PRICE INFORMATION AND TECHNOLOGY ADOPTION DECISIONS
IN TRADABLE PERMIT MARKETS
by
Chao-ning Liao and Hayri Onal*
The authors are Assistant Professor, Department of Economics, National Cheng-Kung
University, Taiwan and Professor, Department of Agricultural and Consumer
Economics, University of Illinois at Urbana-Champaign.
* Corresponding author, Tel.: +886-6-275-7575 x56321; Fax: +886-6-276-6491;
E-mail: [email protected]
1
Price information and technology adoption decisions in tradable permit markets
Whether or not to install new control equipment is an important decision variable for
firms facing a new regulation program such as tradable permits. Therefore, sending the
right price signals could help firms in making decisions concerning technology
adoption and buying/selling permits. By incorporating technology adoption as a
Yes/No binary variable, this research develops an optimization framework to analyze
how control agencies can search out appropriate price signals to enhance their overall
efficiency. This paper also demonstrates that when an equilibrium does exist in a
trading market with discontinuous demand and supply functions, the control agency
could change each firm’s abatement costs by sending different price signals while
keeping the total abatement cost of pollution at the minimum level.
Key words: pollution permit, mixed-integer programming, average shadow price
1
1. Introduction
Pollution permit systems have been widely discussed in recent years and
implemented on several occasions. Using permits as a tool to regulate environmental
pollution is gaining acceptance not only among economists, but also among policy
makers. The idea of using permits to regulate pollution was first introduced by Dales in
the late 1960s. The concept of a permit system is to use a market-based policy to
regulate pollution, starting from restricting the quantity of permits at a predetermined
level that meets a specified environmental standard. Firms are required to hold enough
permits that match their actual emission levels. Theoretically, the permit system can
achieve a specified environmental target at minimum cost (Montgomery, 1972). Under
this market-based instrument, a firm’s behavior is encouraged by market signals rather
than through explicit directives regarding pollution levels or methods - that is, firms
undertake pollution control efforts for their own interests (Stavin, 1999).
For firms under permit regulation, whether or not to install certain control
equipment should be viewed as an important decision variable, because improved
emission reduction technologies are sometimes very expensive to install1 and are
typically lumpy or indivisible. When such binary variables are considered in the model,
the costs per unit abatement depend on the extent to which the emission control
equipment is utilized, which in turn depends on the price of emission permits in the
market. Therefore, the costs per unit of abatement are determined endogenously and
simultaneously with technology adoption, equipment utilization, and permit buy/sell
decisions, all of which are driven by the market price of permits. By incorporating
technology adoption as a decision variable, this research develops an optimization
framework to analyze how control agencies can search for appropriate price signals on
technology adoption decisions by firms through an economic model that is closer to
reality.
Tradable permit systems rely on voluntary participation. Therefore, the price of
permits plays a crucial role when determining firms’ technology adoption and permit
trading behavior. Firms usually try to acquire these price signals through auctions held
by governments, observing actual trading prices in the market and related research
studies on permit price. For example, before the implementation of a trading program
for U.S. SO2, a number of price projections based on cost-minimizing optimization
models were published in the trade press. However, the actual trading prices and
volume turned out to be much lower than expected. Similar results have also been
found in the Emission Reduction Market System in Chicago that started in the year
1
Dunham and Case (1997) presented examples with installation costs over US$1 million.
1
2000. Low price and trading volume, together with some expired permits in the first
few trading years, have come with a high price estimation ($285) by the Illinois
Environmental Protection Agency and other related studies (Evans, 1997; Kosobud, et
al., 2001).
If not due to data inaccuracy, the discrepancy between predicted and actual prices
can be explained by a conservative behavior of firms facing a new regulation program.
The situation might also be affected by high price estimations of related research
studies, which may have encouraged many firms to cut their emissions far below the
required levels before the program was launched. Titenberg (1985) stated that a permit
trading system has the potential to offer enough incentives for producers to adopt
cheaper and more efficient pollution-control technologies. Thus, when an emission
reduction technology (or equipment) is expensive or the expected permit price is low,
many firms will opt to buy emission permits rather than install that technology unless
the equipment has a long lifetime. However, once certain equipment is installed, the
investment is irreversible and an adoption cost is incurred even if that technology is not
used in the future. The control agency under this circumstance could help less informed
firms which want to make technology adoption and permit buy/sell decisions through
sending signals of permit prices.
Firms’ choices of whether or not to adopt new control equipment are not properly
dealt with in the previous literature. Most studies which address tradable permit
systems use the marginal analysis approach and incorporate only variable costs
(including operating costs and annualized fixed costs) per unit abatement, assuming
that each emission abatement equipment is used up to its full potential (e.g., Johnson
and Pekelney, 1996). Although this may be a realistic representation in many cases,
when fixed costs vary significantly among alternative technologies and a particular
technology is not used with its full potential, the traditional marginal cost approach
does not reflect the true costs and may not correctly determine the equilibrium price of
permits. In this case the right choice among alternative technology options is
determined not only by variable costs, but also by the initial costs of investment.
Traditional time series and econometric approaches are unfortunately not useful in
dealing with a new trading program since new programs often do not have a precedent
and there is no historical data on permit prices, market demand, and supply conditions.
In the early stages of the development of a new trading program, buyers and sellers
have very little information about the market-clearing price. Incomplete information
may lead permit trading to not be a cost effective way for pollution control. It is likely
during the first few years of a new trading program that many firms will prefer to wait
and see how the program works instead of making a rational decision if full information
were indeed available. In order to enhance the overall effectiveness of this new program,
2
it is important for control agencies to send the “right” price signals to firms facing a
new regulation. One useful price index for firms is the shadow price from a resource
constraint in optimization problems. However, when integer variables such as Yes/No
decisions for technology adoption are included in the problems, the shadow price may
no longer offer appropriate imputed prices for scarce resources (Gomory and Baumol,
1960). Therefore, seeking out other substitutes is essential and will be discussed in
more details in the next section.
2. Shadow Price and Average Shadow Price as Price Signals
The shadow price concept has been widely used in empirical economic analyses
including environmental control and resource management problems. In a
mathematical programming economic model, an imputed value for each limited
resource is obtained along with the optimum values of endogenous decision variables.
This imputed value reflects the marginal opportunity cost and is often called the
“shadow price” of that resource. This price may or may not be comparable with the
market price (or equilibrium price), if a market exists, and it represents the price that the
user is willing to pay for that resource. This type of information is extremely useful
when using mathematical programming models for economic analysis. For example,
under transferable discharge permit trading, the market is cleared (i.e., total demand =
total supply) if the shadow price of the equilibrium constraint is announced as the price
of tradable permits.
If each individual firm makes its decisions based on the shadow price information,
then the entire system achieves the socially optimum (economically efficient)
allocation of resources automatically. This enables a social planner with full
information to send the right price signals to individual firms in advance to help and
guide less informed firms in their decision making. However, the marginal analysis and
Kuhn-Tucker optimality conditions used to develop the marginal opportunity costs of
binding constraints (shadow prices) fail for the case of discrete optimization such as
mixed integer programming (MIP) problems. This is because the objective function is
neither concave nor convex when the availability of one or more resources changes.
