A Difficulty in Assessing Sustainability

Long-term forecasting of the aggregate production function is essential for assessing sustainability, but very difficult.

Is our current living standard sustainable?  Or does depletion of non-renewable natural resources such as fossil fuels and metal ores mean that it must eventually decline?  And if the latter, what lower standard of living, if any, would be sustainable?  A well-founded sustainability criterion, enabling us to give clear and convincing answers to those questions, could make an important contribution to public debate on economic policy.  In particular, it could help counter the view that growth in GDP is the key indicator of economic performance.

Qualitatively, it is fairly clear how it might be possible to sustain consumption while non-renewable resources are progressively depleted.  We start from two reasonable assumptions:  that production technology will permit a considerable degree of substitution between inputs, and that there must always be at least some input of non-renewable resources. Inputs of non-renewable resources must therefore be ever-decreasing so that reserves are never totally exhausted (in mathematical terms, the stock of non-renewable resources must be asymptotic to zero).

To maintain output and consumption in those circumstances, there must be increasing inputs of reproducible capital (which includes both capital equipment and infrastructure and renewable natural resources).  Furthermore, it will not be sufficient simply to maintain output: output must grow to provide not only for consumption but also for the necessary investment in reproducible capital and for offsetting depreciation of the increasing stock of that capital.

Whether this scenario is feasible will depend on a number of variables: the initial stocks of capital and resources, resource extraction costs, production technology, population and labour inputs, current consumption and (if we are considering the economy of one country rather than of the whole world) opportunities for trade. There are difficulties of measurement and/or forecasting associated with several of these variables. This post will argue that, in particular, the forecasting of production technology presents a major challenge in assessing sustainability.

That might seem a surprising claim.  After all, economists have put much effort into the estimation of aggregate production functions and into economic forecasting using models in which a production function is one component (1).  The problem is that, to assess long-term sustainability, we need to estimate the relation of output to inputs for future periods in which the ratio of reproducible capital to input of non-renewable resources will be vastly greater than at present.  And that is very difficult.

To illustrate the point, I shall consider the assessment of sustainability within a model having the following simplifying assumptions:

  1. Output of a single good which can be either consumed or invested as reproducible capital.
  2. A Cobb-Douglas production function with technical progress Y = AKαRβ(1.01t) , where K is man-made capital, R is use of a non-renewable resource S, extracted at nil cost, t is time in years from the present, and A, α and β are parameters.
  3. Constant population and labour (this is why, although production requires labour, labour does not appear as a variable in the production function).
  4. Depreciation of reproducible capital at 5% per annum.
  5. Initial stocks: 100 units of K and 100 units of S (the respective units need not be the same).

The assumption of technical progress at a constant 1% per annum, implicit in 2 above, is made to facilitate a focus on the rest of the production function and keep the mathematics of the model reasonably simple.  The actual rate of future technical progress is difficult to forecast, but that is a reinforcement, not a criticism, of the argument presented here.

The question I shall explore is whether, within this model, annual consumption of 100 units (the same units as those of K) is sustainable.  This will depend on the parameters of the production function.  Consider the following functions:

F1:  Y = 15K0.3R0.2(1.01t)

F2: Y = 24K0.3R0.1(1.01t)

This pair of functions has the following property: if K approximates to 100 and K/R to one, then they yield very similar values of Y (because 15 x (1000.1) ≈ 24).  This is illustrated, for the case t = 0, by Chart 1 below.

Suppose we had to decide whether the production function is F1 or F2 on the basis of input and output data for recent periods in which K/R is in the vicinity of one.  Given the inevitable random variation in output due to other variables, this would be extremely challenging. Statistical tests would at best point to one function as slightly more likely than the other.

At the very much higher K/R ratios that sustainability requires, however, the difference between these functions becomes very significant, as shown in Chart 2 below (note the log scale on the horizontal axis).

Because the same inputs yield more output with F2 than with F1 whenever K/R exceeds one, we expect that sustainable consumption will be greater given F2. To find out how much difference this makes I set up the model in discrete form as a spreadsheet with one row per year for 5,000 years.  Key features of the spreadsheet are as below:

  • Consumption in each year equals that for year 0.
  • Growth of capital equals output less consumption less depreciation.
  • Marginal product of capital is calculated directly using the standard formula (for a Cobb-Douglas production function) MPK = αY/K.
  • Marginal product of the resource is calculated directly (MPR = βY/R) for year 0, but subsequently using the Hotelling rule according to which the rate of growth of the marginal product of the resource equals the marginal product of capital (2).
  • Extraction and use of the resource after year 0 is calculated backwards from its marginal product and output (to avoid circularity, the output figure used is that for the previous year).
  • Resource stock after year 0 is that for the previous year less extraction and use for the previous year.
  • Consumption and extraction / use of the resource for year 0 are trial values. A pair of trial values is considered feasible if it generates time paths for the variables in which, during years 0 to 5,000, the resource is not exhausted and output is never less than consumption.

