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State ownership, credit risk andbank competition: a mixed oligopolyapproachBibhas Saha a & Rudra Sensarma ba University of East Anglia , Norwich , United Kingdomb University of Hertfordshire , Hatfield , United KingdomPublished online: 01 Mar 2012.

To cite this article: Bibhas Saha & Rudra Sensarma (2013) State ownership, credit risk and bankcompetition: a mixed oligopoly approach, Macroeconomics and Finance in Emerging MarketEconomies, 6:1, 1-13, DOI: 10.1080/17520843.2011.641719

To link to this article: http://dx.doi.org/10.1080/17520843.2011.641719

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State ownership, credit risk and bank competition: a mixed oligopoly

approach

Bibhas Sahaa and Rudra Sensarmab*

aUniversity of East Anglia, Norwich, United Kingdom; bUniversity of Hertfordshire, Hatfield,United Kingdom

(Received 24 August 2011; final version received 11 November 2011)

The recent financial crisis led many governments to buy equity in banks leading tosituations of mixed oligopoly in banking markets. We model such a case where apartially state-owned bank competes with a private bank in collecting deposits.The government is purely a welfare maximizer while the private bank maximizesprofits. Both banks face risks in the loan market. We show that if credit risk issufficiently high and there is limited liability, the state-owned bank mitigatesdepositors’ losses by mobilizing less deposits leading to contraction of aggregatedeposits. This contradicts the standard mixed oligopoly results in the literature.

Keywords: banking; mixed duopoly; credit risk

JEL classification: G21, L13, L33

1. Introduction

During the recent financial crisis policymakers around the world have turned theirattention to state ownership of banks. Ironically as recently as a decade ago therewas an apparent consensus in favour of bank privatization and governments indeveloping countries were compelled to divest a significant proportion of the equityof state-owned banks. This was seen as a cure for non-performing assets, a chronicproblem that was associated with public sector banks. But in a dramatic reversal offortune, it is now the private sector banks that are at the centre stage of the recentmeltdown and have even been blamed in the media for their pursuit of profit. Whileeconomists are divided in their prescription for a solution, policymakers have rushedto bail out distressed private banks, and even nationalized many of them. Oneimmediate benefit of nationalization is recapitalization. But how will it affect bankcompetition in the deposit market and in the loan market? The banking literature hasstudied various issues ranging from financial contagion (Allen and Gale 2000), bankruns and liquidity crisis (Smith 1991; Diamond and Rajan 2005), to competition andrisk-shifting (Boyd and De Nicolo 2005); but it has mainly considered interactionsamong private banks, and therefore may not be adequately helpful in understandingthe outcome of emerging competition between public and private banks. This paperstudies the outcome of bank competition in the context of nationalization.Specifically we show that if credit risk is sufficiently high and there is limited

*Corresponding author. Email: rsensarma@gmail.com

Macroeconomics and Finance in Emerging Market Economies,

1–13, http://dx.doi.org/10.1080/17520843.2011.641719

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liability, then a nationalized bank would try to protect depositors from potentiallosses by mobilizing less deposits leading to contraction of industry deposits.

There is a separate body of work in industrial organization known as mixedoligopoly that exclusively studies competition between public and private firms (DeFraja andDelbono 1989;Matsumura 1998). The underlying view of mixed competitionis that state ownership provides an indirect means of regulating the conduct of privatefirms using market instruments rather than using administrative-judicial interventionslike the US anti-trust system. A general result of this literature is that the publiclyowned firm will act more aggressively and have greater market share than the privatefirms; but social welfare may not always rise, though in most cases will, especially if thepublic firm’s degree of state ownership is optimally chosen. Our aim is to adaptthe mixed oligopoly approach to bank competition and see how state ownership affectsbank interactions. There is a significant difference between banks and ordinary firms.In the industrial organization literature firms mainly interact in the output market andhave no or little interactions in the input market. Banks, on the other hand, have tointeract in both the deposit (input) market, and the loan (output) market; moreover bylending to each other they add another dimension to their relationship. Above all,unlike ordinary firms banks have to deal with credit risk, for example loan default riskand its knock-on effects on different aspects of the banking business. In this paper weexplicitly allow for such features which distinguish banks from standard non-financialfirms that are studied in the industrial organization literature.

