Debt Financing and Output Market Behavior

Transcription

1 Debt Financing and Output Market Behavior Jaideep Chowdhury Hans Haller October 11, 2011 Abstract This paper revisits the seminal article of Brander and Lewis who have shown that exogenously given or strategically chosen debt can lead to more aggressive firm behavior in the output market. We find that (i) debt financing of production costs leads to more aggressive output market behavior when the level of upside risk is higher and (ii) debt financing of production costs leads to more aggressive product market behavior than equity financing without debt. Our findings prove robust across several industry structures: Cournot duopoly, Stackelberg leadership, and monopoly. Keywords: Product Markets, Capital Markets, Limited Liability Effect JEL Classification: L13, G32 We thank Hans Gersbach for an instructive conversation and a co-editor and referee for beneficial critique. Department of Finance and Business Law, James Madison University, Harrisonburg, Virginia 22807, USA. chowdhjx@jmu.edu Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA. haller@vt.edu. 1

2 1 Introduction We investigate the linkage between capital market and product market decisions of firms, which places our study at the interface of theoretical corporate finance and theoretical industrial organization. Traditionally, there has been a fruitful division of labor between financial theory and economic theory. In financial theory, the consequences of financial decisions of the firm are analyzed typically without considering their impact on product market decisions and vice versa. The product market is treated as exogenous and assumed to simply offer a random return that is unaffected by the debt-equity positions of the firms under consideration. In economic theory, the financial situation of firms tends to be ignored when product market decisions (on quantity, quality, prices, etc.) are analyzed. 1.1 The Limited Liability Effect In their seminal contribution, Brander and Lewis (1986) stress and examine important linkages between financial markets and product markets, in particular the effect of leverage on equilibrium output in Cournot duopoly with demand or cost uncertainty. They assume that the firm is a residual claimant to (high) earnings while it is protected from losses by limited liability. The latter causes risk shifting which leads a more leveraged firm to behave in a more aggressive manner in the product market. Brander and Lewis call this the limited liability effect which obtains in their normal or standard case when marginal profit of output is larger in better states. Moreover, a leveraged monopolist (duopoly) will choose higher outputs than an unleveraged monopolist (duopoly). 1.2 Endogenous Debt and Investment On the one hand, we specialize and consider industries with a linear demand curve and constant marginal costs. On the other hand, we amend and extend the Brander and Lewis analysis and, in section 4, qualify and substantiate some of their results. We examine three different industry structures, Cournot duopoly, Stackelberg leadership, and monopoly and find that our main results are robust with respect to industry structure. However, most of the discussion and some of the extensions are confined to Cournot duopoly, the leading case in the literature. In our first innovation in the Brander-Lewis framework, we follow Povel and Raith (2004) and concentrate on endogenous debt incurred by Cournot duopolists to finance their variable costs. But in contrast to Povel and Raith, we refrain from designing sophisticated liquidation schemes. Rather, we explore the possibility that firms can obtain funds in a competitive loan market. In all other respects, our model is very close to the original Brander and Lewis (1986) model. Indeed, our model proves closer to the Brander and Lewis model than their own interpretation suggests: Their formal analysis is based on debt-financed production costs whereas their narrative suggests equity-financed production costs. We find in Proposition 2 that debt financing of production costs leads to more aggressive product 2

