Optimal Bids in Multi{Unit Auctions When Demand is Price Elastic1 Dieter Nautz
Freie Universitat Berlin
2
Elmar Wolfstetter
Humboldt{Universitat Berlin3
March 1996
Research support by the Deutsche Forschungsgemeinschaft (DFG), Sonderforschungsbereich 373, \Quanti kation und Simulation O konomischer Prozesse", Humboldt{Universitat Berlin, is gratefully acknowledged. 2 Institut f. Statistik u. O konometrie, Boltzmannstr. 20, 14195 Berlin, Germany, e{mail:
[email protected] 3 Institut f. Wirtschaftstheorie I, Spandauer Str. 1, 10178 Berlin, Germany, e{mail:
[email protected]{berlin.de 1
Abstract The present paper analyzes optimal bidding in discriminatory and competitive multi{unit auctions when bidders have price elastic demand functions. Distinctions are drawn between bidders who are either rms or consumers, and bidders who are either risk neutral or risk averse. Assuming price{ taking bidders who face a given probability distribution of the stop{out price, closed{form solutions of optimal bid functions are derived. In addition, the paper provides some comparative statics of risk.
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1 Introduction Multi{unit auctions are frequently observed in industrial procurement, in the implementation of monetary policies, and in nancial markets. Every week, the U.S. Treasury auctions{o billions of dollars of bills, notes, and Treasury bonds. Governments and private corporations solicit bids for products ranging from oce supplies to tires or construction jobs. Central banks auction securities repurchase agreements (REPOs) as an essential part of their money market management. And several developing countries regularly auction{o their limited stock of foreign currencies as part of their regulation of foreign exchange markets. Altogether, multi{unit auctions account for an enormous volume of transactions. In multi{unit auctions two formats are commonly used: discriminatory auctions where bidders pay their bid for each successive unit, and competitive auctions, where bidders pay the lowest accepted bid | the so{called stop{out price | for each unit they get. Discriminatory auctions have been used for about fty years by the U.S. Treasury in its Treasury bill auctions. Exxon and Citicorp used competitive auctions to sell bonds and commercial paper, and so did the Bundesbank in its REPO auctions. But competitive auctions have not found general acceptance. For example, in 1988 the Bundesbank switched from competitive to discriminatory auctions.1 The desirability of the two auction rules has been the subject of recurrent debate. An early controversy stirred by Friedman (1964) centered around the issue which auction rule maximizes the auctioneer's revenue. More recently, discriminatory auctions were in the news due to their alleged susceptibility to illegal bidding practices, in the aftermath of the so{called Salomon Squeeze.2 Typically, multi{unit auctions are \multi" in the double sense that the auctioneer puts up several units for sale and bidders demand several units. Therefore, bids generally take the form of demand schedules that indicate how many units a bidder wants to buy, contingent upon how much he has to pay. In spite of their empirical relevance, multi{unit auctions have not received much attention in the literature.3 And most of the few available contributions For an empirical account of the background and consequences of this switch see Nautz (1996). 2A few years ago, Salomon Brothers was charged of \cornering" the security market by buying up to 95 % of a security issue, in violation of U.S. regulations that do not allow a bidder to acquire more than 35 % of an issue. Such \cornering" tends to be pro table because many security dealers engage in short{sales during the time when an issue is announced and the time it is actually issued. See Brady, Breeden and Greenspan (1992). 3For a comprehensive survey of auction theory see Wolfstetter (1996). 1
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worked with the rather restrictive assumptions that bidders demand at most one unit or that they are permitted to submit only one price{quantity bid rather than entire demand schedules. ?) showed that the standard results of single{unit auction theory extend to the multi{unit case, provided bidders demand at most one unit.4 In the same vein, Spindt and Stolz (1989) analyzed the expected stop{out price in a discriminatory auction, Bikhchandani and Huang (1989) explored the role of the resale market in the bidding for Treasury bills, and Simmons (1996) added some revenue comparisons. Tenorio (1993) examined foreign exchange auctions in Zambia, assuming bidders are permitted to submit only one price{quantity bid.5 However, as was already pointed out by Scott and Wolf (1979), single bids are generally not optimal which suggests that bids should be described as complete demand functions. In the present paper we allow bidders to submit complete demand functions, and solve optimal bid functions for price taking bidders whose expectations concerning the stop{out price are described by a given probability distribution. We distinguish between risk neutral and risk averse bidders, and bidders that are either rms or consumers. For most cases closed{form solutions are obtained. In addition, we perform some comparative statics of risk. The plan of the paper is as follows. In Section 2 we explain the model and introduce notation and assumptions. In Section 3 we derive the optimal bid function when the bidder is a rm and the auction is discriminatory, and explore the role of risk aversion and the impact of shifts in the probability distribution of the stop{out price. In Section 4, the analysis is extended to the case when the bidder is a consumer. Section 5 considers the case of a competitive auction. The paper closes with a discussion of the main results and stresses their limitations.
