Loan Default Risk in the Banking Sector and
Macroeconomic Stability
∗
Sebastiaan Pool
†This version: June 23, 2014
Abstract
This paper examines how loan default risk affects bank lending, interest rates, and the business cycle. I estimate a Dynamic Stochastic General Equilibrium model including bad loan provisioning by banks as a proxy for loan default risk using a Bayesian Vector Autoregression model for the Netherlands. The main findings are that: (i) bad loan provisioning is an important driver of credit supply, bank leverage, and business cycle fluctuations; (ii) the effects of bad loan provisioning are very persistent compared to the effects of a conventional monetary policy.
Keywords: Loan default, DSGE, Banking, Bayesian estimation
JEL Code: E44, E51, E52
∗For helpful comments and discussion I would like to thank Jan Jacobs, Jan Marc Berk, Leo de Haan, Laurie Reijnders, Ben Heijdra, Christoph Hanck, and Jakob de Haan. In addition, I would like to thank Martin Admiraal for his assistance with the data.
1
Introduction
The Global Financial Crisis (GFC) reignited macroeconomists’ attention to the interaction between the banking sector and the real economy. Although in the past, various macroeconomists acknowledged that the banking sector had affected business cycle fluctuations, the conventional notion was that business cycle fluctuations were primarily caused by the real economy (Gertler and Kiyotaki, 2010). The GFC, however, was an unambiguous reminder that the banking sector can be a key driver of business cycle fluctuations and that there exist adverse feedback effects between the banking sector and real economy. A vast amount of literature has emerged, see Brunnermeier et al. (2012) for a survey. Nevertheless, there is still a lot of uncertainty about the impact and consequences of these adverse feedback effects and proper identification of financial distress remains difficult (Gilchrist and Zakrajˇsek, 2011).
This study uses a Dynamic Stochastic General Equilibrium (DSGE) model introduced by Gerali et al. (2010). The DSGE model embeds a banking sector in a conventional monetary policy framework and includes bad loan provi-sioning by banks as a proxy for loan default risk. Bad loan proviprovi-sioning represents funds set aside to pay for anticipated loan defaults. The explicit role for bad loan provisioning allows me to analyze and quantify the effects of loan default risk on the real economy. Loan default risk creates financial friction. In this study I show that financial friction can increase persistence in the effects of an initial adverse shock and that financial friction can amplify the effects of an initial adverse shock via the adverse feedback mechanism between the banking sector and real economy (Brunnermeier et al., 2012).
The theoretical model is estimated for the Netherlands using a Bayesian Vector Auto Regression (BVAR) approach for the period 1979Q1:2009Q4. I analyze the Netherlands because, in contrast to most countries, banking data is available from 1979 onwards. More importantly, the Dutch banking sector was under financial distress during the GFC which had severe conse-quences for the real economy. Hence, the Netherlands is a prime example of how instability in the banking sector can destabilize the real economy. The BVAR approach is appropriate because the theoretical model contains many parameters and data series are relatively short. The prior parameter speci-fication is constructed using the structural parameter values of Gerali et al. (2010), Heijdra (2009) pp. 387-390, and data averages. The resulting pos-terior parameter modes and distributions are used to plot impulse response functions.
a positive provisioning shock, the Impulse Response Functions (IRFs) indi-cate a decrease in the loan rate. However, the amount of outstanding loans decreases following a positive provisioning shock. Moreover, not only the amount of outstanding loans decreases, but also the value of bank capital. These results indicate that banks decrease the size of their balance sheets after a positive provisioning shock.
I also compare the effects of a contractionary monetary policy shock to a positive provisioning shock. The value of bank capital decreases sharply after a provisioning shock, while the value of bank capital decreases only shortly after a contractionary monetary policy shock. For this reason, the effects of a provisioning shock appear to be more persistent than the ef-fects of a monetary policy shock. In addition, the results show that after a contractionary monetary policy shock the inflation rate shortly increases. The result is counter intuitive because economic theory predicts downward pressure on the inflation rate after a contractionary monetary policy shock (Sims, 1992). The effect of a positive provisioning shock on the inflation rate is less ambiguous and appears to be positive. Since a positive provisioning shock clearly increases the inflation rate, it may be important for monetary authorities to anticipate the effects of bad loan provisioning on the inflation rate to ensure a stable price level.
risk even if exogenous risk is small. However, these models lack a clear de-scription of a financial sector, more importantly a banking sector. Freixas and Rochet (1997), Gerali et al. (2010), and Gertler and Kiyotaki (2010) do attempt to model an explicit role for the banking sector; yet, in these models there is only a limited role for financial frictions and the adverse feedback effects between the banking sector and real economy. In contrast, this study analyzes how financial frictions affect the banking sector inherently.
Second, Bernanke et al. (1999) and Brunnermeier and Sannikov (2014) model financial frictions using loan default risk which requires borrowing agents to have collateral. However, these studies rest mainly on the repre-sentative agent assumption and therefore actual loan default never happens. Although this study also rests mainly on the representative agent assump-tion, I do include an explicit role for loan default. Loan default risk, i.e., the possibility that a loan goes into default, creates financial friction which destabilizes the banking sector and may have adverse feedback effects on the real economy.
their role by investing in long-term projects and financing themselves with short-term debt, i.e., they perform liquidity transformation. As a result they are exposed to liquidity risk. If during turmoil times the value of loans that go into default is high and collateral values decrease, bank capital deterio-rates. Banks will need additional liquidity to deleverage their balance sheets; however, as bank capital deteriorates they need to pay high interest rates which restrains the ability to rollover their short-term debt. This liquidity mismatch constrains future credit supply (Brunnermeier et al., 2012).
The GFC has shown that the banking sector is an important source of business cycle fluctuations. One of the reasons is the presence of financial frictions which cause persistence. In addition, if the banking sector has liq-uidity problems, these financial frictions have non-linear amplification effects (Brunnermeier et al., 2012). During tranquil times imbalances may build up until a shock reverses the process and causes a persistent wealth destruction. Financial frictions exist because borrowers and lenders have different infor-mation which generates asymmetric inforinfor-mation problems (Christiano et al., 2009, 2011). Lenders face agency costs and more importantly the risk that borrowers default. Borrowers would like to signal their credibility, but face a hard time distinguishing themselves from one another, i.e., distinguishing good borrowers with a low loan default probability from bad borrowers with a high loan default probability. As a consequence, financial friction generated by loan default risk increases yield spreads on interest rates (Gilchrist and Mojon, 2014).
banking sector. Persistence of an initial adverse shocks arises because bor-rowers need to rebuild their net worth through retained earnings to be able to borrow again. Banks respond by decreasing their lending activities and increasing risk premiums. These measures spill-over to the real economy if borrowing for investment tightens which slows down the economy and po-tentially triggers a crisis. During a crisis demand and thus firm profits are low causing more borrowers to go into default which destabilizes the banking sector further. This adverse feedback mechanism potentially causes losses in terms of output, employment, and wealth (Gilchrist et al., 2014), and explains how financial frictions can amplify the initial adverse shock.
Many theoretical explanations have been proposed for the traditional channels of monetary policy. However the empirical evidence on the strength of these channels differs substantially, see Bernanke and Gertler (1995) and references therein. Traditionally, the empirical evidence concerning the ef-fects of monetary policy on the real economy is studied using VAR models, see e.g. Bernanke and Gertler (1995) or Christiano et al. (1999) for a discussion. In addition, DSGE models are often estimated using Bayesian estimation techniques to study the effects of monetary policy on the real economy, see e.g. Bernanke et al. (1999), Christiano et al. (2005), Smets and Wouters (2007), Gerali et al. (2010).
2
Model
The model is based the Gerali et al. (2010) who analyze the interaction between the banking sector and the real economy using a DSGE framework.
2.1
Households and Entrepreneurs
The inhabitants in the economy consist out of two groups of consumers, en-trepreneurs, denoted by the superscript (E), and households, denoted by the superscript (H). The size of both groups is normalized to unity. En-trepreneurs are less patient than households. In equilibrium the enEn-trepreneurs will borrow via the banking sector from the households. Furthermore, en-trepreneurs are able to acquire physical capital and hire labor from house-holds in order to produce.
