The Maximum Duration of Constant Consumption

At what rate should an essential non-renewable resource be depleted to sustain an economy for as long as possible? 

Suppose the inputs of a closed economy consist of produced capital, a non-renewable resource and labour .  Output is of a single good which can be either consumed or added to the stock of produced capital.  The quantity of output is determined by a constant-returns Cobb-Douglas function, implying in particular that if any of the inputs is nil then output is nil.  Technology is constant, as is labour input.  Produced capital depreciates at a constant positive rate .

Question:  Given that (as explained in my previous post) constant consumption cannot be sustained forever, for what length of time can it be sustained at a given rate, and at what rate should the resource be depleted to achieve that maximum time?

Suppose the production function is:

Y(t) = A(K(t))α(R(t))β      (A,α,β > 0; α + β < 1)


Y(t) = output at time t;

K(t) = stock of produced capital at time t;

R(t) = rate of use of a non-renewable resource at time t;

A, α, β are fixed parameters, the value of A reflecting the  technology and the labour input, both of which are assumed constant.  

The initial stocks of produced capital and the resource are K0 and S0, and depreciation of produced capital is at a rate δK(t) (0 < δ < 1). We wish to find the maximum duration T for which consumption can be sustained at a given rate C, and what the associated time path of use of the resource.

The charts below show the optimal time paths when A=1, α = 0.3, β = 0.2, δ = 0.1, C = 2 and K0 = 100, with several values of S0.  I start with S0 = 5, for which the maximum duration of consumption at the given rate is 22, the optimal time path of use of the resource being as shown in Chart 1 below.

Even in this simplest case, the optimal rate of use of the resource is not constant.  Use of the resource increases over time, being always just sufficient for output to equal the required rate of consumption.  There is no additional output to offset depreciation, so capital decreases, which is why use of the resource must increase to sustain output and consumption. 

When the initial stock of the resource is larger, the possibility of setting use of the resource as always just sufficient for output to equal the required rate of consumption may not be optimal, but remains available.  Thus the time until the resource is exhausted when its use follows that path provides a lower bound on the maximum duration.  We can state:

Proposition 1: A lower bound on the maximum duration T is given by:

\qquad T\,\geq \, \dfrac{\beta}{\alpha \delta}\ln\Bigg(\dfrac{\alpha \delta S_0A^{1/\beta}K_0^{\alpha/\beta}+\beta C^{1/\beta}}{\beta C^{1/\beta}}\Bigg)

with strict equality if:

\qquad S_0\,\leq\,\dfrac{C^{(1-\beta)/\beta}}{\alpha A^{1/\beta}K_0^{(\alpha -\beta)/\beta}}

In the case shown in Chart 1 the strict equality condition is satisfied.  We now consider the case  S0 = 30, for which the condition is not satisfied.  In this case Proposition 1 implies that T is at least 33, but it is in fact 34, this slightly longer duration being obtained when the time path of R is as shown in Chart 2.

It can be seen that the time path of use of the resource is slightly kinked at around time t = 26.  After the kink, it follows the simple path described above.  Before the kink, use of the resource is somewhat more than sufficient for output to equal the required rate of consumption, so that there is a little extra output offsetting part of the depreciation.

The timing of the kink depends on the depletion of the resource.  When the initial stock of the resource is sufficiently large, it is worthwhile to use some of it to invest in produced capital, because a larger stock of produced capital enables output and consumption to be sustained with less use of the resource.  When the remaining stock of resource becomes sufficiently small, the maximum time over which the required rate of consumption can be sustained becomes so small that it is not worthwhile to use any of the resource to invest in produced capital.  How small is sufficient in any particular case depends on the values of the various parameters. 

Chart 3 shows the optimal time path when S0 = 500.  The maximum duration is 104, much more than the lower bound implied by Proposition 1, which is 52.  Here the kink, at t = 101, is much sharper and proportionately much closer to the maximum duration.

What is especially striking in Chart 3, as compared with Charts 1 and 2, is the concave shape of most of the optimal path.  It is also notable that use of the resource increases over the range 1-70, then decreases, then increases again after the kink.  To understand this apparently bizarre behaviour we will proceed directly to Chart 4, which shows the optimal path when S0 = 2000.

Here the optimal path is still concave over most of the time period, but almost constant over the range 80-240.  Capital is almost constant over the same period, during which output exceeds the required rate of consumption by almost exactly enough to offset depreciation.

There are many possible combinations of capital and use of the resource having the property of keeping capital constant.  Among these is one which minimizes the rate of use of the resource, and that minimum rate is approximately 7.59, which is the value to which in Chart 4 it approximates over the range 80-240.  Being the smallest such value it enables the required rate of consumption to be sustained for the longest time from a given quantity of resource.  Using a term common in the study of optimisation problems, we may describe the almost constant section of the time path as a turnpike

We can now see why the path in Chart 3 takes the shape it does.  It is trying, as it were, to reach the turnpike, but the stock of resource is not quite sufficient.  Nevertheless, getting close to the turnpike for at least a small part of the time path contributes to maximising the duration.

Returning to the case S0 = 2000 (Chart 4), the lower bound on the maximum duration implied by Proposition 1 is 61, a very poor approximation to its actual value, 302.  However, when the initial stock is sufficiently large that the turnpike occupies a high proportion of the optimal time path, we can obtain a much better approximation by dividing S0 by the value of R during the turnpike.  Hence we arrive at:

Proposition 2: If S0 is large then a reasonable approximation to the maximum duration T is given by:

T=S_0A^{1 /\beta}\Big(\dfrac{\alpha}{\delta}\Big)^{\alpha /\beta}\Big(\dfrac{1-\alpha}{C}\Big)^{(1-\alpha)/\beta}

For S0 = 2000 this yields a value of 264, reasonably close to 302. 

