Bank mergers as scale-free coagulation

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Physica A 336 (2004) 571 – 584

www.elsevier.com/locate/physa

Bank mergers as scale-free coagulation Dmitri O. Pushkina;∗ , Hassan Aref b a Department

of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA b Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0217, USA Received 19 June 2003; received in revised form 15 December 2003

Abstract The asset size distribution of US banks is viewed as the result of a scale-free coagulation process. When two banks merge, the assets of the combined institution equals the sum of the assets of the constituent banks. Analysis of the Smoluchowski coagulation equation suggests the emergence of a steady state, power-law distribution with an exponent that only depends on the degree of homogeneity of the coagulation rate. Bank merger data satis8es such power-law scaling. We develop an underlying theoretical framework for bank mergers quite di9erent from prevailing ideas based on game theory on the one hand, and recent econophysical models on the other. As a corollary we show that in order to avoid the emergence of a mega-bank, the rate of return should decrease with the bank size. Finally, we suggest that stochastic coagulation may provide a unifying description of fast integration processes characteristic of globalization. c 2003 Elsevier B.V. All rights reserved.  PACS: 64.60.−i; 05.20.−y; 47.53.+n; 05.70.−a Keywords: Coagulation; Bank mergers; Bank-size distribution; Power-law; Self-similarity; Mean 8eld; Merger market; Growth; Gelation; Globalization

1. Introduction Globalization has led 8nancial institutions to consolidate. The US banking system provides an especially vivid example. The gradual removal of state and federal restrictions on geographic expansion since the 1980s led to an unprecedented wave of bank mergers [1]. According to Rhoades [2]: ‘From 1980–1998, about 8000 bank mergers took place (equal to 55% of all banks in existence in 1980), involving 2.4 trillion ∗

Corresponding author. Tel.: +1-217-333-0334; fax: +1-217-244-5707. E-mail addresses: [email protected] (D.O. Pushkin), [email protected] (H. Aref).

c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter  doi:10.1016/j.physa.2003.12.056

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dollars in acquired assets.’ The size distribution of banks today reJects this two-decade long wave of mergers. While individual bank mergers are usually studied using game theory [3–6], to provide a description of the evolution of the banking system as a whole, encompassing some 8ve orders of magnitude in assets, we turn to the statistical physics of complex systems (cf., Ref. [7]). The main idea of a statistical description is that the particular system should be replaced with a statistical ensemble of similar systems [8]. In our case we envision an ensemble of bank markets, all obeying the same microeconomic laws. Then, the macroscopic (macroeconomic) description of bank markets is furnished by those characteristics which survive averaging over an in8nite ensemble. Most such characteristics can be expressed as integrals of the form
f(v)n(t; v) dv : (1) Here t is time, v is bank assets in million dollars, f(v) is a function characterizing a single bank of the size v, n(t; v)dv is the average number of banks in the interval (v; v + dv) at the time t. It is, therefore, clear that the main goal of a macroscopic theory is to determine the distribution n(t; v). From this perspective bank mergers fall under the heading of coagulation processes. Coagulation processes are numerous and signi8cant. They include: water droplets coalescencing during rain formation, coagulation of colloidal particles, polymerization reactions, and merging of planetesimals to form planetary embryos. Yet, however di9erent the physical mechanisms leading to mergers may be, the distributions characterizing these processes obey the same laws. In particular, coagulation processes are prominent for evolving very broad, heavy-tailed distributions. They often prove to possess power-law tails n(t; v) ∼ v−

for large v :

(2)

This observation has motivated us to seek and analyze data on large American banks in search of a power-law distribution of their assets [9]. We have discovered that, indeed, the distribution is, essentially, a power-law over about four decades, and the exponent  ≈ 2. We give a detailed exposition of our empirical analysis and 8ndings in Section 2. The notion of a power-law scaling for the distribution of assets is, of course, not new. For example, Pareto [10] found such a distribution for the revenue of companies. Zipf [11,12] considered several examples of power-law scaling observed in the social and economic sciences from a uni8ed viewpoint. His two best known examples may be the frequency distribution of words in English as a function of their length and the distribution of populations in cities. Empirically the exponent  often has a value close to 2. Recently, similar scaling laws have been found for 8rms and other economic organization sizes, e.g. Refs. [13,14]. The physics community, intrigued by a number of examples of power-law distributions that have cropped up in economics and the social sciences, has proposed a variety of models aimed at clarifying how complex micro-interactions can lead to scale-free power-law distributions, see, for example, Refs. [14–19]. Such distributions are conjectured to possess a considerable degree of universality as the values of the power-law

