Bank Capital Regulation and Credit Crunch

Byung-Kun Rhee

Kyungsan University

<Abstract>

With simple dynamic model with bank, we try to show three things. (i) When the bank capital adequacy requirement is increased, the incumbent bank shareholders may be better-off by meeting this requirement by reducing risky loan than by issuing new equities. (ii) If the banker does so, there may be continuing shrinkage of loan. (iii) The extent of the loan reduction is greater when the economy is in bad condition in the sense that the average productivity is lower or the volatility of the output is higher.

I. Introduction.

One of the common economic phenomenon in the Asian countries after the financial crises is the sharp decline in credit. (Alba, Bhattacharya, Claessens, Ghosh, and Hernandez(1999), Ding, Domaç, and Ferri(1999)) [Tables or graphs are to be provided later.] There are some empirical studies on the causes of the decline. Using survey data of commercial banks in Thailand, Ito and Pereira de Silva(1999) suggested that the decline was due to the contraction of credit supply of banks.

Behind the tightening of bank credit supply was the restructuring of banking sector of the crisis-stricken countries. For the restructuring of the banking sector, IMF demanded strict enforcement of capital regulation. [quote IMF Agreement with Asian countries] Peek and Rosengren(1995) suggested empirical evidence about the link between the bank regulatory enforcement and the shrinkage of bank loan with U.S. data.

The issue of capital based regulation of bank was widely studied after the Basle agreement in 1988. Gorton and Winton(1995), Calem and Rob(1996) and Jürg Blum(1999) are among them. However, most of the studies on the effect of capital regulation focused on the bank’s risk taking. Furlong and Keely(1989) is one of the studies that tighter bank capital regulation reduces bank’s risk-taking incentives. However, Calem and Rob(1996) and Jürg Blum(1999) showed that increased capital regulation induce banks to take more risky assets and therefore, tighter capital regulation makes bank more prone to default.

The reason why banks become more riskier is that they may not meet the regulation and go bankrupt. Bad bank in the sense of low capital ratio has to invest in assets with higher return which are riskier. If blessed, they meet the capital requirement and increased capital can generate more investment as Cooke’s ratio. But if not, they can simply exit. However, this type of response is plausible only for marginal bank.

In Asian countries, the elimination of bad financial institutions was done in a first stage of recovery from the crisis. But the credit contraction had been continued even after a couple of years.

Together with restructuring, banks become less willing to make loan because of the tightened regulation. The tightening of the bank regulation was in the form of the increase of capital adequacy requirement. When the authority increase the capital adequacy rate, bank has two options. It may either issue more equity or reduce risky asset. If the bank owners decide what to do, they may prefer reduction of loan instead of eequity issue because further equity issue would reduce their share.

Bank may leave the industry instead of meeting the requirement. But in this paper this option is ruled out.

In this paper, we show that the enforcement of the capital adequacy requirement may reduce the bank credit supply. The reduction of credit supply cause the financing difficulty of firms. Firms unable to borrow may go bankrupt. Their previous loan become non-performing. In addition, workers laid off from the bankrupt firms have to withdraw their deposit. Then banks must decrease the loan payout further because of the loan loss reserve. The purpose of the capital adequacy requirement is to reduce the risk of bank failure. However, the above mentioned story may expedite the bank failure.

This paper suggests a simple dynamic model with consumer, firm and banks. Consumers use bank deposit to smooth their consumption. Firms borrow fixed investment and working capital from bank. If they could borrow required capital, they produce goods. If not, they go bankrupt. Bank accepts deposit and give loans to firms to maximize profit. Bank maximizes profit of owner and is subject to regulation.

In this model we show that tightening of bank regulation may contribute credit crunch. Especially when the economy is in bad state, the tightening of bank regulation is not desirable.

This paper is organized as follows. In section II, model is presented. The results are followed in section III. Section IV concludes.

II. Model

There are four types of agents: consumer/worker, firm, bank, and the government. Since we are interested in the behavior of bank, we use many assumptions to make the behavior of consumer and firm as simple as possible.

