A simple model of economic growth or decline under the influence of resource depletion Bent S0rensen

Niels Bohr Institute, Blegdamsve] 1 7, 2100 Copenhagen ~), Denmark (Received December 19 75)

A simple model is constructed for calculating production development and associated profit, under various assumptions regarding pollution control and availability of raw materials and energy. The purpose is to provide a transparent tool for discussing growth limitations.

I n t r o d u c t i o n

It has been suggested that models of growth well known in plant and animal ecology, models which are characterized by positive and negative feedback mechanisms, could also be applied to socio-economical systems 1'2. Definite choices of system variables, limiting factors and their functional relationships have been suggested 1" 3 and criticized as well 4-7 . Perhaps the most serious objection to the 'world models' of Forrester and Meadows et al. is that they discuss stability and instability in terms of averages between industrialized and non-industrial areas of the world, where in reality the social and economic structure and levels are widely different.

Attempts to approach these problems have recently been initiated 8. Further critiques address the fixed and purely economic control system and question the neglect of technological and scientific innovation factors, that might avert certain types of instability.

We shall adopt the point of view, that a simple model structure without regional differentiation can only be used for a fairly homogeneous society, i.e. a limited region in political geography. We shall try to build such a model, stripped of complications that are not directly associated with the production process, in order to see as clearly as possible the influence on the growth pattern that is brought about by definite physical constraints such as the need for pollution abatement and the depletion of raw ma- terials and certain energy resources. Focusing on the pro- duction process makes this model relevant only for regions in which the industrial sector dominates. Different model structures must be shaped for agricultural and other sectors. A more complete regional or world model would require the juxtaposition of several such models, interacting by mechanisms that reflect the economical and political power structure of the world, in particular the relations between the power structure and physical and human conditions.

A s imple p r o d u c t i o n m o d e l

Consider a group of products manufactured in the quantity p(t) (measured in production units, P.U.) per year or another convenient time unit, at the time t (time units, T.U.). The market value of the yearly production is p(t) (economical units, E.U.). We assume that the products are of a useful nature that can be sold in the quantities produced, at the price 1 E.U. per P.U. in fixed prices. This implies that the products in question follow or determine the price index, which in turn determines the ratio between inflated and fixed prices, the latter being used here throughout. For definiteness we assume that the selling price is composed in the following way. Each production unit requires one unit of raw materials and one unit of process energy. The extraction of one raw material unit requires one additional energy unit. The production cost (labour, investment in machinery etc.) per P.U. amounts to 0.5 E.U., the raw materials extraction cost (minus energy) 0.1 E.U. per raw materials unit and finally the energy cost 0.02 E.U. per energy unit. There remains 0.36 E.U./P.U. for distribution and profits at the different levels.

The very crude assumption of a fixed production cost implies that an increase in capital investment associated with more advanced machinery would be exactly compensated by a drop in the expenditure on labour, or alternatively that the net gain obtained by technological innovation would be spent on reducing working hours and improving working conditions, rather than on cutting the labour force or wages. Such trends have certainly not always been characteristic of the past, but may to some extent be a reasonable picture of the tendencies of present industrial societies with mixed economy and influential labour organizations.

We now introduce the regulating principle, by which it is

24 Appl. Math. Modelling, 1976, Vol 1, June

decided whether to increase or decrease the production level. This may be based on the derivative of the profit re(t) (per T.U.) with respect to production. If dm/dp is positive the production level is increased, if it is negative it is decreased. For simplicity the amount of change may be given by the same derivative, so that the production level for the next time unit is:

1d ] p(t + 1) = p(t) l +~pm(t (1)

The size of the time unit can now be specified as the time necessary to implement a change in production level (building a new production facility, raising capital for the investment, increasing the labour force etc.). Clearly such a time unit would vary according to a number of factors. We do not wish to complicate the model with conditions on capital availability, labour availability etc. We thus treat the time unit as a fixed quantity. Similarly we have assumed that the entire production can always be sold at the fixed price, i.e. we disregard market saturation as a limiting factor. Some of these conditions have been approximately