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exponents often lie in a comparatively narrow range. Typically, these studies postulate a set of simple but plausible rules for modeling micro-interactions and explore the ways a system, which obeys these rules, gives rise to power-law distributions. Due to the assumed universality, the result should not be sensitive to the accuracy of modeling of micro-interactions so long as these belong to the same universality class. Our approach is qualitatively di9erent in that we derive the coagulation model of bank mergers from economic data. We propose a statistical approach motivated by the recent economic analysis of bank mergers [2] and the apparent paradoxes of the integration of the 8nancial services industry [20]. We introduce the probability, K(u; v), that a pair of banks with assets u and v million dollars will merge during a unit time interval. The merger probability function K(u; v) describes the bank merger market, a novel economic concept bearing strong resemblance to the mean 8eld in statistical physics. The knowledge of K uniquely determines the evolution of the bank distribution starting with an arbitrary initial distribution. We derive and discuss the corresponding evolution equation, known as the Smoluchowski coagulation equation [21], in Section 3. The particular form of the merger probability function should be determined from empirical data or micro-economical modeling, 1 and its properties will reJect the micro-economic structure of the merger market. According to recent economic studies [2] the competition between the banks happens similarly in di9erent markets and the bank merger movement has no ‘preferred’ size. Therefore the function K(u; v) cannot have a ‘preferred’ size either. In mathematical terms such a function is called homogeneous, and it can be characterized by a real number , which is called the homogeneity index of K. We have previously shown [22] that the evolution of such systems often leads to power-law distributions, and, quite remarkably, the exponent  depends only on ! We give a detailed exposition of these ideas in Section 4. In Section 5 we return to the empirical data and analyze it in the light of the theoretical ideas developed in the previous sections. We consider time evolution of the bank distributions, deduce the expected value for and discuss the e9ects of statistical Juctuations for the largest banks. In Section 6 we discuss an economically signi8cant phenomenon of capital condensation, which we de8ne as the formation of a mega-bank which possesses a 8nite fraction of the total capital in the market. We see capital condensation as a parallel to the phenomenon of gelation, known from coagulation theory. In order to relate our discussion to the (micro-economical) rate of return we propose a simple plausible model for K. Note that this kind of empirical modeling of K is done at this point for the 8rst time and only for the purposes of the current discussion. We deduce from our analysis that in order for the merger market to be stable against the emergence of a mega-bank, the rate of return must decrease with bank size. 1

We point out, that the problem of determining what micro-interactions underlie K may not even have a unique solution. Indeed, di9erent economic conditions may give rise to similar K’s. Such a picture, in fact, should look rather pleasing to a physicist due to its analogy with equilibrium statistical mechanics where the partition function determines all the macroscopic properties of the system no matter which microscopic interactions had led to its particular form.

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In Section 7 we conclude our discussion by bringing out why we expect stochastic consolidation (coagulation) to be a generic way to describe globalization processes. Following arguments similar to those for banks, the appearance of power laws in sociologic and economic data is to be expected. 2. Power-law distribution of large American banks Let us look at the data, which is available from the National Information Center (NIC) website provided by the Board of Governors of the Federal Reserve System [9]. According to the data, 3672 banks had assets larger $100M on December 31, 2002. The total assets of ‘JP Morgan Chase’—the largest bank at the time—comprised $622,388M. Clearly, a very broad distribution, spanning nearly four decades, has evolved. Fig. 1 presents a histogram of the bank assets obtained by sorting the data into 12 bins covering suitable intervals in the logarithm of asset size. The plot shows the common logarithm (logarithm to base 10) of the number of banks versus the common

3.5 Dec. 2002 Dec. 2001

3

Linear (Dec. 2002) Linear (Dec. 2001)

log10 (Number of banks)

2.5

2

1.5

1

0.5

0 2.16

2.78

3.35 3.89 log10 (Assets,$M)

4.42

4.90

Fig. 1. Histogram of bank assets. Line represents a linear regression 8t to the log–log data. Line slope equals 1 − .

