Efficiency Comparison of carbon allowance auctions of
the European Union ETS and the joint mechanism of
California and Québec
Faculty of Economics and Business
Master of Science Economics
Behavioural Economics and Game Theory (track)
15 ECTS
Rick Schmitz
11088354
Statement of Originality
This document is written by Student Rick Schmitz who declares to take full responsibility for the content.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
In this thesis I theoretically compare two of the largest carbon allowance auction mechanisms, the European Union emission trading scheme (EU ETS) and the joint mechanism of California-Québec, in order to nd out which mech-anism is more e cient and/or gives a higher utility to its seller. The dierence between the two mechanisms is that the California-Québec mechanism uses a reserve price. I show that in most of the cases the California-Québec mecha-nism outperforms the EU ETS in both e ciency and seller utility. However I also demonstrate that if the reserve price is too high, meaning that it would exclude too many bidders from the auction, the EU ETS is more e cient and gives a higher utility to the seller.
Contents
1 Introduction 4 2 Literature review 7 3 The Model 9 3.1 The Game . . . 10 3.2 The bidder . . . 11 3.3 Bidder equilibrium . . . 15 4 Mechanism comparison 16 4.1 Seller comparison . . . 16 4.2 E ciency comparison . . . 22 5 Empirical comparison 29 6 Conclusion 32 7 Appendix 34 References 361 Introduction
Global warming is one of the biggest challenges that our modern society has to face. In order to keep the consequences of global warming controllable, most of the world's governments endeavour to develop strategies to cut down their greenhouse gas emissions (especially carbon) and thereby mitigating global warming. A popular strategy is the instalment of a cap and trade system. A cap and trade system installs a cap on pollution of rms. A cap is a limit over which the pollution cannot rise. To reduce pollution this limit falls over time. The trade part of the system sets up a market on which the rms have to buy rights for their amounts of pollution. The rights that the rms are buying are divided into allowances, each allowance represents the right to pollute one metric ton of CO2. In a cap and trade system a
government gives economic incentives to its local rms to cut carbon emissions and invest in cleaner production. This incentive arises because the rms under such a cap and trade system have to pay for their right to pollute. The government has thus integrated pollution in the cost function of its local rms.
An important part of the carbon emissions allowances is allocated through auc-tions. In these kind of auctions the seller, in this case the government, has not as main objective to maximize revenue from the auction, but to reduce pollution whilst giving an incentive to rms to invest in green energies and minimizing the impact of their cap and trade system on their economy. This feature gives a dierent approach to the analysis of these auctions, leading to dierent measures of utility on the seller side.
As the number of allowances being auctioned is growing, it is of a certain im-portance to compare the two largest existing auction mechanisms (as they serve as benchmark for smaller auction mechanisms) based on the e ciency and seller utility criteria to nd out if one government could improve its e ciency by switching to a dierent mechanism or adapting parts of the other mechanism to his mechanism.
The two largest carbon emissions auctions are the European Union emission trad-ing scheme (EU ETS) and the joint greenhouse gas allowance auction system of Cali-fornia and Québec. The EU ETS was created in 2005 and was the rst system of this kind in the world. In the rst phase of the EU ETS from 2005-2007 the allowances were allocated for free to the rms by the local governments, but from 2008 on the rst countries started to conduct auctions on part of their allowances. Since then the share of auctioned allowances has been growing and it is planed that in 2020 auctioning will be the default method of handing out the allowances. Furthermore it
is planned that by 2020 there will be a single European market, because at the mo-ment some countries1 are still opting out of the common auction platform. The rst
allowance auctions in California were held by the California Air Resources Board in 2012 at the same time the rst auctions in Québec were held by the responsible min-istry. 2014 the platforms of California and Québec decided to merge and to perform their auctions in a joint mechanism. The main dierence between the two auction mechanisms is that the California Québec auction uses a reserve price whereas the EU ETS does not. This dierence leads to the following research questions: which auction mechanism has a higher e ciency, the EU ETS or the California-Québec mechanism? Which mechanism leads to a higher utility for the seller?
This thesis focuses on a theoretical answer to these questions. For this purpose a multiple unit uniform-price auction model2 adapted to the carbon allowance market
will be developed. In the developed model a uniform-price auction allocates Q units to the Q highest bids. The market clearing price payed by the bidders will be the Q lowest winning bid. If the seller sets a reserve price the market clearing price cannot fall below this price. In the case where there less than Q bids above the reserve price, only those above the reserve price will win the auction and pay the reserve price. The seller will keep the remaining objects. It will be assumed that the seller attaches a value larger than zero to the units. A uniform price auction is e cient if the units go to the bidders who value them the most. In an uniform-price auction there is an incentive to shade bids below valuation on the lower valued units, because this will lead to a lower clearing price on all units which could be more important than winning an additional unit. This bid shading gives the possibility to the bidders with a lower valuation to win a bid against bidders with a higher valuation thereby causing an ine ciency. Moreover ine ciency also arises if the reserve price is such that there are less than Q units sold, because there could be a trade bene cial for both parties (if the clearing price is higher than the sellers valuation) on the unsold units. This model will lead to the following nding: an auction functioning with a reserve price will outperform the auction without a reserve price in terms of e ciency as long as the reserve price does not exclude too many bidders.3 The intuition that there is
less ine ciency in an auction with a reserve price is that in this auction, due to the higher boundary set by the reserve price, the bidders use less bid shading and bid closer to their valuation and thereby giving a smaller probability to win to the lower value bidders leading to a higher e ciency. However if the reserve price is set to high
the opposite is true, because with a high reserve price, to many bidders are excluded causing a larger e ciency loss than the shading on the lost units would have caused. In this thesis the seller utility has the traditional revenue maximizing component plus two additional components. One component is a function that captures if an incentive to invest in green energies is given. This function is positive if the clearing price is above the value that the seller attaches to an allowance, negative if the clearing price is below that value, and zero if the clearing price equals that value. The other component is a quadratic function which captures to what extent the objective of reducing pollution whilst minimizing an impact on the economic activity is met. The function is positive if the seller sells a little less allowances than his target, zero if he sells all allowances and negative if a large quantity of allowances is not sold.
The ndings for seller utility go in the same direction as the ndings for e ciency. An auction functioning with a reserve price will outperform the auction without a reserve price in terms of seller utility as long as the reserve price does not exclude too many bidders. The intuition behind this nding is that an auction which sets a reserve price will get a higher utility, because as already mentioned above under a reserve price the buyers will submit higher bids leading to a higher clearing price if all units are sold, which will not only lead to a higher revenue but also give a higher or equal return on the other two components. Moreover if not all units are sold selling a little less valuations at a higher price will give a higher utility. This is due to the special components of the utility being positive and the seller having a positive valuation on the unsold units. Nevertheless if the reserve price is too high the seller will achieve the opposite. A too high reserve price excludes too many buyers from the market and reduces the total revenue for the seller and causes the second additional component of the utility to become largely negative, because the auction has an excessive impact on the economy.