The maximum trial value of consumption consistent with feasibility as defined above is an approximation to maximum sustainable consumption (it’s only an approximation because of the discrete spreadsheet approach and because of the 5,000 year time horizon).  It was found that:

  • With production function F1, constant consumption of 100 units is unsustainable as maximum sustainable consumption is approximately 79.7 units.
  • With production function F2, constant consumption of 100 units is sustainable as maximum sustainable consumption is approximately 117.6 units.

So the assessment of whether consumption of 100 units is sustainable rests on the weak foundation of whether F1 or F2 in Chart 1 above provide a better statistical fit to current and recent data.

A possible objection is that the implicit sustainability criterion, that is, the feasibility of consumption at 100 units per annum forever, or for 5,000 years, is too demanding.  Suppose instead, therefore, that we set the time horizon at just 100 years.  Surprisingly, perhaps, this makes very little difference to the conclusion.  For F1, maximum sustainable consumption increases from 79.7 to 80.0 units.  For F2, it increases by such a tiny amount that on rounding to one decimal place, maximum sustainable consumption is still 117.6 units.

The spreadsheet used to derive the above charts and results may be downloaded here:

Constant Consumption with Different Production Functions Adam Bailey

Notes and References

  1. One example is the NIESR’s NiGEM model: see https://nimodel.niesr.ac.uk/nigem-intro/nigemintro.php?t=1&b=1&w=1
  2. A derivation of this form of the Hotelling rule may be found in Perman R, Ma Y, McGilvray J & Common M (3rd edn 2003) Natural Resource and Environmental Economics Pearson Addison Wesley  pp 660-1, the rule being equation 19.42j.
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A Valuation Case Study: the Great Barrier Reef

A valuation of the Great Barrier Reef illustrates many common issues in the economics of environmental valuation.

A recent report by Deloitte Access Economics (1) estimates the asset value of the Great Barrier Reef at A$56 billion (2).  This is the sum of:

  • Value to local recreational users and tourists from elsewhere in Australia: A$32 billion, estimated by the travel cost method (pp 39 & 40).
  • Value to Australians of the Reef’s existence, regardless of having visited it or intending to do so; A$24 billion, estimated by contingent valuation (p 34).

The main data source was a representative survey of over 1,000 Australians (p 30). The survey also included 500 residents of 10 other countries world-wide.

The report also identifies the following components of value for which it does not offer numerical estimates:

  • Value to tourists from outside Australia: not estimated due to data limitations (p 85).
  • Existence value to non-Australians: not estimated as difficult to allow for contextual cultural factors, language barriers and purchasing power differences in using survey responses from some countries (p 35).
  • The value of certain ecosystem services provided by the Reef such as maintenance of water quality and storm protection for the adjacent coast (pp 40 & 42): not estimated as difficult to separate from ecosystem services provided by neighbouring terrestrial and river ecosystems (p 84).
  • “Traditional owner value” to peoples who have lived in the vicinity of the Reef since before Australia was settled by Europeans, relating to cultural heritage, sacred sites and archaeological sites: not estimated because such unique sites lack the substitutability on which non-market valuation relies (pp 45-7).
  • Brand value, that is, the contribution of the Reef to “Brand Australia”. This is given a whole chapter of the report (pp 50-7) which is not easy to summarise.  One key point is how far the contribution is from the existence of the Reef and how far from perceptions of Australia’s performance as guardian of the Reef (p 56).

Taking the above together the report suggests, reasonably, that the full value of the Reef is much more than A$56 billion.  Some, though not all, of the points below tend to reinforce that conclusion.

There are other components of value which the report does not mention.  One is the possible medicinal use of plants and animals found in the Reef (3).  A negative component – illustrating the general point that natural capital is often associated with natural hazards  –  is its role as a shipping obstacle and hazard (4).

As is common with environmental valuation studies, therefore, the components of total economic value fall into three categories: those that can be measured with the available data; those that are in principle measurable but for which the necessary data is difficult to obtain; and those that are intrinsically unmeasurable (but still perhaps important).

As a contingent valuation sceptic (see this post), I would have been inclined to put the existence value of the Reef in the unmeasurable category.  A specific difficulty with the approach adopted is that respondents were asked how much they would be willing to pay weekly to “guarantee” that the Reef is “protected”, having been told that such a payment would be in the context that all Australians would have to pay (p 80). This is a strange question given that global climate change is a major threat to the Reef.  It seems quite possible that some respondents viewed the payments as to guarantee that the Australian government would do what it could to protect the Reef (eg from risks originating within Australia such as mining activity), while others viewed them as guaranteeing the preservation of the Reef, implying protection against all risks including climate change.

The remainder of this post considers the method used to estimate the value of the Reef to tourists from within Australia. The individual version of the travel cost method was used, given that individual data was available and showed good variation between individuals in visit numbers (p 78), so that it could be expected to give more precise results than the alternative zonal method (5).  The variation between individuals was not just due to luck: it was facilitated by the decision to ask respondents how many times they had visited the Reef in the last five years (not merely the last year).

