License Auctions when Winning Bids are Financed Through Debt Marco A. Haan∗

Linda A. Toolsema†

October 27, 2009

Abstract We study an auction where two licenses to operate on a new market are sold and winning bidders finance their bids on the debt market. Higher bids imply higher debts which affects product market competition. When debt induces firms to compete more aggressively retail prices are lower than in a model without debt, as are auction revenues. When debt induces firms to compete less aggressively retail prices are higher than in a model without debt, and the effect on auction revenues is ambiguous. Net firm profits are always higher than in a model without debt due to endogenous credit rationing.

jel Classification Codes: D44, D45, L13 Keywords: License Auctions, Debt, Oligopoly

∗

IEEF, Faculty of Economics and Business, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands, [email protected]. † COELO and IEEF, Faculty of Economics and Business, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands, [email protected]. We thank three anonymous referees, the editor Yeon-Koo Che, Esther Hauk, Tobias Kretschmer, Allard van der Made, Jos´e Luis MoragaGonz´alez, Sander Onderstal, Bert Schoonbeek, participants of the SOM Workshop Competition and Market Power 2002 in Groningen; IIOC 2003 in Boston; NASM 2003 in Chicago; ESEM 2003 in Stockholm; EARIE 2003 in Helsinki; the Kiel Workshop on the Economics of Information and Network Industries 2003; NAKE Day 2003 in Amsterdam; APEA 2005 in Tokyo, and seminar participants at the universities of Groningen, Amsterdam, and York for useful comments and discussion.

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I

Introduction

Over the last decade, license auctions in the US and Europe sparked a huge interest from both academics and the general public. In the US, the FCC auctioned licenses to use the electromagnetic spectrum for personal communication services. Between July 1994 and July 1998, 16 auctions were held, where 5,893 licenses were sold. Total revenues amounted to $22.9 billion dollars.1 Throughout Europe, licenses for “third generation” (3G) mobile telecommunication (or UMTS) took place during 2000 and 2001.2 These auctions, held in 9 countries, raised over $100 billion, or over 1.5% of GDP. The revenue per inhabitant differed greatly per country.3 Currently, European countries are preparing to auction off the 3G expansion band, amongst others.4 Traditional auction theory5 may not be the most appropriate framework to study these auctions. Indeed, Klemperer [2002b] argues that in analyses of license auctions based on this literature, often too much attention is given to technicalities concerning asymmetric information, and too little attention to market structure and industrial organization aspects. Traditional models typically assume that for each bidder the value of the object that is being auctioned is fixed and given. In a license auction, this is often not the case. Here, firms bid on the right to compete on a market. The willingness to pay for that right will depend on the characteristics of the aftermarket (see e.g. Jehiel and Moldovanu [2003]). One striking aspect of license auctions is that winning firms often have to take on debt to be able to finance their bid (see e.g. The Economist [2002]). From the industrial organization literature, it is well-known that the competitive behavior of a firm, and therefore its profit, is affected by the amount of debt that it holds. More debt induces a firm to compete either more aggressively (Brander and Lewis [1986]), or less

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aggressively (Showalter [1995]), depending on the exact specification of the model. Thus, not only does willingness to pay in the auction depend on the characteristics of the aftermarket, but in case of debt finance the amount paid in the auction also directly affects the characteristics of the aftermarket. In this paper, we model this issue. Two licenses to operate on some new market are being auctioned. Firms have symmetric information. The two winners of the auction will finance their bids on a competitive debt market and then compete on the product market. Before participating in the auction, bidders obtain a line of credit from a bank, that comes with a given credit limit. The bank thereby commits to fund any bid up to that credit limit. Our main results are the following. In a benchmark model in which firms have sufficient internal funds, future retail profits are competed away in the auction. The output market is not affected by the auction. Debt financing, however, does affect retail prices. If debt induces firms to compete more aggressively in the output market, retail prices are lower than they would be without debt financing. Auction revenues will be lower. Still, profits are not fully competed away in the auction, so firms make strictly positive expected net profits. Hence, although debt makes the output market more competitive, it makes the auction less so. Instead, if debt induces firms to compete less aggressively in the output market, retail prices are higher than they would be without debt financing. The effect on auction revenues is ambiguous. Again, firm profits will be higher with debt financing than they are in a world in which firms have sufficient internal funds. The fact that expected net firm profits are higher with debt financing is due to endogenous credit rationing. Winning bidders, even though they make positive expected net profits, will not be outbid. A higher bid would require a higher debt 2

level. With limited liability, a higher debt level will induce firms to make riskier decisions. At some point, a higher debt level therefore reduces the expected repayment to banks. Thus, a firm would be willing to bid more if it could obtain funding for such a bid, but it is not able to. When they have sufficient internal funds, firms bid up to the profits they can make from the license. If debt induces firms to compete more aggressively those profits will be lower, hence auction revenues decrease. If debt induces firms to compete less aggressively those profits will be higher. However, due to credit rationing, firms will not be able to bid up to those profits and the effect on auction revenues is ambiguous. In the popular press, it is often argued that auctions lead to higher retail prices, since firms have to “earn their money back.” That argument boils down to a sunk-cost fallacy (see e.g. Klemperer [2002a]). In our model, auctions do affect retail prices but for an entirely different reason. Indeed, we show that auctions may even lead to lower prices than beauty contests: auctions lead to higher debt, and higher debt may lead to lower prices. The remainder of this paper is structured as follows. After an overview of some related literature in the next section, we present our model in Section III. Section IV derives the equilibrium. In Section V we discuss the implications of debt financing, both when debt induces firms to compete more aggressively (Section V(ii)) and when it induces them to compete less aggressively (Section V(iii)). Section V(iv) studies the special case in which both firms obtain their financing from the same bank, while Section VI gives a numerical example. Section VII concludes.

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II

Related literature

As argued in the introduction, a firm’s bid in a license auction will depend on the characteristics of the aftermarket. A number of papers study auctions in which the utility of the losing bidders depends, either positively or negatively, on the identity of the winner(s) (Jehiel and Moldovanu [1996] and [2000]; Jehiel et al. [1996] and [1999]; Das Varma [2002]). This will affect the equilibrium bids. Maasland and Onderstal [2007] study an auction with financial externalities. In their paper, the utility of the losers depends on the payment of the winner. Other papers explicitly study how license auctions affect the aftermarket. Hoppe et al. [2006] show that auctioning more licenses need not result in a more competitive aftermarket. In Goeree [2003], the signalling of private information during the auction affects the aftermarket. In Janssen [2006] the only equilibrium consistent with forward induction is the one in which firms set collusive prices on the aftermarket and bid collusive profits in the auction. In Janssen and Karamychev [2007], the auction selects the least risk-averse bidder, which affects prices on the aftermarket. In Burguet and McAfee [2009] firms are financially constrained. A higher bid in the auction then implies less funds for production in the aftermarket. In that setting, auctions may still maximize consumer surplus. A number of papers study aspects of the interplay between debt and auctions. Chowdhry and Nanda [1993] also study the strategic role of debt in an auction, but do so in the context of a takeover contest, in which many raiders bid to take over a firm. Clayton and Ravid [2002] study the effect of the initial debt level of a firm on its bidding behavior in the US FCC auctions. They find that, as debt levels increase, firms tend to reduce their bids. However, this paper studies debt levels before the

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auction, while we study debt that firms have to take on due to the auction. In Zheng [2001], bidders differ with respect to the amount of funds that they have. Firms with less funds have to take on more debt upon winning the auction. That implies that these bidders are willing to risk more, and therefore bid more aggressively. RhodesKropf and Viswanathan [2005] study the interplay between asymmetric cash positions and independent private values in a model where auction winners may take on debt.

