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A Dynamic Theory of Multiple Borrowing

Multiple borrowing—a borrower obtains overlapping loans from multiple lenders—is a common phenomenon in many credit markets. We build a highly tractable, dynamic model of multiple borrowing and show that, because overlapping creditors may impose default externalities on each other, expanding financial access by introducing more lenders may severely backfire. Capital allocation is distorted away from the most productive uses. Entrepreneurs choose inefficient endeavors with low returns-to-scale. These problems are exacerbated when investments become more pledgeable or when borrowers have access to more lenders, explaining why increased access to finance does not always improve outcomes.


Introduction
Multiple borrowing-when a borrower takes out overlapping loans from multiple lenders-is a common phenomenon in many credit markets. Consumers hold multiple credit cards, student loans, and other debt secured by homes and vehicles, often financed by different lenders. Firms borrow from multiple banks and can issue multiple public debt securities to a wide range of investors. Hav-ing access to many lenders can be a sign of financial development and competition; yet taking out new loans can affect borrowers' ability and incentive to pay off existing loans. Consequently, multiple borrowing can cause creditors to impose default externalities onto each other.
The goal of this paper is to understand how these distortions affect investment decisions and the allocation of capital. To do this we build a tractable, dynamic model of multiple borrowing where a single borrower takes out overlapping loans from different lenders to finance investment without the ex ante ability to commit to an exclusive lending relationship. We show that lack of ex ante commitment power induces perverse resource misallocation, as more productive projects can receive less investment, and entrepreneurs could deliberately choose inefficient endeavors with low returns to scale. These problems are exacerbated when borrowers have access to more lenders, explaining why increased access to finance does not always improve outcomes. We further analyze how the inefficiency is affected by pledgeability of investment capital, supply side constraints on the amount of debt that can be issued to a single borrower, and bargaining protocols.
We also analyze prudential policies that could alleviate the lack-of-commitment friction.
In the model, an entrepreneur sequentially visits multiple lenders to obtain funds for a new investment opportunity. Crucially, the entrepreneur lacks the ability to commit to exclusive borrowing from a single lender and cannot write loan contracts that are contingent on the terms of those signed with future lenders. The more debt owed the less likely it is to be repaid, giving rise to an externality between lenders; new lenders willingly provide additional investment that existing lenders would not and thereby dilute the value of existing debt. Rational lenders anticipate this additional borrowing and offer loan terms that compensate them for this, making multiple borrowing undesirable ex ante, but without commitment unavoidable ex post.
The model is of a multi-agent dynamic game with multilateral externalities. The potential of future debt dilution implies that, at the time of lending, each lender has to anticipate an entrepreneur's future path of debt accumulation, and the future path is itself an equilibrium outcome between the entrepreneur and future lenders. Despite the apparent difficulty of analyzing such a dynamic game, we devise a recursive formulation of this problem and obtain analytic characterizations in closed form. The analytic solution to the model enables us to sharply characterize the inefficiency due to lack of commitment. First, we show that financial expansion could backfire, as high lender availability can lead to overindebtedness, underinvestment, and lower welfare. Second, because more productive entrepreneurs have a stronger desire to borrow in the future, and thus face worse commitment problems, they can be penalized in equilibrium and end up raising less investment capital, generating an especially perverse form of investment misallocation. Finally, given the choice, entrepreneurs could explicitly pick investment opportunities with lower productivity than others available in order to receive better financing terms.
Interestingly, we show that commitment problems are not necessarily alleviated by relaxing financial constraints. Features that are often understood to strengthen credit markets, such as pledgeability, can have negative implications. Pledgeability of investment allows for leveraged borrowing. This increases the total investment capital entrepreneurs can raise, but also increases the return to borrowing, thus raising entrepreneurs' desire to borrow in the future, increasing the relative severity of commitment problems for more productive projects.
Lower returns to future investment create endogenous commitment power to not borrow from additional lenders. Since entrepreneurs cannot commit ex ante to limit future borrowing, investment opportunities with concave returns, those that generate high output for existing investment but have low marginal returns to additional investment, are especially valuable as these opportunities generate low incentives for future debt accumulation. This logic also explains why collateral increases distortions; the static benefit of pledgeability allowing levered investment has negative dynamic consequences because it raises incentives for future borrowing.
We also explore alternative modeling assumptions and potential policy interventions to better understand the source of our results relating lack of commitment in credit markets to capital allocation. We examine the implications of each lender having limited funds, the observability of borrowing history, and the allocation of bargaining power between borrowers and lenders. Policy interventions, such as uniform borrowing limits and interest rate caps, can improve outcomes but are challenging to implement in practice. We devise a Pigouvian tax that is easily implementable with only information on borrowing history and can induce the full commitment allocation.
The notion of commitment externalities in credit markets is not new to our paper. The economics of the commitment externality was analyzed in the seminal work of Stiglitz (1991, 1993) in the insurance context. There is a large literature that analyzes inefficiencies in credit markets due to lack of commitment, including Bizer and DeMarzo (1992) , Kahn and Mookherjee (1998) , Parlour and Rajan (2001) , Brunnermeier and Oehmke (2013) , Donaldson et al. (2019) , Admati et al. (2018) , and DeMarzo and He (2020) . 1 We provide a detailed review of this literature in Section 5 . Here we note that the closest paper to ours in this literature is Bizer and DeMarzo (1992) . They study the problem of a borrower who can visit multiple lenders to take out loans and smooth consumption. Their environment also features debt dilution and lack of commitment, and they analyze a static equilibrium defined by the satiation point of the borrower so that no future lending happens in equilibrium even if new lenders become available. In contrast, we formulate our model so that the borrower can never be satiated-future borrowing is always ex post desirableand we analyze the full dynamic path of lending outcomes. Our formulation allows us to study a broader set of questions, most notably how lack of commitment affects resource allocation and how it interacts with the nature of investment opportunities. Our dynamic model also allows a tractable analysis of the relationship between commitment and pledgeability, the availability of lenders, and bargaining between borrowers and lenders.