3 market behavior than equity financing without debt. This conclusion shows that the opposite result obtained by Povel and Raith is mainly due to firm-specific debt contracts and liquidation costs and not to endogenous debt per se. The Brander and Lewis result that a more leveraged firm behaves in a more aggressive manner in the product market is rather an assumption than a conclusion in our basic model: By assumption, variable costs are debt financed whereas there is no prior or exogenous debt. In turn, variable costs are increasing in output and, therefore, debt is increasing with output and vice versa. Therefore, while the particular Brander and Lewis result obtains, it is not caused by limited liability in our model. This does not mean, however, that the limited liability effect is not at work at all. To get a better understanding how it works in our context, notice that the limited liability effect can prove sensitive to both the type of imperfect competition and the nature of uncertainty faced by the oligopolists: Showalter (1995) considers Bertrand competition instead of Cournot competition and finds that while debt yields a strategic advantage under demand uncertainty, it does not under cost uncertainty. In our model, firms are quantity setters and face demand uncertainty. A priori, this creates a bias in favor of the limited liability effect. Our second major innovation consists in performing comparative statics with respect to the magnitude of upside risk, in order to further scrutinize the impact of risk on output market behavior when limited liability applies. We find in Proposition 1 that greater upside risk increases the benefits from risk shifting and, consequently, induces the firm(s) to choose more output and debt. 1.3 Related Literature The limited liability effect can have further implications in dynamic scenarios like in Maksimovic (1988), who argues that (again in the standard case) high levels of debt may prevent a firm from credibly committing itself not to engage in disruptive competitive practices; in other words, high levels of debt may prevent the firm from achieving optimal tacit collusion with its rivals. In contrast, Glazer (1994) argues that in the standard case, high levels of debt can render firms temporarily less aggressive and can be conducive to tacit collusion. He analyzes a two-period model where debt is issued for two periods while firms choose output every period. Firms use operating profits to pay back debt before paying dividends. Therefore, high first period profits reduce the remaining (short-term) debt(s) and by the Brander-Lewis result second period output(s). Thus Glazer finds that with long-term debt, firms may prefer lower first period output than without debt, followed by higher second period output. More generally, with long-run debt and several periods, firms may choose low outputs and tacitly collude in the early stages. The limited liability effect becomes obsolete when firms are assumed to maximize total value rather than equity/shareholder value. Most notably, Modigliani and Miller (1958) investigate how the total value of the firm depends on the mix of debt and equity financing of investments and find that under certain assumptions, total value is independent of the 3

4 financial structure. Brander and Lewis (1988) intentionally rule out the limited liability effect by assuming total value maximization rather than equity value maximization. The corporate finance and economic theory literatures have developed models of firmspecific debt contracts. Contractual stipulations can take into account the needs and opportunities of the firm as well as the interests of its creditor(s). In particular, firm specificity allows the contract to condition the probability of bankruptcy and/or liquidation on the amount of actual repayments. Such contracts help avoid or alleviate the moral hazard problem that the firm s unscrupulous management diverts without penalty some or all of its cash flow when cash flow or operating profit is unverifiable by third parties. Such contracts may also alleviate risk shifting and mitigate the limited liability effect. By and large, this literature finds that capital market imperfections and predatory behavior of competitors put financially constrained or distressed firms at a disadvantage and induce more leveraged firms to behave less aggressively in output markets. See, e.g., Fudenberg and Tirole (1986), Poitevin (1989), Bolton and Scharfstein (1990), Maurer (1999), Faure-Grimaud (2000). Most of the debt contract literature assumes that a firm or entrepreneur has to finance a fixed start-up or project cost. Povel and Raith (2004) depart from that tradition and consider a Cournot duopoly where one of the firms, say firm 1, is financially constrained and has to finance all or part of its variable cost by borrowing from an investor. Financing and production decisions are made before the realization of uncertain inverse market demand. Firm 1 has a positive continuation value after the Cournot duopoly takes its course. The continuation value may be reduced if the firm is liquidated. An optimal debt contract minimizes the expected liquidation cost. It assumes the form of randomly liquidating the firm in the case of default. The stipulated probability of liquidation is decreasing in the amount repaid. The contract induces the firm to borrow not more than it needs to, to produce at most the Cournot equilibrium output it would choose as a financially unconstrained firm, and to always repay as much as it can. Thus the optimal contract is designed in such a fashion that the limited liability effect is neutralized by the countervailing threat of liquidation. An obvious question is whether both the firm and the investor would have a full understanding of the firm s financial and economic situation so that a contract could be designed exactly that way and, more importantly, whether the contract provisions (random liquidation) would be carried out as stipulated. 2 Preliminary Analysis The limited liability effect à la Brander and Lewis (1986) has the following explanation: The firm s debt holders are residual claimants in case of bankruptcy. In the absence of bankruptcy, the firm s equity holders are residual claimants. Hence the equity holders of the firm are only concerned with its returns in the good states of nature. The manager of the firm caters to the equity holders of the firm. The 4

