The Macroeconomic Effects of Bail-In Policy
∗Matthijs D.B. Katz† August 11, 2017
Abstract
We examine the macroeconomic effects of bail-in policy in the wake of a debilitating credit crisis. We investigate the effect of a bail-in of creditors, whereby deposits are converted into net worth for banks, on output through the credit transmission channel. We formulate an RBC model with balance sheet-constrained banks that intermediate funds between households and non-financial firms, and a fiscal authority that recapitalizes banks through taxes or a bail-in. Our simulations show that an immediate bail-in effectively ameliorates macroeconomic conditions after a credit crisis. However, if a bail-in is announced before it is implemented, there is no effect on the real economy, as depositors demand a higher return to make up for the future losses they incur. Unlike an anticipated bail-in, a de-layed tax-financed fiscal recapitalization does alleviate the negative effects of a credit crisis.
Keywords: Bail-In, Banking, Financial Frictions, Real Business Cycle JEL Classification: E32, E44, E50
∗
I would like to thank Christiaan van der Kwaak and Ben Heijdra for helpful comments and discussions, and for guiding me through the process of writing this thesis.
†
I. Introduction
This paper investigates the macroeconomic effects of bank bail-in policy. A bail-in occurs when a bank’s creditors recapitalize the institution by writing off a part of the bank’s debt. After the Global Financial Crisis and Eurozone Crisis, which started in respectively 2007 and 2009, large parts of the financial system of multiple countries were either failing or on the brink of failure. These banks were bailed out by governments using taxpayer money, which caused widespread discontent. Moreover, bailouts might lead to moral hazard issues, since banks are aware that they will be rescued by taxpayers, leading to excessively risky behavior. Hence, bail-ins were concocted to make a bank’s insiders rather than taxpayers carry the burden. There have been a number of instances of a bail-in of creditors, but the bail-in executed in
Cyprus proved to be a virtual watershed.1 After the downgrading of Greek and Cypriot
sovereign debt to junk status brought Cyprus’ financial system to the brink of collapse in 2012, the European Commission, European Central Bank, Eurogroup, and International Monetary
Fund agreed to provide ae10 billion bailout of the country if the government recapitalized
the banking system by converting uninsured deposits into net worth, i.e. a bail-in. As a result, 37.5% of uninsured deposits were converted into equity to recapitalize Bank of Cyprus, Cyprus’ largest systemic bank. In 2014 the European Parliament adopted the Bank Recovery and Resolution Directive (BRRD), which set up an effective framework for handling failing banks without relying on supranational authorities. The BRRD places limits on bailouts, and stipulates that banks should be recapitalized through bailing-in its creditors. That is, a bank’s bonds and a share of its deposits that are not covered by the EU Deposit Guarantee Scheme are converted into equity when the resolution authority chooses to do so. This ensures that small savers and taxpayers do not pick up the bill if a bank is in trouble.
Even though the bail-in regime is standard practice in the European Union (EU) now, there has not been any evaluation of the macroeconomic effects of converting bank debt into equity in a formal model. Bail-ins seem attractive on paper, but since bail-ins have been implemented only a few times we can say very little about the macroeconomic implications
of a bail-in. Hence, evaluating a bail-in in a formal macroeconomic mode is a worthwhile exercise for a number of reasons. One reason is that there are substantial forward looking elements associated with a bail-in: a bank’s creditors know that they will lose their assets after a financial crisis. As such, a bail-in regime might cause creditors to liquidate their assets en masse, i.e. causing a run on banks, or lead to creditors demanding a higher return to
compensate for the losses they incur.2 This can affect the ability of banks to attract core
demand deposits as funding, and subsequently affect their ability to finance investment for firms. Both of these anticipation effects might make the whole exercise of a bail-in less effective than otherwise would be the case.
In this paper we investigate the effects of a bail-in of depositors on output through the credit transmission channel after a credit crisis. In line with Gertler and Kiyotaki (2011) we construct a closed economy real business cycle (RBC) model with financial frictions. Banks provide loans to final goods producers and are financed through deposits and net worth. We extend this model by including the possibility to bail-in depositors to recapitalize the banking sector. This is a policy rule that mechanically converts deposits into net worth for the banking sector if a financial crisis occurs. Moreover, we compare the effects of both an immediate and an anticipated bail-in to a tax-financed fiscal recapitalization of the banking sector by the government, i.e. a bailout. Hence, in a bail-in the debts of banks are written down and converted into net worth, whereas the government directly injects new net worth in the banking sector through higher taxes in a bailout. It is important to stress that we do not intend to perform a quantitative modeling exercise, but merely aim to show the main macroeconomic dynamics generated by a bail-in.
To the best of our knowledge, we are the first to assess the effects of a bail-in where deposits are converted into net worth in a formal macroeconomic framework. The model we put forward in this framework is meant as a first pass at implementing a bail-in in a
2 Even though deposits up toe 100,000 are insured by the EU Deposit Guarantee Scheme, firms tend to
macroeconomic model, while describing some of the intricacies that should be dealt with in future research.
We find that an immediate bail-in of depositors is highly effective at amelioriating economic conditions after a credit crisis. A credit crisis imposes large capital losses on banks, which reduces net worth. As a result, banks have to shrink their balance sheets due to the financial friction between depositors and banks. Since banks receive new net worth in a bail-in, they do not have to shrink their balance sheets by as much as in the no intervention case. As such, credit provision to the real economy falls by less, which moderates the fall in output and investment. It has to be said that an immediate bail-in is an unrealistic scenario and never occurs in practice: since a bail-in imposes rather large losses on creditors, actually implementing a bail-in is a politically sensitive affair. This simulation is presented mainly for expositional purposes. A more realistic scenario is that of a delayed bail-in. However, if the bail-in is announced at the start of a credit crisis and implemented with a lag, the effect on output is zero. Since depositors know that they lose a share of their deposits in a bail-in, they form expectations about a bail-in occurring. This is reflected in the realization of the return on deposits. As households anticipate that they will lose a share of their deposits at a certain date, they will demand a higher return to exactly make up for the losses they incur. This erodes the banking sector’s gain in net worth, making aggregate economic conditions identical to the case without policy intervention. We find that a tax-financed fiscal recapitalization of the banking sector is more effective than an anticipated bail-in. However, the neutrality of an anticipated bail-in is driven by our assumption of a perfectly competitive banking sector: since banks take interest rates as given, depositors can simply demand a high return on deposits. This implies that an anticipated bail-in will be less effective than a tax-financed recapitalization in any case where banks take interest rates as given.
risk-taking (Dam and Koetter, 2012). As such, in the long run, bail-ins can reduce risk-taking and moral hazard issues. Hence, our results only apply to the short run.
