Finance and Economics Discussion Series

Divisions of Research & Statistics and Monetary Aairs

Federal Reserve Board, Washington, D.C.
Systemic Risk, International Regulation, and the Limits of
Coordination
Gazi I. Kara
2013-87
NOTE: Sta working papers in the Finance and Economics Discussion Series (FEDS) are preliminary
materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth
are those of the authors and do not indicate concurrence by other members of the research sta or the
Board of Governors. References in publications to the Finance and Economics Discussion Series (other than
acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.
Systemic Risk, International Regulation, and the Limits of
Coordination

Gazi Ishak Kara

September 16, 2013

Abstract
This paper examines the incentives of national regulators to coordinate regulatory policies
in the presence of systemic risk in global nancial markets. In a two-country and three-period
model, correlated asset re sales by banks generate systemic risk across national nancial mar-
kets. Relaxing regulatory standards in one country increases both the cost and the severity
of crises for both countries in this framework. In the absence of coordination, independent
regulators choose ineciently low levels of macro-prudential regulation. A central regulator
internalizes the systemic risk and thereby can improve the welfare of coordinating countries.
Symmetric countries always benet from coordination. Asymmetric countries choose dierent
levels of macro-prudential regulation when they act independently. Common central regulation
will voluntarily emerge only between suciently similar countries.

I am grateful to Anusha Chari, Gary Biglaiser, Peter Norman, Sergio Parreiras, Richard Froyen, Jordi Mondria,
Gunter Strobl, Gregory Brown, S. Viswanathan, Adriano Rampini, M. Ibrahim Turhan, Ahmet Faruk Aysan, Toan
Phan, Mehmet Ozsoy, John Schindler and seminar participants at the NBER International Finance and Macroe-
conomics Fall Meeting, Federal Reserve Board of Governors, Bogazici University, Central Bank of Turkey, WEAI
Conference in San Francisco, and Financial and Macroeconomic Stability Conference in Istanbul for helpful com-
ments and suggestions. All errors are mine. The analysis and the conclusions set forth are those of the author and
do not indicate concurrence by other members of the research sta or the Board of Governors.

Oce of Financial Stability Policy and Research, Board of Governors of the Federal Reserve System. email:
Gazi.I.Kara@frb.gov
1
1 Introduction
The underlying shocks that precipitated the nancial crisis of 2007-2009 quickly spread across global
nancial markets and were amplied at an unprecedented scale. The strikingly global nature of
the crisis has revived interest in the international coordination of nancial regulation. Regulatory
reforms and the strengthening of coordination between national nancial regulators are prominent
items on the international reform agenda. The Financial Stability Board (FSB) was set up by the
G-20 countries during the crisis to create guidelines for regulatory coordination and the supervision
of systemic risk in the international nancial system.
1
This paper, using a game-theoretic model, analyzes the incentives of national regulators towards
international cooperation when there is systemic risk in global nancial markets. In the model,
systemic risk in nancial markets is generated through asset re sales. The model shows that in
the absence of cooperation, independent regulators choose ineciently low regulatory standards
compared to regulation levels that would be chosen by a central regulator. A central regulator
internalizes systemic risk and improves welfare in cooperating countries. The model also demon-
strates that common central regulation will voluntarily emerge only between suciently similar
countries.
Key features of a third generation of bank regulation principles, popularly known as Basel
III, strengthen capital regulations and add new elements to Basel bank regulation principles such
as liquidity and leverage ratio requirements. With Basel III, the objective of regulation aimed at
creating a level playing eld for internationally active banks is supported by an objective of creating
sound regulatory practices that will contain systemic risk in national and international nancial
markets, and prevent pro-cyclical amplication of these risks over time. In this paper, I revisit
the issue of coordinating international nancial regulation in light of recent developments in the
international regulatory infrastructure.
Acharya (2003, 2009) and DellAriccia and Marquez (2006) are notable studies in the literature
on international nancial regulation. All three of these studies focus on the level playing eld
objective of nancial regulation and examine the benets to international coordination of nancial
regulation under externalities that operate through the competition in loan markets. This paper
diverges from the previous literature by focusing on systemic externalities across nancial markets
generated by re sales of assets. I examine the eects of systemic externalities on the nature of
international nancial regulation in the absence of cooperation between regulators, as well as its
eects on the incentives of national regulators towards cooperation. The paper also contributes
to the literature by examining the eects of structural dierences across countries on the choice
of regulatory standards when countries are linked through systemic externalities in international
1
The Financial Stability Board (FSB) was established after the 2009 G-20 London summit in April 2009; it is the
successor to the Financial Stability Forum (FSF). The FSF was founded in 1999 by the G-7 nance ministers and
central bank governors.
2
nancial markets. I show herein that common central regulation voluntarily emerges only between
suciently similar countries.
During times of distress, asset prices can move away from the fundamental values and assets
can be traded at re sale prices. When rms or nancial intermediaries face liquidity shocks, and
debt-overhang, collateral or commitment problems prevent them from borrowing or issuing new
equity they may have to sell assets to generate the required resources. If the shocks are wide
spread throughout an industry or an economy, then potentially deep-pocket outsiders will emerge
as the buyers of the assets. However, some assets are industry-specic: when they are redeployed
by outsiders, they will be less productive, and they will be sold to outsiders at a discount. This
idea, which originated in Williamson (1988) and Shleifer and Vishny (1992), was later employed
by re sales models such as Lorenzoni (2008), Gai et al. (2008), Acharya et al. (2010) and Korinek
(2011).
Industry-specic assets can be physical, or they can be portfolios of nancial intermediaries
because many of these contain exotic tailor-made nancial assets (Gai et al., 2008). The asset
specicity idea is captured in this paper through a decreasing returns to scale technology for
outsiders, similar to the ones proposed by Kiyotaki and Moore (1997), Lorenzoni (2008), Gai et al.
(2008), and Korinek (2011). The less ecient technology of outsiders makes the situation even
worse for distressed intermediaries because they have to accept higher discounts to sell more assets.
Empirical and anecdotal evidence suggests the existence of re sales of physical as well as nancial
assets.
2
These facts indicate that when numerous intermediaries concurrently face the same types of
shocks and sell assets simultaneously, asset prices can fall, which forces the intermediaries to sell
additional assets. Because an individual intermediary takes the market price as given and decides
how much of its assets to sell to continue operating at an optimal scale, each intermediary ignores the
negative externality of its asset sales on others. In a nancially integrated world, intermediaries from
dierent countries sell assets in a global market to potentially the same set of buyers. Therefore,
initial shocks that hit individual countries can be amplied in globally integrated nancial markets.
I consider this systemic externality in this paper, and also seek answers to the following ques-
tions: How do national regulators behave under this systemic externality if the regulators act
non-cooperatively? Would an individual regulator tighten or relax regulation when regulation is
tightened in another country? Would national regulators relinquish their authority to a central
international regulator who would impose the same set of regulatory standards across countries?
2
Using a large sample of commercial aircraft transactions, Pulvino (2002) shows that distressed airlines sell aircraft
at a 14% discount from the average market price. This discount exists when the airline industry is depressed but not
when it is booming. Coval and Staord (2007) show that re sales occur in equity markets when mutual funds engage
in sales of similar stocks. re sales have been shown to exist in international settings as well; for examples, surges
in foreign direct investment into emerging markets have been recorded during Asian and Latin American nancial
crises. In particular, Krugman (2000), Aguiar and Gopinath (2005), and Acharya et al. (2010) show that asset sales
to outsiders during these crises were associated with high discounts, and that many foreigners ipped the assets they
purchased to domestics once the crises abated at very high returns.
3
How do asymmetries across countries aect the nature of regulatory standards and the incentives
of national regulators towards international cooperation?
Briey, I propose a three-period, two-good model that features two countries with independent
regulators. In each country there is a continuum of banks. Banks are protected by limited liability,
and there is deposit insurance. Banks borrow consumption goods from local deposit markets, and
invest in a productive asset in the rst period.
All uncertainty in the model is resolved at the beginning of the second period and one of the
two states of the world is realized: a good or a bad state. In the good state there are no shocks
and banks investments produce net positive returns in the last period. However, in the bad state,
banks investments are distressed and they have to be restructured to produce the normal positive
returns that are obtained in good times.
A continuum of global investors with large resources in the second period can purchase produc-
tive assets in the second period to produce consumption goods in the third and nal period. Assets
in dierent countries are perfect substitutes for global investors. However, global investors are not
as productive as the domestic banks in managing domestic assets and face decreasing returns to
scale from these assets.
I solve the equilibrium of this model by backwards induction. Following the shocks in the interim
period, banks need to sell some fraction of their assets in a global capital market to pay for the
restructuring costs. An asset sale in the bad state is unavoidable because other domestic resources
required to carry out the restructuring process are unavailable. The price of the productive asset
is determined in a competitive market in which banks from the two countries and global investors
meet.
I show that a higher initial investment by banks in either of these countries will lead to a
lower price for the productive asset in this market. If the asset price falls below a minimum
threshold, return to the assets that can be retained by the banks is lower than the value of the
initial investment, and the banks become insolvent. I call this case a systemic failure. Depositors
encounter real losses when a systemic failure occurs because returns to the remaining bank assets
do not cover depositors initial investment.
Regulation in this model can be interpreted as a minimum capital ratio requirement. Each
regulator determines the initial regulatory standard by taking into account the equilibrium in the
asset market in the interim period. Due to the systemic externality discussed above, banks always
leverage up to the maximum by borrowing funds from the local deposit market. In other words,
the minimum capital ratio always binds. Therefore, the initial investment level of banks in a given
country is determined completely by the regulatory standard.
In the rst period, regulators act simultaneously and choose the regulatory standard for their
domestic banks by taking the regulatory standard in the other country as given. I show that when
the countries are symmetric, there exists a unique symmetric Nash equilibrium of the game between
4
the two regulators. Moreover, regulation levels in the two countries are strategic substitutes: if one
regulator tightens the regulatory standard in its jurisdiction, the other regulator optimally loosens
its regulatory standard. The intuition behind this result is as follows: When the rst country
reduces the maximum leverage level (i.e., tightens regulatory standards), the extent of the re sale
of assets in the bad state by banks in that country are reduced, and a higher price is realized for
the assets sold by these banks. This increases the expected returns in the bad state, which allows
the regulator in the other country to relax regulation levels.
I show that, due to this systemic risk, regulatory standards in equilibrium when regulators act
non-cooperatively will be ineciently lax compared to regulatory standards that would be chosen
by a central regulator. A central regulator aims to maximize the total welfare of the two countries,
and internalizes these externalities. I assume that, for political reasons, the central regulator has
to choose the same regulation levels in both countries. If the two countries are symmetric, I show
that forming a regulatory union will increase welfare in both. Therefore, it is incentive compatible
for the independent regulators of symmetric countries to relinquish their authority to a central
regulator.
I also consider the incentives of regulators when there are asymmetries between countries, with
a focus on the asymmetries in the asset returns. In particular, I assume that banks in one country
are uniformly more productive than the banks in the other country in terms of managing the
long-term asset. I also show that cooperation would voluntarily emerge only between suciently
similar countries. In particular, the regulator in the high-return country chooses lower regulatory
standards in equilibrium and is less willing to compromise on stricter regulatory standards.
Interest in the international coordination of nancial regulation is not an entirely recent phe-
nomenon. Arguments in favor of coordination and harmonization of regulatory policies across
countries were made in the 1988 Basel Accord (Basel I) which focused on credit risk and set mini-
mum capital requirements for internationally active banks and was enforced in the G-10 countries
in 1992.
3
However, Basel I did not create an entirely level playing eld for internationally active banks
because countries retained a signicant degree of discretion about dierent dimensions of regulation.
Furthermore, rapid developments in nancial markets, especially more complex nancial products
brought about by nancial innovation, created signicant dierences about the stringency of capital
regulations across countries in practice (Barth et al., 2008). These developments created a challenge
for regulators and paved the way for Basel II.
4
While progress on the implementation of Basel II was slower than expected, the global nancial
3
The intent of Basel I was to strengthen the soundness and stability of the international banking system and to
diminish competitive inequality among international banks by creating a level playing eld Basel (1988).
4
Basel I was updated in 2004 with more sophisticated sets of rules and principles for capital regulation that were
intended to accommodate the developments in global nancial markets. A 2006 survey by the Bank for International
Settlements (BIS) showed that 95 countries (comprising 13 BSBC member countries plus 82 non-BSBC jurisdictions)
had planned to implement Basel II by 2015.
5
crisis renewed urgency about increased cooperation and the better regulation of international -
nancial markets, in part because insucient policy coordination between countries and deciencies
in Basel II regulatory mechanisms were blamed for the severe contagion of the crisis. Most of the
international regulatory mechanisms proposed prior to the crisis had emphasized the soundness
of nancial institutions individually (micro-prudential regulation), but had neglected regulatory
standards that could enhance the stability of the nancial system as a whole by considering sys-
temic risks (macro-prudential regulation). The model in this paper focuses on macro-prudential
regulation in the context of regulating systemic risk in the international banking system.
The paper proceeds as follows. Section 2 contains a brief summary of related literature. Section
3 provides the basics of the model and presents the main results of the paper without resorting
to a particular functional form. International nancial regulation between asymmetric countries is
considered in Section 4. Section 5 investigates the robustness of the results obtained from the basic
model to some changes in the model environment. Section 6 shows the set of parameter ranges
for which systemic failures occur in the uncoordinated equilibrium when countries are symmetric.
Conclusions are presented in Section 7. All proofs are provided in the Appendix.
2 Literature Review
This paper belongs to the international nancial regulation theory that has developed in recent
decades. This paper is closest to Acharya (2003), DellAriccia and Marquez (2006), Acharya (2009),
and Bengui (2011). In particular, DellAriccia and Marquez (2006) investigate the incentives of na-
tional regulators to form a regulatory union in a two-country banking model, where a single bank
from each country competes for loans in both markets in a Bertrand dierentiated products setup.
If one of the banks is allowed to expand its balance sheet, low average returns to bank loans will
be realized in both markets. Banks in this model are also endowed with a costly monitoring tech-
nology. Low average returns reduce incentives of banks to monitor, and hence undermines their
stability. The authors show that, under this externality, independent national regulators will imple-
ment lower capital requirements compared to capital requirements that would be implemented by a
central regulator. They also show that symmetric countries always gain from cooperation, whereas
a cooperation emerges voluntarily only between suciently similar asymmetric countries. The coor-
dination problem for asymmetric countries as presented in this paper is similar to DellAriccia and
Marquez (2006). However, in that model the asymmetry between countries was due to dierences
in regulators exogenously specied tastes and preferences. In this paper I consider asymmetries
that are due to structural dierences across countries, such as dierences in asset returns.
Acharya (2003) shows that convergence in international capital adequacy standards cannot be
eective unless it is accompanied by convergence in other aspects of banking regulation, such as
closure policies. Externalities in his model are in the form of cost of investment in the risky asset.
He assumes that a bank in one country increases costs of investment for itself and for a bank in
6
the other country as it invest more in the risky asset and thereby creates externalities for the bank
in the neighboring country.
In the model considered by Acharya (2009), failure of a bank creates both negative and positive
externalities for surviving banks. Negative externality is the increase in the cost of the deposits for
surviving banks through a reduction in overall available funds. Positive externalities are strategic
benets that arise either through depositor migration from the failing banks to surviving banks,
or through acquisition of the failed banks assets and businesses by surviving banks. He shows
that that if the negative externality dominates positive externalities, banks in dierent regions will
choose their investments to be highly correlated compared to globally optimal correlation levels.
Acharya calls this fact systemic risk shifting.
This paper also diers from previously mentioned studies in terms of its source for the externali-
ties between national nancial markets. I focus on externalities between national nancial markets
that operate through asset markets and asset prices whereas the studies cited above considered
externalities that operate through costs in the loan or deposit markets. In this paper, systemic risk
in international nancial markets arises as banks from two countries experience correlated liquidity
shocks, and nancial amplication eects are triggered due to re sales. In that regard, this paper
is closest to Bengui (2011), but mainly diers from Bengui (2011) by considering the coordination
problem under systemic risk for structurally dierent countries. On the other hand, Bengui (2011)
considers the coordination problem for symmetric countries with risk averse individuals and im-
perfectly correlated shocks across countries. As this paper arms, provision of macro-prudential
regulation is insucient when countries act independently, and regulatory standards are strategic
substitutes across countries. He also shows that risk taking could be higher in nationally regulated
economies compared to the competitive equilibrium, and that starting from a competitive equi-
librium unilateral introduction of a (small) regulation could be welfare reducing for the country
introducing the regulation.
Another branch of this literature considers regulation of a multi-national bank that operates
across two countries. Two notable studies, Dalen and Olsen (2003) and Holthausen and Rnde
(2004), focus on the tension between home and host country regulation of a multi-national bank
where informational asymmetries are the driving force of regulatory competition. Unlike these
studies, my paper focuses on a model in which banks invest in a single country and are therefore
regulated only by their home country, but interact with each other in global asset markets. The
tension between regulators in my model arises from the externalities that banks in dierent countries
create for each other in global asset markets during times of distress.
This study can also be viewed as a part of the broader literature on macroeconomic policy
coordination that was especially active especially from late 1970s through the 1990s. Cooper (1985)
and Persson and Tabellini (1995) provide extensive reviews of this literature. Hamada (1974, 1976)
are the pioneer studies in the application of the game-theoretic approach to strategic interactions
7
among national governments.
Last, this paper is also related to the literature that features asset re sales. The common
theme across these studies is that, under certain conditions, asset prices can move away from the
fundamental values and assets can be traded in markets at re sale prices. One reason for re
sales is the combined eect of asset-specicity and correlated shocks that hit an entire industry or
economy. Origins of this idea can be found in Williamson (1988) and Shleifer and Vishny (1992)
which claim that re sales are more likely when major players in an industry face correlated shocks
and the assets of the indusry are not easily redeployable in other industries. In such a scenario,
a rm needs to sell assets to restructure and continue operations at a smaller scale; however, it
cannot sell its assets at full value because other rms in the same industry are experiencing similar
problems. Outside investors would buy and manage these assets but they are not as sophisticated
as the rms in the industry. Therefore, they would be willing to pay less than the full value of the
assets to the distressed rms. Moreover, unsophisticated investors may face decreasing returns in
the amounts of assets they employ. This possibility makes the situation even worse for distressed
rms because if many of them try to sell assets to outside investors simultaneously, they will have
to accept higher discounts.
The closest papers in this literature to mine are Lorenzoni (2008), Gai et al. (2008) and Ko-
rinek (2011) which essentially address the same question: how do privately optimal borrowing and
investment levels of nancial intermediaries compare to the socially optimal levels under pecuniary
externalities in nancial markets generated through asset re sales? In these studies, the reasons for
re sales are limited commitment on nancial contracts and the fact that asset prices are determined
in a spot market. Lorenzoni (2008) and Gai et al. (2008) consider a single-country, three-period
model with a continuum of banks. Banks borrow from consumers and oer them state-contingent
contracts. In the interim period, banks are hit by shocks and need to sell assets in some states to
restructure distressed investments. These papers show that there exists over-borrowing and hence
over-investment in risky assets in a competitive setting compared to the socially optimal solution.
Because in the competitive setting each bank treats the market price of assets as given when it
makes borrowing and investment decisions in the initial period, it does not internalize the exter-
nalities created for other banks through re sales. The planner considers the fact that a higher
investment will translate into lower prices for capital sold by banks during the times of distress. The
main dierence between my paper and these papers is that they focus on issues in single-country
cases to the exclusion of issues related to the international dimension of regulation.
Asset specicity is not the only reason for re sales. In Allen and Gale (1994, 1998) and
Acharya and Yorulmazer (2008) the reason for re sales is the limited available amount of cash in
the market to buy long-term assets oered for sale by agents who need liquid resources immediately.
The scarcity of liquid resources leads to necessary discounts in asset prices, a phenomenon known
as cash-in-the-market pricing.
8
3 Model
This model contains three periods, t = 0, 1, 2; and two countries, i = A, B. In each country there
is a continuum of banks and a continuum of consumers each with a unit mass and a nancial
regulator. There is also a unit mass of global investors. All agents are risk-neutral.
There are two goods in this economy: a consumption good and a capital good (i.e., the liquid
and illiquid assets). Consumers are endowed with e units of consumption goods at t = 0, and none
at later periods.
5
Banks have a technology that converts consumption goods into capital goods one-to-one at
t = 0. Capital goods that are managed by a bank until the last period yield R > 1 consumption
goods per unit. Consumption goods are perishable, and the capital fully depreciates at t = 2.
Capital goods can never be converted into consumption goods.
6
Banks in each country i = A, B choose the level of investment, n
i
, in the capital good at t = 0,
and borrow the necessary funds from domestic consumers. I consider deposit contracts that are in
the form of simple debt contracts, and assume that there is a deposit insurance fund operated by
the regulator in each country. Therefore, banks can raise deposits from consumers at a constant
and zero net interest rate. I also assume that banks are protected by limited liability.
7
All uncertainty is resolved at the beginning of t = 1: a country lands in good times with
probability q, and in bad times with probability 1 q. In order to simplify the analysis, I assume
that the states of the world at t = 1 are perfectly correlated across countries. In good times no
banks are hit with shocks, therefore no further actions are taken. Banks keep managing their
capital goods and realize the full returns from their investment, Rn
i
, in the last period. They make
the promised payment, n
i
, to consumers, and hence earn a net prot of (R 1)n
i
. However, in
bad times, the investments of all banks in both countries are distressed. In case of distress, the
investment has to be restructured in order to remain productive. Restructuring costs are equal to
c 1 units of consumption goods per unit of capital. If c is not paid, capital is scrapped (i.e., it
fully depreciates).
There are no available domestic resources (i.e., consumption goods) with which to carry out
the restructuring of distressed investment at t = 1. Only global investors are endowed with liquid
resources at this point. Due to a commitment problem, banks cannot borrow the required resources
5
I assume that the initial endowment of consumers is suciently large, and it is not a binding constraint in
equilibrium.
6
I focus on a simple, tractable model where there is no safe asset and the liquidity shock at the interim period has
a degenerate distribution. I conjecture that relaxing the assumption of no safe asset will not change the qualitative
results of this model. If we allow banks to hoard safe assets, and consider a more general distribution of liquidity
shocks, banks will hold some optimum amount of safe assets at the initial period for precautionary reasons. These
precautionary savings, however, will not be sucient to cover liquidity needs under large realizations of shocks. In
these states of the world, asset re sales will be unavoidable, and that inevitability will generate the externality
between countries that is the crucial part of the current model.
7
Limited liability and deposit insurance assumptions are imposed to match reality and to simplify the analysis of
the model. All qualitative results carry on when these assumptions are removed, as shown is Section 6.
9
from global investors. My particular assumption is that individual banks cannot commit to pay
their production to global investors in the last period.
8
The only way for banks to raise necessary
funds for restructuring is to sell some fraction of the investment to global investors in an exchange
of consumption goods.
These capital sales by banks will carry the features of a re sale: the capital good will be traded
below its fundamental value for banks, and the price will decrease as banks try to sell more capital.
Banks in each country will retain only a fraction of their assets after re sales. If the asset price falls
below a threshold, the expected return on the assets that can be retained by banks will be lower
than the value of the initial investment; hence, banks will become insolvent.
9
I call this situation
a systemic failure.
Figure 1: Timing of the Model
Once it is known that banks are insolvent, deposit insurance requires the bank owners to manage
their capital goods to realize the returns in the last period. The regulator then seizes banks returns,
and makes the promised payments to depositors. The deposit insurance fund runs a decit. If re
sales are suciently mild, however, then banks will have enough assets to make the promised
payments to the depositors. In this case banks remain solvent, but compared to good times they
make smaller prots. This sequence of events is illustrated in Figure 1.
Banks are subject to regulation in the form of an upper limit on initial investment levels.
10
Regulatory standards are set, independently, by the individual national regulators at the beginning
of t = 0. The regulator of country i determines the maximum investment allowed for banks in
its jurisdiction, N
i
, while taking into account the regulation in the other country, N
j
, as given.
8
For simplicity, I assume that the commitment problem is extreme (i.e., banks cannot commit to pay any fraction
of their production to global investors). Assuming a milder but suciently strong commitment problem where banks
can commit a small fraction of their production, as in Lorenzoni (2008) and Gai et al. (2008), does not change the
qualitative results of this paper.
9
Because all uncertainty is resolved at the beginning of t = 1, the expected return to capital retained by banks
after re sales, which is certain at that point, is R units of consumption goods per unit of capital.
10
This regulation becomes equivalent to a minimum capital ratio requirement when we introduce a costly bank
equity capital to the model, as shown in Section 5. I abstract from costly equity capital in the basic model in order
to simplify the exposition.
10
Investment levels of banks in country i have to satisfy n
i
N
i
at t = 0. The regulatory standard
in a country is chosen to maximize the net expected returns on risky investments.
3.1 Global Investors
Global investors are endowed with unlimited resources of consumption goods at t = 1.
11
They can
purchase capital, y, from banks in each country at t = 1 and employ this capital to produce F(y)
units of consumption goods at t = 2. For global investors capital supplied by the banks in these
two countries are identical.
12
Let P denote the market price of the capital good at t = 1.
13
Because
we have a continuum of global investors, each investor treats the market price as given, and chooses
the amount of capital to purchase, y, to maximize net returns from investment at t = 2.
max
y0
F(y) Py (1)
The amount of assets they optimally buy satises the following rst order conditions
F

