MANAGERIAL AND DECISION ECONOMICS, VOL.

Optimal Decisions in Foreign Exchange Bidding Markets

Rafael Tenorio College of Business Administration, Uniilersily of Notre Dame, Notre Dame, IN, USA

I characterize optimal bidding decisions in bidding markets where each agent does not perceive she can significantly affect the market outcome. Using a foreign exchange bidding framework to provide a micro-foundation for the shape of a bidder‘s payoff function, I show that (1) in a discriminatory auction a bidder bids for a price that equals the value of the marginal product of her bid quantity, and (2) in a competitive auction a bidder bids for a price that equals the value of the average product of her bid quantity. An example illustrates the comparative properties of these solutions.

INTRODUCTION

Auction markets where bidders bid for both prices and quantities are commonly used to allocate financial and monetary resources. Financial in- struments like bonds, bills and securities, are traded using these auctions. More recently, some countries have also used them in their foreign exchange markets.’ In this paper, I analyze opti- mal bidding decisions by bidders participating in these kinds of auctions. Foreign exchange mar- kets are used as the benchmark, and they provide a micro-foundation for the shape of the bidder’s payoff function under two alternative auction for- mats: discriminatory, where a bidder pays her own bid price for each unit she gets; and competi- tive or marginal pricing, where a bidder pays the maximum rejected price for each unit she gets.

The analytical complexity of multiple unit auc- tions has generally prevented researchers from obtaining closed-form solutions for bidding strategies within a general framework.’ In con- trast, the approach used here allows one to cleanly derive and characterize bidders’ price-quantity strategies under each of the mentioned auction formats. This is accomplished by assuming that the number of bidders is large, and each bidder is small enough so as not to significantly affect the market outcome. This assumption appears rea- sonable in foreign exchange and some financial markets, where a large number of competing bid-

CCC 0143-6570/95/010001-06 0 1995 by John Wiley & Sons, Ltd.

ders - usually hundreds - participate at each auction. As will be shown, using this set-up amounts to taking a ‘decision-analytic’ approach3 to bidding at the auction, where the emphasis is on establishing a bidder’s optimal behavior given some conjectures about the expected conse- quences of her opponents’ aggregate behavior. Smith (1966) used a similar modeling strategy to analyze bidding behavior and outcomes at the US T-Bill auction. Unlike Smith’s model, the present paper emphasizes the derivation of a bidder’s reward from first principles, as well as the analyti- cal link between optimal bid prices and quanti- ties.

In addition to characterizing optimal bidding behavior at each auction, the results also extend the basic properties of unit-demand bidding (Vickrey, 1962) to auctions with endogenous quantity choice. More precisely, it will be shown that bidders underrepresent their average valua- tions in discriminatory auctions and reveal their average valuations in competitive auctions.

THE MODEL

Auctioneer

A single seller auctions D identical units (dollars). The amount D is determined in advance and

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publicly announced prior to the auction. The seller sets a reservation price which, without loss of generality, I assume to equal zero.

Bidders

There are n bidders who are risk-neutral ex- pected utility maximizers: Each bidder is as- sumed to be sufficiently small with respect to the aggregate so that she perceives she cannot sig- nificantly affect the market outcome? Assume further that bidders are symmetric so that the problem can be approached from a generic bid- der i’s standpoint.

Decisions

Minimum quantity constraints and other bidding requirements usually induce bidders to submit ‘lumpy bids’, i.e. bids where several units are demanded at the same price.6 Assume bidder i’s decisions are represented by the pair

where x i E [0, m) is i’s per-unit willingness to pay and qi E [0, D) is the number of units she wants to purchase.

Clearing Rate

The auctioneer receives the n bids, and es- tablishes a decreasing price priority vector (x, , , . . . , x,, . . . , x,), where x,, 2 . . . zx, z . . . r x , . Suppose the bid s, = ( x , , 4,) satisfies Cfq, = D. Then x, is the ‘clearing rate’, i.e. the one that establishes the cutoff between successful ( x , 2 x , ) and unsuccessful ( x , < x , ) bidders. If Zzq, > D, the marginal bidder gets a residual quantity. The smallness assumption implies that bidders i’s ac- tions will not affect the clearing rate in a mean- ingful way, and thus the probability that she places on being exactly the marginal bidder is of mea- sure zero.’

