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 –|
|Total amount of permits to be shared, exogenous.|
|Firms: and denote Competitive and Dominant firm, resp.|
|Marginal abatement cost of firm , where is the emission level.|
|Total abatement cost of firm|
|Total abatement cost of the industry|
|,||Firms’ types, random. The support of belongs to .|
|Expectation of conditional on , assumed linear with coeffs.|
|,||Business as usual and cost-effective emissions of , resp.|
|– Secondary market –|
|Post-market holdings of permits for firm and market price, resp.|
|Realized profit of firm .|
|Realized profit of firm (under eq. #n the secondary market).|
|(the asterisk denotes equilibrium value).|
|Value function for firm (valuation of initially getting permits).|
|– Primary market (auctioning or grandfathering) –|
|auction under complete and incomplete information resp.|
|Initial allocation of permits for firm (under grand. or auctioning).|
|Price of permits at the auction.|
|s share of permits (, under auctioning or grandf., resp.).|
|Monotone transformation of total abatement cost.|
|(incorporates secondary-market behavior).|
|Firm ’s bid (mapping from support of to set of demand functions)|
|, , ,||Auxiliary 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:
If the solution of problem  is interior, it is given by
The first-order condition (FOC) of problem  is
where is given by eq. . Using the expressions for given in eq.  together with the market-clearing condition and rearranging, the FOC can be written as , and using eq.  and rearranging we get eq. , which completes the proof of the Lemma. ■
Assume the dominant firm’s strategy is linear (we prove below that this is necessarily the case) and takes the form . We now show that there exist values of the coefficients , , , such that the market clearing condition has eq.  as a solution. Thus, when and bid and respectively, eq.  is the equilibrium price. Using the expressions for and in the market clearing condition, substituting eq.  for and collecting terms, we get the following equation:
Equating coefficients, we get a system of four equations which has, as a unique solution, , , , , where the signs follow from the fact that, under Assumption 2, only can be equal to (in case 1) or (in cases 2 and 3). The fact that is necessarily linear comes from the fact that both and 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 . Plugging eq.  in the bidding function, we find the equilibrium allocation as given by eq.  in the auction.
The next step is to show that and constitute the unique equilibrium of the game. By construction, eq.  is an equilibrium price and is the best strategy for as it is the result of solving the FOC of the competitive firm. So, we only have to show that is the best strategy for , which requires to prove that the value of given in eq.  solves eq. . To prove this, note that, for a specific value of , solving eq.  in terms of is equivalent to solving it in terms of . Therefore, eq.  provides an allocation for that maximizes for any possible value of , which implies that it also maximizes it on average and, therefore, we have solved eq. . The equilibrium is unique since both and are unique.
B.2 Proof of Corollary 1
The first condition guarantees interior solution for quantities in the secondary market  () and the second does the same for the auction (). 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 and conditions  to , collapse to the following conditions, all of which hold under Assumption 3:
- Case 1
(independent types): ;
- Case 2
( efficient): ;
- Case 3
( efficient): .
B.3 Proof of Proposition 4
where the last inequality follows by setting at its lowest possible value, , and at its highest possible value, . Using this expression for the lower bound and Assumption 3 we conclude . Using eqs ,  and noting that , we conclude
where the first inequality follows by setting , and the second inequality from Assumption 3.
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 under case 2 and we have
from which we obtain , where and the latter condition is true within the triangle delimited by the points , , . The area of this triangle is and dividing by the area of the relevant domain, which in this case is the triangle delimited by the points , and , we obtain the probability of the event :
Since the latter expression is decreasing in and , we can find the infimum value by setting them a their highest possible values compatible with Assumption 3, i. e., . Plugging these values in eq. #39] we get . By setting we obtain as the supremum value.
B.5 Proof of Proposition 8
The strategy of the proof is to show that for all the relevant values of the parameters. In turn, this is done by minimizing in terms of and and showing that the minimum value is positive, which implies that it is positive for any pair . In the three cases included in Assumption 2, 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.
the sign of which is determined by the term in square brackets, which we denote as . For the relevant values of the parameters we have and , which implies that reaches a minimum at.Using these values, we get , which is always positive under the interior solution condition .
where , . Denote as the term in curly brackets, which determines the sign of the whole expression. Now we solve the problem of minimizing subject to , and . There are two candidates that satisfy the first-order Kuhn-Tucker conditions. The first is , and the second .  For the first candidate we have
where the inequality comes from the fact that is increasing in for any (which is a required condition to guarantee interior solution) and replacing by we get . Analogously, for the second candidate we have
where, again, the inequality comes from the fact that is increasing in for any and replacing by we get .
where , . Denote as 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 subject to , and is , . Evaluating for this candidate we get
where the first inequality comes from the fact that is increasing in and then we can use to obtain a lower bound and the second inequality follows simply by using the expression for and rearranging.
B.6 Proof of Corollary 2
Consider an arbitrary value of the (monotone transformation of) total cost , say . For grandfathering, denote as the set of values of the types, , such that the cost under grandfathering is not larger than . Similarly, define for the auction. From Proposition 8 we know that, for any realization of the types, we have . In particular, this will be the case for those types contained in . Then, we conclude that is included in or, in other words, for any value of , where and are the distribution functions of under auctioning and grandfathering respectively, which implies FOSD of over .