The model has the three periods. At t=0, consumers deposit their saving in the bank and bank investments its fund. Firms are established with loan from banks. However, the production does not take place until t=1. The the beginning of t=1, the bank supervisory authority may announce new capital adequacy rate. In that case equity market opens at the beginning of period 1 so that the bank may sell new shares. At later periods(t=1,2), production takes place and consumer consumes some part of their income/wealth. At time t=2, all liabilities are to be liquidated and all assets are compensated.

There exist continuum of many consumers of mass 1. Since they are assumed to have the same utility function, and receive equal unit of endowment (assumed to be one), they are identical at t=0. However, some of them may become unemployed as some firms may go bankrupt. Consumers have two sources of income. One is the endowment of consumption good given at t=0, and the other is wage income. They supply one unit of labor to the firm, and receive wage, w. Each consumer/worker enter an employment contract with one of continuum of firms with wage w and this contract binds for two periods. Since the contract is written before the realization of shocks, the wage rate is the same and the workers and firms are randomly matched. Therefore after the contract, indices of consumers can be used to denote firms.

Consumers’ utility function is given as following,

[eqn: u(c_1 ) ~+~ beta u(c_2 )]                                  (1)

where [eqn: c_1]([eqn: c_2]) is the consumption at t=1(t=2), and β is the discount rate.

We may assume that consumer consumes at t=0, and he/she save the rest of the endowment. Since the consumer's saving decision is  not our interest, we assume it away.

There exist continuum of entrepreneurs with mass one. At t=0, they must build the factory which costs [eqn: B_0]. They borrow the start-up cost from bank as a long term loan. They pay interest at t=1, but both principal and interest at the end of t=2. Each entrepreneur hire one worker for production. Because of a rigidity of labor market, firm-worker enter an employment contract for a predetermined wage level. Even though the contract lasts for two periods, worker is laid-off in case the firm is not able to finance the working capital or hit by a bad shock.

During the period t=0, after the contract is written, each entrepreneur makes  his/her effort which affects the level of output afterward.

In addition to the start-up cost([eqn: B_0]), firm needs working capital of amount [eqn: B_1]to operate at t=1. If a firm is able to finance, it produces output y, and pays out wage and interest. We assume that the profit is consumed by the entrepreneur.  If a firm fails to finance the working capital, the firm cannot produce any output, the worker employed in this firm is laid-off, and the loan is not performed. Even though the firm is suspended from the production process at t=1, it can still produce at t=2 if it succeeds in financing the required working capital. Working capital [eqn: B_1] is borrowed as a short-term loan, which is to be redeemed both interest and principal at the end of each period.

The utility function of entrepreneurs is simply the sum of the level of each consumption. Since the entrepreneurs consume the residual profit, the functional form is not a great matter.

The level of output of a firm is determined by the amount of effort of the entrepreneur, overall economic condition and individual shock. But it is assumed that we cannot decompose the effects into those of individual factors.

The behavior of the bank is as follows. There is only one (representative) bank. It is owned by a banker who pays all the original capital(K). We assume that the banker puts all of his/her wealth to establish the bank. When the authority increases the capital adequacy rate, and the bank needs to increase the capital, the bank has to sell the shares to the public if it does not have retained earning. The capital rate is defined as the ratio of capital(K) and risk-weighted sum of assets. In this model loan to the firm has risk-weight of 1 and government bond 0.

We assume that the banker is the manager of the bank and maximizes his/her own profit. The profit accrued to the banker is consumed. As in the case of entrepreneurs, the utility function of the banker is the sum of the consumption each period.

At time t=0, the bank accepts the consumer deposit which is summed to 1. It has 1+K to be invested to maximize profit. The bank may either lend to firms or buy government bond which pays fixed interest(r). The lending rate is given by [eqn: r_l] and deposit rate by [eqn: r_d]. We assume that the lending rate([eqn: r_l]) is expected to be greater than r.

Since we are analysing the economy with credit market either in excess supply or excess demand, we assume that interest rates are fixed. We are assuming the following economy. The government issues bond with fixed rate of return. Only the bank can buy the bond. Since the operation cost of the bank is null, the deposit rate is same as the return from the government bond. And the borrowing-lending rates spread is fixed.