The ndings mean that from a theoretical point of view a seller of carbon emission allowances should adopt a reserve price a little above his valuation for the allowances in order to achieve a higher e ciency and a higher utility.
The thesis is organized as follows: In a rst part gives review on the current uniform-price auction literature is given. A second part develops the model. The third part examines the bidder's equilibrium strategy. The fourth part will rst compare the seller utility and afterwards the e ciency of the two mechanisms. The fth part will take a brief look at the actual results of carbon emission auction and will be followed by a conclusion.
2 Literature review
The literature on uniform-price auction is rich. However, much of the literature concentrates on nding equilibrium strategies and the thereby appearing strategy of demand shading, i.e. not bidding the true valuation. Engelbrecht-Wiggans and Kahn (1997) characterize the arising Nash-Equilibria of uniform-price auctions. They nd equilibria where demand shading is the best reply and they show that under some conditions it is even optimal to bid zero on the second unit. M. Ausubel et al. (2014) go one step further and begin to examine the eect of demand shading on the e ciency of a uniform-price auction. They show that due to demand shading the uniform-price auction is generally socially ine cient. They prove the following theorem: In any equilibrium of uniform-price auction, with positive probability objects are won by bidders other than those with highest values, which is per se an ine cient situation. They however do not analyse a situation in which a reserve price is installed. Bresky (2013) does this step. He compares an auction with reserve price to an auction without reserve price. He nds that under certain conditions the reserve price does indeed raise e ciency.
The motivation behind my thesis is to apply these ndings to the speci cs of the carbon allowance market, especially the comparison of an auction with and without reserve price. The thesis could be seen as an extension of the paper of M. Ausubel et al. (2014) by considering a situation which is not included in their general proof, and furthermore it also extends the ndings of Bresky (2013) to the special circumstances of the carbon emission certi cates. The thesis examines the situation of the carbon emission market where the buyers can submit multiple bids which is an addition to almost all above mentioned papers in which the buyers can only submit two bids. Moreover I want to introduce the notion of risk-averseness to the utility comparison on the seller side. Hu et al. (2010) include this risk-averseness in single-unit auctions, and show that the higher the degree of risk averseness the lower the reserve price. These ndings are helpful as a certain degree of risk-averseness can be assumed on the seller side in the carbon emissions market.
The main literature on carbon emission auction was written on the beginning of the establishment of the auctions and mostly focus on which auction mechanism to use, as for example Cong and Wei (2010), who show under which situation which auction mechanism is the most e cient. Another paper which focuses on which mechanism should be applied for the sale of emissions is Lopomo et al. (2011). They come to the conclusion that the uniform price auction will be the most e cient. Although the mechanism of uniform price has already been chosen by the markets, the considerations on e ciency in these papers will be of use in the comparison of both emission trading markets.
3 The Model
The model which will be used is similar to the model developed by M. Ausubel et al. (2014) in the way that it examines multiple objects and that the valuations of the bidders are dependent on a signal. The dierence that arises is that in my model there is in some cases a reserve price the seller has multiple objectives and that there is an asymmetry because the valuations from the bidders are drawn from individual distributions rather than from a common distribution. The model is also inspired by the model of Krishna (2002), who characterises demand shading in an uniform-price auction, the addition of my model is that the valuation are depending on a signal and that there are multiple objects opposed to the two objects developed by Krishna (2002). The model is the following:
A seller, in this case a multitude of governments, announces to sell Q identical carbon emission rights. The carbon emission rights are sold to N potential bidders, in this case rms or other entities. To be able to bid the bidders have to be registered and have to provide a guarantee4 prior to the auction. The bidders are assumed to
be asymmetric as each bidder faces a dierent cost/production function depending on his activity. The value of each unit (one allowance) is thus dependent on the marginal revenue it adds to the rm. It will be assumed in the following that the bidders receive their production cost as a private one-dimensional signal si before the
auction. Signals are drawn from individual distributions Fi with support [0; Hi] and
nite density fi, that is strictly positive on (0; Hi). The individual distribution
func-tion Fi and the realization is only known to bidder i. Thereby we assume that only
the bidding entity knows about its costs. Also types rank value, meaning if si > sj
then vi(si) > vj(sj). A bidder maximizes expected pro t E π P vij(s
i) , b
i, b−i, p
, where vij(s
i)is the marginal revenue from the jth certi cate, b
i his own bid function,
b−i the bid functions of his competitors and p the market clearing price.
Unlike conventional auctions where the objective of the seller is to maximize revenue, the seller in the carbon emission auction market has multiple objectives. These objectives are to reduce the overall emissions, to give an incentive to the rms to enlarge their investments in green energies. At the same time minimizing the impact on the economic activity and to maximize revenue under these constraints.
4A guarantee is in this case an amount equal to the bids submitted. In the case of a loss the
The seller's valuation for one allowance is v, this term is constant and known to the seller prior to the auction. The seller is risk averse and his utility function us : R → R is twice continuously dierentiable satisfying u0s > 0 and u00s ≤ 0. The
seller's utility will be analysed in section 4.1.
An assignment of emission allowances auctioned among bidders is said to be ex-post e cient if each unit goes to the bidder who values it the most and no bene cial trade is excluded.
3.1 The Game
The seller uses a sealed-bid uniform-price auction format to allocate the emission allowances. The seller announces the number of identical units for sale, Q units, and in the case of California-Québec market also the reserve price, prior to the auction. Each bidder has then to submit his sealed bids. The bidders will submit their bid function biafter receiving their signal. For the analysis it will be assumed that the bid
function is right-continuous at vij(0) = 0, left-continuous at all vij(s
i) and weakly
decreasing.
When the deadline is reached, the seller establishes the market clearing price following the uniform price mechanism and sells the units to the successful bidders. The uniform price mechanism is the following: the price chosen will be the lowest winning bid and the Q highest bids will win the auction. For the California-Québec market, which functions with a reserve price, an additional pricing rule is established. In the case where less than Q bids are higher than the chosen reserve price, only the bids above or equal to the reserve price will win and the market clearing price will be the reserve price. Additionally article 7 paragraph 5 of the EU regulationComission (2010) on carbon emission auction cancels an auction in which not all the quantity is sold. Paragraph 6 of the same regulation also states that an auction is cancelled if the price of the auction is signi cantly under the price of the secondary market. This cannot not be viewed as a reserve price, because it is not a xed boundary and thus lacks the essential property of a reserve price that below a certain price no trade will be performed. We assume thus that the price of the secondary market is part of the signal which a bidder receives and is thereby part of the equilibrium strategy. In both cases, if the auction is cancelled or if not every certi cate is sold, the seller keeps the allowances, which makes it reasonable to assume that the seller attaches a value to every emission certi cate.
Both markets have a tie-breaking procedure, because both include a randomness in the following for simplicity it will be assumed that a tie will be broken randomly.