The mathematics underlying the above may be found in the following two downloads. (pdf and Excel 2019 format respectively) 

Posted in Macroeconomic Theory, Mathematical Techniques, Sustainability | Tagged , , , , | Leave a comment

An Inconvenient Truth about the Hartwick Rule

The relevance to the Hartwick Rule of depreciation of produced capital is not widely understood.

Suppose the inputs of a closed economy consist of produced capital K, a non-renewable resource R and labour L.  It produces a single good which can be either consumed or added to the stock of produced capital.  The quantity of output Y is determined by a constant-returns Cobb-Douglas function, implying in particular that if any one of K, R or L is nil then Y is nil.  Technology is constant, as is labour input.  Produced capital depreciates at a constant positive rate δK.

Question:  In such an economy, is it possible for consumption at some constant positive level to be maintained forever?

I have the impression that many people familiar with at least some of the vast literature on sustainability would be inclined to answer in the affirmative.  And the reason would be that they have encountered the Hartwick Rule according to which, loosely, sustainability can be achieved by investing the rents from non-renewable resources in produced capital. 

The correct answer, however, is that under the conditions of the question, constant consumption cannot be sustained forever.  In simple terms, this is because use of the non-renewable resource must decline towards (but never reach) zero.  To maintain consumption, produced capital must therefore increase without limit.  Consequently depreciation of that capital must also increase without limit.  So output must become large enough to offset a huge quantity of depreciation, as well as providing for consumption and an increase in the stock of capital.  That requires a certain minimum rate of use of the resource.  Continuing use of the resource at that minimum rate must eventually exhaust any finite initial stock.  A more formal proof will be given below.

I claim no originality for this result (although I have not seen elsewhere the particular proof I set out below).  It can be found in the literature.  Indeed, it can be found in the very place that those who assume that the Hartwick Rule justifies an affirmative answer might appeal to as a source.  Most of Hartwick’s 1977 paper relies on an assumption of no depreciation (1).  In the final paragraph, however, we find the following (2):

“If there is depreciation of reproducible capital at the rate δ per unit capital per unit time, then … our savings investment rule will not [my italics] provide for the maintaining of … consumption constant over time.”

I should emphasise that the reasoning in Hartwick’s paper is entirely correct: what could mislead readers is its balance and tone, with most of the paper devoted to the unrealistic case of zero depreciation, introduced with something of a rhetorical flourish, and only a few sentences on the practically important case of capital depreciating over time.

The result may also be found in a paper by Buchholz, Dasgupta & Mitra (3), which explicitly models depreciation from the outset, albeit via the more general formula δKθ (0 ≤ θ ≤ 1).  For a class of production functions of which the Cobb-Douglas is one example, it is shown that constant consumption forever (an “equitable path” in the paper’s terminology) is impossible when δ > 0 and θ = 1 (4).  That is precisely the case to which my question applies. However, the result is one among many in the paper, some of which are given much more prominence, and could easily be missed by a casual reader. 

A further (and currently open access) source for the result is the PhD thesis of Hamilton.  Here it is at least given prominence early on (5), although most of the thesis reverts to the assumption of zero depreciation (6). 

I am struck however by the fact that, although many discussions of the economics of sustainability refer to the Hartwick Rule, they often fail to mention that it will not enable constant consumption to be sustained forever if produced capital depreciates at a constant positive rate.  Here are a couple of examples from textbooks.

Hanley, Shogren & White’s Environmental Economics in Theory and Practice devotes most of a section on weak sustainability rules to consideration of the Hartwick Rule (7).  Having explained the Rule and in particular the consequential feasibility of non-declining consumption, it identifies four limitations: briefly, the Rule does not hold for all types of production function or for open economies, and where it does hold, it is not necessarily consistent with constant welfare or with ecological sustainability.  But the limitation in respect of depreciation of produced capital is not mentioned.

Common & Stagl’s Ecological Economics: An Introduction refers to the Hartwick Rule in the context of discussing policies that might be adopted by a benevolent dictator in a closed economy with a constant-returns Cobb-Douglas production function using produced capital, non-renewable resource and labour inputs, with constant technology and constant population  (8). A choice of savings rate reflecting the current generation’s preferences as between current and future consumption is shown to be likely to lead to an unsustainable outcome, with consumption initially rising but eventually declining asymptotically towards zero.  It is then stated that there is a savings policy which would ensure constant consumption forever, the policy being the Hartwick Rule.  The one qualification made is that the resource must be depleted efficiently – a requirement also identified in Hartwick’s paper and generally known as the Hotelling Rule (9).  The effect of depreciation of produced capital is not mentioned.

The Hartwick Rule is also commonly referred to in policy-orientated literature relating to sustainability.  A report on an EU-funded project concerning ecosystem restoration in the UK includes the following entry in its glossary (10):

“Hartwick Rule – simple rule of thumb for sustainable development for countries that depend … on non-renewable natural resources: consumption can be maintained … if rents from non-renewable resources are continuously invested rather than used for consumption.”

