A Dynamic Model of Optimal Creditor Dispersion

Borrowing from multiple creditors exposes firms to rollover risks due to coordination problems among creditors, but it also improves firms' repayment incentives, thereby increasing pledgeability. Based on this trade-off, I develop a dynamic debt rollover model to analyze the evolution of creditor dispersion. Consistent with empirical evidence, I find that firms optimally increase creditor dispersion after poor performance. In contrast, cross-sectionally higher-growth firms can support more dispersed creditors. Frequent debt renegotiation paralyzes firms’ ability to increase pledgeability by having more creditors. Finally, holding a cash balance while borrowing from multiple creditors improves firms' repayment incentives uniformly across all future states.


Introduction
Many firms borrow from multiple creditors. 1 Having more creditors risks coordination problems among them, making it more difficult for these firms to restructure their debt. At face value, this seems to suggest that during bad times, firms should consolidate their creditors so that they can renegotiate the distressed debt more easily and avoid bankruptcy. Surprisingly, empirical evidence suggests the opposite. Many firms increase the number of lenders they employ when their performance deteriorates (Farinha andSantos 2002, Rauh andSufi 2010). An obvious explanation for this pattern is that the existing lenders refuse to throw more good money after bad, leaving firms no choice but to borrow from more creditors. However, one question remains: Why are new creditors willing to lend while the incumbents are rushing for the exit?
To avoid the devastating consequences of coordination failures among creditors, many mechanisms are in place to promote efficient renegotiation of distressed debt. However, easier debt renegotiation can potentially alter firms' repayment incentives, which affect their decisions to issue or to refinance debt early on. How do the mechanisms that create an expost efficient debt renegotiation affect firms' ex-ante choice of creditor dispersion, which in turn feeds into the likelihood and outcome of the negotiation? Do such mechanisms always help prolong firms' life as intended?
To study these questions, creditor dispersion needs to be viewed as a dynamic variable that evolves endogenously, instead of a one-time choice. In this paper, I develop a parsimonious dynamic model in which a firm with insufficient internal resources must finance a long-term project by repeatedly rolling over short-term debt. The key friction is that the firm cannot commit to repay its debt at maturity, but needs incentives to do so. A dispersed creditor structure creates coordination problems, which following bad shocks, can result in inefficient liquidation. With a good shock realization, however, the same coordination problems enhance the firm's repayment incentives by making it harder for the firm to opportunistically hold up its creditors. The firm optimally readjusts the number of creditors in each period by trading off the risk of rollover failure against the benefit of better commitment.
In contrast to the literature, where static models have largely been used to examine creditor structure (e.g., Berglöf and von Thadden 1994, Bolton andScharfstein 1996, andDiamond 2004), the dynamic debt rollover framework proposed here is arguably closer to reality. First, many firms use staggered debt structure, and rollover failures can be costly (Almeida et al. 2012). Perhaps more importantly, debt rollover itself is fundamentally a dynamic concept: the ability to roll over debt today depends on whether the firm's new creditors anticipate they can, in turn, roll over their debt in the future, which in turn depends on whether creditors forecast that rollover will be possible even further in the future. Finally, Roberts and Sufi (2009a) show that 94% of the loans with maturity longer than three years are renegotiated, and Roberts (2015) shows that a median renegotiated loan receives four renegotiations in its lifetime. These empirical findings suggest that it is important to treat debt renegotiation as a dynamic process.
Despite my model's parsimony, it generates a rich set of predictions, especially in the time series. First, my model delivers predictions on how many creditors a firm has, as well as when it decides to seek more creditors or consolidate the existing ones. In the time series, I show that firms increase the number of creditors when their performance deteriorates. This is because the required leverage endogenously increases when firms perform poorly. In order to support this higher leverage, firms must expand their pool of creditors such that future repayments are more credible. Next, I contrast the time-series analysis with cross-sectional studies, where I investigate firms with exogenously different growth rates. Better firms with higher growth rates have larger debt capacities, allowing these firms to survive more rounds of renegotiation and therefore support more creditors. Hence, unlike the time-series finding, faster-growing firms can have more creditors on average.
The sharp contrast between the time-series and the cross-sectional predictions reconciles the seemingly contradictory empirical findings. Farinha and Santos (2002) and Rauh and Sufi (2010) find that both lending relationships and the composition of debt become more dispersed when firms' performance deteriorates. In contrast, Houston and James (1996) find that firms with more relationship banks have higher asset growth compared to their single-bank counterparts. Through the lens of my dynamic model, the evidence in Farinha and Santos (2002) and Rauh and Sufi (2010) is consistent with the time-series prediction, whereas Houston and James' (1996) finding is consistent with the cross-sectional prediction.
In addition, the dynamic aspect of my model also helps to connect an important stream of theoretical literature with some stylized facts on the relation between borrowing costs and creditor dispersion. Many classic studies (e.g., Berglöf and von Thadden 1994, Bolton andScharfstein 1996, andDiamond 2004) build on the idea that dispersed creditor structure commits the borrowing firm to pay back its debt. This channel implies that the required interest rate should be lower when the debt is more widely held, because the difficulty renegotiating with dispersed creditors makes repayments more credible. However, Rajan (1994,1995) empirically document a significant positive relation between the number of creditors and the cost of credit. My model suggests that although borrowing from more creditors indeed helps reduce the required interest rate, firms only do so when their performance deteriorates, driving up the observed interest rate in equilibrium.
Second, the dynamic model allows me to study how renegotiation frequency affects pledgeability -the maximum amount of debt capital a firm can raise. I find that when a firm can instantaneously renegotiate its debt, it can no longer pledge any value from a potential better state in the future, even with dispersed creditors to improve its commitment power. 2 This result casts doubt on the traditional view in many two-period models that dispersed short-term debt alleviates the commitment problem stemming from renegotiation. In a fully dynamic world, when there is little time between two rounds of renegotiations, the fundamental is unlikely to change and thus very persistent. In order to pledge the extra value from a better state in the following period, the firm needs to risk inefficient termination should the current (worse) state continue. Because of the persistent fundamental, this termination risk becomes almost a certainty, whereas the benefit of realizing higher commitment power in the better state becomes vanishingly small. Consequently, having dispersed creditors does not improve firms' ex-ante pledgeability when renegotiation becomes very frequent.
Finally, I present a novel role for cash. In conjunction with multiple creditors, cash creates extra commitment power uniformly across all future states. This channel is different from the well-known Hart and Moore (1998) intuition for why firms borrow additional money and hold cash. In Hart and Moore (1998), where cash is unverifiable and cannot be seized by creditors upon liquidation, holding cash allows the firm to essentially buy back a fraction of the project (to reduce partial liquidation) when termination is relatively more inefficient. Unlike Hart and Moore (1998), in my model, cash can be seized by creditors. Consequently, when there is only one creditor, cash plays no role. This is because the firm can achieve the same outcome by simultaneously reducing the face value of debt and the same amount of cash inside the firm. However, when there are multiple creditors, cash is part of the firm's state non-contingent assets, which increases the reservation value for each creditor independent of the fundamental realization. Having multiple creditors therefore serves as a multiplier that magnifies the committable repayment as a result of the cash holding.
My paper is related to a large literature on coordination problems among dispersed creditors. Perhaps the most famous example is a bank (creditor) run. Diamond and Dybvig (1983) show that in a static setting, socially inefficient bank run equilibria generally exist. Goldstein and Pauzner (2005) further characterize the probability of a bank run under a global game framework. He and Xiong (2012a) study the dynamic evolution of a panicbased run on staggered corporate debt.
If borrowing from multiple lenders is subject to costly runs, then why do firms continue this practice? Many scholars believe that such runs can be a disciplinary device that enhances ex-ante efficiency. Berglöf and von Thadden (1994) show that having multiple creditors specialize in lending at different maturities is a superior structure. Short-term creditors can impose externalities on long-term creditors at the debt renegotiation stage, thereby increasing firms' ex-post repayment incentives. The threat of a bank run associated with dispersed demand deposits can incentivize financial intermediaries to return the proceeds to the depositors (Diamond and Rajan 2001) and encourage the depositors to acquire costly information about the banks (Calomiris and Kahn 1991). Diamond (2004) demonstrates that dispersed creditors can solve the "passive lender" problem. A single lender with a large stake has little incentive to discipline a firm because such actions also hurt the lender. Without credible discipline, the firm may misbehave ex-ante. In contrast, the negative externalities of the disciplinary actions among multiple creditors provide higher incentives for lenders to be active ex-post, forcing the borrower to behave properly ex-ante. More recently, Zetlin-Jones (2014) examines an environment with multiple intermediaries and dispersed depositors. These papers share the key insight that the threat of coordination failures associated with multiple creditors acts to discipline the firm and can potentially improve ex-ante efficiency. However, these studies vary the number of creditors exogenously and therefore are silent on when firms endogenously change their creditor structure.
Several attempts have been made to endogenize the optimal number of creditors. Bolton and Scharfstein (1996) is, to my knowledge, the first attempt, and it is closely related to my paper. The firms in their model can strategically default and renegotiate debt even when they have the money to repay. The creditor(s), upon default, can inefficiently sell the project to an outside investor. The benefit of having more creditors is to increase their collective bargaining power against the firm following a strategic default, so that creditors can extract higher repayments. However, in a fundamental-driven default, the stronger bargaining power also makes it less likely the creditors will find an outside investor, making default more costly. From a completely different angle, Detragiache, Garella, and Guiso (2000) study an economy in which bank financing can fail and the quality of the firm's project is unknown. Having multiple bank relationships in this situation makes financing more robust. However, when all relationship banks deny credit, having more banks makes the firm worse off because the uninformed market is more pessimistic about the firm's quality. Bris and Welch (2005) consider a "free-riding" problem that reduces an individual creditor's willingness to expend effort on debt collection. As opposed to the creditor-discipline channel, dispersed creditors lower their collective bargaining power under the free-riding problem. A common feature of these papers is that they are all static: a one-time choice of creditor dispersion. I embrace Bolton and Scharfstein's (1996) idea, but instead bring the trade-off to a dynamic world. More recently, several researchers study the implications of creditor structure and rollover risks on equilibrium debt maturity. Cheng and Milbradt (2012) solve for the optimal debt maturity given the trade-off between risk-shifting incentives and rollover failures. Brunnermeier and Oehmke (2013) show that excessive short-term debt may prevail in equilibrium with multiple creditors, despite the increased rollover risks. Petersen and Rajan (1995) propose a model that illustrates how lenders' market power affects the quality of the financed firms and the cost of credit. They take the lenders' market power as an exogenous parameter. In my model, I endogenize the variation of bargaining power by modeling the game between the firm and its creditors. Ongena and Smith (2000b) provide a comprehensive survey on findings related to the number of creditors.
The effects of debt renegotiation and rollover have been investigated from an asset-pricing perspective. Mella-Barral and Perraudin (1997) and Mella-Barral (1999) study debt pricing when firms can renegotiate and service the troubled debt rather than just defaulting directly, as in Leland (1994). He and Xiong (2012b) investigate how creditors with different maturities strategically interact with each other when they decide whether to roll over maturing debt. Similar to Diamond (2004), creditors' decisions not to roll over debt pose externalities on other incumbent creditors with claims that have not yet matured. Hege and Mella-Barral (2005) examine an economy in which a firm can exchange liquidation rights for coupon concessions on debt. They study how this feature affects the credit risk premia as the number of creditors changes. With the creditor structure exogenously fixed, these authors focus on pricing the debt claims given the possibility of renegotiation or rollover frictions. In contrast, I focus on the optimal choice of creditor dispersion.

