An energy analysis of the investment problem

An energy analysis of the investment problem J. Y Greenman

The central planning model presented here generates optimal resource utilization and investment paths in the case where there is substitution between new and old capital, and where the resource input to capital is decreasing with time. Solutions to the model are found to have a straightforward interpretation in terms of energy analysis. Application of the model to a one-sector, one-resource case identifies bounds on the rates of both investment and resource utilization. These bounds are compared with those suggested by other models.

In a recent series of papers I-’ the implications of the finiteness of essential resources have been studied using a variety of economic models. In particular the question as to whether resources should be exhausted in finite time and the question of the existence of a non-zero limit to per capita consumption have been examined in some detail. The reassuring results obtained arise primarily from the assumption of substitution between resources and other factors of production, an effect largely ignored in the work of Forrester. Some caution, however, has to be shown in the choice of production function to capture this substitution effect. For example, conventional neoclassical production functions may be in conflict with fundamental laws of physics. These laws prohibit substitution wifhouf limit between resource and non-resource factors. To meet this objection this paper develops a vintage model of production which, for simplicity, restricts attention to just one type of substitution: that between new and old capital generated by resource-saving technology. The required resource input is given exogenously as a decreasing function of vintage with a positive lower bound for unit production. This model is embedded in a mature economy with stable population and consumption demand. The question is then asked: ‘For how long can J. V. Greenman University

is with the Department

of Essex, Wivenhoe

the economy maintain an exogenous path of consumption with steady improvement in resource-saving technology with the presently available stock of resources?’ In other words, we wish to know how much time there is in which to develop alternative resource technologies. Posing the problem in this way sidesteps intergenerational issues9vr0 and the question of uncertainty in, for example, future technology. The model is unashamedly didactic. Its usefulness derives from the belief that simple paradigms are of’ importance in gaining insight into the workings of a complex system. The model The economy is divided into (n + 1) sectors, the zeroth being the capital good sector. For the ith sector at time f, the rate of investment, the rate of output, the rate of input from the ith sector, and the rate of input of the kth primary resource are denoted by Ii(t), Xi(t), X,(t), and Rki(t) respectively. If we assume complete linearity in the production process we can write:

xi(t) = 0;

Park, Colchester

CO4 3SQ,

Essex, UK.

&(r - u)li(u)

j

du,

(1)

*&)

of Mathematics,

i=o.

l,...,n

I would

like to thank Dr Richard Eden for hospitality at the Energy Research Group of the University of Cambridge during a sabbatical visit, I would also like to thank Professor Hammond, Dr W. P. Kennedy and Dr J. F. Toland of the University of Essex for useful conversations during the preparation of this paper. Final manuscript

$02.00

=

I

xii(u>p,
0 1979

IPC Business Press

(2)

‘yj(‘)

i=l,...,n;i=O,l

received 28 May 1979.

0140-9883/79/040219-05

X,(t)

,...,n

219

An energy analysis of the investment problem: J. V. Greenman

Rkj(t) =

(3)

Zj(U)Yj(U)

aj(r)

k=

l,...,

r;j=O,

>

i

xii(t) +

Yj(U) G Uj

i=l,...,n

(9

Pj(f - u)Pi(t) dt -

PO(U)

I

Rj(u)

flj(f

-

U)

dt

k=l Rj(u)

+ t

(4)

zj(t)

C&It)Q xdt)3

(8)

where:

i= 1

j=O

f

= 0

Yj(z.4) < 0 I

n

l,...,

where &j(t) denotes the vintage of the oldest capital used in the jth sector in year t, xii(t) is the rate of usage of the ith product, rkj(u) is the kth primary resource per unit of jth sector capital of vintage U, flj(t - U) is a given function converting the stock of capital of vintage u in sector j into the flow of capital services in that sector at time t while oj is the constant factor which converts those services into a flow of output. The intersector production constraints are given by:

XO(t)

with multipliers PO(t), pi(t) and qk. The first order conditions are given by:

I

xij(U)

Pj(f - u)Pdt>dt

>

Rj(u)

and R&U) denotes the periods of use ofjth of vintage u; and by:

sector capital

j=O

where Cj(t) denotes the rate of final consumption of the ith product at time t. The primary resource constraints are given by:

zj(oj(t))Zi(f) Zj(t)

= 0 (9)

G 0 I

where:

T

iolRXi(f) dt
k=l,...,r

(6)

zj(t)

E

ojPj(t)