2 The Model Consider a price{taking participant in a multi{unit auction where many units of a homogenous good are put up for sale. The bidder has a regular, downward sloping demand function, D, (and inverse demand function Z := D?1 ), 4Some of these results are generalized in Maskin and Riley (1989) who otherwise focus on the design of optimal multi{unit auctions, inspired by the authors' earlier analysis of optimal price discrimination schemes in Maskin and Riley (1984) 5Similarly, Nautz (1995) introduced a given grid of permitted prices.
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either because he buys in order to sell at a secondary market or because he buys as a consumer. The auctioneer announces an allocation rule that commits him to serve demand at a certain stop{out price, p, which may be set to clear demand and supply, and stipulates a certain payment rule. In practice one observes two types of auction rules: discriminatory , and nondiscriminatory or competitive auctions; they share the same allocation rule but have di erent payment rules. In both auctions, the bidder is asked to submit a demand function, B , as a function of the stop{out price p (his \bid function"). This function must be monotone decreasing, just like D, because otherwise bidders would have to bid negative quantities at some prices. The auctioneer is committed to award the bidder B (p) units depending upon the stop{out price. In turn, the bidder is committed to make certain payments depending upon the type of auction. Of course, the submitted demand function B may | and typically will | di er from the true demand function D. The bidder's payo function depends upon whether he is either a rm or a consumer and whether the auction is either competitive or discriminatory. If the bidder is a rm (f ), it buys in order to resell (directly or indirectly), and its true demand is a derived demand. In that case, its payo is an increasing, concave and twice continuously di erentiable function U () of the pecuniary gain from trade, : Uf := U ( ); (2.1) Z B (p) Z (x)dx ? C: (2.2) := 0
Thereby C represents the total cost of submitting B which of course depends upon the auction rule. Whereas, if the bidder is a consumer (c), his inverse demand function is his marginal rate of substitution function. His payo function is then simply obtained by integrating over Z , after deducting the price to be paid:6 Uc :=
Z B (p) 0
Z (x)dx ? C:
(2.3)
In a discriminatory auction, the bidder has to make distinct payments for each of the awarded B (p) units, according to his submitted inverse demand This speci cation is commonly used in monopoly and auction theory; see for example Maskin and Riley (1984), Maskin and Riley (1989). Of course, the additive separability assumes the absence of income e ects and the linearity in C assumes risk neutrality in expenditures. For a discussion of the merits of this speci cation see Newbery and Stiglitz (1981), Chapters 5 and 8. 6
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or marginal willingness to pay function ZB := B ?1. Therefore, given p, his total cost of submitting the demand function B , is C (B (p)) :=
Z B (p) 0
ZB (x)dx; ZB := B ?1 (discriminatory auction):
(2.4)
Whereas, in a competitive auction, the bidder pays the stop{out price for each awarded unit. In this case, the total cost of submitting the demand function B is C (B (p); p) := pB (p) (competitive auction):
(2.5)
Finally, bidders are price takers in the sense that they face a given probability distribution of the stop{out price P which re ects the bidder's expectations.7 The associated probability distribution function of P is F : [p; p] ! [0; 1]. F is taken to be log{concave,8 the density function f (p) := F 0(p) is positive everywhere, and p is suciently high to assure that demand vanishes at P = p, i.e. D(p) = 0. We mention that similar assumptions were used by Scott and Wolf (1979) who did however not derive optimal bid functions.
