Some Carbon Tax Scenarios

How does a competitive industry respond to an emissions tax in the short run and the long run?  What if the industry is a monopoly?

In this post I bring together two standard pieces of microeconomic analysis: the effect of an emissions tax to address a pollution externality; and the behaviour of profit-maximising firms in different market structures.  It’s a straightforward exercise, but some of the results may be found a little surprising.

My method here is the exploration of numerical examples.  Therefore the only claim I make for the results is that they demonstrate possibilities: to infer any sort of generalities would be an obvious fallacy.  The numbers from which the examples begin have been chosen for ease of calculation: it is no accident that many of the output and other figures to which they lead are round numbers. 

I consider two industries, one with many firms in perfect competition, and one a monopoly.  The following assumptions are common to both:

  1. Emissions are uniformly mixed and very large in total (as is the case for CO2 and some other pollutants).  Hence the damage due to any one firm’s emissions is independent of its location, and its contribution to total emissions is too small to affect the marginal damage per unit of emission. 
  2. Marginal damage m from the pollutant (in terms of the local currency) is 1 per unit of emission.
  3. In the absence of an emissions tax, with firms taking no particular measures to limit their emissions, the emissions ratio e (the ratio of emissions to output) is assumed to be 4.
  4. The emissions tax is introduced with minimal notice.  Therefore all adjustment to the tax takes place after its introduction.
  5. Firms’ costs are consist of four components: a) a fixed component; b) a component proportional to the square of output (in conjunction with (a) this yields the characteristic U-shaped average cost curve); c) a component reflecting, for any level of output, higher costs for a lower emissions ratio; d) a component for the cost of the tax, where applicable.  I take costs to include ‘normal’ profit: all references below to profit should be understood to mean economic or supernormal profit.

Outcomes are assessed from several points of view, of which perhaps the most important is net welfare, calculated as consumer surplus plus producer surplus minus damage due to emissions plus tax receipts.

An Industry in Perfect Competition

The industry is assumed to be a constant-cost industry, that is, the entry or exit of firms does not affect the cost functions of its firms.  Writing q for output volume and t for the tax rate, the cost function per period of each firm is:

c(q,e)  =  10 + 0.1q^2 + 4q/e + qet

Writing P for the market price and Q for total industry output volume, the market demand function per period (in inverse form) is:

P  =  11 - 0.01Q

Initial Position with No Emissions Tax

We assume that the industry is in equilibrium, with competition having driven the market price to the minimum point of the firms’ average cost (AC) curves so that their profit will be zero.  We have to find the output q of each firm at which average cost is minimised. Price P is then equal to average cost at that point.  Using the market demand function we can then find industry output Q, from which we can infer the number of firms (Q/q) and the total value of the industry’s sales (PQ).  We can also calculate the industry’s total variable costs excluding tax, which is of interest as an indicator of the total employment supported by the industry and its suppliers.

The industry’s emissions are simply Qe.  To calculate producer surplus we need the average variable cost (AVC) of one firm at the output determined above.  We then have everything needed to calculate the components of net welfare.

With Emissions Tax: Short Run

I take the short run to be a period in which no firm has changed its emissions ratio or exited the industry.  Thus any reduction in emissions resulting from the tax must be due to a reduction in output.  In a previous post, I noted that a reduction in emissions in response to a tax could be due to the introduction of abatement technology, to a reduction in output, or to a combination of the two.  Here I consider the implications of the timing of such a combined response: a reduction in output can usually be almost immediate, but the introduction of abatement technology will normally take time. 

In the short run, having not fully adjusted to the tax, firms will not set their output to the minimum point of their average cost curves.  Instead, we must start from the more fundamental principle that they will set their output to the point at which their marginal revenue (the market price P) equals their marginal cost.  So from the cost function we obtain marginal cost in terms of firm output q and set this equal to P: this yields the inverse supply function for a firm.  Since the number of firms is known from the initial position, we can infer the market supply function relating P and Q.  From this in conjunction with the market demand function we can infer the values of P and Q, and hence q.  The remaining calculations are just as for the initial position.

With Emissions Tax: Long Run

I use the term ‘long run’ in a special sense: a period in which all firms have adjusted to the tax as fully as possible by changing their emissions ratio or exiting the industry.  This is not quite the Marshallian long run since the fixed component of the firms’ cost functions is assumed unchanged from the initial position (I leave for another day the important case in which abatement of emissions involves investment in fixed capital). 

The method of calculation is as for the initial position except that the average cost curve now contains two unknowns: firm output q and the emissions ratio e.  So we must find the combination of values of those two variables which minimises average cost. Once we have found that minimum point, yielding q, e and P, the calculations proceed in the familiar way.


Table 1 below sets out the results of the above calculations.  It can be seen that the industry’s emissions are reduced in the short run and further reduced in the long run.  Thus the primary purpose of the tax is achieved.  Also on the positive side, net welfare is increased in the short run and further increased in the long run.

 Initial Position with No TaxEmissions Tax at t = 1: Short RunEmissions Tax at t = 1: Long Run
Output per firm q10610
Profit / – Loss per firm0-60
Number of firms808050
Industry output Q800480500
Price per unit of output P36.26
Industry sales value240029763000
Industry variable costs excluding tax160016001500
Emissions ratio e442
Industry emissions Qe320019201000
Net welfare80014401750
Table 1: Short and Long-Run Effects of an Emissions Tax on a Perfectly Competitive Industry

Output per firm in the long run is the same as in the initial position.  Thus the long run reduction in emissions is achieved via a combination of a lower emissions ratio and a reduction in the number of firms.