Valuations and Payoffs

Entry to the auction is restricted to bidders hold- ing valid import licenses. Assume that bidder i is a producer with technology ki, which uses two inputs to produce a final good to be sold in the domestic market at a price p i . The first input is a

composite fixed input x acquired in the domestic market at a price Pk. The second input is an imported good m purchased in the international market at a dollar-denominated price pm. The function pi is increasing and concave in both arguments. To focus on the problem at hand, assume further that i is a price taker in the markets for the final good and inputs so that p,, P k and pm are given. Thus, i’s constrained profit function is:

where y is the per-unit price i pays for each of qi foreign exchange units, and bars over variables indicate that they are predetermined.’ Substitut- ing Eqn (2b) into (2a):

One can further simplify Eqn (3) by normalizing prn to unity, omitting fixed costs and renaming pi(.,.) as ui(qi), where ui > 0 2 uy. Thus, i’s profit function becomes:

The auction formats considered here are the ones usually employed to sell multiple and homo- geneous units. In a discriminatory auction (Ad), bidder i pays her own bid price for each unit she gets ( y = x i ) . In a competitive auction (A‘) , the price that she pays is the highest rejected one ( y <xi).9 Accordingly, i’s payoffs are:

and

where yd and y , are the maximum rejected bids under Ad and A‘, respectively.

Beliefs

The beliefs that bidder i holds about Y d and y , will crucially determine her bidding decisions at

OPTIMAL DECISIONS IN FOREIGN EXCHANGE BIDDING MARKETS 3

the auctions. Assume that i uses all the informa- tion available to form individual assessments about this variable in each case. These assess- ments are represented by the densities f,'(yd) with c.d.f. F,(yd) for Ad, and gi(y,) with c.d.f. G,(y,) for A'. For simplicity, these densities are as- sumed to be continuous over the real intervals bd, jd] and [y,, j , ] , respectively." By assumption, bidder i acts as if the random variables Yd and y, were independent of her actions.

OPTIMAL BIDDING DECISIONS

Discriminatory Auction

Proposition 1: Bidder i's optimal strategy in Ad is to submit a bid (xf, 4:) such that xy equals the value of the marginal product of 4:. Proof: Bidder i's optimization problem is:

Since the expected payoff function is concave, the necessary and sufficient conditions for a maxi- mum at (x ," , 4:) are:

p,u,(q:)f , (xp) - q:[ F,(xp) + f , < x , " > x , " ] = 0 (8a)

(8b) [ p,u:(q:) -xp]F,<xp> = 0

From Eqn (8b), for any x," > yd, it follows that:

where VMP stands for value of the marginal product. Substitution of Eqn (8c) into (8a) yields:

Once q: is determined in Eqn (8d) as a function of pi and the parameters of ui and f , , xf follows from Eqn (8c). 0

The interpretation of i's optimal decision is straightforward. Bidding for the marginal valua- tion of her quantity is optimal because all the

non-marginal units give i a positive economic surplus, i.e. jjiu:.(qi) >xy V qi < qf. This solution is similar to the optimal choice of a production input in the theory of the firm. However, unlike the standard problem, in this auction the bidder is both a price and a quantity maker.

Note that Ad is the multi-unit version of a first-price auction. It is well established that in a first-price auction a bidder bids below her true valuation to be assured of a surplus if she wins. In Ad this translates into i submitting a price below the value of the average product of her bid quan- tity. Since for a concave function the value of the average product exceeds the value of the marginal product, setting xy = VMP(q,d) guarantees i a non-negative surplus.