-

i

4 krk j(aj(f))

k=l

0

where Rk denotes the initial endowment of the kth resource extractable with the given technology and T the length of time the economy can be maintained at the proposed consumption rates. The problem is to maximize T subject to the constraints (l)-(6) and appropriate non-negativity conditions, supposing that the consumption path and usage rates, Xii(U) and rkj(u), are given exogenously. The problem could be restated in social welfare terms. This is not done here because of the difficulties in choosing a suitable welfare function and the fact that many energy studies are formulated with given consumption paths. The model could also be enlarged by taking the usage rates to be decision variables, on the assumption that capital has the putty-clay property such that input proportions can be chosen at installation but not altered thereafter.” Enlargement of the model in this way will be the subject of a future paper. The first order conditions for the optimal path are obtained by first order variation from the Lagrangean:

L=J(l

~PO(r)(~ozjO~xOO)

+

i

i= 1

Pdt)xij(oli(t))

)

Conditions (8) and (9) have direct economic meaning when we interpret pi(t) and qk as the relative prices of the ith sector product and kth resource respectively. Zj(f) is the rate of profit in using one unit of capital of vintage oj(t) at time c in the jth sector. The oldest vintage to 3e used is essentially that which yields zero profit. Similarly Yj(U) denotes the net life-cycle profit of an extra unit of investment at time u.

-ilPAr)

0 X

i - k=li

qk

f Rkj(t) ]dt j=O

Vmtoge, u

(7) Figure

1. Production

ENERGY

region.

ECONOMICS

October

1979

An energy analysis of the investment problem:

As a first step in solving the set of relations (l)-(6), (8) and (9) we eliminate the shadow prices pi(t). From (9) with Zi(q(t)) assumed non-zero we obtain: PiCt)= i

i=O,

4k”kiCt)7

1,. . . ,n

(10)

k=l

where M(t) G B(t)(S - A(t))-’ and S, B(t) and A(t) are the matrices with elements Sij = oraij, B(t)kj = rkj(Oi(t)) and A(t)ij = xij(aj(t)) respectively, defining Xoj as zero. Substituting Equation (10) in (8) we obtain with Zj(U) non-zero:

J. V. Greenmm

relative to the energy used by otherwise idle capital. The elimination of prices between relations (8) and (9) therefore yields an equation system where the accounting unit is energy.12 This system does not appear to be easy to handle mathematically, including as it does coupled non-linear integral equations. It has, however, a relatively simple underlying structure which we now illustrate by considering the simplest case where there is only one resource and only one sector.

The one-sector, one-resource case Oi

&(t - U) i

I

qk”kjct)

When I(u), I(a(t)) are both non-zero the four equations we have to solve become:

dt

k=l

Rj(u)

qkMkO@) -

+

1 i,

i k=l

fij(t

qkrk,@)

-

U)

s

(I(+))

dt

s Rj(u)

1 - T(u)) dr = -
R(u)

z(t) + C(t) = u + f xij(u> i= 1

&(f

-

U> k$,

qkMki(t)

(12)

dt)

= 0

s Rj(u)

s

Z(u) du

(13)

=frf t

T (11)

The economic meaning of this statement is straightforward in the case where there is only one resource, say energy, with k = 1, which is exhausted at doomsday. In this case qk = 0 for k > 1 and the left hand side of Equation (11) becomes the net energy savings obtained through the introduction of one extra unit of capital in the jth sector in year u. This assertion is justified by the following observations: Using the expansion: M(t) = B(t)S-’

+ B(t)S-lA(t)S-’

t B(t)S-‘A(t)S-‘A(t)S-r

+...

it is clear that the second term in Equation (11) is the total direct and indirect energy needed to manufacture this extra unit of capital. The input quantities, Xii and occurring in the expansion are evaluated at the vintage times &j(t) because the extra capital is produced by bringing into operation at each stage in the production chain capital that is not already being utilized. The third term in Equation (11) is the total energy used directly to operate the extra capital over its lifetime and the fourth term is the total indirect operating energy. The sum in curly brackets is therefore the total energy cost of introducing and operating the extra capital. The first term in Equation (11) gives the gross energy that would have been used if the new capital had not been introduced and production had been maintained with old capital.

rkj,

Equation (11) tells us that any investment made must save enough energy during its lifetime to pay for the energy used in its production, the savings being measured

ENERGY ECONOMICS October 1979

dt s 0

dur (u)Z(u) = R

(14)

s a(t)

p(t) = 4 r@(t))