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10000 Dec. 2002 Regression Slope= -1 Slope= -1

Rank

1000

100

10

1 100

1,000

10,000

100,000

1,000,000

Assets, $M Fig. 2. Zipf plot of bank asset data.

logarithm of assets size. A power-law distribution, n(v) ˙ v− , appears as a straight line with the slope 1 −  on this plot. A linear regression 8t to the data shows n(v), to vary as a power-law with  ≈ 1:9. A comparison to the analogous data from February 15, 2001 shows that although the distribution has grown taller, its slope hardly changed. As we shall see, this kind of scaling follows naturally from a coagulation model of the merger process. Let us denote the ‘rank’ of a large bank of the size v as R(v). The plot R versus v is called Zipf plot, as it was used by Zipf to determine if the data contains a power-law dependence. The Zipf plot allows a detailed study of the bank distribution for very large banks, where the data are sparse. It can easily be shown that when the distribution density is a power-law with exponent , the ‘Zipf plot’ has a power-law tail with exponent  − 1. Fig. 2 shows the Zipf plot for the data from December 31, 2002. It can be immediately seen that the distribution is approximately a power law over the whole range of data; the value of the exponent  is close to 2. This conclusion agrees with the one drawn from the histogram analysis. A detailed statistical study, however, allows to distinguish three regions of altering behavior. A piecewise power-law regression provides an excellent 8t of data with  = 2 for 100 6 v 6 1100$M and  = 1:7 for 1100 6 v 6 37; 000$M. For the larger v the distribution is subject to Juctuations, but the value of  remains close to 2. We will explain this behavior in the light of a coagulation model for bank mergers in Section 5.

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1.6 1.4

Hill estimator, H(k)

1.2 1 0.8 0.6 0.4 0.2 0 0

500

1000

1500

2000 k

2500

3000

3500

4000

Fig. 3. Hill plot of bank asset data.

Finally, we corroborate our analysis with the Hill plot for the data [23]. The Hill estimator H (k) is the maximum likelihood estimator of ( − 1)−1 based on k + 1 order statistics [24]. Fig. 3 shows H (k) for k = 1; : : : ; N − 1. The plot attests strongly to the overall power-law behavior of the current distribution with the exponent  ≈ 1:9.

3. Homogeneous merger market model Several features of bank mergers suggest that a statistical physics approach is appropriate. First, bank mergers are not con8ned to banks of a certain size. The total assets of a bank participating in a merger can be as little as several million dollars and as much as several hundred billion dollars. Second, merger activity is not limited to a particular market or state. Although competition between banks takes place in many di9erent markets, the merger process happens similarly in all of them. Third, the overall dynamics of the bank market is only weakly dependent on the particulars of bank interactions. When a new bank forms as the result of a merger, its total assets equal the sum of the assets of the acquiring and the acquired banks. Let K(v; u) denote the probability that a pair of banks with assets v and u million dollars, respectively, will merge. Assuming this quantity to be well de8ned, and assuming that bank mergers are independent, the average number of mergers creating banks of size v during a small time interval dt

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around time t is given by
1 v−v0 K(v − u; u)n(t; v − u)n(t; u) du dt : 2 v0

577

(3)

Here v0 is the asset size of the smallest banks participating in mergers, and n(t; u) du is the average number of banks with total assets between u and u + du million dollars at time t. Similarly, the average number of mergers removing banks of asset size v during the same time interval is
∞ n(t; v) K(v; u)n(t; u) du dt : (4) v0

Hence, dividing the change in n(t; v) by dt, we arrive at the following balance equation:
9n(t; v) 1 v−v0 = K(v − u; u)n(t; v − u)n(t; u) du 9t 2 v0
∞ − n(t; v) K(v; u)n(t; u) du : (5) v0

Macroscopic (macroeconomic) equation (5) determines the time evolution of the bank size distribution n(t; v). It is of the type 8rst written by von Smoluchowski [21] as a theoretical description of coagulation of gold particles in suspensions. Since then it has been applied to numerous aggregation phenomena: coagulation of smoke, smog, and dust particles in aerosols [25], coalescence of water droplets in clouds [25], polymerization [26], algal aggregation [27], and the formation of planets [28]. Somewhat unexpected applications have also appeared, such as the use of (5) to study random graphs [29] and use of a similar equation to study the structure of the World Wide Web [30]. The present paper provides yet another ‘unorthodox’ application of Smoluchowski’s coagulation equation. We have treated bank mergers as a stochastic process, whereas in reality such events only happen after much planning and deliberation. It has been shown that although most competition between banks happens in local markets, the merger process is global [2]. Two banks in di9erent geographic markets may never design to merge. In fact, they may not even know of each other’s existence. Nevertheless, they participate in the global merger process by competing in local markets that are not independent. This also explains why it is meaningful to consider the US banks as the totality of banks taking part in the merger process, and to refer to them as the bank merger market. The data supports this assumption. At this level of description the interaction between any pair of banks depends only on their size, i.e., no bank merger market structure is reJected by Eqs. (3)–(5). Therefore, we might call this model the homogeneous merger market model. It ignores Juctuations in the bank merger market, which is in accord with the statistical observation that during the period 1980–1998 mergers tended to increase market concentration, as measured by the Her8ndahl–Hirschman Index (HHI), in relatively unconcentrated markets and decrease it in relatively highly concentrated markets [2]. HHI is de8ned as the sum of the squares of individual market shares of all 8rms operating in a particular market. This behavior means that if there were a local Juctuatuation in bank density at a certain