Model Setup
In Subsections 2.1 and 2.2, I introduce the baseline model, featuring a firm's optimal choice of creditor dispersion in a dynamic debt rollover framework. Then, I discuss key modeling assumptions in Subsection 2.3 and define the equilibrium concept in Subsection 2.4.

The Project and Financing
Time t is discrete and the discount rate is r > 0. At time t = 0, a risk-neutral firm with no cash starts a long-term project that needs an upfront investment I 0 . The project generates no interim cash flow but only an observable liquidating dividend at a random project maturity τ π . 3 At the beginning of each period t, the project matures with probability π, generating the final dividend Y t . If the project does not mature, then a new fundamental Y t+1 = Y t z t+1 realizes. I assume that the shock process z t+1 ∈ {1, 1 + µ} follows a Markov chain with z 0 = 1. The transition probabilities of z t are given as follows: In other words, the project starts in a bad state and the fundamental stays constant over time, until the bad state switches to a good one at some random date. Starting from this point, the fundamental Y t perpetually grows at a rate of µ. With a slight abuse of notation, when z t appears in superscripts, I denote by G(ood) and B(ad) the growth state z t = 1 + µ and the no-growth state z t = 1, respectively. The expected value of the final dividend at the end of period t − 1 (after the realization of z t ) can be calculated as follows, with the detailed mathematical derivation in the Appendix: the linear first-best values in (1). Throughout the paper, If the project does not mature in period t, it can be sold prematurely for λY t+1 . When λ is sufficiently small, in particular, smaller than the expected value of the project in a bad state, then selling the project prematurely is never optimal.

Dynamic Debt Rollover Game
In order to model debt rollover in a simple fashion, I assume that the firm can only issue one-period debt to short-lived creditors. Since the project does not generate any interim cash flow, the firm must repeatedly issue new debt to finance the repayments to the maturing creditors. Figure 1 presents the timeline of the debt rollover game. The firm enters period t with shock z t , fundamental Y t , and N t ≥ 1 incumbent creditors, each holding a maturing debt contract with a face value Ft Nt . Hence, the total face value outstanding is F t . First, the project matures with probability π, generating a final liquidating dividend Y t . All incumbent creditors split the final dividend equally up to the promised face value, each receiving 1 Nt min(F t , Y t ). The equity receives the residual max(Y t −F t , 0). 4 If the project does not mature, then the next period's fundamental Y t+1 is realized and is publicly observed. Because the project does not generate any interim cash flow, the firm must refinance (roll over) the maturing debt into the next period. One of the main frictions in this model is that the firm cannot commit to repaying creditors. Instead, the firm always attempts to renegotiate the actual repayment. Specifically, the firm meets each creditor sequentially and makes a take-it-or-leave-it offer S i t to the ith negotiating creditor, who is also the ith lender who provided the money in period t − 1. The offer history is public information.
If the ith creditor is the first to decline the offer, I assume that the firm can avoid an immediate default by selling the project for λY t+1 in order to honor the repayment of Ft Nt . Selling the project can be more broadly interpreted in practice as downsizing investment, spinning off assets, or fire sales. Since the company has not yet defaulted, automatic stay or equal treatment among creditors does not apply. To simplify strategic actions, I assume all subsequent creditors automatically refuse any renegotiation and demand the full face value Ft Nt , which is paid from the total proceeds λY t+1 sequentially until depletion. In addition, creditors j < i who have already accepted the offers are now at the rear of the collection queue. Therefore, creditors' payoff in the event of a renegotiation failure features "sequential service," as commonly assumed in the large literature following Diamond and Dybvig (1983). 5 Mathematically, if the ith creditor declines the offer, the i + kth (k = 0, 1, ..., N t − 1) creditor receives repayment: and the remaining max(λY t+1 − F t , 0) goes to the firm. Here, I use the convention that if i + k > N t , it denotes the i + k − N t th creditor. The first case in (3) captures the full repayment to a creditor when the resale proceeds λY t+1 are still sufficient to honor this repayment. The second case characterizes the payoff to the last creditor who exhausts the (remaining) proceeds, and everyone to follow receives 0 as in the final case. A particularly relevant payoff is that the first rejecting creditor receives a repayment of min Ft Nt , λY t+1 . On the other hand, if every incumbent creditor accepts the offer, then renegotiation is successful. The firm needs to pay back the total renegotiated offers Nt i=1 S i t by refinancing 5 Alternatively, I can endow subsequent creditors with strategic actions to accept or decline their renegotiation offers. In addition, instead of a sequential priority, one can assume the non-negotiating creditors (j = i) have equal priority on the remaining proceeds. The equilibrium outcome remains robust to these modifications. As will soon be clear, the exact same coordination problem emerges, as long as there is an advantage to being the first creditor to reject the renegotiation offer. them into the next period. Specifically, the firm chooses a new aggregate face value F t+1 , number of creditors N t+1 , and then sequentially offers new debt contracts maturing in period t + 1 to N t+1 new creditors, each with face value F t+1 N t+1 . 6 In exchange, the firm asks for a loan of D j t+1 from the jth creditor (j ≤ N t+1 ), who will also be the jth creditor to negotiate in period t + 1 when the debt matures. This arrangement can be interpreted as staggered maturity structure within a period. Define the total value of debt to be D t+1 ≡ N t+1 j=1 D j t+1 . For simplicity, I assume the firm refinances the exact amount of repayment: The upfront investment is financed by debt issuance at period 0, that is D 1 = I 0 . In Section 6, I show that in the current setup, the firm never borrows more money than necessary to refinance the maturing debt and keeps the excess cash inside the firm. Put differently, condition (4) can be established as a result rather than an assumption. Each of the N t+1 new creditors can accept or decline the debt issuance. If all these creditors accept, debt rollover is successful in period t. The N t incumbent creditors are paid according to the negotiated schedule: {S i t |i ≤ N t }, and period t + 1 begins with new state variables (z t+1 , Y t+1 , N t+1 , F t+1 ). On the other hand, if any creditor refuses the debt issuance, the firm must sell the project for λY t+1 and the game ends. In this case, the proceeds λY t+1 are shared equally among incumbent creditors: Nt min(F t , λY t+1 ), and the firm gets max(λY t+1 − F t , 0). It is worth pointing out that this scenario does not occur in equilibrium. If incumbent creditors do not expect the firm can refinance the (renegotiated) repayments, they would have rejected the renegotiation offers in the first place.

Remarks on the Modeling Assumptions
The payoff structure in the renegotiation game, warrants further discussion. Similar to the sequential-service condition in Diamond and Dybvig (1983), the payoff function (3) gives priority to creditors whose debt matures sooner. By declining their renegotiation offers, these creditors can secure partial repayment at the cost of other creditors, thereby creating a coordination problem that is at the heart of the model. The main predictions of the model remain valid as long as there is a coordination problem among creditors (even for some other reasons unrelated to the split of repayment) that is more severe with a larger pool of creditors. Such coordination problems are widely documented empirically and can emerge theoretically as an outcome of an efficient mechanism design. 7 One feature that could alleviate the coordination problem in practice is the legal arrangement known as "avoidable preference," which allows creditors to claw back unusual payments made to other "preferred" creditors within ninety days prior to default. However, the legal actions required to enforce this rule are expensive. In addition, creditors are arguably not deterred from accepting "preference," simply because scrambling to demand repayment is still a dominant strategy. As Countryman (1985)  Perhaps most seriously, the avoidable preference rule can in fact create an incentive for creditors to run. Knowing that payments seized ninety days prior to bankruptcy can be safely retained, creditors in a distressed firm have an incentive to demand payments (in the model, refuse to accepting renegotiation offers) as early as possible to stay out of the ninety-day window. Indeed, this concern is a major reason why law experts question the effectiveness of such a policy. For example, McCoid (1981) argues that "a creditor frequently will . . . accept the preference and hope for a success [to retain the payment]. . . Moreover, . . . there is a companion incentive to act as early as possible and increase the odds of effecting the transfer out of the ninety-day period." In summary, the simple payoff structure (3) captures the essence of coordination problems that arise with multiple creditors. 7 Using a natural experiment, Hertzberg, Liberti, and Paravisini (2011) show that creditors reduce lending when they anticipate that other incumbent creditors will learn negative information about the firm. Gilson John, and Lang (1990) and Brunner and Krahnen (2008) show that creditor dispersion adversely affects the probability of a successful workout for distressed firms. From a theoretical perspective, Calomiris and Kahn (1991), Berglöf and von Thadden (1994), Bolton and Scharfstein (1996), Diamond (2004), and Zetlin-Jones (2014) all show that coordination failure can naturally stem from optimal financing contracts.

