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

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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  https://www.westcumbriamining.com/wp-content/uploads/West-Cumbria-Mining-Statement-5th-March-2021.pdf
  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 https://en.wikipedia.org/wiki/Price_elasticity_of_demand#Selected_price_elasticities
  4. Truby J (2012) Strategic behaviour in international metallurgical coal markets  EWI Working Paper No. 12/12  http://hdl.handle.net/10419/74411  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  https://www.ewi.uni-koeln.de/en/energy/coal/#papers-2015  p 14
  6. Lawrence K & Nehring M (2015) Market structure differences impacting Australian iron ore and metallurgical coal industries  Minerals Vol 5 https://www.mdpi.com/2075-163X/5/3/473 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 https://www.worldsteel.org/steel-by-topic/raw-materials.html
  10. Wikipedia – Direct reduced iron https://en.wikipedia.org/wiki/Direct_reduced_iron
  11. Hoffmann C, Van Hoey M & Zeumer B (3/6/2020) Decarbonization challenge for steel  McKinsey & Company https://www.mckinsey.com/industries/metals-and-mining/our-insights/decarbonization-challenge-for-steel  p 5
  12. SSAB https://www.ssab.com/company/sustainability/sustainable-operations/hybrit
  13. H2GS https://www.h2greensteel.com/
  14. Siderwin https://www.siderwin-spire.eu
  15. Boston Metal Boston Metal | A world with no pollution from metals production
  16. Carbon Sequestration Leadership Forum https://www.cslforum.org/cslf/Projects/AlReyadah
  17. Thyssenkrupp https://www.thyssenkrupp.com/en/newsroom/content-page-162.html
  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  https://www.nature.com/articles/s41467-018-06886-8
  19. Thyssenkrupp https://www.thyssenkrupp.com/en/stories/the-path-of-green-hydrogen; Tata Steel https://www.tatasteeleurope.com/ts/sustainability/carbon-neutral-steel
  20. The UK used to belong to the EU Emissions Trading System, but following Brexit it now has its own system: Wikipedia https://en.wikipedia.org/wiki/UK_Emissions_Trading_Scheme
  21. West Cumbria Mining Ltd – How will materials be transported  https://www.westcumbriamining.com/what-is-the-plan/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 https://www.worldsteel.org/en/dam/jcr:5001dac8-0083-46f3-aadd-35aa357acbcc/Steel%2520Statistical%2520Yearbook%25202020%2520%2528concise%2520version%2529.pdf  Table 1 pp 1-2
  23. World Bank Carbon Pricing Dashboard https://carbonpricingdashboard.worldbank.org/map_data (Use the dropdown box under “Information on carbon pricing initiatives selected” to look for details re individual countries.)
  24. Metal Bulletin  https://www.metalbulletin.com/Article/3868471/EU-ministers-approve-granting-of-free-carbon-allowances-to-steel-other-metal-sectors-until-2030.html
  25. Ember – Daily EU ETS carbon market price (euros)  https://ember-climate.org/data/carbon-price-viewer/
  26. Carbon Pricing Leadership Coalition – Report of the High Level Commission on Carbon Prices  https://www.carbonpricingleadership.org/report-of-the-highlevel-commission-on-carbon-prices  p 3

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Dynamic Optimisation: A Fully Worked Example

In a previous post, I referred to the importance in environmental and natural resource economics of the technique of dynamic optimisation, also known as optimal control.  However, the technique is difficult, and worked examples in textbooks or on the web often seem to pass over key points.  Here I present my own example, which I describe as fully worked because it shows every step from the largely verbal statement of the problem to the optimal paths of the key variables and the maximum value of the objective functional, identifying some options and pitfalls along the way.  It is intended for readers familiar with elementary algebra, calculus and static optimisation who have at least begun to study dynamic optimisation.

The Problem

Capital K, is the only factor of production and is not subject to depreciation. The initial capital stock is 100,.  Output is at a rate 0.5K,, and may be used as consumption C, or investment I,, the latter being added to K,.  The instantaneous utility function U_t is \ln(C_t) . We are required to maximise social welfare W, from time t = 0, to 10,, where social welfare is defined as the integral of instantaneous utility subject to a continuous discount factor of 10\%, per time period.