Setup
An entrepreneur attempts to finance a new investment opportunity with the constraints that they cannot commit to exclusive borrowing from a single lender and that there is limited enforcement of debt repayment, i.e., the borrower only repays if the cost of default is high enough.
The model has two stages ( t = 1 , 2 ) and contains two sets of agents: a single entrepreneur and an infinite sequence of potential lenders. All parties are risk neutral and have no discounting. The entrepreneur is endowed with a variablescale investment opportunity that returns R ( K ) = zK  deterministically with z > 1 at t = 2 for any K ≥ 0 invested at t = 1 . The returns to this investment opportunity are observable but not pledgeable (we relax this assumption in Section 4.1 ). After the project's, returns are realized, the entrepreneur learns of their cost of defaulting on their debt and only repays if the default cost exceeds the amount of debt owed. Specifically, we assume default costs ˜ c are drawn from a distribution p ( is also the probability of repayment and is decreasing in D . Fig. 1 summarizes the timing of the model. We model the entrepreneur raising capital from the lending market at t = 1 as a dynamic game of complete information in which the entrepreneur sequentially and stochastically contacts a potentially infinite number of lenders. Each lender's opportunity cost of funds is the risk-free rate, which we normalize to one. Upon meeting a lender, the entrepreneur makes a take-it-or-leave-it offer for a simple debt contract ( k i , d i ), specifying the amount borrowed k i and the promised repayment amount d i . The lender, who can observe the history of the entrepreneur's borrowing from previous lenders, chooses whether to accept or reject this offer, and if accepted, the funds are exchanged. We study alternative bargaining protocols in Section 4.4 .
After signing a contract with each lender, the entrepreneur loses access to the lending market with probability 1 − q . Otherwise, with probability q , the entrepreneur meets a new lender and the process described above is repeated until eventually access to the lending market is lost. After losing access to the lending market, the entrepreneur invests the aggregate financing they raised in the new project, and the model progresses to stage t = 2 .
We assume no lender is visited more than once. The parameter q measures the availability of lenders relative to the stochastic fund raising deadline of the investment opportunity. The model implies that the probability of meeting at least N lenders is q N−1 . Fig. 2 provides a graphical exposition of the lending market game of stage t = 1 .

Model discussion
A crucial feature of our model is that loan contracts cannot be contingent on the terms of loans issued by subsequent lenders. Importantly, by modeling default as a binary decision, we are able to abstract from security design and limit our analysis to the study of debt contracts. Financing arrangements that allow more sensitivity between realized output and repayment could in principal help borrowers internalize the effects of obtaining subsequent financing. While such contingent contracts may be available in some segments of well-developed credit markets, loans without exclusivity clauses prevail in many less developed credit markets. We study the value of exactly this type of contingent contracting and its role in ensuring the efficient allocation of capital across projects.
The assumption of sequential borrowing from a possibly large number of banks is adopted for analytic tractability and need not be taken literally. While the sequential formulation we model implies observability of previous lending, this is not important for our results. What drives our findings is that lack of commitment to an exclusive lending relationship leaves room for the externalities between lenders and perversely affects equilibrium outcomes. While it is possible to instead model borrowing from multiple lenders as simultaneous so that lenders do not observe other borrowing and obtain broadly similar results, such a model is analytically less tractable and subject to the standard critique in the simultaneous contracting literature that the results are sensitive to the specification of agents' off-equilibrium beliefs ( Segal and Whinston, 2003 ). Further, our model of stochastic sequential lending provides a parsimonious way to vary the degree of limited commitment.
We formulate the endogenous choice of default via a random default cost that is realized ex post. The formulation is similar to a random shock to an outside option, as in Aguiar et al. (2019) , and reflects forces that generate ex ante indeterminacy in the ex post costs of defaulting on debt. These forces can include weak institutions or limited contractual enforcement. The Appendix provides a microfoundation of the random default through moral hazard.
Section 4 introduces a number of extensions to the model, including partial pledgeability of the investment project ( Section 4.1 ), limiting the amount of investment that each lender can provide ( Section 4.2 ), adding concavity to the investment project ( Section 4.3 ), studying the equilibrium under alternative bargaining protocols between the entrepreneur and lenders ( Section 4.4 ), and regulatory policies that could alleviate the lack-ofcommitment friction ( Section 4.5 ).

Illustrating the commitment problem
First, suppose the entrepreneur meets exactly one lender and makes a take-it-or-leave-it offer ( K, D ), funding K into the investment project and issuing debt face value D . The lender accepts any loan offer that is weakly profitable in expectation. Under the mutual knowledge that the borrower will not take out additional loans from future lenders, the optimal lending contract solves: The maximization problem can be rewritten as The solution is characterized by the first order condition: where we have used the property that Eq. (1) characterizes the debt issuance (and p ( D ) D the investment level) of the model if the entrepreneur could commit not to take out any additional loans from future lenders. At the optimum, the costs and benefits of issuing an additional unit of debt must be equalized. The marginal cost is the increase in expected debt repayment costs, which is simply the probability of actually repaying the marginal unit of debt as shown on the right hand side of Eq. (1) . 2 The marginal gain from issuing additional 2 The right hand side of the first order condition before simplifying is The term in brackets captures that repayment debt is shown on the left hand side and is equal to the project's productivity z times the marginal investment that can be raised from the additional debt issuance. The term p ( D ) represents the value of a marginal dollar of promised repayment. Because extra repayment reduces the value of all debt claims, the value of inframarginal debt D falls; this is reflected in the term p ( D ) D . Hence, after having issued debt D to the first lender, the additional investment the entrepreneur can raise by issuing an additional unit of debt to the first lender is To illustrate the source of the commitment problem, suppose the borrower who has issued debt D to the first lender gets to meet a second lender. The entrepreneur can raise investment of size p ( D ) by issuing one unit of debt to the second lender, whereas if they had tried to promise a marginal dollar of repayment to the first lender, the lender would have only lent p ( D ) + p ( D ) D < p ( D ) . In other words, the second lender offers better marginal interest rates than the first lender. Because the second lender does not internalize that its lending dilutes the value of preexisting debt claims. The same logic holds for all future lenders, and therefore the borrower would always choose to take out additional loans from future lenders if it had the opportunity to do so. Thus, in any equilibrium lenders must anticipate all potential future borrowing and charge higher interest rates that compensate them for value they expect to lose. We now formally study the equilibrium of this model.

Equilibrium definition and solution
Our solution concept is Markov Perfect Equilibrium (MPE). The relevant state variable, which both borrower and lenders' strategies are conditioned on, is the cumulative debt that has been issued to all preexisting lenders. We now proceed to characterize the stationary MPE. We show at the end of this section that the equilibrium we identify captures the limiting outcome of the unique subgame perfect equilibrium (SPE) of finite-lender versions of the model as the number of lenders grows large.
Throughout the paper, let ( k i , d i ) denote the contract that the borrower offers to lender i , where k i is the investment that the borrower requests and d i is the face value of debt issued. Let K i ≡ i j=1 k j and D i ≡ i j=1 d j respectively denote the cumulative debt and investment after contracting with lender i .
All proofs are in the Online Appendix.

Lender best responses
Lenders have rational expectations of future borrowing and thus only accept contracts that yield nonnegative expected returns, taking the strategies of the borrower and other lenders as given. Specifically, suppose the borrower has previously obtained total face value of debt D to all preexisting lenders and is proposing to issue d i to lender costs change indirectly though changes in the probability of repayment. But because the borrower is optimizing the default decision ex post, this indirect effect is zero.
i . Denote ˜ p i ( D + d i ) to be lender i s perceived repayment probability. Lender i s optimal strategy is to accept the loan offer if and only if it is weakly profitable in expectation, i.e., k i ≤˜ p i ( D + d i ) × d i . We focus on a stationary equilibrium in which lenders' best response can be characterized by a function ˜ p ( ·) such that ˜ p i ( ·) = ˜ p ( ·) ∀ i . In equilibrium, rational expectation implies that ˜ p ( ·) has to be consistent with the borrower's future path of debt accumulation.