2.1.1 Households
The representative household maximizes its expected utility ΛH0 (i):
ΛH0 (i) ≡ E0 ∞
X
t=0
(βH)tαCln CtH(i) − (1 − αC) ln LHt (i) , (1)
where CtH(i) denotes consumption of agent i at time t, LHt (i) denotes hours worked, βH is the household discount factor, E
0 is an expectation operator,
αC denotes the weight of consumption and (1 − αC) denotes the weight of
leisure, and ln(·) denotes the natural logarithm. The representative house-hold maximizes expected utility subject to the budget constraint:
CtH(i) + Dt(i) = wtLHt (i) +
1 + rd t−1
πt
where Dt(i) denotes deposits, wtis the real wage, (1+rdt−1)Dt−1(i)/πtdenotes
interest income on last period’s deposits, and πt ≡ Pt/Pt−1 is the inflation
rate. In Appendix A I solve the household optimization problem.
2.1.2 Entrepreneurs
The representative entrepreneur maximizes its expected utility ΛE 0(i) by
choosing consumption CE
t (i), physical capital Kt(i) and loans from banks
Bt(i): ΛE0(i) ≡ E0 ∞ X t=0 (βE)tln CtE(i) , (3) subject to the budget constraint:
CtE(i)+1 + r
b t−1
πt
Bt−1(i)+wtLHt (i)+Kt+1(i) = (1−δk)Kt(i)+Yt(i)+Bt(i), (4)
where (1 + rb
t−1)Bt−1(i)/πt denotes interest payment on last period’s loans,
δk is the depreciation rate of physical capital, and Yt(i) denotes revenue from
producing the wholesale good. Entrepreneurs produce an unique variety of a wholesale good Yt(i) according the following production function:
Yt(i) = A[Kt(i)αLHt (i)
1−α], (5)
where A is a Hicks-neutral technology parameter. I assume a constant tech-nology parameter because this study does not focus on technological progress. Entrepreneurs produce Yt(i) using capital, and labor supplied by the
house-hold sector. The law of motion for the capital stock is denoted as:
where It(i) denotes investment. The amount of resources banks are willing
to lend to entrepreneurs is constrained by the amount of collateral the en-trepreneur has. Collateral is given by the enen-trepreneur’s holdings of physical capital. Hence, entrepreneurs can borrow subject to the following balance sheet constraint:
(1 + rbt)Bt(i) ≤ mtEt[πt+1(1 − δk)Kt(i)], (7)
where mt denotes the loan-to-value ratio (LTV). Entrepreneurs, producing
the wholesale good Yt(i), compete in a monopolistic competitive market.
Therefore I adopt a Dixit and Stiglitz (1977) framework. To introduce price stickiness entrepreneurs cannot change their price each period. They can only change their price with the exogenous probability (1 − η). Hence, I adopt Calvo (1983) pricing. In Appendix A I solve the entrepreneur’s optimization problem (Heijdra, 2009, pp. 394-395).
2.1.3 Loan and deposit demand
of market power. Berger et al. (2004) link market concentration to market power and the interest rate setting behavior of banks. They find evidence that low market concentration in the banking sector increases market power of banks. Other studies report contestability and regulatory restrictions as a source of market power, e.g. Demirguc-Kunt et al. (2004). Several empir-ical papers confirm the presence of market power in the banking sector, see Berger et al. (2004) and Degryse and Ongena (2008) for a discussion.
To model market power in the banking sector I also use a Dixit-Stiglitz framework for the credit market. Each bank produces a unique variety of loans Bt(j) and deposits Dt(j). Accordingly, loan demand by entrepreneurs
and deposit demand by households at bank j are given by, see Appendix A for the derivations:
Bt(j) = rb t(j) rb t µb Bt, (8) Dt(j) = rd t(j) rd t µd Dt, (9)
where µb and µddenote the elasticity of substitutions for loans and deposits,
respectively.
2.2
Defaults
Entrepreneurs default on their loan repayment if their net worth Nt drops
below the threshold value Nt. I assume that if net worth drops below Nt
func-tion. The net worth of the entrepreneur is given by:
Nt(i) ≡(1 − δk)Kt(i) + Yt(i) + Bt(i) − CtE(i) −
1 + rb t−1
πt
Bt−1(i)−
wtLHt (i) − Kt+1(i). (10)
Real profits, RPt(i) = Yt(i) − wtLHt (i) − rtbKt(i), might be negative if the
entrepreneur is not able to change its price for a long period. In a frictionless economy negative profits are no problem; on average the entrepreneur earns positive profits and repays the loan and the entrepreneurs do not need col-lateral at all. However, if the entrepreneur only pays interest if Nt ≥ Nt, a
financial friction arises. Note that Brunnermeier and Sannikov (2014) adopt a similar approach by constraining agents to have positive net worth; other-wise the agent go into default and obtains zero utility.
Since there is only a probability (1 − η) that the entrepreneur is able to change its prices, see Appendix A, it is possible that RPt(i) becomes so
negative that Nt< Nt. In this case the entrepreneur defaults. At the macro
level, given a Poisson distribution of η and the law of large numbers, see Calvo (1983), a constant percentage θd of the entrepreneurs default on their
loan repayments:
Qt= Θ(η)Bt≡ θdBt−1, (11)
2.3
The banking sector
Banks act as intermediaries for all financial transactions between households and entrepreneurs in the model. Households save deposits to smooth income over time and entrepreneurs borrow to finance production. Following Gerali et al. (2010) the banking sector is modelled as a collaboration between three branches, i.e., two retail branches and a wholesale branch. The two retail branches are responsible for the collection of deposits and the allocation of loans and set interest rates is a monopolistic competitive fashion. The wholesale branch manages the capital position of the banking entity.
2.3.1 Wholesale branch
The wholesale branch operates under perfect competition and combines bank capital Kb
t and wholesale deposits Dt on the liability side and issues loans Bt
on the asset side to maximize its profits. In addition, the wholesale branch anticipates that each period a number of entrepreneurs go into default and do not pay interest. In anticipation they provision an amount equal to EtQt+1
on their balance sheet to pay for these expected losses in the next period. Hence, each wholesale branch has a balance sheet constraint which is given by:
EtQt+1+ Bt= Dt+ Ktb. (12)
The law of motion for bank capital evolves according to:
Ktb = (1 − δb)Kt−1b + ωbJt−1b , (13)
where Jb
t are overall bank profits of the retail and wholesale branches, (1−ωb)
denotes the dividend policy of the bank, and δb denotes resources used to
Moreover the wholesale branch faces adjustment costs, denoted by the parameter κw, whenever the capital-to-assets ratio Ktb/Bt deviates from the
optimal capital-to-asset ratio νw. Note that, Ktb/Bt denotes bank leverage.
The wholesale branch maximizes the discounted sum of future cash flows by choosing the appropriate loan and deposit levels subject to the balance sheet constraint: max Bt,DtE 0 ∞ X t=0 (1 + rwbt )Bt− Bt+1+ Dt+1− (1 + rtwd)Dt+ ∆Kt+1b −κw 2 Kb t Bt − νw 2 Ktb , subject to EtQt+1+ Bt = Dt+ Ktb, (14) where rwd
t and rtwb denote respectively the deposit rate and the loan rate
charged by the wholesale branch to the corresponding retails branches, and ∆ denotes the change in a variable over time. Using Equation (11) and the balance sheet constraint at time t and t + 1 in the objective function, Equation (14) can be rewritten as:
max Bt,Dt rtwbBt− rwdt Dt+ θd∆Bt+1− κw 2 Kb t Bt − νw 2 Ktb. (15)
The FOCs link the wholesale rates on loans and on deposits to the degree of leverage Ktb/Bt. In addition, I assume that banks are not finance constrained
because they can always borrow from the central bank at rate rt. For this
reason arbitrage opportunities ensure that rwdt = rt. Using these results the
FOCs can be rewritten as:
or: Stw ≡ rwb t − rt= −κw Kb t Bt − νw Kb t Bt 2 , (17) defining a spread Sw
t between lending and borrowing activities
2.3.2 Retail branches
The retail branches are monopolistic competitors on both the loan and de-posit markets, i.e., both the loan branch as well as the dede-posit branch pro-duces a differentiated product.