A Note on Notation

A widely used convention is that the subscript t,, as in C_t, indicates discrete time, and that a variable in continuous time should be written as in C(t), .  I find however that it saves a little keying time, and results in less cluttered formulae, to use the subscript approach for continuous time, and sometimes to omit the t, altogether when it is clear from the context.  More conventionally, I use the notation \dot C, to indicate a time-derivative, and \ddot C, for a second time-derivative.

I use Latex to display mathematical symbols and formulae. However, using Latex within a WordPress blog is not entirely straightforward, one problem being to obtain a satisfactory vertical alignment of symbols within text paragraphs. The commas which follow some symbols are a workaround which corrects vertical alignment in many (though not all) cases and seem to me preferable to the alternative of displaying symbols – like K for example – with their base lower than that of the surrounding text.

Writing the Problem in Mathematical Formulae

Our problem statement above contains the symbols K, C, I, U, W, t.  The first question we should consider is whether we need all these for a precise mathematical formulation.  It is clear that we can dispense with U, and relate W, directly to C,, writing the objective functional as:

\textrm{Maximise }W=\int_0^{10}(\ln(C_t))e^{-0.1t}dt\qquad(1)

We need K, which is clearly the state variable, but what is the control variable?  Since C + I = 0.5K,, either of C, or I, determines the other.  Nothing in the problem statement indicates that one is a choice variable and the other a residual.  Either could be the control variable, but we do have to choose (because the method requires maximisation of the Hamiltonian or Lagrangian with respect to the control variable).  Let us choose C, as the control variable (but Alternative 1 below will show that choosing I, leads to the same results).  We therefore write the equation of motion as:

\dot K=0.5K-C\qquad(2)

We also have the boundary conditions:

K_0=100\ \textrm{and }K_{10}\ \textrm{free}\qquad(3)

Does that complete the formulation of the problem?  No!

Pitfall 1

If we rely on the formulation above, there is nothing to prevent negative consumption, with investment \dot K, exceeding output and W, undefined (because the log of a negative quantity is undefined).  There is also nothing to prevent negative investment.  Thus the above formulation allows a time path in which capital is initially accumulated, but towards the end of the time period is run down to zero, enabling consumption to exceed output.  That could be a desirable scenario if the capital is in the form of a good which can also be consumed.  More typically, however, capital cannot be consumed and therefore consumption cannot exceed output, and the above formulation will therefore lead to erroneous results by permitting more consumption than is feasible.  Indeed, there is nothing in the formulation to rule out the combination of infinite consumption and infinite negative investment.

We therefore add two constraints and, to prepare for writing the required Lagrangian function, rewrite each as a quantity to be less than or equal to a constant, in these cases zero:

C_t \geq 0\ \forall t \in [0,10]\ \textrm{and so } -C_t \leq 0\qquad(4)

C_t \leq 0.5K_t\ \forall t \in [0,10]\ \textrm{and so } C_t-0.5K_t \leq0\qquad(5)

Although we also require that capital should not be negative, we need not specify this as a further constraint since it is is implied by the combination of K_0=100 and \dot K\geq 0,, the latter following from the equation of motion together with constraint (5).  Indeed, these imply the stronger condition K_{10} \geq 100. The combination of (1) to (5) completes the mathematical formulation of the problem.

The Value of W for Two Naïve Solutions

Before applying the method of optimal control, let us consider a couple of simple and feasible time paths for consumption and calculate the implied values of W,.  The results will provide a benchmark against which we can compare our final result.  Suppose first that there is no investment and all output is consumed.  Then capital is always 100, and consumption is always 0.5(100) = 50,.  Hence:

W=\int_0^{10}(\ln50)e^{-0.1t}dt

W=3.912\left[-10e^{-0.1t}\right]_0^{10}=3.912(-10e^{-1}+10)=\mathbf{24.73}  

Now suppose that output is always divided equally between consumption and investment.  Before we can calculate W, we need to find the time path of capital by solving the differential equation:

\dot K =0.5(0.5K)=0.25K\qquad(6)

Making the standard substitution K, = e^{bt} so that \dot K,= be^{bt} we have:

be^{bt}=0.25e^{bt}\ \textrm{and so } b=0.25\qquad(7)