Borrower best response
Conditional on meeting a lender, and given lender strategies represented by ˜ p ( ·) , the borrower makes a loan offer that maximizes expected continuation utility of potentially being able to meet and borrow from more lenders. Solving for the borrower's best response function thus involves a dynamic optimization problem. Taking lender pricing as given, the entrepreneur forms strategies that maximize continuation utility at each value of the state variable D .
The borrower knows that lenders will accept any loan they expect to be profitable, so to maximize their own utility they will only offer loans on which lenders expect to make exactly zero profit. If the entrepreneur has cumulative debt D upon meeting the lender and leaves the lender with cumulative debt D = D + d, then the maximum amount of new capital the lender would provide is given by ˜ p D d, the probability of repayment times the face value of new debt issuance. Thus, taking loan pricing as given the entrepreneur solves: (2) The first term on the right hand side of Eq.
(2 )is the marginal payoff from the new investment opportunity associated with obtaining additional investment k = ˜ p D d. With probability 1 − q the entrepreneur loses access to the lending market, invests total capital raised K into the investment opportunity, and either repays debt D or defaults and pays the default cost ˜ c . Finally, with probability q the entrepreneur does not lose access to the lending market and will receive continuation utility V ( D ) from future borrowing.
Denote a policy function that solves the dynamic programming problem in Eq. (2 ) by g ( · ). Conditional on arriving to a new lender with aggregate face value of debt D , the borrower will leave with aggregate face value of debt D = g ( D ) . Following this strategy generates a sequence of total aggregate face values of debt that have been accumulated up to a given lender: { g (0), g ( g (0)), g 3 (0), ....}. The aggregate face value of debt obtained in the lending market is a random variable, denoted by D agg , that realizes a particular value of this sequence depending on how many lenders the borrower is able to visit before the lending market ends.

Equilibrium definition
The equilibrium is a fixed point in the pair of functions, g ( · ) and ˜ p ( ·) . Given lender strategies ˜ p ( ·) , the borrower's optimal strategy of debt accumulation is the policy function g ( · ) that solves the dynamic programming problem in Eq. (2) . Given the probability distribution of total aggregate face value of debt D agg induced by g ( · ), each lender forms rational expectations over the probability that the borrower will repay, denoted by ˜ set of strategies forms an MPE if lender strategies given by ˜ p ( ·) are rational given g ( · ), and the borrower strategy g ( · ) is optimal given lender strategies embodied in ˜ p ( ·) .
Definition 1 . A symmetric Markov Perfect Equilibrium of the lending game is a pair of functions ˜ p ( ·) and g ( · ) such that: 1. g ( · ) solves the dynamic programming problem in Eq.
2. Lenders' perceived expected repayment probabilities ˜ p ( ·) used to form accept/reject strategies are correct taking g ( · ) as given: ˜

Closed form solution
The MPE is defined by a pair of functions that solve an infinite-dimensional fixed point problem, which is in general difficult to characterize. We make the following simplifying assumption on the distribution of default costs, under which the dynamic value function V ( · ) becomes quadratic in the state variable, and the model admits a closed form solution. Proposition 1 . For 1 > q ≥ 0, the unique stationary MPE in linear strategies can be characterized by two scalars b * ≤ 1 and * ≤ 1 : The two equilibrium scalars are the fixed point of the pair of The closed form solution to the two equations is Ex ante borrower welfare, given by .
There is an intuitive interpretation for both and b . First, if the borrower arrives to a lender with current debt D , its remaining debt capacity is 1 − D . 3 When leaving this lender the borrower will have pledged a total of g ( D ) and thus the borrower will have remaining debt capacity 1 − g ( D ) . Therefore b * is the fraction of current borrowing capacity that remains after visiting a lender.
Second, the variable * captures the equilibrium degree of future debt dilution. Recall that with commitment, the repayment probability is exactly ˜ which corresponds to * = 1 . Thus, * < 1 corresponds to the expected repayment probability relative to the case with commitment. When * is low, the lenders charge high interest rates because they expect the borrower to accumulate substantially more debt from future lenders.
Finally, there is also a static interpretation for the dynamic equilibrium. Taking the lender's strategy parameterized by as given (which determines ˜ p ( D ) and the interest rates), the entrepreneur chooses how much debt to issue when meeting each lender. This specifies a best response b = B ( ) for the entrepreneur in Eq. (4) , which can be thought of as representing a loan demand schedule. Since higher interest rates induce the borrower to take out debt less aggressively, the loan demand schedule is downwardsloping ( B ( ) < 0). Given the aggressiveness of the entrepreneur's borrowing behavior parameterized by b , the lenders are able to determine the distribution of the face value of total borrowing. This maps into the distribution of the value of their own debt claims. Lenders set their decision rule such that they only accept loans they expect to be weakly profitable, and thus specify a best response = L ( b ) as the loan supply schedule in Eq. (3) . The interest rates lenders have to charge in order to break even increases as the entrepreneur takes out loans more aggressively; hence the loan supply schedule is upward-sloping The equilibrium ( b * , * ) is then the intersection of the "loan demand" equation B ( ) and the "loan supply" equation L ( b ), which is depicted in Fig. 3 .

Describing the equilibrium path
Proposition 1 enables us to iterate on the policy function to compute the equilibrium sequence of debt issuance . . } and to compute the equilibrium sequence of interest rates using the function ˜ p ( ·) . We find that, in equilibrium, each subsequent lender lends less than previous lenders while charging a higher interest rate.
The equilibrium lending path can be visualized in Fig. 4 . The solid lines denote the sets of loan contracts ( k, d ) that each lender i believes will have zero profit given the cumulative level of debt D i and expectations about future borrowing (i.e., the curves trace out for various D i ). The marked point on each curve represents the loan ( k i , d i ) chosen in equilibrium (recall d i = 3 Given that ˜ c ∼ U [ 0 , 1 ] , the borrower would default on all debt with certainty if it were to acquire additional debt of more than 1 − D beyond current outstanding debt D .  D i +1 − D i ). As the borrower visits more lenders, borrowing amounts fall ( d 1 > d 2 > and k 1 > k 2 > ) and interest rates d i / k i , represented by the inverse of the slope of dotted lines, rise.
Note that, despite charging higher average interest rates, subsequent lenders charge lower marginal interest ratesas discussed in Section 2.3 -because they do not internalize the effect of their lending on the repayment of previously issued debt. The result, that new lenders provide better marginal interest rates, can also be seen from the figure: the slope of the contract curve for the first lender at ( D 1 , k 1 ) is lower than the slope of the curve for the second lender at ( D 1 , 0), indicating that the second lender is willing to provide more capital for a marginal unit of debt.
Why do the sets of zero profit loan contracts have an inverted U shape? Holding repayment probability fixed, higher face value translates into higher present value. However, a higher face value of debt also reduces the probability of repayment. Thus, at each lender the borrower faces a "debt Laffer curve": the value of debt is initially increasing in the amount of debt pledged, but for sufficiently high face values of debt, promising to repay an additional dollar of debt actually decreases the total amount of financing. Given the contract for any D i , the borrower would always choose contracts to the left of the peak investment as shown in Fig. 4 . This is because issuing debt is costly, and, for any ( k, d ) to the right of the peak, there exists another contract with the same level of investment k and a strictly lower level of debt issuance. The borrower always prefers the contract with lower debt. The debt Laffer curve plays an important role in the comparative statics of how investment changes with entrepreneur productivity, as will be explained in the next section.

