An artificial market for pollution permits

APPENDICES
Appendix 1: Mathematical Statement of the Model Appendix 2: Calculations for Estimating Parameters for more Realistic Firms C:\MY DOCUMENTS\RACHEL\WORD\REPORTS\BLUE CIRCLE\BC INDUSTRIES\BC EMISSIONS TRADING 1199 - BCR.DOC There is an intense debate about how best to control emissions of the gases whichcontribute to the greenhouse effect. The UK has signed up to a target for reductions inthese emissions and a number of measures are open to the government in meeting thesetargets. One method of control is to allocate permits without which emissions cannot bemade. Since this is a very restrictive method, one way of making it more flexible is to allowfor a market in such permits.
However, there are concerns about whether such a market is possible and the UKgovernment is currently suggesting that any scheme should be voluntary. Othergovernments are also investigating the potential for trading. The worry is that if only a fewfirms are involved, or permits are misallocated, the market may be unstable or trading maybe extremely thin. This report looks at whether these worries are well founded or not.
In the report, we set out an initial framework of analysis for understanding how well amarket for permits might function. In particular, we address the questions: how many firms trading at what sort of level appear to be necessary for stability? It is not possible to study the evolution of an existing market for permits in the UK oranywhere else, for the obvious reason that such markets do not exist. Trading does takeplace in markets for power and some of the power brokers offer options for dealing inemissions. However, these markets are rather different from the market in a productwhich has yet to exist. In the case of power trading, there are real markets, while existingemissions trading is for a product which has yet to have any real standing.
To cope with this, we investigate the problem by creating an artifical economy within acomputer, populated by firms following straightforward rules of behaviour. Many different'histories' of how firms learn to operate once a permit market is created can be generated,and the evolution of the market examined.
Of necessity, any model must be a considerable simplification of reality. This is noexception. We assume that firms are very familiar with their own businesses, and with thecosts of abating pollution. However, we assume, deliberately, that firms are initially almostcompletely ignorant of how to behave in the permit market. They use very simple learningrules as the market develops. In this way, we can focus entirely on how the permit marketdevelops.
Because the model is very simple, it can be solved analytically for the 'equilibrium' price ofpermits. At this price, the permit market can be said to be generating an efficientdistribution of output and pollution.
Initially, we assume that firms are very similar to each other. They therefore have littleincentive to trade. They also follow simple rules of learning about the value of permits Under these assumptions, prices in the permit market are at first very volatile, as firms learnhow to operate in the market. But, in general, even though the number of trades carriedout is small, the market is an orderly one : price volatility in the permit market soon drops dramatically the market price for permits then moves around close to its theoretical equilibriumlevel In other words, even in a market which by construction is 'thin', the permit price showsvery considerable stability, and moves around the theoretically 'efficient' price (the price,like any market in reality, never settles for ever at a single level, but changes from period toperiod).
These results hold even when as few as six firms are postulated to take part in the market.
We also develop a more realistic version of the model, in which firms differ in terms ofsize, of the amount of pollution per unit of output, and of the costs they face in carryingout abatement of pollution.
A crucial finding is that the greater the number of ways in which firms differ, the greaterthe number of trades which are carried out in the permits market.
There are a number of ways in which the model can be developed still further. Thelearning rules, for example, can be made more sophisticated. The impact of various typesof taxes and/or regulations can be incorporated into the model, and so on. But theprototype system reported here generates some very interesting findings.
Not so long ago, the production of the products now known as greenhouse gases camefree. While environmentalists had voiced concern for years about their effect onatmosphere and consequent rises in temperature, reactions to these warnings werepiecemeal and halfhearted. There was no way to put a value (or disvalue) on the production of such gases. The Kyoto Protocol, agreed on 11th December 1997, changed all that. It was devised as a route to controlling global warming by setting targets forproduction of these commodities – though it has yet to be ratified by the US. In settingsuch targets, for the first time there is a way of putting a cost on these gases.
