Journal of Banking and Finance 106 (2019) 180–194

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Journal of Banking and Finance

journal homepage: www.elsevier.com/locate/jbf

Regulatory competition in capital standards: a ‘race to the top’ result �

Andreas Haufler a , b , ∗, Ulf Maier a

a University of Munich, Seminar for Economic Policy, Akademiestr. 1, 80799 Munich, Germany b CESifo, Seminar for Economic Policy, Akademiestr. 1, 80799 Munich, Germany

a r t i c l e i n f o

Article history:

Received 26 October 2018

Accepted 5 June 2019

Available online 12 June 2019

JEL classification:

G28

F36

H73

Keywords:

Regulatory competition

Capital requirements

Bank heterogeneity

a b s t r a c t

Several countries have recently introduced national capital standards exceeding the internationally coor-

dinated Basel III rules, which is inconsistent with the ‘race to the bottom’ in capital standards found in

the literature. We study regulatory competition when banks are heterogeneous and give loans to firms

that produce output in an integrated market. In this setting capital requirements change the pool quality

of banks in each country and inflict negative externalities on neighboring jurisdictions by shifting risks to

foreign taxpayers and by reducing total credit supply and output. Non-cooperatively set capital standards

are higher than coordinated ones, and a ‘race to the top’ results, when governments care equally about

bank profits, taxpayers, and consumers.

© 2019 Elsevier B.V. All rights reserved.

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1. Introduction

The regulation of banks, and in particular the setting of capi-

tal adequacy standards, is arguably one of the most important pol-

icy issues in the aftermath of the 2007/08 financial crisis. In many

countries large, commercial banks needed to be recapitalized with

public funds in recent years. In several countries, such as Ireland

or Iceland, the public bailout was so massive that it threatened

the entire state of public finances. The new Basel III capital stan-

dards, which foresee the ratio of common equity (Tier 1 capital) to

risk-weighted assets to rise to 7% until 2019, are therefore widely

believed to represent a critical step forward in ensuring more re-

silient banking sectors around the world.

The financial sectors of many countries have grown dramati-

cally in recent decades and represent an important source of value

� We thank two referees, the editor and an associate editor for their very help-

ful and constructive comments. This paper was presented at seminars and confer-

ences in Bremen, Burnaby (Simon Fraser), Dubai, Exeter, Glasgow (Strathclyde), Lux-

embourg, Munich, Oxford, Uppsala, Würzburg and Zurich. We thank Steve Bond,

Pierre Boyer, Michael Devereux, Sunil Mohanty, Alan Morrison, Bernd Rudolph,

Klaus Schmidt, Tim Schmidt-Eisenlohr and Michael Stimmelmayr for many helpful

comments and Tobias Hauck and Bernhard Kassner for excellent research assistance.

Financial support by the German Research Foundation (DFG) through CRC TRR 190

and Grant No. HA 3195/9-1 is gratefully acknowledged. ∗ Corresponding author.

E-mail addresses: Andreas.Haufler@econ.lmu.de (A. Haufler),

Ulf.Maier@econ.lmu.de (U. Maier).

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https://doi.org/10.1016/j.jbankfin.2019.06.001

0378-4266/© 2019 Elsevier B.V. All rights reserved.

dded, highly paid jobs, and - in good times - tax revenue. 1 There-

ore, an important concern in policy discussions is that the national

etting of higher capital adequacy standards will not distort in-

ernational competition between the banking sectors of different

ountries, and maintain a ‘level playing field’. The existing liter-

ture on regulatory competition (to be discussed below) focuses

n the increased costs that tighter capital regulation imposes on

omestic banks vis-à-vis their foreign competitors. This literature

nanimously finds that national capital standards will be set too

ax in the process of regulatory competition, and a ‘race to the bot-

om’ will therefore result. In short, the literature so far views cap-

tal regulation as being fundamentally similar to capital taxation,

here a ‘race to the bottom’ is both the widely accepted theoreti-

al standard, and it is also empirically observed. 2

This paradigm is unable, however, to explain several important

mpirical facts about capital regulation in the banking sector. First,

any countries have enacted capital standards that substantially

xceed the internationally negotiated Basel III rules. Switzerland,

or example, introduced a core capital ratio of 10% for its largest

anks, well above the Basel III standards, and it did so earlier

han implied by the Basel schedule. Similarly, the United States

1 Auerbach et al. ( 2010 , Fig. 9.5) document the increasing fiscal importance of

he financial sector in the United States and the United Kingdom. In both countries,

orporate tax revenues from financial corporations made up more than 25% of total

orporate tax revenues before the financial crisis. 2 See Keen and Konrad (2013) for a survey of the theoretical literature on capi-

tal tax competition, and Devereux and Loretz (2013) for a survey of the empirical

evidence.

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 181

Fig. 1. Credit shares of banks in five European countries, 2007–2015.

Source: Bank for International Settlements, Credit statistics 2016, Table F2.4: Bank credit to the private non-financial sector; http://stats.bis.org/statx/srs/table/f2.4 . Credit

shares are fractions total credits given by banks in 22 European countries.

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emands a (non-risk weighted) leverage ratio of 5% from its largest

nd systemically relevant banks, significantly above the Basel III

tandard of 3%. In the European Union, British plans to impose

ational capital standards above the Basel III standards met with

tern resistance from most EU partners. 3 The final compromise was

hat the United Kingdom was allowed to implement national capi-

al standards ahead of the Basel III schedule, but that it would not

xceed the capital standards in other EU member states.

One important reason for why countries have enacted tight reg-

lation policies is to protect national taxpayers. The latter effec-

ively pay for bank failures when governments make discretionary

ecisions to bail out individual financial institutions, but they are

lso involved more generally because virtually all developed coun-

ries have national deposit insurance schemes. 4 It is therefore no

oincidence that many of the countries that have adopted capital

tandards above the Basel III rules have large banking sectors, rel-

tive to the country’s GDP. And indeed, the EU Commission explic-

tly mentions a possible ‘race to the top’ scenario to motivate why

apital standards among EU members must be strictly harmonized

t the level of the Basel III accord: “It is uncertain what the po-

ential impact in terms of costs and growth would be in case of

igher capital requirements in one or more Member States, poten-

ially expanded through a ‘race to the top’ mechanism across the

U” ( European Commission, 2011 , p. 10).

A second observation is that capital regulation has important

tructural effects on the banking sector that are not captured in

tandard models of market competition between homogeneous

anks. In Europe, in particular, the number of credit institutions

as fallen significantly after the financial crisis. In the Euro area

his decrease amounted to 25% in the period from 2008 to 2016

European Central Bank, 2017 , Chart 2.1). While several factors

3 See “European Leaders to weigh new capital requirements for banks”, The New

ork Times, 1 May 2012. 4 This argument is stressed explicitly in the communication with which the

oard of Governors of the Federal Reserve System (2014) motivated higher leverage

atios for systemically relevant banks: “Higher capital standards for these institu-

ions place additional private capital at risk before the federal deposit insurance and

he federal government’s resolution mechanism would be called upon, and reduce

he likelihood of economic disruptions caused by problems at these institutions.”

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re responsible for this development, empirical analyses show that

ighter capital requirements have been a significant factor explain-

ng the decline in the number of credit institutions ( Buch et al.,

014 ). Simultaneously, concentration in the banking sector – as

easured by the market share of the five largest credit institutions

also increased in most (though not in all) Euro area countries,

nd also in the Euro area average ( European Central Bank, 2017 ,

hart 2.10).

Finally, a third empirical observation is that capital regulation

ay benefit large banks not only in the competition against their

maller domestic rivals, but also in the international competition

or market shares. In Switzerland, for example, high capital re-

uirements were partly introduced to restore faith in the Swiss

anking system, after one of Switzerland’s largest banks, UBS, had

ncurred huge losses in the US subprime loan market and needed

o be saved with large public loans. 5 Interestingly, Swiss banks do

ot seem to have been hurt by the higher capital requirements im-

osed by Swiss regulators. Fig. 1 plots the market shares of Swiss

anks in the European market for bank credits to the private sec-

or for the period 2007–2015, and compares it to those of its main

uropean competitors. The figure shows that the market share of

wiss banks has continuously risen during this period, whereas

ess strictly regulated banks in Germany, for example, have lost

arket shares at the same time.

In this paper we aim to set up a model of regulatory competi-

ion in capital standards that is able to explain these stylized facts.

pecifically, our analysis introduces two new features that jointly

ffer a motivation for why tighter capital standards can benefit a

ountry’s banks, and why regulatory competition may even lead to

‘race to the top’ in capital regulation.

First, our model allows for banks that are of heterogeneous

uality and differ in their probability of failure. When individual

anks are unable to signal their quality themselves, higher capital

tandards act as a signal of average quality in the national bank-

ng sector. This is because higher capital standards drive the weak-

5 See “How Switzerland saved its banking industry”, Newsweek Maga-

ine, 27 December 2010. http://europe.newsweek.com/how- switzerland- saved- its-

anking- industry- 68855?rm=eu .

182 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

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6 Recently, Boyer and Kempf (2019) have analyzed non-cooperative banking regu-

lation in a setting where banks can costlessly choose the jurisdiction in which they

operate. In their framework, liquidity regulation to limit risk-taking and taxes on

bank profits emerge endogenously as the efficient regulatory instruments. In the

non-cooperative equilibrium, a ‘race to the bottom’ results with respect to profit

taxes, though not with respect to liquidity standards.

est banks from the market, thus improving the pool quality of the

remaining banks. Loan-taking firms anticipate the increase in av-

erage bank quality and are willing to pay higher loan rates in ex-

change for the added security gained (for example, through more

reliable access to credit). For low levels of capital requirements,

we show that this selection effect of capital standards can be suf-

ficiently strong to overcompensate the higher cost of capital, thus

increasing the market share of banks in the more strictly regulated

economy.

A second distinguishing feature of our model is that we con-

sider governments that include taxpayers and consumers in their

welfare function, in addition to the profits of the banking sector.

Our model incorporates competitive firms that use bank credit to

produce output for an integrated market. Changes in the availabil-

ity and the price of credit thus have consequences for the real

economy, and these spill over to the foreign country through the

integrated output market. Moreover, we explicitly incorporate the

fact that taxpayers have to come up for the losses of failed banks

due to the existence of a deposit insurance scheme.

In the Nash equilibrium, we show that tighter capital controls

in one country reduce this country’s aggregate loan volume while

increasing the average quality of its banks. These changes increase

aggregate profits in the foreign banking sector, but they simulta-

neously exert negative externalities on foreign consumers and tax-

payers. Foreign consumers lose because the reduced loan volume

caused by tighter capital standards reduces aggregate output in

the integrated market, and hence consumer surplus. Foreign tax-

payers lose because the reduced loan supply from the country im-

posing tighter capital controls will, in equilibrium, draw additional,

and lower-quality, banks into the foreign banking sector. This ex-

poses foreign taxpayers to additional default risks, due to both the

higher aggregate loan volume and the lower average quality of

their banks. Hence, imposing tighter capital controls can serve as

an instrument to shift the default risks arising from the banking

sector from domestic to foreign taxpayers.

The main result of our analysis is that when governments care

equally about bank profits, consumer surplus, and expected tax

revenue losses, the negative externalities that tighter capital re-

quirements impose on foreign consumers and taxpayers will domi-

nate the positive externality on the profits of foreign banks. Hence,

the non-cooperative setting of capital standards leads to higher

capital requirements than is optimal from a global welfare perspec-

tive. This implies a ‘race to the top’ in capital regulation that is very

different from the established patterns for the taxation of capital.