Although the tax reduces the volume of output and increases its price per unit, these may be regarded as necessary side-effects of the emissions reduction.  However, the fact that both these changes slightly overshoot in the short run may be considered to impose an unnecessary (albeit temporary) detriment on consumers.  The need for the losses incurred by firms in the short run is questionable: by providing an incentive for firms to exit the industry they hasten the arrival of long-run equilibrium with few firms and profits restored to nil, but perhaps that process could be facilitated by other means.  These features of the short-run position after introducing a tax with minimal notice suggest that there could be advantage in giving a longer period of notice allowing firms to adjust before the tax comes into effect.  However, the way in which firms would respond during such a notice period would be difficult to predict.  It would depend on, among other things, the degree of certainty with which firms believe that the tax will be introduced, and the judgments firms make as to how many of their competitors will exit the industry. 

It is important to note that the industry will not leave the short run one day and arrive at the long run the next.  Between the two is a transitional process in which some firms introduce abatement technology and others exit the industry.  Again, firms’ behaviour during this period is difficult to predict.  Perhaps some firms will make an early strategic decision to exit.  Alternatively, all firms may begin incurring the extra costs of abatement technology, and only as losses accumulate will some firms decide to leave the industry.

How does the tax effect employment in the long run?  To the extent that industry variable costs excluding tax are a good proxy for the employment supported by the industry, the direct effect is only a small reduction. Although many firms leave the industry, the effect on employment is largely offset by the extra costs per firm of reducing their emissions (staff made redundant by exiting firms may be re-employed by other firms).  Taking a broader view, however, the significant increase in industry sales value implies, given constant aggregate demand, a corresponding reduction in demand for other goods, adding to any reduction in employment.  Much therefore depends on how the government uses the tax receipts. If it uses them in ways which raise employment, either via government expenditure on goods and services, or via a cut in another tax, then the overall effect on employment could be neutral or even positive.    

A Monopoly

The single firm’s cost function is:

C(Q,e) = 800 + 0.01Q^2 + 4Q/e + Qet

Its inverse demand function is:

P = 13 - 0.01Q

Initial Position with No Emissions Tax

Here e = 4 and t = 0.  Using the demand function we can express profit Pr as a function of Q only and then find the level of Q that maximises profit.  Price P, sales value and profit follow immediately.  We can also calculate variable costs, emissions (Qe), and then the components of net welfare.

With Emissions Tax at Rate Equal to Marginal Damage: Long Run

For this industry I omit the short-run analysis and proceed directly to the long run.  Here t = 1 while e, along with Q, is an unknown to be found.  So we find the levels of Q and e which maximise profit.  The only other difference from the calculations for the initial position is that we need both total variable costs (in order to calculate producer surplus) and variable costs excluding tax (as an indicator of employment). 

With Emissions Tax at a Rate Less Than Marginal Damage: Long Run

We take the case t = 0.7.  The method of calculation is exactly as for t = 1.


Table 2 shows the results of the above calculations.  As expected, the tax reduces emissions, partly by reducing output and partly by reducing the emissions ratio, and the higher tax rate reduces emissions by more. 

 No Emissions TaxEmissions Tax at t = 1: Long RunEmissions Tax at t = 0.7: Long Run
Output Q300225249
Profit Pr1000213363
Price per unit of output P1010.7510.51
Sales value PQ300024192617
Variable costs excluding tax12009561037
Emissions ratio e422.39
Emissions Qe1200450596
Net welfare105012661295
Table 2: Effects of an Emissions Tax at Different Rates on a Monopoly

The tax considerably reduces the firm’s profits, but they are still positive, and a reduction in the profits of a monopoly may be considered of little concern.  The small increase in price represents only a modest additional burden to consumers.  Since the reduction in sales value exceeds that in variable costs excluding tax, the net effect on employment may well be positive, even before consideration of how the government uses the tax receipts.

Net welfare is increased at either of the two tax rates, but is slightly higher when the rate is somewhat lower than the rate of marginal damage.  The reason for this is that, leaving aside the emissions damage, the initial position is sub-optimal relative to what could be achieved if output were set to equate price and marginal cost, rather than restricted so as to maximise the monopolist’s profit.  The theory of second best implies that a policy measure that would otherwise be optimal to address a market failure may not be optimal if another form of market failure is also present (1).  For a theoretical treatment of taxes to address externalities in the context of monopoly see Barnett (1980) (2).

A policy-maker selecting a tax rate in this situation might nevertheless want to look not only at net welfare but also at its separate components.  These are shown in Table 3 below.

 No Emissions TaxEmissions Tax at t = 1: Long RunEmissions Tax at t = 0.7: Long Run
Consumer surplus450253310
Producer surplus180010131164
Damage due to emissions-1200-450-596
Tax receipts0450417
Net welfare105012661295
Table 3: Effects of an Emissions Tax at Different Rates on a Monopoly, showing Components of Net Welfare

It can be seen that the extra net welfare at the lower tax rate is due to an increase in producer surplus plus a smaller increase in consumer surplus, offset by an increase in damage due to emissions and a reduction in tax receipts.  The increase in producer surplus is exactly reflected in increased profits.  A policy-maker might reasonably conclude that, although it does not maximise net welfare, the tax rate equal to the rate of marginal damage is to be preferred.