3 Optimal Bidding when Bidders are Firms In this section we assume that the bidder is a rm that buys in order to sell, either directly because it is engaged in retailing, or indirectly because it bids for an intermediate product that serves as an input in its production activity. We characterize the optimal bid in a discriminatory auction, examine the e ect of risk aversion, and perform some comparative statics of risk. The analysis of the competitive auction is postponed up to Section 5. If the bidder submits the demand function B , the outcome is random since the stop{out price of the auction is random. Therefore, the payo associated with B is the expected utility E [Uf ]. In order to translate the optimization problem into an optimal control problem, we rewrite the bidder's expected utility, using some elementary 7We stick to the convention of writing random variables in capital letters and realizations in lowercase. 8A function F is log{concave if ln(F ) is concave. Most important distribution functions are log{concave; for example, the normal, the uniform, and the exponential distribution. Log{concavity is frequently used in economics. A detailed list of log{concave functions is provided by Bagnoli and Bergstrom (1989).
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rearrangements as follows. The rst step uses integration by parts (also note that: (p) = 0), and the second step uses (2.2) and (2.4): E [Uf ] : =
Z p p
=? =
U ( (B (p)))dF (p)
Z p p
U 0 ( (B (p))) 0(B (p))B 0(p)F (p)dp
Z p U 0 ( (B (p))) [p ? Z (B (p)] B 0(p)F (p)dp: p
(3.1) (3.2) (3.3)
The optimal bid is that monotone decreasing function B that maximizes E [Uf ]. We de ne the control variable, u(p), and the state variables x1 (p), x2(p):9 u(p) := B 0(p) x1(p) := (B (p)) x2(p) := B (p):
(3.4) (3.5) (3.6)
Then, using the objective function in the form (3.3), one nds the optimal bid function by solving the control problem: max s.t.
Z p p
U 0 (x1(p)) [p ? Z (x2(p))] u(p)F (p)dp
(3.7)
x01(p) = [Z (x2(p)) ? p]u(p) x02(p) = u(p) x2(p) = D(p ) x1(p) = (D(p)):
(3.8) (3.9) (3.10) (3.11)
Thereby, the initial conditions (3.10), (3.11) are due to the fact that, at P = p, it is obviously optimal to bid one's true demand, x2(p) = B (p) = D(p). Therefore, the Hamiltonian is H (p; x1 ; x2; u; 1 ; 2 ) :=U 0 (x1(p)) [p ? Z (x2(p))] u(p)F (p) + 1(p)[Z (x2(p)) ? p]u(p) + 2(p)u(p):
(3.12)
And one obtains the following conditions for a maximum (when there is no 9
For ease of access we stick to the notation used in Kamien and Schwartz (1991).
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risk of confusion, the arguments of functions are omitted):10 0 = @H = (p ? Z ) (U 0F ? 1) + 2 @u
@H = U (Z ? p) F u @x1 @H = (F U 0 ? 1) Z 0u: 02 = ? @x2 01 = ?