Hence for some constant c,:

K_t=ce^{0.25t}\qquad(8)

Since K_0 = 100 we can infer that c=100, and so:

K_t=100e^{0.25t}\ \textrm{and so } C_t=0.5(0.5K_t)=25e^{0.25t}\qquad(9)

Hence:

W=\int_0^{10}(\ln(25e^{0.25t})e^{-0.1t}dt=\int_0^{10}(3.219+0.25t)e^{-0.1t}dt\qquad(10)

W=\left[-(32.19+2.5t+25)e^{-0.1t}\right]_0^{10}

W=-82.19e^{-1}+57.19=\mathbf{26.95}

As we might expect, allocating half of output to investment, allowing capital to accumulate and increase output as time goes on, yields a higher W, than simply consuming all output.  But there is no reason to expect that this value of W, is the maximum.

Necessary Conditions for a Solution

From (1) and (2) we obtain the Hamiltonian, introducing a costate variable \lambda_t:

H=(\ln C)e^{-0.1t}+\lambda (0.5K-C)\qquad(11)

This is a present value Hamiltonian because it retains the discount factor in the objective functional and so converts \ln C_t at any time to its present value, that is, its value at time 0,.  An alternative approach will be considered below.  Because we have two inequality constraints, we must extend the Hamiltonian to form a Lagrangian, introducing two Lagrange multipliers \mu_t and \nu_t:

\mathcal{L}=(\ln C)e^{-0.1t}+\lambda (0.5K-C)+\mu C+\nu (0.5K-C)\;(12)

The expressions in brackets after the Lagrange multipliers are from the inequality constraints (4) and (5) with signs changed. The general rule here is that given a constraint g, \leq k and writing \theta_t for the associated multiplier, the term to be included in the Lagrangian is \theta_t(k-g).

Applying the maximum principle, we have to maximise the Lagrangian with respect to the control variable C_t at all times.  In this case, the Lagrangian is differentiable with respect to C_t, so we can try to use calculus to find a maximum.  But we also need to consider whether there might be a corner solution, that is, a solution at either of the limits of the constrained range of C,, which are 0, and 0.5K,.  We can rule out the possibility of a maximum at t = 0,, since \ln 0, equals minus infinity.  But there is no obvious reason why there should not be a maximum at C, = 0.5K for at least some values of t,, so we should keep this possibility in mind.  Setting the derivative with respect to C, of the Lagrangian equal to zero we have:

\dfrac{\partial \mathcal{L}}{\partial C}=\dfrac{e^{-0.1t}}{C}-\lambda+\mu-\nu=0\qquad(13)

The maximum principle also requires the conditions:

\dot K=\dfrac{\partial \mathcal{L}}{\partial \lambda}=0.5K-C\qquad(14)

\dot {\lambda}=-\dfrac{\partial \mathcal{L}}{\partial K}=-0.5\lambda-0.5\nu\qquad(15)

Although the effect of (14) is merely to repeat the equation of motion (2) it is standard practice to write it out at this point in the working.  We also require the Kuhn-Tucker conditions in respect of the two inequality constraints, conditions (17) being known as the complementary slackness conditions.

\mu \geq 0\ \textrm{and } \nu \geq 0\ \forall t \in [0,10]\qquad(16)

\mu C=0\ \textrm{and }\nu (0.5K-C)=0\ \forall t \in [0,10]\qquad(17)

Finally, there is the transversality condition.  With a fixed terminal time, but terminal capital free subject to the implied condition K_{10} \geq 100, we have the situation known as a truncated vertical terminal line.  Therefore we provisionally adopt the condition:

\lambda_{10}=0\qquad(18)

However, we will have to check that the resulting solution is consistent with the condition K_{10} \geq 100 (and if not we must recalculate the solution with K_{10} fixed at 100,).  (12) to (18), with the provisos noted, constitute the necessary conditions for a maximum.

Sufficiency of the Necessary Conditions

We will test whether the Mangasarian conditions are satisfied.  The basic conditions are:

(A) The integrand of the objective function, (\ln C)e^{-0.1t},, must be differentiable and concave in the control and state variables, C, and K,, jointly.