With the individual method, the first step is to regress individual visit numbers against individual travel costs to the Reef and other variables that may influence visit numbers.  The report does not state the full regression model, but from the survey questions (pp 76-82) it appears that the other variables included age, gender and education.  Variables that seem not to have been included are income and travel costs to one or more substitute sites.  Although income is mentioned in discussion of the method (p 85), there is no indication that income data was collected.  This may have been because it was anticipated that such data would be difficult to obtain (US studies often collect income data but it may be that willingness to disclose one’s income varies between cultures).  The issue of substitute sites may have been ignored because of the complexity of allowing for many possible substitutes, and lack of consensus in the literature as to how this should be handled.  Whatever the reasons, omission of these variables can result in omitted variable bias leading to a biased estimate of the travel cost coefficient and hence of the site value, although the magnitude and direction of bias will depend on the circumstances (6).

Mention of substitute sites may seem surprising given the unique nature of the Great Barrier Reef.  But a substitute, in economics, does not have to be a perfect substitute.  One definition is that two goods are substitutes if their cross-elasticities of demand are positive.  Applied to two tourist sites, that would be the case if a higher travel cost to either were associated with a higher demand for visits to the other – a plausible scenario.

The decision effectively to ignore travel time (p 85) is surprising. Common practice is to include the value of time as well as expenditure on transport and accommodation within travel cost to the study site.  The report cites three Australian studies which applied a zero value to travel time, but this is rather selective even among Australian studies (7), and certainly not representative of the global literature including US and UK studies.  Admittedly there is no consensus on how the value of time should be determined.  But to assume a nil value is likely to result (other things being equal) in under-estimation of the value of the Reef.  A fairly simple and still conservative alternative would have been to value time at a suitably small fraction (say a quarter) of the average hourly wage, and to include the effects of different fractions in the sensitivity analysis.

The statistical methods used to estimate the trip-generating function and then derive the value (consumer surplus) per visit (A$662) are not described in full, but appear from the outline provided (pp 85-6) to have been appropriate.  As is common in individual travel cost studies, negative binomial regression was used because of overdispersion (p 86), which is a slightly misleading term for a feature (not a defect) often found in the distribution of individual visit numbers, namely that the variance is greater than the mean.

I would have liked to see some consideration of what potential visitors might have done instead if the Reef had not existed.  Many would probably have visited other tourist sites, increasing the value of those sites and offsetting to some degree the lost value associated with the Reef.  A case can be made that the most useful concept of value for a recreational site is not the gross site value but the contribution which the site makes to the total value of all such sites (8).

To obtain the total annual value to tourists, value per visit was multiplied by the annual number of visits to the Reef, which was estimated as 2.3 million (p 86).  This number was inferred from Tourism Research Australia (TRA) data on numbers of visits within Australia to regions adjacent to the Reef.  However, these numbers include visits to the regions but not to the Reef itself, and some crude round-number assumptions, claimed to be conservative, were made to eliminate such visits (p 86).  An alternative approach would have been to include in the survey a question designed to distinguish visits to the Reef itself from visits to the adjacent regions, and to use that data to estimate, for all visitors to those regions, the proportion who visit the Reef itself.

The final step was to capitalise the stream of annual values so as to obtain the asset value.  The result is heavily influenced by the choice of time horizon and discount rate, as is illustrated by the sensitivity analysis (p 88).  A time horizon of 33 years was adopted, one stated reason being the severe threats to the future health of the Reef (pp 87-8).  Within that time frame, annual consumer surpluses as calculated from the survey data were discounted at a rate of 3.7%, determined using the Ramsey formula (p 87):

Social discount rate  =  Rate of time preference + [Annual growth rate of consumption

                        x Minus the elasticity of marginal utility with respect to consumption]

Given assumptions of a very low rate of time preference (0.05%) and an elasticity of 1, the discount rate largely reflects the growth rate of consumption, which was assumed to equal the average GDP growth rate over the previous 30 years (which can be inferred to have been 3.65%).

While the particular figures used in the capitalisation could be challenged, there are some more fundamental issues here.  One could value the Reef on the assumption that it will degrade along a defined path, or on the basis of its remaining in its current condition.  The key question here is not which of these scenarios is more likely, but which basis of valuation will yield more useful information.  If what we are interested in is the value which is potentially at risk from degradation of the Reef, then it is valuation on  the ‘current condition’ basis which is more useful, and the ‘degradation’ argument for the 33-year time horizon does not apply.

Whether future economic growth is likely to be at a similar rate to that of the previous 30 years can be debated.  My instinct would be to calculate the central estimate of value on a zero-growth basis, and to consider the effect of positive rates within the sensitivity analysis.  If, however, a positive growth rate is assumed, then for consistency it needs to be considered that, since expenditure on tourism is discretionary, demand for tourism is likely to be highly income-elastic.  Subject to the condition of the Reef,  and to the adequacy of infrastructure to accommodate visitors, demand for visits might be expected to grow even faster than GDP.  The report, however, does not consider whether the number of visits to the Reef may change in future.  The implicit assumption is that annual visits and annual consumer surpluses will remain as estimated (9).

Notice that the combination of a very low rate of time preference and a zero rate of economic growth will yield, via the Ramsey formula, a very low discount rate.  In conjunction with a long time horizon this could imply an asset value many times higher than the report’s A$56 billion.