00
(3.13) (3.14) (3.15)
Totally di erentiate (3.13) with respect to p and one obtains, using (3.14) and (3.15), U 0 [f (Z ? p) ? F ] = ?1 :
(3.16)
Stated in terms of B in lieu of the x0s and u, this is U 0 ( (B (p))) [f (p)(Z (B (p)) ? p) ? F (p)] = ?1 (p):
(3.17)
Of course, in a discriminatory auction it cannot be optimal to submit a marginal willingness to pay function ZB that exceeds the true marginal willingness to pay Z (otherwise the bidder could obviously improve his bid). Therefore, for all p Z (B (p)) p which is equivalent to
B (p) D(p):
(3.18)
Also, notice that the initial condition B (p) = D(p) implies p = Z (B (p)). Combined with (3.17) this entails 1 (p) = 0:
(3.19)
With these preliminaries, we can now characterize optimal bidding.
Proposition 1 Suppose the bidder is risk neutral. Then, the optimal bid
function in the discriminatory auction is:
!
F (p) B (p) = D p + : f (p)
(3.20)
It exhibits bid \shadeing" almost everywhere: B (p) < D(p), for all p 2 (p; p). See Kamien and Schwartz (1991), Part II, where necessary and sucient conditions are discussed in detail. 10
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Proof Risk neutrality entails U 0. Hence, (3.14) implies 01 0. Since 00
1 (p) = 0, by (3.19), 1 must vanish everywhere 1 (p) 0:
(3.21)
Therefore, by (3.17) Z (B (p)) = p +
F (p) : f (p)
(3.22)
Finally, apply the function D := Z ?1 to (3.22), and one has (3.20), as asserted. Incidentally, the log{concavity of the probability distribution function F assures the monotonicity of the bid function B . It is obvious that in a discriminatory auction the bidder can only gain if the demand is understated somewhere. However, Proposition 1 shows that it is optimal to understate everywhere, except at the lower boundary of the stop{out price p. An immediate implication is that, with probability one, the bidder is awarded less units than he would like to acquire at the realized stop{out price.
Example 1 Suppose the stop{out price is uniformly distributed, with support [0; 1], and let D(p) := 1 ? p. Then, under risk neutrality, B (p) = D(2p) = 1 ? 2p, and hence ZB (x) = 1 ? 21 x < 1 ? x =: Z (x). A plot of these functions
is displayed in Figure 1. It illustrates the extent of optimal bid shadeing.
Next we show that if the gap between true and revealed demand shrinks at each p then the random stop{out price must have increased stochastically in the sense of rst{order stochastic dominance (FSD).11
Proposition 2 Consider two probability distribution functions of the stop{ out price, F1 and F2, and the associated optimal bid functions BF1 and BF2 of a risk neutral bidder in a discriminatory auction. Then, BF2 (p) BF1 (p) 8p ) F2(p) F1(p)
8p:
(3.23)
Proof By Proposition 1 and the fact that D is monotone decreasing, one has
BF2 (p) BF1 (p) 11
()
F1(p) f1 (p)
Ff 2((pp)) () 2
f1 (p) F1 (p)
Ff2((pp)) : 2
(3.24)
Recall, FSD is de ned as follows: F2 FSD{dominates F1 if F2 (p) F1(p) 8p.
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Z; ZB
1 6Z
1=2 p
1=3
.... ....... ......... . . . .... .................. ............... .................. . . . . . . ..... .............................. ........................ . . . . . . . . . ........ ....... .......................... .. .... . ..... .................. .... ...... .................... .... . ...... . . . . . . . . . .. .. .. ..... . .......................... ...... .... ... .................. . . ..... . . . . . . . . . . ...... .... .. . . . .......................... ..... ........ .................. .... . . . ..... . . . . . . . . . ....... .... . . . . .......................... ........ ..... . . . . . . . . . .... ............................ ........ . . . . . . . . ..... ....................... .... ....... ........... .... ....... . . . . ..... ............. .... ........ . ..... . . ......... ......... ......... ............. .... ...... ..... . . ....... .... ........ . ..... . ..... ...... ........ .... . . ...... ........ ..... . ...... . .... ......... ...... . .. ..... . ...... ...... .... ...... ........ ..... ...... . ...... .... ........ . . ...... ..... . ........ ...... .... ...... ........ ..... . ........ .......... . .. . . ........ ........ ....... . .................. ...... ....................... . ........... . . . ..
