Banking Competition and Economic Stability: a simpleapproach

R. FischerU. de Chile

N. InostrozaNorthwestern U.

F. J. Ramrez *

9 de enero de 2015

Resumen

We study the eect of banking competition on stability in a 2-period economy. If a rst periodreal shock occurs, a fraction of rms default on their loans. Banks face capital adequacy cons-traints and depositors are rst-priority creditors, so second period lending falls, amplifying theshock. The model admits prudent (banks survive) and imprudent equilibria (banks collapse). Wend existence conditions for a prudent equilibrium. Competition increases rst period eciencyat the expense of higher variance in second period outcomes; imprudent equilibria become moreattractive. We also examine the role of banking regulation and of anticipated regulatory forbea-rance.

Keywords: Bank competition, stability, eciency, forbearance.

JEL: E44, G18, L16.

*Center for Applied Economics (CEA) and Finance Center (CF), Department of Industrial Engineering, Universityof Chile, Av. Repblica 701, Santiago, Chile. E-mail: rfischer@dii.uchile.cl. The authors acknowledge thesupport of Fondecyt Project # 1110052 and of the Instituto Milenio de Sistemas Complejos en Ingeniera. F. Ramirez andN. Inostroza acknowledge the support of the Centro de Finanzas of the Department of Industrial Engineering, Universityof Chile. Emails: rfischer@dii.uchile.cl, felipram@gmail.com, NicolasInostroza2018@u.northwestern.edu.

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1. Introduction

This paper analyzes a simple two-period model in which a banking system amplies real eco-nomic shocks. We focus on the interaction between the amplication eect and the intensity ofcompetition in the banking sector. A shock aects the economy through the banking channel: aninitial systemic shock to productivity leads some rms to default on short term loans. Since deposi-tors have rst priority, banks are weakened after repaying short term deposits. This initial reductionin the capital base leads to a reduction in lending in the next period, because of capital adequacyrestrictions. Thus the real eects of the initial shock are amplied by the banking system. As inKiyotaki and Moore (1997) a credit multiplier amplies the eect of a real shock.

We link this eect to competition in the nancial market, because in a more competitive market,rates are lower, leading to more borrowing and to increased leverage. As the banking market beco-mes more competitive, the amplication eect becomes larger, even though the economy is moreecient, so competition creates a tradeo between eciency and stability.

There is an extensive literature on the relationship between nancial market stability and com-petition, much of it reviewed in Vives (2010), and which we cover in the next section. Briey, fromthe point of view of theory, the predictions are ambiguous. For example, Boyd and Nicol (2005) notethat reduced competition raises interest spreads, which tempts borrowers to choose riskier projects,so the loan book of banks becomes more fragile. On the other hand, in the so called charter valueapproach, a less competitive banking system means that banks are more valuable and owners areless willing to risk them, so they transfer risks to borrowers, see Beck (2008) for references. Alterna-tively, with more competition, there are fewer rents from screening and relationship banking (Allenand Gale, 2004), leading to more instability. Beck (2008) shows that there is corroborating empiricalevidence for these contrasting arguments.

Note furthermore that there are two kinds of nancial fragility: rst, fragility leading to bankruns due to sunspots and a second form in which the banking system amplies the eect of an initialreal shock, by reducing lending and thus magnifying its economy-wide eects. In this paper we exa-mine the relationship between competition and this second type of fragility using the balance-sheetchannel. After an initial economic shock banks need to contract their lending in order to improvetheir balance sheet, which is weakened by the default of borrowers, in what Tirole (2006) denotesa credit crunch. Often the improvement in the balance sheet is required by regulatory authorities,which may even impose more stringent capital adequacy restrictions.1

Note that in our simple model, the shock aects all banks equally. This is complementary tomodels which stress the interconnectedness of the banking system (Acemoglu et al. (2013); Allen

1Switzerland has imposed a stringent set of capital adequacy rules for Sistemically Important Financial Institu-tions (SiFis) that will constrain lending by banks. See Swiss urge capital boost for banks Financial Times, Octo-ber 4, 2010. http://www.ft.com/intl/cms/s/0/4a24a1c8-cf26-11df-9be2-00144feab49a.html#axzz2SWZy2mnj.

et al. (2010); George (2011), among others) as a reason for fragility. In those models the fragilityis given by the interconnectedness and the initial exposure of banks, so a shock can aect bansasymmetrically. RF: In this paper, we are interested not in the fragility of the nancial sector per se,but on how it aects economic activity more generally by increasing the response to real shocks.We show that bank competition, by overextending banks, increases their exposure to shocks andthus the impact they have on economic activity. This is why our model diers from, for instanceHellman et al. (2000) who study the eectiveness of capital adequacy ratios to restrain the eects ofcompetition on the fragility of the banking system.

Bank regulation reduces the eects of the associated moral hazard problem by imposing capitaladequacy restrictions. At the beginning of our rst period, banks lend to rms (think of it as lendingfor capital investment) using funds that are provided by short term deposits and their own capital.At the end of the period, if there is no shock, rms generate revenue to repay loans, and the bankcan repay depositors with this income. If there is a shock, some rms are unable to repay their loans,so the banks end up with less capital and reserves. At the beginning of the second period the rmsask the banks for working capital loans, and the banks lend by obtaining new deposits.

To simplify the analysis of the second period, we assume relationship banking. In general, thismeans that banks that lend in the rst period have an advantage over other banks, and therefore canobtain a rent from their relationship, and will be able to retain the client against the competition.To simplify matters, we assume that the bank keeps all the prots obtained in the second period.2

In the second period there are no shocks and therefore no risk of failure. This implies that banksalways want to lend to every rm in the second period, but they are restricted by capital adequacyrestrictions. Thus the second period has no strategic behavior nor risk. All the action occurs in therst period, which we solve by backwards induction. Note that we dier from much of the literature,which examines the eects of competition on the risk banks by having them choose the risk-returnprole of individual bank loans. We study fragility by examining the eects of the interaction bet-ween systemic productivity shocks and the intensity of competition on the balance sheet of banks.

In the rst period banks are imperfect competitors, and they maximize prots over the twoperiods, considering the probability of a shock. We compare Collusive, Cournot and Bertrand equi-libria.3 Banks start out with some initial capital and can go to the market to request short term fundsfrom depositors. Since deposits are protected by deposit insurance, depositors are willing to lend atthe risk-free rate.

There are many potential entrepreneurs, who own no assets except for the idea of a project.All projects are equally protable and equally risky. Agents are dierentiated by the value of their

2Otherwise we would have to add a parameter describing the share of second period prots each party to the rela-tionship obtains, with no change to the results.

3We model competition via conjectural variations parameter. It is a convenient mechanism of mapping our threecompetitive alternatives, by using dierent values of a single parameter. For more discussion on the competitive assum-ptions, see section 7 on leverage.

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outside option, which follows a distribution with a continuous density. In the rst period, agentswhose expected return from the project exceeds their outside option approach banks for loans tocarry out their projects. Banks fund entrepreneurs with short period loans which must be returnedat the end of each period.4

If there is no shock, all entrepreneurs pay back their rst period loans. The income stream recei-ved by banks is large enough to allow them to pay back their deposits, and to have enough capitalto fully satisfy the second period loan demand, given the capital adequacy constraint. Thereforeall agents that received a rst period loan will also obtain the working capital loan for the secondperiod. However, in the case of a productivity shock, things are dierent. The shock wipes out therst period returns for a fraction of rms.5 Those rms are unable to repay the bank and since thebank must repay depositors, at the end of the rst period it has less capital and reserves than in thecase of no shock. As banks they lend a multiple of their own capital (though they must satisfy capitaladequacy restrictions), the banking system amplies economic shocks. The reduction in own capitalfollowing a shock leads to a larger reduction in loans, a well known proerty of fraactional reservebanking. Some rms have to cease operations because they have no working capital for the secondperiod. The intensity of the shock is conveniently measured as the fraction of rms unable to repaytheir rst period loan.

There is another possible outcome, which occurs when banks are overextended and repaymentof deposits in case of a shock wipes out the capital and reserves of the bank (leading to an imprudentequilibrium). In that case there is a collapse of the banking system and no second period lending.6

Since all banks are assumed identical, symmetry implies that there is a simultaneous banking co-llapse. In the prudent equilibrium, banks are judicious and choose to lend quantities that even in thecase of a shock, will allow them to survive. In the imprudent equilibria, banks are improvident inthe sense that they choose to lend larger amounts than when behaving prudently, and if the shockhappens, the banking system collapses.