(y) = P (2)
The rst order condition above determines global investors (inverse) demand function for the
capital good. Using this, we can dene their demand function D(P) as follows:
y = F

(P)
1
D(P) (3)
We need to impose some structure on the return function of global investors and the model param-
eters in order to ensure that the equilibrium of this model is well-behaved.
3.2 Basic Assumptions
Assumption 1 (CONCAVITY).
F

(y) > 0 and F

(y) < 0 for all y 0, with F

(0) R.
11
The assumption that there are some global investors with unlimited resources at the interim period when no
one else has resources can be justied with reference to the empirical facts during the Asian and Latin American
nancial crises. Krugman (2000), Aguiar and Gopinath (2005), and Acharya et al. (2010) provide evidence that,
when those countries were hit by shocks and their assets were distressed, some outside investors with large liquid
resources bought their assets.
12
The assumption that capital goods in the two countries are perfect substitutes for global investors is for simplicity.
The externality that is central to this model is due to the fact that supply conditions in one country aect the prices
that the banks in the other country can obtain for their assets in distress times. A milder assumption of imperfect
substitutes would generate the same externality at a cost of higher complexity.
13
Price of capital at t = 0 will be one as long as there is positive investment, and the price of capital at t = 2 will
be zero because capital fully depreciates at this point.
11
Assumption CONCAVITY says that although global investors return is strictly increasing the
amount of capital employed (F

(y) > 0), they face decreasing returns to scale in the production of
consumption goods (F

(y) < 0), as opposed to banks that are endowed with a constant returns to
scale technology as described above. F

(0) R implies that global investors are less productive

than banks at each level of capital employed.
Concavity of the return function implies that the demand function of global investors for capital
goods is downward sloping (see Figure 2). Global investors will require higher discounts to absorb
more capital from distressed banks at t = 1. This assumption intends to capture that distress
selling of assets is associated with reduced prices. Using a large sample of commercial aircraft
transactions Pulvino (2002) shows that distressed airlines sell aircraft at a 14% discount from the
average market price. This discount exists when the airline industry is depressed but not when it is
booming. Coval and Staord (2007) show that re sales exist in equity markets when mutual funds
engage in sales of similar stocks. Furthermore, Krugman (2000), Aguiar and Gopinath (2005), and
Acharya et al. (2010) provide signicant empirical and anecdotal evidence that during Asian and
Latin American crises, foreign acquisitions of troubled countries assets were very widely spread
across industries and assets were sold at sharp discounts. These evidence suggests that foreign
investors took control of domestic enterprises mainly because they had liquid resources whereas the
locals did not, even though the locals had superior technology and know-how to run the domestic
enterprises. Further support for this argument comes from the evidence in Acharya et al. (2010)
that many foreigners eventually ipped the assets they acquired during the Asian crisis to locals,
and usually made enormous prots from such trades.
The idea that some assets are industry-specic, and hence less productive in the hands of out-
siders, has its origins in Williamson (1988) and Shleifer and Vishny (1992). Examples of industry-
specic assets include oil rigs and reneries, aircraft, copper mines, pharmaceutical patents, and
steel plants. These studies have claimed that when major players in such industries face correlated
liquidity shocks and cannot raise external nance due to debt overhang, agency, or commitment
problems, they may have to sell assets to outsiders. Outsiders are willing to pay less than the value
in best use for the assets of distressed enterprises because they do not have the specic know-how
to manage these assets well and therefore face agency costs of hiring specialists to run these assets.
The decreasing returns to scale technology assumption captures the ineciency of outsiders, similar
to Kiyotaki and Moore (1997), Lorenzoni (2008), Gai et al. (2008), and Korinek (2011). It is also a
reduced way of modeling that global investors rst purchase assets that are easy to manage, but as
they purchase more assets they will need to buy ones that require sophisticated management and
operation skills.
Assumption 2 (ELASTICITY).

