62488 The Economics of Pollution Trading and Pricing under Regulatory Uncertainty by Odin K. Knudsen and Pasquale L. Scandizzo Abstract In this paper, we explore the effects of uncertainty on pricing of pollution permits, through the use of a dynamic model of pollution markets. We consider two major sources of uncertainty – that arising from the volatility of demand for the underlying resource and that coming from the regulatory environment. Both sources of uncertainty are common in pollution permit trading as not only does the market respond to the volatility of fundamentals but also to the vagaries of the institutional structure, created by public policy and enforced through regulation. The paper shows that even in the presence of strategic behavior on the part of the agents involved, the trading of permits effectively reduces emissions, and pricing does reflect opportunity costs and environmental objectives. Furthermore, and somewhat paradoxically, the higher uncertainty, the greater the impact of regulation. Introduction Pollution permits and trading are becoming increasing important as a market friendly instrument to control pollution at lower costs. Although such schemes have had their birth with sulfur dioxide trading in the United States, they really did not hit international prominence until the Kyoto Protocol came into force. By building into Protocol carbon emission trading and with the emergence of the European Trading System (ETS), pollution trading became a multi-billion dollar market. Despite their growth, the economics underlying these pollution markets are not well understood. Although it is assumed that these markets promote least cost means of meeting targets on carbon emissions, the economics of pricing of permits 1 and penalties are not well understood, along with a host of other issues associated with important policy decisions, including regulatory uncertainty. Because the markets for an externality such as pollution are essentially artificial markets, created by legislation, an additional form of uncertainty is added to the normal randomness of prices: the vagaries of regulatory enforcement. This regulatory uncertainty is quite evident in the carbon emissions trading of the Clean Development Mechanism (CDM) of the Kyoto Protocol and under the ETS with a complex and judicial system of enforcement. Under the Kyoto Protocol and the Marrakesh Accords, three forms of emissions trading were permitted: 1. The trading of Certified Emission Reductions (CERs) under the CDM; 2. The trading of Emission Reduction Units ERUs) under the Joint Implementation mechanism; and 3. The trading of Assigned Amount Units (AAUs) under International Emissions Trading. Each one of these mechanisms for carbon trading face high regulatory uncertainty. The CDM as the regulations are enforced by a semi-political CDM Executive Board which has made inconsistent decisions and later reversed others. With each change the market has responded with a major variation in prices. Furthermore the Executive Board has relied on Designated Operating Entities (DOEs) to enforce the regulations and standards set by the EB. Since the DOEs are private, standards of validation and verification differ between DOEs. A project developer may find that depending on the particular DOE, even though certified by the EB, a different intensity of enforcement. The other two markets face similar regulatory uncertainty but of different forms. The ERUs in JI market depends on a supervisory body similar to the EB of yet unknown dimensions and rigor. The market for AAUs while in theory the simplest, depending only on governments to trade a relatively known instrument, has the uncertainty of not only how many AAUs does a country actually possess but also on the political demand by buying countries that the AAUs be greened, that is associated with some other environmental investment scheme of unknown dimensions and rules. 2 These regulatory uncertainties are coupled with the normal market drivers of carbon, e.g. energy prices, industrial activity, economic growth etc. All these uncertainties are focused on the ETS market which accepts for compliance purposes EUAs, CERs, ERUs and indirectly AAUs. The rules of this market are administered by the European Commission (EC) and depend on the allocation of EUAs to the market and which industries will fall under the ETS and which will not. Furthermore, when there is a miscalculation as with the May 2006 collapse of price of EUAs of 2006 vintage because of an overallocation of EUAs, politics quickly emerges to try and adjust enforcement or standards. Enforcement mechanisms on industries which receive the EUAs also are uncertain. The EC has the weapon to enforce compliance but only at the national level through the European Court of Justice where fines can be imposed on member states for non-compliance with EC regulations. However the process is laborious, usually taking many years and