Welfare analysis of bank capital requirements with endogenous default

Economic Modelling xxx (2018) 1–15

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Welfare analysis of bank capital requirements with endogenous default Fernando Garcia-Barragan, Guangling Liu * Department of Economics, Stellenbosch University, South Africa

A R T I C L E

I N F O

JEL: E44 E47 E58 G28 Keywords: Bank capital requirement Default Welfare

A B S T R A C T

This paper presents a tractable framework with endogenous default and evaluates the welfare implication of bank capital requirements. Using a dynamic general equilibrium model we analyze the social welfare response to a negative technology shock under different capital requirement regimes, Basel II and III. In Basel III, we consider alternative indicators, such as output gap and credit-to-output gap. We then consider the scenario where the default rate is augmented in different capital requirement regimes. We show that it is welfare improving by including the default rate as an additional indicator for all capital requirement regimes. A more aggressive reaction to default can effectively mitigate the negative effect of the shock on welfare and this attenuation effect works through the bank funding channel.

1. Introduction This paper develops a dynamic general equilibrium model with endogenous default, and investigates the welfare implication of bank capital requirements when the default rate is considered as an additional indicator. Van den Heuvel (2008) argues that it is critical to understand the welfare implication of bank capital requirements as one would simply raise capital adequacy ratio to 100% if there were no costs of implementing capital requirements. In this paper we argue that changes in the loan default rate can have a significant impact not only on financial intermediaries’ bank capital, but also on borrowers’ balance sheet and future borrowing capability. It is, therefore, critical for regulatory authorities to consider the effect of default on capital requirements and the implications for social welfare. Our welfare analysis of capital requirements is related to Van den Heuvel (2008). Using a general equilibrium growth model with liquidity-creating banks, Van den Heuvel (2008) investigates the welfare cost of Basel I and II, and shows that it is equivalent to a 0.1%–1% permanent loss in consumption. Angeloni and Faia (2013) report similar findings on Basel II: risk-weighted capital requirements amplify the cycle and are welfare deteriorating. Basel III, on the other hand, is welfare improving. The above mentioned two articles, however, do not take into the consideration of default when investigating the welfare cost of capital requirements.1

The Bank for International Settlements (BIS) has been consistently emphasizing the critical role of default played in the bank capital requirement decision makings (e.g., BCBS, 2009, 2010, 2011). On the academic front, among others, Geanakoplos (2011) and Goodhart et al. (2013) argue that for a long time mainstream macro-models have ignored financial frictions and point out it is important to consider default in macro-models and policy analysis. Catarineu-Rabell et al. (2005) evaluate three possible scenarios where risk weights assigned to bank assets are constant, or depend positively or negatively on the probability of default. The authors find that setting risk weight positively to default is desirable from the regulation point of view. The countercyclical capital buffer of Basel III requires banks to increase their holdings of capital during economic booms. This precautionary regulation aims to curtail credit booms that might end in financial crises. However, not all credit booms lead to crises (see, Bakker et al., 2012). The probability of default can be a good candidate for correcting this potential error: non-beneficial reductions in bank loans. In this paper, we augment the default rate in the capital requirement rule and study the implications for welfare. Which indicator should be used when implementing the countercyclical capital buffer of Basel III is an empirical question. The BIS suggests that the difference between the aggregate credit-to-output ratio and its long term trend can be a good candidate (BCBS, 2009). There is, however, no general consensus on this. Some studies criticize

* Corresponding author. 1

E-mail addresses: [email protected] (F. Garcia-Barragan), [email protected] (G. Liu). For literature on bank capital requirement in general see, e.g., Repullo (2013); Rubio and Carrasco-Gallego (2014, 2016); Angelini et al. (2015); Kanngiesser et al. (2017).

https://doi.org/10.1016/j.econmod.2018.03.002 Received 9 October 2017; Received in revised form 9 February 2018; Accepted 3 March 2018 Available online XXX 0264-9993/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: Garcia-Barragan, F., Liu, G., Welfare analysis of bank capital requirements with endogenous default, Economic Modelling (2018), https://doi.org/10.1016/j.econmod.2018.03.002

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15

this indicator (credit-to-output gap) proposed by the BIS. For instance, Drehmann and Tsatsaronis (2014) argue that the credit-to-output gap is not useful as a warning indicator of banking crises, especially for emerging market economies. Repullo and Saurina (2011) suggest that regulatory authorities should use output growth as the indicator when implementing the countercyclical capital buffer, instead of the creditto-output gap. Some studies suggest that excessive credit growth is a valid indicator for potential banking crises (e.g., Lowe and Borio, 2002); some suggest the aggregate credit contains information of the likelihood of future financial distresses (e.g., Schularick and Taylor, 2012); and some suggest both deviations of aggregate output and credit from their steady states should be considered (e.g., Resende et al., 2013). In this paper, we consider output gap and credit-to-output gap as potential candidates and compare their performance in terms of welfare. The contribution of this paper is three-fold. First, to the best of our knowledge, this is the first attempt to study the welfare implication of capital requirements with default, where default is endogenously embedded in the model economy and the default rate is considered as an additional indicator in the capital requirement rule. Second, we investigate whether the proposed countercyclical capital buffer of Basel III does a better job than Basel II in terms of welfare. Third, using social welfare as the criterion, we evaluate different potential indicators, namely credit-to-output gap and output gap, for the implementation of the countercyclical capital buffer. To study the welfare implication of different capital requirement regimes with endogenous default, we develop a real business cycle model (RBC) with banking, in which borrowers may default on their financial obligations. We introduce endogenous default along the lines of Shubik and Wilson (1977) and de Walque et al. (2010), where borrowers may default on the loans borrowed from the previous period upon paying a penalty cost. We then examine the social welfare response to a negative technology shock under different capital requirement regimes with and without default, that is, a capital requirement regime is responding to the default rate, or otherwise. By augmenting the default rate in a capital requirement rule we introduce a stabilizer not only in the financial sector but also in the real sector. First, the imposed penalty costs provide firms (the borrower) incentives not to, or default less on bank loans. Second, banks benefit from the augmented capital requirement rule as banks are more profitable and better capitalized with a lower default rate. Banks are, therefore, able to accumulate more funds and supply more credit to firms. This is, in turn, beneficial for production. Last, households can consume and invest (in the form of deposits) more with a higher production. Exante, a capital requirement rule responding to the default rate is welfare improving. The capital requirement regimes studied in this paper are as follows. Following Angeloni and Faia (2013), we assume a fixed rate of bank capital requirement for Basel I. Both Basel II and III evolve as a Taylor-type rule. In the case of Basel II, the capital requirement reacts negatively with respect to output gap. There are two specifications for Basel III. For the first specification, namely Basel III, the capital requirement reacts positively to output gap; and for the second specification, namely Basel III credit-to-output, the capital requirement reacts positively to the credit-to-output gap. We then augment the default rate gap (deviation from its steady state) into Basel II, Basel III, and Basel III credit-to-output.2 The results of our welfare analysis are the following. First, introducing default in Basel II, Basel III, and Basel III credit-to-output is welfare improving in all cases. It is through the bank funding channel that introducing default in the capital requirement rule attenuates the negative effect of the shock on welfare. Moreover, a more aggressive reaction