3.2 The bidder
Let c = (c1, c2, . . . , cQ) represent the ordered bids of the opponents of bidder i.
Further I de ne G = F1×F2×. . .×Fi−1×Fi+1×. . .×FN, the marginal distribution
of the highest competing bid, with associated density g = G0. G
i(bj)is the probability
that the i highest competing bid is below bj G1(bQ) = P rob (c1 < bQ) and Fji is the
marginal density of bid j of bidder i, for example Fi
j (R) = P rob (bj < R).
The expected pro t for a bidder in an auction without a reserve price is thus:
E(π) =´c:c 1≤bQ PQ j=1vij(s i) − Q · b Q g1(c) dc +´c:c 2≤bQ−1∧c1>bQ PQ−1 j=1 vij(s i) − (Q − 1) · min {c 1, bQ−1} g2(c) dc +´c:c 3≤bQ−2∧c2>bQ−1 PQ−2 j=1 vij(s i) − (Q − 2) · min {c 2, bQ−2} g3(c) dc + . . . +´c:c Q≤b1∧cQ−1>b2 (vi 1(s i) − min {c Q−1, b1}) gQ(c) dc
The equation above is the sum of the expected payos of each possible outcome of the auction for one bidder. In the following I will try to break down this equation to facilitate the analysis. I denote P (bQ = c1)the probability that two bids are equal
and α represents the probability of winning a tie.
The expected pro t without a reserve price becomes thus
E (π) = (G1(bQ) + α · P (bQ = c1)) PQ j=1vij(s i)− Q · b Q + (G2(bQ−1) − G1(bQ) + α · P (bQ−1 = c2)) PQ−1 j=1 vij(s i) − (G2(bQ−1) − G1(bQ−1)) · (Q − 1) bQ−1 − (G2(bQ−1) − (1 − G1(bQ−1))) · (Q − 1) c1− αP (bQ−1 = c2) · (Q − 1) bQ−1 + . . . + (GQ(b1) − GQ−1(b2) + αP (b1 = cQ)) (vi1(s i)) − (G Q(b1) − GQ−1(b1)) · b1 − (GQ(b1) − (1 − GQ−1(b1))) · cQ−1− αP (b1 = cQ) · b1 (1)
The equation (1) again shows the expected gains minus the expected price of that outcome each multiplied by their respective probabilities. Each bidder sets up its bid following his signal and in order to maximize expected pro t max
bi
Now I dierentiate (1) with respect to bQ without considering the tie cases, ∂π ∂bQ = 0 0 = g1(bQ) PQ j=1vij(s i)− g 1(bQ) PQ−1 j=1 vij(s i)− g 1(bQ) Q · bQ− Q · G1(bQ) 0 = g1(bQ) viQ(s i) − b Q − g1(bQ) (Q − 1) bQ− Q · G1(bQ)
If I set bQ = viQ(s), then the equation becomes
0 = g1 viQ(s i) viQ(s i) − v iQ(s i) − g 1 viQ(s i) (Q − 1) v iQ(s i) − QG 1 viQ(s i) 0 = −g1 viQ(s i) (Q − 1) v iQ(s i) − QG 1 viQ(s i) ⇒ ∂π ∂bQ |bQ=viQ(si)≤ 0
From this equation we see that in order to maximize pro ts each bidder has to weigh between marginal bene ts of winning one more unit, g1 viQ(s)
viQ(s) − bQ
and the cost of that win which is a higher total cost. Additional to this traditional weighing a strategical component comes into play in the uniform price auction. As the lowest winning bid determines the market clearing price the strategical compo-nent is that each bidder has an incentive to shade his bid in order to decrease the eventual market clearing price. This is especially the case in an auction where a large quantity is sold. The marginal bene t of winning the last unit will be small but the reduction in cost due to a lower total price will be a larger. The maximization problem demonstrates that in a uniform-price auction bidding the true value does not maximize payo. This con rms the ndings of chapter 13 of Krishna (2002), in a multiple units setting, which is that bids in an uniform price auction are lower than their marginal value.
Similar reasoning can be applied for the bidders who face a seller applying a reserve price. Their expected payo is:
E(π) =´c:R≤c 1≤bQ PQ j=1vij(s i) − Q · b Q g1(c) dc +´c:R≤c 2≤bQ−1∧c1>bQ PQ−1 j=1 vij(s i) − (Q − 1) · max {R, min {c 1, bQ−1}} g2(c) dc +´c:R≤c 3≤bQ−2∧c2>bQ−1 PQ−2 j=1 vij(s i) − (Q − 2) · max {R, min {c 2, bQ−2}} g3(c) dc + . . . +´c:R≤c Q≤b1∧cQ−1>b2 (vi 1(s i) − max {R, min {c Q−1, b1}}) gQ(c) dc
As in the case without reserve price this equation is the sum of the expected payos of each possible outcome of the auction for one bidder. Again I will break down the formula
E (π) = 1 − Fi Q(R) (G1(bQ) + α · P (bQ = c1)) PQ j=1vij(s i)− Q · b Q + 1 − FQ−1i (R) (G2(bQ−1) − G1(bQ) + α · P (bQ−1 = c2)) PQ−1 j=1 vij(s) − 1 − Fi Q−1(R) (G2(bQ−1) − G1(bQ−1)) · (Q − 1) bQ−1 − (1 − G1(R)) (G2(bQ−1) − (1 − G1(bQ−1))) · (Q − 1) c1 −G1(R) 1 − FQ−1i (R) − 1 − FQi (R) (Q − 1) R − α · P (bQ−1 = c2) · (Q − 1) bQ−1 + . . . + (1 − F1i(R)) (GQ(b1) − GQ−1(b2) + αP (b1 = cQ)) (vi1(s i)) − (1 − Fi 1(R)) (GQ(b1) − GQ−1(b1)) · b1− (1 − GQ(R)) (GQ(b1) − (1 − GQ−1(b1))) · cQ−1 −GQ(R) (1 − F1i(bQ−1)) R − αP (b1 = cQ) b1 (2)
The equation (2) shows the expected gains minus the expected price of that outcome each multiplied by their respective probabilities. I will dierentiate equation (2) with respect to bQ.