III

The model

There are N > 2 bidders, which participate in a license auction where two licenses to operate on a new market are sold. In what follows, we will often refer to them as firms. In the auction the two highest bidders win and have to pay their own bid. To focus on the effect of debt financing we assume that firms are a priori identical, which implies that allocative efficiency is not an issue in this auction. Firms do not have any funds. Before they can participate in the auction, they have to find a bank that is willing to finance their bid. Auctioneers often require such a bank guarantee from bidders. We consider a four-stage model. In the first stage, the financing stage, potential bidders go to a competitive debt market to find a bank that is willing to finance their bid in the auction. The bank decides on a credit limit, that is, a maximum amount that the firm is allowed to bid in the auction and that the bank is willing to fund. The bank also provides the auctioneer with a guarantee that the bidder will indeed be able to pay a bid up to that maximum. In the second stage, the auction stage, the firms compete in a sealed-bid license auction, where winning bidders pay their own bid. Without loss of generality, the highest bidder will be denoted firm 1 and 5

the second highest bidder will be denoted firm 2. Their bids are denoted b1 and b2 . In the case of ties, winners will be decided by coin toss. In stage 3, the debt stage, each winning firm i ∈ {1, 2} negotiates a debt level di with its bank, in return for an amount bi to finance its bid.6 The debt level di is the amount that firm i promises to repay at the end of the game. Should the firm and its bank not reach an agreement, the bank that gave the guarantee has to pay a penalty Φ > 0, and the license will be awarded to the next-highest bidder. The winning bidders have full bargaining power in this stage, as they still have the option to go to a different bank. In stage 4, the competition stage, the two winning firms compete in the output market, where they face uncertainty. After firms have chosen their strategic actions (i.e., have set price or quantity), uncertainty is resolved, consumers make their purchase decisions and – if possible – debts are repaid. The remainder of this section describes in detail the four stages of the model: the financing stage, the auction stage, the debt stage, and the competition stage. We describe these stages in the same order as we solve the model: by backward induction, starting with the last stage. We then give a formal definition of the equilibrium concept, and end with some technicalities. Stage 4: the competition stage In this stage, the two firms that have won the auction in stage 2 compete in the output market. Products are substitutes. Demand functions are downward sloping. Firms compete by simultaneously choosing an action a, which could be either price p or quantity q. Let ai ≥ 0 denote the action chosen by firm i, i = 1, 2. Firm i’s retail profits will depend on the actions chosen by both firms, and the realization of some common random variable ω. The uncertainty reflected by ω may concern e.g. the marginal cost firms will incur, or the level of 6

demand that they will face. Firm i’s retail profits are denoted πi ≡ π(ai , aj , ω), which is continuous and strictly concave in ai . Similarly, firm j’s retail profits are given by πj ≡ π(aj , ai , ω). We thus assume that the profit function π is the same for both firms. Profits depend on the action chosen by this firm, which is the first argument of the profit function, and the action chosen by the other firm, which is the second argument, as well as on the realization of the random variable ω. This notational convention is used throughout the paper. We use the convention that higher values of ω correspond to less favorable states of the world in which, ceteris paribus, retail profits are lower: ∂πi < 0. ∂ω If the uncertainty concerns marginal cost, low ω thus reflects low marginal cost. If the uncertainty concerns demand, low ω reflects high demand. The realization of ω is drawn from a continuously differentiable, strictly positive probability density function f (ω) with domain [ω, ω ¯ ]. For expositional convenience, we also assume that πi ≥ 0 for all relevant ai and aj . Firms face limited liability. Firm i can just repay its debt di if ω is such that π (ai , aj , ω) = di . In that case, retail profits are just sufficient to cover the promised repayment di . Denote the value of ω for which this equality is satisfied as ω ˆi ≡ ω ˆ (ai , aj , di ). If ω ≤ ω ˆ i firm i is able to fully repay its debt. Its bank then receives di . If ω ∈ (ˆ ωi , ω ¯ ], the firm earns positive retail profits, but these are insufficient to repay the debt di . In that case, all retail profits will be paid to the bank, and the firm’s net profits are zero. We denote the expected net profits of firm i, i.e. after the repayment of its debt, 7

as Πi ≡ Π(ai , aj , di ). We thus have Z (1)

ω ˆ (ai ,aj ,di )

Πi ≡ Π(ai , aj , di ) =

(π (ai , aj , ω) − di ) f (ω) dω. ω

For the competition subgame, we make the usual assumptions that Πi is continuous, strictly concave in ai and that |∂ 2 Πi /∂a2i | > |∂ 2 Πi /∂ai ∂aj |. This is sufficient for uniqueness of the equilibrium of this subgame (see e.g.

Tirole [1988],

p. 226). We will denote the unique equilibrium of the competition subgame as (a∗1 , a∗2 ) ≡ (a∗ (d1 , d2 ), a∗ (d2 , d1 )), using asterisks to refer to equilibrium values. We define the continuation profits for firm i in the competition subgame as πi∗ ≡ π ∗ (di , dj , ω) ≡ π(a∗ (di , dj ), a∗ (dj , di ), ω). Also, we define expected retail profits of firm i as: Z (2)

ω ¯

∗

E(π (di , dj )) ≡

π ∗ (di , dj , ω)f (ω) dω.

ω

Consider the game described in Brander and Lewis [1986], in which firms choose debt levels purely for strategic reasons. We will refer to this as the strategic debt game. Firm i then sets di to maximize E(π ∗ (di , dj )), taking dj as given. Sufficient for uniqueness of an equilibrium in the strategic debt setting game is that E(πi∗ ) is strictly concave in di and |∂ 2 E(πi∗ )/∂d2i | > |∂ 2 E(πi∗ )/∂di ∂dj |. We will also assume this to be the case. We denote the symmetric equilibrium of the strategic debt game as dBL . We assume that, at that debt level, the expected net profits of each firm are strictly positive. Hence, we require that the strategic effect of debt is not so strong that profits are completely dissipated if both firms use debt for strategic reasons only. As is common in the strategic debt literature, we make the implicit assumption that

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for relevant d, it is not feasible for one firm to preempt the other firm by defecting to a debt level that is so high that it becomes a monopolist. Stage 3: the debt stage In the debt stage, the two firms that have won the auction negotiate the terms of their loan with their bank. A winning firm has submitted a bid bi ∈ [0, Ci ], with Ci its credit limit. A debt contract can be represented by (bi , di ): firm i receives the amount bi now, in return for the promise to repay di at the end of the game. For ease of exposition, we will write ω ˆ (di , dj ) ≡ ω ˆ (a∗ (di , dj ), a∗ (dj , di ), di ). The expected repayment to the bank of firm i, Ri , can then be written as Z (3)

ω ¯

Ri ≡ R(di , dj ) = Pr(ω ≤ ω ˆ (di , dj ))di +

π ∗ (di , dj , ω)f (ω) dω.

ω ˆ (di ,dj )

The expected net profits to the bank now equal Ri − bi . The expected net profits of firm i at this stage can be written as (4) Π∗i

Z ∗

∗

ω ˆ (di ,dj )

∗

≡ Π (di , dj ) ≡ Π(a (di , dj ), a (dj , di ), di ) =

(π ∗ (di , dj , ω) − di ) f (ω) dω.

ω

Taking dj as given, firm i sets di such that its expected net profits are maximized given what will occur in the competition stage and subject to the constraint that the expected net profits to the bank are at least 0. One might be tempted to argue that the bank would be willing to accept any contract that yields net expected profits of at least −Φ. But the bank would only be willing to do so if rejection would imply that the firm would not pay its bid, as only in that case the bank has to pay Φ. If the bank would reject the contract, however, the firm would have an incentive to go to an alternative bank. Of course, the outside option of that bank is 0 rather than −Φ, as it has not given a guarantee. The firm can offer a contract that yields the alternative 9

bank nonnegative profits and that it is willing to accept. Offering a contract that yields −Φ to the original bank is based on the implicit threat that it has to pay a penalty if it does not accept the contract. That, however, is not a credible threat. Stage 2: the auction stage In the auction stage, N firms submit bids to obtain a license. Firm k submits bid Bk , k ∈ {1, . . . , N }. A firm cannot submit a bid that is higher than its credit limit: Bk ∈ [0, Ck ], ∀k ∈ {1, . . . , N }. Expected net profits of firm k at the auction stage can then be denoted ΠA k (B1 , . . . BN ). The highest bid is b1 ≡ max{B1 , . . . , BN }, the second highest bid b2 ≡ max {B1 , . . . , BN } \{b1 }. In case ties occur, we define T as the number of firms that have submitted the same bid as the second-highest bidder: T ≡ #{k|Bk = b2 }. Note that the highest bidder may also be among these. Given the vector of bids, the probability of obtaining a license for firm k now equals

Pk (B1 , . . . , BN ) =

   0 if Bk < b 2       1 if Bk = b2 < b1 T  2  if Bk = b1 = b2  T      1 if B = b > b k 1 2

and we have

(5)

∗ ∗ ∗ ΠA k (B1 , . . . , BN ) = Pk (B1 , . . . , BN ) · Π (d (Bk , B−k ), d (B−k , Bk )),

with B−k = max {B1 , . . . , BN } \ {Bk } the highest bid of all bidders other than k and d∗ (Bi , Bj ) the equilibrium debt level that will be negotiated if this firm has submitted a bid Bi and the other winning firm has submitted a bid Bj .