The workings supporting the above results may be downloaded below (MS Word 2010 format).


  1. Wikipedia Theory of the Second Best
  2. Barnett, A H (1980) The Pigouvian Tax Rule under Monopoly  American Economic Review 70(5) pp 1037-41

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Covid-19 and Household Size

Differences in national rates of Covid-19 infection may be partly due to differences in household sizes.

While many questions about the Covid-19 virus are currently unanswered, one point on which there has been wide agreement is that transmission is more likely indoors than outdoors (1,2).  If therefore we are to explain differences in national rates of infection, an obvious place to look is differences in indoor environments.  A plausible hypothesis is the following:

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

The thinking behind this is simple.  If one member of a household becomes infected, and unless there is effective self-isolation within that household, then it is quite likely that their infection will transmit to other members. The larger the household, the more people they can infect.  The hypothesis does not imply that household size is the sole or main reason for differences in rates of infection, merely that it is one contributory factor.

If correct, the hypothesis suggests a possible link between housing policy and rates of Covid-19 infection.  Countries (such as the UK) with restrictive planning policies that have limited the supply of land for building new homes will have fewer homes than they would otherwise have. This reduced supply of housing will lead to higher costs (whether for ownership or renting).  As a consequence, fewer people will be able to afford their own home, and (other things being equal) average household sizes will be larger: young adults, for example, will tend to stay longer with their parents before setting up their own home.  Larger households in turn will create more scope for transmission of infection.

But is the hypothesis correct?  An ideal test would require large sample data on household size and numbers infected at individual household level.  Here I present the results of a ‘quick and dirty’ test based on data currently available at national level. 

At the present time, reliable data on total rates of infection since the start of the outbreak is not available.  National totals of confirmed cases are incomplete because many cases have not been confirmed by testing, and international comparisons of those totals reflect differences in rates of testing as much as in rates of infection.  I therefore used national rates of death from Covid-19 as an, admittedly imperfect, proxy for rates of infection.  Even such death rates are unlikely to be perfect for international comparison, since practice in recording the cause of death of patients with multiple conditions may vary.  As a proxy for rates of infection, death rates suffer from the limitation that they are also influenced by differences in health systems between countries.  Nevertheless, it seems reasonable to assume, at least for the developed countries of Western Europe, that official figures on deaths from Covid-19 are at least of the right order of magnitude. 

Average household size was calculated from national statistics for population and numbers of households.

A regression was estimated for the model:

DP  =  C  +  (B x PH) + E

where:  DP is death rate from Covid-19 per million population; C is the regression constant; B is the slope coefficient; PH is average population per household; and E is the error term.  The regression was run on data for 14 Western European countries: Austria, Belgium, Denmark, France, Germany, Italy, Ireland, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, United Kingdom. 

Estimation of the regression was by weighted least squares, with weighting by population (implying that the fitting of the regression line takes more account of data points for countries with larger propulations).  The justification for the weighting is that a local random factor affecting the death rate within a region with a population of a few million could have a large effect on the overall death rate of a country with a smaller population.  Within a larger country, however, the effect of such a local factor would be less, and different random factors within different regions of the country would probably tend to offset each other.  It is proper to record that the choice of weighted least squares, rather than ordinary (ie unweighted) least squares, makes a large difference to the result.

The estimated regression line was:

DP  =  -1,334 + (770 x PH)

The precise values of the estimated coefficients, which rather implausibly imply a nil death rate at a household size of about  1.7, are not important. What does matter is that the estimated slope coefficient is positive, consistently with the hypothesis (and is sufficiently large that the null hypothesis that its true value is zero or less is rejected at the 5% significance level (3)).

I would describe this result as ‘interesting’. But no more conclusion should be drawn than that the hypothesis merits further research.

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

Notes and References

  1. Sandhu, S (11/5/2020) Why you are less likely to catch coronavirus outside than indoors, according to experts  i
  2. Moffitt, M (28/4/2020)  China study suggests outdoor transmission of COVID-19 may be rare  SFGATE
  3. This can be inferred from the fact that the 95% confidence limits of the estimated slope coefficient are both positive.


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Urban Wildlife – An Exploration

In many cities around the world, there are wild animal species whose presence is enjoyed by some and a nuisance to others. Management of such species should be informed by economic analysis as well as ecological, ethical and animal welfare considerations.

I had better say this right away: this post was prompted not by the current Covid-19 pandemic, but by repeated sightings last summer of a fox around my London home.  Nevertheless, the pandemic does show that the proximity of wild animals can in certain circumstances have enormous adverse consequences for humans.  At the time of writing, the precise source of the initial outbreak in Wuhan, China, is not known, but it appears that the virus was transmitted from wild animals to humans at a market.  Quite possibly, however, it was a wild animal farmed outside the city and brought to market for sale, and not therefore urban wildlife in the sense considered in this post. 

Human attitudes to wildlife depend greatly on its nature, location and behaviour.  Large carnivores in what remains of their natural habitats are widely considered worthy of conservation efforts, but if such a creature should be on the loose in a city, as happens occasionally (1), then most people would  support its killing in the interests of public safety if other options are impractical or ineffective.  Very small animals encountered in or near people’s homes may evoke little affection or sympathy: many who would never harm larger animals will happily swat a fly or lay a mouse trap. 