We also consider several modifications of our benchmark

model. We show that the regulatory ‘race to the top’ is further in-

tensified when the banking sector of each country is partly owned

by foreign shareholders. Moreover, we study the robustness of our

main result in a situation where banks are able to perfectly signal

their quality to firms. While the negative externalities of capital

regulation on the foreign country are reduced in this setting, our

main result of a ‘race to the top’ in capital standards is maintained.

Our analysis relates to several strands in the literature. A first

set of papers analyzes the effects of capital regulation in the pres-

ence of moral hazard in the banking sector and shows that it

curbs risky behaviour ( Rochet, 1992; Hellman et al., 20 0 0; Repullo,

2004 ). A few papers in this literature also incorporate bank het-

erogeneity. Morrison and White (2005) set up a model where the

regulator uses both screening and capital requirements to address

simultaneous moral hazard and adverse selection problems. As in

our model, capital requirements improve the quality of the surviv-

ing banks in their framework, and hence the average loan qual-

ity. Similar results are obtained in Kopecky and VanHoose (2006) .

These papers do not consider competition between regulators.

The literature on regulatory competition in the banking sector

is small. Sinn (1997, 2003) draws an explicit parallel between cap-

tal regulation and capital taxation. He models both as examples

f the more general phenomenon of ‘systems competition’, which

ndermines nation states’ tax as well as regulatory functions.

charya (2003) introduces competition between bank regulators

hat simultaneously choose the level of capital requirements and

he bailout policy when banks become insolvent. Dell’Ariccia and

arquez (2006) analyze regulatory competition in a framework

here duopolistic banks make loans either in their home or the

oreign market. They derive non-cooperative capital standards by

rading off the losses to the domestic banking sector against the

enefits of increased financial stability. All these papers arrive at

he conclusion that national capital standards will be set inef-

ciently lax in the non-cooperative policy equilibrium. 6 Finally,

orrison and White (2009) focus on regulatory competition be-

ween two countries that differ with respect to the quality of their

ational regulators and ask whether a level playing field is desir-

ble in this setting. None of these papers explicitly incorporates

rms that use bank loans to produce real output, and none of them

erives the effects of capital regulation on either taxpayers or con-

umers.

The heterogeneity of firms incorporated in this paper is an

mportant topic in the recent international trade and tax litera-

ure. Tax policy competition in a framework with heterogeneous

anufacturing firms has been analyzed, for example, by Davies

nd Eckel (2010) , Krautheim and Schmidt-Eisenlohr (2011) and

aufler and Stähler (2013) . As this literature shows, the ‘race to

he bottom’ result with respect to capital tax competition is gen-

rally maintained in the presence of firm heterogeneity. In the in-

ernational trade literature, Buch et al. (2011) show a close em-

irical link between size, productivity and international activity in

he banking sector that is similar to the well-established patterns

or the manufacturing sector. Niepmann (2015) introduces a frame-

ork of cross-border banking based on international trade theory

nd Niepmann (2016) extends this model to account for heteroge-

eous monitoring ability of banks. These papers do not consider

egulatory policies, however.

Finally, a small strand in the recent public economics literature

ocuses specifically on the financial sector. Keen (2011) stresses the

ualitative similarities between regulation and taxation of the fi-

ancial sector and Lockwood (2014) analyzes the optimal taxation

f financial intermediaries. This literature has also provided em-

irical evidence that recent bank levies have been effective in in-

reasing the equity ratio of European banks Devereux et al. (2013) .

selection of papers dealing with the taxation and regulation of

he financial sector is collected in de Mooij and Nicodème (2014) .

This paper is set up as follows. Section 2 presents our bench-

ark model. Section 3 analyzes nationally optimal regulation poli-

ies. Section 4 turns to the central issue of whether decentralized

apital standards are set higher or lower than is globally optimal.

ection 5 discusses and analyzes the robustness of our main result.

ection 6 concludes.

. The model

.1. Banks

Our benchmark model considers a region of two countries i ∈ {1,

}, which are symmetric in all respects. The symmetry assumption

nsures that governments face the same incentives in our model,

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 183

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9 This is true even though bank size is a sign of quality in our model, and

is observable by firms. The empirical evidence, however, shows similar increases

hus allowing clear-cut answers to the question of whether cap-

tal standards race to the top or to the bottom. In each country

here are a large number of heterogeneous banks, which competi-

ively lend funds to producing firms in an integrated regional loan

arket. Banks in each country operate under the authority of a na-

ional regulator who imposes a capital requirements k i , which we

efine as the ratio of equity capital to total assets, for all banks

ithin his jurisdiction. The number of active banks in each coun-

ry, and the volume of loans distributed by each bank, are endoge-

ous.

Banks differ exogenously in their quality, where the quality in-

ex q summarizes the technology available to a bank in a similar

ay as is known from the literature on heterogeneous (manufac-

uring) firms. In our setting, the bank’s quality q corresponds to

he likelihood that the investment financed by the bank’s loans are

uccessful. Therefore, as in related literature ( Dell’Ariccia and Mar-

uez, 2006; Allen et al., 2011 ), the bank’s quality directly deter-

ines the success probability of the firms to which it lends. Given

ts interpretation as a success probability, the quality index q is

istributed in the interval [0,1] and we assume, for simplicity, that

his distribution is uniform.

There are several ways in which the quality of a bank can im-

rove the success probability of borrowing firms. A first argument

ocuses on the monitoring capacity of banks in situations where

he manager of the borrowing firm faces a situation of moral haz-

rd. In this interpretation, the index q therefore represents a mon-

toring quality (or an inverse monitoring cost parameter) of the

ank. 7 Managers are identical ex ante, but adjust their effort con-

inuously to the differential monitoring qualities of their banks.

he effort level provided by the manager in turn affects the prob-

bility that the firm’s investment is successful (cf. Besanko and

anatas, 1993; Holmstrom and Tirole, 1997 ).

A second argument focuses instead on the lending capacity

f banks. After the initial loan contract has been signed, firms

ay face random liquidity shocks during the process of produc-

ion. These shocks will force them to terminate the project unless

hey can flexibly draw on additional credit lines of their bank. As

hown by Boot et al. (1993) , the ability of banks to offer these

exible, discretionary financial contracts will depend on the qual-

ty with which banks manage the liquidity pool of their portfo-

ios. A similar argument is made in the analysis of Inderst (2013) ,

here the expected payoff of projects depends on the ability of

anks to roll over loans. In this interpretation, the index q there-

ore stands for the quality of the bank’s financial management. The

mportance of this effect is empirically confirmed by Ivashina and

charfstein (2010) , who show that banks with better access to de-

osit financing and less reliance on short-term debt had to cut

heir lending less in the 2008 financial crisis. Similarly, Popov and

dell (2012) show that firms were more likely to be credit con-

trained during the crisis, if they were dealing with banks that had

xperienced a decline in equity, or losses on their financial assets.

Since firm owners directly benefit from the quality of a bank

hat lends to them, they are willing to pay a higher loan rate for

loan from a higher quality bank. However, the quality index q is

rivate information to each bank. Signalling this quality to firms

s hindered by the ‘opaqueness’ frequently attributed to banks in

he literature, which makes it difficult for outside parties to draw

uality inferences from banks’ balance sheets. 8 The empirical evi-

ence furthermore suggests that the opaqueness of banks is more

7 Since q is exogenous in our setup, we do not endogenize the monitoring deci-

ion of banks. See Niepmann (2016) for a similar modeling approach in a setting

ith heterogeneous banks. 8 Dang et al. (2017) provide a rationale for the opaqueness of banks. They show

hat preventing third parties from aquiring private information on a bank’s loans

llows the bank to efficiently share risks between borrowers and lenders.

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ronounced in periods of financial crisis. Flannery et al. (2013) and

lau et al. (2017) compare the trading efficiency for the stocks of

anks and those of non-financial firms in ‘normal’ periods and in

eriods of ‘crisis’. In ‘normal’ times, the comparison between bank

tocks and non-bank stocks yields no unambiguous conclusions. In

crisis’ times, however, both studies find a significantly lower trad-

ng efficiency for bank stocks as compared to non-bank stocks, and

ttribute this to the increased opaqueness of banks in times of

risis.

In our benchmark model, we will therefore focus on a situation

f ‘crisis’ and assume that banks are unable to signal their quality

o firms. 9 In this situation, government capital regulation acts as an

mperfect substitute for banks signalling their individual quality. As

e will show below, higher capital requirements eliminate lower

uality banks from the market and increase the average pool qual-

ty of the remaining banks. This increase in average quality leads

o a higher loan rate being paid to all banks in that country. In

ection 5.3 we then consider a ‘normal’ period, in which banks are

ble to perfectly signal their quality to firms. Our main result is

hown to carry over to this setting. 10

Banks fund themselves either through equity capital or through

xternal funds, which we take to be saving deposits of individuals.

n line with common practice in virtually all developed countries,

e assume that the savings deposits are fully insured by the gov-

rnment of the country in which the bank is located. 11 Hence, and

mportantly for our model, the (expected) costs of bank failures are

artly borne by the taxpayers of the bank’s residence country. Be-

ng fully insured against failure, depositors demand a competitive

eturn on their savings, which we normalize to unity. In contrast,

nd following a standard assumption in the literature, the bank’s

ost of equity includes a risk premium and is exogenously given

y ρ > 1 (cf. Hellman et al., 20 0 0; Dell’Ariccia and Marquez, 20 06;

llen et al., 2011 ). Moreover, equity holders receive all excess prof-

ts of banks, in return for sharing in the risk of bank failure.

Given that savings deposits are implicitly subsidized by tax-

ayers through the deposit insurance scheme, profit-maximizing

anks will never choose to hold costly equity capital in excess of

he minimum level k i stipulated by the national regulator. Hence

he only decision taken by heterogeneous banks in our model con-

erns their volume of lending, denoted by l . The scale of operations

f each bank is limited by transaction costs that are rising more

han proportionally when the bank’s level of operation rises. One

ypical justification for this assumption is that banks must spend

xtra effort s to find good-quality customers when their loan vol-

me is expanded (see Acharya, 2003 ). For simplicity, we assume

hat transaction costs are quadratic in the volume of an individual

ank’s loan volume l , and given by (1/2) bl 2 , with b > 0.

With these specifications the expected pure profits of a bank in

ountry i with quality q that chooses to distribute a total number

f l loans are given by

i (q, l) = q [ R i − (1 − k i )] l − ρk i l −1

2

bl 2 ∀ i ∈ { 1 , 2 } . (1)

n Eq. (1) , R i is the return per unit of the bank’s loans. In our

enchmark model of ‘crisis’, this depends on the capital standards

n opacity for larger and smaller banks during the 20 07–20 09 financial crisis

Flannery et al., 2013 , Table 3). This suggests that a larger bank size does not solve

he fundamental uncertainty about the valuation of banks’ assets in times of crisis. 10 We are grateful to a referee for this suggestion, and for the interpretation of the

ifferent scenarios. 11 The main argument in favor of deposit insurance schemes is that they pre-

ent bank-runs and thus stabilize the banking system ( Diamond and Dybvig, 1983 ).

arth et al. (2006) give an overview of deposit insurance schemes around the world,

nd discuss its benefits and costs.