Apart from the above-mentioned theoretical issues, there are good empiricalreasons as to why banking should be modelled using the mixed oligopoly approach.Despite widespread banking deregulation, entry of private and foreign banks has byand large remained a subject of state control in many countries. In fact, statepresence is a common feature in banking systems all over the world. See Barth et al.(2001) for evidence on European Union (EU) countries, Sherif et al. (2003) fortransition economies and Shirai (2002) for China and India. It is, therefore,reasonable to expect that the banking industry in many economies (especiallyemerging) resembles mixed oligopoly with strategic interactions among banksoccurring at many dimensions. There have been only a few articles that haveconsidered mixed ownership or non-profit maximizing behaviour of banks. Purroyand Salas (2000) studied competition between a private bank and a savingsinstitution, and Saha and Sensarma (2004) between a private bank and a publicbank. Both articles have demonstrated that the mixed duopoly approach can beuseful; but they share a common limitation of focussing only on deposit competitionand not credit risk, which we aim to overcome in this paper.

In modelling public banks, our first hurdle is to define the objective of a publicbank. Following the mixed oligopoly literature we assume that it maximizes socialwelfare, which is the sum of the payoffs earned by all participants in the bankingindustry, which principally are banks’ shareholders, borrowers (who are entrepre-neurs seeking funds for risky projects) and depositors. But by no means is this theonly possibility. One can add other objectives, specifically linked to loan or depositmarkets that might better describe the circumstances and motivation for stateintervention in a particular country.

Throughout our analysis we will consider only a duopoly setup. We model depositcompetition between a partially state-owned bank and a private bank.We introduce aloan market and allow uncertainty in the form of default risk. The government’sobjective in this model is entirely social welfare maximization and the private bank is

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a profit-maximizer. Both banks have equity capital along with deposit liabilities. Wefirst consider the case where the banks bear the entire risk; they pay depositors the dueamount, even if the borrowers have defaulted. In this case the public bank actsaggressively, and this leads to greater deposit mobilization. Then we introduce limitedliability, which essentially puts the depositors at risk; if borrowers default, depositorslose money as banks (like their borrowers) are protected by limited liability. We showthat the public bank in this case takes into account the depositors’ loss, and if thedefault risk is significant it will act conservatively and collect less deposits. Thoughthis will benefit the private bank in their deposit mobilization, aggregate deposits canfall below the pure duopoly level (which corresponds to the case of competitionbetween two private banks). This result is quite opposite to what standard mixedduopoly models predict.

Standard mixed duopoly models, which usually focus on output competition,predict more aggressive behaviour of the public firm largely because it puts a positiveweight on social welfare, and in the usual way greater output leads to greater welfare.This argument largely holds for bank competition as well, if depositors do not face anyrisk (due to either lack of limited liability or complete deposit insurance). But as hasbeen seen in recent crises, limited liability on the part of the entrepreneur-borrowersand banks put depositors at risk. A social welfare oriented public bank takes intoaccount the potential loss that depositors might face when default occurs. If this risk isnot small, the public bank will try to restrict its deposit mobilization below a profit-maximizing level. The overall outcome may be a contraction of total deposits.

Our result can be related to a well established literature that has studied the effect ofbank competition on risk taking. Both the theoretical and empirical results are mixed,though policymakers tend to believe that competition has destabilizing effects on risk-taking by the bankers. Boyd and De Nicolo (2005) provide an overview of thisliterature, and argues that the negative view of competition is largely based on thetheories that consider mainly single market interactions, viz. the deposit market (seealso Hellman et al. 2000). If a loan market is introduced, they argue, bank competitionwill reduce the interest rate on loans and thus reduce the risk of default. In our paper,the state owned bank has a similar effect on aggregate deposit as competition does in asetup of private banks. We show that if the probability of default is below a criticallevel, then the Boyd and De Nicolo (2005) type of positive effect emerges, but if theprobability of default is above this critical level, then the negative effects may dominate.

The rest of the paper is organized as follows. Section 2 provides an empiricalmotivation for the paper by drawing on recent data from Indian banking. Section 3sets up the theoretical model of deposit competition with default risk in the lendingside. In section 4 we consider the benchmark case of unlimited liability where thedepositors are not exposed to the effects of credit risk. Section 5 explores the case oflimited liability, which shifts the risk onto depositors. Here we derive our mainresults on the effects of state ownership. Section 6 concludes.