∂π ∂bQ = 0 0 = g1(bQ) 1 − FQi (R) PQ j=1vij(s i)− Q · b Q + (−Q) 1 − Fi Q(R) G0(bQ) −g1(bQ) 1 − FQ−1i (R) PQ−1 j=1 vij(s i) 0 = g1(bQ) 1 − FQi (R) PQ j=1vij(s i)− Q · b Q − 1 − Fi Q−1(R) PQ−1 j=1 vij(s i) −Q (1 − FQ(R)) G1(bQ) 0 = g1(bQ) FQ−1i (R) − FQi (R)PQ−1 j=1 vij(s i)+ g 1(bQ) 1 − FQi (R) viQ(s i) − b Q −g1(bQ) 1 − FQi (R) (Q − 1) bQ− Q 1 − FQi (R) G1(bQ) If I set bQ = viQ(s i) 0 = g1 viQ(s i) FQ−1i (R) − FQi (R)PQ−1 j=1 vij(s i) +g1 viQ(s i) 1 − Fi Q(R) viQ(s i) − v iQ(s i) −g1 viQ(s i) 1 − Fi Q(R) (Q − 1) bQ− Q 1 − FQi (R) G1 viQ(s i) By de nition Fi
Q−1(R) ≤ FQi (R) because the bids are ordered the probability
the Qth bid is below R has to be at least as high as the probability that Q − 1th bid is below R. ⇒ g1 viQ s i FQ−1i (R) − FQi (R) Q−1 X j=1 vij s i !! ≤ 0 and thus ⇒ ∂π ∂bQ |bQ=viQ(si)< 0
This means that in a reserve price setting there is also an incentive to shade bids, because bidding the true value does not maximize expected pro t. Nevertheless the incentive is given to a lesser extent, as a bidder in a reserve price setting has to consider the limit of shading which is the reserve price. This could be an indication for an auction with reserve price to be more eective, because bidders have an incentive to bid closer to their signalled value especially if the value is close to the reserve price.
3.3 Bidder equilibrium
In the following I will try to form an equilibrium strategy for a bidder participating in a carbon emission auction. As we have seen in the analysis of the maximization problem it is not an optimal strategy to bid the true valuation according to the received signal. However proposition 13.4 of Krishna (2002) demonstrated that for any strategy in a uniform price auction it is optimal to bid the true value on the rst unit. This nding can also be applied in my multiple units model. The reasoning is quite intuitive, if the bidder bids below his valuation on the rst unit and loses, he gets a payo of 0. He can increase his utility by bidding his true valuation, because even if he has to pay a cost equal to his valuation he ends up having won one unit at the same payo instead of having won no unit. It is reasonable to assume that this strategy has a greater utility. Thus an equilibrium strategy for the carbon emission auction takes these two aspects into account
bi1 vi1 s i , si, R = v i1 s i bij vij s i , si, R = 0 0 ≤ vj(si) ≤ R βij vij(s i) , si, R R < vj(si)
In the case without a reserve price a similar strategy is adopted
bi1 vi1 s i , si = v i1 s i bij vij s i , si = β ij vij s i , si In both cases the function βij vij(s
i) , si
shades the bid to some extend below the true valuation.
Nevertheless, equilibrium strategies are not the main theme of this thesis. For a more detailed description of how to determine equilibrium strategies see M. Ausubel et al. (2014) and Engelbrecht-Wiggans and Kahn (1997), these two papers establish conditions for strategies to form an equilibrium. For our analysis it is of importance to know that the equilibrium strategy uses demand shading and that βij vij(s
i) , si is a mathematically well-de ned function, continuous in vij(s
i) , si, which at each
valuation equals expected marginal pro t of that bid to the expected marginal cost of that bid.
Theorem 1: If the bidders follow the above explained strategy bi(v, si) = βij vij(s
i) , si
then there exists an equilibrium in which bid shading occurs.
The proof of proposition 1 is provided in the appendix.
4 Mechanism comparison
4.1 Seller comparison
The most novel approach of this thesis is the seller's utility function with special components for the carbon allowance auction. As mentioned in the model section in the carbon emission auction the seller has multiple objectives. I will now examine the utility function of the seller, which I constructed in order to account for the multiple objectives of the seller, which are to reduce the overall emissions, to give an incentive to the rms to enlarge their investments in green energies. At the same time minimizing the impact on the economic activity and to maximize revenue under these constraints. The utility function of the seller is:
us(Qa) = us(Qv) + ψ (Qa)
if he does not sell all the allowances or
us(p, Qa) = us(QaE (p)) + γ (E (p)) + ψ (Qa) + us((Q − Qa) v)
if he sells all the allowances. Qa is the quantity sold, which in the case without
reserve price is always equal to the quantity put up to sale Q5, but in the case with a
reserve price Qa can be smaller than Q. The function γ (p) captures to what extent
the sellers target of giving an incentive to invest in green energies is met. γ (p) > 0 p > v γ (p) = 0 p = v γ (p) < 0 p < v
The function becomes negative if the price of the auction falls below the valuation which the seller attributes to an allowance and positive if the price is above that
value. In the case of an auction functioning with a reserve price I assume that a risk-averse seller sets the reserve price in the interval (v, H) with H being the highest valuation possible. Furthermore the function has diminishing marginal returns.
The function ψ (Q) captures the emission reduction target. The function is posi-tive if the seller sells a little less allowances, meaning that the pollution is lower than his target. However if Qa becomes to low, the function ψ (Q) becomes negative,
be-cause this is a sign for the auction having a too large impact on the cost for the rms (a low Qa means low activity), which is not compliant to one of the sellers objectives
(impact on the economic activity). The function is zero if the seller sells as much certi cates as the targeted.
The function ψ (Qa)is bounded at Qa= Q and quadratic of the form
ψ (Qa) = 0 Qa= Q ψ (Qa) = − (Q − Qa) 2 + ε Qa< Q ψ (Qa) = ` Qa> Q
Qa> Q is impossible because the seller cannot sell a higher quantity than he put up
to the auction. ε is a constant re ecting a targeted minimal impact on the economic activity. The other terms in the utility capture the utility that a seller gets from the revenue earned in the auction us(QaE (p)) and the utility which the seller receives
from keeping some units us((Q − Qa) v).
The expected price is equal to the lowest winning bid of the auction, the Q highest bit. In the auction without a reserve price the seller has no in uence on the price whereas by introducing a reserve price the seller gives an incentive to bid higher to the bidders and thereby in uencing the price.
E (p) = E
βiQ¯
viQ¯ (s
i) , siwith ¯Q being the lowest winning bid or E (p) = R
I will divide the comparison in three dierent cases.
Case 1: In both mechanisms all Q emission certi cates are sold.
Case 2: Not all quantity is sold in the California Québec auction but all quantity is sold in the EU ETS market.
Case 3: Not all quantity is sold in the California-Québec auction and no sale in the EU ETS market.
I will start with the analysis of case 1.
Lemma 1: In case 1 a seller operating with a reserve price has a higher expected utility than a seller operating without a reserve price.
Proof:
The quantity sold are Q emission certi cates. The essential argument of this proof is that the expected price is higher in a setting with a reserve price. The expected price is in both cases the lowest winning bid. The lowest winning bid is the Q highest overall bid, in this proof βiQ¯ vij(s
i) , si
. I need thus to show that the highest winning bid in an auction with a reserve price is higher and will lead to a higher expected utility in the reserve price auction, i.e.
EusR(R, Q) > EusN(R, Q)
usR(QE (pR)) + γ (E (pR)) > usN(QE (pN)) + γ (E (pN))
Because of a higher price usR(QE (pR)) > usN(QE (pN))and γ (E (pR)) > γ (E (pN)).