When capital adequacy restrictions are loose, so that in the prudent equilibria under a shockbanks are close to bankruptcy, they choose the imprudent equilibria, requiring the need for pru-dential regulation to avoid these outcomes. The banking regulator can exclude imprudent equilibriathrough judicious use of capital adequacy restrictions but an inappropriate application of these con-ditions, or the (correct) belief that these may be loosened in case of a negative real shock, may lead

4In principle, the model could allow long term deposits and long term lending (though it would lead to cumbersomemodications). In that case, the fraction of banks assets that are covered by these long term contracts would no besubject to the problems raised in this paper. However, since banks provide short term nance using short term funds,the problem described in the paper does not go away. In fact, in our interpretation, second period lending is for workingcapital. Working capital is usually short term and variations in the amount of working capital available to rms explainmuch of the variability in their output.

5Though they remain viable for the second period if the bank can fund them.6Note that because of deposit insurance, there is no possibility of a bank run a la Diamond and Dybvig (1983).

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to imprudent equilibria.7

Note that the correct application of capital adequacy regulations, by precluding the collapse ofthe banking system, implies that there is no need for deposit insurance, so providing it is costless tosociety. However, even without a banking system collapse, increased competition leads to increasedvariance in economic outcomes. Basically, as competition increases, banks charge a lower interestrate and lend more. In the case of no shock, there is more economic activity. On the other hand, whenthere is a crisis, a larger mass of entrepreneurs fail to pay their loans, leading to a larger reductionin bank capital. This, in turn, reduces second period lending. Hence, second period activity is morevariable as competition increases.

We also examine the eect of capital adequacy rules. In response to a shock, governments usuallyrelax the capital adequacy rules, at least in the short run. This reduces the magnitude of the shock.We show that this emergency response works only when banks do not expect the rule to be relaxed.If banks have perfect foresight about the future capital adequacy rule, this is incorporated in theirlending decisions. Hence the eects of an expected future relaxation in the capital adequacy rules inresponse to a shock are ambiguous. The eects on pre-shock lending and therefore on the varianceof post-shock GDP depend on parameter values.

The main result of this paper is that independently of the type of competition, and even con-sidering the possible switch from a prudent to an imprudent equilibrium, an increase in the degreeof competition in the banking sector increases the variance of post shock activity and hence thevariance of GDP. This is the rst paper which obtains these results without using a reduced formfor competition.

One nal issue is that the model is not a general equilibrium model, in the sense that it relieson the existence of deposit insurance and the supply of deposits is perfectly elastic. Using depositinsurance is not uncommon in the literature, as in Allen and Gale (2004) and others. That paperdoes, however, include a cost of insurance to banks that is independent of individual riskiness, butwhich covers the aggregate cost of deposit insurance. In our model, when the prudential equilibriumis chosen and there is no banking collapse, the real cost of insurance is zero. Our model could beadapted to accommodate a at insurance rate, with no change in the main results, and with additionaldiculties, to an increasing supply schedule for deposits. However, guaranteeing that the insurancerate is actuarially fair would complicate the calculations.

The next section is a literature review, followed by a section describing the model. Section 4 de-rives the equilibria in the model by backwards induction. Section 5 analyzes existence and unique-ness. Next we study the comparative statics of the equilibrium and regulatory forbearance. Section7 analyzes extensions to other means of increasing leverage, such as nancial liberalization, and

7The model allows that for certain parameter values, an imprudent equilibrium is chosen by banks, and it leads tohigher welfare than without a bank collapse. This occurs if the probability of a shock is small but when it happens theshock is very deep, so many rms fail to repay their loans. The cost in terms of reduced lending (and therefore reducedeconomic activity) of trying to prevent the rare occurrence of a banking collapse may be too high in this case.

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section 8 contains the conclusions.

2. Literature review

There is a large literature on the relationship between stability and competition in nancialmarkets, so this review will highlight important contributions, under the proviso that it will leavemany relevant papers unmentioned. As mentioned above, Vives (2010) provides a comprehensivereview of the empirical and theoretical literature.

On the theoretical side, Allen and Gale (2004), following an earlier tradition, use a static model tostudy the relationship between competition and stability. In their model, rms can choose the returnand riskiness of loans, and competition leads to riskier lending. This also implies that eciency isnot attained with competitive markets, because of excessive risk taking.

There is an alternative literature, which follows from Stiglitz and Weiss (1981) in which the riskand interest rates charged on loans are positively related. Hence, as in Boyd and Nicol (2005), therisk taking behavior of borrowers increases as interest rates go up due to less intense competition.This view is further explored in De Nicolo, Boyd, and Jalal (2009), which uses an elegant model thatincludes a safe asset to make the point that there is no one-to-one relationship between stability andcompetition. Hakenes and Schnabel (2011) make a similar point, in a model in which banks haveequity, so that a capital adequacy ratio can be used to regulate risk taking. The authors obtain thesurprising result that limiting the leverage of banks increases entrepreneurial risk taking, since lesscompetition (due to a higher capital adequacy ratio) translates into higher interest rates on loans.Very recently, Carletti and Leonello (2012) describe a two period model where competition leadsto increased stability because when there is competition, banks prots from lending are low andkeeping large reserves is not expensive, so banks do not default. With less competition they obtaina mixed equilibrium with some banks choosing a risky strategy and others choosing a safe strategy.Hence the banking system is less stable as competition decreases.

In a recent paper, Martinez-Miera and Repullo (2010) show that the results of Boyd and Nicol(2005) depend crucially on having perfect correlation of loan defaults. They note that when loansare not perfectly correlated, more competition reduces the return on loans that do not default, so thetotal eect of competition on stability depends not only on the reduced riskiness of loans but also onthe reduced margin on loans that do not default. Using an imperfectly competitive model, Martinez-Miera and Repullo (2010) establish that the second eect is dominant under perfect competition andthat in less competitive markets there is a U-shared relationship between competition and stability.In our model, only the margin eect is present.

Wagner (2010) uses an alternative argument to the same purpose by noting that even though in-creased competition leads to lower rates and therefore to borrowers that choose less risky projects,the banks can also inuence the level of risk of their loans. When facing lower return due to compe-

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tition, they will choose borrowers with riskier projects and higher returns, and this will counteractthe stabilizing eect of competition of Boyd and Nicol (2005).

A recent unpublished monograph by Freixas and Ma (2012) develops a more tractable modelthat obtains the results of Martinez-Miera and Repullo (2010) but incorporates the possibility ofbank runs of the Diamond and Dybvig (1983) type.8 They have banks that use two types of funding:insured deposits or uninsured money market funds. They dierentiate between portfolio, liquidityand solvency risk and show that the conditions under which competition reduces risk depends ona simple condition involving the fraction of insured deposits in bank liabilities, the productivity ofprojects and the interest rate. When productivity is low and banks are funded with insured deposits,competition increases total credit risk. They argue that their more detailed model allows them tointerpret the dierent results obtained in the empirical literature, which they review in detail.

We have mentioned before the large and growing literature theoretical and empirical on in-terconnectedness among banks and its eects on the stability of a banking system, as for instanceAllen and Gale (2004) or Cifuentes et al. (2005) and more recently Allen et al. (2010). This literature iscomplementary to our paper in the sense that it examines the sensibility to shocks of diering ban-king systems with dierent degrees of interconnectedness, but do not examine its interaction withcompetition.9 A recent interesting paper along these lines is Acemoglu et al. (2013), who examine amodel in which there is a range in which increasing interconnectedness makes the banking systemmore resistant to a real shock, while beyond this range increasing interconnectedness increases thesensibility to real shocks.

The empirical evidence of the relation between competition and economic stability is mixed.Early studies of the eects of bank liberalization in the USA Keeley (1990), Edwards and Mishkin(1995) and others showed that liberalization lowered the charter values of banks and this increasedrisk taking. For Spain, Saurina-Salas et al. (2007) found that liberalization and increased competitionwas associated to higher risk, measured as loan losses to total loans.

In cross country studies, diverse studies show that increase competition contributes to stability.This is the case of Schaeck, Cihak, and Wolfe (2009), who use the Panzar and Rosse H-statistic tostudy the probability of a crisis using 41 countries. They also point out that bank concentration isassociated to higher probability of crisis, so concentration and competition capture dierent aspectsof the fragility of banking systems. Similarly, in a recent working paper, Anginer, Demirguc-Kunt,and Zhu (2012) use a sample of 63 countries to look at the eects of competition (measured by the

8Even in the case of bank runs there are two dierent approaches: the multiple equilibria-sunspot view of Diamondand Dybvig (1983) (the expectation of a collapse, coupled to the maturity mismatch leads to runs) and those where runsare triggered by the deterioration of fundamentals. There is an alternative approach to the same problem, as for examplein Rochet and Vives (2004), the bank fails because the fundamentals are weak and this leads to a higher probability of arun.