Since the lending rate is fixed, the only risk of the loan is the possibility of default.

During the period, the bank collects information about the entrepreneurs (effort). It is denoted by x, which can be interpreted similar credit rating of each firm. The accuracy rate of the information is denoted by the correlation between x and y, which is denoted by [eqn: rho_xy]. Even after the realization of the first period output level, the bank keeps the same credit rating for each firm. This is the same as assuming that the bank know the effort level of each firm and regards the deviation from the credit rating as individual shock.

Even though the value x denotes the possible level of output, we call it ꡐcredit ratingꡑ for convenience.

We assume that the level of output at each period([eqn: y_t , t=1,2]) is correlated with the credit information(x). The two variables ([eqn: x],[eqn: y_t]) are assumed to follow bivariate normal distribution with correlation coefficient [eqn: rho _ xy]. For the convenience of the presentation, we assume that the probability that either [eqn: y_t] is less than 0 or greater than Y is negligible.

The expected output at t of a firm with credit x is given by,

[eqn: mu _ t + rho_xy ( sigma_t over sigma_x ) (x - mu _x )]

where [eqn: mu_x] and [eqn: sigma_x] are mean and variance of the credit rating. And [eqn: mu_t] and [eqn: sigma_t] are the mean and variance of output of firms at time t.. These parameters ([eqn: mu_t] , [eqn: sigma_t] ) denote the overall (market) condition of the economy. We assume that they are revealed at the beginning of each period.

At the beginning of period t=1, the bank reshuffles the portfolio. Using the credit information, it decides how much to lent to firms and how much to buy government bond. When the authority changes the capital adequacy rate, it changes the portfolio according to the new regulation.

At the end of period 1, bank receives the interest from the asset. Since the output of each firm is subject to a shock, some firms may fail to pay the interest. Then the loan becomes bad and appropriate reserve must be placed.

We assume that loan loss reserve can be held in the form of risk-free asset which is government bond in this model.

As for the bad loan, bank simply accepts the remained output of the firm. Firms pay the wage bill first, and turn everything they produce to the bank in case they cannot meet the obligation. Naturally, the possibility of false report of a firm arises. We simply assume the myopia of the firm and they reveal their production honestly since we want to keep the firms' behavior simple.

Also the bank may face the withdrawal of deposit, especially from the workers who became unemployed during the period.

At the beginning of period 2, the bank reshuffles the portfolio again. And the end of the period, the bank receives all the principal and interest and return the deposit with interest.

The government does two things: supplying the government bond, and determining the capital adequacy rate of bank.

(1) Optimizations of consumers and entrepreneurs

Consumer maximizes the lifetime utility (1) subject to the following budget constraint,

[eqn: c_1 ~+~ c_2 over {1+r_d} ~=~ 1 ~+~ r_d ~+~ w I_1 ~+~ w over {1+r_d} I_2]

where [eqn: I_1] and [eqn: I_2] are indices for the employment status of period 1 and 2.

For the simplicity, we assume that the time preference of the consumer is equal to the inverse of the gross deposit rate, i.e. [eqn: beta (1+r_d ) = 1]. Since the time preference is fully compensated by the deposit interest rate, consumer  smooths his/her consumption completely, i.e. [eqn: c_1 = c_2] holds ex ante. Of course, the ex post level second period consumption([eqn: c_2]) depends on the realization of shock, and is not the same as [eqn: c_1].

In this model the only uncertainty the consumer faces is the possibility of layoff. Since the consumption takes place at the end of each period, the employment status at t=1 is known. The solutions are different according to the period 1 status.

Exactly speaking, it must be the possibility of being unpaid. Because of the employment contract written at the beginning of period 0, all workers are employed.

Let p denote the possibility that the employed at t=1 become unemployed at t=2 and q the possibility the unemployed become employed. Because of the completely smooth consumption, the consumptions of the worker employed(paid) at t=1 become

In fact, the possibilities are different according to the level of the realized output at t=1. The worker employed in a firm with higher output has lower probability of being laid off next period because the bank uses previous period output as a credit rating. To avoid the complexity, we assume that workers are ignorant of the level of output. They are only informed whether they are paid or not.