Under the agreement, the European Union is committed to reducing emissions ofgreenhouse gases (six gases, including carbon dioxide) to 8% below 1990 levels by theperiod 2008 to 2012. As its share of the EU commitment, the UK has accepted – in June1998 – a legal target of a 12.5% reduction on a 1990 base. This means finding an additionalreduction of 5 million tonnes of carbon above that which is expected to be delivered byexisting policies. The government has also suggested a more exacting domestic target of a20% reduction in carbon dioxide (CO2) by 2010, though this is still being discussed.
The Kyoto protocol itself proposes the establishment of an international trading scheme aswell as two mechanisms whereby countries can use savings elsewhere to compensate fortheir own emissions. The international trading scheme is still in its infancy and is notexpected to be sorted out until the 6th Conference of Parties at the end of next year. Eventhen, issues may not be fully resolved.
In the meantime a number of countries have been working on their own schemes. Withinthe European Union, for example, Denmark is developing a domestic emissions tradingscheme. They have introduced a tradable quota scheme for their electricity producers tobegin operating from the year 2000, and targets have been set to 2003. Admittedly, thescope of the scheme is currently very limited but the intention is to expand the scheme toinclude other electricity producers across Scandinavia.
This could include Norway, where the Parliament has set up a commission to design adomestic emissions trading scheme. This is due to report by the end of the year. Thetrading system is expected to apply to those sectors which are exempt from Norway’s CO2tax – which includes much of heavy industry - and other sectors could also be included.
In Australia, a consultation exercise is under way on a number of the difficult design issuesassociated with a domestic trading scheme, including allocation and coverage. Twodiscussion papers have already been published, and two more are in the pipeline. Althoughthis is very much “work in progress”, it seems that the intention is for as comprehensive ascheme as possible. This could involve a combined upstream and downstream approach.
The pace at which trading schemes are being developed is best illustrated by Canada andthe United States, where trading in carbon is already underway. GERT – the GreenhouseGas Emissions Reduction Trading Pilot – was established in Canada in June 1998. Sincethen there have been a number of trades in emissions reduction credits. These credits canbe generated by reducing existing emissions; avoiding an increase in emissions; or throughcarbon sequestration. In the United States a shadow market in CO2 is already up andrunning. This is without any rules or arrangements for carbon trading actually being inplace. But there is proposed legislation in Congress which could formalise this carbonmarket.
In the UK, a report published by the government in November 1998 and prepared by LordMarshall, concluded that a trading scheme offered a number of advantages in meetingemissions targets Trading schemes give firms legal targets to reduce emissions. But they allow companies that can reduceemissions more easily to go further, and to sell the excess to companies finding it more difficult or expensiveto meet their targets. In this way emissions reductions take place where it is cheapest, allowing targetsoverall to be reached more cost-effectively. This attractive flexibility for individual firms is combined withcertainty for the regulator. With a fixed number of permits in circulation, provided that the complianceregime is robust, the regulator knows in advance what overall minimum reduction in emissions will result. 1 “Economic Instruments and the business use of energy”, A Report by Lord Marshall, November 1998, para 44, p11. However, despite recognising these clear advantages in principle, Marshall was scepticalabout the speed at which an emissions trading system can be introduced, and about howwide its coverage can be. Nonetheless, government remains keen on developing a tradingscheme as the following quotations, taken from a recent speech by Ian Coates at the DTI,make clear.
“I firmly believe that emissions trading will form a crucial element in our long-term strategy to reducegreenhouse gas emissions.” “I believe such a trading system can both encourage business find the most cost-effective ways of reducingemissions, and give invaluable practical experience of a trading system ahead of the introduction of aninternational trading system.” “The development of a domestic scheme offers us the opportunity to make significant progress in reducing ouremissions, but in a way that allows the most cost-effective approaches to be employed across industry.” These are quotations from John Battle – then Minister of State at DTI – Patricia Hewitt –then Economic Secretary at the Treasury – and Michael Meacher – who remains theMinister for the Environment.
The desire to introduce a trading scheme is not just confined to the government. A groupconvened by ACBE (Advisory Committee on Business and the Environment) andCBI(Confederation of British Industry) has been working since the middle of 1998 toorganise an emissions trading scheme for the UK. This effort has been accelerated by thegovernment’s proposals for the Climate Change Levy and the negotiations with energyintensive sectors to set targets for emissions reduction alongside a possible reduction in thetax rate.