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market at some time moment, the collective dynamics of mergers would lead to its dissipation. This justi8es the mean 8eld assumption, underlying (5), that there is no local correlation build-up due to mergers. In coagulation theory such systems are called ‘mixing’. 2 We should also comment on why we can introduce the bank merger probability, K(v; u), conventionally called the coagulation kernel, and assume it to be independent of the prevailing distribution of bank asset sizes. The hypothesis is that no individual merger can change the bank merger market signi8cantly. Two arguments lead to this hypothesis. First, the total assets of the acquired and the acquiring banks are negligibly small compared to the total merger market assets, except for the rare cases of mega-bank mergers. Second, any merger or acquisition that would substantially lessen competition is prohibited by the Clayton Antitrust Act of 1914 (Section 7, 15 U.S.C. 18) and applicable state laws. A recent study [2] showed that, somewhat unexpectedly, the average local market concentration as measured by the Her8ndahl–Hirschman Index (HHI) based on commercial bank deposits hardly budged during 1980–1998, in spite of the intensity of the merger wave. This apparent paradox highlights that the incentive to merge comes from the opportunity to increase eSciency (usually measured by the pro8t X-eSciency, which refers to how close a 8rm’s actual pro8ts are to the pro8ts of a best-practice 8rm producing the same outputs) rather than the ability to exercise market powers by raising prices and, hence, to a9ect the bank merger market [20]. This hypothesis is qualitatively di9erent from the assumptions of the usual oligopoly models in game theory. 4. Power-law bank-size distributions Let us now proceed to some conclusions of the model. If the size distribution of institutions that evolve according to the homogeneous merger market model is known at a given time, the solution of (3) determines the distribution for all subsequent times. Generally speaking, di9erent time-dependent distributions correspond to di9erent initial distributions. However, it was realized in the 1960s that time-dependent distributions of some coagulating systems, after proper re-scaling, approach a 8xed form, which is independent of the initial conditions [34]. Such distributions are called self-preserving. They have been extensively studied both numerically and analytically [34,35]. However, although such distributions have been well understood for particular coagulation kernels, a general result as to when they arise has been lacking [36]. Recently, we have used an alternative approach that studies coagulating systems with a constant source of the smallest elements [22], i.e., substitutes a boundary value 2

Global interactions between the banks and a multi-dimensional structure of the bank market could have led the physicist to anticipate the validity of the mean 8eld assumption on theoretical grounds. Consider, for example, the following heuristic argument: the mean 8eld assumption is known to be exact when the dimension of a system, d, is higher than the upper critical dimension dc , see, for example, [33]. Let the interacting banks be represented as vertices of a graph, with the edges corresponding to possible mergers. If any pair of banks may merge, the graph is complete and d = ∞. Therefore, d ¿ dc for any 8nite value of dc .

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problem for the initial value problem discussed thus far. Situations where such a model is appropriate often arise in nature: vapor nucleation in clouds provides the smallest droplets that later coalesce into raindrops; fresh reagents may be constantly introduced into an evolving mix, and so on. In the present application the smallest banks in the merger market emerge from the ‘pool’ of newly chartered banks. Due to the ‘forcing’ at small scales we show [22] that the distribution converges to a steady-state power-law distribution, n(v) = Av− . The exponent in the power-law and suScient criteria for its existence have been worked out [22]. Thus, given rather general information about the bank merger probability, K(v; u), one can predict whether a power-law distribution will arise, and one can 8nd the value of the exponent  without having to solve Eq. (5). Of course, the power-law distributions observed in real data are quasi-stationary. The steady state of the theory, reaching to in8nite asset sizes, occurs only asymptotically. Power laws arise when a system ‘forgets’ about its history. In our case, at large enough sizes the distribution of banks formed by mergers does not ‘remember’ the smallest bank size. Part of the reason for this is that the banking laws do not di9erentiate between banks based on the absolute size of their assets. Correspondingly, the bank merger probability function K(v; u) should have no ‘preferred’ size. In mathematical terms this amounts to the requirement that K be a homogeneous function of its arguments: K(v; u) =  K(v; u)