[eqn: c_1^1 ~=~ E`(c_2^1 `) ~=~ {(R_d + w) R_d} over {1+ R_d} ~+~ (1-p) w over {1+R_d}]              (2)

where [eqn: R_d] denotes gross deposit interest rate, [eqn: 1+r_d]. Depending on the magnitude of consumption([eqn: c_1^1]) and wage w, they withdraw some of their deposit or make new deposit.

The consumptions of the unpaid workers at t=1 are

[eqn: c_1^0 ~=~ E`(c_2^0 `) ~=~ {R_d^2 + q`w} over {1+ R_d}]                         (2')

Since they do not have current income, they have to retrieve this amount from their deposit.

The optimization problem of entrepreneur is very simple. They borrow [eqn: B_0] at t=0, and make some effort during the period. At period 1, they try to borrow [eqn: B_1] from bank. The entrepreneur who is not able to finance it, he/she cannot do anything other than waiting for next period. Entrepreneurs who borrow [eqn: B_1] produce output. Firms with realized output less than w inform their workers that they cannot pay wage. Those with output greater than w but not enough to pay interest and the principal of the short-term loan([eqn: B_1]) turn in all of their output to the bank after paying wage to workers. The rest of the firms realize positive profit which is consumed by entrepreneurs. The behavior at period 2 is the same as at t=1 except that the firms have to pay the long term principal([eqn: B_0]) at the end of the period. We assume that the scrap value of each firm is [eqn: (1-delta)B_0]. Therefore, firms whose period 2 output are not big enough to redeem the debt use the scrap value.

We assume that bank has to place [eqn: delta B_0] as reserve when the loan turns bad. The reserve rate is the same for short-term loan even though we assume that it is used as working capital which does not have any scrap value.

(2) Bank profit maximization

The bank behavior at t=0 is simple. Since each firm has to borrow [eqn: B_0] at t=0, loan demand is [eqn: B_0]. The bank makes loan of [eqn: B_0] and buys [eqn: 1+K-B_0] of bond.

At the beginning of period t=1, the bank reshuffles the portfolio. Since [eqn: B_0] is lent as a long-term loan, it has [eqn: 1+K-B_0] of bond to be reshuffled. Different from the loan decision at t=0, only firms with expected output higher than [eqn: w + r_l B_0 + (1+r_l ) B_1] are expected to liquidate their debt. Therefore the investment strategy of the bank at the beginning of period 1 is as follows.

Let [eqn: under{x}] denotes the level of x such that the expected output at t=1 is equal to the minimum level required to pay the wage and interest, i.e.,

[eqn: mu _ 1 + rho_xy ( sigma_1 over sigma_x ) (`under{x} - mu _x )] = [eqn: w + r_l B_0 + (1+r_l ) B_1].

If [eqn: 1+K-B_0 ~<~ {Y-under{x}} over Y B_1] is true, the bank allocates the available fund to firms according to the order of their credit rating. But if [eqn: 1+K-B_0 ~>=~ {Y-under{x}} over Y B_1], the bank gives loan to all firms with high enough expected output. The amount of  new loan is [eqn: {Y-under{x}} over Y B_1] and the amount of government bond reduces to [eqn: 1+K-B_0 - {Y-under{x}} over Y B_1]. We assume that the latter is the case.

At the beginning of period t=2, the bank does the same thing except that the credit rating x is replaced by period 1 output [eqn: y_1].

III. Results

(1) Reference equilibrium

The above described model with many simplifying assumptions is easily solved. Since the prices are all assumed fixed, we cannot exactly define an equilibrium. We can only track the evolution of the economy.

Admitting his problem, we call an equilibrium an allocation that maximizes each agent's utility given the fixed prices. We think of an equilibrium which is similar to the steady state. Even though this model has only three periods, it can be similar to the steady state when we extend to model to infinite time setup. This equilibrium will be compared to the dynamics of the model when the government enforces the capital regulation of the bank.