9A partial exception is Cohen-Cole et al. (2011), who use a Cournot model where competition reduces total protsand interconnectedness aects the banks that are closely linked to each other. A local shock reduces loans of closelylinked banks but tends to increase loans of those that are not closely linked.

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Lerner index). They incorporate the (co-)dependency among bank risks, in order to examine syste-mic nancial fragility, rather than at the level of individual banks. They nd a stabilizing eect ofcompetition. On the other hand, in a recent article Beck, Jonghe, and Schepens (2013), who incorpo-rate the regulatory framework and nancial market characteristics as an explanatory variable in thecountry cross sections, nd a positive association between market power and a measure of nancialfragility.

After reviewing the evidence, Vives (2010) concludes:

Theory and empirics point to the existence of a trade-o between competition andstability along some dimensions. Indeed, runs happen independently of the level ofcompetition but more competitive pressure worsens the coordination problem of in-vestors/depositors and increases potential instability, the probability of a crisis and theimpact of bad news on fundamentals.

3. The basic model

We consider an economy with three dates (t = 0, 1 and 2) and two periods. There is a continuumof risk neutral entrepreneurs with zero assets, where we denote an entrepreneur by z [0, 1]. In therst date, t = 0, agents decide whether to undertake a risky project which lasts two periods, or toexercise an outside option. Although the risky project is the same for all entrepreneurs, these agentsare dierentiated by the value of their outside option, which yields a safe return uz for entrepreneurz at the end of the second period.10 The distribution of uz is given by G(), which has a continuousdensity g() and full support [0,U].

The risky project requires one unit of investment capital at t = 0 that the agent must borrowfrom a bank. The project provides returns at t = 1 and at t = 2. The returns in t = 1 depend onthe state of the economy, denoted by s, which s can take two values, high (h) and low (l). In stateh, which occurs with probability p, the economy has a high productivity shock, i.e., all projects aresuccessful, in which case they return y1. In state l, the economy suers a low productivity shock,in which case each project succeeds and returns y1 with probability q (0, 1) and fails (returns 0)otherwise. Figure 1 shows the events over time:

At t = 1 all rms (even those that were unsuccessful) can apply for a working capital loan from banks, in order to operate in the second period.11 In the second period there are no shocks and

10This is similar to the assumption in Boyd and Nicol (2005) and used for the same purpose: to dierentiate amongentrepreneurs and thus obtain a demand curve for loans.

11An alternative is to allow only successful rms to be able to ask for loans, and we have examined this case. It ismore complex, because failure will aect both the demand and the supply of loans, whereas in the present case, onlyloan supply is aected. On the other hand, we believe our formulation is reasonable if we interpret the rst loan as oneof initial investment plus working capital and the second one as a loan of working capital and for maintenance costs.

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t = 0 t = 2t = 1

y1 y21p

No shock, firms re-pay, banks repay de-posits, get dividends

Shock, fraction q offirms do not repay,banks return de-posits using capital,no dividends.

All firms obtain 2ndperiod financing.

py1 y2

0

Some firms do notobtain second periodfinancing.

Figura 1: Scheme of returns over time.

all rms that obtain the working capital loan receive a return of y2 at the end of the period. Thefollowing timeline shows the relevant variables at the dierent points in time:

The economy has two other clases of risk neutral agents: depositors and banks. Depositors lendto banks each period and receive their money back at the end of the period. Their supply is perfectlyelastic at a risk-free rate that we normalize to zero, for notational simplicity. Depositors do not askfor more than the risk-free rate because the government insures deposits at failed banks. This impliesthat depositors play a passive role in the model.12

Dene = (1 + )1 as the discount factor associated with the cost of capital . We need thefollowing assumption:

Assumption 1 py1 1 > 0,

Assumption 1 implies that the expected net present value of the rst stage of a project is positive,even in the state of nature s = l.

Banks are the nancial intermediaries of this economy, specialized in channeling funds frominvestors to entrepreneurs. There are N identical banks. To fund their projects, entrepreneurs borrowfrom banks, and banks compete to attract entrepreneurs. At date t, each bank extends loans lt thatare nanced by deposits dt and inside equity et . Hence the budget constraint for a representativebank at date t is:

lt dt + et (1)Each bank is run by a single owner-manager who provides the equity et ; the owners opportunitycost of capital is > 0, so that equity nancing is more expensive than deposit nancing. This

12At this stage, this assumption is for simplicity only. As we will see later, banks behave prudently and never fail.Thus there is no need for insurance, and depositors do not face systemic risk.

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assumption is typically assumed in the literature.13

Banks compete for borrowers in the rst stage and have an ongoing relationship with the borro-wer in the second period, so they can extract all rents from borrowers in the second stage.14

We assume that entrepreneurs do not use internal nancing in period 2. This simplication isstandard in relationship-banking models.15 Moreover, if entrepreneursrst-period prots are smallrelative to the amount of the working capital loan, the eects of relaxing this assumption would benegligible (Repullo (2012)) RF: : No pude

encontrar estareferencia.

. Also, following Repullo and Suarez (2013), we assume that it is impossibleto recapitalize a bank at date t = 1. Their argument, which we adopt, is that the dilution costs of anurgent equity raise could be high for a bank with opaque assets in place.

Finally, there is a nancial regulator that imposes capital adequacy requirements that limit theamounts that banks can lend to xed multiple of their capital. For banks, this implies that:

l0 e00, 0 0 1 (2)

l1 e1s1, 0 s1 1, s {h, l} (3)

where (0, h1, l1) indicates the capital adequacy requirement in period 0, in period 1 at state hand in period 1 at state l respectively.

We assume that is xed by the regulator and there exists commitment no to change it in thefuture. However, the regulator can set l1 h1 , i.e., it can announce that in case of a shock, thecapital adequacy requirement will be relaxed, by a predetermined amount.

4. Equilibrium

Since this is a two period model, we solve it by backwards induction.

4.1. Equilibrium at t = 1

We assume, at seems reasonable, that only in the state of the nature l there is the possibilityof a credit crunch16, in the sense that some protable projects cannot get nancing even with nouncertainty about their protability because banks do not have enough equity (capital plus reser-

13See Berger and Ofek (1995) for a discussion of this issue; and Gorton and Winton (2003), Hellman et al. (2000) andRepullo (2004) for a similar assumption.

14Another simplifying assumption, that allows us to obtain rst period expressions for demand that are analyticallysimple. Having competition in the second stage would not add much to the model, since any small cost advantage to theincumben bank allows it to make a deal to share this advantage and maintain the relationship. For relationship lendingand rents see Yafeh and Yosha (2011), Hauswald and Marquez (2006) and Degryse and Ongena (2008).

15See, for example, Sharpe (1990).16A credit crunch is dened as a situation in which there is a reduction in the general availability of credit or a sudden

tightening of the conditions required to obtain a loan from the banks.

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ves) to nance them, given the capital adequacy restriction. When the state of the nature is h, allprojects succeed and managers pay back their rst period loans. Thus banks have enough capitaland reserves, after returning the deposits, to nance all applications for loans in period 1, and allagents know this.17

Suppose that the realized state of nature was s. At date t = 1, the following variables are taken asgiven by agents: (1) the equilibrium interest rate charged on rst period loans, r0; (2) the bankscapitalin the state s, es1; (3) the total amount of credit given by the representative bank to nance projectsin the rst period, l0; (4) the number of entrepreneurs that obtained funding in the rst period, G(u),where u is the utility cuto for entrepreneurs.

Entrepreneurs, even if their projects fails in the rst period, can ask for a loan of amount fromthe same bank from which they asked their original loan, because of our assumption of an ongoingrelationship.18 Given that rms cannot apply for loans from other banks, the incumbent bank canextract all the rents from its borrowers at this stage, and the entrepreneursparticipation constraintwill be binding in equilibrium.19 This means that banks cannot expect failed entrepreneurs to pay apenalty fee for having defaulted on their rst period loans. Hence, if entrepreneur z gets nancing,we obtain the interest rate charged to that entrepreneur:

y2 (1 + r1) = 0 1 + r1 = y2

(4)

As the second stage of the project is protable for banks, banks will want to nance the maximumnumber of these projects they can. The demand for second period loans is l0, where we have usedthe fact that rst period loans are of size 1. Thus, in state s, the bank solves the following problemat t = 1:

Max{ls1,ds1,Divs }

[(1 + r1)l s1 ds1

]+ Divs

s.t. es1 + ds1 l s1 = Divses1 Divs s1l s1

l0 l s1Divs, l s1, d

s1 0

17We can always adjust the parameters of the model in particular, the magnitude of the shockto have this case instate l. That is, we do not examine the case of a capital constrained banking system.