The generation of dual variables and their economic interpretation in integer
programming problems were first addressed by Gomory and Baumol (1960). They
showed that the dual variables of an integer programming problem may not accurately
represent the marginal contribution of the inputs and in some cases shadow prices
associated with some goods may be zero although they may not be considered as free
goods. In other words, the total value of the final good may not be equal to the total
imputed costs of all inputs to produce the output. Geoffrion and Nauss (1977) also
pointed out that due to the character of multiple discontinuities caused by changes in
3
the value of integer variables, integer programming models can behave in an erratic and
unpredictable manner.
Kim and Cho (1988) proposed a new concept of the shadow price, based upon
average rather than marginal analysis, in pure integer programming problems and they
interpreted it as the average shadow price (ASP). The ASP of a constraint in integer
programming is defined as the average rate of change of the objective function and as
the decision maker’s maximum willingness to pay for an extra unit of that resource.
They demonstrated the existence and uniqueness of this new concept and showed that
the ASP satisfies a version of the complementary slackness theorem in integer
programming. Crema (1995) followed the same idea and further proved that this new
concept can be extended to mixed integer programming problems as well.
As a counterpart to conventional shadow prices, the ASP is associated with a
resource constraint can perhaps be utilized for this purpose, since ASP is linked with a
resource constraint and is interpreted as the maximum price a decision maker would
like to pay for one additional unit of that resource. Although this interpretation is
similar to that of conventional shadow prices, the ASP’s calculation is fundamentally
different from conventional shadow prices, which are obtained directly from the
optimum solution of the “primal” problem. Liao and Onal (2001) showed that the ASP
could be the equilibrium price only if (1) firms with a technology adoption from the
social planner’s model remove all their emissions and (2) these firms’ MAC must be
lower than firms without technology adoption. Therefore, the function of the average
shadow price as a signal for technology adoption decisions may be limited. In this study
the concept of ASP, together with another price signal derived from an economic model
proposed below, will be explored.
3. Economic Model
The empirical framework consists of two optimization models. The first
optimization model assumes that a social planner wants to use a tradable permit policy
for pollution regulation and its objective is to minimize the total emission control cost,
including variable costs of technology use and fixed costs of installing the equipment
by producers. Each producer can either choose to install an efficient technology to
comply with its emission reduction requirement and sell excess permits or can buy the
required permits from other participants in the permit market. The model assumes that
all these decisions are controlled by the social planner who has full information about
the individual producers’ cost structure. This means implicitly that all participants
cooperate among themselves and with the social planner to adopt the socially optimum
solution. Clearly, this is not a true representation of reality, but the purpose here is to
determine a socially optimum solution that provides a benchmark against other
4
alternatives, particularly the free market solution where individual producers act
independently to minimize their compliance costs. The model is described below.
3.1 Social Planner’s Model
Min   VC FiT  X FiT     FC FiT  D FiT  , such that:
(1)
B Fi   emis FiT  X FiT  S Fi  emislim Fi for all Fi
(2)
 B Fi   S Fi
(3)
i T
i T
T
i
i
 emis FiT  X FiT  bFi
for all Fi
(4)
T
X FiT  const  D FiT for all Fi and T.
(5)
Here, Fi and T denote firm i and technology, respectively; VC FiT is the variable cost of
installing technology T by firm Fi; FC FiT is the fixed cost of installing technology T by
firm Fi; B Fi and S Fi are the respective amounts of permits bought and sold by firm Fi;
emis lim Fi is the required reduction of emission by firm Fi; emis FiT is the emission
reduction if technology T is used by firm Fi; bFi is the baseline emission (or historical
emission) level of firm Fi; X FiT is the utilization rate (i.e., number of days) of
technology T by firm Fi; D FiT is a binary variable indicating whether or not technology
T is adopted by firm Fi; const is a certain constant number that is used with the binary
variable D FiT ; and utilization rate X FiT represents the fact that a piece of equipment
can only be used after installation.
Equation (1) is the objective function and represents the total cost of emission
control. The first part of the equation represents the total variable costs resulting from
the use of all technologies adopted by the firms, and the second part shows the total
5
fixed costs of installing the required equipment. Variable costs are defined as costs per
ton of pollution reduction.
Equation (2) regulates the annual emission level for each firm. Each firm must
have enough permits on hand to match its seasonal emission level. Permits can be
generated by installing cleaner technology or by purchasing it through market
transactions. These two sources constitute the supply side of permits. The right-hand
side of equation (2) is the demand for permits. For any firm, this includes the amount of
permits sold and used by the firm to cover the required emission reduction by the social
planner. If the total supply of permits is greater or equal to the total demand for permits,
then the emission standard is met.
Equation (3) implies that the total demand and total supply of permits have to be
balanced. This is also the constraint for deriving the average shadow prices. Constraint
(4) implies that producers cannot generate more permits than their baseline emission
level. Equation (5) is a technical constraint which ensures that equipment can be used
only after it is installed.
In equilibrium, firms’ behavior regarding technology adoption and emission
reduction can be discussed in two ways. Since the average abatement cost decreases as
the amount of emission reduction increases (all firms experience increasing returns to
scale with control equipment), the social planner has an incentive to induce firms to
remove all their emissions once certain equipment is adopted. However, there will be at
most one firm that might not be able to remove all its emissions due to the equilibrium
constraint. Since the social planner’s objective is to minimize both the total social costs
and maintain the equilibrium constraint, it will assign firms to adopt new control
equipment to meet the requirement. As the required reduction level is greater than zero,
at least one firm needs to adopt a certain piece of equipment. When the required
reduction level is high, there will be more than one firm with technology adoption in
equilibrium.
3.2 The Firm-level Model
The second optimization model is called the firm level model in which firms are
assumed to be rational and minimize their cost of compliance by choosing the optimum
technology adoption and marketing decisions. The structure of the firm-level model is
similar to the social planner’s model except that the permit price and cost/benefit of
permit trading enter into the objective function. The model assumes that firms are all
price takers in the trading market and make their buy and sell decisions individually.
The purpose of the firm-level model is to check whether the firms’ responses match
with what the social planner wants once each firm receives the price information from
the social planner. Given the price signals, if firms’ choices of technology adoption and
6
permit buy/sell decisions are exactly the same as that from the social planner’s model,
then that price signal successfully leads the firms’ decision making. Using the same
notation described in the social planner’s model, the decision model for a particular
firm F is described below.
 VCT  X T    FCT  DT   price  B  S  , such that
Min
T
(6)
T
B   emisT  X T  emislim  S
(7)
 emisT  X T  emislim
(8)
X T  const  DT for all T
(9)
T
T
S , B  0, DT =0, 1.