Equilibrium choice
The linear-stationary MPE we have identified is closely related to the unique subgame-perfect equilibrium of a finite lender version of the model. Hence, our infinite lender model and particular solution concept are merely abstractions for algebraic tractability.
Formally, we define the finite N -lender game by modifying the infinite lender game as follows. After meeting the i -th lender, the borrower gets to meet the (i + 1) -th lender with probability q if and only if i + 1 ≤ N, and with probability zero otherwise. Each finite N -lender game has a unique subgame-perfect equilibrium. The following proposition demonstrates that MPE identified in Proposition 1 of the infinite lender game is the limit of the sequence of SPEs of the finite N -lender games as we take N to infinity.
Proposition 3 . Fix i. As N → ∞ , lender i's strategies in the sequence of SPEs of the finite N-lender game converge uniformly to ˜ p ( ·) .

Comparative statics
We now analyze how equilibrium outcomes depend on availability of lenders q and productivity z . Because the outcome of the lending game depends on the stochastic number of lenders the borrower meets, we will focus our attention on ex ante, expected outcomes .

More lenders leads to worse allocations
The expected number of lenders a borrower visits is 1 1 −q , which increases with q . Higher availability of lenders (higher q ) leads to a worsening of the commitment problem as we formalize in the following proposition.
Proposition 4 . When more lenders are available (higher q), in expectation, for any z > 1 and q ≥ 0 : 1. Debt issuance is smoothed out over multiple lenders. That is, the cumulative debt D i at each stage i of the lending 5. The entrepreneur's welfare decreases in q and, in the limit as q → 1, converges to zero, the level that would be obtained if the entrepreneur does not have access to the lending market at all. In this limit, E [ D agg ] → z−1 z . Higher q raises the expected debt issuance as the entrepreneur is more able to smooth out debt issuance over multiple lenders to exploit the fact that future lenders offer better interest rates for marginal debt. Higher debt accumulation thus raises the probability of default, and in order for lenders to break even, the equilibrium interest rate increases.
Despite higher debt issuance, total investment declines as q increases, and the entrepreneur's ex ante welfare also declines. In fact, as q → 1, the commitment problem becomes so severe that the credit market effectively shuts down: even though there is positive borrowing and investment, all surplus from investment is offset by costly default. 4 These results illustrate the fact that dynamic commitment problems can completely unravel the ability of an agent to capture or generate surplus.

Better opportunities, worse commitment problems
Entrepreneurs who have more productive investment technology also have greater incentives to borrow in the future; hence, the commitment problem is more severe for highly productive entrepreneurs. The next result shows that nonexclusivity generates a perverse form of misallocation: high-productivity entrepreneurs may not be able to raise as much capital for investment as lower productivity entrepreneurs.

Debt and interest rates are both increasing in z :
3. The more lenders in the lending market, the lower is the cutoff productivity z ( q ) for investment declines: ∂ z ( q ) ∂q < 0 .

More lenders (q) lower investment especially for high-
Borrower welfare is globally increasing in productivity z: These results can be visualized in Fig. 5 , which plots equilibrium expected investment as a function of z for three different values of q . For all levels of q , the level of investment that the entrepreneur gets to raise in expectation first increases in z and then decreases. Entrepreneurs with better opportunities could be facing tighter constraints. The third and fourth part of Proposition 5 shows that such endogenous misallocation of resources is more severe in markets with more lenders. First, when q is higher, the productivity z ( q ) that maximizes equilibrium investment is lower. More generally, the negative effect of increased availability of lenders q on equilibrium investment is more severe the higher is productivity z . Despite these distortions, borrower welfare is always increasing in the productivity level z . We show in Section 4.3 that this result is a consequence of imposing a linear investment technology; for general nonlinear production functions, projects that deliver more output for any level of investment can deliver lower welfare than less productive projects that have more rapidly diminishing returns to scale.
Why can better opportunities be associated with lower levels of investment? When commitment problems are present, it is possible that the equilibrium involves using available debt capacity inefficiently. To better understand the intuition, recall that the present value of repayments to creditors is not uniformly increasing in the face value of claims issued, because higher levels of debt increase expected dilution. This gives rise to a Laffer curve as discussed above and seen in Fig. 4 . Of course, when facing an individual lender, no borrower would propose to borrow so much that they could receive more capital from that lender if they reduced their promised repayment. However, without commitment, the entrepreneur will want to borrow from any future lenders they get to meet. This means that lenders expect high future borrowing and charge high interest rates, and, from an ex ante perspective, the expected total equilibrium debt and investment could endogenously be on the wrong side of the aggregate debt Laffer curve. Proposition 5 shows that this happens precisely when productivity z is high. If an entrepreneur with a high productivity could have committed to a lower level of debt issuance, they could have received more investment. But because entrepreneurs with the highest returns to investment have the strongest desire to borrow, they are expected to accumulate more debt and thus end up bor-rowing at such unfavorable interest rates that they receive lower investment than low-productivity borrowers.
Beyond the fact that investment can decline in productivity, Proposition 5 also shows that higher productivity projects are especially sensitive to lack of commitment. Thus, from an aggregate perspective, the more severe the lack of entrepreneurs ability to commit to exclusive borrowing, the more investment is distorted into relatively lower productivity projects. This intuition can be formalized by considering aggregate productivity of capital in an economy with a distribution of productive projects given by F ( z ). We can then define the economy-wide average productivity on invested capital as Proposition 6 . For any distribution of productive projects F ( z ), the average productivity on invested capital ˜ Z is decreasing in q.
Proposition 6 is consistent with empirical evidence, discussed in detail in Section 6 , that increasing access to borrowing does not seem to increase observed returns to investment in environments where commitment to exclusive borrowing may be difficult.
Taken together, these results are surprising and run against the common intuition that more productive projects induce higher levels of investment despite the presence of financial constraints. In most classical theories of inefficient investment choice, investment occurs if and only if the net present value of a project exceeds a certain threshold, which can be above or below zero. Such models do not generate the stark patterns of resource misallocation that plague developing countries. The commitment friction we explore in this paper, however, can generate these distortions. In fact, we show in Section 4.3 that this force is so strong that it can also influence the investment opportunity chosen by entrepreneurs.

Model implications
In this section, we discuss implications of our framework through several distinct model extensions. Section 4.1 introduces partial-pledgeability of the investment project to the model. Standard intuition suggests that pledgeability should relax financial constraints, yet we show that without commitment it exacerbates debt accumulation and may actually worsen allocations. Our baseline model abstracts from any benefits of future lenders because the entrepreneur could have obtained as much capital as desired from the initial lender so long as the promised repayment is credible. Section 4.2 instead analyzes a case in which each lender can only provide limited funds to the entrepreneur. We argue in Section 4.3 that the value of exclusivity depends critically on the returns to scale of the investment project. We show that the lack of exclusivity can induce entrepreneurs to choose investment projects with lower levels of returns compared to other options if they also feature more rapidly diminishing returns. In Section 4.4 , we show that shifting bargaining power from the entrepreneur to lenders actually worsens the commitment problem and exacerbates debt accumulation. Finally, Section 4.5 considers prudential policies that could alleviate the problem of nonexclusivity. We show that debt limits or interest rate ceilings, such as those adopted in several credit markets prone to loan-stacking, could help prevent future lending but are challenging to implement. We also describe a Pigouvian credit tax that restores the full-commitment outcome and can be implemented practically.

