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Auctioning Emission Permits with Market Power

Francisco Alvarez and Francisco J. André


We analyze emission permit auctions in a framework in which a dominant firm enjoys market power both in the auction and in the secondary market while its competitor behaves in a competitive way. We obtain linear equilibrium bidding strategies for both firms and a unique equilibrium of the auction, which is optimal ex-post for the dominant firm. Under specific distributional assumptions we conclude that the auction always awards less permits to the dominant firm than the cost-effective amount. Our results serve as a warning about the properties of auctioning under market power. Under interior solution the auction allocation is dominated by grandfathering in terms of aggregated cost with probability one. As a policy implication, the specific design of the auction turns out to be crucial for cost-effectiveness. The chances of the auction to outperform grandfathering require that the former is capable of diluting the market power that is present in the secondary market.

Funding statement: This paper has greatly benefited from comments and fruitful discussion with Michael Finus and Carmen Arguedas. Francisco J. André acknowledges financial support from the research Projects ECO2012-39553-C04-01 (Spanish Ministry of Science and European Regional Development Fund, ERDF), SEJ 04992 and SEJ-6882 (Andalusian Regional Government).

Appendix A: Notation

– Fundamentals –
QˉTotal amount of permits to be shared, exogenous.
i,jC,DFirms: C and D denote Competitive and Dominant firm, resp.
MCie=αiβeMarginal abatement cost of firm i, where e is the emission level.
TCiTotal abatement cost of firm i
TC=TCD+TCCTotal abatement cost of the industry
αi, α=αC,αDFirms’ types, random. The support of α belongs to Ω:=θ,θ+σ2.
E{αi|αj}=μj+λjαjExpectation of αi conditional on αj, assumed linear with coeffs. μj,λj.
eiBAU, eiCEBusiness as usual and cost-effective emissions of i, resp.
Θ,ΘD,ΘCAuxiliary coefficients
– Secondary market –
qi1,p1Post-market holdings of permits for firm i and market price, resp.
Πiqi1,p1,αRealized profit of firm i.
πiqi0,α:=Πiqi1,p1,αRealized profit of firm i (under eq. #n the secondary market).
(the asterisk denotes equilibrium value).
viqi0,αiValue function for firm i (valuation of initially getting qi0 permits).
– Primary market (auctioning or grandfathering) –
ACI,AIIauction under complete and incomplete information resp.
qi0Initial allocation of permits for firm i (under grand. or auctioning).
p0Price of permits at the auction.
δ:=qC0QˉC's share of permits (δA, δG under auctioning or grandf., resp.).
hδ,αMonotone transformation of total abatement cost.
(incorporates secondary-market behavior).
qibαi,p0Firm i’s bid (mapping from support of αi to set of demand functions)
m0, m1, m2, m3Auxiliary coefficients of the leader’s optimal demand.

Appendix B: Proofs

B.1 Proof of Proposition 3

The first step is to prove the following lemma:

Lemma 1

If the solution of problem [24] is interior, it is given by


The first-order condition (FOC) of problem [24] is


where qC(αC) is given by eq. [20]. Using the expressions for πD given in eq. [12] together with the market-clearing condition and rearranging, the FOC can be written as 25qC8Qˉ5+3αD+2αC5β95βp0=0, and using eq. [20] and rearranging we get eq. [33], which completes the proof of the Lemma. ■

Assume now that the equilibrium is interior (which we check below). Using eq. [22] in eq. [33] we get the following value for the optimal price:


Assume the dominant firm’s strategy is linear (we prove below that this is necessarily the case) and takes the form qDp0=m0+m1Qˉ+m2αD+m3p0. We now show that there exist values of the coefficients m0, m1, m2, m3 such that the market clearing condition qCp0+qDp0=Qˉ has eq. [33] as a solution. Thus, when D and C bid qCAIIp0 and qDAIIp0 respectively, eq. [33] is the equilibrium price. Using the expressions for qCAIIp0 and qDAIIαD in the market clearing condition, substituting eq. [34] for p0 and collecting terms, we get the following equation:


Equating coefficients, we get a system of four equations which has, as a unique solution, m0=3λDβλD+110, m1=3λDλD+110, m2=6λD+214βλD+110, m3=18λD+94βλD+110, where the signs follow from the fact that, under Assumption 2, λD only can be equal to 0 (in case 1) or 1/2 (in cases 2 and 3). The fact that qDAIIp0 is necessarily linear comes from the fact that both qCp0 and p0 are linear and hence, in the systems of equations set above, any non-linear term must be zero for the market-clearing condition to hold for any pair αD,αC. Plugging eq. [34] in the bidding function, we find the equilibrium allocation as given by eq. [23] in the auction.

The next step is to show that qCAIIp0,αC and qDAIIp0,αD constitute the unique equilibrium of the game. By construction, eq. [33] is an equilibrium price and qCAII is the best strategy for C as it is the result of solving the FOC of the competitive firm. So, we only have to show that qDAIIp0 is the best strategy for D, which requires to prove that the value of qD0 given in eq. [23] solves eq. [21]. To prove this, note that, for a specific value of αC, solving eq. [24] in terms of p0 is equivalent to solving it in terms of qD0. Therefore, eq. [25] provides an allocation for D that maximizes πD for any possible value of αC, which implies that it also maximizes it on average and, therefore, we have solved eq. [21]. The equilibrium is unique since both qCAIIp0,αC and qDAIIp0,αD are unique.

B.2 Proof of Corollary 1

Using eqs [9], [10], [11], [34] and [23], and rearranging, we conclude that the occurrence of an interior solution w.p.1 requires that the following conditions hold:


The first condition guarantees interior solution for quantities in the secondary market [15] (0qi1Qˉ) and the second does the same for the auction (0qi0Qˉ). The third and fourth conditions ensure that the price is non-negative in the secondary market and the auction respectively. To ensure that these inequalities hold w.p.1 we check that each of them holds true for the most adverse realizations of the types. Under the three cases considered in Assumption 2, using the relevant expressions for μDand λD conditions [35] to [38], collapse to the following conditions, all of which hold under Assumption 3:

  1. Case 1

    (independent types): 2σβQˉ63θ44+σ22;

  2. Case 2

    (D efficient): 2σβQˉ63θ44;

  3. Case 3

    (C efficient): σβQˉ63θ44+σ22.

B.3 Proof of Proposition 4

In case 1, using the definition of δ, eqs [7], [26] and [23] we obtain a lower bound for the difference between δA and δCE:


where the last inequality follows by setting αD at its lowest possible value, θ, and αC at its highest possible value, θ+σ. Using this expression for the lower bound and Assumption 3 we conclude δA>δCE. Using eqs [23], [26] and noting that δG=0.5, we conclude


where the first inequality follows by setting αD=θ, αC=θ+σ and the second inequality from Assumption 3.

In cases 2 and 3 the procedure is similar, using the adequate values for μD and λD as given in eqs [27] and [29] together with the relevant values of δG according to eq. [28], [30] and [31].

B.4 Proof of Proposition 6

Given the symmetry of cases 2 and 3, it is sufficient to prove the result for case 2. Using the expressions for δG under case 2 and δCE we have


from which we obtain δGδCEαCαD+χ, where χ:=βQˉσ6θ+3σ1 and the latter condition is true within the triangle delimited by the points θ+χ,θ, θ+σ,θ, θ+σ,θ+σχ. The area of this triangle is σχ22 and dividing by the area of the relevant domain, which in this case is the triangle delimited by the points θ,θ, θ+σ,θ and θ+σ,θ+σ, we obtain the probability of the event δGδCE:


Since the latter expression is decreasing in σ and βQˉ, we can find the infimum value by setting them a their highest possible values compatible with Assumption 3, i. e., 2σ=βQˉ=6344θ. Plugging these values in eq. #39] we get 19723920.68>1. By setting σ=βQˉ=0 we obtain 1 as the supremum value.