18It is possible to consider as an alternative assumption that rms that fail in the rst period go out of the market,and we have also examined this case (which has the added complication that in the bad state, both the demand and thesupply of loans depends on the fraction 1q of failing rms). However, in an interpretation of the original investment asincluding initial investment plus working capital, and a second period in which only working capital is needed, becausethe project is not a failure but has not met initial expectations, the interpretation we include is more appropriate.

19This is not essential; the borrower could split the second period surplus with the bank, the division of the surplusreecting the ease of substitution with other banks. However, including this possibility would have added a parameterto the model without materially changing our results.

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For each bank and in each state s = h, l, the decision variables are the total amount of credit toprovide, l s1, the total amount of deposits to raise, ds1 and the rst period dividends policy, Divs. Theobjective function is the discounted utility of the representative bank at date t = 1, and it consistsof two terms. The rst term

[(1 + r1)l s1 ds1

]is the net present value of the banks net prots of

date t = 2, where (1 + r1) is determined as in (4). The second term Divs is the cash left over, whichis used to pay dividends to shareholders at t = 1.

The rst (equality) restriction is the time t = 1 budgetary restriction of the bank. The secondrestriction is the capital adequacy restriction, which applies after dividends are paid and determinesthe loanable funds. The third restriction requires that total loan supply must be smaller than loandemand in each state (otherwise the cost of loans is zero). The following proposition characterizesthe equilibrium at date t = 1:

Proposition 1 At date t = 1, each bank makes loans of l s1 = mn{es1s1, l0

}, takes deposits of ds1 =

(1 s1)l s1 and pays dividends Divs = es1 s1l s1, s = h, l.Proof: See Appendix.

This result links the stock of capital of banks at t = 1, es1, to the supply of credit that each bankprovides for the second period, l s1. In particular, note that if the capital es1 is suciently low therewill be a credit crunch, as banks will not be capable of meeting the eective demand for loans, l0.This happens because the capital adequacy restriction limits the quantity of credit that banks cansupply, and this may result in an unmet demand for credit given by Max

{l0 e

s1s1, 0

}. Notice that

there is a credit crunch only in state l if el1 < l1l0 eh1 , i.e., if banks do not have enough internalcapital in that state.

RF: For later analysis, note that if there were an unexpected lowering of the capital adequacyratios at t = 1 in the bad state of the world, it would be possible to eliminate the credit crunch.However, as we will see in section 6.1 below, forbearance is guaranteed to be eective in reducing thevariance of second period results only if the change is unexpected, and cannot be an equilibrium.20

Consider now the value of es1. In the state of nature s = h, we know that all projects succeed,so each entrepreneur has the resources to pay his debt at the end of the rst period. Therefore, the

20Temporary forbearance of capital adequacy requirements is common when banks are distressed. For instance, inMitra, Selowsky and Zalduendo, Turmoil at Twenty: Recession, Recovery and Reform in Central and Eastern Europeand the Former Soviet Union, World Bank 2010. we nd:

Some previous episodes of systemic banking distress, such as Argentina 2001, Bulgaria 1996, Ecuador1999, Indonesia 1997, Korea 1997, Malaysia 1997, Mexico 1994, the Russian Federation 1998, and Thailand1997 have also seen regulatory forbearance. Specically, to help banks recognize losses and allow corpo-rate and household restructuring to go forward, the government might exercise forbearance either on lossrecognition, which gives banks more time to reduce their capital to reect losses, or on capital adequacy,which requires full provisioning but allows banks to operate for some time with less capital than prudentialregulations require.

11

capital of the representative bank at date t = 1 is (before paying dividends):

eh1 = (1 + r0)l0 d0 (5)

where d0 = l0 e0 are the deposits that the bank must repay at the end of the rst period, justbefore t = 1. As we have assumed that in the state h the representative bank has enough capital tonance all the entrepreneurs who ask for a loan, i.e., there is no credit crunch, then it must hold thath1 l

h1 =

h1 l0 eh1 .

Using the results of Proposition 1, we obtain the net present value of the bank at date t = 1 whenthe state of the nature is s = h:

h1 = [(1 + r1)lh1 dh1

]+

[eh1 + d

h1 lh1

]=

[r0 + e0/l0 + (y2 ) h1 (1 )

]l0 (6)

where we have used the interest rate r1 obtained in equation (4), (5) and the results of proposition 1.Now we study the case when the state of the world is s = l. Recall that in this case, from the

point of view of t = 0, each project succeeds with probability q and fails with probability 1 q. Bythe Law of Large Numbers, exactly a fraction 1 q of entrepreneurs fail, so in this economy thereis no aggregate uncertainty. Therefore, ql0 entrepreneurs succeed and pay their debts. On the otherhand, we have assumed that all agents, including those who fail in the rst period and are unableto repay their loans, ask for a working capital loan to continue their projects in the second period.Now, the capital of the representative bank at date t = 1 is:

el1 = Max {q(1 + r0)l0 (l0 e0), 0} (7)

where the Max operator arises due to limited liability of banks. Recall that if this capital is zerothen the bank fails at date t = 1, and its depositors are paid with the residual value of the bank(q(1 + r0)l0) plus the compensation made by the government from deposit insurance. This happenswhen the probability of success q satises:

q < q 1 (e0/l0)1 + r0 (8)

If the fraction of rms that manage to repay after the shock are q q, then banks do not failin the event of a crash, though their second period capital shrinks. As in the previous case, banksalways want to nance as many projects as possible. As we have mentioned before, to make thingsinteresting, we assume that in state l banks cannot nance all the entrepreneurs who ask for a second

12

period loan, so that21

l l1 =e1l1

< l0 (9)

From the discussion above, only a fraction [0, 1) of the demand for credit l0 is going to besatised:

=l l1l0

=q(1 + r0) (1 (e0/l0))

l1(10)

The variable measures the ratio of the second period economy under a shock to the size of theeconomy without the shock, i.e., when it is close to one, the economy is able to resist the shockwithout many ill eects. Similarly, 1 is the fraction of entrepreneurs rationed by banks at datet = 1, and can be interpreted as the magnitude of the credit crunch; and 1 q can be interpretedas the magnitude of the shock, as it represents the fraction of entrepreneurs that cannot repay theirloans.

A nal observation: as a consequence of proposition 1, in the case of a shock banks do not paydividends in the rst period because reinvesting all repayments into loan renewals is more protable.

The discounted utility of the representative bank at t = 1, after a shock can be written as:

l1 = [(1 + r1)l l1 dl1

]+

=0Divl

=

l1

[y2 (1 l1)

]Max {q(1 + r0)l0 (l0 e0), 0} (11)

which should be compared to the prots at t = 1 in the case of no shock, given by equation (6). Thenext step is to proceed to the analysis of the prot maximization problem at date t = 0.

4.2. Equilibrium at t = 0

In the rst period, given an aggregate demand for loans L(r0), banks choose the prot-maximizingvolumes of deposits (d0), and loans (l0). This automatically denes the equilibrium interest rate r0charged to entrepreneurs.

4.2.1. The demand for credit

Given the rst period interest rate r0 charged by banks, we dene u(r0) [p + (1 p)q] [y1 (1 + r0)] as the expected net future value (at the end of the second period) thatthe entrepreneur will obtain if he undertakes the two-stage risky project. Observe that the entrepre-neur gets no rents from operating the rm in the second period because the banks extract all prots.

21Otherwise the case with a shock can be treated as if it were the case without a shock and nothing happens after theshock.

13

An entrepreneur z will be willing to embark in this venture rather than stay with the safe optiononly if u(r0) uz . These participation constraints implicitly dene an aggregate loan demand thatis decreasing in the interest rate at t = 0, given by:

L(r0) =

u(r0)0

g(u)du = G([p + (1 p)q][y1 (1 + r0)]) (12)

with L(r0)r0 = g(u(r0))u(r0)r0

< 0. As usual, it will be more convenient to work with the inversedemand function, r0(L). We can rearrange the last equation to obtain:

1 + r0(L) = y1 G1(L)

p + (1 p)q (13)

where j=Nj=1 L j L. This expression denes explicitly a downward sloping inverse demand of loans.We make the following standard assumption:

Assumption 2 The distribution function of outside options G(z) is twice-continuously dierentiable,positive, and concave for all L (0, 1)

4.2.2. The banks optimization problem

At the beginning of the rst period, each bank chooses the volume of its deposits (d0), and loans(l0). Given the balance sheet identity, l0 = d0 + e0, only one of these variables can be chosen inde-pendently. Recalling that 0 is the capital adequacy constraint at t = 0, the representative bank att = 0 solves:

Max{l0}

0 [ph1 + (1 p)l1

] e0

s.t. l0 (e0/0) (14)

where the objective function is the expected net present value of prots of the bank while the res-triction corresponds to the capital adequacy condition at t = 0.