(15)

dropping all unnecessary suffiies and taking f3= 1. The effect we capture in this special case is the investment response to increasing efficiency in the resource utilization of capital as described by the monotonically decreasing function r(u). We impose on this function the additional condition that r(t() > r. > 0. This models the physical constraint that the resource used to produce one unit of output cannot ordinarily be reduced to an arbitrarily small amount. To understand the properties of the Equations (12) and (13) it is helpful to use the two-dimensional plot of Figure 1. A point (u, t) in the shaded area indicates that capital of vintage u is used in year t in production. A horizontal cut, AB, therefore gives the spectrum of capital used in year t and a vertical cut, CD, the time span over which capital of vintage u is used. Equation (I 2) therefore involves integration over a vertical cut to yield the savings in the resource over the lifetime of the capital. Equation (13) on the other hand involves integration over a horizontal cut to yield the total production in year t. In attempting to solve the set of Equations (12)-(15) the first thing to note is that Equation (12) cannot hold when u is close to T since as u approaches T the left hand side of Equation (12) approaches zero with the range of integration whereas the right hand side is bounded away from zero by the constraint r(u) 2 r. > 0. Hence there must exist a T1 < T such that the programming alterna-

221

An energy analysis of the investment problem: J. V. Greenman tive to Equation (12), I(u) = 0, holds in the interval T1 < u < T. In economic terms this just means that close to doomsday there is not enough time to recoup the amount of resource used in producing the extra capital by savings in the use of that capital. Before describing one method of solution for the set of Equations (12)-(15) two more simplifying assumptions are made. First, consumption is taken to be constant, with c(t) = C’,. With this condition Equation (12) implies that a(t) = (Y(T& - Tz in the interval T, < t d T and: T, Co = u

I(u) du s TZ

for u <0

I(u) = re +rr e-61( = r0

for 2420

l

(T2-%TI)

+(T,6

a=,;(1

s-

G e"ct

-T,o))+&

while the optimal rate of utilization U(t), is given by:

I 40

of the resource,

r(u)Z(u) du = E, + E, e-Of - E, ear

(19)

for 0 < I < T, where E, i E, and E3 are three constants

given by:

E,

E

Jr, 6-l

e(T,o-T8 ),

E32!! cl

An examination of the forms of Equations (18) and (19) indicates that both Z(t) and u(t) are strictly monotonically decreasing functions in the interval 0 < t =GT,. The shadow price of the resource relative to the capital good, however, initially increases exponentially for small values of the ratio r,,/r, with the productivity coefficient u playing the role of interest rate:t3vi4 G ‘Ii 1 ,ot

with rl E rl e(TI~-TG)

t

“)Z(cu(u))dv

(17)

0

for 0 < t < T, where Z(cu(v)) is read off from the given vintage stock of capital available at the start of the optimal programme, and the initial investment I(0) is determined by substituting Equation (17) in Equation (13) with t= 0. To find T, T1 and Tz when its value is less than zero use Equation (17), the condition I(T1) = 0 and the equation obtained from Equation (12) by putting t= T1. Use Equation (15) to obtain the relative shadow price. The rate of depletion of the resource and the rate of investment both depend on the historic stock of capital.

222

where

4”(t) = 4/P(T) = (+(t)))-’

Differentiate Equation (13) and solve the resulting first order linear differential equation for I(t) to obtain :

Z(t) = Z(0) cur -

(18)

(16)

Differentiate Equation (12) to obtain (Y‘(t) = o/S for 0 < t < T1 and hence: +

cot

I

We are clearly trading some realism for mathematical tractability. The immediate implication of Equation (16) is that the oldest vintage used, Tz, at the cessation of investment, T1, must be non-positive since there is no advantage in not using capital incorporating all technological improvements. The model can now be solved as follows:

ol(t)=;t

Z(t) = J p - A

U(t) =

since no investment is being made in this interval of time. Second, it is optimistically assumed that all resourcesaving improvements are made at a stroke. Expressing this precisely, it is supposed that:

l

If, for example, investment has been constant at rate J then the optimal rate of investment for 0 < f < T, takes the form:

As c increases i(t) deviates from the exponential to saturate at value r&l. The marginal unit profit, d(q(/)/dU, on the other hand, behaves initially like (E4 ear - Es e20r) before saturating out also at r;‘. If it is assumed that investment behaviour was different in the past then future optimal investment strategies will be different. For example with cyclical historic investment we will obtain cyclic future investment. Expressing this precisely, with: /(t)=JO+J1

sinwt

for t < 0 and with 0

an investment

schedule

J(t) = F, + F, sin(wcu(t) + 4) - F3 ear for 0 =Gt < T,, where F,, F, and F3 are three constants and 4 is a lag factor. U(t) is also cyclical but the relative price q(t) retains its monotonically increasing form. If, against the spirit of the above, we put r. = 0 then if T2 turns out to be negative the golden age where no resources are necessary cannot be reached without relaxing the consumption constraint. This is because

