Aggregate Output Quantity
Period
0 48
10 24 32
Aggregate Emissions
Period
0 48
0 24 32
Allowance Price
Period
0 48
8 56
Emission Rate by Firm
Period
TypeA TypeB
TypeC TypeD
0 48
0 1 2 3
This section uses a simple representative agent modeling technique to compute the equilibrium conditions under the different pollution abatement institutions. Underlying the entire economic model is the fact that firm’s production of output creates a pollution externality, which causes environmental damage. The firm faces a downward sloping inverse demand curve for its output:
P(Q). The firm’s costs are derived from two sources. The unit capacity cost is a cost of
production infrastructure which has a constant marginal value and is a decreasing weakly convex function of the firm’s emission rate, r: c(r). The unit variable cost, w, is a constant variable cost embodying the value of labour and raw materials that are needed to create a unit of output, and it is not a function of the firm’s emission rate. The environmental damages are assumed to be an increasing and weakly convex function of total emissions: D(rQ).
Before analyzing the emission permit trading institutions, it is important to point out the optimal equilibrium conditions embodied by the traditional Pigouvian tax scenario. A social planner would want to choose emission rate and output in order to maximize total welfare, and so the social planner’s welfare maximization problem is
. (A1)
The two first order conditions are
(A2)
and
, (A3)
as long as output is greater than zero. The second order conditions are detailed at the end of this section. The second order conditions of this problem are sufficient to prove that the critical values implied by the above solution are consistent with a relative maximum. These two familiar first order conditions say that the emission rate should be set so that the marginal abatement cost is equal to the marginal damage, and that the output should be set such that the marginal benefit of another unit of output equals it marginal cost, including the indirect damage cost. A Pigouvian emission tax set equal to D'(r*Q*) will be optimal in the sense that it will elicit an emission rate of r* and an output of Q* from the profit maximizing representative firm. The optimal total
emissions in the system will be E* = r*Q*.
At this point it is useful to specify the equilibrium conditions for an unregulated profit maximizing representative firm for comparison. The firm’s profit maximizing problem and its solution look very similar to the optimal case above only it is devoid of any reference to the external damage
. (A4) The two first order conditions are
(A5)
as long as output is greater than zero, and
. (A6) The latter is the zero profit condition and the former results from the fact that emission rates are uncontrolled and thus maximized in order to decrease the unit capacity cost which is a decreasing weakly convex function of emission rate. Equation A2 and A5 together imply that r0>r*, and the fact that c(r0)<c(r*) and that the marginal damage term in equation A3 is positive means that equation 6 leads to P(Q0)< P(Q*) and thus Q0>Q*. Total uncontrolled emissions will be greater than optimal, E0= r0Q0 > E* = r*Q*.
A cap-and-trade emission permit scheme with a fixed allocation of permits can be shown to achieve the optimal total emissions. Under this scheme a regulator assigns a fixed number of emission permits to each firm, which we will name ‘allowances’ so as to reduce confusion with permits used in other permit trading institutions. Any unused allowances can be sold, and any needed ones can be bought in an allowance market. Let Pa represent the price of allowances under cap-and-trade and ‘A’ be the firm’s allotment of allowances. Assuming perfect competition in this industry implies that the firm does not believe its own behaviour will impact the industry price or output level and so the firm’s profit maximization problem is just
. (A7)
The two first order conditions are
(A8)
if output is greater than zero, and
These two conditions say that profit will be maximized when the marginal abatement cost is equal to the price of allowances and the price of output is equal to the marginal cost of output. If the regulator sets the aggregate allowance allocation equal to the optimal level of emissions, then in equilibrium rQ = A = E*. Comparing the cap-and-trade first order conditions with the optimal ones illustrates that the price of allowances will be equal to the marginal damage in equilibrium, Pa=D’(r*Q*), since equations A5 and A6, and A8 and A9 are of the same form. This also implies that the emission rate and output under the cap-and-trade scheme will be identical to those of the optimal policy: ra=r*, Qa=Q* and P(Qa)=P(Q*). This implies that the total emissions under the cap-and-trade scheme will be optimal, Ea= raQa = E* = r*Q*.