2. An empirical motivation

Our theoretical model can draw motivation from the situation in Indian bankingwhich has, for a long time, been characterized by competition between public sectorbanks (most of whom are partially owned by the government) and private sectorbanks. We examine data on deposits and gross non-performing assets (NPAs) ofpublic and private sector banks in India for the period 1999–2000 to 2008–2009. We

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obtained the deposit data from various issues of a Reserve Bank of India (RBI)publication viz. Basic Statistical Returns of Scheduled Commercial Banks in India.The NPA data were collated from various issues of another RBI publication viz.Report on Trend and Progress of Banking in India. Figure 1 panel A plots depositgrowth of the two bank groups that indicate that public banks have been moreconservative in deposit collection than private banks over the past decade. Thetraditional explanation for this phenomenon is that public banks operate in asaturated market whereas private banks have been able to continuously expand theirdeposit base through exploration of new markets as well as innovative strategies.Our model offers an alternative explanation in terms of credit risk which has beenhitherto missing in the policy debates. More specifically, we argue that higher creditrisk would lead the public banks to collect fewer deposits than the private bankssince the former are more concerned with protecting depositors’ losses.

To cross-check the validity of the above argument we calculated the simplecorrelation coefficient between deposits and credit risk (proxied by NPAs) of publicand private banks. The first row of Table 1 shows that deposits are negatively

Table 1. Correlation of deposits and NPAs in Indian banks.

Public sector Private sector

NPA (t) 70.745 0.748NPA (t71) 70.913 0.446

Figure 1. Deposit and NPA growth in Indian banks.

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correlated with NPAs for public banks but not so for private banks. Clearly publicbanks seem to reduce their deposits when credit risk is high. In order to study acausal relation between the two variables we also look at the correlation betweendeposits and past NPAs (second row of Table 1). Once again the correlation is highlynegative for public banks, which indicates that they adopt a conservative approachto deposit collection as a result of higher credit risk. However this does not seem tobe the case for private banks whose concern is for profits and not depositors’ losses.

In fact when we look at the growth in NPAs (see Figure 1 panel B) it is apparentthat private banks are faced with increasing credit risk in most years. Yet it is thepublic banks who respond to higher credit risk (for instance in 2001–2002 and 2003–2004) with lower deposit growth. This is also the prediction of our theoretical model.However, our model does not match with the situation in Indian banking during therecent global financial crisis as Indian banks were unaffected by the crisis (note thefalling credit risk levels in the last couple of years in Figure 1 panel B).

3. The model

The public bank is indexed 0, and the private bank indexed 1. Depositors earninterest rate rD by the following rule

rD ¼ bðD0 þD1Þ; b > 0;

where D refers to deposits collected by a bank and b is the slope of the aggregatedeposit supply curve. The public bank is jointly owned by the government and aprivate partner, and the choice of the volume of deposit is made by the bank’s boardof management consisting of a government representative and the private partner. Itmaximizes the following objective function

Z ¼ ySWþ ð1� yÞp0; ð1Þ

where y (y 2 [0,1]) is the degree of public ownership,1 SW is social welfare (defined asthe sum of bank profits, depositor surplus and entrepreneur-borrower surplus) and p isprofit. The resultant deposit choice gives rise to the public bank’s reaction function.

There is an alternative approach to modelling a public firm suggested byFershtman (1990), in which the same weights (y and (17y)) can be applied to a fullypublic bank’s and a fully private bank’s reaction function to arrive at the partiallypublic firm’s reaction function. Saha and Sensarma (2004) followed this approach.Both approaches offer qualitatively similar conclusions. In fact, Kumar and Saha(2008) have shown that both approaches are formally equivalent.

The private bank’s manager chooses D1 to maximize p1. The manager’s choice ofD1 gives the private bank’s deposit reaction function.