A set of proposals has now been developed which are practical and workable. They werepresented to ministers on October 27th and have been extensively discussed with officialsof the (Department of Environment, Transport and the Regions (DETR), Department ofTrade and Industry (DTI) and the Treasury. They have also been reflected in theChancellor’s Pre Budget Report on November 9th this year.
Many details remain to be worked out – but now that the principles have been set, a newworry has emerged. It is clear that it is possible to set up a market, but this simply begs thequestion of whether the market itself will work. If only a few companies trade, or they areignorant of how to assess costs and benefits, the market may be far too volatile or thin.
This report looks at these problems. It uses innovative techniques to look at how a marketin permits for emissions could develop, even under very restrictive conditions. It showsthat firms are capable of learning how to trade, even when they have short memories andthe allocation of permits has uncertainties attached to it.
This is obviously not the final word. The proposed scheme has a number of specialfeatures which we are working on incorporating into the model. These include the abilityto bank permits for specified periods as well as the existence of firms with allocations ondifferent bases – those who have accepted absolute targets compared to those with energyefficiency targets for example.
We consider a simple situation in which firms produce one good and ‘pollution’. They havecosts associated with producing their output and they can also reduce pollution byincurring costs. They have an initial allotment of pollution permits for one period whichthey may buy and sell before making their production and pollution level choices. Theylearn about the profitability of their behaviour in the permit market and use theirexperience to modify their choices.
In order to illustrate the process of how a market in permits evolves, we assume to beginwith that all firms in the model are identical. But even under this highly simplifyingassumption, as a result of the operation of the permit market, production and profit levelssubsequently differ across firms. We set out the results of the evolution of both the marketprice for permits and the number of trades which are are carried out.
Next, we show the results when the model is made more realistic. We look at whathappens when firms differ in size, in their allocation of permits, in their overall costfunctions, and in their costs of reducing pollution (the abatement function). The model isset up to be as realistic as possible in terms of scales of output and pollution, usinginformation in the Marshall Report, Economic instruments and the business use of energy,published in November 1998. A description of how we used this information is set out inAppendix 2.
The formal mathematical statement of the model is given in Appendix 1.
We assume that firms can sell all of their output at a constant price. In other words weassume that the demand for the final product is perfectly elastic. The same applies to inputmarkets (apart from the pollution permits). We make these simplifying assumptions in orderto focus on the market for permits. Whilst they are unrealistic over the whole range ofpossible production levels of actual firms, in terms of marginal variations in output aroundexisting levels, they are not completely unreasonable.
Each firm knows the cost of producing any amount of its commodity . It knows howmuch pollution is generated by any given level of production and also knows the cost ofabating pollution by any amount. The firm does not know the structures of costs andabatements of other firms. Again, in terms of small movements around existing levels ofproduction, these are not completely unrealistic assumptions.
The amount of pollution a firm can generate is limited by its holdings of pollution permits.
Each single permit allows a firm to produce one unit of pollution. Permits are valid for justone period.
The model is essentially a single period model. Permits are issued which are valid for afixed amount of time. An initial sequence of trading takes place, as a result of which firmsdecide their output and abatement levels for the whole of the period.
In this prototype model, the steps of the sequence are somewhat artifical, in order to focuson the learning process. A similar type of sequence is used widely in economic theory, and can be thought of in the following way. Each firm chooses an initial set of prices at whichit would be willing to buy and sell permits. No actual trading takes place at these prices.
Instead, they are submitted to a central agency or broker. This agency or broker theninforms firms about what would happen to their bids, and what the price established in themarket would be if trade were to take place on the basis of the initial bids.
Firms then choose what their output and abatement levels would be if trade were to takeplace on this basis, and compute the consequences for their profitability. They then submita revised set of buy-and-sell prices. The agency or broker then informs them of theconsequences of these new bids, and again the firms decide what output and abatementwould be on the basis of these new prices, and the resulting consequences for profitability.
This whole process is then repeated a large number of times.