to default can effectively mitigate the negative effect of the shock. Second, compared with Basel II, the countercyclical capital buffer (both Basel III and Basel III credit-to-output regimes) is slightly welfare deteriorating. Last, there is no clear evidence on either credit-to-output gap or output gap is a better candidate for implementing the countercyclical capital buffer. These conclusions are obtained based on the analysis of both the first and second moments of social welfare in response to a negative productivity shock, and complemented by the analysis on the transmission mechanisms through which introducing default in the capital requirement rule attenuates the negative effect of the shock on welfare. The results of sensitivity analysis and significance test suggest our findings are robust. The rest of the paper is structured as follows. section 2 describes the model. section 3 presents the functional forms and parameters values in the model. section 4 discusses the cyclical properties of our models, and the results of the social welfare analysis and sensitivity analysis. section 5 concludes. 2. The model The model economy is inhabited by households, firms, banks, and a government. Banks intermediate credit between borrowers (firms) and savers (households), facing capital requirement regulation imposed by the government. We introduce endogenous probability of default into the model as follows. In contrast to de Walque et al. (2010), we assume firms accumulate physical capital with own profits and bank loans. We view this is more in line with the institutional environment, as opposed to the assumption in de Walque et al. (2010) whereby physical capital is accumulated by bank loans only. In each period, firms may default on a fraction of loans borrowed from previous period upon paying a penalty cost. We assume households demand liquidity (deposits) and its usage yield utility in the spirit of Sidrauski (1967). We introduce deposits as households’ assets and one kind of banks’ liabilities, and model the preferences in a less restrictive way, not depending on modeling choices Van den Heuvel (2008).3 For simplicity, we assume banks do not default on deposits. Banks supply loans to firms and finance these loans with deposits and own funds (capital).4 We further assume banks can recover a fraction of defaulted loans upon paying an insurance premium to the government. In our welfare analysis, we consider various types of bank capital requirement regimes that banks face, which are explained in the introduction. 2.1. Households There is a continuum of identical households with mass one. In each period households consume consumption goods, Ct , and hold bank deposits, Dt . Households supply labor, Ht , inelastically to firms and receive a real wage of Wt .5 Households are subject to lump-sum taxes, Tt . Households maximize their expected discounted utility as: max∞

{Ct ,Dt }t=0

𝔼0

∞ ∑

𝛽 t U (C t , D t ) ,

(1)

t =0

subject to the budget constraint: Ct + Dt + Tt = Rdt−1 Dt −1 + Wt Ht , where 𝛽 ∈ (0, 1) is the discount factor and return on deposits.

(2) Rdt

is the real gross rate of

3 We acknowledge that choices of utility function affect welfare analysis results. This is, however, beyond the scope of the current study. 4 Since the study focuses on the welfare analysis of bank capital requirements with default, it is sufficient to have a stylized banking sector in the model. 5 For simplicity we normalize labor supply to 1.

2 We only consider the case where the capital requirement reacts positively to the default gap since the otherwise makes no intuitive sense.

2

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Economic Modelling xxx (2018) 1–15

Considering 𝜆ht as the Lagrange multiplier of the budget constraint, the first order conditions (FOCs) for households are as follows: ( ) Ct ∶ Uct (Ct , Dt ) + 𝛽 𝔼t UCt ct +1 , Dt +1 = 𝜆ht , (3) Dt ∶ Ud (Ct , Dt ) = 𝜆ht − 𝛽 𝔼t 𝜆ht+1 Rdt ,

f

𝜋t ∶

2.2. Firms Firms produce output, Yt , using a standard Cobb-Douglas function, Yt = At F (Kt −1 , Ht ), where function F is homogenous of degree one. The technology shock, At , follows a logarithmic AR(1) process. The size of firms in the economy is normalized to one. In each period firms demand an amount of loans, Lt , from banks and invest in physical capital, Kt . The law of motion for physical capital is defined as follows: ( ) f Kt = (1 − 𝛿) Kt −1 + Lt + 𝜈f 1 − 𝜉f 𝜋t , (5)

𝜒=

t =0

(

{

(8)

subject to the law of motion of capital (Equation (5)) and the firm’s profits (Equation (7)). Denoting 𝜆at and 𝜆bt as the Lagrange multipliers of Equation (5) and Equation (7), the first order conditions are: ( ) 𝜒t ∶ 𝜆bt Rlt Lt−1 = 𝛽 𝔼t 𝜆bt+1 z𝜒 𝜒t , Rlt , Lt−1 , (9) (10)

( ) Kt ∶ 𝛽𝜆bt+1 At +1 FK Kt , Ht +1 = 𝜆at − 𝛽 (1 − 𝛿) 𝔼t 𝜆at+1 ,

(11)

( ) ( ) Lt ∶ 𝜆at = 𝛽 𝔼t 𝜆bt+1 1 − 𝜒t +1 Rlt +1 + 𝛽 2 𝔼t 𝜆bt+2 zL 𝜒t +1 , Rlt +1 , Lt ,

(12)

6 7

fraction of defaulted loans through insur-

max Dt ,Ftb ,Lt ,𝜋tb

}∞ t =0

𝔼0

∞ ∑

(

)

𝛽 t log 𝜋tb ,

(16)

t =0

subject to the balance sheet (Table 1), bank’s profits (Equation (15)), and the following restrictions:

t =0

Ht ∶ At FH (Kt −1 , Ht ) = Wt ,

(14)

ance. The function q is non-negative and once differentiable, and the partial derivatives with respect to its arguments, q𝜒 , qRl , and qL , are non-negative. The representative bank’s maximization problem is defined as follows:

)

𝛽 t log 𝜋tf ,

,

recover q 𝜒t −1 , Rlt −1 , Lt −2

The representative firm maximizes the discounted sum of profits as: 𝔼0

1

𝛽𝜔z Rl L

where the notation [x]+ is max{x, 0}, which guarantees that banks’ profits are non-negative. Following Van den Heuvel (2008) we assume that deposits by the government. Banks are able to ( are fully insured )

z 𝜒t −1 , Rlt −1 , Lt −2 , where Rlt is the real gross rate of return on loans.

}∞

(13)

As in the case of firms, there is a continuum of banks with mass one in each period. Table 1 shows the representative bank’s balance sheet, where loans (Lt ) are on the asset side, and deposits (Dt ) and own funds (Ftb ) are on the liability side. The representative bank’s profits are defined as follows: ( [ )]+ 𝜋tb = (1 − 𝜒t ) Rlt Lt−1 − Rdt−1 Dt−1 + q 𝜒t−1 , Rlt−1 , Lt−2 , (15)

The function z is assumed to be non-negative, and at least twice differentiable on 𝜒t , Rlt , and Lt . Moreover, the partial derivatives of function z with respect to its arguments, z𝜒 , zRl , and zL , are non-negative. The representative firm’s profits are defined as follows: ( ) 𝜋tf = At F (Kt−1 , Ht ) − Wt Ht − (1 − 𝜒t ) Rlt Lt−1 − z 𝜒t−1 , Rlt−1 , Lt−2 . (7)

f 𝜒t ,Ht ,Kt ,Lt ,𝜋t

)

2.3. Banks

In each period firms may choose to default a fraction 𝜒t of loans borrowed from the previous period. Following de Walque et al. (2010), we assume firms have to pay a penalty cost (or reputation losses) in( the following period based on the amount of loans defaulted, i.e., )

max

(

+ 𝜈f 1 − 𝜉f 𝜆at = 𝜆bt ,

where 𝜔z is the parameter that measures the size of the penalty costs.8 Equation (14) implies that the size of default depends negatively on the penalty costs of default, which makes intuitive sense – the higher the penalty costs are, the less probability that firms are going to default on loans.

where 𝛿 ∈ (0, 1) is the depreciation rate of physical capital. We assume a fraction 𝜉f of profits are used as administrative expenses, and only a fraction 𝜈f of net profits are reinvested in physical capital.6 We introduce these frictions to ensure that in equilibrium firms cannot retain enough earnings so that they are completely self-financed. In equilibrium, firms borrow a positive amount of loans to finance the gap between capital depreciation and the amount of net profits that are reinvested in capital7 : ( ) L = 𝛿 K − 𝜈f 1 − 𝜉f 𝜋 f . (6)

{

𝜋tf

where FK and FH are the partial derivatives of the production function with respect to capital and labor. Equation (9) equalizes the marginal cost of defaulting with its possible benefit. Equation (10) gives the optimal real wage. Equation (11) shows the marginal product of capital as the difference between its shadow value today and its discounted future shadow value tomorrow. Equation (12) equalizes the shadow value of loans to its costs, which equals to its discounted rate of return when there is no default (𝜒t +1 = 0). When default occurs the shadow value of loans today equals to the sum of its discounted rate of return and the marginal cost of penalty. From Equation (9) we can obtain the steady state default rate:

(4)

where Uc and Ud are the partial derivatives of the utility function with respect to consumption and deposits. Equation (3) is the consumption Euler equation and Equation (4) is the asset pricing equation for deposits.