Loan branch. The loan branch maximizes its profits by lending to en-trepreneurs while financing these lending activities by borrowing from the wholesale branch at rate rtwb. The loan branch maximizes its profits by choosing the appropriate lending rate rb
t(j) facing quadratic interest rate
adjustment costs denoted by the parameter κb:
max rb t(j) E0 ∞ X t=0 " (rtb(j) − rwbt )Bt(j) − κb 2 rtb(j) rb t−1(j) − 1 2 rtbBt # , (18)
subject to the loan demand schedule, Equation (8). The loan branch does not take into account the possibility of loan default; this is done by the wholesale branch.
Imposing a symmetric equilibrium, i.e., rbt(j) ≡ rbt, for all j and log-linearizing the loan rate setting equation, see Appendix A for the loan branch optimization problem and Appendix B for the log-linearization, gives:
denotes a log-linearized variable. Equation (19) states that the loan rate depends on the loan rate in the previous period, the expected loan rate in the next period and the borrowing costs charged by the wholesale branch. If the loan rate is perfectly flexible (κb = 0), the maximization problem
simplifies to:
rbt = µb µb − 1
rtwb, and r˜tb = ˜rwbt . (20)
Hence each branch simply sets the loan rate as a mark-up over its marginal costs.
Deposit branch. Similarly to the loan branch, the deposit branch of bank j collects deposits Dt(j) from households and passes these to the wholesale
branch which compensates them at rate rt= rwdt (the interest rate set by the
central bank). In addition, the deposit branch also faces quadratic interest rate adjustment costs which are denoted by the parameter κd. The deposit
branch maximization problem becomes:
max rd t(j) E0 ∞ X t=0 " (rt− rtd(j))Dt(j) − κd 2 rd t(j) rd t−1(j) − 1 2 rtdDt # , (21)
subject to deposit demand, Equation (9). Again I impose a symmetric equi-librium and log-linearize the solution around the steady state, see Appendix A for the deposit branch optimization problem and Appendix B for the log-linearization. The solution to the deposit branch optimization problem is denoted by:
˜
where ζd 1 ≡ κd µd+1+(1+βH)κd, ζ d 2 ≡ βHκ d µd+1+(1+βH)κd, and ζ d 3 ≡ µd+1 µd+1+(1+βH)κd.
Sim-ilar to the loan branch, the deposit rate depends on the deposit rate in the previous period, the deposit rate in the coming period, and the lending rate offered by the wholesale branch. If the deposit rate is perfectly flexible (κd = 0) the maximization problem simplifies to
rdt = µd µd− 1
rt, and r˜dt = ˜rt. (23)
Bank profits. Finally, combined real profits of the wholesale, loan, and deposit branches are equal to:
Jtb = rbtBt− rdtDt− κw 2 Kb t Bt − νw 2 Ktb− Qt− Ctb, (24) where Cb
t are the adjustment costs for changing the interest rates at the retail
level. Hence, bank profits are determined by interest income on loans minus interest expenses on deposits, deviations from the optimal capital-to-asset ratio, default losses, and adjustment costs for changing the interest rates.
2.4
Closing the model
The goods market equilibrium (GME) is given by:
Yt= Ct+ It, (25)
where Ct= CtH+ CtE. Using the GME I can express the household
consump-tion Euler and entrepreneur consumpconsump-tion Euler equaconsump-tions as funcconsump-tions of Ct,
see Appendix B Equation (B.42).
All entrepreneurs produce an unique product variety, Yt(i), and maximize
pricing. Aggregating the solution to the firm optimization problems to the macro level and log-linearizing the aggregate firm optimization solution, see Appendix B, gives a New Keynesian Phillips Curve (NKPC):
˜
πt= γ ˜Yt+
1
1 + ρEEtπ˜t+1. (26)
Finally, a simple monetary policy rule is postulated which takes the form of a Taylor rule:
˜
rt= δππ˜t+ δyY˜t. (27)
Equation (27) argues that the monetary authority has two objectives, a stable inflation rate and a stable growth rate. Usually, a NKPC and a Taylor rule include the output gap variable instead of the output variable. However, there is no output growth in the model; as a consequence, potential output is a constant. Hence, percentage changes in the output gap variable are equivalent to percentage changes in the output variable.
2.5
Structural VAR representation
Appendix B summarizes the model and log-linearizes the model around its steady state. The linearized model can be represented as a structural Vector Autoregression (VAR). Using the auxiliary variable definition gLXt ≡ ˜Xt−1
I can write all variables dating t as gLXt+1 = ˜Xt. The reduced form VAR
representation of the model is given by:
where I multiplied both sides by Γ−10 to get rid of Γ0 on the left hand side.
Iterating all variables one period backward in time gives:
Zt= Γ−10 Γ1Zt−1+ Γ−10 Γ2Zt−2+ Γ−10 Υ0εt. (28)
where Zt = h ˜Qt, ˜Ktb, ˜πt, ˜Kt, ˜Ct, ˜Yt, fLwt, gLDt, gLBt, ˜rdt, ˜rtb, ˜rt
i0
, Γ0 is a
coeffi-cient matrix specifying the contemporaneous response of each variable to all variables, Γ1 and Γ2 are respectively the coefficient matrices specifying the
response of each variable to the first and second lag of all variables, Υ0 is
a coefficient matrix specifying the contemporaneous response of each vari-able to the shock term, and εt is a vector containing two shock terms, i.e.,
the monetary policy shock εMt and the provisioning shock εQt. The mone-tary policy shock represents deviations of the policy rate from the predicted policy rate specified in Equation (27). An unexpected change in bad loan provisioning denotes a provisioning shock and is represented by a deviation of the level of bad loan provisioning from the steady state level of bad loan provisioning specified in Equation (11).
3
Methodology
3.1
Bayesian Vector Auto Regressions (BVAR)
To quantify the effects of loan default risk on economic activity, the VAR model is estimated using Bayesian estimation techniques. First, consider the following reduced form VAR:
Xt= Φ(L)Xt−1+ vt, (29)
here Xt is a vector of endogenous variables introduced in Section 2, see
Equation (28), Φ(L) is a lag polynomial Φ(L) ≡ Φ1+ Φ2L1 + · · · + ΦpLp−1,
and vt is a vector of reduced form residuals, assumed to be iid ∼ (0, Σv).
3.2
Minnesota prior
The prior specification is the Minnesota (or Litterman) prior which assumes a Normal (N ) Inverse-Wishart (IW ) distribution. The Normal Inverse-Wishart distribution is represented as follows:
Σv ∼ IW (Ψ; d), (30)
Φ|Σv ∼ N (b, Σv⊗ Ω). (31)
where b is a assumed to be a stacked vector of zeros of dimension p×2, where
denotes the number of variables and p denotes the number of lags; Σv is the
variance-covariance matrix of dimension p2 × 2× 2, multiplied by a scalar
matrix Ω; d are the degrees of freedom of the Inverse Wishart distribution and is defined as the dimension of Φ plus 2, which is the minimum degrees of freedom that guarantees that the prior is proper (Nydick, 2012); Ψ is a diagonal × symmetric scale matrix.
variance-covariance matrix Σv, the Inverse-Wishart distributions is the conjugate prior
for the multivariate normal distribution
The Minnesota prior assumes a Normal distribution for Φ and an Inverse-Wishart distribution for Σv. First, let Φ = Φ + νx, νx ∼ N (0, Σx), where
Σx is fixed variance-covariance matrix. The Minnesota prior uses prior
in-formation about Φ and Σv to estimate the VAR coefficients, i.e., it is
possi-ble to specify the prior mean of the coefficient denoted by Φ and the prior (co)variance of the coefficients denoted by Σv. I calculate the prior values of
the reduced form coefficients, Φ, using the values of the structural parame-ters specified by Gerali et al. (2010), Heijdra (2009) pp. 387-390, and data averages. The values of the structural parameters are presented in Table 2 in Appendix C. The resulting prior coefficient matrices are also specified in Appendix C. Moreover, the Minnesota prior assumes that Σv is a function
of a small number of hyperparameters denoted by ψ. More specifically, Σv
is a diagonal matrix and its σı,p element corresponding to lag p of variable
in equation ı has the form:
σı,p = ψ0 h(p) if ı = , ∀p. ψ0× h(p)ψ1 × σ σı 2 otherwise when ı 6= , ∀p
where ψ0 and ψ1 are hyperparameters,
σ
σı
2
is a scaling factor and h(p) a deterministic function of p. More specifically, ψ0 denotes the relative prior
tightness of a variables own lags; ψ1 denotes the relative prior tightness of
the other variables; and h(p) is the relative tightness of the variance of lags other than the first one. I follow Canova (2007) and set ψ0 = 0.2, ψ1 = 0.3,
and assume a harmonic decay h(p) = pψ2 and set ψ
2 = 1. Estimation
hyperparameters.