Notes and References

  1. Deloitte Access Economics (2017) At what price? The economic, social and icon value of the Great Barrier Reef  https://www2.deloitte.com/content/dam/Deloitte/au/Documents/Economics/deloitte-au-economics-great-barrier-reef-230617.pdf   All page references are to this report.
  2. A$ = Australian dollars. A$56 billion is equivalent to GB£33 billion or US$43 billion.
  3. Bruckner A (2002) Life-Saving Products from Coral Reefs Issues in Science and Technology XVIII(3) http://issues.org/18-3/p_bruckner/
  4. See for example The Guardian (6/4/2010) The Great Barrier Reef scandal https://www.theguardian.com/world/2010/apr/06/great-barrier-reef-ship-aground
  5. King D M. & Mazzotta M Ecosystem Valuation – Options for Applying the Travel Cost Method  http://www.ecosystemvaluation.org/travel_costs.htm#OPTIONS
  6. Re omission of substitute site variables see Caulkins P, Bishop R & Bouwes N (1985) Omitted Cross-Price Variable Biases in the Linear Travel Cost Model: Correcting Common Misperceptions Land Economics 61(2) pp 182-7
  7. Two Australian studies which applied positive values to travel time are: a) Lansdell N & Gangadharan (2003) Comparing Travel Cost Models and the Precision of their Consumer Surplus Estimates: Albert Park and Maroondah Reservoir Australian Economic Papers 42(4) p 403  b) Whitten S & Bennett J (2001) A travel cost study of duck shooting in the Upper South East of South Australia  Private and Social Values of Wetlands Research Reports No. 7  https://crawford.anu.edu.au/pdf/staff/jeff_bennett/wtlndrr07.pdf
  8. This point is briefly noted in Bateman I (1993) Valuation of the environment, methods and techniques: revealed preference methods, in Turner R (ed) Sustainable Environmental Economics and Management: Principles and Practice Belhaven Press, London  pp 218-9
  9. This is shown by the fact that a stream of consumer surpluses of A$662 x 2.3 million = A$1.523 billion annually for 33 years, discounted at 3.7% per annum, approximates to the A$29 billion stated on p 39.
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Energy and Environment in China

Arthur Kroeber’s book on China’s economy includes an excellent section on energy but a rather selective account of its environmental issues.

Arthur Kroeber’s book China’s Economy in Oxford University Press’s What Everyone Needs to Know series (1) deserves a wide readership.  Admittedly it’s rather dry: those who like their reading on serious and important topics to be spiced with anecdotes or cultural references had better look elsewhere.  But for the general reader (in English) who wants to understand China’s development and possible future – to go beyond journalistic impressionism and simplistic or politically-motivated judgment – I doubt whether there is anything better.  This is a properly researched book at a well-chosen level, a serious piece of description and analysis which avoids over-technicality.  The chapter on ‘Changing the Growth Model’, for example, makes extensive use of key ratios such as the capital-output ratio while avoiding the complexities of, say, total factor productivity.  The tables and charts are helpful and not overdone. Also helpful is the division of each chapter into sections headed by questions.  Sensitive issues such as corruption and possible exchange rate manipulation are treated in a fair-minded and temperate manner.

A full review would be beyond the scope of this blog, but I offer here some comments on Chapter 8 entitled Energy and the Environment.

Starting with energy, I commend the book for using consistent, well-defined and sensible units (p 150).  Writings on energy often confuse matters by switching between different units, using vague units (“enough to power a million homes”) or, worst of all, failing to distinguish between units of energy and units of power (2).  By contrast, Kroeber sets out very clearly the main facts of China’s energy use. The total in 2014 was 22 billion barrels of oil equivalent, as compared with 17 for the US, 12 for the European Union and 95 for the whole world. Of China’s 22, 14 are from coal (which is about half of world coal consumption) and another 5 from other fossil fuels, underlining the importance of China in worldwide efforts to address climate change by limiting greenhouse gas emissions.

Kroeber also presents two important per unit measures of energy use.  As might be expected given China’s huge population, its per capita energy use is not especially high: much less than the US, less than the EU, and only slightly above the world average. More surprisingly, perhaps, China’s energy use per unit of output (GDP), also termed energy intensity, is more than twice that of the US and the EU, and almost twice the world average.  This is despite considerable improvements in energy intensity already achieved, eg a 19% improvement during 2005-2010 after the government had set energy efficiency targets for large firms in heavy industry (p 160).