with uncertain results both on rulings and penalties. At the national level, each EU government finds its own means of enforcement at the industry or entity level. This uncertainty creates a gamesmanship between the EC, the EU states and the industries that eventually have to face the imposition of regulation and possible fines. Furthermore the EC must pursue its enforcement in a political environment and sometimes without the complete capacity to deal with all the legal filings, documentation and defenses. Finally, the master stroke of uncertainty is no one knows for sure if the markets will continue and if they do, what form they are likely to take. The Kyoto Protocol expires in 2012 and the EC has not announced the 2008 allocations or coverage. Meanwhile trading of all these carbon emissions is taking place at a frenzy pace, and with a great deal of fluctuations in prices. The World Bank reports that trading of all Kyoto instruments has exceeded $20 billion in 2006. To model such a market in any detail would create such a black box that analytic light is unlikely to emerge. Instead our purpose in the paper is to explore in simple abstract models how regulatory uncertainty could affect the market and prices of permits. Even though several authors have offered a basic treatment (Field 1997, Kahn 1998 Tietenberg 2000, Weber 2002), economic and regulatory issues behind the properties and use of these 3 innovative market instruments still need to be explored, particularly when markets are dynamic and the fundamental drivers are themselves uncertain. In order to approach the problem gradually, we present a model that focuses on the link between pollution abatement penalties and demand and supply of permits when market demand is stochastic and regulation is uncertain. Trading permits under uncertainty allows firms to behave strategically, by optimally deciding when to exercise opportunities and managing threats of penalties from regulators. From the policy perspective, this approach to pollution trading under uncertainty brings forth the effect of a pollution penalty on the market for permits and on the price of output, how the transaction costs of the regulator affect the price of permits, and how increased level of uncertainty in general affects the market. In doing this, we are not attempting to model exactly the complexity of any single market such as the ETS but to build an approximation that yields insight into the effect of various policy parameters on the market for permits and output. We model the behavior of the regulator as an agent that extracts penalties on firms that exceed their allowances supplemented by market purchased permits but does so only when it is able to cover the transaction costs of enforcement and when the violation is not caused by a transitory increase in output demand of the firm. On the industry side, the firm knows that the regulator will not attack at any violation but only when they suspect that the violation is more permanent, in a sense, imbedded into the fundamentals of the firm and market. But the firm does not know how the demand for output will emerge over the future and may find itself in the position of polluting beyond its allowances and be forced into the market for permits when their prices are high to avoid the imposition by the regulator of penalties. On the other hand, it may find that demand for its output has fallen and that it is in a position to sell to the market excess allowance. In a dynamic market and regulatory regime, the firm has to decide whether to be short or long in permits and by how much to buffer against the uncertainty of the market and the behavior of the regulator. In turn, the policymaker has to decide what penalties to impose on violations and how overall allocation of permits will affect the industry and the price of output. The Basic Model The basic model is based on the idea that trading permits under uncertainty allows firms to behave strategically, by accounting for price 4 uncertainty and anticipating the regulator’s strategy in implementing the regulation. We begin from an industrial sector base, such as the power sector, where demand is assumed to be exogenous and stochastic and output adjusts to demand in every period. Specifically the output (and demand) of the industrial sector Q is assumed to be a random variable following a stochastic process of the Brownian motion variety: (1) dQ = αQdt + σQdz dz being a random variable with mean zero and variance equal dt . The parameters α and σ2 represent respectively the drift or trend in demand and the variance. Within the sector, firms (depicted by the subscript i) are technologically heterogeneous with emissions y i assumed for simplicity to be proportional to their output Qi : (2) yi = Qi u i The firm has a share wi of the sectorial output and therefore is responsible for wi ui Q of the sector’s emissions where ui is the emissions per unit of output of the ith