ZB (1)
ZB (2)
B (p)
1
-s
Figure 1: Inverse bid functions in Examples 1, 2 If this holds, then 0
Z p p
!
f2 (y ) f1 (y ) ? dy = ln(F1(p)) ? ln(F2(p)) F2(y ) F1(y )
(3.25)
and therefore F2(p) F1(p). Note that the converse of Proposition 2 does not hold true. In fact, F1(p)=f1 (p) F2(p)=f2 (p) holds everywhere if and only if F2 (p)=F1 (p) is always increasing. But this cannot be assured by rst{order stochastic dominance alone. However, in the following example we consider a particular class of rst{order stochastic dominance shifts where this monotonicity property is satis ed, so that the implication stated in Proposition 2 goes in both directions.
Example 2 Consider rst{order stochastic dominance shifts of F , generated
by
F (p) := F (p) ; 2 (0; 1]: 1
(3.26)
(Note, the stop{out price gets stochastically larger i decreases.) Then, by Proposition 1, the risk neutral bidder's optimal bid function in a discriminatory auction is !
F (p) : B (p) = D p + f (p)
(3.27)
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Hence, the gap between true and revealed demand gets smaller everywhere if the stop{out price increases stochastically. As an illustration consider the assumptions of Example 1. There, one obtains ZB (x; ) := 1 ? (1+ )x. This function is plotted in Figure 1 for = 1 (see ZB1 ) and for = 21 (see ZB2 ).
Having explored the impact of changes in bidders expectations, as re ected in shifts of the probability distribution of the stop{ou price, we now look into the e ect of a change of the bidder's attitude towards risk. Proposition 3 Suppose the bidder is risk averse. Then, optimal bidding in a discriminatory auction is pointwise higher than under risk neutrality ! F (p) : (3.28) B (p) D p + f (p)
Proof Since Z (B (p)) p and U
00
< 0, (3.14) implies
01 (p) 0
8p
(3.29) for any monotone decreasing bid function. Together with (3.19) this entails 1 (p) 0 8p: (3.30) Using (3.17), we conclude, for all p F (p) 1 (p) F (p) Z (B (p)) = p + ? p+ : (3.31) 0 f (p) f (p)U (x1 (p) f (p) Therefore, B (p) D p + Ff ((pp)) , everywhere.
4 Optimal Bidding when Bidders are Consumers Now assume the bidder is a consumer who bids for consumption goods. Then, by (2.3), his expected utility of submitting the demand function B is E [Uc ] : =
=
Z p Z B (p)
p Z p p
0
!
Z (x)dx ? C (B (p)) dF (p)
(B (p))dF (p):
(4.1) (4.2)
Evidently, it coincides with the payo function of the risk neutral rm. An immediate implication of this coincidence is:
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Proposition 4 Suppose the bidder is a consumer. Then, his optimal bid-
ding in a discriminatory auction is the same as that of a risk neutral rm, which was already characterized in Proposition 1. Accordingly, its comparative statics of risk is already stated in Proposition 2.
This coincidence should, however, not be misinterpreted. Changes in the consumer's attitude towards risk in consumption do a ect his payo function, since that attitude is already re ected in the shape of his true demand function. Risk aversion concerning a consumption good is equivalent to a monotonically decreasing true demand function, while changes in the degree of risk aversion re ect in changes in its curvature. How does a change in the consumer's attitude towards risk a ect his optimal bidding? We address this question assuming utility functions that exhibit either a constant degree of absolute risk aversion (CARA) or a constant degree of relative risk aversion (CRRA), : 1 U (x; C ) : = ? e? x ? C; > 0 (CARA) (4.3) U (x; C ) : =
1 x1? ? C; 2 (0; 1] (CRRA): 1?
