Financial intermediaries lend funds obtained from households to non-…nancial …rms. In
addition, …nancial intermediaries in this model are meant to capture the entire banking
sector, i.e., investment banks as well as commercial banks (Gertler and Karadi, 2011).93 Let
Sjt be the quantity of …nancial claims on non-…nancial …rms that the intermediary holds; Qbt the relative price of each claim;Njtb the net worth that intermediary j has at the end of period t; and Bjt+1the amount of deposits the intermediary obtains from households. The …nancial intermediary’s balance sheet is then given by
QbtSjt =Njtb +Bjt+1 (3.37)
Household deposits with the intermediary at timetpay the non-contingent real gross return Rt+1att+1. And the intermediary earns the stochastic returnRkt+1on the assets over this period. Both Rkt+1 and Rt+1will be determined endogenously. Then the intermediary’s net worth is given by
Njtb+1 = Rkt+1QtbSjt Rt+1Bjt+1 (3.38)
= (Rkt+1 Rt+1)QtbSjt+Rt+1Njtb (3.39)
Any growth in net worth above the riskless return depends on the premiumRkt+1 Rt+1 the intermediary earns on his assets, as well as his total assets,QbtSjt. Since the intermediary will not fund assets with a discounted return less than the discounted cost of borrowing,
for the intermediary to operate in periodt, this constraint can be expressed by
Et i t;t+1+i(Rkt+1+i Rt+1+i) 0 (3.40)
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where i t;t+i is the stochastic discount factor which the intermediary at t applies to earnings att+iand i 0. With perfect capital markets, the relation always holds with equality. So the risk adjusted premium is zero. With imperfect capital markets, however,
the premium may be positive due to limits on the intermidiary’s ability to obtain funds.
So long as the intermediary can earn a risk adjusted return that is greater than or equal to
the return the household can earn on its deposits, it pays for the banker to keep building
assets until exiting the industry. Thus, the objective function of the intermediary can be
given by Vjt = maxEt 1 X i=0 (1 ) i i+1 t;t+1+i Njtb+1+i = maxEt 1 X i=0 (1 ) i i+1 t;t+1+i h (Rkt+1+i Rt+1+i)QtbSjt+Rt+1+iNjtb+i i (3.41)
whereVjt is the expected terminal wealth of the bankj. To the extent the discounted risk adjusted premium in any period, i t;t+i (Rkt+1+i Rt+1+i), is positive, the intermediary will want to expand its assets inde…nitely by borrowing additional funds from households.
Following Gertler and Karadi (2011), we introduce the moral hazard problem between the
bankers and households. We assume that after collecting deposits, the banker can choose to
divert some of funds for his own consumption. Speci…cally, the banker can divert fraction
0 b 1 of funds. In this case, the depositor can force the intermediary into bankrupcy
and recover the remaining fraction1 b of funds. This implies that lenders are willing to
supply funds to the banker if the following incentive constraint is satis…ed:
In the above inequalityVjt is what the banker would lose by diverting a fraction of funds and bQtSjt means the gain from doing so.
From the equation (3.41) we can expressVjt as follows:
Vjt= tQbtSjt+ tNjtb (3.43)
with
t=Etf(1 ) t;t+i(Rkt+1 Rt+1) + t;t+i xt;t+1 t+1g (3.44)
t=Et (1 ) t;t+iRt+1 + t;t+i zt;t+1 t+1 (3.45)
where xt;t+1 Qbt+1Sjt+1=QbtSjt, is the gross growth rate in assets between t and t+ 1, and zt;t+1 Njtb+1=Njtb is the gross growth rate of net worth. The variable t means the expected discounted marginal gain to the banker of expanding assets QbtSjt by a unit, holdingNjtb constant, and while thas the interpretation of the expected discounted value of having another unit ofNjtb, holding the assetsQbtSjt constant. Using (3.43) we can show the incentive constraints (3.42) as
tQbtSjt+ tNjtb bQbtSjt (3.46)
If this constraint binds, we can obtain following relationship between the bank’s assets and
the bank’s net worth:
QbtSjt = b t t
where tis the banker’s leverage ratio.94 Holding constant net worth, expanding the assets
raises the banker’s incentive to divert funds. The equation (3.47) limits the intermediaries’
leverage ratio to the point where the banker’s incentive to cheat is exactly balanced by the
cost. Therefore, the moral hazard problem yields an endogenous capital constraint on the
intermediary’s ability to expand the assets.
Combining (3.47) with (3.39) allows to express the evolution of the banker’s net worth as
Njtb+1 = [(Rkt+1 Rt+1) t+Rt+1 ]Njtb (3.48)
In addition, it follows that
zt;t+1=Njtb+1=Njtb = (Rkt+1 Rt+1) t+Rt+1 (3.49)
xt;t+1=Qbt+1Sjt+1=QbtSjt = t+1= t Njtb+1=Njtb = t+1= t zt;t+1 (3.50)
All the components of t depend only on economy wide variables. This allows for total
aggregation across the intermediaries, obtaining
QbtSt= tNtb (3.51)
whereQb
tStdenotes the aggregate quantity of intermediary assets andNtb re‡ects aggregate intermediary net worth. In the general equilibrium of our model, variation in Ntb, will induce ‡uctuations in overall asset demand by intermediaries.95
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An interpretation of this condition is as follows (Gertler and Karadi, 2011). With frictionless competi- tive capital markets, intermediaries will expand borrowing to the point where rates of return will adjust to ensure tis zero. However, the moral hazard problem between the banker and household may place limits
on this arbitrage. Speci…cally, the intermediary’s loans are constrained by its net worth.
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This mechanism of …nancial frictions at the bank level also arises if there is a shock that a¤ects the bank’s net worth. If a shock has a negative impact on the net worth, the bank has to decrease the loans
Nb
t consists of the net worth of existing intermediaries, Netb;and the one of new bankers, Nb
nt.
Ntb =Netb +Nntb (3.52)
Netb is given by
Netb = (Rkt Rt) t 1+Rt Ntb 1 (3.53)
because bankers in business in period t 1 did not die at t with the ratio : We also suppose that the funds the household gives its new banker equal to a small fraction of
the value of assets that exiting bankers had intermediated in their …nal operating period.
Assuming that the exit probability is i.i.d., the …nal period assets of exiting bankers att is(1 )QbtSt 1. Thus, we suppose that each period the household gives =(1 ) of this value to its entering bankers.96 Accordingly, in the aggregate,
Nntb = QbtSt 1 (3.54)
Combining equations (3.53) and (3.54) gives the equation of motion for Ntb.
Ntb= (Rkt Rt) t 1+Rt Ntb 1+ QbtSt 1 (3.55)
3.3.3.3. Intermediate goods …rms
The optimisation problems of intermediate goods …rms follow the above two models, de-
to the …rms. The leverage ration expands the contraction of the loans. This fall of the loans will lower investment, thus output.
scribed by equations (3.12) - (3.15), (3.26).
The …rm …nances its capital acquisition by obtaining funds from intermediaries. To
acquire the funds, the …rm issuesStclaims equal to the number of units of capital acquired Kt+1 and prices each claim at the price of a unit of capital Qbt. That is, QbtKt+1 is the value of capital acquired andQbtSt is the value of claims against this capital as follows:
QbtKt+1 =QbtSt (3.56)