Auction Theory in Practice:The U.K. Carbon Dioxide Auction

E. Maskin

Harvard University

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• In March, 2003, U.K. government ran world’s first 2" auction"CO

• Part of Britain’s effort to reduce greenhouse gas emissions (to comply with Kyoto Protocol)

• Design of auction raised interesting theoretical questions

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The U.K.’s 3-step approach to reducing

emissions:2CO

• (3) Compel reductions by law

• (2) Induce other firms to reduce by using financial incentives

• (1) Solicit voluntary pledges by firms to reduce carbon dioxide pollution in production

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• Steps (1) and (3) are straightforward

• Step (3)– measure of last resort– politically costly

• Step (1)– firms make pledges to improve public relations– not sufficient

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– most interesting– idea: pay firms to reduce emissions– payments needed because reductions are costly

• Step (2)

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• Answer: Hold an auction– gets firms to compete for incentive money– what form should auction take?– none of the “standard” auctions is well suited

• Government makes budget of £ B available in incentive money (in fact, £ B = £ 300 million)

• Problem: How to induce as much emission reduction as possible while staying within the budget?

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(2) government selects firms with lowest bids (reverse of normal auction) until budget is exhausted

• Suppose that each firm can reduce emissions by up to 10 tons

• Discriminatory Auction (has been used to sell U.S. Treasury notes)

(1) each firm bids a price p p = how much the firm is to be paid per

ton of emission reduction

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firm 2 selected and paid £4 10

firm 1 selected and paid £4.5 10

firm 3 partially selected and paid £6 2.5

firm 4 rejected

Suppose budget = £100 andfirm 1 bids £4.5firm 2 bids £4firm 3 bids £6firm 4 bids £8

then

Total reduction = 22.5 tons

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Then selection rule maximizes emission

reduction subject to

(i) staying within budget

(ii) ensuring firms break even

Auction’s good point:

Suppose firms make bids equal to costs

In our example, this means thatfirm 1’s cost/ton = £4.5firm 2’s cost/ton = £4firm 3’s cost/ton = £6firm 4’s cost/ton = £8

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Auction’s bad point:

Consider firm 2 with cost/ton = £4– if bid p = £4, will only break even– if bid p > £4, will make a profit if selected– so will bid p > £4

Firms will not make bids equal to costs

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But if all firms think this way, total reduction can be

no greater than100 16.67 tons6

Firm 2’s bid depends on what it thinks others will do:if it believes others will bid more than £6, then will bid

p ≥ £6

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Would like auction in which firms are induced to make bids equal to costs

Second-price (Vickrey) auction has this property

(1) each firm bids a price

(2) firm with lowest bid selected

(3) selected firm paid second-lowest price

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For example, if

firm 1 bids £4.5

firm 2 bids £4

firm 3 bids £6

firm 4 bids £8

then, firm 2 selected and paid £4.5/ton

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Claim: In a second-price auction, a firm will bid a

price equal to its cost of reduction.

Suppose firm has cost/ton = £4 but bids p = £6 ratherthan £4

– if lowest bid by others is £7, then firm’s profit is the same with bids £4 or £6 (is paid £7 in either case)– if lowest bid by others is £3, then firm loses with bids £4 or £6– if lowest bid by others is £5, then firm loses with £6 and wins with £4

So better off bidding p = £4

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(i) may violate budget– suppose second-lowest bid = £11

– then winner should be paid £11 10 > £100

– minor problem: change rule so that if second-lowest bid exceeds £10, the winner is paid only price £10

Drawbacks of second-price auction

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supposefirm 1 bids £4.5firm 2 bids £4firm 3 bids £6

firm 4 bids £8

(ii) Much of budget may be wasted

then firm 2 selected and paid£4.5 10 = £45

£55 left over in the budget

But this achieves reduction of only 10 tons.

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Change rules:

Want to make multiple winners possible

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For example, if, as before,

firm 1 bids £4.5

firm 2 bids £4

firm 3 bids £6

firm 4 bids £8

If two bids are less than £5 (and rest are more than £5), select both the low bidders pay each £5/ton

select firms 1 and 2 and pay them £5/ton each

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– if at least three bids are less than £5, e.g.,

firm 1 bids £3

firm 2 bids £4

firm 3 bids £2

firm 4 bids £6

– select two lowest bidders (firms 1 and 3 in this case)

– pay them each the third-lowest bid (£4 in this case)

– two lowest bidders should not be paid £5 in this case, because then firm 2 would have incentive to underbid.

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– Modification increases total pollution reduction from 10 to 20 tons (near upper bound of 22.5 tons)

if cost < £5, firm doesn’t gain from setting p > cost (since not paid by p anyway) and may suffer (if

there are two other firms that bid less than p);

firm doesn’t gain from setting p < cost (if underbidding caused firm to win, it would be paid less than cost)

Claim: still in a firm’s interest to bidp = cost/ton

if cost > £5, then modification won’t affect firm—it will lose anyway

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– at most £66.67 is spent

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3Another way budget might be wasted:

three (or more) firms with bids less than £

– even with above modification, only 2 firms

selected

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Further modification:

– lowest three bidders selected– each is paid either

£3.33 (if all others bid above £3.33) or

fourth-lowest bid (if this is less than £3.33)

If three (or more) firms bid each less than £3.33then

This delivers a reduction of 30 rather than 20 tons

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We continue similarly:– if at least four firms bid less than £2.5, select

four lowest– if at least five firms bid less than £2, select five

lowest, and so on

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To summarize, the auction has the following rules(1) each firm bids a price(2) if no more than one bids below £5, low bidder is selected paid the lower of £10 and second-lowest bid(3) if more than one bids below £5 but not more than two bid below £3.33 two lowest bidders selected paid the lower of £5 and the third-lowest bid(4) if more than two bid below £3.33 but not more than three bids below £2.5 three lowest bidders selected paid the lower of £3.33 and the fourth-lowest bid

etc.

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Auction may look rather complicated but actually simple in sense that each firm should just bid its cost—no “strategic” behavior need be contemplated

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Auction can be implemented by “reverse clock”:Auctioneer starts by setting p = £10 and gradually lowersprice. All firms are initially “in,” but can drop “out” at anytime.1. If all but one firm drops out at p = £10, remaining firm is

paid £10 per ton2. If only one firm left when price reaches p between £5

and £10, remaining firm paid p per ton3. If only two firms left when price reaches £5, remaining

firms both paid £5 per ton4. If only two firms left when price reaches p between

£ 3.33 and £5 remaining two firms each paid p per ton 5. If only three firms left when price reaches £3.33,

remaining three firms each paid £3.33 per ton etc.

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Theorem: Among all auctions in which firms

are induced to bid their costs and in which

payments do not exceed the budget, the

described auction maximizes (expected)

emissions reduction (assuming firms are

more likely to have low rather than high

costs)

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– Analysis has assumed that all firms can make the same emissions reduction—10 tons.

– Auction can readily be extended to the more realistic case in which different firms can reduce by different amounts

– But that would take another lecture to show

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Hope that you’re convinced that auctions are not only important as policy tools but also interesting theoretically