firm. The more clean technologically the firm, the lower is the emissions factor ui . The government is assumed to have implemented a cap and trade system where each firm is allocated an emissions allowance Yi that it can supplement through market purchases or sell to other firms in the industry at a market determined price p . The quantity of the permit qi acquired in the market adds to its overall allowances or if sold detracts from it. To deter firms from exceeding their emission allowances, the regulator is authorized to impose a fine at a rate of γ of the amount of emissions that exceed the allowance plus the net amount of permits derived from the trading of emissions. However, imposing a sanction on the ith firm has an implementation cost to the regulator of Vi . Because of this implementation cost, the regulator in attempting to limit emissions relies not only on the collection of the fine when it is worth doing, but also on the deterrent value of the threat to impose this sanction. 5 Furthermore the regulator realizes that because the output of emissions is stochastic, a temporary or random violation of the allowance limit may not be worth the transaction costs of implement the procedures to collect the fine or penalty. But if the allowance is exceeded by a significant enough margin, the regulator is prepared to impose the penalty. 1 The firm on its side is attempting to maximize its net worth as measured by its net present value as any other company but also must decide whether to purchase emission permits to cover the possibility of an increase in emissions as demand for example for electricity randomly increases due to fluctuations in weather. It also must manage the threat of the regulator imposing potentially stiff penalties on its excessive emissions. Over the longer term, the firm can invest in cleaner technologies but in the short term it faces both the vagaries of the market for permits and the possibility of fines from the regulator. The firms is therefore trying to maximize its expected net present value Π i : Pwi Q (3) Π i = − C ( wi ) − pqi − F (Qi ) , δ where P is price of output, C ( wi ) is a concave cost function 2 dC d C ( ≥ 0, > 0 ), C indicating irreversible investment and operating d ( wi ) d ( wi2 ) costs appropriately discounted, and F (Qi ) is the liability threat coming from the regulator. The discount factor δ is the difference between the risk free rate of interest r and the drift α of the stochastic process or δ = r − α . The term pqi represents the value of permits acquired by the firm in pollution trading. The last term of (3), F (Qi ) , requires a longer explanation. Because of uncertainty on both pollution levels and the regulator’s behavior, the value of the term depends on the circumstances. If the regulator is expected to impose a fine on the firm, it will have the value of the expected penalty; but 1 Note that in many regulatory or legal situations, enforcement is not absolute. A simple reference to vehicle speeding where the police may decide to let some unknown violation of speed limits take place or in over law enforcement, eg taxes or recreational drugs, where certain violations may not be worth the effort of imposing the sanction. 6 obviously is zero if the regulator is expected to decide that the violation is minor and/or will not persist and therefore is not worth the enforcement costs, either directly to the regulator or indirectly to society. If there is a violation that could engage the regulator at some future time, then the option has a value in between these two extremes. To the firm, the possible imposing of the fine therefore is a contingent liability, whose value depends on the option held and/or exercised by the regulator. The value of this option depends on uncertainty, the period of the option and the underlying asset value, in this case the possible fine, along with other parameters such as the interest rate. In turn, this liability option is a call option representing an asset option for the regulator, who will exercise it when the value of the emission is high enough that the value of the contingent liability option equals the expected amount of the fine that can be collected minus the costs of enforcement.2 Public policy and analysis set the rate of the fine for excess emissions at a level that reflects the marginal costs to society of pollution above the targeted amount.3 As we will demonstrate later, there exists a fine that corresponds to each pollution target and vice versa. The regulator therefore does not pursue fines against the polluter at the point when the expected amount of the fine is equal to the costs of enforcement, but waits until the violation has become excessive enough to justify not waiting any longer. At this point, the marginal increase in the option value is just equal to the marginal expected present value of the net collection of the fine or of the rate of the fine if this is constant.4 We assume that firms not only differ in technology and emission levels, but also, idiosyncratically, in the amount of transaction or economic costs that they generate when the regulator tries to sanction them. At any point in time, the option is taking on a value that depends on this expected net marginal fine and on the time when the firm’s liability option held by the regulator is expected to be exercised. 