184 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

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12 Market entry costs are difficult to estimate, because they can be identi-

fied only through their effects on market participation patterns. One example is

Das et al. (2007) , who estimate entry costs for three Colombian manufacturing in-

dustries in a dynamic framework with firm heterogeneity.

set by the bank’s home country i , but not on the individual quality

of the bank. From this gross loan rate the bank must deduct the

costs of savings deposits (1 − k i ) , which are paid back by the bank

only with its success probability q . Following the related literature

(e.g. Dell’Ariccia and Marquez, 2006 ) we assume that the returns

to all loans of a bank are perfectly correlated. Therefore, when a

bank’s loans fail, the bank’s return is zero and the bank will go

bankrupt. Savers will then be compensated by payments from the

national deposit insurance fund, whereas equity holders lose all

their investment. Total equity cost is k i l ρ , where equity capital in

the bank is k i l and the opportunity cost of one unit of equity is ρ .

Finally, the bank must deduct the transaction costs (1/2) bl 2 for its

lending. All net profits, and all uncovered losses, accrue to equity

holders as residual claimants.

We assume that loan markets are competitive and pure profits

to banks arise only from the heterogeneity of national banking sec-

tors. Hence all banks take R i as given when choosing l . The optimal

loan volume l ∗ for each bank in country i is then given by

l ∗ =

qφi − k i ρ

b ∀ i , (2)

where we have defined the short-hand notation

φi ≡ R i − (1 − k i ) ∀ i (3)

to indicate the return per unit of loans for each bank in country i ,

net of the funding costs for savings deposits. This term therefore

represents the expected increase in a bank’s profits when the suc-

cess probability of its loans increases.

From Eq. (2) , the loan volume of a bank is an increas-

ing function of its quality q . Thus, a better bank is also larger

in equilibrium. This corresponds to the empirical evidence in

Buch et al. (2011) , showing that bank productivity and bank size

are positively correlated. Moreover, the loan volume is an increas-

ing function of the return R i and a decreasing function of the cap-

ital adequacy ratio k i , both of which are specific to the country in

which the bank is located.

Substituting (2) in (1) gives the optimized profits of a bank of

quality q in country i :

π ∗i (q ) =

(qφi − k i ρ) 2

2 b ∀ i. (4)

The equilibrium number of banks is determined by the condition

that the marginal bank, denoted by the cutoff quality level ˆ q i , re-

ceives zero expected profits. From Eq. (1) and noting that l ∗ = 0

holds for the critical bank [see Eq. (2) ], this condition is

ˆ q i φi − k i ρ = 0 ∀ i. (5)

Consequently, only banks with q ≥ ˆ q i will be active in the market.

Active banks obtain positive expected profits in equilibrium, as the

market loan rate R i must be high enough for the cutoff bank ˆ q i to

break even. Therefore, despite being price-takers in the loan mar-

ket, banks in each country earn rents as a result of their heteroge-

neous quality.

Eq. (5) further shows that capital standards in country i directly

affect the cutoff quality level ˆ q i by increasing the cost of capital for

all banks. As low-quality banks benefit most from limited liabil-

ity and cheap deposit funding, they are hit hardest by an increase

in capital standards. Without any capital requirements (k i = 0) , all

banks will be active in the market ( q i = 0) . In contrast, full equity

financing of banks ( k i = 1 ) results in ˆ q i = ρ/R i . Hence, the condi-

tion for a positive number of banks to stay in the market even

with full equity financing is that the opportunity cost of equity ρis lower than the equilibrium return on loans, R i . We make this

assumption in the following.

It remains to determine the aggregate loan volume L i of all ac-

tive banks in country i . Normalizing the exogenously given number

f potentially entering banks to unity and integrating over the op-

imal loan volumes (2) of all active banks gives

i =

∫ 1

ˆ q i

l(q ) dq =

(1 − ˆ q i )(φi − k i ρ)

2 b =

( 1 − ˆ q i ) 2 φi

2 b ∀ i. (6)

ere (1 − ˆ q i ) is the measure of active banks in country i , and (φi − i ρ) / 2 b gives the average loan volume per active bank. The second

tep in (6) then uses (5) to simplify the resulting expression.

.2. Firms and consumers

One of the features of our model is that we explicitly incorpo-

ate firms that use bank loans to produce consumer goods. In the

ollowing sections this will allow us to study the welfare effects of

apital standards on banks, taxpayers and consumers.

We assume that there are a large number of identical, potential

roducers in an integrated final goods market, which do not have

ny private sources of funds. The potential producers compete for

redit in the international loan market, where each firm can obtain

redit from either the domestic or the foreign banking sector. Note

hat the location of firms is irrelevant in our model, because all

rms are identical and the output market is integrated. Each firm

hat enters the market in equilibrium demands one unit of credit

o produce one unit of output. Total output in the integrated mar-

et therefore depends on the expected number of successful loans

rom banks in both countries. The expected output produced with

oans from banks located in country i is

i =

∫ 1

ˆ q i

ql(q ) dq = L i q e i , q e i ≡

(2 +

ˆ q i 3

). (7)

q. (7) shows that changing the cut-off quality of banks ˆ q i has

mbiguous effects on aggregate output in our model. On the

ne hand, it reduces the total loan volume of country i ’s banks

rom (6) . At the same time, however, it also increases the average

uality of country i ’s banking sector. This is shown by the higher

xpected success rate q e i , where the specific formula for q e

i derives

rom the assumption of a uniform distribution of bank qualities.

Next we determine the loan rate that firms are willing to pay

or bank loans from each country i in the competitive equilibrium.

ll potential entrants in the final goods sector have to incur a uni-

orm fixed cost c for their projects, which can be thought of as

arket entry costs. 12 Further, as firms can not observe the quality

f the contracting bank, they have to form expectations about the

verage quality of loans distributed by all active banks that reside

n a specific country. This is given by the expected success rate q e i

efined in Eq. (7) . If the investment is successful, the firm sells its

roduct in the integrated market for the homogeneous consumer

ood at a price P . Firms will not repay the loan if their project

ails, but the entry cost c has been incurred nevertheless. Allowing

or unrestricted, but costly, entry of firms into the output market,

he zero profit condition for entering, risk-neutral firms implies

e i (P − R i ) = c ∀ i. (8)

ince producing firms are identical, they also make zero expected

rofits in the aggregate. Hence the entire surplus generated in the

oan market is transferred to the banking sector via the loan rate

i .

The price of the homogeneous output good, P , is determined

rom an inverse demand function P = A − y, where A measures the

ize of the integrated market and y ≡ y 1 + y 2 is the total expected

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 185

o

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13 Our analysis abstracts from insurance funds paid by the banking sector. In-

troducing a deposit insurance premium for banks to cover their potential losses

would reduce banking sector profits in (12) , and simultaneously reduce tax losses

to consumer-taxpayers in (13) . Existing insurance funds paid by the banking sector,

like the EU’s ‘resolution fund’ phased in since 2016, are built up only gradually and

with a moderate overall target volume. 14 See Niepmann and Schmidt-Eisenlohr (2013) and Beck and Wagner (2016) for

analyses of international regulatory coordination when bank failures in one country

have adverse effects on banks in the other country.

utput financed with bank loans from both countries. Substituting

nto (8) gives

i = A − c

q e i

− y = A − 3 c

2 +

ˆ q i − y ∀ i. (9)

q. (9) shows that the loan price is decreasing in total output,

nd in the amount of firms’ entry costs c . Moreover, loan rates

re country-specific and depend positively on the expected quality

f the banking sector in country i . Consequently the price of bank

oans differs systematically between the two countries whenever

heir capital requirements differ, with bank loans from the country

ith the higher capital requirement receiving a higher return.

.3. Market equilibrium and welfare

To derive the market equilibrium, we substitute the loan

ate (9) into (5) and, together with banks’ lending volumes (2) ,

nto (7) . This yields a system of three equations:

ˆ 1

[ A − 3 c

2 +

ˆ q 1 − y − 1 + k 1

] = ρk 1 , (10a)

ˆ 2

[ A − 3 c

2 +

ˆ q 2 − y − 1 + k 2

] = ρk 2 , (10b)

= y 1 + y 2 =

1

b

∫ 1

ˆ q 1

[ q 2 (A −y −1 + k 1 ) −qk 1 ρ − q 2

(3 c

2 +

ˆ q 1

)] dq

+

1

b

∫ 1

ˆ q 2

[ q 2 (A − y − 1 + k 2 ) − qk 2 ρ − q 2

(3 c

2 +

ˆ q 2

)] dq. (10c)

Eqs. (10a)–(10c) jointly determine the cutoff qualities of banks,

ˆ 1 and ˆ q 2 , and the aggregate output level y , all as functions of

he capital requirements k 1 and k 2 imposed by the two countries.

hese core variables then determine the total level of loans from

ach country from (6) and the country-specific loan rate from (9) .

We consider a national regulator in each country who sets cap-

tal requirements so as to maximize national welfare. Our wel-

are measure is broader than that used in the existing literature

n regulatory competition, covering all agents in country i whose

ncome is affected by capital regulation. Hence we include bank

rofits �i , which equal the sum of all gains and losses accruing to

quity holders in the banking sector of country i . We also incor-

orate the welfare effects on consumer-taxpayers, however, which

re twofold. First, consumers are affected by the negative tax rev-

nues T i , which incorporate the expected costs to resident tax-

ayers when banks fail and savings depositors are compensated

hrough the deposit insurance fund. Second, by affecting the sup-

ly of loans, capital standards also affect aggregate output and

ence consumer surplus S i in each country. These effects cover all

elevant welfare changes arising from capital regulation. Savings

epositors can be ignored in the welfare function, because they

lways receive the fixed return of unity. Moreover, all producing

rms make zero profits from Eq. (8) .

We introduce an aggregate measure C i to capture the welfare

f consumer-taxpayers and take the government’s objective to be

weighted sum of bank profits and consumer welfare. This gives:

i = α�i + γC i , C i = S i +

β

γT i , α, β, γ ≥ 0 . (11)

ence, in the aggregate measure of consumer welfare, γ is the

eight for consumer surplus S i and β is the weight for (negative)

ax revenues T i .

The components of national welfare can be directly calculated

rom the equilibrium in the loan market. In our benchmark analy-

is we assume that all equity holders of country i ’s banks are also

esidents of country i . We will relax this assumption in Section 5.2 .

otal profits in the banking sector of country i are given by aggre-

ating (4) over all active banks. This yields

i =

∫ 1

ˆ q i

(qφ − k i ρ) 2

2 b dq =

6 by 2 i

(2 +

ˆ q i ) 2 (1 − ˆ q i ) ∀ i, (12)

here we have used (6) and (7) to express �i as a function of the

utput produced with loans from country i ’s banks ( y i ), and of the

utoff quality of banks in i ( q i ).

The expected losses borne by taxpayers in country i arise from

he deposit insurance scheme. 13 These losses are determined by

he share of deposit financing, the aggregate loan volume, and the

verage failure probability of country i ’s banks. Moreover, we ab-

tract from international contagion effects and assume that the

osses from failed banks arise only in the country in which the

ank is located. 14 Aggregating and using (6) and (7) in the second

tep gives

i =

−(1 − k i )

b

∫ 1

ˆ q i

(1 − q )(qφi − k i ρ) dq =

−(1 − k i )(1 − ˆ q i ) y i (2 + ˆ q i )

∀ i.

(13)

inally, since the output market is regionally integrated and the

odel is symmetric, consumers in each country receive one half of

he total consumer surplus in the integrated market. The consumer

urplus measure in each country i is therefore

i =

1

2

(A − P ) y

2

=

(y 1 + y 2 ) 2

4

∀ i. (14)

rom (12)–(14) we can determine the effects of capital require-

ents on national and regional welfare, as well as its components.

