## UK Climate Change Policy – An Outline

Anyone who has followed the news in the UK over the last year will have heard plenty about COP26 and Extinction Rebellion, and about extreme heat, forest fires and melting ice in various parts of the world.  UK policy on climate change, despite its implications for everyone’s lives in the decades ahead, has received rather less attention.  Here I present, in Q & A form, an outline of the legal framework, current policy, progress to date, and ideas which seem likely to influence future policy.

What is the UK’s long-term target for reducing its greenhouse gas emissions as a contribution to mitigating global climate change?

The government has adopted a target of reducing net emissions to zero by 2050.

Which gases count towards this target?

All significant greenhouse gases, including carbon dioxide, methane and others.

Which emissions are considered to be the UK’s emissions?

Its territorial emissions consisting, broadly, of those emissions generated within the UK.  These exclude emissions from production overseas of goods imported to the UK, sometimes termed “consumption emissions”.  Emissions from international shipping and aviation were initially excluded, but more recently the UK’s share of such emissions has been considered part of its territorial emissions.

What is the significance of “net” emissions?

Deductions against total emissions are allowed for removals of greenhouse gases from the atmosphere and funding emissions-reducing projects abroad.

Wasn’t there a target of reducing emissions by 80%?

That was the target set in the Climate Change Act 2008.  It was changed to 100% (ie reduction to zero) in 2019 (1).

What else did the Climate Change Act do?

It required the government to set “carbon budgets” for the UK covering all the main greenhouse gases  for 5 year periods starting from 2008-12, and to ensure that these budgets are not exceeded.  Budgets from 2023-27 were to be set at least 12 years in advance.  It also required the government to make 5-yearly assessments of the risks to the UK from climate change and to prepare adaptation policies to address those risks.  It established a Committee on Climate Change to advise the government on climate change matters including the setting of carbon budgets.

Where are we now in terms of the 5 year carbon budgets?

The first and second 5 year periods have been completed and we are now in the third (2018-22).  Carbon budgets for the fourth and fifth periods (2023-32) were set some years ago.  The carbon budget for the sixth period (2033-37) has a particular importance as it was the first to be set after the 2050 target had been amended to net zero.

Are the UK’s carbon budgets consistent with international agreements on climate change?

The UK has generally set its carbon budgets to go beyond its commitments under international agreements:

• For the Kyoto Protocol’s second commitment period (2013-20), the UK’s target reduction was 20% (all reductions quoted are relative to a 1990 baseline) (2).  Its carbon budget for 2018-22 is a reduction of 34% (3,4).
• Following the Paris Conference (2015), countries were required by 2020 to submit their plans, known as nationally-determined contributions, for reaching the goal of limiting global warming to below 2 and preferably to 1.5 degrees Celsius above pre-industrial levels.  The UK’s submission committed it to a reduction in emissions of at least 68% by 2030 (5).  This did mean that its emissions would need to fall by more than the 57% that had previously been budgeted for the UK’s fifth period (2028-32), but for the sixth period (2033-37) its budgeted reduction was set at a much more challenging 78% (6,7)

So the UK has been a good citizen internationally in respect of climate change?

That’s debatable. On the one hand:

• The UK can reasonably claim to have exercised international leadership in its early adoption of a legally binding target via the Climate Change Act.

On the other hand:

• In focusing on carbon budgets which exclude consumption emissions, the UK is arguably measuring its performance against inappropriate benchmarks.
• Some argue for even faster emissions reductions by developed countries including the UK because of the risks climate change poses to life in the tropics and in low-lying islands.  They also point to the responsibility of developed countries, via their historic emissions, for the vast majority of climate change to date.
• The Copenhagen agreement (2009) included a pledge by developed countries to transfer US$100 billion annually to poor countries by 2020 to help fund mitigation of and adaptation to climate change (8). The UK pledged to transfer £5.8 billion over the 5 years beginning April 2016 (9), equivalent to about US$ 1.5 billion annually, a very small contribution towards the US$100 billion target, although the contributions of other developed countries have also fallen well short. How have the UK’s actual emissions compared with its carbon budgets? The budgets for the first two periods were comfortably met, that is, actual emissions were below the budgeted levels (10). Results for the third period should be available in 2023. So the UK has made a good start on the road to net zero emissions? Again, that’s debatable. Those budgets were set before the net zero target had been adopted. The post-2008 recession helped to reduce emissions simply by reducing economic activity. More fundamentally, it is much easier to reduce emissions from some sources than from others. Much of the reduction was achieved by switching from coal to natural gas to generate electricity, but burning natural gas still produces about half the emissions of coal. A net zero economy will need to generate very large quantities of zero-carbon electricity. Where is it likely to be especially difficult to reduce emissions to zero? Difficult sectors include: • Agriculture, especially livestock production (a major source of methane). • Shipping and aviation (unlike road and rail transport, these cannot be electrified to take advantage of electricity from renewable sources). • Cement production, for use in concrete (this currently involves burning limestone, a process which produces carbon dioxide even if the fuel source is zero-carbon). What measures does the Climate Change Committee propose to reduce emissions by 78% as required by the sixth carbon budget (2033-37)? The Committee proposes (11): • Conversion to low-carbon technologies in industry, transport and the home, to be achieved mainly by electrification or use of hydrogen fuel. • Expansion of low-carbon electricity from 50% now to 100% by 2035 using various renewable sources, the largest contribution coming from offshore wind. • Reduction of demand for carbon-intensive goods and services. • Changes in land use, with more woodland and biofuel crops, and restoration of peatlands. Of these four areas, the first is projected to have the largest effect, but all are needed to achieve the budgeted 78% reduction. Is it true that the UK already gets 50% of its electricity from low-carbon sources? Yes, approximately. Almost 50% of electricity in 2020 was from low-carbon sources, comprising wind (25%), nuclear (15%), solar (4%) and hydro (3%) (12). But the low-carbon share of total energy, including not only electricity but also gas for home heating and oil for transport, is still much less than 50%. Has the government accepted the Committee’s proposals on mitigation? Yes, broadly. Over the last two years the government has published a number of detailed documents setting out strategies and specific policies to reduce emissions: There are also various policies which have been in place for some years which will continue with at most minor changes. These strategies and policies to a large degree reflect the Committee’s proposals. Most of them apply to the whole of the UK, although as the last two documents illustrate there are some exceptions in respect of matters devolved to Scotland, Wales and Northern Ireland. What kinds of policy tools is the government using to reduce emissions? A combination of project funding, taxes, regulation, and what might loosely be described as other financial incentives. What funding is the government providing? Funding commitments include (in rounded £ Million): • £2,500M over 2020-25 to help public sector bodies fund heat decarbonisation and energy efficiency measures. • £1,750M to fund energy efficiency upgrades in social housing and low-income households. • £2,000M over 5 years for schemes to facilitate cycling and walking (and so reduce car use). • £1,300M to accelerate the roll-out of electric vehicle charging infrastructure. • £1,000M to build an electric vehicle supply chain. • £600M to provide grants to help buy electric vehicles. • £300M to try to make electric vehicle batteries 95% recyclable by 2035. • £600M over 2020-22 for zero-emission buses (mainly electric but some powered by hydrogen). • £1,000M by 2025 for investment in carbon capture, utilization and storage (CCUS), a technology offering the promise of emissions-free use of fossil fuels and biofuels in electricity generation and industry. • £1,000M for the Net Zero Innovation Portfolio providing funding for low-carbon technologies and systems. • £450M over 3 years for the Boiler Upgrade Scheme to provide grants to help replace boilers by electric heat pumps (replacing the former Renewable Heat Incentive Scheme). • £240m to support hydrogen production projects. • £500M