2 This is referred to as the value matching condition. 3 Although the rate of the fine should be part of a complex analysis of benefits and costs, in practice, the regulator’s estimation is driven by a wide set of conflicting positions from industry, the general population, and civil society. After navigating these political and economic considerations, the European Union has set the rate of fine for the second phase (2008 thru 2012) of the ETS at Euro 100 per ton of CO2. 4 This is called the smooth pasting condition. 7 We can express these concepts analytically. Following Dixit and Pindyck (1994[OK1]) the value of the contingent asset or option for the regulator F (Qi ) at time T can be expressed as:  âˆ?  (4) F (Qi ) = sup EQ  ∫ e − Ï? ( s −τ ) γ ( wi Qs u i − Yi − qi )ds − Vi     Ï„ +T    For i= 1 to N. In (4) γ is the ad valorem rate of the fine, while Yi is the pollution cap imposed by the regulator on the ith firm and Vi the cost of implementing the sanction against the ith firm.5 Notice that the fine is raised on the difference between the expected present value of the firm’s emission and total value of the individual emission cap and of the quantity of permits (expressed in emission tons) owned by the firm. In other words, we assume that an irreversible investment in implementation costs has to be made by the agency in charge, in order to recover the fine, once a threshold has been exceeded by the firm. The regulator acts on the expected emissions over a period. Because output is assumed to follow a geometric Brownian motion and the emissions are proportional to output, expected emissions are the same as point emissions. The value of the contingent asset that the regulator holds (Dixit and Pyndick, 1994) is: Q (5a) F (Qi ) = γ ( wi u i − Yi − qi ) − Vi if wi Q ≥ Qi∗ δ when the firm in its expected emissions has exceeded or is just at the trigger point of the regulator Qi∗ ui . For the firms that have not exceeded this trigger point but may be expected to exceed it some time in the future, the value of the contingent asset is: wi Qui β1 Qi° (5b) F (Qi ) = ( ∗ ) [γ ( ui − Yi − qi ) − Vi ] if wi Q < Qi∗ Qi u i δ 5 It is reasonable to assume that the threshold level of emissions Yi is fixed with reference to the expected present value of emissions although some other criteria could be used by the regulator. 8 β1δ V w Qu where δ = α − Ï? , Qi∗u i = (Yi + qi + i ) and ( i ∗ i ) β = Ee − rt is the 1 i β1 − 1 γ Qi u i expected discount factor corresponding to the (stochastic) time of enforcement ti for the ith firm. Consider the value of the firm under the threat of sanction in the case of expected action by the regulator some time in the future (the second case above). Substituting expression (5b) into (3), we obtain: Vi Yi + qi + Pwi Q wi Qu i γ Πi = − C ( wi ) − pqi − γ [ ] β1 ( ) (6) δ β1 Vi β1 − 1 (Yi + q i + ) β1 − 1 γ The expected present value of the firm is affected by the regulation as a contingent liability (the fourth term in equation (6)). Because β1 in equation (6) is inversely proportional to volatility of emissions and asymptotically approaches 1 as volatility goes to infinity (see Dixit and Pyndyck), the contingent liability approaches the certainty fine, that is a fine equal to the amount of emissions times the rate of fine[OK2]. As volatility approaches zero or certainty ( β1 ), the contingent liability also becomes zero (by assumption the firm’s emissions are below the trigger point with certainty in this case). Obviously when emissions are above the trigger point of the regulator’s action, the sanction is applied and the fine becomes equal to the rate of fine times the amount of emissions above the allowance. We now explore the behavior of the firm in the face of this contingent liability. We assume that each firm maximizes net present value by selecting the firm output level as a share of the industrial sector output wi and the number of permits qi which it will buy. Market prices for output and for the permits are determined by the conditions that supply and demand must be equal in both markets or: ∑ wi = 1 and ∑ qi = 0 . i i We first find the value of the number of permits as long or short positions that maximize the value of the firm, by taking the first derivative of (6) with respect to qi and equating it to zero: 9 ∂Π i βδ V Q (7) = −p + γ[ 1 (Yi + qi + i )] − β1i ( wi u i ) β1 = 0 ∂qi ( β 1 − 1) γ δ Equation (7) is a condition for a maximum since, as it can be easily ∂ 2Π i checked, < 0 for all non zero values of q i . Solving for the amount of ∂q i2 permits