. Nationally optimal capital standards

In this section we analyze the effects of capital standards that

re set in a nationally optimal way. In Section 3.1 we first discuss

he effects that capital requirements have on the equilibrium in the

oan market. Section 3.2 then turns to the conditions for a sym-

etric Nash equilibrium in capital standards.

.1. Capital standards and the loan market

In a first step we derive the effects that a unilateral increase

n country i ’s capital requirement k i has on the equilibrium in the

oan market. To save on notation, we omit country subscripts in

he following when no confusion is possible, invoking the symme-

ry of our model. The changes in the endogenous variables ˆ q i , ˆ q j ,

i and y j are derived in Appendix A.1 and are given by

∂ q i ∂k i

=

(ρ − ˆ q ) + ρ(φ +

ˆ c q )(2 +

ˆ q )(1 − ˆ q ) 2

2(φ +

ˆ c q )�> 0 ,

∂ q j

∂k i =

ˆ q (1 − ˆ q ) κ

2(φ +

ˆ c q )�, (15)

186 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

Fig. 2. The effects of a small capital requirement in country i .

e

F

c

t

l

t

t

u

l

s

y

i

s

i

∂y i ∂k i

=

(1 − ˆ q )κ

12 b(φ +

ˆ c q )�,

∂y j

∂k i =

−2 φ(1 − ˆ q )(1 − ˆ q 3 ) κ

12 b(φ +

ˆ c q )�,

∂y

∂k i =

(1 − ˆ q ) κ

2�, (16)

where we have introduced the short-hand notations

� ≡ 3 b(φ +

ˆ c q ) + 2 φ(1 − ˆ q 3 ) > 0 ,

≡ � + 3 b(φ +

ˆ c q ) > 0 , ˆ c ≡ 3 c

(2 +

ˆ q ) 2 , (17)

and

κ = −φ[3(ρ − 1)(1 +

ˆ q ) + (1 + 2

q )(1 − ˆ q ) ]︸ ︷︷ ︸

(1)

+ 3 cρ(1 − ˆ q )

(2 +

ˆ q ) ︸ ︷︷ ︸ (2)

<> 0 .

(18)

The first term in Eq. (15) shows that an increase in country i ’s

capital requirement unambiguously raises the quality of the cutoff

bank in this country, ˆ q i . This is due to both the higher opportu-

nity cost of equity in comparison to savings deposits, and to the

reduced volume of implicit taxpayer subsidies as a consequence of

the higher equity ratio. Hence, by raising the cost of finance for all

banks, higher capital requirements k i drive the weakest banks in

country i from the market.

The second term in (15) and the terms in (16) all depend on

the size of κ , as given in (18) . It is thus critical for our analysis to

discuss the effects summarized by κ in detail. As shown in (18) ,

the effect of a higher capital requirement on the total level of per-

forming loans can be decomposed into two parts. The first term

is unambiguously negative, as capital standards raise the costs of

refinancing for all banks. We label this the cost effect of higher

capital standards. The second term in (18) is positive, however. It

captures the positive effect that higher capital requirements have

on the pool quality of banks in country i . The rise in ˆ q i induced

by a higher capital requirement results in a higher loan rate that

firms are willing to pay for loans from banks based in country i , as

they face a lower probability of losing their entry cost c . In the fol-

lowing we will refer to this effect as the selection effect of capital

standards. In sum, we can therefore not sign κ , in general. 15

Fig. 2 illustrates the two cases corresponding to κ < 0 and κ > 0,

respectively, for the case of a small capital requirement in coun-

try i . Eqs. (6) and (7) , together with (3) , yield an inverse supply

function R S ( y i ) that describes y i as a positive function of the loan

rate R i when y j is held constant. At the same time, P = A − y j − y i gives the price that competitive firms achieve in the output mar-

ket, as a function of country i ’s volume of successful loans. From

this, the demand for loans from banks in country i, R D ( y i ), can be

derived as a parallel shift of the demand function in the output

market. The vertical intercept of the loan demand function is de-

termined by the firms’ entry cost c and the inverse of the expected

success probability q e i

[see Eq. (9) ].

In the absence of any capital requirements, the loan supply

curve for country i ’s banks, R 0 S , starts at per-unit refinancing costs

of unity. This represents the case of pure deposit finance. A small

capital requirement k i shifts the loan supply curve upward ( cost

effect ). The associated increase in the cutoff quality of country i ’s

banks also leads to a parallel upward shift of the initial loan de-

mand curve R 0 , by lowering the firms’ probability of losing their

D

15 Introducing a deposit insurance premium for banks that is adjusted to changes

in the capital ratio k i (cf. footnote 13 ) would lead to a smaller cost effect of cap-

ital requirements, as the increase in the cost of capital would be partly offset by

a lower deposit insurance premium. However, since a higher capital ratio k i also

forces a switch from deposits to more expensive equity (with ρ > 1), it would raise

banks’ capital costs even if the insurance premium covered all expected losses to

taxpayers.

i

i

r

o

r

c

c

o

ntry costs ( selection effect ). In Case A, given in the upper panel of

ig. 2 , the entry cost c is small and the shift in the loan supply

urve dominates the shift in the loan demand curve. As a result

he equilibrium shifts from E 0 to E 1 and the volume of successful

oans given by country i ’s banks is reduced from y 0 i

to y 1 i . This case

hus corresponds to κ < 0. In Case B, shown in the lower panel of

he figure, the firms’ entry costs c are sufficiently large so that the

pward shift in the loan demand curve dominates the shift in the

oan supply curve. Hence the equilibrium shifts from E 0 to E 2 , re-

ulting in an increase in successful loans by country i ’s banks from

0 i

to y 2 i . This corresponds to the case κ > 0.

The implications for country j then follow from the equilibrium

n the loan market. If κ < 0, a rise in k i reduces the aggregate loan

upply of banks in country i . This raises the loan rate for banks

n country j . The higher profitability will draw additional banks

n country j into the market, thus lowering ˆ q j [the second term

n (15) ]. Moreover, the aggregate loan volume in country j will

ise, and with it the output y j generated from these loans [the sec-

nd term in (16) ]. Hence a unilateral increase in country i ’s capital

equirement shifts business from banks in country i to banks in

ountry j . If κ > 0, all effects are reversed. In this case, a higher

apital standard in country i will boost the aggregate loan supply

f banks in country i . The expansion of loans from country i will

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 187

t

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17 Behn et al. (2016) show, for example, that the Basel II capital regulations, which

hen reduce the loan price for banks in country j , raising ˆ q j and

educing y j .

.2. Nash equilibrium in capital standards

In a second step, we use the effects on the loan market equilib-

ium variables, as given in (15) and (16) , to determine the effects

f capital standards on each country’s welfare and derive the Nash

quilibrium in the regulatory policies k i . Differentiating the welfare

unction (11) and its components (12)–(14) gives

∂W i

∂k i = α

∂�i

∂k i + β

∂T i ∂k i

+ γ∂S i ∂k i

, (19)

here

∂�i

∂k i =

18 by 2 i

ˆ q i

(1 − ˆ q i ) 2 (2 +

ˆ q i ) 3 ∂ q i ∂k i

+

12 by i (1 − ˆ q i )(2 +

ˆ q i ) 2 ∂y i ∂k i

, (20)

∂T i ∂k i

=

(1 − ˆ q i ) y i (2 +

ˆ q i ) +

3(1 − k i ) y i (2 +

ˆ q i ) 2 ∂ q i ∂k i

− (1 − k i )(1 − ˆ q i )

(2 +

ˆ q i )

∂y i ∂k i

, (21)

∂S i ∂k i

=

y

2

∂y

∂k i . (22)

e first evaluate equations (20)–(22) at an initial capital stan-

ard of k i = 0 . Hence, we ask how welfare in country i is affected

y the introduction of a small capital standard when, in the ini-

ial equilibrium, banks’ funding needs can be fully met by cheap

and insured) savings deposits. Note that an initial capital standard

f k i = 0 implies ˆ q i = 0 from (5) . Turning first to the effects on

he profits of country i ’s banking sector in (20) , the first term in

his expression vanishes when ˆ q i = 0 initially. Hence the effects on

ank profits are exclusively determined by the change in the ag-

regate level of successful loans (i.e., output), as given by the sec-

nd term. The induced output change also determines the change

n country i ’s consumer surplus, as given in (22) .

The effects on tax revenues in (21) are threefold. The first ef-

ect gives the direct, positive effect on tax collections (i.e., a reduc-

ion in expected subsidy payments) by decreasing the bank’s re-

iance on deposits backed by a tax-financed insurance mechanism.

oreover, increasing the critical bank quality ˆ q i , and hence rais-

ng the average success rate of loans, additionally reduces the ex-

ected burden on taxpayers by the second effect. The sign of the

hird effect is ambiguous, however, as it depends on the change in

he aggregate volume of loans offered by banks in country i , and

ence on the sign of κ .

In Appendix A.2 we derive sufficient conditions under

hich (20)–(22) are all positive when evaluated at k i = 0 initially,

nd the introduction of a small capital standard strictly increases

elfare in country i . The sufficient conditions are given by 16

(A − 1) <

[3 b(2 ρ − 1) + 2 ρ/ 3

b(3 ρ − 2)

]c and (A − 1) >

[ 15

8

+

1

4 b

] c.

(23)

he first inequality in (23) is just the condition for κ to be pos-

tive at k = 0 . This requires that the firms’ entry costs c must be

ufficiently large in relation to the market size parameter A , which

etermines the profit margin of banks. If this condition is fulfilled,

he selection effect of capital standards dominates the cost effect

hen both are evaluated at an initial capital adequacy ratio of zero.

he second inequality in (23) states, in contrast, that the firms’

xed cost, and hence the induced expansion of bank loans is not so

16 Note that the two conditions in (23) are not mutually exclusive. For example, if

= 1 . 2 and b = 4 / 3 , both conditions are simultaneously fulfilled when 3 c > A − 1 >

33 / 16) c.

g

w

m

p

S

arge as to overcompensate the positive first two effects of a small

apital standard in the tax revenue expression (21) . We summarize

hese results in:

roposition 1. When both conditions in (23) hold, then introducing a

mall capital standard k i > 0 simultaneously increases aggregate bank-

ng sector profits and the welfare of consumer-taxpayers in country i,

or any combination of α, β , γ ≥ 0 .