Strategies and Equilibrium Definition
A pure strategy profile includes the following items: the firm's renegotiation strategy (S i t ) at date t against the ith incumbent creditor; the firm's refinancing strategies (D j t+1 , N t+1 , F t+1 ) at date t from the jth new creditor maturing at date t + 1; each incumbent creditor's acceptance strategy (s t,i | i≤Nt ∈ {a(ccept), d(ecline)}) in response to the negotiation offer S i t ; and finally each new creditor's acceptance strategy (r t,j | j≤N t+1 ∈ {a, d}) in response to the refinancing offer D j t+1 , N t+1 , F t+1 . There are three components in the expected payoff to the creditors maturing in period t: a repayment of 1 Nt min (F t , Y t ) when the project matures with probability π, a renegotiated repayment S i t if rollover is successful, and a termination repayment X i t , given by (3), should rollover fail. Formally, given any pure strategy profile S i t , D j t+1 , N t+1 , F t+1 , s t,i , r t,j , the value of debt issued to the ith creditor valued at the end of date t − 1 is: where 1 RO ≡ Π i≤Nt 1 s t,i =a Π j≤N t+1 1 r t,j =a is the indicator function of a successful rollover, meaning that all incumbent creditors accept debt renegotiation and all new creditors accept debt issuance. The expectation is taken over all possible realizations of z t+1 conditional on state z t at the beginning of date t (or equivalently at the end of t − 1). The total firm value at the end of period t − 1 is the present value of either the final dividend Y t at the project maturity τ π or the resale value λY t when debt rollover fails at τ L : Throughout the paper, I focus on Markov perfect equilibria In equilibrium, the creditors' acceptance/rejection strategies (s * t,i , r * t,j ) must be rational, given other players' strategies and the debt pricing rule (5). Whenever the creditors are indifferent, I assume they accept the offers. The firm's renegotiation/refinancing strategies (S i * t , D j * t+1 , N * t+1 , F * t+1 ) maximize the expected equity value E t (V t+1 − D t+1 ) at the time the respective decisions are made. When the firm is indifferent between different refi-nancing strategies, I assume it prefers fewer creditors (N t ) and then a lower face value (F t ) of debt. This assumption can be microfounded by adding an arbitrarily small cost to maintain each creditor relationship and to renegotiate down the maturing debt. Finally, when solving the model, I relax the integer restriction on the number of creditors N t ≥ 1.

Model Solution
In this section, I construct a Markov equilibrium (5) and (6)) depend only on state variables (z t , Y t , N t , F t ). I start by defining the key quantity -debt capacity -in Subsection 3.1. Next, in Subsection 3.2, I describe firm's renegotiation strategies (S i * t ) and creditors' acceptance strategies (s * t,i , r * t,j ) in equilibrium, which allow me to characterize the debt capacity in Subsection 3.3. Here, I also discuss the tradeoff associated with more creditors: the benefit of stronger commitment power against the cost of higher rollover risk. Finally, in Subsection 3.4, I describe the firm's refinancing strategies (D j * t+1 , N * t+1 , F * t+1 ), which also govern the evolution of creditor dispersion. I show that the firm chooses the number of creditors to simultaneously maintain sufficient commitment power and reduce rollover risks.

Introducing Debt Capacity
Aggregating the value of all debt contracts in (5) issued at the end of period t − 1, I have the total value of debt D t = D zt Nt (F t , Y t ): The interpretation of the above expression is similar to that of (5). The term min (F t , Y t ) captures the standard debt payoff when the project matures with probability π. Alternatively, when the project does not mature, if debt rollover is successful (1 RO = 1), then the total renegotiated payment Nt i=1 S i t is honored, otherwise the total repayment upon rollover failure is Nt i=1 X i t = min (F t , λY t+1 ) from (3).
I can now naturally define debt capacity DC t+1 = DC z t+1 (Y t+1 ) at the refinancing stage in period t depending on the realized shock z t+1 (or equivalently, the fundamental Y t+1 ): Intuitively, the debt capacity is the maximum level of debt that can be raised in period t by appropriately choosing the face value F t+1 and the number of creditors N t+1 .

Basic Equilibrium Strategies
In this subsection, I characterize some of the basic equilibrium strategies. The jth new creditor accepts the refinancing offer (D j t+1 , N t+1 , F t+1 ) (i.e., r * t,j = a) if and only if he can at least break even on the newly issued debt. On the other hand, the firm would like to issue new debt at the highest possible price. Therefore, in equilibrium, the prices of new debt contracts must be given by (5): The ith incumbent creditor accepts the renegotiation offer S i t (i.e., s * t,i = a) if and only if the offered repayment S i t is weakly higher than the payoff as the first rejecting creditor min( Ft Nt , λY t+1 ) from equation (3). Consequently, the firm's take-it-or-leave-it renegotiation offer must be exactly the creditor's reservation payoff from rejecting the offer: In addition, for the incumbent creditors to accept the offer, the total renegotiated repayment must be credible, meaning that it can be refinanced into period t + 1: where DC t+1 is the debt capacity given by (8).

Characterizing Debt Capacity
Using the equilibrium strategy (10) and rollover condition (11), I can rewrite debt capacity (7) and (8) as the invariant function to the following problem: The trade-off associated with more creditors N t becomes apparent in (12). On the one hand, having more creditors improves the firm's commitment power. Recall from (10), when renegotiation results in a reduction of the original face value (i.e., when Ft Nt > λY t+1 ), the firm must pay back the resale value of λY t+1 to each one of the creditors. Therefore, having more creditors commits the firm to repay a higher multiple of the resale value in the event of a renegotiation: a total of N t λY t+1 to all creditors. Of course, instead of a renegotiation, the firm can just make the full repayment (i.e., when Ft Nt ≤ λY t+1 ), which is also more likely to occur with more creditors (N t ). In all, the dispersed creditor structure strengthens the firm's commitment power, and the total (renegotiated) repayment conditional on a successful debt rollover -min(F t , N t λY t+1 ) -weakly increases. On the other hand, this higher renegotiated repayment is less likely to be refinanced, because the rollover condition (11) fails more often. The higher chance of inefficient termination reduces the ex-ante value of debt. Now, I introduce a technical assumption purely for expositional purposes: Effectively, I assume the resale value λY t cannot be refinanced with a single creditor in the bad state. 8 The essence of assumption (13) is to rule out having a single creditor N t = 1 8 The right-hand side of (13) is the borrowing capacity from a single creditor (N t = 1) in a bad state (z t = 1), which is attained by having a sufficiently large face value. Consider the expression for debt capacity (12). If the project matures, then min (F t , Y t ) = Y t is paid to the creditor. If the project does not mature, since N t = 1, the repayment is min (F t , λY t+1 ) = λY t+1 regardless of whether debt rollover is successful. Therefore, as an absorbing state. Should (13) fail, the maximum repayment offered to a single creditor λY t can be refinanced with a single creditor again in the next period. This property creates a region of creditor dispersion (when N ≥ 1 is sufficiently low) such that the firm value is always the first best, and the firm's choice of creditor dispersion becomes irrelevant. The following proposition analytically characterizes the debt capacity that is linear in Y t .
The debt capacity in the bad state DC B (Y t ) ≡ κ B Y t is given by: Proposition 1 is intuitive. In the good state, fundamental Y t deterministically grows at a rate of µ every period. The entire firm value can be pledged to creditors by choosing a sufficiently dispersed creditor structure: If the project matures, the creditors receive all the final dividend Y t , otherwise they receive the firm's entire continuation value v G F B (1 + µ) Y t . In the bad state, the firm can potentially borrow the maximum amount in two ways, based on the aforementioned trade-off between the benefit of commitment power and the cost of rollover risk. First, the firm can choose a large pool of creditors N t = κ G λ and promise a high repayment F t = max Y t , κ G (1 + µ) Y t to pledge the entire firm value from the good state (i.e., when z t+1 = 1 + µ). However, this high commitment power is a double-edged sword. If the state remains bad (i.e., when z t+1 = 1), the firm is unable to write off a sufficient amount of the maturing debt and thereby suffers rollover failure. Mathematically, the necessary repayment after a successful renegotiation (min Consequently, debt rollover fails in this case, leaving creditors a total payoff of λY t . Taken together, expression (15) gives the ex-ante value of debt.
Alternatively, the firm could borrow from a smaller pool of creditors N t = κ B λ and set a lower face value F t = max Y t , κ B (1 + µ) Y t . With fewer creditors, debt renegotiation the maximum pledgeable income with a single creditor is 1 becomes easier: In the bad state, the total renegotiated repayment of λN Y t = κ B Y t can still be refinanced. The firm can therefore completely avoid the inefficient rollover failure. However, the lower commitment power due to easier renegotiation means that the high value of the project in the good state can never be fully pledged to the creditors. Mathematically, with N t = κ B λ creditors, the maximum repayment in the good state is κ B (1 + µ) Y t , which is strictly less than the debt capacity κ G (1 + µ) Y t . In this case, the ex-ante value of debt is: Comparing (16) with (15), one can clearly see that by choosing a more concentrated creditor structure, the firm sacrifices the commitment power in the good state (i.e., The proof of Proposition 1 in the Appendix shows that κ B Y t given by (15) always dominates (16). I end this subsection with a briefly discussion on the possibility of multiple equilibria. As is common in many dynamic games, the equilibrium is not unique in general. In this model, multiplicity can emerge from two potential sources. First, there might be equilibria with time-dependent strategies. I rule out these strategies by focusing on Markov perfect equilibria. Second, debt capacities DC z (Y ) have a self-fulfilling feature. Specifically, the fixed-point problem in (12) that determines debt capacities may have multiple solutions. A low debt capacity tomorrow reduces the odds of a successful debt rollover, resulting in a low debt capacity today, and vice versa. Despite this theoretical possibility, the stochastic process specified in Section 2 indeed guarantees a unique solution characterized by Proposition 1.