(6)

for any positive . Here is another 8xed exponent, called the homogeneity index of K. Homogeneity of the merger probability function does not by itself guarantee that the distribution of large banks ‘forgets’ the size of small banks. Thus, not all homogeneous bank merger functions will lead to power-law distributions. We have shown [22] that power-law distributions develop when the merger probability function K(u; v) satis8es an additional ‘locality’ requirement on the interaction between banks very di9erent in size: Assume that K(v; u) varies as u v − for vu. Then, for a power-law distribution to develop, we must have

− 2 + 1 ¿ 0 :

(7)

The resulting distribution, n(v) = Av− , will then have an exponent that depends only on : 3+

= : (8) 2 5. Discussion If we had an independent, microeconomic derivation of a bank merger probability function K(v; u), we would be able to predict detailed time evolution of bank distribution. Finding K is the subject for a separate economic study, which would lie beyond the scope of the present article. The framework we have outlined, however, allows one to understand how the observed power-law distribution can arise from perfectly reasonable assumptions concerning the aggregate of bank mergers. Summarized brieJy, in a scale-free market

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1 Dec. 1998 Feb. 2001 Dec. 2002 Slope=-1

Normalized rank

0.1

0.01

0.001

0.0001 100

1,000

10,000 Assets, $M

100,000

1,000,000

Fig. 4. Normalized Zipf plot.

dominated by mergers, with a constant supply of newly charted banks, the bank distribution has a propensity to evolve to become a power law. In order to verify this tendency we study the bank distribution at three di9erent times: December, 1998; February, 2001; and December, 2002. Fig. 4 shows the corresponding Zipf plots, where for comparison sake the ‘rank’ R(v) is normalized by the total number of banks under consideration, R(100), for each set of data, i.e., we cut o9 at banks with assets less $100M. It can be readily seen that since 1998 the distribution has generally grown straighter, signifying that similarity has been developing due to bank mergers. The ever diminishing deviations of the distribution from a straight line are the remainders of the initial (not scale-free) distribution that existed prior to the merger wave. The distribution slope tends to −1. Thus,  = 2, in accord with the conclusions of Section 2. It immediately follows from (8) that K(v; u) should be homogeneous with index ≈ 1. Fig. 5 shows the Zipf plots for the 50 largest banks with assets in the range $104 – 106 M, and the 50 largest bank holding companies with even larger assets. (Data as of December 2002.) Despite poorer statistics, it is evident that the plots align. This proves that both banks and bank holding companies belong to the same scale-free merger market. The universal character of the merger market is in accord with the ideas of Section 3.

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100 Banks

Rank

Bank Holding Companies

10

1 10,000

100,000

1,000,000

Assets, $M Fig. 5. Zipf plot for top 8fty banks and bank holding companies.

Finally, we would like to discuss deviations from the macroscopic law due to 8nite statistics. Until now we have assumed that the system we are dealing with is macroscopic. While for many coagulation processes, that may involve numbers of agents as large as Avogadro’s number, 6 × 1023 , this appears a safe assumption, for bank mergers certain reservations are appropriate. Physicists have evolved a number of ways of dealing with statistical Juctuations. They include, for example, the master equation of Marcus–Lushnikov [31], which describes coagulation as a Markovian process; or the van Kampen’s ‘-expansion’, which may be used to correct (5) in mesoscopic approximation [32]. However, while conceptually simple, the Smoluchowski equation captures the substance of merger processes. It has, therefore, been proven to provide a remarkably good description even for many 8nite (mesoscopic) coagulating systems. Another closely related, but di9erent, question pertains to comparison of the predicted distribution n(v) with the observed data. By de8nition, n(v) dv is the average number of banks in the small interval (v; v + dv). When banks are spaced densely, n(v) can be estimated as the observed number of banks in a suitable interval of assets, divided by the length of this interval. And thus, n(v) can simply be identi8ed with the observed bank distribution, and dubbed a continuous description.

