Optimal loan supply under regulation

4.3.2.3 Aggregate loan supply without regulation

Summing over all banks in (4.3) yields aggregate loan supply by all banks with-out regulation L^U.

L^U =aF(a d₀) +

b dZ 0

a d0

(ei+d₀)f(ei)dei+b[1 F (b d₀)] (4.4)

where

a = M B

r_I

1 1

b = M B

r_I

1 1

where a is the optimal loan supply to …rms by banks which are interbank bor-rowers,F(a d0)is the proportion of interbank borrowers,b is the optimal loan supply to …rms by banks which lend in the interbank market and1 F(b d₀)is their proportion. The proportion of banks that are interbank borrowers, lenders or not active in the interbank market is entirely determined by banks’holdings of capital because debt is assumed to be the same for all banks. As stated in the beginning, these are simplifying assumptions to be able to exclusively focus on the role of variable risk-weights for the cyclical behaviour of aggregate loan supply. Figure 4.1 illustrates aggregate loan supply to …rms across banks with di¤erent capital holdings. In the upper part, banks with capital below a d₀ are interbank borrowers because their available fundsei+d0 fall short of desired lendingaat the re…nancing costr_I. Interbank borrowing for each of these banks is the di¤ence betweena and e_i+d₀, depicted by the triangularl_i . Banks with capital above b d₀ are interbank lenders because their available funds exceed desired lending b at the opportunity cost r_I. Interbank loans for each of these banks are the di¤erence between e_i +d₀ and b, depicted by the triangular l⁺_i . Banks with capital abovea d₀ but belowb d₀ are not active in the interbank market and lend the sum of their capital and debt,e_i+d₀, to …rms. In the lower part a hypothetical cumulative distribution function of bank capital is graphed, from which the proportions of each type of bank can be read o¤.

Figure 4.1: Aggregate loan supply to …rms across banks without a bank capital constraint

Risk-weightsw depend on aggregate risk with w=w( ) w⁰ = @w( )

@ <0

Risk-weights depend only on aggregate risk, more speci…cally the probability that production in each sector of the economy is positive. In practice, the risk-weights also depend on borrower speci…c characteristics. We abstract from these, however, to focus on the presumption that as the economy experiences a downswing or recession all types of borrowers appear riskier than before and receive a higher risk-weight. Moreover, the idiosyncratic risk is diversi…able by banks. Under the Basel I Accord risk-weights were constant, whereas the innovation in the revised framework, Basel II, is to make risk-weights dependent on borrower riskiness, part of which is aggregate risk. Precisely this innovation sparked the debate about potential procyclical aggregate credit supply. The capital constraint can be written as

ei cw( )Li

where c is some value …xed by the regulator. From bank pro…t maximisation one can show that the capital constraint is binding if

e_i < ~h (4.5)

~h cw( ) M B

~ r_I

1 1

=cw( ) ~a

Variables with a tilde denote variables in the scenario with a capital constraint.

The introduction of a capital constraint changes the equilibrium interbank rate and therefore the opportunity cost of lending to …rms. Condition (4.5) says that banks with capital below~hare constrained from lending as much as they would like to. In addition the budget constraint is binding too. Also it is necessary to assume that there are some interbank borrowers left who are willing to absorb funds in the interbank market. Therefore we look at the case where the criti-cal value ~h does not exceed the threshold at which a bank ceases to borrow in the interbank market. As a consequence, in the model, only interbank borrow-ers are potentially constrained and interbank lendborrow-ers are always unconstrained.

Optimal loan supply by constrained banks L^c_i is then given by

L^c_i = _{cw( )}^ei if e_i <~h (4.6)

Loan supply is determined by the capital constraint and is equal to a multiple

1

cw( ) of capital. Furthermore, the cyclical properties depend on the sensitivity of risk-weightswwith respect to aggregate risk. As rises risk-weights fall and loan supply rises. The more risk-weights react to changes in the stronger is the e¤ect on loan supply by constrained banks. Unconstrained banks under regulation behave essentially as in the unregulated case, except that under regulation they might face a di¤erent interbank rate.

4.3.3.2 Aggregate loan supply under regulation

Summing over all constrained banks as well as the remaining unconstrained banks yields the aggregate loan supply under regulation L^R.

L^R = 1

cw( )

~h

Z

0

e_if(e_i)de_i+ ~ah

F(~a d₀) F ~h i

+

~b dZ 0

~ a d0

(e_i +d₀)f(e_i)de_i+ ~bh

1 F ~b d₀ i

The …rst term is loan supply to …rms by constrained banks with capital below~h, the second term is loan supply to …rms by the remaining unconstrained interbank borrowers. The third and fourth term are lending by banks not active in the interbank market and by interbank lenders. Figure 4.2 illustrates the situation after the introduction of bank capital regulation.The fairly steep upward sloping line is loan supply to …rms by constrained banks⁹. Banks with capital below or equal to ~h supply loans according to a multiple _{cw( )}¹ of their capital. Among these banks some of them are interbank lenders because their (constrained) lending to …rms is lower than their available funds e_i +d₀. Conversely, some constrained banks are interbank borrowers because their (constrained) lending exceeds available funds e_i + d₀. Finally, by assumption a part of interbank borrowers remain unconstrained. These banks have capital above h~ but below

~a. Note that the introduction of bank capital regulation changes the proportion of interbank borrowers and lenders. Therefore the equilibrium interbank rate is changed under a capital constraint. Comparing …gures 4.1 and 4.2 one can notice that desired interbank borrowing is smaller under regulation than without regulation, given the same distribution function for bank capital¹⁰. Therefore the interbank rate should be lower with a capital constraint,r~_I < r_I. Intuitively the

9Its slope is larger than1 for reasonable values for c andw( ). Under the Basel Accord c= 0:08such that for risk weights up to12:5the slope is larger than one.

10For now the distribution of bank capital is held …xed. In a later section the e¤ects of a changing distribution of bank capital will be discussed.

Figure 4.2: Aggregate loan supply to …rms with a bank capital constraint

reason is that some banks that originally borrowed from the interbank market are forced by the capital constraint to borrow less or even lend funds in the interbank market. As a result lending to …rms by interbank borrowers and lenders is larger under regulation,a > a~ and ~b > b.