For the description of the equilibrium, we need to define a few notations. Let [eqn: lambda] be  the proportion of firms which are not able to finance working capital at the beginning of period 1. That is,

[eqn: lambda = int_0^{under{x}} phi(x) dx = Phi left( {under{x}-mu_x} over sigma_x } right)]

where [eqn: phi] and [eqn: Phi] denote the p.d.f. and c.d.f. of normal distribution respectively. Let [eqn: gamma] denote the proportion of firms which are able to get loan at t=1, but unable to pay wage, and [eqn: theta] the proportion of firms which are able to pay wage but unable to liquidate loan, i.e.,

[eqn: gamma = int_0^w h(y``| x>= under{x} ) dy],

[eqn: theta = int_w^{w+r_l B_0 + R_l B_1} h(y``| x>= under{x} ) dy].

where [eqn: h] denotes the conditional p.d.f. of bivariate normal distribution.

Before we describe the equilibrium, we present the last assumption. We assume that the credibility of the bank credit rating is same as the that of previous output. For this we assume [eqn: rho_xy = rho_12].

Given capital adequacy rate([eqn: k]), suppose the following three conditions hold.

(i) [eqn: k = K over {B_0 + (1-lambda) B_1 }]

(ii) [eqn: delta lambda B_0 + delta (1-lambda)(gamma+theta)(B_0 +B_1 ) = 1+K-B_0 - (1-lambda)B_1]

(iii) [eqn: c_1^1 < w] and [eqn: (1-lambda)(1-gamma)(w-c_1^1 ) = [lambda+(1-lambda)gamma]c_1^0]

(iv) [eqn: p = gamma] and [eqn: q = {(1-lambda) gamma} over {lambda+(1-lambda)gamma} (1-gamma)].

(v) [eqn: mu_1 = mu_2] i.e. the average productivity does not change overtime, and [eqn: mu_x = mu_1]  i.e. the bank knows the average productivity when they forecast the firms credit.

Equilibrium dynamics

period 1

At the beginning of the period, bank gives [eqn: (1-lambda)B_1] of loan to firms with credit rating higher than [eqn: under{x}], and buys government bond with the rest of the fund [eqn: 1+K-B_0 -(1-lambda)B_1]. The amount of loan to firms is [eqn: B_0 +(1-lambda)B_1]. By condition (i), the bank is satisfying the capital requirement. But 100[eqn: lambda] percent of the firms become bankrupt.

At the end, 100[eqn: gamma] percent of the firms cannot pay even wages with their output, and 100[eqn: theta] pay wages but cannot pay the bank and [eqn: lambda B_0 + (1-lambda)(gamma+theta)B_1] of loan turns bad. The bank has to place reserve for the non performing loan and it is already holding the appropriate amount by condition (ii).

Consumers form the expected probabilities [eqn: p] and [eqn: q]. If condition (iv) is true, consumers have the correct expected probabilities as explained below. Those who are paid consume [eqn: c_1^1] in equation (2). And the unemployed consume [eqn: c_1^0] in equation (2') withdrawing some of their deposit. However, condition (iii) above means that there is no net change in total amount of bank deposit. Those who are paid save exactly the same amount of the unpiad's withdrawal.

At the beginning of period 2, the bank has the same amount of fund to be reshuffled by the condition (iii). Condition (v) and the assumption that the bank uses the same credit ratings imply the bank gives loan to the same firms as in the previous period. At the end of the period, the proportions of firms which are unable to pay wage and unable to liquidate the bank loan are the same by the law of large number.

The unemployment rate is [eqn: lambda+(1-lambda)gamma] in both period. The proportion [eqn: lambda] of workers keep unemployed in both period. Among the rest of workers, but [eqn: 100(1-lambda)] percent of workers are paid, but [eqn: 100 lambda] are not. Therefore the probability that the employed at t=1 become unemployed at t=2 is [eqn: gamma] and the possibility that the unemployed become employed is [eqn: {(1-lambda) gamma} over {lambda+(1-lambda)gamma} (1-gamma)]. It is the  multiple of two probabilities. One is the probability that one of the unemployed belongs to a firm which joined the production but unable to pay wage by a bad shock. The other is that the firm produces output large enough to pay wage. Condition (iv) means that consumers have the correct probabilities. Therfore the economy is in equilibrium.