The Dynamic of Creditor Dispersion
I now characterize the last equilibrium object: the firm's refinancing strategy (N * t+1 , F * t+1 ), which also governs the evolution of creditor dispersion. Because the total (renegotiated) repayment min (F t , N t λY t+1 ) is independent of the firm's choice of refinancing strategy, the firm's objective to maximize equity value is equivalent to total firm value maximization. The firm value (6) can be recursively written as: The next proposition captures the essence of the firm's refinancing strategy. The analytical expressions for (N * t+1 , F * t+1 ) are relegated to the Appendix.

Proposition 2
The firm always chooses the lowest level of creditor dispersion N * t+1 , such that the firm can refinance the total (renegotiated) repayment min (F t , N t λY t+1 ).
Intuitively, the number of creditors is the mechanism through which the firm allocates actual repayments across states. On the one hand, firms can choose a more dispersed creditor structure, and therefore cannot easily renegotiate repayments based on the fundamental realization. This feature results in more equalized repayments across states. For instance, when λN t+1 Y t+1 ≥ F t+1 , it is impossible for the firm to renegotiate any debt regardless of the realized state z t+2 , and the required repayment is a constant, min (F t+1 , λN t+1 Y t+2 ) = F t+1 . On the other hand, when the firm chooses a more concentrated creditor structure, renegotiation becomes easier, and the actual repayment depends more heavily on the realized state, in particular, more repayments in the good state. For example, if λN t+1 (1 + µ) Y t+1 ≤ F t+1 , then the firm can renegotiate the actual repayment to λN t+1 Y t+2 in both states, which is higher if the state becomes good, z t+2 = 1 + µ.
In equilibrium, the firm finds it optimal to allocate as much repayment as possible to the good state, so that the entire first-best firm value can be pledged to the creditors without rollover risk (recall Proposition 1). The resultant lower repayment in the bad state improves the firm value since refinancing a lower repayment is easier. Hence, the firm maximizes the repayment inequality across states by borrowing from as few creditors as possible. As a corollary of the firm's refinancing strategy in Proposition 2, I next characterize the evolution of the number of creditors over time.
Corollary 1 The number of creditors decreases in a good state and increases in a bad state.
To understand this result, it is useful to focus on the firm's leverage: Note from (12) that the value of debt D t is homogeneous of degree one in (F t , Y t ). Hence, the maximum amount of debt that can be refinanced from N creditors is linear in Y t : for some constant κ z N . From Proposition 2, we know that the firm always uses the lowest number of creditors to refinance the repayment. Therefore, the number of creditors N * t+1 in equilibrium must be just enough to support the required leverage. Mathematically, N * t+1 is the smallest N , such that: κ Consequently, the evolution of creditor dispersion N t is driven by the leverage process d t in equilibrium. In the good state, fundamental Y t grows by 1 + µ every period. The value of debt in (12) can be rewritten recursively as: The proof in the Appendix shows that D G t grows by less than 1 + µ per period. For example, when debt is risk free (D G t < 1 1+r Y t ), then D G t+1 = F t = (1 + r)D G t grows only by the discount rate 1 + r < 1 + µ. 9 Thus, the fundamental grows more rapidly than the value of debt, and both leverage d t and the number of creditors N t decrease in the good state.
In the bad state, the relevant case is that the firm survives even when the bad state persists in the next period. Conditional on survival, the value of debt can be recursively written as state switches to good Over time, the value of debt must increase, i.e., D B t+1 > D B t , to compensate for both the time discount 1 1+r and a possible default upon project maturity if F t > Y t . Since the fundamental Y t does not change in the bad state, both leverage d t and creditor dispersion N t increase. Figure 2 shows the main features of Proposition 2 and Corollary 1. I plot the evolution of creditor dispersion for a firm with the following parameters: the state switching probability p = 0.3, the maturing intensity π = 0.1, the resale value λ = 0.6, the discount rate r = 0.3, the growth rate µ = 0.39, and the initial leverage D 1 Y 1 = 0.95 * λ 1+r . Different curves in the figure represent different sample paths of the number of creditors N t conditional on whether the state turns good at periods 2, 3, 4, 5, and 6. I choose this initial leverage such that the first debt contract maturing in period 1 is risk free and the sample paths are most typical. The firm in this example starts in a bad state (z t = 1). Since there is no growth in the bad state, the face value of debt F t and the leverage Dt Yt increase gradually. In order to support this higher leverage, the firm must increase creditor dispersion N t and commit a bigger fraction of the total value to its lenders. This effect is captured by the steadily increasing solid dark line in Figure 2. If the bad state persists for sufficiently long (6 periods in the example), then the firm exhausts its debt capacity and faces termination in period 6. Otherwise, when the state turns good (z t = 1 + µ) at some point prior to period 6, the fundamental Y t grows and leverage decreases. Consequently, the firm consolidates creditors gradually, captured by the dashed curves in the figure.
It is worth noting that the firm's ability to pledge the entire firm value in the good state without any rollover risk relies on the specification that the good state (z t = 1 + µ) is absorbing. This specification greatly simplifies the model's exposition. In general, if a good state can also switch to a bad one, then scheduling repayments in the good state may incur the default risk: Project maturing leads to a repayment Y t strictly less than the promised face value F t . The proof in the Appendix shows that even in this case, D t+1 < (1 + µ) D t , and leverage still decreases. subsequent rollover risks as well. Therefore, the total firm value in the good state is no longer first best and becomes sensitive to the amount of repayment in that state. Hence, there is a tradeoff between expected rollover risk in the good and bad states, and the firm may choose to limit the repayment in the good state by having more creditors. Although an analytical solution is not generally available, numerical analysis suggests that despite the alternative theoretical possibility, the intuitions in Proposition 2 and Corollary 1 remain robust.

Creditor Dynamics and Firm Characteristics
In the time series, Proposition 2 and Corollary 1 show that creditor dispersion is negatively driven by a firm's fundamental shocks z t : A firm increases creditor dispersion following bad shocks. In this section, I compare different firms with exogenously different qualities. In Subsection 4.1, I analyze the comparative dynamics of creditor dispersion with respect to the firm's growth rate µ. With a slight abuse of language, I call it the "cross section" of creditor dispersion. In contrast to the time-series findings, a firm's average creditor dispersion over its lifetime can be positively correlated with the firm's quality: A high-growth firm can have more creditors on average. The sharp contrast between the time-series and cross-sectional analysis highlights the importance of using a dynamic model to analyze firms' creditor structure. These predictions also help reconcile the seemingly contradictory empirical findings and link them to classic theories on creditor dispersion, as discussed in Subsection 4.2.

Comparative Dynamics with Respect to Growth Rate
Consider firms with exogenously different growth rates µ. Firms with higher growth rates are ex-ante better, and therefore have higher debt capacity. In addition, these firms can pledge a higher first-best firm value in the good state, so the required leverage and the number of creditors increase (decrease) more slowly (rapidly) in a bad (good) state. The following result as a corollary to Propositions 1 and 2 summarizes these properties.
1. The firm with a higher growth rate µ 2 has a higher borrowing capacity in both states: 2. Suppose both firms need to support the same leverage D t+1 Y t+1 at the refinancing stage in period t, then the higher-growth firm chooses fewer creditors: To understand statement 2 in Corollary 2, note that the same debt contract issued by a higher-growth firm at period t is more valuable. This is because the (renegotiated) repayment in the good state (i.e., min(F t+1 , N t+1 λ (1 + µ) Y t+1 ) from condition (10)) is higher when the debt matures at t + 1. Consequently, the firm with a higher growth rate µ can choose a smaller number of creditors and still achieve the same leverage.
The above intuition suggests that better firms with higher asset growth increase creditor dispersion more slowly and therefore should have fewer creditors on average, similar to the time-series predictions in Corollary 1. (Recall: Firms reduce creditor dispersion following good shocks.) However, this is only half of the story. With the dynamic aspect of creditor dispersion, the prediction can be reversed. Statement 1 in Corollary 2 suggests that better firms have higher borrowing capacity. Hence, they can support a larger amount of creditors without rollover failure. Specifically, type-µ 2 firm can survive with creditors in the bad state, while the low growth µ 1 firm would have suffered a rollover failure. If one calculates the time-series average number of creditors (defined in (21) below) for both types of firms, then high-growth firms could have more creditors, even though it may take longer for these firms to expand the borrowing pool to any given size. 10 I conclude this subsection with a numerical example to demonstrate the two opposing channels discussed above. The parameters in the example are (p, π, λ, r) = (0.3, 0.1, 0.6, 0.3), and I vary the growth rate µ between 0.32 and 0.44. As µ changes, the first-best value of the project changes too. To ensure that all firms start equally indebted and that their creditor structure is equally dispersed, I choose the initial leverage as D 1 , which is the highest leverage with a single creditor, N * 1 = 1. 11 In this example, I focus on the bad state in which the number of creditors gradually increases. In Figure 3, I plot the dynamics 10 In Appendix B, I present a static version of the model, which also predicts the number of creditors in a high-growth firm increases less rapidly (statement 2 in Corollary 2). However, this static model cannot capture the dynamic intuition: a high-growth firm can survive longer due to its larger debt capacity (statement 1 in Corollary 2). 11 Note that the selected d 1 is the debt capacity with a single creditor on the right-hand side of (13). I do not choose a lower initial leverage because even though the firm also has a single creditor in this case, the choice of leverage can be arbitrary and therefore lacks discipline. I can instead choose a higher d 1 such that N * 1 > 1 is the same (there is indeed a one-to-one mapping between d 1 and N * 1 in this case). The qualitative result is exactly the same. Therefore, the maximum leverage with N * 1 = 1 is a natural starting point.
of creditor dispersion N * t for firms with µ = 0.32, 0.36, 0.39, and 0.41. The trajectories for better firms are further to the right, longer (as predicted by statement 1 in Corollary 2), and flatter on the common domain (as predicted by statement 2 in Corollary 2). The number of periods the firm can survive in a bad state ranges from T = 2 (for µ = 0.32) to T = 16 (for µ = 0.44), as plotted in the left panel of Figure 4.
Next, for a given µ, I calculate the following time-series average number of creditors: where τ L is the time of rollover failure if the state remains bad for sufficiently long. The weights p(1 − p) t−1 (1 − π) t are the unconditional probabilities that the state turns good exactly at period t and the project does not mature at or before period t. The rationale is that if firms only enter the sample in the period when the good shock realizes, then E (N * t ) defined in (21) is the observed average number of creditors for firms with growth rate µ. I repeat this calculation for µ = 0.32, 0.33, ... , and 0.44, and plot the results in the right panel of Figure 4. The average number of creditors broadly increases as the growth rate µ increases. The driving effect is that better firms have bigger debt capacity κ B , which enables them to support more creditors and survive longer, as suggested by statement 1 in Corollary 2. However, whenever µ increases without affecting the longevity of the firm, one can easily prove that the average dispersion decreases, reflecting statement 2 in Corollary 2: N t increases more slowly in better firms. The left panel of Figure 4 shows that when µ ∈ {0.32, 0.33, 0.34}, µ ∈ {0.35, 0.36}, or µ ∈ {0.37, 0.38}, the firm can survive 2, 3, and 4 periods, respectively. Within each set, the average creditor dispersion E (N * t ) decreases with µ in the right panel of Figure 4.