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Stage 1: the financing stage In the financing stage, each firm that aims to participate in the auction tries to secure a line of credit from some bank. The banking market is competitive. Banks decide on the line of credit to which they are willing to commit. Each firm chooses the bank that offers the line of credit that allows it to make the highest possible expected net profit. If firms would have access to an unlimited supply of credit, any symmetric equilibrium would collapse. By increasing its bid and thereby its debt level, one firm could then commit to become so aggressive on the aftermarket that it becomes infeasible for the other auction winner to find financing, leaving a monopoly position for the defecting firm. As the purpose of our analysis is to study how competition in the aftermarket affects bidding in the auction and vice-versa, this is not an appropriate set-up. If banks have to commit to a credit line, we do find a symmetric equilibrium in which pre-emption is infeasible and both licenses are sold. Moreover, this set-up reflects the practice that we often observe in real-world license auctions.7 We will explicitly assume that each bidder chooses a different bank to obtain a line of credit. When we assume that firms can go to the same bank, the analysis changes slightly, as we will show in Section V(iv). For simplicity, we assume that each bank strictly prefers financing a winning bidder to being inactive, even if it yields the same profits.8 Similarly, we assume that firms strictly prefer winning the auction to not doing so, even if both events yield the same profits.9 More formally, denote by Ck the credit limit that firm k can secure. The bid that firm k will submit in the auction is a function of the credit limits secured by all firms, so we can write Bk = Bk∗ (C1 , . . . , CN ). Consider a bank that is offering firm k a credit limit Ck0 . The firm will not take up this offer if a different bank offers some Ck00 > Ck0 , and if Ck00 allows it to obtain higher expected net profits ΠA k . There will be such a 11

bank as long as offering Ci00 still allows a bank to at least break even. If firm k is among the winners of the auction, then the debt level that this firm will ∗ obtain in the debt stage will be a function of both Bk and B−k , so d∗k = d∗ (Bk∗ , B−k ),

where we have omitted the arguments of Bi∗ for ease of exposition. Similarly, the ∗ , Bk∗ ). The bank debt level of its product market competitor can be denoted as d∗ (B−k ∗ ∗ , Bk∗ )) ≥ Bk∗ , as ), d∗ (B−k offering Ck is able to at least break even if R(d∗ (Bk∗ , B−k

Bk∗ is the amount of credit that the firm will ultimately use. Given the credit limits secured by all other firms, firm k will thus secure the line of credit that maximizes its expected net profits ΠA k , subject to this constraint. Equilibrium concept Putting together all the elements of the four subgames, we can now define the subgame perfect Nash equilibrium. Definition 1 A subgame perfect Nash equilibrium of the game described above con∗ sists of credit limits (C1∗ , . . . CN∗ ) and bids (B1∗ , . . . BN ) for all bidders, and debt levels

(d∗1 , d∗2 ) and actions (a∗1 , a∗2 ) for the two highest bidders, such that we have: 1. Equilibrium at the competition stage:

(6)

a∗i = arg max Π(ai , a∗j , di ), ai

∀di , for i = 1, 2 and j 6= i, and Π(ai , aj , di ) as defined in (1); 2. Equilibrium at the debt stage: d∗i ∈ arg max Π∗ (di , d∗j ) di

s.t. R(di , d∗j ) ≥ bi , 12

∀bi , for i = 1, 2 and j 6= i, R(di , dj ) as defined in (3), and Π∗ (di , d∗j ) as defined in (4); 3. Equilibrium at the auction stage: ∗ ∗ ∗ ∗ Bk∗ ∈ arg max ΠA k (B1 , . . . , Bk−1 , Bk , Bk+1 , . . . , BN ) Bk

s.t. Bk ≤ Ck , ∀Ck , for k ∈ {1, . . . .N }, and ΠA k (B1 , , . . . , BN ) as defined in (5). 4. Equilibrium at the financing stage: ¡ ∗ ¢ ∗ ∗ ∗ ∗ Ck∗ ∈ arg max ΠA (B C , . . . C , C , C , . . . C k k 1 1 k−1 k+1 N ,..., Ck

¡ ∗ ¢ ∗ ∗ ∗ BN C1 , . . . Ck−1 , Ck , Ck+1 , . . . CN∗ ) ¡ ¢ ∗ ∗ ∗ ∗ s.t. R(d∗ (Bk∗ , B−k ), d∗ (B−k , Bk∗ )) ≥ Bk∗ C1∗ , . . . Ck−1 , Ck , Ck+1 , . . . CN∗ , for k ∈ {1, . . . .N } and B−k ≡ max{B1 , B2 , . . . , BN }\{Bk }. Further technicalities There are cases in which a firm cannot possibly make strictly positive net profits. As the amount of debt di that a firm holds increases, there is a point where it is impossible for firm i to make strictly positive net profits regardless of the action ai that it chooses, even if we are in the most favorable state of the world, so if ω = ω. Formally, for any dj there is a d˜i ≡ d˜(dj ) such that π ∗ (d˜(dj ) , dj , ω) = d˜(dj ). For firm i to be able to make strictly positive net profits, ˜ j ). If this is not satisfied, then any action ai will yield a zero we need di < d(d net profit. For technical convenience, we assume that for any di ≥ d˜(dj ), we have 13

∗ a∗ (di , dj ) = limD%d(d ˜ j ) a (D, dj ). Thus, the firm will choose the same action that it

would choose with the highest di that could still yield positive net profits. It will also prove useful to consider the case in which both firms have the same level of debt dc , and to study the effect of an increase in that common debt level. Similar to d˜i , we can define d˜c as that value of the common debt level for which firms are not able to make positive net profits, regardless of the value of ω. Thus, π ∗ (d˜c , d˜c , ω) = d˜c . Again, we assume that if dc increases beyond d˜c the behavior of the firms does not change. Thus for any dc ≥ d˜c , we have a∗ (dc , dc ) = limD%d˜c a∗ (D, D). Note that we have assumed that equilibrium debt levels in the strategic debt game of Brander and Lewis [1986] are such that expected net profits are strictly positive. This implies that dBL < d˜c .

IV

Solving the model

We now turn to the equilibrium of the model. Remaining proofs are in the appendix. Theorem 1 The unique symmetric equilibrium of our model has all firms obtaining credit limit C ∗ , bidding B ∗ , obtaining debt level d∗ , and setting the strategic variable a∗ , with 1. C ∗ = R(d∗ , d∗ ); 2. B ∗ = C ∗ ; 3. d∗ = arg maxd R(d, d∗ ); 4. a∗ = arg maxa Π(a, a∗ , d∗ ).

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Intuitively, this equilibrium can be understood as follows. Consider a simultaneousmove game in which two banks unilaterally decide on a debt level di with the aim to maximize the expected repayment Ri . The equilibrium of that game has d∗ = arg maxd R(d, d∗ ). Our model has the same equilibrium debt level, for the following reason. Banks have an incentive to allow firms to submit bids that are as high as possible. This entails setting credit limits that are as high as possible. As banks’ profits are driven to zero in equilibrium, this in turn implies that expected repayments should be set as high as possible, given the equilibrium behavior of other banks. But that game is exactly equivalent to one in which two firms each unilaterally decide on a debt level di to maximize expected repayment Ri . Firms then bid up to the maximum amount allowed by their bank: B ∗ = C ∗ . The credit limits that banks offer are such that banks expect to break even, so C ∗ = R(d∗ , d∗ ).

V

The implications of debt financing

V(i)

Introduction

In the previous section, we derived the equilibrium of our model. In this section, we compare that outcome to that in a model in which firms do not have to take on debt in order to finance their bids. That is, we consider a case in which all firms have sufficient internal funds to finance their bids. This allows us to study the equilibrium effects of debt financing on retail prices, auction revenues, and firm profits. In order to do so, we need to put some additional structure on the competitive process. The strategic effect of debt may work in two opposite directions. More debt may result in more aggressive competition, that is, higher quantities and lower prices

15

(Brander and Lewis [1986]). However, it may also be the case that more debt results in less aggressive competition, that is, in lower quantities and higher prices (Showalter [1995]). This distinction is important for our results. In Section V(ii), we study the effect of debt financing when more debt results in more aggressive competition. Section V(iii) considers the opposite case, where more debt results in less aggressive competition.