This paper is structured as follows. Section II presents the model, and section III details the calibration employed. The results of the numerical simulations are presented in section IV, and section V concludes.
II. Model
We construct an infinite horizon stochastic RBC model of a closed economy with a banking sector à la Gertler and Kiyotaki (2011) and Gertler and Karadi (2011). Banks provide loans to final goods producers, who use these loans to purchase capital from capital goods producers. These loans are financed by net worth and deposits. These banks are balance-sheet constrained due to an agency problem between banks and depositors (Gertler and Kiyotaki, 2011).
The other agents considered in our model are households, non-financial firms, and a fiscal authority. Households consist of workers and bankers, who respectively supply labor to a final goods producer and operate a bank. There are two non-financial firms: final goods producers and capital goods producers. Final goods producers borrow funds from a bank to purchase capital from capital goods producers and hire labor from households. Capital goods buy final goods and refurbish used capital, which they sell to final goods producers. The final good is consumed by households and sold to capital goods producers for investment purposes. The fiscal authority can engage in a tax-financed recapitalization of the banking sector.
The main difference between our model and earlier models with financial frictions lies in the possibility to bail-in depositors. We introduce a policy rule that converts deposits held by households into net worth for financial intermediaries when the economy suffers a credit crisis. This allows us to examine the anticipation effects of a bail-in in a relatively straightforward manner.
A. Households
a wage, whereas bankers manage a bank and transfer their profits back to the household if they cease operating a bank. We assume that there is perfect consumption insurance between the
members of the household. The household gains utility from consumption Ct, and disutility
from supplying labor Lt. The income the household receives can come from two sources:
either by receiving income from labor Wt or from repayment of deposits Dt−1 with interest
RDt−1. Investing in deposits in t − 1 yields a repayment of the principal Dt−1 plus an interest
rate RDt−1in the next period. However, in case the economy suffers a credit crisis, it is possible
that a share Ψt of the household’s deposits is converted into net worth for the banking sector
in a bail-in one period after the funds have been deposited. However, Ψtmight include lags,
such that depositors know in period t − 1 the share of deposits that will be converted into net worth in period t. The functional form of this policy measure will be specified in Section II.D. Households are the owners of the economy’s firms, and therefore also receive a lump-sum
transfer from the profits earned from owning both banks and non-financial firms Πt. Income
is spent on consumption goods Ct, depositing funds Dt at a bank that is operated by another
household, and lump sum taxes Tt that are only levied when the fiscal authority engages
in a recapitalization of the banking sector. We follow Christiano et al. (2005) by assuming that households have consumption habit formation. As such, the household maximizes the following utility function:
max {Ct+s,Lt+s,Dt+s}∞s=0 Et (∞ X s=0 βs 1 1 − σ(Ct+s− hCt+s−1) 1−σ − χ 1 + ϕL 1+ϕ t+s ) , β ∈ (0, 1) , h ∈ [0, 1), ϕ ≥ 0,
subject to the budget constraint:
Ct+ Dt+ Tt= WtLt+ Πt+
1 + RDt−1− Ψt
Dt−1,
supply. Maximizing the utility function subject to the budget constraint yields the following first order conditions:
Ct: λt= (Ct− hCt−1)−σ− βhEt(Ct+1− hCt)−σ, (1) Lt: λtWt= χLϕt, (2) Dt: Et h βΛt,t+1 1 + RDt − Ψt+1 i = 1, (3)
where λt is the household’s marginal utility of consumption, and βΛt,t+1 ≡ βλt+1
λ is the
household’s stochastic discount factor. Of these first order conditions, equation (1) is the increase in marginal utility from consumption due to a relaxation of the budget constraint, and (2) is the intratemporal first order condition for labor, where the disutility from supplying an extra unit of labor is equated to the increased utility from additional labor income. Finally, (3) is the household’s Euler equation, which relates the decrease in utility due to lower consumption today to higher income received tomorrow due to investing in deposits at a bank. Households will take it into account that they might lose a share of their deposits in a bail-in
when deciding on how many funds to deposit at their bank. As such, the policy rule Ψt+1
features in the household’s first order condition for deposits. Hence, it affects the equilibrium
interest rate on deposits RDt : if Ψt+1 increases, then RtD must also increase.
B. Non-financial firms
The non-financial firms populating the economy come in two varieties: there are standard final goods-producing firms and capital goods firms that produce, refurbish and sell capital to final goods firms for use in production.
B.1. Final goods producers
Yi,t = At(ξtKi,t−1)αL1−αi,t , (4)
log(At) = ρAlog(At−1) + εA,t,
log(ξt) = ρξlog(ξt−1) + εξ,t,
where Ytis aggregate output, Atis a Hicks-neutral exogenously determined level of total factor
productivity, ξtis the quality of capital, Ki,t−1 is the capital stock, and Li,t is hours worked
by households. Both At and ξtare lognormal AR(1) processes. These are driven respectively
by the stochastic shocks εA,t ∼ N (0, σ2
A) and εξ,t ∼ N (0, σξ2). Every period the firm hires
labor Li,t from households. The labor force is paid a wage rate Wt. Since the markets for
labor and capital are perfectly competitive, workers get paid their marginal product:
Wt= (1 − α)
Yi,t
Li,t
. (5)
After production has taken place and the labor force has been paid, the firm sells the net
of depreciation capital stock to capital goods producers for a price QKt , and thus receives
(1 − δ)ξtQKt Ki,t−1in return. What the firm receives from capital goods producers for their
net capital plus the net return on capital is used to pay back the bank:
1 + RKt = α Yi,t Ki,t−1 + (1 − δ)ξtQ K t QKt−1 . (6)
The capital quality shocks ξt capture economic obsolence of the capital stock as in Gertler
and Kiyotaki (2011), and drive the model’s asset price dynamics. These drops in asset prices are the driving force behind credit crises in the model. At the end of period t − 1 final goods
producers receive a loan Si,t−1K from banks to purchase capital Ki,t−1 from capital goods
producers for a price QKt−1. Moreover, final goods producers credibly pledge all future profits
after the wage bill to the bank. The newly purchased capital is used in production in period
t. Since the capital quality shock ξt materializes at the beginning of period t, the effective
capital stock per firm is ξtKi,t−1. If a negative ξt shock hits the system, a part of the capital
a negative realization of the capital quality shock ξt affects the return on capital through
two different channels. First, it reduces the effective capital stock, thereby lowering α Yi,t
Ki,t−1.
Second, it leads to a lower price of capital QKt . These lower asset prices will in turn cause the
return on capital RtK to fall further.
