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Journal of Banking and Finance 106 (2019) 180–194
Contents lists available at ScienceDirect
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
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.”
q
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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-
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
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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
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B
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
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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
W
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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.
o
i
c
s
c
i
c
β
a
e
i
‘
i
t
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-
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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