qi yields: 1 γ β −1 Q V (8) qi = ( ) β ( 1 ) wi ui − Yi − i , 1 p β1 δ γ The firm will display a positive or zero demand for permits ( qi ≥ 0 ) if : β 1 Q p V (9) wi u i ≥ ( ) β1 [( 1 )(Yi + i )] δ γ β1 − 1 γ Note that in (9) the term on the left hand side of the equation is the trigger point for the regulator in the absence of the holding by the firm of permits. If the price of permits is just equal to the ad valorem fine ( γ = p ), the firm will be induced to buy permits only if the expected present value of its emissions at the permit price is higher than the threshold that the regulator will enforce the sanction. On the other hand, if the price of permits p is greater than the tax rate γ , the firm will be induced to buy permits only if its production is above the predictable threshold of sanction, while the opposite will be true if the tax exceeds the price. For the firm selling permits ( qi < 0 ), on the other hand, a price higher than the tax rate will be an incentive to sell, with respect to the indifference case (emission level equal to the threshold so that qi = 0 ). In equilibrium, short and long positions should cancel each other N (demand equals supply), i.e. ∑q i =1 i = 0 for N firms. This implies the following equilibrium price: 10 U U / N β1 (10) p* = γ [ ] β1 = ( ) γ β1 V U* /N (Y + ) β1 − 1 γ In (10), the numerator of the expression in parenthesis N 1 Q U/N = ∑ wi ui denotes the expected emissions of an average firm, while N δ i =1 β1 V the denominator of the same expression, (Y + ) / N is the average β1 − 1 γ value of the threshold of intervention, i.e. the value of emissions that would prompt the regulator to intervene for an average firm without any permits.6 In particular, it can be easily shown (Dixit and Pyndyck, p…) that: U / N β1 (11) ( * ) = Ee −rt U /N Where Ee − rt is the expected value of the discount factor for the stochastic time of the regulator’s intervention for the average firm that is not holding permits. Since holding permits would postpone the day of reckoning, the time in (11) is the earliest time that the average firm will be sanctioned. Expression (10) thus predicts that the equilibrium price of permits will equal the expected present value of the fine ad valorem rate (expressed as a value per unit of output) for an average firm. As expression (11) shows, the discount factor is a function of the ratio between average emission level and the level at which the regulator would impose a fine on a representative firm producing the industry output with an average technology. If this ratio were unity, i.e. the regulator were exactly at the threshold for the average firm, the equilibrium price of permits would be equal to the fine ad valorem rate. While a ratio higher than unity would not be permitted by a rational regulator, its possible occurrence because of the regulator’s negligence will cause the price of permits to exceed the fine, because more firms will be non compliant and regulator action will appear overdue. Vice versa, for values of the thresholds above expected emissions, intervention by the regulator will 6 Note that Y , the pollution cap, can also be interpreted as an amount that can be deducted from the fine base of emissions U . This may include permits issued by the government or any other element that causes emissions to decline (such as, for example, the adoption of a specific abatement technology). 11 appear more remote and price of permits will tend to fall below the value of the fine. An increase in uncertainty, since it will imply an increase in the threshold value of intervention, will be associated to a lower value of the price of permits. Table 1 and figure 1 show how the equilibrium price would evolve V Y+ γ under alternative values of β and g = , if the value of the fine were U fixed at 100 Euro per ton.7 Note that the pollution ratio g is a hybrid term between the ratio of the targeted emissions cap Y relaxed by the aggregate bill (converted to emission tons by the ad valorem fine rate) to the total emissions U of the sector. This pollution ratio is likely always to be less than one unless the emissions gap is not very stringent and the enforcement costs are high. Also recall that the term β is inversely proportional to the uncertainty of output and therefore emissions of the industry. 7 This value is the fine under the second phase of the European Trading Scheme. However, under the ETS the fine is supplemented by the need to cover the gap in emissions to allowances by a market purchase. Also, purchases are allowed from the mechanisms of Clean Development and Joint Implementation. We have abstracted from these complications for simplicity of exposition. 