Product market competition and access to credit

Abstract

In this paper, we unveil a disregarded benefit of product market competition for firms. We introduce the probability of bankruptcy in a simple model where firms compete à la Cournot and apply for collateralized bank loans to undertake productive investments. We show that the number of competitors and the existence of outsiders willing to acquire the productive assets of distressed incumbents affect the equilibrium share of investment financed by bank credit. Using a sample of Italian manufacturing firms, mostly small- and medium-sized enterprises (SMEs), we found evidence showing that the degree of product market competition is positively correlated with the share of investment financed by bank credit only when outsiders are absent.

Appendices

Appendix A: Proofs

1.1 A.1 Triopoly

To simplify the notation, we anticipate that $q^{\ast }\left (3\right )$ is the equilibrium capacity set by each rival at date 0. The representative firm A’s expected profit function at date 0, U _A , is thus given by

$$\begin{array}{@{}rcl@{}} U_{A}&=&p\left( -r_{A}+P_{3}q_{A}\right) +p\left[ 2p\left( 1-p\right) \frac{1}{2}P_{3}q^{\ast}\left( 3\right)\right.\\ &&\left.+\vphantom{\frac{1}{2}}\left( 1-p\right)^{2} P_{3}2q^{\ast}\left( 3\right) \right] +\left( 1-p\right) 0-M. \end{array} $$

(12)

When firm A is healthy—with probability p —at date 1 it repays r _A to bank A, produces at the maximum capacity q _A without additional production costs and it earns P ₃ q _A , where $P_{3}=1-\left [ q_{A}+2q^{\ast }\left (3\right ) \right ]$ indicates the price of the homogeneous good when the total production is equal to the total capacity, $q_{A}+2q^{\ast }\left (3\right )$, regardless of the allocation of PAs among healthy firms. Moreover, when one rival, either firm 2 or firm 3, fails—this occurs with probability $2p\left (1-p\right )$ —both firm A and the healthy rival are willing to purchase the failing firm’s PAs; firm A acquires these PAs with probability $\frac {1}{2}$ and gets the extra-revenue $P_{3}q^{\ast }\left (3\right )$. When both rivals fail—this occurs with probability (1−p)² —firm A is the only potential buyer of the two rivals’ PAs and acquires them with probability 1; its extra-revenue is $P_{3}2q^{\ast }\left (3\right )$. By contrast, if firm A fails—with probability (1−p) —it earns nothing. Finally, M denotes the opportunity cost of firm A’s own funds.

The expected profit function of bank A, V _A , is given by

$$\begin{array}{@{}rcl@{}}V_{A}=p\left[ r_{A}-2p\left( 1-p\right) \frac{1}{2}P_{3}q^{\ast}\left( 3\right) -(1-p)^{2}2\varepsilon\right] \\+(1-p)\left[ p^{2}P_{3} q_{A}+2p(1-p)\varepsilon\right] -\left( cq_{A}-M\right) . \end{array} $$

When firm A is successful—this occurs with probability p —bank A receives r _A . Moreover, when only one rival fails—with probability $2p\left (1-p\right )$ —bank A lends an expected extra amount $\frac {1}{2}P_{3}q^{\ast }\left (3\right )$ to firm A, which offers $P_{3} q^{\ast }\left (3\right )$ to buy the PAs of the failing rival, $\frac {1}{2}$ being the probability that firm A obtains the PAs on sale and actually pays the offered price. With probability $\left (1-p\right )^{2}$, bank A funds the amount 2ε offered by firm A to acquire the PAs of both failing rivals. By contrast, firm A fails with probability 1−p. When both rivals are healthy—with probability p ² —bank A sells firm A’s PAs at price P ₃ q _A ; with probability $2p\left (1-p\right )$ only one rival, either 2 or 3, is healthy and buys at price ε. Finally, the last term, c q _A −M, is the opportunity cost of the amount lent to firm A, since we assume zero risk-free interest rate. Note that the expected cost of the extra credit in the event that firm A is the only healthy one, −p(1−p)²2ε, cancels with the expected value recovered from the sale of firm A’s PAs in case only one rival is healthy, $2\left (1-p\right )^{2}p\varepsilon$.

To obtain the firm A’s expected profits at date 0, we first calculate the expected repayment p r _A required by bank A to break even:

$$ pr_{A}=\left( cq_{ A}\!-M\right) +p^{2}\left( 1-p\right) P_{3}q^{\ast}\left( 3\right) -\left( 1-p\right) p^{2}P_{3}q_{A}. $$

(13)

The above value is positively affected by the opportunity cost of lending, $\left (cq_{A}-M\right )$, and by the expected extra-credit to firm A when the firm competes with an healthy rival to buy the failing firm’s PAs, $2p^{2}\left (1-p\right ) \frac {1}{2}P_{3}q^{\ast }\left (3\right )$. On the contrary, p r _A is negatively affected by the expected value recovered by bank A from the sale of firm A’s PAs when the two rivals are healthy, $\left (1-p\right ) p^{2}P_{3}q_{A}$. Plugging Eq. 13 into Eq. 12 gives Eq. 4 where $q_{B}+q_{C}=2q^{\ast }\left (3\right )$.

1.2 A.2 Entry by outsiders

Duopoly Firm A’s expected profit function at date 0 is given by (1). Bank A’s expected profit function is instead affected by the new expected liquidation value of PAs and given by

$$\begin{array}{@{}rcl@{}} \begin{array}{c} V_{A}=p\left[ r_{A}-(1-p)\left( P_{2}q_{B}+\varepsilon\right) \right] +\\ \left( 1\,-\,p\right) \left[ p\left( P_{2}q_{A}-E+\varepsilon\right) +\left( 1-p\right) \left( P_{2}q_{A}\!-E\right) \right] -\left( cq_{A}-M\right) \text{.} \end{array} \end{array} $$

The bank’s break-even condition is

$$pr_{A}\,=\,p(1-p)P_{2}q_{B}-\left( 1\,-\,p\right) P_{2}q_{A}+\left( 1-p\right)^{2}E+\left( cq_{A}\,-\,M\right) \!. $$

Plugging this value into Eq. 1 yields Eq. 5.

Triopoly A similar reasoning can be invoked to compute Eq. 6 under triopoly.

1.3 A.3 Self-financed firms

To compute the representative self-financed firm A’s expected profit, we rely on the following reasoning. With probability p firm A actually competes in the product market by gaining P ₂ q _A . Moreover, when firm A is the only buyer because firm B is failing—probability $p\left (1-p\right )$ —firm A’s extra-profit is $p\left (1-p\right ) \left [ P_{2}q_{B}-\varepsilon \right ]$, where P ₂ q _B is the extra-revenue thanks the additional capacity q _B and ε is the offer to acquire the rival’s PAs. With probability $\left (1-p\right )$ firm A is in distress, in which case it cashes the expected liquidation value of its PAs, $\left (1-p\right ) p\varepsilon$. Finally, firm A incurs the cost c q _A of installing the capacity q _A . Overall, the expected profit of firm A in case it is self-financed amounts to

$$p\left[ P_{2}q_{A}+\left( 1-p\right) \left( P_{2}q_{B}-\varepsilon\right) \right] +\left( 1-p\right) \varepsilon-cq_{A}. $$

Note the the above expression is equivalent to Eq. 4 after rearrangement. The result follows.

Appendix B: Additional table

In Table 3 we report the estimates of the parameters of Eqs. 7–9.