3.3
Gibbs sampling
To estimate the posterior parameter distribution I use a Gibbs sampling method (Geman and Geman (1984); Gelfand and Smith (1990); Kim and Nelson (1999)). Gibbs sampling is convenient because given a multivariate distribution it is simpler to sample from a conditional distribution than to marginalize by integrating over a joint distribution.1 The Gibbs sampling
method works as follows. First, a vector Ξ = (ξ0, ξ1, . . . , ξn) that contains all
n parameters of the model is defined to obtain random draws from a joint den-sity p(ξ0, ξ1, . . . , ξn). The `th sample is denoted as Ξ`= (ξ`0, ξ1`, . . . , ξn`).
Sub-sequently, arbitrary initial values are chosen denoted by Ξ0. For each sample
` ∈ {1; z} where z are the number of draws, the Gibbs sampler draws ξ` from the conditional distributions, i.e., ξ`
∼ p(ξ|ξ1`, . . . , ξ−1` , ξ `−1
+1, . . . , ξn`−1).
Hence, the Gibbs sampler draws each coefficient from the distribution of that coefficient conditional on the most recent values of all other variables. Geman and Geman (1984) show that the distribution of the resulting Gibbs sequence Ξ` converges to the true joint density of a geometric rate in z. The Gibbs
sampling method is estimated using 2, 500 burn-in iterations ensuring conver-gence. The remaining draws, Ξ` = (ξ`
0, ξ1`, . . . , ξn`), for ` ∈ {2, 500; 25, 000},
are random draws from the joint density p(ξ0, ξ1, . . . , ξn).
4
Data
I estimate the BVAR for the Netherlands using data from 1979Q1:2009Q4. The data include the following variables: output measured by gross domestic product in real terms; private consumption in real terms; the capital stock measured as fixed capital accumulation; bank capital measured as capital and reserves, and borrowing from the Central Bank; inflation using the price deflator for private consumption; wages measured as the real minimum wages rate; the nominal policy rate represented by the three month money mar-ket interest rate; nominal interest rate on loans measured as the bank rate that meets the short- and medium-term financing needs of the private sector; nominal interest rate on deposits measured as the rate paid by commercial or similar banks for demand, time, or savings deposits; loans measured as out-standing bank loans to the private sector; deposits measured by the aggregate amount of customer deposits; and provisioning measured by aggregate pro-visioning of the banking sector, Table 6 in Appendix D lists all variables and their composition. Moreover, Table 3-5 in Appendix D present all variables, parameters, indices, and operators used throughout this study.
Bad loan provisioning, banking capital, the loan rate, the deposit rate, and wages are reported on an annual basis while all other data series are reported on a quarterly basis. To be able to estimate the BVAR on quar-terly data I interpolate these variables using the quadratic match average conversion method.2
Expectations about the value of loans that go into default are not directly observed, therefore I use aggregate bad loan provisioning. Provisioning is an item on the balance sheet representing capital reserved for anticipated future
losses. Bad loan provisioning represents funds set aside to pay for anticipated losses due to loans that go into default. Banks may use bad loan provisioning to smooth their income over time. During economic booms the value of loans that go into default is lower than expected and provisioning increases; accordingly, during crises the value of loans that go into default is higher than expected and provisioning decreases. Hence, bad loan provisioning is expected to be pro-cyclical.
Bad loan provisioning is a net flow variable. A disadvantage of a flow variable is that one cannot distinguish inflows and outflows; only the resulting net change is observed. It is, however, possible to distinguish two causes that change the level of bad loan provisioning. First, if many firms go into default, banks decrease bad loan provisioning levels to cover for the losses. Second, if banks anticipate more loans to go into default, an increase in loan default risk, they increase their level of bad loan provisioning today. In the former case the level of bad loan provisioning decreases while in the latter case the level of bad loan provisioning increases.
Bank capital is defined as the sum of capital, reserves, and borrowing from the Central bank. The asset side of the bank balance sheet states loans and securities. The liability side of the bank balance sheet states deposits, capital, reserves, and borrowing from the Central Bank. The difference between loans and deposits equals bank capital and is therefore represented by the sum of capital, reserves and borrowing from the Central Bank.
Table 1: Summary statistics endogenous variables
Variable Transformation Obs. Mean St. Dev. Median Yt ln Yt 123 10.41 0.24 12.86 Ct ln Ct 123 12.18 0.20 12.15 Kt ln Kt 123 11.33 0.27 11.28 Ktb ln Ktb 123 10.41 0.96 10.39 πt ln πt 123 4.71 0.21 4.17 wt ln wt 123 10.00 0.06 9.99 Bt ln Bt 123 13.12 0.78 12.93 Dt ln Dt 123 12.71 0.96 12.73 Qt ln Qt 123 7.37 0.89 7.28 rtd ln rdt 123 1.27 0.27 1.24 rb t ln rtb 123 1.90 0.56 2.04 rt ln rt 123 1.55 0.55 1.60
5
Results
This section analyzes the effects of a monetary policy shock and a shock to bad loan provisioning. Figure 1 shows the IRFs after a monetary policy shock represented by an one standard deviation increase in the policy rate, and Figure 2 shows the IRFs after a provisioning shock represented by an one standard deviation increase in bad loan provisioning.
In Section 4 I stated that bad loan provisioning is a net flow variable which changes due to two causes: actual loan defaults and changes in expec-tations about the value of loans that go into default. The value of loans that go into default is, however, correlated with the business cycle; in the Nether-lands the correlation coefficient between the growth rate in the number of firms going into defaults and the growth rate in output is −0.48.3 Hence, including output as endogenous variable controls indirectly and partly for the actual value of loans that go into default. As a consequence, any unexpected
change in the bad loan provisioning series is primary determined by changes in expected loan default risk and not by actual loan default.
5.1
Monetary policy shock
A contractionary monetary policy shock tightens liquidity constraints in the banking sector. The bottom-center panel in Figure 1 shows that the mone-tary policy shock increases the loan rate for almost two years. The increase is not one-for-one, but peaks at about a quarter of the initial policy rate in-crease. The deposit rate, presented in the bottom-left panel, is not affected by a monetary policy shock. Apparently, banks only adjust the loan rate after a monetary policy shock and leave the deposit rate unaffected. The result is in contrast with the no-arbitrage rule stated in Section 2, rwd
t = rt.
The contractionary monetary policy shock also tightens liquidity con-straints in the real economy. Banks increase the loan rate and decrease the amount of outstanding loans. Consequently, borrowing from the banking sector to finance investment in the capital stock is constrained. The second-row-left panel in Figure 1 shows that the capital stock starts to decreases about a year after the monetary policy shock. In the model I assume that entrepreneurs produce using the production factors capital and labor. There-fore the decrease in the capital stock should cause production to slow down via the production function. The IRF in the second-row-right panel indeed show a decrease in output after a monetary policy shock. The results indicate that the real economy faces a contraction of funds available and slows down. However, output starts to decrease immediately after the monetary policy shock, while the capital stock starts to decrease after the first year. The cap-ital stock is predetermined and adjustment takes time. Entrepreneurs facing higher interest rates can, however, immediately cut back production if they expect a demand contraction .
are less capable to perform their role in the intermediation process. For this reason fiat money becomes more expensive, i.e., inflation. Fisher inflation may offset the deflation caused by the monetary contraction.