The subscript R denotes a seller with a reserve price and the subscript N denotes a seller without a reserve price.
Thus I need to show that EβiQ¯
viQ¯(s i) , si, R > Eβ iQ¯ viQ¯(s i) , si. The
proof can be found in the appendix.2
Now that I have examined case 1, I will move on to case 2, where the utility functions evolve as follows:
EusR(R, Qa) = us(QaR) + γ (R) + ψ (Qa) + us((Q − Qa) v)
and usN (p, Q) = us(QE (p)) + γ (E (p)).
In this case it is not clear which mechanism outperforms the other. The level of the reserve price will determine which mechanism generates a higher utility for the seller, leading to the following proposition.
Lemma 2: Under the assumption that the outcome of the auction without reserve price is higher than v, in case 2 there exists a reserve price at which the inequality EusR(R, Qa) > EusN(R, Q) changes to
EusR(R, Qa) < EusN(R, Q). This change is due to a increasing R.
Proof: First is it important to see that, if we are in case 2, the seller has set the reserve price so high that at most Q − 1 valuations are above R, but overall there are still more than Q bids in the auction so that all emission allowances could be traded. I will now examine the inequality:
EusR(R, Qa) > EusN(R, Q)
usR(QaR) + γ (R) + ψ (Qa) + usR((Q − Qa) v) > usN(QE (p)) + γ (E (pN))
usR(QaR) + usR((Q − Qa) v) − usN(QE (pN)) + γ (R) − γ (E (pN)) + ψ (Qa) > 0
(3)
First it is clear that γ (R)−γ (E (pN)) > 0, because E (pN)has to be smaller than
Rbecause there are less than Q valuations above R and no bidder has an incentive to bid above his value. Thus the inequality holds by construction of the function γ (p). Next I examine usR(QaR)+usR((Q − Qa) v)−usN(QE (p)). I will take the limits
of these terms with respect of R going to the highest valuation.
lim
R→HusR(QaR) = usR(R)
The limit is usR(R) because if R is equal to the highest valuation, only the bidder
having this valuation submits a truthful bid, which is R. The total utility that the seller gains from auctioning one unit is usR(R).
lim
R→HusR((Q − Qa) v) = usR((Q − 1) v)
The limit is usR((Q − 1) v) because if R is equal to the highest valuation than only
the bidder having this valuation submits a truthful bid, which is R, all the other bidders do not participate in the auction. The seller keeps (Q − 1) units which gives him a utility of usR((Q − 1) v).
lim
R→HusN(QE (p)) = usN(QE (p))
usN(QE (p)) is not aected by the reserve price.
Thus under the assumption E (pN) > v a high R does not make up the dierence
usR((Q − 1) v) − usN(QE (p)), especially if Q is large.
Thus lim
R→HusR(QaR) + usR((Q − Qa) v) − usN(QE (p)) < 0.
If E (p) < v then the opposite holds lim
R→HusR(QaR) + usR((Q − Qa) v) − usN(QE (p)) > 0.
The last argument of the inequality (3) is the function ψ (Qa), here it is clear to
see that
lim
R→Hψ (Qa) = − (Q − 1) 2
+ ε
because if the reserve price reaches the highest valuation only one unit is sold. As the function γ (p) has diminishing returns it can also be assumed that at the limit the gain in incentive is dominated by the loss in economic activity due to the too high reserve price.
The total limit is thus
lim
R→HusR(QaR) + usR((Q − Qa) v) − usN(QE (p)) + γ (R) − γ (E (p)) + ψ (Qa) < 0
I have thus shown that for an high R the utility for the seller is higher in the auction without a reserve price, unless the outcome of the auction is E (p) < v. In this case the opposite is true because this outcome means a negative utility for the seller without a reserve price.
For lemma 2 to be true it remains to be shown that for a lower R
EusR(R, Qa) > EusN(R, Q). To demonstrate this I will examine the case where
the seller sets a reserve price close to v, so that Q − 1 valuations are above R. This means that ψ (Qa) is positive because selling one allowance less does not aect the
economic activity. It is also clear that usR((Q − 1) R) + usR(v) − usN(QE (p)) > 0
as the seller gains on Q − 1 units and loses only on 1 unit. This means that
usR(QaR)+usR((Q − Qa) v)−usN(QE (p))+γ (R)−γ (E (p))+ψ (Qa) > 0which
implies that EusR(R, Qa) > EusN(R, Q).
I have thus proven that for a low, sensible to the valuations, reserve price the auction with the reserve price gives a higher level of utility in case 2 .2
The last case that remains to be examined is case 3. Here I establish the follow-ing proposition.
Lemma 3: In case 3 the expected utility is larger for a seller operating with a reserve price.
Proof:
Again in this case the reserve price is so high that at most Q − 1 valuations are above R, additionally in this case overall there are less than Q bids in the auction so that all emission certi cates cannot be traded and according to the mechanism the auction without reserve price is cancelled.
This proof is straightforward. The expected utility for the auction with reserve price is in case 3:
EusR(R, Qa) = us(QaR) + γ (R) + ψ (Qa)
The expected utility for the auction without reserve price is in case 3:
EusN(R, Qa) = ψ (Q) + us(Qv)
⇒ EusR(R, Qa) > EusN(R, Qa)
usR(QaR) + γ (R) + ψ (Qa) + usR((Q − Qa) v) > ψ (Q) + usN(Qv)
us(QaR) + usR((Q − Qa) v) − usN(Qv) + γ (R) + ψ (Qa) − ψ (Q) > 0
By the construction of the respective functions γ (R) ≥ 0 , ψ (Q) < 0, ψ (Qa) >
ψ (Q). If ψ (Qa) is also negative then |ψ (Qa)| < |−ψ (Q)|.
Furthermore it is clear that us(QaR) + usR((Q − Qa) v) − usN(Qv) > 0. 2
Thus I have shown that, if the seller sets a relatively low sensible reserve price above v, the expected utility is higher for the mechanism with a reserve price in all three cases. Only if the reserve exceeds a certain level, the auction without the reserve price gives a higher utility. A seller should thus adopt a mechanism with a reserve price in order to increase his utility. Finding the optimal reserve price is beyond the scope of this paper but would be an interesting topic for further research.
4.2 E ciency comparison
The next objective of this paper is to nd out which mechanism has the highest e ciency. First I will recall the de nition of e ciency in an auction, which is, that an auction is e cient if the units auctioned go to the bidders who have the highest valuation for them and no bene cial trade is excluded. I will use again the three cases of section 4.1 to compare the e ciency of the two mechanisms. Furthermore, like in Bresky (2013), who measures e ciency in a symmetric uniform-price auction where bidders only bid for two objects, I will de ne two dierent kinds of e ciency losses: exclusion loss (EL) and shading loss (SL). I will extent the analysis of Bresky (2013) to an asymmetric bidder uniform-price auction in which each bidder submits multiple bids.6 The exclusion loss arises if the reserve price is at a level at which less
than Q valuations are above it, meaning that some units remain unsold. The shading loss is the loss in e ciency due to bid shading. This loss arises if a bidder wins a certi cate although he does not poses the highest valuation for it.