A paper by van der Ploeg entitled Challenges and Opportunities for Resource Rich Economies, though acknowledging that the Hartwick Rule is “hotly debated”, appears to accept that given a Cobb-Douglas production function the Rule can make possible constant consumption in the absence of technical progress (11).  The one exception it notes is for open economies, arguing that resource-exporting countries can sustain constant consumption by investing less than implied by the Hartwick Rule (12), an assertion which would surely not be made if the relevance of depreciation were understood? 

The extent of the influence of the Hartwick Rule was described by Ottenhof in a piece written for the 40th anniversary of Hartwick’s 1977 paper (13).  It states:

“the Hartwick Rule has gone on to become a pillar of sustainability economics, forever changing the way we think about the concept of sustainability.”

While it also refers to opposition among the ecological community to the weak sustainability approach associated with the Hartwick Rule, this clearly relates to arguments as to whether, or to what extent, produced capital can substitute for natural resources, and not to the effect of depreciation.

Why then is the fact that depreciation of capital undermines the significance of the Hartwick Rule not more widely recognised?  One reason may be a perception that depreciation is a minor technical issue that can safely be ignored with little consequence.  In some economic contexts such a perception would be valid. If one is considering the short-term response of an economy to a change in fiscal or monetary policy, with a focus on the effects on activity and employment, then it could be entirely reasonable to ignore depreciation.  But the long-term scenario suggested by the Hartwick Rule, with ever-increasing quantities of produced capital offsetting ever-reducing use of a renewable resource, is a context in which to ignore depreciation would be seriously misleading. 

Another reason may be that the Hartwick Rule, if taken to provide a basis for sustainability, suggests many avenues for further research.  For what types of production function does the Rule hold?  Can it be extended to cases of many non-renewable resources?  How much consumption can be sustained forever?  What are the implications for measurement of national income?  How large are the rents from non-renewable resources in particular countries, and how do they compare with those countries’ investments in produced capital?  By contrast, acceptance that the Rule is not of much practical importance because capital depreciates may seem, from a research perspective, as something of a dead end.

A further reason may be an assumption that the problem with depreciation can be simply overcome by working in terms of a net rather than a gross production function.  This calls for a little explanation.  A gross production function expresses gross output – output before depreciation of capital – as a function of inputs.  Similarly, a net production function expresses output net of depreciation as a function of inputs.  The relation between the two is simple: if the gross production function is G(K,R,L), the net production function is F(K,R,L), and depreciation is δK, then:

G(K,R,L) – δK  =  F(K,R,L)              (A)

There is nothing wrong in itself in using a net production function: in some contexts it can simplify matters to do so.  The potential pitfall however is to assume that standard assumptions about the functional forms of gross production functions will simply carry over to net production functions.

Surprisingly, this fallacy can be found in Hartwick and Olewiler’s The Economics of Natural Resource Use (14).  Unlike the textbooks mentioned above, it includes depreciation in its discussion of the economics of sustainability.  It introduces a production function Q = F(K,R,L) and, since this is followed by the statement that consumption equals Q – I, where I is net investment, it is clear that this is a net production function (15).  Subsequently it is stated that, by following the Hotelling Rule and (though not referred to by name) the Hartwick Rule,  consumption can be maintained indefinitely at a constant positive level if the production function F has the Cobb-Douglas form KαRβL1-α-β (and subject to certain conditions on α and β). 

The problem with this lies in the assumption of a net production function with Cobb-Douglas functional form.  From (A) above this implies that the corresponding gross production function is:

G(K,R,L)  =  KαRβL1-α-β + δK              (B)

This is an implausible form for a production function.  The implication that some output can be obtained without use of a non-renewable resource or labour is not necessarily a problem.  But it implies something much stronger, namely, that the productivity of produced capital in the absence of other inputs, indicated by the coefficient δ, is precisely what we know to be the rate of depreciation – an amazing coincidence. 

Note what (B) is not saying.  There is some plausibility in a production function which divides produced capital into two parts, call them K1 and K2, the former yielding output only in conjunction with non-renewable resource inputs, and the latter yielding output without them (think of coal-fired power stations and solar panels) and with a production coefficient reflecting the actual productivity of K2.  So we might write something like:

G(K1,K2,R,L)  =  K1αRβL1-α-β + θK2             (C)

But that is not what (B) does.  It treats produced capital as homogeneous, yet capable of producing so much output on its own – precisely enough to offset depreciation -, and more in conjunction with other inputs.  I know of no reason why a production function might take such a form. 

I conclude with a more formal statement and proof, in discrete time, of the above result.

Theorem The output of a closed economy in any period consists of a quantity of a single good, any part of which is either consumed within the period or added to the stock of produced capital for the next period.  Once added to the stock of produced capital it cannot subsequently be consumed. The production function is:

Y_t = AK_t^{\alpha}R_t^{\beta}\quad (\alpha,\beta > 0;\,\alpha + \beta < 1)


Y_t = output in period t;

K_t = stock of produced capital in period t;

R_t = quantity of a non-renewable resource used in period t;

A,\alpha,\beta are fixed parameters, the value of A\, reflecting both the constant technology and the constant labour input. 

The stock of produced capital in period t\, is subject to depreciation of \delta K_t\,(\delta > 0).  Given finite initial stocks of produced capital and the resource, no positive quantity of consumption per period can be sustained forever.