P,y
=
y
P
P
y
=
F

(y)
yF

(y)
> 1 for all y 0
12
Assumption ELASTICITY says that global investors demand for the capital good is elastic.
This assumption implies that the amount spent by global investors on asset purchases, Py = F

(y)y,
is strictly increasing in y. Therefore we can also write Assumption ELASTICITY as
F

(y) +yF

(y) > 0
If this assumption was violated, multiple levels of asset sales would raise a given amount of
liquidity, and multiple equilibria in the asset market at t = 1 would be possible. This assumption
is imposed by Lorenzoni (2008) and Korinek (2011) in order to rule out multiple equilibria under
re sales.
14
Assumption 3 (REGULARITY).
F

(y)F

(y) 2F

(y)
2
0 for all y 0
Assumption REGULARITY holds whenever the demand function of global investors is log-
concave, but it is weaker than log-concavity.
15
In order to see this, let (y) F

(y) denote the

(inverse) demand function of global investors. We can rewrite Assumption REGULARITY as
(y)

(y) 2

(y)
2
0.
We can show that the demand function is log-concave if and only if (y)

(y)

(y)
2
0. Log-
concavity of demand function is a common assumption used in the Cournot games literature (see
Amir (1996)); it ensures the existence and uniqueness of equilibrium in a simple n-player Cournot
game. Therefore, I call it a regularity assumption on F(). Clearly Assumption REGULARITY
holds whenever the demand function is log-concave. However, Assumption REGULARITY is
weaker than log-concavity and may also hold if the demand function is log-convex (i.e., if (y)

(y)

(y)
2
0).
Assumption REGULARITY will ensure that the objective functions of regulators are well-
behaved. It will be crucial in showing that the equilibrium of this model exists and it is unique.
Many regular return functions satisfy conditions given by Assumptions CONCAVITY, ELAS-
TICITY and REGULARITY. Here are two examples that satisfy all three of the above assump-
tions:
Example 1 F(y) = Rln(1 +y)
Example 2 F(y) =
_
y + (1/2R)
2
14
Gai et al. (2008) provides the leading example where this assumption is not imposed and multiple equilibria in
the asset market is therefore considered.
15
A function is said to be log-concave if the logarithm of the function is concave.
13
The following example satises Assumption CONCAVITY, but not Assumptions ELASTIC-
ITY and REGULARITY.
Example 3 F(y) = y(R 2y) where 2y < R for all y 0.
Assumption 4 (RANGE).
1 + (1 q)c < R 1/q
Assumption RANGE says that the return on investment for banks must not be too low because
if they are, equilibrium investment levels will be zero. Nor they must be too high; if they are,
equilibrium investment levels will be innite. This assumption, while not crucial for the results,
allows us to focus on interesting cases in which equilibrium investment levels are neither zero nor
innite.
3.3 Equilibrium with Symmetric Countries
In this section I consider only symmetric countries and solve the model by backwards induction.
First, I analyze the equilibrium at the interim period in each state of the world, for a given set of
investment levels; then I solve the game between the regulators at t = 0. Note that, if good times
are realized t = 0, no further actions need to be taken by any agent. Therefore, at t = 1 we need
only to analyze the equilibrium of the model for bad times.
I solve the model without resorting to some particular functional form. The results of this
paper hold for any functional form that satisfy Assumptions CONCAVITY, ELASTICITY, and
REGULARITY.
3.3.1 Crisis and re sales
Consider the problem of a bank in country i if bad times are realized at t = 1. The bank reaches
t = 1 with a level of investment equal to n
i
which was chosen at the initial period. The investment
is distressed and must be restructured using liquid resources. The investment will not produce
any returns in the last period if it is not restructured.
16
The bank cannot raise external nance
from global investors because it cannot commit to pay them in the last period. Therefore, the
only way for the bank to raise the funds necessary for restructuring is to sell some fraction of the
investment to global investors and use the proceeds to pay for restructuring costs, whereby it can
retain another fraction of the investment.
At the beginning of t = 1 in bad times, a bank in country i decides what fraction of capital to
restructure (
i
) and what fraction of restructured capital to sell (1
i
) to generate the resources
for restructuring. Note that
i
will then represent the fraction of capital that a bank keeps after re
16
For example, if the assets are physical, restructuring costs can be maintenance costs or working-capital needs.
14
sales.
17
Thus the bank takes the price of capital (P) as given, and chooses
i
and
i
to maximize
total returns from that point on
max
0
i
,
i
1

i
= R
i

i
n
i
+P(1
i
)
i
n
i
c
i
n
i
(4)
subject to the budget constraint
P(1
i
)
i
n
i
c
i
n
i
0. (5)
The rst term in (4) is the (certain) total return that will be obtained from the unsold part of
the restructured assets, which are
i
n
i
, in the last period. The second term is the revenue raised
by selling a fraction (1
i
) of the restructured assets, which are
i
n
i
, at the given market price
P. The last term, c
i
n
i
, gives the total cost of restructuring. Budget constraint (5) says that the
revenues raised by selling capital must be greater than or equal to the restructuring costs.
By Assumption CONCAVITY, the equilibrium price of capital must satisfy P F

(0)
R, otherwise global investors will not purchase any capital. Later on, I will show that in equilibrium
we must also have c < P. For the moment, we will assume that the equilibrium price of assets
satises
c < P R (6)
Now, consider the rst order conditions of the maximization problem (4) while ignoring the con-
straints

i
= [R
i
+P(1
i
) c]n
i
(7)

i
= (R P)
i
n
i
(8)
From (8) it is obvious that
i
is increasing in
i
because P R by (6): when the price of capital
goods is lower than the return that banks can generate by keeping them, banks want to retain a
maximum amount. Choosing
i
as high as possible implies that the budget constraint will bind.
Hence, from (5) we obtain that the fraction of capital goods retained by banks after re sales is

i
= 1
c
P
(9)
The fraction banks retain after re sales (
i
) is increasing in the price of the capital good (P) and
17
Following Lorenzoni (2008) and Gai et al. (2008), I assume that banks have to restructure an asset before selling
it. Basically, this means that bank receive the asset price P from global investors, use a part, c, to restructure
the asset, and then deliver the restructured assets to global investors. Therefore banks will sell assets only if P is
greater than the restructuring cost, c. We could assume, without qualitatively changing our results, that it is the
responsibility of global investors to restructure the assets that they purchase. However, the model is more easily
solved using the current story.
15
Figure 2: Equilibrium in the Capital Goods Market
decreasing in the cost of restructuring (c). From (9) we can also obtain the total capital supply of
a bank in country i as
S
i
(P, n
i
) = (1
i
)n
i
=
c
P
n
i
(10)
for c < P R. This supply curve is downward-sloping and convex, which is standard in the re
sales literature. A negative slope implies that if there is a decrease in the price of assets banks
have to sell more assets in order to generate the resources needed for restructuring. This is because
banks are selling a valuable investment at a price below the fair value for them due to an exogenous
pressure (e.g., paying for restructuring costs).
On the other hand, using (9) we can write the rst order condition (7) as

i
= R
i
n
i
0 (11)
Equation (11) shows that revenues are increasing in
i
at t = 1. Therefore, banks will optimally
choose to restructure the full fraction of the investment (
i
= 1). In other words, scrapping of
capital will never arise in equilibrium.
Note that if the capital price is greater than R, banks want to sell all the capital goods they
have because they can get at most R per unit by keeping and managing them. If the price is lower
than c, however, they will optimally scrap all of their capital (
i
= 0). As discussed above, prices
above R and below c will never arise in equilibrium. The total asset supply curve of banks from
the two countries is plotted in Figure 2 for an initial total investment in the two countries of N
0
.
16
3.3.2 Equilibrium in the Capital Market at t=1
Equilibrium price of capital goods, P

, will be determined by the market clearing condition

E(P

, n
A
, n
B
) = D(P

) S(P

, n
A
, n
B
) = 0 (12)
The condition above says that the excess demand in the capital market, denoted by E(P, n
A
, n
B
),
is equal to zero at the equilibrium price. D(P) in Equation (12) is the demand function of global
investors which was obtained from the rst order conditions of global investors problem as shown
by (3). S(P, n
A
, n
B
) is the total supply of capital goods. We can obtain it as
S(P, n
A
, n
B
) =
c(n
A
+n
B
)
P
(13)
by adding the individual supply of banks in each country given by (10).
This equilibrium is illustrated in Figure 2. Note that the equilibrium price of capital at t = 1 will
be a function of the initial investment level in the two countries. Therefore, from the perspective
of the initial period I denote the equilibrium price as P

(n
A
, n
B
).
How does a change in the initial investment level in one of the countries aect the price of
capital at t = 1? Lemma 1 shows that if investment into the risky asset in one country increases
at t = 0, a lower price for capital will be realized in the re sales state at t = 1.
Lemma 1. P

(n
i
, n
j
) is decreasing in n
i
for i = A, B and j = i under Assumptions CONCAVITY
and ELASTICITY.
Proof. See the appendix.
Lemma 1 implies that higher investment in the risky asset in one country (i.e., a higher n
i
)
increases the severity of the nancial crisis for both countries by lowering the asset prices. This eect
is illustrated in Figure 3. Suppose that initial investment level in country A increases, increasing
the total investment in the two countries from N
0
to N
1
. In this case, banks in country A will
have to sell more assets at each price, as can be seen from individual supply function given by
(10). Graphically, the total supply curve will shift to the right, as shown by the dotted-line supply
curve in Figure 3, which will cause a decrease in the equilibrium price of capital goods. Lower asset
prices, by contrast, will induce more re sales by banks in both countries in the bad state due to
the downward-sloping supply curve. This additional result is formalized in Lemma 2.
Lemma 2. Equilibrium fraction of assets sold in each country, 1

(n
i
, n
j
), is increasing in n
i
for i = A, B under Assumptions CONCAVITY and ELASTICITY.
Proof. See the appendix.
17
Figure 3: Capital Goods Market: Comparative Statics
Together lemmas 1 and 2 imply that a higher initial investment in the risky investment in one
country creates negative externalities for the other country by making nancial crises more severe
(i.e., via lower asset prices according to Lemma 1) and more costly (i.e., more re sales according
to Lemma 2).
3.3.3 Banks Problem
Each bank in country i at t = 0 chooses the investment level, n
i
, to maximize expected prots
given by
max
0n
i
N
i

i
(n
i
) = q(R 1)n
i
+ (1 q) max [Rn
i
n
i
, 0] (14)
where is the equilibrium fraction of rescued assets by a bank in country i which is equal to 1c/P
as shown by (9). A bank borrows from the local deposit market at a constant zero interest and
invests in the productive asset. With probability q, that both countries are in good times, the
investment produces returns as expected Rn
i
. Banks make the promised payments to depositors,
n
i
, leading to a net prot of (R 1)n
i
. With probability 1 q, both countries are in the bad
state. Banks from the two countries face restructuring costs and hence are forced to re sale their
assets. Since banks are price-takers in the capital market, the fraction of capital that they can
save from the re sales, = 1 c/P, is exogenous to them. In other words, because each bank is
small compared to market size, it does not take into account the eect of its investment choice at
t = 0 on the equilibrium price (P), and thus on the fraction of assets retained after re sales in
equilibrium ().
Banks undoubtedly earn net positive returns in the good state since q(R 1) > 0. Moreover,
18
due to the limited liability, they never receive negative prots in the bad state. Because banks do
not internalize the eect of their initial investment on the stability of the nancial system at t = 1,
there is no counter-eect that osets the positive returns on investment. Therefore, a banks net
expected return from risky investment at t = 0 is always positive, and a bank always makes itself
better o by investing more. Therefore, the regulatory upper limit on the risky investment at t = 0
will bind (i.e., banks will choose n
i
= N
i
at t = 0).
Fire sales will be severe for some parameters and banks may become insolvent in equilibrium,
as analyzed in Section 5. I assume that when banks are insolvent after re sales they are required
by law to manage remaining assets until the last period and to transfer asset returns into the
deposit insurance fund. This is a reasonable assumption because banks are the only sophisticated
agents in our model domestic economy that can manage those assets. Furthermore, in practice, the
dissolution process of insolvent banks usually does not happen immediately. It is a time-consuming
process because, for example, loans have to be called-o or sold to third parties to make payments
to debtors. This assumption also captures this time dimension of the dissolution process.
3.3.4 Regulators Problem
Regulators of the two countries simultaneously determine the regulatory standards for the banks
in their own jurisdictions before banks make their borrowing and investment decisions at t = 0.
Regulation in each country i = A, B takes the form of an upper limit, N
i
0, on the investment
level allowed for domestic banks. Banks in country i have to abide by the regulation by choosing
their investment levels as n
i
N
i
. As they set the standards, regulators anticipate that banks will
choose initial investment levels that are as high as possible, and incorporate this fact into their
decision problem.
The objective of an independent national nancial regulator is to maximize the net expected
social welfare of its own country. Social welfare is dened as the expected return to the risky
investment minus the cost of the initial investment. Therefore, regulator i chooses N
i
0, to solve
max
N
i
0
W
i
(N
i
, N
j
) = max
N
i
0
qRN
i
+ (1 q)R

(N
i
, N
j
)N
i
+ (e N
i
) (15)
while taking the regulatory standard in the other country, N
j
, as given. Let (

N
A
,

N
B
) denote the
Nash Equilibrium of the game between the regulators at t = 0 whenever it exists. I assume that
the initial endowment of consumers (e) is suciently large, and that it is not a binding constraint
in equilibrium.
Social welfare given by (15) incorporates the fact that banks investment level, n
i
, equals N
i
,
the regulatory upper limit. With probability q, the good state is realized when banks in country
i obtain a total return of RN
i
. With probability 1 q both countries land in the bad state. In
the bad state, banks perform asset sales as described previously, and manage the remaining assets,
19

(N
i
, N
j
)N
i
until t = 2, to obtain a gross return of R per unit. Therefore the return to the
investment in the bad state in country i is R

(N
i
, N
j
)N
i
. The cost of the initial investment (N
i
)
is subtracted to obtain net returns to the investment. Each regulator takes into account the eect
of both countries regulatory standards on the price of capital in the bad state. This is why the
fraction of assets that banks can keep after the re sales,

(N
i
, N
j
) = 1 c/P

(N
i
, N
j
), is written
as a function of the regulatory standards in the two countries.
The following equation gives the rst order conditions of regulator is problem in (15)
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)R
_

(N
i
, N
j
)
N
i
N
i
+

(N
i
, N
j
)
_
1 (16)
By rearranging the terms, we can write this rst order condition as a sum of the marginal benet
and marginal cost of increasing N
i
, the regulatory standard:
W
i
(N
i
, N
j
)
N
i
= {qR + (1 q)R

(N
i
, N
j
)} +
_
(1 q)R

(N
i
, N
j
)
N
i
N
i
1
_
(17)
When regulator i increases N
i
, there will be more investment in the risky asset. The rst curly
brackets give the expected gross marginal benet from increasing N
i
: with probability q, the good
state is realized and a total return of R is obtained from the additional unit of investment. With
probability 1 q, the bad state is realized and a total return of R

(N
i
, N
j
) is obtained from the
additional investment. In the bad state, the return can be obtained only from a fraction,

(N
i
, N
j
),
of the original investment, because another fraction of the investment is sold to global investors.
The second curly brackets in (17) give the expected marginal cost of increasing N
i
. Because

(N
i
, N
j
) is decreasing in N
i
, as implied by Lemma 2, a smaller fraction of assets will be retained
by banks if the bad state is realized for higher initial investment levels. The rst term captures
this fact: with probability 1 q the bad state is realized, and an additional unit of investment will
decrease the fraction of capital that can be retained by banks of country i by

(N
i
, N
j
)N
i
, causing
a total loss of (

(N
i
, N
j
)/N
i
)N
i
. Last, -1 in the second curly brackets gives the marginal cost
of funds required for the risky investment.
3.3.5 An alternative formulation of regulators objective function
We can alternatively write the regulators objective function in a way that explicitly shows the
returns to the investment and costs of re sales. Start by substituting 1 c/P for

() function,
using the derivation obtained in (9) to write (15) as
W
i
(N
i
, N
j
) = qRN
i
+ (1 q)R
_
1
c
P

(N
i
, N
j
)
_
N
i
+ (e N
i
)
20
Add and subtract cN
i
to the expression above to get
W
i
(N
i
, N
j
) = qRN
i
+ (1 q)R
_
RN
i

Rc
P

(N
i
, N
j
)
N
i
+cN
i
cNi
_
+ (e N
i
)
Last, rearranging the terms inside the curly brackets gives
W
i
(N
i
, N
j
) = qRN
i
+ (1 q)
_
RN
i
[R P