Let me de ne the two dierent losses
EL (R) = N X i=1 Q X j=1 R ˆ 0 vij s i H Q−j vij s i dG Q vij s i
The integral goes from 0 to R in order to account for the loss caused by the re-serve price. In this equation HQ−j vij(s
i)
is the probability that only Q − j values are above vij(s
i), this includes only the valuations which belong to the Q highest
valuations7. All other valuations are excluded from the EL because they are not part
of an e cient outcome. A dierence to Bresky (2013) is that I have to sum up the expected exclusion loss of each bidder in order to get the total expected exclusion loss, because as the bidders are asymmetric I cannot multiply by N the expected EL of one bidder. EL(R) is the EL of the auction with a reserve price. EL(0) is the EL of the auction without the reserve price which is only present in the special case in which the auction is cancelled in this case the EL(0) is equal to the sum of all valuations of all bidders because all the bidders are excluded from the auction. In all other cases the EL(0) is zero because no bidder is excluded from the auction.
6An other dierence is that I changed the name of the losses in Bresky (2013) SL is called
marginal loss and EL is called utility loss.
7The fact that the bids are ordered assures that only the Q highest valuations are accounted in
Next I will de ne SL. As mentioned earlier in an e cient outcome the Q highest values should win the auction. Thus if the equilibrium bids are such that a bidder with a less than Q highest value wins and a bidder with a Q highest value loses, this outcome contributes to SL. The loss in e ciency is thus the dierence in valuation between the value that lost and the value that won. In the de nition of SL v is the lower value that wins and ev the higher value that loses. HQ
^ vij(s
i) is the
probability that ^vij(si)is among the Q highest values and 1 − gQ βij vij(s
i) , si is the probability that the equilibrium bid loses to a valuation not among the Q highest. SL (0) =PN i=1 PQ j=1 ´H 0 ´v^ij(si) βij vij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(si) SL (R) =PN i=1 PQa j=1 ´H R ´v^ij(si) βij vij(si),si,R ^ vij(s i) − v −ik(s −i)H Qa ^ vij(s i) · 1 − gQa βij vij(s i) , si, R dG Q v−ik(s −i)dG Qa ^ vij(si)
SL(0) is the SL for the auction without reserve price and SL(R) the SL for the auc-tion with reserve price. The rst integral goes from the amount that the bidder bid on the jth unit to the actual valuation the bidder has for this unit, because only in this space SL can arise. The second integral goes from 0 to H, because the valuation of the second bidder can take any value from 0 to the highest. In the case of the auction with reserve price the second integral starts at R, because bidders with a valuation lower than R are excluded from the auction. In both equations again a dierence to Bresky (2013) is that I sum up the expected shading losses for every bidder, and not multiply the expected SL of one bidder, in order to get the total expected shading loss.
I will start with the analysis of case 1 (In both mechanisms all Q emission al-lowances are sold).
Lemma 4: If the outcome of the auction is case 1 then the mechanism with a reserve price has a higher expected e ciency. The inequality EL (R) + SL (R) < EL (0) + SL (0) holds.
means that I only have to show that SL (R) < SL (0) . SL (0) − SL (R) = PN i=1 PQ j=1 ´H R ´v^ij(si) βij vij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(si) −PN i=1 PQ j=1 ´H R ´v^ij(si) βijvij(si),si,R ^ vij(si) − v−ik(s −i)H Q ^ vij(si) · 1 − gQ βij vij(s i) , si, R dG Q v−ik(s −i)dG Q ^ vij(s i) +PN i=1 PQ j=1 ´R 0 ´v^ij(si) βijvij(si),si ˜ vij(s i) −v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) (4)
From the proof of lemma 1 and under the assumption that the bidders bid according to their equilibrium strategy, I know that the bidders have an incentive to bid higher and closer to their valuations in the setting with a reserve price. This is su cient to guarantee that SL (0) − SL (R) > 0, because as all bidders bid closer to their value the probability of winning for example the Qth unit is higher if they have the Q highest valuation. gQ βij vij(s i) , si, R > g Q βij vij(s i) , si ⇒ 1 − gQ βij vij(s i) , si > 1 − g Q βij vij(s i) , si, R (5) As the other values of the integral are the same in SL(0) and SL(R). (5) means that the dierence between the product of the rst two rows and the product of the third and fourth row of equation (4) is always positive. The remainder of (4) is by construction always positive.2
In case 2 (not all quantity is sold in the California Québec auction but all quantity is sold in the EU ETS market) the analysis is less straightforward. In the auction with the reserve price only Qaunits are sold.
Lemma 5: Under the assumption that the clearing price in the auction without reserve price is above the sellers valuation, in case 2 there exists a reserve price at which the dierence SL (0)−SL (R)−EL (R) which is decreasing in R changes from positive to negative. This means that there exists a reserve price at which the auction without reserve price starts to be more eective than the auction with a reserve price. Before that reserve price the auction with reserve price is more e cient.
Proof:
Let us examine the dierence SL (0) − SL (R)
SL (0) − SL (R) = PN i=1 PQa j=1 ´H R ´v^ij(si) βijvij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) −PN i=1 PQa j=1 ´H R ´v^ij(si) βijvij(si),si,R ^ vij(si) − v−ik(s −i)H Qa ^ vij(si) · 1 − gQa βij vij(s i) , si, R dG Q v−ik(s −i)dG Q ^ vij(s i) +PN i=1 PQ j=1 ´R 0 ´v^ij(si) βij vij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) +PN i=1 PQ j=Qa ´H 0 ´v^ij(si) βij vij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) (6)
I proved already in lemma 4 that the dierence in SL is bigger if the same quan-tities are sold, which means that the dierence between the product of the rst two rows and the product of the third and fourth row of equation (6) is always positive. The remaining rows of equation (6) are by construction always positive. This means that in case 2 SL is greater in the auction without a reserve price.