4 /

2

12

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tY

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Yea rs

Figure 1 Production curves assuming: A, pol lut ion control or recycling, arbitrary amounts of energy and raw materials at f ixed prices; B, same but with pollut ion kept below a f ixed level; C, as B but raw materials cost is increasing. No recycling is implemented and energy cost is still f ixed. The approximate scale of years indicated below the time units is based on product ion figures for Denmark 1 9 0 0 - 1 9 7 0

A simple mode/of economic growth or decline: Bent Sarensen

fulfilled for the affluent industrialized countries, at least during certain periods. A more rigorous treatment of capital, labour and market mechanisms would follow approximately the line of the physical types of limiting factors considered in the next paragraphs, but would in addition depend on non-physical aspects of the society, which make them harder to model. Since the main effect of these constraints would be to alter the rate at which the production level could be changed, much of their effect can be absorbed into the definition of the time unit, and when looking for parallels between the model behaviour and the actual production figures for a given country one should keep in mind the elasticity of the time unit.

The assumptions made so far imply a constant profit derivative dm[dp = 0.36, which leads to an exponentially increasing production level (Figure 1). This is indeed the type of curve characterizing past production development in most industrial countries. By comparison with total industrial production level, e.g. for Denmark, one finds that 1 T.U. equals 4-5 years during most of the period 1900- 1970. Some products have, of course, :reached a saturation level, but the observed exponential growth applies to totals of many products.

Pol lu t ion con t ro l

By pollution we shall understand all side effects of the production cycle, accidents at work, releases with adverse health effects and environmental impacts, including those associated with resource extraction and product usage and disposal. It seems reasonable to assume that the pollution potentially created (independently of whether it is curbed or not) is proportional to the production p(t). We shall now assume that authorities will enforce maximum total pollution limits, that should not be exceeded independently of the production level, and we put this limit at a pollution level of one pollution unit, corresponding to the pollution created by 1 P.U. (production unit). We shall further assume that a fixed cost of X, times 0.01 E.U. (labour and machinery) plus 0.5 energy units (which costs another 0.01 E.U. according to the assumptions above), is associated with reducing the pollution associated with 1 P.U. by a factor X. At the production level p the pollution control thus costs p2 times the unit cost, since the pollution from p P.U. must be reduced by p pollution units each. One can also understand this in terms of the maximum permitted concentration of pollution at the emission stage, which must be put at p - i if the combined total level should remain constant. Superfluous pollution, i.e. pollution which does not cost money to remove, is assumed to have already been dealt with.

The model is now in a form familiar from ecological systems, and the growth curve (Figure 1) becomes S-shaped, with an asymptotical production level of 9 times the level that defined the maximum permitted pollution level. If the maximum pollution level is accepted as similar to the present level, the model predicts that a 9-times increase in production can be sustained if appropriate pollution abatement is enforced. This control mechanism will be kept together with those extentions of the model presented below.

In demanding a ceiling on total pollution level we emphasize growth constraints that were not effective in the previous studies 1" 3. By 'pollution control' they understand a lower, but fixed pollution per P.U., in which case pollution

App l . Math . Mode l l i ng , 1976, Vo l 1, June 25

A simple model o f economic growth or decline: Ben t Sorensen

catastrophies occur under a number of circumstances. In contrast, the pollution control mechanism introduced here, under influence of the standard production po]icies, leads to a stable situation i f no other limiting factors are encountered.

It would seem unrealistic to assume weaker pollution regulations than ours, which correspond to the intentions underlying the best of current environmental control laws.