We now introduce a loan market where the interest rate is determined by thefollowing loan demand curve rL ¼ �c� cL; c > 0; where L is the total loans made andc is the slope of the aggregate loan demand curve. The banks give out loans from notonly the deposits raised but also from equity (E) raised from the capital market.Total lending by a bank equals the sum of deposits and shareholder equity, that isL¼DþE. Left to themselves the shareholders might not put in enough of their ownfunds into the bank leading to a highly leveraged bank (where the leverage ratio is D/L). Therefore, through capital regulation the regulator has to enforce a minimum

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level of shareholder equity (Milne 2006). Let g be the capital to asset ratio imposedby the regulator (similar to risk weighted capital adequacy ratio, e.g. 8% underBasel-I norms) such that g¼E/L. We can then rewrite the balance sheet equality asL¼ kD where k ¼ 1

1�g.We also assume that the entrepreneur-borrowers face uncertainty in the

realization of the value of their project. There are two states of nature: good andbad. The good state occurs with probability p1 and bad state with probability p2.In the good state the borrowers are able to pay back both principal and interest,but in the bad state they pay back nothing. However, the bank can liquidate thefirm and recover the principal, but the interest is lost. Alternatively the loan can beassumed to be collateralized. In the bad state, as the bank loses interest earnings,its ability to repay the depositors is affected. The depositors get back the principal,but lose out on the interest payments. However, since the bank has equity (k4 1)it will be able to meet the interest obligations to the extent it has funds. Thus, thedepositors stand to lose in the bad state. Note that if there is complete depositinsurance or if the banks are sufficiently capitalized (k large) then the depositors donot lose. But we do not consider deposit insurance here and instead allow thedepositors to bear the risk by allowing limited liability on the part of the banksand entrepreneur-borrowers.2

4. Bank competition: unlimited liability case

As a benchmark case we consider here deposits as debt obligations that are honouredin all states of nature.

Bank i lends Li (¼ kDi) and earns at the end of the year

Epi ¼ p1½ð1þ rLÞkDi � ð1þ rDÞDi� þ p2½kDi � ð1þ rDÞDi�;¼ ½~k� ~dðD0 þD1Þ�Di; ð2Þ

where ~k ¼ k� 1þp1kc and ~d ¼ p1k2cþ b. Sum of the two banks’ expected profits is

Ep ¼ Ep0 þ Ep1 ¼ ½~k� ~dðD0 þD1Þ�ðD0 þD1Þ: ð3Þ

From the area under the positively sloping deposit supply curve, we find the surplusthat depositors expect to enjoy

E½DS� ¼ rDðD0 þD1Þ � bðD0 þD1Þ2

2¼ bðD0 þD1Þ2

2: ð4Þ

We have another group – entrepreneur-borrowers – whose expected surplus, basedon the negatively sloping loan demand curve, is

E½BS� ¼ p1

��ckD0 þD1Þ � ck2

ðD0 þD1Þ2

2� rLkðD0 þD1Þ

�;

¼ p1ck2 ðD0 þD1Þ2

2: ð5Þ

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Expected social welfare of this economy is then defined as SW¼EpþE[DS]þE[BS]. Adding Equations (3), (4) and (5) we get

E½SW� ¼ ~k�~dððD0 þD1ÞÞ

2

" #ðD0 þD1Þ: ð6Þ

The public bank pursues an objective function Z¼ yE[SW]þ (17y)Ep0, where y isthe degree of public ownership. Maximizing Z we derive the public bank’s depositreaction function as

D0 ¼~k

ð2� yÞ~d� D1

2� y:

Similarly, the private bank’s reaction function is

D1 ¼~k

2~d�D0

2:

The reaction curves of the two banks are shown in Figure 2. Two thick curves,denoted as RF0 and RF1 are drawn with the assumption that y 2 [0,1]. Thedownward slopes indicate that the deposits are strategic substitutes.

If the private bank chooses zero deposit, the public bank will choose itsmonopoly deposit as

~kð2�yÞ~d, and similarly, if the public bank chooses zero deposit the

private bank’s manager will choose D1 ¼ ~k2~d: Conversely, if the private bank chooses

D1 ¼ ~k~d, the public bank will simply close down, and similarly, the public bank’s

choice of D0 ¼ ~k~dwill force the private bank to close down. Thus, the monopoly and

entry-deterring levels of deposits of each bank can be defined in the usual way asquantity setting firms’ outputs are defined. The equilibrium deposits are given bypoint M comprising of D0 and D1, which we obtain as

D�0 ¼~k

~dð3� 2yÞ; ð7Þ

Figure 2. Deposit reaction functions: unlimited liability.