∞ ∑

1

Ftb = (1 − 𝜉b − 𝜍) Ftb−1 + 𝜈b 𝜋tb ,

(17)

Ftb ≥ 𝜏t Lt ,

(18)

where Equation (17) is the law of motion for bank own funds and Equation (18) is the minimum bank capital requirement constraint. The representative bank needs to pay a fraction 𝜍 of own funds to the insurance

Table 1 Balance sheet. Assets Loans

The remaining profits are consumed by firms. A letter without time subscript represents its corresponding steady state.

8

3

Liabilities Lt

Deposits Funds

See section 3 for the details of the functional forms.

Dt Ftb

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Economic Modelling xxx (2018) 1–15

authority (the government). A fraction 𝜉b of profits are being used to cover the administrative expenses, whereby a fraction 𝜈b of net profits are kept as retained earnings.9 We introduce the administrative expenses to pin down insurance premium. More importantly, we calibrate parameters 𝜉b , 𝜍 , and 𝜈b such that a positive Lagrange multiplier of the bank capital requirement constraint exists, implying a binding capital requirement constraint in equilibrium. The minimum bank capital requirement rule 𝜏t will be discussed in details in the following section. f g Denoting 𝜆dt , 𝜆et , 𝜆t and 𝜆t as the Lagrange multipliers of the balance sheet, bank’s profits, flow of bank funds, and the minimum bank capital requirement constraint, the first order conditions are as follows: Dt ∶ 𝜆dt = 𝛽 𝔼t 𝜆et+1 Rdt , f

Table 2 Parametrization.

(19)

g

f

Ftb ∶ 𝜆t − 𝜆dt − 𝜆t = 𝛽 (1 − 𝜉b − 𝜍) 𝔼t 𝜆t +1 ,

Description

Parameter

Value

Capital share in production Discount factor Depreciation Habit persistence Utility coefficient of deposits Penalty cost coefficient (z-function) Insurance coverage (q-function) Technology persistence Bank capital requirement Insurance payment of banks Banks administrative costs Firms administrative costs Banks reinvestment rate Firms reinvestment rate

𝛼 𝛽 𝛿

1/3 0.99 0.03 0.5 0.0015315 31.1908 0.85 0.8 0.10 0.07 0.07 0.07 0.137732 1/4

(20)

( ) ( ) g Lt ∶ 𝛽 𝔼t 𝜆et+1 1 − 𝜒t +1 Rlt +1 + 𝛽 2 𝔼t 𝜆et+2 qL 𝜒t +1 , Rlt +1 , Lt = 𝜆dt + 𝜆t 𝜏t ,

𝜋tb ∶

1

𝜋tb

j

𝜙d 𝜔z 𝜔q 𝜌A 𝜏 𝜍 𝜉b 𝜉f 𝜈b 𝜈f

+ 𝜈b 𝜆ft = 𝜆et ,

Table 3 Main steady state ratios.

(21)

Description

Value

(22)

Consumption to output Taxes to output Capital to output Deposits to loans

C∕Y = 0.665338 T ∕Y = 0.00232762 K ∕Y = 8.28242 D∕L = 0.9

where qL is the partial derivative of function q with respect to loans. Equation (19) equalizes the shadow value of deposits to its returns. Equation (20) shows that the shadow value of funds equals to its discounted future value after paying insurance premium and administrative costs. Equation (21) gives the shadow value of loans, which equals to the return of loans plus insurance premium. The bank capital requirement (Equation (18)) is always binding in steady state. In steady state, Equation (21) becomes: [ ] 𝛽𝜆e (1 − 𝜒) Rl + 𝛽𝜔q 𝜒 Rl − Rd 𝜆g = (23) ,

Making use of the bank’s balance sheet, the necessary condition for bank profits to be positive is: 1−𝜏 Rl > . Rd 1 − 𝜒(1 − 𝜔q )

(28)

Based on our calibration, the right-hand-side of Equation (28) is 0.9,

𝜏

and

From Equation where 𝜔q ∈ (0, 1) is the insurance coverage (23), we see that the Lagrange multiplier of the capital requirement constraint (𝜆g ) is positive so long as 𝜆e is positive and the following condition holds:

Rl Rd

> 1 and, hence, banks profits are positive.

ratio.10

Rl Rd

>

1 ( ). 1 − 𝜒 1 − 𝛽𝜔q

2.4. Government The government plays three roles in the economy: (i) setting lumpsum taxes, (ii) collecting and manages insurance fund, and (iii) acting as bank capital regulation authority. For the first two tasks, the government sets lump-sum taxes to meet the need for the insurance repayment: ( ( ) ) (29) Tt = q 𝜒t −1 , Rlt −1 , Lt −2 − z 𝜒t −1 , Rlt −1 , Lt −2 − 𝜍 Ftb−1 .

(24) l

With a positive default probability in equilibrium, we have RRd > 1. In other words, there is always a positive spread between the real rates of loans and deposits. From the steady state condition of Equation (22), we have:

𝜆 = 1∕𝜋 + 𝜈b 𝜆 , e

b

f

That is, the government finances insurance repayment with the insurance premium paid by banks and the penalty costs paid by firms, whereby if there is a shortfall a lump-sum tax, Tt , is levied on households. And for the third task, we consider the following types of capital requirement regimes:

(25)

𝜆e

which indicates that is positive as long as bank profits are positive and 𝜆f ≥ 0. By equalizing Equation (20) and Equation (21), we can solve for 𝜆f : [ e ] 𝛽𝜆 (1 − 𝜒) Rl + 𝛽 2 𝜆e 𝜔q 𝜒 Rl + 𝜆g (1 − 𝜏) [ ] 𝜆f = , (26) 1 − 𝛽 (1 − 𝜉b − 𝜍)

(

𝜏t = 𝜏

which is positive as long as the minimum bank capital requirement is less than 100% (𝜏 < 1), and the default rate is less than one (𝜒 < 1). That is, so long as conditions 𝜏 < 1, 𝜒 < 1 and 𝜋 b > 0 hold in steady state, bank capital requirement constraint is always binding. In steady state, bank profits are always positive. This can be seen from the steady state condition of bank profits:

𝜋 b = (1 − 𝜒) Rl L − Rd D + 𝜔q 𝜒 Rl L.

9 10

(

𝜏t = 𝜏

Yt Y

)𝜙 y (

Lt ∕Yt L∕Y

𝜒t 𝜒

)𝜙 l (

)𝜙 𝜒

𝜒t 𝜒

,

)𝜙 𝜒

(30)

.

(31)

As pointed out in the introduction, we consider the government follows a Taylor-type of rule, adjusting the capital requirement ( ) ( ratio ) in L ∕Y

Yt Y

, or the credit-to-output gap Lt ∕Y t . In ( ) addition, we consider the default rate gap 𝜒𝜒t as an additional indi-

response to output gap (27)

cator in different capital requirement regimes. Variables without a time subscript denote their respective values. ( )steady-state ( ) ( ) The coefficients

As in de Walque et al. (2010), the remaining profits are consumed by banks. See section 3 for the details of the functional forms.

𝜙y , 𝜙l , and 𝜙𝜒 are related to 4

Yt Y

,

Lt ∕Yt L∕Y

, and

𝜒t 𝜒

, respectively.