R a w mate r i a l s d e p l e t i o n

Raw materials may be grouped in classes of fairly substitutable materials (conductors, insulators, etc.), several of which would no longer be available in easily accessible deposits, after 30-100 years of usage at present rate 9. Although the earth is finite, the absolute abundance of any group of elements is usually so large that depletion is of little practical relevance. However, extraction of e.g. minerals from plain rock or seawater requires more and more input in terms of labour and energy, as the concentration approaches the average natural one for the mineral in question. The rate at which the efforts have to be increased as function of decreasing grade of ore depends on techno- logical innovation, but it is hardly disputable that the optimum effort and certainly the energy input has to raise as more new raw materials become required for the production process. The apparent decline in extraction costs for certain raw materials in the past has been associated with passing from less to more optimized processing (technological innovation) of ores of not too different grade. When the process approaches optimum no further improvements are possible. As an example the energy required to extract a minor constituent from plain rock must always exceed the minimum value given by the physical forces that trap the desired mineral (crushing the rock and breaking the chemical bindings). Upon this background it seems reasonable to define a raw materials cost which depends on the amount of raw material already used and dispersed (recycling will be dealt with in the next paragraph), and starts at the fixed value given above:

labour cost of one raw material unit =

0.1 1 + a i= r( i E.U. (2)

energy requirement for one raw material unit =

[l+a~r(i)]energyunitsi=, (3)

By choosing a = 0.1 the cost will double when 10 raw material units have been used (10 T.U. with a usage rate r(0 of 1 raw material unit/T.U, corresponding to 1 P.U./T.U.). The unit energy cost is still kept fLxed. The resulting production curve (Figure 1) has a maximum and decreases again. A similar curve has been derived by King Hubbert a°, for the use of a finite resource. We have shown that uniformly increasing raw materials cost gives qualitatively the same growth pattern as the depletion of a truly finite resource. This is an important result, since Meadows et al. 3 have been heavily criticized for assuming a finite, initial supply of raw materials 4' 7.

R e c y c l i n g

When the raw materials concentration in the discarded products exceeds the concentration at the deposits where

extraction of new raw materials is carried out, recycling becomes feasible. The economic feasibility of recycling also requires that the extra cost o f recollecting used products and separating the useful parts is warranted. For definiteness we shall assume that the labour and energy required for recycling is twice the initial cost for extracting new raw materials. However, the recycling cost remains constant whereas the price o f new raw materials increase. As a criterion for increasing or decreasing the recycling percentage T(t) one can take the derivative o f the fraction o f the selling price going into providing the raw materials with respect to the recycling percentage. The recycling percentage may be decreased or increased with the same derivative, if it can be kept within physical limits (maximum possible recycling Zm~),

T(t) -- d@ [by (t) + br(t)]

T(t + 1) = 0 if negative (4)

Tma~ if above Tm~,

The recycling and new raw material fractions o f the selling price are:

by(t) = p ( t - 1)T(t)[0.2 + 2Ce(t)]/p(t) (5)

br(t) = r(t) 1 + 0.1 r(i) [0.1 + Ce(t)]/p(t) (6)

where Ce(t) is the energy cost presently f x e d at 0.02 E.U./ energy unit.

If the production is decreased so much, that less raw materials are needed than provided by the recycling fraction from the previous time period, then a raw materials stock, s(t), is built up, that can be utilized later on:

s(t) = max[0, p(t -- 1)T(t) + s(t -- 1) - -p ( t ) ] (7)

The need for new raw materials amounts to:

r(t) = max[O,p(t)--p(t-- 1)T(t)--s(t-- 1)] (8)

The profit and energy use become:

m(t) = 0.5p(t) -- 0.01p(t) 2 -- 0.2p(t -- 1 ) r ( t ) -

0.1 1 +0 .1 r(i) r(O-e(t)Q(t) (9) -=

e(t) = p(t) + 0.5p(t) 2 + 2p(t-- 1)T(t) +

[1+ 0.1 ~ r(i)] (10)

Instead of basing the production and recycling policy (1) and (4) on instantaneous derivatives, one may base the regulation on the present value (capital value) o f the profit that would accumulate if the production and recycling percentage were kept constant during N time units, i.e. (neglecting interest):

din(t) 1 d -~ ~-~-.~ rn(t + i, p and Tfixed) dp

(11) d b ( t ) ~ 1 d N + i , p a n d

dT N = -- - ~ ~ b(t T fixed)

where b(t) = by(t) + br(t). As we shall see, planning based on a large value of N may spare the society for fluctuations