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D�1 ¼~kð1� yÞ

~dð3� 2yÞ: ð8Þ

The aggregate deposit in this model is

D�0 þD�1 ¼~kð2� yÞ

~dð3� 2yÞ:

It can be readily checked that both the public bank’s deposit and aggregate depositare increasing in the degree of nationalization. On the other hand, the private bank’sdeposit is decreasing in the degree of nationalization. In fact, when y¼ 0 and bothbanks are profit-maximizers we would have a pure (or equivalently private) duopoly.The public bank’s reaction curve would then swing inward and we have the Cournotdeposits as D0 ¼ D1 ¼ ~k

3~d. This is given by point N in Figure 2. Since point M lies

south-east of point N, it is clear that the mixed duopoly generates greater deposit forthe public bank but lower deposit for the private bank than the private duopoly.

At the other extreme, when the public bank is fully public, y¼ 1, the privatebank’s deposit is reduced to zero (point O in Figure 2). This is the case where thepublic bank crowds out the private bank, a phenomenon common in mixedoligopoly models.3

In this case, when debt obligations are honoured in all states of nature thestandard results of mixed duopoly models continue to hold, in which the public bankacts more aggressively (in raising deposits) and the private bank acts moredefensively. In other words, the standard profit-shifting motive of public ownershipcontinues to hold even if there are some states of nature forcing defaults. It can alsobe checked that if the government was to choose optimal privatization, it wouldchoose y¼ 1. Total deposit under optimal (zero) privatization would be D0 ¼ ~k

3~dand

the expected profit of the public bank will be zero. That is, its expected loss in thebad state would be equal to its expected profit in the good state. The private bank isdriven out of business.

We summarize the above results from the no limited liability case in the followingproposition.

Proposition 1 When debt obligations are always honoured by banks, the public bankacts more aggressively than the private bank, i.e. D04D1. Moreover the public bank’sdeposit and aggregate deposit will increase with the degree of nationalization while theprivate bank’s deposit will fall. That is, @D0

@y > 0; @D@y > 0 and @D1

@y < 0.

5. Bank competition: limited liability case

Now we assume that the banks are insufficiently capitalized, so that in the bad statethey cannot meet their full obligations. This will be the case if k¼ 1 or k is slightlygreater than 1 or the bank is not able to recover the full amount of the principal fromthe entrepreneurs. For simplicity we will assume that the full amount of the principal isrecovered, but k is not large enough to cover the interest charges payable to thedepositors in the equilibrium. Due to limited liability, depositors get back what isavailable with the bank, kD, when kD5 (1þ rD)D. Since rD increases with the totaldepositD, it is quite possible that the depositors may not lose at all if the total deposit isnot large relative to k. That is, if k4 (1þ rD) then even in the bad state the depositorsare safe. For our story to hold, we need to focus on the case where k5 (1þ rD).

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Under limited liability banks’ expected profit is given only by the good state.

Epi ¼ p1½ð1þ rLÞkDi � ð1þ rDÞDi�;¼ p1½k� � dðD0 þD1Þ�Di; ð9Þ

where k� ¼ k� 1þ kc and d ¼ k2cþ b:From the above one gets the aggregate expected profits as

Ep ¼ Ep0 þ Ep1 ¼ p1½k� � dðD0 þD1Þ�ðD0 þD1Þ: ð10Þ

The depositors now stand to lose in the bad state. Their surplus is

E½DS� ¼ p1ð1þ rDÞðD0 þD1Þ þ p2kðD0 þD1Þ �D� bðD0 þD1Þ2

2;

¼ 1

2p1bðD0 þD1Þ þ p2ðk� 1Þ � p2b

ðD0 þD1Þ2

� �ðD0 þD1Þ: ð11Þ

Note that with the restriction kD5 (1þ bD)D or k715 b(D0þD1), (so that limitedliability applies) E[DS] is now strictly less than b

ðD0þD1Þ22 , which was the expected

depositor surplus under the ‘no limited liability’ case.The expected borrower surplus remains unchanged from Equation (5). Adding

the expressions in Equations (5), (10), and (11) we derive the expected socialwelfare as