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15

Table 4 Cyclical properties: US data and models under different Bank capital requirement regimes. Variable (x)

Model: Bank capital requirement rule (Basel regime) US Data

Production Y Consumption C Deposits rate Rd Deposits D Bank funds* Fb Loans L Loans rate Rl Default**

𝜒

𝜎y

cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y) 𝜎x 𝜎x /𝜎y cor (x, y)

0.01485 1 0.01215 0.82133 0.87678 0.00703 0.47364 0.24109 0.01908 1.28448 0.04980 0.02069 1.92112 0.04185 0.02696 1.81474 0.02512 0.00803 0.54078 0.00787 0.03592 3.32111 −0.26737

II (𝜙𝜒 =)

III (𝜙𝜒 =)

III credit-to-output (𝜙𝜒 =)

0

0.25

0.75

0

0.25

0.75

0

0.25

0.75

0.0129 1 0.0103 0.7984 0.9072 0.0061 0.4728 −0.8882 0.0277 2.1472 0.9806 0.0208 1.6124 0.9652 0.0270 2.09302 0.9796 0.0110 0.8527 0.0044 0.0244 1.8914 −0.9889

0.0129 1 0.0103 0.7984 0.9084 0.0063 0.4883 −0.8846 0.0273 2.1162 0.9821 0.0138 1.0697 0.9325 0.0260 2.0155 0.9803 0.0072 0.5581 −0.2061 0.0250 1.9379 −0.9844

0.0129 1 0.0102 0.7906 0.9114 0.0073 0.5658 −0.8471 0.0257 1.9922 0.9894 0.0075 0.5813 −0.3951 0.0229 1.7751 0.9862 0.0070 0.5426 −0.7296 0.0264 2.0465 −0.9664

0.0128 1 0.0104 0.8125 0.9071 0.0061 0.4765 −0.8881 0.0278 2.1718 0.9795 0.0348 2.7187 0.9870 0.0285 2.2265 0.9805 0.0190 1.4843 0.1705 0.0233 1.8203 −0.9915

0.0128 1 0.0104 0.8125 0.9079 0.0061 0.4765 −0.8887 0.0276 2.1562 0.9796 0.0281 2.1953 0.9844 0.0277 2.1640 0.9802 0.0153 1.1953 0.1210 0.0237 1.8515 −0.9913

0.0129 1 0.0103 0.7984 0.9104 0.0066 0.5116 −0.8754 0.0266 2.0620 0.9843 0.0140 1.0852 0.9290 0.0253 1.9612 0.9832 0.0066 0.5116 −0.2099 0.0248 1.9224 −0.9842

0.0128 1 0.0103 0.8046 0.9067 0.0067 0.5234 −0.8614 0.0273 2.1328 0.9844 0.0360 2.8125 0.9773 0.0282 2.2031 0.9836 0.0183 1.4296 0.1663 0.0233 1.8203 −0.9932

0.0128 1 0.0103 0.8046 0.9073 0.0067 0.5234 −0.8642 0.0272 2.1250 0.9847 0.0289 2.2578 0.9699 0.0274 2.1406 0.9834 0.0143 1.1171 0.1077 0.0238 1.8593 −0.9928

0.0128 1 0.0102 0.7968 0.9099 0.0073 0.5703 −0.8449 0.0261 2.0390 0.9889 0.0138 1.0781 0.8546 0.0248 1.9375 0.9862 0.0053 0.4140 −0.3833 0.0250 1.9531 −0.9833

Second moments of the US data and that of the model under different Basel regimes when implementing simple (𝜙𝜒 = 0) and extended Basel rules (𝜙𝜒 = 0.25 and 0.75). 𝜎x : standard deviation of variable x; cor (x, y): correlation between variables x and y. Sources: The Federal Reserve Bank of St Louis (production, consumption, bank funds, population, deposits rate, loans rate, loans, deflator of GDP), Bureau of Economic Analysis (deposits) and the Bureau of Labor Statistics (default). The data series are quarterly over the period 1960:Q1 - 2017:Q1, unless specified. Notes: (*): The data for bank funds is only available over the period 1984:Q1 - 2017:Q1; (**): The data for default rate is only available over the period 1992:Q3 - 2016:Q4. 𝜙𝜒 = 0 implies a bank capital regime does not include default in its policy rule.

3. Functional forms and parameter values In this section we discuss function forms and parameter values being used for our welfare analysis. The function forms of households’ preference, penalty costs, insurance coverage, production technology, and technology shock process are as follows: U (Ct , Dt ) = log (Ct − jCt −1 ) + 𝜙d log (Dt ) ,

(32)

( ) 𝜔 ( )2 z 𝜒t −1 , Rlt −1 , Lt −2 = z 𝜒t −1 Rlt −1 Lt −2 , 2

(33)

( ) q 𝜒t −1 , Rlt −1 , Lt −2 = 𝜔q 𝜒t −1 Rlt −1 Lt −2 ,

(34)

Yt = At F (Kt −1 , Ht ) = At (Kt −1 )𝛼 (Ht )1−𝛼 ,

(35)

log (At ) = (1 − 𝜌A ) log (A) + 𝜌A log (At −1 ) + 𝜉tA .

(36)

Equation (32) gives the instant utility function, U (Ct , Dt ), considering satisfaction from consumption with habit persistence (j ∈ (0, 1)) and deposits, where 𝜙d > 0 is a parameter. In Equation (33), 𝜔z > 0 captures the size of the penalty costs charged to firms in case firms default. In Equation (34), 𝜔q ∈ (0, 1) is the fraction of the amount of defaulted loans that banks are able to recover from insurance. 𝛼 ∈ (0, 1) is the capital-output share in the production function. 𝜌A governs the persisFig. 1. Social welfare IRFs following a negative technology shock of size 1%: Comparison of simple Basel rules. Solid line: Basel I; dashed line: Basel II; dotted line: Basel III; crossed line: Basel III credit-to-output. The coefficients corresponding to output and credit-to-output gaps (𝜙y and 𝜙l , respectively) are set to 0.5.

5

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 2. Social welfare IRFs following a negative technology shock of size 1%: Comparison of extended Basel rules. Solid line: Basel II; dashed line: Basel III; dotted line: Basel III credit-tooutput. The coefficients corresponding to output and credit-to-output gaps (𝜙y and 𝜙l , respectively) are set to 0.5, whereas the coefficient associated to the default deviation (𝜙𝜒 ) is set to 0.25.

Fig. 3. Social welfare IRFs following a negative technology shock of size 1%: Comparison of extended Basel rules under different values of the coefficients related to output and credit-to-output gaps (𝜙y and 𝜙l , respectively): 0.5 (solid line), 1.0 (dashed line) or 1.5 (dotted line). The coefficient related to default deviation (𝜙𝜒 ) is set to 0.25 for all Basel regimes.

tence of the productivity shock. 𝜉tA is the productivity shock, which is normally distributed with zero mean and variance 𝜎A2 . In general, we use standard values found in the literature to conduct our welfare analysis. We consider the period length to be one quarter. Table 2 lists the values of the parameters and Table 3 shows the main steady state ratios. The discount factor 𝛽 = 0.99. The habit persistence parameter j is fixed at 0.5. This value is just as the one used in Gerali et al. (2010), and close to the estimated value of 0.73 in Boldrin et al. (2001) and the one (0.65) reported in Christiano et al. (2005). The discount factor together with the deposits utility coefficient (𝜙d = 0.0015315) imply an annualized real return on deposits of 2%, such that Rd = 1.005, and a 5.5% annualized real return on loans, such that Rlt = 1.01375. Both rates assigned here are in line with the literature (e.g., de Walque et al., 2010; Iacoviello, 2015). In steady state, the ratio of consumption to output is 66%, which is the same as in de Walque et al. (2010). The ratio of total taxes to output is 0.23%, which is slightly lower than that (0.4) in de Walque et al. (2010) due to the model setup difference. On firms’ side, following de Walque et al. (2010), we assume the steady state default rate of non-financial corporate loans 𝜒 = 5%, which is based on the weighted average of US corporate bond defaults. We calibrate the coefficient of penalty cost for default 𝜔z = 31.1908 to meet certain steady state conditions. The reinvestment rate of firms 𝜈f = 1∕4. The administrative expenses count 7% of firms’ profits, 𝜉f = 0.07. As discussed in the previous section, these frictions are introduced to pin down firms’ reinvestment rate so that in equilibrium firms are not completely self-financed. Following the real business cycle literature, the depreciation rate of physical capital, 𝛿 , is set at 3% and the capitaloutput share in the production function, 𝛼 , is set at 1∕3. In steady state, the capital-output ratio is approximately 8.28.

