Managing Natural Capital

A critical analysis of Dieter Helm’s ‘Natural Capital: Valuing the Planet’.



The Asset-Based Approach

The Importance of Natural Capital

Valuing Natural Capital

Natural Goods – and Bads

Sustainability and Renewable Natural Capital

Sustainability and Non-Renewable Natural Capital

National Accounting and the Capital Maintenance Charge




This book (1) addresses big and important issues about the relation between the economy, the environment and the well-being of future generations.  Its starting point is the scale of the challenge we face from environmental degradation associated with past and likely future economic growth.   To address this, it advocates an approach to economic policy-making designed to produce sustainable growth.  Central to this approach is what Helm terms the aggregate natural capital rule.

The book was presumably intended for a variety of readerships: environmentalists; economists; policy-makers; and ‘the educated layperson’.  What readers make of it may well depend on where they are coming from.  Although the book is very readable, with many examples to illustrate its general arguments, and no formulae or diagrams, I did not find it straightforward to understand how the elements of Helm’s approach fit together. The nearest he comes to attempting an overall summary of his position is in his description of three steps, according to which the aggregate rule is to be applied only after applying cost-benefit analysis to evaluate potential projects and correcting prices to reflect environmental costs (pp 132-3). Whether this is consistent with his emphasis elsewhere (pp 8, 241) on the central role of the aggregate rule is not entirely clear.

Readers with little economic background may wonder about the origins of the aggregate natural capital rule – especially if they google that precise phrase and find that the relatively few hits all lead back to Helm (2).  In fact (as the notes to Chapter 3 indicate), the rule has been developed within a considerable body of literature on the economics of sustainability, notable contributors being Robert Solow, who proposed that capital, including resources, should be “maintained intact” (3, 4), and David Pearce, who advocated “constant capital”, defined as including natural and man-made capital (5).  Solow, in turn, made his proposal in the context of earlier literature on the properties of long-term economic models in which it can be shown that consumption can be maintained indefinitely if investment follows what is known as the Hartwick Rule (6). Specifically, he showed that to maintain intact an “appropriately defined stock of capital” was equivalent to following the Hartwick Rule.

Economists will not be surprised to find that Helm sees a large role – much larger than at present – for prices in providing incentives to encourage conservation and discourage environmental degradation (pp 117-130 & 139-168). These might take the form of compensation payments, taxes or prices of tradeable permits: what is important is that they should reflect the environmental costs (externalities) of the activities of developers, producers and consumers.  Given sufficient information in a case of, say, pollution, the price can be set to induce an optimal balance between the damage done by the pollution and the costs of reducing that damage (p 164).  At the level of individual markets and particular environmental problems, then, Helm advocates policies designed so far as possible to optimise, to maximise economic efficiency with environmental costs included in the assessment of efficiency.  Since I broadly agree with this approach, which is a mainstream view among economists who study environmental policy, I shall say little more about it, focusing instead on other parts of the book.  Whether the public can be persuaded to accept environmental pricing on the scale proposed by Helm is open to question, but he certainly does a service in presenting the case for pricing to a wide audience.

When he considers the economy and the environment in aggregate and addresses the issue of sustainable growth, Helm’s approach is much more pragmatic.  Although informed both by economic analysis and ethical values, it is also influenced by considerations of political and practical feasibility.  He is not concerned, except in some brief critical comments (p 56), with the sort of economic models that consider levels of capital stock, resources and consumption for many ahead and seek to identify sustainable or optimal paths.  The aggregate natural capital rule is presented not as an ideal policy that would lead to some sort of optimum in balancing the interests of the current and future generations, but as “a line in the sand” (pp 8 & 241) – a challenging yet achievable target which would be a big improvement on the status quo, and one which we should be able to go beyond (p 73).  The rule is an example of what Solow calls a “rule of thumb” intended to ensure that long-term interests are not neglected (7).

Despite its sub-title and many references to the importance of valuation, the book does not discuss in any detail the methods commonly used by economists to value non-market environmental goods, such as the travel cost, hedonic and contingent valuation methods.  These are mentioned only briefly (pp 124-6) and not by name.  Given my interest in the travel cost method, I found this slightly disappointing, but most readers will probably be content to be spared such technicalities.

Those familiar with company accounting may be intrigued by Helm’s frequent references to the need for balance sheets within national accounts (pp 86ff).  It is clear that these would include a nation’s assets, including natural assets.  But I could find no answer in the book to the question of what would be the balancing items on the other side of such a balance sheet.  What Helm means, it seems, is a single-sided account showing the values of assets and changes thereto, perhaps better described as an asset account.

One final introductory point.  Helm rejects, for good reasons, the idea that we should aim for zero economic growth in order to ease pressures on resources (pp 37-8).  But once he has made that point, economic growth features in the book as a background assumption, something that technological progress will more or less inevitably make possible.  Although there is a chapter entitled “Sustaining Economic Growth”, there is little discussion of policies to achieve growth.  It is no criticism – given that one book cannot cover everything – to point out that technological progress is taken for granted, and that the focus is on sustainability, in the sense of ensuring that growth is not undermined by depletion of natural capital.


The Asset-Based Approach

The aggregate natural capital rule is an example of a sustainability rule.  Helm presents it as a sensible intermediate position between two other rules that have been widely discussed (p 63).  Advocates of strong sustainability hold that we should maintain all natural assets, and that man-made assets can never adequately substitute for natural assets (consequently they are opposed to economic growth and favour what they might term ‘living in harmony with nature’).  Weak sustainability, on the other hand, permits depletion or degradation of natural assets, provided that this is compensated for by appropriate investment in other assets.

Before considering the merits of different sustainability rules, it is worth noting something they have in common, namely, a focus on assets.  Helm regards this as a matter of considerable importance, describing his approach as “asset-based”. This has two aspects.  One is the extension of national accounting to include the asset account.  Whether that is worthwhile depends partly on the feasibility of valuing assets, which is considered below.

The second aspect concerns planning for future generations. In this context, Helm is at pains to explain why an asset-based approach is better than an alternative which he calls an “income or consumption-based approach” (pp 55-8). His target here appears to be a type of long-term economic modelling which makes detailed assumptions about the preferences of  people in the future and tries to plan for their happiness.  Such assumptions, he suggests are both impracticable (“extremely informationally demanding”) and inappropriate (interfering in “the personal domain”).  Noting the Brundtland definition of sustainable development which refers to “the ability of future generations to meet their own needs”, Helm suggests that what the next generation needs is not a detailed plan, but a set of assets that will enable them to achieve their own conception of a good life (p 57).

At one level this argument is correct.  It can be read simply as an application to the long term of well-known criticisms of detailed central planning.  As an argument against any form of planning for future consumption, however, it is unconvincing.  Admittedly we cannot know what preferences people in 30 or 50 years’ time will have in respect of, for example, lifestyles, fashion, entertainment and holidays, nor what new sorts of devices and technologies consumers will have available.  We can however make reasonable assumptions about their basic needs.  Indeed, on the very next page (p 58) Helm refers to the needs of future generations for what he terms “social primary goods” – including health, housing, education and electricity – together with infrastructure and systems to deliver those goods.  If we reject, as we should, the idea of comprehensive planning for the needs and preferences of future people, the obvious alternative is a much more modest form of planning focusing on such primary goods and leaving plenty of flexibility in other respects.

The real reason why we should focus on assets, which Helm never states, is simply this.  The main way in which one generation can affect the circumstances of the next is through the assets it passes on.  Can income be passed on?  Yes, but before it can be passed on it must be saved, and the very act of saving converts income, a flow, into a stock of value, or asset.  Can current research on solar power help the next generation to avoid an energy shortage?  Yes, but only in so far as that research leads to the passing-on of suitable infrastructure, installations, skills or knowledge, all of which are assets of sorts.  Indeed, if assets are broadly understood to include all forms of capital – physical, financial, natural, human, social, institutional – then it becomes a tautology that anything of value that can pass to the next generation is by that very fact an asset.

While Helm’s focus on assets, or capital, is entirely appropriate, therefore, it is not quite as substantive a point as he suggests; nor does it have much to do with respecting future generations’ right to their own conception of a good life.  The crucial issues are which kinds of assets are most important or most at risk, to what extent different kinds of assets can substitute for each other, and what guidelines and policies should be adopted to maintain and manage assets.

A balanced approach to planning for future generations should also have regard to population growth. We might consider population an asset in its role as the source of labour and a repository of knowledge and skills.  But when we think of people as ends rather than means, as consumers, and as putting pressure on nature by virtue of their numbers and needs, we adopt a different perspective.  Another way in which one generation can affect the circumstances of the next, therefore, is through the extent to which it allows its population to grow, increasing future needs for goods of all kinds.  Helm does discuss the impact of population trends (pp 23-4), but is I suggest too dismissive of policies to moderate population growth as a way to improve the lives of future generations (p 246).  Such policies need not imply coercion: for poor countries with rapidly growing populations, as in sub-Saharan Africa, a case can be made for aid to support education in birth control.  A limitation of any approach to sustainability which focuses exclusively on assets, therefore, is that it ignores the potential trade-off, in terms of marginal benefits, between money spent on conservation projects and on family planning programmes.