In this equilibrium, the bank portfolio at t=1 and t=2 are the same and expected consumption level at t=2 is the same as that of t=1. In addition, unemployment (unpaid) rate, firm bankruptcy rate and the rate of bad loan are the same in both period. This is why we call it similar to the steady state

Consumption levels are as given in equation (2) or (2') according to the employment status in period 1. Realized consumption at t=2 is different from that of t=1 depending on their employment status at t=2. Entrepreneurs' (expected) profit is the realized output minus wage and interest payment. The banker's profit at t=1 is

[eqn: pi = r_l (1-lambda)[1-(gamma+theta)](B_0 +B_1 ) + int_w^{w+r_l B_0 +R_l B_1} (y-w)h(y`|x>=under{x})dy]

[eqn: + r [delta lambda B_0 + delta (1-lambda)(gamma+theta)B_1 ] - r_d]

The first term is the interest received from the good firms and the second income collected from the bad firms. The third is the retirn from the safe asset and the last deposit interest payment.

(2) Dynamic effect of the increase of capital adequacy rate

Suppose the economy is in steady state and at time t government increases capital adequacy rate to [eqn: k' `` (k<k')].

When the banker faces the new regulation, he/she has to decide whether it is better to issue new equity losing the share or to reduce the risky asset (loan to the firms in this model) getting less profit.

Result 1

If the decision of the bank is made by the incumbent owners, the bank has an incentive to reduce the risky loan instead of issuing the new equity when they face higher capital requirement.

[Sketch of proof]  (Formal proof is to be provided later)

Instead of the formal proof of the result, here, we present a stripped down version of the above model. Let's denote the bank accounting identity as [eqn: K+D = L+B], where K denotes bank capital, D the deposit, L the risky loan, and B the risk-free government bond. Initial capital regulation is satisfied, i.e. [eqn: K over L = k]. Profit of the banker who owns all the equity is [eqn: pi = r_l L + r B - r_d D]. Now, there is an increase in capital adequacy rate to [eqn: k ' (k<k')]. Let's use [eqn: Delta] to denote the equity issue when the bank decides to meet the requirement by issuing the new equity. Since this equities must be acquired by consumers in this model, deposit has to be reduced by the same amount. Let A denote the amount of risky loan to be reduced when the bank decides to meet the new requirement by decreasing the loan. Then the following holds,

[eqn: K+Delta over L = K over L-A = k'].

Profit returned to the incumbent shareholder is denoted by [eqn: pi^E] when they issue new equity, and by [eqn: pi^L] when they reduce risky loan.

Suppose there is no default. They become

[eqn: pi^E = [r_l L + r B - r_d (D-Delta)] ( K over {K+Delta})]

[eqn: pi^L = r_l (L-A) + r (A+B) - r_d D].

When they issue equity, they lose some of the profit share, but total profit increases by the decrease of deposit interest payment. If they use loan reduction to keep all profit share, they have less profit by the amount of reduced loan times the interest spread.

With some manipulation, we can easily get,

[eqn: pi^E - pi^L = (1-{k over k'}) (K+D)(r_d - r)].

The sign is the same as that of the difference between the deposit rate and risk-free rate. If we assume that the deposit rate is equal to the risk-free rate, the incumbent banker gets the same profit regardless of how to meet the requirement.

Now suppose that the constant amount of loans are made to continuum of firms lined from 0 to 1. Suppose that [eqn: lambda] of firms from the bottom go bankrupt. Then,

[eqn: pi '^E = [r_l (1-lambda) L + r B - r_d (D-Delta)] ( K over {K+Delta})]

When the bank reduces loans to firms, it should cut the loan to low quality firms and reduce the default rate. By the decrease of the total loan, the bank can lend to only [eqn: {L-A over L} = k over k'] of good firms. If [eqn: {k over k'} > 1-lambda], the bank is able to make loan to good firms and some bad firms. Then the profit becomes,

[eqn: pi '_1^L = r_l (1-lambda) L + r (A+B) - r_d D].

In this case, the first terms of two profit equations are equal. However, in no default case, [eqn: pi ^E = pi^L] is true where first term of [eqn: pi^E , ~r_l L,] is greater than that of [eqn: pi^L , ~r_l (L-A).] Therefore, [eqn: pi ' ^E < pi '_1 ^L] must be true.