Empirical Relevance
The model's predictions relate to the empirical literature in three aspects. First, the model predicts that firms often renegotiate their debt before declaring bankruptcy, which is supported by the empirical evidence in Roberts and Sufi (2009a,b) and Roberts (2015). These authors show that in the event of a technical default, instead of an immediate termination of the lending relationship, 62.6% of borrowers receive waivers from creditors, 32.2% of loan contracts are renegotiated, and only 4.4% of loan contracts are terminated. In aggregate, 94% of the loans with maturity longer than three years are renegotiated, and a median renegotiated loan receives four renegotiations in its lifetime. In addition, related to the theoretical prediction that renegotiation lowers the debt that the firm carries forward, Roberts and Sufi (2009b) show that borrowers reduce net debt issuance following renegotiations.
Second, the sharp contrast between the time-series (see Subsection 3.4) and the crosssectional (see Subsection 4.1) predictions reconciles the seemingly contradictory empirical findings. On the one hand, the time-series prediction of creditor dispersion is consistent with empirical evidence documented by Farinha and Santos (2002), who show that firms are more likely to abandon a single creditor structure when historical performance declines. 12 More recently, Rauh and Sufi (2010) show that as credit quality deteriorates, firms introduce more heterogenous debt capital. Specifically, fallen angels (firms being downgraded from investment grade to speculative grade) move from having only senior unsecured debt to secured bank debt and subordinated bonds, which are more difficult to renegotiate. On the other hand, echoing the cross-sectional prediction that firms with higher growth can support more creditors, Houston and James (1996) find that firms with multiple relationship banks have significantly higher asset growth compared with their single-bank counterparts.
Third, the dynamic aspect of my model also helps to connect some stylized facts on the cost of borrowing with the theoretical idea that creditor dispersion induces repayment incentives. Indeed, many classic papers (e.g., Berglöf and von Thadden 1994, Bolton and Scharfstein 1996, and Diamond 2004) build on the idea that dispersed creditor structure commits the borrowing firm to pay back its debt. This channel implies that the required interest rate should be lower when the debt is more dispersedly held, because the difficulty renegotiating with dispersed creditors makes the repayments more credible. On the contrary, Rajan (1994,1995) empirically document a significant association between more creditors and a higher cost of credit. My model suggests that although borrowing from more creditors should lower the required interest rate, firms only do so when their performance deteriorates, driving up the observed interest rate in equilibrium.
The regularity conditions rule out a tedious special case, which is discussed in the Appendix, together with the proof of this prediction. In equilibrium, leverage d t+1 determines the number of creditors N * t+1 as in (20): More creditors are needed to support a higher leverage. On the other hand, leverage increases as the fundamental stays bad, driving up the cost of borrowing. Therefore, the optimal creditor dispersion and the equilibrium interest rate are positively correlated.

Renegotiation Frequency
It is well accepted that having dispersed debt ownership is a mechanism to alleviate firms' commitment problems and thereby increase their debt capacity (e.g., Berglöf and von Thadden 1994, Bolton and Scharfstein 1996, Hart and Moore 1998, Diamond 2004). However, it is unclear how effective this mechanism is when commitment problems become severe in a dynamic world. In particular, if firms can frequently renegotiate their debt, can dispersed creditors still create extra pledgeability by allowing firms to borrow against better outcomes in the future? My dynamic model is a natural venue to investigate this question. I find, perhaps surprisingly, that when renegotiation is frequent, having a dispersed creditor structure can no longer increase firms' pledgeability.
I carry out the analysis in two steps. First, I hold the renegotiation frequency fixed and vary the number of creditors N t exogenously in Subsection 5.1. Next, in Subsection 5.2, I let the renegotiation frequency shrink to zero, holding the number of creditors N t fixed. This approach controls for the degree of coordination problems and isolates the effect of renegotiation frequency. In the limit, the debt capacities in the bad state converge to the resale value λY uniformly for any creditor dispersion N .

Debt Capacity with Exogenous Creditor Dispersion
I start by modifying the baseline model in Section 2 to allow for more frequent renegotiation. Specifically, I assume that each period lasts ∆t < 1, and the firm needs to roll over one-period debt with ∆t maturity until the project matures. The firm can renegotiate the repayment each time the debt matures. 13 In each period, the project matures with probability π∆t; a bad state switches to a good one with probability p∆t; the discount rate is r∆t; and finally the growth rate of the fundamental process Y t is µ∆t. All other model ingredients are the same. When ∆t = 1, this specification is the same as in the baseline model. 14 Here, I am interested in the limiting case when ∆t → 0 as it proxies for very frequent renegotiation.
Similar to (1), the first-best firm value in the good state after the ∆t-modification is: When ∆t → 0, this first-best value approaches its continuous-time limit π r−µ+π . The main objective of this subsection is to characterize the debt capacity with any exogenously fixed number of creditors, that is N t ≡ N for all t and some constant N . The key difference relative to the baseline model is that the firm cannot adjust the number of creditors over time. Therefore, the renegotiated repayment min (F t , N λY t+1 ) must be refinanced by the same number of creditors N . Hence, similar to but different from (11) in the baseline model, debt rollover is successful whenever: where the debt capacityDC z t+1 N (Y t+1 ) with exogenously fixed N is given by: The linear debt capacityDC zt N (Y t ) =κ zt N Y t is characterized in the following proposition. 13 Alternatively, the firm can have multiple renegotiations before maturity. Results are the same qualitatively, with added mathematical complexity because renegotiation dates and maturity dates are different.
14 One can also use the continuous-time analogue of the variables. For example, maturing intensity 1 − e −π∆t , discount factor e −r∆t , state switching intensity 1 − e −p∆t , and growth rate e µ∆t − 1. When ∆t is small, these numbers converge to the linearization I used in the main text. Proposition 3 Suppose N t ≡ N for all t and κ G ≥ 1, then debt capacities are given by: The debt capacity in the good state is straightforward. When creditor dispersion is fixed at a high level (N > κ G λ ), the entire first-best firm value is pledgeable, as in Proposition 1 in the baseline model. When creditor dispersion is small (N ≤ κ G λ ), the firm's repayment incentive is bounded by N λ (1 + µ∆t) Y t . In the Appendix, I also verify that this repayment can indeed be refinanced by having N creditors again, thereby establishing (25).
The bad state is more interesting. When the firm cannot adjust creditor dispersion N , the debt capacity with any N is always achieved by having rollover failure if the bad state persists. This result is counterintuitive. Recall from the baseline model that if N is small enough (N ≤ κ B λ ), debt rollover is always successful: The renegotiated repayment of λN Y t ≤ κ B Y t can always be refinanced. Why does it fail to hold when the number of creditors is exogenously fixed? This is because in order to raise the renegotiated payment, the firm must increase creditor dispersion, making future repayments more credible (recall Corollary (1)). When this option is infeasible, the firm has no choice but to suffer rollover failure in the bad state. Put differently, the payment as a result of a renegotiation with N creditors cannot be refinanced by N creditors again, i.e., λN Y t >κ B N Y t . Essentially, there is no meaningful renegotiation in the bad state when the number of creditors is exogenously fixed, a consequence of the dynamic nature of the model.
It is worth pointing out that many bankruptcy procedures designed to coordinate creditors may backfire. 15 While promoting the efficient outcome in the bankruptcy state, these procedures also reduce firms' commitment power ex-ante. In the model, imposing an efficient ex-post renegotiation is equivalent to having a single creditor exogenously. The calculation of debt capacity in (26) shows that no meaningful renegotiation can occur in the bad state even with a single creditor, and firms are terminated sooner. This is an unintended side effect when ex-post coordination failure is eliminated.

Vanishing Debt Capacity with Frequent Renegotiation
When a firm only borrows from a single creditor, N t = 1, the maximum repayment if the project does not mature in period t is λY t+∆t , which approaches λY t when the period length is very short ∆t → 0. Because in the limit, both the discount rate and the probability that the project matures are negligible, thus the debt capacity with a single creditor approaches λY t (see equation (26) with N = 1 and ∆t → 0). Proposition 4, which is also the main result of this section, shows that when debt renegotiations are frequent, having dispersed creditors becomes irrelevant.