B.2. Capital goods producers
Capital goods producers purchase the net effective capital stock (1 − δ)ξtKt−1from final goods
producers for a price QKt together with final goods It for investment purposes. They combine
previous period’s capital stock with It units of the final good to produce new, refurbished
capital. Afterwards, the new capital Kt is again sold to final goods firms for a price QKt . Like
final goods producers, capital goods producers face a perfectly competitive market structure. Following Christiano et al. (2005), we assume that capital goods producers face convex adjustment costs in changing the level of gross investment used in the production of capital. The law of motion for the aggregate capital stock is then:
Kt= (1 − δ) ξtKt−1+ " 1 −γ 2 I t It−1 − 1 2# It, (7)
where γ is a parameter that affects investment adjustment costs. Profits for capital goods producers are:
ΠKt = QKt Kt− QKt−1(1 − δ)ξtKt−1− It. (8)
The capital producers’ maximization problem is then to maximize the sum of current and
expected discounted future profits by finding the optimal path for investment It. Substituting
the law of motion for capital into (8) we find that the optimization problem is given by:
where the household’s stochastic discount factor is used to discount future profits, since the household owns the capital goods producing firm. Maximizing (9) yields an expression for the
price of capital QKt : 1 QKt = 1 − γ 2 I t It−1 − 1 2 − γ I t It−1 − 1 I t It−1 + Et " βΛt,t+1 QKt+1 QK t γ I t+1 It − 1 I t+1 It 2# (10) C. Banking sector
The structure of the model’s financial sector follows Gertler and Kiyotaki (2010) and Gertler and Karadi (2011). The banking sector acts as an intermediary between savers and borrowers, and consists of a continuum of banks j ∈ [0, 1] that engage in perfect competition and are each managed and owned by a different household. The asset side of a bank’s balance sheet
consists of private loans to final goods producers Sj,tK, which have price QKt .3 In turn, the
bank funds its operations through net worth Nj,t and through risk-free deposits attracted
from households Dj,t. However, in order to avoid banks engaging in self-financing, we assume
that banks can only attract deposits from other households.4 The balance sheet of a bank j
is therefore given by:
QKt Sj,tK = Nj,t+ Dj,t, (11)
where the left-hand side of the identity is a bank’s assets, and the right-hand side is a bank’s
liabilities. Private loans pay a net return RKt+1 in period t + 1, and banks pay a deterministic
return on deposits RDt in period t + 1. The difference between the return on assets and funding
costs is new net worth. Banks also receive net worth in a bail-in Ψt+1, where household
deposits are converted into new net worth. Banks can also receive new net worth through
3
These loans can also be interpreted as securities issued by final goods firms to buy capital. These securities are subsequently bought by banks. Each security is then priced at the price of a unit of capital QK
t , since
Sj,tK = Ki,t.
4 Without this assumption, bankers would in principle be able to attract funding from their own household,
a tax-financed recapitalization by the fiscal authority Nj,t+1G = Tt+1N Nj,t, which is relative to previous period’s net worth. We assume a recapitalization is linear in the amount of an individual bank’s net worth, since the fall in net worth after a credit crisis is larger if banks have more net worth before the crisis. Net worth decreases if the government support banks
received earlier has to be paid back: ˜Nj,t+1G = ˜Tt+1N Nj,t. Both are proportional to previous
period’s net worth. These variables are discussed in section II.D. The law of motion for net worth of a typical bank j is given by:
Nj,t+1 = 1 + Rt+1K QKt Sj,tK −1 + RtD− Ψt+1 Dj,t+ Nj,t+1G − ˜Nj,t+1G =RKt+1− RDt + Ψt+1 QKt Sj,tK +1 + RDt − Ψt+1 Nj,t + Tt+1N Nj,t− ˜Tt+1N Nj,t. (12)
Furthermore, we assume that banks face a probability of forced exit 1 − ϑ that is exogenous and i.i.d. across time and in the cross-section. If a bank exits, its retained earnings are paid out to the household. With probability ϑ the bank does not exit, and is allowed to continue intermediating funds in the next period. Each period new bankers start operating to keep the number of banks constant. Since the household owns the bank, future outcomes are
discounted by the household’s stochastic discount factor βΛt,t+1. The bank’s objective is then
to maxizimize expected future discounted profits:
V Sj,t−1K , Dj,t−1 = max Et n βΛt,t+1 h (1 − ϑ) Nj,t+1+ ϑV Sj,tK, Dj,t io .
We introduce a principal-agent problem between banks and depositors to generate financial frictions (Gertler and Kiyotaki, 2011). Households know that there is a possibility that banks divert their assets. This can occur in the transition from the current to the next period. If this happens, the household can force the bank to declare bankruptcy in order to recoup the assets it lost. However, letting the intermediary go bankrupt implies that the household
can only receive an exogenously determined share λK of its assets, with 1 − λK going to the
constraint, as depositors will only finance the bank if the bank’s continuation value is greater than or equal to what the bank will gain by diverting assets. This gives rise to the following incentive compatibility constraint:
V Sj,t−1K , Dj,t−1
≥ λKQKt Sj,tK. (13)
A typical bank j’s optimization problem can therefore be characterized by the following Bellman equation and constraints:
Vj,t= max {SK j,t,Dj,t} Et{βΛt,t+1[(1 − ϑ) Nj,t+1+ ϑVj,t+1]} , s.t. Vj,t≥ λKQKt Sj,tK, QKt Sj,tK = Nj,t+ Dj,t, Nj,t= 1 + RtK+ TtN− ˜TtNQKt−1Sj,t−1K −1 + Rt−1D − Ψt+ TtN − ˜TtNDj,t−1, where Vj,t ≡ V Sj,t−1K , Dj,t−1
. To solve the problem characterized by the Bellman equation, we conjecture that the value function is linear and later verify whether this is indeed the case:
Vj,t= νtKQKt Sj,tK + ηtNj,t, (14)
where νtKand ηtare respectively the shadow values of private loans and net worth. Substitution
of the conjectured solution (14) into the incentive compatibility constraint (13) yields the following expression:
QKt Sj,tK ≤ φtNj,t, φt=
ηt
λK− νtK
, (15)
where φt is the ratio of a bank’s assets to its net worth, i.e. the bank’s leverage constraint.