26 Appl. Math. Modelling, 1976, Vol 1, June

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Figure2 Pollut ion control and recycling opt ion included. Cost of raw materials increasing, but energy cost still f ixed independently of demand. The product ion and recycling pol icy is based on present value calculations of the pro f i t during N fo l lowing t ime units. - - , N = 1; . . . . . , N = 20T.U.

in production or profit levels, e.g. associated with overshoot modes. Figure 2 shows the results of allowing for recycling in the model, up to Tmax = 99%. The rapid decline and closing down of industrialization have been avoided, the production and profit curves are again S-shaped with a nearly constant equilibrium production of 6 P.U., a low profit of 12% and an energy expenditure of nearly 36 energy units, or about 6 per P.U. as compared to 2.5 per P.U. initially. Depending on the foresight in planning (N), the recycling starts to be economically feasible after 15 to 20 T.U. and rapidly reaches its maxinmm value. If 1 P.U. and 2.5 energy units are taken as representative of present day industrial nations, this growth mode requires an energy use in the industrial (including recycling industry) sector o f over 14 times the present value. Even if other sectors do not increase their energy consumption correspondingly, the size of the necessary energy conversion with its associated heat pollution may pq:-'e serious questions as to the desirability of this growth mode ~, which closely reflects the assumptions made by a majority of economic planners today. In addition the problem of acquiring so much energy at fixed prices must be considered.

Non-renewable energy resource depletion

Non-renewable energy resources are distinguished from many other raw materials by not being recyclable. If all energy is to be obtained from such sources, a price increase similar to the one used for raw materials above can be expected. The time interval during which prices would double would be very short for oil, natural gas and 23Su, considerably longer for coal. The estimate is complicated in the case of oil by the fact that extraction costs have been negligible in the past and prices hence set according to substitution value. Thus the increased cost of deep-sea drilling or extraction from low grade shale and tars may in principle not affect prices seriously, but the total amount of oil in any concentration on earth seems to be so limited that sharp cost increases cannot be far away. The cost of fission reactor fuel is also uncertain and depends strongly on the success in improving the characteristics of breeder reactors and making them acceptable to societies. If coal were to be used in large quantities, a cost raise would

A simple model o f economic growth or decline: Bent Serensen

mostly derive from averting the environmental disturbances and improving the safety associated with mining, and from pollution abatement at the burning stage. Fusion is a big unknown. If successful, the cost would probably be high, but constant in time.

Consequently it cannot be excluded at present, that energy costs will raise steadily as more of the non- renewable resources are used, and we shall run our model with the same price mechanism attached to energy as was used for raw materials:

[ ;d Ce(t) = 0.02 1 + 0.1 E.U. (12)

The results are again catastrophical, as shown in Figure 3, with extinction of industrial production following continued economic losses (N = 1 T.U.) or vanishing profit (N = 2X) T.U.).

We are thus brought to qualitatively the same conclusions as Meadows et al., but for a different reason. Due to our inclusion of strict pollution control and explicit allowance for recycling of raw materials, the main break-down modes of the Meadows study have been adequately averted. This places the focus on non-renewable energy sources as the likely cause of a catastrophical development. It is thus in the energy sector, that alternative planning nmst be considered.

Use o f c o n t i n u o u s e n e r g y sources

In view of the conclusions derived in the previous section, which depend on the uncertain future cost of energy from non-renewable sources, and in view of their associated

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Appl. Math. Modelling, 1976, Vol 1, June 27

A simple model o f economic growth or decline: Bent Sprensen

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Figure 4 A certain amount of energy (4 units} is available f rom cont inuous sources, at f ixed cost, rest as if Figure 3. The curves marked S include a smoothing mechanism in the production planning, in order to avoid the short term oscillations otherwise present in the case of short range planning ( - - , N = 1 ). Even w i thou t the damping mechanism the N = 20 curve ( . . . . . ) is practically without oscillations

problems of pollution in a broad sense, it should be remembered that the earth possesses a number of continuous energy sources, chiefly associated with solar radiation, which can be utilized at fixed prices and without serious pollution, in particular with httle disturbance of the earth's energy balance. Environmental impact can be considerable, especially in connection with use of falling water, currents and ocean temperature gradients, less so for solar collectors and windmills, the landscape disturbance of which is reversible