(4.4)
Proposition 5 Suppose the consumer has either a CARA or a CRRA utility
function (4.3), (4.4). Then, in a discriminatory auction, a higher degree of risk aversion makes the consumer bid closer to his true demand function.
Proof Assuming CARA the true marginal willingness to pay function is Z (x) = e? x . Therefore, for p 2 (0; 1] one obtains the true demand function 1
By Proposition 4 one has
D(p) = ? ln(p):
(4.5) !
F (p) D(p) ? B (p) = D(p) ? D p + f (p) ! ! 1 F (p) = ln p + f (p) ? ln(p) :
(4.6) (4.7)
Hence, the gap between true and revealed demand is decreasing in , as asserted. Similarly, assuming CRRA one obtains the true demand function 1 D(p) = p? : (4.8)
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Therefore,
! 1 F (p) ? B (p) = p + ; f (p)
(4.9)
and the assertion follows immediately. Altogether, Propositions 3 and 5 show that risk aversion makes both rms and consumers bid closer to their true demand.
5 Optimal Bidding in a Competitive Auction Now consider a competitive auction where the bidder has to pay the stop{out price p for each and every unit. Therefore, if the bidder is a rm, its expected utility associated with a bid function B is, by (2.5), E [Uf ] :=
Z p p
U
Z B (p) 0
!
(Z (x) ? p) dx dF (p):
(5.1)
And again, if the bidder is a consumer, his payo is obtained as a special case of (5.1) by replacing U with the identity map. Proposition 6 In a competitive auction it is always optimal to reveal one's true demand B (p) D(p): (5.2) This applies to rms as well as to consumers. Proof In a competitive auction, the bidder's optimization problem is a straightforward variational problem. The corresponding Euler conditions are ! Z B (p) 0 (Z (x) ? p)dx = 0 8p: (5.3) f (p) (Z (B (p)) ? p) U 0
These conditions are necessary and sucient since U is (weakly) concave and Z is monotone decreasing. From (5.3) it follows Z (B (p)) p; hence, B (p) D(p), as asserted. Notice, the optimal bid function is independent of the probability distribution of the stop{out price, no matter whether the bidder is either a rm or a consumer. Moreover, rms' optimal bidding is independent of their attitude towards risk. But the latter does not extend to consumers because their true marginal willingness to pay function re ects their risk preference.12
Using B (p) D(p), one can see from (4.5), (4.8) that B increases at each p in the degree of risk aversion (either absolute or relative). 12
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Apparently, this indicates two advantages of a competitive auction rule: simplicity (no need to compute the optimal amount of bid shadeing), and robustness (independence of expectations and, in the case of the rm, also of the attitude towards risk).
6 Discussion The present paper has derived closed form solutions of optimal bid functions in multi{unit auctions when bidders demand function is price elastic. Bids were not restricted to single price{quantity pairs, as it is often assumed in the literature. Instead, we allowed bidders to submit any monotonically decreasing demand function, in accordance with common practice in nancial markets. We distinguished between bidders that are either rms or consumers and auctions that are either discriminatory or competitive. Among other results we showed that rms' bidding problem is qualitatively the same as that of consumers if and only if rms are risk neutral. We showed that in a competitive auction it is optimal to bid one's true demand, for each possible stop{out price, whereas in a discriminatory auction it is optimal to engage in bid shadeing everywhere. In addition, we explored the impact of shifts in the probability distribution of the stop{out price and of changes in bidders' attitude towards risk. In the analysis of single{unit auctions it has often been observed that second{price auctions have many advantages but are apparently far less widespread than rst{price auctions. Similarly, our results indicate that competitive auctions, which can be viewed as a generalization of the second{price auction, are favorable but have again found only limited acceptance in real world applications. In the auction literature one has suggested several possible explanations for this surprising predominance of rst{price auctions,13 which have also a bearing