Our model therefore shows that introducing a small capital

tandard may be in the overall interest of the country’s banking

ector when the latter is heterogeneous. By raising the costs of do-

ng business, the capital standard drives the weakest banks from

he market and high-quality banks will benefit from this market

xit via a higher loan rate. When firms value the increase in the

ool quality of banks sufficiently, as measured in our model by

heir entry cost c , then the higher profits of infra-marginal banks

ominate the profit losses of marginal, low-quality banks. These

edistributive effects between heterogeneous banks may thus ex-

lain why large and productive banks do not generally oppose

ew capital regulations, and in some cases even actively advocate

hem. 17

We now turn to the other extreme case and evaluate (20)–

22) for an initial capital ratio of k i = 1 . This case implies that all

oans must be financed by (expensive) equity. For k i = 1 , the first

erm in the tax revenue expression (21) is positive, whereas the

ther two terms are zero. Since the first term in the profit expres-

ion (20) is also positive and the remaining terms in (20) and the

onsumer surplus term (22) are positive multiples of κ , it follows

irectly that κ| k =1 < 0 is a necessary condition for ∂W i ( k i )/ ∂k i to

e negative at k i = 1 , and hence for an interior optimum in capital

tandards. When κ| k =1 < 0 holds, the effects of a rise in k i on firm

rofits in (20) are generally ambiguous. Therefore, a first sufficient

ondition for ∂W i ( k i )/ ∂k i < 0 to hold at k i = 1 is that the marginal

ffect of an increase in k i on aggregate banking sector profits �i is

egative. This condition is derived in Appendix A.3 and given by:

∂�i

∂k i

∣∣∣∣k =1

< 0 ⇐⇒ 3(ρ − 1) − ρ

8 b >

c

6

. (24)

ondition (24) implies that the cost effect of capital standards

measured by ρ − 1 ) dominates the selection effect (which depends

n the firms’ fixed cost c ) at the maximum capital ratio of unity.

oreover, when condition (24) is met, κ is sufficiently negative so

hat the negative effect of k i on the aggregate volume of successful

ank loans in country i [the second term in (20) ] dominates the

ositive effect of a higher k i on the average profitability of active

anks [the first term in (20) ].

In addition to the effect on aggregate banking sector profits, a

ise in k i raises tax revenues in country i , but reduces consumer

urplus, when evaluated at k i = 1 [ Eqs. (21) and (22) ]. Therefore, a

econd sufficient condition for ∂W i ( k i )/ ∂k i < 0 to hold at k i = 1 is

hat the welfare weight of tax revenues ( β) is not too large, relative

o the welfare weight of consumer surplus ( γ ).

We can now turn to the Nash equilibrium in capital standards

n our model. A symmetric Nash equilibrium exists, if the wel-

are function W i ( k i , k j ) is continuous in both k i and k j and strictly

uasi-concave in k i . Continuity is guaranteed in our setting, be-

ause all components of W i are continuous functions of k i and k j .

efore discussing the second-order condition in our model, it is

ave banks the option to introduce a model-based approach for calculating risk

eights, significantly increased the loan market share of the largest banks in Ger-

any, at the expense of their smaller competitors. For a theoretical study of com-

etition between banks of different size under the Basel II rules, see Hakenes and

chnabel (2011) .

188 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

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important to determine the sign of κ in the symmetric (candidate)

Nash equilibrium. We show in Appendix A.4 [ Eq. (A.15) ] that κ is

monotonously falling in the capital standard k i , and only κ < 0 is

consistent with the first-order condition for a Nash equilibrium.

Hence, in any Nash equilibrium, the cost effect of higher capital

standards must dominate the selection effect .

It remains to discuss the sufficient second-order condition

∂ 2 W i /∂k 2 i

< 0 and the uniqueness of the symmetric Nash equilib-

rium. The first-order condition ∂ W i /∂ k i = 0 [ Eqs. (19)–(22) ] is too

complex to derive the second-order condition in full. The second-

order condition becomes analytically tractable, however, when we

ignore the second derivatives of y i ( k i , k j ) and ˆ q i (k i , k j ) , thus treat-

ing ∂ y i / ∂ k i and ∂ q i /∂ k i as constants. With this simplifying as-

sumption, Appendix A.5 derives the second-order condition and

discusses the conditions under which it is negative. It then ana-

lyzes the conditions under which the symmetric equilibrium is also

unique, and it discusses why uniqueness can generally be expected

in our model. These expectations are confirmed by a systematic

numerical evaluation of our model. The numerical examples have,

in all cases, led to a unique, symmetric equilibrium.

In the following we assume that the sufficient second-order

condition for the symmetric Nash equilibrium and the condition

for uniqueness are indeed met. When W i is continuous and strictly

quasi-concave in k i the set of conditions (23) , which is sufficient

for ∂ W/∂ k i | k =0 > 0 , implies that the symmetric Nash equilibrium

has strictly positive capital requirements k ∗i

= k ∗j > 0 . If the suffi-

cient conditions for ∂ W/∂ k i | k =1 < 0 discussed above are also met,

then we further know that the symmetric Nash equilibrium will be

interior, i.e. k ∗i

= k ∗j < 1 . We summarize our results in:

Proposition 2. ( i ) When conditions (23) are met, a symmetric Nash

equilibrium with strictly positive levels of capital standards k ∗i

= k ∗j >

0 exists for all levels of α, β , γ ≥ 0 . ( ii ) The sign of κ in (18) is neg-

ative in the symmetric equilibrium, and the cost effect dominates the

selection effect.

Proof. See Appendix A.4 . �

By Proposition 2( ii ), tighter capital regulation in one country

shifts aggregate loan supply to the other country in the Nash equi-

librium. In this respect, the implications of our model do therefore

not contradict the results in the previous literature. However, even

if tighter capital regulation raises banking sector profits abroad,

this need not lead to a ‘race to the bottom’ in capital standards

when governments simultaneously care about consumer-taxpayers.

This is the issue to which we turn now.

4. ‘Race to the bottom’ or ‘race to the top’?

In the previous section, we have studied the properties of the

Nash equilibrium in capital standards in our model. We are now

ready to address the core issue of our analysis and study the ef-

ficiency properties of this decentralized policy equilibrium. Since

countries are symmetric in our benchmark model, we can w.l.o.g.

define regional welfare as the sum of national welfare levels

W

= W i + W j ∀ i, j ∈ { 1 , 2 } , i = j, (25)

where W i is given in Eq. (11) . Choosing k i so as to maximize ag-

gregate welfare, Eq. (25) would imply ∂ W W

/∂ k i = 0 . The nationally

optimal capital standards derived in the previous section are in-

stead chosen so that ∂ W i /∂ k i = 0 . Hence, any divergence between

nationally and globally optimal capital requirements is shown by

the effect of country i ’s policy variable k i on the welfare of coun-

try j ( j = i ). If ∂ W j / ∂ k i > 0, then the capital requirements chosen at

the national level are ‘too lax’ from an aggregate welfare perspec-

ive, as an increase in k i would generate a positive net externality

n the foreign country’s welfare. The reverse holds if ∂ W j / ∂ k i < 0.

n this case the overall externality on the foreign country is nega-

ive and nationally chosen capital requirements are ‘too strict’ from

n overall welfare perspective.

Differentiating W j with respect to k i gives (see Appendix A.6 ):

∂W j

∂k i = α

∂� j

∂k i + β

∂T j

∂k i + γ

∂S j

∂k i =

−κy j (1 − ˆ q )

2�(φ +

ˆ q c )

⇒ sign

(∂W j

∂k i

)= sign ( ) ,

= (α − γ ) φ︸ ︷︷ ︸ (1)

− 3 γ ˆ q c ︸ ︷︷ ︸ (2)

− β(1 − k j )(1 + 2

q )

(2 +

ˆ q ) ︸ ︷︷ ︸ (3)

. (26)

rom Proposition 2(ii), κ < 0 must hold in the Nash equilibrium.

ence the sign of ∂ W j / ∂ k i equals the sign of in (26) . The sign of

is in turn determined by the sum of three terms, which are all

ssociated with the reduction in country i ’s aggregate loan supply

ollowing a rise in k i .

The first term in isolates the cost effect of higher capital stan-

ards k i . The higher cost of capital for country i ’s banks improves

he competitive position of country j ’s banking sector and causes

ggregate banking sector profits in j to rise, due to the higher equi-

ibrium loan rate. At the same time, however, total expected output

alls and this loss is transmitted to consumers in country j through

he integrated output market. Recall, moreover, that changes in the

quilibrium loan rate R are directly tied to changes in the con-

umer price P by the zero profit condition of competitive firms

Eq. (8) ]. For this isolated effect, the rise in the profits of coun-

ry j ’s banking sector is therefore just equal to the loss in con-

umer surplus for j ’s residents. Hence, if bank profits and consumer

elfare are weighed equally in the government’s objective function

α = γ ), this first term equals zero.

The second term in is unambiguously negative. This term

solates the selection effect of higher capital standards, which leads

o a divergence in the loan rates for banks in countries i and j .

herefore, a rise in the loan rate R i caused by this isolated effect

oes not simultaneously increase R j , and therefore does not ben-

fit country j ’s banking sector. However, consumers in both coun-

ries are still hurt by the increase in country i ’s loan rate, and the

esulting fall in the equilibrium loan volume.

Finally, the third effect in is also unambiguously negative.

his effect gives the change in expected tax subsidies that taxpay-

rs in country j have to pay for their failing banks. These tax sub-

idies will unambiguously increase, because the aggregate level of

ank loans rises in country j [see Eq. (16) ]. Moreover, the average

ailure probability also rises in country j ’s banking sector, due to

he entry of low-quality banks [ Eq. (15) ].

Summing up, we see that tighter capital regulation in coun-

ry i will, on net, cause a negative externality on country j when-

ver consumer welfare is weighed at least as high as bank prof-

ts ( γ ≥α). Capital standards will then be ‘too strict’ in the non-

ooperative regulatory equilibrium. Moreover, we can directly in-

er from (26) that the negative externality of capital standards is

igher, the higher is each country’s welfare weight on tax revenues

β), and the larger is the selection effect of capital standards (i.e.,

he higher are entry costs c ). This is summarized in our main re-

ult:

roposition 3. When governments weigh consumer welfare at least

s high as bank profits ( γ ≥α), then non-cooperatively set capital

tandards exceed those that maximize aggregate welfare in the union

nd a ‘race to the top’ in capital standards occurs. This ‘race to the

op’ is more pronounced, if ( i ) the valuation of taxpayers’ losses in the

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 189

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overnment’s objective function is large ( β is high), and ( ii ) if the ‘se-

ection effect’ of capital standards is strong (firms’ entry costs c are

arge).

Proposition 3 is in direct contrast to the results in the exist-

ng literature, which have found that the non-cooperative setting

f capital standards leads to a ‘race to the bottom’, or to a ‘com-

etition of laxity’ (see Sinn, 2003; Acharya, 2003; Dell’Ariccia and

arquez, 2006 ). 18 Effectively, these contributions have focused on

he effect that capital requirements have on the profits of national

anking sectors. The same effect is also present in our analysis,

nd it corresponds to the positive component (weighed by α) in

he first term in in Eq. (26) . However, our model adds two new

ffects to this analysis that reverse the direction of the net exter-

ality in equilibrium.

First, bank loans produce real output in our model, and the out-

ut markets of the two countries are integrated. Changes in the

verall availability of credit in country i thus affect consumer sur-

lus in both countries. Therefore, while banks in country j benefit

rom a tighter capital regulation in country i , consumers in coun-

ry j simultaneously lose. Moreover, as we have discussed above,

he loss in consumer surplus will be larger than the gain in bank

rofits when banks are heterogeneous and a loan premium exists

or a better pool quality of banks ( selection effect ).

The importance of this international transmission mechanism

as been clearly shown in the recent financial crisis. Since 2007,

orldwide cross-border lending by banks has fallen steeply, and

ore strongly than domestic lending. This development, known as

retrenchment’, has been particularly pronounced in Europe (see

he Economist, 2012 ). Empirical studies have shown that this fall

n cross-border lending has, at least in part, been caused by tighter

egulation ( Buch et al., 2014; Bremus and Fratzscher, 2015 ). More-

ver, there is recent empirical evidence showing that the decline

n cross-border lending has raised the borrowing costs of Euro-

ean firms and has thus been transmitted to the real economy

Bremus and Neugebauer, 2018 ). For the integrated European mar-

et, in particular, there is thus substantial empirical evidence sup-

orting our result that capital regulation in one country imposes

egative externalities on consumers in neighboring jurisdictions.