(B) The equation of motion formula, 0.5K-C,, must be differentiable and concave in C, and K, jointly.

(C) If the equation of motion formula, 0.5K-C,, is non-linear in either C, or K,, then in the optimal solution we must have \lambda_t \geq 0 for all t,.

Considering these in turn:

Condition (A) is satisfied since, applying a calculus test for concavity:

\dfrac{\partial((\ln C)e^{-0.1t})}{\partial C}=\dfrac{e^{-0.1t}}{C}\ \textrm{ so  }\dfrac{\partial^2((\ln C)e^{-0.1t})}{\partial C^2}=-\dfrac{e^{-0.1t}}{C^2} \leq 0\ \forall t\quad(19)

We need not consider K, here since it does not occur in the integrand.

Condition (B) is satisfied since the formula 0.5K-C, is linear in both C, and K, and therefore concave, linearity being sufficient for concavity (there is no requirement for strict concavity).

Condition (C) is satisfied since, again, the formula 0.5K-C, is linear in both C, and K,.

For our problem, a further condition is needed for each of the inequality constraints, the general rule being that if a constraint is represented in the Lagrangian by the expression \theta (k-g) where k, is a constant, the required condition is that g, be jointly convex in the control and state variables:

(D) -C, must be convex in C, and K, jointly.

(E) C-0.5K, must be convex in C, and K, jointly.

These conditions are satisfied since the functions are linear (again there is no requirement for strict convexity). 

Thus the Mangasarian conditions are satisfied, so we can conclude that the necessary conditions (12) to (18) are also sufficient for a maximum (and need not consider the more complex Arrow conditions).

Inferences from the Necessary Conditions

Using a common approach to simplification, we differentiate (13) with respect to time and then use (15) to substitute for \dot{\lambda},:

\dfrac{-0.1e^{-0.1t}C-\dot Ce^{-0.1t}}{C^2}-\dot{\lambda}+\dot{\mu}-\dot{\nu}=0\qquad(20)

\dfrac{-0.1e^{-0.1t}C-\dot Ce^{-0.1t}}{C^2}+0.5\lambda+0.5\nu+\dot{\mu}-\dot{\nu}=0\qquad(21)

Using (13) again we can eliminate \lambda, and \nu, (but not \dot{\nu},):

\dfrac{-0.1e^{-0.1t}C-\dot C e^{-0.1t}}{C^2}+\dfrac{0.5e^{-0.1t}}{C}+0.5\mu+\dot{\mu}-\dot{\nu}=0\qquad(22)

-0.1e^{-0.1t}C-\dot Ce^{-0.1t}+0.5e^{-0.1t}C+(0.5\mu+\dot{\mu}-\dot{\nu})C^2=0\,(23)

Collecting the terms in C, and using the complementary slackness condition (17) \;\mu C=0, (which, since C, can never be zero as \ln 0, equals minus infinity, implies \mu =0, and therefore \dot{\mu}= 0, for all t,):

0.4e^{-0.1t}C-\dot Ce^{-0.1t}-\dot{\nu}C^2=0\qquad(24)

Using the equation of motion (2) to substitute for C,:

0.4e^{-0.1t}(0.5K-\dot K)-(0.5\dot K-\ddot K)e^{-0.1t}-\dot{\nu}(0.5K-\dot K)^2)=0\quad(25)

Collecting terms in e^{-0.1t}, we have the differential equation:

e^{-0.1t}(\ddot K-0.9\dot K+0.2K)-\dot{\nu}((0.5K)^2-K\dot K+(\dot K)^2=0\qquad(26)

Before proceeding we will explore two alternative approaches.

Alternative 1: Investment as the Control Variable

Suppose we take investment I, rather than consumption C, to be the control variable.  The utility function is still \ln C, which we will now have to write as \ln(0.5K-I), , so the objective functional will be:

\textrm{Maximise }W=\int_0^{10}(\ln(0.5K-I))e^{-0.1t}dt\qquad(A1)

The equation of motion will be simply:

\dot K=I \qquad(A2)

This is not tautologous since it implies that investment is the only cause of change in capital, eg there is no depreciation.  The inequality constraints become:

I-0.5K\leq 0\;\; \textrm{and }\;-I\leq 0\qquad(A3)