Recall from the comments on equation (8) that if q < q, banks go bankrupt in the low state(el1 = 0). In that case, l1 = 0 and there are positive prots only in the good state (h1 > 0). No-ting from the denition of q that a reduction in l0 leads to a reduction in q, banks, by lending lesscould have remained solvent and thus would maximize over both the good and bad states of theworld.22 This corresponds to what we denote by prudent behavior, leading to a symmetric prudent

22Observe thatsgn

(dqdl0

)= sgn

(e0l20

(1 + r0) dr0dl0(1 e0

l0

))> 0.

14

equilibrium. Conversely, behavior leading to bankrupt bank is imprudent behavior, and leads to animprudent equilibrium. This is the type of equilibrium behavior in which banks could be accused ofprivatization of prots and socialization of losses.

Hence, there are two dierent expressions for the prot function, depending on whether q > q,and banks survive the shock, and the case in which the inequality is reversed and banks fail. Thuswe dene two functions associated to prots in the two states:

h(L) = p((

y1 G1(L)

p + (1 p)q) 1 + (y2 ) h1 (1 )

)l (L) =

2(1 p)l1

(y2 (1 l1)

) (q

(y1 G

1(L)p + (1 p)q

) 1

)(15)

Prots at t = 0 depend on whether the bank fails in period 1 in the case of a shock, i.e., if el1 = 0.Observe that even when banks follow an imprudent lending policy, the proportion of rms q thatfail under a shock is relevant. The reason is that a fraction 1 q of entrepreneurs make prots in therst period under a shock, and this possibility has an eect on the demand for loans. We have thattotal expected prots for a bank at t = 0 are:

0(l0) =h(L)l0 + (p 1)e0 if el1 = 0(h(L) +l (L))l0 +

((p 1) + 2(1p)

l1

(y2 (1 l1)

))e0 if el1 > 0

(16)

There are two points to make about this expression for bank prots. First, banks maximize protssubject to the capital adequacy restriction l0 e0/0. If an imprudent equilibrium is chosen thiscondition is binding, because p 1 < 0, and therefore h(L) has to be strictly positive or theimprudent equilibrium would have negative prots. Since the imprudent equilibrium is linear in l0,the capital constrain must be binding. Second, observe that the two prot functions are dierent andcannot be transformed into one another via a continuously dierentiable transformation, becauseof the non-negativity constraint on prots if the bank collapses after a shock.

Notation: We dene the following notation which will be useful in the following:

y1 G1(L?)

p + (1 p)q [1 + (N 1)v]l?0

(p + (1 p)q)G(G1(L?)) , N : number of banks. (17)

H 2(1 p)l1

(y2 (1 l1)

)(18)

1 (y2 ) + h1 (1 ) (19)

15

where the term (N 1)v is the aggregate conjectured response of the other banks to a marginalincrease in l0 by a bank.23 Observe that H is the contribution to prots of one additional unit ofcapital at time 1 in the bad state of the world, weighed by the probability of a shock and discountedto time t = 0. We will use the following important assumption:

Assumption 3 p + H > 1

The assumption means that the average value of an additional unit of bank capital at t = 1, discoun-ted to t = 0, is bigger than one, i.e., it is protable on average to have more period 1 capital (Seeappendix).

In order for imprudent equilibria to have a chance of being chosen, we must ensure that theprudent equilibria is interior to the capital adequacy constraint l0 = (e0/0) (otherwise there canbe no imprudent equilibria , since they involve more lending than prudent equilibria). We need thefollowing assumption:

Assumption 4p + H > 0

This is a sucient condition for the prudent equilibria to be interior to the capital adequacyconstraint (proof in the appendix). We can rewrite this condition as

p + H = p(1 (y2 ) + h1 (1 )

)+ H > 0. (20)

which we use later. This condition ensures that projects are not so protable that 1 + r0 0, i.e.,that in a prudent equilibrium banks are unwilling to give away money in the rst period even underBertrand competition.

5. Existence and uniqueness of equilibrium

As there are two potential prot functions, corresponding to prudent and imprudent behaviorof banks, there are potentially two families of equilibria. We use the Pareto optimality criterion tochoose among symmetric equilibria with the same starting capital e0, and we show that there is aneighborhood of p = 0 in which the prudent equilibrium is chosen for all intensities of competition.24

Our procedure is as follows: rst, we show that there is a unique symmetric equilibrium to bothprudent and imprudent behaviors by banks. Next we show that as competition decreases, the gapbetween the prots at the prudent equilibrium and the imprudent equilibrium increases. Then we

23Here v is the conjectural variation parameter, corresponding to the beliefs of rm i of its rivalsreaction to its ownloan supply choices. We assume that v is identical for all rms. When v = 1/(N 1), 0, 1 we reproduce the Bertrand,Cournot and collusive equilibria.

24Optimality can be justied as a selection mechanism if communication is allowed or by evolutionary arguments.

16

show that under Bertrand competition, when p = 0 (i.e., the shock is a certainty) the prudent equi-librium has strictly positive prots while the imprudent equilibrium has negative prots. Hence theresult continues to hold in some neighborhood of p = 0 for competition that is less intense. Thuswe have shown that there is a set of positive measure in which the only equilibrium is the prudentequilibrium, and that in general, the Pareto criterion assures us that only one equilibrium will bechosen by banks.

Lemma 1 There is a unique equilibrium of each type (prudent, imprudent) to the game among banksfor any intensity of competition. In the non-Bertrand case we have

2i

l2i< 0 and

2il2i2ilil j

> 1

Proof: We examine the case of prudent equilibria here and the appendix contains the very si-milar analysis of imprudent equilibria. We begin by noting that the Bertrand case must be trea-ted separately, because in that case L is presumed constant by banks. Hence, banks face a linearmaximization problem, leading to bang-bang solutions in which either all banks do not lend (if(h(L) + l (L)) < 0) or they lend up to the capital adequacy constraint if the sign is positive.Interior solutions are possible only if (h(L) + l (L)) = 0. There is only one interior symmetricalequilibrium, since we require Nl0 = L, where L satises

0 = h(L) +l (L)) = (p + Hq)(1 + r0) (p + H)

Since 1+ r0 is strictly decreasing in L, there is a single solution L and therefore a single symmetricallevel of rst period lending l0 under Bertrand competition.

In non-Bertrand cases, the prot functions satisfy the standard conditions for existence and uni-queness of equilibria. We consider the case of prudent equilibria:

ili

= (p + Hq) (p + H) (21)

and thus

2i

l2i= (p + Hq)

li= (p + Hq)

(2

PeG(G1(L)) liG

(G1(L))PeG3(G1(L))

)< 0

and:2ilil j

= (p + Hq)

l j= (p + Hq)

(1

PeG(G1(L)) liG

(G1(L))PeG3(G1(L))

)

17

from which we derive:2il2i2ilil j

=

(2

PeG(G1(L)) liG(G1(L))PeG3(G1(L))

)(

1PeG(G1(L))

liG(G1(L))PeG3(G1(L))

) > 1

l i0

pii

piipr

piiimp

piiimp

piipr

l i0

pii

(a) (b)

Figura 2: Equilibrium congurations

Consider diagram 2, which shows the possible congurations of the prot function for a singlerm, given that the choices of the other rms are in a symmetric equilibrium. In general there willbe two equilibria: a prudent equilibrium and an imprudent equilibrium. Each curve describes allpossible deviations of the bank given what its rivals are choosing as their rst period lending l0 inthe corresponding symmetric equilibrium. In gure (a) There is no incentive for the bank to jumpto the lending associated to the imprudent equilibrium (given that the other rms are playing theprudent equilibria), since the point at which the curves cross is where el1 = 0 and its prots are lowerby switching. In gure (b) the imprudent equilibrium is selected by the Pareto criterion. Note thatprudent prots are not dened beyond the crossing, since el1 < 0 at those points and the prudentprot function is not dened there.

If the world were to resemble (b), then the role of regulation is to restrict l0 so that the impru-dent equilibrium cannot be attained. To see this last point, consider the case of gure 3. In the gure,the vertical line corresponds to the lending limit dened by the rst period capital adequacy condi-tion (14) and limits rst period lending of any bank to that level, so that even though the imprudentequilibrium is preferred by banks, it cannot be chosen and rms prefer the prudent equilibrium totheir other (symmetric) options.25

The next step in the proof is to show that as competition decreases, the dierence between theprots at the prudent and the imprudent equilibria increase.