Between these extremes are medium-sized wild animals present in and adapted to life in cities about which people may have differing or complex attitudes – animals not large or dangerous enough to present a major physical risk to human life, but capable of being a considerable nuisance while also evoking human sympathy and the protection of custom or law.  Table 1 below lists some examples (this has been compiled from numerous online sources which it would be tedious to list in full).

Trying to be objective, I have avoided any reference to animals “attacking” humans, as it is apparent that what one source considers an attack, another may view as defensive action in response to human provocation or to protect young.  In the same spirit, I have not stated that any of the animals “spread” human diseases.  Undoubtedly many urban wild animals carry pathogens that are potentially harmful to humans (2).  The questions then are the likelihood of transmission to humans, and whether transmission can be prevented by simple precautions such as washing hands after contact or requires more elaborate preventive measures.  It is also pertinent to consider whether or not the risks to humans are greater than those already encountered from pets and domestic animals.

There seems to be little literature on the application of economics to the management of such urban wildlife.  What literature there is on the economics of wildlife management focuses mainly on rural wildlife, as is apparent from the highlighting of costs from harm to livestock and crops, and benefits from hunting and recreation (11).  Nevertheless, the insight that there may exist an optimum population size which maximises benefits less costs (12) appears relevant to urban as well as rural wildlife.

Before pursuing this argument, I will briefly address some lines of thought that seem unlikely to be helpful in informing management strategies for species of the sort listed in the table.  Firstly, conservation is not normally an issue for such species.  It may be that their adaptation to urban environments is to some extent a response to loss of their natural habitats.  But all the listed species are classified by the International Union for the Conservation of Nature as of least concern (13), implying that their populations are large enough that they are far from any risk of extinction.  Secondly, and as a consequence of the above, a focus on such species’ contribution to biodiversity, and on the economic value of that contribution, is unlikely to yield useful information.  There is no foreseeable risk that that contribution will be lost.

Any attempt to measure the benefits and costs to humans of an urban wildlife species would require application of a judicious selection of non-market environmental valuation techniques.  Benefits from observing animals might be measured by adapting the travel cost method, typically used in valuing parks and other recreational sites, to allow for the fact that the animals are not concentrated at one site but spread out across an urban environment.  Thus the relevant travel costs are not those incurred in getting to a particular site but those incurred in getting to wherever, at a particular time, an animal can conveniently be observed.  The “travel” required to make such an observation when at home might be no more than getting up and coming to a window when alerted to an animal’s presence by a family member.  The fact that such “travel” may involve very little cost – the value of perhaps just a minute or two of a person’s time – does not mean that the benefit is correspondingly low.  This is because the travel cost method measures value in terms of consumer surplus – in simple terms the difference between the cost a person would have been willing to incur to view an animal and the cost they actually incur.  The cost they would have been willing to incur might be assessed from the costs actually incurred to view an animal by other people who happen to live where such animals are not present.

Benefits from controlling species regarded as harmful might be measured using a combination of ecological and economic analysis.  Ecological analysis would attempt to quantify the effect of the population of the beneficial species on the population of the harmful species, and the relation between the population of the harmful species and specific forms of harm (theft of unprotected food, dispersion of pathogens, dropping of faeces, chewing of materials, etc).  It would then be for suitable economic analysis to attempt to value the benefit from a reduction in the relevant forms of harm.

The costs attributable to urban wildlife include numerous components.  Theft of food that would otherwise have been available for human consumption can as a minimum be costed at its market value or, where prepared at home from bought-in ingredients, at the market value of the ingredients plus the opportunity cost of the preparation time.  The cost of overturned bins might be taken to be the opportunity cost of the time spent in cleaning up, and that of disturbance to gardens and damage to buildings the cost of reinstatement or repair.  In the case of damage to overhead power lines, the cost of the power outage should be added to the repair costs.  Power outage costs will depend on how many users are affected and what they use electricity for, but can be very large (14).  The cost of injury or ill-health attributable to urban animals should include the costs of associated healthcare and of lost income (to employees) or lost production (to employers) due to time off work.  Given suitable information about the effect of education on children’s employment and earnings prospects, one might also include a cost element for time off school.  Estimating the costs of ill-health of retired people raises issues beyond the scope of this post (15).

Where nuisance attributable to animals is recurrent, further costs may arise. Various forms of avertive behaviour may be adopted in an effort to minimise the nuisance.  These may include: never eating out of doors; keeping windows closed (even in hot weather); buying larger or more secure waste bins; fitting spikes where birds might want to perch; setting deterrents  such as motion-sensitive lights, chemical sprays and imitation snakes; and going out of one’s way to avoid certain locations.  All these have costs: either direct monetary outlays, or time costs, or costs of restrictions on life-style. The last-named costs might be estimated using the hedonic property method, requiring comparison of house prices in locations with and without the need for such restrictions.

None of this is easy.  Even an apparently simple case such as theft of purchased food  requires reliable data on the quantities and types of stolen items, and the species responsible for theft.  Although the economics of environmental valuation has made great strides in recent decades, its application to the full range of benefits and costs of particular species, and collection of the necessary data, would present enormous challenges.