2 Economists will recognise this as being analogous to the process of tatonnement in general equilibrium theory The focus of the model is upon the evolution of the market price for permits during thisprocess. Does the market price converge to its theoretical equilibrium level? If it does,how many steps of the process are required before the price gets close to this level? If theprice does get close to this level, but the process of submitting bids is allowed to continuefor many more steps, does the price remain close to this level? We assume that a firm knows its business well, and will choose the optimal output level(with corresponding optimal pollution abatement level) given its actual permit holdings.
The question, then, is how a firm chooses its permit holdings. Each step, firms do thefollowing: given their currently allocated permits, their target determines their demand orsupply for permits and they then send this to the market together with a price if a firm would have been successful in its previous buy attempt, it will decrease itsprice according to a simple rule. If it would have been unsuccessful, it will increaseits price, again according to a simple rule. These statements hold in reverse if thefirm wants to sell.
We analyse a double auction. This lines up all bids for permits in descending order of priceand all offers to sell in increasing order of price. Given these bids and offers submitted bythe firms, a market clearing price is determined. This is the price at which the number ofpermits that individuals are prepared to sell are just equal to the number of permits thatother individual firms are ready to buy. We assume that all trades would take place at thatprice. Since permits can only be exchanged by one unit at a time (i.e. are non-divisible) the"equilibrium price" may not exactly clear the market and some rationing might occur. Thisis taken care of by putting the traders in a random order, and allocating the availablepermits on a first-come first served basis.
At the outset, firms understand very little either about the permit market, or about thevalue of permits. When the market for permits is introduced, each firm chooses, atrandom, the price at which it is prepared to buy and the price at which it is prepared to sellpermits, from a uniform distribution over a specified price range which is common to allfirms. In other words, firms initially value permits at random from within a specified rangeof values.
Learning about permits takes place according to the profits which firms would experienceif actual trade were to take place at the price established during any particular step of thetrading sequence. Given a firm's allocated permits and the outcome of the market forpermits, it would end up with a certain permit level, which (given the production, pollution,and abatement costs) eventually determines its profits.
In other words, during the trading sequence, a firm is informed of a sequence of permitslevels, one for each step of the process and the corresponding profits. We assume thatfirms modify their offers to buy or sell permits in the light of what happened in theprevious step. If profits would have improved, they continue to change in the samedirection. Slightly more formally, each firm uses a hill climbing algorithm to determine itstarget permit levels. That is, a firm reviews what its profits would have been over theprevious k steps. It identifies the step at which its profits would have been highest, andsets its target permit level equal to its actual level at this step, plus a small random number.
This rule is based upon a straightforward methodology used in search algorithms. If thenumber of steps, k, which the firm reviews is small, it forgets information very quickly.
But even under these conditions, with k set as low as 2, the results obtained from themodel are not qualitatively different to those reported here. The results set out are basedupon a value of k equal to 7, which allows for a more realistic, albeit still rather naïve,learning process.
In order to focus explicitly upon the evolution of the market for permits, we initiallyassume that all firms are identical in terms of size, the amount of pollution they produce,their overall cost functions, and their costs incurred in abating pollution.
This may be thought to be so unrealistic a set of assumptions that it is not worth exploring.
But in fact it yields very valuable information. By assumption, before trading in permits isintroduced, firms by definition are identical. In such circumstances, there is as littleincentive to trade as possible. Yet, to anticipate, even with a very small number of firms,an orderly market in permits does develop.
The model has been simulated many times on the computer, and a large number of artifical'histories' developed under a range of assumptions. The outcome of this very basic modelis most sensitive to variations in the number of firms which are involved, and weconcentrate on the impact of this in reporting the results.
It may be useful, however, to have some feel for the various scales of operation of theresults of the basic model reported here. The initial range of prices at which firms valuepermits is set to be between 0 and 2000. Each firm receives an allocation of 100 permitsin each period. The revenue from producing and selling an extra unit of output is 1000.
The cost structure is such that it is never profitable for a firm to produce more than 500units of output.
Variations in these assumptions can be made. For example, the wider the range of pricefrom which firms form their initial views on the values of permits, the higher is the initialdegree of volatility in the price of permits. The fewer permits are issued - in other wordsthe more severe the degree of pollution control - the more volatile are the initialmovements in price. But it is the number of firms to which the results of the model aremost sensitive.
Figure 1 shows a typical evolution of the price of permits over the steps of the tradingsequence when there are 40 firms operating in the model.