12 Table 1 Equilibrium price (Euro/ton), according to expression (11) under alternative hypotheses on uncertainty and pollution targeting (basic fine value γ = 100 Euro / ton ) Inverse volatility parameter : β Pollution 1.1 1.3 1.5 2 3 4 10 Ratio g 0.95 7.5 15.6 20.2 26.2 31.1 33.2 36.6 0.83 8.6 17.8 23.1 30 35.6 38.0 41.8 0.77 9.3 19.3 25.0 32.5 38.5 41.1 45.3 0.71 10.0 20.8 27.0 35.0 41.5 44.3 48.8 0.67 10.7 22.3 28.9 37.5 44.4 47.4 52.3 0.62 11.4 23.8 30.8 40.0 47.4 50.6 55.8 0.59 12.2 25.3 32.7 42.5 50.4 53.8 59.3 0.50 14.3 29.7 38.5 50.0 59.2 63.3 69.7 13 Figure 1 Equilibrium price (Euro/ton), according to expression (11) under alternative hypotheses on uncertainty and pollution targeting (basic fine value γ = 100 Euro / ton ). 70 60 1.1 50 1.3 40 1.5 30 2 3 20 4 10 10 0 3 10 95 77 67 1. 0. 0. 0. 59 1. 1 5 0. 14 As the figure shows, for plausible values of the pollution ratios and the uncertainty parameter, the price of the permit will tend to be a fraction of the value of the fine. It will only slowly converge to such a value as the reduction target is stepped up and/or the uncertainty decreases. Substituting p * in (8), we find the expression for the demand (supply) of permits for each level of production for the i-th firm: ωiV − Vi (12) qi* = (ω iY − Yi ) + ( ) γ wi ui Q wi ui where ωi = = is the i-th firm’s share of the emission U ∑ wi ui i levels of industry. Equation (12) states that in equilibrium firm demand ( supply) of permits will not depend on stochastic demand Q of the industry but only on the extent to which its emission cap (i.e. its deductible) and the implementation costs that it generates are smaller (greater) than its share, respectively, of pollution caps and implementation costs, based on its contribution to total industry emissions. More specifically, we can distinguish eight different possibilities: Table 2 Alternative determinants of equilibrium demand and supply of permits (ω i Y − Yi ) (ω iV − Vi ) Demand of Permits Demand of Permits ≥0 ≥0 ≥0 ≥0 ≥0 <0 ≥ 0 if < 0 if Vi − ω iV V − ω iV (ω i Y − Yi ) ≥ ( ) (ω i Y − Yi ) < ( i ) γ γ <0 <0 <0 <0 <0 ≥0 ≤ 0 if > 0 if ω iV − Vi ω V − Vi (Yi − ω i Y ) ≥ ( ) (Yi − ω i Y ) < ( i ) γ γ 15 In the four pure demand and supply cases (respectively, first and third row of Table 2 ), demand of permits is positive or negative (i.e. supply is negative or positive), because the actual assignment of the pollution cap and the transaction costs to implement the regulation for the ith firm, fall short (for positive demand) or exceed (for positive supply) its theoretical share on the basis of its contribution to total pollution and to transaction costs. In the four mixed cases (third and fourth row of Table 2), on the other hand, the firm will demand (supply) permit, depending on whether its potential demand (supply) on the basis of its pollution level is greater (smaller) than its potential demand (supply) on the basis of its contribution to implementation costs. In other words, a firm may demand permits because it values pollution more than the regulator does (in terms of allocated caps and implementation costs) or because the costs to recover a fine from it are sufficiently higher than the average costs or both. Substituting (12) back into (5b), we find that the output level that will trigger the regulator’s action against the ith firm is: Qi* Q* βω V (13) u i = wi u i = 1 i (Y + ) δ δ β1 − 1 γ Using expressions (12) and (10), we can also derive the expression for the expenditure for permits of the i-th firm: ωiV − Vi (14) p * qi* = Ee −rt γ [(ωiY − Yi ) + ( )] γ Therefore in equilibrium, each firm will spend (earn) in buying (selling) permits an amount equal to the expected value of the fine for the average firm multiplied by the positive (negative) deviation of its (output based) shares from actual levels of emission and implementation allowances. The Firm’s Output level and Market Equilibrium We now go back to expression (6) and find the profit maximizing value of market share wi . We assume that we are dealing in a regulated or semi-regulated industry like with the power utilities. We hypothesize that the 16 industry bases its output on the expected price for output. Each firm targets its share of stochastic market demand, taking into account the forecast of stochastic demand, the administered price that it will be allowed to charge, investment costs plus the costs involved by the threat of pollution fines and/or of emission permits. If the bids for production are too low (the sum of the shares offered by the firms falls short of unity) as compared to demand, the government increases the administered price, while it lowers it if the output exceed expected demand. Market equilibrium is thus reached when the market price for the output is such that expected demand is equal to the productive capacity allocated and each firm produces its optimal share of total output. Differentiating profit in (6) with respect to wi , using (13), equating to zero and solving for wi , we obtain: PQ0 U* − c1 − ( Ee γ )( )Q0 u i − rt (15) wi = δ U c2 U β1 where Q0 is expected demand at the time of the bid, Ee −rt = ( ) , and we U* have used a quadratic expansion to approximate the cost function, so that: 1 C = c0 + c1 wi + c 2 wi2 . 