V(ii)

Debt leads to more aggressive competition

Consider the case in which more debt induces firms to compete more aggressively. Following Brander and Lewis [1986], we will show that that is implied by the following assumption: Property 1 Marginal retail profit is strictly decreasing in ω. That is,

∂ 2 πi ∂ω∂qi

< 0, with

qi the quantity sold by firm i. This is true e.g. with Cournot competition and uncertainty about either marginal costs or demand, with Bertrand competition with differentiated products and uncertainty about demand, and with Hotelling competition with uncertainty about marginal costs (see Showalter [1995]) . Denote the best-reply function of firm i as βi ≡ β(aj , di ). We then indeed have: Lemma 1 If Property 1 is satisfied, having more debt induces a firm to compete more aggressively:

     >0 ∂β(aj , di )  <0  ∂di     =0

˜ j )], if ai ≡ qi and di ∈ [0, d(d ˜ j )], if ai ≡ pi and di ∈ [0, d(d ˜ j ). if di > d(d 16

Thus, with quantity competition, more debt induces a firm to compete more aggressively, in the sense that its best-reply function shifts upwards. With more debt, a given output of the competitor will lead to a higher output for this firm. Also, with price competition, more debt induces a firm to compete more aggressively, but that now implies that its best-reply function shifts downwards. With more debt, a given price of the competitor will lead to a lower price for this firm. Using the definition of expected retail profits from (2), we can show: Lemma 2 If Property 1 is satisfied, an increase in the common debt level decreases the expected retail profits of the firms:   dE(π ∗ (dc , dc ))  < 0  ddc  =0

if dc ∈ [0, d˜c ], otherwise.

The following result is also convenient in what follows: 10 Lemma 3 Consider the joint level dR c that maximizes expected revenues of the bank:

dR c = arg max R(dc , dc ). dc

If Property 1 is satisfied, this level is strictly lower than d˜c . Intuitively, this can be seen as follows. Suppose that we have dc = d˜c . In that case, all retail profits flow to the bank. Now suppose that we decrease dc slightly to d˜c − ε. Retail profits then increase for all possible realizations of ω. This is good news for the bank.11 This implies that R increases.

Place Figure 1 approximately here 17

In Figure 1, we depict a firm’s expected retail profits E (π ∗ (dc , dc )) as a function of the common debt level dc . Note that E (π ∗ ) is decreasing in dc , using Lemma 2. For simplicity, in the figure we have drawn E (π ∗ ) as a linear function, but of course that does not have to be the case. Out of expected retail profits, the amount R flows to the bank. The firm is left with expected net firm profits Π, which equals the difference between E (π ∗ ) and R. From the definition of d˜c , we have E[π ∗ (d˜c , d˜c , ω)] = R(d˜c , d˜c ). As firms’ behavior does not change beyond d˜c , we have that for all dc > d˜c , E(π ∗ (dc , dc , ω)) = R (dc , dc ) = E(π ∗ (d˜c , d˜c , ω)). In the figure, R is decreasing for dc slightly smaller than d˜c , which is implied by Lemma 3. The equilibrium of our model has d∗ > dR c . If joint bank revenues are maximized, then the expected revenues of a single bank are still increasing in its own debt,12 which implies that d∗ > dR c . It is also possible to show13 that d∗ < d˜c , as reflected in the figure. From the proof of Lemma 2, we have that the effect of debt financing on prices is as follows: Lemma 4 Assume that Property 1 is satisfied. In the competition subgame, if both firms have the same debt level and that debt level increases, equilibrium retail prices decrease: dp∗ (dc , dc ) < 0. ddc This immediately implies: Theorem 2 If debt leads to more aggressive competition, then in an auction with debt financing retail prices are lower than in an auction where firms finance their bids with internal funds. This is easy to see. Debt induces firms to compete more aggressively. In an auction with debt financing, debt levels are obviously14 higher than in an auction where firms 18

pay their bids with internal funds. Hence, retail prices with debt financing are lower. Occasionally, licenses are allocated through a beauty contest rather than an auction, In a beauty contest, firms submit business plans to a government committee which awards the licences for free to those candidates that best meet a set of published criteria (see e.g. Binmore and Klemperer, 2002). Hence, in a beauty contest there is no need for firms to take on debt.15 This immediately implies that an auction with debt financing also yields lower retail prices than a beauty contest. These results are opposite to what is often argued in the popular press: that auctions lead to higher retail prices, since firms have to “earn their money back”. That argument boils down to a sunk-cost fallacy (see e.g. Klemperer, 2002a). In this model, auctions lead to lower prices than beauty contests: auctions lead to higher debt, and higher debt leads to lower prices. It is also interesting to study the effect of debt financing on auction revenues: Theorem 3 If debt leads to more aggressive competition, then in an auction with debt financing auction revenues are lower than in an auction where firms finance their bids with internal funds. This can be seen directly from Figure 1. With internal financing, we have dc = 0. Firms are thus willing to bid up to the point where the function E(π ∗ ) intersects the vertical axis, which is B0 . With debt financing, they only bid up to B ∗ . As debt financing induces firms to compete more aggressively, equilibrium profits are lower as a result. Hence, even if the auctioneer were able to capture the full equilibrium retail profits16 , equilibrium bids would now be lower. Finally, consider the effect on the equilibrium profits of firms:

19

Theorem 4 If debt leads to more aggressive competition, then in an auction with debt financing firms’ expected profits are higher than in an auction where firms finance their bids with internal funds: whereas these profits are competed away in an auction with internal funds, they are strictly positive with debt financing. In an auction with internal funds, profits are competed away in Bertrand-like fashion. In this auction with debt, however, this is not true. This can be seen as follows. First consider the equilibrium debt level of the strategic debt game, dBL . In this game, the firm chooses its debt level to maximize expected retail profits. Our assumption of strictly positive expected net profits at dBL implies that dec > dBL . Hence, at dec a firm can increase its expected net profits by decreasing its debt level. In our model, it is not the firm, but the bank lending to it that decides on the debt level. The bank chooses the debt level to maximize expected repayment. Suppose that firm j faces a debt dec . In the limit as di approaches dec the expected repayment equals expected retail profits. As expected retail profits can again be increased by decreasing the debt level, so can expected repayment. This implies that the equilibrium debt level in our model, d∗ , falls below dec , and thus has positive expected net profits for the firm. Hence, lowering the debt level from d˜c benefits both banks and firms. As firms earn positive expected net profits in equilibrium, they would be willing to face a greater debt level in exchange for a credit limit higher than C ∗ . Suppose that bank i would offer such a limit. At d∗ , any change in di yields a lower expected repayment, so expected bank profits would then decrease. Since these are zero in (B ∗ ; d∗ ), this implies that the bank would make a loss by offering such a higher credit limit.

20

In that sense, we thus have a backward-bending credit supply curve.17 In general, such a curve may be caused by adverse selection or moral hazard. With moral hazard, as the interest rate r increases, firms are inclined to take on riskier projects. At some point, this effect becomes so strong that the expected repayment to the bank decreases with an increase in r, causing the supply of credit to decrease. A similar argument applies in our model. As the winning firms take on more debt, they compete more aggressively, which implies that expected retail profits decrease, at some point leading to a lower expected repayment to the bank. To be able to submit a higher bid, a firm would need to be able to offer a higher expected repayment to the bank, which is simply not feasible. Hence firms would still be willing to submit higher bids, but are not able to do so since they are denied access to the credit market when they do. Thus, although debt makes the output market more competitive, it makes the auction less competitive in the sense that expected net profits are higher.

V(iii)

Debt leads to less aggressive competition

Now suppose that more debt induces firms to compete less aggressively. Showalter [1995] shows that this is the case with Bertrand competition with differentiated products and uncertainty about demand. We study that issue in this section, and thus replace Property 1 with the following: Property 2 Marginal retail profit is strictly increasing in ω. That is,

∂ 2 πi ∂ω∂qi

> 0.

We will discuss our results solely in terms of price competition, as we are not aware of any models with quantity competition for which Property 2 is satisfied. The proofs of Lemmas 5 and 6 below are straightforward variations on the corresponding lemmas in the analysis above, and are therefore omitted. 21

Lemma 5 If Property 2 is satisfied, having more debt induces a firm to compete less aggressively:

  ∂β(pj , di )  > 0  =0 ∂di 

˜ j )] if di ∈ [0, d(d ˜ j ). if di > d(d

Thus, restricting attention to price competition, more debt induces a firm to compete less aggressively in the sense that its best-reply function shifts upwards. Lemma 6 If Property 2 is satisfied, an increase in the common debt level increases the expected retail profits of the firms:   dE(π ∗ (dc , dc ))  > 0  ddc  =0

if d ∈ [0, d˜c ], otherwise.