Typical Evolution of Market Price of Permits The chart show the qualitative properties of the evolution of price with 40 firms veryclearly. There is a relatively short initial number of steps during which firms learn about thevalue of permits, and in which price is volatile. But the market settles rapidly to movearound its theoretical equilibrium value of some 1000. The market never becomescompletely static, and trading continues to take place. But even in what is, by deliberateconstruction, a thin market in which few permits are traded relative to the overall scale ofoutput, price converges to around its equilibrium level. With more than 40 firms, theperiod of high price volatility becomes somewhat shorter, but even with 200 firms it isnever eliminated. But once there are as many as 40 firms in the market, the addition offurther firms makes little difference to the results.
The number of trades which would be carried out in each step of the trading process is setout for a typical solution of the model in Figure 2.
During an initial period of intensive learning about the permits market, the number oftrades per period is of the order of 50, but then settles down to between 10 and 20.
Reducing the number of firms below 40 leads to a slight lengthening of the period of highvolatility, and to a marginally higher degree of volatility even when the model has run formany steps. But convergence to equilibrium is still attained. Figure 3 illustrates a typicalsolution of the model with only 15 firms.
Typical Evolution of Market Price of Permits Comparing the results with those of figure 1, it is apparent that they are qualitativelysimilar. The number of trades carried out is also similar, as Figure 4 shows.
A different kind of behaviour is observed once the number of firms is reduced evenfurther. A certain degree of more erratic behaviour is sometimes observed with 12 firms,and with as few as 10 firms, a typical evolution of the permit price is plotted in Figure 5.
Typical Evolution of Market Price of Permits The price does converge around its equilibrium value, although more slowly than in theprevious charts. But the relatively modest fluctuations in price which then take place arepunctuated by rather more frequent, short bursts of large, erratic movements. The gaps inthe solid line on the chart indicate periods when no trading takes place.
With only 10 firms, the number of trades which are carried out in a typical solution is lowerthan with 15 or 40 firms. This is exactly as one might expect, given the small number offirms in this example.
When the number of firms is reduced below 10, the price does still eventually converge toaround its theoretical equilibrium level. But the degree of volatility can be considerablyhigher, and in a number of periods no trading at all takes place. It is less easy tocharacterise a typical solution in these circumstances, but Figure 7 sets out one example ofthe evolution of price when the number of firms is as few as 6. Again, the gaps in the solidline on the chart indicate periods when no trading takes place. As one would expect thisoccurs more frequently as the number of firms decreases.
Typical Evolution of Market Price of Permits Figure 8 confirms the point about the number of trades when the number of firmsparticipating in the market is very small. Even at its peak level, the number of trades in anygiven period is barely into double figures.
In summary, we have presented results in this section of the paper showing the evolutionof a market for permits using a set of assumptions under which there is little incentive totrade. Deliberately, the number of firms postulated to take part in the market is very small.
They are assumed to be almost wholly ignorant of the potential value of a permit beforethe market is introduced, and they follow a rather naive procedure for learning about thevalue of a permit.
But even in these circumstances, an orderly market in traded permits does develop. After ainitial period of fluctuation, which tends to be larger the fewer the number of firmsinvolved, the market settles down. Futher, it settles around its theoretical equilibriumprice. This means that resources are allocated in an efficient way.
In this section, we report on what happens when a greater degree of realism is introducedto the nature of the firms in the model. It must be emphasised that this is not intended byany means to be an exact description of reality, either in the description of the firms, or inthe particular trading scheme. The intention is still to explore at a rather general level theimplications of introducing a market in which permits can be traded.
A key difference with the previous section is that firms differ in size. The largest firm isaround fifteen times larger than the smallest. The degree to which firms contribute topollution varies even more, with the heaviest polluter contributing some forty times moreper unit of output than the lightest.
The Marshall report contains data on the estimated gross output of industrial sectors, andtheir estimated CO2 emissions. We concentrated on the 22 industrial sectors, coveringindustries such as mining, paper, chemicals. Information on the number of firms in eachsector is available in the August 1999 DTI Publication SME Statistics for the United Kingdom1998, which despite its title contains data on firms of all sizes. We could therefore obtainthe average in each industrial sector of the size of each firm and the level of pollutionassociated with its output.