Pledgeability
In the baseline model, investment capital raised through borrowing is not pledgeable. If the investment can be (partially) recouped by lenders in case of default, the entrepreneur can increase leverage and raise more investment for any given quantity of risky debt. Therefore, it seems reasonable to expect that pledgeability of investment capital could reduce inefficiency in the lending market. This intuition is incomplete. Leverage induced by partial pledgeability also effectively raises marginal returns of issuing risky debt, and thus there is a channel through which increased pledgeability exacerbates the dynamic incentive to dilute debt with subsequent borrowing.
We explore the implications of this insight through a simple extension of the baseline model. Specifically, suppose a fraction θ of the investment is pledgeable. Thus, if the borrower issues risky debt to a lender in exchange for investment capital k , the borrower can credibly pledge to repay an amount θ × k with certainty. Thus, if the borrower has already promised total risky claims D to previous lenders and is issuing additional risky debt d , the current lender's expected repayment is θ k + borrower's value function can now be expressed as: For any risky debt offering d , the lender is willing to supply The multiplier 1 1 −θ can be interpreted as the leverage made possible by pledgeability. Because the borrower has to pay back θ fraction of the investment, the return on investment becomes ( z − θ ) k . Defining ˆ z ≡ z−θ 1 −θ ≥ z as the effective productivity under pledgeability, Eq. (5) becomes isomorphic to Eq. (2) . This formulation shows that pledgeability exacerbates the incentive to borrow; the effective marginal returns to borrowing, captured by ˆ z , are increasing in θ . This leads to the following proposition.

In expectation, cumulative debt, investment, and borrower
welfare are globally increasing in pledgeability θ . The first part of the proposition illustrates that allowing partial pledgeability of investment capital does not qualitatively change the implications of lack of commitment for equilibrium outcomes. More interestingly, increasing the extent to which investment capital can be pledgeable has nuanced implications. Welfare is globally increasing in θ because pledgeability effectively raises the returns to all borrowing, current and future. However, under higher θ , investment declines faster with q , and the level of productivity that maximizes equilibrium investment becomes lower. This is because higher marginal returns to future borrowing generate greater incentives for dilutive debt issuance, and pledgeability raises effective marginal returns to borrowing, especially for more productive entrepreneurs. The solid line in Fig. 6 illustrates the level of productivity that maximizes investment as θ varies.
The borrower is able to raise more investment under higher θ . However, because pledgeability raises the desire to borrow disproportionately for high-productivity entrepreneurs, it especially exacerbates their commitment problem, and a perverse effect might kick in: similar to the effect of higher q , higher pledgeability θ might reduce the average productivity of invested capital ˜ Z , as described in Proposition 6 , because higher θ might cause relative resource outflow from high-productivity entrepreneurs and inflow to low-productivity entrepreneurs. A higher θ does not always generate this effect; numerical explorations suggest that this effect happens when the productivity distribution is skewed toward lower productivity projects.

Limited funds
In the baseline model, higher lender availability q unambiguously worsens overindebtedness and underinvestment. An important assumption for these results is that lenders have unlimited funds so that the entrepreneur could have obtained as much capital as desired from the initial lender as long as the promised repayment is credi- ble. Hence, the availability of any subsequent lenders can only deteriorate allocations.
In this section, we relax that assumption and instead assume that each lender can only provide limited investments up to F to the entrepreneur. Such a constraint could arise, for instance, because of policy regulation or because each lender hopes to limit their exposure to an individual borrower's default risk. The constraint could also arise in settings where lenders simply have limited capital. We then analyze the trade-off between the two implications of increased lender availability: deteriorating commitment and overcoming per-lender funding limits.
Specifically, we impose the additional constraint k ≤ F in the entrepreneur's optimization problem: The additional constraint breaks the stationarity of the model and we are no longer able to provide closed form solutions. However, we are able to numerically solve the model and analyze the equilibrium path. Debt issuance d to each new lender is a decreasing function of the cumulative outstanding debt D ; hence, for sufficiently high levels of cumulative debt D , the additional debt issuance becomes sufficiently low such that the constraint k ≤ F ceases to bind. On any equilibrium path, the cumulative debt D increases as the entrepreneur meets more lenders; hence, the limited investment constraint k ≤ F binds only for a finite number of initial lenders. For subsequent lenders the equilibrium policy and value functions coincide with the baseline model. Fig. 7 demonstrates the equilibrium path of debt issuance given the realized number of lenders the borrower meets. Cumulative investment first increases linearly with the number of initial lenders from whom the entrepreneur raises the maximum investment F ; after sufficient investment is raised, the constraint no longer binds for all future lenders.
We numerically investigate the welfare implications of the limited investment constraint k ≤ F . We find that, if the constraint is tight ( F lower than the full-commitment level of investment), then raising lender availability q could raise entrepreneur welfare, as subsequent lenders are able to provide additional productive investment that the initial lender could not.
Interestingly, this does not imply that, for a given lender availability q , lowering the per-lender funding limit F increases welfare. In fact, lowering F does not increase entrepreneur welfare for any set of parameters we have investigated. Our intuition for this is that the funding constraint affects welfare through two channels. On the one hand, the constraint on future lenders lowers the scope of future borrowing, thereby alleviating the commitment problem; on the other hand, a binding constraint implies that too little investment is available from the initial lender. In other words, lowering F for future lenders raises welfare, whereas lowering F for the initial lender lowers welfare. Because the level of investment raised declines at each round of financing, the constraint k ≤ F is always more binding for the initial lender than for subsequent lenders; consequently, a uniform reduction in F for all lenders seems to always lower welfare.