The Importance of Natural Capital

Having made a case for focusing on assets, Helm proceeds to argue for the special importance of natural capital, defined (p 3) as “elements of nature [that] directly or indirectly produce value to people”.  I find his distinction between ethical and instrumental arguments (pp 58, 60) puzzling: ethical and instrumental considerations surely need to work together, the former to justify objectives, and the latter to show the importance of natural capital to their achievement?  His best argument, and it is a powerful one, is that elements of natural capital are essential factors of production that future generations will need to provide themselves with social primary goods (pp 58-61).  He refers in particular to the brick and tarmac of cities as derived from nature, and the dependence of agriculture on soil ecosystems.  He might equally have referred to water and energy sources as vital factors of production.

A slightly separate argument relates to the physical, mental and spiritual benefits of access to nature (pp 59-60).  This relates only to some aspects of nature: not for example to coal or oil reserves. Moreover the benefits are to some extent a matter of individual preference, culture or circumstance (if we live in a city with polluted air we may value more the cleaner air of the surrounding countryside).  Whether access to nature should be considered a social primary good is debatable.  Although this argument is valid, it is much less important than the factors of production argument: even for those like me who love to walk in the countryside, in our hierarchy of needs it is well below food and a home.


Valuing Natural Capital

The aggregate natural capital rule requires that elements of natural capital be valued: economic value is the only possible basis for aggregating, say, hectares of forest and cubic metres of fresh water.  There is, admittedly, one circumstance in which the rule can be applied without values, namely, when any loss or depletion is compensated for on a like-for-like basis.  But as Helm recognizes (p 155) like-for-like replacement is rarely possible.  Although he suggests that valuation is unnecessary if we aim simply at  “holding … the aggregate line” (p 90), and only needed if we aim to improve or expand the asset base, I cannot see any basis for that.  If we attempt to hold the line by substituting B for A, we need to know whether B has the same value as A.  A comprehensive asset account would also require valuation of elements of natural capital.

Some would argue that to value nature in economic terms is wrong in principle.  Helm quite properly counters this by pointing out that, since we do not have the resources to conserve all natural capital, values can help select priorities for conservation (pp 4, 116).  But that only shows that it would be useful to be able to value natural capital: whether and to what extent it is possible are further questions.

The practicality is that, depending on the type of asset, or service provided by an asset, our ability to estimate economic values using techniques from a large economic literature on the valuation of environmental assets ranges from fairly good to very poor. Consider a forest, sustainably managed with growth of new trees offsetting logging of mature ones.  In the fairly good category we can place the value of the timber, which has a market price, although even in that case there is scope for argument about how that price may change in future, and about discounting of future revenues.  At the other extreme, suppose the plants in the forest include an endangered species, present only in a few sites around the world, of no known practical value, but which might one day be found to have some use for medicinal or other purposes. Any attempt to value the presence of the plant would be highly speculative.  Somewhere between these extremes is the value of the forest in sequestering carbon: the mass of carbon sequestered can be estimated fairly accurately, but there is plenty of scope for argument about the appropriate carbon price per unit mass, reflecting the benefit of carbon sequestration in moderating climate change.  Also between the extremes is the recreational value of the forest, which in principle can be estimated from the costs incurred by visitors in travelling to the forest, but which in practice raises many questions including the value to be placed on visitors’ travelling time and the most appropriate statistical methods for estimating a value from survey data (8).

There is a further problem.  To a large extent, what valuation techniques do is value specific environmental assets in their actual current circumstances.  Suppose for example that a country has many similar forests.  Recreationists who decide to visit a forest will mostly choose the one that is nearest to them and least costly to visit.  If the forests were fewer and further apart, then some of those visitors would choose to make longer, more costly trips (just as some Europeans and Americans make expensive trips to Africa to see animals that they cannot see in the wild in their own countries).  A valuation of a forest derived from actual visit costs will not capture that willingness to incur higher costs. That may be fine if the valuation is to be used in appraising a proposal to convert that one forest site for alternative use, while retaining all the other forests.  But it will not do if the aim is to find the aggregate value of all the forests.  For that purpose, we need to include the full willingness of potential visitors to incur costs of travelling to a forest, and that cannot be inferred from their current behaviour if their nearest forest is relatively close.  Visitors could instead be asked how much they would pay to visit a forest if there were far fewer forests (as in contingent valuation (9)), but how reliable their responses might be given the extremely hypothetical nature of that scenario is open to question.

Helm recognizes that estimating values for natural assets is often difficult (pp 125-6, 136), but  offers several arguments in defence of valuation.  Approximate values, he suggests, are often good enough (p 127).  If the cost of preserving an ecological site is known, then a decision to preserve the site can be based on knowledge that the value of the site is more than that cost, and it does not matter whether it is twice the cost or ten times the cost.  But that is in the context of evaluating a possible project.  For the aggregate rule to be workable, a degree of approximation would be acceptable, but if many of the larger items added together are subject to a wide range of uncertainty, the overall figure will not be very meaningful.  Another argument is that valuations, even if imperfect, are to be preferred to judgments by experts or regulators subject to lobbying by interest groups (p 136).  That seems dubious, since application of valuation techniques itself often requires judgments (eg in choosing between alternative statistical techniques).

An argument more relevant to the aggregation rule is that it does not require valuation of every element of natural capital, because to monitor any change in the aggregate value it would suffice to consider just the value of changes of those elements that have changed.  This is not explicitly stated by Helm, but is a possible interpretation of several passages (pp 10, 90, 104).  The example of a sustainably managed forest illustrates both the strength and the limitations of the argument.  In the absence of a relevant change, nothing about the forest will have contributed to any change in the aggregate value of natural capital, and there is therefore no need for any valuation in respect of the forest.  If, in a period, most assets satisfy the ‘no relevant change’ condition, then the scale of the valuation problem will be greatly reduced.  The difficulty here is what sort of changes should be considered relevant.  Even if the forest itself does not physically or biologically change, any of the following might be considered grounds for treating its value as having changed: a change in the market value of timber; a change in the worldwide population of the endangered plant species, increasing or reducing its risk of extinction; a change in the appropriate carbon price following new scientific evidence on climate change; or a change in the number of recreational visitors due to a change in population in the surrounding area.  In any year, the proportion of natural assets subject to some such change might be high.  There are complex issues here which, so far as I can see, Helm does not address.

Whatever the merits in principle of the aggregate natural capital rule and of a comprehensive asset account, I conclude, obtaining the valuations that their application would require would be extremely difficult, rather more so than Helm seems to suggest.  This is not to deny the importance of environmental valuation: its main roles, however, are in the appraisal of potential projects, enabling the monetary value of changes to environmental assets to be included in cost-benefit analyses, and in the assessment of environmental externalities as a basis for determining the appropriate level of policy instruments such as taxes, subsidies and compensation payments.


Natural Goods – and Bads

Among Helm’s examples of natural capital are some that can cause harm: species, which include pests and pathogens; land, which can be earthquake-prone; air, which can bring hurricanes; and oceans, which can bring surge tides.  It would be unfair to suggest that Helm ignores the downside of nature, which is clearly recognised in his discussion of flood defence (pp 186-8).  Nevertheless, the book conveys, to me at least, an impression of nature as largely benign to the human race.  For those of us who live in southern England, with its relatively mild climate, moderate year-round rainfall and rarity of serious natural disasters, that may seem a plausible view.  Inhabitants of many other regions of the world may incline more to a view of nature as indifferent to humans, to be coped with and managed.  China’s environmental statistics, for example, record (in each case figures are for the latest available year): 623 deaths from 20 earthquakes; 482 deaths from other geological disasters (eg landslides); 19,000 hectares of forest destroyed by fire, with 112 casualties; 3,000,000 hectares of total crop failure due to extreme weather conditions, flood and drought; and 52,000,000 hectares of grassland harmed by rodent and insect pests (10).

Given that there exist, from a human perspective, ‘natural bads’ as well as ‘natural goods’, it needs to be considered how they will be handled within any sustainability rule and any scheme for valuing nature. One approach, to bring natural bads within an asset account as a basis for applying a sustainability rule, would be to treat every bad as merely the absence of the corresponding good.  Thus an asset described as ‘absence of serious earthquakes’ would appear in England’s asset account, but not in Japan’s.  That might seem a neat way to justify an exclusive focus on goods, but it does not really work.  England then has in its account an ‘asset’ which, so far as is known, is a permanent feature of its geology, not at risk and not requiring any maintenance – a rather pointless piece of record-keeping.  Japan, on the other hand, has a serious natural risk which it mitigates by measures such as appropriate construction of buildings, but there is no entry in its account to indicate a need for such mitigation.  Unless we recognise both the goods and the bads in our assessment of nature, and the dual role of investment in man-made capital in mitigating the bads as well as complementing and substituting for the goods, we will have only a partial view of the relation between the economy and the environment.

