9 For a discussion of population and economic growth see
7 3.
When there is a locally stable equilibrium we saw that the long-run values of capital and pollution can be affected by the assumed value of the savings ratio. In fact, thriftier
communities have more capital and more pollution in equilibrium. This may partially explain why the pollution levels in the more industrialized countries are higher than in less industrialized countries even if the same technology exists in both. In the case of Model 3 we note that if the equilibrium is unstable then a large decrease in the savings ratio may result in a locally stable
equilibrium being established.
The assumptions of the above analysis were very severe.
There was no attempt to control pollution and no technical change. The savings behaviour of the community was simple and fixed over time. In assuming no pollution control and a constant savings rate s we "lock" the system into a given trajectory. If there is pollution control and/or if s is permitted to vary then the
trajectories are altered and we wish to find those trajectories which are ’’optimal" in accordance with some measure of social economic welfare. We now turn to a consideration of this problem of optimal control.
CHAPTER IV
OPTIMAL CONSUMPTION PLANNING IN A POLLUTED ENVIRONMENT *
The economic models examined in Chapter III were descriptive rather than prescriptive. We observed the evolution of an economy whose motion was described by a system of differential equations. The centre of attention was the stability of the growth process. Most economic processes can be controlled. For example, in the previous chapter investment could have been controlled by varying the savings ratio. In a controlled model we focus upon the
optimality of the control. Faced with many values for controls, the economic planner desires that control which gives the "best" result in accordance with some given performance criterion.
This chapter starts an investigation into how an economy with a pollution problem should be controlled. The purpose of the following models is to illuminate the central issues in pollution control and economic activity. The central issue is the pure consumption/pollution trade-off. In order to focus on this basic problem we assume away problems of natural resource exhaustion, population growth, technical progress, and capital accumulation. We thus assume that there is a fixed supply of factor inputs which produce a fixed amount of output in each period. We may think of this as the natural state of the economy, or we may think that a policy of "zero" growth in output has been decided upon by the central decision makers.
* A version of this chapter has been accepted for publication in the Economic Record.
75.
The society derives utility from consumption but in consuming output, pollution is generated which yields disutility. By foregoing present consumption the amount of pollution in the future may be reduced. This is the dilemma. Maximizing the utility of consumption is not consistent with maximizing social economic welfare. The
problem for the central planner is to determine the optimal consumption plan.
Keeler, Spence and Zeckhauser [1972, Model II], Plourde [1972] and Smith [1972] investigate various aspects of the pollution problem in similar models. The work of this chapter is closest in spirit to that of Plourde. Smith investigates recycling while Keeler, Spence and Zeckhauser examine the choice of a production technique.
The basic model is formulated more rigorously in section 4*1. Pontryagin's Maximum Principle is used in section 4*2 to discover the nature of the optimal solution. In section 4*3 we consider the effects on equilibrium values of changes in the parameters of the model. Section 4*4 re-examines the problem of 4*2 but with a change
in the assumption regarding the disutility of pollution. Section 4*5 presents the central conclusions of the analysis. Appendix 4A states without proof the necessary conditions for the control problem
4•1 The Mathematical Model
It is assumed that social economic welfare is measured by a
strictly concave utility function of current consumption and the
current stock of pollution. The marginal utility of consumption is
positive but diminishing. The marginal utility of pollution is
negative and decreasing. This disutility may be due to aesthetic
considerations, health, or discomfort. As the pollution level
rises these effects are aggravated. Formally the conditions on the
utility function U are
U = U(C,P) U(C,P) e C v uc > 0 ’ u c c 0 , C > 0 Up < 0 * u pp 0 , P > 0 u ccu pp u cp - °* (la) (lb)
(
2)
For purposes of the basic model we assume that an increase in the
stock of pollution reduces the marginal utility of consumption. That is,
UCP < °- (3)
This is economically reasonable: a rise in the dust particle level
in a picnic area will drive down the enjoyment derived from an
additional sandwich.
The following limit conditions are imposed on the partial derivatives of U:
lim U r (C,P) = 00 for all P > 0, (4)
O o L
and
lim U (C,P) = 0
P+o ^
77
Condition (4) ensures that no optimal policy will entail a zero level of consumption. This assumption is standard in the optimal growth literature. Condition (5) states that a small deviation from a clean environment does not reduce welfare. A few wisps of smoke from a
single automobile are unlikely to result in a public outcry. This condition is sufficient to ensure that no optimal trajectory moves the economy to a clean environment as will be seen below. For many forms of pollution, however, (5) is not reasonable. This assumption is relaxed in section 4*4 below.
A fixed output <j>° is produced in each time period. This may be interpreted as output net of replacement investment. This output
is allocated to consumption C or to pollution control E so that
Posing the problem in this way allows us to focus solely on the consumption - pollution trade-off.