Competitive Auction

Proposition 2: Bidder i's optimal strategy in A' is to submit a bid ( x i , 4:) such that x ; equals the value of the average product of 4:. Proof: In this case, bidder i's optimization prob- lem is:

Necessary and sufficient conditions for a maxi- mum at (x ," , 4:) are:

From Eqn (lOa), for any gi(x,") > 0, we have:

where VAF' stands for value of the average product. Substituting Eqn ( 1 0 ~ ) into (lob):

Once qf is determined in Eqn (10d), x: follows from Eqn (10~). 0

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To understand this result, observe that a bidder’s expected payoff has two components: the total surplus and the probability of winning. Concavity of u, ( . ) implies that for any 4,“ 2 0, VAP(q,C) 2 VMP(q,E), thus if i sets x: = VMNq;), she will reduce her probailiity of winning without gaining any surplus. This is because i’s per-unit outlay is y, < x : , which implies that it pays for her to bid higher than VMP(qf). However, in raising her bid above this value, she will not go beyond VAP(q:), because her probability of winning will only in- crease at the expense of experiencing an economic loss.

It is a well-established result that bidders reveal their true valuations in second-price or Vickrey auctions with unit demands (Vickrey, 1962; Or- tega-Reichert, 1968). The results presented here extend this revelation property to multi-unit auc- tions where quantity choice is endogenous. In this modified set-up, this property is restated as truth- fully revealing one’s average valuation for the bid quantity.

AN EXAMPLE

Although the solutions to bidder 1’s decision problems yield clear price-quantity relationships in both formats, a direct comparison of outcomes is not possible without a parameterization of the belief density and utility functions. For instance, one intuitively expects i to bid higher in A‘ than in Ad, but it is not clear under which format she will bid for more units. The following example allows such a comparison.

Suppose that bidder i’s utility function is lo- garithmic, and that the density she assesses on yd is uniform over [O, 11.” Namely,

Substituting Eqns (11) and (12) into Eqns (8c) and (8d) yields

where e is the base of the natural logarithms.

Since i expects a typical bidder to bid higher in A‘ than in Ad, she will likely expect the mean of y, to exceed the mean of yd. Accordingly, assume that G,(y,) is also uniform but with support [ S , 1 + S], where S > 0. Namely,

G;(x,) = x i - 8, vx; E [ S , 1 + S],

Vjj; E [6qi/ln(qi), (1 + S)q,/ln(q,>l (14)

Once again, for any small S, substituting Eqns (11) and (14) into Eqns ( 1 0 ~ ) and (10d):

&1n(q:)[q,FS2 + 2 - ln(q:)] - ( q , “ ) * S 2 = 0 (15)

It can be shown from Eqn (15) that, for any 6 > 0, q,“ > e2 . Thus it follows that q,C > qf and xi‘ > x4.l’

Why does xi“ exceed x f ? When bidder i evalu- ates the optimality of submitting a higher price in Ad, she trades off a higher probability of winning for a lower per-unit surplus. This tradeoff is absent in A‘, since i places zero probability on the event of paying her own bid price. Thus, submitting a higher bid price is a more rewarding strategy under competition.

The interpretation of qf > qf is related to the concavity of bidder i’s payoff function. As stated above, a tradeoff between i’s probability of win- ning and per-unit payoff exists only under Ad. Since concavity implies that each additional unit i gets is less valuable than the previous one, she will find it optimal to use her bid quantity as a hedging device against this tradeoff. The absence of the same tradeoff under A‘ induces her to pursue the ‘riskier’ strategy of demanding a unit as long as it yields a surplus, and “gamble” on the fact that ( x f -yc) is positive.

The latter result is in contrast to those obtained for auctions in which bidders have market power. If bidder i can meaningfully affect the outcome, she may benefit from bidding for fewer units in A‘ than inAd. A strategy of reducing her bid quantity in A‘ increases the probability that lower-valuing bidders will enter the set of win- ners, which in turn reduces her expected per-unit outlay (see Tenorio, 1993b).

CONCLUSIONS

This paper analyzes individual behavior in multi- ple unit auctions where bidders are small and place non-linear valuations on the auctioned units. Foreign exchange auctions are used as bench-

OPTIMAL DECISIONS IN FOREIGN EXCHANGE BIDDING MARKETS 5

mark and motivation, but the analysis could apply to other bidding markets as well.