We further assume that only one in every thousand firms takes part in the market forpermits. This still gives a total of 271 firms which participate in the theoretical model - amuch higher figure than the more artificial examples set out above.
Firms also differ in respect of their cost of abatement functions. The basic shape of thefunction is the same for all firms: the cost of reducing pollution by a single unit increases,the more the total size of the reduction. In other words, costs rise quite sharply. But thespeed with which this happens is allowed to vary across firms. We do not pretend to haveknowledge of the actual costs of abatment which any firm faces in reality. But real firmsdo differ in this respect, and our model now reflects the existence of such differences.
In Section 3.1, we reported a typical result obtained when the model is solved. In thissection, we report the results of 500 repeated solutions of the model. This enables a richerview of the range of solutions to be obtained, even though the charts themselves looksomewhat more complicated.
Results were obtained under three sets of assumptions about the allocation of permits: The number of permits that are allocated to each firm is exactly in proportion totheir sizes, so that a firm which produces twice as much as another receives twiceas many permits The same principle is used as in (1), but the actual number allocated to each firmcontains a certain amount of random variation. This means that the number ofpermits received by 95 per cent of all firms in the model is within a range of plus orminus 20 per cent of the number they would have received purely on the basis oftheir relative sizes.
As with (2), but the range of variation around the allocation which would havebeen made purely on the basis of relative size is much larger, at plus or minus 40per cent.
With firms operating on the basis of the assumptions made above, there is far moreincentive to trade in the permits market. It is precisely the fact that firms are different which givesthem more reason to trade. We examined this proposition thoroughly, by investigating what happens to the number oftrades carried out when the assumptions of section 3.1 are relaxed both on a one-by-onebasis and in various combinations. So, for example, we assumed that firms are identicalexcept for the amount of pollution associated with each unit of output. We then restoredthe assumption that they are identical in this respect, but made their relative sizes different.
Detailed results on this are available from Volterra Consulting. But, quite clearly, the greaterthe differences between firms, the more incentive they have to trade, and the more trades are carried out.
This can be illustrated by comparing Figures 9 and 10. Figure 9 shows the number oftrades carried out in 500 solutions of the model assuming firms are identical. This isexactly as in section 3.1 above, except that the number of firms is the same as in the resultsreported when firms are different. In other words, rather than involving just 40, or 15 or10 firms, Figure 9 shows the results for 271 identical firms.
The chart needs a little explanation. The middle of the three lines - the dotted line in thechart - shows the average number of trades carried out in each period when the model issolved 500 times. The upper line shows the maximum number which are carried out in eachperiod in any of the 500 different solutions. In other words, it is not the outcome of thesingle solution which on average has the highest number of trades, but rather more thanthis. It shows the highest number carried out in each period in any of the individualsolutions - the maximum of the maximums, as it were. The bottom line shows, similarly,the minimum of the minimums.
The number of trades carrried out is larger than with a smaller number of identical firms,but not dramatically so. Figure 2 on page x shows that with 40 firms, the number of tradesper period settles down at between 15 and 20. Figure 9 shows that with approximately sixtimes this number of firms, the number of trades is approximately six times higher onaverage, at around 100.
Figure 10 reports results over 500 solutions of the model for 271 firms which are differentin the ways described above. The numbers of permits are allocated exactly in proportionto the relative sizes of the firms. The number of trades is substantially higher, settlingaround 320.
Figure 11 shows the results when the number of permits is allocated in proportion to thesizes of firms, plus the larger error term. This means that for 95 percent of firms, thenumber of permits allocated is plus or minus 40 per cent of they would have been allocatedjust on the basis of relative size,. Perhaps surprisingly, this does not make a great deal ofdifference to the overall number of trades carried out, with the number per period settlingat just under 400.
Trading Volume with Errors in Allocation of Permits Finally, Figures 12 and 13 show the level of market price with 271 heterogenous firms.
Figure 12 reports results with permits allocated exactly in proportion with output, andFgure 13 shows them allocated with error as in Figure 11. The middle, dotted line is againthe average price of 500 repeated solutions of the model.