Secondly, we incorporate expected tax revenue losses in our

odel, which result from existing deposit insurance schemes when

anks’ loans default. Capital regulation in one country increases

axpayer risks in the foreign country, because foreign banks will

ncrease their aggregate volume of lending in equilibrium. Bank

eterogeneity adds a further effect because lower-quality banks are

rawn into the foreign banking sector, thus increasing the average

efault risk of banks there. In sum, our model shows that higher

apital standards can be used to shift risks from domestic to for-

ign banks and thus, via the national deposit insurance funds, from

omestic to foreign taxpayers. 19

18 Morrison and White (2009) find a ‘race to the top’ in one of their extensions on

egulatory learning (section VI.E). In their model, the ‘race to the top’ arises because

he integration of national economies facilitates (by way of learning from other reg-

lators) and incentivizes effort s to improve the quality of national regulation. In this

etup the governments’ choice variables are national screening technologies, how-

ver, rather than capital ratios on which we focus here. Moreover, Morrison and

hite (2009) use the term ‘race to the top’ in a positive way, describing the in-

rease in regulatory quality from a closed economy benchmark. Instead we employ

he term ‘race to the top’ normatively, and use the globally efficient level as the

asis for comparison. 19 Note the important difference to the ‘financial stability’ argument that

ell’Ariccia and Marquez (2006) introduce in the government’s objective function to

erive positive equilibrium levels of capital regulation. In their model, tighter capi-

al requirements in country i increase financial stability in this country, but have no

dverse effects on financial stability in country j . In contrast, in our model the re-

uced risks for taxpayers in country i are associated with higher risks for taxpayers

n country j , due to the changed equilibrium in the international loan market.

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This last effect also explains the difference in results to the tax

ompetition literature, which almost universally finds a ‘race to the

ottom’ with respect to capital taxes (see Keen and Konrad, 2013 ,

or a synthesis). In this literature, productive firms typically make

eterministic profits and thus represent a source of positive tax

evenue for national governments. Therefore, when higher taxes in

ne country cause firms to move abroad, the resulting increase in

ts tax base represents a positive externality for the foreign coun-

ry. With capital regulation of banks and a tax-backed deposit in-

urance scheme, this externality is reversed in sign: since the tax

n banks is negative in expected value, a stricter capital regula-

ion in country i that increases the tax base in country j imposes a

egative externality on this country’s taxpayers. 20

The shifting of taxpayer risks is explicitly mentioned in the Eu-

opean Commission’s explanatory memorandum motivating why

U member states are not permitted to set national capital stan-

ards above the internationally coordinated Basel III standards:

Inappropriate and uncoordinated stricter requirements in indi-

idual Member States might result in shifting the underlying ex-

osures and risks (...) from one EU Member State to another”

European Commission, 2011 , p. 10). By showing that capital reg-

lation may impose negative externalities on foreign countries, on

et, the results of our model lend support to the EU’s policy of

armonizing the upper bound of national capital standards at the

evel of the Basel III agreement.

. Discussion and robustness

In this section, we first discuss the role of different welfare

eights for our main result in Proposition 3 (Section 5.1). The fol-

owing sections then analyze the robustness of our main result

ith respect to two changes in assumptions. In Section 5.2 we

sk which additional effects arise when banks in each coun-

ry are partly owned by residents of the other country. Finally,

ection 5.3 considers the case where banks are able to perfectly

ignal their quality to firms, and national capital requirement k i ccordingly lose the signalling role they have in our benchmark

odel.

.1. The role of different welfare weights

We first discuss the robustness of our main result by changing

he welfare weights in the government’s objective function accord-

ng to different criteria discussed in the regulatory as well as the

ax literature.

In the regulatory literature, there has been a long debate about

hether regulatory policies should be based on a total welfare

tandard (i.e., the sum of producer and consumer surplus), or on

consumer surplus standard (see Farrell and Katz, 2006 , for an

verview). While this literature has typically focused on antitrust

ssues, it is nevertheless instructive to apply it to our setting of

apital regulation. A total welfare standard implies that producer

urplus (or profits) and consumer surplus are weighed equally. This

orresponds to setting the weights α = γ in our analysis. As stated

n Proposition 3 , the ‘race to the top’ result will hold under this

ondition, for any positive valuation of tax revenues (i.e., for any

≥ 0).

If a consumer welfare standard is used instead, it implies to put

higher weight on consumer welfare, as compared to profits, or

ven a zero weight on profits. The rationale for differential weights

20 This result would be modified, if we allowed for a positive tax on bank profits

n case of success. It is questionable, however, whether bank profit taxes paid in

good times’ overcompensate the implicit and explicit bailout costs for banks arising

n times of crisis. See e.g. Admati and Hellwig (2013) for a pronounced argument

hat banks represent a net liability for national taxpayers.

190 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

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on consumer and producer surplus is usually based on the argu-

ment that producers are more powerful to advance their interests

in the political process as compared to consumers. Hence, adher-

ing to a consumer welfare standard acts as a corrective response

( Laffont and Tirole, 1991; Neven and Röller, 2005 ). In our setting,

this implies γ > α or even α = 0 , if a pure consumer surplus stan-

dard is used. It is then obvious from Eq. (26) and Proposition 3 that

our ‘race to the top’ result will emerge a fortiori .

So far, we have not discussed the weight of tax revenues β in

our government objective. The tax literature distinguishes between

settings where lump-sum taxes are available, and those where

tax revenues have to be raised by means of distortionary tax in-

struments. In the first case, which often serves as a theoretical

benchmark, there is no reason to weigh tax revenues differently

from consumer or producer surplus. However, real-world tax sys-

tems generally do not include lump-sum tax instruments. In this

case the (expected) losses to taxpayers must be weighed by the

marginal cost of public funds (MCF). This concept measures the

loss in private purchasing power needed to collect one unit of tax

revenue, and it typically exceeds unity. 21 This implies an increase

in the welfare weight β , relative to both consumer and producer

surplus. Since the externality working through higher expected tax

losses is always negative in our model, a higher marginal cost of

public funds (a higher level of β) always implies that the ‘race to

the top’ result is strengthened [cf. Proposition 3( i )].

We can also ask how our results would be changed, if we ex-

clude the consumer surplus effect from our model, thus setting

γ = 0 . This could be relevant, for example, when goods markets

are less integrated than we assume in this model, and hence re-

duced lending volumes of banks in country i have only minor ef-

fects on the real activity in country j . From the above discussion, it

is obvious that the ‘race to the top’ result can survive even in this

case. First, when a pure consumer welfare standard is assumed,

then α = 0 would hold along with γ = 0 . Hence, only the negative

tax revenue externality remains and, for any β > 0, a higher capi-

tal requirement in one country would still hurt foreign consumer-

taxpayers. Second, even if a total welfare standard is postulated

and profits enter national welfare functions with a positive weight

( α > 0), this positive externality can still be outweighed by the

negative externality on foreign tax revenues (weighed by β). As

discussed above, this occurs when β is sufficiently high, due to a

high marginal cost of public funds.

5.2. Foreign ownership of banks

It is straightforward to extend our analysis to the case where

residents in each country own a fraction of the banks in the neigh-

boring country and hence participate in the profits of the foreign

banking sector. International cross-ownership of banks is an em-

pirically important phenomenon. 22 To maintain symmetry, let res-

idents of each country own a share σ of their own resident banks,

and a share (1 − σ ) of the foreign banks. The welfare function of

country j then changes to W j = α[σ� j + (1 − σ )�i

]+ βT j + γ S j .

21 Barrios et al. (2013) have recently estimated the marginal cost of public funds

(MCF) for the EU countries to be 1.90 in the EU average, if a labour tax is used

as the marginal source of tax revenue. Their country estimates range from an MCF

of around 1.4 in the Baltic countries to MCFs around 2.3 in Denmark, France and

Sweden. 22 To give two examples, foreigners held 47% of the largest German commercial

bank, the Deutsche Bank, in 2017 ( http://www.db.com/ir/de/shareholder-structure.

htm ). The share ownership of the French BNP Paribas included 32% non-

European institutional investors in 2017, and the French ownership share is in-

tegrated with that of other European investors ( https://invest.bnpparibas.com/en/

share-ownership ).

c

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ifferentiation with respect to k i yields

∂W j

∂k i = ασ

∂� j

∂k i + α(1 − σ )

∂�i

∂k i + β

∂T j

∂k i + γ

∂S j

∂k i . (27)

n comparison to the previous section [ Eq. (26) ], two changes oc-

ur in the analysis of dW j / dk i . First, the positive effect of k i on the

rofits of the banking sector in country j , �j , is now weighed with

factor σ < 1 and is thus diminished. Secondly, through their par-

ial ownership of banks in country i , residents of country j are now

lso affected by changes in country i ’s banking sector profits. The

ffect of an increase in k i on aggregate profits in country i ’s bank-

ng sector is ambiguous, in general, due to counteracting effects of

he reduction in the aggregate loan volume and the concentration

f loans among the more profitable banks [see Eq. (20) ]. When the

ost effect of capital standards is sufficiently larger than the se-

ection effect, i.e., ρ is large relative to c [cf. condition (24) ], aggre-

ate profits will fall due to the reduced overall loan volume. In this

ase, the additional externalities on the foreign country introduced

y foreign bank ownership are therefore unambiguously negative.

his negative externality caused by foreign firm ownership is well

nown from the tax competition literature as a ‘tax-the-foreigner

ffect’. 23 Using the results from Proposition 3 , we can then sum-

arize:

roposition 4. If aggregate bank profits in country i fall after an in-

rease in k i [ ∂ �i / ∂ k i < 0 in Eq. (20) ], then foreign bank ownership

dds a negative externality of country i’s capital standards on the wel-

are of country j. When governments weigh consumer welfare at least

s high as bank profits, and Proposition 3 holds, the ‘race to the top’

s therefore intensified.

.3. Quality signalling by banks

In our main model, we have assumed that banks cannot signal

heir individual quality to firms, due to the opaqueness of their

alance sheet information in periods of ‘crisis’ (cf. Section 2.1 ). As

result, capital standards set by the banks’ host countries served

s a second-best instrument to signal the average quality of the

ountry’s banking sector. In this section, we assume instead that

anks are able to perfectly signal their individual quality to firms.

n other words, we now consider the effects of capital regulation in

normal’ times. The core issue is then whether a ‘race to the top’

till results in the Nash equilibrium, given that capital regulation

oses its role as an (imperfect) quality signal to firms.

The main difference to our benchmark analysis is that, with

erfect signalling by banks, the loan rate that firms are willing to

ay depends on the quality of an individual bank, rather than on

he average quality of a country’s entire banking sector. Hence the

ero-profit condition for firms changes from (8) in the benchmark

odel to

[ P − R (q )] = c ⇒ R (q ) = P − c

q . (28)

q. (28) implies that the equilibrium loan rate R ( q ) is rising in the

uality q of the bank issuing the loan, due to the free market entry

f identical and risk-neutral firms. Characterizing variables in this

hanged model setup by a tilde symbol, the optimal bank size for

bank of quality q [cf. Eq. (2) ] then becomes

˜

∗ =

q φi − (k i ρ + c)

b , ˜ φi ≡ P − (1 − k i ) = A − y − (1 − k i ) .

(29)

omparing (29) to (9) in the benchmark model shows that the cut-

ff quality ˆ q no longer affects the net return per unit of loans,

23 See Huizinga and Nielsen (1997) for a theoretical derivation of this effect and

uizinga and Nicodème (2006) for an empirical quantification.