2 It is easy to check from (6) that the second derivative of profit w.r.t. wi is always negative, so that expression (15) characterizes a relative maximum. Imposing the condition ∑w i i = 1 and considering only the n ≤ N firms for which expression (16) yields a positive value, we obtain the expression for the equilibrium market price of output: c2 U* Q − c1 + + E (e −rt γ )( ) u (16) P* = n U δ Q0 / δ 17 − ∑u i where u = i . n According to (16), output market price will equal average cost of production plus the expected present value of the fine multiplied by the target level of pollution corrected for the inequality in the distribution of emission technologies. Substituting (16) into (15), we find: 1 1 U* Q − (17) wi* = − E (e −rt γ )( ) (ui − u ) . n c2 U δ In equilibrium, the optimal share of production for the ith firm will be larger, the smaller is its threshold emission level per unit of output compared with the average emission level that would prompt the regulator to impose the fine. While the traditional criticism against the market for permits is that it allows polluting firms to pollute more, the result in (17) shows that the system may actually work. With pollution permit trading, the firms that are the more technologically efficient in terms of reduced pollution per unit of output over the industrial average will prevail in the market. The more severe the pollution cap U* the more this efficiency will be reflected in market share for any level of uncertainty. Likewise the higher the fine the more is the market share from cleaner technologies for any level of uncertainty. Since β1 is inversely related to uncertainty of underlying demand or output, higher levels of uncertainty reduce the expected discounted value of the fine, thereby reducing the amount of impact that technological advantage has on market share. Ceteris paribus, the markets with the highest levels of output uncertainty will have the lowest environmental benefit of improved technologies under a regime of emissions trading and regulatory fines. 4. The industry- wide policy problem Multiplying wi* by u i and summing over all i, we find: 18 _ U* Q 2 (18) ∑w ui * i i = u − Ee −rt γn U δ σu − 1 1 _ where u = ∑ n i u i and σ u = ∑ (u i − u ) 2 . 2 n i For β1 ≠ 2 , expression (18) is a non linear equation that has no general analytical solution. We can gain some insights in its implications, however, by considering the linear case, i.e. β1 = 2 . In this case, average “optimalâ€? pollution level is: V 2δc 2 (Y + ) γ _ (19) ∑w u * i i = V u i 2δc 2 (Y + ) + γnQ02σ u 2 γ As expression (19) shows, for any given level of Y and V , the average level of pollution is always less than the arithmetic mean, i.e. average pollution under equal sharing, and so much so, the larger the level of the pollution tax γ . The average level of pollution depends negatively on the degree of diversification of producers’ emission, measured by its industry- wide variance. In other words, a mean preserving spread of the distribution of polluting firms reduces total pollution, since the threat of the fine is more effective in determining a differential incentives for “lowâ€? versus “highâ€? polluters. On the other hand, realized average pollution level does not depend on how the pollution threshold and/or the transaction costs are distributed across the firms. Even though expression (19) cannot be solved explicitly in terms of total pollution, it can be interpreted as a relationship between one policy target, i.e. the average level of pollution ∑ wi*u i and two instruments: the i pollution threshold Y and the tax γ . Denoting the target level of pollution with R , we can solve (19) to derive the relationship between the two policy instruments for any given value of the target: V Y+ − γnQ0 (u − R) β1 −1 1 γ (20) =[ ] RQ0 / δ c2 t 2 19 − u− R where t = can be considered, in analogy with the Student t statistic, a σu measure of the significance of the gap between the pollution that would be expected without any public action and the pollution target set by the government. From expression (20), we may conclude that it is not only the gap between the target and average pollution to determine the threshold and/or the ax level, but also its size relative to the degree of diversification of the industry. The more diversified the industry, the easier, coeteris paribus, is to achieve a sizable reduction of pollution by favoring low emission firms. Substituting the value of Y of (20) into (17), we find: _ _ 1 (u − u )(u − R) (21) w = [1 − i * ] σ u2 i n Expression (21) shows a remarkable result: once the government has fixed its pollution target, the ensuing optimal share of market capacity for the ith firm does not depend on the value of the demand forecast (independent of the level of the stochastic variable Q), but only on the degree of emissions of the firm, the size of the government target as compared with the industry average