Lemma 7 If Property 2 is satisfied, bank revenues are maximized in the common ˜ debt level at dR c = dc . Intuitively, this can be seen as follows. Suppose that we have dc = d˜c . When we decrease dc from d˜c to some lower value, firms will compete more aggressively. This hurts the bank. Even if firms are able to repay their debt, the bank receives a lower amount. Hence, R decreases. The outcome is depicted in Figure 2. Again, a firm’s expected retail profits E (π ∗ (dc , dc )) are shown as a function of a common debt level dc . Now we have that E (π ∗ ) is increasing in dc , using Lemma 6. Out of expected retail profits, the amount R flows to the bank. The firm is left with expected net firm profits Π, which equal the difference between E (π ∗ ) and R. From the definition of d˜c , we have E(π ∗ (d˜c , d˜c , ω)) = R(d˜c , d˜c ). As firms’ behavior does not change beyond d˜c , we have that for all dc > d˜c , E(π ∗ (dc , dc , ω)) = R (dc , dc ) = E(π ∗ (d˜c , d˜c , ω)). 22

Place Figure 2 approximately here The effects of debt financing are now easy to establish: Theorem 5 If debt leads to less aggressive competition, then in an auction with debt financing retail prices are higher than in an auction where firms finance their bids with internal funds. Debt now induces firms to compete less aggressively. Hence, retail prices with debt financing are higher, also when compared to a beauty contest. Theorem 6 If debt leads to less aggressive competition, then the effect of debt financing on auction revenues is ambiguous. This can be seen directly from Figure 2. There are two countervailing effects at work. With internal financing, firms bid up to expected retail profits E(π ∗ ), while with debt financing, they bid up to expected bank revenue R, as explained in Section IV. For given dc , the former are higher than the latter. Hence, for given dc , bids are higher with internal financing. But debt levels are higher with debt financing, and higher debt now implies higher revenues for the bank, rendering the net effect ambiguous. More formally, equilibrium bids with internal financing are E(π ∗ (0, 0)), while with debt financing they equal R(d∗ , d∗ ). As R(d∗ , d∗ ) < E(π ∗ (d∗ , d∗ )) but E(π ∗ (d∗ , d∗ )) > E(π ∗ (0, 0)), it is a priori unclear which bids are higher.18 Finally, consider the effect on the equilibrium profits of firms. Theorem 7 If debt leads to less aggressive competition, then in an auction with debt financing firms’ expected profits are higher than in an auction where firms finance their bids with internal funds: whereas these profits are competed away in an auction with internal funds, they are strictly positive with debt financing. 23

Proof. Similar to that of Theorem 4. Again, we have that debt financing increases the equilibrium profits of firms. The intuition for this result is the same as that given in Section V(ii). In this case, the equilibrium of the strategic debt game is dBL = 0 (Showalter [1995]), which is clearly below d˜c , the debt level that yields zero profits to both firms. Hence, lowering the debt level from d˜c again benefits both banks and firms. Winning bidders, even though they make positive expected net profits, will not be outbid. A higher bid would require a higher debt level. With limited liability, a higher debt level will induce firms to make riskier decisions. At some point, a higher debt level therefore reduces the expected repayment to banks. Thus, a firm would be willing to bid more if it could obtain funding for such a bid, but it is not able to.

V(iv)

Extension: a common bank

In our analysis so far, we have assumed that each bidder does business with a single bank. Our analysis changes, however, if several bidders obtain a line of credit from the same bank. From the proof of Theorem 1 it is easy to see that in case the same bank finances both winning bidders we would have equilibrium debt levels equal to ∗ ∗ dR c = arg maxd R(d, d) rather than d = arg max R(d, d ). That is, the common bank

maximizes its revenue from both firms rather than maximizing its revenue from just one firm, given that the other firm has obtained d∗ . ¢ ¡ R (see Figures 1 and , d In this set-up, the equilibrium bids are given by R dR c c 2). As firms can submit a higher bid if they go to the same bank, they clearly have an incentive to do so. That common bank will internalize the negative effect that a higher debt level has on the other firm, and hence is able to let firms bid higher than

24

a single bank would. In the case that debt leads to more aggressive competition, our qualitative results do not change: debt financing still decreases retail prices, lowers auction revenues, and increases net firm profits. When debt leads to less aggressive competition, however, our qualitative results do change. Debt financing again implies higher retail prices, but from Figure 2 we immediately have that auction revenues now unambiguously increase, while net firm profits are driven to zero. Of course the analysis becomes more complicated if there is a number of banks that each provides a line of credit to more than one firm. Suppose, for example, that there are four firms and two banks that each provide a line of credit to two firms. In equilibrium, each firm will submit the same bid. But then the probability that the two winners of the auction are financed by the same bank is 1/3, while the probability that they are financed by different banks is 2/3. Banks will have to take this into account when offering their line of credit. In equilibrium, all firms would of course choose the same bank, provided it is feasible to do so.

VI

A numerical example

In this section, we give a numerical example. We consider a Hotelling model with cost uncertainty. Suppose that the two winning firms are located on a Hotelling line of unit length, one firm being located at 0, the other at 1. A mass of consumers, normalized to 1, is uniformly distributed on the line. Transportation costs are normalized to 1 per unit of distance. The willingness to pay is v for every consumer, with v high enough so the market is always covered. Assume that marginal costs c are constant and equal across firms, and drawn from a uniform distribution on [0, 1]. Obviously, retail profits are lower if marginal costs turn out to be higher. We can thus interpret c 25

as the random variable ω in our analysis. We have f (ω) = 1 and [ω, ω ¯ ] = [0, 1]. From the analysis in Showalter [1995], we immediately have that firms have no strategic reason to take on debt, so dBL = 0. Retail profits of firm i now equal 1 π(pi , pj , ω) = (1 + pj − pi )(pi − ω). 2

(7)

Note that ∂ 2 π/∂pi ∂ω = 1/2, which implies that Property 1 holds. Also

ω ˆ i = pi −

2di . 1 + pj − pi

In stage 3, firm i’s expected net profits equal Z

ω bi

Πi = 0

µ

¶ 1 (1 + pj − pi ) (pi − ω) − di dω. 2

Plugging in ω ˆ i yields

(8)

Π(pi , pj , di ) =

(pi (1 + pj − pi ) − 2di )2 . 4 (1 + pj − pi )

Taking the first-order condition yields19 1 5 pi = (1 + pj ) − 6 6

q (1 + pj )2 + 24di .

From (7), we have that, evaluated in equilibrium,

(9)

dπi∗ 1 dp∗ 1 = (1 + p∗j − p∗i ) i + (p∗i − ω) ddj 2 ddj 2

26

µ

dp∗j dp∗ − i ddj ddj

¶ .

Necessarily, both p∗i and p∗j are decreasing in dj .20 Also note that dj does have a direct effect on the reaction function of firm j, but not on that of firm i. We thus have q  2 5 (1 + p ) + 24d − (1 + p ) j i j dp∗ dp∗ 1  j < 5 j. q = = ddj ∂pj ddj 6 ddj 6 ddj (1 + pj )2 + 24di 

dp∗i

∂p∗i

dp∗j

This implies that the term dp∗j /ddj − dp∗i /ddj in (9) is strictly positive. Since we require that πi∗ > 0 for all ω, we have p∗i > ω. Suppose both firms have the same level of debt d. We can then solve for equilibrium prices to find p∗ (d, d) = 2−2d and π ∗ (d, d) = 12 (2−2d−ω). For the expected repayment to the the bank, we have Z

Z

ω ¯

R(di , dj ) = Pr(ω ≤ ω ˆ )di +

∗

1

π (di , dj , ω)f (ω) dω = ω ˆ i di + ω ˆi

π ∗ (di , dj , ω) dω.

ω ˆi

With common d, ω ˆ = p∗ − 2d = 2 − 4d. Note that the highest possible value for ω is 1. Hence, when d ≤ 1/4, the firms are always able to repay their debt. The restriction that πi∗ > 0 requires d < 1/2. We thus have:    R(d, d) =

=

d

if d ≤

R   (2 − 4d) d + 1 1 (2 − 2d − ω) dω if 2−4d 2    d if d ≤ 1

1 4

1 4


1 2

4

  3d − 4d2 −

1 4

if

1 4

< d < 12 .

Solving numerically we find that d∗ = 0.4167, b∗ = R(d∗ , d∗ ) = 0.3055 and p∗ = 27

1.1666. Firms earn strictly positive expected profits, which equal 0.02799. In a model with internal funds, equilibrium retail prices are p∗ (0, 0) = 2, which is higher than prices in the model with debt. The same retail prices would prevail with a beauty contest. With internal funds, bids equal bint = E(π ∗ (0, 0)) =

3 4

which is higher than

in the model with debt, and expected firm profits are driven to zero.

