For an o p t i m a l s o l u t i o n it is r e q u i r e d that there exist a
c o n t i n u o u s f u n c t i o n tp(t) such that
113.
Ui( C ) = U 2 f f = * (17)
i - {p+ä - (||/||) - 4>'(K)}i|>. (18)
Condition (17) is a short-term equilibrium condition which
3P
states that the marginal utility of pollution control must
everywhere equal the marginal cost of pollution control in terms of foregone consumption, U|(C). These in turn must equal the shadow
price of capital ip (t ). More simply, the marginal social values of output in its alternative uses must be equated. These results are intuitively appealing. If the marginal utility of pollution control exceeded the marginal utility of consumption, welfare could be
increased by redistributing output in favour of pollution control.
(17) also gives the derived demand equation for pollution control as an implicit function of the levels of consumption and capital E = E(C,K) (19) with 3_E ac -u i
-[U2'(f > 2+U2 § ]
> 0 and 3E 3K-[U2<I>2+U2
> 0. (20a) (20b)Equation (18) is the time rate of change of the shadow price xp(t ). Using (17) and (18) we express the equations of motion for
6 = üj{p+6 - <fHI> -
K = <J) (K)-C-E(C ,K)-6K , K(0) = Kq
with the planning authority free to select C(0) = C .
(
21)
(
22)
We now investigate the behaviour of the system in the (K,C) phase plane. Consider first the behaviour of the capital stock.
Define N(C,K) = <j) (K)-C-E(C,K)-6K = 0 Nr = <{)’ - {| | + 6 } ^ 0 as K ^ K*
Nc =
-{1 + f } < 0
dC dK K=0- {f +
u + •££}
> < 0 as K - K*> (23)N(C,K) is increasing to some value K* and decreasing thereafter.
For fixed levels of K, K = <f> (K)-C-E(C ,K)-6K is a decreasing function of C. Hence K > 0 below N(C,K) = 0 and K < 0 above it.
Let K in Figure 1 correspond to the stationary capital stock
2
locus of the standard optimal growth model. F°r any given K, the gross investment 4>—C—E required to maintain that capital stock is the same as in the neoclassical model <j>-C. However, since E > 0 this means that the consumption level must be lower. Therefore, the K = 0 curve lies below the K = 0 locus.
2 See for example Dorfman [1969], or Intriligator [1971, Chapter 16].
115.
Turning now to the consumption level we define
M(C,K) 5 p+6 -
(ff/ff) - *'(K) “ 0
2 3 P “k 3P 3K 2 3 P ,2 ‘ 9K; 4>M > 0 3P 3K 32P 3E 3C > 0 (I
?)2 V3E; dC dK ‘m k < 0,c=o
(24)The locus M(C,K) = 0 is downward sloping to the right in the
(K,C) phase space. For fixed levels of consumption C is a decreasing function of the level of the capital stock. Hence C < 0 to the right of C = 0 and C > 0 to the left.
Also since
= p - (f£/f§) > P (25)
the locus C = 0 lies entirely to the left of C = 0 - the locus associated with the non-polluted model. These results are shown in Figure 1.
Figure 5*1
00 00 \ For completeness we demonstrate that the equilibrium (K ,C ) is a saddlepoint. It is sufficient to show that the Jacobian
00 00
determinant of the system (21), (22) evaluated at (K ,C ) is negative.
det J
ac
9C
9C 9K
9K * 9C (26)
00 00
where these derivatives are evaluated at (K ,C ). Using this fact and the information obtained above, we can express (26) as
det J
u i u i
NKM C M KN C <
(
27)
117.
As in the neoclassical optimal growth model, the path leading to the equilibrium is the optimal trajectory. That this choice is sufficient for a maximum is shown in the Appendix to this chapter. The Central Planning Authority selects an initial level of
consumption C(0) = which places the economy on the stable branch of the saddlepoint. With K(0) = Kq given and E(0) fixed by relation (17), the evolution of the economy is determined by the
00
equations of motion (21) and (22) . Suppose Kq < K , then capital and consumption increase over time. This implies that E increases over time as well.
In equilibrium the condition on the capital stock is that it satisfy it /ir \ r _ 9P /8P ^ ^ ^ 6 P 9K^8E or equivalently, 4>'(K)-6 . dE P + dK > P P=constant (28) (28’)
The marginal product of capital net of depreciation no longer equals the social discount rate, but rather the social discount rate plus the marginal rate of transformation of pollution control for capital in maintaining a constant level of pollution. Since
dE
dK 0
P=constant
it follows that the net marginal product of capital is greater than the social discount rate, and consequently (given the strict concavity of 4>) the endpoint capital stock is lower than the modified golden rule capital stock of the standard model.
Since K = 0 lies everywhere below K = 0, the consumption level in the polluted model is lower than in the neoclassical model. These results can be seen in Figure 1.