Proposition 4
The debt capacities in the bad state converge to the resale value uniformly as renegotiation becomes very frequent. Mathematically, for any exogenously fixed N , To understand Proposition 4, note that the probability of switching to a good state (p∆t) converges to 0. Consequently, from (26), if the firm attempts to pledge any extra value from the good state in the next period, it risks termination should the bad state continue, a risk that almost certainly occurs. Since the upside only materializes with a negligible probability, the debt capacity in equation (26) converges to λ. Intuitively, the first-best firm value in the bad state contains the expected value of all future value improvements at every period when the state can potentially turn good. The benefit of having dispersed creditors is to allow the firm to borrow against these improvements. However, out of all future dates, the firm can only borrow against the value improvement once, because debt rollover fails if the state remains bad. Therefore, as ∆t → 0, the expected value of a single value improvement shrinks to 0, so does the benefit of a dispersed creditor structure.
I would like to emphasize that the essence of Proposition 4 crucially relies on the dynamic nature of debt rollover. In the simplified static model in Appendix B, one can also let the probability of realizing the good state approach zero, thus the firm's debt capacity ex-ante declines similar to Proposition 4. However, this comparative static exercise is fundamentally different. Reducing the likelihood of the good outcome in a static model necessarily lowers the quality of the project. Therefore, the resultant lower debt capacity is mechanical because the project is not as valuable to begin with. In contrast, throughout the dynamic analysis in this section, the first-best firm value is held constant in the limit and does not shrink to the resale value. Another way to interpret Proposition 4 is that even in the limit when renegotiation occurs very frequently, the (expected) upside is still the same, but the firm cannot borrow against it by having multiple creditors.
Since troubled firms renegotiate debt at or before its maturity, the renegotiation cycle must be shorter than debt maturity in practice. Therefore, if debt maturity is shorter, renegotiations necessarily occur more often. The above analysis suggests that for a debt instrument with very short maturity, such as a repo agreement, incurring rollover risks and coordination failures cannot improve borrowing capacity ex-ante. This may be one reason why repo lenders often receive super-seniority in bankruptcy proceeds.
It is also interesting to relate my analysis to the classic work of Diamond and Rajan (2001), who show that coordination problems among short-term dispersed depositors (creditors) can improve the bank's (borrower's) pledgeability. My paper differs in two important aspects. First, instead of having a static model as in Diamond and Rajan (2001), the dynamic model in this paper separates two crucial ingredients in the Diamond-Rajan's mechanism: the coordination problem among creditors and the short maturity of debt. While the coordination problem creates a termination threat that improves pledgeability (Proposition 3), the short maturity, on the other hand, can actually lead to more frequent renegotiation, thereby exacerbating the commitment problem. Second, the stochastic fundamental is distinctively different from Diamond and Rajan (2001), who essentially has a deterministic fundamental process. In a dynamic model with a stochastic fundamental, frequent renegotiation makes it likely that a bad state will persist before the next renegotiation. This higher persistency of the bad state, which is absent in a static model, dramatically increases rollover risk, which in turn severely limits the commitment power brought by dispersed creditors.
In the model, I analyze exogenous renegotiation frequency, whereas in practice, a borrowing firm can renegotiate debt at any time before its maturity. While the full interaction between creditor dispersion and endogenous renegotiation timing is left for future research, I conjecture that firms have incentives to renegotiate instantaneously if they cannot commit to an exogenous debt renegotiation schedule. Because debt renegotiation is beneficial for the borrowing firm ex-post, without commitment, the firm may have incentives to abuse it. Therefore, the limiting case discussed in this section represents the outcome when firms cannot commit to the timing of renegotiations.

The Role of Cash
So far, I have assumed that the firm always refinances the exact amount of debt repayment and never carries cash across periods. In this section, I relax this restriction and study the role of cash when the firm borrows from multiple creditors. In Subsection 6.1, I extend the baseline model in Section 2 to allow for cash savings by the firm. As a benchmark, I show in Subsection 6.2 that the ability to save cash is irrelevant whenever the firm borrows from a single creditor. In Subsection 6.3, I demonstrate a novel insight: Cash combined with multiple creditors can increase state non-contingent commitment power. Consequently, firms only keep cash if debt capacity has a positive state non-contingent component.

Extended Model with Cash Savings inside the Firm
I begin with a modified version of the baseline model that accommodates cash. Denote by C t the firm's cash holding upon entering period t, which is the extra state variable. Assume the interest rate on cash is the same as the discount rate r. If the project matures in period t, then the total firm value, including both the project's payoff and cash, is Y t + C t . If sold prematurely, the total firm value is L t (Y t+1 ) + C t , where L t is the resale value of the assets. 16 As a result of the extra cash balance in a renegotiation, the firm must offer: 16 The assumption on the firm's resale value is the crucial difference from Hart and Moore (1998). They assume the liquidation value is unaffected by the firm's cash holding, whereas I assume the cash is part of the assets that can be seized by creditors. similar to the logic of (10). Therefore, the value of debt in (7) becomes: successful rollover At the refinancing stage in period t, the debt capacity net of the cash balance is: Different from definition (8) in the baseline model, here I subtract the cash balance C t+1 1+r that is carried over to period t + 1 from the proceeds of debt issued in period t. The net amount 1+r is what can be used to repay the maturing creditors in period t. Together with the existing cash C t , the total resources available for repayment in period t are DC t+1 + C t . Similar to (11), debt rollover is successful (1 RO = 1) if the total renegotiated repayments from (27) can be honored: At the refinancing stage in period t, the firm chooses cash C t+1 , number of creditors N t+1 , and face value F t+1 to maximize the equity value V t+1 , which is recursively defined as: where (C t+1 , N t+1 , F t+1 ) satisfy: The evolution of cash (32) states that the present value of the cash carried into the next period C t+1 1+r comes from the newly issued debt D t+1 and the cash balance C t , net of the payout to the maturing creditors, min (F t , N t (L t + C t )). The equilibrium variables of interest are the firms' refinancing decisions C * t+1 , N * t+1 , F * t+1 that govern the evolution of cash balance, creditor dispersion, and the face value.

Absence of Cash with a Single Creditor
I first show that cash is redundant whenever the firm borrows/refinances from a single creditor. The result, summarized in Proposition 5, is very robust in that it does not depend on a specific fundamental process Y t nor a functional form of the resale value L t (Y t+1 ).

Proposition 5
In any period t, if the firm chooses a single creditor N t ≡ 1, then equilibrium can be implemented with no cash: C t = 0.
The intuition for Proposition 5 is that both the firm and the creditor are indifferent about who holds the cash: The marginal loan granted in period t − 1 for the extra cash balance of Ct 1+r is risk free, for any C t . Since the cash balance grows to C t in period t, which is part of the liquidation value that can be seized by the creditor, the creditor is happy to provide the present value of this repayment Ct 1+r in period t − 1. Alternatively, the firm can choose not to borrow the extra cash and lowers the face value by C t , and the creditor's payoff is also not affected. Hence, the ability to borrow extra cash is inconsequential with a single creditor.
The following example illustrates the logic of Proposition 5. In addition, this example also shows that cash combined with multiple creditors indeed improves pledgeability.
Example 1: There are three periods: t = 0, 1, 2 and no time discount: r = 0. The project matures at t = 2 with certainty, and the final dividend Y 2 is deterministic. The firm issues one period debt in period 0 to finance the upfront investment I 0 and rolls over the maturing debt at period 1. The resale value of the project at the interim date is L 1 = 0. In this example, if the firm cannot borrow more than I 0 and keep some cash, the total pledgeable income is 0. No project with positive investment (I 0 > 0) can be financed. This is because L 1 = 0, and each creditor receives min F 1 N 1 , L 1 = 0 in a renegotiation. Any one-period debt issued in period 0 will be completely renegotiated away in period 1, despite the firm's ability to refinance Y 2 . In anticipation of this outcome, the creditors refuse to lend regardless of creditor dispersion N 1 and the face value F 1 .
When the firm borrows from a single creditor N 1 = 1, even if cash savings are allowed, the pledgeable income is still 0. To see this, suppose the firm borrows I + C 1 by promising some face value F 1 . When the debt matures in period 1, the repayment is renegotiated to min(F 1 , L 1 + C 1 ) ≤ C 1 . Therefore, a single creditor can never break even.
Once the firm can borrow extra cash from multiple creditors, N 1 > 1, any positive NPV project with I 0 ≤ Y 2 can be financed. Specifically, in period 0, the firm can raise I 0 + C 1 risk-free debt by promising a total repayment of F 1 = I 0 + C 1 to N 1 creditors, where: After making the investment I 0 , the firm keeps C 1 as cash. In period 1, the firm must offer: to each creditor. 17 Therefore, from (34), the firm has the required commitment power to payback each creditor I 0 +C 1 N 1 , and thereby a total of I 0 + C 1 to N 1 creditors. This repayment can be honored by paying back the cash balance C 1 and refinance the remaining I 0 ≤ Y 2 into the final period.
Example 1 shows that cash in the firm improves each creditor's reservation payoff from 0 to C 1 . As a result, if renegotiated in period 1, the total committed repayment to the N 1 creditors is N 1 C 1 , which is higher than the available cash C 1 inside the firm. This wedge (N 1 − 1) C 1 , which equals I 0 + Y 2 in Example 1, creates pledgeability because it commits the firm to borrow against the final dividend and credibly repay it to the initial creditors. This observation suggests that the role of cash in this debt rollover model is very different from that in Hart and Moore (1998), where cash is used to reduce partial liquidation in the states when liquidation is relatively more inefficient.