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 191

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˜ i . This feature significantly simplifies the analysis, relative to the

enchmark model. The comparative static effects of a change in

ountry i ’s capital ratio k i [ Eqs. (15) and (16) in the benchmark

odel] are derived in Appendix A.7 and are given by:

∂ q i ∂k i

=

(ρ − ˆ q )[6 b + (1 − ˆ q 3 )] + ρ(1 − ˆ q )(3 − ˆ q − ˆ q 2 − ˆ q 3 )

˜ φ ˜ �> 0 ,

∂ q j

∂k i =

ˆ q μ

˜ φ ˜ �< 0 , (30)

∂y i ∂k i

=

μ[6 b + 2(1 − ˆ q 3 )]

6 b �< 0 ,

∂y j

∂k i =

−2 μ(1 − ˆ q 3 )

6 b �> 0 ,

∂y

∂k i =

μ

˜ �< 0 , (31)

μ = −(1 − ˆ q )[3(ρ − 1)(1 +

ˆ q ) + 1 +

ˆ q − 2

q 2 ] < 0 ,

˜ = 6 b + 4(1 − ˆ q 3 ) > 0 . (32)

n contrast to the term κ in the benchmark analysis [cf. Eq. (18) ],

he corresponding term μ in this section is always negative; hence,

he responses of ˆ q i and y i to changes in the capital ratio k i can all

e signed unambiguously. An increase in k i always increases the

uality of the domestic cutoff firm ˆ q i and it reduces the aggregate

olume of performing loans, y i . This reduces the supply of loans in

he integrated market and raises the loan rate for firms from coun-

ry j . Hence, y j increases, whereas ˆ q j falls. Finally, the total num-

er of performing loans y = y i + y j will always fall, in equilibrium,

hen one country increases its capital requirement.

The welfare effects caused by the induced changes in y i and ˆ q i re structurally the same as in our benchmark model [ Eqs. (20) –

22) ]. Substituting (30) and (31) into these expressions and simpli-

ying terms gives

∂ ˜ W j

∂k i =

−μy j ˜ φ ˜ �

⎡

⎢ ⎢ ⎣

(α − γ ) φ︸ ︷︷ ︸ (1)

− β(1 − k j )(1 + 2

q )

(2 +

ˆ q ) ︸ ︷︷ ︸ (2)

⎤

⎥ ⎥ ⎦

. (33)

omparing these welfare effects with those in our benchmark

odel [ Eq. (26) ] shows that the negative second effect in (26) is

issing in (33) . This is because the selection effect of higher cap-

tal requirements in country i disappears when signalling occurs

hrough individual banks, rather than through regulatory policies.

owever, the remaining negative effects of higher capital require-

ents in country i on consumer-taxpayers in country j remain un-

ffected by this change. In particular, consumers in country j are

till hurt by the reduced aggregate volume of performing loans,

hich reduces consumer surplus in the integrated market [the

egative part of the first effect in (33) ]. Moreover, country j ’s tax-

ayers continue to be hurt by the higher volume of loans made by

ountry j ’s banks, and by the lower average bank quality in coun-

ry j [the negative second effect in (33) ].

In sum, the incentives for a ‘race to the top’ are weaker under

his modification (and, therefore, in ‘normal’ periods), as compared

o our benchmark model (in periods of ‘crisis’). However, under

he conditions stated in Proposition 3 , a ‘race to the top’ in cap-

tal standards remains the equilibrium outcome, even if banks can

ignal their individual quality to firms.

. Conclusions

This paper has studied international competition in capital

tandards in a symmetric two-country model where banks differ

xogenously in their quality, and hence in the likelihood that their

oans will succeed. In this setting national capital standards act

s a positive signal for the pool quality of banks in the regulat-

ng country and imply higher financing costs, but also higher loan

ates for the resident banks. In the Nash equilibrium, the higher

ost of capital must dominate, implying that each country’s cap-

tal standard imposes a positive externality on the foreign coun-

ry’s banking sector. At the same time, however, capital standards

hift taxpayer risks from the more regulated to the less regulated

ountry and they also reduce consumer surplus in the neighboring

ountry by lowering the availability of credit. These negative exter-

alities on the foreign country will dominate when national gov-

rnments weigh all components of national welfare equally, imply-

ng that the non-cooperative setting of capital standards leads to

‘race to the top’. Therefore, broadening the welfare objective of

overnments reverses the ‘race to the bottom’ result derived in the

xisting literature.

Our model can thus explain why countries such as Switzerland

r the United States, which are characterized by large banking sec-

ors and accordingly a high risk exposure of national taxpayers, in-

roduce capital adequacy rules that exceed internationally coordi-

ated standards. At the same time, our model offers a motivation

or why the European Union has insisted on a strict harmoniza-

ion of capital standards among its member states at the levels

greed upon in the Basel III accord. The consumer surplus exter-

ality that arises from capital standards in our model provides an

rgument for why the imposition of an upper bound on national

apital standards is especially relevant in an integrated market like

he European Union.

Our model can be extended in several ways. A first relevant

xtension is to introduce an endogenous monitoring decision of

anks while maintaining heterogeneity in monitoring costs. In this

etting capital standards would play a further role in reducing

oral hazard in the banking sector, in addition to their roles of sig-

alling the quality of the national banking sector and of protecting

ational taxpayers, on which the present analysis has focused. An-

ther relevant extension would be to introduce imperfect compe-

ition in the banking sector. In such a setting, the effects of capital

tandards on bank profits would likely gain increased prominence.

owever, the negative externalities on foreign consumers and tax-

ayers will continue to exist. Therefore, a ‘race to the top’ is likely

o remain a possible outcome in such a model, though possibly

nder more restrictive conditions. We leave these and other issues

or future work.

ppendix

.1. Derivation of Eqs. (15) and (16)

To analyze the effects of an increase in k i on aggregate output

nd the cutoff qualities ˆ q i in the two countries, we totally differen-

iate the equation system (10a)–(10c) to get

A − y − 1 + k i − 2

c ]d q i =

ˆ q i dy + (ρ − ˆ q i ) dk i , (A.1)

A − y − 1 + k j − 2

c ]d q j =

ˆ q j dy, (A.2)

y =

3(1 − ˆ q 3 ) c

[3 b + 2(1 − ˆ q 3 )](2 +

ˆ q ) 2 (d q i + d q j )

− [3 ρ(1 − ˆ q 2 ) − 2(1 − ˆ q 3 )]

6 b + 4(1 − ˆ q 3 ) (d k i + d k j ) , (A.3)

here we have used the short-hand notations for φ [ Eq. (3) ] and

ˆ [ Eq. (17) ], Eq. (5) has been used to simplify terms, and (A.3) has

sed symmetry after differentiation.

192 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

A

H

i

�

R

q

�

�

A

�

t

e

a

H

�

E

s

T

q

i

a

A

6

ε

F

u

a

E

m

s

k

[

This equation system can be simplified by substitut-

ing (A.3) into (A.1) and (A.2) . This yields the two-equation

system {( q c + φ)[6 b + 4(1 − ˆ q 3 )] − 2

q (1 − ˆ q 3 ) c }

dq i = 2

q c (1 − ˆ q 3 ) dq j

+

{(ρ − ˆ q )[6 b + 4(1 − ˆ q 3 )] − [3 ρ ˆ q (1 − ˆ q 2 ) − 2

q (1 − ˆ q 3 )] }

dk i

(A.4){( q c + φ)[6 b + 4(1 − ˆ q 3 )] − 2

q (1 − ˆ q 3 ) c }

dq j

= 2

q c (1 − ˆ q 3 ) dq i − ˆ q [3 ρ ˆ q (1 − ˆ q 2 ) − 2

q (1 − ˆ q 3 )] dk i (A.5)

Solving the system (A.4) and (A.5) gives (15) in the main text. Sub-

stituting these results back into (A.3) yields

∂y

∂k i =

(1 − ˆ q ) κ

2 φ�, (A.6)

where � and κ are given in (17) and (18) . Finally, differentiat-

ing (7) gives

dy i =

1

6 b

{−2(1 − ˆ q 3 i ) dy + 2(1 − ˆ q 3 i ) c d q i

− [3 ρ(1 − ˆ q 2 i ) − 2(1 − ˆ q 3 i )] dk i }

(A.7)

Substituting (15) along with (A.6) into (A.7) gives (16) in the main

text.

A.2. Derivation of conditions (23)

From (20) and (22) and using (16) , a positive effect of capital

standards on bank profits and consumer surplus, evaluated at k =0 initially, requires that κ > 0 in (18) . Evaluating κ at k = 0 and

noting that ˆ q = 0 for k = 0 from (5) , this condition is

κ| k =0 =

3 ρc

2

− (R i − 1)(3 ρ − 2) > 0 . (A.8)

The endogenous variable (R i − 1) can be substituted using (9) to-

gether with (6) and (7) . This yields

(R i − 1) | k =0 =

3 b

(3 b + 2)

(A − 3 c

2

− 1

). (A.9)

Substituting (A.9) in (A.8) , the condition for κ| k =0 > 0 is

3

2

ρc − (3 ρ − 2)3 b

3 b + 2

[ A − 3 c

2

− 1

] > 0 .

Collecting the terms for c gives the first condition in (23) .

A positive effect on taxpayers results when the positive first

two effects in (21) dominate the third effect, which is negative for

κ > 0. Substituting in from (15) and (16) , evaluating at k = ˆ q = 0

and using y | k =0 = (R i − 1) / 3 b from (6) and (7) gives

∂T i ∂k i

∣∣∣∣k =0

=

(R i − 1)

6 b +

3 ρ

12 b − κ

12 bφ> 0 .

Ignoring the positive first term and noting that φ| k =0 =(R i − 1) | k =0 gives, as a sufficient condition

∂T i ∂k i

∣∣∣∣k =0

> 0 ⇔ 3 ρ(R i − 1) − κ > 0 . (A.10)

Using (A.8) and (A.9) yields

∂T i ∂k i

∣∣∣∣k =0

> 0 ⇔

12 b(2 A − 3 c − 2)(3 ρ − 1)

(3 b + 2) >

3 ρc

2

. (A.11)

Noting that (3 ρ − 1) ≥ 2 ρ and collecting terms gives the second

condition in (23) .

.3. Derivation of condition (24)

Substituting (15)–(17) into (20) gives in a first step

∂�i

∂k i =

6 by i �

(1 − ˆ q )(2 +

ˆ q ) 2 (φ +

ˆ c q )�,

� ≡ 3 y i q [(ρ− ˆ q ) + ρ(φ+

c q )(1 − ˆ q ) 2 (2 +

q )]

2(1 − ˆ q )(2 +

ˆ q ) +

(1 − ˆ q )κ

6 b .

(A.12)

ence the sign of ∂ �i / ∂ k i equals the sign of �. Replacing κ us-

ng (18) and rearranging gives

= {(ρ − 1)[18 y i q b − 6 φ(2 +

ˆ q )(1 − ˆ q ) 2 (1 +

ˆ q )]

+ (1 − ˆ q )[18 y i q b − 2 φ(1 − ˆ q ) 2 (2 +

ˆ q )(1 + 2

q )

+ 6 cρ(1 − ˆ q ) 3 ] }

+ 18 y i q bρ(φ +

ˆ c q )(1 − ˆ q ) 2 (2 +

ˆ q ) .

ewriting the last term using in (17) and 6 by i = φ(1 − ˆ q ) 2 (2 +ˆ ) from (6) and (7) yields

= (1 − ˆ q ) 2 (2 +

ˆ q )�1 − 6 y i q ρφ(1 − ˆ q ) 2 (2 +

ˆ q ) 2 (1 − ˆ q 3 ) ,

1 ≡ −φ

[3(ρ − 1) + (1 − ˆ q ) − ˆ q ρ(1 − ˆ q ) 3

2 b

]+ 6 cρ

(1 − ˆ q ) 2

(2 +

ˆ q ) 2 .