If [eqn: {k over k'} < 1-lambda], the bank lends to only good firms and no default happens. This case, the profit is

[eqn: pi '_2^L = r_l (L-A) + r (A+B) - r_d D]

which is equal to the profit in no default case, i.e. [eqn: pi '_2^L = pi^L]. However, [eqn: pi '^E < pi^E] holds. Again, [eqn: pi ' ^E < pi '_2^L] is true. In both case, loan reduction returns higher profit to the incumbent shareholder than new equity issue.  ■

Result 2

When the government increases the capital adequacy rate at the beginning of period 1 (or end of period 0) and if the bank meets new requirement by reducing risky loan, continuing loan contraction is plausible.

[Sketch]

To meet the new regulation, the bank has to reduce the risky loan to [eqn: B_0 + (1-lambda')B_1] where [eqn: lambda'] is defined so that [eqn: K over {B_0 + (1-lambda')B_1} = k']. Let's define [eqn: under{x}'] as the level of credit rating x such that [eqn: lambda' = int_0^{under{x}'} phi(x) dx], i.e. [eqn: under{x}'] is the lowest level of x among the firms which receives loan from the bank.

At the beginning of period 1, bank loan decreases by [eqn: (lambda'-lambda)B_1], increasing the risk-free bond by the same amount. At the end of the period, among the firms which are able to get working capital, [eqn: gamma'] of firms are not able to pay wage, and [eqn: theta'] of them cannot pay the bank where

[eqn: gamma' = int_0^w h(y``| x>= under{x}' ) dy],

[eqn: theta' = int_w^{w+r_l B_0 + R_l B_1} h(y``| x>= under{x}' ) dy].

The amount of bad loan is [eqn: lambda' B_0 + (1-lambda')(theta'+gamma')(B_0 +B_1 )] and [eqn: 100 delta] percent of the amount has to be put aside as loan loss reserve.

The unemployment rate changes to [eqn: lambda'+(1-lambda')gamma'].  By the same story as in the reference case, the probability that an employed worker become unemployed([eqn: p']) becomes [eqn: gamma'] and [eqn: q'] becomes [eqn: {(1-lambda')gamma'} over {lambda' + (1-lambda') gamma'} (1-gamma')]. Since the employed has less probability to be unemployed next period, their consumption([eqn: c'_1^1] ) is greater than [eqn: c_1^1]. The consumption of the unemployed ([eqn: c'_1^0]) is greater or less than [eqn: c_1^0] if [eqn: q'] is greater or less than [eqn: q].

If the unemployment rate [eqn: lambda'+(1-lambda')gamma'] is greater than [eqn: lambda+(1-lambda)gamma] which is more likely because of the credit contraction, the equality in (iii) does not hold. The amount of new deposit by the employed [eqn: (1-lambda')(1-gamma')(w-c'_1^1 )] is likely to be less than the withdrawal of the unemployed. Then there is net ouflow of bank deposit, leaving the bank less fund to be invested. Let G denote the amount of the decreased deposit. In addition, above the amount of loan loss reserve has increased as mentioned. Therefore the bank may not be able to make the same amount of loan in period 2 and there is a further loan contraction.

Result 3

The amount of the shrinkage of loan becomes bigger when the overall economic condition turns bad in the sense that the average productivity decreases or the volatility of the output increases.

[Sketch]

The amount of the decreased loan in period 1 depends on the relative change of capital adequacy rate [eqn: k] and [eqn: k']. However, when the average productivity decreases or the volatility of the output increases, the unemployment rate and bad loan increases. Then the amount of deposit withdrawal and loan loss reserve increases, and the bank fund to be lent to firms decreases. Therefore  loan contraction is bigger.

III. conclusion and policy implication

With simple dynamic model with bank, we try to show that the increase in bank capital adequacy rate may cause credit contraction. [Formal proof is not given yet, though.] If this is true, the enforcement of bank regulation has to be imposed with caution. Especially when the economy is in a bad state, the increased bank capital adequacy rate may cause credit crunch.

References

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