Although after the monetary policy shock deposit levels are unaffected and output declines, consumption increases, see the second-row-center panel. Nonetheless, as consumers can either save their income or consume it, these results seem contradictory. The result may be explained by looking at the IRF of the wage rate, the third-row-left panel. The wage rate appears to increase a bit about one and a half year after the monetary policy shock. The model predicts an increase in the use of the factor labor and a decrease in the use of the factor capital if the cost of capital, the loan rate, increases. Moreover, the inflation rate increases also after the monetary policy shock. If entrepreneurs substitute capital for labor, labor income may increase while the increase in the inflation rate induces households to consume more and save less. The decrease in output can be explained by a decrease in invest-ment. If investment decreases more than consumption increases, aggregate output still declines.
5.2
Provisioning shock
bottom-Figure 1: IRFs BVAR estimation following a monetary policy shock.
Figure 2: IRFs BVAR estimation following a provisioning shock.
center and the bottom-left panels respectively. In addition, the amount of outstanding loans decreases sharply, while the amount of outstanding de-posits is unaffected, see the third-row-right and the third-row-center panel respectively. Presumable banks decrease the amount of outstanding loans by restricting credit supply. If banks only accept the least risky loans, they can lower the loan rate because loan default risk decreases. It also appears that banks keep the spread between the loan rate and the deposit rate more or less constant as both rates drop simultaneously. The difference between the loan rate and the deposit rate determines bank profitability, see Equation (17). If banks focus on a constant spread between the loan and deposit rate, than a drop in the loan rate would lead to a corresponding drop in the deposit rate. Although theoretically arbitrage opportunities arise if the deposit rate is different from the policy rate, the results show that in practice both rates do not move one-for-one. Hence, it appears that financial frictions, e.g. loan default risk, decrease effectiveness of monetary policy because the policy rate and the bank rates do not move one-for-one.
bank capital. As such banks try to keep their leverage ratio as close as pos-sible to the optimal leverage ratio. The results suggest that banks indeed focus on a constant leverage ratio because the decrease in the amount of outstanding loans is followed by a decrease in the value of bank capital. The adjustment path of bank capital is, however, much more pronounced after a provisioning shock than after a monetary policy shock. After a monetary contraction the amount of outstanding loans decreases faster than the value of bank capital; as a consequence, banks deleverage. After the provisioning shock the decrease in bank capital is much more consistent with a constant leverage ratio. An advantage of not adjusting the level of bank capital is that banks are able to increase the amount of outstanding loans sharply once the economy recovers. Yet, if the value of bank capital declines after the provi-sioning shock, banks need to rebuild bank capital before they can increase the amount of outstanding loans. Hence, the results suggest that the effects of a provisioning shock are much more persistent than the effects of a monetary policy shock.
to decrease the size of their balance sheet, i.e., banks lower the amount of outstanding loans and they decrease the value of their bank capital. Banks remain commercial entities which focus on profits. During times in which bad loan provisioning levels are high, they might aim for relatively safe in-vestments. If the capital stock is a relatively safe asset and interest rates are low, the capital stock might be able to grow despite a decrease in the amount of outstanding loans. Note, the wage rate is not affected by the provisioning shock suggesting no change in the demand for labor, see the third-row-left panel. Moreover, output decreases a little after the provision-ing shock suggestprovision-ing that although the capital stock increases, production does not, see the second-row-right panel. Hence, apparently firms build up their capital stock, but are not able to increase production. In the model there are no capital adjustment costs; as a consequence, the price of capital is always equal to one. However, it may be an interesting area for future research to analyze how the price of capital is affected to determine why the capital stock is increasing while output decreases.
anticipating increases or decreases in bad loan provisioning appears to be important in determining the strength of monetary policy.
6
Conclusion
This study analyzed the effects of bad loan provisioning on the banking sector and real economy. I argued that after controlling for the business cycle via changes in output, unexpected changes in bad loan provisioning may be interpreted as changes in loan default risk. I built a structural macro model embedding a banking sector to analyze how bad loan provisioning affects the banking sector and the real economy. In the model the banking sector uses bad loan provisioning to prevent large fluctuations in income caused by loan default. The explicit role for bad loan provisioning enabled me to analyze how loan default risk affects the banking sector and real economy.
provisioning shock banks decrease the value of their bank capital much more than after the monetary policy shock. The result indicates that the effects of a provisioning shock are much more persistent than the effects of a monetary policy shock.
The IRFs also suggest that the capital stock starts to increase following the bad loan provisioning shock. Firms either finance their capital through retained earnings and/or by issuing equity, or banks increase their lending for investment in the capital stock. In the latter case this would imply that the capital stock is a relatively safe asset. However, a proper explanation is missing because the price of capital in the model is by assumption equal to one. An interesting area for future research is to analyze how the price of capital is affected.
References
Berger, A. N., Demirg¨u¸c-Kunt, A., Levine, R., and Haubrich, J. G. (2004). Bank concentration and competition: An evolution in the making. Journal of Money, Credit and Banking, 36(3):433–451.
Bernanke, B. S. and Gertler, M. (1995). Inside the black box: The credit channel of monetary policy transmission. Journal of Economic Perspec-tives, 9(2):27–48.
Bernanke, B. S., Gertler, M., and Gilchrist, S. (1999). The financial accel-erator in a quantitative business cycle framework. Handbook of Macroeco-nomics, 1(C):1341–1393.
Brunnermeier, M. K., Eisenbach, T. M., and Sannikov, Y. (2012). Macroe-conomics with financial frictions: A survey. Technical report, National Bureau of Economic Research, No. 18102.
Brunnermeier, M. K. and Sannikov, Y. (2014). A macroeconomic model with a financial sector. the American Economic Review, 104(2):379–421.
Calvo, G. A. (1983). Staggered prices in a utility-maximizing framework. Journal of Monetary Economics, 12(3):383–398.
Canova, F. (2007). Methods for applied macroeconomic research, thirteenth edition. Princeton University Press, Princeton, New Jersey.
Christiano, L. J., Eichenbaum, M., and Evans, C. L. (2005). Nominal rigidi-ties and the dynamic effects of a shock to monetary policy. the Journal of Political Economy, 113(1):1–45.
Christiano, L. J., Motto, R., and Rostagno, M. (2009). Financial factors in economic fluctuations (preliminary). Technical report, Citeseer Working paper NO 1192.
Christiano, L. J., Motto, R., and Rostagno, M. (2014). Risk shocks. the American Economic Review, 104(1):27–65.
Christiano, L. J., Trabandt, M., and Walentin, K. (2011). Introducing finan-cial frictions and unemployment into a small open economy model. Journal of Economic Dynamics and Control, 35(12):1999–2041.
Degryse, H. and Ongena, S. (2008). Competition and regulation in the bank-ing sector: A review of the empirical evidence on the sources of bank rents. Handbook of Financial Intermediation and Banking, 1:483–554.
Demirguc-Kunt, A., Laeven, L., and Levine, R. (2004). Regulations, market structure, institutions, and the cost of financial intermediation. Journal of Money, Credit and Banking, 36(3):563–583.
Diamond, D. W. and Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. the Journal of Political Economy, 91(3):401–419.
Dixit, A. K. and Stiglitz, J. E. (1977). Monopolistic competition and opti-mum product diversity. the American Economic Review, 67(3):297–308.
Gelfand, A. E. and Smith, A. F. (1990). Sampling-based approaches to calcu-lating marginal densities. Journal of the American Statistical Association, 85(410):398–409.
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. Pattern Analysis and Machine Intelligence, IEEE Transactions on, (6):721–741.
Gerali, A., Neri, S., Sessa, L., and Signoretti, F. M. (2010). Credit and banking in a DSGE model of the euro area. Journal of Money, Credit and Banking, 42(1):107–141.
Gertler, M. and Kiyotaki, N. (2010). Financial intermediation and credit policy in business cycle analysis. Handbook of Monetary Economics, 3(11):547–599.
Gilchrist, S. and Mojon, B. (2014). Credit risk in the Euro area. Technical report, National Bureau of Economic Research No. w20041.
Gilchrist, S., Schoenle, R., Zakrajˇsek, E., et al. (2014). Inflation dynamics during the financial crisis. Technical Report 826, Boston University Mimeo.
