Pollution Control and Output

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

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

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

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

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


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


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


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


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


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

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

A couple of consequences of the above are worth stating:

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

Notes and References

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

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

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

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

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

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

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

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

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

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

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

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

For the resource:

\dfrac{dS}{dt} = -R

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

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

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


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

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


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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The results are summarised in the chart below:


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

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

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

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

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


 Notes and References

  1. Further information on the Dasgupta-Heal-Solow-Stiglitz model may be found in Pezzey J C V & Toman M A (2002) The Economics of Sustainability: A Review of Journal Articles  Resources for the Future Discussion Paper 02-03 pp 5-10  http://www.rff.org/files/sharepoint/WorkImages/Download/RFF-DP-02-03.pdf
  2. See for example the title and opening sentence of Hartwick J M (1977) Intergenerational Equity and the Investing of Rents from Exhaustible Resources American Economic Review 67(5) pp 972-4.
  3. Solow R M (1974) Intergenerational Equity and Exhaustible Resources The Review of Economic Studies Vol 41 pp 29-45.  See p 37 and Appendix B p 43.
  4. Hartwick, as 2 above, p 973. A simpler proof by J Pezzey may be found in Perman R, Ma Y, McGilvray J & Common M (3rd edn 2003) Natural Resource and Environmental Economics Pearson Addison Wesley pp 660-2.
  5. Buchholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model Scandinavian Journal of Economics 107(3) pp 547-561. See pp 549 & 558.
  6. Buchholz et al, as 5 above, p 553 state this formula but (because they start from a Cobb-Douglas production function without my B term), without the term in B. The fact that B is raised to the power 1/(1-β)  was derived mathematically by me and confirmed, for my chosen initial and parameter values,  by spreadsheet calculations with C in the vicinity of the implied Cmax.
  7. Solow, as 3 above, pp 37 & 39, which suggest β ≈05 and α ≈ 3β.
  8. ILO / OECD (2015) The Labour Share in G20 Economies https://www.oecd.org/g20/topics/employment-and-social-policy/The-Labour-Share-in-G20-Economies.pdf. See charts p 4.
  9. World Bank http://data.worldbank.org/indicator/NY.GDP.PCAP.CD
  10. World Bank http://data.worldbank.org/indicator/NY.GNS.ICTR.ZS?view=chart
  11. Johansson A, Guillemette Y & 7 Others (2013) Long Term Growth Scenarios Economics Department Working Papers No 1000 http://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=ECO/WKP(2012)77&docLanguage=En   See charts in Fig 11 p 23.  Although these charts suggest an average ratio somewhat above 2, they relate to selected countries only.  Ratios are probably lower in many poor countries with predominantly agricultural economies.
  12. Wikipedia: Oil Reserves https://en.wikipedia.org/wiki/Oil_reserves. .  These figures relate to proven reserves, and (as has happened in the past) additional reserves will probably be discovered.  On the other hand the effect of carbon emissions on climate will probably limit the extent to which reserves can be exploited.
  13. IEA FAQs-Oil https://www.iea.org/about/faqs/oil/
  14. Tilton J E (2006) Chapter 3 Depletion and the Long-Run Availability of Mineral Commodities, in Doggett M D & Parry J R (ed) Wealth Creation in the Minerals Industry: Integrating Science, Business and Education Society of Economic Geologists Special Publication No. 12 http://www.bbk.ac.uk/ems/research/Seminar_info/Wealth%20Creation  See Table 1 p 63 listing 11 minerals, of which (at zero production growth) 5 have life expectancy less than oil and 5 more.
  15. The distinction between reserves and resource base is explained in Tilton (as 14 above) p 62.
  16. Buchholz et al, as 5 above. Modelling of depreciation is introduced on p 551 and the stated result is on p 553 (case θ = 1 and δ > 0).
  17. OECD Consumption of Fixed Capital as % of GDP 2013 (latest year available) Total (of OECD countries) 16.5% (http://www.keepeek.com/Digital-Asset-Management/oecd/economics/national-accounts-at-a-glance-2015/consumption-of-fixed-capital_na_glance-2015-table49-en#.WD63Z9WLTIU), assumed similar for non-OECD countries.
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A Key Technique of Environmental Economics