VII

Discussion and conclusion

In this paper, we considered license auctions in which winning firms have to take on debt in order to finance their bids. Since debt has a strategic effect in the aftermarket, it will also affect the outcome of the auction. In addition, the outcome of the auction determines the level of debt firms will take on. We argue that when debt induces firms to compete more aggressively, there is a negative relation between retail prices and the fees paid at the auction. Thus, higher fees imply lower prices for consumers. In the equilibrium of our model with debt financing, firm profits are higher than they would be with internal funds. This is due to credit rationing. Winning bidders, even though they make strictly positive expected profits, will not be outbid. Any higher bid would yield a debt level that implies negative expected net profits for the bank, and therefore financing cannot be obtained. Thus, if debt makes the output market more competitive, then it makes the auction less competitive in the sense that expected net profits are higher. Our results change, however, when debt induces firms to compete less aggressively. In that case, expected firm profits are again higher than in the case without debt financing. Hence, also if debt makes the output market less competitive, it makes the auction less competitive in the sense that expected net profits are higher. Yet, 28

retail prices are higher than they are with a beauty contest, or with an auction with internal funds. The effect on auction revenues is ambiguous. If firms can go to the same bank to finance their bids, they choose to do so. The common bank internalizes the negative external effects that firms have on each other, allowing them to ultimately bid higher. When debt induces firms to compete less aggressively, this implies that net firm profits will be driven to zero. Also, auction revenues will now be unambiguously higher than without debt. Our results suggest that in auction design it is important to take into account how winners will finance their bids. When debt finance is used results from standard auction theory, implicitly based on internal finance, need not apply.

Appendix Proof of Theorem 1 We first show that C ∗ is a symmetric equilibrium at the financing stage, and then establish uniqueness. To see that C ∗ is a symmetric equilibrium, suppose that each bank offers such a credit limit. It is then an equilibrium for all firms to bid up to their limit in the debt stage. This can be seen as follows. By bidding less an individual firm will surely not be among the winners of the auction. When bidding C ∗ , there is a probability that it is. Hence no firm will defect to a bid lower than C ∗ . Second, it is easy to see that all banks are indeed willing to offer C ∗ . Suppose that a bank offers C ∗ and firm i takes up that offer. Moreover, suppose that i is among the winners of the auction. Both winners have submitted a bid C ∗ in the

29

auction. The equilibrium of the debt stage then has d∗ ∈ arg max Π∗ (d, d∗ ) d

s.t. R(d, d∗ ) ≥ C ∗ .

By construction, this yields a unique equilibrium d∗ such that R(d∗ , d∗ ) = C ∗ , which implies that a bank is indeed willing to offer C ∗ .21 Next, we show that no bank has an incentive to defect from C ∗ . Suppose that, given that all other banks offer credit limit C ∗ , firm i’s bank offers some Ci < C ∗ . In that case, firm i is not able to win in the auction, hence firm i would prefer one of the banks that does offer C ∗ . Suppose that a bank offers Ci > C ∗ . By offering such a line of credit, it can be sure that there is a firm that takes up this offer and bids at least b = C ∗ + ε in the auction: by doing so it wins the auction for sure. A necessary condition for the bank to be willing to offer this is that it yields a nonnegative profit. By construction, this is not feasible as d∗ already maximizes R(d, d∗ ) and R(d∗ , d∗ ) = C ∗ . The bank will thus make a loss of either ε if it does proceed to finance the bid, or a loss of Φ if it decides not to do so. Naturally, it is in the firm’s best interest if the bank does finance its bid, which implies that it will submit a bid b ∈ (C ∗ , C ∗ + Φ).22 Hence, by offering such a higher credit limit to a firm, the bank would definitely make a loss. Therefore, it will choose not to do so. To establish uniqueness of the symmetric equilibrium, suppose that all banks offer some C 0 < C ∗ . Again, it is a symmetric equilibrium in the auction stage for all firms to bid up to their credit limit. Suppose now that a bank defects from the tentative equilibrium and offers credit limit C 00 ∈ (C 0 , C ∗ ). A bidder taking up that offer can bid ε higher than its competitors. As this leads to a discrete increase in the probability 30

that it wins the auction, it is definitely willing to do so for ε infinitesimally small. Moreover, the bank is also willing to make the offer. This can be seen as follows. With C 0 + ε < C ∗ , by construction, there are (d∗1 , d∗2 ) with R(d∗1 , d∗2 ) ≥ C 0 + ε and R(d∗2 , d∗1 ) ≥ C 0 . This implies that the bank will at least break even when offering such a defection. Hence, C 0 is not a symmetric equilibrium.23 Finally, it is trivial that no equilibrium exists in which all banks offer some C 0 > C ∗ . In that case, banks would surely make losses, as no debt levels exist such that bids can be financed. We have thus established that in the unique equilibrium of the finance stage, each bank offers a credit limit C ∗ as defined in the theorem. The equilibrium values B ∗ , d∗ , and a∗ then follow directly. ˜ j ). The first-order Proof of Lemma 1 Consider the case in which di ≤ d(d condition (FOC) for firm i in the competition stage is given by

(10)

∂Πi = 0. ∂ai

Totally differentiating this expression yields ∂ 2 Πi ∂ 2 Πi ∂ 2 Πi da + da + ddi = 0. i j ∂a2i ∂aj ∂ai ∂di ∂ai A similar equality holds for firm j. This system of two equations can be solved using Cramer’s rule to give ∂ 2 Πi ∂ 2 Πj

(11)

∂di ∂ai ∂a2j ∂ai ∂βi ≡ =− , ∂di ∂di H

31

where H≡

∂ 2 Πi ∂ 2 Πj ∂ 2 Πi ∂ 2 Πj − . ∂a2i ∂a2j ∂aj ∂ai ∂ai ∂aj

Our assumption |∂ 2 Πi /∂a2i | > |∂ 2 Πi /∂ai ∂aj | directly implies that H > 0. The secondorder conditions (SOCs) are ∂ 2 Πi /∂a2i < 0. From (11) we then have that the sign of ∂ai /∂di is the same as the sign of ∂ 2 Πi /∂di ∂ai . To sign this expression, consider the FOC given by (10). From (1), we have ∂Πi = ∂ai (12)

Z Z

ω ˆi

∂π (ai , aj , ω) ∂ω ˆi f (ω) dω + (π(ai , aj , ω ˆ i ) − di )f (ˆ ωi ) ∂ai ∂ai

ω ˆi

∂π (ai , aj , ω) f (ω) dω = 0, ∂ai

ω

= ω

where the second equality follows from the fact that

(13)

π(ai , aj , ω ˆ i ) = di .

From (12), we obtain

(14)

∂ 2 Πi ∂π (ai , aj , ω ˆi) ∂ω ˆi = · f (ˆ . ωi ) · ∂di ∂ai ∂ai ∂di

This derivative equals the product of three terms. We consider the sign of each in turn, starting with the last one. Differentiating (13) with respect to di yields ˆi ∂πi ∂ ω = 1, ∂ω ˆ i ∂di so ∂ ω ˆ i /∂di = [∂π (ai , aj , ω ˆ i ) /∂ ω ˆ i ]−1 < 0, hence the last term of (14) is negative. The second term is clearly positive: f (ˆ ωi ) > 0, provided ω ˆ i ∈ [ω, ω ¯ ]. Finally, consider the

32

sign of the first term, ∂π (ai , aj , ω ˆ i ) /∂ai . From Property 1, we have   ∂ 2 πi  < 0 if ai ≡ qi , ∂ω∂ai   > 0 if a ≡ p . i

i

Thus, ∂π (ai , aj , ω) /∂ai is decreasing in ω in case of quantity competition (ai ≡ qi ) and increasing in ω in case of price competition (ai ≡ pi ). Consider the integral in (12). With quantity competition, we have that the integrand is decreasing, and the integral equals zero. This necessarily implies that the integrand is negative when evaluated at the upper limit of the integral, hence ∂π (ai , aj , ω ˆ i ) /∂ai < 0. From (14), we then have ∂ 2 Πi /∂di ∂ai > 0. In turn, this implies from (11) that indeed ∂ai /∂di > 0. With price competition, the integrand in (12) is increasing, while the integral is zero. This implies that the integrand is positive when evaluated at ω ˆ i , so ∂π (ai , aj , ω ˆ i ) /∂ai > 0. From (14), we then have ∂ 2 Πi /∂di ∂ai < 0. In turn, this implies from (11) that indeed ∂βi /∂di < 0. ˜ j ), the result is trivial. By assumption, an increase in di does not For di ≥ d(d affect ai , hence it does not affect the reaction function βi . Proof of Lemma 2 First consider the case of price competition. From Lemma 1, an increase in firm i’s debt level shifts its best-reply function downwards. Since best-reply functions are upward sloping, firms are symmetric, and di = dj = dc , an increase in dc must decrease equilibrium prices for both firms: dp∗ (dc , dc )/ddc < 0. Next, define pm c ≡ arg max E(π(pc , pc )). pc