(A.13)

sufficient condition for � < 0 is that �1 < 0, as the last term in

is negative. A sufficient condition for �1 < 0 is obtained by set-

ing the term (1 − ˆ q ) in the squared bracket equal to zero and by

valuating the negative last term in the bracket at its maximum in

bsolute value. The latter is ρ/8 b , which is obtained for ˆ q = 1 / 4 .

ence a sufficient condition for �1 < 0 is

1 < 0 ⇐⇒ φ[

3(ρ − 1) − ρ

8 b

] > 6 cρ

(1 − ˆ q ) 2

(2 +

ˆ q ) 2 . (A.14)

valuating (A.14) at k i = 1 gives φi = R i from (3) on the left-hand

ide of the inequality, and ρ = ˆ q i R i from (5) on the right-hand side.

his allows to cancel R i . Finally the maximum of the term ˆ q (1 −ˆ ) 2 / (2 + ˆ q ) 2 on the right-hand side of the inequality is 1/36, which

s reached for ˆ q = 1 / 4 . This gives condition (24) in the main text as

sufficient condition for ∂ �i /∂ k i | k =1 < 0 .

.4. Proof of Proposition 2(ii)

Differentiating κ in (18) with respect to k i and using φ = by i / [(1 − ˆ q ) 2 (2 + ˆ q )] from (6) and (7) gives:

dκ

dk i = ε

∂ q i ∂k i

− 6 b[3(ρ − 1)(1 +

ˆ q ) + (1 + 2

q )(1 − ˆ q )]

(1 − ˆ q ) 2 (2 +

ˆ q )

∂y i ∂k i

,

(A.15)

=

−9 ρc

(2 +

ˆ q 2 ) − 6 by

(1 − ˆ q ) 2 (2 +

ˆ q ) 2

{3(ρ − 1)[5(1 +

ˆ q ) + 2

q 2 ]

+ (1 − ˆ q )(5 + 2

q + 2

q 2 ) }

< 0 .

rom the positive effect of k i on ˆ q i in (15) the first term in (A.15) is

nambiguously negative. Moreover, the second term in (A.15) is

lso negative when κ > 0 initially and hence dy i / dk i > 0 [see

q. (16) ]. Therefore, as long as the value of κ is non-negative, κust be unambiguously falling in k i .

To see that κ < 0 must hold in the Nash equilibrium, start by

etting κ = 0 , with a corresponding capital standard k 0 . Starting at

0 , a marginal increase in k i has a zero effect on consumer surplus

from (22) and (16) ], but a positive effect on bank profits and tax

A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194 193

r

f

t

h

e

A

q

t

E

m

a

t

fi

t

o

o

s

s

H∣∣∣∣

W

(

a

T

t

t

a

t

e

m

m

[

r

c

e

b

t

s

i

i

A

W

D

S

q

A

q

a

i

q

q

y

2

evenues [from (20) and (21) , together with (15) and (16) ]. There-

ore, if a Nash equilibrium exists, k i must be increased from its ini-

ial level k 0 . Since the negative relationship between κ and k i also

olds at k i = k 0 , this implies that κ < 0 must be true in the Nash

quilibrium. �

.5. Second-order condition and uniqueness of equilibrium

Multiplying the first-order condition in Eqs. (19)–(22) by (2 +ˆ ) , differentiating with respect to k i under the simplifying assump-

ions that η ∈ { ∂ q i /∂ k i , ∂ y i /∂ k i } = const. and rearranging gives

∂ 2 W i

∂k 2 i

=

{[36 αby i q

(1 − ˆ q ) 2 (2 +

ˆ q ) 2 +

3 β(1 − k i )

(2 +

ˆ q )

]∂ q i ∂k i

+

12 αb

(1 − ˆ q )(2 +

ˆ q )

∂y i ∂k i ︸ ︷︷ ︸

(+)

+ 2 β(1 − ˆ q ) ︸ ︷︷ ︸ (1)

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

∂y i ∂k i

+

⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

⎡

⎢ ⎢ ⎣

18 αby 2 i (2 +

ˆ q +

ˆ q 2 )

(1 − ˆ q ) 3 (2 +

ˆ q ) 2 ︸ ︷︷ ︸ (+)

−3 βy i (1 − k i )

(2 +

ˆ q ) 2

⎤

⎥ ⎥ ⎦

∂ q i ∂k i

− βy i

(1 +

3

(2 +

ˆ q )

)︸ ︷︷ ︸

(2)

+

γ

2

∂y

∂k i

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

∂ q i ∂k i

+

[12 αby i (1 + 2

q )

(1 − ˆ q ) 2 (2 +

ˆ q ) 2 + β(1 − k i )

]∂y i ∂k i

∂ q i ∂k i

+

γ (2 +

ˆ q )

2

(∂y

∂k i

)2

︸ ︷︷ ︸ (+)

. (A.16)

valuating (A.16) in a Nash equilibrium with κ < 0 implies that the

ultipliers for the terms can be signed by ∂ y i / ∂ k i < 0, ∂ q i /∂ k i > 0

nd ∂ y / ∂ k i < 0. Hence, all terms in (A.16) are negative, except for

he three terms marked by a ( + ) symbol. Hence, in this simpli-

ed version, the second-order condition will be fulfilled, if these

erms are dominated by the remaining, negative terms. Since none

f the positive terms involves a welfare weight of β , the second-

rder condition will be met, if the level of β is sufficiently high.

Uniqueness of the Nash equilibrium is guaranteed when the

lope of the best response function does not exceed one in ab-

olute value for any capital standard k i (cf. Vives, 2005 , p. 441).

ence,

∂ 2 W/∂ k i ∂ k j ∂ 2 W/∂k 2

i

∣∣∣∣ < 1 ∀ k i , k j . (A.17)

e differentiate the first-order condition ∂ W i /∂ k i = 0 [ Eqs. (19)–

22) ] with respect to country j ’s capital ratio k j , with analogous

ssumptions as made above. This gives

∂ 2 W i

∂k i k j =

{[36 αby i q

(1 − ˆ q ) 2 (2 +

ˆ q ) 2 +

3 β(1 − k i )

(2 +

ˆ q )

]∂ q i ∂k i

+

12 αb

(1 − ˆ q )(2 +

ˆ q )

∂y i ∂k i

+ β(1 − ˆ q ) ︸ ︷︷ ︸ (1)

⎫ ⎬

⎭

∂y i ∂k j

+

{[18 αby 2

i (2 +

ˆ q +

ˆ q 2 )

(1 − ˆ q ) 3 (2 +

ˆ q ) 2 − 3 β(1 − k i ) y i

(2 +

ˆ q ) 2

]∂ q i ∂k i

− βy i ︸︷︷︸ (2)

+

γ

2

∂y

∂k i

⎫ ⎬

⎭

∂ q i ∂k j

+

[12 αby i (1 + 2

q )

(1 − ˆ q ) 2 (2 +

ˆ q ) 2 + β(1 − k i )

]∂y i ∂k i

∂ q i ∂k j

+

γ (2 +

ˆ q )

2

∂y

∂k i

∂y

∂k j . (A.18)

here are two differences between (A .16) and (A .18) . First, the mul-

ipliers for the terms are now ∂ y i / ∂ k j > 0 and ∂ q i /∂k j < 0 . Condi-

ion (A.17) is therefore more likely to be met, if | ∂ y i / ∂ k i | > | ∂ y i / ∂ k j |nd | ∂ q i /∂k i | > | ∂ q i /∂k j | , i.e., the effects of changes in k i on coun-

ry i variables are larger in absolute value than the corresponding

ffects on the variables in country j . The first of these conditions

ust necessarily be fulfilled since (∂ y i /∂ k j ) = (∂ y j /∂ k i ) from sym-

etry and ∂ y/∂ k i = (∂ y i /∂ k i ) + (∂ y j /∂ k i ) < 0 follows from κ < 0

cf. Eq. (16) ].

The second difference between (A.16) and (A.18) is that the di-

ect effects of changes in k i [see Eq. (21) ] enter the second-order

ondition (A.16) , but not the best response function (A.18) . These

ffects are incorporated in the terms marked by (1) and (2) in

oth (A.16) and (A.18) , which differ in the two equations. Since

hese direct effects are both negative, they contribute to the ab-

olute value of ∂ W

2 /∂ k 2 i , but not to the value of ∂ W

2 / ∂ k i ∂ k j . This

s a second reason for why uniqueness can generally be expected

n our model.

.6. Derivation of equation (26)

Using (12) –(14) , we can write welfare in country j as

j =

6 αby 2 j

(1 − ˆ q j )(2 +

ˆ q j ) 2 − β(1 − k j )(1 − ˆ q j ) y j

(2 +

ˆ q j )

+

γ (y i + y j ) 2

4

i = j.

ifferentiating with respect to k i gives, in a first step

∂W j

∂k i =

12 αby j

(1 − ˆ q )(2 +

ˆ q ) 2 ∂y i ∂k i

+

18 αby 2 j

q

(1 − ˆ q ) 2 (2 +

ˆ q ) 3 ∂ q j

∂k i

− β(1 − k j )(1 − ˆ q )

(2 +

ˆ q )

∂y j

∂k i +

3 β(1 − k j ) y j

(2 +

ˆ q ) 2 ∂ q j

∂k i

+

γ (y i + y j )

2

∂y

∂k i . (A.19)

ubstituting in from (15) and (16) , using φ j = 6 by j / [(1 − ˆ q ) 2 (2 +ˆ )] and collecting terms gives Eq. (26) in the main text.

.7. Derivation of equations (30) and (31)

With individual signalling by banks, the cutoff productivities

ˆ 1 and ˆ q 2 and the total volume of performing loans y = y 1 + y 2 re derived from the optimal loan volume of an individual bank

n (29) . This yields the equation set

ˆ 1 [ A − y − 1 + k 1 ] = (ρk 1 + c) ,

ˆ 2 [ A − y − 1 + k 2 ] = (ρk 2 + c) ,

=

1

b

∫ 1

ˆ q 1

[q 2 (A − y − 1 + k 1 ) − q (k 1 ρ + c)

]dq

+

1

b

∫ 1

ˆ q

[q 2 (A − y − 1 + k 2 ) − q (k 2 ρ + c)

]dq. (A.20)

194 A. Haufler and U. Maier / Journal of Banking and Finance 106 (2019) 180–194

D

D

E

F

F

H

H

H

I

I

K

K

K

L

d

M

M

N

NN

N

P

R

R

S

S

T

V

Totally differentiating (A.20) yields the equation system ⎡

⎣

1 0 0

− ˆ q ˜ φ 0

− ˆ q 0

˜ φ

⎤

⎦

[

dy d q 1 d q 2

]

=

[

μ/ �(ρ − ˆ q )

0

]

dk i . (A.21)

Solving (A.21) yields Eqs. (30) and (31) in the main text, where

∂ q i ∂k i

=

(ρ − ˆ q ) � +

ˆ q μ

˜ φ ˜ �

is an intermediate step.

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