The main results are that a bidder (1) will submit a bid price equal to the value of the marginal product of her bid quantity in a dis- criminatory auction, and (2) will submit a bid price to the value of the average product of her bid quantity in a competitive auction. An example is presented where bidders find it optimal to submit a lower price and bid for fewer units under discrimination than under competition.

Since the analysis is not based on full-blown equilibrium models of the auctions, the conclu- sions we can extract about market outcomes are somewhat limited. For instance, the revenue that the auctioneer expects to receive from each for- mat cannot be assessed without resorting to highly restrictive assumptions. On the other hand, the efficiency of the market outcomes can be easily evaluated. Since lumpiness implies that bidders do not show their willingness to pay for every unit and their valuation functions are concave, it is possible that a marginal unit awarded to a win- ning bidder may be less valuable that an intra- marginal unit not awarded to a losing bidder. Thus, the resulting allocations may not necessar- ily be Pareto-efficient.

Finally, one of the main advantages of this decision-analytic approach to bidding is that it yields clear characterizations of the bidders’ strategic decisions under fairly general conditions. In contrast, more comprehensive equilibrium analyses generally need several simplifying as- sumptions to be able to generate solutions, and even then these solutions may only be partially characterized. The analyst will likely need to bor- row from both approaches when studying the workings of these auctions.

Acknowledgements I thank Edi Karni, Joe Harrington Jr, and participants at the Microeconomics Seminar at Johns Hopkins University for useful comments on an earlier version of this paper. I remain responsible for any errors or omissions.

NOTES

1 . T-Bill auctions have been studied by Smith (1966) and Ortega-Reichert (1968), among others. Bikhchandani and Huang (1993) provide an excel- lent survey. FOREX auctions are a younger institu- tion. See, for instance, Krumm (19851, Quirk et al. (1987) and Tenorio (1992, 1993a).

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Full-fledged strategic analyses of multi-unit bidding are carried out within the framework of incomplete information games. See, for instance, Vickrey (19621, Wilson (1979), Maskin and Riley (1989) and Tenorio (1993b). The idea behind this approach is that the re- searcher’s task is to advise a player (the ‘client’) about how to choose a strategy in a game. Unlike the conventional game-theoretic approach, this method first assesses the expected behavior of the opponents, and then solves the client’s decision problem. Myerson (1991) discusses the virtues and limitations of this approach. See also Kadane and Larkey (1982). Although the bidder is risk-neutral with respect to monetary payoffs, later in this section she will be defined as risk-averse with respect to the auctioned units. Given the aim of this paper, I state the smallness property intuitively rather than discussing the for- mal issues involved in this definition. Formally, each bidder would not make any difference in the aggregate if the number of bidders approaches infinity (see Wilson, 1977). Alternatively, one could assume that the bidders set is countable infinity, so that each bidder has atomless measure (see Feld- man and Gilles, 1985). Lumpy bids have been extensively documented in the literature (see references in note 1). In addition to minimum quantity constraints, auctioneers do not accept bids in the form of demand schedules specifying contingent amounts that would be bought at various prices. Together, these two conditions are sufficient to generate lumpiness. The probability of being the marginal bidder can be stated as the distribution of the order statistic of the marginal price (see Ortega-Reichert, 1968). It can be shown that this probability collapses to zero as the number of bidders goes to infinity. The liquidity constraint is set as an equality be- cause, by auction rules, bidders that win the foreign exchange must use it to pay for their imports. In some cases a format has been used where the winning bidders pay the lowest accepted price rather than the highest rejected one for each unit they get. With a large number of bidders one would expect that both formats yield similar behavior. These densities may be based on both past and contemporaneous information. Among the most informative factors are bid averages and disper- sions in previous auctions, past and current quanti- ties supplied and numbers of bidders, and parallel market prices. Since most of these factors are pub- lic information, it is likely that the assessments about the rates will not vary widely across bidders. Thus, one can conjecture that bidders will largely agree on the parameters of the densities f and g. Simulations that use other functional forms for ui as well as more complicated densities for the bid- der’s beliefs yield similar results. Smith (1966) obtains comparable results using dif- ferent functional forms for the payoff functions and belief densities.

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