Motivated by the Blackorby-Schworm (1993) observation that market outcomes may differ from those originating in market-actor optimization, this paper claims that the number of banks in the market is larger than the number justified by bank profit maximization alone or in combination with bank depositor welfare maximization. This claim is made within the context of bilateral monopoly banks and intertemporal utility maximization by bank depositors. The basic policy implication towards bank population rationalization is a minimization of the deviation away from the optimal interest rate margin at every stage of the business cycle. It is meant to be an acyclical policy though the target of optimal bank population is attainable by active countercyclical policy as well. The nature of this policy issue makes the use of macroprudential measures imperative, jointly perhaps with a fiscal-monetary policy mix. A dynamic version of the model in a Cournot environment is akin to the modeling of Minsky's hypothesis of financial fragility.

The differing empirical findings of the voluminous banking literature make one think that one reason for the differences may be the neglect of some notion of socially optimal bank population as the basis for discussion. Such a study task may appear to be superfluous when urgent policy issues are debated in view of bank markets with a limited number of sizeable banks, and when even then competition among them may be fierce given the multiproduct character of this market. For example, Bikker and Spierdijk

More precisely, this paper elaborates upon the bare essentials of such a model. The key observation is that bank-borrower relations are mostly personal, and the market loan demand and supply do not coincide with those coming out of some notion of representative bank/borrower supply/demand (

The optimality of competition intensity comes as a by-product of this optimal number, which is also the optimal concentration. Any discussion of the banking sector is made within this strictly microeconomic framework, assuming away the industrial economics of the topic under investigation (factors like those associated with financial market failures). There is much literature on this economics but with no point of reference for the arguments advanced. The proper questions that should be raised methodologically are: Given this or that difference between the actual and socially desirable number of banks, what does this or that industrial economics hypothesis implies about it?^{[1]} How sensitive are these implications to alternative model assumptions and/or extensions of the basic modeling? Which of these implications are validated or refuted by empirical evidence? An extensive line of research may thus be put forward in the expectation of obtaining a better policy-related understanding of the behavior of the banking system.

Indeed, the focus is not on microeconomics and industrial economics

The next section combines the typical textbook approach to the loan market with bank profit maximization in an imperfectly competitive banking system, which is also a monopsonist of bank deposits; that is, the system is modeled as a bilateral monopoly. Each individual bank is a monopsonist too, given that deposits are homogeneous goods and all banks provide typically the same deposit rate regardless of the bank market structure (

Section 3 points to the procyclicality of the banking system, which in the context of the analysis of section 2 is attributable to the large number of banks. It next contemplates upon the policy implications of the discussion in section 2 or the same, of procyclicality, by arriving at an acyclical interest-rate-spread rule as a means of fostering acyclical bank population, that is, a bank number at each stage of the business cycle that does not exceed the socially optimal one, the number justified by optimization on the part of banks and depositors. The spread rule towards the rationalization of bank population might be pursued by the central bank either directly, by imposing to the market the bank profit margin dictated by the national income statistics of the particular moment of the business cycle, or indirectly, by adopting the tax rate and money supply policy that will induce the market to the desirable profit rate. The efficacy of this fiscal-monetary policy mix is influenced by the fraction of nonperforming loans. For the same reason, efficiency, this policy mix should be combined with macroprudential measures as well, given the large information requirements upon which the fiscal-monetary policy mix has to be based. The introduction in the discussion of a balanced government budget elucidates this policy mix even further. Section 3 also discusses the attainment of the goal of bank population rationalization as a by-product of broader counter-cyclical policy.

Section 4 presents a dynamic version of the basic equation for bank population, complete with a second one about national income, as it comes out of a dynamic variant of the Cournot theorem for the banking sector. That is, there is a system of two differential equations, acknowledging the influence that loan market concentration and competition can exert on bank numbers and vice versa. According to the theoretical and empirical literature on concentration & competition, the standard thesis is that bank competition is conducive to growth, but the stability of the banking system is better served by market power (

To sum up other noteworthy results, a key condition, in order to have a positive number of banks in the market, is that loan supply should be more sensitive than loan demand to lending rate changes. Increased sensitivity of loan supply to output fluctuations relative to the sensitivity of loan demand is also found. Bank population increases/decreases during output expansion/contraction, enhancing/diminishing thereby bank competition and lessening/rising, in turn, the interest rate margin. Banking is procyclical because of the presence of too many banks. There is a sharp increase and decrease in bank numbers during the course of the business cycle, but there are banks at the trough of the cycle that refuse to leave the industry. The fiscal-monetary policy mix towards bank population rationalization by minimizing the interest rate margin deviation away from its optimal value, involves an inverse co-movement in the tax rate and money supply, whose prerequisite is (i) a minimum loan performance rate, and (ii) a minimum quantity of money below which borrowers may not be able to continue servicing their loans. In the dynamic analysis, the fixed point is consistent with at least three banks, and once the long-run equilibrium is disturbed, anything can happen. Finally, the Cournot approach to the dynamic version of the model is akin to the modeling of Minsky's hypothesis of financial fragility. Section 5 concludes this paper by looking at the intertemporal aspect of the policy, possible extensions of the model, and prospects for empirical work.

