1. Auction Theory in Practice: The U.K. Carbon Dioxide Auction E. Maskin Harvard University

2. 2 In March, 2003, U.K. government ran world’s first Part of Britain’s effort to reduce greenhouse gas emissions (to comply with Kyoto Protocol) Design of auction raised interesting theoretical questions

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

4. 4 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

5. 5 –most interesting –idea: pay firms to reduce emissions –payments needed because reductions are costly Step (2)

6. 6 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?

7. 7 (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

8. 8 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 and firm 1 bids £4.5 firm 2 bids £4 firm 3 bids £6 firm 4 bids £8 then Total reduction = 22.5 tons

9. 9 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 that firm 1’s cost/ton = £4.5 firm 2’s cost/ton = £4 firm 3’s cost/ton = £6 firm 4’s cost/ton = £8

10. 10 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

11. 11 But if all firms think this way, total reduction can be no greater than 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

12. 12 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

13. 13 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

14. 14 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 rather than £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

15. 15 (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

16. 16 suppose firm 1 bids £4.5 firm 2 bids £4 firm 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.

17. 17 Change rules: Want to make multiple winners possible

18. 18 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

19. 19 –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.

20. 20 –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 bid p = cost/ton if cost > £5, then modification won’t affect firm—it will lose anyway

21. 21 –at most £66.67 is spent Another way budget might be wasted: three (or more) firms with bids less than £ – even with above modification, only 2 firms selected

22. 22 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 £3.33 then This delivers a reduction of 30 rather than 20 tons

23. 23 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

24. 24 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.

25. 25 Auction may look rather complicated but actually simple in sense that each firm should just bid its cost—no “strategic” behavior need be contemplated

26. 26 Auction can be implemented by “reverse clock”: Auctioneer starts by setting p = £10 and gradually lowers price. All firms are initially “in,” but can drop “out” at any time. 1.If all but one firm drops out at p = £10, remaining firm is paid £10 per ton 2.If only one firm left when price reaches p between £5 and £10, remaining firm paid p per ton 3.If only two firms left when price reaches £5, remaining firms both paid £5 per ton 4.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.

27. 27 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)

28. 28 –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

29. 29 Hope that you’re convinced that auctions are not only important as policy tools but also interesting theoretically