(N
i
, N
j
)]
cN
i
P

(N
i
, N
j
)
cN
i
_
+ (e N
i
) (18)
Consider this alternative objective function in detail. It is composed of two main terms as
in (15): net expected returns in both the good state and the bad state. Because the rst term,
q(R 1)N
i
, that gives the net expected return in the good state is clear, I focus on the latter.
The rst term inside the curly brackets gives the net total return that could be obtained from the
investment if there were no re sales. The second term is the cost of re sales: cN
i
\P

(N
i
, N
j
) is
the amount of assets sold in re sales as given by (10), where banks receive P

< R from these

assets instead of R. The last term inside the curly brackets, cN
i
, is the total cost of restructuring.
Because the two versions of the regulators objective function are the same, I will use the
rst formulation in the rest of the paper for the sake of analytical convenience, even though the
alternative formulation could be more intuitive.
3.3.6 Regulatory Standards in the Uncoordinated Equilibrium
Having analyzed the problem of regulators, we can turn to investigating the equilibrium of the
game between regulators at t = 0 when they act independently. The aims are to show that there
exists a unique symmetric equilibrium of this game, and then to perform comparative statics. I
start by analyzing the properties of the best response functions of regulators. The following lemma
establishes that independent regulators have a unique best response to each regulatory standard
choice by the opponent country.
Lemma 3. Under Assumptions CONCAVITY, ELASTICITY and REGULARITY, each regula-
tors best response is unique valued.
Proof. See the appendix.
An interesting question in this setup concerns how that unique best response behaves as regu-
lation in the opponent country changes. Suppose that the regulator of country B decides to tighten
regulation (i.e., to reduce N
B
). How would the regulator of country A optimally react? The next
proposition shows that regulator A optimally chooses to relax its regulatory standard (i.e., increase
N
A
), as regulator B imposes stricter regulations. In other words, the optimal regulatory standards
in the two countries are strategic substitutes.
21
Proposition 1. Under Assumptions CONCAVITY, ELASTICITY and REGULARITY, optimal
regulatory standards in the uncoordinated equilibrium are strategic substitutes.
Proof. See the appendix.
The intuition for this result is as follows: If regulator B tightens its regulatory standard by
reducing the upper limit on the investment level for its banks, there will be less distressed assets at
t = 1 in the bad state; hence, a higher asset price will be realized. Therefore, fewer assets will be
sold in equilibrium as shown by Lemma 2, which means that banks in both countries will be able
to retain a higher fraction of their initial investment after re sales. This retention will increase the
marginal return to investment and initially allow regulator A to optimally choose a higher upper
limit on the investment level (i.e. relax its regulatory standard).
The next lemma shows that in order to have nite and strictly positive equilibrium investment
levels in the two countries, banks return from investment R should not be too low or too high. The
exact condition on R is given by Assumption RANGE, which states that 1 +c(1 q) < R 1/q.
Lemma 4. The best responses of each regulator satisfy 0 < N

i
< for i = A, B if Assumption
RANGE holds, i.e. if 1 + (1 q)c < R 1/q.
Proof. See the appendix.
1+(1q)c is the net expected cost of the investment: each unit of investment requires one unit
of consumption good initially. With probability 1 q bad times are realized, in which case banks
have to incur an extra restructuring cost of c units of consumption goods per unit of investment.
If the return on the investment, R, is less then this expected cost, 1 + (1 q)c, booth countries
social welfare will be higher without any investment at all. Therefore, if R < 1 + (1 q)c, then
equilibrium investment levels are zero in both countries. But if R > 1/q, then the expected return
to the investment in the good state alone will be higher than initial cost of investment, which is 1.
In this case, even when the entire initial investment is expected to be lost in the bad state, the net
expected return to investment will always be positive. For suciently high initial endowment levels,
this case leads to a corner solution in which social welfare is maximized by having all endowments
invested in the risky asset. I also impose qR 1 in order to rule out these inconsequential details
and focus on the interesting cases.
Now we are ready to examine the existence of a Nash equilibrium in the game between regulators.
The nice features of the objective functions of regulators established above help us to show that a
Nash equilibrium exists.
Proposition 2. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE,
at least one pure strategy Nash Equilibrium exists in the game between two nancial regulators at
t = 0. Moreover there exists at least one symmetric pure strategy NE.
22
Proof. See the appendix.
The next natural question is whether there are multiple equilibria or there is a unique equilib-
rium. Fortunately, under the previously stated conditions there is a unique symmetric equilibrium
of the game between the regulators, as shown by the following proposition.
Proposition 3. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE,
there exists a unique symmetric Nash Equilibrium of the game between the regulators at t = 0.
Proof. See the appendix.
3.3.7 Comparative Statics for the Uncoordinated Equilibrium
What happens to the unique regulatory standards of the uncoordinated equilibrium in the two
countries as good state becomes more likely, or if banks per unit return from investment, R,
increases? The next proposition shows that in both cases, regulatory standards in the two countries
are relaxed (i.e., regulators increase the upper limit on the risky investment).
Proposition 4. Regulatory standards in the uncoordinated symmetric equilibrium become more lax
as q and R increase.
Proof. See the appendix.
This result is quite intuitive because as the good state becomes more likely (i.e., as q increases),
regulators will face the cost of re sales less often and will allow more investment in equilibrium.
But as R increases, returns to investment in both good and bad states also increase, making the
investment socially more protable.
I conclude this section by showing that in equilibrium, price of the capital good at t = 1 in
bad times must be greater than restructuring costs (c). I tentatively assumed this while discussing
banks optimal re sales decisions at t = 1 after they receive bad shocks. Now it is time to prove
this claim formally. Under this result, as I have shown, banks optimally restructure all assets in
equilibrium. In other words, as previously stated, scrapping of capital never arises in equilibrium.
Lemma 5. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE, the
equilibrium price of the capital good at t = 1 in bad times satises P

> c.
Proof. See the appendix.
Lemma 5 holds because if regulators allow the investment level in their country (N
i
) to be too
high, they know that they will drive down the equilibrium price below the cost of restructuring,
in which case banks do not restructure any assets. Therefore, it is never optimal for any of the
regulators to allow such high investment levels, independent of the choice of the competing regulator.
23
3.4 Internationally Coordinated Regulation
Suppose that there is a higher authority, call it the central regulator, that determines optimal
regulatory standards in these two countries. In practice, the central regulator could be an interna-
tional nancial institution such as the International Monetary Fund or the Bank for International
Settlements, or it could be an institution created by a binding bilateral agreement between the
two countries. I assume that, for political reasons, the central regulator must choose the same
regulatory standards for both countries. The question that I address in this section is as follows:
Suppose that at the beginning of t = 0, national regulators can either set regulatory standards
independently or simultaneously relinquish their authority to the central regulator. Would they
choose the latter?
I dene the central regulators problem as follows: it chooses the regulatory standards in coun-
tries A and B, (N
A
, N
B
), to maximize the sum of expected social welfare of these countries as given
below
max
N
A
,N
B
0
GW(N
A
, N
B
) = max
N
A
,N
B
0

i=A,B;j=i
{q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
]} (19)
In other words, the central regulator maximizes the sum of the objective functions of individual
regulators. For symmetric countries, it is natural to assume that each country receives an equal
weight in the central regulators objective function.
18
I denote the internationally optimal common
regulatory standards by (

N,

N) and compare them to the regulatory standards in the uncoordinated
symmetric equilibrium, (

N,

N).
Another way to state the central regulators problem for symmetric countries involves thinking of
the central regulator as choosing the total investment level across the two countries, N = N
A
+N
B
,
to maximize their overall welfare. After determining the optimal total investment level N, it
imposes

N
i
= N/2 for i = A, B, where
N = arg max
N0
q(R 1)N + (1 q)[R

(N)N N] (20)
It is easy to see that the two alternative problems for the central regulator given by (19) and (20)
are the same due to the countries symmetry.
Now we can compare the internationally optimal regulatory standards to the standards that
arise as a result of strategic interaction between regulators. The following proposition shows that
a central regulator will impose tighter regulatory standards (i.e., a lower N) compared to what
would have been chosen by independent regulators.
18
Note that the central regulator does not consider the welfare of the global investors. However, the results of the
paper are robust to this generalization.
24
Proposition 5.

N <

N, i.e. the central regulator chooses tighter regulatory standards compared
to the standards chosen by independent national regulators in the uncoordinated equilibrium.
Proof. See the appendix.
Proposition 5 shows that due to the systemic risk caused by asset re sales, standards chosen
by independent national regulators are ineciently lax compared to regulatory standards that
would be chosen by a central regulator. A central regulator maximizes the total welfare in the
two countries and hence internalizes the systemic externalities that arise from re sales. A central
regulator takes into account the fact that allowing more investment in the risky asset by relaxing
regulatory standards in one country reduces the welfare of the other country due to higher numbers
of re sales during distress times.
3.4.1 Is voluntary cooperation possible?
We see that risky investment levels will be higher in both countries if regulators act strategically.
But will countries ever benet from relinquishing their regulatory authority to a central regulator
that imposes tighter standards in both countries? The following proposition shows that symmetric
countries always benet from relinquishing their authority to a central regulator.
Proposition 6. If the countries are symmetric then both regulators prefer to deliver their authority
to a central systemic risk regulator, i.e., W
i
(

N
i
,

N
j
) W
i
(

N
i
,

N
j
) holds for i = A, B.
Proof. See the appendix.
When regulators act independently, each allows investment into the risky asset up to the point
where the expected marginal benet from the risky investment is equal to the expected domestic
marginal cost of the investment. However, at this level of investment, the marginal total cost
across the two countries far exceeds the sum of the marginal benets. This happens because
neither regulator considers the adverse eect of increasing investment level on the welfare of the
other country. Yet, the central regulator can choose a total investment level in the risky asset where
the total marginal benet is equal to the total marginal cost, and hence can improve the overall
welfare of the two countries. Therefore, it is in the interest of the regulators of symmetric countries
to simultaneously surrender their authority to a central regulator.
4 Asymmetric Countries
In the previous section, we saw that regulators of symmetric countries are always better o by
relinquishing their authority to a central regulator. Can a similar argument be made for countries
that are asymmetric in some dimensions? In other words, if there are dierences across countries,
would national regulators still benet from relinquishing their authority to a central regulator?
25
In this section, I answer this and the following questions that arise when there are asymmetries
across countries: How would the asymmetries aect regulation levels in the two countries in equilib-
rium? How do central regulation levels compare to regulation levels chosen by national regulators
independently? Which countries are more likely to accept a common central regulation?
I focus on dierences in returns on the risky investment across countries. In particular, I assume
that banks in country A are uniformly more productive than banks in country B. In terms of the
parameters of the model, this assumption can be stated as R
A
> R
B
.
19
Furthermore, to simplify the following analysis, I also assume that F

(0) 1 in this section.

20
Under this assumption, global investors will purchase capital only if the price of capital falls below
one. This assumption also rules out possible multiple equilibria in the capital goods market at t = 1
when there are return dierences between countries. Note that from global investors perspectives,
the capital goods in the two countries are still identical at t = 1.
The next proposition shows that when regulators act independently, the regulator of the high-
return country chooses lower regulatory standards (i.e., a higher N) in equilibrium. This result
complies with Proposition 4 in the previous section where we have seen that equilibrium investment
levels increase in the return to investments given by R.
Proposition 7. If R
A
> R
B
, then

N
A
>

N
B
in the uncoordinated equilibrium.
Proof. In the appendix.
Now we can compare common central regulatory standards to uncoordinated regulation levels
when there are asymmetries between the countries. The next proposition shows that in order for a
common central regulation to be acceptable to both regulators, it must require stricter regulatory
standards in both countries compared to the uncoordinated regulatory standards.
Proposition 8. There exists no central regulation level N > min{

N
A
,

N
B
}.
Proof. See the appendix.
The proof of Proposition 8 makes use of the envelope theorem to show that welfare of a country
is decreasing in the investment level of the other country. In order to forego the authority to
19
This assumption is justied when there is segregation between the investment markets of the two countries.
There is both a theoretical and a practical reason for making this assumption. From the theoretical perspective, this
assumption shuts down the externality channel that operates through the competition between banks from dierent
countries in loan markets and allows me to focus on regulatory spillovers that operate through asset prices during
times of distress. The previous literature considered the regulatory spillovers operating through competition in loan
and deposit markets, which shows us when cooperation is justied under those extarnalities (e.g., DellAriccia and
Marquez (2006)). From a practical point of view, there are well documented return dierences across countries
and a large body of literature explains those dierences based on levels of technology and human capital as well
as institutional factors. I just take the return dierences across countries as given and examine the desirability of
coordination of macro-prudential policies in a world characterized by those structural dierences.
20
This assumption simplies the analysis by making the demand function of global investors independent of the
return dierences between the countries.
26
independently and optimally choose regulatory standards as a response to the regulatory standards
chosen by the other country, each regulator must be compensated by a stricter regulatory standard
(i.e., a lower N) in the other country. Therefore, any common regulation level above

N
B
, which
is the minimum of the two regulation levels given the assumption that R
A
> R
B
, will always be
rejected by regulator A.
This discussion implies that if a common regulation level is accepted by the regulator of the
high-return country, it will always be accepted by the regulator of the low-return country. This
happens because common regulation reduces investment levels in both countries, as shown by
Proposition 11. However, it reduces investment levels more in the high-return country compared to
the low return country. Therefore, if the regulator of the high-return country is willing to accept a
common regulation level, it will necessarily be accepted by the regulator of the low-return country,
as shown by the following lemma.
Lemma 6. For any common regulation level N such that W
A
(N, N) > W
A
(

N
A
,

N
B
) we have
W
B
(N, N) > W
B
(

N
B
,

N
A
).
Proof. See the appendix.
Lemma 6 allows us to focus on the welfare of country A in search of mutually acceptable common
regulatory standards. We may dene N
m
as the regulatory standard that maximizes the welfare
of country A if it is uniformly imposed in both countries. Formally, I dene N
m
as follows:
Denition 1. N
m
arg max
N
W
A
(N, N)
Given this denition, we can write the net maximum benet from common central regulation
to country A as W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
). The next proposition shows that this net maximum
benet decreases as the dierences between the countries become larger.
Proposition 9. Suppose that F

(0) 1. Let s R
A
R
B
> 0. Then for any R
A
, there exists s
(0, R
A
1) such that W
A
(N
m
, N
m
)W
A
(