Hence the Lagrangian is:

\mathcal{L}=(\ln(0.5K-I))e^{-0.1t}+\lambda I+ \mu (0.5K-I)+\nu I\qquad(A4)

From the Lagrangian we derive the conditions:

\dfrac{\partial\mathcal{L}}{\partial I}=\dfrac{-e^{-0.1t}}{0.5K-I}+\lambda -\mu +\nu=0\qquad(A5)

\dot K=\dfrac{\partial\mathcal{L}}{\partial\lambda}=I\qquad(A6)

\dot {\lambda}=-\dfrac{\partial\mathcal{L}}{\partial K}=\dfrac{-0.5e^{-0.1t}}{0.5K-I}-0.5\mu\qquad(A7)

We also have the complementary slackness conditions:

\mu(0.5K-I)=0\;\;\textrm{and  }\nu I=0\qquad(A8)

Differentiating (A5) with respect to time, using (A7) to substitute for \dot{\lambda},, and substituting \dot K, for I,:

\dfrac{0.1e^{-0.1t}(0.5K-\dot K)+(0.5\dot K-\ddot K)e^{-0.1t}}{0.5K-\dot K)^2}-\dfrac{0.5e^{-0.1t}}{0.5K-\dot K)}-0.5\mu -\dot{\mu}+\dot{\nu}=0\quad(A9)

e^{-0.1t}(-\ddot K+0.4\dot K+0.05K)-0.5e^{-0.1t}(0.5K-\dot K)+(-0.5\mu - \dot{\mu}+\dot{\nu})(0.5K-\dot K)^2=0\quad(A10)

Collecting terms in e^{-0.1t}, and using the first complementary slackness condition to eliminate \mu , and \dot{\mu},  we have:

e^{-0.1t}(-\ddot K+0.9\dot K-0.2K)+\dot{\nu}((0.5K)^2-K\dot K+\dot K^2)=0\quad(A11)

It can be seen that this is equation (26) above with signs reversed, so thereafter we can proceed as in the main line of reasoning.

Alternative 2: the Current Value Hamiltonian

When the objective functional contains a discount factor, an alternative method is to use the current value Hamiltonian.  Where there are inequality constraints, this leads to a current value Lagrangian, which for our problem can be written:

\mathcal{L}_C=\ln C+\rho (0.5K-C)+\sigma C+ \tau (0.5K-C)\qquad(A12)

where the multipliers \rho ,\sigma ,\tau are equal respectively to the original multipliers \lambda ,\mu ,\nu each multiplied by e^{-0.1t},.  In the necessary conditions, the equivalent of (13) is slightly simplified by the absence of the discount factor:

\dfrac{\partial\mathcal{L}_C}{\partial C}=\dfrac{1}{C}-\rho +\sigma -\tau =0\qquad(A13)

On the other hand the equivalent of (15) requires an extra term 0.1\rho , (the discount rate being 0.1,):

\dot{\rho}=-\dfrac{\partial\mathcal{L}_C}{\partial K}+0.1\rho =-0.5\rho-0.5\tau+ 0.1\rho =-0.4\rho-0.5\tau\qquad(A14)

The difference between the coefficients in the terms 0.5\lambda , in (15) and 0.4\rho , in (A14) may seem trivial, but it leads to additional complexity later in the reasoning.  The equivalent of (24), which I re-write here for ease of reference:

0.4e^{-0.1t}C-\dot Ce^{-0.1t}-\dot{\nu}C^2=0

is found to be:

0.4e^{-0.1t}C-\dot Ce^{-0.1t}-(\dot{\rho}-0.1\rho )C^2=0\ \quad(A15)

The more complex coefficient of C^2, in turn makes it slightly more complicated to solve what below I call Case 2.  This is not to argue against the current value approach, still less to suggest that it represents a pitfall.  But whether on balance it simplifies matters, as is often suggested, seems to depend on the type of problem.

Solving the Differential Equation

Our differential equation (26) looks rather intractable, but we can simplify matters by considering separately the two cases \nu = 0, and \nu \neq 0.  To be more precise, we consider:

Case 1: \nu = 0,  over some time interval.

Case 2: \nu \neq 0  over some time interval.