It is plausible to suppose that a higher urban population density of a wild species will not imply proportionately higher benefits to humans.  The principle of diminishing marginal utility suggests that someone who enjoys seeing a fox or a monkey is unlikely to obtain ten times as much enjoyment from seeing ten foxes or monkeys, or from seeing just one ten times as often.  That could be tested empirically by observing how much time and effort a person incurs to view such animals, and how that time and effort varies with the number of known opportunities for views.   

The costs to humans of a higher urban population density of a wild species, however, may be more than proportionately higher.  One overturned waste bin in a street may be seen as an exception. Many such bins not only create proportionately more clean-up work, but may also be perceived as lowering the tone of the neighbourhood, and may prompt efforts to address the problem by deterrent measures or obtaining more animal-proof bins. Occasional animal noises may be a minor issue, but persistent noise, especially at night when it may disturb sleep, can be a serious problem.  Similarly, occasional theft of food may be tolerable, but routine thieving may force changes in life-style, such as never eating out of doors, keeping windows closed even in hot weather, and abandoning businesses selling food at outdoor market stalls.

If these suggestions are broadly correct, then we can represent the situation in terms of the sort of diagram familiar from, for example, the economics of pollution control.

As in other contexts, net benefit is maximised when marginal benefit equals marginal cost.

However, such an approach should not be the sole determinant of how a species should be managed.  It is also important to consider, from the perspectives of effectiveness, cost and ethics, the means of getting from the current to the optimum population.  More fundamentally, an animal rights perspective would suggest that an approach based solely on benefits and costs for humans is to be rejected as an example of human supremacism (16). 

What would be missing, in an approach that only takes account of benefits and costs for humans, is consideration of animal welfare.  That implies, as a minimum, avoidance of direct cruelty to animals including, in particular, avoidance of practices such as blocking of dens and use of poisons that subject animals to lingering and unnecessarily painful deaths.  The importance of animal welfare in this narrow sense is widely recognised: in the UK, for example, use of such practices to kill foxes is illegal (17).

However, appropriate management of urban wildlife requires a much broader conception of animal welfare that has regard to the extent to which animals lead worthwhile lives with a positive balance of well-being over suffering, and to how humans and the urban environment they create can affect that balance.  Positive features of an urban habitat may include a plentiful food supply and an absence of natural predators.  Negative features may include intense competition for territory and the risk of death or injury in road accidents.  The balance might be expected to vary between species and locations.  

Ideally, management of an urban species should be informed by knowledge not only of the actual welfare of its members but also of how their welfare might change if its population density were to increase or reduce, or if features of its environment were to change.  But rarely if ever do we have such knowledge (18).  Even our knowledge of actual welfare is very limited. 

There is a further difficult and contentious issue.  Philosophers have explored, in respect of humans, the ethics of policies that affect the size of the future population.  How should we choose between scenarios A and B, if people in A have greater well-being but people in B are more numerous?  Totalism asserts that we should maximise total well-being, calculated as population multiplied by average quality of life (19).  Averagism holds that we should maximise average quality of life, without regard to size of population.  Both these positions, and others, have been shown to have counter-intuitive implications.  A considerable literature in this field has not led to anything approaching a consensus.  The point here is this: the same ethical issue arises in respect of actions or policies which affect the size of an animal population. 

A plausible assumption is that the average well-being of an animal species is greatest when its population density is neither too small nor too large.  If it is too small, then it may be very difficult for individuals to learn behaviour from others or to find mates.  If it is too large, then competition for territory may become severe, diseases may spread more easily, and the available food supply may be inadequate.  Rather more speculatively, it might be argued that if population expands beyond the level that its food supply can support, then average well-being will decline so rapidly as to render the debate between totalism and averagism irrelevant (because total welfare will fall despite the extra population).  If so, then the relation between population density and welfare, according to whatever is our preferred measure of overall welfare of an animal population, could be represented as in the diagram below.

There is still much that this analysis leaves out.  It passes over the issue of what in an animal’s environment we are regarding as held constant as its population density varies.  Whatever management method might be chosen – for example shooting, poisoning or sterilization for a reduction in population, and providing extra food or nesting opportunities for an increase – would in itself amount to a change in the environment.  It also omits the ecological consequences of a change in the population density of an animal species.  Through food chains and in other ways, there are likely to be effects on the populations of other animal species, and so in turn on the welfare of those species.

Nevertheless, it is I believe of interest to consider the implications of putting together the economic analysis summarised in Diagram 1 and the animal welfare analysis summarised in Diagram 2.  I assert the following:

Proposition 1: There is no reason to suppose that the population density of an urban animal species that optimises its net benefit to humans is the same as, or even close to, the population density that optimises its own overall welfare.

The justification for Proposition 1 is simply that the bases of the optimum for humans and the optimum for the animal species are entirely different.  It would be entirely coincidental if these optima happened to be the same or close. 

For the next proposition it is convenient to represent the actual current population density of an urban animal species in a particular location as A, its optimum density for humans as OH, and its optimum density for animal welfare as OA.  We can then state:

Proposition 2:  For any urban animal species, there are 6 possible orderings by increasing animal population density of A, OH and OA.  For 4 of these orderings, there is available what might be termed a Pareto improvement (20), a change in population density that would yield a net benefit for humans and raise the overall welfare of the animal species.  For the other 2 orderings, a change in animal population density that was advantageous to humans would be disadvantageous to the animal species, and vice versa.