For the banking sector, we set the steady state bank capital requirement at 10%. This implies that, in steady state, the ratio of bank funds over loans F b ∕L = 10%. We assume banks need to spend 7% of own funds as administrative expenses to pin down insurance premium to 7% of own funds and their re-investment rate to 𝜈 b = 0.137732. The recovery rate and insurance premium are calibrated in line with de Walque et al. (2010), where banks can recover 85% of defaulted loans, upon paying 6% of their own funds as insurance premium. 4. Welfare analysis In this section we first look at the cyclical properties of alternative models. We then study the welfare implication of different capital requirement regimes with and without endogenous default. We further extend our welfare analysis with different parametrization for each different capital requirement regimes. Lastly, to complement our welfare analysis, we investigate the transmission mechanisms through which bank capital requirement regimes with default impact the negative effect of the shock on welfare. The Basel capital requirement regimes considered in the study are as follows. By setting 𝜙y = 0 and 𝜙l = 0 in equations (30) and (31), we obtain Basel I. As for Basel II, we consider a negative value of 𝜙y , which mimics the pro-cyclical capital requirement of Basel II. We reverse the sign of 𝜙y , to obtain Basel III. This captures the counter-cyclical capital buffer regulation proposed in Basel III, as in Angeloni and Faia (2013). For Basel III credit-to-output we set 𝜙l > 0. Finally, by setting 𝜙𝜒 = 0, we have the scenario of no default in capital requirement regimes, or the simple Basel rule, whereas a positive value of 𝜙𝜒 implies otherwise, or the extended Basel rule. To quantify and numerically evaluate the welfare implication of each case we define the social welfare as: SW0 ≡ 6

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 4. Social welfare IRFs following a negative technology shock of size 1%: Comparison of simple and extended Basel rules under different values of the coefficient related to default deviation (𝜙𝜒 ): 0 (solid line), 0.25 (dashed line) and 0.75 (dotted line). The coefficients related to output and credit-to-output gap (𝜙y and 𝜙l , respectively) are set to 0.5 for all Basel regimes.

Fig. 5. Social welfare variance with different values of 𝜙y , 𝜙l and 𝜙𝜒 . Left panel: Basel II and III; right panel: Basel III credit-to-output.

∑∞ t =0

𝛽 t U (Ct , Dt ).

some extended Basel rules (with default) mimic the pro-cyclical property of bank loans rate very well and some produce counter-cyclical bank loans rate. Lastly, our models are able to mimic both the second moments and the counter-cyclicality of default. The results are, however, not robust – underestimated standard deviation and overestimated countercyclicality. Nonetheless, the extended Basel rules give mixed results – for some variables, including default in the Basel rule improves model fit and the reverse holds for others.

4.1. Cyclical properties Table 4 reports the cyclical properties of the models under different capital requirement regimes. Column 3 reports the standard deviation of a particular variable (x), its standard deviation relative to the same of output (𝜎x /𝜎y ), and its correlation with output (cor (x, y)).11 Columns 4–13 report the same for different models discussed above. Our models perform very well in terms of mimicking the second moments as a real business cycle model. That is, the standard deviations of output, consumption, and the risk-free rate (deposits rate) generated from model simulations are matching those observed in the data. In addition, our models reveal that, as shown in the data, both consumption and the risk-free rate are less volatile than output. All models successfully reproduce the high pro-cyclical property for consumption. The model, however, fails to reproduce the pro-cyclical risk-free rate observed in the data. Finally, these findings are consistent across all models with different capital requirement regimes, both simple and extended Basel rules. For financial variables, in general, our models perform reasonably well in terms of reproducing the second moments. This is especially the case for bank loans and bank funds, whereby models overestimate the standard deviation for deposits. Our models, however, overestimate the pro-cyclical property of all quantitative financial variables and give mixed results for the correlation between bank loans rate and output –

4.2. Simple and extended rules: Basel II vs III We start our welfare analysis by first looking at the social welfare dynamics when implementing the simple Basel rule, where there is no default in bank capital regimes. Fig. 1 reports the dynamics of social welfare in response to a negative technology shock of size 1%.12 Compared with Basel I (solid line), social welfare declines lesser under Basel II (dashed line). As for Basel III (dotted line), and Basel III credit-tooutput (starred line), under both cases social welfare declines slightly more than that under Basel II. We, therefore, conclude that the welfare cost of Basel III is higher than Basel I and II. We now investigate, compared to Basel II, whether the proposed extended rule of Basel III is welfare improving. Fig. 2 plots the social welfare response to the same shock under different capital requirement

12 We set 𝜙y = −0.5 for Basel II, 𝜙y = 0.5 for Basel III, and 𝜙l = 0.5 for Basel III creditto-output, respectively. All impulse response functions (IRFs) are in percentage deviation from steady state.

11

See Appendix Appendix B for the detailed discussion on data source and transformation. 7

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 6. Bank capital requirement IRFs following a negative technology shock of size 1%: Comparison of simple and extended Basel rules. Solid line: simple Basel rule; dashed line: extended Basel rule with 𝜙𝜒 = 0.25; dotted line: extended Basel rule with 𝜙𝜒 = 0.75.

Fig. 7. Main variables’ IRFs: Basel II, III and III credit-to-output (simple rule vs. extended rule). Left column: Basel II; middle column: Basel III; right column: Basel III credit-to-output. Solid line: simple Basel rule (𝜙𝜒 = 0); dashed line: extended Basel rule (𝜙𝜒 = 0.75).

regimes with default, or the extended Basel rule.13 As the figure indicated, the same conclusion emerges. That is, the welfare cost under both Basel III and Basel III credit-to-output is slightly higher than that of Basel II. We show that, for both simple and extended Basel rules, there is no improvement for welfare under Basel III compared to its predecessor. This result contradicts the findings of Angeloni and Faia (2013), in which the authors show that social welfare deteriorates more severer under Basel II. On the other hand, Repullo and Saurina (2011) and Repullo and Suarez (2013) argue that Basel III has a potential to exacerbate the procyclicality of its predecessor. This is especially the case for using credit-to-output gap as the indicator when implementing the counter-cyclical buffer, because it is negatively correlated with the

business cycle. If this is the case, it is unlikely that the implementation of Basel III will improve social welfare, regardless of the simple or the extended Basel rule. 4.3. Different parametrization of the extended Basel rule In this section we take our investigation one step further and compare the social welfare response under different capital requirement regimes with default. Firstly, we consider different values for 𝜙y or 𝜙l , while keeping 𝜙𝜒 constant (so as 𝜙𝜒 = 0.25). The results are reported in Fig. 3, where the numbers in the description represent different values of 𝜙y or 𝜙l , depending on the Basel regime specification. We find that a more aggressive capital requirement regime of Basel II mitigates the negative effect of the shock on welfare, while a more aggressive Basel III and III credit-to-output deteriorate the social welfare. These findings are consist with those of Resende et al. (2013), in which the authors conclude that the counter-cyclical capital requirement is optimal when

13 We set 𝜙𝜒 = 0.25 and the values of the parameters of 𝜙y and 𝜙l are the same as in the previous exercise. For the rest of our analysis we only focus on Basel II and the two Basel III policy regimes and exclude Basel I.