Now I need to examine the dierence SL (0) − SL (R) − EL (R) = PN i=1 PQa j=1 ´H R ´v^ij(si) βijvij(si),si ^ vij(si) − v−ik(s −i)H Q ^ vij(si) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) −PN i=1 PQa j=1 ´H R ´v^ij(si) βijvij(si),si,R ^ vij(s i) − v −ik(s −i)H Qa ^ vij(s i) · 1 − gQa βij vij(s i) , si, R dG Q v−ik(s −i)dG Q ^ vij(s i) +PN i=1 PQ j=1 ´R 0 ´v^ij(si) βij vij(si),si ^ vij(s i) − v −ik(s −i)H Q ^ vij(s i) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(si) +PN i=1 PQ j=Qa ´H 0 ´v^ij(si) βijvij(si),si ^ vij(si) − v−ik(s −i)H Q ^ vij(si) · 1 − gQ βij vij(s i) , si dG Q v−ik(s −i)dG Q ^ vij(s i) −PN i=1 PQ j=1 ´R 0 vij(s i) H Q−j vij(s i) dG Q vij(s i) (7)
If the result of (7) is greater than 0 the auction with reserve price is more eective. The dierence between the product of the fth and sixth row and ninth row of equa-tion (7) will always be negative, because in the product of the fth and sixth row of equation (7) the loss is only a dierence between values, but in the ninth row of equation (7) the losses are always the total valuation of a unit. Additionally it is reasonable to assume that HQ−j vij(s
i) > H Q ^ vij(si) 1 − gQ βij vij(s i) , si because HQ ^ vij(s
i) is multiplied by a term smaller than 1 and additionally in the
LHS of the equation the valuation only has to be higher than Q − j valuations rather than Q. Furthermore the product of the seventh and eighth row of equation (7) is by construction always positive. Thus the auction with a reserve price is more e cient as long as the gain on lower shading between the sold certi cates, in both auctions, plus the gain on lower shading on the unsold certi cates outweighs the loss that arises on the excluded valuations.
Next I am going to show that this dierence is decreasing in R. I am taking limits of EL and SL. lim R→HEL (R) = N X i=1 Q X j=1 vij s i ! − viH
lower are the number of units won because more and more values drop below the reservation price. EL reaches its limit when only one value is above or equal to the reserve price and thus at this limit all valuations but the highest are excluded from the auction and contribute to EL.
lim
R→HSL (R) = 0
The higher the reserve price gets the lower is the SL because only the bidder with the highest valuations remain. The limit is reached when the certi cates are sold only to those who value them the most. At this point there is 0 SL because the possibility to lose against a lower valuation is zero.
SL (0)does not change, because there is simply no reservation price in that mech-anism. ⇒ limR→HSL (0) − SL (R) − EL (R) = SL (0) − PN i=1 PQ j=1vij(s i)− v iH < 0
This limit is smaller than 0, because the shading loss cannot exceed the total of all but one valuation on the market. I thus showed that SL (0) − SL (R) − EL (R) is indeed decreasing in R, which means that for a high R that auction without a reserve price is more e cient. To complete the proof I need to show that for a low reserve price this dierence is positive. I consider the case in which Q − 1 certi cates are sold, which means that only the Q highest valuation contributes to EL. It reasonable to assume that the gain of lesser shading on Q−1 units outweighs the loss of one unit in terms of e ciency, especially in cases of the carbon emission auctions in which Q is high compared to the price and there is a large possibility to shade.2
To nish the e ciency comparison I will now examine case 3 (not all quantity is sold in the California-Québec auction and no sale in the EU ETS market).
Lemma 6: If the outcome of the auction is case 3 then the mechanism with a reserve price has a higher expected e ciency EL (R) + SL (R) < EL (0) + SL (0).
Proof:
First I will analyse the exclusion loss. In an auction without reserve price the auction is cancelled, which makes it clear that the exclusion loss is higher in that
auction. EL (R) < EL (0) PN i=1 PQ j=1 ´R 0 vij(s i) H Q−j vij(s i) dv ij(s i)<PN i=1 PQ j=1vij(s i)
This inequality always holds because the LHS adds the expected values lost due to a reserve price, whereas the RHS adds the total of all valuations. Furthermore it needs to be shown that EL (R) + SL (R) < EL (0), because there is no shading loss if no trade takes place.
EL (0) − EL (R) − SL (R) > 0 PN i=1 PQ j=1vij(s i) −PN i=1 PQ j=1 ´R 0 vij(s i) H Q−j vij(s i) dG Q vij(s i) −PN i=1 PQa j=1 ´H R ´v^ij(si) βijvij(si),si,R ^ vij(si) − v−ik(s −i)H Qa ^ vij(si) · 1 − gQa βij vij(s i) , si, R dG Q v−ik(s −i)dG Q ^ vij(s i)> 0
This dierence can never be negative, because the expected loss in e ciency cannot be higher than the total loss of valuations due to no sale. I already proved that
lim
R→HSL (R) = 0 . Meanwhile EL never equals the total of valuations as long as
allowances are traded. Basic economic reasoning also tells us that the overall e ciency has to be higher in the auction with a reserve price in case 3, because a situation in which a trade takes place in which both parties gain from that trade, this trade is more e cient (proof of proposition 4 shows that the seller is better o in the reserve price case) than no trade.2
To sum up the ndings of this section it can be stated: as long as the reserve price is reasonably set and does not exclude too many buyers, the auction mechanism with a reserve price has a higher e ciency than the mechanism without a reserve price.
5 Empirical comparison
In this section I will brie y compare the actual outcomes of the carbon emission auc-tion on the California-Québec market with those of the EU ETS. Even though some countries of the EU ETS started their rst auctions in 2008 and the rst California only auction was in 2012, I will only compare the outcomes of the auctions since the 19 November 2014 because at this date the rst joint auction of California and Québec was held.
The following table8Board (2016) shows the results of every auction performed since
2014 on the California-Québec market.
Auction Date Total Current Vintage Allowances Oered Total Current Vintage Allowances Sold Current Auction Settlement Price Total Future Vintage Allowances Oered Total Future Vintage Allowances Sold Advance Auction Settlement Price 05/2016 67,675,951 7,260,000 $12.73 10,078,750 914,000 $12.73 02/2016 71,555,827 68,026,000 $12.73 10,078,750 9,361,000 $12.73 11/2015 75,113,008 75,113,008 $12.73 10,431,500 10,431,500 $12.65 08/2015 73,429,360 73,429,360 $12.52 10,431,500 10,431,500 $12.30 05/2015 76,931,627 76,931,627 $12.29 10,431,500 9,812,000 $12.10 02/2015 73,610,528 73,610,528 $12.21 10,431,500 10,431,500 $12.10 11/2014 23,070,987 23,070,987 $12.10 10,787,000 10,787,000 $11.86
The next table9Board (2016) shows the reserve prices for these auctions. Auction Date Reserve Price Current
Vintage Allowances
Reserve Price Future Vintage Allowances 05/2016 $12.73 $12.73 02/2016 $12.73 $12.73 11/2015 $12.10 $12.10 08/2015 $12.10 $12.10 05/2015 $12.10 $12.10 02/2015 $12.10 $12.10 11/2014 $11.34 $11.35
We can see that apart from the last two auctions, which see a dramatic drop in quantity sold, each auction sells almost all the emissions up to sale at a clearing price close to or above the reserve price. This means that the exclusion loss is low in this auction mechanism. The dramatic drop in the last auction is di cult to explain as there was no severe recession in California or in Québec during May 2016 and February 2016. One possible explanation could be that the bidders were able to reduce their pollution to a level where they did not need a large quantity of allowances.