25If the crossing between the two curves occurs to the left of the maximum of the curve corresponding to the prudentequilibrium, one cannot use the FOC to characterize the equilibrium. In this case a prudent equilibrium exists only if thecapital adequacy restriction lies to the left of the crossing, where (ipr/l i0) > 0.

18

pii

l i0(0)= e i0/0 l i0

piiimp

piipr

Figura 3: A prudential limit on overlending

Proposition 2 As competition decreases, the prudent equilibrium becomes more attractive comparedto the imprudent equilibrium ( (

P?I?)v > 0).

Proof: See appendix.The intuition for this result is that with more competition a given shock leaves a bank with

smaller values of second period capital in case of shock (eli (lPr0 ; q)). This means that the prudent

equilibrium is less attractive, since banks can nance fewer rms in the second period. The impru-dent equilibrium, which foregoes nancing rms in the second period in case of shock, becomesrelatively more attractive.

The last stage in the proof is to nd conditions under which the prudent equilibrium is preferableto the imprudent equilibrium in the Bertrand equilibrium. By proposition 2, this means that for anylower degree of competition, the prudent equilibrium continues to be chosen. More generally, if thereis any level of competition for which under specied conditions the prudent equilibrium is preferredto the imprudent equilibrium, then this continues to hold true for any lower degree of competition.

The last result we need is to show that there is a region in parameter space where prudent equi-libria exist and are preferred to imprudent equilibria.

Proposition 3 There is a neighborhood of p = 0 in which prudent equilibria are preferred for allintensities of competition.

Esta demostracion sirve aunque l1 , h1 .Proof: We proceed by showing that there is a neighborhood where prudent equilibria are preferredin the Bertrand case, and therefore by proposition 2, will also be preferred in less competitive bankingsystems. Recall that under Bertrand competition, an interior equilibrium can exist only if h(L) +l (L) = 0. In that case, the banks prots are no problem. Prove others.

B = (p + H 1)e0 > 0,

19

The other case, in which h(L) + l (L) > 0 (if this term is negative there is no lending), leadsto l0 = e0/0 in the prudent equilibrium. This is the same amount of lending as in the imprudentequilibrium. Since the outcomes will be the same in the bad state, this is inconsistent with at leastone of the two equilibria existing: second period capital cannot be strictly positive and zero at thesame time. Thus only interior equilibria exist in prudent equilibria. RF: Agregue esta

explicacion.Note that if p = 0, the shock always hits. The imprudent equilibrium always leads to bankruptcyof the bank and zero prots. In that case, we cannot be in a prudent equilibrium where lending isbound by the capital adequacy constraint since it would be inconsistent with the denition of aprudent equilibrium.

Thus only interior prudent equilibria are viable and we showed above that prots in the Bertrandprudent equilibrium are strictly positive. By continuity of the prot functions, there is a neighbor-hood of p = 0 in which prudent equilibria are also chosen. Note also that by proposition 4 , lowerintensity of competition leads to less lending, and therefore lower probabilities of collapse.

It is interesting to note that there is also a range for which the imprudent equilibria are preferredfor any intensity of competition. Consider the case where p 1 and q 0, so there is a smallprobability of a shock, but when it occurs avoiding a banking collapse does not provide an advantagebecause most rms fail; el1 > 0 but small in the prudent equilibrium. In this case, it is easy to showthat an imprudent equilibrium is preferred (in the limit, banks do not lend in the prudent equilibrium).Essentially, making an eort to avoid an improbable collapse, and not gaining much by it, leads toan inferior equilibrium, given the cost of constraining lending in the highly probable state with noshock. This result becomes clearer when we examine the expressions for Expected GDP that aredeveloped below.

6. Comparative statics

Having shown the existence ranges of Pareto selected prudent equilibria for certain parametercongurations, we can proceed to examine the comparative statics in this nancial system. Thiscorresponds to the case in which the economy is subject to relatively small shocks that do not en-danger the banking system, or alternatively, that the banks are very well capitalized; or nally, thatthe value of the parameter 0 restricts lending as in gure 3.26. In order to do comparative staticswe recall Assumption 3, which is used to show the following important result:

Proposition 4 Increased competition in banking (lower v) increases rst period lending (and reducesthe rst period interest rate) in both prudent and imprudent equilibria. Moreover el1 .

26In this last case we need, in addition, that q < q so the prudential equilibrium is viable, i.e., that the crossing of piiprand piiimp occurs to the right of the highest point in pi

ipr . Otherwise the banking system is inherently unstable and we

cannot perform comparative statics.

20

Proof: Note that

20l0v

= *..,p +

2(1p)l1

(y2 (1 l1)

)q

(p + (1 p)q)+//-

ddl0

*,l0 Lv

G(G1(L))+-

The denominator of the rst expression in the RHS is positive by Assumption 3, and

ddl0

*,l0 Lv

G(G1(L))+- =

Lv + l0

2Ll0v

G(G1(L))

(l0Lv

Ll0

) (G(G1(L))G(G1(L))3

)> 0

where the last inequality is implied by the fact that G() is increasing and concave. Thus prots haveincreasing dierences in (l0,v). We can use Topkis Lemma, which implies that rst period lendingl0 is decreasing in v. For the case of imprudent equilibria, see the appendix.

This result implies that a more competitive banking system leads to increased economic activityin the rst period, and to more eciency in that period. There are more entrepreneurs that carryout projects given the lower interest rates. On the other hand, banks become riskier, because theyare more leveraged. In case of a shock, a larger fraction of the bankscapital will be wiped out; thatis, the banksloan book is riskier. 27 Thus, it is not clear that the expected second period product ishigher as competition increases.

The expected value of GDP over the two periods is:

Y P = p[(y1 1) + (y2 )]l?0 + (1 p)[q(y1 1)l?0 +el?1l1

(y2 )] + GmaxU (r?0 )

udG(u)

Proposition 5 When the risk of a shock is low, increased competition raises expected GDP and secondperiod activity in a prudent equilibrium.

Proof: See appendix.28

The reason for the qualier is that increased competition raises rst period activity as well assecond period activity if there is no shock, but lowers it otherwise. When the probability of a shock islow (p 1), increased competition raises expected second period activity, and thus more competitionunambiguously raises expected GDP.29 When the probability of a shock is high (p 0), competition

27Moreover individual loans are riskier for the bank: since the probability of renewing a loan is smaller when thefraction of the banks capital that is lost increases, loans become riskier and therefore have a lower return for the bank,in addition to the fact that interest rates are lower.

28Note that in the event of a banking collapse, deposit insurance is a transfer (from taxpayers to depositors) so it doesnot alter GDP. Income is aected by the failure of projects after the shock, and that is reected in the expression forGDP, both in the case of a banking collapse in which there is no output from any investments, and in the case withouta collapse, in which output after a shock decreases.

29Observe that in an imprudent equilibrium, there is second period activity only if there is no shock, and without ashock, competition increases activity. Thus increased competition always increases expected GDP in imprudent equili-

21

reduces expected second period activity.Note that in the case p = 1 and q = 0, there is no leverage in the prudent equilibrium (i.e., e0 = l0)

and this provides strictly less GDP than the imprudent equilibrium, where banks do use leverage.Therefore, there is a neighborhood in which expected output is higher under imprudent equilibria.As mentioned in the previous section, the eort to avoid an outcome which is unlikely (a bankingcollapse following the very rare shock), specially when the shock is very severe and therefore fewrms can pay back their loans imposes too severe a constraint on lending, and it is preferable torisk the low probability shock. Thus, there is a range in parameter space where imprudent equilibriaare preferred by banks and lead to higher welfare than the prudent equilibria.

The next result shows the risks associated to increased nancial competition: it shrinks the rangeof shocks for which the prudent banking equilibrium is valid. Recall that in (8) the value q is thefraction of rms that survives a shock that leaves the banks on the threshold of failure in the case ofa prudent equilibrium. An increase in q means that a smaller shock endangers the system. We have:

Lemma 2 Increased banking competition (lower v) decreases the range of shocks (q [q, 1]) for whichthe prudent equilibria (el1(l

Pr0 ; q) > 0) are well dened.

Proof: See appendix.

Next we show that even for prudent equilibrium, so there are no banking crisis, increased com-petition increases risk in the economy, because the magnitude of the sudden stop in lending in thesecond period after a shock is larger. Recall that (l l1/lh1 ) (see (10)) measures how much lendingthere is after a shock compared to lending without a shock, and that lending is directly associa-ted to economic activity. Moreover, a fall in increases the variance of second period lending, andtherefore the variance of second period economic activity.30

Proposition 6 Consider a prudent equilibrium. Increased banking competition (lower v) leads to largerreductions in lending in the second period in the case of a shock.