Alvarez, F., and F. J. André. 2015. “Auctioning vs. Grandfathering in Cap-and-Trade Systems with Market Power and Incomplete Information.” Environmental and Resource Economics 62:873–906. Search in Google Scholar
Alvarez, F., and C. Mazón. 2012. “Multi-Unit Auctions with Private Information: An Indivisible Unit Continuous Price Model.” Economic Theory 51:35–70. Search in Google Scholar
Antelo, M., and L. Bru. 2009. “Permits Markets, Market Power, and the Trade-Off Between Efficiency and Revenue Raising.” Resource and Energy Economics 31:320–33. Search in Google Scholar
Burtraw, D., J. Goeree, C. A. Holt, E. Myers, K. Palmer, and W. Shobe. 2009. “Collusion in Auctions for Emission Permits: An Experimental Analysis.” Journal of Policy Analysis and Management 28:672–91. Search in Google Scholar
Cong, R. G., and Y. M. Wei. 2012. “Experimental Comparison of Impact of Auction Format on Carbon Allowance Market.” Renewable and Sustainable Energy Reviews 16 (6):4148–56. Search in Google Scholar
Cramton, P., and S. Kerr. 2002. “Tradeable Carbon Permit Auctions: How and Why to Auction Not Grandfather.” Energy Policy 30:333–45. Search in Google Scholar
Ehrhart, K. M., E. C. Hoppe, and R. Löschel. 2008. “Abuse of EU Emissions Trading for Tacit Collusion.” Environmental and Resource Economics 41:347–61. Search in Google Scholar
Ellerman, A. D., F. J. Convery, and C. de Perthuis. 2010. Pricing Carbon: The European Emissions Trading Scheme. Cambridge: Cambridge University Press. Search in Google Scholar
Godby, R.. 1999. “Market Power in Emission Permit Double Auctions.” In Research in Experimental Economics (Emissions Permit Trading), Vol. 7, edited by R. M. Isaac and C. Holt, 121–62. Greenwich, CT: JAI Press. Search in Google Scholar
Godby, R.. 2000. “Market Power and Emissions Trading: Theory and Laboratory Results.” Pacific Economic Review 5:349–64. Search in Google Scholar
Grimm, V., and L. Ilieva. 2013. “An Experiment on Emissions Trading: The Effect of Different Allocation Mechanisms.” Journal of Regulatory Economics 44:308–38. Search in Google Scholar
Hahn, R. W.. 1984. “Market Power and Transferable Property Rights.” The Quarterly Journal of Economics 99 (4):753–65. Search in Google Scholar
Haile, P. A.. 2000. “Partial Pooling at the Reserve Price in Auctions with Resale Opportunities.” Games and Economic Behavior 33:231–48. Search in Google Scholar
Hepburn, C., M. Grubb, K. Neuhoff, F. Matthes, and M. Tse. 2006. “Auctioning of EU ETS Phase II Allowances: How and Why?.” Climate Policy 6:137–60. Search in Google Scholar
Hinterman, B. 2011. “Market Power, Permit Allocation and Efficiency in Emission Permit Markets.” Environmental and Resource Economics 49:327–49. Search in Google Scholar
Ledyard, J. O., and K. Szakaly-Moore. 1994. “Designing Organizations for Trading Pollution Rights.” Journal of Economic Behavior & Organization 25:167–96. Search in Google Scholar
Montero, J. P. 2009. “Market Power in Pollution Permit Markets.” The Energy Journal 30 (special issue 2):115–42. Search in Google Scholar
Montgomery, W. D. 1972. “Markets in Licences and Efficient Pollution Control Programs.” Journal of Economic Theory 5 (3):395–418. Search in Google Scholar
Mougeot, M., F. Naegelen, B. Pelloux, and J. L. Rullière. 2011. “Breaking Collusion in Auctions Through Speculation: An Experiment on CO2 Emission Permit Markets.” Journal of Public Economic Theory 13 (5):829–56. Search in Google Scholar
Muller, R. A., S. Mestelman, J. Spraggon, and R. Godby. 2002. “Can Double Auctions Control Monopoly and Monopsony Power in Emissions Trading Markets?” Journal of Environmental Economics and Management 44:70–92. Search in Google Scholar
Reeson, A. F., L. C. Rodriguez, S. M. Whitten, K. Williams, K. Nolles, J. Windle, and J. Rolfe. 2011. “Adapting Auctions for the Provision of Ecosystem Services at the Landscape Scale.” Ecological Economics 70 (9):1621–7. Search in Google Scholar
Sturn, B. 2008. “Market Power in Emissions Trading Markets Ruled by a Multiple Unit Double Auction: Further Experimental Evidence.” Environmental and Resource Economics 40:467–87. Search in Google Scholar
Wang, J. J. D., and J. F. Zender. 2002. “Auctioning Divisible Goods.” Economic Theory 19:673–705. Search in Google Scholar
Wilson, R.. 1979. “Auctions of Shares.” Quarterly Journal of Economics 93:675–98. Search in Google Scholar
de Castro, L. I., and A. Riascos. 2009. “Characterization of Bidding Behavior in Multi-Unit Auctions.” Journal of Mathematical Economics 45:559–75. Search in Google Scholar
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