5 manager pursues risky strategies which provide higher return in the good states of nature because the firm does not have to pay back the entire debt in case of bankruptcy. The firm is said to be protected by the limited liability effect of debt financing. The limited liability protection induces the levered firm to undertake more risky projects and behave more aggressively in the output market. In the Brander-Lewis model, firm i is assumed to have returns of the form R i (z; q i, q i ) D i (1) where z is an ex ante uncertain state of nature, q i 0 is firm i s output and D i 0 is i s exogenously given debt. The standard assumptions are that z belongs to an interval [z, z] and R i / z > 0. In the standard or normal case, 2 R i / q i z > 0. q i stands for the output of the other firm (duopolist) if there is one and can be omitted or put equal to 0 otherwise. It is implicitly assumed that there exist an inverse market demand P (z; q) at industry output q = q i + q i and a cost function C i (z; q i ) so that R i (z; q i, q i ) = P (z; q i + q i )q i C i (z; q i ). A further crucial assumption is that for each (q i, q i ), there exists a unique state = (q i, q i ) (z, z) such that R i (, q i, q i ) D i = 0. (2) Limited liability means that equity holders are protected from making losses. That is, their return is R i (z; q i, q i ) D i if z and 0 if z. Therefore, a manager who maximizes equity or shareholder value, maximizes [R i (z, q i, q i ) D i ]F (dz) (3) with respect to q i, given q i, where F is the distribution of z. In the case of a monopolist, q i = 0. In the case of Stackelberg leader i, q i = q j (q i ) where q j (q i ) is the reaction function of the follower j. The gothics D i and R i always refer to the respective terms in the Brander- Lewis model. In principle, the firm might incur a total debt of C i (z; q i ) + D i if its costs of production are debt-financed. Brander and Lewis (1986) and subsequently Maksimovic (1988) and Glazer (1994) presume that the firm s production costs are equity-financed. However, we beg to disagree: The formal analysis in Brander and Lewis (1986) relies on debt-financed production costs. We refer to subsection 4.3 for a detailed discussion of this discrepancy. Povel and Raith (2004) assume that the firm finances production internally to the extent that it can and resorts to debt financing for the rest. In our basic model in section 3, we assume absence of exogenous debt (for simplicity), zero fixed, set-up or sunk costs, and marginal cost of production c 0. We further assume that the firm issues debt to finance its operating cost so that it will have debt equal to D i = cq i. (4) 5

6 Thus there is an direct linkage between the financial decision and the output decision. Remark 1. The cost of production (4) includes interest payments. The firm pays its suppliers kq i, the market value of debt, for factors of production. Constant marginal costs assume constant returns to scale and competitive markets for factors of production. For instance, if there is only one factor of production, the production function is of the form f(y) = ay for input y 0 and the factor price is p, then k = p/a. The firm, if it debt-finances its input requirements, owes its creditors the amount cq i, the face value of debt, where c = (1 + r)k and r is the rate of interest. As creditors, we think of banks, although supplier or trade credit also plays an important role in practice. When production is equity financed, cq i is the firm s opportunity cost, if the interest rates for borrowing and saving (alternative investments) are the same. Propositions 2 and 4 are shown under this assumption. If the rates are different but the spread between borrowing and saving rates is not too large, the conclusions of Proposition 2 and 4 still hold. For details, we refer to Remark 5. Remark 2. In models of the credit market, two polar market structures of the banking (creditor) sector are pre-dominant. On one end of the spectrum, a borrower deals with a monopoly lender. On the other end, borrowers deal with a perfectly competitive banking sector. Needless to say, there exist models of imperfectly competitive credit markets as well. Here we assume an exogenously given market interest rate r. We refrain from endogenizing r by imposing a break-even constraint on banks. The rationale is that banks may not be fully informed about the model parameters k and θ, z, z below. They may lend to many similar but not identical firms. While a bank need not break even on the loan to a particular firm, it can still break even on its loan portfolio. The latter condition could be used to endogenize r which would conceivably include a market risk premium. We indicate in Appendix E how this might be done. Incomplete information about industry parameters in addition to competition among creditors and other impediments to complete contracts is also a reason why lenders cannot always impose firm-specific contracts à la Povel and Raith (2004). Thus we presume that banks use a relatively unsophisticated rating procedure so that many different firms end up in the same rating category and are charged the same interest rate. The application of sophisticated rating and risk management tools by commercial banks to credit markets is a recent development, mainly driven by regulatory requirements of the Basle accords and recommendations. Now this sort of banking regulation is not concerned with the pricing of loans per se but rather with adequate loan loss provisions. Yet if the sophisticated techniques were costless, they would be widely used even beyond regulatory requirements, since more detailed rating and loan assessment allows a more differentiated pricing of loans. However, advanced statistical and computational tools do not make up for poor data quality. The accurate assessment of the risk of specific commercial loans and individual firms requires costly industry-specific expertise and front office input. Therefore, while the trend is towards more sophistication, cost-benefit considerations may still favor traditional simpler rating procedures that refrain from too detailed investigations of individual loans and companies. For an elaboration on these issues, see Part III of Basle Committee on Banking Supervision (1999). 6