on the lack of popularity of competitive auctions. However, some recent experience suggests that these auction formats are less and less used simply because they are judged as unfair. For example, the government of New Zealand came under heavy criticism when it was publicized that some participants in their auctions of spectrum rights for radio, television and cellular{telephone use were awarded particular frequencies at absurdly low prices due to the absence of a signi cant second bid.14 Similarly, when the Bundesbank ran it's REPO sale as a competitive auction, it See for example Rothkopf, Teisberg and Kahn (1990). In one case that made the headlines in the newspapers, a rm that bid NZ $100,000 for a television license paid the second highest bid of NZ$ 6. On these and other incidents see McMillan (1994). 13 14
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was strongly criticized particularly by large banks. Among other criticism, they claimed that this auction format induced small banks to bid at astronomically high interest rates, at no risk of ever paying these high rates, just in order to assure the allocation of the desired central bank credit.15 Finally, the main limitation of the present paper is that we solved optimal bid functions for a given probability distribution of the stop{out price but did not address the equilibrium problem of nding a self{con rming probability distribution. To address the equilibrium problem would require a full edged market model, with many bidders, and explicit assumptions about bidders' beliefs concerning their rival bidders' true demand functions.
Interestingly, when the Bundesbank switched to a discriminatory auction, the German nancial press welcomed this decision as a change towards an \auction for adults" (\Zinstender fur Erwachsene" | so the headline in the Borsen{Zeitung from November 27, 1992). 15
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References Bagnoli, M. and Bergstrom, T. (1989). Log{concave probability and its applications, Working paper, University of Michigan. Bikhchandani, S. and Huang, C.-F. (1989). Auctions with resale markets: An exploratory model of Treasury bill markets, Journal of Financial Studies 2: 311{339. Brady, N. F., Breeden, R. C. and Greenspan, A. (1992). Joint Report on the Government Securities Market, U. S. Government Printing Oce. Friedman, M. (1964). Comment on `Collusion in the auction market for treasury bills', Journal of Political Economy 72: 513{514. Harris, M. and Raviv, A. (1981). Allocation mechanisms and the design of auctions, Econometrica 49: 1477{1499. Kamien, M. I. and Schwartz, N. L. (1991). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, second edn, North{Holland. Maskin, E. and Riley, J. G. (1984). Monopoly with incomplete information, Rand Journal of Economics 15: 171{196. Maskin, E. and Riley, J. G. (1989). Optimal multi{unit auctions, in F. Hahn (ed.), The Economics of Missing Markets, Information, and Games, Clarendon Press, pp. 312{335. McMillan, J. (1994). Selling spectrum rights, Journal of Economic Perspectives 8: 145{162. Nautz, D. (1995). Optimal bidding in multi{unit auctions with many bidders, Economics Letters 48: 301{306. Nautz, D. (1996). How auctions reveal information | a case study on German REPO rates, Journal of Money, Credit, and Banking (forthcoming). Newbery, D. and Stiglitz, J. E. (1981). The Theory of Commodity Price Stabilization. A Study in the Economics of Risk, Oxford University Press. Rothkopf, M. H., Teisberg, T. J. and Kahn, E. P. (1990). Why are Vickrey auctions rare?, Journal of Political Economy 98: 94{109.
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Scott, J. and Wolf, C. (1979). The ecient diversi cation of bids in Treasury bill auctions, Review of Economics and Statistics 60: 280{287. Simmons, P. (1996). Seller surplus in rst{price auctions, Economics Letters 50: 1{5. Spindt, P. A. and Stolz, R. W. (1989). The expected stop{out price in a discriminatory auction, Economics Letters 31: 133{137. Tenorio, R. (1993). Revenue equivalence and bidding behavior in a multi{ unit auction: An empirical analysis, Review of Economics and Statistics 75: 302{314. Wolfstetter, E. (1996). Auctions: an introduction, Journal of Economic Surveys (in print) pp. 1{65.