Nonexclusivity distorts project choice
The value of exclusivity depends critically on the returns to scale of the investment project. For projects with concave returns, obtaining sufficiently large amounts of capital from early lenders would lower the marginal returns to future borrowing, therefore limiting borrowers' ex post incentives to contract with additional lenders and lowering ex ante interest rates. Thus, concavity of investment returns can be desirable because it embeds endogenous commitment power. Nonexclusivity may discourage the entrepreneur from starting projects with the highest level of returns and encourage instead those with lower and more concave returns.
To see this, consider investment projects with linear returns up to a certain size of investment and zero marginal return for any additional investment with output being z × min K , K . Clearly, the entrepreneur would never borrow further after having raised investment K . This implies that when K is sufficiently low, lack of exclusivity becomes irrelevant, as the entrepreneur does not borrow from any lenders beyond the initial one in equilibrium. Moreover, an appropriate level of K could even restore the second-best level of investment, the level that would have been obtained under full-commitment for a linear-return project with productivity z , and strictly raise entrepreneur welfare.
Because concavity generates endogenous commitment power, if entrepreneurs can select from a menu of projects to finance, they may prefer an investment opportunity that is strictly dominated by another in terms of having strictly lower and more concave returns for any level of investment (see Fig. 8 ).
Proposition 8 . Consider investment opportunities R L ( K ) and R H ( K ) such that R H ( K ) > R L ( K ) for all K. Absent commitment, an entrepreneur could prefer to undertake the domi-  Proposition 8 generates a strong empirical prediction. If entrepreneurs have scope to choose between projects that can scale significantly and those that cannot, increasing the availability of finance through lender entry in an environment with limited ability to commit could result in lower growth or the failure of increased access to credit to facilitate growth. As discussed in Section 6 , this prediction is also consistent with the growing consensus that microfinance has not provided wide-scale poverty relief as once hoped. However, this prediction is difficult to test empirically, as it would require detailed data on the returns to scale of economic opportunities as well as variation in the degree of lender availability that is uncorrelated with the nature of projects requiring financing.

Lender bargaining power
In the baseline model, the borrower makes take-it-orleave-it offers to lenders, which effectively gives all bargaining power to the borrower. How does the distribution of surplus and who makes offers affect our results? We now extend the model so that contractual terms are determined by generalized Nash bargaining. Relative to an outside option of zero, each lender is able to capture fraction β of the surplus from the contract. With β = 1 we can also interpret the model as lenders making offers instead of borrowers. Borrower surplus is defined with respect to the borrower's outside option of taking no loan from this lender and instead proceeding to the next stage of the lending market and possibly meeting another lender.
In developing this analysis, it is helpful to reformulate the value function representation of the borrower's problem in terms of its outside option. Define to be the expected continuation value to the entrepreneur after signing a contract bringing total promised repayment to D , but before the uncertainty about the next lender arrival is realized. It is easy to see that our baseline model's debt accumulation policy can be solved using the recursive value function for W ( D ) as Under generalized Nash bargaining, the debt accumulation solves (6) subject to the constraint Proposition 9 . Consider the generalized Nash bargaining extension with β > 0 . In equilibrium, the lender's assessment of repayment probability ˜ p ( ·) and the borrower's debt accumulation policy g ( · ) can be summarized by the endogenous scalars b Nash ( z, q, β) and Nash ( z, q, β) : To understand the proposition, note that Eq. (7) characterizes the probability of repayment as a function of the borrower's aggressiveness in future debt accumulation. This function is the same as in the baseline model (3) . On the other hand, Eq. (8) characterizes the borrower's policy function as a function of lenders' perceived probability of repayment. Notice that Eq. (8) is similar to Eq. (4) of the baseline model except that q switches to ˆ q ≤ q . In other words, for a given , debt accumulation policy under generalized Nash bargaining coincides with the policy in a baseline model with a lower likelihood of future lender arrival. Consequently, for a given , b Nash is strictly decreasing in the lender's bargaining power β, meaning that shifting bargaining power to the lender leads to strictly more aggressive debt accumulation and worsens the commitment problem.
How can this be? To understand this, note that the source of inefficiency lies in default externalities across lenders. When the entrepreneur holds all the bargaining power and lenders exactly break even, the entrepreneur takes their own lack of commitment into account and makes contractual choices to maximizes the total welfare, which is equal to the entrepreneur's own payoff. On the other hand, when lenders hold some bargaining power, a fraction of total surplus will be captured by future lenders. Such future surplus is not accounted for when the current contracting parties bargain over contractual choice. This makes the contracting parties effectively more myopic; the parties jointly issue a debt contract as if the probability of meeting future lenders is lower than the actual probability, ˆ q < q, thereby causing more aggressive debt accumulation.

Policy implications
Regulatory tools can improve outcomes in the model. Intuitively, any prudential policy that discourages borrowing in the future would alleviate the commitment problem. We show that such policies can take the form of interest rate caps, borrowing limits, or simply limiting the number of lenders from which a borrower can obtain loans. Because the constrained-optimal investment levels depend on entrepreneurs' productivity z , the optimal prudential interest rate caps and borrowing limits must also be borrower-specific.
We also solve for the optimal Pigouvian credit tax, levied on new lenders and transferred to preexisting lenders. The optimal credit tax can restore the fullcommitment outcome and is easily implementable: it can be calculated solely based on equilibrium debt issuance and interest rates; thus the regulator does not need to know any structural parameters of the environment.

Limits on the interest rate, total debt, and number of lenders
Prudential regulations, such as limits on the interest rate, total debt, and the number of lenders, can improve welfare because they embed commitment power. For instance, interest rate caps ensure that debt beyond a certain level is never issued, as a binding interest rate cap prevents the issuance of loans that are especially dilutive. Early lenders can then be assured that such future borrowing will not occur, and can provide initial loans at better interest rates. Consequently, interest rate caps can increase investment and improve welfare as long as they are not too severe. Further, for a given investment opportunity the interest rate cap can be set to be the interest rate that would prevail in the full-commitment equilibrium. This resolves the inefficiencies induced by nonexclusivity and induces the full-commitment level of debt, interest rate, and investment. Likewise, debt limits and directly restricting the number of lenders can improve welfare in similar ways.
We denote the equilibrium when q = 0 as the fullcommitment outcome.
Setting R = 2 z−1 z restores the full-commitment outcome. (Debt limit) Consider the baseline model with an additional constraint on the total outstanding debt: D agg ≤D . The debt limit affects borrower's ex ante welfare as follows: Setting D = z−1 2 z−1 restores the full-commitment outcome.
Both debt limits and interest rate caps were adopted for microfinance loans in India in 2011 ( Subbarao, 2011 ), where multiple borrowing was prevalent and posed a threat to the microfinance industry. However, in general, regulators should be cautious when implementing marketwide prudential policies and limit debt issuance and interest rates, as caps that are too low can restrict productive investment. Welfare-enhancing prudential policies depend on the distribution of entrepreneurs' productivities, information on which might be difficult for the regulators to collect.

Pigouvian credit tax
We now show that a simple Pigouvian tax on new loans can restore the full-commitment equilibrium. The tax is levied on subsequent loans and rebated to previous lenders so that (1) subsequent lenders internalize the incremental default risk on preexisting loans, and (2) the current interest rate does not price in future debt dilution because current lenders expect to be compensated by future tax rebates. Most importantly, implementing this tax only requires publicly observable information on current and prior loan terms, and, unlike in the case of interest rate caps and debt limits, the planner does not need to know entrepreneur-specific characteristics such as productivity.
Specifically, let τ n be the tax levied on the n -th lender, who is willing to provide investment k n = p ( D n ) d n − τ n .
At the origination of this loan, each prior lender i < n is compensated by τ i n . Government budget balance requires Proposition 11 . (Optimal credit tax) The following set of taxes and transfers restore the full-commitment level of investment, debt issuance, and interest rate outcomes: Under the optimal credit tax, the borrower obtains all financing from the first lender.
These taxes and transfers are chosen such that 1) the tax levied on the n th lender exactly equals the impact of the loan d n on the value of outstanding loans D n −1 Proposition 11 , these taxes and transfers are stated in terms of borrowing history and can be implemented without direct knowledge of how debt accumulation affects repayment probabilities. This is because lenders charge competitive pricing for loans; consequently, equilibrium interest rates reveal repayment probabilities, and the difference in loan terms between sequential lenders reveals changes in repayment probabilities. The tax instruments we propose are economically similar to the "performance pricing " covenant observed in the broadly syndicated corporate loan market. Performance pricing covenants specify that the interest rate of any existing loan is a prespecified function of the total debt of the borrower, such that the interest rate on an existing loan can increase if the borrower subsequently issues additional debt. Proposition 11 highlights that, even if implementing such a contingent contract is not feasible, the same effect can be achieved through regulatory policy as long as prior borrowing is observable.