The notation price used here is the price “expected” by producers. For example, an
average shadow price derived from the social planner’s model can be substituted into
the firm level models to see if the firms’ optimum responses and total abatement cost
derived from the firm level models coincide with the solution obtained from the social
planner’s model.
3.3 Equilibrium Price in the Firm-level Model
Firms’ buy/sell decisions concerning permits depend on their average cost of
pollution abatement and the perceived price signals. Mathematically, the average
abatement cost of a certain control equipment T for a firm Fi that removes all of its
emissions bFi with fixed cost FC FiT and variable cost VC FiT is:
AC Fi 
FC FiT  bFi  VC FiT
bF
.
(10)
i
To simplify the analysis, we assume that each firm’s choice of control equipment is
limited to its own best available control technology (BACT). Therefore, the subscript T
in each variable can be deleted.
When a given price P is announced, each firm in a competitive tradable permit
market compares this price signal with its average cost of abatement shown in (10). If
the price is less than the average cost (i.e. P  AC Fi ), then firm Fi ’s best strategy for
pollution control is to buy all the required permits, which is emislim Fi in the market. On
7
the other hand, if P  AC Fi , then firm Fi adopts new equipment and supplies??
bFi  (1  a Fi ) units of pollution permit, where a Fi is the required reduction ratio for firm
Fi from its baseline emission level. When P  AC Fi , firm Fi is indifferent towards
buying all the required permits in the market and conducting pollution abatement itself
by adopting new control equipment. Thus, for any given price P , the total supply is



 (bFi  2  emislim Fi ) , where Fi  Fi / P  ACFi and the total demand is  emislim Fi' ,
'
i
Fi


where Fi '  Fi ' / P   ACF ' .
i
In the firm-level model, different prices are announced to check if the trading
market (i.e.  bF '  (1  a F ' ) =  emislim Fi ) can be cleared. Since the objective function
i
i
i
i
is not continuous, the equilibrium price may not exist or there may be more than one
equilibrium price that clears the market. When the equilibrium price does not exist, the
social planner’s problem turns into finding a certain price P such that the gap between
total supply and total demand under that price is minimized as in the following:
(  bFi  (1  a Fi )   emislim F ' ).
Min
*
i
i
P
(11)
i
Instead of minimizing the absolute value of the above objective function, we only
look for the price that assures the total supply is higher than the total demand (i.e. firms
have over-complied with government standards). This is because total demand higher
than total supply implies that environmental quality has been under-achieved. Both
over-complying and under-complying with the environmental standards generate costs
to society. The cost from over-complying can be measured as the cost of generating the
amount of excess supply of permits. The cost of under-complying is the damage to
environment quality.
When the equilibrium price does not exist, there must be a certain price P or a
price range P1 to P 2 that minimizes the gap represented by:
 bFi  (1  a Fi )   emislim Fi' .
i
i
If the control agency feels that the cost of over-compliance is high, then it can release a
price signal that is lower than P or P1 to P 2 . The trading market will then be in excess
demand. On the other hand, a high price signal should be released to induce investment
into control equipment.
8
3.4 Average Shadow Price and Equilibrium Price
To illustrate the relationship between ASP and equilibrium price graphically, a
trading market that considers only 5 firms is used. However, the result of this simplified
example can be extended to n firms. Figure 1 shows each firm’s total cost curves for
emission reduction. When no equipment is adopted, the total cost is zero. However,
once certain equipment is adopted and used, firms have to pay both fixed and variable
costs. Since a constant unit variable cost is assumed, the total cost curve is a straight
line starting from a given fixed cost level (shown by the small circles in the figure).
When these firms reduce their emission levels to 0, the total costs are TC F1 , TC F2 , TC F3 ,
TC F4 , and TC F5 ,respectively.
The slopes of the OF1 to OF5 curves, shown in descending order, are the minimum
average abatement cost (MAC) of each firm. To make the figure clearer, these total cost
curves are not crossed with each other and each firm has a different MAC. Again, this
assumption does not affect our conclusion. For any given firm, the average abatement
cost level at rFi , where 0  rFi  bFi , must be higher than its MAC. This is because
firms will experience an increasing return to scale (IRTS) from emission reduction.
In the 5-firm example, if the value of ASP is between the slopes of OF3 and OF2
(for example, the slope of OC), then firms 3 to 5 will remove all their emissions and
become permit sellers. On the other hand, firms 1 and 2 will become buyers. By adding
up the total demand and total supply, the ASP can be tested to see if it is an equilibrium
price. Based on this rationale and possible firms’ cost structure shown from Figures 1 to
14, we will discuss several scenarios to simulate firms’ possible reactions to ASP.
From Liao and Onal (2001), ASP could be the equilibrium price only when certain
conditions??? hold. Therefore, we first demonstrate a case where the ASP could be the
equilibrium price. Using the cost structure shown in Figure 1, we assume that firms 3 to
5 are chosen for technology adoption by the social planner and reduce their emissions
to zero in order to meet the environmental standard at minimum cost C 5 (i.e.:
C 5  TC3  TC4  TC5 ). The total emission reduction and permit supply are
5
5
 bFi (=  emislim
i 3
i 1
5
Fi
) and  (bFi  emislim
2
(i.e., firms 1 and 2) will buy  emislim
i 1
5
constraint (3),  (bFi  emislim
i 3
Fi
i 3
Fi
) , respectively???. The remaining firms
units of permits. Due to the equilibrium
2
Fi
) and  emislim
i 1
9
Fi
are equal. The initial ASP, which is
also a concept of the weighted average cost of firms with technology adoption, is
C5
.
5
 bFi
i 3
Thus, the slope of the OA curve must lie somewhere between firms??? OF3 and OF5.
Whether the OA curve is close to OF3 or OF5 depends upon the cost structure and
the weight (i.e., firms’ emission reduction level). As long as the ASP is lower than the
MAC of any firms with technology adoption, the social planner has an incentive to buy
extra permits by paying the ASP rather than removing emissions by itself. The iteration
process for updating a new ASP will not stop until the final ASP is at least equal to or
higher than the slope of OF3 as shown in Figure 1, leading???? the objective value back
to C 5 . In this case, the final ASP is the MAC of firm 3. When this price is released to
2
the firm-level model, the total demand for firms 1 and 2 is  emislim
5
supply from firms 4 and 5 is  (bFi  emislim
i 4
Fi
and the total
Fi
i 1
).
Firm 3 under the above circumstance is indifferent between being a permit buyer
5
or seller. If it decides to be the latter, then the total supply becomes  (bFi  emislim
i 3
Fi
)
and the market is in equilibrium. The ASP is the equilibrium price. If it does not adopt
3
the equipment, then the total demand and total supply are respectively  emislim
i 1
5
 (bFi  emislim
i 4
Fi
Fi
and
) , and the market is in excess demand. Since firm 3 would also be
willing to adopt at any price that is higher than its MAC, but lower than MAC of firm 2,
this price range reaches the equilibrium, too. Therefore, we have more than one
equilibrium price in this case.