loans νtK indicates a higher value from holding an additional unit of loans, ceteris paribus increasing the bank’s continuation value. This reduces the incentive for bankers to divert
assets. A higher shadow value of net worth ηt implies a higher continuation value from an
additional unit of net worth. Depositors will therefore allow the bank to increase its leverage
ratio. An increase in the fraction of assets that bankers can divert λK will incentivize bankers
to divert assets instead of continuing operating the bank. As such, ceteris paribus households will decrease their holdings of deposits and force the bank to decrease its leverage ratio. The full solution of the typical bank’s optimization problem can be found in the mathematical appendix. The first order conditions are:
ηt= Et h Ωt,t+11 + RtD− Ψt+1+ Tt+1N − ˜Tt+1N i, (16) νtK = Et h Ωt,t+1 RKt+1− RDt + Ψt+1 i , (17) where Ωt,t+1≡ βΛt,t+1h(1 − ϑ) + ϑνt+1K φt+1+ ηt+1 i
is the household’s stochastic discount factor augmented to incorporate the financial friction. Equation (16) shows the shadow value
of an additional unit of net worth ηt. This is equal to the sum of the gross return on deposits,
minus the share of deposits that are converted into net worth in a bail-in, and net worth injections by the fiscal authority and subsequent repayments, discounted by the augmented stochastic discount factor. Equation (17) is the first order condition for loans to final goods producers. The presence of a binding bank balance sheet constraint limits the ability of banks to arbitrage away the difference between the expected rate of return on private loans and deposits, since banks cannot perfectly elastically expand their balance sheet. Since the
leverage ratio φt does not rely on individual bank characteristics, we can simply aggregate
across all individual banks to find the macroeconomic leverage constraint:
QKt StK = φtNt, (18)
at the end of each period a share 1 − ϑ of banks will leave the financial services industry. In case of forced exit, the remaining net worth is paid out to the operating household as dividends. If the bank does not go out of business, its retained earnings become new net worth. Hence, the law of motion for banks that continue operating equals:
No,t = ϑ h RKt − RDt−1+ Ψt QKt−1St−1K +1 + RDt−1− Ψt+ Nt−1 i ,
where the subscript o stands for old. Since 1 − ϑ banks leave the financial services industry, a fraction (1 − ϑ) f 0f bankers become workers again. To keep the number of banks in the aggregate constant, the same amount of workers wil become bankers. In turn, households
provide startup funds to its family members that start a new bank by transferring χF
1−ϑ of
the exited bankers’ assets QKt−1SKt−1 to new bankers. Hence, the expression for aggregate net
worth for new bankers is:
Nn,t= χFQKt−1St−1K ,
where the subscript n stands for new. The aggregate law of motion for net worth is then equal to the sum of net worth of old bankers, net worth of new bankers, and government financial sector support:
Nt= ϑ h RKt − RDt−1+ Ψt QKt−1St−1K +1 + RDt−1− Ψt Nt−1 i + χFQKt−1St−1K + NtG− ˜NtG. (19) D. Fiscal authority
The fiscal authority raises revenue by levying lump-sum taxes Tt on households. The only
expenditure of the fiscal authority in the model is a tax-financed recapitalization of the banking
sector NtG if the economy suffers a credit crisis. The fiscal authority also raises revenue from
Tt+ ˜NtG= NtG. (20)
The fiscal recapitalization variables take the following functional forms:
NtG= TtNNt−1, (21)
TtN = ζεξ,t−n, ˜
NtG= ωNt−vG (22)
Hence the government injects new net worth into the banking sector relative to aggregate net worth in period t − 1 in case a quality of capital shock hits the economy, i.e. if ζ < 0. The fiscal recapitalization can be carried out on impact (n = 0) or with a lag of one or more periods (n > 0). The parameter ω indicates whether the support is a gift (ω = 0), a zero interest loan (ω = 1), or if the banking sector has to pay interest for the recapitalization it received from the fiscal authority (ω > 1). In turn, v indicates whether the net worth injection needs to be paid back immediately or with a lag. In addition, banks can also receive
an injection of new net worth through a bail-in that converts a fraction Ψt of deposits into net
worth after during a credit crisis. This can be seen as a policy measure that is imposed by some sort of regulator, e.g. the Single Resolution Board (SRB) that oversees the resolution of banks in EU Member States. The bail-in variable is parametrized as follows:
Ψt= %εξ,t−l. (23)
E. Market clearing
In equilibrium, the amount of loans extended by the banking sector (StK) must equal the
aggregate capital stock (Kt):
StK= Kt. (24)
The aggregate resource constraint is given by:
Yt= Ct+ It. (25)
F. Recursive competitive equilibrium
The definition of the model’s recursive competitive equilibrium and a summary of all the relevant first order conditions can be found in Appendix B.
III. Calibration
The model is calibrated on a quarterly frequency and largely follows Gertler and Kiyotaki’s (2011) calibration for the United States. As such we use the same parameter values for the share of capital in output α, the household’s subjective discount factor β, the parameter for investment adjustment costs γ, the habit formation parameter h, the inverse Frisch elasticity of labor supply ϕ, and the household’s relative risk aversion parameter σ The value of the utility of weight of labor parameter ψ is chosen to be such that steady state labor supply is equal to 1/3. Most of the values assigned to these parameters are standard in the literature. However, the value for ϕ is lower than that which is commonly used. This is to compensate for the lack of labor market frictions in the model (Gertler and Kiyotaki, 2011).
We also follow Gertler and Kiyotaki (2011) for the calibration of the banking sector’s parameters. As such, we set the aggregate leverage ratio to 4, and define an auxiliary credit
spread variable Γt≡ RK
t − RDt−1 which is set to 25 basis points on a quarterly frequency in the
transfer to new bankers χF and the share of assets that bankers can divert λK are calibrated such that the previously mentioned targets are hit. Following Gertler and Kiyotaki (2010) and Gertler and Karadi (2011) we initiate a credit crisis by allowing for a shock to the quality
of capital ξt of 5%, with an autocorrelation coefficient ρξ of 0.66.
Ta b l e 1 Model parameters
Parameter Value Definition
Households
β 0.99 Subjective discount factor
σ 1 Risk aversion
h 0.5 Habit persistence parameter
χ 2.9027 Relative utility weight of labor
ϕ 0.1 Inverse Frisch elasticity
Banking sector
λK 0.3863 Fraction of divertable assets
χF 0.0021 Transfer to entering bankers
ϑ 0.972 Survival rate of bankers
Non-financial firms
α 0.33 Capital share in output
γ 1.728 Investment adjustment cost parameter
δ 0.025 Depreciation rate
Autoregressive components
ρA 0.95 Autoregressive component of capital quality
ρξ 0.66 Autoregressive component of productivity
Shocks
σA 0.01 Standard deviation of productivity shock
IV. Numerical simulations
As an appetizer to our main results, we first present the effects of a capital quality shock in a model with financial frictions and one without these frictions in order to present the mechanisms that drive a credit crisis. Next, we explore how policy makers can alleviate the recession a credit crisis causes. Specifically, we analyze the effects of several financial sector policies that aim to reduce the negative effects of a credit crisis. We present a bail-in of depositors that is implemented immediately after the start of a crisis, a bail-in that is executed four quarters after it is announced, and both immediate and delayed direct recapitalizations by the fiscal authority.