8

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 8. Main variables’ IRFs: Basel II, III and III credit-to-output (simple rule vs. extended rule). Left column: Basel II; middle column: Basel III; right column: Basel III credit-to-output. Solid line: simple Basel rule (𝜙𝜒 = 0); dashed line: extended Basel rule (𝜙𝜒 = 0.75).

either it is relatively aggressive or it is fixed (a-cyclical). Clerc et al. (2015) also show that business cycle fluctuations are mitigated with a more aggressive Basel III. We now repeat the same experiment but considering different values of the parameter for default, while keeping 𝜙y and 𝜙l constant.14 Fig. 4 reports the response of social welfare with different values of the parameter 𝜙𝜒 : 0 (solid line), 0.25 (dashed line), and 0.75 (dotted line). The results show that, compared to the simple Basel rule, the extended Basel rule mitigates the negative effects of the shock on welfare. The results also indicate that the higher the parameter 𝜙𝜒 is, the lesser the social welfare will decline in response to a negative productivity shock. This implies that it is welfare improving if a capital requirement rule reacts more aggressively to default. More importantly, this finding holds for all three bank capital requirement regimes. To have a better understanding of the welfare response with different parameter values, we report the second moments of social welfare in Fig. 5.15 The left panel displays the variance of social welfare under Basel II and Basel III and the right panel shows the same under Basel III credit-to-output. It is clear that for Basel II a lower variance of social welfare is obtained with a more aggressive policy (a greater value of 𝜙y in absolute term). The opposite is true for Basel III and Basel III credit-to-output: a more aggressive policy leads to a higher social welfare variance (greater values of 𝜙y and 𝜙l ). In contrast to these mixed results, a higher value of the policy parameter of default (𝜙𝜒 ) helps to reduce the variance of social welfare under all regimes.16

4.4. The transmission mechanisms: simple vs extended rule In order to have a better understanding of why the use of the extended Basel rule is welfare improving, we investigate the transmission mechanisms through which it mitigates the negative effects of the shock. A negative productivity shock results in an increase in the default rate (see the last row in Fig. 7). Since capital requirement is increasing in default, the regulatory authority will increase the capital requirement accordingly based on the extended Basel rule. Fig. 6 reports the IRFs of capital requirement ratio in response to the shock under different capital requirement regimes. Under Basel II, the use of the extended Basel rule reinforces bank capital regulation, especially with a more aggressive response to default. This, however, can potentially exacerbate the pro-cyclicality of Basel II.17 Under Basel III, on the other hand, the counter-cyclical capital buffer is offset considerably with the extended Basel rule. This is especially the case for a capital requirement regime responding more aggressively to default (𝜙𝜒 = 0.75). The same holds for Basel III credit-to-output. Figs. 7 and 8 report the IRFs of the key variables to a negative technology shock. The solid-line represents the IRFs under the capital requirement regime specified at the top of each column in its simple Basel rule form (𝜙𝜒 = 0), and the dashed-line represents the IRFs when implementing the extended Basel rule (𝜙𝜒 = 0.75). The results show that it is through the bank funding channel that introducing endogenous default in a capital requirement regime improves social welfare. As discussed earlier, banks benefit from the extended Basel rule as banks will be more profitable and better capitalized with a lower default rate (especially in the Basel III credit-to-output). As evidenced in Fig. 7, this results in a significant recovery in bank funds after the shock, compared with the case under the simple Basel rule. Banks can, therefore, supply more credit to firms. This, in turn, mitigates the negative effects of the

We set both 𝜙y and 𝜙l to 0.5 for the two Basel III regimes and 𝜙y = −0.5 for Basel II. We use Dynare to approximate the model around its steady state up to the second order and calculate the second moments of the social welfare by solving the model at each point on a grid of 1353 points for parameters 𝜙y ∈ [−4, 0) and 𝜙𝜒 ∈ [0, 1] for Basel II, 𝜙y ∈ (0, 4] and 𝜙𝜒 ∈ [0, 1] for Basel III, and 697 points for 𝜙l ∈ (0, 4] and 𝜙𝜒 ∈ [0, 1] for Basel III credit-to-output. 16 In Table C.5 Appendix Appendix C, the variance ratio test indicates that the social welfare variance from the extended Basel rule is significantly smaller than that from the simple Basel rule. 14 15

17 For studies on the pro-cyclicality of Basel II see, e.g. Repullo and Suarez (2013) and Liu and Seeiso (2012).This is, however, not the focus of the current study.

9

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 9. Social welfare IRFs following a negative technology shock of size 1% with different insurance coverage ratios (𝜔q ): Basel II (left column), Basel III (middle column) and Basel III credit-tooutput (right column). Solid line: simple Basel rule; dashed line: extended Basel rule.

shock and helps the economy to recovery quicker and, hence, welfare improving. Fig. 8 suggests that the extended Basel rule attenuates the negative effects of the shock directly through financial aggregates, not the interest-rate channel. This is evidenced that, under the extended Basel rule, the decline in bank loans and deposits is less pronounced, despite almost identical IRFs of their corresponding rates.18 With a binding bank capital requirement, any reduction in loans is transferred into a 𝜏 fraction of reduction in bank funds and a 1 − 𝜏 fraction of reduction in deposits, shrinking banks’ balance sheet. In this regard, the extended Basel rule mitigates the shrinkage of banks’ balance sheet.

of social welfare under the extended Basel rule is lower than that under the simple Basel rule.19 In principle, a higher insurance coverage ratio enhances this attenuation effect as banks are able to recover more from the defaulted loans and extend more new loans for production and, hence, consumption increases. This is, however, not the case. As revealed in Fig. 9, there is no significant difference in the responses in welfare when the insurance coverage ratio increases from 0.65 to 0.95 under each Basel regime. This implies that changes in an important parameter like insurance coverage ratio do not change the findings, either qualitatively or quantitatively. Fig. 10 depicts the social welfare responses with different capital requirement levels: 5%, 10% and 15%. Similarly, we see that the extended Basel rule mitigates the negative effect of the shock on welfare under all Basel regimes and associates with relatively lower standard deviation of the social welfare.20 In contrast to the insurance coverage ratio, we see this attenuation effect slightly increases as the capital requirement level increases. Nonetheless, changes in capital requirement ratio do not alter the results significantly. Finally, we check whether the findings are sensitive to the persistence of the shock. Following the literature that the technology shock is normally assumed to be persistent, We choose three different values for the persistence of the shock, 0.75, 0.80 and 0.95.21 As revealed by Fig. 11, the attenuation effect of the extended Basel rule diminishes slightly as the persistence of the shock increases. This is understandable, as when a more persistent negative real shock hits the economy, the implication of the shock on welfare becomes more severe and, hence, reduces the attenuation effect of the extended Basel rule.

4.5. Sensitivity analysis Having established that it is welfare improving with the extended Basel rule and the transmission mechanisms through which it reduces welfare losses, in this section we perform the sensitivity analysis by comparing the welfare under the same Basel regime but with different parameterizations. Parameter values are chosen such that all constraints are binding in equilibrium. For the sake of gravity, we choose three parameters for the sensitivity analysis: insurance coverage ratio (𝜔q ), bank capital requirement rate (𝜏 ) and the persistence of the shock (𝜌). Indeed, these three parameters are all relevant and important for the welfare analysis, which will be discussed individually below. Fig. 9 reports the responses of social welfare where insurance coverage ratio is calibrated to 0.65 (solid lines), 0.8 (dashed lines) and 0.95 (dotted lines). The results show that, compared to the simple Basel rule, the extended Basel rule mitigates the negative effect of the shock on welfare under all Basel regimes. Moreover, the standard deviation

19

See Table C.6 the variance ratio test results in Appendix Appendix C. See Table C.7 the variance ratio test results in Appendix Appendix C. 21 These values are close to the ones from the literature, i.e., 0.822 in Smets and Wouters (2003) and 0.93 in Iacoviello and Neri (2010). 20

18 There is only a temporary response difference for bank loans and deposits rates after the shock occurs.

10

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15 Fig. 10. Social welfare IRFs following a negative technology shock of size 1% with different levels of capital requirement ratio (𝜏 ): Basel II (left column), Basel III (middle column) and Basel III credit-to-output (right column). Solid line: simple Basel rule; dashed line: extended Basel rule.