7 Further, there is an inverse market demand of the affine-linear form P (z; q) = θ + z q for q θ + z where θ > c is a positive constant and z is a random parameter. We further assume that z is uniformly distributed on a non-degenerate interval [z, z] with constant density f = 1/(z z). Then the firm has operating revenue and operating profit R i (z; q i, q i ) = (θ + z q i q i )q i (5) R i (z; q i, q i ) D i = (θ + z c q i q i )q i. (6) Now the counter-part of (2) is R i D i = 0 or, if q i > 0, provided that c θ + q i + q i (z, z). = c θ + q i + q i (7) Remark 3. Suppose (θ c) (<) z. Then c θ + q i + q i (<) z for q i + q i = 0. Assuming (θ c) > z rules out that possibility and guarantees that c θ + q i + q i > z for all q i + q i 0. Suppose next that (θ c) z. Then c θ + q i + q i z for all q i + q i 0 in which case the firm can never serve its debt and makes zero expected profit. To rule out that possibility, we are going to assume (θ c) < z. Accordingly, we make the additional assumption z < (θ c) < z. (8) Remark 4. Note that (8) implies that c θ+q i +q i has range [c θ, ) for q i +q i 0, with c θ (z, z). In particular, (8) does not rule out that c θ +q i +q i z for sufficiently large values of q i + q i. But whenever c θ + q i + q i z, the firm cannot serve its debt in any state of nature and makes zero expected profit. In that case, we can set = z so that the maximand (9) becomes zero. Then with limited liability, the firm s manager maximizes with the same qualifications as (3). [R i (z; q i, q i ) D i ]fdz, (9) 3 Demand Uncertainty and Endogenous Debt We assume that the firms are issuing debt to finance production thereby interlinking debt with output as expressed by equation (4). Further, we assume that the firms perfectly foresee the switching state of nature which is given by equation (7) if c θ + q i + q i (z, z) and by = z if c θ + q i + q i z. A firm with perfect foresight should know that is a 7

8 function of q = q i + q i and hence treat as endogenous. In the sequel, we are considering three market structures: Cournot duopoly, Stackelberg leadership, and monopoly. cournot duopoly For firm i = 1, 2, the manager maximizes the firm s equity value (9), that is, solves max q i V i = (R i D i )fdz = where R i is the profit of firm i and debt D i is given by equation (4). The Cournot Duopoly solution is given by Proof : See Appendix A. q c 1 = q c 2 = q c = θ c + z 4 (θ + z q 1 q 2 c)q i fdz (10) (11) stackelberg solution We assume, without loss of generality, that firm 1 is the leader and the firm 2 is the follower. Firm 1, the leader, operates on the reaction function of the follower. Both firm 1 and firm 2 have perfect foresight. For both, and debt D i = cq i are endogenous variables. The Stackelberg solution is given by Proof : See Appendix B. q s 1 = θ c + z 3, q s 2 = 2(θ c + z). (12) 9 monopoly output A monopolist with perfect foresight solves the following problem: max V = q (R D)fdz = (θ + z q c)qfdz where R is the revenue of the monopoly firm and debt D is given by (4): D = cq. The monopoly output is given by Proof : See Appendix C. qm = θ c + z. (13) 3 8