A. Financial crisis in model without financial frictions vs. model with financial frictions
To understand how financial frictions have macroeconomic effects, we first compare the responses of the model economy without (blue, solid) and with (red, slotted) financial frictions to a negative capital quality shock of 5% as presented in Figure 1. In both models the shock destroys the existing capital stock and output and consumption fall as a result. Restricting our analysis first to the model without financial frictions, the simulation shows that the ex-post return on capital falls on impact. However, the lower capital stock leads to an increase in the marginal product of capital, which increases the expected return on capital. The higher expected return causes investment to increase, which slowly leads to a higher capital stock due to the presence of investment adjustment costs. The marginal product of labor falls together with the capital stock, which decreases wages. Since part of household wealth is destroyed by the capital quality shock, labor supply increases and consumption falls. All in all, the fall in output is relatively modest.
final goods producers. Since banks reduce the size of their balance sheet, they demand less deposits. This in turn depresses the return on deposits, and causes credit spreads to increase. The reduction in lending triggers a fall in the price of capital, which further depresses the ex-post return on loans. Furthermore, the lower capital price and return on loans causes the expected return on capital to increase, which increases credit spreads. The increase in the credit spread in turn further reduces final goods producers’ demand for loans. The lower demand for loans triggers a fall in the price of capital, which further reduces the return on loans and imposes more capital losses on banks, again leading to a fall in net worth and further increasing aggregate leverage. Since final goods producers require a loan from a bank to acquire capital, the lower demand for loans leads to lower investment.
0 10 20 30 40 −6 −4 −2 0 Capital quality: ξt R el . ∆ fr o m s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −4 −2 0 Output: Yt R el . ∆ fr o m s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −4 −2 0 Consumption: Ct R el . ∆ fr o m s. s. in p er ce n t Quarters 0 10 20 30 40 −20 −10 0 10 20 Investment: It R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −1500 −1000 −500 0 500
Ex post return on capital: RK t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −6 −4 −2 0 Wages: Wt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −10 −5 0 5 Price of capital: QK t R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −2 0 2 4 Labor: Lt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 10 20 30 40
Expected return on capital: Et(RKt+1)
Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( a ) 0 10 20 30 40 −14 −12 −10 −8 −6 −4 −2 0 Capital: Kt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Banking sector leverage: φt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −60 −50 −40 −30 −20 −10 0
Banking sector net worth: Nt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 20 40 60 80 100 120 140 Credit spread: Γ t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −100 −80 −60 −40 −20 0 20
Ex ante return on deposits: RD t−1 Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 Deposits: Dt Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( b )
B. No policy intervention vs. immediate bail-in
Motivated by the introduction of the BRRD in the EU in 2014, we investigate whether an immediate bail-in of depositors is capable of mitigating the adverse effects of a credit crisis. The results of this simulation are presented in Figure 2. We compare the response of the model economy to a capital quality shock of 5% without policy intervention (blue, solid) to the response when a bail-in immediately occurs after the shock arrives (red, slotted). That is, upon impact of the shock deposits equal to 1.25% of annual steady state output are converted
into net worth for the banking sector.5
An immediate bail-in of deposits clearly has positive effects on the economy: it relaxes the banking sector’s balance sheet constraints by adding to their stock of net worth. Output still falls on impact, but the trough is lower than in the no intervention case. Consumption increases slightly faster, and investment falls by less and recovers more quickly. Net worth falls by less on impact, and the credit spread does not increase as much. Because banks immediately receive new net worth their balance sheet constraints are alleviated, and they can expand their balance sheet and continue lending to final goods producers. Moreover, both the return on deposits and the total amount of deposits fall by slightly more than in the no policy case: since banks have more net worth after a bail-in they can rely less on deposits for funding, and therefore do not have to attract as much deposits as in the no policy case. Hence, an immediate bail-in is quite effective at alleviating a credit crisis due to the smaller decrease in lending to final goods producers. However, the quantitative effects are relatively small.
Although our results show that an immediate bail-in is effective at ameliorating macroe-conomic conditions after a credit crisis, we must admit that this is a relatively unrealistic scenario: a bail-in imposes large losses on creditors, and therefore is politically sensitive. As such, it is more realistic to assume that there is a lag between the arrival of the shock and the implementation of bail-in. Hence we explore the effects of a delayed, anticipated bail-in in the next section.
5 This corresponds to a conversion of approximately 0.8% of deposits into net worth, which explains why the
0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Policy as % of annual output
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −5 −4 −3 −2 −1 0 Output: Yt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −5 −4 −3 −2 −1 0 Consumption: Ct R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −15 −10 −5 0 5 10 Investment: It R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −14 −12 −10 −8 −6 −4 −2 0 Capital: Kt Ab s. ∆ fr om s. s. in p er ce n t. Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 2 Price of capital: QK t R el . ∆ fr om s. s. in p er ce n t Quarters ( a ) 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Banking sector leverage: φt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −60 −50 −40 −30 −20 −10 0
Banking sector net worth: Nt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 20 40 60 80 100 120 140 Credit spread: Γ t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −100 −80 −60 −40 −20 0 20
Ex ante return on deposits: RD t−1 Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 Deposits: Dt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Expected return on capital: Et(RKt+1)
Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( b )
C. No policy intervention vs. anticipated bail-in
Although we showed that an immediate bail-in can improve economic consequences after a credit crisis, policy makers such as governments or regulators face legal and political constraints that might make this option impossible in practice. As such, a more realistic scenario is one where the bail-in is implemented several quarters after the start of a crisis. Therefore we will analyze the effects of an anticipated bail-in that is implemented four quarters after the onset of a credit crisis. The results of this simulation are presented in Figure 3. In this policy experiment we compare the no policy case (blue, solid) to an anticipated bail-in (red, slotted). In the anticipated bail-in, deposits equal to 1.25% of annual steady state output are converted into net worth for the banking sector four quarters after the shock arrives. The future implentation of the policy measure is announced when the shock arrives.
The impulse response functions show that a delayed bail-in of the banking sector virtually wipes out any positive effects that an immediate bail-in has on the macro-economy. Since the bail-in is anticipated, households know that their deposits will be converted into net worth for banks four quarters after the shock arrives. As the bail-in term features in the household’s Euler equation (equation (3)), households will incorporate this information in their decision-making. Moreover, since the banking sector engages in perfect competition, banks have no market power and thus take interest rates as given. As such, households will simply demand a higher interest rate on their deposits to make up for the losses they incur when they lose their deposits. The higher interest rate on deposits drives down the credit
spread Γt(which excludes returns resulting from the bail-in) in the period of the bail-in, which
reduces the profitability of banks, and thus makes the change in net worth compared to the no intervention case zero. Figure 4 compares a simulation with an immediate bail-in (blue, solid) to one with an anticipated bail-in (red, slotted), where both interventions are of the same size as in previous simulations. This simulation also shows that any beneficial effects of a bail-in are lost when it is anticipated.
by demanding higher interest rates. However, there is ample empirical evidence that banks do in fact have some degree of market power (e.g. Salas and Saurina, 2003). Moreover, we also assume that all deposits have the same maturity structure and have to be rolled over after each period, whereas banks generally offer deposits with different maturity structures. As such, our results will not completely carry over to the real world, but they do indicate that an anticipated bail-in will be less effective when an economy’s financial sector is closer to perfect competition. Moreover, if households cannot charge higher rates, they do have the possibility of withdrawing their deposits, thereby starting a bank run.