Consider another possible cost structure where some firms have a low MAC, but
are not picked by the social planner for technology adoption due to “capacity”
considerations. Since firms cannot generate more permits than their initial allocation, it
may not be efficient for firms to pay a fixed cost and only supply a limited amount of
permits even when their MAC is low. In Figure 2 we assume that the MAC of firm 2 is
now lower than the MAC of firm 3. However, firms 3 to 5 are still picked for
technology adoption and will remove all their emissions. The total demand (i.e.:
5
2
 emislim
i 1
Fi
) and total supply (i.e.:  (bFi  emislim
i 3
Fi
) ) will be balanced in the social
planner’s model, but the ASP is not the equilibrium price. This is because the ASP will
be at least higher than firm 3’s MAC. When this price is released, not only will firms 3
to 5 do so, but also firm 2 will adopt control equipment. The total permit supply must
10
5
equal at least  (bFi  emislim
i 2
Fi
) and is higher than what the market needs.
Figures 3 to 5 still consider the cases where firms 3 to 5 are chosen for technology
adoption, but one firm with technology adoption only removes part of its emissions in
the social planner’s model in order to satisfy the equilibrium constraint (3). Since all
firms have the property of an increasing return to scale (IRTS), there will usually only
be one firm that reduces part of its emissions. The only exception is when we have
multiple solutions in the social planner’s model. Figure 3 is such??? an example. When
firms 3 and 4 both have technology adoption and have the same unit variable costs (or
their total cost curves are parallel), then the social planner may be indifferent in
assigning the reduction responsibility to these two firms. However, this will not affect
our conclusion derived from Figures 3 to 5.
In Figure 3 we assume firm 4 only removes rF4 units of emissions ( rF4  bF4 ), and
then the supply by firm 4 is (rF4  emislim
F4
) in the social planner’s model. The
equilibrium constraint will make the total supply, which is
(bF3  emislim
F3
)  (rF4  emislim
F4
)  (bF5  emislim
2
be equal to the total demand, which is  (emislim
i 1
Fi
F5
),
).
By the definition of ASP again, the final ASP is no less than the MAC of firm 3.
This price will make firm 4 in the firm-level model supply (bF4  emislim
F4
) units of
permits. If the ASP is found to be greater than MAC of firm 3 and lower than MAC of
5
firm 2, then the total supply of permits (i.e.:  (bFi  emislim
i 3
2
demand (i.e.:  emislim
i 1
Fi
Fi
) is higher than the total
). If it is even higher than firm 1’s MAC, then the market will
have no demand in the firm-level model.
Instead of firm 2, Figures 4 and 5 assume that it is firm 3 that only removes rF3
units of emissions ( rF3  bF3 ) and supplies (rF3  emislim
F3
) units of permits. Compared
with Figure 3, these two figures only differ in the location of the total cost curve of firm
2 and can be analyzed by the same rationale. The final ASP in Figures 4 and 5 must be
greater than the slope of the OB curve which corresponds to the average cost at the
reduction level rF3 . Since the average cost at rF3 must be higher than that at bF3 , firm 3
11
will supply more permits (i.e.: bF3  emislim
F3
) than the market needs under the ASP.
We now focus on cases where some firms have the same MAC. As shown in
Figure 6, this situation may occur when two firms have exactly the same baseline
emission levels and control equipment, such as the MAC of Fa and Fb represented by
the slope of OE, or when certain combinations of these two happen to result in the same
MAC, such as the slopes of OD, OF, and OG. If we still assume that firms 3 to 5 are
always the firms with technology adoption in these cases, then we can explore the ASP
and equilibrium price issue by simply modifying the figures used previously. Figures 7
to 11 focus on the cases where the ASP is not an equilibrium price and Figures 12 to 14
present the equilibrium cases.
Consider first the cases discussed from Figures 2 to 5 above where the ASP is not
the equilibrium price and both firms 1 and 2 are buyers in the trading market. Figure 8
depicts a case where firms 1 and 2 have the same MAC. Since the MAC is lower than
that of firm 3, the final ASP must cause these two firms to change from being buyers in
the social planner’s model to sellers in the firm-level model. The permit market will
hence be in excess supply. Figures 9 and 10 consider a situation where one firm
removes part of its emission. By the same rationale, the market will definitely be in
excess supply. Similar to Figure 8, the cost structure assumed in Figure 11 leads to no
permit demand when the ASP is announced in the firm-level model.
In the end, firms with the same MAC are considered in the case where all firms
having technology adoption not only remove all their emissions, but also have lower
MAC than firms without technology adoption. In Figure 12 we assume firm 3 has the
same MAC as firm 4. Since both of these firms adopt equipment and remove all their
emissions, their MAC will equal the final ASP. If these two firms adopt equipment
when this price is announced in the firm-level model, then the market will be in
equilibrium. On the other hand, if either or none of them choose not to adopt, then the
market will be in excess demand.
Figure 13 is the case where firm 2 has the same MAC as firm 3, but only the latter
is picked for technology adoption by the social planner. The ASP will be the MAC of
firm 3. Whether this price is the equilibrium price depends on firms 2 and 3’s decision
in the firm level model. If one of them chooses to adopt, then the market is in
equilibrium. If both of them respond to the price in the same way, then the market is not
in equilibrium. In Figure 14, firms 4 and 5 have the same MAC. The ASP in this case
will be the MAC of firm 3 and could be the equilibrium price.
4. Simulation and Results
An artificial tradable permit market with data shown in Table 1 is used to simulate
12
the above 5-firm example. Using the optimization software GAMS for simulation, this
research considers three different scenarios to see how price signals can be sent for
inducing technology adoption. In the first scenario, we assume that there exists a certain
price that can clear the trading market. The second scenario replicates a case in which
the equilibrium does not exist due to the discontinuity of decision variables. The last
scenario considers a market where a command and control policy (C&C) is used
together with permit trading. The purpose of this scenario is to show that C&C, which is
often thought as a less flexible policy for firms, does not necessarily increase the total
social abatement costs when used with a permit policy. Furthermore, firms that face
both C&C and a trading policy may become better off through technology adoption
when a certain price signal is released.
In scenario 1 we first assume that the firms initially face a less stringent regulation
where firms 1 to firm 5 are each asked to reduce 30%, 30%, 30%, 50%, and 70%
pollution from their baseline emission levels, respectively. Therefore, the total required
reduction level is 26 tons as shown in Table 1. These artificial numbers are set to ensure
that the ASP from the model could be the equilibrium price. The social planner’s model
is first solved to find the total abatement costs, firms’ technology adoption decisions,
permit trading volume, average shadow price, and the equilibrium condition. The
results in Table 2 indicate that the ASP is $186 per ton. Firms 4 and 5, which have lower
average abatement costs than the other firms, are chosen by the social planner to adopt
control equipment and remove all their emissions. These two firms are also the only
two suppliers of permits in the market. The total abatement cost for the entire system,
assuming implicitly full cooperation between the social planner and the firms, is $2,216.