Proof:  We proceed by reductio ad absurdum.  Suppose consumption of C\, per period (C > 0) can be sustained forever from finite initial stocks K_0 of capital and S_0 of the resource.  Then for some S\, such that 0 < S \leq S_0:

\sum_{t=1}^{\infty} R_t = S\qquad(P1)

From this we can infer (16):

\lim_{t\rightarrow \infty}R_t=0\qquad(P2)

Hence given any \epsilon > 0, there exists a positive integer N such that R_t < \epsilon for all t > N.  For such t:

AK_t^{\alpha}\epsilon^{\beta} > AK_t^{\alpha}R_t^{\beta}>C\qquad(P3)   

and therefore:

K_t > \dfrac{(C/A)^{1/\alpha}}{\epsilon^{\beta /\alpha}}\qquad(P4)

Since this holds for any \epsilon > 0, however small, we must have:

\lim_{t\rightarrow \infty}K_t=\infty \qquad(P5)

But growth of K in any one period is finite (since the production function can only yield finite output from finite inputs).  Hence there must be an infinite number of periods in which K_t is both larger than K_0 and growing.  K_t can grow in a period only if output exceeds depreciation, so for each of those infinite periods we must have:

AK_t^{\alpha}R_t^{\beta} > \delta K_t\qquad(P6)

and therefore:

R_t > (\delta/A)^{1/\beta}K_t^{(1-\alpha)/\beta} > (\delta/A)^{1/\beta}K_0^{(1-\alpha)/\beta}\qquad(P7)

Since P7 applies to an infinite number of periods, we have:

\sum_{t=1}^{\infty}R_t > \sum_{t=1}^{\infty}(\delta/A)^{1/\beta}K_0^{(1-\alpha)/\beta} = ((\delta /A)^{1/\beta}K_0^{(1-\alpha)/\beta} \infty\,\,\,(P8)

\sum_{t=1}^{\infty}R_t  = \infty > S_0\qquad(P9)

Thus our supposition leads to a contradiction.  QED.

Notes and references

  1. Hartwick J M (1977)  Intergenerational Equity and the Investing of Rents from Exhaustible Resources  The American Economic Review 67(5) pp 972-4.  The assumption of no depreciation is in the middle of the first paragraph on p 972.
  2. Hartwick, as 1 above, p 974.
  3. 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.
  4. Buchholz et al, as 3 above.  The depreciation formula is introduced on p 551 and this result is on p 553.
  5. Hamilton K (1995)  Sustainable Development and Green National Accounts  PhD thesis accessible at  pp 2 & 7-8.
  6. Hamilton K, as 5 above, see final sentence p 9.
  7. Hanley N, Shogren J F & White B (2nd edn 2007)  Environmental Economics in Theory and Practice  Palgrave Macmillan  pp 19-21
  8. Common M & Stagl S (2005)  Ecological Economics: An Introduction  Cambridge University Press  pp 350-1
  9. Common & Stagl, as 8 above, pp 351-2
  10. Bright G  Natural Capital Restoration Project Report  p 191
  11. Van der Ploeg F (2006)  Challenges and Opportunities for Resource Rich Economies  EUI Working Papers RSCAS No. 2006/23  p 17
  12. Van der Ploeg, as 11 above, p 18
  13. Ottenhof N (2017)  Hartwick’s Rule continues to influence sustainable development after 40 years
  14. Hartwick J M & Olewiler N D (2nd edn 1998)  The Economics of Natural Resource Use  Addison-Wesley
  15. Hartwick & Olewiler, as 14 above, p 399
  16. I am grateful to Thomas and GEdgar, participants in Mathematics Stack Exchange, for confirming the validity of this step
Posted in Macroeconomic Theory, Sustainability | Tagged , , , , , , | Leave a comment

Getting to Zero Emissions

A short review of Bill Gates’ How to Avoid a Climate Disaster

When someone famous for their achievements in one field of human endeavour offers opinions on some unrelated topic, it is wise to treat their views with a degree of scepticism.  So it was with some caution that I approached Bill Gates’ new book (1).  How much credence, I wondered, should be given to the views on climate change of a software entrepreneur and philanthropist?

Having read the book, I have no hesitation in recommending it as a survey for the lay reader of the problem presented by climate change and, as its subtitle puts it, the solutions we have and the breakthroughs we need.  And I would add that this is a topic on which everyone is in some respects a layperson: no one could possibly be an expert in all of the relevant fields, which include climate science,  energy, engineering, agriculture, economics and behavioural science.  Although I have some criticisms, I commend the book for its broad focus and for its judicious combination of science and common sense.

Central to the book is the claim that developed countries should aim for net zero greenhouse gas emissions by 2050, and middle-income countries as soon as possible thereafter (p 35).  The case for zero is set out in detail in Chapter 1.  In essence, failure to reduce emissions to zero will mean that the world will progressively get hotter (pp 18-24), and that would have all kinds of dire consequences (pp 25-34), and indeed be disastrous if zero is not achieved by 2050 (pp 35 & 196).  The case for not aiming for zero earlier than 2050 is simply that it isn’t feasible (p 196): supporting argument is implicit in much of the rest of the book which shows that to get to zero we need not only to achieve numerous technological breakthroughs but also to implement them on a very large scale.