Fig. 11. Social welfare IRFs following a negative technology shock of size 1% with different persistence of the shock (𝜌): Basel II (left column), Basel III (middle column) and Basel III credit-tooutput (right column). Solid line: simple Basel rule; dashed line: extended Basel rule.

11

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Economic Modelling xxx (2018) 1–15

In summary, based on the sensitivity analysis conducted here together with the cyclical property analysis of the model in subsection 4.1 and the welfare analysis with different parameterizations of the extended Basel rule in subsection 4.3, we show that our findings are robust. In addition, results from social welfare variance significance test are consist with the findings of the sensitivity analysis.22

ment regimes. Our results show that including default in the capital requirement rule is welfare improving. In addition, a more aggressive reaction to default mitigates the negative welfare effect of the shock. Lastly, in terms of welfare, there is no clear evidence on either creditto-output gap or output gap is a better candidate for implementing the countercyclical capital buffer.

5. Conclusions

Acknowledgements

To study the welfare implication of capital requirements with default, we develop an RBC model with banking, in which borrowers may default on their financial obligations upon paying a penalty cost. We propose an extended Basel rule, in which we augment the default rate in the bank capital requirement rule. We then examine the welfare response to a negative technology shock under different capital require-

We thank the journal editor and an anonymous referee for valuable comments and suggestions. The financial assistance of the National Research Foundation (NRF) of South Africa towards this research is hereby acknowledged (Grant number: 90577). Financial support from Economic Research Southern Africa (ERSA) is also greatly acknowledged. The usual disclaimer applies.

Appendix A. Maximization problems Appendix A.1. Households The Lagrangian of the representative household reads as follows: h = 𝔼0 [

∞ ∑

𝛽 t {log (Ct − jCt−1 ) + 𝜙d log (Dt )

t =0

+ 𝜆ht Rdt−1 Dt−1 + Wt Ht − Tt − Ct − Dt

]}

(A.1)

.

(A.2)

Defining 𝜆ht as the Lagrange multiplier, the first order conditions with respect to {Ct , Dt }∞ t =0 are: Ct ∶

1

(Ct − jCt−1 ) Dt ∶ 𝜙d

1 ) = 𝜆ht , Ct +1 − jCt

− 𝛽 j𝔼t (

(A.3)

1 = 𝜆ht − 𝛽 𝔼t 𝜆ht+1 Rdt . Dt

(A.4)

Appendix A.2. Firms The Lagrangian of the representative firm reads as follows: f = 𝔼0

∞ ∑

{

(

)

[

(

)

𝛽 t log 𝜋tf + 𝜆at (1 − 𝛿) Kt−1 + Lt + 𝜈f 1 − 𝜉f 𝜋tf − Kt

]

t =0

[

+ 𝜆bt At Kt𝛼−1 Ht1−𝛼 − Wt Ht − (1 − 𝜒t ) Rlt Lt−1 − { The first order conditions with respect to

𝜒t , Ht , Kt , Lt , 𝜋tf

}∞ t =0

)2 𝜔z ( 𝜒t−1 Rlt−1 Lt−2 − 𝜋tf 2

.

(A.5)

are: (

𝜒t ∶ 𝜆bt Rlt Lt−1 = 𝛽 𝔼t 𝜆bt+1 𝜔z 𝜒t Rlt Lt−1

)2

,

Ht ∶ (1 − 𝛼) At Kt𝛼−1 Ht−𝛼 = Wt ,

𝛼𝛽 𝔼t 𝜆bt+1 At+1 Kt𝛼−1 Ht1+−𝛼 1

= 0, ( )2 ( ) Lt ∶ 𝜆at − 𝛽 𝔼t 𝜆bt+1 1 − 𝜒t +1 Rlt +1 − 𝛽 2 𝔼t 𝜆bt+2 𝜔z 𝜒t +1 Rlt +1 Lt = 0, ( ) f f 𝜋t ∶ 1∕𝜋t − 𝜆bt + 𝜈f 1 − 𝜉f 𝜆at = 0. Kt ∶

]}

− 𝜆at

22 In Appendix Appendix C, we report the results from the social welfare variance significance test, in which we test whether the variance of the social welfare under the simple Basel rule is statistically different from that under the extended Basel rule in conjunction with the sensitivity analysis. The results show that the standard deviation under the simple Basel rule is statistically significantly higher than that under the extended Basel rule. This holds for all scenarios with different values of the selected three parameters that are being used for the sensitivity analysis.

12

+ (1 − 𝛿) 𝛽 𝔼t 𝜆at+1

(A.6) (A.7) (A.8) (A.9) (A.10)

F. Garcia-Barragan and G. Liu

Economic Modelling xxx (2018) 1–15

Appendix A.3. Banks The Lagrangian of the representative bank reads as follows: b = 𝔼0

∞ ∑

{

(

)

(

𝛽 t log 𝜋tb + 𝜆dt Dt + Ftb − Lt

) (A.11)

t =0

[

+ 𝜆et (1 − 𝜒t ) Rlt Lt−1 − Rdt−1 Dt−1 + 𝜔q 𝜒t−1 Rlt−1 Lt−2 − 𝜋tb [

+ 𝜆ft (1 − 𝜉b − 𝜁) Ftb−1 + 𝜈b 𝜋tb − Ftb [

+ 𝜆gt Ftb − 𝜏t Lt

]}

]

]

.

{ }∞ The first order conditions with respect to Dt , Ftb , Lt , 𝜋tb t =0 are: Dt ∶ 𝜆dt = 𝛽 𝔼t 𝜆et+1 Rdt ,

𝜆dt + 𝜆gt

− 𝜆ft

(A.12)

f (1 − 𝜉b − 𝜍) 𝔼t 𝜆t+1

Ftb ∶ +𝛽 = 0, ( ) g Lt ∶ 𝛽 𝔼t 𝜆et+1 1 − 𝜒t +1 Rlt +1 + 𝛽 2 𝔼t 𝜆et+2 𝜔q 𝜒t +1 Rlt +1 = 𝜆dt + 𝜆t 𝜏t ,

𝜋tb Taxes:

∶

1∕𝜋tb

− 𝜆et

f + 𝜈b 𝜆t

= 0.

(

𝜏t = 𝜏

Yt Y

)𝜙 y (

𝜒t 𝜒

)𝜙 𝜒

(

, or 𝜏t = 𝜏

Lt ∕Yt L∕Y

(A.14) (A.15)

) ) ( ( Tt = q 𝜒t −1 , Rlt −1 , Lt −2 − z 𝜒t −1 , Rlt −1 , Lt −2 − 𝜍 Ftb−1 .

Capital requirement rule:

(A.13)

)𝜙 l (

𝜒t 𝜒

)𝜙 𝜒

(A.16)

.

(A.17)

Technology shock: log (At ) = (1 − 𝜌A ) log (A) + 𝜌A log (At −1 ) + 𝜉tA .