We shall investigate the implications of assuming the availability of a modest, fixed amount of energy from continuous sources. The limit chosen, 4 energy units, would correspond to installing a solar panel area or windmill swept area of a few promille of the land area, according to the interpretation of the model energy units used earlier. Energy above this limit would have to be bought at the increasing prices (12). The resulting growth pattern of production and other variables is shown in Figure 4. Using the instantaneous regulating criteria (1) and (4) an over- shoot phase is followed by oscillations about average values corresponding to the ceiling on the continuous energy sources. The oscillations can be suppressed by requiring that the change in production level at a time t be made so small that it does not by itself trigger a change of opposite sign at time t + 1 (N = 1,S curves in Figure 4). Such a regulation is in the realm of current industrial or political planning. The initial overshoot phenomenon can also be avoided by using the present value criterion (11) for time intervals of sufficient length (N = 20 T.U. case shown). This requires planning on a larger time scale than the one characteristic of current efforts by industry or by national governments.

Although the overshoot mode is not catastrophical, it may have the appearance of economic depressions experienced in the past (and to some extent at present). It is thus of strong importance to introduce the mentioned larger time horizons in present and future planning.

The model results including utilization of continuous energy sources exhibit further advantages. The equilibrium profit amounts to 22%, i.e. twice as much as in the case (Figure 2) of unlimited energy at fixed cost. The production level stabilizes at just 10% above the unit value, and energy

use hardly exceeds the assumed ceiling on permitted exploitation of continuous energy flows.

Conclus ions Through stepwise elaboration on a very simple model of industrial production we have identified conditions that lead to various types of growth pattern, typical of the ongoing discussion of growth limitations. However, in some cases the causes of defimite growth modes have been shown to deviate considerably from those appearing in earlier studies, and the specific role played by the energy sector has been clearly displayed.

The correspondence between assumptions and results have been made very transparent, in order that the experience gained may easily be used to construct more elaborate models, by extending the number of geographical regions and the subdivision of the societies into sectors, and by modifying the numerical parameters entering the model, in accordance with the accumulation of more empirical data.

Nomenc la tu re a proportionality factor between the accumulated

amount of raw materials used and the relative cost increase of new raw materials

b(t) fraction of selling price going into raw materials, at time t

br(t) fraction of selling price which at time t goes into new raw materials

by (t) fraction of selling price which at time t goes into recycling

Ce(t) cost of one energy unit at time t e(t) energy usage at time t E.U. economical unit (money value corrected for

inflation) m(t) profit at time t N number of time units, over which profits are

maximized, in the selection of production and recycling levels

p(t) production at time t (the amount or value of production)

P.U. production unit r(t) usage rate of raw materials at time t s(t) stock of raw materials at time t T.U. time unit T(t) recycling percentage at time t Tmax maximum recycling percentage

References 1 Forrester, J.W. 'World Dynamics', Wright-Allan Press,

Cambridge, 1971 2 Odum, H.T. 'Environment, Power and Society', Wiley, New

York, 1971 3 Meadows, D.H., Meadows, D.L., Randers, J. and Behrens III,

W.W. 'The Limits to Growth', Potomac Associates, New York, 1972

4 Oerlemans, T.W., Tellings, M.M.J. and de Vries, H. Nature 1972, 238, 251

5 Salerno, J. Nature 1973, 244,488 6 Boyle, T.J. Nature 1973, 245,127 7 Science Policy Research Unit, 'Thinking about the Future',

Sussex University Press, London, 1973 8 Mesarovic, M. and Pestel, E. 'Mankind at the Turning Point',

Dutton, New York, 1974 9 Dunham, K. Ambio 1974, 3,126

10 King Hubbert, M. Sci. Am. 1971, 224, 61 11 Wilson, C. and Matthews, W. (Eds.), SCEP report, 'Man's

impact on the global environment', and SMIC report, 'Inadvertent climate modification', MIT Press, Cambridge, 1971

28 Appl . Math. Modell ing, 1976, Vol 1, June