to increase planting of new woodland to over 100 square miles annually (13). • Numerous smaller sums, including funding for various competitions designed to stimulate innovation in low-carbon energy and emissions reduction. What about taxes? The government’s approach has been to use taxes to change business behaviour while avoiding climate-related taxes directly on households. Businesses can however pass on to households the costs of these taxes. The main taxes are: • The Climate Change Levy, paid by electricity generators on their fossil fuel inputs and by other businesses on their electricity and fossil fuels, albeit with reduced rates for certain energy-intensive businesses (to mitigate loss of international competitiveness). • The Green Gas Levy, a tax on natural gas suppliers. The money raised is used to fund the Green Gas Support Scheme providing incentives for the production of biomethane for injection into the gas grid. • The Landfill Tax, designed to discourage disposal of waste by landfill. Introduced in 1996, this tax has contributed to an 80% reduction in landfill emissions (14). There are also some tax breaks designed to encourage emissions reduction: • Exemption from vehicle excise duty (car tax) for electric cars and vans. • Electric vehicles are also exempt from fuel duty, a tax on petrol and other liquid fuels used in trasnport. • Enhanced capital allowances for businesses on the purchase of certain new energy-efficient or low-carbon equipment. What regulation is in place or planned? Regulations in place include: • The Renewable Transport Fuel Obligation requiring fuel suppliers to supply a certain percentage of renewable biofuels, currently 9.6% and planned to be increased to 14.6% by 2032. Among planned regulations are some which will have very direct effects on households and individuals: • The sale of new petrol and diesel cars and vans will be banned from 2030. Sale of hybrid vehicles will be allowed until 2035, from which date all new cars and vans will have to be zero-emission. • All new heating systems will have to be net zero compatible by 2035, implying that the installation of new natural gas boilers will be banned. In the meantime, policy will aim to ensure that electric heat pumps become an attractive and cost-effective alternative. • The sale of peat for garden use is likely to be banned, to help preserve peatlands. What are the “other financial incentives”? There are three large and complex schemes: • The UK Emissions Trading Scheme is a “cap and trade” scheme which replaced, following Brexit, the UK’s participation in the EU Emissions Trading Scheme. It applies to electricity generation, energy-intensive industries including steel and cement production, and aviation on some routes. The government sets a cap on the total emissions allowed by businesses within the scheme, and issues emissions allowances to businesses to the amount of the cap. These allowances can be traded which encourages businesses to reduce emissions where this can be done at least cost. The cap will be gradually reduced as the UK moves towards net zero. For electricity generation only, the Scheme applies in conjunction with a Carbon Price Floor ensuring that the effective cost of emissions reaches a cerain rate, currently £18 per tonne of carbon dioxide. For other sectors the cost per tonne has at times been much lower. • The Contracts for Difference Scheme is designed to support low-carbon electricity generation from sources which, at a certain stage of development, are too expensive to compete with electricity from conventional sources. Generators submitting successful bids for new low-carbon capacity are paid over a 15 year period at a rate calculated to reflect their extra cost over the average UK price of electricity. • The Renewables Obligation closed to new renewable generating capacity in 2017, but continues to require electricity suppliers either to obtain a certain proportion of their electricity from certain renewable sources accredited prior to closure, or else suffer a financial penalty. What is the long-term vision for energy supply? It is envisaged that the UK will need much more electricity than now, partly because of normal growth in demand, but also to meet new demands for electricity including heat pumps, electric vehicles and hydrogen production. The majority of electricity will be from renewables, mainly onshore and offshore wind and solar, also biomass, with energy storage in the form of hydrogen to overcome the intermittency problem of wind and solar. There will also be significant contributions from nuclear, and from gas with carbon capture and storage. Transport will be powered mainly by electricity or hydrogen, with some use of biofuels especially for shipping and aviation. Is carbon capture and storage a proven and cost-effective technology? Around the world, there are currently some 40 carbon capture facilities in operation (15). However, some of these use the carbon (eg for enhanced oil recovery) rather than storing it. The effectiveness of storage – whether there is a long-term risk of leakage to the atmosphere – is hard to prove and may depend on the storage site. It is probably fair to say that capture has been proven to be feasible, but that whether capture and effective long-term storage can be delivered at a cost comparable with other low-carbon technologies remains to be demonstrated. It may be however that, as with wind and solar over the last decade, costs will fall as experience