N
A
,

N
B
) 0 if s s, and W
A
(N
m
, N
m
)W
A
(

N
A
,

N
B
) <
0 otherwise.
Proof. See the appendix.
Proposition 9 provides the main result of this section: if the return dierences between the
two countries are above a threshold, then at least the high-return country will be worse o if a
common regulation level is imposed across the two countries, even if the common regulation is
chosen such that it maximizes the welfare of the high-return country. Large return dierences will
imply that such a common regulation level is too strict compared to the regulatory standard that
would be chosen by the high-return country in the uncoordinated equilibrium. Therefore, welfare
of the high-return country will fall if it decides to accept common regulatory standards in the face
of high return dierences between countries. In other words, Proposition 9 shows that voluntary
27
cooperation can exist only between suciently similar countries. If the dierences across countries
are suciently high, then at least one of them will be worse o by accepting common central
regulation.
5 Extensions: Discussion of Assumptions
In this section I examine the robustness of the main results with regard to changes in some of the
assumptions in the basic model. I revisit the assumptions of deposit insurance, limited liability
for bank owners, and nonexistence of initial equity capital for bank owners, and show that the
qualitative results do not change when these assumptions are relaxed.
5.1 Deposit Insurance
With a deposit insurance fund, banks are able to borrow at constant and zero net interest rate from
consumers because consumers are guaranteed by the fund that they will always recover their initial
investment. If banks do not have sucient funds to make the promised payments to consumers
following a bad state, the deposit insurance fund steps in and pays consumers the decit between
the promised payment and the resources available to a bank.
What happens if there is no deposit insurance? The answer depends on the competitive struc-
ture of the deposit market. I consider two polar cases: rst, each bank is a local monopoly in the
deposit market; and second, there is perfect competition between banks in the deposit market. I
begin here with the local monopoly case and discuss the perfect competition case in Section 5.1.1.
When each bank is a local monopoly as in the basic model, the interest on deposit contracts will be
just enough to induce risk-neutral consumers to deposit their endowments with them. In technical
terms, the individual rationality condition for consumers will bind. I restrict attention to deposit
contracts that are in the form of simple debt contracts.
21
A bank in country i will choose the amount to borrow and invest in the risky asset, n
i
, and the
interest rate on the deposits, r, to maximize the net expected prots:
max
r,n
i
0
q(R r)n
i
+ (1 q) max{(1 c/P)Rn
i
rn
i
, 0} (21)
subject to
qrn
i
+ (1 q) min{R(1 c/P)n
i
, rn
i
} n
i
(IR) (22)
n
i
N
i
(23)
21
There are two justications for this restriction. First, this assumption is realistic: the deposit contracts are in the
form of simple debt contracts in practice. Second, debt contracts can be justied by assuming that depositors can
observe banks asset returns only at a cost. According to Townsend (1979), in the case of costly state verication,
debt contracts will be optimal.
28
where 1 c/P = is the fraction of assets retained by banks at t = 1 after re sales (which,
as before, banks take as given). The bank has to satisfy the individual rationality constraint of
consumers given by (22): expected return to deposits must be greater than n
i
, the initial deposit
of a consumer. A consumer will receive a gross return of rn
i
in the good state which happens with
probability q. In the bad state, which arises with probability 1 q, he will obtain the minimum of
the promised payment, rn
i
, and the returns available to the bank after re sales Rn
i
. If Rn
i
< n
i
the consumer will experience a loss in the bad state.
As before, the bank is also subject to the maximum investment regulation n
i
N
i
. Because
the problem of a bank is still linear, it will yield a corner solution as before: there will be either
a maximum investment (n
i
= N
i
) or no investment at all (n
i
= 0). We can examine the choice of
the investment level (n
i
) and the choice of deposit rate (r) separately. First, consider the choice
of optimal r for a given investment level. We can see from the problem of banks above that for a
given P there are two cases to consider:
Case 1 R(1 c/P) > 1. In this case, banks have sucient resources to cover the initial
borrowing from depositors even in the bad state. Therefore, they will oer zero net interest to
consumers. Banks will set r = 1, and the IR condition will be satised with equality. Because banks
make net positive prots in both states of the world, they want to invest as much as possible. Banks
will borrow and invest in the risky asset until the regulatory requirement binds (n
i
= N
i
). Given
that banks invest as much as possible, regulators will choose the same standards in equilibrium as
in the basic model. Therefore, the symmetric equilibrium of Section 3 and its qualitative results
will prevail.
Case 2 R(1 c/P) < 1. In this case, returns on the assets retained by banks after re sales
are not sucient to cover the initial borrowing from depositors because R(1 c/P)n
i
< n
i
. Banks
have to oer positive net interest rate to consumers in the good state to compensate for their losses
in the bad state. For the IR condition of consumers to be satised, r has to be such that
r
1 (1 q)R(1 c/P)
q
r

(24)
This can be seen by rearranging the IR condition (22), and noting that min{R(1 c/P)n
i
, rn
i
} =
R(1 c/P)n
i
in this case. Banks will oer consumers the lowest r that satises (24) to maximize
their prots, and hence will set r = r

. I check if there is an equilibrium where banks make

maximum investment and regulators choose the same standards as before for such r

. Suppose that
regulators choose their standards assuming that banks will make the maximum allowed investment.
We know, from the analysis in Section 3 that in this case there will be unique symmetric equilibrium
regulatory standards given by (

N,

N). Banks will indeed make the maximum investment under
these regulatory standards if their expected prot is positive. Because in this case banks receive
zero returns in the bad state, their expected prot is equal to q(R r

)n
i
, as can be seen from
(21). The expected prot is positive if R > r

when P = P

N,

N). Using the denition of r

in
29
(24) this condition can be written as
P >
c(1 q)R
R 1
(25)
Because = 1 c/P, this condition can be restated as

c(1 q)R
R 1
(26)
In order to see that this is indeed the case in the symmetric equilibrium obtained in Section 3
(when =

N,

N)), rearrange the FOCs of the regulators problem given by (16) to get

N,

N) =
1 qR
(1 q)R

(

N,

N)
N
i
N
i
>
1 qR
(1 q)R
(27)
since (

N,

N)/N
i
< 0 as shown in the proof of Lemma 3. Therefore, the symmetric equilibrium
obtained under the deposit insurance will prevail when this assumption is removed.
5.1.1 No deposit insurance and perfectly competitive deposit markets
Now, instead of assuming that each bank is a local monopoly in the deposit market, I assume
that the deposit market is perfectly competitive and analyze the robustness of the results to this
change in the environment. If the deposit market is perfectly competitive banks will earn zero
prots because consumers will get all of the returns on the risky investment. Each bank in country
i will choose the amount of investment in the risky asset (n
i
) to maximize the expected utility of
a representative depositor:
max
0n
i
N
i
0
qRn
i
+ (1 q)Rn
i
n
i
(28)
With probability q, the consumers will receive a gross return of Rn
i
, and with probability 1 q,
they will receive (1 c/P)Rn
i
, which is the return on the assets retained by their bank after re
sales. The cost of the initial investment, n
i
, is subtracted to obtain net expected return to deposits.
For consumers who choose to deposit their endowments with the bank, the net expected return
must be greater than zero, and if it is greater than zero, consumers will choose to invest everything
they have. Hence, the regulatory requirement,n
i
N
i
will bind. But if regulators assume that
banks will make the maximum investment, we know from Section 3 that there will be a unique set
of regulatory standards given by (

N,

N). Last, we have to check whether banks will indeed chose
maximum investment if (N
i
, N
j
) = (

N,

N). Rearranging (28) shows that the expected net utility
of a representative depositor will be greater than zero if
P >
c(1 q)R
R 1
(29)
30
This is the same condition as (25). We know from the analysis above that this condition is sat-
ised in the symmetric equilibrium of Section 3, i.e. when P = P

N,

N). Therefore, we can
conclude that the symmetric equilibrium obtained under the deposit insurance will prevail when
this assumption is removed regardless of whether the deposit market is competitive or each bank
is a local monopoly in the deposit market.
5.2 Limited Liability
In the basic model, I assumed that banks are protected by limited liability. Limited liability
assumption means that bank prots are (weakly) positive in each state of the world. If returns to
the assets of a bank fall short of its liabilities, then the bank owners receive zero prots. Banks
have always wanted to make unlimited investment in the risky asset under this assumption. Now
instead, suppose that bank owners have some wealth or endowment at the last period that can
be seized by depositors if the returns on assets are not enough to cover the promised payments to
depositors.
22
When there is no limited liability, a bank in country i chooses 0 n
i
N
i
to maximize the
expected prots:
max
0n
i
N
i
0
qRn
i
+ (1 q)R(1 c/P)n
i
n
i
(30)
where P is the price of capital in the re sale market in the bad state at t = 1. Each bank takes
this price as given. This problem is essentially the same as the problem of banks when there is no
deposit insurance and the deposit market is perfectly competitive. This can be seen by comparing
problems (21) and (30).
The rst order condition for the problem of banks is

n
i
= qR + (1 q)R(1 c/P) 1 (31)
The rst order condition will be positive if and only if
P >
c(1 q)R
R 1
P (32)
In other words, as long as P P, banks will still want to make unlimited investment in the risky
asset. But if regulators expect banks to set n
i
= N
i
, they will choose the same set of regulatory
standards as in the case with limited liability. Note that as long as regulators internalize the losses
of bank owners due to re sales, their objective function will be the same as (15). Therefore, in order
to show that equilibrium regulatory standards do not change when the limited liability assumption
is removed, we have to check whether the price of capital in the uncoordinated equilibrium satises
22
Instead, the negative utility of bank owners in this case can be interpreted as the disutility of legal punishment
for bankruptcy.
31
P

N,

N) P. This is again the same condition as (25). The analysis in Section 5.1 showed
that this condition indeed holds in equilibrium. Therefore, the symmetric equilibrium and the
qualitative results obtained under the limited liability assumption will prevail when we remove this
assumption.
5.3 Initial Bank Equity Capital
In the basic model, I also assumed that banks have no initial endowment of their own that they
can invest in the risky asset. Because banks raised necessary funds for investment from the deposit
market, the liability side of their balance sheets contained only debt and not any equity capital.
23
In this section, I assume that banks have an initial endowment of E units of consumption good
which they have to invest in the risky asset. This equity is costly: the opportunity cost of equity
to bank owners, , is greater than one, the cost of insured deposits. These two assumptions are a
common way of introducing equity capital to a banking model (see DellAriccia and Marquez (2006),
Hellmann et al. (2000), and Repullo (2004) among others). The assumption that the amount of
equity capital is xed captures the fact that it is dicult for banks to raise equity capital at short
notice.
When there is bank equity capital, regulation will take the form of a minimum capital ratio
requirement. Let k
i
= E/n
i
be the actual capital ratio of a bank in country i. In this case,
regulation will require banks to have k
i
K
i
, where K
i
is the capital adequacy requirement in
country i.
Given its equity, and the price of capital goods in the bad state of t = 1, each bank chooses
how much to invest in the risky asset (i.e., n
i
as before) to maximize expected prots:
max
0n
i
N
i
0
q(Rn
i
(n
i
E)) + (1 q)max{Rn
i
(n
i
E), 0} E (33)
subject to the capital regulation
k
i
=
E
n
i
K
i
(34)
Note that n
i
E is the amount of funds borrowed from the local deposit market. We can write the
capital ratio requirement condition as
n
i

E
K
i
N
i
(35)
This analysis shows that there is one-to-one mapping from capital regulations to the form of regu-
lation used in the main text. The banks problem does not change: they still want to invest in the
23
The term equity capital should not be confused with the capital good. Any initial endowment of bank owners will
still be in the form of consumption good. I use the term equity capital to refer to bank owners own endowments
that they invest in the bank.
32
risky asset as much as possible, as long as the net expected return is positive. The minimum capital
requirement binds (i.e., banks will choose n
i
= E/K
i
in an equilibrium with positive investment
levels).
Consider the regulators problem after equity is introduced to the model. Regulators will
anticipate that for a given capital ratio requirement, K
i
, banks will choose their total investment
level such that this requirement binds: n
i
= E/K
i
. Because banks will raise E/K
i
E = (1
K
i
)E/K
i
units of consumption goods from the local deposit market, we can write regulators
objective function as
W
i
(K
i
, K
j
) = q
_
R
E
K
i

E
K
i
_
+ (1 q)
_
R
_
E
K
i
,
E
K
j
_
E
K
i

E
K
i
_
(36)
The function () is the same as the function () except that it is dened over the minimum capital
ratios (K
i
, K
j
), not over the total investment levels. It represents the fraction of initial assets that
a bank retains after re sales. If we dene N
i
E/K
i
we can express the objective function above
as
max
N
i
0
W
i
(N
i
, N
j
) = max
N
i
0
q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
] (37)
This objective function is exactly the same as the regulators problem in the main text. Therefore,
all qualitative results in the main section will carry on when we introduce costly bank equity and
redene regulation as a minimum capital ratio requirement.
Note that when we introduce costly equity, the net expected return on the risky investment
must be suciently large to cover the opportunity cost of internal bank equity, E, for banks.
Otherwise, banks will choose not to invest in the risky asset at all. For this reason, the set of
parameters where we have strictly positive investment in equilibrium is smaller under costly equity.
6 Systemic Failures in Regulated Economies
In this section I examine systemic failures when the two countries are symmetric. By systemic
failures I refer to the fact that all banks in the two countries become insolvent after re sales.
Systemic failures will occur if the asset prices in the crisis state are so low that the returns from
investments that could be retained by banks after re sales are not enough to cover the promised
return to depositors, which is simply equal to the initial value of the investment. Systemic failures
might occur even in regulated economies. Because countries are symmetric and we assume perfectly
correlated shocks across countries, systemic failures, if they occur, will happen in both countries at
the same time. We can write the systemic failure condition in equilibrium as
R

N,

N)

N <

N (38)
33
where

N denotes symmetric equilibrium investment levels. The left hand side is the (expected)
return from investments that could be retained by banks after the re sales, and the right hand
side is the promised payments to depositors, which are simply the initial value of the investment.
For the rest of the analysis I will work with a particular functional form for which I can obtain
a closed-form solution for equilibrium investment levels. The technology of global investors is given
by: F(y) = Rln(1 +y). I solve the model for this particular functional form in Appendix A. Using
this closed form solution, we can show that the systemic failure condition given by (38) above is

N >
R
2c
_
R 1 c
R 1
_
N
c
(39)
where N
c
is dened as the critical equilibrium investment level beyond which banks fail in the bad
state (i.e., if

N > N
c
then banks in the two countries become insolvent in the bad state). We have
already seen that

N is increasing in q. This helps to prove the following result.
Proposition 10. Let F(y) = Rln(1 + y). If 1 + c < R <

R then there exists a q (0, 1/R) such
that for all q q we have that

N(q) N
c
. In other words, for such R, if the probability of the good
state is higher than q, banks fail in the bad state in the uncoordinated equilibrium. If R 1 + c
then banks always fail in the bad state, and if R

R then banks never fail in the bad state where

R
1
2
_
2 +c +

8 +c
_
(40)
Proof. See the appendix.
By Proposition 4 we have already seen that equilibrium investment level is increasing in q and
R. Proposition 10 shows that if R is suciently high, then systemic failures do not occur. In
order to prove this, I show that the dierence N
c

N is monotonically increasing in R, and that
this dierence is positive for any value of q if R is suciently high. Remember that banks fail if
N
c
<

N, which means that they will not fail as long as the dierence N
c

N is positive. But
if R has moderate values, given by 1 + c < R <

R, then banks fail in the bad state only if the
probability of good state, q, is suciently high. For moderate values of R, a suciently high q leads
to systemic failures because

N is increasing q, whereas N
c
is independent of q as can be seen from
(39). Hence, for any value of R such that 1 +c < R <

R, there is a suciently high q such that the
dierence N
c

N is negative. Last, if R is suciently low, given by R < 1 + c, then total return
from maintained assets after re sales is never enough to cover the initial value of the investment,
because 1 +c is the marginal cost of funds for the investment if the bad state is expected to occur
with certainty. In order to prove this, I show that for these low values of R, the dierence N
c