It is worth noting that the long-run equilibrium condition is independent of the utility function of pollution. The modification from the neoclassical version depends only upon the technical
relationship between pollution control and polluting capital. The condition
U ’(C) = U ’(P) | |
must hold of course but this must be satisfied everywhere along the optimal path.
5•3 Summary
Before stating the conclusions of the above analysis let us consider the standard optimal economic growth model. This has been used as a yardstick against which we compare our results.
Implicit in the standard model is the assumption that there are no wastes produced by the economic process, or alternatively, that if any wastes are generated they can be disposed of at no cost to society. The path which approached the equilibrium (Kg,Cg) was the optimal trajectory.
On the other hand, if wastes are generated and, as Mishan contends, are ignored by policy makers they will operate so as to
max
,c o
e"ptU 1 (C)dt o
(
29)
119.
In a competitive system the individual agents take the level of pollution as given exogenously. Their independent maximizing behaviour
3
may move the economy along the trajectory to (K^,Cg) in the same manner as in a centralized system where pollution was ignored.
For these reasons, the comparison of our results with those of the standard model seems very reasonable. The analysis of this chapter shows that in the long-run, if we take into account the pollution generated by production then we approach a smaller capital stock and have a lower consumption level than if pollution is
ignored. Is Mishan [1967] right and Parish [1972] wrong? Not quite. If the capital stock is initially below the equilibrium level, then we still desire growth and as we grow not only does
consumption rise but also the level of pollution control expenditure. This of course assumes that the optimal policy is being followed.
We turn in Chapter VI to an investigation of the growth problem when pollution is a state variable. The problem of
implementing the optimal policy in a decentralized society will also be considered.
3 This problem is the subject of current debate. See Hahn [1966,1970], Kurz [1968], and Shell-Stiglitz [1967].
APPENDIX 5A
In order to prove the sufficiency theorem for the problem in this chapter we need to show that the utility function is concave in C, E and K. That it is strictly concave in C follows from the original assumptions (10). From (12) we know that U 2 is strictly concave in P, but P in turn depends upon E and K. Thus we want the second order partial derivatives of U 2 with respect to E and K.
u 2 = u2 (p(k,e)) 1 » , a2p ap 2 — 2~ = U2 ^ 4 + U2 (H > * 0 3E2 2 3E2 2 3E
th. =
£l + „« (if)2 < o
3K2 2 3K2 2 3K 2 3 U2
|P . |P > o
3E3K 2 3E ' 3Kth. th.. aV,
3E2 3K2 ^ 3E3K^(
1)
(
2)
(3) (4) [U2 4 + U2 ( f > 2^ U2 4 + U2 # ) 2^ t U2 I * I ] ' 3E 9K (u.)2 4 4 + D X , +'2 ' 3E2 3K2 2 2 3E2 3K 3E' 3K2
121.
Conditions (2), (3) and (5) imply that is strictly concave in
E and K.
Let U(C,P(K,E)) and <f> (K) be strictly concave functions. Let
(C*,I*,E*,K*) be a feasible policy satisfying the necessary conditions (16) and (17), and also the transversality condition
lim e pti|;(t) = 0. t-x»
(
6
)We shall show that if (C,I,E,K) is any other feasible policy then
e~pCU(C*,P(K*,E*))dt > e"ptU(C,P(K,E))dt (7)
with strict inequality unless K*(t) = K(t), C*(t) = C(t), E*(t) = E(t)
From the concavity of U,
e p t {U(C*,P*)-U(C,P)}dt
e p t {U^(C*)(C*-C)+l^(P*) ||^-(E*-E)+U£(P*) ||^(K*-K)}dt
e p t {^* (C*-C)+i|j* (E*-E)+[ (p+6-(j>1 (K*) ] (K*-K) }dt
(e P {C*+E*-C-E+(6-<j) ’ (K*)) (K*-K)}dt
(p\p*e pt-e pti*) (K*-K)dt
(e pti|)*) {C*+E*-C-E+(S-<j>1 (K*) ) (K*-K)+K*-K}dt- (K*-K)e pt> *
(e pt>*){C*+E*+I*-(C+E+I)-<t>’ (K*) (K*-K)}dt
(e pt>*){c}>(K*)-<f>(K)-V (K*) K*+<f>' (K*)K}dt
(e pt> * M (<|> (K*)-4>T (K*)K*)- (cj) (K)-cJ) ’ (K*)K)}dt
>_ 0 from the concavity of <J) (K) .
Thus since U and $ are strictly concave then (7) is met with strict inequality unless K*(t) = K(t), C*(t) = C(t), E*(t) = E(t).
Since the path leading to the equilibrium (K ,C ) satisfies all the necessary conditions and also lim e pP^(t) = 0 this path is
t-x» the optimal path.
123. CHAPTER VI