E½SW� ¼ p1 k� � dðD0 þD1Þ

2

� �ðD0 þD1Þ þ p1b

ðD0 þD1Þ2

2p2ðD0 þD1Þðk� 1Þ

� bðD0 þD1Þ2

2: ð12Þ

Following the same procedure as before we get the following two reaction functions

D0 ¼p1k

� þ yp2ðk� 1Þp1dð2� yÞ þ yp2b

� p1dþ yp2bp1dð2� yÞ þ yp2b

D1; ð13Þ

D1 ¼k�

2d�D0

2: ð14Þ

Solving the above system of equations we get the following Cournot-Nash deposits

D�0 ¼p1k

�dþ yp2fðk� 1Þ2d� bk�gd½p1dð3� 2yÞ þ yp2b�

; ð15Þ

D�1 ¼p1k

�dð1� yÞ þ yp2fbk� � ðk� 1Þdgd½p1dð3� 2yÞ þ yp2b�

: ð16Þ

These result in the aggregate deposit

D�0 þD�1 ¼p1k

�ð2� yÞ þ yp2ðk� 1Þp1dð3� 2yÞ þ yp2b

:

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At this point we can make the following observations. Neither D0 nor D1 areunconditionally monotonic in y. In particular, it is striking that even if y¼ 1, theprivate bank’s deposit does not become zero. It remains strictly positive. Moreimportantly, the aggregate deposit under the mixed duopoly (with the public bankbeing fully public) may not necessarily be greater than that under pure duopoly. Ifboth banks were privately owned (i.e. y¼ 0), aggregate deposit is

Dðy ¼ 0Þ ¼ 2k�

3d:

With full public ownership in the public bank, aggregate deposit is

Dðy ¼ 1Þ ¼ p1k� þ p2ðk� 1Þp1dþ p2b

:

By comparing these expressions and individual deposits of the banks we arrive at thefollowing result.

Proposition 2 If p24 dk�

kð3dc�2k�kcÞ, aggregate deposit under full public ownership of the

public bank will be less than the aggregate deposit under pure duopoly. That is, D(y¼ 1)5D (y¼ 0); moreover, D0(y¼ 1)5D0(y¼ 0)¼D1(y¼ 0)5D1(y¼ 1). On theother hand, if p25 dk�

kð3dc�2k�kcÞ, we have D (y¼ 0)5D(y¼ 1), and D1(y¼ 1)5

D0(y¼ 0)¼D1(y¼ 0)5D0(y¼ 1).

One implication of proposition 2 is that when the likelihood of the bad state isabove a critical level, the depositors’ loss matters a lot and the publicly owned bankcollects deposit very conservatively. However, this defensive act encourages the privatebank to increase its deposit, because the two banks’ deposits are strategic substitutes.However, the aggregate deposit falls short of the fully private or pure duopoly level andthus the depositors’ loss is restricted to an optimal level. On the other hand, if the badstate is less likely, then the government’s social welfare maximization objectiveencourages the public bank to mobilize greater deposits and the market is expandedwell beyond the pure duopoly level. In the latter case, we have a similar outcome as ifthere was no uncertainty. But in the former case, public ownership leads to acontraction in the market, quite in contrast to standard mixed duopoly models.

We may relate this to the finding of Boyd and De Nicolo (2005) where greatercompetition reduces default risk by lowering the cost of borrowing (as the loan ratefalls) and also by making more low-risk-low-profit projects viable. But our modelsuggests that this finding may not hold unconditionally. Though we considered onlyexogenous risk, it can be argued that in the presence of a public bank, results can goeither way. If the default risk is significant, then the public bank’s conservativeapproach will lead to a contraction in total loans causing interest rate to rise. In theview of Boyd and De Nicolo then entrepreneurs will shift to more risky projects. Ifwe allow endogenous default probability, we will be able to see how the public bankreacts and how the government’s choice of public ownership is affected.