The 6 possible orderings are: 1) A < OA < OH ; 2) A < OH < OA  ; 3) OA < OH < A ; 4) OH < OA < A ; 5) OA < A < OH ; 6) OH < A < OA .

For ordering 1, an increase in population density from A to OA would optimise the overall welfare of the animal species, but also bring the net benefit for humans closer to that at OH.  For ordering 2, an increase from A to OH would optimise for humans but also raise overall welfare for the animal species.  For orderings 3 and 4, a suitable reduction in population density would be beneficial for both humans and the animal species.  It is only for orderings 5 and 6, in which A is between OA and OH, that a change in density can be beneficial for humans or for the animal species, but not for both.

Proposition 2 should not be taken to imply that the 6 orderings are equally likely: such a claim would be way beyond what can be supported by current knowledge.  What it does suggest, however, is that, in the management of an urban animal species there is not necessarily a conflict between what is good for humans and what is good for the species.  Where an urban animal species is, on balance, a nuisance to humans, it is possible that a reduction in its population would also be good for the species. 

However, that is just a possibility.  It is also possible that a reduction in the population of an urban animal species would be good for humans but bad for the species.  Much more research is needed to enable us to make well-supported decisions on the management of urban animals.

Notes & References

  1. The Guardian (29/9/2016) Lion shot dead at Leipzig zoo after escaping from enclosure
  2. “Pathogen” is a general term for micro-organisms that can cause disease, including viruses, bacteria, fungi, protozoa and worms.
  3. Metro (8/9/2018)  Inside the secret world of London’s urban foxes
  4. Aberdeen City Council  Living with Urban Gulls
  5. Reuters (11/12/2018)  Monkeys run amok in India’s corridors of power
  6. East Coast Radio (30/8/2017) Monkey-ing around – harmless or menace?
  7. San Francisco Animal Care and Control  Coyotes
  8. The Humane Society of the United States  Coyotes and people: what to know if you see or encounter a coyote
  9. The Guardian (3/9/2019)  Swooping magpie shot by Sydney council after ‘particularly aggressive’ attacks
  10. The Guardian (10/12/2017)  Magpie edges out white ibis and kookaburra as Australian bird of the year
  11. See for example Gren I-M, Haggmark-Svensson T, Eloffson K & Engelmann M (2018)  Economics of Wildlife Management – an overview  European Journal of Wildlife Research 64:22  – start of Introduction
  12. See for example Gren et al as 12 above, p 3
  13. This can be confirmed at the IUCN Red list website, entering in turn the names of the species.
  14. Trotti, J 9/7/2018 The Human and Economic Costs of Power Cuts and Blackouts
  15. See for example Huter K et al 2016 Economic evaluation of health promotion for older people – methodological problems and challenges BMC Health Services Research 16 (Suppl 5)
  16. Positive.News 8/3/2017  Are you a human supremacist?
  17. UK Government  Foxes, moles and mink: how to protect your property from damage
  18. Hecht, L (2019) Optimal population density: trading off the quality and quantity of welfare   See especially the final paragraph.
  19. Wikipedia  Population Ethics
  20. I use the term Pareto improvement here in a specialised sense.  It is not implied that such a change would leave each individual human and each individual animal no worse off, only that overall welfare for humans and overall welfare for animals would both be improved.
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Mitigable Public Bads

The economic theory of public goods is sometimes assumed to be adaptable in a straightforward manner to public bads.  Here I consider some implications of the fact that such bads are usually mitigable.

Consider a region around an airport, subjected to noise from flights.  Noise arrives at every home in the region, but any household can choose, at a cost, to moderate the noise level inside its home by installing double glazing.  Such a choice by a household has no effect on the noise level inside any other home.

Suppose the sole water supply to a poor village is polluted.  Everyone in the village will be risking their health if they drink that water as supplied.  But (for some types of pollutant) any household can treat its water to make it safer to drink, perhaps by filtering or chemical treatment.  Such a choice by a household has no effect on the safety of the water consumed by any other household.

Similarly, suppose a region’s agriculture is affected by a drought.  All farms in the region receive less rainfall than normal but some, because of the crops they have chosen to grow or their cultivation methods, are better able than others to withstand the effects of the drought. Such choices by a farm have no effect on the resilience of other farms.

These are all examples of what I call mitigable public bads.  The idea of a public bad is derived from that of a public good, commonly defined as a good which is both rival and non-excludable. By non-rival is meant that one person’s consumption of or benefit from the good does not reduce the amount available to others. Non-excludable means that neither the provider of the good nor any other agent is able to pick and choose which individuals within the scope of provision of the good can consume it (hence the free-rider problem consisting in the fact that those who decline to pay for the good cannot be excluded from benefiting from it).  Street lighting in a city, for example, is non-excludable because no one can determine that certain individuals passing through the city’s streets will have their way illuminated and others will not. The fact that the provider could turn off the lights, excluding everyone, is irrelevant, as is the fact that only those who live in or visit the city can benefit from its street lighting.