is gained. How much will it cost to get to net zero? The Climate Change Committee estimates that the annual investment needed to deliver its proposals will rise to about £50 billion by 2030 and remain fairly stable thereafter (16). That’s a 12% increase on current total annual investment of about £400 billion. The required investment can be delivered largely by the private sector provided that the government creates a stable long-term policy framework. The Committee also estimates that there will be a net decrease in annual running costs, largely due to fuel cost savings in industry, and that these savings will steadily rise, so that by around 2040 they will more than offset the required annual investment cost. Households are being faced with large increases in energy costs right now. How far is that due to climate change policies? It is true that household energy costs are higher than they would be without policies to reduce emissions. Several of these policies increase costs to energy suppliers, and some at least of these extra costs are passed on to households. However, policies such as the Renewables Obligation and the Climate Change Levy have been in place for some years, and have not changed dramatically over the last year. The main reason for the sudden increase in energy costs is the combination of: • The UK’s heavy reliance on gas which heats 85% of homes and is the source of 35% of its electricity. • An increase over the last year in world demand for gas. Of these factors, the second is largely outside the UK’s control, but the first is partly due to UK policies over the past decades (although alternatives such as a slower removal of coal from the energy mix or greater investment in nuclear would also have had disadvantages). In the long run, further expansion of wind and solar should make UK energy costs less dependent on world markets. What is the UK’s long-term target for adaptation to climate change? There is no clear target. Climate change presents many different risks, and the extent of future warming and other changes in climate is far from certain. Unlike mitigation, adaptation is not usefully framed as working towards a single numerical target. What are the main conclusions of the Climate Change Committee’s latest assessment of the risks from climate change? Its Independent Assessment of UK Climate Risk, published in 2021, states starkly that “adaptation action has failed to keep pace with the worsening reality of climate risk” (p 11) and that the government “has not heeded our past advice” on setting and adequately resourcing “a framework of targets, incentives and reporting” (p 23). It identifies eight risk areas judged to require urgent action: • Risks to natural ecosystems from increased temperatures and extreme events such as droughts and wildfire. • Risks to soil health from increased flooding and drought. • Risks to natural carbon stores such as soil, trees, saltmarsh and underwater kelp forests. • Risks to crops, livestock and commercial trees from multiple hazards including heat stress, drought, flooding, fire, pests, diseases and invasive non-native species. • Risks to the supply of food, goods and vital services due to climate related impacts on supply chains and distribution networks. • Risks to electricity supply from climate-related hazards including flooding, water shortages, increased tenperatures and wildfire, sea level rise and storms. • Risks to human health, well-being and productivity from increased exposure to heat in homes and other buildings. • Multiple risks from climate change effects overseas which could lead to cascading impacts across sectors and countries. What specifically does the Committee recommend? Its most prominent recommendations take the form of principles for good adaptation. It asks the government to: • Set out a vision of a well-adapted UK. • Integrate adaptation to climate change into policies on a wide range of matters (rather than treating it as a self-contained topic). • Take early action where necessary, eg to prevent irreversible changes to ecosystems or avoid the need for expensive retrofitting of buildings. • Prepare for extreme weather events and not just for average temperature rises. • Assess interdependencies such as the human and economic effects of climate-related failure of electricity supply. • Address climate-related inequalities, eg low-income households being more exposed to flood risk. What does the government see as its role in respect of adaptation? The second National Adaptation Programme was published in 2018 (the government has not yet published a third programme responding to the Climate Change Committee’s latest assessment). The Programme includes a very long list of actions, but it also includes passages which suggest that the government sees for itself a limited role. It states for example that infrastructure operators are private businesses responsible for their own business continuity measures, and that the government’s responsibility is to ensure that no policy or regulatory barriers prevent them from managing their climate risks (p 31). Of the actions in the Programme, many refer vaguely to, for example, “supporting” or “encouraging” initiatives, “working with” partners, and “monitoring” progress. The only major funding commitment highlighted in the Programme is £2,600 million over 6 years to reduce flood and coastal erosion risk. So the government is not actually doing all that much about adaptation? That’s debatable. There are