0 10 20 30 40 −0.5 0 0.5 1 1.5
Policy as % of annual output
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −5 −4 −3 −2 −1 0 Output: Yt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −5 −4 −3 −2 −1 0 Consumption: Ct R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −15 −10 −5 0 5 10 Investment: It R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −14 −12 −10 −8 −6 −4 −2 0 Capital: Kt Ab s. ∆ fr om s. s. in p er ce n t. Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 2 Price of capital: QK t R el . ∆ fr om s. s. in p er ce n t Quarters ( a ) 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Banking sector leverage: φt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −60 −50 −40 −30 −20 −10 0
Banking sector net worth: Nt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 20 40 60 80 100 120 140 Credit spread: Γ t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −100 −80 −60 −40 −20 0 20 40
Ex ante return on deposits: RD t−1 Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 Deposits: Dt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Expected return on capital: Et(RKt+1)
Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( b )
0 10 20 30 40 −0.5 0 0.5 1 1.5
Policy as % of annual output
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −5 −4 −3 −2 −1 0 Output: Yt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −5 −4 −3 −2 −1 0 Consumption: Ct R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −15 −10 −5 0 5 10 Investment: It R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −14 −12 −10 −8 −6 −4 −2 0 Capital: Kt Ab s. ∆ fr om s. s. in p er ce n t. Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 2 Price of capital: QK t R el . ∆ fr om s. s. in p er ce n t Quarters ( a ) 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Banking sector leverage: φt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −60 −50 −40 −30 −20 −10 0
Banking sector net worth: Nt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 20 40 60 80 100 120 140 Credit spread: Γ t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −100 −80 −60 −40 −20 0 20 40
Ex ante return on deposits: RD t−1 Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 Deposits: Dt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Expected return on capital: Et(RKt+1)
Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( b )
D. Anticipated bail-in vs. anticipated fiscal recapitalization
We have already seen that an anticipated bail-in is ineffective because households will demand higher interest rates to make up for future losses that they expect to incur. The question remains whether an anticipated fiscal recpitalization fares any better. To this end, we compare an anticipated bail-in (blue, solid) to an anticipated tax-financed fiscal recapitalization (red, slotted) of the financial sector in Figure 5. In both cases, the banking sector receives new net worth equal to 1.25% of annual steady state output after four quarters. The intervention itself is announced the instant the shock arrives. We have relegated a simulation with an immediate bail-in and recapitalization to Appendix D.
In contrast to an anticipated bail-in, a delayed fiscal recapitalization of the banking sector does have positive macroeconomic effects. The anticipation of the intervention itself affects the economy: since banks know that they will be recapitalized, the entire banking sector’s continuation value increases, which alleviates the balance sheet constraint. As such, both investment and the price of capital are higher before the policy is implemented. This makes the fall in output smaller on impact of the shock. Moreover, leverage and credit spreads increase by less, and net worth falls by less. When the recapitalization is implemented after four quarters, the leverage ratio, deposits, and credit spreads all fall, whereas net worth increases. Finally, output, investment, and the price of capital continue to increase.
0 10 20 30 40 −0.5 0 0.5 1 1.5
Policy as % of annual output
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −5 −4 −3 −2 −1 0 Output: Yt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −6 −5 −4 −3 −2 −1 0 Consumption: Ct R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −15 −10 −5 0 5 10 Investment: It R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −14 −12 −10 −8 −6 −4 −2 0 Capital: Kt Ab s. ∆ fr om s. s. in p er ce n t. Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 2 Price of capital: QK t R el . ∆ fr om s. s. in p er ce n t Quarters ( a ) 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Banking sector leverage: φt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 −60 −50 −40 −30 −20 −10 0
Banking sector net worth: Nt
R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 20 40 60 80 100 120 140 Credit spread: Γ t Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −100 −80 −60 −40 −20 0 20 40
Ex ante return on deposits: RD t−1 Ab s. ∆ fr om s. s. in b as is p ts . Quarters 0 10 20 30 40 −10 −8 −6 −4 −2 0 Deposits: Dt R el . ∆ fr om s. s. in p er ce n t Quarters 0 10 20 30 40 0 5 10 15 20 25 30 35 40
Expected return on capital: Et(RKt+1)
Ab s. ∆ fr om s. s. in b as is p ts . Quarters ( b )
V. Conclusion
Motivated by the Cypriot bail-in in 2013 and the adoption of the BRRD in the EU in 2014, we present a first pass at implementing a bail-in of depositors to recapitalize banks in a macroeconomic model. We construct a closed economy RBC model enriched with a balance-sheet-constrained banking sector à la Gertler and Kiyotaki (2011) to investigate the effects of a bail-in of depositors after a credit crisis on output through the credit transmission channel of the banking sector. We allow for two different policy interventions after a credit crisis: a bail-in of creditors that converts household deposits into net worth for the banking sector, and a direct tax-financed recapitalization of the banking sector by the fiscal authority.
Our numerical simulations show that an immediate bail-in of depositors to recapitalize the banking sector is highly effective compared to the case without any policy intervention, and equivalent to an immediate tax-financed recapitalization. The bail-in reduces the fall in net worth, which alleviates banks’ balance sheet constraint. As banks do not have to tighten their balance sheets as much, they can continue supplying credit to the real economy. This reduces the drop in investment, and thereby alleviates the fall in output.
about the effectiveness of a bail-in, even if our model is relatively stylized and leaves out some of the intricacies of the real world.
On the other hand, we must acknowledge that the real economy neutrality of an anticipated bail-in is driven by our assumption of a perfectly competitive banking sector. If banks would have market power and engage in monopolistic competition as in Gerali et al. (2010), which is a more realistic assumption, they would at least be able to influence interest rates to some degree. However, unless banks would be able to completely set interest rates there will always be some response from depositors demanding higher rates. However, in reality agents would simply withdraw their funds if they cannot influence interest rates, thereby causing a bank run. Another way to overcome the neutrality of an anticipated bail-in is to introduce deposits with different maturity structures, such that deposits are not rolled over every period. Therefore, further research is required to give the definitive answer whether a bail-in is an effective instrument to combat credit crises.