ENERGY

ECONOMICS

October

1979

An energy analysis of the investment problem: J. V. Greenman there are insufficient resources to manufacture the capital that can make use of the remarkable advance in technology. With a larger endowment of resources the golden age can be reached with T2 = 0, sufficient capital being transformed into zero resource-input capital. If we relax condition (16) by taking: r(u)=ro +rl e --sU (20) for all u, then the solution procedure becomes a little more complicated because of the possibility that Tz is positive, some new capital being subsequently replaced. If this happens then Equation (12) when differentiated, has to be solved in a sequence of steps defined by the intervals: Tt+l E o(Tt) < u < Tt,

i=1,2,...,p

where p is the first integer such that To is non-positive. Similarly Equation (13) has to be solved in succession for the intervals: Q-1 < t < Ui E Cf-l(Uj_l) for i = 1, . . , q where 4 is the first integer such that U, > Tz. From a simple sign analysis of Equation (12) one can obtain the inequality a’(t) <;

(21)

the difference between the two sides of the inequality increasing with t. In the less optimistic case, definition (20) the rate of economic obsolescence is therefore slower. Too great a rate of replacement of capital leads to a less than optimal solution because: l l

older capital with larger resource input has to be used to produce the new capital; and new capital produced instead at a later date would have operated with less resource.

One can also show from Equation (21) that the resource shadow price q(t) is still bounded above by a multiple of the exponential cur.

Conclusions The model developed here for optimizing resource utilization yielded solutions in the form of dynamic equations, which have a straightforward interpretation in generalized ‘energy analysis’ terms. In the simplest case of one sector and one resource the dynamic equations reduce to delay-differential equations

ENERGY

ECONOMICS

October 1979

equivalent to a finite sequence of standard first order linear differential equations. Analysis of these equations yielded bounds on the optimal rate of investment and the optimal utilization rate of the resource. A second paper will deal with the investment interactions of two- and three-sector models, in particular with: l l l

the capital and non-capital good sectors; the energy supply sector; and sectors with substitutable products.

The model will also be extended to include putty-clay substitutability and production lags. The purpose of the study is to establish whether our economic intuition is sufficiently robust to withstand the complexities of disaggregation. It is also hoped to use the theory of hierarchical systems15 to examine the problem of achieving the central planner’s optimal paths in a decentralized economy with market imperfections and uncertainties.

References 1 P. Dasgupta and G. Heal, ‘The optimal depletion of exhaustible resources’, Review of Economic Studies: Symposium on the Economics of Exhaustible Resources, May 1974, pp 3-28. R. M. Solow, ‘International equity and exhaustible resources’, ibid, pp 29-46. T. C. Koopmans, ‘Proof for a case where discounting advances the doomsday’, ibid, pp 117- 120. J. Stiglitz, ‘Growth with exhaustible natural resources: efficient and optimal growth paths’, ibid, pp 123138. 5 J. Stightz, ‘Growth with exhaustible natural resources: the competitive economy’, ibid, pp 139-I 52. 6 R. M. Solow, ‘The economics of resources or the resources of economics’, The American Economic Review. Vol 64. Mav 1974. DD 1-14. 7 R. J. Gilbert, ‘&t&al deple%on of an uncertain stock’, Review of Economic Studies, Vol46, No 1, 1979, pp 47--58. 8 J. W, Forrester, World Dynamics, Wright-APen Press lnc, Cambridge, MA, 1971. 9 R. M. Solow, op tit, Ref 2. R. M. Solow, op cit. Ref 6. :c: C. Bliss, ‘On putty-clay’, Review of Economic Studies, Vol35,1968, pp 105132. 12 ‘The economics of energy analysis revisited’, Energy Policy, Vol5, No 2, June 1977, pp 158-61. R. M. Solow, op tit, Ref 6. :: H. HoteIIing, ‘The economics of exhaustible resources’, Journal of Political Economy, Vol 39, April 1931. PP 137~-175. 15 M. D. Mesa&c et al. Theory of Hierarchical Multilevel Systems, Academic Press, New York, 1970.

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