Public bads have sometimes been defined as “goods” which are non-rival and non-excludable but which tend to lower rather than raise welfare (1).  For a bad to be non-rival is usually clear enough: my disturbance by noise from flights does not render my neighbours’ disturbance any less.  But non-excludability in relation to a bad is not so clear.  No one can be expected to pay for a bad, so a “provider” such as a factory causing air pollution has no interest in excluding non-payers. And whereas it can be assumed that few would wish to exclude themselves from a public good, sufferers from a public bad will certainly so wish if they can do so at reasonable cost. 

Usage in this context seems not to be settled. But I find it convenient to use the term public bad for any welfare-lowering “good” which is both non-rival and, in the sense that no one can pick and choose who within its broad scope suffers from it and who is excluded, non-excludable. If affected individuals can take action to moderate the effects of the bad on themselves, then I shall say that the public bad is mitigable. Environmental economists sometimes refer in this context to defensive expenditure or avertive expenditure, but as an adjective qualifying public bad, my sense is that mitigable is more appropriate than defensible or avertable.  Mitigable public bads are thus a subset of public bads, but an important one, as the above examples suggest.  Indeed, it seems plausible that most public bads are mitigable, at least for some individuals and to some degree. 

Mitigation need not be limited to individuals acting alone. In the case of installing double glazing, a decision by a household, or by a group of households making a bulk purchase to obtain a discount on the installation cost, can still be regarded as mitigation.  But for most public bads a clear distinction can be made between mitigation, action by potential sufferers to protect themselves from the bad, and what I shall term reduction, consisting in action at or close to the source of the bad to limit its scale or scope.  Examples of reduction would be a ban by an airport authority on flying to and from the airport in a certain direction, limiting noise for all households in that direction, or installation of equipment by a smoke-emitting factory to capture pollutants in the smoke, improving air quality for everyone in its vicinity.   

Two more definitions will be useful. I shall refer to the state of a public bad as its condition, across the whole of the affected region, after any reduction but before consideration of possible mitigation. By an individual’s exposure I shall mean the condition of the bad as experienced by that individual after any mitigation they have undertaken.  Thus exposure can differ between individuals both because some parts of the affected region may be affected more badly than others, and because the mitigation they have undertaken may differ.

A well-known result concerning public goods can be stated informally as below:

Proposition 1 The level of provision of a public good is optimal if the marginal cost of providing the good equals the sum over individuals of their marginal benefit from the good.

An individual’s marginal benefit from a public good can be interpreted as their marginal rate of substitution of the good for private goods.  This is equivalent to the slope of an indifference curve connecting combinations of goods which yield the individual the same utility. Since an individual’s utility function defines a whole set of indifference curves, the question then is which curve’s slope should be included in the sum when we are considering a possible re-allocation of production between private goods and the public good. The answer I will rely on here is to take the status quo distribution of private goods between individuals and assume that, in any re-allocation of production, individuals’ quantities of private goods would all change pro rata to the total quantity of private goods.  Any quantity of the public good together with an individual’s implied quantity of private goods would then define a point identifying a single indifference curve of that individual.  An alternative approach is to assume that all the indifference curves of any one individual have the same shape and therefore the same slope at any one quantity of the public good, regardless of the individual’s quantity of private goods.  This is equivalent to assuming that individual utility functions are quasilinear in the private goods (2), and is plausible if the quantity of private goods that individuals would have to forgo in return for the public good is small in relation to the total private goods.  Under this assumption it does not matter which curve of each individual we take: the sum of the slopes of one curve per individual will be the same whichever curves we take.

A point quite properly highlighted in most discussions of public goods is the sharp contrast between Proposition 1 and the optimality condition for a private good, which requires that the marginal cost of provision equal each individual’s marginal benefit. From this it follows that a public good will be under-provided in a free market (3).  However, other features of the proposition are sometimes overlooked. Firstly, the idea of optimal provision makes no sense for natural public goods like sunshine, except to the extent that they can be manipulated by human intervention. Secondly, for those cases where optimal provision does make sense, it is often a gross oversimplification to assume that the level of provision can be adequately characterised as a number of units on a single scale.  Just consider national defence, commonly given as an example of a public good. Thirdly, the relevant marginal cost of providing the public good is marginal social cost.  In the case of street lighting, for example, marginal cost should include the marginal social cost due to any greenhouse gases emitted in generating the electricity to supply the lights.  

Having noted these points, which apply equally to public bads, we may ask what is the equivalent of Proposition 1 for a mitigable public bad. In this case there are two kinds of decision to be made: the extent of mitigation by each individual; and the extent of action at source to limit or reduce the bad. 

Given the state of the bad, a rational individual will undertake mitigation up to but not beyond the point at which their marginal cost of mitigation (MCM) equals their benefit from a marginal lessening of exposure (BMLE). This is assuming normally shaped curves, that is, marginal cost increases and marginal benefit falls with additional units of mitigation.  For the purpose of optimal mitigation by individuals, it is not important how units of mitigation and exposure are defined: the only requirement is that, for any one individual, marginal cost and marginal benefit are measured with respect to the same units.  Note that, even if mitigation is possible for an individual, they will not undertake any mitigation if their marginal cost at zero mitigation exceeds their marginal benefit at that point.  In symbols, the condition under which an individual will undertake mitigation is:

MCM0  <  BMLE0                                  (A)

where the zero subscript indicates that the marginal quantities are measured at the state of the bad.