I now take a look at the data of the EU ETS. The following table10ETS (2016)
shows the detailed results from the auctions performed in 2016 and 2015 and the average results from 201411.
9The data is taken from the summary report of the California Air Resources Board May 2016. 10The data is taken from the o cial summary report of the EU ETS March 2016.
Auction Date Total auction volume Auction clearing price 03/2016 51,577,000 $5.70 02/2016 48,051,000 $6.07 01/2016 30,825,000 $7.63 12/2015 23,244,500 $9.51 11/2015 40,789,500 $9.75 10/2015 37,934,000 $9.60 09/2015 40,788,000 $9.27 08/2015 17,508,000 $9.27 07/2015 40,788,000 $8.89 06/2015 37,934,000 $8.53 05/2015 29,116,000 $8.54 04/2015 35,016,000 $8.12 03/2015 43,706,000 $7.77 2014 335,052,000 $8.14
From that table I can infer that the actual outcomes are compatible with the theoretical ndings. The price is lower in the auction mechanism without the reserve price, which could mean that the bidders in the EU ETS have shaded their bids more than the bidders on the California-Québec market. From these table I cannot infer something about the loss of e ciency. However I can see clearly that the bidders in the California-Québec market have a greater incentive to invest in cleaner production as they have to pay for their emission much higher than a European rm in the EU ETS. This could mean that the utility of the European governments is lower due to this low incentive, but as I mentioned earlier, I do not know the valuations neither the utility of the seller. The only valid conclusion that I can draw is that in an auction with a reserve price the bidders do submit higher bids on average.
6 Conclusion
The main ndings of this thesis are that the auction mechanism with the reserve price12 outperforms the mechanism without a reserve price13 completely in case 1 (in
both mechanisms all Q emission certi cates are sold) and case 3 (not all quantity is sold in the California-Québec auction and no sale in the EU ETS market) and as long as the reserve price does not exclude too many buyers also in case 2 (not all quantity is sold in the California Québec auction but all quantity is sold in the EU ETS market) in both e ciency and seller utility. This nding is due to a change in the bidder equilibrium strategy, a reserve price causes the bidder to shade his bids less. There are two reasons for this change. First the bidder has a boundary under which he cannot shade his bid, and second, as bidders with a lower valuation are excluded from the market, the bidder knows that he faces tougher competition and that he, in order to win the same amount, cannot shade as much as in an auction with a less strong competition. By shading less the expected price rises, which increases seller utility not only from a revenue point of view, the seller gains also from the increased incentive of the bidders to adopt a cleaner production or to consume less energy.
The reduced shading also increases the e ciency of the uniform-price auction in case 1 and 3, because due to higher bidding less e ciency is lost from bidders with lower values outbidding bidders with a higher valuation. M. Ausubel et al. (2014) already showed that a uniform-price auction is generally ine cient.
In case 2 the condition for the auction with the reserve price to be more e cient is that the reserve price is not set too high and does not exclude too many bidders. Here lies one limitation of the thesis. I did not search for the optimal reserve price, because to calculating the optimal reserve price would require serious mathematical calculations which are beyond the scope of this master thesis. However this would be an interesting subject for further research.
The model simpli es, also for reasons of feasibility, the bidder equilibrium. This can be seen as a further limitation of this thesis as for example the production cost and level due to a certain amount won could be more integrated in the valuations of the bidders, to achieve a setting closer to the real strategy components of the bidders. This would also be helpful in nding the optimal reserve price.
Keeping these limitations in mind, the numbers of the actual emission auctions support the theoretical ndings of lower bid shading and higher bids submitted in
12In this thesis represented by the California-Québec mechanism. 13In this thesis represented by the EU ETS mechanism.
a setting with a reserve price. As there is quite a gap between the prices, we could draw the conclusion that seller utility is higher in the California-Québec mechanism. A recommendation to the EU ETS could be that in order to achieve its targets of the carbon allowance auction it could be bene cial to adopt a reserve price.
7 Appendix
Proof of Theorem 1:
By construction of the bidding formula βij vij(s
i) , si
bidders following this for-mula will lead to an equilibrium, because this function ful ls the requirement that at each realization marginal pro t equals marginal cost and thus a bidder cannot not improve his position by switching strategy. As shown earlier a strategy that max-imizes pro t has to use bid shading, in order to ful l the equilibrium property the strategy βij vij(s
i) , si
has thus to contain bid shading.2 Proof of EβiQ¯ viQ¯(s i) , si, R> Eβ iQ¯ viQ¯ (s i) , si
We know from the bidder equilibrium that the strategy βij is such that for each
quantity j the expected marginal pro t equals the expected marginal cost.
Let us examine the maximization problem in section 3.2 following strategy βiQ
the problem becomes
0 = g1 βiiQ viiQ(si) , si viQ(s i) − β iiQ viiQ(si) , si −g1 βiiQ viiQ(si) , si (Q − 1) βiiQ viiQ(si) , si − Q · G1 βiiQ viiQ(si) , si
which following to the de nition of βiQ
viQ¯ (s
i) , si has to equal to 0.
If we take a look at the maximization problem of the auction with a reserve price
0 = g1(bQ) Fi Q−1(R) − FQi (R) PQ−1 j=1 vij(s i)+ g 1(bQ) 1 − FQi (R) viQ(s i) − b Q −g1(bQ) 1 − FQi (R) (Q − 1) bQ− Q 1 − FQi (R) G1(bQ)
and we transform this equation to
0 = g1(bQ) Fi Q−1(R) − FQi (R) PQ−1 j=1 vij(s i) + 1 − FQi (R) g1(bQ) viQ(s i) − b Q − g1(bQ) (Q − 1) bQ− QG1(bQ) 0 = g1 bQ 1−FQ(R) FQ−1i (R) − FQi (R) PQ−1 j=1 vij(s i) + g1(bQ) viQ(s i) − b Q − g1(bQ) (Q − 1) bQ− QG1(bQ)
it becomes clear that the bidder faces a dierent maximization problem and thus uses a dierent bidding strategy. I can show that the strategy in an auction with a reserve price is to bid closer to the valuations, as it is assumed throughout the
general auction literature. To see this consider the case of marginal bidder i whose valuation of his Q unit is viQ(si) = R. Without his reserve price he will shade his
bid according to βiQ
viQ¯(s
i) , si < R. If the reserve price is enforced he will now
have to bid βiQ
viQ¯ (s
i) , si, R = R in order to win the unit. Shading below R is
not reasonable because the price cannot fall below R. Thus with the instalment of a reserve price the bidder has to raise his price for the marginal unit equal to R. Additionally we can imagine that he will raise his bid on the valuations above viQ(si)
because even though he does not know the valuations of the other bidders, he knows that only bidders with valuations above R will participate in the auction. Meaning that there is a competition with a higher valuation forcing the bidder to bid higher in order to have the same probability to win a unit. 2
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