Gilchrist, S. and Zakrajˇsek, E. (2011). Monetary policy and credit supply shocks. IMF Economic Review, 59(2):195–232.
Greenbaum, S. I., Kanatas, G., and Venezia, I. (1989). Equilibrium loan pricing under the bank-client relationship. Journal of Banking & Finance, 13(2):221–235.
Howells, P. G. and Bain, K. (2008). The economics of money, banking and finance: a European text, fourth edition. Pearson Education, Edinburgh Gate.
Kim, C.-J. and Nelson, C. R. (1999). State-space models with regime switch-ing: classical and Gibbs-sampling approaches with applications, first edi-tion. the MIT Press, Cambridge MA.
Kiyotaki, N. and Moore, J. (1997). Credit chains. the Journal of Political Economy, 105(21):211–248.
Nydick, S. W. (2012). The Wishart and inverse Wishart distributions. Tech-nical report, Journal of Statistic, 2012 - tc.umn.edu.
O’Hara, K. (2012). BMR: Bayesian macroeconometrics in R. R package version 0.1.1.
Sharpe, S. A. (1990). Asymmetric information, bank lending, and implicit contracts: A stylized model of customer relationships. the Journal of Fi-nance, 45(4):1069–1087.
Sims, C. A. (1992). Interpreting the macroeconomic time series facts: The effects of monetary policy. European Economic Review, 36(5):975–1000.
A
Appendix: model solution
A.1
Household maximization problem
The households maximize their utility subject to the budget constraint by choosing CH
t (i), (1 − LHt (i)) and Dt(i):
Lt≡ E0 ∞ X t=0 (βH)t αCln CtH(i) − (1 − αC)ln(1 − LHt (i)) + λt wtLHt (i) + 1 + rt−1d πt
Dt−1(i) − CtH(i) − Dt(i)
.
(A.1)
Notice (βH)t≡ 1 1+ρH, ρ
H is the patience parameter. The FOCs are:
∂Lt ∂CH t (i) = (βH)tEt αC CH t (i) − λt = 0, (A.2) ∂Lt ∂(1 − LH t (i)) = (βH)tEt 1 − αC 1 − LH t (i) − λtwt = 0, (A.3) ∂Lt ∂Dt(i) = (βH)tEt −λt+ λt+1 1 + ρH 1 + rd t πt+1 = 0. (A.4)
Iterating Equation (A.2) forward 1 period and substituting the results and Equation (A.2) in Equation (A.4) gives the consumption Euler Equation for households: 1 CH t (i) = 1 + r d t (1 + ρH)(π t+1)E t 1 CH t+1(i) . (A.5)
A.2
Entrepreneur maximization problem
The entrepreneurs maximize their utility subject to the budget constraint, borrowing constraint and capital accumulation identity by choosing CE
t (i),
Kt(i), LHt (i) and Bt(i):
Lt≡ E0 ∞ X t=0 (βE)t " ln CtE(i) + λ1t
(1 − δk)Kt(i) + RPt(i) + Bt(i)
− CE t (i) −
1 + rb t−1
πt
Bt−1(i) − wtLHt (i) − Kt+1(i)
+ λ2t mtEt(πt+1(1 − δk)Kt(i)) − (1 + rbt)Bt(i) # , (A.7)
where I used the law of motion for the capital stock to substitute out invest-ment:
Kt(i) = It(i) + (1 − δk)Kt−1(i). (A.8)
The FOCs are:
∂Lt ∂CE t (i) = 1 CE t − λ1 t = 0, (A.9) ∂Lt ∂LH t (i) = λ1twt= 0, (A.10) ∂Lt ∂Kt(i) = λ1t(1 − δk) + λ2tmtEtπt+1(1 − δk) − λ1t−1(1 + ρ) = 0, (A.11) ∂Lt ∂Bt(i) = λ1t − λ2t(1 + rbt) − λ 1 t+1 1 + ρ 1 + rb t πt+1 = 0. (A.12)
Similarly as above, I derive the entrepreneur Euler Equation using Equation (A.9) at time t − 1, t, and t + 1:
A.2.1 Firm maximization problem
Entrepreneurs own firms therefore the maximization problem of firm i is part of the entrepreneurs decision problem. Here I first derive a relationship between demand for variety Yt(i) produced by entrepreneur i and the price
set by the entrepreneur, Pt(i). Next I determine the optimal capital labor
input mix such that given a particular output level, profits are maximized. Finally, I determine which price level the entrepreneur sets given demand for variety Yt(i).
First I postulate that the final good sector produces a homogenous good Yt using a CES production function, see Dixit and Stiglitz (1977):
Yt= Z 1 0 Yt(i)1−1/θdi 1/(1−1/θ) , (A.14)
where Yt(i) is an unique input variety produced by entrepreneur i and θ is
the elasticity of substitution in production. The final good sector minimizes costsR0∞Pt(i)Yt(i) subject to the CES production function, Equation (A.14).
Hence, the firm optimization problem becomes:
Lt ≡ Z 1 0 Pt(i)Yt(i)di + λt " Yt− Z 1 0 Yt(i)1−1/θdi 1/(1−1/θ)# . (A.15)
The solution defines unit costs Pt and demand for Yt(i):
Pt ≡ Z 1 0 Pt1−θdi 1/1−θ , (A.16) Yt(i) =Yt Pt(i) Pt −θ . (A.17)
goods sector. From Equation (5) I derive the intermediate firm optimization problem (the decision is made by the entrepreneur). This is an intermediate step to determine the optimal capital-labor mix:
min
Kt(i),LHt (i)
rtbKt(i) + WtLHt (i),
subject to Yt(i) = At[Kt(i)αLHt (i)
1−α]. (A.18)
From the Lagrangian I derive the following FOCs:
rbt = λtα Yt(i) Kt(i) , Wt = λt(1 − α) Yt(i) Lt(i) , (A.19)
where λtare the Lagrangian multipliers for the production constraint. Using
Equation (A.19) and Equation (5) I can rewrite λt as:
λt= 1 At rb t α α Wt 1 − α 1−α . (A.20)
Substituting the FOC in the total costs function T Ct(i), while noticing λt,
and using the total cost function in the nominal profit function N Pt(i), I
obtain:
T Ct(i) ≡ rtbKt(i) + WtLHt (i) = λtYt(i), (A.21)
N Pt(i) ≡ Pt(i)Yt(i) − T Ct(i), (A.22)
= Pt
Yt(i)
Yt
−1/θ
Yt(i) − M CtYt(i), (A.23)
where I used the Dixit-Stiglitz demand function for output variety Yt(i) and
defined λt ≡ M Ct (noticing that each firm has the same marginal costs) in
Firms maximize expected firm value by setting prices Pt(i). First I define
the real profit function RPt(i):
RPt(i) ≡ N Pt(i) Pt = Pt(i) Pt − mct Yt Pt(i) Pt −θ , (A.24)
where mct≡ M Ct/Pt. Firms maximize their real profits RPt(i) by choosing
the price level Pt(i).