A baseline-and-credit emission permit scheme with an allocation of permits proportional to output can also be used to control emissions. In this context we refer to permits as being ‘credits’ as this scheme can be thought of as an emission reduction credit system where firms face an emission standard. Emission rates above the standard require firms to buy credits and emission rates below the standard generate credits which can be sold. Let Pc represent the price of credits under a baseline-and-credit scheme and rs be the emission standard. It is assumed that the emission standard is binding and that regulators have chosen a standard tighter than the uncontrolled rate, so rs<r0. Again, assuming that the firm is a price taker under perfect competition the
representative firm’s profit maximization problem is
, (A10)
where Q(r - rs) is the firm’s net demand for credits. The two first order conditions are
(A11)
if output is non-zero and
(A12)
Notice that the marginal revenue equals marginal cost condition now involves a new term, -rsPc, due to a subsidy inherent in the setting of an emission rate standard. Net demand for permits must be non-positive and assuming that the performance standard is binding in equilibrium we have rc=rs. This leaves equation A12 to be P(Qc)=c(rc)+w. This new formulation has the same form as
since we are assuming r=r<r , and so c(r)>c(r). This means that P(Q)>P(Q) and so Q<Q . In order to assess how the baseline-and-credit institution compares to the optimal solution or the cap-and-trade case we must make an assumption regarding the emission standard. Suppose the regulator sets the performance standard equal to the socially optimal rate, which is also the cap-and-trade equilibrium rate, then rc=rs=ra=r*. First, equations A8 and A11 testify that the price of allowances in equilibrium will equal the price of credits in equilibrium, Pc=Pa. Comparing the marginal revenue equals marginal cost equations A9 and A12 show that when rc=ra that price of output under the credit scheme will be lower than under the allowance scheme due to the effect of the emission standard as a subsidy on output result, P(Qc)<P(Qa). This means that output will be greater under the baseline-and-credit scheme that under the cap-and-trade scheme or the optimal policy, Qc>Qa=Q*. Consequently aggregate emission rates under the baseline-and-credit scheme will exceed optimal levels, Ec = rcQc > E* = r*Q*.
The solutions to all of the optimization problems above are all relative maxima. Since the proof is similar for all the above cases, we will only illustrate the proof for the socially optimal case below. To recap, the welfare maximization problem is
. (A13)
The two first order conditions are
(A14)
and
. (A15)
It is obvious that the Wr condition can be satisfied if Q* equals zero or if the bracketed term equals zero. Since we are only interested in checking whether the positive output solution is a relative welfare maximum, we assume that the top first order condition implies that -c’(r*) = D’(r*Q*). The necessary and sufficient second order conditions for a relative maximum at this critical point are that Wrr<0, Wqq<0 and WrrWqq-Wrq2>0. In our welfare maximization context, based on our assumptions on the cost and environmental damage functions, these bivariate conditions are met. Using the fact that r*>=0, Q*>0, c’(r)<0, c’‘(r)>0, D’(rQ)>0, D’‘(rQ)>0,and P’(Q)<0 we can show that
, (A16)
. (A17) For the last second order condition we must use the fact that
(A18)
Notice how the cross derivative simplifies down to one term when evaluated at the critical point, specifically when Wr=0 is substituted in. Now it is easy to show that the last second order
condition for a welfare maximum is met, WrrWqq-Wrq2>0. This must be true if r*>=0, Q*>0, c’‘(r*)>0, D’‘(r*Q*)>0,and P’(Q*)<0.
The final second order condition must be met because every term in the final line above must be positive. Therefore the critical r* and Q* values defined by the first order conditions are indeed consistent with a welfare maximum.
Table of theoretical predictions
Variable Predictions
Emission rate r* = ra = rc < r0 Output quantity Q* = Qa < Qc < Q0
Output price P(Q0) < P(Qc) < P(Qa) = P(Q*) Total emissions E* = Ea < Ec < E0
Emission permit price Pc = Pa = D’(r*Q*)
With the same functional assumptions as in the last section, this section uses a model with ‘n’
firms with possibly different emission rates, output, and variable and capacity cost functions in order to illustrate the impact that differential firms have on the theoretical predictions of the model. Again, we will commence with the optimal case. A social planner would want to choose emission rate and output for each firm in order to maximize total welfare, and so the social planner’s welfare maximization problem is
, (B1)
where qi is the output of firm i and . The first order conditions are
(B2)
when qi is greater than zero and
. (B3)
The first set of conditions says that the optimal planner would set emission rates, the ri*s, such that each firm’s marginal abatement cost must equal the marginal damage of aggregate emissions.