VI. References
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and Econometrics, Tenth World Congress of the Econometric Society. Cambridge University
press: New York, NY.
Brunnermeier, M.K., and Y. Sannikov (2014). A Macroeconomic Model with a Financial Sector. American Economic Review 104, no. 2: 379-421.
Christiano, L.J., M. Eichenbaum, and C.L. Evans (2005). Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy. The Journal of Political Economy 113, no. 1: 1-45. Curdia, V., and M. Woodford (2011). The Central Bank Balance Sheet as an Instrument of Monetary Policy. Journal of Monetary Economics 58, no. 1: 54-79.
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European Central Bank (2016). Financial Stability Review, May 2016.
URL https://www.ecb.europa.eu/pub/pdf/other/financialstabilityreview201605.en.pdf Gerali, A., S. Neri, L. Sessa, and F.M. Signoretti (2010). Credit and Banking in a DSGE Model of the Euro Area. Journal of Money, Credit and Banking 42, no. 1: 107-141.
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A. Mathematical appendix
A. Households
The household’s optimization problem is to maximize expected, discounted lifetime utility:
max {Ct+s,Lt+s,Dt+s}∞s=0 Et (∞ X s=0 βs 1 1 − σ(Ct+s− hCt+s−1) 1−σ − χ 1 + ϕL 1+ϕ t+s ) , (26) β ∈ (0, 1) , h ∈ [0, 1), ϕ ≥ 0, (27)
subject to the budget constraint:
Ct+ Dt+ Tt= WtLt+ Πt+
1 + RDt−1− ΨtDt−1. (28)
Using the utility function and the household budget constraint, we set up the Lagrangian:
Solving the household’s optimization problem yields the following first order conditions: Ct: λt= (Ct− hCt−1)−σ− βhEt(Ct+1− hCt)−σ, (30) Lt: λtWt= χLϕt, (31) Dt: Et h βΛt,t+1 1 + RDt − Ψt+1 i = 1. (32) B. Non-financial firms B.1. Final goods producers
The constant returns to scale production technology available to final goods firms is:
Yi,t = At(ξtKi,t−1)αL1−αi,t . (33)
Taking the first order condition with respect to labor, we get that the wage rate equals:
Wt= (1 − α)
Yi,t
Li,t
. (34)
The representative final goods firm’s profits are equal to:
ΠFi,t = Yi,t+ QKt (1 − δ)ξtKi,t−1− (1 + RKt )QKt Ki,t−1− WtLi,t, (35)
which states that profits are equal to production plus the income gained after selling previous period’s capital stock net of depreciation times the price of capital, minus the cost of capital and the wage bill. Since firms operate in a perfectly competitive environment, profits are zero. We plug the expression for the wage rate into the final goods firm’s profit function, set the profit function to zero, and solve for the return on capital. This equals:
B.2. Capital goods producers
Final goods producers sell their used capital stock to capital producers. They sell the capital
stock net of depreciation and what is left after the shock, i.e. (1 − δ)ξtKt−1 for a price QKt .
Capital producers use the final good It to refurbish the depreciated capital stock. They face
convex adjustment costs, which depend on the change in investment Itrelative to the previous
period. The law of motion for capital is then:
Kt= (1 − δ) ξtKt−1+ " 1 −γ 2 I t It−1 − 1 2# It. (37)
Profits for capital goods producers are then the revenue they make from selling refurbished capital, minus the costs they incur when purchasing old capital and final goods used in the refurbishing process:
ΠKt = QKt Kt− QKt−1(1 − δ)ξtKt−1− It. (38)
The capital producers’ maximization problem is then to maximize the sum of current and
expected discounted future profits by finding the optimal path for investment It:
max {It+s}∞s=0 Et (∞ X s=0 βsΛt+s,t+1+s " 1 −γ 2 I t+s It−1+s − 1 2! QKt+sIt+s− It+s #) , (39)
where the household’s stochastic discount factor is used to discount future profits, since the household owns the capital goods producing firms. Maximizing the objective function by
differentiating with respect to It and rewriting yields an expression for the price of capital
B.3. Aggregation of non-financial firms
To find expressions for aggregate supply, we aggregate (33) for all firms i. Aggregation over both the left hand and right hand sides yields:
Z 1 0 Yi,tdi = Yt, Z 1 0 At(ξtKi,t−1)αL1−αi,t di = Atξtα Z 1 0 Ki,t−1α L1−αi,t di = At(ξtKt−1)αL1−αt .
Hence we find aggregate output equals:
Yt= At(ξtKt−1)αL1−αt . (41)
Aggregating the factor prices across all firms, we find that:
Wt= (1 − α) Z 1 0 Yi,t Li,t di = (1 − α)Yt Lt , (42) 1 + RKt = Z 1 0 α Yi,t Ki,t−1+ Q K t (1 − δ)ξt QKt−1 di = α Yt Kt−1 + Q K t (1 − δ)ξt QKt−1 (43) C. Banking sector C.1. Optimization problem
A typical bank’s optimization problem is characterized by:
where Vj,t≡ V
SKj,t−1, Dj,t−1
. We set up the following Lagrangian:
L = (1 + µt) EtnβΛt,t+1 h (1 − ϑ)1 + RKt+1+ Tt+1N − ˜Tt+1N QKt Sj,tK −1 + RDt − Ψt+1+ Tt+1N − ˜Tt+1N Dj,t + ϑV Sj,tK, Dj,t io − µtλKQKt Sj,tK + χt n 1 + RtK+ TtN − ˜TtNQKt−1Sj,t−1K −1 + Rt−1D − Ψt+ TtN − ˜TtNDj,t−1+ Dj,t− QKt Sj,tK o (44)
where µt is the Lagrange multiplier on the incentive compatibility constraint, and χt is the
Lagrange multiplier on the balance sheet identity. The financial intermediary maximizes with
respect to Sj,tK and Dj,t. The first order conditions for these two variables are:
Sj,tK : (1 + µt) Et ( βΛt,t+1[(1 − ϑ) [(1 + RKt+1+ Tt+1N − ˜Tt+1N )QKt ] + ϑ ∂Vj,t+1 ∂SK j,t ] ) − µtλKQKt − χtQKt = 0 (45) Dj,t : (1 + µt) Et ( βΛt,t+1[− (1 − ϑ) h 1 + RDt − Ψt+1+ Tt+1N − ˜Tt+1N i + ϑ∂Vj,t+1 ∂Dj,t ) + χt= 0. (46)