Using Calvo Pricing (Calvo, 1983), and denoting the probability that a firm is able to change its price by the probability (1−η) I obtain the expected value of the firm that has just received a ”green light”, i.e., the firm is allowed to change it price in period t, and has set a new price Ptn(i):
EtVt≡Et Pn t (i) Pt − mct Yt Pn t (i) Pt −θ + η rd t+1 Pn t (i) Pt+1 − mct+1 Yt+1 Pn t (i) Pt+1 −θ η2 rd t+1rt+2d Pn t (i) Pt+2 − mct+2 Yt+2 Pn t (i) Pt+2 −θ + . . . , =Et ∞ X τ =0 ητSt+τ Pn t (i) Pt+τ − mct+τ Yt+τ Pn t(i) Pt+τ −θ , (A.25)
where St+τ is the discount factor. Entrepreneurs use the cost of capital to
discount, i.e., St+τ ≡ 1/(1 + rbt), ∀τ . Maximizing Equation (A.25) w.r.t. the
new price Ptn(i) and rewriting gives:
Ptn(i) = Ptn= θ θ − 1Et P∞ τ =0η τS t+τPt+τθ Yt+τmct+τ P∞ τ =0ητSt+τPt+τθ−1Yt+τ . (A.26)
Log-linearizing Equation (A.26) gives Equation (B.52) in the main text. To derive the relationship between the aggregate price level and the new price level set s periods ago I rewrite the aggregate price level as:
Pt1−θ = Z 1
0
Subsequently, using the law of large numbers we know that (1 − η)ηs is the fraction of firms that has changed its price s periods ago. Hence, Equation (A.27) can be rewritten as:
Pt1−θ = (1 − η)(Ptn)1−θ+ (1 − η)η(Pt−1n )1−θ+ η2(Pt−2n )1−θ+ . . .. (A.28)
Using Equation (A.27) for Pt−11−θ I can rewrite Equation (A.27) as:
Pt1−θ = (1 − αp)(Ptn)1−θ+ αpPt−11−θ. (A.29)
A.2.2 Loan and deposit demand
The demand functions for loans is derived in a similar fashion as demand for product Yt(i). Entrepreneurs minimize their loan payments subject to the
aggregation technology, see again Dixit and Stiglitz (1977):
Lt≡ Z 1 0 rbt(j)Bt(j)dj + λt " Bt− Z 1 0 Bt(j)1−1/µbdj 1/(1−1/µb)# . (A.30)
Solving this problem by aggregating all FOCs across all entrepreneurs gives loan demand at bank j:
Bt(j) = rb t(j) rb t µb Bt. (A.31)
In a similar fashion the deposit demand function is derived. Households max-imize interest revenue from deposits subject to the aggregation technology:
Solving this problem by aggregating all FOCs across all household gives de-posit demand at bank j:
Dt(j) = rd t(j) rd t µd Dt. (A.33)
A.2.3 Retail Branch
The retail branch of bank j consist out of 2 parts, a loan branch and a deposit branch. Both maximize their profits subject to the demand schedules by choosing the appropriate interest rates. Substituting demand for loans, Equation (8), in Equation (18) gives the following maximization problem:
max rb t(j) E0 ∞ X t=0 Λp0,t " (rbt(j) − rtwb) r b t(j) rb t µb Bt− κb 2 rb t(j) rb t−1(j) − 1 2 rtbBt # . (A.34) The solution to the problem is:
rb t(j) rb t µb Bt+ (rtb(j) − r wb t ) µbB t rb t rb t(j) rb t µb−1 − κbr b tBt rb t−1(j) rb t(j) rb t−1(j) − 1 + βtHEt κb rb t+1(j) rb t(j) − 1 rbt+1Bt+1 rb t+1(j) (rb t(j))2 = 0. (A.35)
Rewriting Equation (A.35) gives:
Imposing a symmetric equilibrium rbt(j) ≡ rtb, ∀j: 1 − µb+ µbr wb t rb t + κb 1 − r b t rb t−1 rb t rb t−1 + βtHEt rb t+1 rb t − 1 r b t+1 rb t 2 Bt+1 Bt = 0. (A.37)
Notice that if Equation (A.37) is log-linearized we obtain Equation (19). In a similar fashion the deposit branch maximizes its profits subject to the deposit demand schedule. Substituting demand for deposits, Equation (9), in Equation (21) gives the following maximization problem:
max rd t(j) E0 ∞ X t=0 Λp0,t " (rt− rdt(j)) rd t(j) rd t µd Dt− κd 2 rd t(j) rd t−1(j) − 1 2 rdtDt # . (A.38) The solution, after rewriting in a similar fashion as in Equation (A.36), is:
rd t(j) rd t µd Dt − 1 + µd− µd rt rd t(j) + κd 1 rd t−1(j) − r d t(j) (rd t−1(j))2 rdtDt + βtHEt rd t+1(j) rd t(j) − 1 r d t+1(j) rd t(j) 2 Dt+1 = 0. (A.39)
Imposing again a symmetric equilibrium rd
t(j) ≡ rtd, ∀j: −1 + µd− µdrt rd t + κd 1 − r d t rd t−1 rdt rd t−1 + βtHEt rd t+1 rd t − 1 r d t+1 rd t 2 Dt+1 Dt = 0. (A.40)
A.2.4 Model solution
The model is closed by defining the equilibrium in the final good sector as:
Yt= Ct+ It, (A.41)
where Ct ≡ CtH + CtE. The labor market and the capital market are in
equilibrium if: Lt = Z 1 0 LHt (i)di, Kt = Z 1 0 Kt(i)di. (A.42)
Moreover, the link between aggregate production and the aggregate factor inputs Lt and Kt can be defined as follows:
Yta ≡ Z 1 0 Yt(i)di = AtKtαL 1−α t , (A.43) where Ya
t is an alternative output measure. Defining an alternative price
index Pa t: (Pta)−θ ≡ Z 1 0 Pt(i)−θdi −1/θ , (A.44)
Using the alternative price index I can link aggregate output Yt to the
ag-gregate factor inputs:
B.2
Log linear model
Consumption Euler and labor supply The Euler equations:
1 CH t = 1 + r d t (1 + ρH)(π t+1) 1 CH t+1 , ˜ Ct+1H = ˜CtH + ρ H 1 + ρHr˜ D t + ˜πt+1. (B.40) Assuming mt = m, ∀t: 1 CE t+1 =1 + ρ E CE t 1 m + πt+1 1 + rb t − (1 + ρ E)2 (1 − δk)mCt−1E , ˜ Ct+1E =(1 + ρE) 1 m − 1 1 + (rb t)∗ ˜ CtE− 1 + ρ E 1 + (rb)∗π˜t+1+ (rb)∗(1 + ρE) [1 + (rb)∗]2 ˜r B t − (1 + ρE)2 (1 − δk)m ˜ Ct−1E . (B.41) Consumption: Ct=CtH + C E t , ˜ Ct= CH C ˜ CtH +C E C ˜ CtE, ˜ Ct+1= CH C ˜ Ct+ ρH 1 + ρH˜r D t + ˜πt+1 +1 − C H C (1 + ρE) 1 m − 1 1 + (rb t)∗ ˜ Ct− 1 + ρE 1 + (rb)∗π˜t+1+ (rb)∗(1 + ρE) [1 + (rb)∗]2 r˜ B t − (1 + ρE)2 (1 − δk)m ˜ Ct−1 , (B.42)
where I used that in the steady state CCH + CCE = 1 and CtH = CCHCt and
. Output Yt= Ct+ It, ˜ Yt= C Y ˜ Ct+ I Y ˜ It. (B.44) Investment: It= Kt− (1 − δk)Kt−1, ˜ It= 1 δk ˜ Kt− 1 − δk δk ˜ Kt−1. (B.45)
Substituting the log-linearized investment equation in the log-linearized out-put equation gives:
˜ Yt= C Y ˜ Ct+ I Y 1 δk ˜ Kt− 1 − δk δk ˜ Kt−1 . (B.46)
Capital stock and labor demand First I determine the borrowing rate and wage rate to obtain the capital stock and labor supply:
Bank Capital Bank capital evolves according: Ktb = (1 − δb)Kt−1b + ωbJt−1b , Kb t − Kb δbKb = (1 − δb)(K b− Kb t−1) δbKb + ωb (Jt−1b − Jb) Jb , ˜ Ktb = (1 − δb) ˜Kt−1b + δbJ˜t−1b . (B.49)
Bank profits evolve according:
Jtb =rtbBt− rtdDt− κw 2 Kb t Bt − νw 2 Ktb− Qt− Ctb, ˜ Jtb = 1 (rb)∗− ρH(1 + θ d− νw) − θd (rb)∗(˜rbt+ ˜Bt)− ρH(1 + θd− νw)(˜rtd+ ˜Dt) − θdQ˜t) , (B.50)
where I used that in the steady state Cb = 0, i.e., there are no adjustment costs. Combining bank capital and bank profits gives:
where γ ≡ 1−αα 1−α+ρ1+ρEE
1− ¯LH
1−LH > 0 (Heijdra, 2009, pp. 387-390). Equation
(B.52) is a New Keynesian Philips Curve. Note, however, that I substitute the output gap ( ˜Yt− ˜Ytp) for output ˜Yt. In the model there is no endogenous
growth in output; for this reason, potential output is a constant. Hence, the growth rate in the output gap is equal to the growth rate in output.