9 Main Conclusions As an immediate conclusion of the foregoing equilibrium analysis we obtain the following Proposition 1 For all three market forms, Cournot, Stackelberg and Monopoly, the debt financed firm(s) produce(s) more output at a higher level of upside demand risk measured by z. Ceteris paribus, an increase of z expands the good states of nature which reinforces the limited liability effect. More detailed analysis shows: Proposition 2 For all three market forms, Cournot, Stackelberg and Monopoly, debt financed firms will choose higher outputs than equity financed firms. Proof: See Appendix D. 4 Reexamination of Brander and Lewis In this section, we first determine the Cournot equilibrium outputs when firms do not assume any debt and then for firms à la Brander and Lewis (1986). The ensuing equilibrium outputs allow us to reassess the Brander and Lewis findings. 4.1 No Debt In case of no (exogenous or endogenous) debt, the firm solves the following problem: max q i V i = z (R i C i )fdz = Firm i s first order condition is given by The reaction function of firm 1 is given by z (θ + z q 1 q 2 c)q i fdz = (θ + (z + z)/2 q 1 q 2 c)q i. θ + (z + z)/2 c 2q i q i = 0. The Cournot solution is q i (q i ) = θ c 2 q c i = θ c 3 + z + z 4 q i 2. (14) + z + z, i = 1, 2. (15) 6 9

10 4.2 Exogenous Debt In the Brander and Lewis case, where debt is exogenous, the manager of firm i solves max q i V i = where is defined by (R i D i )fdz = (θ + z q 1 q 2 c)q i fdz D i fdz (16) D i = R i ( ; q 1, q 2 ) (2) and R i is the operating profit of firm i, with analogous stipulations as in Remarks 4 and 5. Maximizing (16) with respect to q i, we get the first order condition as follows: V i = 1 q i z z R i dz = 1 q i z z (z )(θ 2q i q i c)+ 1 (z )(z +) = 0. (17) 2(z z) Notice that V i / q i comprises the additional term [R i ( ; q 1, q 2 ) D i ] / q i which is zero because of (2). It follows that the reaction function of firm i is given by q i (q i ) = θ c 2 + z + 4 With D 1 = D 2 = D, equilibrium outputs are given by q i = θ c 3 q i 2. (18) + + z, i = 1, 2. (19) 6 For q i > 0, equation (2) amounts to (θ + (q i + q i ) c)q i = D i or = c θ + (q i + q i ) + D i /q i. With D 1 = D 2 = D, q 1 = q 2, and (18), the equilibrium output satisfies 0 = [ 4qi 2 (θ c + z)q i D or qi = θ c + z + ] (θ c + z) D /8. (20) 4.3 Reexamining Brander and Lewis Comparison of (15) and (19) yields that firms in the Brander and Lewis setting with exogenous debt D 1 = D 2 = D choose higher outputs than firms that do not incur any debt. This follows from (z, z). The explicit determination of Cournot equilibrium outputs in the Brander and Lewis setting with exogenous debt D 1 = D 2 = D yields expression (20), which confirms the Brander and Lewis finding that more leveraged firms take a more aggressive output stance. Moreover, expression (20) converges to expression (11) as D tends to 0 which shows: Proposition 3 A duopoly with D > 0 produces higher equilibrium outputs than a duopoly with D = 0. 10