Even in England, much investment in man-made capital is wholly or partly to mitigate natural bads.  Coastal and flood defences are examples, but so is any investment in the production of goods intended (perhaps as one of various functions) to provide comfortable temperatures and protection from the elements.  That includes housing, heating and air-conditioning (both appliances and the energy they use), and clothing.  Also of this nature is investment in health services, in so far as it is to address diseases of natural origin.

In considering possible sustainability rules, therefore, we should ask how, if at all, they will apply to natural bads.  Most fundamentally, should we regard the mitigation of natural bads as broadly desirable, or should we reject all such mitigation on the grounds that it is a form of interference with nature?  I take it that most people would accept the former, and that, for example, very few would argue that the appropriate response to flood risk is always either to live with the risk or if possible to move home, and never to build defences.  There are then questions about substitution.  Can the mitigation of a natural bad be an acceptable substitute for depletion or degradation of a natural asset?  Suppose for example it were possible to eliminate malaria from a region, but at the price of damage to other elements of the region’s ecosystems.  Would that be an acceptable project, subject only to the benefit being shown to exceed the cost?  Or should it be subject to the further requirement that the damage be compensated for by equivalent enhancement of other natural assets?  If moreover we formulate a sustainability rule in terms of maintaining an aggregate, can we simply net the natural bads against the natural assets?  These sorts of questions are not considered by Helm, but are I suggest quite fundamental ones for his approach.


Sustainability and Renewable Natural Capital

So far as renewable natural assets are concerned, Helm’s sustainability rule has only one version: the aggregate level should be kept at least constant (p 64).  But what exactly is renewable natural capital?  Helm refers to its “potentially infinite yield at zero cost” and its ability to renew itself or reproduce (pp 3, 50).  His examples – pansies, peat bogs, fish and trees – perhaps suggest that renewable natural capital must be biological.  And when he argues that depletion of renewables is “the real concern” (more serious than depletion of non-renewables) because they are subject to thresholds below which they cannot renew and regenerate themselves (p 35), it is clearly biological assets – species and ecosystems – that he means.  But sunlight, rainfall and wind are surely also renewable natural capital – both as essential support for the biosphere and as renewable energy sources?  They are certainly not non-renewable: the only alternative would be to have a third category, perhaps described as permanent natural capital.

Renewable natural capital is thus a very diverse category, and that presents a difficulty for Helm’s sustainability rule.  Why should we focus on the aggregate level of renewable natural capital, unless we consider that one element or type of such capital can substitute for another?  Helm gives the example of substitution at the margin of habitats for newts by habitats for nightingales (p 155).  However, there is nothing in the rule that limits substitution to marginal changes, to species of little practical value or concern to most people, or even to the biosphere as a whole. Prima facie, it suggests that it would be acceptable to substitute, for example: squid for cod; oak trees for skylarks; micro-organisms for elephants; or forests for lakes.

Within Helm’s overall approach, there are several considerations which to some extent limit the permitted substitutions.  Firstly, the sustainability rule is not supposed to be considered on its own: rather, it is preceded by cost-benefit analysis of potential conservation projects (pp 132-3).  In any particular set of circumstances, therefore, many possible substitutions via projects would be rejected because, in cost-benefit terms, they were less advantageous than others.  Secondly, Helm notes the importance of keeping the level of a species above the critical threshold below which it would no longer be able to reproduce itself (p 10), an example of a sustainability constraint.  Thirdly, the practical unit for such thresholds is often a habitat or ecosystem rather than a single species (pp 51, 104-6).  Hence many one-species-for-one-species substitutions are simply impracticable, because changes in levels of those species would also affect other species within their respective ecosystems.

While the effect of these considerations is significant, they do not go far enough in limiting the sort of substitutions that should be considered acceptable.  Explicit regard should be had to the functions or services provided by the respective assets.  If for example ashwoods (woods in which ash trees are the predominant species) are dying from ash dieback, oakwoods are in several respects a good functional substitute.  Both ash and oak (like all trees) sequester carbon; being deciduous and forming airy canopies, both allow light for plants on the woodland floor, especially in spring; and both yield hardwood timber.  Woods of spruce, a species which is evergreen, forms a dense canopy and yields softwood, would be functionally a less good substitute; while many other types of renewable natural capital, especially animals and non-biological assets, would be even less good.  Our focus, I suggest, should be less on very broad aggregates such as renewable natural capital, and more on multiple smaller aggregates of assets with similar functions.  This would counteract the tendency which broad aggregates have to encourage unacceptable substitutions.  An important additional advantage would be that, within any one functional group, aggregation could be to a much larger extent on the basis of physical measures, reducing reliance on questionable monetary valuations.


Sustainability and Non-Renewable Natural Capital

For non-renewable natural assets, Helm states two versions of the aggregate rule.  The weak version requires that depletion of non-renewables be compensated by investment in general capital, that is, in some combination of man-made and renewable natural capital (p 64).  That is quite close to the rules of Solow and Pearce, and requires valuation both of the depletion and of the compensating additional capital.

The main objections to the weak rule are the same as those identified above for the aggregate rule as it applies to renewable natural capital. It places too much reliance on valuations, and not enough on functions, in determining acceptable substitutions of one asset for another.  Indeed, these objections apply with even greater force, simply because of the huge variety of assets embraced by ‘general capital’.  Consider for example investments in entertainments, tourism and works of art, assets which (though they contribute to a broad view of a good life) have little or no impact on the provision of social primary goods.  Investing in these sorts of assets is fine, but it would be fallacious to regard it as compensating for the depletion of non-renewable natural capital.  Such investments do have monetary value, and that value may help to ensure that the aggregate value of natural and man-made capital is maintained. But they cannot substitute, functionally, for the depletion of non-renewable factors of production such as fossil fuels and other minerals.

The strong version of the aggregate rule differs from the weak rule in two respects.  Firstly, it does not permit depletion of non-renewables to be compensated for by investment in man-made capital: the compensation has to be in renewable natural capital.  Secondly, the terms of compensation are not to be based on asset valuations: instead, the amount of the compensating investment is to equal the economic rents from depletion of non-renewables (p 64).  Economic rent in this context, in simple terms, is the return to the owner of a resource arising from the difference between the market value of the resource and its extraction cost.  A possible advantage of this rule is that it could be applied without having to consider the sort of valuation issues identified above for the case of forests: for example, should the valuation of fossil fuel reserves be adjusted in response to new research findings on greenhouse gases and climate change?  Economic rents, based on current market prices and current costs, can be more straightforward to identify.

I shall argue however that the strong rule – and specifically, it’s requirement for compensating investment to be in renewable natural capital – is too restrictive.  Consider how it would apply to the depletion of fossil fuels.  Given such depletion (and to limit greenhouse gas emissions), we need to invest in developing alternative energy sources.  What is less clear is which alternative sources should be chosen, and the best mix will vary between countries and regions, with solar, wind and various biofuels among the options.  Solar and wind energy are often described as renewable, but that does not mean that to invest in them is to invest in renewable natural capital.  The  investment is in equipment such as solar panels and wind turbines – forms of man-made capital – to capture energy from those renewable (or permanent) sources.  On any normal approach to valuation, the investment adds to the value of man-made capital, but there is no addition to the value of renewable natural capital, which is what the strong rule requires.  A possible rejoinder is that the aggregate value of renewable natural capital should include a value for sunlight and wind, and furthermore that this value should be adjusted to reflect increases in our ability, via more and better solar panels and wind turbines, to capture their energy.  Barring that very problematic approach, the strong rule would discourage investment in solar and wind energy, and encourage investment in those renewable sources where the investment really would increase renewable natural capital – biofuels, perhaps.  And this would be regardless of the relative efficiency of these different energy sources, or of their respective implications for land use, air pollution and greenhouse gas emissions.

Let’s pursue the argument a little further, on the supposition that we have been thus encouraged to opt for biofuels as an alternative energy source.  There is then a question of what type of biofuel should be chosen, and some biofuel crops will be more efficient sources of energy than others, having regard to the climate and soil in particular locations.  The strong rule, however, would tend to encourage the selection of perennial crops (eg sugar cane and oil palms), which have a capital value based on their expected stream of future harvests, rather than annuals (eg wheat and corn), again regardless of their relative efficiency.