China’s high energy intensity calls for explanation, and Kroeber identifies three causes (pp 150-2 & 161).  One is the structure of its economy, with industry accounting for a high proportion of output, agriculture smaller proportionally than in many poorer countries, and services as yet smaller proportionally than in most developed countries.  Because industry is more energy-intensive than agriculture or services, and because a high proportion of China’s industry consists of especially energy-intensive heavy industry such as steel and cement manufacturing supporting its housing and infrastructure boom, its overall energy-intensity is high.  A second cause is China’s unusually high reliance on coal, which is a less efficient energy source than natural gas for generating electricity.  This is a consequence of the geographical accident that it has large reserves of coal but much less oil and gas.  The third cause relates to the efficiency with which China uses its energy sources. Here the picture is mixed.  Many of China’s coal-fired power stations have been built relatively recently to modern standards, and are somewhat more efficient than older power stations in the US.  Its fuel efficiency standards for vehicles compare reasonably well with those in developed countries. However, the energy efficiency of homes and offices is often poor, and many old, unprofitable and energy-intensive industrial plants have been kept open by local governments seeking to maintain employment and tax revenues.  Energy prices, though not especially low by international standards, are subject to controls which can reduce incentives to make energy-saving investments.

Kroeber does not attempt to quantify the overall effect of these causes, but it does seem plausible that together they go a long way towards explaining China’s high energy intensity.  A point he might have added is that the annual temperature range in much of China is such that homes need both heating in winter and air-conditioning in summer.

Like many countries, China has sought to diversify its energy sources in order to reduce its reliance on coal which is both a major source of local air pollution and a major contributor to global greenhouse gas emissions (pp 152-4).  It has become a major oil importer (although the IEA’s statistics do not seem to support Kroeber’s claim that in 2013 it became the world’s largest (3)).  It also imports natural gas.  Over the last decade, China has more than doubled its output of nuclear power and hydropower, and increased its output of electricity from renewables almost twentyfold.  However, the effect of all this in reducing coal’s share of China’s energy mix has been relatively small. In absolute terms coal consumption has continued to grow (from which it may be inferred that China’s overall energy use has also been growing).  Its coal consumption may now be close to peaking, although Kroeber advises caution on this point, both because of the short-term effect of macroeconomic fluctuations on energy demand, and because of possible under-reporting of output by smaller coal mines.

Kroeber states, correctly, that China produces over 90% of the coal it uses, and that its coal imports are only a modest proportion of its total use.  It might be added that in the context of world trade in coal, China is nevertheless a major player, and was the largest importer in 2014 (4).  Because its imports are the difference between two huge numbers (its demand and its domestic production), there is considerable potential for fluctuations in its imports to have a major impact on the pattern of trade in coal.

Because of its huge energy consumption and reliance on fossil fuels, China is the world’s largest emitter of greenhouse gases, accounting in 2012 for 24% of the global total.  Kroeber considers, but seems to me to do less than justice to, the fact that some of China’s emissions relate to the production of goods for export, and arguably should be attributed to the importing countries in any international apportionment of responsibility for climate change (pp 154-5).  I cannot see why he links the issue to that of multinational companies moving their production to China, as if exports produced by Chinese companies are irrelevant in this context.  He also states that most of China’s emissions relate to heavy industries supporting domestic construction and not to export industries.  It would have been useful to have quantified or given a source for this claim, and to have noted that domestic construction includes construction of factories producing goods for export and transport links to carry such goods.

Turning to other environmental issues, the book focuses mainly on the much-publicised issue of air pollution, treating other issues only indirectly via an environmental performance index.  Understandably perhaps given the broad scope of the book, it says little about soil and water other than noting their “extreme degradation” due to industrialisation (p 155).  A fuller treatment would have considered each of the following, and efforts to address them: soil erosion (5), soil pollution (6), reduced river flows (7), depletion of groundwater (8), and water pollution (9).  These are not minor or merely local issues.  Unless effectively addressed, they have the potential to constrain China’s food production and so increase its demand for food imports with impacts on world food prices (10); and the costs of addressing them are likely to divert significant resources from elsewhere in the economy.

As in other countries, air pollution in China includes both gases – notably sulphur dioxide – and particulates of various sizes.  Over 50% is attributable to burning of coal, 15-20% to vehicle emissions, and the remainder to other sources (pp 159 & 161).  Although Kroeber seems to suggest that the problem is most serious in Beijing and other northern cities (pp 155 & 161), he does not offer a systematic description of the geographical pattern of air pollution.  If, as seems likely, the air is cleaner in much of the countryside, then that surely needs to be taken into account in any assessment of rural-urban inequality (a topic discussed by Kroeber elsewhere in the book (pp 30-5))?

China has made some progress in addressing air pollution in that emissions of sulphur dioxide have been reduced, although concentrations of small particulates have continued to rise (p 161).  What progress there has been seems to have been achieved largely via improvements in energy efficiency and some diversification away from coal as described above.

Kroeber rightly rejects the idea that China’s environmental problems are “uniquely attributable” to its growth model or political system, pointing out that Japan, the UK and the US all experienced severe air pollution in the mid-twentieth century (p 156).  He argues however that its problems are particularly severe for a country at its stage of development.  As evidence for this he presents a version of the Environmental Kuznets Curve (a formulation of the tendency for countries to give a higher priority to environmental issues as they become richer), plotting scores on Yale University’s Environmental Performance Index (EPI) (11) against gross national income for 30 of the world’s most important countries (p 157).  This shows a fairly clear relation between EPI and income, albeit with, as is to be expected, some spread of points about the line of best fit.  China’s EPI  score is some 14% less than might be predicted from its income level.