Levels of Market Price for More Realistic Firms Levels of Market Price with Errors in Allocation of Permits In summary, the results of this section re-inforce those of section 3.1. A market inemissions permits is likely to be orderly, in the sense that a reasonably large number ofpermits will be traded and, after an initial period of learning about the value of permits, theprice of permits does not in general exhibit large fluctuations.
This simple market model enables us to study the behaviour of firms who have noexperience of the market for permits and who use very simple rules to modify their bidsand offers. In general, despite the fact that firms are deliberately given naive rules oflearning about the value of permits, and that by construction the level of trading is thin, theprice of permits converges rapidly around its theoretical equilibrium level. In the contextof the model, the values of both output and pollution are at their optimal levels.
Even with a very small number of firms who, by assumption, have little incentive to trade,a market does emerge which is orderly. The price converges to around its equilibriumlevel, even though the number of trades actually carried out is small.
The greater the degree of difference between firms, in terms of their sizes, the degree atwhich they pollute, and the costs they incur in abating pollution, the higher the number oftrades which will be carried out in the permits market.
This model, despite its simplicity, provides a simple and flexible tool for testing a numberof interesting hypotheses about the evolution of prices on such a market and on theproduction and pollution levels attained. Many questions of interest to the publicauthorities involved in this area can be examined in extensions of the model.
Appendix 1:Mathematical Statement of the Model We assume that the firms are identical in the structure of their costs and revenue. They only differ in terms of parameter values (α, β, γ). The cost of producing y units of output is and the revenue from selling y units of output is In the simulations carried out we set ρ = 1000. The costs of production increase quadratically but the revenue generated rises more slowly.
Each unit of production is assumed to generate α units of pollution, i.e. if P(y) is the level of pollution associated with the production of y units of output then The parameter α was given the value 0.5.
The cost of abatement was also taken to increase quadratically, i.e. the cost of reducing thelevel of pollution by z units is If a firm is able to produce z0 units of pollution as a result of holding the necessary permits,then the cost of reducing its level of pollution to z0 when producing y units of output isgiven by (This assumes that the amount of pollution arising from this level of output is greater thanz0.) Combining the costs of production and abatement, we can find the marginal cost ofproducing a unit of output, MC(y) = 2 y / γ + 2α (β / γ) (α y - z0) In the first case the level of pollution is below the permitted level z0, in the second case itwould otherwise be above z0 and so incurs an abatement cost.
The level of output of a firm, given a certain number of permits can then be calculated byequating the marginal revenue and marginal cost curves and solving for y.
Similarly, if the firm has a number of permits, we can work out the price that it will bewilling to pay for an extra permit. This will be just equal to the amount of extra profit thatwill be gained from the extra output produced as a result of having the extra permit.
Explicit details can be obtained from the authors.
The data in the report on homogeneous firms was generated with identical costs andrevenues for each firm. The parameter values were: and each firm was given 100 permits. These parameters mean the firms should value anextra permit at 944 (which we can see that they do from the results in Figures 1,3,5 and 7).
Appendix 2:Calculations for Estimating Parameters for moreRealistic Firms The left hand side of the table below shows data taken from the DTI: ‘Small and MediumEnterprise (SME), Statistics for the United Kingdom, 1998’. Together with data from theMarshall Report: ‘Economic instruments and the business use of energy’, November 1998.
The right hand side shows the estimates used in our model of more realistic firms(explanation below).
Actual Data
Estimates for Model
* Pollution per unit of output (tonnes of CO2 per £00s of output)** Average size of firm £mn Number of Firms in the model was calculated by dividing the actual number of firms byone thousand and rounding to the nearest integer, therefore it is as if one in every thousandfirms decides to trade. For each classification sector, alpha was calculated by dividingestimated CO2 emissions in 10,000s of tonnes by Gross output in millions of tonnes, then multiplying by ten. This gives an average amount of pollution per unit of output. Gammawas calculated by dividing Gross output by the actual number of firms, to give an averagesize of firm in each classification sector. No data is currently available on the cost to firmsof abating pollution. Thus we estimated the beta value for each firm by drawing it from alognormal distribution with mean 15 and standard deviation 0.7, giving a minimum of 3and a maximum of 80.

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