The upshot is that the strong rule tends to discourage a sensible energy policy, unless we either adopt the radical course of putting a monetary value on sunlight and wind, or else let the rule be over-ridden by other considerations.  There isn’t a strict incompatibility here, since we could always ensure compliance with the rule by some other investment in renewable natural capital, unrelated to energy – but that would involve costs.  The crucial point is that the discouragement of a sensible energy policy would be for no good purpose: if a part of non-renewable natural capital has no significant function other than as an energy source, and if the best alternative source of an equivalent amount of energy happens not to require investment in renewable natural capital, then there can be no harm in allowing the depletion of that particular part without any compensating investment in renewables.

The scope of this argument is not limited to fossil fuels.  Take copper, a mineral used in plumbing and wiring (and for various other purposes), present in abundance in the Earth’s crust, but with only limited reserves economically viable given current prices and technologies (11).  It would be sensible to invest in technologies for extracting copper at lower cost from less accessible or lower concentration reserves, and in alternatives to copper for use in plumbing and wiring.  It is quite difficult to see how investment in renewable natural capital might make an effective contribution here: investment in suitable man-made equipment or in extraction of alternative minerals appears much more promising.  Whatever the merits of different investments to address a potential copper shortage, the choice should not be influenced by a requirement to compensate for depletion of copper reserves by investment in renewable natural capital.  Similar considerations apply to other metallic minerals.  As in the case of energy, adherence to the strong rule tends to distort the allocation of resources, including capital investment, for no good purpose.

Can we perhaps argue that the strong rule is nevertheless a useful “line in the sand”: a rule which will usually point us in the right direction, even if cases can be identified in which it would lead to sub-optimal decisions?  No, because fossil fuels and metallic minerals are not minor exceptions, they are among the most important examples of non-renewable natural capital.

The other side of this is that the strong rule would tend to lead to over-investment in renewable natural capital. How can that be?  Surely there is no shortage of potentially beneficial projects to restore and enhance the natural environment?  Indeed there are, and that could be an attraction of such a rule for some.  For environmental groups campaigning to clean up polluted sites, protect habitats, and save endangered species, the idea that the depletion of fossil fuels and other minerals should have to be compensated for by correspondingly large investments in renewable natural capital must be very exciting.  But as Helm points out (pp 4, 119), we do have to make choices.  We cannot afford to do everything, and most people, conscious of the importance of primary goods, would regard investments to maintain energy supplies and electrical and water infrastructure as an essential part of our response to depletion of non-renewables.  This is not to deny the importance of renewables, or even to reject Helm’s assertion (p 35) that depletion of biological renewables is potentially the more serious threat to growth and sustainability. The point is simply that our response to depletion and degradation of ecosystems should be decided on its own merits, and not artificially linked to depletion of minerals.


National Accounting and the Capital Maintenance Charge

The key features of Helm’s proposed approach to national accounting  are an asset account (‘balance sheet’) and a capital maintenance charge.  The asset account would be a list of assets (natural and man-made) and their values, though for reasons of practicality it might initially be limited to the most important and most at-risk assets (p 89).  Presumably it would also record changes in those assets including depletion, degradation, natural growth, maintenance and enhancement.  A number of countries including the US (12) have explored the preparation of such accounts, but few if any have adopted them as a routine part of their national accounting (13).  Given the difficulty, described above, of valuing natural capital, this is perhaps not surprising.

The capital maintenance charge has two aspects.  It is an adjustment to GDP, to ensure that it reflects damage to the environment and depletion of natural resources.  In this aspect it is one of various adjustments to GDP that have been proposed by environmental economists. The second aspect is that it would mean raising a huge sum to pay for capital maintenance via some combination of pollution taxes, higher utility bills, ending of perverse subsidies, rents from depletion of non-renewables, and reductions in other public expenditure (pp 92 & 223-8).

Whatever their merits, these two aspects do not have to go together: one could adjust GDP, as a measure of economic performance and to inform policy-making, without any cashflow or tax implications.  For Helm, it seems, national accounting is not just data-gathering and publication of aggregate information, important though that is, but also accounting in the sense of national revenue collection, perhaps more properly termed public finance.  I shall say no more about this, since such large-scale fund-raising and investment raises broader questions about the role of the state and the balance between the public and private sectors that are not specific to environmental and natural resource issues.

Let’s return now to the capital maintenance charge as an accounting adjustment.  Environmental adjustments to GDP may have either of two motivations.  One is to provide a more comprehensive measure of current welfare.  Conventional GDP does not reflect the harm done by pollution, except to the extent that it affects the sorts of output that GDP measures.  So the current welfare perspective suggests a deduction for the estimated value of the current harm due to pollution.  From this perspective, however, there is no reason for a deduction for depletion of non-renewable resources, because such depletion does not reduce current welfare.

The second motivation is quite different, in that it is concerned with the future as well as the present.  It aims to measure, not actual current welfare, but the level of current welfare that could be sustained in the future, sometimes termed Net National Product or Green GDP.   This implies a deduction for depletion of non-renewable resources, because such depletion potentially impacts on future welfare.  So far as pollution is concerned, it implies a deduction for emissions of cumulative pollutants which will eventually cause harm, even if their current levels in the environment are low enough to be harmless.  This has been the motivation of most economists who have explored environmental adjustments to GDP, including for example Solow (14), and it also seems to be Helm’s motivation in proposing that GDP be reduced by the capital maintenance charge, which he links to depletion of non-renewable assets (p 92).

Whether this sort of adjustment to GDP is feasible and useful is a matter of controversy among economists.  This partly concerns valuation: theory suggests that depletion of mineral reserves should be valued at prices that, putting it simply, make full allowance for future needs (15).  Current market prices are unlikely to meet that requirement. So one focus of dispute is whether the ‘right’ prices can be identified with sufficient accuracy, either by showing that market prices, perhaps with adjustments for gross distortions such as the effect of OPEC, provide a reasonably good approximation to the ‘right’ prices, or by showing that appropriate adjustments to market prices can be calculated from other data such as estimates of future needs for factors of production.

A second difficulty concerns the fact that man-made capital tends to wear out, or need repair, a phenomenon generally described as depreciation.  If we consider that income and welfare can be sustained, even without technical progress, because depletion of non-renewable resources can be offset by sufficiently large investment in other capital including man-made capital, then it is incumbent upon us to consider the depreciation of the ever-increasing stock of man-made capital.  Given the standard assumption that annual depreciation is a constant proportion of capital, it must eventually exceed income, leaving no part of income for consumption to support welfare, or for investment in additional capital (16).  Without technical progress, therefore, there would be no level of welfare that could be sustained indefinitely.

That leads to the third and in my view most fundamental difficulty with adjusting GDP to try to achieve a measure of sustainable welfare. To avoid the above conclusion, which would result in adjusting GDP to zero, we have to make one or more arbitrary assumptions.  One possibility is to set a time horizon of X years, and find the level of welfare that could be sustained for that period.  Another would be to make an assumption about future technical progress – for example that the output achievable from given inputs will increase annually by n% – and find the welfare level that could be sustained indefinitely on that assumption.  Population growth is another consideration where a more or less arbitrary assumption is likely to be needed: even if world population eventually stabilizes, it makes a significant difference to the feasibility of a given average welfare level whether it settles at, say, 9 billion or 10 billion. It is fine for economists to make assumptions on these matters and see what they imply.  But they are not the sort of matters on which there is any prospect of reaching a consensus.  It is not going to be possible, with such assumptions, to arrive at a single adjusted GDP figure which can be considered a correct measure of sustainable welfare and which will command, or merit, credibility with policy-makers and the general public.



There is much in this book with which I can agree.  I agree that much greater use should be made of taxes and other price-based policy instruments to address environmental externalities. I agree that valuation of specific natural assets has an important role to play in setting such policies, and in estimating the costs and benefits of proposed projects that impact on the environment.

Most fundamentally, I agree that depletion and degradation of natural capital is a serious and growing problem, and that the economy should be managed in a way which gives a much more central place to environmental and resource issues, and has much more regard to the interests of future generations.  However, I am not persuaded that adherence to the aggregate natural capital rule would point the economy in a more sustainable direction.  The strong version of the rule would encourage a misallocation of investment resulting in  sub-optimal adaptation of the economy to depletion of fossil fuels and other minerals.  The weak version would place too much reliance on questionable valuations and, because it would permit bizarre substitutions of one natural asset for another, would have to be supplemented by other rules to produce sensible results.  Moreover, neither version embraces investment to mitigate natural bads.

My proposed alternative would include the following elements:

  1. A greater focus on targets relating to functional groups of assets, with less focus on aggregation over the very diverse category of natural capital as a whole. Some possible examples of functional groups are: sources of fresh water; sources of energy; waste sinks; and food crop pollinators (there would need to be many more than that).  Functional groups would not be limited to those relating directly to human needs, but would also include, for example, supporting ecosystems.