Proof: See appendix.Note that this proposition, at its heart, has the notion that the equity of banks after a shock is

smaller as competition increases. The next result is expected: a reduction in the size of the shocks(q ), i.e., a reduction in risk, leads to higher rst period lending.

Proposition 7 Consider a prudent equilibrium. Assume that under Bertrand competition, rst periodlending is interior to the capital adequacy constraint. Then less risk (higher q) leads to more lending inthe rst period (higher l0).

Proof: See appendix.

bria.30Since Var(l1) = p(1 p)(l0 ll1)2 = p(1 p)(1 )2(l0)2.

22

6.1. Regulatory forbearance

We have seen in section 4 that unexpected regulatory forbearance on the capital adequacy cons-traints can eliminate the eects of shock on second period activity (in the case of prudent equilibria).In many cases, banks anticipate that the regulator will exercise forbearance after the shock, and thisshould alter their behavior. Note that this is equivalent to studying the eect of reducing the valueof the future capital adequacy parameter l1 in the bad state on the bankers problem at t = 0.31 Thefollowing results show the eects of regulatory forbearance:

Proposition 8 (Regulatory forbearance) In a prudent equilibrium, if the nancial regulator prac-tices forbearance in the bad state (l1 <

h1 ), the variance of the second period outcome decreases because

l0 decreases, i.e., l0/l1 < 0.

Proof:First, note that expected prots for banks in a prudent equilibrium at t = 0 can be written as(see 16):

maxl0

(h(L) +l (L)

)l0 + constants

After some rearranging, the FOC become:

p( ) + H (q 1) = 0

the rst and second terms correspond, respectively, to the marginal prots in the good and the badstate. Moreover, > 0, reecting that ex post, the rm would have preferred to lend more inthe rst period since in the good state, it has more capital than it needs to nance second periodprojects. Similarly, q1 < 0, reecting the fact that ex post, the bank would have preferred to havelent less in the rst period, so that she would have had more capital in the second period, allowingit to have more second period lending.

Now note that this division of the FOC isolates the j1, j = l, h parameters, with l1 appearingonly in H, that is in the condition applicable in case of shock. It is easy to show that H/l1 < 0.Hence, forbearance (a reduction in l1 relative to h1 ) decreases lending by making it more valuableto have capital in case of a shock.

Observation: Note that a change in l1 has no eect on an imprudent equilibrium, because in theequilibria, in case of shock el1 = 0.

31Note that when there is no shock, the capital adequacy condition does not bind so relaxing only aects secondperiod behavior when there is a shock.

23

To get additional intuition about this result, note that Var(l1) = p(1 p)(l0 l l1)2. From thedenition of el1 in (7) we have that el1 = q(1 + r0)l0 (l0 e0), and l l1 = el1/l1. Therefore:

dl l1dl1

= el1

l21Static Eect 0 , so the strategic term in (22) is also negative. Counterintuitively, the eect of knowing thatthe government will relax the capital adequacy conditions after a shock reduces rst period lending,and thus leads to reduced variance in second period outcomes due to both a strategic and a directeect. We have shown:

Corollary 1 The eect of expected forbearance is to increase second period lending in the bad stateand thus to decrease the eect of the shock and the variance in economic outcomes.

It is interesting to examine briey the case in which the regulator commits itself to an incon-sistent behavior. Assume for instance that the regulator announce that it will change the capitaladequacy parameter act so as to eliminate the eects of a shock, in the case of prudent equilibria.Thus second period lending in the bad state satises: RF: Esto lo

agregu

lh1 = ll1 =

el1l1

Then, in the case of Bertrand competition there is no prudent equilibrium, because banks always gainby increasing their loans slightly (they increase their prots in both states of the world), so long asel1 > 0. Thus only imprudent equilibria survive under Bertrand competition when governments usethis inconsistent policy.

In the general case, for smaller intensities of competition, the inconsistent behavior of the regu-lator will lead to increased lending (in a prudent equilibrium), with respect to the case of a consistentregulator, which sets the capital adequacy conditions independently of the ex post situation. For aproof, see the appendix.

24

6.2. Comparative statics between types of equilibria

We have been working under the assumption that the equilibrium do not involve a collapse of thebanking system (i.e., we are not in an imprudent equilibrium). However, under certain conditions, acollapse of the banking system in the bad state of the world may be convenient for rms, becausethey do so much better in the good state of the world. That is, under certain conditions it may beconvenient for bank owners to bet the bank on the non-occurrence of the bad state of the world.We have shown before that it is possible to use the capital adequacy conditions to exclude thispossibility, forcing them to be more conservative. However, the regulator may not always apply theseconditions, or the regulator may be incapable of supervising the banks compliance with the rule.For this reason, we explore the case in which imprudent equilibria are allowed by the regulator. Inparticular, we are interested on the possibility that competition may lead banks to become imprudent.We now proceed to the main result of the paper.

Proposition 9 Increased banking competition always leads to increased variance in second period eco-nomic outcomes. This occurs within and among types of equilibria.

Proof: Consider rst the case of a prudent equilibrium. Proposition 6 shows that if we are in therange in which increased competition leads to a prudent equilibrium, so there is no switch to animprudent equilibrium, second period economic results have increased variance.

Now consider imprudent equilibria. Since competition implies higher lending in these equilibria,economic activity without a shock is higher. On the other hand, when there is a shock, lendingand the associated economic activity is always zero. Hence the variance of second period economicactivity increases.

Finally, note that by proposition 2, when competition increases, the equilibria can go from pru-dent to imprudent, and never in the other direction. We can decompose the eects of increasedcompetition and a switch in type of equilibria as an increase in competition among prudent equili-bria, and a switch between types of equilibria, keeping constant the intensity of competition. Therst eect increases the variance of second period economic activity. Furthermore, the jump froma prudent to an imprudent equilibrium, keeping constant the intensity of competition increases thevariance of economic results, since loans are larger under the imprudent equilibrium and thus se-cond period economic activity is higher when there is no shock, and the eects of the shock are alsomore severe.

25

7. Extension: Leverage

The results on stability depend on competition through its eects on the leverage (or gearing)chosen by banks. At a fundamental level, it is the increase in leverage that causes instability. Thus,even though this paper only compares among Collusive, Cournot and Bertrand equilibria, one coulduse other models to alter the intensity of competition, such as banks located in a Salop (1979) circlemodel, and the main messages should continue to hold.32

Noting that leverage is key to understanding the amplication eect raises the question of whet-her there are other policies that increase leverage and therefore tend to increase instability. Considerthe following two examples.

7.1. Financial liberalization

Up to now we have assumed that the cost of funds for banks is zero, but we now examine thecase of a positive cost of funds in a closed nancial market, This would not aect our qualitativeresults, but would reduce lending and leverage with respect to our baseline model. Now assume thatthis closed economy liberalizes its nancial markets and savings can ow in or out of the economy,but there is no change in the structure of the banking market foreign banks cannot operate in thecountry. We also assume that the probability of a shock and its depth remain the same as beforeliberalization.

Then, if the domestic rate cost of funds prior to liberalization was higher than the internationalrate, the reduction in the cost of funds after liberalization will increase lending and bank leverage.Moreover, since the pass-through of a cost reduction to rates on loans is higher in a more competitivemarket, the positive impact of nancial liberalization on leverage will be higher in more competitivebanking markets. Hence the impact of liberalization on second period instability is reinforced whenthe banking sector is more competitive. This reasoning may explain the observation that in severaldeveloping countries, initial nancial liberalization was accompanied by nancial instability.

7.1.1. Improved creditor protection

Consider improved creditor protection as described in Balmaceda and Fischer (2010). Assumethat in the case of a shock, rm that do not obtain a second period loan still have a residual valuefor the bank. This liquidation value of the project is received in period 2, but banks cannot inclu-de it as part of rst period capital.33 Thus these liquidation values cannot be used to increase thesupply of second period loans. However, since projects are more protable overall, leverage increa-ses. Moreover, if this residual value increases with the quality of creditor protection, leverage will

32For example, we have examined the eect of increasing the number of rms in Cournot competition and the mainresults continue to hold.

33Perhaps because these liquidation values have a probability of being zero.

26

increase. In this case, we get the somewhat surprising result that second period instability increaseswith increased creditor protection.

8. Conclusions

This paper has examined the eects of increased banking competition in two period model wherea rst period shock to economic activity leads to defaults on loans. These defaults lower the capitaland reserves of banks, reducing their lending in the second period. Thus the rst period shock isamplied by the banking system. We study the eects of varying degrees of competition in thissetting.