6. Conclusion

This paper explores partial nationalization and credit risk in the framework of a‘mixed oligopoly’. We show that if there is no limited liability or if there is complete

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deposit insurance, then in the event of loan default the banks lose money, butdepositors remain safe. But the banks’ loss is equal to depositors’ gain. Therefore, interms of social welfare this loss does not matter. The public bank acts aggressivelyand mobilizes greater deposits leading to an expansion of aggregate deposits, thoughin the process the private bank’s operation is squeezed. This is very much in line withthe predictions of standard mixed duopoly models. But if the banks are protected bylimited liability and deposits are only partially insured, then depositors are exposedto risk, and they cannot pass on this risk to anybody else. Therefore, in the socialwelfare calculations the expected loss of the depositors matters. The public banktakes into account this loss, and changes its behaviour in the following way. If thedefault risk is significant, it will mobilize less deposits resulting in a contraction ofaggregate deposits, though the private bank will be able to profit from this defensiveact. Essentially, the public bank will try to borrow less and lend less. This is areversal of the standard mixed duopoly result. However, if the risk of default issmall, the public bank returns to aggressive competition.

Finally we note some of the limitations of our analysis. While we have assumedthat capital regulation is the only form of bank regulation, in practice regulation ismore complex and multidimensional. For instance, regulators may impose reserverequirements on banks to control liquidity. While we have not explicitly allowedsuch a requirement it can be easily assumed that a fraction of deposits is held by thecentral bank while the rest of it is free to be loaned out. Another type of regulation,especially in developing countries, is restrictions on lending whereby the regulatormay specify targets on loans to certain priority areas. This would require us toinclude different types of borrowers (priority but high risk and non-priority but lowrisk) in the model. With a pool of different types of borrowers we could also look atadverse selection by the banks. We could analyse moral hazard by allowing theborrowers to default even in good states of nature. Lastly we have assumed linearfunctions for the supply of deposits and demand for loans. Generalizing thesespecifications and addressing the aforementioned issues can be considered in futureresearch.

Notes on contributors

Bibhas Saha is a senior lecturer in economics at the University of East Anglia. His researchinterests include economics of corruption, development economics, education and labourmarket. Previously he taught at the Indira Gandhi Institute of Development Research, India.

Rudra Sensarma is a senior lecturer in finance at the University of Hertfordshire. His researchinterests include banking, financial markets, monetary policy and development economics.Previously he worked at the Reserve Bank of India, the University of Birmingham and theIndian Institute of Management (Lucknow).

Notes

1. While the ownership exceeds 50% is very important, it cannot be denied that any changein y will have some effect on the bank behaviour.

2. Exogenously given schemes of deposit insurance can be introduced in our frameworkwithout much complication. But in principle the insurer should be allowed to monitor thebehaviours of the banks, which is a non-trivial exercise and we do not pursue it here.

3. However, in standard mixed oligopoly models for this to happen constant marginal cost isrequired. In our banking model, neither the marginal return curve nor the marginal costcurve is constant. But on both dimensions they depend on aggregate deposits. This is acrucial feature of banks, as opposed to standard firms.

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Appendix

Detailed derivation of proposition 1

It is straightforward to compare D0 and D1 from Equations (7) and (8) and concludethat D04D1. Next, we derive the following expressions:

@D0

@y¼ 2

ð3� 2yÞ2> 0:

@D1

@y¼ 1

ð3� 2yÞ2< 0:

@D

@y¼ � 1

ð3� 2yÞ2> 0

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Detailed derivation of proposition 2

We demonstrate the results graphically. Consider Figure A1 where the depositreaction functions of the two banks under limited liability are plotted under theassumption that the probability of default is above the critical level. While theequilibrium deposits are given by point M as before, our interest is in comparing thiswith the case when y¼ 1. This is indicated by the shaded reaction function of thepublic bank RF0 whereas the private bank’s reaction function RF1 is unaffected. Theequilibrium point O under y¼ 1 lies north-west of point M. This indicates that fullnationalization leads to lower deposit collected by the public bank but higher depositcollected by the private bank. Figure A2 shows how it is the reverse when theprobability of default is below the critical level. In this case, point O lies south-east ofpoint M, that is the public bank’s deposit level rises while the private bank’s depositlevel falls which is similar to what was observed in the case of unlimited liability. Inboth Figure A1 and Figure A2, as before, points N denote the case of y¼ 0.

Figure A1. Deposit reaction functions: limited liability and high default risk.

Figure A2. Deposit reaction functions: limited liability and low default risk.

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