If zero is our goal for 2050, how should we measure progress towards that goal, and what intermediate targets should we set?  These questions are briefly but tellingly addressed in Chapter 11 (pp 196-7).  For a country which generates lots of electricity from coal, replacing coal-fired power stations by gas-fired ones is an effective way of reducing its emissions.  What it is not is a step towards zero emissions (2).  It also risks diverting funds away from investment in zero-emission technologies, and creating pressure to allow the gas-fired power stations to continue operating beyond 2050 so as to obtain a satisfactory return on investment.  For zero emissions, electricity will need to be obtained without reliance on fossil fuels (or just possibly from fossil fuels with full carbon capture and storage).  Progress towards zero should therefore be measured in terms of the development and implementation of zero-emission technologies.  Although Gates does not develop the point, there seems to be the basis here for a critique of international agreements which set short-term country-level targets for reductions in emissions.

Discussions of climate change abound in figures and comparisons which, even if accurate, are presented without enough context to make them meaningful.  Gates uses figures well, stating at the outset that the world’s annual greenhouse gas emissions are currently 51 billion tons (p 3), and using that figure to put proposed means of reducing emissions into context (p 53).  He shows a healthy scepticism based on rough but reasonable calculation for ideas which, however desirable in themselves, are unlikely to make more than a very small contribution to meeting the goal of zero emissions.  While accepting that we should plant more trees, for example, he calculates that to plant enough trees to absorb the emissions produced by the population of the US would require about half the world’s land area (p 129). 

Chapters 4 to 8 consider in turn the difficulties and possible solutions in getting to zero emissions in respect of electricity generation, production of goods and infrastructure, agriculture, transport, and heating and cooling.  These chapters achieve a good balance between readability and inclusion of just enough technical detail to show how complex the issues are and how big the challenges.  Gates evidently relishes learning about the detail: he talks of “following closely” a company developing molten oxide electrolysis for emission-free steel production (p 110), and of a visit to a fertilizer distribution centre in Tanzania as a “kind of trip I love” (p 121).  The main conclusion is stark.  Although we have some of the technologies needed to get to zero, we need to invent many new technologies (p 158) and make them affordable for middle-income countries (p 199).

Making a personal selection from the book’s list of nineteen necessary technologies (p 200), I will mention:

  • Emission-free hydrogen production for use in storing electricity (pp 93-4) and in transport (p 139);
  • Emission-free cement production for use in concrete, an essential product in much infrastructure (pp 98-100);
  • Plant- and cell-based meat and dairy food to reduce emissions from agriculture (pp 119-121);
  • Improved nuclear reactor designs, because electricity from nuclear fission is proven to work, emissions-free, and does not suffer from the intermittency of sources such as wind and solar (pp 84-87).

To improve our chances of achieving the technological breakthroughs we need, the book advocates a massive increase in relevant R&D, with developed-country governments making big bets on high-risk high-reward projects and leaving safer investments to the private sector. 

Deployment of emissions-reducing technologies, whether existing or new, is also crucial, and to encourage this the book recommends a combination of standards and market-based incentives (pp 206-8).  In respect of standards for clean electricity and clean fuel, it makes the important point that standards should be technology-neutral, that is, they should specify a goal (eg that utilities must obtain so much of their electricity from emissions-free sources) but allow any technologies that delivers that goal.  In advocating a carbon price, it emphasizes that the purpose is to raise the price of fossil fuels and other products that generate emissions so as to make emissions-free alternatives more competitive, with both the choice between a carbon tax and cap-and-trade and the use to which the resulting revenues are put being somewhat secondary issues.  Most economists would I think broadly agree on these points. 

Perhaps as a consequence of Gates’ enthusiasm for technological matters, the book seems to me to underplay the seriousness of the behavioural and political issues involved in getting to zero.  Regarding plant- and cell-based-meat, he notes that many US states have tried to ban these products from being labelled as “meat”, and concludes that there will be a need for “healthy public debate” about their regulation, packaging and sale (pp 120-1).  I wouldn’t like to predict what the outcome of such debate might be.  And if, to get to zero, we need not only to develop lots of new technologies and then implement them at scale, but also to find time for public debates along the way, that’s quite a lot to fit in to the 29 years to 2050.  Another example is his suggestion of border carbon adjustment as a policy towards countries refusing to join international agreements on climate change and  (p 215).  Such a policy has difficulties in its own right (3) but, more fundamentally, would also be subject to the need to look at relations between countries – which may involve a variety of trade, security and political issues – in the round. 

I would also like to have seen some discussion of population growth as a contributory factor in increasing emissions.  It is true that the highest growth rates are in poor countries with low levels of emissions (4), but those countries may not always be so poor.  Many countries with significant emissions levels also have growing populations, and could consider financial incentives for smaller families as a climate change policy (the emissions due to people who are never born are zero).  Moreover addressing climate change should not be at the price of letting poor countries remain poor so that their emissions will remain low.  We need to address both climate change and poverty, and aid programmes offering improved access to family planning in poor countries can surely make a contribution to both?

These are minor criticisms.  If you are only going to read one book on how we should address climate change, or if you are a librarian who can only afford one such book for your public or school library, this would be an excellent choice. 

Notes and references

  1. Gates, Bill (2021) How to Avoid a Climate Disaster: the Solutions we Have and the Breakthroughs we Need   Penguin Random House LLC
  2. Combustion of coal which is mainly carbon produces mainly CO2, while combustion of natural gas which is mainly methane (CH4) produces a mixture of CO2 and water.
  3. See for example Cosbey A (2012)  It Ain’t Easy: The Complexities of Creating a Regime for Border Carbon Adjustment
  4. For population growth rates by country see

Posted in Climate change, Energy | Tagged , , , , | Leave a comment

Climate Change and a Proposed Coal Mine

A proposed new coal mine in Cumbria, England has prompted vehement arguments for and against.  The underlying problems are a flawed policy framework with insufficient international coordination.