The stock of pollution increases as a result of the consumption process. The flow increases at an increasing rate with respect to consumption. The stock of pollution is subject to decay at a
constant exponential rate a. In the absence of anti-pollution activity the stock of pollution evolves over time according to
4>° = C + E
(
6)
P = g(C) - aP where g(0) = 0 g f > 0 , g M > 0 C > 0 (7) and lim g ’ (C) = 0 C->o(or hasten the decline) of pollution by devoting some expenditure to anti-pollution activities. The amount of pollution cleaned up will be a function h of the amount of pollution control expenditure E. This function satisfies the following conditions
h(E) e C (2) , h(0) = 0 h' > 0 , hM < 0 , E > 0, and
lim h ’ (E) = °°. E-nd
Either the activity annihilates the pollution or it transforms it to some form which can be disposed of at no cost to society
(other than through E). Thus the communities net contribution to the flow of pollution is measured by
g(C) - h(E).
From (6) the level of anti-pollution activity can be determined solely by the choice of the consumption level
E = <f>° - C.
The net contribution then is a function of the consumption level. Define
Z(C) = g(C) - h(t(>0-C). Z has the following properties:
Z' (C) = g' + h ’ > 0, and
ZM (C) = gM - h" > 0.
The flow of pollution increases at an increasing rate with respect to consumption. Note also that from (8)
7 9. l i m Z ' ( C ) = l i m g ’ (C) + l i m h ' (4>°—C) C-*cj)0 C-*<J>° C-*<j>° = 00 # L e t C b e t h e s o l u t i o n o f Z(C) = 0 . Then o Z(C) < 0 f o r C < C o and Z(C) > 0 f o r C > C . o F o r C < Cq t h e communit y i s n e t p o l l u t i o n a b a t i n g ; f o r C > Cq , i t i s n e t p o l l u t i n g . Cq i s t h e l e v e l o f c o n s u m p t i o n w h i c h w i l l j u s t s u s t a i n a c l e a n e n v i r o n m e n t . T h e s e r e s u l t s a r e shown i n F i g u r e 1. F i g u r e 4^1
Z(C) may be thought of as the pollution control function. By selecting the consumption level the community uniquely determines the amount of pollution it generates in net terms. The control function has two components. One is an active control represented by h(E), where existing pollution is cleaned up. The other
component is a passive control given by g(C). Consider transferring a unit of output from the consumption sector to the anti-pollution sector. In terms of pollution control, society gains in a two-fold manner. First, with more anti-pollution control expenditure more pollution can be cleaned up. Secondly, since the consumption level is lower, less pollution is being generated.
With expenditure on pollution control, the accumulation of waste is governed by
4•2 The Optimal Solution
We assume the existence of a Central Planning Authority who seeks to maximize the discounted flow of utility. In order to give consideration to generations not yet born, the time horizon is left infinite. Future utility is discounted at a constant exponential rate p > 0. The formal planning problem is to
P = Z(C) - aP.
(
10)
max e ptU(C,P)dt p > 0
(ID
C J o
subject to
P = Z(C) - aP , P(0) = Pq , P(oo) free o
81.
This is a fixed (infinite) time free right-hand endpoint
optimal control problem. There is one state variable P and one
control variable C. Pq is the historically given initial pollution
l e v e l .
The performance functional (11) is an improper integral. H o w e v e r , ,00 e' ptU(C,P)dt <_ J o c°° e""ptU (tj)° ,0)dt ■' o U(4>°,0) P which is finite
and hence a maximum of the integral exists.
Using the Arrow and Kurz [1970] formulation of Pont r y a g i n ’s Maximum Principle (see Appendix 4A) the necessary conditions for a
solution to the above problem (11) are:
there exist a function \p(t) such that
H = U(C,P) + \ p ( Z ( C ) - aP)
and functions r(t) and q(t) such that
L = H + r(Z(C) - aP) + q(<t>° - C)
3,
-u-cc.p)
9C - 0 , or ty+r - Z T ( C ) + Z'(C) • p) T ip = pip - — , or ip = ( p + a H - U p (C,P) + ra r _> 0 , rP = rP = 0 , q(<t>°“ C) = 0.(
12)
(13) (14) q >_ 0 (15)A policy of all-out consumption is non-optimal since from (9) and (12) q-U (C,P)
llm c_z^Tc)— > = 0
0(f)0 => lim (^+r) = 0. But for C = <|)0 , P > 0 or P > 0 = > r = 0and hence ip = 0, which can only exist for an instant since
ip
= -Up (C,P) > o.For an interior solution (P > 0, 0 < C < 4>°) the necessary conditions are
It is well known in control theory that the costate variable
ip
has the interpretation of a shadow price of the corresponding state variable if the objective function has the dimension of an economic value.^ From (12’) it is seen that the shadow price of pollution is negative. (12’) also gives the derived demand for consumption as a function of the shadow price and the level of pollution. We can
1 See Arrow and Kurz [1970, Chapter 2] or Intriligator [1971 Chapter 14].
-u c(c,p)
(
12’)
and
83.