Note that we do not yet have an expression for the partial derivatives of the next period’s value function in the first order condition. By applying the envelope theorem we can get expressions for these partial derivatives:
∂Vj,t ∂Sj,t−1K = χt 1 + RKt + TtN − ˜TtNQKt−1, (47) ∂Vj,t ∂Dj,t−1 = −χt 1 + RDt−1− Ψt+ TtN − ˜TtN . (48)
Sj,tK : EtnβΛt,t+1[(1 − ϑ) + ϑχt+1](1 + RKt+1+ Tt+1N − ˜Tt+1N ) o = λK µ t 1 + µt + χt 1 + µt (49) Dj,t: Et n βΛt,t+1[(1 − ϑ) + ϑχt+1](1 + RDt − Ψt+1+ Tt+1N − ˜Tt+1N ) o = χt 1 + µt . (50)
Next, we define the following variables:
ηt= χt 1 + µt , (51) νtK= λK µ t 1 + µt + χt 1 + µt = λK µ t 1 + µt + ηt. (52)
Finally, we use the auxiliary variables we defined in (51) and (52) in (49) and (50), and subtract (50) from (49) to end up with the following first order conditions for respectively loans and net worth:
νtK = EtnβΛt,t+1[(1 − ϑ) + ϑ (1 + µt+1) ηt+1] RKt+1− RD t + Ψt+1 o (53) ηt= Et n βΛt,t+1[(1 − ϑ) + ϑ (1 + µt+1) ηt+1] 1 + RDt − Ψt+1+ +Tt+1N − ˜Tt+1N o . (54)
Next, we check whether our conjectured solution to the value function is correct. We guessed that the solution to the value function would be linear in loans and net worth:
Vj,t= νtKQKt Sj,tK + ηtNj,t. (55)
Assuming that the incentive compatibility constraint is binding, we substitute it into (56), slightly rewrite it and plug in the law of motion for net worth to get that:
Vj,t= Et n βΛt,t+1 h (1 − ϑ) + ϑνt+1K φt+1+ ηt+1 i Nj,t+1 o = EtnΩt,t+1hRKt+1− RD t + Ψt+1 QKt Sj,tK +1 + RDt − Ψt+1+ Tt+1N − ˜Tt+1N Nj,t io , (57) where Ωt,t+1 ≡ βΛt,t+1 h (1 − ϑ) + ϑνt+1K φt+1+ ηt+1 i
. If we compare this to our conjectured solution, we find the following first order conditions:
νtK = EthΩt,t+1RKt+1− RDt + Ψt+1i (58) ηt= Et h Ωt,t+1 1 + RtD− Ψt+1+ Tt+1N − ˜Tt+1N i , (59)
which coincide with expressions (53) and (54) we found using the Lagrangian method.
C.2. Aggregation of banking sector
A typical bank’s leverage constraint is given by φt= ηt
λK−νtK
. Since this does not depend on any idiosyncratic characteristics of a single bank j, we can simply aggregate over the incentive compatibility constraint to obtain the aggregate economy-wide leverage constraint:
QKt StK ≤ φtNt. (60)
The law of motion for old banks, i.e. those that did not go out of business, is:
No,t = ϑ h RKt − RD t−1+ Ψt QKt−1SKt−1+1 + RDt−1− Ψt+ TtN − ˜TtNNt−1 i , (61)
where the subscript o stands for old. The expression for aggregate net worth for new bankers is:
where the subscript n stands for new. The aggregate law of motion for net worth is then equal to the sum of net worth of old bankers, net worth of new bankers, and government financial sector support:
Nt= ϑ h RKt − RDt−1+ Ψt QKt−1St−1K +1 + RDt−1− Ψt Nt−1 i + χFQKt−1St−1K + NtG− ˜NtG. (62) B. Equilibrium conditions
A. First order conditions
The household’s first order conditions are:
λt= (Ct− hCt−1)−σ− βhEt(Ct+1− hCt)−σ, (63)
λtWt= χLϕt, (64)
1 = EthβΛt,t+1
1 + RDt − Ψt+1i. (65)
The first order conditions and equations for non-financial firms are:
The first order conditions for the banking sector are: QKt StK = Nt+ Dt, (71) νtK = Et h Ωt,t+1 RKt+1− RDt + Ψt+1 i , (72) ηt= Et h Ωt,t+1 1 + RDt − Ψt+1+ Tt+1N − ˜Tt+1N i , (73) Nt= ϑ h RKt − RD t−1+ Ψt QKt−1St−1K +1 + RDt−1− ΨtNt−1 i + χFQKt−1St−1K + NtG− ˜NtG, (74) φt= ηt λK− νtK , (75) φtNt= QKt StK, (76) where Ωt,t+1 ≡ βΛt,t+1h(1 − ϑ) + ϑνK t+1φt+1+ ηt+1 i
. The evolution of fiscal variables is given by: Tt+ ˜NtG= NtG, (77) NtG= TtNNt−1, (78) TtN = ζεξ,t−n, (79) ˜ NtG= ωNt−vG , (80) ˜ TtN = ˜ NtG Nt−1 , (81) Ψt= %εξ,t−l. (82)
The markets for goods and securities clear:
Yt= Ct+ It, (83)
Stochastic processes obey the following laws of motion:
log(At) = ρAlog(At−1) + εA,t, (85)
log(ξt) = ρξlog(ξt−1) + εξ,t. (86)
B. Recursive competitive equilibrium
Let Xt =
h
Ct−1, St−1K , Kt−1, Dt−1, Nt−1, It−1, Yt−1, At, ξt i
be the state vector. A recursive
competitive equilibrium is a sequence of quantities and prices Ct, λt, Lt, Dt, Kt, RDt , RtK,
Wt, QKt , It, Yt, At, ξt, StK, Nt, νtK, ηt, φt, Ψt, Tt, NtG, ˜NtG, TtN, and ˜TtN that satisfies: • The first order conditions for the household: (63) - (65).
• The first order conditions for non-financial firms: (66) - (70). • The first order conditions for the banking sector: (71) - (76). • The time path for fiscal variables: (77) - (82).
• The markets for goods and financial assets clear: (83) and (84).
C. Steady state conditions A. Targeted moments ¯ L = 1/3 (87) ¯ φ = 4 (88) ¯ Γ ≡ ¯RK− ¯RD = 0.0025 (89) ¯ A = 1 (90) ¯ ξ = 1 (91)
B. Steady state values
D. Immediate bail-in vs. immediate fiscal recapitalization
0 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 35 40 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 5 10 15 20 25 30 35 40 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 5 10 15 20 25 30 35 40 -12 -10 -8 -6 -4 -2 0 2 4 6 8 0 5 10 15 20 25 30 35 40 -14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 35 40 -8 -7 -6 -5 -4 -3 -2 -1 0 1 ( a ) 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 0 5 10 15 20 25 30 35 40 0 20 40 60 80 100 120 0 5 10 15 20 25 30 35 40 -100 -80 -60 -40 -20 0 20 0 5 10 15 20 25 30 35 40 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 ( b )