N
is negative for any value of q. Therefore in this case, systemic failures will surely happen in the
bad state.
The region of parameters for which systemic failures occur in the bad state is illustrated in the
34
left panel of Figure 4. The horizontal axis in Figure 4 measures q, the probability of success, from
0 to 1, and the vertical axis measures R, the return to investment, from 1 to 2. Since we assume
that Rq 1, we should ignore the region where Rq > 1 in Figure 4. This region is shaded by grey.
The blue region shows the set of R, q pairs for a given c, for which systemic failures occur in the
bad state. Technically, in the blue region we have that

N > N
c
. There are two horizontal red lines
in the left panel of Figure 4. The lower one shows R = 1 + c, and it is clear from the graph that
banks fail for any value of q if R 1 + c. The higher red line shows R =

R, and it is again clear
from the graph that systemic failures never occur if R

R. Last, if R is between the two red lines
(i.e., if 1 + c < R <

R), then for any such R there exists some q (0, 1/R) such that systemic
failures occur if q q, as claimed in Proposition 10.
Figure 4: Systemic Failures
It is clear from the analysis above that systemic failures are more likely when the initial invest-
ment level is high. Because central regulation reduces investment levels in both countries, we can
claim that moving to a central regulation can eliminate systemic failures. This can be observed
from the right panel in Figure 4 where the parameter values for which systemic crisis occurs under
the common central regulation are shown in blue. The parameter set for which systemic crisis
occurs in the uncoordinated equilibrium is the sum of the colored regions (the same area as in the
left panel). It is clear from this right panel that when countries move to common central regulation,
the parameter set for which systemic failures occur in the bad state shrinks. The following lemma
shows the parameter values under which systemic crisis does occur in the bad state in the uncoor-
dinated equilibrium and moving to a common central regulation eliminates the crisis. Therefore, a
common central regulation improves not only the social welfare but also the nancial stability of
coordinating countries.
Lemma 7. For any given R <

R, there exists some q > q, where q is as dened in Proposition
10, such that if q ( q, q] moving to a central common regulation from the symmetric uncoordinated
35
equilibrium eliminates the systemic failures in the bad state.
Proof. See the appendix.
7 Conclusion
I have examined the incentives of national regulators to coordinate regulatory policies in the pres-
ence of systemic risk in global nancial markets, using a two-country, three-period model. Banks
borrow from local deposit markets and invest in risky long-term assets in the initial period. They
may face negative shocks in the interim period that force them to sell assets. Asset sales of banks
feature the characteristics of a re sale: assets are sold at a discount, and the higher the number
of assets sold, the lower the market price of assets is. The asset market in the interim period is
competitive. Each bank treats the asset price as given, and therefore neglects the eects of its sales
on other banks. Due to this externality, correlated asset re sales by banks generate systemic risk
across national nancial markets.
If the regulatory standard is relaxed in one country, banks in this country invest more in the
risky asset in the initial period. If the bad state arises in the interim period, these banks are forced
to sell more assets, causing the asset price to fall further. A lower asset price will increase the cost
of distress for the banks in the other country as well. Banks may even default in equilibrium if the
asset prices fall below a threshold.
I have shown that, in the absence of cooperation, independent regulators choose ineciently low
regulation compared to regulatory standards that would be chosen by a central regulator. A central
regulator takes the systemic risk into account and improves welfare in cooperating countries by
imposing higher regulatory standards. Therefore, it is incentive compatible for national regulators
of symmetric countries to relinquish their authority to a central regulator.
I have also considered the incentives of regulators when there are asymmetries between countries
with a focus on the asymmetries in asset returns. In particular, I have assumed that banks in one
country are uniformly more productive than the banks in the other country in terms of managing
long-term assets. I have shown that cooperation would voluntarily emerge only between suciently
similar countries. In particular, the regulator in the high-return country chooses lower regulatory
standards in equilibrium and is less willing to compromise on stricter regulatory standards.
8 Appendix
8.1 Functional Form Examples
Example 1 F(y) = Rln(1 +y)
36
For this return function we obtain the (inverse) demand function as
P = F

(y) =
R
1 +y
and hence y = F
1
(P) =
R P
P
D(P) (8.1)
This demand function is clearly downward slopping and convex as seen below
D

(P) =
R
P
2
< 0 and D

(P) =
2R
P
3
> 0
F() satises Assumption CONCAVITY since
F

(y) =
R
(1 +y)
2
< 0 and since F

(0) = R
Lets check whether this functional form satises the conditions given by Assumption ELASTICITY
and Assumption REGULARITY, respectively.
F

(y) +yF

(y) =
R
1 +y
y
R
(1 +y)
2
=
Ra
(1 +y)
2
> 0
Clearly Assumption ELASTICITY is satised. Below we see that this function satises Assumption
REGULARITY as well:
F

(y)F

(y) 2F

(y)
2
=
R
(1 +y)
2R
(1 +y)
3
2
_
R
(1 +y)
2
_
2
= 0
From above we can also see that this return function induces a log-convex demand function since
we will have F

(y)F

(y) F

(y)
2
> 0
Example 2 F(y) =
_
y +a
2
For this example the demand function will be obtained as
P = F

(y) =
1
2
_
y +a
2
and hence y = F
1
(P) =
1
4P
2
a
2
D(P)
This demand function is also downward slopping and convex as seen below
D

(P) =
1
2P
3
< 0 and D

(P) =
3
2P
4
> 0
Assumption CONCAVITY is satised since
F

(y) =
1
4 (y +a)
3/2
< 0 and F

(0) = R implies that

1
2a
= R or a =
1
2R
37
We can easily show that Assumption ELASTICITY is satised:
F

(y) +yF

(y) =
1
2 (y +a)
1
2
y
1
4 (y +a)
3
2
=
y + 2a
4 (y +a)
3
2
> 0
Likewise we can show that this function satises Assumption REGULARITY as well:
F

(y)F

(y) 2F

(y)
2
=
1
2 (y +a)
1
2
3
8 (y +a)
5
2
2
_
1
4 (y +a)
3
2
_
2
=
1
16 (y +a)
3
< 0
Note that in contrast to the rst example this functional form induces a log-concave demand
function since we can show that F

(y)F

(y) F

(y)
2
< 0
8.2 Symmetric Countries: An example
In this section, I obtain closed form solutions for non-cooperative equilibrium regulation levels and
regulation levels under cooperative benchmark for the particular functional form choice for the
global investors technology given by Example 1 above.
For analytical convenience suppose that the technology of global investors is given by the fol-
lowing logarithmic function as investigated by Example 1 above
F(y) = Aln(a +y) (8.2)
where the amount of assets the global investors optimally buy satises the following rst order
conditions
F

(y) =
A
a +y
= P (8.3)
which will induce a downward slopping demand function
y = F

(P)
1
=
AaP
P
D(P) (8.4)
Imposing Assumption CONCAVITY on this functional form gives
F

(0) = R
A
a
= R or A = aR (8.5)
It is shown in the previous section that this functional form satises the conditions given by
Assumptions ELASTICITY and REGULARITY. Since this functional form satises all sucient
conditions, we can proceed with solving for the equilibrium. We start solving the model backwards
as in the previous section. Therefore we rst nd the equilibrium price at t = 1 using the market
38
clearing condition.
D(P) = S(P, N
A
+N
B
)
AaP
P
=
c(N
A
+N
B
)
P
(8.6)
Hence, we get the equilibrium price of assets at t = 1 as
P

=
Ac(N
A
+N
B
)
a
(8.7)
which is clearly decreasing in the investment levels in both countries. Note also that equilibrium
price is determined only by the total investment level in the two countries. Exact division of the
total investment between the countries will not aect P

. This property of the equilibrium price of

assets is very helpful in the analysis of the model.
We can also obtain equilibrium fraction of assets retained by banks after re sales as a function
of initial investment levels in each country by plugging the equilibrium price given by equation
(8.7) into equation (9) that denes this fraction as a ratio of market price, which will give that

(N
A
, N
B
) = 1
c
P

(N
A
, N
B
)
= 1
ac
Ac(N
A
+N
B
)
(8.8)
Remember that regulator i

s objective function is
max
N
i
0
W
i
(N
i
, N
j
) = q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
] (8.9)
Substituting for

(N
i
, N
j
) from (8.8) gives
max
N
i
0
W
i
(N
i
, N
j
) = q(R 1)N
i
+ (1 q)
_
R
_
1
ac
Ac(N
i
+N
j
)
_
N
i
N
i
_
(8.10)
where FOCs can be obtained as
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)R
_
1
ac(AcN
j
)
[Ac(N
i
+N
j
)]
2
_
1 = 0 (8.11)
Solving for N
i
gives the best response function of regulator i
N

i
(N
j
) =
AcN
j

_
ac(AcN
j
)
c
(8.12)
where we dene

(1 q)R
R 1
(8.13)
We can use the best response functions to solve for the symmetric equilibrium investment level.
39
After some algebra we can obtain that for i = A, B

N
i
=
4Aac
_
8Aac + (ac)
2
8c
(8.14)
Note that by Assumption CONCAVITY we impose that A = aR. Substituting for A using this
identity gives

N
i
=
4R c
_
8Rc + (c)
2
8c
(8.15)
8.2.1 Central Regulation
Lets consider the central regulators problem where the central regulator chooses the total invest-
ment level in both countries.
max
N0
W(N) = q(R 1)N + (1 q)[R

(N)N N] (8.16)
Denote the solution to this global problem by N. Central regulator will impose maximum
investment level in each country as

N
i
= N/2 as discussed before. N will be characterized by the
FOCs of the problem above which we could derive as
W(N)
N
= qR + (1 q)R
_
1
acA
(AcN)
2
_
1 = 0 (8.17)
Solving for N and substituting

N
i
= N/2 gives the globally regulated investment level in each
country as

N
i
=
A

Aac
2c
(8.18)
8.3 Proofs Omitted in the Text
8.3.1 Proofs for the Symmetric Countries Case
Lemma 1. P

(n
i
, n
j
) is decreasing in n
i
for i = A, B under Assumptions CONCAVITY and
ELASTICITY.
Proof. Applying the IFT on the MC condition gives
dP

dn
i
=
E()/n
i
E()/P
=
S(P

, n
i
, n
j
)/n
i
D

(P

) S(P

, n
i
, n
j
)/P
(8.19)
First, note that using the expression for the total supply function given by (13) we obtain
S()
n
i
=
c
P
> 0 (8.20)
40
Hence, we can write the derivative in (8.19) as
dP

dn
i
=
c
P

(P

) P

[S(P

, n
i
, n
j
)/P]
(8.21)
The following equivalence will help us to write this derivative using only return function of global
investors return function, F(), and its derivatives
P

S(P

, n
i
, n
j
)
P
= P

c(n
i
+n
j
)
P
2
=
c(n
i
+n
j
)
P

= S(P

, n
i
, n
j
) (8.22)
Using this equivalence we can express the derivative given by (8.21) as
dP

dn
i
=
c
P

(P

) +S(P

, n
i
, n
j
)
(8.23)
Let y

S(P

, n
i
, n
j
) denote the total volume of equilibrium re sales. In equilibrium we will
have P

= F

(y

) from the demand curve. Therefore we can obtain

D

(P

) =
1
F

(y

)
(8.24)
where we make use of the fact that D

(P) F

(P)
1
as given by (3). Hence, we can rewrite the
denominator of the expression (8.23) above as
P

(P

) +S(P

, n
i
, n
j
) =
F

(y

)
F

(y

)
+y

(8.25)
which we can write equivalently as
P

(P

) +S(P

, n
i
, n
j
) =
F

(y

) +y

(y

)
F

(y

)
< 0 (8.26)
This expression is negative since F

(y) < 0 by Assumption CONCAVITY and

F

(y) +yF

(y) > 0 (8.27)

by Assumption ELASTICITY. Therefore we conclude that dP

/dn
i
< 0.
Lemma 2. Equilibrium fraction of assets sold in each country, 1

(n
i
, n
j
), is increasing in n
i
for i = A, B under Assumptions CONCAVITY and ELASTICITY.
Proof. Using (9) we can write banks asset sales in equilibrium as
1

(n
i
, n
j
) =
c
P

(n
i
, n
j
)
(8.28)
41
Note that

n
i
=

dP

dn
i
< 0 (8.29)
since /P = c/P
2
> 0 from (9) and by Lemma 1 we have that dP

/dn
i
< 0 for i = A, B.
Therefore, equilibrium fraction of assets rescued after re sales (

) is decreasing in n
i
for i = A, B.
Since equilibrium fraction of assets sold in each country is given by 1

(n
i
, n
j
), we obtain
that this fraction is increasing in n
i
for i = A, B.
Lemma 3. Under Assumptions CONCAVITY, ELASTICITY and REGULARITY, each regula-
tors best response is unique valued.
Proof. For this proof I refer to the conditions given by Assumptions CONCAVITY, ELASTICITY
and REGULARITY. I show that if the global investors return function, F() satises these con-
ditions, then the objective functions of independent regulators are concave. This also tells us that
rst order conditions of the regulators problem, which also implicitly denes their best response
functions, is monotone and decreasing. Therefore, there is a unique solution to these rst order
conditions or in other words best response of each regulator is unique valued.
Lets reproduce regulators objective function here for convenience
max
N
i
0
W
i
(N
i
, N
j
) = max
N
i
0
q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
] (8.30)
FOCs for regulator of country is problem will be given by
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)R
_

(N
i
, N
j
)
N
i
N
i
+

(N
i
, N
j
)
_
1 (8.31)
Lets dene the following function for convenience
v
i
(N
i
, N
j
)

(N
i
, N
j
)
N
i
N
i
+

(N
i
, N
j
) (8.32)
Hence, we can write the FOCs simply as
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)Rv
i
(N
i
, N
j
) 1 (8.33)
We will show that Under Assumptions CONCAVITY, ELASTICITY and REGULARITY we have
v
i
1
(N
i
, N
j
) < 0, hence the objective function is concave. This also means that the best response
functions are unique-valued. Note that in Lemma 1 we have obtained
dP

dN
i
=
c
P

(P

) +S(P

, N
i
, N
j
)
(8.34)
which is negative as we have shown there. Since D(P

) = S(P

, N
i
, N
j
) by the market clearing
42
condition, we can also express this derivative as
dP

dN
i
=
c
P

(P

) +D(P

)
(8.35)
We will use this expression in the derivative of

with respect to N
i
below

(N
i
, N
j
)
N
i
=

(N
i
, N
j
)
P

dP

dN
i
(8.36)
=
_
c
P

_
2
1
D(P

) +P

(P

)
< 0
Hence we can obtain the second derivative as

(N
i
, N
j
)
N
2
i
=
_
c
P

_
2
G(P

)
dP

dN
i
(8.37)
where we dene
G(P

)
2D(P

) + 4P

(P

) +P
2
D

(P

)
P

[D(P

) +P

(P

)]
2
(8.38)
Note that the derivative of v
i
(), which was dened by (8.32), with respect to the rst argument is
equal to
v
i
1
(N
i
, N
j
)
v
i
(N
i
, N
j
)
N
i
=

2

(N
i
, N
j
)
N
2
i
N
i
+ 2

(N
i
, N
j
)
N
i
(8.39)
Put the ndings above together to get this derivative as
v
i
1
() =
_
c
P