By analogy with the case of a public good, we expect that for a state S of a bad to be optimal, the marginal cost of reducing S must equal the sum over individuals of some sort of marginal quantity.  But what exactly?  The benefit to an individual from a marginal reduction in the state (BMRS) will depend upon whether, at S, they will undertake mitigation. If inequality A is not satisfied so that they do not undertake mitigation, then their benefit from a marginal reduction in S is simply their benefit from a marginal lessening of exposure:

BMRS(MCM0 > BMLE0)  =  BMLE0                    (B)

If however inequality A is satisfied so that they undertake mitigation, then their net benefit from mitigation at the margin will be BMLE minus MCM. Hence their benefit from a marginal reduction in S is the benefit from a marginal lessening of exposure less the benefit they could instead have obtained themselves from mitigation, the difference being the marginal cost of mitigation:

BMRS(MCM0 < BMLE0)  =  BMLE0 – (BMLE0 – MCM0)  =  MCM0           (C)

Putting the above together we have:

Proposition 2 Provided individual mitigation behaviour is rational, the state of a mitigable public bad is optimal if the marginal cost of reducing the bad equals the sum over individuals of the lower of a) their benefit from a marginal lessening of exposure and b) their marginal cost of mitigation.

However, extreme care is needed to ensure consistency in the units of measurement of the various marginal quantities.  Suppose we have a defined scale on which to measure the state of a bad.  For each individual, units of exposure must be such that the harm suffered from u units of state together with sufficient mitigation to limit exposure to u – 1 units must be the same as the harm suffered from u – 1 units of state with no mitigation.  A unit of mitigation is then simply that quantity of mitigation which will reduce exposure by one unit.

This has the important implication that both the physical requirements for and the cost of a unit of mitigation may vary greatly between individuals according to their circumstances.  Suppose the state of noise in the region around an airport is measured by average loudness at a defined location near the airport.  A given number of units will then be associated with more noise at some locations than others.  Suppose, as may be the case, that noise falls off with distance from the airport.  A reduction of one unit in the state of the noise may be quite significant for someone living close to an airport, but barely noticeable for someone on the edge of the affected region. Consequently the latter would need to do less than the latter, in physical terms, to achieve one unit of mitigation, and their marginal cost of mitigation will be lower.

Notwithstanding the above, a person living further from the airport will be less likely than someone nearer to it to undertake mitigation if, as is likely, benefit from a marginal lessening of exposure falls even faster with distance than marginal cost of mitigation, eventually reaching a point at which further lessening is imperceptible and offers no benefit.

The public policy implications of Proposition 2 are that those mitigable public bads which are due to human activity tend to be over-provided in a free market, but also that the extent of government action to correct that market failure should have regard to the availability and cost of mitigation by individuals.  If for example a tax is the preferred policy instrument, then it would be sub-optimal if the chosen tax rate per unit of state exceeds the sum over individuals as defined in Proposition 2.

An important question now is how the optimal state of a mitigable public bad compares with what it would be if no mitigation were possible.  This is illustrated in Chart 1 below.

The horizontal axis shows the state of the bad, reducing (ie improving) from left to right.  The vertical axis shows the quantity of a composite good representing all private goods.  It is assumed that there are no other public bads and no public goods.  The production possibility frontier PPF is the outer boundary of the possible combinations of the public bad and private goods, its concave shape reflecting the standard assumption of a diminishing marginal rate of transformation between goods.  The status quo is assumed to lie somewhere on the PPF.

S is the optimal state on the assumption that mitigation is impossible, and line I is the sum of individual indifference curves on the same assumption, chosen as explained above.  The slope of I at S will be the sum over individuals of their BMLE’s at S, and given the status quo assumption, I will be tangent to the PPF.

The red line IM is a sum of indifference curves passing through the intersection of PPF and I, on the assumption that mitigation is available. If inequality A above were not satisfied for any individual so that no mitigation would be undertaken, the slope of IM at S would also be the sum over individuals of their BMLE’s at S, and IM would be coincident with I in the vicinity of S, which would still be optimal despite the availability of mitigation.  If however A is satisfied for at least some individuals so that mitigation is undertaken, then the slope of IM will be a sum which includes MCM for at least some individuals, and therefore less than the slope of I.  Hence there is a region to the left of S within which PPF meets curves higher than IM, and the optimal state, given mitigation, will lie within this region, perhaps at SM.  So we have:

Proposition 3 Provided individual mitigation behaviour is rational, the optimal state of a public bad for which mitigation is available is greater (ie worse) than or equal to what the optimal state would have been if mitigation were not available, and is strictly greater if at least some individuals would have undertaken mitigation at the latter state.

In developing here a theory of mitigable public bads, I have passed over several points suggesting that Propositions 2 and 3 are in need of qualification.  Without following through the implications of each, I will just mention that an individual’s private cost of mitigation may not equal the social cost (for example the manufacture of double glazing may involve external costs not reflected in its price), that mitigation expenditure may be taxed or subsidised, and that rational mitigation behaviour may involve an element of gamesmanship if individuals believe that mitigation may lead to less action by government to reduce a bad.  It also needs to be borne in mind that optimality does not imply an acceptable income distribution: sometimes considerations of equity can properly take precedence over optimality.

Notes and References

1. Wikipedia  Public bad

2. Wikipedia Quasilinear utility

3. Wikipedia  Public good – Free rider problem

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