Cash with Multiple Creditors
In this subsection, I examine the role of cash when there are multiple creditors. From (27), the total payment after renegotiation is min (F t , N t (L t + C t )), and the committable repayment 17 Note that a simple rearrangement of (33) gives N t L t + N t C t contains two components. The first component N t L t (Y t+1 ) depends on the realization of Y t+1 , thereby increasing the firm's state-contingent commitment power. In contrast, the second component from the cash balance N t C t is independent of Y t+1 , thereby increasing the state non-contingent commitment power. Example 2 shows that firms choose to borrow extra money and carry a cash balance when the debt capacity has a positive state non-contingent component.
Example 2: There are three periods: t = 0, 1, 2. The project matures at t = 2. There are two equally likely states in period 1, with the corresponding resale value L z 1 and final dividend Y z 2 (z = G, B) as in the following table.
Case 1: X = 10. In this case, the debt capacity at period 1 (Y z 2 ) is proportional to the resale value, Y z 2 = 5L z 1 , and there is no state non-contingent component in Y z 2 . Hence, the firm can pledge the entire final dividend Y z 2 in period 0 by just using dispersed creditors without cash: By borrowing from N 1 = Y z 2 L z 1 = 5 creditors and promising a total face value of F 1 = 10, the firm can pledge the entire final dividend: mathematically, min (N 1 L z 1 , F 1 ) = N 1 L z 1 = Y z 2 . Therefore, no cash is needed. Case 2: X = 7. 18 In this case, I show that the firm can design a pair of (N 1 , C 1 ) to pledge the entire final dividend. To do so, the total renegotiated payment in period 1 must be exactly the final dividend plus the cash balance C 1 : Plug in the numbers for each state z = G, B and solve for (N 1 , C 1 ) = (2, 3). In order to borrow the entire final dividend = 6 at period 0, the firm borrows a total of 6 + C 1 = 9 from N 1 = 2 creditors and keeps C 1 = 3 as cash. The total face value promised to the creditors is Y G 2 + C 1 = 10. In the good state, there is no renegotiation because the reservation value of each creditor is L G 1 + C 1 = 5. To repay 10, the firm uses up the cash balance 3 and refinances the remaining 7, which coincides with the final dividend Y G 2 . In the bad state, the actual repayment to each creditor is renegotiated to L B 1 + C 1 = 4. The a total repayment of 8 is honored by using cash C 1 and final dividend Y B 2 . At period 0, the expected repayment is indeed 10+8 2 = 9, making the creditors break even. From a more general perspective, the debt capacity in this case can be decomposed into state contingent and non-contingent components: Compared with (35), one can see that the state-contingent component 2L z 1 is pledged by having N 1 = 2 creditors, and the state non-contingent component 3 is pledged by having cash savings inside the company.
Case 3: X > 10. In this case, and there is no role for cash: Any outcome can be implemented without keeping cash. Take X = 15 for instance. Similar to (36), the final dividend Y z 2 can be decomposed two components: In order to pledge the entire final dividend the cash savings must be negative. In fact, solving (35) gives (N 1 , C 1 ) = (10, − 5 9 ). Hence, any positive cash balance reduces pledgeability. Example 2 helps explain why firms save cash. If debt capacity contains a positive stateindependent component, then firms can borrow extra cash to increase commitment power uniformly across all states, and pledge this component to the creditors. On the other hand, if the debt capacity is more state-sensitive than the resale value, there may not be a positive state-independent component and the firm never borrows extra cash. For instance, in the baseline model, firms can refinance up to the debt capacity κ z Y t z, which is more statesensitive than the resale value λY t z: Hence, assuming away cash savings in the baseline model is with out loss of generality, a result formally established in the following proposition.
Proposition 6 Firms never borrow extra cash in the baseline model, even when do so is allowed, i.e., C * t = 0.
Despite being a nonexistence result, Proposition 6 points out a new possibility for future research to investigate firms' cash holding: adopting a more sophisticated fundamental process. For example, with a mean-reverting process, the project's expected profitability can be negatively correlated with the current state. Consequently, condition (37) could fail and cash savings can emerge in equilibrium. Many interesting questions are waiting to be answered. For example, when do firms keep cash and how does it vary with cash flow distribution? How does cash holding interact with creditor dispersion? How do firm characteristics affect the dynamics of cash accumulation?

Extension -Uneven Allocation of Face Value
So far, I have assumed that the firm must issue debt with identical face value to all creditors. However, the model's intuition does not depend crucially on this assumption. In this section, I show that when debt is more asymmetrically distributed among creditors, the outcome is closer to having fewer creditors.
To see the intuition, consider two creditors with different face values F 1 ≤ F 2 . To make renegotiation offers acceptable to both, the firm must repay a total of: min(F 1 , λY ) + min(F 2 , λY ).
Clearly, given a total face value of F = F 1 + F 2 , the maximum ex-post repayment is 2 min( F 2 , λY ), achieved by having identical face values, i.e., F 1 = F 2 = F 2 . This is also the case analyzed in the baseline model. The smaller the F 1 , the smaller the total repayment. In the extreme case, when F 1 = 0, the repayment reduces to the single creditor case min(F, λY ). Intuitively, renegotiating debt with a smaller creditor is more difficult, whereas forcing concession from a larger one is easier. In the limit, if a large creditor holds almost the entire outstanding debt, then the outcome approaches the single creditor case.