The model allows us to understand several phenomena in the interaction between banking com-petition, economic activity and regulation. We have shown that there are two types of symmetricequilibria, which we denote by prudent and imprudent equilibria. Equilibria of the rst type amplifythe initial shock but do not cause the collapse of the banking system and the breakdown of lendingactivity, as occurs in the second type of equilibria. Both types of equilibria can be Pareto Optimalunder dierent circumstances, such as the prevalence of shocks and their magnitude.

We have a series of results related to the eects of increasing competition among banks. First,as competition increases, the imprudent equilibria become relatively more attractive to banks. Mo-reover, even when we consider only prudent equilibria, increased competition means that the am-plication of the initial shock is larger, because banks tend to lend more and therefore a shock leadsa larger reduction in capital and reserves after paying back depositors. This leads to less lendingin the second period, because banks are restricted by capital adequacy parameters. We also showthat when the risk of a shock is low, increased competition raises GDP (in expectation), as well asexpected second period activity.

The paper also examines the role of the banking regulator. In the model, capital adequacy rulescan be used to exclude imprudent equilibria. This is consistent with the observation that requiredcapital ratios have risen after the experience of the 2008 nancial crisis. We also show that unantici-pated regulatory forbearance in the aftermath of a shock can be used to reduce or even to eliminatethe amplication eect. However, we also show that when banks predict that there will be regulatoryforbearance after a shock, the eects of forbearance on economic activity are ambiguous. Parado-xically, it is possible that anticipated forbearance increases the variance of second period activityby encouraging rst period lending and thus a larger amplication of the initial shock. Our mainresult is to show that within and between types of equilibria, increased competition always leads toincreased variance in second period economic activity.

In the extensions we show that other ways of increasing leverage may also lead to increasedinstability. In particular, banking competition may add to the the instability caused by nancialliberalization of domestic nancial markets.

27

A worthwhile extension of this approach would be to have entrepreneurs dierentiated by theircapital endowments, and have credit rationing driven by informational asymmetries or legal de-ciencies. Such a model would allow us to study the interaction between the legal protection for len-ders and the eects of banking competition, or the interaction between the distribution of wealth,competition and stability.

Another extension is to endogenize banking capital. One possibility is to x a minimum capitalstock to have a bank and then allow free entry (or, have an exogenous xed cost of setting a bank). Inthat case regulatory policy, by setting the minimum capital to establish a bank would determine theintensity of competition in the case of Cournot competition. Alternatively, we could x the numberof banks and allow capital to ow into them until the expected return on investment in banks is thesame as the return in other sectors (zero in our case).

Finally, we can consider alternative policies in the event of a productivity shock leading to acollapse. A possibility is the government recapitalizing banks as was done to some banks in the2008 nancial crisis. If the banks becomes the property of government (and managed by controllersnamed by it), from the point of view of the bans controllers, at t = 0, the strategic problem is thesame as the one we have analyzed before. There is no additional incentive to be imprudent. Theonly dierence is that, if the lending policies of the recapitalized bank do not change, the eects ofthe initial shock are neutralized or even eliminated altogether. If the initial controllers keep someownership rights, the eect is to increase incentives to be imprudent.

28

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31

A. Appendix: For Online Publication

Proposition 1 At date t = 1, each bank makes loans of l s1 = mn{es11, l0

}and takes deposits of

ds1 = (1 s1)l s1 and pays dividends of Divs = es1 1l s1.

Proof: Rearranging terms and using the equality condition es1 + ds1 l s1 = Divs plus 1 + r1 = y2 , theproblem of the bank is:

Maxls1,Div

s

(y2 1

)l s1 + (1 )Divs + es1

s.t. es1 1l s1 + Divsl0 l s1

Now, note that by Assumption 2 the rst term in the objective function is positive. Moreover, As-sumption 2 ensures that

(y2 1

)> (1 ), so the objective function increases more with l s1 than

it increases with Divs. Hence, Divs is positive only if the second restriction is binding. Therefore, itis direct that in equilibrium:

l s1 = mn{es11, l0

}

Lemma 1 (Imprudent equilibria) There is a unique equilibrium of each type (prudent, imprudent)to the game among banks, i.e.,

2i

l2i< 0

and2il2i2ilil j

> 1

Proof: Case of imprudent equilibrium:

ili

= p( )

and thus2i

l2i= p

li= p

(2

PeG(G1(L)) liG

(G1(L))PeG3(G1(L))

)< 0

and2ilil j

= p

l j= p

(1

PeG(G1(L)) liG

(G1(L))PeG3(G1(L))

)

32

From which we derive:2il2i2ilil j

=

(2

PeG(G1(L)) liG(G1(L))PeG3(G1(L))

)(

1PeG(G1(L))

liG(G1(L))PeG3(G1(L))

) > 1

Proposition 2 As competition decreases, the prudent equilibrium becomes more attractive comparedto the imprudent equilibrium ( (

P?I?)v > 0).

Proof:: Assume rst that the solution to both problems is interior. Then dene the following opti-mization program:

W (x) Maxl00

xP + (1 x)I

which is the convex combination of the banksproblem in the two types of equilibrium. Rewriting,and using expressions 1519 we obtain

W (x) Maxl00

x [p((1 + r0) )l0 + H (q(1 + r0) 1)l0 + (p + H 1)e0]

+ (1 x)[p((1 + r0) )l0 + (p 1)e0]

Using the Envelope Theorem and the denition of el1 from equation 7:

W (x) = H[(q(1 + r?0 (x)) 1)l?0 (x) + e0] Hel?1 (x)

Thus:P? I? = W (1) W (0) =

10

Hel?1 (x)dx

By taking derivatives with respect to v we obtain:

Wv

= H 1

0

el?1 (x)v

dx > 0

We prove that el?1 (x)v > 0 (x). In eect:

el?1 (x)v

= (q(x) 1) l?0

v

The derivative on the RHS is negative by proposition 4 below. For the other term in the RHS, notethat from the FOC of the rst period banks problem,

[p + Hqx](x) = p + Hx

33

and thus(x) =

p + Hxp + Hqx

Finally, it is easy to check that:34

1 q(x) = p(1 q)p + Hqx

> 0

nally, consider the case in which the capital adequacy conditions constrain lending in the imprudentequilibrium. Figure 4 shows the possible congurations and it is clear that the result continues tohold in this case.

Pii

v

Ii

i

v

Ii

Pi

Figura 4: Prot dierences increase when the imprudent equiilibrium is credit constrained for arange of competition parameters

Proposition 4 We show that as competition increases in banking (lower v), rst period lending increa-ses in the case of an imprudent equilibrium.

Proof: Note that:

2I0l0v

= p(p + (1 p)q)

ddl0

*,l0 dLdv

G(G1(L))+-

Where,

ddl0

*,l0 dLdv

G(G1(L))+- = *.,

dLdv + l0

d2Ldl0dv

G(G1(L))+/- *,

l0 dLdvG(G1(L))

G(G1(L))+-

34This result is true when the FOC hold with equality. At v = 0 (Bertrand) there is the possibility of corner solutions.where the result does not necessarily hold, because solutions are of the bang-bang type.

34

Given that G() is assumed to be increasing and concave we conclude that the RHS of the ex-pression above is strictly positive and therefore we get the result.

Proposition 5 When the risk of a shock is low, increased competition raises expected GDP and secondperiod activity in a prudent equilibrium.

Proof: The expected value of GDP over the two periods is:

Y P = p[(y1 1) + (y2 )]l?0 + (1 p)[q(y1 1)l?0 +el?1l1

(y2 )] + GmaxU (r?0 )

udG(u)

The eect of an increase in the degree of competition among banks can be written as:

dY P

dv=

{[p+ (1 p)q](y1 1) + p(y2 )} dl?0dv + (1 p)l1 (y2 )del?1dvU (r?0 )G(U (r?0 )) *.,

dl?0dvPeG(G1)(L?)

+/-Where the rst and third terms are strictly negative, while the second term is positive, so the sign

of the expression is ambiguous. The eect of competition on second period activity can be writtenas:

(y2 )[pdl0dv

+(1 p)1

de1dv

]Evaluating at the polar cases p = 0, 1 we see that when the risk of a shock is low (p 1) increasedcompetition is benecial and raises second period activity by proposition 4 (and therefore GDP isunambiguously higher). On the other hand, when the risk of a shock is large (p 0), increasedcompetition decreases second period GDP.

Lemma 2 Increased banking competition (lower v) decreases the range of shocks (q [q, 1]) for whichthe prudent equilibria (el1(l

Pr0 ; q) > 0) are well dened.

Proof: From the denition in (8),

q 1 e0l?0

1 + r0(L(l?0 ))(23)

By implicit derivation, we obtain:

dqdv

=

(e0

l?20 (1 + r0)

)dl?0dv *.,

1 e0l