  2. More reliance on appropriate physical and biological measures, and less on valuations, in defining targets for particular functional groups of assets, and in determining the terms of acceptable substitutions of one asset for another within groups. The starting point in setting targets would be an assumption that for each functional group we should at least maintain a suitably defined aggregate, although in some cases more or less demanding targets might be justified by particular circumstances.  Targets should have regard to the likely needs for social primary goods of future generations.

  3. A greater focus on natural bads and investment to mitigate them, with clear recognition that investment in long-term mitigation of natural bads is an important part of what we leave for future generations. Examples include: long-lasting flood and tidal defences together with building of new homes in areas not at risk of flooding;  development of more drought-resistant food crops; and permanent medical advances (such as the eradication of smallpox).

  4. Extension of national accounting to include an asset register recording natural assets (and bads) in physical and biological terms (but in most cases without valuations), changes thereto, risk assessments, and classification by functional group (with many assets contributing to more than one group). The register would provide the detailed data needed to monitor performance against the targets for functional groups.

I readily accept that the above is far from fully worked out, but suggest that a framework along these lines would be less likely than the valuation-based aggregate natural capital rule to lead to mis-allocation of investment, and more likely to carry credibility with decision-makers.

Notes and References

  1. Helm D (2015) Natural Capital: Valuing the Planet  Yale University   All page references above are to this book.  Dieter Helm is Professor of Energy Policy at Oxford University and Chairman of England’s Natural Capital Committee.
  2. Based on my search on 4 January 2017.
  3. Solow R M (1986) On the Intergenerational Allocation of Natural Resources Scandinavian Journal of Economics  88(1) pp 141-9.  See Abstract p 141.
  4. Solow R M (1992) An Almost Practical Step Toward Sustainability p 169  (This paper is recommended as a non-mathematical way in to the literature on the economics of sustainability.)
  5. Pearce D (1992) Green Economics  Environmental Values 1(1)  pp 3-13   See Fig 1 p 6.
  6. Hartwick J (1977) Intergenerational Equity and the Investing of Rents from Exhaustible Resources American Economic Review 66 pp 972-4.
  7. Solow, as 3 above, p 148.
  8. See for example the Ecosystem Valuation website’s account of the Travel Cost Method:
  10. National Bureau of Statistics of China China Statistical Yearbook 2015   The figures quoted are derived from tables 8-27, 8-30, 8-31, 8-32 & 8-35.
  11. Wikipedia Copper
  12. US Bureau of Economic Analysis (1994) Integrated Economic and Environmental Satellite Accounts. See Table 1:
  13. Some countries, eg Canada, Germany and the Netherlands, publish “environmental accounts” which are essentially compilations of environmental statistics with few if any monetary valuations of non-market assets.
  14. Solow, as 4 above, pp 162-3.
  15. Solow, as 4 above, p 169. Solow’s argument is also described in Perman R, Ma Y, McGilvray J & Common M (3rd edn 2003) Natural Resource and Environmental Economics  Pearson Addison Wesley p 632.
  16. Buchholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model Scandinavian Journal of Economics 107(3) pp 547-61.  Modelling of depreciation is introduced on p 551 and the relevant result is on p 553 (case θ = 1 and δ > 0).  See also my post Constant Consumption with Resource Depletion.
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Pollution Control and Output

Discussions of pollution control often focus on abatement technologies. But that’s only half the picture.

Suppose a profit-maximising firm is producing and selling a good, but the production process emits a pollutant. In the absence of policy intervention, it will produce the quantity at which its marginal revenue equals its marginal production cost, making no effort to limit its emissions. Now suppose the government introduces a per unit tax on emissions. How will the firm react? The standard answer is that it will take suitable measures to abate its emissions, reducing them to the point at which its marginal abatement cost equals the tax.

Whether that answer is correct depends on what we mean by abatement. Discussions of abatement often focus on technologies to reduce, capture or degrade polluting by-products. For the above answer to be correct, however, we need a broader definition of abatement including both technological measures to reduce emissions and reductions in the scale of production. Such a definition may be found in Common & Stagl (1), but is not common in the literature, perhaps because it could be difficult in practice to distinguish reductions in output intended to reduce emissions from reductions undertaken for other reasons.

The important point here, however we choose to define abatement, is that the firm’s response to an emissions tax – or indeed to other policy instruments such as standards and marketable permits – is likely to include both technological measures to reduce emissions and a reduction in output. Let’s explore this in terms of diagrams.

Diagram 1 shows, for a single-product firm, the conventional marginal revenue (MR) and marginal cost (MC) curves. It doesn’t matter for present purposes whether the firm is a price-taker (as shown) or has some degree of market power. The profit-maximising volume Q*. is at the intersection of the curves.


Diagram 2 presents the marginal revenue and marginal cost curves in a different way, the horizontal axis showing output measured as a reduction from the profit-maximising quantity Q*.  Thus the marginal revenue and marginal cost curves are the mirror images of those in Diagram 1, and show the reductions in revenue and cost respectively from marginal reductions in output.  Also shown is part of the marginal profit (MP) curve.  This is simply the difference between marginal revenue and marginal cost or, equivalently, the cost in terms of lost profit of a marginal reduction in output.  Given the standard continuous convex form of the marginal cost curve (a consequence of a typical U-shaped average cost curve), marginal profit will be zero when output is Q*, and the marginal profit curve will be continuous and concave (as shown).


Diagram 3 shows the marginal cost of reducing emissions with output constant at Q*, so that the reduction in emissions is entirely due to technological measures.  The upward-sloping form of the curve reflects the reasonable assumption that unit reductions in emissions become progressively more costly as total reductions increase.  Whether the curve is linear (as shown), convex or concave will depend on circumstances.  Hanley, Shogren & White suggest that it will normally be convex, but also note the possibility of economies of scale in the abatement technology, in which case it could be concave (2).


Diagram 4 shows the marginal cost of reducing emissions with constant technology, so that the reduction in emissions is entirely due to a reduction in output.  In this case the cost is the lost profit, so the marginal cost curve is derived from the marginal profit curve in Diagram 2 together with the relation between output and emissions.  It seems likely that in many cases the latter relation will be approximately linear.  Hence the marginal cost curve is likely to resemble the marginal profit curve, albeit enlarged or reduced in the horizontal dimension to reflect the output-emissions relation.


Diagram 5 brings together the marginal cost curves of Diagrams 3 and 4.  Suppose now that a per unit tax on emissions is levied at the level shown.  A profit-maximising firm, applying the equimarginal principle, will reduce emissions by a combination of a reduction in output and technological measures.  Relying on either method alone would mean that opportunities to reduce emissions at lower cost than paying the tax were being left unexploited.  The total reduction in emissions will be that at which the marginal cost of emissions with both output and technology free to vary (the third curve in Diagram 5) equals the tax.


As Diagram 5 is drawn, a profit-maximising firm will reduce emissions by R* in total, of which R1 will be due to a reduction in output and R2 to technological measures.  The important point here is the combination: whether more of the reduction in emissions will be due to technological measures (as shown) or more will be due to output reduction will be determined by the positions and shapes of the curves and amount of the tax, and so depend on circumstances.

There is admittedly a simplification in the above analysis.  It treats the effects of output reductions and technological measures as independent, whereas in fact they influence each other.  If for example output is reduced by say 10%, and this reduces emissions by 10%, then the effect of technological measures will probably be less than if output had not been reduced.  This sort of interdepence is hard to show in simple diagrams. A more exact analysis might express costs and emissions as functions of both output and technology, and use differential calculus to find the profit-maximising combination for a given rate of emissions tax.  Nevertheless, the implicit assumption of independence is a good approximation provided that only fairly small reductions in emissions are under consideration.  And even in circumstances where the approximation is less good, the point remains that profit-maximisation in response to an emissions tax requires a combination of output reduction and technological measures.

A couple of consequences of the above are worth stating:

  1. A larger reduction in emissions is likely, given profit-maximisation, to involve a higher proportion of the emissions reduction due to output reduction. This is a likely consequence of the concave form of the marginal cost of emissions reduction curve with technology held constant (Diagram 4).  It will only not be the case if, loosely, the marginal cost of emissions reduction curve with variable technology and constant output is even more concave than that with variable output and constant technology.
  2. A well-known piece of analysis shows that market-based instruments such as emissions taxes are more efficient than standards because they result in firms with lower marginal abatement costs contributing more to the overall reduction in emissions than those with higher marginal abatement costs (3). In the light of the above it can be seen that this principle should be broadened to relate to costs of reducing emissions with both variable technology and variable output.  In particular, if two firms have available the same technology and the same output-emissions relation, then one with lower costs of reducing output, that is, the more gently sloping marginal cost and marginal profit curves, will contribute more to the reduction in emissions.

