Relationship Lending and Liquidation Under Imperfect Information

Abstract

Using a model of a competitive credit market, we study a firm’s choice between financing a production project using a transaction loan and a relationship loan. The project itself is characterized by uncertainty, with regards to both the amount and the timing of revenue. While the transaction lender enjoys a relatively lower cost of funds, the relationship lender’s advantage lies in being able to make a relatively more informed decision about the continuation value of the project in the event that the firm misses its initial payment obligation. In this setting, we make two important findings. First, we document how the firm’s optimal choice of loan type is dependent on both the liquidation value of the project and how accessible transaction credit is to distressed firms. Second, we investigate an opportunity for a lender to improve the quality of its lending relationship with the firm, and find that, under imperfect information, the lender may choose to decline certain welfare improving innovations.

Appendix

Proof Proof of Proposition 1

Both banks employ strategies where they each offer the competitive contract that the firm most prefers. The firm then randomly selects one of these contracts. To identify the firm’s preferred contract, consider the following two cases.

• Case 1. Assume that $\frac {p -q}{1 -q}R \geq 1$. This means that if the firm takes a transaction loan, the firm’s payoff is pR − 1. In this case, the firm prefers the relationship loan over the transaction loan if

  $$ qR +\alpha (p -q)R +[1 -q -\alpha (p -q)]X -1 -c \geq pR -1 $$

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  $$ X \geq \frac{(1 -\alpha )(p -q)R +c}{1 -q -\alpha (p -q)} $$

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  Say that X satisfies this inequality. We now establish whether the bank’s plan for a relationship loan at t = 1 is credible. From earlier, we know that credibility holds as long as

  $$ X \geq \frac{(1 -\alpha )(p -q)\beta R}{1 -q -\alpha (p -q)} $$

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  Thus, when the firm prefers the relationship loan over the transaction loan, the bank’s plan under the relationship loan is indeed credible if the following is true

  $$ \frac{(1 -\alpha )(p -q)R +c}{1 -q -\alpha (p -q)} >\frac{(1 -\alpha )(p -q)\beta R}{1 -q -\alpha (p -q)} $$

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  This clearly holds.

• Case 2. Now assume that $X \geq \frac {p -q}{1 -q}R$. In this case, if the firm takes the transaction loan, the firm’s payoff is qR + (1 − q)X − 1. Hence, the firm prefers the relationship loan over the transaction loan when

  $$ qR +\alpha (p -q)R +[1 -q -\alpha (p -q)]X -1 -c \geq qR +(1 -q)X -1 $$

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  $$ \frac{\alpha (p -q)R -c}{\alpha (p -q)} \geq X $$

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  Given that $X \geq \frac {p -q}{1 -q}R$ for case 2 of our proof, to establish the credibility of the bank’s plan, it is sufficient to show that

  $$ \frac{p -q}{1 -q}R \geq \frac{(1 -\alpha )(p -q)\beta R}{1 -q -\alpha (p -q)} $$

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  $$ \frac{1 -q -\alpha (p -q)}{(1 -q)(1 -\alpha )} \geq \beta $$

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  Since β < 1 by assumption, it is sufficient to show that the left-hand side of this inequality is not less than 1, i.e.,

  $$ \frac{1 -q -\alpha (p -q)}{(1 -q)(1 -\alpha )} \geq 1 $$

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  $$ 1 \geq p $$

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Proof Proof of Proposition 2

As part of the equilibrium, we assign beliefs for the firm such that if all relationship loan contracts contain a repayment, L₁, where $L_{1} \geq L_{1}^{RL}$, then the firm believes each bank is equally likely to be the innovating bank.⁶ If one or more banks offer a repayment where $L_{1} <L_{1}^{RL}$, then the firm believes the bank with the lowest offer is the innovating bank.

Working backwards, suppose that the bank did choose to innovate. We then show the banks’ strategies for making loan offers in the subgame is Nash behavior.

First, consider deviations by the non-innovating bank. Under the conditions given in the proposition, the firm prefers the relationship loan over the transaction loan, so there is no reason to offer a transaction loan. Given the profile of strategies, the non-innovating bank earns zero. A deviation where this bank offers $L_{1}^{RL}$ with probability 1 earns the bank zero. An offer below $L_{1}^{RL}$ results in negative profit, and any pure strategy offer above $L_{1}^{RL}$ earns the bank zero.

Second, we show that the innovating bank has no profitable deviations. Any deviation where this bank offers a relationship loan with a repayment below $L_{1}^{RL}$ or above $L_{1}^{RL} +\eta$ will not raise the bank’s current profit, namely $\pi _{i}(L_{1}^{RL})$.

Consider a δ where 0 < δ < η and suppose the innovating bank offers $L_{1} =L_{1}^{RL} +\delta$. The probability that the other bank will offer a relationship loan with a lower repayment is then δ/η. Thus, the expected profit for the deviating bank is

$$ \frac{\delta }{\eta }0 +(1 -\frac{\delta }{\eta })\pi_{i}(L_{1}^{RL} +\delta ) $$

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$$ (1 -\frac{\delta }{\eta })[q(L_{1}^{RL} +\delta ) +\alpha_{h}(p -q)\beta R +(1 -q -\alpha_{h}(p -q))X -1 -c -z] $$

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$$ (1 -\frac{\delta }{\eta })[q\delta +\pi_{i}(L_{1}^{RL})] $$

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To ensure that the bank does not have a incentive to deviate, we require that

$$ (1 -\frac{\delta }{\eta })[q\delta +\pi_{i}(L_{1}^{RL})] <\pi_{i}(L_{1}^{RL}) $$

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$$ \delta [q -q\frac{\delta }{\eta } -\frac{1}{\eta }\pi_{i}(L_{1}^{RL})] <0 $$

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$$ \eta <\delta +\frac{1}{q}\pi_{i}(L_{1}^{RL}) $$

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$$ \eta <\delta +\frac{1}{q}[qL_{1}^{RL} +\alpha_{h}(p -q)\beta R +(1 -q -\alpha_{h}(p -q))X -1 -c -z $$

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$$ \eta <\delta +\frac{1}{q}(p -q)(\alpha_{h} -\alpha )[\beta R -X] -\frac{1}{q}z $$

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This holds by assumption.

Finally, given that (α_h − α)(p − q)[βR − X] > z, if the bank deviates and declines the innovation the bank will then earn zero and hence, be worse off. □

Proof Proof of Proposition 3

Under the conditions given, the bank chooses to innovate only if

$$ (\alpha_{h} -\alpha )(p -q)[\beta R -X] >z $$

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To determine when an innovation is welfare improving, we can assume the bank innovates and then calculate the relationship loan contract that earns the bank zero expected profit:

$$ L^{\ast } =\frac{1}{q}[1 +c +z -\alpha_{h}(p -q)\beta R -(1 -q -\alpha_{h}(p -q))X] $$

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At this repayment, we now examine exactly when the firm is better off due to innovation. In this case, the firm is better when

$$ q[R -L^{\ast }] +\alpha_{h}(p -q)(1 -\beta )R >q[R -L_{1}^{RL}] +\alpha (p -q)(1 -\beta )R $$

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$$ \begin{array}{@{}rcl@{}} &&-z +\alpha_{h}(p -q)\beta R -\alpha_{h}(p -q)X +\alpha_{h}(p -q)(1 -\beta )R\\ &>&\alpha (p -q)\beta R -\alpha (p -q)X +\alpha (p -q)(1 -\beta )R \end{array} $$

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$$ (\alpha_{h} -\alpha )(p -q)[R -X] >z $$

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