11 This is also the formal result of Corollary 1 in Brander and Lewis (1986). However, contrary to the interpretation of Brander and Lewis, this result by itself does not prove that a completely equity-financed industry will produce a lower output than the corresponding leveraged industry. The reason is that D = 0 does not mean that the industry is equityfinanced. Namely, the model we analyze in section 3 exhibits zero exogenous debt, D = 0, whereas production is debt-financed. Hence our model is the limit case, for D 0, of the Brander and Lewis model with exogenous debt. To substantiate our claim, let us reexamine the Brander-Lewis assumption that the payoff to equity holders is max{0, R i (z, q i, q i ) D i }. With our model specification, their assumption means that equity holders receive max{0, R i (z, q i, q i ) [D i + D i ]} where D i is the endogenous cost of production, given by (4). That is, limited liability applies to both exogenous debt D i and the cost of production, D i = cq i. Thus, the Brander and Lewis analysis incorporates both exogenous and endogenous debt and obtains our model with zero exogenous debt yet debt-financed production as a limit case. Nevertheless, the combination of Propositions 2 and 3 does yield the Brander and Lewis assertion as does direct comparison of (15) and (20): Proposition 4 A completely equity-financed industry will produce a lower output than the corresponding leveraged industry. The analysis underlying (20) still applies when D becomes negative, that is, when the firms have some positive financial reserves to begin with and debt finance only part of their production costs. Hence: Proposition 5 There exist D < 0 < D such that symmetric Cournot duopoly equilibrium output is increasing in D [ D, D ]. Remark 6 (Interest rate differentials). Suppose the opportunity cost of a selffinancing firm i producing output q i is c q i = (1 + r )kq i where r < r is the rate of interest or rate of return on funds put to alternative uses. If r rather than r is used in (20), then the comparison between (15) and (20) can be reversed and Proposition 4 need no longer hold. To see this, put D = 0 in (20) and consider the following numerical example: k = 1, θ = 5/4, c = 9/8, c = 33/32, z = 0, z (θ c ) = 7/32 so that r = 1/8, r = 1/32. Then (20) amounts to qi = (θ c + z)/4 = (1/8)(1/4) = 1/32 and (15) with c instead of c amounts to qi c (θ c )/6 = (1/6)(7/32) = (7/6)(1/32). This yields qi c > qi which means that without exogenous debt, the completely equity-financed duopoly produces a higher output than the corresponding debt-financed duopoly. However, Proposition 4 still holds if r < r is sufficiently close to r. In the example, let us replace c and r by c = 35/32 and r = 3/32, respectively. Then (15) becomes qi c (7/6)(1/32) (1/3)(2/32) = (1/2)(1/32) and qi c < qi. More generally, the conclusion of Proposition 4 persists as long as c > c > c (θ c + z)/2 and z (θ c ). Notice that θ c + z > 0 because of (θ c) < z. Further, z (θ c ) implies (θ c )/3 + (z + z)/6 (θ c + z)/6 and c > c > c (θ c + z)/2 implies (θ c + z)/6 < (θ c + z)/4. Hence c > c > c (θ c + z)/2 and z (θ c ) yield qi c < qi when c is replaced by c in (15). 11

12 5 Proofs Remark 5. The expression in (26) is a cubic in the monopoly output q and goes to infinity as q tends to infinity. However, the relevant domain for q is [0, z + θ c], since the monopolist makes zero profit and we can set = z if q z + θ c; see Remark 4. The first and second order conditions show that (26) has a local minimum at q = z + θ c and a local maximum at q = (z + θ c)/3. Hence the latter is the monopoly output. Mutatis mutandis, this remark also applies to the analysis in 4.2, 5.1, and Appendix A: Cournot Duopoly For firm i = 1, 2, the firm s equity holder or manager solves the following problem: max q i V i = (R i D i )fdz = (θ + z q 1 q 2 c)q i fdz (21) where R i is the revenue of firm i and D i = cq i. Using (6) and replacing by c + q θ, we get V i = z z z [(θ c q)q i + z + 2 q i] = z z z [(θ c q)q i + z + c + q θ q i ] 2 = z 2(z z) [θ c q + z]q i = 1 2(z z) (z )2 q i = Hence firm i s best response is the solution of the problem 1 2(z z) (θ c q + z)2 q i. (22) max q i (θ c q + z) 2 q i. (23) The first order condition is (z )(θ c + z q i 3q i ) = 0. V i = 0 at z = ; hence the latter can be ruled out at the maximum and i s best response is given by q i (q i ) = (θ c + z q i )/3. (24) Checking the SOC confirms that V i is maximized when (24) holds and minimized when z =. The reaction functions (24) imply (11). 5.2 Appendix B: Stackelberg Solution Firm i = 1, the leader, solves problem (23) while q 2 (q 1 ) is given by (24). Then the firm s objective function becomes (θ c q + z) 2 q 1 = (θ c (q 1 + q 2 ) + z) 2 q 1 = (4/9)(θ c q 1 + z) 2 q 1 12