_
2
_
G(P

)N
i

2
c
_
dP

dN
i
(8.40)
again where G(P

) is as dened by (8.38) above.We will show that G(P

) is negative under As-

sumptions CONCAVITY, ELASTICITY and REGULARITY, and hence v
i
1
() < 0.
Note that using D(P) F

(P)
1
and P = F

(y) we can obtain

(i) D(P) = y, (ii) D

(P) =
1
F

(y)
and (iii) D

(P) =
F

(y)
F

(y)
3
(8.41)
Hence, we can write the expression in the numerator of G(P

) as
2D(P

) + 4P

(P

) +P
2
D

(P

) = 2y

+
4F

(y

)
F

(y

)

F

(y

)
2
F

(y

)
F

(y

)
3
(8.42)
Re-arranging the RHS of (8.42), we obtain
2y

(y

)
3
+ 4F

(y

)F

(y

)
2
F

(y

)
2
F

(y

)
F

(y

)
3
(8.43)
43
Note that the denominator of the last expression is negative by Assumption CONCAVITY. Re-
arrange the numerator to write it as
2F

(y

)
2
_
y

(y

) +F

(y

. .
(+) by ELASTICITY
F

(y

)
_
F

(y

)F

(y

) 2F

(y

)
2

. .
> 0
() by REGULARITY
(8.44)
The expression in (8.44) is positive under Assumption ELASTICITY and REGULARITY as shown
above. This implies that
2D(P

) + 4P

(P

) +P
2
D

(P

) < 0 (8.45)
i.e. the numerator of G(P

) which was given by (8.42) is negative. Putting these results together

we obtain that v
i
1
(N
i
, N
j
) < 0.
The analysis above shows that FOCs of each regulator is monotone and decreasing. Therefore, we
conclude that their best response functions are unique valued as desired.
Proposition 1. Under Assumptions CONCAVITY, ELASTICITY and REGULARITY, optimal
regulatory standards in the two countries are strategic substitutes.
Proof. Optimal regulatory standards in the two countries are strategic substitutes if and only if the
best response functions of regulators are downward slopping. We can apply the Implicit Function
Theorem (IFT) on the rst order conditions to obtain the sign of the slope of best response functions.
This sign is shown to be equal to the sign of the cross derivative of the objective function due to
the results in Proposition 1. In order to show that the sign of the cross derivative of the objective
function is negative, I again refer to the technical conditions given by Assumptions CONCAVITY,
ELASTICITY and REGULARITY. I show that for any (induced) demand function that satises
Assumptions CONCAVITY to REGULARITY, this sign is negative and hence optimal investment
levels are strategic substitutes.
If N
i
and N
j
are strategic substitutes we must have
2
W
i
(N
i
, N
j
)/N
i
N
j
< 0. Remember that
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)Rv
i
(N
i
, N
j
) 1 (8.46)
where
v
i
(N
i
, N
j
)

(N
i
, N
j
)
N
i
N
i
+

(N
i
, N
j
) (8.47)
hence

2
W
i
(N
i
, N
j
)
N
i
N
j
= (1 q)R
v
i
(N
i
, N
j
)
N
j
(8.48)
= (1 q)R
_

(N
i
, N
j
)
N
i
N
j
N
i
+

(N
i
, N
j
)
N
j
_
< 0
44
The sing of equation (8.48) is negative since
(i)

(N
i
,N
j
)
N
j
< 0 as shown by eq (8.36) and
(ii) for the cross derivative of

(N
i
, N
j
) we know that

(N
i
, N
j
)
N
i
N
j
=

2

(N
i
, N
j
)
N
2
i
(8.49)
since

(N
i
, N
j
) is determined only by the sum of the two investment levels, not by their individual
values. Therefore we get an equation similar to the one obtained in the proof of Lemma 3
v
i
2
()
v
i
(N
i
, N
j
)
N
j
=
_
c
P

_
2
_
G(P

)N
i

1
c
_
dP

dN
i
(8.50)
which is negative under Assumptions 1 to 3 as shown in the proof of Lemma 3. Hence, the best
response functions are downward slopping which implies that N
i
and N
j
are strategic substitutes.
Lemma 4. The best responses of each country satisfy 0 < N

i
< for i = A, B if Assumption
RANGE holds, i.e. if 1 +c(1 q) < R 1/q.
Proof. Part 1 If R 1/q then N

i
< .
Lets rst dene M = N
A
+N
B
such that
P

(M) = c (8.51)
Fix some N
j
and consider two exhaustive cases:
Case 1 N
j
< M.
Dene
N
i
= M N
j
(8.52)
Consider regulators objective function as N
i
N
i
lim
NiN
i
W
i
(N
i
, N
j
) = lim
NiN
i
q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
] (8.53)
Now note that
lim
NiN
i

(N
i
, N
j
) = lim
NiN
i
_
1
c
P

(N
i
, N
j
)
_
= 0 (8.54)
since lim
Ni,N
j
M
P

(N
i
, N
j
) = c by denition in (8.51). Therefore
lim
NiN
i
W
i
(N
i
, N
j
) = lim
NiN
i
(qR 1)N
i
0 (8.55)
45
since qR 1 by Assumption RANGE. Hence, it is never optimal to choose N

i
N
i
in this case.
Case 2 N
j
M.
By denition of M this implies that P

(N
i
, N
j
) < c for any N
i
> 0. In this case banks optimally
discard all capital at t = 1, i.e.

= 0 which implies that

(N
i
, N
j
) = 0. Hence social welfare in
country i will be given by
W
i
(N
i
, N
j
) = (qR 1)N
i
0 (8.56)
since qR 1 by Assumption RANGE. Hence, N

i
= 0 in this case. Therefore we conclude proof of
Part 1 by showing that N

i
< for i = A, B as long as qR 1.
Part 2 For the second part of the proof I will show that welfare in country i is always
decreasing in N
i
when R < 1+c(1q), and hence the best responses will be given by N

i
= N

j
= 0.
Remember
W
i
(N
i
, N
j
) = q(R 1)N
i
+ (1 q)[R

(N
i
, N
j
)N
i
N
i
] (8.57)
Note that the highest value of

(N
i
, N
j
) will be obtained as N
i
, N
j
0. Therefore
If W
i
(0, 0) < 0 then W
i
(N
i
, N
j
) < 0 for all N
i
, N
j
> 0 (8.58)
Consider rst
lim
Ni,N
j
0

(N
i
, N
j
) = lim
Ni,N
j
0
_
1
c
P

(N
i
, N
j
)
_
= 1
c
R
(8.59)
since lim
Ni,N
j
0
P

(N
i
, N
j
) = R. Using this we can write
lim
Ni,N
j
0
W
i
(N
i
, N
j
) =
_
q(R 1) + (1 q)[R
_
1
c
R
_
1]
_
N
i
(8.60)
from which we can obtain lim
Ni,N
j
0
W
i
(N
i
, N
j
) < 0 as long as
q(R 1) + (1 q)[R
_
1
c
R
_
1] < 0 (8.61)
Re-arranging this inequality gives
qR + (1 q)R + (1 q)c 1 < 0 (8.62)
where further simplication implies R < 1 +c(1 q).
Hence, we conclude that N

i
(N
j
) = 0 for i = A, B as long as R < 1 +c(1 q).
Proposition 2. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE,
at least one pure strategy Nash Equilibrium exists in the game between two nancial regulators at
46
t = 0. Moreover there exists at least one symmetric pure strategy NE.
Proof. For this proof, I make use of a theorem due to Debreu (1952) which states that Suppose
that for each player the strategy space is compact and convex and the payo function is continuous
and quasi-concave with respect to each players own strategy. Then there exists at least one pure
strategy NE in the game.
I establish below that this game satises all three conditions stated in this theorem.
(i) Following Lemma 4 we can restrict strategy space for each regulator to [0, M] which is
compact and convex.
(ii) Continuity of the objective function is obvious.
(iii) For concavity we evaluate the second derivative of the objective function with respect to
the own action:

2
W
i
(N
i
, N
j
)
N
2
i
= (1 q)R
v
i
(N
i
, N
j
)
N
i
< 0 (8.63)
as shown by Lemma 3 above. Hence Nash Equilibria equilibria exist. Existence of a symmetric
Nash Equilibrium equilibrium is implied by the symmetry of the game.
Proposition 3. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE,
there exists a unique symmetric NE of the game between the regulators at t = 0.
Proof. I will make use of the theorem that states: If the best response mapping is a contraction
on the entire strategy space, there is a unique Nash Equilibrium in the game.
In two-player games best response functions are contraction everywhere if the absolute value of
their slopes are less than one everywhere. In order to show this, I make use of the nice feature of
equilibrium price function that it is determined only by the sum of the investment levels in the two
countries.

i
(N
j
)
N
j

2
W
i
(N
i
,N
j
)
N
i
N
j

2
W
i
(N
i
,N
j
)
N
2
i

< 1 (8.64)
which can be equivalently stated as

2
W
i
(N
i
, N
j
)
N
i
N
j

<

2
W
i
(N
i
, N
j
)
N
2
i

(8.65)
Using the expressions for these derivatives given before this corresponds to

(N
i
, N
j
)
N
i
N
j
N
i
+

(N
i
, N
j
)
N
j

<

(N
i
, N
j
)
N
2
i
N
i
+ 2

(N
i
, N
j
)
N
i

(8.66)
47
Note that derivative on the right hand side is negative by Lemma 3 and the derivative on the
left hand side is negative by Proposition 1. Moreover,

(N
i
, N
j
) and its derivatives are determined
only by the sum of the two investment levels, not by their individual values. This implies that

(N
i
, N
j
)
N
j
N
i
=

2

(N
i
, N
j
)
N
2
i
and

(N
i
, N
j
)
N
j
=

(N
i
, N
j
)
N
i
(8.67)
which implies that |LHS| < |RHS| in (8.66) and hence the slope of best response functions is less
than one everywhere on the domain.
Proposition 4. Non-cooperative symmetric equilibrium investment levels are increasing in q and
R.
Proof. Using Cramers rule on FOCs we get

N
i

2
W
i
N
i

2
W
j
N
2
j

2
W
i
N
i
N
j

2
W
j
N
j

2
W
i
N
2
i

2
W
j
N
2
j

2
W
i
N
i
N
j

2
W
j
N
j
N
i
(8.68)
where {q, R, c} is a parameter of the model. First note that

2
W
i
N
2
i
< 0, and

2
W
j
N
j
N
i
< 0 for i = A, B (8.69)
by Lemma 3 and Proposition 1. Moreover in the proof of Proposition 3 we have shown that

2
W
i
(N
i
, N
j
)
N
i
N
j

<

2
W
i
(N
i
, N
j
)
N
2
i

(8.70)
which implies that the sign of the denominator above is positive. Moreover, in a symmetric equi-
librium we will have

2
W
i
N
i

=

2
W
j
N
j

and

2
W
j
N
2
j
=

2
W
i
N
2
i
(8.71)
which allows us to write the derivative as

N
i

2
W
i
N
i

()
..
_

2
W
i
N
2
i

2
W
i
N
i
N
j
_

2
W
i
N
2
i

2
W
j
N
2
j

2
W
i
N
i
N
j

2
W
j
N
j
N
i
. .
(+)
(8.72)
the term inside the brackets in the numerator is again negative by the inequality (8.70). Therefore
the sing of the derivative above will be equal to the sign of
2
W
i
/N
i
. To obtain this sign
48
consider the FOCs of regulators problem
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)Rv
i
(N
i
, N
j
) 1 (8.73)
which will imply that in equilibrium
v
i
(

N
i
,

N
j
) =
1 qR
(1 q)R
(8.74)
Therefore we can obtain that
W
i
(N
i
, N
j
)
N
i
R
= q + (1 q)v
i
(N
i
, N
j
) (8.75)
= q +
1 qR
R
=
1
R
> 0
in equilibrium, using equation (8.33). Hence we conclude that

N
i
/R > 0.
For comparative statics with respect to q consider
W
i
(N
i
, N
j
)
N
i
q
= R Rv
i
(N
i
, N
j
) (8.76)
which in equilibrium, using equation (8.33) we can write as
W
i
(

N
i
,

N
j
)
N
i
q
= R
_
1
1 qR
(1 q)R
_
(8.77)
=
R 1
1 q
> 0 (8.78)
hence we can also conclude that

N
i
q
> 0 for i = A, B
i.e. equilibrium investment levels in both countries are increasing as the probability of good state
rises.
Lemma 5. Under Assumptions CONCAVITY, ELASTICITY, REGULARITY and RANGE, equi-
librium price of assets satisfy P

> c.
Proof. By Proposition 4 we have established that equilibrium investment levels are increasing in
both q and R. Since Assumption RANGE restricts qR 1, for a given set of other parameters
we will obtain the highest investment in equilibrium if qR = 1. Consider the FOCs of regulators
49
problem evaluated at equilibrium regulation standards for qR = 1
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)R
_

N
i
,

N
j
)
N
i

N
i
+

N
i
,

N
j
)
_
1 = 0
= (R 1)
_

N
i
,

N
j
)
N
i

N
i
+

N
i
,

N
j
)
_
= 0
which implies that

N
i
,

N
j
)
N
i

N
i
+

N
i
,

N
j
) = 0
Lemma 3 has shown that

(N
i
, N
j
)
N
i
< 0
Therefore for the FOCs above to hold we need

N
i
,

N
j
) = 1
c
P

N
i
,

N
j
)
> 0 (8.79)
which implies that P

N
i
,

N
j
) > c as needed.
Proposition 5.