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

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

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

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

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

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

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

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


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

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

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

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

\dfrac{dS}{dt}= -R

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

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

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

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

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

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

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

Notes and References

  1. Stranlund J K Lecture 8 Dynamic Opimization      http://people.umass.edu/resec712/documents/Lecture8DynamicOptimization.pdf
  2. Perman R, Ma Y, McGilvray J & Common M (3rd ed’n 2003) Natural Resource and Environmental Economics  Pearson Addison Wesley  pp 480-505, 512-7, 548-53 & 574-81.
  3. Wikipedia: Pontryagin’s Maximum Principle https://en.wikipedia.org/wiki/Pontryagin%27s_maximum_principle
  4. Buchhholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model Scandinavian Journal of Economics 107(3) pp 547-561 (Modelling of depreciation is introduced on p 551 and the stated result is on p 553 (case \theta = 1 and \delta > 0).
  5. Wolfram Alpha Widgets: General Differential Equation Solver http://www.wolframalpha.com/widgets/view.jsp?id=e602dcdecb1843943960b5197efd3f2a
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In Defence of the Linear Demand Function

Presentations of microeconomic analysis often assume linear demand functions, but rarely justify them in terms of utility theory. However it can be done, and without making outrageous assumptions.

Originally posted 11/2/2016.  Re-posted following site reorganisation 21/6/2016.

Run a Google search on “assume a linear demand function”, and you get thousands of hits. The straight-line demand function is much used in elementary economic analysis and student exercises, presumably because the complexities of curves can distract from a topic’s economic substance, perhaps even just because it’s easy to draw. But is there more to be said for it than that? Could a real demand function be linear?

The answer to the question might seem to lie in empirical investigation, and certainly that could be helpful. For the most part, we would have to rely on natural experiments in which the price and other variables have changed through the normal working of an economy, and use statistical methods to try to isolate the effect of the price. Given some unexplained variation in the data points, the form of the demand function may be far from clear.

Furthermore, the range of prices observed may be quite narrow. The fitted demand function may be of doubtful value if we want to predict the effect of a price outside that range, or to use the consumer surplus (the area under the whole demand function above the actual price) as a welfare measure. The latter point is especially important in the economic valuation – from revealed preferences in surrogate markets – of environmental goods for which the actual price is often zero. The travel cost and hedonic methods often require measurement of the area under the whole demand function for an environmental good, right down to zero price, so that it becomes crucial to consider the shape of the demand function even at very low prices.

There remains therefore a considerable role for theory, in particular utility theory. While theory cannot tell us the form of the demand function for a particular good, it can provide some insight into what forms are possible, and in what sort of circumstances they might occur. A general principle of econometrics is that model-building should be supported by theory (1), and the identification of demand functions is no exception.

The derivation of a demand function from a utility function together with a budget constraint is a straightforward application of constrained optimisation. The converse – to find the utility function that will result in a given form of demand function – needs more mathematical sophistication. A detailed treatment, including the case of linear demand, may be found in Border (2). The utility function developed below, leading to a linear demand function, is an adaptation of that derived by Border. The aim here is to show that such a utility function is not just a mathematical curiosity, but in certain circumstances has economic plausibility.

Let’s consider some utility functions, for the case of two goods, X and Y. Take the functions to relate to a normal person, neither rich nor with an unusually modest desire for goods. To relate the two-good case to the reality of a world of many goods, take X to be a particular good in whose demand function we are interested, and Y to be a composite good representing all other goods. Suppose further that X is an inessential, a leisure good for example. Let P_X and P_Y be the prices of the goods, with P_Y assumed fixed, and let M be the person’s income.

The assumption that more of a good is normally preferred to less suggests two basic types of utility function, multiplicative and additive. To allow some degree of flexibility, we may include coefficients in either type, say a and b, leading naturally to the two forms below:

U(X,Y) = X^aY^b.......(E1)

U(X,Y) = aX + bY.......(E2)

The multiplicative approach results in the familiar Cobb-Douglas utility function (E1). A feature of this function is that it exhibits a diminishing marginal rate of substitution, represented by indifference curves that are convex to the origin. In other words, as we increase the quantity of one good, say X, the ‘value’ of given increments gradually decreases, where value is measured in terms of the quantity of Y that must be forgone to keep utility constant. A standard piece of analysis (3) shows that it leads to curvilinear demand functions with demand inversely proportional to price, implying that there is no limit on quantity demanded as price falls to zero. These properties suggest that the Cobb-Douglas function may be plausible in some circumstances. For our case, however, it is inappropriate because it implies that utility will be nil when consumption of X is nil, even if consumption of Y is high, which contradicts the assumption that X is an inessential.

What is needed for our case, therefore, is an additive utility function. However, the basic additive form (E2) exhibits perfect substitutability between the two goods, represented by straight-line indifference curves. If we increase the quantity of X in this case, the value of given increments, in terms of the quantity of Y that must be forgone to keep utility constant, does not decrease. Since this is not very plausible, let us refine (E2) by replacing aX and bY by functions f(X) and g(Y):

U(X,Y) = f(X) + g(Y)......(E3)

To ensure a diminishing marginal rate of substitution, we require that, as X and Y respectively increase from zero, these functions initially increase, but at a decreasing rate (in terms of calculus their first derivatives are positive and their second derivatives negative).