The latter set say that output levels must be set such that each firm’s marginal cost including the externality must be equal to the price of output at Q*. Aggregate emissions will be
. Since equation sets B2 and B3 do not contain qi* as an isolated term, only the toal Q* and E* is identified in the solution, then any combination of qi*s that sum to Q*, and that sum to E* when multiplied by the corresponding ri*, is a possible solution to the welfare
maximization problem. This means that output quantity distribution is not only indeterminate between like firms, but is not even determined between firms with different cost structures.
For the remaining analysis we will assume that each firm’s cost function and variable cost is exactly the same as was implicitly assumed in the optimal planners problem, that way we can compare each firm’s behaviour under different scenarios. Each unregulated competitive firm’s profit maximizing problem and solution looks very similar to our earlier analysis and again provides a useful benchmark:
The two first order conditions are
(B5)
as long as output is greater than zero, and
. (B6) Equation B5, which states that each firm sets its marginal abatement cost to zero, will ensure that each firm chooses an emission rates higher than it’s optimal rate, ri0 > ri*, since -ci’(ri0
)=0<-ci’(ri*)=D’(E*) and marginal abatement cost is monotonically decreasing in ‘r’. Since ci(ri0) < ci(ri*) and equation B6 is missing the positive marginal damage term that the social planner’s first order condition contains in equation B3, we know that the right hand side of equation B6 must be smaller than that of equation B3 and so the uncontrolled output price must be smaller than the optimal output price, P(Q0)< P(Q*). This directly implies that uncontrolled output is greater than optimal output, Q0>Q*. Since only the total output is identified in the uncontrolled equilibrium, any set of qis that sum to Q0 will be equilibrium quantities. Notice in this general case that there are no conditions on aggregate emissions, the sum of the individual ‘ri0qi0' firm emissions is
unknown since the distribution of output between firms is unknown. Since every firm has a higher uncontrolled emission rate than their optimal rate, we do know that aggregate uncontrolled emissions must be higher than optimal emissions, no matter how this higher level of aggregate output is distributed. However if the firm’s cost functions have the property that each firm’s marginal abatement cost is equal to zero at the same emission rate, rmax0=ci’-1(0) for all ‘i’, then aggregate emissions must be E0=rmax0Q0.
Under a cap-and-trade scenario firm ‘i’s profit maximization problem is just
. (B7)
The two first order conditions are
(B8)
(B9) Equation B8 ensures that each firm’s emission rate is set such that their marginal abatement cost is equal to the price of allowances. Equation B9 makes certain that each firm earns zero profit.
As in the representative agent model, as long as the regulator allocates the optimal total number of allowances, , then the cap-and-trade equilibrium looks just like the optimal equilibrium above, where the price of allowances equal the optimal marginal damage (equations B2 and B3, and B8 and B9 are of the exact same form) . This means that under a cap-and-trade system each firm will set their emission rate to the optimal rate and aggregate output, output price and aggregate emissions will also be at their optimal levels. Again, the equilibrium distribution of output quantity between firms is unidentified, but all possible equilibrium distributions are optimal.
Lastly, we focus again on a price-taking baseline-and-credit firm, only now with the possibility that different firms have different cost structures. Firm ‘i’s profit maximization problem is
, (B10)
where qi(ri - rs) is firm ‘i’s net demand for credits. Assuming that qi is greater than zero, the two first order conditions are
(B11)
and
(B12)
Equation B11 is the usual marginal cost of abatement equals to the price of credits condition and equation B12 is the usual zero-profit condition. Let us assume that the regulator sets the emission rate standard equal to the average emission rate under the social planner scenario, rs=E*/Q*. If the emission standard is binding and gross demand for credits is equal to gross supply then
. Substituting in for the emission standard we can calculate that
in the same in proportion to aggregate output in the baseline-and-credit case as it was in the optimal case, then each firm choosing its optimal emission rate will satisfy our baseline-and-credit equilibrium condition. Focusing on the baseline-and-credit equilibrium in which each firm chooses their optimal emission rate, one notices that equation B12 is similar to the optimal equation B3 (since equation B11 togther with the assumption that ric=ri* implies that Pc=D’(E*)) except for an extra negative term in equation B12 that does not exist in equation B3. This will cause baseline-and-credit output price to be less than the optimal output price, implying that the aggregate output in this case will be larger than aggregate output chosen by the social planner. Each firm choosing its optimal emission rate combined with a proportionately larger than optimal aggregate output will result in aggregate emissions exceeding the optimal amount under this baseline-and-credit trading scheme.