Since Case 1 implies that \nu ,  is constant over the relevant interval, we can infer that \dot{\nu}= 0,  over that period.  Equation (26) therefore simplifies to:

\ddot K-0.9\dot K+0.2K=0\qquad(27)

The standard method for this type of differential equation is to make the substitution K=e^{xt},  implying \dot K=xe^{xt},  and \ddot K=x^2e^{xt},.  After dividing through by e^{xt},  we are left with the equation:

x^2-0.9x+0.2=0\qquad(28)

By factorisation or by the quadratic equation formula, this is neatly solved by x=0.4, \textrm{ or }0.5.  Hence the solution to the differential equation (27) is:

K=c_1e^{0.4t}+c_2e^{0.5t}\qquad(29)

where c_1,c_2 are constants to be found (generally a second order differential equation requires two constants of integration).  Differentiating (29) with respect to time we can infer:

\dot K=0.4c_1e^{0.4t}+0.5c_2e^{0.5t}\qquad(30)

C=0.5K-\dot K=0.1c_1e^{0.4t}\qquad(31)

Pitfall 2

Having obtained equations (29) to (31) it is tempting to think that our work is almost complete.  Putting t=0, in (29) we have:

100=c_1+c_2\qquad(P1)

Since investment right at the end of the time period can do nothing to increase consumption within the time period, we can infer that \dot K=0, at t=10,.  Hence, putting t=10, in (30):

0=0.4c_1e^4+0.5c_2e^5\qquad(P2)

-0.4c_1=0.5e(100-c_1) \quad(P3)

c_1(0.5e-0.4)=50e\qquad(P4)

c_1=\dfrac{50e}{0.5e-0.4}=141.7 \textrm{  and }c_2=100-c_1=-41.7\quad(P5)

Substituting into (31):

C=14.17e^{0.4t}\qquad(P6)

Therefore:

W=\int_0^{10}(\ln (14.17e^{0.4t})e^{-0.1t}dt=\int_0^{10}(2.651+0.4t)e^{-0.1t}dt\quad(P7)

W=\left[-(26.51+0.4t+40)e^{-0.1t}\right]_0^{10}=-70.51e^{-1}+66.51=\mathbf{27.33}

As expected, this yields a higher value of W, than either of the naïve solutions considered above.  Nevertheless, this is not the time path that maximises W,.  The fallacy here is the assumption that our Case 1 applies to the whole period t=[0,10].  Just because \dot K=0, at t=10, , it does not follow that \dot K\neq 0, at all t<10,

We must also consider Case 2, \nu \neq 0, . Using the second complementary slackness relation (17), this implies that:

0.5K-C=0\qquad(32)

Thus Case 2 is what we described above as a corner solution.  Using the equation of motion (2) this implies that, within the relevant time range, \dot K= 0,  and therefore \ddot K=0,.  Hence the differential equation (26) reduces to:

e^{-0.1t}(0.2K)-\dot{\nu}(0.5K)^2=0\qquad(33)

0.2e^{-0.1t}-0.25K\dot{\nu}=0\qquad(34)

\dot{\nu}=\dfrac{0.8e^{-0.1t}}{K}\qquad(35)

Integrating with respect to t,, noting that K,  can be treated as a constant since \dot K= 0,:

\nu =-\dfrac{8e^{-0.1t}}{K}+c_3\qquad(36)

Which Case is Terminal?

We will now show that, as time approaches t=10,, the system must be in Case 2, with K,  constant.  This is what we would expect from economic reasoning, since there must be a time beyond which the effect of further investment in making possible higher output and consumption in the remainder of the time period is too small to compensate for the consumption that would be forgone in making that investment.  To show this using the method of optimal control, we start from the transversality condition (18), \lambda_{10}= 0.  We can therefore reduce (13) at t=10,  to:

\dfrac{e^{-1}}{C_{10}}+ \mu_{10}-\nu_{10}= 0\qquad(37)

Given the first complementary slackness relation (17), \mu C,, this further simplifies to:

\dfrac{e^{-1}}{C_{10}}-\nu_{10}= 0\qquad(38)

This implies that \nu_{10}\neq 0 (otherwise C_{10}  would be infinite which is impossible given the problem data).  So the system cannot be in Case 1 at t=10, and must be in Case 2.