Notes and References

  1. Common M & Stagl S (2005) Ecological Economics: An Introduction Cambridge University Press  p 415.
  2. Hanley N, Shogren J F & White B (2nd edn 2007) Environmental Economics in Theory and Practice Palgrave Macmillan  pp 132-3.
  3. See for example Hanley, Shogren & White, as 2 above, pp 133-4.
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Constant Consumption with Resource Depletion

I present here a defence of the following proposition: to maintain consumption per person in the long run as non-renewable resources are increasingly depleted, the world needs technical progress so that we can obtain more output from available resources.  The alternative suggested by the Hartwick rule, relying solely on investment in man-made capital to substitute for natural resources, is extremely unlikely to work.

That the world needs technical progress to maintain per capita consumption as non-renewable natural resources are depleted may seem non-contentious. Admittedly, it’s questionable whether sufficient technical progress is possible, and arguable that those of us who are well-off should to some degree adapt to resource scarcity by consuming less.  But if one accepts that some essential resources are non-renewable, and accepts the objective of maintaining consumption, then surely it’s undeniable that the development and use of improved technologies is the only approach that offers any chance of success?

There is, however, a line of thought which suggests that consumption could be maintained without technical progress.  It has three building blocks.  The first starts from the fact that man-made capital can to some extent substitute for non-renewable resources (recycling plants, for example, can facilitate re-use of metals, reducing demand for mining of metal ores, and better-insulated homes can reduce demand for fossil fuels for heating). But  it asserts much more: that although the use of non-renewable resources can never be eliminated altogether, there is no minimum quantity of non-renewable resource needed to produce a given level of output. However little non-renewable resource is being used at any time, it would always be possible to use even less, by substituting for it even more man-made capital.

The second building block is that depletion of non-renewable resources could take place in such a way that the rate of extraction and use progressively declines but never reaches zero.  At any future time, therefore, there will remain some small quantity of non-renewable resource which, together with sufficiently large and increasing quantities of man-made capital, may support sufficient output to maintain consumption into the even more distant future.  The third building block is that the composition of output – how much is for consumption, and how much for investment in man-made capital – could include sufficient investment at all times that man-made capital would increase rapidly enough to offset the declining use of non-renewable resources.  How all this might be brought about –whether it would be best achieved via market forces or central planning – is beyond the scope of this post.

Taken together, these building blocks offer a plausible scenario in which long-run constant consumption without technical progress might be possible.  But whether it really is possible, at what level, and under what conditions, are further questions which I address below. My argument is not a proof or demonstration – it could be challenged in various ways -, but I offer it as a fair-minded assessment based on relevant theory and available data.

A useful framework for exploring these questions is the Dasgupta-Heal-Solow-Stiglitz model (1), of which we will consider a simple version.  We assume production of a single good which can be either consumed or used as (man-made) capital.  Output (Y) of the good requires inputs of man-made capital (K), a non-renewable natural resource (annual rate of extraction and use R, total reserves S), and labour (L).  Extraction of the resource is at zero cost, and labour which (along with population) is assumed constant.  We assume a constant returns Cobb-Douglas production function:

Y = AK^{\alpha}R^{\beta}L^{\gamma}    (\alpha+\beta+\gamma=1)

Here A, \alpha, \beta, \gamma are assumed to be fixed parameters.  Because the function is multiplicative, even if R is very  small Y can be very large if K is sufficiently large, but if R is zero then Y is zero.  This production function is therefore a more precise form of the first building block.  Because labour and population (P) are constant, it is convenient to rewrite the function in per capita terms and include the effect of labour within a new constant B = A(L/P)^{\gamma}:

\dfrac{Y}{P} = A\Big(\dfrac{K}{P}\Big)^{\alpha}\Big(\dfrac{R}{P}\Big)^{\beta}\Big(\dfrac{L}{P}\Big)^{\gamma} = B\Big(\dfrac{K}{P}\Big)^{\alpha}\Big(\dfrac{R}{P}\Big)^{\beta}

There are state equations for each of the variable inputs.  For capital, reflecting the division of output between consumption C and investment, and for the time being ignoring depreciation of capital, we have:

\dfrac{dK}{dt} = Y-C

For the resource:

\dfrac{dS}{dt} = -R

Given initial capital K_0 and initial reserves of the resource S_0, we want to know whether constant consumption is possible, and if so what level is possible, and what rates of capital investment and use of the resource will maximise that level.  There is a considerable literature addressing these questions, prominent in which are two rules: the Hotelling rule and the Hartwick rule.  The Hotelling rule is essentially a more precise form of the second building block, although it explicitly concerns not the declining rate of use of the resource but the associated increase in the marginal product of the resource (which can be written as \partial Y/\partial R or Y_R).  The two are associated via the formula (which follows directly from the production function):

R=\dfrac{\beta Y}{Y_R}

Writing Y_K for the marginal product of capital, the Hotelling rule can be expressed as:


Since the marginal product of capital will be positive, the rule implies that the marginal product of the resource should rise and therefore the rate of use of the resource should fall.

Similarly, the Hartwick rule is a more precise form of the third building block, although it explicitly concerns not the division of output between investment and consumption, but the growth of the capital stock (which equals investment if depreciation is ignored).  It can be stated as:


These rules have often been expressed rather differently from the above, using the concept of economic rent (2).  The assumption, often implicit, is a competitive economy in which returns to factors equal their marginal products.  For our purposes it is simpler to avoid that assumption and state the rules directly in terms of marginal products, as above.

From the literature may be extracted the propositions below.  Unless otherwise stated, constant consumption means a stream of consumption at a constant rate extending into the future with no time limit.

Proposition 1:  Constant consumption is only possible if \alpha > \beta (Solow 1974 (3)).

Proposition 2:  If the Hotelling and Hartwick rules are both satisfied, then output Y and consumption C will both be constant, with C = (1-\beta)Y (Hartwick 1977 (4)).

Proposition 3:  Any two of the following imply the third: a) satisfaction of the Hotelling rule; b) satisfaction of the Hartwick rule; and c) constant consumption (Buchholz, Dasgupta & Mitra 2005 (5)).

Note however that constant consumption can occur with neither of the rules being satisfied.  This can be shown by starting from a scenario satisfying both rules, and then considering the effect of reducing consumption to a lower constant level.  Then, compared to the original scenario, K will grow faster (while the time path of R will be unchanged), implying that Y will grow.  Hence Y_R will grow faster while Y_K will be less, so the Hotelling rule will not be met.  Hence the Hartwick rule also will not be met (otherwise Proposition 3 would imply a contradiction).

Proposition 4:  To maximise constant consumption from given initial capital K_0 and initial reserves of the resource S_0, the Hotelling and Hartwick rules must both be satisfied.  Maximum constant consumption C_{max} is given by (cf Buchholz, Dasgupta & Mitra 2005 (6)):

C_{max} =  (1-\beta)B^{1/(1-\beta)}((\alpha-\beta)S_0)^{\beta/(1-\beta)}K_0^{(\alpha-\beta)/(1-\beta)}

An obvious question is how this maximum constant consumption compares with actual consumption today.  To explore this, I took oil to be a representative non-renewable resource, and used the following assumptions, which are round numbers (to avoid spurious precision) intended to correspond reasonably well to reality:

  • \alpha=0.3; \beta=0.1 (consistent with a constant-returns Cobb-Douglas production function when labour is explicitly included with parameter 0.6), based on Solow (1974)(7) and allowing for a subsequent fall in the labour parameter based on income shares to c 0.6 (ILO / OECD (2015)(8)).
  • Current world GDP per capita = $10,000 (all monetary values are in current US$) (World Bank (9)).
  • Current world Consumption per capita = $7,500, based on GDP per capita as above and gross savings ratio 25% (World Bank (10)).
  • Current world Capital per capita = US$ 20,000, based on GDP per capita as above together with an average capital-output ratio of 2, inferred from charts in Johansson et al (2013) (11).
  • Current world oil reserves per capita = 300 barrels, based on total reserves for 17 largest countries 1,324 x 109 (12) rounded up to 2,000 x 109 for smaller countries, divided by world population 7 x 109.
  • Annual world oil extraction / use per capita = 5 barrels, based on total demand 35 x 109 barrels (IEA (13)) divided by world population 7 x 109.

The choice of oil as representative is based both on its importance and on the estimated life expectancy of its reserves being towards the middle of the range of life expectancies of different minerals (Tilton 2006 (14)).  The choice of reserves rather than the much larger resource base (15) as the appropriate measure of stock in this context is because exploitation going beyond reserves is likely to require either technical progress or, if relying on current technology, high extraction costs which would make constant consumption much harder to achieve.