To its supporters it’s a no-brainer.  The mine will produce coking coal, an essential input in the production of steel from iron ore.  And a modern economy needs steel for a myriad of purposes, not least in the construction of wind turbines to reduce dependence on fossil fuels.  What’s more, it will reduce Europe’s imports of coking coal from the US, saving more than 20,000 tonnes of CO2 equivalent per annum in emissions from shipping fuel.  It will also create jobs in a relatively poor region of the UK.

Its opponents are equally adamant.  To address climate change and meet the widely-accepted target of  net zero carbon emissions by 2050, the use of coking coal needs to be phased out because, like all coal, it emits CO2 when burnt.  Already, nearly 30% of world steel production uses no coking coal.  Allowing a new coal mine would undermine the UK’s credibility as host of the next  UN Climate Change Conference (Glasgow, November 2021).  

The circumstances have been widely reported in the UK, but for readers elsewhere here is a summary.  West Cumbria Mining Ltd (“WCM”) is a company formed to exploit coal reserves in the Cumbria region of north-west England.  Before it can develop and operate a mine it requires planning permission from Cumbria County Council (“the Council”).  Environmental campaigners asked the UK government to intervene, using reserve powers under which the  Secretary of State for Housing, Communities and Local Government can “call in” a matter considered to be nationally significant and impose his own decision whether or not to grant permission. The Secretary of State has so far declined to exercise that power in this case, and in October 2020 the Council resolved to grant permission.  However, the Council informed WCM on 9 February 2021 that it would reconsider its decision.  At the time of writing the outcome of the Council’s reconsideration is awaited and WCM is preparing to take legal action against it (1)

[Update 13 March 2021. The Secretary of State has now, after all, decided to “call in” the planning application by WCM. This means there will now be a public inquiry, which may take many months, with the Secretary of State rather than the Council making the final decision.]

In my view both sides overstate their case.  Let’s start with the saving of emissions from shipping fuel.  20,000 tonnes of CO2 equivalent may seem a lot, but it’s a tiny fraction of the emissions from use of the coal the mine would supply.  WCM estimate annual supply from the mine at 3 million tonnes.  Its use in steel production will yield almost 9 million tonnes of CO2 emissions (2).  That’s more than 400 times the saving on emissions from shipping fuel.  What we should be considering is the net increase in emissions if the mine goes ahead.  But that’s hard to estimate because it depends on the extent to which the supply from the mine adds to total world use of coking coal.

Economic analysis can help here.  Total world use of coking coal will depend on the market equilibrium point where its supply and demand curves intersect.  The extra coking coal from the Cumbria mine will shift the supply curve to the right in the standard price-quantity diagram (see Box 1).  How this affects the equilibrium quantity depends on the elasticity of supply (ES)and elasticity of demand (ED), the relevant formula being (see Box 1):

\frac{\text{Net increase in equilibrium quantity}}{\text{Quantitative shift in supply curve}}=\frac{-E_D}{E_S-E_D}

There are two ways in which we can try to make very rough estimates of the elasticities so as to estimate the value of the above fraction.  One is to apply what economists know to be true for the elasticities of supply and demand for most goods.  It is rare for demand to be either perfectly elastic (ED = minus infinity) or completely inelastic (ED = 0).  Elasticities of demand for broadly defined goods (not for example particular brands) are typically within the range -0.2 to -2.0 (3).  In the case of elasticity of supply, it is especially important to consider the time scale over which changes are being considered. If an industry is already producing at full capacity, it will take time to increase its output since extra equipment will need to be installed and additional workers recruited and trained.  For many goods, therefore, supply is inelastic in the short term (ES < 1) but more elastic in the longer term (ES > 1).  For our purposes, it is long-term elasticity which is relevant, since the mine is expected to have a long operating life. 

The other way to try to estimate the elasticities is via a literature search for empirical estimates of the elasticities of supply and demand for coking coal.  Unfortunately, there seem to have been few relevant studies, and some of those are quite old.  Truby (2012) cited a study by Ball & Loncar (1991) estimating elasticity of demand for Western Europe in the range -0.3 to -0.5, and also a study by Graham, Thorpe & Hogan (1999) estimating elasticity of demand at -0.3 (4).  Lorenczik & Panke (2015) estimated elasticity of demand in the international market at between -0.3 and -0.5 (5).  For elasticity of supply, Lawrence & Nehring (2015) estimated 0.30 for Australia and 0.73 for the US in 2013 (6): the specification of a particular year suggests that these estimates are of short-term elasticity. 

Taking all the above into account, it might be reasonable to estimate elasticity of demand at   -0.4 and elasticity of supply at 2.0.  Putting these values into the above formula yields a fraction of 0.17.  That would imply that the extra 3M tonnes per annum from the Cumbria mine would increase world use of coking coal by 510,000 tonnes. The net increase in CO2 emissions, allowing for the savings on shipping from the US, would be 1,476,000 tonnes annually (7).  I offer that as one plausible scenario, not a prediction.  The more fundamental point is that the fraction is certainly not going to be zero.  That would require either zero elasticity of demand (completely inelastic demand) or infinite elasticity of supply (perfectly elastic supply).  Neither of those are remotely plausible.  Even if the fraction were just 0.01, an implausibly low figure, world use of coking coal would increase by 30,000 tonnes per annum, increasing net emissions by 68,000 tonnes (8). 