When Does the System Switch from Case 1 to Case 2?

Taking our Case 2 equation (36) at t=10,, and using (38) to substitute for \nu_{10} :

\dfrac{e^{-1}}{C_{10}}=-\dfrac{8e^{-1}}{K_{10}}+c_3\qquad(39)

From the equation of motion (2), and since \dot K=0, in Case 2, we can substitute 2C_{10} for K_{10}:

\dfrac{e^{-1}}{C_{10}}=-\dfrac{8e^{-1}}{2C_{10}}+c_3\qquad(40)

c_3=\dfrac{e^{-1}}{C_{10}}+\dfrac{4e^{-1}}{C_{10}}=\dfrac{5e^{-1}}{C_{10}}\qquad(41)

Substituting for c_3 in (36):

\nu =-\dfrac{8e^{-0.1t}}{K}+\dfrac{5e^{-1}}{C_{10}}\qquad(42)

While the system is in Case 2, K, is constant, so we can replace it by K_{10} and therefore by 2C_{10}:

\nu =-\dfrac{8e^{-0.1t}}{2C_{10}}+\dfrac{5e^{-1}}{C_{10}}=\dfrac{5e^{-1}-4e^{-0.1t}}{C_{10}}\qquad(43)

Since Case 2, by definition, has \nu \neq 0,, and since from (38) \nu_{10}>0, the system will be in Case 2 while:

5e^{-1}-4e^{-0.1t}>0\qquad(44)

1.25>e^{1-0.1t}\qquad(45)

1-0.1t<\ln 1.25=0.223\qquad(46)

t>7.77\qquad(47)

So we can infer that the system is in Case 1 during t=[0,\;7.77] and in Case 2 during t=(7.77,\;10] .

Solving Case 1

Having found the time period over which Case 1 applies, we can now determine the constants c_1,c_2 in equations (29) to (31).  Taking (29) at t=0, we have:

100=c_1+c_2\qquad(48)

Since the system switches to Case 2 at t=7.77, with \dot K=0,, from (30) we have:

0=0.4c_1e^{0.4(7.77)}+0.5c_2e^{0.5(7.77)}\qquad(49)

0=0.4c_1+0.5c_2e^{0.777}\qquad(50)

\dfrac{c_2}{c_1}=-0.8e^{-0.777}=\dfrac{100-c_1}{c_1}\qquad(51)

-c_1(0.8e^{-0.777}-1)=100\qquad(52)

c_1=-\dfrac{100}{0.8e^{-0.777}-1}=158.2\qquad(53)

c_2=100-158.2=-58.2\qquad(54)

Substituting into (29 to (31), we have the time paths of the key variables over t = [0, 7.77] :

K=158.2e^{0.4t}-58.2e^{0.5t}\qquad(55)

\dot K=63.3e^{0.4t}-29.1e^{0.5t}\qquad(56)

C=15.8e^{0.4t}\qquad(57)

Although not essential to solve the problem, it may be of interest to note the time paths, over the same period, of the various multipliers.  From the first complementary slackness relation (17) and because C, can never be zero, we can infer that \mu = ,0, and from the definition of Case 1 we have \nu =0,.  Substituting these values into (13):

\dfrac{e^{-0.1t}}{C_t}-\lambda_t=0\qquad(58)

\lambda_t=\dfrac{e^{-0.1t}}{15.8e^{0.4t}}=0.0633e^{-0.5t}\qquad(59)

The value of \lambda , can be interpreted as the shadow price of the state variable, capital, that is, the amount by which W, could be increased if an extra unit of capital were available at time t,.  It can be seen that this value at t=0, is 0.0633,, which may seem surprisingly small given the extra consumption over the whole period which an extra unit of initial capital would make possible, but can be shown to be correct given that W, depends on the log of consumption.