The premises of the theoretical discussion are quite simple in comparing loan market quantities consistent first with bank profit maximization, and then with depositor-consumer welfare maximization. Let

Given that modeling loan demand in terms of the output level,

Consider also the following empirically-valid

where

which, given that loan supply is a multiple of bank deposits, is equal to the elasticity of deposit supply with respect to

where

We insert Eqs. (2) and (4) in Eq. (3), and solve for

equate next Eq. (5) with Eq. (7) and solve for

Eq. (6) captures the quantities desired by the typical bank, whereas Eq. (3) is an

where

Note that having equated the profit-maximizing loan rate to the market ones, quantities

where ^{[2]} Letting

where

Now, noting that all quantities are in current prices, one might set:

which when in turn is solved for

Social welfare maximization follows intuitively from the fact that although bank-depositor deals are taking place within a bilateral monopoly environment, banks are owned by depositors and have to satisfy their owners. But, when Eq. (9') is inserted as

While as far as Eqs. (8) and (8') are concerned, they become:

or, in view of Eq. (11):

Given that

The example of uniform fluctuations is illustrated in

The higher steepness of the supply curve in the loan market relative to the slope of the demand curve suggests the presence of Marshallian, short-run quantity stability, despite the assumption of imperfect competition (

Equally disturbing are the trends in banking numbers depicted by

Trends in all three diagrams are in line with the stylized fact of procyclical banking (

In general, short-run quantity stability in the loan market does not necessarily connote optimality of the number of banks, because presumably of interest rate instability. Our algebra above suggests that the problem with bank numbers might be tackled by trying to have

or since

where:

The general rule to reduce the part of procyclicality coming out of the presence of too many banks is that the spread should be in tune with the course of national income and money, or with a combination of the two, like the one given by Eq. (17):

It is a rule that prescribes direct intervention to the spread driven by the momentum of the cycle,

These parameters are those related to loan demand and supply, and to consumer preferences, which and thereby

According to Eq. (19), the response of

since the fraction in Eq. (20) is less than 1 (see

As follows:

Now, given the stage of the business cycle, Eq. (22) suggests quite schematically that equal-spread loci in the positive quadrant generated by

More insight into policymaking may be gained by introducing into the discussion the government budget constraint;

given

At a given stage, it is a rectangular hyperbola whose slope,

The design of the proper fiscal-monetary policy mix towards the desirable policy-wise spread can prove to be quite a formidable task, because it stumbles on the many information details it presupposes; and if it is not successful, it will strengthen instability. At least medium-term instability will always be present, and improper policymaking can reinforce it, prolong it, and perhaps, perpetuate it. The same destabilizing can be the policy mix if there are other competing policy goals, whose presence is almost certain. For example, the average long-term loan rate has been found to be less volatile than the average deposit rate in the US (

Note that the above interest-rate rule is not a policy against this cycle, but one that takes the cycle for granted and seeks to downsize bank population to the benefit of banks and its customers. The analysis has been static and hence, without explicit reference to the time considerations that would allow feedback on the state of economic activity on the number of banks. Such feedback is made explicit through the dynamics attempted in the next section given the business cycle context within which the main arguments of the present discussion are being advanced. Bearing this in mind, one can still note from now about policy that the target of bank population rationalization might be tackled successfully, indirectly as a by-product, so to speak, of a more general counter-cyclical policy as follows.

What is sought policy-wise is the counter-cyclicality of money supply and at least acyclicality of tax policy, as these would be conducive to, among other things, bank-numbers rationalization along with some macroprudential policy, targeting the number of banks directly. Although knowledge of the macroprudential tools is still limited, and there may be coordination problems with microprudential policies, evidence suggests that these tools do reduce procyclicality (

Such macroprudence can complement a fiscal-monetary policy mix against procyclicality in general and not against particularly banking procyclicality. In industrial economies at least, tax policy is already acyclical and in many instances countercyclical (

In order to have a glimpse of the dynamics associated with the topic under examination, market concentration and competition have to be introduced in the discussion as follows. To account for the role of the loan performance rate too, the expression of

where:

specifies an inverse relationship between output growth and the number of banks in the system. This is because increased bank competition reduces profitability, while increased loan losses make banks vulnerable to bankruptcy. More bank competition lessens market power, diminishes profit margins, and encourages at the same time risk-taking, extending loans on "easier" terms during upturns in economic activity and coming back to trouble banks as precarious loans during downturns (

To obtain a similar differential equation for the time derivative of

where

The discrepancy

Next, inserting Eq. (26) in Eq. (25) and solving for

The derivative:

specifies the rate of increase of the number of banks as a positive function of the level of output. The argument is that during the upturn of economic activity, an increasingly more optimistic outlook prevails, encouraging, among other things, entry into the banking industry.