Conclusion
In this paper, I construct a dynamic model in which the firm must repeatedly roll over short-term debt contracts and can renegotiate repayment. Having more creditors results in the disadvantage of coordination problems, which, following bad shocks, make it harder for a firm to restructure its debt to avoid liquidation. With a good shock realization, however, these same coordination problems enhance repayment incentives by making it harder for a firm to opportunistically hold up its creditors. In the model, the firm actively chooses the number of creditors over time by optimally trading off commitment power with the liquidation probability.
The analysis shows that in the time series, firms increase their number of creditors after poor performance, whereas in the cross section, better firms with higher growth rates can have more creditors on average. If the firm can renegotiate debt very frequently, the extra pledgeability from multiple creditors vanishes. Holding a cash balance while borrowing from multiple creditors can improve a firm's ability to pledge state non-contingent borrowing capacity.
Finally, I offer several potential directions for future research. First, one could endogenize renegotiation timing (or similarly maturity profile) together with creditor structure. I offer some preliminary conjectures on this question at the end of Section 5, but a more comprehensive investigation can be informative. Second, how a firm's cash holding evolves remains an open question. The analysis in Section 6 points out that one must consider a more general fundamental process. The answer to this question could deliver new insights about how a firm's cash holding varies over time and how it is affected by the firm's creditor structure. Lastly, throughout the paper, I assume that the resale value of the project λY is sufficiently small, so that continuing the project is always efficient. With a different specification, abandoning the project could be optimal in certain states of the world. Creditor structure could affect firms' incentive to continue the project (e.g., risk shifting and debt overhang). How does creditor structure affect firms' investment decisions? I look forward to future research that can shed light on these topics.
[36] Roberts, Michael R. "The role of dynamic renegotiation and asymmetric information in financial contracting." Journal of Financial (A-1) We conjecture and verify a linear solution: (A-2) Solving the above system for (v G F B , v B F B ) gives (1).
Proof of Proposition 1: As discussed in the main text, the proof focuses on the bad state.
I first show that the debt capacity κ B Y t can only be achieved by having either N t = κ G λ or N t = κ B λ . Suppose N t > κ G λ , then the repayment cannot be renegotiated. Otherwise, if F t ≥ λN t Y t+1 , then the renegotiated repayment cannot be refinanced because Therefore, one can lower creditor dispersion to N t < N t such that N t > κ G λ still holds. Such an adjustment does not affect the equilibrium outcome. If the project is terminated under N t , it is still terminated under N t . On the other hand, if the firm can pay back F t , then F t ≤ κ θ t+1 Y t+1 < λN t Y t+1 . Hence, the firm still has incentive to make the repayment F t with N t creditors.
Suppose κ G λ > N t > κ B λ . In this case, the firm can always refinance in the good state. If debt capacity κ B is attained with termination in the bad state, then one can increase number of creditors to N t = κ G λ and set the face value F t = λN t (1 + µ) Y t . In this case, the project is still terminated in the bad state, but the payoff to creditors in the good state is strictly higher. Hence, the new structure N t , F t can generate a higher level of debt, contradicting the definition of κ B . If debt capacity κ B is attained without termination, then F t ≤ κ B Y t < λN t Y t . One can then lower N t to N t = Ft λYt < N t . This adjustment does not affect the actual repayment F t , but contradicts the minimality of N t .
Finally, if N t < κ B λ , then similar to the previous case, the firm can increase the creditor dispersion to N t = κ B λ and set the face value F t = λN t (1 + µ) Y t without creating termination in any state. As a result, the repayment is strictly higher under the new structure N t , F t in both states: contradicting the definition of κ B . Therefore, the two possibilities (15) and (16), associated with setting N t = κ G λ and N t = κ B λ , are the only possibilities to achieve κ B Y t . Next, I show that κ B Y t is attained by (15) instead of (16). Suppose otherwise, κ B Y t is attained by (16): Subtracting the above equation from (13) yields inequality (A-3) implies that κ B < λ, which is in turn smaller than κ G . However, these results then imply (16) is strictly dominated by (15). Contradiction! Hence, condition (15) indeed gives the debt capacity in the bad state. Completing the proof.
The more precise statement of Proposition 2: Assume λ (1 + µ) < 1. Let D t+1 = min (F t , N t λY t+1 ) be the amount of renegotiated repayment that must be refinanced as in (18). If the realized state is good: z t+1 = 1 + µ, then the new number of creditors N * t+1 and the total face value F * t+1 are given by (A-4) If the realized state is bad: z t+1 = 1, then Proof of Proposition 2: First, consider the good state. Since the firm is never prematurely terminated in this state, the objective is to minimize the number of creditors and then the face value of debt. The smallest number of creditors is 1. Therefore, when D t+1 ≤ λY t+1 (1+µ) 1+r , the firm can issue risk-free debt with a face value F t+1 = (1 + r) D t+1 . This contract clearly minimizes the face value.
When D t+1 ≤ π+(1−π)λ(1+µ) 1+r Y t+1 , it is still possible to borrow from a single creditor. Solving for the face value F t+1 from the following debt valuation condition gives the smallest face value in the proposition.
When D t+1 > π+(1−π)λ(1+µ) 1+r Y t+1 , then it must be N * t+1 > 1. To minimize N * t+1 , the face value F * t+1 must be at least Y t+1 to minimize the repayment that has to be refinanced. Therefore N * t+1 solves and the face value F * t+1 = max Y t+1 , N * t+1 λY t+1 (1 + µ) . Next, consider a firm that is currently in a bad state. Since the firm value in the good state is first best, it is optimal to repay as much as possible in the good state and as little as possible in the bad state next period. The scenarios when N * t+1 = 1 are very similar to the previous case when the state is good. The debt is risk free if D t+1 ≤ λY t+1 1+r . Without a renegotiation, it is clearly impossible to further reduce repayment to lower than (1 + r) D t+1 . When debt is not risk free with a single creditor, the face value must be higher than the reservation value of the single creditor in a bad state λY t+1 , which is also the actual repayment. The conditions to pin down the face value F * t+1 are depending on whether the repayment is renegotiated in the good state F t+1 > λY t+1 (1 + µ).
The firm can pledge at most 1 1+r {π + (1 − π) [p (1 + µ) + 1 − p] λ} Y t+1 to a single creditor, so when D t+1 exceeds this level, N * t+1 > 1. If N * t+1 ≤ κ B λ , then rollover is successful in both the good and bad states. In this case, minimizing the repayment in the bad state is equivalent to minimizing N * t+1 , which solves then the conclusion N * t+1 = 1 ≤ N t trivially holds. On the other hand, if (A-6) fails, then Together with the fact that D t+1 = min(F t , N t λY t+1 ) ≤ N t λY t+1 , we have Next, we study the bad state z t+1 = 1. Clearly, if N t = 1, the result again trivially holds. Now consider N t > 1. Because debt rollover is still possible in period t with z t+1 = 1, Proposition 2 suggests that at the end of period t − 1, it must fall into the fourth case in condition (A-5). Hence, we have and where D t+1 = min(F t , λN t Y t ) = λN t Y t is the actual repayment in the bad state in period t.
There are two possibilities, which I consider in turn. Case 1: If the project is not sold in period t + 1 following a bad state (z t+2 = 1), Because we are fourth case in condition (A-5), the lower bound of this case implies that Denote byd the solution tô Condition (13) then implies that λ >d. As a result, and (A-10) implies D t >dY t .
It then follows from (A-11) and (A-9) that , we have D t < D t+1 and therefore N * t+1 > N t . Case 2: If the project is sold in period t + 1 following a bad state, then this period follows the fifth case in (A-5) and . From the lower bound of the fifth case in (A-5) and condition (A-7), we have This completes the proof.
Proof of Corollary 2: From Proposition 1, the debt capacities in (14) and (15) are increasing in µ, establishing the first statement in the corollary. Next, I prove the second statement. There are two cases depending on whether the shared refinancing leverage D t+1 Y t+1 falls into the same category in equations (A-4) and (A-5). Case 1: For both firms, D t+1 Y t+1 falls into the same category in equations (A-4) and (A-5). It is easy to check that within each of these categories, N * t+1 is decreasing in µ. Case 2: D t+1 Y t+1 falls into different categories in equations (A-4) and (A-5). Because the cutoffs between cases are increasing in µ, it must be that D t+1 Y t+1 (µ 2 ) falls into an earlier category. Because N * t+1 is weakly increasing across the categories. Therefore, N * t+1 (µ 1 ) ≥ N * t+1 (µ 2 ).
Proof of Prediction 1: First, note that if N * t+1 ≥ 1 λ(1+µ) , then in a good state, (A-4) falls into the third case and F * t+1 = N * t+1 λ (1 + µ) Y t+1 ≥ Y t+1 . Similarly in a bad state, (A-5) falls into the fourth and fifth cases and F * t+1 = N * t+1 λ (1 + µ) Y t+1 ≥ Y t+1 again holds. Next, in each one of the aforementioned cases, the equilibrium N * t+1 is increasing in d t+1 = D t+1 Y t+1 , and the interest rate F * t+1 D t+1 is also increasing in d t+1 . Therefore, the creditor dispersion N * t+1 covaries positively with the interest rate Proof of Proposition 3: First, consider the good state z t = 1 + µ∆t at period t. If the project matures, then the total repayment is at most Y t . If the project does not mature, then with N creditors, the maximum repayment that can be refinanced is Hence, the ex-ante value of debt at time t is bounded by There are two cases, depending on the size of N . Case 1: N > κ G λ . In this case, setting the face value at F t = κ G Y t (1 + µ∆t) ∈ [Y t+∆t , N λY t+∆t ), the firm has incentive to pay F t if the project does not mature. The value of debt is which by (22) is κ G Y t . Since κ G is already the first-best firm value, the debt capacity is indeedκ G N = κ G . Case 2: N ≤ κ G λ . I first show that λN ≤κ G N , meaning that the renegotiated payment of N λY t+∆t can be refinanced into period t+∆t with N creditors. Suppose otherwise λN >κ G N . By setting the face value F t = max 1,κ G N (1 + µ∆t) Y t , the value of debt can achieve which, by the definition of debt capacity, is bounded byκ G N Y t : κ G N ≥ 1 1 + r∆t π∆t + (1 − π∆t)κ G N (1 + µ∆t) .
Next, consider the bad state z t = 1. I first show that in order to achieve the debt capacitŷ κ B N , debt rollover must fail should the bad state persist, i.e. z t+∆t = 1. Suppose otherwise, the debt capacity is attained by having successful rollover in both states: Clearly, for rollover to be successful, it must be Hence, the repayment in the good state must be bounded by Therefore, we havê Parameter condition (13) implies that: λ > π 1 + r − (1 − π) (µp + 1) > π 1 + r − (1 − π) = π r + π , which in turn implies that for all sufficiently small ∆t, the following condition holds: Taking the difference (A-14)−(A-15), and using the fact that 1+r∆t > (1 − π∆t) (1 + µ∆t), we haveκ B N < λ. Since rollover must be successful per assumption, condition min(N λY t , F t ) ≤ κ B N Y t < λY t holds in the bad state. However, the ex-ante value of debt can be strictly improved by raising F t = Y t , and the payoff in the bad state can improve to λY t through termination. This is a contradiction! Hence, the debt capacityκ B N Y t must be achieved by having termination in the bad state.
Clearly, the maximum repayment if the project matures is Y t , which is achieved by any face value F t ≥ Y t . Next, I determine the repayment to the debt holders when the bad state switches to a good one, i.e. z t+∆t = 1 + µ∆t.
As shown before, when N ≤ κ G λ , the renegotiated repayment of λN Y t (1 + µ∆t) ≤ κ G N Y t (1 + µ∆t) can be refinanced by N creditors. Therefore, the debt capacity is achieved by setting F t = max (1, (1 + µ∆t) N λ) Y t and On the other hand, when N > κ G λ , the repayment that can be refinanced in the good state is bounded by κ G Y t (1 + µ∆t), which is attained by having the face value set at the same level. As a result, the debt capacity is given bŷ Proof of Proposition 4: As established by expression (22), the limit of κ G as ∆t → 0 is finite (specifically, π r−µ+π ). Hence, it follows immediately from (26) thatκ B N → λ as ∆t → 0.
Proof of Proposition 5: Suppose N t = 1. I show that the modified refinancing strategieŝ C t = 0 andF t = F t − C t implement the same payoff to both the firm and the creditor in every period. First, the net amount that must be refinanced in period t is So the continuation game (32) in period t is not affected. Second, the debt rollover outcome in period t − 1 is not affected by the modification. This is because the modified value of debt (28) iŝ D t = 1 1+r π min F t , Y t + (1 − π)E t 1 RO min F t , L t + (1 − 1 RO ) min F t , L t = D t − Ct 1+r .
So the net payout to creditors maturing in period t − 1 is the same, that iŝ and the condition (32) in period t − 1 stays the same. Finally, the equity valueV t at the end of period t − 1 is not affected by the modification. By definitionF and therefore the expression in (31) stays the same. In summary, the equilibrium can be implemented by setting C t = 0, completing the proof.
Proof of Proposition 6: DefineF t = F t − C t andD t = D t − Ct 1+r . The value of debt in (28) becomeŝ with the rollover condition (30): First, in equilibrium, it must be that Otherwise, the firm can lowerF t to N t λ (1 + µ) Y t + (N t − 1) C t without affecting the actual debt repayment, contradicting with the minimality ofF t . As a result of (A-18), the net repayment in the good state isF t . Second, I show that having termination in the good state is not optimal, that isF t ≤ DC G t+1 . Suppose otherwise, from (A-18), we know If termination happens also in the bad state, then obviously the values of debt and equity can both be improved by reducingF t to DC G t+1 . SinceF t > DC G t+1 which in turn dominates DC B t+1 , it must be DC B t+1 ≥ N t λY t + (N t − 1) C t . (A-20) Hence, κ B ≥ λN t . Taking the difference between (A- 19) and (A-20), we have The contradiction implies thatF t ≤ DC G t+1 . Finally, consider a new set of refinancing strategies in period t − 1: N t = N t + (Nt−1)Ct F t =F t , and no cash C t = 0. It is easy to verify that Hence, the new pair of N t ,F t , C t does not affect the actual payout to the creditors in both states, and equilibrium outcome can be implemented without cash.

B A Static Benchmark
In this section, I present a static version of the dynamic model in Section 2 to show that statement 2 in Corollary 2 is effectively static, and this static model cannot produce some of the key dynamic intuitions. There are three dates: t = 0, 1, 2 and no time discount. At date 0, the firm chooses the face value of debt F 1 and creditor dispersion N 1 to finance a project. The upfront investment is I 0 , and the project generates a final dividend only at date 2. At date 1, the state Y 1 ∈ {1, µ > 1} realizes with P rob (Y 1 = µ) = p and the resale value L 1 = λY 1 . The renegotiation game at date 1 is exactly the same as in the dynamic model in Section 2. The When the upfront investment I 0 is low, the debt is risk free. As I 0 increases, the required pledgeability increases, as does N * 1 . In this static model, the growth rate µ negatively correlates with the required number of creditors N * 1 for any given financing size I 0 . The intuition is similar to statement 2 in Corollary 2: Better firms can pledge more to each creditor because of a higher final dividend in the good state, and therefore do not need as many creditors as worse firms do. One may be tempted to conclude that higher-growth firms should have fewer creditors. This conclusion can be misleading. In the dynamic model, better firms can survive longer and support more creditors in the long run due to the higher debt capacity. Hence, the equilibrium can feature a positive correlation between the number of creditors and the firm's growth rate.
Finally, consider a reduction in the probability of the good outcome, p. This exercise is the static version of the dynamic analysis in Subsection 5.2. Similar to the finding in Proposition 4, when p decreases, the debt capacity µ 2 p + (1 − p) λ as in (A-21) also decreases because the termination payoff λ is more likely to occur. However, unlike the model in the main text, where the total firm value is held constant as the renegotiation frequency changes, here the ex-ante firm value µ 2 p + 1 − p is decreasing in p. Hence, in this static model, the reduction of the debt capacity is a mechanical result of the worsening firm quality.    in the bad state. In the right panel, I calculate the average number of creditors E (N * t ) weighted by the unconditional probability that the state switches to good (z t = µ) in period t: p(1 − p) t−1 (1 − π) t . I plot these values for firms with growth rates µ = 0.32, 0.33, 0.34, ... 0.44.