Turning to the opponents’ case, it would certainly help towards the target of net zero carbon emissions by 2050 if the use of coking coal in steel production could be phased out.  Whether that is feasible at reasonable cost, however, is far from certain.  The main reason why 30% of current production uses no coking coal is that its input material is not iron ore but recycled scrap steel which can be processed into new steel in an electric arc furnace (9).  The use of recycled steel can probably be increased, but in a growing economy demand for new steel is always likely to exceed the supply of recycled scrap. 

The main hope for ending the use of coking coal is therefore the development of new technologies for producing steel from iron ore.  One promising approach is to use hydrogen to produce direct reduced iron (DRI, also known as sponge iron) which can then, like scrap steel, be processed into new steel in an electric arc furnace (10). If the hydrogen is “green hydrogen”, produced by electrolysis of water using electricity from a renewable source, and if the electricity powering the furnace is also from such a source, then the whole process is emissions-free.  McKinsey reports that all main European steelmakers are currently building or testing hydrogen-based production processes (11).  Development appears to be most advanced in Sweden, where steelmaker SSAB has a joint venture with iron ore producer LKAB and energy company Vattenfall to produce steel using a technology known as HYBRIT (12), and the H2GS (H2 Green Steel) consortium plans a large steel plant using a similar technology (13). 

Another approach to steel production without coking coal involves reducing iron ore to iron by means of electrolysis.  Again, if all the electricity is from a renewable source then the whole process will be emissions-free.  Steelmaker ArcelorMittal is leading a project which has proved the potential of the technology (14), and Boston Metal is offering to tailor what it calls Molten Oxide Electrolysis (MOE) for customers producing steel and other metals (15).

However, phasing out the use of coking coal is not the only way in which carbon emissions from steel production might be reduced to zero or very low levels.  The alternative is to continue using coking coal but with carbon capture, utilisation and storage (CCUS), and around the world there are a number of CCUS initiatives relating to the steel industry. Al Reyadah, a joint venture between Abu Dhabi National Oil Company and clean energy company Masdar, captures CO2 from an Emirates Steel plant and injects it into nearby oil fields for enhanced oil recovery (16).  Steelmaker Thyssenkrupp has a project called Carbon2Chem which uses CO2 from steel production as a raw material in the production of fuels and fertilisers (17).  Another possibility, although apparently only at the proposal stage, is the retrofitting of conventional steel plants to permit a process known as calcium-looping which uses CO2 to react with limestone and produce lime fertiliser: Tian et al (2018) make the remarkable claim that this could allow decarbonised steel production at relatively low cost as early as 2030 (18).

Which of these various technologies will prove successful is difficult to predict.  It is noteworthy that some large producers, including Thyssenkrupp and Tata Steel, are hedging their bets by exploring both hydrogen-based and CCUS approaches (19).  This uncertainty in turn creates a problem for producers of coking coal.  If technologies based on hydrogen or electrolysis come to dominate the steel industry, then demand for coking coal will eventually fall to zero.  The speed of the fall will partly depend on how climate change policies and other considerations influence firms’ decisions on whether to continue operating existing conventional steel plants for their full working life.  It seems possible that demand for coking coal will fall gradually over several decades, with lower-cost mines continuing to find buyers as others cease production. If however CCUS approaches become dominant, then the outlook for coking coal producers will be much brighter.  It’s also possible that more than one technology will be successful, resulting in some ongoing demand for coking coal.  I conclude that the opponents’ main argument against the mine – that the use of coking coal needs to be phased out to address climate change – is not proven.

A company like WCM which chooses to make a substantial investment in a new coking coal mine is taking a big risk.  To make a worthwhile return on its investment it will need to be able to sell its coal at a good price for many years, but if demand for coking coal rapidly declines due to technological change in the steel industry, then it will not be able to do so.  So investors have to make a judgment as to whether they can accept their perceived risk-return pattern.  The key issue then is the policy context within which they make that judgment.

It is appropriate that the mine should require approval by the Council in respect of what might be termed “normal planning matters” such as effects on the local economy, possible disturbance to residents, impacts on the local environment, and restoration and after-care when the mine reaches the end of its operational life.  Any approval would very likely be conditional on measures to limit local impacts.  It is also appropriate that the Council should have regard to climate change policy in making decisions on its own activities, such as the heating of its schools and offices.  What is more dubious is a local government body accountable primarily to its local electors being left to take a decision which has national and international implications because of the extra carbon emissions the coal produced in the mine would generate.  It is a flawed policy framework which places this burden (or opportunity, depending on one’s point of view) on the Council.

A better way to ensure that the decision whether to proceed with the mine has due regard to its climate change implications would be to ensure that WCM will bear the full social cost of its coal production. In economic jargon, there is a market failure in the form of an externality: the emissions from use of its coal would have a cost to society which it would not bear.  The standard economic prescription to correct such a market failure is to internalize the externality.  That could be achieved by pricing CO2 emissions via either a carbon tax or an emissions trading system.  The direct effect on WCM would be small as the emissions from the mine itself would not be large. Much more important would be the indirect effect arising from making steel producers bear the full social cost of their operations.  Unless their steel was produced in an emissions-free way, the carbon price would add to their costs and lower the price they could afford to pay for their inputs including coking coal.  Thus the potential returns from the mine would be reduced, and the risk of loss would be increased.