The market is in equilibrium and the total trading volume is 9 tons.
Now assume that this price is announced as the market price by the social planner
and the firms consider this as the market price when making their decisions - namely,
the firms take for granted that they will pay or receive this price when they generate and
trade pollution permits. When each individual solves its own optimization problem in
the firm-level model, the total abatement cost is calculated as $2,216. Therefore, the
firm-level model and the social planner’s model reach the same total abatement costs
under the permit price of $186. However, the results in Table 3 show that the trading
market may or may not be cleared when $186 is announced. This is because the ASP
from the social planner’s model is the same as firm 4’s MAC. Thus, firm 4 is indifferent
between adopting the control equipment and buying all the required permits from the
market under this price. The trading market cannot be cleared and is short of 3 units if
firm 4 does not adopt the equipment. Therefore, the ASP is the equilibrium price only if
firm 4 decides to adopt control equipment.
Since the ASP may not lead to market equilibrium, the firm-level model is used to
13
derive firms’ permit buy/sell decisions at each price level. That is, different permit
prices, from low to high, are announced one at a time in the model, and then the
individual firm’s permit demand and supply under each price is added up to see if the
total supply is greater, smaller, or equal to the total demand. The price at which the total
permit demand equals total supply can also represent the equilibrium price. The results
shown in Table 5 indicate that when the price ranges from $186 to $195, the permit
market could reach equilibrium and the total abatement costs will stay at $2,216. This is
because all firms’ best responses to this price range are the same. Even buyers may lose
by paying a higher announced permit prices, but sellers can also benefit from increased
revenue while keeping the total abatement costs unchanged. But when the price is
higher than $195, then the total supply of permits increases due to firm 3’s technology
adoption. The market hence is in excess supply.
Whether $186 can be interpreted as the equilibrium price depends on firm 4’s
technology adoption decision. If the control agency wants firm 4 to adopt new
technology to maintain environmental quality, then it should release a price signal that
is slightly above $186, but lower than $195. The MAC of firm 4 and the next higher
MAC, such as firm 3 in this case, form an equilibrium price range. Figure 15 shows the
firms’ behavior from a firm-level model simulation. Since the binary variable is
included, the demand and supply become step functions. When the price is within the
range of $186 and $195, the demand and supply curves coincide. Thus, the market is in
equilibrium.
In the same scenario we now allow for firms without technology adoption to have
lower average abatement costs than firms with adoption. To simplify the analysis,
assume that firm 1’s fixed cost and variable cost are changed to $1000 and $20 per ton,
respectively. Therefore, its minimum average abatement cost is $120 per ton. After
solving the social planner’s model, firms 4 and 5 are still the two firms with technology
adoption and they remove all their emissions. The total abatement cost is $2216 and the
ASP is $186. Both are the same as the previous case. However, when $186 is
announced in the firm-level model, firm 1 also adopts and becomes another permit
supplier. The market is then in excess supply. The ASP in this case is definitely not the
equilibrium price.
Figure 16 depicts the supply and demand for permits based on the revised dataset
in the firm-level model. It indicates that when the price is from $121 to $186 per ton, the
smallest gap between supply and demand is 4 tons. No equilibrium price exists in this
case. Even when firm 1 has a lower average abatement than firm 4, it is not picked
initially by the social planner for technology adoption. The reason is similar to that
discussed in Figure 2.
In the second scenario, assume that the government sets a higher emission
14
standard and asks firm 1 to reduce more from its baseline (from 30% to 40%). Table 5
indicates that the total abatement costs and trading volume of reaching this strict
standard have increased to $2,915 and 10 tons. Firms 3 and 5 are now picked by the
social planner to adopt control equipment. However, firm 3 in this scenario does not
remove all of its emissions. The ASP from the social planner’s model in this case is
$699 and is higher than all firms’ MAC. Therefore, if the ASP is released to the
firm-level model, then all firms choose to adopt control equipment. The market as such
is in a situation of excess supply. Over-compliance with the emission standards by firms
creates efficiency loss.
To find a better price signal, the firm-level model is used again to derive prices that
can clear the market. The iteration result shown in Table 6 indicates that when the price
is within the range of $187-$195, the total demand is 10 and total supply is 9. Firms 4
and 5 are willing to adopt control equipment here. Being short one unit of permits
implies that the environmental standard has not been achieved. Even though the market
is not cleared, the supply and demand curves in Figure 17 show that this price range
minimizes the gap between total demand and total supply. If the damage caused by the
unit of pollution is large, then the social planner might want to release a higher price
signal such as $196-$200. When any price is picked from this range, firms 3, 4, and 5
will adopt control equipment. The total demand then is 7 and total supply is 16.
However, the excess supply of permits indicates that the society has put too many
resources on control equipment investment. The social planner under this circumstance
needs to compare the costs from pollution damage and over-investment in control
equipment. If it weighs less on the latter, then the social planner should release a price
signal that induces more firms to adopt equipment.
In scenario 1, whether the market will reach an equilibrium depends upon the
behavior of firm 4. In scenario 2, the firm’s behavior will also affect the equilibrium in
the firm-level model. Only when this firm installs the equipment and removes part of its
emissions will the market reach equilibrium. Since all firms are price takers in a
competitive market, they will believe they can sell all of their permits under the price P
and no one will think that they themselves would end up with extra permits on hand.
One way to prevent the efficiency loss is to restrict firms’ behavior on emission control.
For example, the social planner may force firms to reduce their emissions to what the
society wants or instead install control equipment.
To ensure that environmental quality can be met, we consider the last scenario
where the government not only pre-announces an estimated permit price to firms for
decision making, but also intervenes in the market through a command and control
policy. In this scenario we assume that the control agency requires firm 4, which has the
lowest fixed cost and medium variable costs, to adopt the control equipment. Thus, the
15
social planner with the dataset in Table 1 is solved again with this newly-added
constraint. The results in Table 9 indicate that the ASP is $85.2 and the total abatement
costs are maintained at the same level as that without further government. However, the
individual firm’s costs have changed. Firm 4, which only incurred $558 in the previous
case, now has to pay $860 if the permit price is at $85.2.
In the firm level model, we find that when a command and control policy is
imposed on firm 4, the equilibrium price range shown in Table 10 and Figure 18 has
been changed to $55 - $195 while the total abatement costs stay the same. Figure 19
depicts the relationship between each possible equilibrium permit price ($55-$195) and
the total abatement costs from the firm-level model for firm 1, firm 4, and firm 5. Firm
4 and Firm 5 with technology adoption can benefit from a higher released price signal.
Firm 4’s corresponding costs at permit price $55 and $195 are $951 and $531,
respectively. Thus, if the government can announce both a higher price and use the
C&C policy, then firm 4 could be better off than before and firm 5 can even gain
positive profit from permit sales when the price is greater than $186. Again, firms with
technology adoption can benefit from a higher released price signal no matter if this
technology adoption decision is voluntary or forced by the government. However,
permit buyers might be worse off under high prices.