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The situation is di9erent when banks are spaced more sparsely. Then the observed bank distribution is likely to be subject to substantial statistical deviations from n(v). For instance, when the average number of banks in some interval of interest equals 0.5, one cannot expect to observe this number in actual statistics. A rough estimate, telling these cases apart, can be obtained as follows: the number of banks in a small size interval (v; v + Vv) is, approximately,  n(v)Vv. The Juctuations of this number due to the 8nite statistics can be estimated as n(v)Vv. Therefore, the Juctuations are relatively unimportant if n(v)Vv1, i.e., on the scale Vv1=n(v). On the other hand, a continuous description is contingent on the scale separation condition: Vvv. Thus, n(v)v1 :

(9)

Let us apply the above criterion to our data. A power-law distribution (with  ¿ 1), satisfying the normalization condition
Vmax n(v) dv = N ; (10) Vmin

where N is the total number of banks, Vmin and Vmax are, respectively, the sizes of the smallest and largest banks under consideration, may be cast in the form   Vmin −1 n(v) = N ( − 1)Vmin : (11) v It follows from (9) and (11) that the continuous description is e9ective for v=Vmin  N 1=(−1) . With  = 2, Vmin ≈ $102 M, and N ≈ 103 we obtain v$105 M. This estimate 8nds excellent agreement with our visual judgement on continuity of the bank distribution presented in Fig. 4.

6. Capital condensation The present statistical physics framework suggests an interesting phenomenon of capital condensation. It is well known from coagulation theory that if ¿ 1, the phenomenon of gelation occurs. Translated into our current bank merger terminology this means that in a 8nite time the kinetics of consolidation will lead to the formation of a mega-bank which possesses a 8nite fraction of the total capital in the market and, thus, has exceptional powers of control. This should probably be considered as averse to a healthy and stable bank merger market. In this view the estimate value of ≈ 1 obtained in the previous section is marginal and should raise concern for economists! If we had a micro-economic model for the kernel, we would also have a relation between micro-economic parameters and the stability of the merger market. For now we present a simple empirical argument. Let the incentive to merge be proportional to the expected pro8t, i.e., set K(v; u) ˙ P(v + u) − P(v) − P(u) :

(12)

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Here P(v) is the expected pro8t from a bank with capital v. Since P is assumed scale-free, we have P(v) =  P(v) with some exponent . Thus, K(v; u) ˙ (v + u) − v − u :

(13)

Absence of capital condensation (the analog of gelation) implies  ¡ 1. Thus, in order for the merger market to be stable against the emergence of a mega-bank, the pro8t function should grow less than linearly, and the rate of return P(v)=v ˙ v−(1−) should decrease with v. As this conclusion appears quite remarkable from the economical perspective, we re-summarize it as follows: when the rate of return per every dollar of assets increases with the total assets size, and when the probability of a bank merger is proportional to the expected pro8t from the merger, the collective dynamics of consolidation will lead to formation of a mega-bank in 8nite time. 7. Globalization and power laws A similar framework may be invoked to understand power laws resulting from the dynamics of globalization in many di9erent 8elds where a total asset is conserved. The rapid advancement of globalization can partly be attributed to the ease and bene8ts of forming new links between already existing institutions. The typical time scale of consolidation turns out to be much shorter than the time scale of ‘conventional’ development. Under such circumstances stochastic consolidation proves to be the generic process. Based on arguments similar to those given here for banks, the appearance of power laws in the data is to be expected. Acknowledgements DOP thanks Sergey Knysh for discussion and Susanne Aref for assistance in data analysis. This work was performed under the auspices of the Center for Simulation of Advanced Rockets (CSAR) at the University of Illinois, Urbana-Champaign. CSAR is supported by DoE as part of the ASCI program. References [1] A.N. Berger, R. DeYoung, H. Genay, G.F. Udell, Globalization of 8nancial institutions: evidence from cross-border banking performance, Brookings-Wharton Papers on Financial Services 3 (2000). [2] S.A. Rhoades, Bank mergers and banking structure in the United States, 1980–98, Sta9 Studies, Vol. 174, Board of Governors of the Federal Reserve System, Washington, 2000. [3] R. Deneckere, C. Davidson, Incentives to form coalitions with Bertrand competition, RAND J. Econ. 16 (1985) 473–486. [4] M.K. Perry, R.H. Porter, Oligopoly and the incentive for horizontal merger, Am. Econ. Rev. 75 (1985) 219–227. [5] J. Farrell, C. Shapiro, Horizontal mergers: an equilibrium analysis, Am. Econ. Rev. 80 (1990) 107–126.

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