decrease, otherwise the equilibrium condition for harvest cannot hold, i.e., z: cz*, and ez c”X’ References Arrow, K.J. and K. Kurz, 1970, Public investment, the rate of return, and optima) fiscal policy, resources for the future (The John Hopkins Press, Baltimore, MD). Brock. W.A., 1977, A polluted golden age, in: V.L. Smith, ed., Economics of natural and environmental resources (Gordon and Breach Science Publishers, London). Clark, C.W., 1976, Mathematical bioeconomics. The optimal management of renewable resources (Wiley, New York j. Dasgupta, P. and G. Heal, 1974, The optimal depletion of exhaustible resources, Review of Economic Studies Symposium. 3- 28. Keeler. E., M. Spence and R. Zeckhauser. 1972, The optimal control of pollution, Journal of Economic Theory 4, 19-34. Maler, K.-G.. 1974. Environmental economics: A theoretical inquiry (John Hopkins, Baltimore, MD). Romer, P.M., 1986. Increasing return:; and long run growth, Journal of Political Fconomy 94, 1003-1037.

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Seierstad, A. and K. Sydsaeter, 1987, Optimal control theory with economic applications (North-Holland, New York). Sorger, G., 1989, On the optimality and stabtlity of competitive paths in continuous time growth models, Journal of Economic Theory 48.526547. Tahvonen, 0.. 1989, On the dynamics of renewable resource harvesting and optimal pollution control, dissertation, Publications of the Helsinki School of Economics A, 67. Tahvonen, 0. and J. Kuuluvainen, 1990, The existence of steady states in growth models with renewable resources and pollution. Discussion papers no. 299, Department of Economics, University of Helsinki, Helsinki.