Solving Case 2

A feature of Case 2 is that K,  remains constant.  To find at what level it remains constant, we have simply to find its level at t=7.77,, when Case 1 switches to Case 2.  Substituting into (55):

K=158.2e^{0.4(7.77)}-58.2e^{0.5(7.77)}=3540-2832=708\qquad(60)

This is the value of K,  over the period (7.77,\;10] , and enables us to confirm that K_{10}\geq 100  and therefore to accept the condition (18), \lambda_{10}=0, without qualification.  Over the same period, \dot K=0,  and:

C=0.5(708)=354\qquad(61)

Turning to the multipliers, \mu =0,  for the same reason as during Case 1.  Substituting for C,  in (43):

\nu =\dfrac{5e^{-1}-4e^{-0.1t}}{354}=0.0052-0.0113e^{-0.1t}\qquad(62)

Thus \nu ,  increases gradually from 0,  at t=7.77,  to 0.0010,  at t=10,.  The positive values of \nu , when t>7.77,  indicate that if the constraint C_t\leq 0.5K_t were relaxed then W,  could be increased.

To obtain \lambda .  over the same period, we use (61) and (62) to substitute for C,  and \nu ,  respectively in (13):

\dfrac{e^{-0.1t}}{354}-\lambda_t-0.0052-0.0113e^{-0.1t}=0\qquad(63)

\lambda_t=-0.0052+0.0141e^{-0.1t}\qquad(64)

Thus \lambda ,  falls from 0.0013,  at t=7.77,  to, as expected, 0,  at t=10,, at which point an extra unit of capital would have no effect within the time period on C,  or W,

Table 1 below shows the values of all the variables at integral time points over the whole period [0,\;10] , convering Cases 1 and 2.

The Optimal Value of W

It remains to check that the optimal paths we have now identified do indeed result in a larger W,  than our best so far – the 27.33,  obtained from our Pitfall 2.  Summing the relevant integrals over the Case 1 and Case 2 periods we have:

W=\int_0^{7.77}(\ln(15.82e^{0.4t}))e^{-0.1t}dt+\int_{7,77}^{10}(\ln(354))e^{-0.1t}dt\qquad(65)

W=\int_0^{7.77}(2.761+0.4t)e^{-0.1t}dt+\int_{7.77}^{10}5.869e^{-0.1t}dt\qquad(66)

W = \left[-(27.61+4t+40)e^{-0.1t}\right]_0^{7.77}+\left[58.69e^{-0.1t}\right]_{7.77}^{10}\qquad(67)

W=(-98.69e^{-0.777})-(-67.61)+(-58.69e^{-1})-(-58.69e^{-0.777})\qquad(68)

W=-45.38+67.61-21.59+26.98=\mathbf{27.62}>27.33

Reference

The main source used in preparing this post was:

Chiang, A (1999)  Elements of Dynamic Optimization  Waveland Press, Illinois

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Covid-19 and Household Size – An Update

An updated analysis of national data remains consistent with the hypothesis that Covid-19 infection rates are higher in larger households .

In a post in May 2020 I presented the results of a regression analysis on data from 14 Western European countries tending to support the following hypothesis:

Rates of Covid-19 infection will be higher, other things being equal, in larger households, that is, households with more occupants.

With Western Europe now well into a second wave of Covid-19 infection, it is timely to assess whether an updated analysis continues to support the hypothesis.

A reminder of some key features of the analysis:

  • Rates of death from Covid-19 are used as a proxy for rates of infection, actual rates of infection being difficult to measure.  Published statistics on confirmed infections are heavily dependent on differences in testing arrangements at different times and between countries.
  • Data used are at national level, with no allowance for variations within countries.
  • Estimation of the regression is by weighted least squares, with weighting by population.

The regression model is:

DP  =  C  +  (B x PH) + E

where:  DP is cumulative death rate from Covid-19 per million population; C is the regression constant; B is the slope coefficient; PH is average population per household; and E is the error term. 

The estimated regression line based on data to 3 December 2020 was:

DP  =  -2,004 + (1,209 x PH)

The precise values of the estimated coefficients are not important. Nor is it surprising that the slope coefficient is higher than estimated in May: this is to be expected since cumulative death rates have increased while average population per household is stable.  The important point is that, as in May, the estimated slope coefficient is positive, consistently with the hypothesis (and is sufficiently large that the null hypothesis that its true value is zero or less is rejected at the 5% significance level (1)). 

A spreadsheet containing the underlying data and full regression output may be downloaded here:

Note

  1. This can be inferred from the fact that the 95% confidence limits of the estimated slope coefficient are both positive.
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