B.5 Proof of Proposition 8

The strategy of the proof is to show that h(δA,α)>h(δG,α) for all the relevant values of the parameters. In turn, this is done by minimizing h(δA,α)h(δG,α)in terms of αD and αC and showing that the minimum value is positive, which implies that it is positive for any pair α. In the three cases included in Assumption 2, h(δA,α)h(δG,α) is a continuous and bounded function defined on a compact set and, we can use the Weierstrass theorem to state that there exists a minimum. The strategy of the proof is common but the development is slightly different for each case, so we consider them separately.

CASE 1: αD and αC are not correlated. Using eqs [15], [23], [26] and [32], the difference between total cost under auctioning and grandfathering can be written as


the sign of which is determined by the term in square brackets, which we denote as Δ1α. For the relevant values of the parameters we have ΔαD>0 and ΔαC<0, which implies that Δαreaches a minimum atαD,αC=θ,θ+σ.Using these values, we get Δ1θ,θ+σ=40σ230σβQˉ+25β2Qˉ2, which is always positive under the interior solution condition 2σ<βQˉ.

CASE 2: αDαC. Using δG=3θ+2σ6θ+3σ together with eq. [15], [23], [27] and [32], the difference of total cost between both systems can be written as


where X:=6θ+3σ, Y=3θ+2σ. Denote as Δ2α the term in curly brackets, which determines the sign of the whole expression. Now we solve the problem of minimizing Δ2α subject to αDθ, αCθ+σ and αDαC. There are two candidates that satisfy the first-order Kuhn-Tucker conditions. The first is αD=θ, αC=θ+σ and the second αD=αC=θ+σ. [16] For the first candidate we have


where the inequality comes from the fact that Δ2θ,θ+σ is increasing in βQˉ for any βQˉ>2σ (which is a required condition to guarantee interior solution) and replacing βQˉ by 2σ we get Δ2θ,θ+σ>σ2334θσ+98σ2>0. Analogously, for the second candidate we have


where, again, the inequality comes from the fact that Δ2θ+σ,θ+σ is increasing in βQˉ for any βQˉ>2σ and replacing βQˉ by 2σ we get Δ2θ+σ,θ+σ>σ2576θ2+95σ2+414θσ>0.

CASE 3: αCαD. Using δG=3θ+σ6θ+3σ together with eq. [15], [23], [29] and [32], the difference of total cost between auctioning and grandfathering can be written as


where X:=6θ+3σ, Z=3θ+σ. Denote as Δ3α the term in curly brackets, which determines the sign of the whole expression. We conclude that the only candidate that satisfies the Kuhn-Tucker first-order conditions to minimize Δ3α subject to αCθ, αDθ+σ and αCαD is αD=θ+σ, αC=θ. Evaluating Δ3α for this candidate we get


where the first inequality comes from the fact that Δ3θ,θ+σ is increasing in βQˉ and then we can use βQˉ=2σ to obtain a lower bound and the second inequality follows simply by using the expression for X and rearranging.

B.6 Proof of Corollary 2

Consider an arbitrary value of the (monotone transformation of) total cost h, say h˜. For grandfathering, denote as ΦGh˜:=α/h(δG,α)h˜ the set of values of the types, αD,αC, such that the cost under grandfathering is not larger than h˜. Similarly, define ΦA(h˜):={α/h(δA,α)h˜} for the auction. From Proposition 8 we know that, for any realization of the types, we have h(δG,α)h(δA,α). In particular, this will be the case for those types contained in ΦAh˜. Then, we conclude that ΦAh˜ is included in ΦGh˜ or, in other words, CAh˜CGh˜ for any value of h˜, where CA and CG are the distribution functions of h under auctioning and grandfathering respectively, which implies FOSD of G over A.


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Published Online: 2016-9-8
Published in Print: 2016-10-1

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