Kroeber suggests that this can be explained in terms of China’s political system, its ‘East Asian’ approach to development, with an unusually high premium on maximising economic growth, and its aspiration to be a superpower (pp 157-8).  This seems questionable.  A possible alternative explanation starts from the fact that environmental improvement is usually a gradual process. This is for various reasons: some pollutants have a finite life over which they gradually degrade; fish stocks take time to recover from a pollution incident; newly planted trees take many years to mature; and so on.  When a country initiates the sort of environmental improvements typical of its income level, therefore, it is likely to take some years for the full benefit to be realised.  If the country’s economy has grown rapidly, as China’s has, then this time lag may result in a lower EPI score than that of another country which has a similar income level but has reached that level more gradually.  If for example countries A and B have similar income levels but A’s economy has grown annually at 8%  and B’s at 1%, then an average time lag of about 2 years would be sufficient to give A a score 14% below B.

Looking to the future, addressing air pollution is now a stated priority of the Chinese government (p 159).  The main policy instruments likely to be used are stricter environment laws and stricter enforcement.  Other approaches used in western countries, such as emissions trading schemes and class-action lawsuits against polluting companies, Kroeber suggests, are unlikely to be successful in the Chinese context (p 158).  On the other hand, the fact that a high proportion of emissions are from a small number of heavy industries may make the problem easier to address, especially, it might be added, as some of the companies in those industries are state-owned (p 100).  At any rate, Kroeber is optimistic that the next few years will see accelerated progress against air pollution.

Notes and References

  1. Kroeber, A R (2016) China’s Economy: What Everyone Needs to Know Oxford University Press.  Page references in the text are to this book.
  2. Difference Between Energy and Power http://www.differencebetween.net/science/difference-between-energy-and-power/
  3. International Energy Agency Key World Energy Statistics 2015  p 11 ftp://ftp.energia.bme.hu/pub/energetikai_alapismeretek/KeyWorld_Statistics_2015.pdf
  4. International Energy Agency, as 3 above, p 15
  5. Xinhuanet (15/3/2017) Central China Province to Spend 2 Billion Yuan on Erosion Control http://news.xinhuanet.com/english/2017-03/15/c_136131474.htm
  6. Xinhuanet (18/1/2017) China Sets Up Lifelong Accountability System to Control Soil Pollution http://news.xinhuanet.com/english/2017-01/18/c_135994290.htm
  7. Earth Observatory Yellow River Delta https://earthobservatory.nasa.gov/Features/WorldOfChange/yellow_river.php
  8. Qiu J (13/7/2010) China Faces Up to Groundwater Crisis Nature News  http://www.nature.com/news/2010/100713/full/466308a.html
  9. Xinhuanet (22/4/2014) China’s Underground Water Quality Worsens: Report http://news.xinhuanet.com/english/china/2014-04/22/c_126421022.htm
  10. OECD-FAO (2013) Agricultural Outlook 2013-2022 Chapter 2 Feeding China: Prospects and Challenges in the Next Decade  See especially Risks and Uncertainties pp 83-7  http://www.oecd.org/berlin/OECD-FAO%20Highlights_FINAL_with_Covers%20(3).pdf
  11. Yale University Environmental Performance Index  http://epi.yale.edu/
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Net National Product and Sustainability

National product, measured net of a deduction for depletion of natural resources, can in certain conditions provide some indication of whether current consumption is sustainable.  But the conditions are stringent, and even when they are met, other indicators may perform better.

When gross national product (GNP) and related economic aggregates were first developed by Kuznets and others in the 1930’s and 1940’s, there was debate as to whether the aim should be to measure activity and output, or welfare and well-being.  Against a background of mass unemployment and then World War II, the debate was won by those who wanted the former (1).  To this day, GNP as calculated in most countries remains a measure of activity and output, and (as many critics have pointed out) it is easy to find examples of activities which raise GNP but do not enhance and may even lower welfare.

It has always been recognised that net national product (NNP), which equals GNP less an allowance for depreciation of capital assets due to wear and tear, is in some ways a more meaningful measure, and most countries publish estimates of NNP as well as GNP.  Nevertheless, most economic discussion focuses on gross aggregates, including GDP (gross domestic product, which is similar to GNP but excludes certain international income flows).  This seems to be partly because of the short-term link between activity and employment, and partly because of difficulties – both conceptual and practical – in measuring depreciation (2).

In the 1970’s, growing interest in environmentalism and concerns regarding resource depletion (3) led some economists to explore long-term macroeconomic models in which the essential inputs to production include a non-renewable natural resource.  This led to the idea that NNP might be adapted – by including a suitable deduction for resource depletion –  to provide an indicator of sustainability.  Much of the academic literature on this topic stems from a paper published by Weitzman in 1976 (4).  The paper was also  an important influence on Nature’s Numbers (1999), a report commissioned by the US government on expanding the US national accounts to “include the environment” (5).

To understand what Weitzman did, we need some definitions.  Given a long-term model including assumptions about the rates of investment in man-made capital and of extraction and use of a non-renewable resource, together with initial quantities of capital and the resource, we can infer the time paths of the variables, including the rate of consumption.  Generally the rate of consumption will vary over time. Given also a discount rate, we can find the present value of the implied stream of consumption.   We can also find the shadow price of an input by finding how much that present value increases if the initial quantity of the input is increased by one unit.