Slotting data above into our production function to find the implied value of B we have:


Hence B = 440 (rounded) so the implied production function is:

\Big(\dfrac{Y}{P}\Big) = 440\Big(\dfrac{K}{P}\Big)^{0.3}\Big(\dfrac{R}{P}\Big)^{0.1}    (units: Y,K  in US$; R in barrels)

We now have everything we need to estimate maximum constant consumption using the formula in Proposition 4:

C_{max} = (1-0.1)\cdot440^{1/(1-0.1)}\cdot((0.3-0.1)300)^{0.1/(1-0.1)}\cdot20000^{(0.3-0.1)/(1-0.1}

C_{max} = 0.9\cdot440^{10/9}\cdot60^{1/9}\cdot20000^{2/9}

C_{max}= 0.9\cdot865.3\cdot1.576\cdot9.032

\bf{C_{max} = \$11086}

This may seem quite promising.  It suggests that it might be possible, without technical progress, to sustain indefinitely a level of per capita consumption well above the current average level of $7,500.

However, the above argument ignores depreciation of man-made capital.  In some contexts that might be a reasonable simplification.  Here, however, it is particularly inappropriate.  This is partly because we are concerned with the long term, in which the cumulative effect on the capital stock of even a moderate rate of depreciation becomes very large.  It is also because, in a scenario which relies on ever-increasing man-made capital to substitute for depletion of the natural resource, the cost of replacement capital to offset depreciation becomes ever larger.  That creates a potential problem: can output continue, indefinitely, to be sufficient to provide for replacement capital, as well as consumption and additions to capital?

It can in fact be shown that output cannot so continue. If we assume a fixed proportional rate of depreciation, however small, then constant consumption is impossible.  Instead of Propositions 2, 3 and 4, we have (given all other assumptions of our model):

Proposition 5: If capital depreciates at a constant positive proportional rate, then constant consumption is impossible (Buchholz, Dasgupta & Mitra (2005)(15)).

Given this proposition, our focus naturally switches to this question: if constant consumption forever is impossible, then for how long is it possible?   If it were found that constant consumption could continue for, say, 500 years, then all but fanatical advocates of zero discount rates would probably accept that that would be almost as good as forever.  However, the maximum duration of constant consumption will depend upon the rate of consumption and the rate of depreciation of capital \delta.  For the latter, I used 8%, based on 16.5% of GDP (OECD (16)) divided by capital-output ratio 2 and rounded.

For various rates of consumption, I identified the maximum sustainable duration by the following method.  Application of dynamic optimisation as described in this post showed that maximisation requires that the time path of use of the resource follow this variant of the Hotelling rule (Y_R being the marginal product of the resource and Y_K that of capital):

\dfrac{dY_R/dt}{Y_R} = Y_K-\delta

A spreadsheet was set up applying the production function, the state equations for capital (adjusted to allow for depreciation) and the resource, the above rule, and a fixed rate of consumption. To ensure that the discrete approach implicit in use of a spreadsheet was not a source of material inaccuracy, each row represented a period of just 0.01 years.  When a trial value was entered for use of resource in the first period, values of all the variables were automatically calculated for all subsequent periods. These values were inspected to identify 1) the period at which the resource became exhausted, and 2) the period (if any) at which the rate of output fell below that of consumption. The trial value was repeatedly adjusted until a value was found at which periods 1 and 2 were the same, indicating the maximum duration at the chosen rate of consumption.  For comparison purposes, and also as an independent check of the formula in Proposition 4, this method was used with depreciation at both 8% and zero.

The results are summarised in the chart below:


With or without  depreciation, maximum years constant consumption are very low for annual per capita consumption above $12,000.  With no depreciation, maximum years increase rapidly below $11,500, to about 50 years at $11,200, 150 years at $11,100, and 300 years at $11,090.  At $11,080, maximum years were found to exceed 500, consistently with the formula in Proposition 4 which implies infinite maximum years for consumption below $11,086.

With 8% depreciation, however, maximum years constant consumption increase more gradually at lower rates of consumption.  At $11,000, maximum years are just 8 (rather than infinite when there is no depreciation); at $10,000, 17 years; at $9,000, 47 years; at just over $8,000, 100 years; and at $7,000, about 400 years.

On the assumptions made, consumption could be maintained at about its current level ($7,500 per person) for about 250 years.  That might not seem such a bad prognosis, except that the assumptions on which it depends are so stringent.  In reality, resource extraction costs are not zero, and they tend to increase as the more accessible sources become exhausted and exploitation shifts to more difficult locations.  In reality, it is unlikely that, without technical progress, man-made capital can substitute for non-renewable resources to the extent implied by the Cobb-Douglas production function.  In reality, population will continue to grow for some time, even if it eventually stabilizes.  What’s more, a scenario which provides only for maintaining per capita consumption, with no hope of growth, and no means of raising the living standards of the poor other than redistribution from the better off, would be a gloomy prospect.  Leaving aside any wider social and political consequences, it would almost certainly lead to powerful demands to raise consumption in the short term at the price of reducing investment and so reducing consumption in the longer term.

The conclusion I draw is that the approach suggested by the Hartwick rule would be unlikely to sustain consumption per person for more than a few years, and would be extremely unlikely to do so for longer (100 years, say).  It is important to be clear as to the nature of the inference here.  Although I have presented a model, based on assumptions, and drawn out some of its implications, I have not claimed that what happens in the model is likely to happen in reality – an obvious fallacy.  Instead, the premise of the inference is that, even under what might be considered the ideal conditions of the model (ideal, that is, for constant consumption without technical progress as resources deplete), maintaining consumption in the long term at its current level would be only just about possible.  Since actual circumstances are, in several important respects, likely to be very far from ideal, the above conclusion follows.

The spreadsheet used to obtain the above results may be downloaded here:


 Notes and References

  1. Further information on the Dasgupta-Heal-Solow-Stiglitz model may be found in Pezzey J C V & Toman M A (2002) The Economics of Sustainability: A Review of Journal Articles  Resources for the Future Discussion Paper 02-03 pp 5-10
  2. See for example the title and opening sentence of Hartwick J M (1977) Intergenerational Equity and the Investing of Rents from Exhaustible Resources American Economic Review 67(5) pp 972-4.
  3. Solow R M (1974) Intergenerational Equity and Exhaustible Resources The Review of Economic Studies Vol 41 pp 29-45.  See p 37 and Appendix B p 43.
  4. Hartwick, as 2 above, p 973. A simpler proof by J Pezzey may be found in Perman R, Ma Y, McGilvray J & Common M (3rd edn 2003) Natural Resource and Environmental Economics Pearson Addison Wesley pp 660-2.
  5. Buchholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model Scandinavian Journal of Economics 107(3) pp 547-561. See pp 549 & 558.
  6. Buchholz et al, as 5 above, p 553 state this formula but (because they start from a Cobb-Douglas production function without my B term), without the term in B. The fact that B is raised to the power 1/(1-β)  was derived mathematically by me and confirmed, for my chosen initial and parameter values,  by spreadsheet calculations with C in the vicinity of the implied Cmax.
  7. Solow, as 3 above, pp 37 & 39, which suggest β ≈05 and α ≈ 3β.
  8. ILO / OECD (2015) The Labour Share in G20 Economies See charts p 4.
  9. World Bank
  10. World Bank
  11. Johansson A, Guillemette Y & 7 Others (2013) Long Term Growth Scenarios Economics Department Working Papers No 1000   See charts in Fig 11 p 23.  Although these charts suggest an average ratio somewhat above 2, they relate to selected countries only.  Ratios are probably lower in many poor countries with predominantly agricultural economies.
  12. Wikipedia: Oil Reserves .  These figures relate to proven reserves, and (as has happened in the past) additional reserves will probably be discovered.  On the other hand the effect of carbon emissions on climate will probably limit the extent to which reserves can be exploited.
  13. IEA FAQs-Oil
  14. Tilton J E (2006) Chapter 3 Depletion and the Long-Run Availability of Mineral Commodities, in Doggett M D & Parry J R (ed) Wealth Creation in the Minerals Industry: Integrating Science, Business and Education Society of Economic Geologists Special Publication No. 12  See Table 1 p 63 listing 11 minerals, of which (at zero production growth) 5 have life expectancy less than oil and 5 more.
  15. The distinction between reserves and resource base is explained in Tilton (as 14 above) p 62.
  16. Buchholz et al, as 5 above. Modelling of depreciation is introduced on p 551 and the stated result is on p 553 (case θ = 1 and δ > 0).
  17. OECD Consumption of Fixed Capital as % of GDP 2013 (latest year available) Total (of OECD countries) 16.5% (, assumed similar for non-OECD countries.
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A Key Technique of Environmental Economics

Dynamic optimisation, also known as optimal control, is one of the most important techniques of environmental and natural resource economics.  Although it’s difficult,  the basics should be a compulsory element of graduate-level courses in these fields.