Loans rate wholesale level
rwbt = θd+ rt− κw Kb t Bt − νw Kb t Bt 2 , ˜ rwbt = 1 ρH(µ d− 1) + θdµd [ρH(µd− 1)]˜rt+ [κwµd(νw)3]( ˜Bt− ˜Ktb) . (B.53) Loan rate 1 − µb+µb rwbt rb t + κb 1 − r b t rb t−1 rtb rb t−1 + βtHEt rb t+1 rb t − 1 r b t+1 rb t 2 Bt+1 Bt = 0, ˜ rt+1b =µb− 1 + (1 + β H)κ b βHκ b ˜ rtb− κb βHκ b ˜ rbt−1− µb− 1 βHκ b(ρH(µd− 1) + θdµd) [ρH(µd− 1)]˜rt+ [κwµd(νw)3]( ˜Bt− ˜Ktb) . (B.54)
where I substituted Equation (B.53) for rtwb.
Deposit rate −1+µd− µd rt rd t + κd 1 − r d t rd t−1 rdt rd t−1 + βtHEt rd t+1 rd t − 1 r d t+1 rd t 2 Dt+1 Dt = 0, ˜ rt+1d = µd+ 1 + (1 + β H)κ d βHκ d ˜ rtd− κd βHκ d ˜ rdt−1− µd+ 1 βHκ d ˜ rt, (B.55)
Loans The budget constraint of the entrepreneur at the macro level deter-mines the amount of loans:
Bt= Kt+1− (1 − δ)Kt− RPt(i) + CtE+ 1 + rb t−1 πt Bt−1+ wtLHt , Bt= It+ wtLHt − Yt+ CE C Ct+ 1 + rb t−1 πt Bt−1, ˜ Bt= I B ˜ It+ CE C C B ˜ Ct+ wLH B ( ˜L H t + ˜wt) + (rb)∗r˜Bt−1+ (1 + (rb) ∗ )( ˜Bt−1− ˜πt). (B.56)
Deposits The budget constraint of the household at the macro level de-termines the amount of deposits:
Dt= wtLHt − CtH + 1 + rd t−1 πt Dt−1, ˜ Dt= wLH D ( ˜wt+ ˜L H t ) − CH C C D ˜ Ct+ ρHr˜t−1D + (1 + ρ H )( ˜Dt−1− ˜πt), (B.57)
where I used that EtQt+1= Qt.
Loan default Loan default is specified in Equation (11) and log-linearization gives:
Qt= θdBt−1,
˜
C
Appendix: priors
Table 2: Prior parameter values and shares.
Parameters Value Shares Value η 0.25 CCH 0.8 αC 0.35 YC 0.5 βH 0.9943 1− ¯1−LLHH 1.45 βE 0.975 1−LLHH 0.5 α 0.25 wLDH 0.25 δk 0.025 BI 0.5 θ 6 wLBH 2 m 0.35 νw 0.09 µd −1.46 µb 3.12 δb 0.1049 θd 0.003 κw 11.5 κb 10 κd 3.62 δπ 2 δy 0.5
Source: Gerali et al. (2010). Note, 1− ¯LH
1−LH = 1.45 is stated
in (Heijdra, 2009, pp. 387-390) and for θd, L
H 1−LH, wLH D , I B, and wLH
D
Appendix: variables
Table 3: Notation and definitions
Variables Denoting ΛHt Utility of households ΛE t Utility of entrepreneurs CH t Consumption of households CtE Consumption of entrepreneurs
Ct Total Consumption of households and entrepreneurs
LH
t Hours worked by households
Dt Deposits hold by households
Dt Loans hold by entrepreneurs
wt The real wage rate
rdt The deposit rate rb
t The loan rate
rwd
t The deposit rate charged by the wholesale branch
rwbt The loan rate charged by the wholesale branch rt The policy rate
πt The inflation rate
Pt Price index
Pn
t New price index
Pa
t Alternative price index
Kt The physical capital stock
Yt Production, in equilibrium equal to GDP
Ya
t Alternative output measure
It Investment
Nt Net worth of the entrepreneur
RPt Real profits
N Pt Nominal profits
T Ct Total costs
Table 3 continued: Notation and definitions
Variables Denoting Vt Firm value
St Stochastic firm discount factor
Qt Bad loan provisioning and loan default
Kb
t Bank capital
Jb
t Overall bank profits
Stw Bank spread between lending and borrowing activities Cb
t Bank entity’s total adjustment costs for changing interest rates
˜
Lt Auxiliary variable definition to change time indices
Zt A vector containing al endogenous variables
εt A vector containing the shock terms
εM
t Monetary policy shock
εQt Bad loan provisioning shock
Xt A vector containing al endogenous variables
vt Vector containing reduced form residuals
Lt Lagrangian
λt Lagrangian multipliers
Table 4: Notation and definitions
Parameters Denoting
βH Households’ discount factor
βE Entrepreneurs’ discount factor
ρH Households’ discount rate ρE Entrepreneurs’ discount rate
αC Share of consumption in the household utility functions
δk Depreciation rate of physical capital
A Technology parameter
α Weight of capital in the production function m Loan-to-value ratio
η Entrepreneurs’ probability of changing their price αp Share of the old price in the price index
θ Elasticity of substitution in production µb Elasticity of substitution between loans
µd Elasticity of substitution between deposits
Table 4 continued: Notation and definitions
Parameters Denoting
δb Resources used to manage bank capital
ωb Banks’ percentage of retained earnings
νw Optimal capital-to-asset ratio
κw Capital-to-asset ratio adjustment costs
κb Loan rate adjustment costs
κd Deposit rate adjustment costs
ζb
1 Auxiliary parameter in the log-linearized loan rate
ζb
2 Auxiliary parameter in the log-linearized loan rate
ζ3b Auxiliary parameter in the log-linearized loan rate ζd
1 Auxiliary parameter in the log-linearized deposit rate
ζd
2 Auxiliary parameter in the log-linearized deposit rate
ζ3d Auxiliary parameter in the log-linearized deposit rate γ Auxiliary parameter in the NKPC
δπ Weight of inflation in the Taylor rule
δy Weight of output in the Taylor rule
Γ0,1,2 Coefficient matrices
Υ0 Coefficient matrix
Φ(L) Lag polynomial
Φ Reduced form coefficient matrix Σv Variance-covariance matrix
Ψ Diagonal symmetric scale matrix b Stacked vector of zeros
d Degrees of freedom of the IW Ω A scalar matrix
Θ Precision matrix parameter
νx Error term for the coefficients’ mean
ψ Hyperparameters
σı,,p Variance-covariance elements
h(p) Relative tightness of the variance of lags Ξ Vector containing all parameters
Table 5: Notation and definitions
Indices and operators Denoting
Et Expectation operator at time t
H Households index E Entrepreneurs index
t Time index
τ Future time index s Past time index
i Agent index for household or entrepreneur
j Bank index
p Lag index
Variable index ı Equation index n Parameter index ` Gibbs sample index
∼ A tilde on a variable denotes a log-linearized variable ln (·) Natural logarithm of a variable
Table 6: Variables and sources
Variable Source Composition
Output OECD National Gross domestic product. Accounts Constant prices 1990.
Consumption OECD National Private consumption. Accounts Constant prices 1990.
Capital Stock OECD National Gross fixed capital Accounts formation
Bank Capital Bank Profitability National currency in millions. Statistics OECD Aggregate bank balance sheet.
Inflation OECD National Price deflator for private Accounts consumption. Index 1990=100.
Wages OECD Labour Real minimum wages
Loans IMF International Bank credit to the private Financial Statistics sector. Constant prices 1990.
Deposits Bank Profitability National currency in millions. Statistics OECD Aggregate bank balance sheet.
Bad Loan Bank Profitability National currency in millions. Provisioning Statistics OECD Aggregate bank balance sheet.
Policy rate IMF International Three-month money market Financial Statistics interest rate (%).
Loan rate The World Bank Rate paid by commercial banks for demand, time, or savings deposits.
Deposit rate The World Bank Bank rate that meets the