Connection to theoretical literature
We now discuss how our paper relates to the broader literature on nonexclusive contracting in credit markets and highlight where differences in modeling assumptions generate different results. First, Parlour and Rajan (2001) , Bisin and Guaitoli (2004) , and Attar et al. (2019) study models in which lenders, not borrowers, make offers, and find that equilibria are substantially different from those in models of multiple borrowing in which borrowers make offers, such as Bizer and DeMarzo (1992) and Kahn and Mookherjee (1998) . Specifically, they find that if moral hazard is particularly severe, a monopolistic outcome can be attained despite free entry of lenders. Another important feature, as identified by Segal and Whinston (2003) , is that allowing the proposal of menus of contracts affects equilibrium outcomes. In particular, Attar et al. (2019) highlights that allowing lenders to propose menus of contracts can induce constrained efficient outcomes. While lenders in our model cannot offer menus of contracts, our Nash bargaining model in Section 4.4 does nest the case in which lenders make offers, and we find that lenders' bargaining power strictly worsens allocations.
Our paper also speaks to the role of debt covenants in solving commitment problems between financial claimants of a firm, discussion of which dates back to Smith and Warner (1979) . Matvos (2013) and Green (2018) show empirically that corporate debt covenants create value. Covenants that limit indebtedness add value in our model by mechanically serving as an explicit form of commitment, and these are often observed in credit agreements of large firms in developed countries. Attar et al. (2019) also investigate the role of covenants in credit markets without commitment and come to a different conclusion. 5 They study a model in which lenders make menus of offers and show that the ability to offer contracts covenants (contingent on total financing raised) can fail to resolve com-mitment externalities. Their insight is that in an expanded contracting space there is an unraveling of the force that allows nonexclusive competition to induce constrained efficient outcomes. In contrast to this result, we argue in Section 4.5 that in our model covenants (or an equivalent tax) that allow interest rates to be contingent on subsequent financing do restore the constrained efficient allocation. This happens in our model, but not that of Attar et al. (2019) , because lenders in our model would only write covenants to protect against dilution and do not have the scope to consider how such covenants could be used strategically to crowd out other lenders and increase profits. Szentes (2015) shows more generally that the ability to contract on contracts in common agency settings generates substantial indeterminacy in equilibria by providing principals with the scope to collude arbitrarily, including the ability to implement a monopolist outcome.
We also build on the literature studying the concept of debt dilution: that issuing additional debt can reduce the value of a borrower's existing debt claims. 6 The simplest mechanism is that issuing debt of equal priority to existing debt can reduce existing debt' s value if the new debt derives some of its value from pre-existing cashflows. Thus, seniority arises as a natural resolution of debt dilution ( Hart and Moore, 1995 ). However, as noted in Bizer and DeMarzo (1992) and discussed in the literature on sovereign debt (e.g. Chatterjee and Eyigungor, 2015 ), debt dilution can also occur with seniority if junior issuances increase the likelihood of costly default on senior claims, for example through moral hazard. 7 The approach in our model blurs the distinction between these mechanisms. Dilution in our model arises from moral hazard, though we do not explicitly model new debt issues as subordinate to existing debt. Our model abstracts from seniority, as the cashflows backing the loan are binary. In a version of the model with richer state-contingent payoffs, seniority could dampen the dilution associated with sequential borrowing but cannot eliminate it completely due to moral hazard. 8 The process of debt accumulation in our model is similar to the temporal dynamics of leverage without commitment in Admati et al. (2018) and DeMarzo and He (2020) . With a complete lack of commitment [the limiting case of q = 1 in our model, and assumed throughout in Admati et al. (2018) and DeMarzo and He (2020) , borrowers are unable to obtain any value from increasing debt but do so anyway. The reason, as explained in Section 3.2 , 6 The definition of "debt dilution" is varied in the literature. Some papers consider dilution to mean the loss of value associated with issuing additional pro-rata claims. Other papers, including this one, define debt dilution more broadly as reduction in the value of existing debt arising directly or indirectly from the issuance of any new debt claims. 7 Also related, Donaldson et al. (2018) study the nuanced relationship between seniority, collateral, and negative pledge covenants when borrowers cannot commit ex ante to a future debt issuance policy and negative pledge covenants are imperfect in preserving the seniority of unsecured debt. 8 This is similar to Bisin and Rampini (2006) , who study an environment in which a primary lender can use the bankruptcy code to effectively establish itself as senior to other creditors, and find that this cannot completely overcome the distortions that arise from nonexclusive contracting.
is that lenders price in the fact that borrowers cannot commit to avoid taking debt issuance too far in the future, which dilutes the value of current debt.