I managed to get through my MSc in Applied Environmental Economics (University of London 2009-2011) despite little understanding of dynamic optimisation.  Ah, you may think, he revised selectively, and luckily no question on the topic came up in his exams.  But no, I was a diligent student.  The syllabus included simple optimisation using elementary calculus, and constrained optimisation using Lagrange multipliers.  So I could apply the techniques needed, for example, to maximise utility subject to a budget constraint, or to minimise (for a non-cumulative pollutant and given the relevant functions) the sum of pollution damage and abatement costs.  But dynamic optimisation, in which the aim is to identify the optimal time path of a variable, was not in the syllabus as a general technique.  I learnt some of the results of applying the technique to particular topics, such as the Hotelling rule for optimal extraction of a mineral, but I did not learn how, in general, to solve a dynamic optimisation problem.

It was only some while after completing the course that this struck me as odd, and I judged it important to learn about dynamic optimisation. The main sources I used were online lecture notes by Stranlund (1) together with relevant sections of Perman, Ma, McGilvray & Common (2).  I’m not especially recommending these – there are many others, and what is most useful will depend on what a student already knows – but for me they served well.

Why might it be argued that a graduate-level course in environmental and natural resource economics should include the technique of dynamic optimisation?  Firstly, because many issues in environmental and especially natural resource economics are inherently dynamic, and cannot be adequately treated in a static framework.  Consider for example:

  1. Optimal harvesting of a fishery: the benefit from catching fish now must be balanced against the benefit from leaving the fish to grow and reproduce and perhaps yield greater harvests in future. Similarly for a forest.
  2. Optimal extraction of a mineral: the benefit from being able to use mineral now must be balanced against the benefit of leaving it to be extracted at a later date when its market value may be higher. Similarly for extraction of groundwater in locations where it is not replenished by rainfall.
  3. Optimal abatement of a stock pollutant: the benefit from a faster reduction in concentration of pollutant via a drastic reduction in polluting activities must be balanced against the benefit of allowing those activities to continue at a somewhat higher level with a slower reduction in concentration (for a specific case see this post). Similarly where the practical choice is between different rates of mitigation of increase, as in the important case of greenhouse gases and climate change.
  4. Optimal management of an ecologically important river basin: the benefits of abstraction of water for human use now must be balanced against possible long-term effects on wildlife populations.
  5. Optimal management of a whole economy involving policies to influence both rates of investment in man-made capital and rates of extraction and use of non-renewable natural resources, an important question being the feasibility of substituting capital for natural resources to sustain at least non-declining consumption.

Secondly, because dynamic optimisation using the maximum principle of Pontryagin (3) is a general technique that can be applied to many optimisation problems in dynamic settings.  It is a unifying principle that, once understood, is transferable from one dynamic problem to another.  For example, I am currently working on this problem (a special case, which I haven’t encountered in the literature, of 5 above):

Suppose an economy has a single good which can be either consumed or used as capital K, and a single non-renewable resource R extracted at zero cost, with a Cobb-Douglas production function Y = K^{\alpha}R^{\beta}     \ ( \alpha > \beta) and depreciation of capital at a rate \delta K\  (\delta > 0).  It can be shown that constant consumption cannot be maintained indefinitely at any level (4).  Given therefore initial capital K_0 and an initial stock of resource S_0, what is the maximum possible duration of constant consumption at a given level C?

Without dynamic optimisation, I would have no idea how to approach this problem, other than more or less random experimentation with spreadsheets in a discrete framework. Applying the maximum principle, however, it was fairly straightforward to derive this efficiency condition (a variant of the Hotelling rule, Y_R being the marginal product of the resource and Y_K that of capital):


Solution still required an element of trial and error, but in a spreadsheet set up to meet the above condition, reducing the variations to be considered to a very manageable level (some results will be presented in a future post).

So what is dynamic optimisation?  In outline, the elements of a dynamic optimisation problem are one or more state variables, one or more control variables, an objective function to be maximised (or minimised) containing some or all of those variables, a time horizon, and conditions on the initial and final values of the state variables.  In my problem, for example, there are two state variables, capital K_t and the remaining stock S_t of the resource, and one control variable, the rate of use of the resource R_t. The essence of a dynamic optimisation problem is to find the time path(s) of the control variable(s) that optimises the objective function: often, the time path(s) of the state variable(s) are also inferred.  In my problem, a convenient formulation of the objective is minimisation of the total use of the resource within a fixed time horizon (showing that that minimisation problem yields the efficiency condition above being the key to solving the original maximum duration problem).

Whereas simple or static optimisation problems may or may not include constraints (conditions a solution must meet), dynamic optimisation problems invariably include constraints in the form of state equations defining the rates of change of state variables.  In my problem, the state equation for capital relates growth of capital to income, consumption and depreciation, while that for the resource relates depletion of the stock of resource to use of the resource:

\dfrac{dK}{dt}=K^{\alpha}R^{1-\alpha}-C-\delta K

\dfrac{dS}{dt}= -R

So far so good.  But that’s just setting up the problem.  It’s the solving that can be difficult, and is presumably the reason why the topic did not feature in my course syllabus.  After all, universities need to recruit students, and those wishing to study environmental economics at graduate level may have taken various first degrees in various disciplines – economics, environmental science, agriculture, etc –  not always with a strong mathematical content.

In outline, the method of solution is this.  Drawing on the objective function and state equations, you set up an expression known as a Hamiltonian, which will contain one or more additional variables known as costate variables.  Using the Hamiltonian, you derive various necessary or first-order conditions that any solution must satisfy.  Tests must then be applied to assess whether these necessary conditions are also sufficient.  If they are, these conditions will define a unique solution, but further work is then needed to derive, in explicit form, the time paths of the control and other variables.

Why is this difficult?  Firstly, it isn’t (to me at least) intuitive.  In simple optimisation without constraints, it’s fairly easy to grasp the ideas that the first derivative of a function is its gradient, that the gradient must be zero for a maximum or minimum, and that the second derivative is needed to determine which.  But I could not make such a statement about dynamic optimisation.  I can apply the maximum principle, but I would not claim to understand, either intuitively or more formally, why it works.  Secondly, it isn’t easy to interpret formulae containing costate variables.  I know that costate variables in an economic context represent shadow prices, but in a dynamic setting one encounters formulae containing rates of change of shadow prices, which seem one step further removed from ‘real’ economic variables like capital or income.  Thirdly, when as is often the case the objective function concerns the discounted present value of some stream of values, care is needed to ensure consistency in using either present (discounted) or current (non-discounted) values, the latter requiring a current-value Hamiltonian and adjustments to the normal conditions, and interpretation of the results accordingly.  Fourthly, the necessary conditions relating to the time horizon and values of state variables at that time, known as transversality conditions, need to be carefully considered – not especially difficult, perhaps, but one more issue on top of everything else, and one which can highlight any vagueness in the original formulation of the problem.  Fifthly, the tests for sufficiency, such as the Mangasarian and Arrow conditions, involving consideration of whether functions are concave, can be complex to apply: it can be tempting to bypass them and just assume that the necessary conditions are sufficient, but that risks major error if, for example, the optimum is a corner solution.

Finally, when it comes to deriving explicit time paths from the necessary conditions, the devil is in the detail.  Rarely is the derivation a matter of elementary algebra.  Sometimes it requires solution of differential equations (online solvers such as that in Wolfram Alpha Widgets (5) can be useful, but cannot handle all such equations).  Often, especially for problems with multiple state or control variables, there is no exact analytic solution, and approximate numerical methods such as my trial and error with spreadsheets are all that is available.

The issue then for designers of graduate-level courses in environmental economics is this.  Omitting the technique of dynamic optimisation is unsatisfactory. But a requirement to be able to solve dynamic optimisation problems unaided would be asking too much of most students, and probably deter applicants.  The sensible solution, I suggest, is to require a knowledge of the technique that stops short of an ability to solve dynamic optimisation problems unaided.  For example, students might be expected to be able to:

  1. Identify the sort of problems that require the technique, and give examples from environmental and natural resource economics.
  2. Formulate mathematically a dynamic optimisation problem given in words.
  3. Identify the state and control variables in a given problem.
  4. Derive, for simple cases, the Hamiltonian and necessary conditions for a given problem.
  5. Draw inferences from the necessary conditions in cases where this is a matter of simple algebra.
  6. Discuss the complications that can arise in deriving an explicit solution from the necessary conditions.

A more in-depth treatment of dynamic optimisation might also be included in a course as an option, perhaps combined with other techniques such as econometrics under a heading such as ‘quantitative techniques’.

Notes and References

  1. Stranlund J K Lecture 8 Dynamic Opimization
  2. Perman R, Ma Y, McGilvray J & Common M (3rd ed’n 2003) Natural Resource and Environmental Economics  Pearson Addison Wesley  pp 480-505, 512-7, 548-53 & 574-81.
  3. Wikipedia: Pontryagin’s Maximum Principle
  4. Buchhholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model Scandinavian Journal of Economics 107(3) pp 547-561 (Modelling of depreciation is introduced on p 551 and the stated result is on p 553 (case \theta = 1 and \delta > 0).
  5. Wolfram Alpha Widgets: General Differential Equation Solver
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