N <

N, i.e. independent national regulators choose a higher investment in
equilibrium compared to the cooperative benchmark.
Proof. For this proof I use the alternative formulation of central regulators problem which I re-
produce here for convenience
max
N0
W(N) = q(R 1)N + (1 q)[R

(N)N N] (8.80)
The FOCs for this problem will be given by
W()
N

N
= qR + (1 q)R v(N) 1 = 0 (8.81)
where N is the optimal total investment level in the two countries. Rearranging the FOCs gives
that
v(N)

(N)N +

(N) =
1 qR
(1 q)R
(8.82)
where v() is similar to the function v
i
(N
i
, N
j
) except that it is dened over the total investment
level in the two countries. The same is true for the function

(). Now using the fact that globally

optimal investment level in each country will satisfy

N = N/2 we can write
v(N) = v(2

N) =

(2

N)(2

N) +(2

N) (8.83)
50
Note that equilibrium fraction of rescued assets,

() , is determined by the sum of the invest-

ment levels in the two countries. In other words, exact division of global investment between the
two countries do not aect

() and hence its derivatives. This property allows us to write v(N)

as (with a slight abuse of notation)
v(N) = v(2

N) =

(2

N)(2

N) +

(2

N)
= 2

1
(

N,

N)

N +

N,

N)
= v(

N,

N) +

1
(

N,

N)

N (8.84)
On the other hand, remember that each independent regulators FOCs will give us
W
i
()
N
i

(

N
i
,

N
j
)
= qR + (1 q)Rv
i
(

N
i
,

N
j
) 1 = 0 (8.85)
from which we get that in equilibrium
v
i
(

N
i
,

N
j
) =
1 qR
(1 q)R
(8.86)
Comparing (8.86) and (8.82) we see that v(N) = v
i
(

N,

N) where (

N,

N) are the symmetric
Nash equilibrium investment levels. Using this together with equality (8.84) we can write
v(

N,

N) = v(N)
1
(

N,

N)

N
= v
i
(

N,

N)
1
(

N,

N)

N (8.87)
we have previously shown that

(N
i
, N
j
) < 0 which implies that v(

N,

N) > v(

N,

N). Lemma 3 has
shown that v
i
(N
i
, N
j
) is decreasing in N
i
, and Proposition 1 has shown that v
i
(N
i
, N
j
) is decreasing
in N
j
. Since v
i
(N
i
, N
j
) is decreasing in both arguments, we can conclude that

N <

N.
Proposition 6. If the countries are symmetric then both regulators prefer to deliver their authority
to a central systemic risk regulator, i.e. W
i
(

N
i
,

N
j
) W
i
(

N
i
,

N
j
) holds for i = A, B.
Proof. First, lets dene

W
i
W
i
(

N
i
,

N
j
) as the welfare of country i = A, B under central regula-
tion and

W
i
W
i
(

N
i
,

N
j
) as the welfare of country i = A, B under the symmetric non-cooperative
equilibrium. Note that by symmetry we have

W
A
=

W
B
and

W
A
=

W
B
Since central regulation levels, (

N
A
,

N
B
), maximize the total welfare of the two countries, W
A
+W
B
,
by denition we know that

W
A
+

W
B


W
A
+

W
B
51
which implies that

W
i


W
i
for i = A, B.
8.3.2 Proofs for the Asymmetric Countries Case
Proposition 7. If R
A
> R
B
, then

N
A
>

N
B
in non-cooperative equilibrium.
Proof. Remember the FOCs of regulators problem
W
i
(N
i
, N
j
)
N
i
= qR + (1 q)Rv
i
(N
i
, N
j
) 1 (8.88)
which will imply that in equilibrium we have
v
i
(N
i
, N
j
) =
1 qR
i
(1 q)R
i

i
(8.89)
Note that
i
is decreasing in R. Since R
A
> R
B
by assumption, we will have that
A
<
B
.
Remember that by v
i
(N
i
, N
j
) is dened as
v
i
(N
i
, N
j
)

(N
i
, N
j
)
N
i
N
i
+

(N
i
, N
j
) (8.90)

A
<
B
implies that v
A
(

N
A
,

N
B
) < v
B
(

N
B
,

N
A
). Using the expression above this is equivalent
to

(N
A
, N
B
)
N
A

N
A
,

N
B

N
A
+

N
A
,

N
B
) <

(N
B
, N
A
)
N
B

N
B
,

N
A

N
B
+

N
B
,

N
A
) (8.91)
Since

(), and its derivatives determined only by the sum of the regulation levels, we will have

N
A
,

N
B
) =

N
B
,

N
A
) (8.92)
and

(N
A
, N
B
)
N
A

N
A
,

N
B
=

(N
B
, N
A
)
N
B

N
B
,

N
A
(8.93)
Using these two equalities in (8.91) gives

(N
A
, N
B
)
N
A

N
A
,

N
B
_

N
A


N
B
_
< 0 (8.94)
From Lemma 3 we know that

(N
A
, N
B
)
N
A

N
A
,

N
B
< 0 (8.95)
Therefore, for inequality (8.94) to be true we need to have

N
A


N
B
> 0, or equivalently

N
A
>

N
B
.
52
Proposition 8. There exists no central regulation level N > min{

N
A
,

N
B
}.
Proof. By Envelope Theorem we will have that
dW
i
(N

i
(N
j
), N
j
)
dN
j
=
W
i
(N
i
, N
j
)
N
j

N
i
=N

i
(N
j
)
< 0 (8.96)
By assumption we have

N
A
>

N
B
. Consider a central regulation N =

N
B
. From above we get that
W
B
(

N
B
,

N
B
) > W
B
(

N
B
,

N
A
). However, by denition W
A
(

N
B
,

N
B
) < W
A
(

N
A
,

N
B
), i.e. a central
regulation with N =

N
B
is rejected by Regulator A. Now consider N >

N
B
. By Envelope Theorem
we have that
W
A
(N

A
(N), N) < W
A
(

N
A
,

N
B
) (8.97)
Moreover, by denition
W
A
(N, N) < W
A
(N

A
(N), N) (8.98)
Hence, any common regulation such that N >

N
B
will also be rejected by regulator A.
Lemma 6. For any common regulation level N such that W
A
(N, N) > W
A
(

N
A
,

N
B
) we have
W
B
(N, N) > W
B
(

N
B
,

N
A
).
Proof. Suppose that W
A
(N, N) > W
A
(

N
A
,

N
B
) for some N. From Proposition 8 we know that
such N must satisfy N < N
B
. Note that we can obtain
W
A
(

N
A
,

N
B
) > W
A
(N

A
(

N
A
),

N
A
) > W
A
(

N
B
,

N
A
) (8.99)
where the rst inequality is follows from Envelope Theorem given by (8.97), and the second is by
denition of optimality. Remember that
W
i
(N
i
, N
j
) = qR
i
N
i
+ (1 q)R
i

(N
i
, N
j
)N
i
N
i
(8.100)
Hence, W
A
(N, N) W
A
(

N
B
,

N
A
) will be given by
qR
A
[N

N
B
] + (1 q)R
A
[

(N, N)N

N
B
,

N
A
)

N
B
] [N

N
B
] > 0 (8.101)
Which we can re-arrange and write as
(1 qR
A
)[

N
B
N] + (1 q)R
A
[

(N, N)N

N
B
,

N
A
)

N
B
] (8.102)
Now consider W
B
(N, N) W
B
(

N
B
,

N
A
) which will be equal to
(1 qR
B
)[

N
B
N] + (1 q)R
B
[

(N, N)N

N
B
,

N
A
)

N
B
] (8.103)
53
Now, lets compare (8.102) and (8.103). First note that the rst terms are positive in both of them.

N
B
N > 0 as argued above and it receives a higher weight in (8.103) since 1 qR
B
> 1 qR
A
due to our assumption that R
A
> R
B
. Now, if the second terms are also positive, (8.103) will be
positive and we are done. However, if second terms are negative, we know that this second term
receives a higher weight in (8.102). Hence, if (8.102) is positive in spite of this higher weight on the
negative term, (8.103) will necessarily be positive, since it carries a lower weight on the negative
term.
Proposition 9. Suppose that F

(0) = 1. Let s R
A
R
B
> 0. Then for any R
A
, there exists s
(0, R
A
1) such that W
A
(N
m
, N
m
)W
A
(

N
A
,

N
B
) 0 if s s, and W
A
(N
m
, N
m
)W
A
(

N
A
,

N
B
) <
0 otherwise.
Proof. Lets see how this dierence changes with s
d[W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
)]
ds
(8.104)
this derivative will be given by
W
A1
N
m
s
+W
A2
N
m
s
+
W
A
s

N
m
,N
m
W
A1
d

N
A
ds
W
A2
d

N
B
ds

W
A
s

N
A
,

N
B
(8.105)
Applying envelope theorem reduces this to
W
A
s

N
m
,N
m
W
A2
d

N
B
ds

W
A
s

N
A
,

N
B
(8.106)
Now note that since
W
A
(N
A
, N
B
) = q(R
A
1)N
A
+ (1 q)[R
A

(N
A
, N
B
)N
A
N
A
] (8.107)
we will have that
W
A
s

N
m
,N
m
= 0 and
W
A
s

N
A
,

N
B
= 0 (8.108)
Therefore
d[W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
)]
ds
= W
A2
d

N
B
ds
(8.109)
First note that
W
A2

W
A
(N, N)
N
B
= (1 q)R

(N
A
, N
B
)
N
B
N
A
< 0 (8.110)
Moreover, by Proposition 4 we have shown that

N
i
is decreasing in R for i = A, B. This will
imply that as s increases (or R
2
decreases), regulator 2 will choose a lower

N
B
in equilibrium, i.e.
54
d

N
B
/ds < 0.Therefore,
d[W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
)]
ds
= W
A2
d

N
B
ds
< 0 (8.111)
Hence, the benet from cooperating is decreasing in the dierence between the two countries.
Therefore if W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
) = 0 for some s it will be negative for s s.
We will show that such an s exists. It is clear that for s = 0 we have W
A
(N
m
, N
m
)
W
A
(

N
A
,

N
B
) > 0. On the other hand, as R
B
1 + c(1 q), we can argue that

N
B
will nec-
essarily become zero. From Proposition 11 we know that for central regulation be acceptable by
regulator 1, we must have N
m
<

N
B
. However, as

N
B
0, it wont be possible to reduce

N
B
suf-
ciently to compensate regulator 1. Therefore, we will have W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
) < 0 for
suciently large s. By continuity an s such that W
A
(N
m
, N
m
) W
A
(

N
A
,

N
B
) = 0 must exist.
8.3.3 Proof of the Systemic Failure Exercise
Proposition 10. Let F(y) = Rln(1 + y). If 1 + c < R <

R then there exists a q (0, 1/R) such
that for all q q we have that

N(q) N
c
.In other words, if the probability of the good state is
higher than q, banks fail in the bad state in non-competitive equilibrium. If R 1 + c then banks
always fail in the bad state, and if R

R then banks never fail in the bad state where

R is given
by

R
1
2
_
2 +c +

8 +c
_
(8.112)
Proof. By Proposition 4 we have already shown that equilibrium investment level is increasing in
q and R. Fix some R. Note that, if

N(R, 1/R, c) < N
c
, then

N(R, q, c) <

N
c
for all q [0, 1/R]
since

N is increasing in q.
I will rst show that when q = 1/R, the dierence N
c

N is monotonically increasing in
R. Moreover, it is negative when R is small and positive for suciently high R. Therefore, we
will establish using Intermediate Value Theorem that there exists some

R such that N
c
(

R, c)

N(

R, 1/

R, c) = 0. Using denitions of N
c
and

N we can write this dierence as
N
c

N =
aR
2c
_
R 1 c
R 1
_

a
_
4R c
_
8cR + (c)
2
_
8c
(8.113)
where
=
(1 q)R
R 1
as dened before. Evaluating this dierence at q = 1/R gives that
N
c

N =
aR
2c
_
R 1 c
R 1
_

a
_
4R c

8cR +c
2
_
8c
(8.114)
55
We can determine the behavior of this dierence as R changes by looking at the derivative with
respect to R which will be given by

_
N
c

N
_
R
=
aR
2c
_
(R 1)
2
+c
(R 1)
2
1 +

8cR +c
2
c
_
> 0 (8.115)
This derivative is clearly positive, which means that the dierence is monotonically increasing in
R. Moreover, note that given q = 1/R
lim
R1+c
_
N
c
(R, c)

N(R, q, c)
_
< 0 (8.116)
This follows since N
c
(R, c) 0 as R 1 + c, which can be clearly seen from the denition of
N
c
(R, c). We know that

N(R, q, c) > 0 as R 1+c since by Proposition 3 we have established that
equilibrium investment levels are positive as long as R > 1 + c(1 q). To complete the argument,
I will show that when q = 1/R
lim
R
_
N
c
(R, c)

N(R, q, c)
_
> 0 (8.117)
In order to see this we can expand the dierence N
c

N given by (8.114) to write it as
N
c

N =
aR
2c
_
c
R1

c
4
+

8cR+c
2
4
_
(8.118)
Now, note that as R becomes large the rst term inside the parenthesis goes to zero whereas
the last term goes to innity. Therefore this dierence is denitely positive for large R. Now,
using the Intermediate Value Theorem we can conclude that there exists some

R > 1 + c such
that

N(

R, 1/

R, c) = N
c
(

R, c). Using the denitions of

N, and N
c
from equations (39) and (39)
respectively we can solve for

R as follows

R =
1
2
_
2 +c +

8 +c
_
(8.119)
Case 1 R

R
The analysis above implies that for R

R we have that

N(R, 1/R, c) N
c
(R, c

). Moreover,
since for any given R, the highest value of

N is obtained when q = 1/R, we will also have that for
R

R we have

N(R, q, c) N
c
(R, c) for all q [0, 1/R]. In other words, if R

R, banks never
fail in the bad state for any value of q [0, 1/R].
Case 2 1 +c < R <

R.
56
In this case we can show that
lim
q0

N(R, q, c) < N
c
< lim
q1/R

N(R, q, c) (8.120)
Hence by the Intermediate Value Theorem we will conclude that there exists q (0, 1/R) such that

N( q) = N
c
. Therefore, banks will fail in the bad state if q q, and they will survive otherwise.
First, I will consider the rst part of the inequality in (8.120). It will be useful to note that
lim
q0
=
R
R 1
and lim
q1/R
= 1 (8.121)
Therefore lets consider

N when q = 0 and check whether it is less than N
c
a
_
4R
cR
R1

_
8Rc
R
R1
+
_
R
R1
_
2
_
8c
<
?
aR
2c
_
R 1 c
R 1
_
(8.122)
where left hand side of the inequality is

N evaluated when q = 0, and right hand side is N
c
given
by (39). Simplifying both sides reduces this comparison to
4aR
acR
R 1
a

8Rc
R
R 1
+
_
R
R 1
_
2
<
?
4aR
4acR
R 1
(8.123)
simplifying further yields
c
R 1
<
?
1 (8.124)
which is true as long as 1 +c < R as we proposed for this case.
For the second part of the inequality (8.120) consider

N when q = 1/R and check when it is
greater than N
c
a
_
4R c

8Rc +c
2
_
8c
>
?
aR
2c
_
R 1 c
R 1
_
(8.125)
where the left hand side of the inequality is

N evaluated when q = 1/R, and right hand side is N
c
given by (39). Simplifying reduces this comparison to
4aR ac a
_
8Rc +c
2
>
?
4aR
4acR
R 1
(8.126)
Solving for R yields that this inequality is true as long as
R <
1
2
_
2 +c +

8 +c
_


R (8.127)
57
Therefore we can conclude that if 1 + c < R <

R,there exists q (0, 1/R) such that

N(q) > N
c
for q q . In words, when 1 +c < R <

R banks fail systemically in the bad state if q q (R), and
they survive if q < q (R).
Case 3 1 +c(1 q) < R 1 +c
We know that when 1 + c(1 q) < R 1 + c we will have N
c
(R, c) = 0 by denition and

N(R, q, c) > 0 from Proposition 3. Therefore, we will have that

N(R, q, c) N
c
whenever 1+c(1
q) < R 1 +c for all q [0, 1/R]. Therefore banks always fail in the bad state in this case.
Lemma 7. For any given R <

R, there exists some q > q, where q is as dened in Proposition
10, such that if q ( q, q] moving to a central common regulation from the symmetric uncoordinated
equilibrium will eliminate the systemic failure in the bad state.
Proof. We know that systemic crises happen when initial investment levels are high. If initial
investment levels are close to the critical borders beyond which systemic crises occur, i.e.

N() is
slightly above N
c
(), then moving to global regulation can reduce investment levels in both countries
to

N < N
c
, and eliminate systemic failures in the bad state.
We can follow the similar lines in the proof of Proposition 10, and show that if 1 +c < R <

R
then there exists a q (0, 1/R) such that for all q q we have that

N(q) N
c
. In other words,
for such R, if the probability of the good state is higher than q, banks fail in the bad state under
common central regulation. If R 1 + c then banks always fail in the bad state, and if R

R
then banks never fail in the bad state where

R
1
2
_
2 +c +

4 +c
_
(8.128)
It is clear that

R <

R (1/2)
_
2 +c +

8 +c
_
. We can also show that 1/R > q > q which
will complete the proof.
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