If the mine is made to bear its full social cost in this way, so that its private costs and benefits are aligned with its costs and benefits to society, then a commercial decision by WCM as to whether investment in the mine would be worthwhile will also reach the correct decision from society’s point of view.  In that case there would be no need for the Council or the UK government to become involved in assessing the climate change implications of the mine.  With the market failure corrected, the matter could be left to the market (subject to planning approval in respect of genuinely local considerations). 

Although the EU and the UK have emissions trading systems (20), this does not mean that the mine will bear all of its social costs.  One reason is that WCM plans to export coal to the EU and beyond (21).  Countries just beyond the EU with sizeable steel industries include Turkey (34Mt), Iran (26Mt) and Ukraine (21Mt) (22).  Of these, Ukraine is considering an emissions trading scheme, but prior to legislating on such a scheme has just began a three year period in which large industrial installations are required to collect data on emissions (23).  Turkey is reported to be considering an emissions trading scheme, but without recent developments.  There appear to be no significant moves towards an emissions trading scheme (or carbon tax) in Iran.  Thus there is a significant chance that, for the next few years and perhaps beyond, some of WCM’s coal would be exported to countries with no carbon price at all.

A second reason is that, although steel production is within the scope of the EU emissions trading system, it is likely to continue to receive some of its carbon allowances for free until 2030 at least (24).  The EU’s understandable concern is that, since many steel products can readily be traded internationally, there is a risk that a stricter emissions regime could lead producers to transfer their operations to countries with laxer policies.  Nevertheless, free allocation of allowances means that the steel industry, and the mines which supply its coking coal, are not bearing all their social costs. 

A third reason is that it is questionable whether the market price of carbon allowances within the EU trading system is and will be high enough.  The EU sets an annual cap on the number of allowances, the number being slightly reduced each year, and the caps have a major influence on the market price of allowances.  Arguably they should be lower so that the market price will be higher.  Admittedly the price has risen in recent years, from very low levels during 2012-2018 to around €20 in 2019 and almost €40 in early 2021 (25).  Whether the price will remain at around that level remains to be seen.  The High Level Commission on Carbon Pricing (2017) concluded that, to achieve the 2015 Paris Agreement’s aim of limiting global average temperature to well below 2°C above pre-industrial levels, the carbon price should be at least US$ 40-80 by 2020 (26).  The current €40 (equivalent to $48) is towards the lower end of that range.

Each of those reasons underlines the need for international coordination on climate change and therefore the importance of the coming Glasgow Conference.  A successful conference could put pressure on countries which have not established a carbon price to move towards setting one, or accelerate existing initiatives.  An expectation that carbon pricing will become more widespread would weaken the argument that free allowances are needed to avoid the risk of producers relocating abroad.  And a successful conference could agree tighter national caps on emissions leading to higher market prices for emissions allowances.

Notes and references

  1. West Cumbria Mining Statement 5/4/2021
  2. Coking coal is used in steel production both to reduce iron ore to iron and as fuel, but both processes generate CO2.  The atomic mass of carbon is 12 and that of oxygen 16, so 1 tonne of carbon yields (12 + (2×16))/12 = 44/12 tonnes CO2.  If the coal is 80% carbon, then 3M tonnes coal yields 3M x 0.8 x 44/12 = 8.8M tonnes CO2.
  3. Wikipedia – Price elasticity of demand – Selected price elasticities
  4. Truby J (2012) Strategic behaviour in international metallurgical coal markets  EWI Working Paper No. 12/12  p 13
  5. Lorenczik S & Panke T (2015) Assessing market structures in resource markets – An empirical analysis of the market for metallurgical coal using various equilibrium models  EWI Working Paper No. 15/02  p 14
  6. Lawrence K & Nehring M (2015) Market structure differences impacting Australian iron ore and metallurgical coal industries  Minerals Vol 5 p 483
  7. 510,000 tonnes x 0.8 x 44/12 (as per Note 2 above) = 1,496,000 tonnes, less 20,000 tonnes shipping fuel.
  8. 30,000 tonnes x 0.8 x 44/12 (as per Npte 2 above) = 88,000 tonnes, less 20,000 tonnes shipping fuel.
  9. World Steel Association – Raw materials
  10. Wikipedia – Direct reduced iron
  11. Hoffmann C, Van Hoey M & Zeumer B (3/6/2020) Decarbonization challenge for steel  McKinsey & Company  p 5
  12. SSAB
  13. H2GS
  14. Siderwin
  15. Boston Metal Boston Metal | A world with no pollution from metals production
  16. Carbon Sequestration Leadership Forum
  17. Thyssenkrupp
  18. Tian S, Jiang J, Zhang Z & Manovic V (2018) Inherent potential of steelmaking to contribute to decarbonisation targets via industrial carbon capture and storage  Nature Communications 9, Article No. 4422/2018
  19. Thyssenkrupp; Tata Steel
  20. The UK used to belong to the EU Emissions Trading System, but following Brexit it now has its own system: Wikipedia
  21. West Cumbria Mining Ltd – How will materials be transported  The statement about “EU and beyond” is at the bottom of the factsheet.
  22. World Steel Association – Steel statistical yearbook 2020 concise version  Table 1 pp 1-2
  23. World Bank Carbon Pricing Dashboard (Use the dropdown box under “Information on carbon pricing initiatives selected” to look for details re individual countries.)
  24. Metal Bulletin
  25. Ember – Daily EU ETS carbon market price (euros)
  26. Carbon Pricing Leadership Coalition – Report of the High Level Commission on Carbon Prices  p 3

Posted in Climate change, Minerals | Tagged , , , , , | Leave a comment