33

Thus, pm c is the price that, if set by both firms, maximizes their expected retail profits. Note that expected retail profits are equal to firm profits in the case of zero debt: E(π(pc , pc )) = Π(pc , pc , 0). Using symmetry between the two firms and the condition for stability of the Nash equilibrium, we then have that Eω (π(pc , pc )) is increasing in pc for pc < pm c . Obviously, the equilibrium price in a duopoly is strictly lower than the price that maximizes joint profits,24 hence p∗ (0, 0) < pm c = arg max Π(pc , pc , 0). With ∗ dp∗ (dc , dc )/ddc < 0, this implies that p∗ (dc , dc ) < pm c for all dc , and hence dE(π (dc , dc ))/ddc <

0. For the case of quantity competition, we have that best-reply functions are downward sloping and, from Lemma 1, that an increase in dc leads to an upward shift in a firm’s best-reply function. Hence, equilibrium quantities q ∗ (dc , dc ) are increasing in dc : dq ∗ (dc , dc )/ddc > 0. Define qcm ≡ arg max E(π(qc , qc )). qc

Thus, qcm is the price that, if set by each individual firm, maximizes the firms’ expected retail profits. Concavity of π(qc , qc ) implies that E(π(qc , qc )) is decreasing in qc for qc > qcm . Obviously, the equilibrium quantity a firm sets in a duopoly is strictly higher than the quantity each firm sets when joint profits are maximized25 , hence q ∗ (0, 0) > qcm ≡ arg max Π(qc , qc , 0). With dq ∗ (dc , dc )/ddc > 0, this implies that q ∗ (dc , dc ) > qcm for all dc , and hence dE(π ∗ (dc , dc ))/ddc < 0.

34

Proof of Lemma 3 If d1 = d2 = dc , the bank lending to firm i has expected revenue Z

ω ¯

R(dc , dc ) = Pr(ω ≤ ω ˆ (dc , dc ))dc +

π ∗ (dc , dc , ω)f (ω) dω.

ω ˆ (dc ,dc )

Using Leibniz’s rule and dropping the arguments of ω ˆ , we have ∂ Pr(ω ≤ ω ˆ) ∂ω ˆ dR(dc , dc ) = dc + Pr(ω ≤ ω ˆ ) − π ∗ (dc , dc , ω ˆ )f (ˆ ω) ddc ∂d ∂dc Z ω¯ ∗c dπ (dc , dc , ω) + f (ω) dω. ddc ω ˆ Since π ∗ (dc , dc , ω ˆ ) = dc and f (ω) = ∂ Pr(ω ≤ ω ˆ )/∂ω, the first and third terms cancel out, so we have

(15)

dR(dc , dc ) = Pr(ω ≤ ω ˆ) + ddc

Z

ω ¯ ω ˆ

dπ ∗ (dc , dc , ω) f (ω) dω. ddc

Consider the case in which dc = 0. Then, ω ˆ (dc , dc ) = ω ¯ , so we have

(16)

¯ dR(dc , dc ) ¯¯ = Pr(ω ≤ ω ¯ ) = 1. ¯ ddc dc =0

Consider the case in which dc = d˜c . Then, ω ˆ (dc , dc ) = ω, so we have

(17)

¯ Z ω¯ ∗ dE(π ∗ (dc , dc )) dR(dc , dc ) ¯¯ dπ (dc , dc , ω) = f (ω) dω = < 0, ¯ ˜ ddc ddc ddc ω dc =dc

where the inequality follows from Lemma 2. Combining (16) and (17), by continuity there must be at least one dˇc ∈ (0, d˜c ) where dR(dc , dc )/ddc evaluated at d = dˇc equals zero, and moreover d2 R(dc , dc )/dd2c evaluated at dc = dˇc is negative. Combined with 35

˜ (17), this implies that indeed dR c < dc . Proof of Theorem 4 First of all, it is easy to see that profits are zero with internal financing: all firms would then be willing to bid up to B0 . Profits would thus be completely competed away in Bertrand-like fashion. The analysis for debt financing is more complicated. From (3), using Leibniz’s rule and dropping the arguments of ω ˆ , we have, along the same lines as in the proof of Lemma 3: ∂R(d1 , d2 ) = Pr(ω ≤ ω ˆ1) + ∂d1

Z

ω ¯ ω ˆ1

∂π ∗ (d1 , d2 , ω) f (ω) dω. ∂d1

Evaluate this for d1 = d2 = d˜c . With some abuse of notation, we can write Z ω¯ ∂R(d˜c , d˜c ) ∂π ∗ (d˜c , d˜c , ω) ∂E(π ∗ (d˜c , d˜c )) = f (ω) dω = . ∂d1 ∂d1 ∂d1 ω

(18)

By construction ∂E(π ∗ (dBL , dBL )) = 0. ∂d1 We then have ∂E(π ∗ (d˜c , d˜c )) ∂E(π ∗ (d˜c , d˜c )) ∂E(π ∗ (dBL , dBL )) = − ∂d1 ∂d1 ∂d1 ¶ Z d˜c µ 2 ∂ E(π ∗ (d, d)) ∂ 2 E(π ∗ (d, d)) = + dd. ∂d21 ∂d1 ∂d2 dBL ¯ 2 ∗ ¯ 2 ∗ ¯ ¯ Strict concavity implies that ∂ E(π∂d2(d,d)) < 0. The assumption that ¯ ∂ E(π∂d2(d,d)) ¯ > 1 1 ¯ 2 ∗ ¯ ¯ ∂ E(π (d,d)) ¯ ¯ ∂d1 ∂d2 ¯ then implies that the integrand is negative for any value of d, which immediately implies that the integral is negative as well. Combined with (18), we 36

have thus established that ∂R(d˜c , d˜c ) < 0. ∂d1 But this implies that if firm 2 would set d2 = d˜c , then this firm would set some d1 < d˜c . Hence, we cannot have that d∗ = d˜c . Therefore necessarily d∗ < d˜c , establishing the result. Proof of Lemma 7 By construction, R(dc , dc ) = E(π ∗ (dc , dc )) − Π(dc , dc ) (see e.g. Figure 2). From Lemma 6, dE(π ∗ (dc , dc ))/ddc > 0 ∀dc ∈ [0, d˜c ]. Moreover, note that Π(0, 0) > 0 whereas Π(d˜c , d˜c ) = 0. Strict concavity then implies dΠ(dc , dc )/ddc < 0 ∀dc ∈ [0, d˜c ]. Hence h i dR(dc , dc ) dE(π ∗ (dc , dc )) dΠ(dc , dc ) = − > 0 ∀dc ∈ 0, d˜c ddc ddc ddc which establishes the result.

References Binmore, K. and Klemperer, P., 2002, ‘The Biggest Auction Ever: The Sale of the British 3G Telecom Licences’, Economic Journal, 112, pp. C74–C96. B¨orgers, T. and Dustmann, C., 2003, ‘Awarding Telecom Licenses: The Recent European Experience’, Economic Policy, 36, pp. 215–268. Brander, J. and Lewis, T., 1986, ‘Oligopoly and Financial Structure: The Limited Liability Effect’, American Economic Review, 76, pp. 956–970.