Since X is an inessential good, we expect that the rate of increase of f(X) will decrease quite rapidly, so that f(X) will reach a maximum at quite a low threshold value of X, and will remain at that maximum when X exceeds the threshold. For the composite good Y, however, we expect that the rate of increase of g(Y) will decrease much more slowly, and that g(Y) will still be increasing at the maximum Y the individual can afford from their income, that is, M/P_Y.

The indifference map will look something like that below, although the relative sizes of the horizontal and vertical dimensions will depend on the units in which the goods are measured and the scales of the axes.

Linear Demand Indifference Map

Any expenditure on X above that needed to achieve the threshold will not add to the utility derived from X, and by reducing expenditure on Y will reduce the utility derived from Y. A rational individual will therefore spend most (or all) of their income on Y and at most a very small proportion q on X.

The possible range of expenditure on Y will therefore be small, implying a narrow range for consumption of Y:

M(1-q)/P_Y \leq Y \leq M/P_Y.......(E4)

Since the rate of increase of the Y component g(Y) of U decreases only very slowly, a straight line can provide a very good approximation of g(Y) within the narrow range defined by (E4), narrow because q is very small. Hence we can find parameters, say s and t, such that within the defined range:

s + tY \approx g(Y)

We can therefore rewrite (E3), to a very good degree of approximation within the relevant range, as:

U(X,Y) = f(X) + s + t(Y).......(E5)

It remains to consider the function f(X). A simple way to achieve the necessary properties – increasing in X, but at a decreasing rate – is:

f(X) = aX - cX^2.......(E6)

It can be seen (eg using differentiation) that f is increasing when X is less than a/2c, and comes to a maximum of a^2/4c at that threshold (when X exceeds the threshold we assume that it rermains at that maximum, rather than declining as (E6) suggests).

There are other functions that share the same properties. For example, the square in (E6) could be replaced by a cube, or some other power, and if desired the parameters a and c could be calibrated to yield the same maximum as (E6). However, such functions differ in their curvature, a simple indicator of which is the ratio R of their value at half the threshold to their value at the threshold (their maximum). For (E6) we have:

R = \displaystyle \frac{a(a/4c) - c(a/4c)^2}{a(a/2c) - c(a/2c)^2} = \frac{3a^2/16c}{a^2/4c} = 0.75

It can be shown that the higher the power replacing the square in (E6), the lower R becomes. For a cube, for example, it is about 0.69. While there cannot be said to be a correct value of R, which may vary between individuals and between goods, a value of three-quarters for a leisure good – implying that a person gets three-quarters of the possible enjoyment from half the quantity at which they are sated – seems entirely plausible.

For the circumstances we have described, therefore, a plausible form of utility function within the relevant range is:

U(X,Y) = aX - cX^2 + s + tY.......(E7)

We now use the standard constrained maximisation technique to find the demand function for X. The budget constraint is:

XP_X + YP_Y = M

Hence the Lagrangian expression is:

L(X,Y,\lambda) = aX - cX^2 + s + tY + \lambda(XP_X + YP_Y - M)

Taking partial differentials and equating to zero:

\partial L / \partial X = a - 2cX + \lambda P_X = 0

\partial L / \partial Y = t + \lambda P_Y = 0


\lambda = (2cX - a)/P_X = -t/P_Y

2cX = a - tP_X / P_Y

X = (a/2c) - (t /2cP_Y)P_X.......(E8)

Since a, c, t and by assumption P_Y are all constant, the demand function (E8) is linear in P_X.

A caveat. The above is an individual demand function, and like all demand functions is subject to the overriding condition that demand cannot be negative. Because of that condition, a market demand function built up by summing linear individual demand functions is not necessarily linear along its whole length. This is only so where each individual demand function has the same choke price (lowest price at which demand is zero). Otherwise, the market demand function is linear over the price range from zero up to the lowest choke price of any of the individual functions, but will be kinked at that choke price (and at all higher individual choke prices).

Notes and References

1. Gujarati D N (International Edition 2006) Essentials of Econometrics McGraw Hill International p 336

2. Border K C (2003) The “Integrability” Problem http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf   For linear demand see pp 7-8.

3. See for example Nicholson W (9th edn 2005) Microeconomic Theory: Basic Principles and Extensions Thomson South-Western pp 102-3

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