Whatever the present value of the consumption stream may be, there must exist a rate of consumption which, if maintained constant forever, has the same present value.  Weitzman calls this the stationary equivalent of future consumption (others have called it, more conveniently, constant-equivalent consumption).  Finally, by properly calculated NNP we mean consumption plus or minus adjustments for any change in man-made capital and any depletion of the resource, valued at their respective shadow prices.

We can now state Weitzman’s main conclusion as follows: if the present value of consumption is optimised (by suitable choice of rates of investment in capital and of use of the resource), then (subject to some technical assumptions) properly calculated NNP will equal the stationary equivalent of future consumption (6). I shall refer to this as Weitzman’s equality (Nature’s Numbers calls it the output-sustainability correspondence principle).

How exactly does this relate to sustainability, taken here to mean the possibility of maintaining consumption indefinitely at a given rate?  Constant-equivalent consumption, after all, is merely a mathematical construct: it cannot be assumed (and Weitzman did not claim) that constant consumption at that rate is feasible within the parameters of the model.  Moreover, it is a construct that depends on the discount rate, whereas the feasibility of constant consumption at a given rate will depend on the production function and initial quantities of inputs, but should have nothing to do with the discount rate.

The link can be made as follows. For a given model and given initial values, let OC(r) be the feasible consumption stream with optimal present value PVOC(r) at discount rate r.  Let CE(r) be constant-equivalent consumption with present value PVCE(r) given r. Let NNP(r) be properly calculated initial NNP for the optimal scenario, using shadow prices consistent with r. Lastly, let CC* be the maximum feasible rate of constant consumption, and PVCC*(r) its present value given r. Then, from the definition of constant-equivalent consumption, we have PVOC(r) = PVCE(r).  Since OC(r) is optimal given r, we must have PVCC*(r) ≤ PVOC(r) and therefore PVCC*(r) ≤ PVCE(r). Because CC* and CE are both constant rates, we can infer that CC* ≤ CE(r). Assuming Weitzman’s equality, this implies CC* ≤ NNP(r).

Importantly, the argument does not depend on the value of r.  If correct, therefore, it implies the following partial sustainability indicator (to be understood in the context of a model as outlined above):

Sustainability Indicator 1

Take a selection of discount rates  and find properly calculated NNP consistent with the optimal consumption time path at each rate.  If a putative rate of constant consumption CC* exceeds NNP at any one of these discount rates, then it is not sustainable forever.  But if CC* is less than NNP at all of the discount rates, then it may be sustainable.

It is a merit of this indicator that it does not rely on a single discount rate. Thus it avoids the need to address the vexed question of what is the appropriate discount rate, if any, to apply to the welfare of future generations.

An important limitation however is that its application requires identification of optimal time paths, not just of consumption but also of capital and the resource, in order to obtain the correct shadow prices and properly calculate NNP.  There is no basis here for the tempting thought that sustainability might be assessed from conventional NNP less a deduction for actual depletion of non-renewable resources valued at their market prices.

To assess the reliability of this indicator, and (consistently with my interest in the replicability of scientific research as discussed here) to explore the conditions within which Weitzman’s equality is valid, I set up a long-term model in spreadsheet form with one row per year.  This implies a discrete approach, with some ad hoc devices to avoid circular dependencies, and therefore with results only approximating to those of a continuous time model.  It has the potential however to highlight ‘awkward’ cases which may not fit the assumptions (eg of smoothly differentiable curves) on which continuous models sometimes rely.

The assumptions of my model were:

  1. Output of a single good which can be either consumed or invested as man-made capital.
  2. A Cobb-Douglas production function Y = K0.3R0.1, where K is man-made capital and R is use of a non-renewable resource S, extracted at nil cost (reasons for these particular parameters are given in this post).
  3. Constant population, labour and technology.
  4. No depreciation of man-made capital.
  5. An exogenous discount rate, unrelated (given no assumption of a competitive economy) to the marginal product of capital.
  6. Initial stocks: 100 units of K and 100 units of S (the respective units need not be the same).

The model is admittedly a gross simplification of any real economy: the point is that if the indicator should not work well under what might be considered ideal conditions, then it would hardly be likely to work well in application to a real economy.

Optimal scenarios were identified for six different discount rates, the largest being 4% and the smallest 0.5%.  Although in principle the time horizon was infinity, the time paths of the variables were calculated for 5,000 years, the present value of consumption beyond that date even at 0.5% being insignificant.  Optimal time paths were found by judicious trial and error in respect of use of the resource in the first period and allocation of output between consumption and investment, together with application of the Hotelling rule (an intertemporal efficiency condition) for use of the resource after the first period.

To find the initial shadow price of capital, the optimal time paths were also found on the assumption of one extra unit of initial capital (ie 101 units of K and 100 units of S).  The shadow price (in terms of the present value of consumption as numeraire) was then calculated as the difference between the optimal present value of consumption given 101 units of K and that given 100 units.  The initial shadow price of the resource was found similarly.