13 which is maximized at q 1 = (θ c + z)/3 and minimized when z =. (12) follows from q 1 = (θ c + z)/3 and (24). 5.3 Appendix C: Monopoly Output For the monopolist, the firm s equity holder or manager solves max V = q (R D)fdz = (θ + z q c)qfdz. (25) After replacing by c + q θ, finding monopoly output can be reduced (similar to the above treatment of duopoly) to solving the problem max (θ c q + z) 2 q (26) q with corresponding first order condition (θ c q + z)(θ c 3q + z) = 0. Checking the SOC confirms that V is maximized at q = (θ c + z)/3 and minimized when z =. Hence (13) as asserted. 5.4 Appendix D: Proof of Proposition 2 cournot duopoly Condition (9) and comparison of (11) and (15) show the assertion. stackelberg solution In the case of no debt, firm 2 s reaction function is given by (14): Hence firm 1 maximizes q 2 (q 1 ) = θ c 2 + z + z 4 q 1 2 = 1 2 [θ c + (z + z)/2 q 1). (θ c + (z + z)/2 q 1 q 2 )q 1 = 1 2 (θ c + (z + z)/2 q 1)q 1 and the first order condition yields (14) and (27) yield q 1 = θ c 2 q 2 = θ c 4 + z + z 4. (27) + z + z 8. (28) 13

14 Comparison of (12) with (27) and (28) proves the assertion. monopoly output With equity financing, the monopolist maximizes (θ c(z + z)/2 q)q. The first order condition yields monopoly output (θ c)/2 + (z + z)/4. Comparison with (13) yields the assertion. 5.5 Appendix D: How to Endogenize the Interest Rate Here we indicate how the cost coefficient c and, consequently, the interest rate r might be endogenized. For the sake of expositional simplicity, we consider the monopoly case. In our model, monopoly output is given by (13), qm = (θ c+z)/3. Let z 0 be given by the condition R i = 0, given the monopoly output. Then θ + z 0 qm = 0 or z 0 = qm θ = (z (2θ + c))/3. Moreover, at the monopoly output, = c θ + qm = (z 2(θ c))/3. In the the states z [, z], the bank (creditor) receives full payment of D i = cqm. In the states z [z 0, ], the bank is residual claimant to R i = (θ + z qm)q m 0. In states z < z 0, the bank receives zero payment from the monopolist. Hence the bank has the expected gross return R b = 1 z z [z ] cq m + 1 z z z 0 (θ + z q m)q mdz. Suppose the creditor pays interest at the rate ρ (or has to guarantee a rate of return ρ) for the funds it raises. Then its cost of financing the amount kq m it lends is C b = (1 + ρ)kq m. The creditor breaks even on the loan to the monopolist if R b = C b. If the bank extends credit to many firms, say local monopolists, with varying values of k, θ, z, z, then R b and C b ought to be replaced by the respective expected values R b and C b. The values of c such that R b = C b holds are the candidates for cost coefficients at which the bank breaks even. References [1] Basle Committee on Banking Supervision, Credit Risk Modelling: Current Practices and Applications. [2] Bolton P, Scharfstein DS, A Theory of Predation Based on Agency Problems in Financial Contracting. American Economic Review, 80, [3] Brander JA, Lewis TR,1986. Oligopoly and Financial Structure: The Limited Liability Effect. American Economic Review, 76, [4] Brander JA, Lewis TR, Bankrupcy Costs and the Theory of Ologopoly. Canadian Journal of Economics, 21,

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