Empirical evidence
How do the implications of our paper, and the broader theoretical literature of multiple borrowing, connect with what we observe in reality? The broadest empirical prediction we generate is that commitment is important for the efficiency of lending outcomes. We thus discuss the relevant empirical evidence in two contexts: in the developing world, where weak contracting environments make explicit commitment difficult, and in more sophisticated credit markets, where we would not expect commitment externalities to be as large of a problem.
The focus on commitment is well grounded in the reality of financing in developing countries around the world, which as it expands is becoming increasingly associated with the narrative of multiple borrowing crises (e.g., see Faruqee and Khalily, 2011 ). The quintessential example of multiple borrowing is the spectacular boom and bust of the nascent microfinance industry in the 1990s in the Indian state of Andhra Pradesh, where thousands of lenders allegedly entered the market to supply credit throughout the state with much of it in the form of "overlapping " loans from many lenders to the same borrower. Borrowers accumulated large debt balances from many lenders, who individually had no way of observing or controlling a borrower's total indebtedness. This ultimately proved unsustainable and culminated in a default crisis and near collapse of the industry in 2010. Multiple borrowing crises have occured around the world in other areas where microfinance has grown rapidly.
Several abstract facts have emerged from studies of multiple borrowing in microfinance. First, it appears that episodes of crisis associated with multiple borrowing are common in places where microfinance has experienced rapid growth; such as similar episodes and concerns about multiple borrowing have been described in Peru, Guatemala, Bolivia ( de Janvry et al., 2003 ), Uganda ( McIntosh et al., 2005 ) and Bangladesh ( Khalily et al., 2016 ). At the peak of its microfinance boom in 2009, surveys estimated that 84% of rural villagers in Andhra Pradesh had loans from multiple lenders, with the median household holding four loans from all sources ( Johnson and Meka, 2010 ). The same study also suggests that some borrowers in Andhra Pradesh borrowed from multiple lenders because they could not obtain sufficient financing from individual lenders. This is true in the equilibrium of our model; no lender will provide a very large amount of financing for fear that its claims will be diluted by other lenders. Of course, it is unclear if in practice limitations on how much lenders will lend to an individual borrower are a consequence of or impetus for multiple borrowing behavior.
Second, consistent with our model, there is some evidence of a positive relationship between the intensity of competition between microfinance institution (MFI) lenders and interest rates. Porteous (2006) explores the relationship between MFI competition and interest rates they charge by examining the cases of Bolivia, Bangladesh, and Uganda, all of which are considered to have very competitive microfinance markets. The study shows that the difference between MFI and formal bank interest rates declined significantly in Bolivia following substantial consolidation, while this spread had remained flat in Bangladesh and Uganda, where such consolidation has not yet occurred.
Third, debt limits and interest rate caps are often discussed as a response to problems arising from multiple borrowing and competition among microfinance lenders. In Bangladesh the microfinance regulator imposed rate caps, and the central bank considered broader interest rate limits ( Porteous, 2006 ). In response to the Andhra Pradesh crisis, in 2011 the Reserve Bank of India imposed new regulations on the banking industry and stated that they were in part meant to address multiple borrowing. A debt limit (USD 790) and interest rate cap (26%) were imposed ( Subbarao, 2011 ). As shown in Section 4.5 , these policies work by explicitly limiting the scope of commitment externalities in lending markets.
It is important to note that while these patterns can be explained by the mechanisms highlighted in our model, they could also be attributed to "growing pains" in new and rapidly expanding industries-the inexperience of new borrowers and lenders, and an intense desire to establish market share at the expense of sound lending practices ( Armendáriz and Morduch, 2005;Porteous, 2006 ). Also, while there is broad evidence that multiple borrowing is viewed as a problem by microfinance lenders and policymakers, empirical evidence is less clear. For example, Khalily et al. (2016) study documents correlational evidence that, within Bangladesh, households engaging in multiple borrowing are actually better off and do not have excessive levels of debt.
There is, to our knowledge, only indirect evidence on the predictions of our model related to investment productivity and the ability to obtain financing. 9 The relation we characterize between financial constraints and productivity generates a particularly stark form of misallocation: a negative correlation between the level of investment in a project and its productivity. This is consistent with a growing body of evidence from micro and experimental studies conducted in developing economies where productive firms can be especially credit constrained. McKenzie and Woodruff (2008) study microenterprise in Mexico and use a randomized experiment to estimate returns to investment capital. In addition to finding very high returns on average, they also find a positive relation between the returns to investment capital of borrowers and the financial constraints they face, indicating that it is the projects with the best economic fundamentals that are most underserved. Similarly, Banerjee and Duflo (2014) , using policy changes in credit access programs in India, find similar evidence of a negative selection effect. Their analysis of the relation between loan growth and profit growth suggests that it is the least productive firms that are acquiring the most financing.
While this evidence is consistent with our results, that higher productivity projects receive less investment, it does not specifically shed light on our prediction that lack of commitment in credit markets could cause entrepreneurs to purposefully pursue opportunities with lower returns to scale than otherwise available to them. However, this result provides at least a potential explanation for the emerging consensus that microfinance has failed to achieve the widespread poverty reduction once thought possible. Despite the widely held view that expanding credit access would promote entrepreneurial activity and in turn lead to growth, randomized control trials of micro-credit expansions find little impact on borrower income and business size ( Banerjee, Karlan and Zinman, 2015 ).
Turning to more developed credit markets, multiple and sequential borrowing is common, but there is less systematic concern about its negative implications. In consumer and small and medium enterprise credit markets, lenders are concerned about the possibility of default driven by multiple borrowing, but only in the context of fraud, in which inauthentic credentials are used to open and draw on a large number of credit accounts before informationsharing systems can detect it ( Phelan, 2016 ). Why does multiple borrowing not seem to be an explicit concern here outside of cases of pure fraud? It is possible that reputational mechanisms associated with the availability of a credit registry play an important role in aligning borrower incentives and preventing excessive borrowing. 10 Ivashina and Kovner (2011) find evidence for this in the market for large corporate debt contracts, showing that stronger relationships between a private equity owner of a firm and banks are associated with better pricing and less stringent debt covenants on the firm's debt. Prilmeier (2017) similarly finds that covenants are stronger earliest in the bank-borrower relationship. Among the most common covenants in corporate loans and bonds are limitations on both subsequent debt issuance and overall leverage and interest coverage ratios of a firm ( Billett, King and Mauer, 2007;Bradley and Roberts, 2015 .) These covenants not only establish the seniority of existing debt, but also prevent the issuance of junior or subordinate debt. Consistent with the discussion in Section 5 , this suggests that seniority alone cannot ameliorate externalities arising from multiple borrowing. In the syndicated loan market, "performance pricing" covenants tie the interest rate to be paid on debt to the leverage or total debt of a firm. 11 This contractual feature, similar to the Pigouvian tax we analyze in Section 4.5 , can force borrowers to internalize the effect that subsequent borrowing has on existing creditors.

Conclusion
This paper argues that commitment problems in lending markets can explain emerging empirical evidence that 10 Liberman (2016) finds that borrowers are willing to pay substantially to eliminate defaults from their credit record. 11 See Asquith et al. (2005) for a detailed description of performancebased pricing covenants. the rapid expansion of credit access can have perverse effects. When borrowers cannot commit to exclusive contracting, increasing the availability of lenders makes markets appear less competitive as interest rates rise and entrepreneur investment and welfare fall. More importantly, commitment problems can result in better projects receiving less investment than worse projects. This force can be so severe that what look like good opportunities are passed over for inferior investment technology. Finally, we show how simple regulatory tools such as interest rate ceilings and debt limits can improve outcomes and ameliorate the distortive forces we highlight.
The intuition for these results is that the externalities the lenders impose on each other when commitment or contingent contracting is not possible can prevent the borrower from being able to use pledgeable cash flows efficiently. When explicit commitment is impossible, there is value in any implicit commitment mechanism that attenuates the demand for further borrowing. Thus, the return profile of an investment opportunity itself is an important driver of the severity of commitment distortions.
Since commitment is less of a problem for projects with lower marginal returns, when given the choice entrepreneurs will endogenously choose investment opportunities that are everywhere less productive than other available opportunities, as long as they are sufficiently more concave. Thus, our model provides a new micro-foundation for the idea that commitment problems in lending markets can induce substantial misallocation in capital investment and can explain observations both of low growth and of economic activity below the technological frontier. While we have augmented our study with a sample of the growing anecdotal evidence that multiple borrowing is problematic, there is much more to learn. Formally testing the empirical validity of the mechanisms we highlight in explaining the failure of increased access to finance to significantly improve outcomes is an important topic for future research.