From the result of the scenarios, we know that as long as the released prices are
higher than $186, firm 4 will adopt the equipment automatically. The government
seems to have no need to use the C&C policy, but after comparing the equilibrium range
from these two scenarios, we find that the range is further reduced to $55-$195 when
the C&C policy is imposed. When an “equity” issue arises among large and small firms
in the same trading program, not only changing the initial allocation but also releasing
low price signals could lower smaller firms’ burden in pollution control.
In the last scenario we only focus on the situation where the equilibrium exists in
the market. However, it is still possible for the control agency to use both an appropriate
price signal and C&C policy when there is no equilibrium. For example, when a certain
firm only removes part of its emissions, the solution from the social planner’s model
can ensure that the market is cleared (like scenario 2). The C&C policy can also be
imposed on that firm with a partial reduction in the social planner’s model. Again,
sending a higher price signal may make the firm that fully cooperates with the social
planner better off. Letting the firm realize that unsold permits may turn into a loss is
another way to persuade the firm to adopt BACT (WHAT is “BACT”?*), but remove
part of its emission.
When more available control technologies and firms become involved in the
model, we will have enough variations in our data structure. Under this circumstance,
the demand and supply curves are closer to the usual downward- and upward-sloping
16
shapes. The possibility of having a unique equilibrium is high. However, the BACT for
firms within the same industry or for certain pollutants is usually the same. When
trading programs do not include many participants or offer less variability in firms’ cost
structure, we may still not have the usual supply and demand curves for achieving the
equilibrium.
5. Conclusion
For some firms under tradable permit systems, certain equipment for pollution
control is expensive to install and typically lumpy or indivisible. Therefore, the prices
of permits play a crucial role when determining firms’ technology adoption and permit
trading behavior. An important feature that makes this study unique is the incorporation
of discrete (binary) decision variables - namely, technology adoption decisions - in an
optimization model (a mixed integer program) that simulates the firms’ decisionmaking behaviors. The model is a more realistic representation of the actual decision
problem than the conventional modeling approach seen in the permit trading literature
where abatement costs involve variable costs only based on the simplifying assumption
that once adopted the abatement technologies will be utilized at full capacity. In reality,
the average cost of abatement under alternative technology options is endogenously
determined, depending upon the firms’ decisions regarding the number of permits
generated, purchased, or sold - all of which are determined by permit prices over the
duration of the emission trading program.
The concept of average shadow price, which is introduced as a counterpart to
conventional shadow prices when working with mixed integer programming models,
may offer a practical tool to resolve this problem. Although the two concepts have
similar interpretations, the empirical results show that the average shadow price may
not lead to the market equilibrium condition. The price range derived from the
firm-level model is a more useful index that can guarantee both technology adoption
and environmental quality.
When a binary variable is incorporated, the demand and supply become step
functions. In other words, we may have more than one price that can clear the market.
Which price we should pick as a price signal could depend on empirical needs. If the
purpose of control agency is to reward permit sellers for adopting new equipment, then
a higher price can be released. If one wants to lower buyers’ burdens in the trading
market, then a lower price can be picked. When equilibrium (total supply=total demand)
exists in the MIP models, the command and control policy (C&C) can be used together
with permit trading to assure that the environmental standard is met. The total
abatement cost, however, stays the same. The firm regulated by the C&C and tradable
permit can be compensated by a higher price signal released by the social planner. All
17
firms with technology adoption in fact benefit from a high released price. The increased
benefit comes directly from permit buyers, but the ASP may not always perform as a
good price signal for technology adoption. On the other hand, an iteration process from
the firm-level model serves a better way for finding approximate price signals.
Market equilibrium may not exist in the MIP model. The control agency under this
circumstance should consider the cost and benefit from excess demand and supply in
trading markets. What this means is that if the cost of over-investment on equipment is
higher than that of under-achievement in environmental quality, then the control agency
should release lower price signals and induce fewer firms for technology adoption. On
the other hand, higher price signals should be sent if the permit system is designed for
regulating toxic materials.
18
Appendix: Derivation and Economic Interpretation of Average Shadow Prices
The general mixed integer linear programming (MILP) model can be defined as:
Max Q : cx
s.t. Ax  b ,
(A.1)
where x  S  {x : x  (x i , x j ), Bx  s, x  0, x i are integer variables; x j are real
variables i, j  I } ; b , c , and s are vectors; A and B are matrices with conformable
dimensions; and I represents the index set for integer variables.
Consider the following right-hand side parametric programming problem:
(A.2)
Max Q : cx

s.t. Ax  b  d , x  S
where  is a scalar and d  0 is a unit vector ( d  1 ) that has the same dimension as
b.
Let H be an optimization problem, F(H) represent the set of all feasible solutions,
and v(H) be the optimal solution value. Assume that F(Q) is not empty and S is a
bounded set. Define f ( )  v(Q )  v(Q) ,   0 . The average shadow price (ASP),
denoted by q, relative to the direction d is then defined by:
q  inf p  0 : f ( )  p  0,   0.
(A.3)
It is shown that q has a finite value. Equation (A.3) is equivalent to
p
v(Q )  v(Q)

.
(A.4)
The value of p obtained from equation (A.4) is a measure of the average change in the
objective value resulting from a small change in the right-hand side.
For many economic questions, Ax  b represents the resource constraints such as
total labor supply or capital availability. The decision maker may be interested in
questions like: can the objective value be possibly increased by using more of these
resources? If yes, then what is the optimal resource quantity (  )? Crema defined the
critical point of any given resource as:
C
*
where p1 

  :   0 and  1 ,  2 such that 0   1     2 :

p  p  ,
1
2
f ( )  f ( 1)
f ( 2)  f ( )
and p 2 
. Because S is a bounded set, C * is a
 1
 2 
finite set.
19
In order to find the ASP and  , define a net profit function as:
e( p)  sup f ( )  p :   0 p  0 ,
where e(p) measures the maximum additional profit we can obtain from buying an extra
unit of the resource at price p. From this definition we know that (1) if p  q , e( p) is:
e(.)
p1
e(0)
e( p1)
 

Quantity of resource ( )
Figure A.1. Net Profit Function and Average Shadow Price
zero, and (2) for any non-negative p, q=0 if and only if e( p) =0. This net profit function,
together with equation (A.3), gives us the basic tool for finding the ASP. Crema (1995)
suggested the following algorithm to find q by solving a finite sequence of MILPs.
The Algorithm
Step1: Find e(0)  sup f ( ) :   0 .
Step2: If e(0)  0 , stop. q=0 is the solution.
Step3: Find  1  min  :   0, f ( )  e(0).
Step4: Let p1  f ( 1)  1 and r=1.
Step5: Find e( p r ) .
Step6: If e( p r ) =0, stop. q  p r is the solution.
Step7: Find  r 1  min  :   0, e( pr )  f ( )  pr  .