and the variance of industry emission. The condition for wi to be greater than or equal to zero can be derived from (21) as: _ σ u2 (22) u i ≤ u + _ (u − R) which can be written in percentage terms as follows: ui CV (23) _ ≤ 1+ _ _ u (u − R) / u _ where CV is the coefficient of variation σ / u . From (23), we can infer, for 2 u 2 example, that for a reduction target equal to 5 percent of the expected 20 pollution level, and for a coefficient of variation of 10%, a firm would have to expect a level of emissions not greater than twice the average to enter the market. This upper limit raises to 4 times the average in the case of a CV equal to 20% and to 20 times the average for a CV equal to 100% . On the cost side, considering a quadratic approximation of the cost function, substituting the value of wi obtained from (21) : 1 (24) C = nc0 + c1 + c 2 (1 + t 2 ) 2n Total production cost (the total level of production is equal to total demand and thus is assumed to be exogenous and random) is thus positively related to the square of the reduction of average emissions that the government wants to achieve. As equation (24) shows, however, cost is uniquely determined by the pollution target and the industry structure (its degree of diversification) and cannot be affected by a separate policy instrument. The only feasible instrument to reduce the costs determined by the pollution target would thus be an improvement in productivity (i.e. a reduction of the values of the parameters of the cost function). Conclusions In this paper, we have explored the effects of uncertainty on pricing of pollution permits through the use of a dynamic model of pollution markets. We have brought in two major sources of uncertainty – that arising from uncertain demand for the underlying resource and that coming from regulatory uncertainty. Both sources of uncertainty are common in pollution permit trading as not only does the market respond to the volatility of fundamentals but also to the vagaries of the institutional structure being created by public policy and enforced through regulation. The dynamic uncertainty inherent in pollution permit markets and the strategic decision-making that is demanded of participants in the market both on the part of firms and the regulator create market behavior that is not evident from simple static models of supply and demand. As we have shown using a real options model operating under dynamic uncertainty, the effect of regulation on permit pricing is not straightforward. The regulator operating also under uncertainty had two instruments at its disposal: the rate 21 of the fine and the timing of the imposition of the fine. The firm on the other hand has several instruments: the amount of output or market share, the amount of pollution permits it secures from the market and the efficiency by which it uses technology to reduce polluting emissions. We have found that under uncertainty the combination of the threat of the sanction and the market for permits may be effective in reducing the emission levels by shifting the competitive advantage in favor of less polluting firms. This will occur both because of the reduction of firm value to the potential imposition of the sanctions and because less polluting firms will be able to sell part or all of their allowances to the more polluting ones. Uncertainty, however, tends to reduce the value of the market price of permits, since in equilibrium this is simply equal to the expected present value of the fine. Thus, higher uncertainty will require, for the regulation to be effective, comparatively higher fines. Even under uncertainty of regulation and demand for output, the effect of pollution permit trading is positive to achieving a cleaner industrial base. Firms that are more technologically efficient in reducing pollution will tend to acquire larger market shares, with the exact effect depending on the uncertainty of demand for output and the severity of the fine. Using this type of real options approach, we believe that avenues of research are open. For example, through relatively simple analytic models other issues with respect to permit trading can be explored, for example, the effect on new entry into the market – when will new firms with cleaner technologies enter the market when demand is uncertain and the behavior of the regulator uncertain. We will explore this issues and others in later papers. References  Dixit, A. K. and Pindyck, R. S. 1994. Investment under Uncertainty. Princeton Princeton University Press.  Field, B. C. 1997. Environmental economics: An introduction. 2nd ed. Hightstown, N.J.: McGraw-Hill. 22  Kahn, J. R. 1998. The economic approach to environmental and natural resources. 2nd ed. Fort Worth, Tex.: Dryden.  Tietenberg,T. 2000. Environmental and natural resource economics, 4th ed. New York: Harper-Collins.  Weber, D. W. 2002. Pollution permits: a discussion of fundamentals. The Journal of Economic Education 23