The maximum feasible rate of constant consumption was calculated using a formula (for the Cobb-Douglas case) found by Solow (7) and restated in a slightly simpler form by Buchholz, Dasgupta & Mitra (8).

The results are set out in Table 1 below.

From now on I take the words “properly calculated” as read. It can be seen from Table 1 that NNP at each discount rate exceeds maximum constant consumption.  Thus the results are consistent with Sustainability Indicator 1.  However, comparison of NNP with constant-equivalent consumption shows Weitzman’s equality holding only at 1.1% and higher rates.

Why does Weitzman’s equality not hold at all discount rates?  The reason, in simple terms, is that the proof in his paper assumes that the time paths of the variables are smooth (differentiable) curves (9).  This is a valid assumption when the discount rate is sufficiently high, in which case there is nothing to be gained by investment of any part of output. The optimal scenario then has constant capital and consumption of all output throughout, resulting in smooth time paths of all variables.  At lower discount rates, however, investment of the whole of output is found to be worthwhile for a finite initial period, and then the optimal time path of consumption switches abruptly to zero investment, with consumption of the whole of output.  In the jargon of dynamic optimisation, this is known as a bang-bang solution, and what makes it possible is that the problem of maximising the present value of consumption subject to the constraints of the model leads to a Hamiltonian which is linear in consumption (10).  In my discrete approach, this takes the form (as the allocation of output row in Table 1 shows) of a number of years with all output invested, then one transitional year with part of output invested, and then all subsequent years with all output consumed.  At low discount rates, therefore, there is a time at which the path of consumption and consequently of some other variables is not smooth.

The optimal consumption path takes the bang-bang form when the initial shadow price of capital exceeds one, implying that, at the margin, investment of output will contribute more than immediate consumption to the present value of the consumption stream.  As Table 1 shows, that point is reached when the discount rate is between 1% and 1.1% (with different assumptions it might be reached at some other rate).

One other feature of the bang-bang solution should be noted.  It was stated above that use of the resource after the first period was obtained via the Hotelling rule.  When no investment is taking place, so that capital is constant, the effect of spreading use of the resource between years is to spread output and hence consumption between years, so the required version of the rule is that the marginal product of the resource should grow at the discount rate.  When however the whole of output is being invested, the effect of spreading use of the resource is to spread investment between years, the requirement then being that the marginal product of the resource should grow at a rate equal to the marginal product of capital.  My spreadsheet was designed to use, in each year, the appropriate one of these two versions of the Hotelling rule.

Although the above results are consistent with Sustainability Indicator 1, they suggest that it could be improved by making use of the apparent implication that NNP consistent with an optimal consumption time path will be minimised when the discount rate is such that the shadow price of capital is one.  But we can do better than that.  Since the lowest constant-equivalent consumption (3.61 at 0.5%) is less than the lowest NNP (3.66 at 1.1%), it would be better still to ignore NNP and refer directly to constant-equivalent consumption (which is also easier to find as it does not require the shadow prices).  A possible formulation is:

Sustainability Indicator 2

Select a low discount rate, eg 0.5%, and find constant-equivalent consumption (CE) for the optimal consumption time path at that rate.  If a putative rate of constant consumption CC* exceeds CE, then it is not sustainable forever.  But if CC* is less than CE, then it may be sustainable.

For my model this works quite well, in that the difference between constant-equivalent consumption (3.61) and maximum constant consumption (3.49) is fairly small.  But further work would be needed to explore whether it would work well in a wide range of circumstances.  And importantly, it does not avoid the need to identify the optimal consumption time path for the discount rate.

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

NNP & Sustainability Spreadsheet Adam Bailey

Notes and References

  1. Coyle, D (revised and expanded edition 2014) GDP: A Brief but Affectionate History Princeton University Press  pp 12-16
  2. OECD (second edition 2009) Measuring Capital: OECD Manual Ch 5 pp 43-51  https://www.oecd.org/std/productivity-stats/43734711.pdf
  3. See for example Meadows D H et al (1972) The Limits to Growth Universe Books  https://www.clubofrome.org/report/the-limits-to-growth/
  4. Weitzman M L (1976) On the Welfare Significance of National Product in a Dynamic Economy  The Quarterly Journal of Economics  90(1) pp 156-162
  5. Nordhaus W D & Kokkelenberg E C (eds) (1999) Nature’s Numbers: Expanding the National Economic Accounts to Include the Environment p 188  https://www.nap.edu/catalog/6374/natures-numbers-expanding-the-national-economic-accounts-to-include-the
  6. Weitzman, as 4 above, p 160.
  7. Solow R M (1974) Intergenerational Equity and Exhaustible Resources The Review of Economic Studies  Vol 41 p 39
  8. Buchholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model  Scandinavian Journal of Economics  107(3) p 553
  9. Weitzman, as 4 above, p 157, which states assumptions about the existence of certain partial differentials.
  10. Wikipedia – Bang-bang control https://en.wikipedia.org/wiki/Bang%E2%80%93bang_control
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