Step8: Let pr 1  f ( r 1)  r 1 , r=r+1 and return to Step 5.
Figure A.1 illustrates the above algorithmic steps graphically. In the figure the
20
y-axis is the value of e when the price is p and the x-axis is the quantity of the resource
under consideration. Thus, the net profit function e at price zero becomes a step
function. This function reaches its maximum value when the amount of the extra
resource equals  * , implying that any additional resource after  * does not increase the
net profit for p=0. If the resource is not free, then any point beyond  * reduces the net
profit function e(.) by increasing the cost of purchasing that resource. If e(0)=0, then it
means that any additional amount of the resource cannot increase the net profit even if
this resource is “free”. The ASP under this case is zero. If e(0) is greater than zero, as
for the case in Figure A.1, from Step 3 and the figure, then we can obtain  * as the
minimum amount of resource that maximizes the net profit function. Step 4 calculates
the initial ASP by using the formula p1  v(Q *)  v(Q)  *  f ( *)  * .
After obtaining the initial ASP, we can draw a total cost line for purchasing the
extra resource when the price is p1 and a new net profit function e( p1 ) is required by
Step 5. Unlike the previous case, producers now have to pay price p1 for extra units of
the resource. Thus, the magnitudes of the steps in the step function e(0) and the total
cost line p1  represent possible profits (e( p1 ) in Figure A.1). In the same way, a new
minimum amount of the resource, which is  * * in Figure A.1, can be found to maximize
as e p1  f  **  p1  ** . Starting with step 6 in the algorithm, the ASP is updated. This
procedure continues until we find e pr  =0, where pr is the solution.
21
References:
Atkinson, S.E., Lewis, D.H., 1974. A cost effective analysis of alternative air quality
control strategies. Journal of Environmental Economics and Management 1,
237-250.
Crema, A., 1995. Average shadow price in a mixed integer linear programming
problem. European Journal of Operational Research 85, 625-635.
Dales, J.H., Pollution, Property and Prices, Toronto: University of Toronto Press,
1968.
Dunham, S., Case, C., 1997. Emissions trading opportunities in northeastern Illinois.
Paper presented at the annual meeting of Illinois Economic Association, Chicago,
IL.
Evans, D.A., 1997. An Analysis of the Emissions Reduction Market System.
Unpublished Master’s Thesis, Department of Agricultural and Consumer
Economics, University of Illinois, Urbana-Champaign.
Geoffrion, A.M. and R. Nauss. 1977. “Parametric and postoptimality analysis in integer
linear programming.” Management Science. 23:453-466.
Gomory, R.E. and W.J. Baumol. 1960. “Integer Programming and Pricing.”
Econometrica. 28(3):521-550.
ICF Resources Incorporated, 1989. Economic analysis of Title V(sic) (Acid Rain
Provisions) of the Administration’s Proposed Clean Air Act Amendments (HR
3030/S 1490). Prepared for the United States Environmental Protection Agency.
Illinois Environmental Protection Agency, 1996a. Emissions Reduction Market System
Adoption of 35 III. Adm. Code 205 and Amendments to 35 Illinois Administrative
Code 106: Statement of Reasons. Springfield, IL.
Illinois Environmental Protection Agency, 1996b. Technical Support Document for
VOM Emissions Reduction Market System. Springfield, IL. Bureau of Air, IEPA.
Illinois Environmental Protection Agency, 2001. Annual Performance Review Report –
2000, Emission Reduction Market System. Springfield, IL.
Illinois Environmental Protection Agency, 2002. Annual Performance Review Report –
2001, Emission Reduction Market System. Springfield, IL.
Illinois Environmental Protection Agency, 2003. Annual Performance Review Report –
2002, Emission Reduction Market System. Springfield, IL.
Johnson, S.L., Pekelney, D.M., 1996. Economic Assessment of the Regional Clean Air
Incentives Market: A new emissions trading program for Los Angeles. Land
22
Economics 72, 277-297.
Kim, S., Cho, S., 1988. A shadow price in integer programming for management
decision. European Journal of Operational Research 37, 328-335.
Kosobud, R.F., Stokes, H.H., Tallarico, C.D., 2001. Emissions trading to reduce urban
ozone: The Chicago area ERMS program. working paper, Department of
Economics, University of Illinois at Chicago.
Krupnik, A.J., 1986. Cost of alternative policies for the control of nitrogen dioxide in
Baltimore. Journal of Environmental Economics and Management 13, 189-197.
Liao, C.N., nal, H., 2002. Is average shadow price equilibrium price? working paper,
Department of Agricultural and Consumer Economics, University of Illinois at
Urbana-Champaign.
Maloney, M.T., Yandle, B., 1984. Estimation of the cost of air pollution control
regulation. Journal of Environmental Economics and Management 11, 244-263.
Montgomery, W.D. 1972. “Markets in Licenses and Efficient Pollution Control
Programs.” Journal of Economic Theory. 5:395-418.
Stavins, R.N. 1995. “Transaction Costs and Tradable Permit Markets.” Journal of
Environmental Economics and Management. 29:133-148.
U.S. Council on Environment Quality, 1978. Environmental Quality: The Ninth Annual
Report of the Council of Environmental Quality. U.S. Government Printing
Office, Washington D.C.
United States Environmental Protection Agency, “Clean Air Market Programs: Acid
Rain Program SO2 Allowances Fact Sheet.”
http://www.epa.gov/airmarkets/arp/allfact.html
23
Fixed Costs
($)
Firm 1
Firm 2
Firm 3
Firm 4
Firm 5
1,500
1,500
1,500
900
700
Table 1: Data for Simulation
Unit Costs of
Minimum Average
Baseline Emission
Operation
Abatement Cost
Level (tons)
($/ton)
($/ton)
55
10
$205
50
10
$200
45
10
$195
36
6
$186
20
20
$55
24
Table 2: Results from the Social Planner’s Model
Baseline Required
Minimum Average
Initial
Control
Emission Reduction
Buy Sell
Abatement Cost ASP
Allocation
Cost
Levels
Rate
($/ton)
$205
Firm 1
10
0.3
7
3
$558
$200
Firm 2
10
0.3
7
3
$558
$195
Firm 3
10
0.3
7
3
$558
$186
Firm 4
6
0.5
3
3
$558
$55
Firm 5
20
0.7
6
6
$-16
Total
56
30
9
9
$2,216
$186
25
Table 3: Results from the Firm-level Model when ASP=$186/ton is Announced
Outcome 1
Firm 1
Firm 2
Firm 3
Firm 4
Firm 5
Total
Buy (tons) Sell (tons)
3
3
3
3
6
12
6
Outcome 2
Control
Cost
$558
$558
$558
$558
$-16
$2,216
26
Buy (tons) Sell (tons)
3
3
3
3
6
9
9
Control
Cost
$558
$558
$558
$558
$-16
$2,216
Table 4: Demand and Supply Schedules
Total Abatement
Costs
0 - 55
26
0
55 - 186
12
6
186 - 196
9
9
$2,216
196 - 201
6
16
201 - 205
3
23
2050
30
Note: The price with “ ” indicates that the price is not included.