Eqs. (24) and (28) form a system of two nonlinear differential equations. Setting the former equal to zero, yields:

since

Eqs. (29) and (30) define the only fixed point of the system,

one concludes that the linearization is a center (stable but not asymptotically stable), and the nonlinear system might be either stable or unstable, given

where

The empirical evidence that bank concentration & competition affects financial fragility is thus corroborated. One link between the two is the influence of competition on bank liquidity. Of course, the hypotheses about this link are two opposing ones (

As far as the impact of policy on bank concentration & competition is concerned, the policy combination discussed in the last section towards bank population rationalization is either "business-cycle neutral", that is − acyclical, or counter-cyclical. In the former instance, the macroprudential-fiscal-monetary policy is not supposed to influence the trends in market concentration and competition accompanying the course of the cycle, only to alleviate them. This is a policy alternative regardless of the fact that it has been advanced through an analytical framework in which market structure considerations have been assumed away. Such considerations become meaningful only as a result of the dynamics of the present section. Note, in particular, that any equation describing bank population is of the same format, which implies that the derived Minsky-like cyclical pattern of the system should be attributed to the Cournot modeling of the loan market. In any case, countercyclical policymaking is sensible methodologically only within a dynamic environment. And, as far as policymaking is concerned, the impact of macroprudential and fiscal-monetary policies will be in the same direction by policy design.

For example, capital buffers are characterized by counter-cyclicality over the business cycles, and they are related inversely to market concentration, implying that a decline in market concentration would strengthen the countercyclical operation of capital buffers (

Yet, the monetary authority comes to compete with banks in terms of stabilization, which would be welcome only if it was more effective than banks in pursuing this goal; and, it does so at the expense of market concentration and competition, which might be against growth (

A summary of the results has been already provided in the last paragraph of the introductory section. Here, the intertemporal aspect of the proposed policy, possible extensions of the model, and directions for empirical work are being discussed. The intertemporal utility, assumed for simplicity is strongly additive, and hence, with zero intertemporal risk aversion and substitution elasticity. Also, a unit discount rate has been assumed; tomorrow's utility is the same as the present utility. Under these circumstances, no further remarks than those made earlier as to the welfare consequences of policy may be advanced. But, in general, a change in the deposit rate can induce, as a matter of straightforward logic, more or less consumption during this period depending on the direction of change, the value of the non-unit discount rate, and how risks at different times interact. This is something that the bank population rationalization policy has to offset. Moreover, if money enters the utility function for the current period, say, the decline of it, that would accompany a tax rate increase within the context of the proposed fiscal-monetary policy mix will reduce present utility, and there ought to be policy provision against this reduction.

These remarks predispose how the model above can be qualified and/or extended, and there can certainly be many. For example, intertemporally dependent preferences dampen consumption volatility, facilitating the operation of the proposed policy. In any case, the aspect of household borrowing, borrowing constraints, and (in) voluntary bequests, could alter results significantly. The incorporation of public debt and optimal intertemporal taxation would be a more realistic approach, too. Optimal policymaking is greatly influenced in the presence of debt because it can substitute for taxes or money supply (

Finally, empirical work might be motivated by the dynamics examined above as one more version of Minsky's financial instability hypothesis, especially when not much of such work has been undertaken on the subject (

This research received no external funding.

Acknowledgments to anonymous referees' comments and editor's effort.

The authors claim that the manuscript is completely original. The authors also declare no conflict of interest.

The derivative:

will be positive if

Now, the total differential of

In so far as the sign of Eq. (19) is concerned, its denominator will be positive if:

Applying to this inequality the rules of proportions, if

we obtain that:

where the left-hand side is negative given

To find

Next, insert Eq. (26) in

equate (A1) and (A2), and solve for

Finally, insert

Microeconomics is quite clear on how to handle the difference in causality between market agents and market outcomes. Industrial economics, and more so empirical industrial economics, is not. The structure-conduct-performance paradigm wants by its very definition the market to be dictating behavior, while the hypothesis of quiet life forgets depositors and wants bankers dancing on the beat of the markets dazzled by the success of their businesses. The "new" industrial organization wants firms in general to be shaping the market but in terms of entry competition on incumbent firms preferably within a game-theoretic setting. The efficient-structure hypothesis according to which superior efficiency including efficiency in monopoly-seeking activities, increases market concentration even to the point of ending up to "socially desirable" monopolies, while according to the contestability hypothesis, monopolies may still opt to have this social profile in fear of new entry (Shepherd 1990). And, finally, there is empirical industrial economics based on econometric modeling tailored to the question and industry being studied. It is clear that although each of these strands in industrial economics has its own merit, none of them starts from the basics of whether firm behavior shapes or is shaped by market outcomes and of comparing these outcomes in line always with standard microeconomics. The non-price factors that industrial economics is supposed to introduce into the microeconomics of the firm and the market may be contemplated once this comparison has been made. ↑

The associated saving supply elasticity with respect to

The cyclical relationship between N, sinY (vertical axis), and Y (horizontal axis).

The cyclical relationship between N, sinY/Y (vertical axis), and Y (horizontal axis).

The cyclical relationship between N, 0.6cos3Y-0.3cos7Y (vertical axis), and Y (horizontal axis).

Equal-spread Loci and the Government Budget.