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Burguet, R. and McAfee, R. P., 2009, ‘License Prices for Financially Constrained Firms’, Journal of Regulatory Economics, 36, pp. 178–198. Chowdhry, B. and Nanda, V., 1993, ‘The Strategic Role of Debt in Takeover Contests’, Journal of Finance, 68, pp. 731–745. Clayton, M. J. and Ravid, A. S., 2002, ‘The Effect of Leverage of Bidding Behavior: Theory and Evidence from the FCC Auctions’, Review of Financial Studies, 15(3), pp. 723–750. Cramton, P. and Schwartz, J. A., 2000, ‘Collusive Bidding: Lessons from the FCC Spectrum Auctions’, Journal of Regulatory Economics, 17, pp. 229–252. Das Varma, G., 2002, ‘Standard Auctions with Identity-dependent Externalities’, RAND Journal of Economics, 33, pp. 689–708. Freixas, X. and Rochet, J.-C., 1997, Microeconomics of Banking. MIT-Press, Cambridge, Mass. Goeree, J. K. 2003, ‘Bidding for the Future: Signaling in Auctions with an Aftermarket’, Journal of Economic Theory, 108, pp. 345–364. Halper, M., 2007,

‘A Battle for Europe’s Airwaves’,

Fortune, Available at

http://money.cnn.com/magazines/fortune/fortune archive/2007/06/25/100116284/. Hoppe, H.; Jehiel, P. and Moldovanu, B., 2006, ‘License Auctions and Market Structure’, Journal of Economics and Management Strategy, 15, pp. 371–396. Janssen, M. C., 2006, ‘Auctions as Coordination Devices’, European Economic Review, 50, pp. 517–532. 38

Janssen, M. C. and Karamychev, V., 2007, ‘Selection Effects in Auctions for Monopoly Rights’, Journal of Economic Theory, 134, pp. 576–582. Jehiel, P. and Moldovanu, B., 1996, ‘Strategic Nonparticipation’, RAND Journal of Economics, 27, pp. 84–98. Jehiel, P. and Moldovanu, B., 2000, ‘Auctions with Downstream Interaction Among Buyers’, RAND Journal of Economics, 31, pp. 768–791. Jehiel, P. and Moldovanu, B., 2003, ‘An Economic Perspective on Auctions’, Economic Policy, 18, pp. 269–308. Jehiel, P.; Moldovanu, B. and Stacchetti, E., 1996, ‘How (Not) to Sell Nuclear Weapons’, American Economic Review, 86, pp. 814–829. Jehiel, P.; Moldovanu, B. and Stacchetti, E., 1999, ‘Multidimensional Mechanism Design for Auctions with Externalities’, Journal of Economic Theory, 85, pp. 258– 283. Klemperer, P., 1999, ‘Auction theory: A Guide to the Literature’, Journal of Economic Surveys, 13, pp. 227–286. Klemperer, P., 2002a, ‘How (Not) to Run Auctions: The European 3G Telecom Auctions’, European Economic Review, 46, pp. 829–845. Klemperer, P., 2002b, ‘What Really Matters in Auction Design’, Journal of Economic Perspectives, 16, pp. 169–189. Maasland, E. and Onderstal, S., 2007, ‘Auctions with Financial Externalities’, Economic Theory, 32, pp. 551–574. 39

McAfee, R. P. and McMillan, J., 1996, ‘Analyzing the Airwaves Auction’, Journal of Economic Perspectives, 10, pp. 159–175. McMillan, J., 1994, ‘Selling Spectrum Rights’, Journal of Economic Perspectives, 8, pp. 145–162. Rhodes-Kropf, M. and Viswanathan, S., 2005, ‘Financing Auction Bids’. RAND Journal of Economics, 36, pp. 789–815. Showalter, D. M., 1995, ‘Oligopoly and Financial Structure: Comment’, American Economic Review, 85, pp. 647–653. The Economist, 2002, ’Too Many Debts; Too Few Calls’, July 18. Van Damme, E., 2002, The European UMTS-auctions, European Economic Review, 46, pp. 846–858. Wolfstetter, E., 1996, ‘Auctions: An Introduction’, Journal of Economic Surveys, 10, pp. 367–420. Zheng, C. Z., 2001, ‘High bids and Broke Winners’, Journal of Economic Theory, 100, pp. 129–171.

Notes 1. Cramton and Schwartz [2000]; for more on the design of these auctions, see e.g. McAfee and McMillan [1996], or McMillan [1994]. 2. See Van Damme [2002], or B¨orgers and Dustmann [2003].

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3. Klemperer [2002a]. 4. See e.g. Halper [2007]. 5. For a survey see Wolfstetter [1996], or Klemperer [1999]. 6. Since the prospects of a firm will depend not only on its own winning bid but also e.g. on the amount of debt that the other winner obtains, it will be hard to fully specify the required repayment ex ante. Still, if it were possible to write such a complete contract, it would not affect our analysis. 7. One example available online is the “auction of 1800 MHz spectrum right” in Singapore in 2008; see http://www.ida.gov.sg/doc/Policies%20and%20Regulation/Policies and Regulation Level3/20060425153817/1800MHz3BankGuarantee.pdf 8. As firms have all bargaining power in the debt stage, banks almost always make zero profits. If banks would be indifferent between being active and being inactive, they would also be indifferent as to which line of credit they offer, as long as it does not induce a loss. To avoid such a multiplicity of uninteresting equilibria, we make this assumption. It is easy to justify, for example by assuming that banks do have ε bargaining power, or by assuming that being the financier of the auction winner yields reputation benefits that banks can exploit on other markets. 9. The equilibrium of our model turns out to have auction winners making strictly positive profits, which implies that this assumption is indeed satisfied. 10. Should the solution to this maximization problem not be unique, we choose dR c as the lowest dc for which the maximum is reached. 11. On the other hand, for ω such that ω < ω ˆ , bondholders no longer receive all retail profits. For very small ω, this implies that they now receive less than in the previous case, but this is only a second-order effect. 12. At dR c , we have ∂R1 /∂d1 + ∂R1 /∂d2 = 0. With ∂R1 /∂d2 < 0, this implies ∂R1 /∂d1 > 0. 13. See the proof of Theorem 4. 14. Suppose the equilibrium has d∗ = 0. As firms have an incentive to use at least some credit, that can only be consistent with credit limits C ∗ = 0. A bank could then profitably defect by offering some C ∗ > 0. Hence, the equilibrium has d∗ > 0.

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15. We ignore the costs of preparing the business plans. 16. In Theorem 4, we show that this is not the case, which is an additional reason that equilibrium bids are lower. 17. See e.g. Freixas and Rochet [1998, section 5.2]. 18. Note that in the case where debt leads to more aggressive competition we unambiguously have R(d∗ , d∗ ) < E(π ∗ (d∗ , d∗ )) < E(π ∗ (0, 0)). 19. In fact, the first-order condition yields four solutions, but it is easy to show that this is the only feasible one. N 20. Suppose that firm j faces an infinitesimally small increase in its debt level from dO j to dj , so ¢ ¢ ¡ ¡ O O N ˜ < pj pi , dO dN j . Firm i’s bestj > dj , while dj < dj (di ). From lemma 1, we have that pj pi , dj

reply function is left unaffected by the change in firm j’s debt level. Denote the initial equilibrium ¢ ¡ N N¢ ¡ O of the competition stage by pO i , pj , and the new equilibrium by pi , pj . Since both best-reply O N O functions are upward sloping, we must have pN i < pi and pj < pj .

21. Note that, by assumption, firms will not preempt each other by defecting to a d0 that is so high that the other firm cannot profitably operate. 22. Note that, if this bank decides not to finance the bid, it can be sure that the firm cannot find another bank willing to finance its bid, hence this bank will still have to pay the penalty Φ. Also note that, should this bank choose to incur the penalty rather than finance the bid, the license goes to one of the other firms that submitted a bid C ∗ . This also implies that the other firm never has an incentive to choose a debt level different from d∗ , even if its competitor did submit a higher bid. 23. The assumption that firms cannot preempt each other also implies that (C 0 , C 0 ) cannot be supported as an equilibrium by the implicit threat that, say, firm 1 would preempt firm 2 should firm 2 accept some C 00 > C 0 . 24. Formally, note that pd ≡ p∗ (0, 0) solves ∂Π(pi , pd , 0) = 0. ∂pi

42

Plugging this into the FOC of joint-profit maximization yields ∂Π(pd , pd , 0) ∂Π(pd , pd , 0) + > 0, ∂pi ∂pj as ∂Πi /∂pj > 0, since an increase in j’s price will increase i’s profits. Strict concavity then implies d that the joint-profit maximizing price pm c is strictly higher than p .

25. Formally, note that q d ≡ q ∗ (0, 0) solves ∂Πi (qi , q d , 0) = 0. ∂qi Plugging this into the FOC of joint-profit maximization yields ∂Π(q d , q d , 0) ∂Π(q d , q d , 0) + < 0, ∂qi ∂qj as ∂Π/∂qj < 0, since an increase in j’s quantity will decrease i’s profits. Strict concavity then implies that the joint-profit maximizing quantity qcm is strictly lower than q d .

43

profits B0

E (π ∗ )

Π B∗

R

∗ ˜ dR c d dc

dc

Figure 1: Solving the model when debt leads to more aggressive competition.

profits

E (π ∗ ) B∗ B0

Π

R

d∗

˜ dR c = dc

dc

Figure 2: Solving the model where debt leads to less aggressive competition.