Price Range($/ton) Total Demand (tons)
27
Total Supply (tons)
Table 5: Data for Simulation
Firm 1
Firm 2
Firm 3
Firm 4
Firm 5
Fixed Costs
($)
Unit Costs of
Operation
($/ton)
Baseline Emission
Level (tons)
1,000
1,500
1,500
900
700
20
50
45
36
20
10
10
10
6
20
28
Average Cost of
Abatement with
Zero Emission
($/ton)
$120
$200
$195
$186
$55
Table 6: Demand and Supply Schedules (Firm 1’s Cost Structure)
Total Abatement
Price Range($/ton) Total Demand (tons) Total Supply (tons)
Costs
0 - 55
26
0
55 - 120
12
6
120 - 186
9
13
186 - 195
6
16
195 - 200
3
23
2000
30
Note: The price with “ ” indicates that the price is not included.
29
Table 7: Results from the Social Planner’s Model when Firm 1’s
Required Reduction Rate is Increased.
Baseline Emission Required Reduction Initial
Buy Sell Control Cost ASP
Levels
Rate
Allocation
Firm 1
10
0.4
6
4
546.8
Firm 2
10
0.3
7
3
410.1
Firm 3
10
0.3
7
4
1268.2
Firm 4
6
0.5
3
3
410.1
Firm 5
20
0.7
6
6
279.7
Total
56
29
10 10
2915
699
30
Table 8: Demand and Supply Schedules when Firm 1
is Forced to Reduce More (ASP=699)
Price Range($/ton) Total Demand (tons)
Total Supply (tons)
Total Abatement
Costs
0-5527
0
55-186
13
6
186-195
10
9
195-200
7
16
200-205
4
23
2050
29
Note: The price with “ - ” indicates that the price is not included.
31
Table 9: Results from the Social Planner’s Model when C&C is Imposed on Firm 4
Baseline Emission Required Reduction Initial
Buy Sell Control Cost ASP
Levels
Rate
Allocation
Firm 1
10
0.3
7
3
256
Firm 2
10
0.3
7
3
256
Firm 3
10
0.3
7
3
256
Firm 4
6
0.5
3
3
860
Firm 5
20
0.7
6
6
588
Total
56
30
9 9
2216
85.2
Note: The control costs of each firm are based on a permit price of $85.2.
32
Table 10: Demand and Supply Schedules when Firm 4 is Forced
to Adopt the Control Equipment (ASP=85.2)
Price Range($/ton) Total Demand (tons)
0-5555-195195-200200-205205-
23
9
6
3
0
Total Supply (tons)
3
9
16
23
30
Note: The price with “ - ” indicates that the price is not included.
33
Total Abatement
Costs
$2,216
cost
TCF3
TCF4
A
C
TCF3  TCF4  TCF5
F3
F1
F4
F2
TCF5
F5
O bF1 bF2
bF3
bF4
bF5
emission reduction
Figure 1
34
R5
cost
A
TCF3  TCF4  TCF5
TCF3
F3
F1
F4
TCF4
F2
TCF5
F5
O
bF3
bF4
bF5
emission reduction
Figure 2
35
R5
cost
A
TCF3  TCF4  TCF5
F1
F3
TCF3
F2
TCF4
F4
B
TCF5
F5
O
rF4
bF4bF3
bF5
emission reduction
Figure 3
36
R5
cost
A
TCF3  TCF4  TCF5
F3
F1
TCF3
TCF4
F4
F2
B
F5
TCF5
O
rF3
bF3
bF4
bF5
emission reduction
Figure 4
37
R5
cost
A
TCF3  TCF4  TCF5
F3
F1
TCF3
TCF4
B
F4
F2
TCF5
F5
O
rF3
bF3
bF4 bF5
emission reduction
Figure 5
38
R5
cost
Fh
G
Fg
F
Ff
Fe
E
Fa  Fb
D
Fd
Fc
O
emission reduction
Figure 6
39
cost
A
TCF3  TCF4  TCF5
TCF3
F1  F2
F3
F4
TCF4
TCF5
F5
O bF1  bF2
bF3
bF4
bF5
emission reduction
Figure 7
40
R5
cost
A
TCF3  TCF4  TCF5
TCF3
F3
F4
TCF4
F1  F2
TCF5
F5
O
bF3
bF4
bF5
emission reduction
Figure 8
41
R5
cost
A
TCF3  TCF4  TCF5
F1  F2
F3
TCF3
B
TCF4
TCF5
F4
F5
O
rF4 bF4 bF3
bF5
emission reduction
Figure 9
42
R5
cost
A
TCF3  TCF4  TCF5
F3
F1  F2
TCF3
F4
TCF4
B
F5
TCF5
O
rF3
bF3
bF4
bF5
emission reduction
Figure 10
43
R5
cost
A
TCF3  TCF4  TCF5
F3
B
TCF3
TCF4
F4
F1  F2
TCF5
F5
O
rF3
bF3
bF4 bF5
emission reduction
Figure 11
44
R5
cost
A
TCF3  TCF4  TCF5
TCF3
F1
F2
F3  F4
TCF4
TCF5
F5
O bF1
bF2  bF3 bF4 bF5
emission reduction
Figure 12
45
R5
cost
A
TCF3  TCF4  TCF5
TCF3
F1
F2 F3
F4
TCF4
TCF5
F5
O bF1
bF2  bF3 bF4 bF5
emission reduction
Figure 13
46
R5
cost
A
TCF3  TCF4  TCF5
TCF3
F1
F2
F3
TCF4
F4  F5
TCF5
O bF1
bF2
bF3 bF4  bF5
emission reduction
Figure 14
47
R5
250
price/ton
200
150
permit supply curve
permit demand curve
100
50
0
0
10
20
30
40
tons
Figure 15: Demand and Supply Curves from the Firm-level Model
When the Equilibrium Exists
48
250
price/ton
200
150
permit supply curve
permit demand curve
100
50
0
0
10
20
30
40
tons
Figure 16: Demand and Supply Curves from the Firm-level Model
When the Equilibrium Does Not Exist
49
$250
price/ton
$200
$150
permit supply curve
permit demand curve
$100
$50
$0
$0
$10
$20
$30
$40
tons
Figure 17: Demand and Supply Curves from the Firm-level Model
(Firm 1 Has to Reduce More)
50
250
price/ton
200
150
permit supply curve
permit demand curve
100
50
0
0
10
20
30
40
tons
Figure 18: Demand and Supply Curves from the Firm-level Model
When C&C is Imposed
51
$1,200
$1,000
Firm 4
Total abatement costs
$800
$600
$400
Firm 1
$200
Firm 5
$0
$0
$50
$100
$150
$200
-$200
Permit price
Figure 19: Firms’ Abatement Costs under Different Permit Prices
When C&C is Imposed
52
$250