(A.18)

Appendix B. Data description and sources The second moments of the data are calculated using the US data over the period 1960Q1 - 2017Q1, taken from the Federal Reserve of St Louis, Bureau of Economic Analysis and the Bureau of Labor Statistics. The data for default rate is only available from 1992Q1 and the data for bank funds is only available from 1984Q1. All variables are transformed into real term using GDP deflator and aggregate variables are converted to per capita using the Population series. We use the Hodrick and Prescott (1997) filter to extract the cyclical components from the logged series. The data description and sources are as follows: Production: Real Gross Domestic Product per capita. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve Bank of St Louis (A939RX0Q048SBEA). Consumption: Real personal consumption expenditures per capita, Chained 2009 Dollars, Quarterly, Seasonally Adjusted Annual Rate. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve Bank of St Louis (A794RX0Q048SBEA). Bank funds: Total Equity Capital for Commercial Banks in United States, Thousands of Dollars, Quarterly, Not Seasonally Adjusted. Time period: 1984:Q1 - 2017:Q1. Source: Federal Reserve Bank of St Louis (USTEQC). Deposits: Households and nonprofit organizations; checkable deposits and currency; asset, Flow. Time period: 1960:Q1 - 2017:Q1. Source: Bureau of Economic Analysis (FA153020005.Q). Deposits rate: 3-Month Treasury Bill: Secondary Market Rate, Percent, Monthly, Not Seasonally Adjusted. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve of St Louis (TB3MS). Loans: Nonfinancial Corporate Business; Total Liabilities, Level, Billions of Dollars, Quarterly, Not Seasonally Adjusted. Time period: 1960:Q1 2017:Q1. Source: Federal Reserve Bank of St Louis (TLBSNNCB). Loans rate: Bank prime loan rate. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve of St Louis (MPRIME). Default: The size of companies closing in the private industr (BDS0000000000000000120006RQ5). Time period: 1992:Q1 - 2017:Q1. Source: Bureau of Labor Statistics. GDP implicit price deflator: Gross Domestic Product: Implicit Price Deflator, Index 2009 = 100, Quarterly, Seasonally Adjusted. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve Bank of St Louis (GDPDEF). Population: Working Age Population: Aged 15 and Over: All Persons for the United States, Persons, Quarterly, Seasonally Adjusted. Time period: 1960:Q1 - 2017:Q1. Source: Federal Reserve Bank of St Louis (LFWATTTTUSQ647S).

Appendix C. Significance test: Variance ratio test The main finding of the study is that it is welfare improving by including the default rate as an additional indicator for all capital requirement regimes. In other words, compared to the simple Basel rule, the extended Basel rule attenuates the negative effect of the shock on social welfare (see Fig. 4). In this section, we use the variance ratio test to assess the significance of this main finding. The hypothesis of the variance ratio test is that the standard deviation from experiment (or sample) A is the same as the standard deviation from experiment (or sample) B. Tests are carried out in Stata 13 (“sdtest”) with 1000 social welfare IRF simulations obtained from Dynare for 13

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Economic Modelling xxx (2018) 1–15

each experiment. We first report the variance ratio test results for the comparison between the simple Basel rule and the extended Basel rule. To complement the sensitivity analysis, we also report the test results with alternative values for insurance coverage ratio, capital requirement ratio and the persistence of the shock. Table C.5 shows that the standard deviation of the social welfare under the simple rule is greater than that under the extended rule.23 With a more aggressive response to default in the capital requirement regime (𝜙𝜒 = 0.75), the F-statistic indicates the results are statistically significant at 0.001 significance level, whereby the results are insignificant when capital requirement responds to default less aggressive (𝜙𝜒 = 0.25). This holds across different regimes, Basel II, III and III credit-to-output. Tables C.6, C.7 and C.8 report the same but with different parameterizations for the three selected parameters that are being used for the sensitivity analysis in subsection 4.5. We see that the same conclusion emerges with all parameter values, except for the persistence of the shock 𝜌 = 0.95. Based on the variance ratio test results, we conclude that, overall, the main findings of the study are statistically significant. The significance test also confirms the finding that a more aggressive response to default can effectively mitigate the negative effect of the shock on welfare. Table C.5 Variance ratio test: simple vs. extended Basel rule. Basel regime

Simple rule vs. Extended rule (𝜙𝜒 = 0.25)

Basel II Basel III Basel III credit-to-output

Extended rule (𝜙𝜒 = 0.75)

f-stat

Pr(F>f)

f-stat

Pr(F>f)

1.0804 1.0671 1.0759

0.1109 0.1524 0.1238

1.3146*** 1.2530*** 1.3043***

0.0000 0.0002 0.0000

The null hypothesis is that both standard deviations are equal (Ho ∶ 𝜎(simple rule)∕𝜎(extended rule) = 1). An alternative hypothesis is the standard deviation from the simple rule is greater than that from the extended rule (Ha ∶ 𝜎(simple rule)∕𝜎(extended rule) > 1). Notes: (***) indicates that the null hypothesis is rejected at 0.001 significance level.

Table C.6 Variance ratio test: Different insurance coverage ratios (𝜔q ). Basel regime

Insurance coverage 𝜔q

Basel II

0.65 0.80 0.95 0.65 0.80 0.95 0.65 0.80 0.95

Basel III

Basel III credit-to-output

Simple vs Extended rule (𝜙𝜒 = 0.75) f-stat

Pr(F>f)

1.3068*** 1.3126*** 1.3186*** 1.2493*** 1.2521*** 1.2550*** 1.2960*** 1.3022*** 1.3086***

0.0000 0.0000 0.0000 0.0002 0.0002 0.0002 0.0000 0.0000 0.0000

The null hypothesis is that both standard deviations are equal (Ho ∶ 𝜎(simple rule)∕𝜎(extended rule) = 1). An alternative hypothesis is the standard deviation from the simple rule is greater than that from the extended rule (Ha ∶ 𝜎(simple rule)∕𝜎(extended rule) > 1). Notes: (***) indicates that the null hypothesis is rejected at 0.001 significance level.

Table C.7 Variance ratio test: Different capital requirement ratios (𝜏 ). Basel regime

Basel II

Basel III

Basel III credit-to-output

Bank capital req. 𝜏

0.05 0.10 0.15 0.05 0.10 0.15 0.05 0.10 0.15

Simple vs Extended rule (𝜙𝜒 = 0.75) f-stat

Pr(F>f)

1.1236* 1.3146*** 1.6682*** 1.1144* 1.2530*** 1.4145*** 1.1236* 1.3043*** 1.6071***

0.0328 0.0000 0.0000 0.0436 0.0002 0.0000 0.0328 0.0000 0.0000

The null hypothesis is that both standard deviations are equal (Ho ∶ 𝜎(simple rule)∕𝜎(extended rule) = 1). An alternative hypothesis is the standard deviation from the simple rule is greater than that from the extended rule (Ha ∶ 𝜎(simple rule)∕𝜎(extended rule) > 1). Notes: (***) indicates that the null hypothesis is rejected at 0.001 significance level.

23

An F − statistic > 1 implies the alternative hypothesis of Ha ∶ ratio > 1 is true (see the “sdtest” function in Stata 13). 14

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Economic Modelling xxx (2018) 1–15

Table C.8 Variance ratio test: Different shock persistence levels (𝜌). Basel regime

Shock persistence 𝜌

Basel II

0.75 0.85 0.95 0.75 0.85 0.95 0.75 0.85 0.95

Basel III

Basel III credit-to-output

Simple vs Extended rule (𝜙𝜒 = 0.75) f-stat

Pr(F>f)

1.4442*** 1.2056*** 1.0485 1.3463*** 1.1699*** 1.0417 1.4226*** 1.2013*** 1.0477

0.0000 0.0016 0.2274 0.0000 0.0066 0.2593 0.0000 0.0019 0.2307

The null hypothesis is that both standard deviations are equal (Ho ∶ 𝜎(simple rule)∕𝜎(extended rule) = 1). An alternative hypothesis is the standard deviation from the simple rule is greater than that from the extended rule (Ha ∶ 𝜎(simple rule)∕𝜎(extended rule) > 1). Notes: (***) indicates that the null hypothesis is rejected at 0.001 significance level.

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