a number of areas in which climate change adaptation has been integrated into broader policies. To give a couple of examples: • Farmers and others can obtain funds via the Countryside Stewardship Scheme and similar schemes for projects to improve the rural environment. Guidance includes climate change adaptation among the outcomes supported (p 7). • The National Planning Policy Framework sets out law and guidance to be followed by local planning authorities in England in determining planning applications for new housing and other developments. It states that plans should take a proactive approach to adaptation to climate change, “taking into account the long-term implications for flood risk, coastal change, water supply, biodiversity and landscapes, and the risk of overheating from rising temperatures” (p 45). However, it is easy for the government to add wording about climate change adaptation into policy documents. The Climate Change Committee clearly considers that there is much more to be done in terms of funding and delivery. This post is intended to be largely factual. I plan in due course to post a critical analysis of UK climate change policy. Notes and References 1. The Climate Change Act 2008 (2050 Target Amendment) Order 2019 https://www.legislation.gov.uk/ukdsi/2019/9780111187654 2. Wikipedia: Kyoto Protocol – Emissions Cuts https://en.wikipedia.org/wiki/Kyoto_Protocol#Emissions_cuts 3. Climate Change Committee (2008) Building a Low Carbon Economy https://www.theccc.org.uk/publication/building-a-low-carbon-economy-the-uks-contribution-to-tackling-climate-change-2/ The 34% reduction in emissions in the 3rd carbon budget, implying a budget of 2,570 MtCO2e, is on p xix. 4. The Carbon Budgets Order 2009 https://www.legislation.gov.uk/uksi/2009/1259/article/2/made The Order shows that the 3rd budget was set at 2,544 MtCO2e, very close to the Committee’s recommendation. 5. DBEIS The UK’s Nationally Determined Contribution under the Paris Agreement https://www.gov.uk/government/publications/the-uks-nationally-determined-contribution-communication-to-the-unfccc#:~:text=On%2012%20December%202020%2C%20the,2030%2C%20compared%20to%201990%20levels. 6. Climate Change Committee (2020) The Sixth Carbon Budget UK Carbon Budgets – Climate Change Committee (theccc.org.uk) The 78% reduction in emissions is on p 38, and can be seen from the chart on p 39 to imply a budget of c 1,000 MtCO2e. 7. The Carbon Budget Order 2021 The Carbon Budget Order 2021 (legislation.gov.uk) The Order shows that the 6th budget was set at 965 MtCO2e, very close to the Committee’s recommendation. 8. House of Commons Library (2021): COP26: Delivering on$100 billion climate finance https://commonslibrary.parliament.uk/cop26-delivering-on-100-billion-climate-finance/
9. HM Government UK International Climate Finance https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1029990/icf-brochure-2021.pdf  The £5.8 billion figure is on p 4.
10. Cambridge Econometrics (2019) How the UK met its carbon budgets  https://www.theccc.org.uk/publication/how-the-uk-met-its-carbon-budgets/ p 5
11. Climate Change Committee, as 6 above, p 25
12. These percentages are derived from the following figures in the Digest of UK Energy Statistics Table 5.6 Electricity fuel use, generation and supply https://www.gov.uk/government/statistics/electricity-chapter-5-digest-of-united-kingdom-energy-statistics-dukes  In row 504 (All generating companies, supplied gross) (in GWh) Wind 75,380, Nuclear 45,668, Solar 13,158, Hydro 6,636 + 1,397, All Sources 297,683.
13. HM Government (2021) The England Trees Action Plan 2021-24  https://www.gov.uk/government/publications/england-trees-action-plan-2021-to-2024  The figure given (p 3) is 30,000 hectares, equal to c 116 square miles (1 square mile = 259 hectares).
14. DBEIS Final UK greenhouse gas emissions national statistics 1990 to 2019 https://www.gov.uk/government/collections/final-uk-greenhouse-gas-emissions-national-statistics  Table 5.1 Estimated territorial greenhouse gas emissions by end user category, UK, 1990=2019, row 111 Landfill.
16. Climate Change Committeem as 6 above, pp 20-1

## 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)

where:

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)

## 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:

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

where:

$Y_t =$ 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:

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

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

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 https://core.ac.uk/download/pdf/16221331.pdf  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  https://circabc.europa.eu/sd/a/d4510f50-76ec-4332-9598-3cd762f21c64/UK-2015-Natural-capital.pdf  p 191
11. Van der Ploeg F (2006)  Challenges and Opportunities for Resource Rich Economies  EUI Working Papers RSCAS No. 2006/23  https://cadmus.eui.eu/bitstream/handle/1814/6254/RSCAS_2006_23.pdf?isAllowed=y&sequence=3  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  https://economicsandpolicy.ca/2017/06/19/hartwicks-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  https://math.stackexchange.com/questions/4139757/if-sum-limits-t-1-inftyr-t-is-finite-with-r-t-geq-0-does-lim-t-rig

## 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 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2163203
4. For population growth rates by country see https://www.indexmundi.com/g/r.aspx?v=24