D ESARROLLO EN EL ÁMBITO MUNDIAL DEL SISTEMA CONCESIONAL

This section is similar to Gertler and Karadi (2011) (hereby GK2011). Having

one more financing instrument makes up for few differences.In this Appendix I will:

(a) set up a one-to-one relationship between leverage and excess return on assets for any fixed moral hazard parameterΘB,

(b) derive expression for these excess returns, and

(c) show that banks’ choice on leverage and excess returns does not depend on bank sepcific factors, therefore allowing for aggregation.

The value of the bank is homogenous to degree one in net worth and given the linear relationship in the balance sheet equation 3.40, then VB

i,t should be homogenous of degree one inBi,t, ABi,t as well asLi,t andSi,t.

V_iB_,t(Li,t,Bi,t,Si,t) =νLi,tLi,t+νsi,tQtSi,t−νBi,tBi,t (B.59)

where QtSi,t =κAi,t and Li,t = (1−κ)AB_i,t. νLt, νst and νBt are the marginal values of respective asset as of end of period. Using balance sheet equation

3.38 the value function is:

V_iB_,t(Li,t,Ni,t,Si,t) =µLi,tLi,t+µsi,tQtSi,t+νBi,tNBi,t (B.60)

- the incentive constraint eq. 3.42: VB

i,t>ΘAi,t - equationB.60: VB

i,t =µLi,tLi,t+µsi,tQtSi,t+νBi,tNBi,t The Lagrangean`=V_iB_,t+λB_t(V_iB_,t−ΘB(AB_i,t))becomes:

`=µL_i,tφB_i,t(1−κ) +µs_i,tφB_i,tκ+νB_i,t(1+λB_i,t) −λB_i,tΘ_BφB_i,t (B.61)

λB_i,t > 0if the constraint (IC) binds, orλB_i,t=0if it does not.

First order conditions with respect (φi,t, κ and λi,t KT1 and KT2) when IC constraint binds are:

foc1. φB i,t: 0= (1+λB_i,t)(µL_i,t(1−κ) + (µL_i,tκ)) −λB_i,tΘB (B.62) foc2. λ_i,t: 0=µL_i,tφB_i,t(1−κ) +µs_i,tφB_i,tκ+νB_i,t−Θ_BφB_i,t (B.63) KT 2: λB_t>0 (B.64)

My interest is when the IC binds λB_i,t > 0, the KT1 condition B.63 leads to

an expression relating leverage to bank premium:

φB_i,t= ν B i,t

ΘB−µL_i,t(1−κ) −κµs_i,t

where φB_i,t = A B i,t

NB_i,t. For non-binding IC, i.e. λ B

i,t = 0, then the 1−st FOC implies µL_i,t=0. Hence, (1−κ)µL_i,t=max " 0,ΘB− νB_i,t φB_i,t −κµ s i,t # (B.66)

Now I need to relate the µL_i,t and µS_i,t to respective returns of Li,t and Si,t financing instruments. In terms of leverage equationB.60is:

V_iB_,t =µL_i,t(1−κ)φB_i,t+κµ_is_,tφB_i,t+ν_iB_,tNB_i,t (B.67)

whereµL_i,t=νL_i,t−νB_i,t is the excess value of all loans over cost of depositsR_t+1

µs_i,t=νs_i,t−νB_i,tis the excess value of S-type financing above risk free return Rt+1.

Rewrite equation 3.41:

V_iB_,t=Et

P_∞

τ=1(1−σB)στ_B−1Λt,t+τNB_i,t+τ

in recursive form (shown in subsection B.1.7)

V_iB_,t=EtΛt,t+1

(1−σB)NBi,t+1+σBViB,t+1

(B.68)

whereNB_i,t+1 is defined as in the income statement equation3.40.

Plug equationB.67shifted att+1into equationB.68to get:

V_iB_,t=EtΛt,t+1Ωi,t+1NBi,t+1 (B.69)

where,

Ωi,t+1= (1−σB) +σB

is the shadow value of a unit of net worth and can be re-written making use of equation (B.65) (when IC binds):

Ωi,t+1= (1−σB) +σBΘBφBi,t+1 (B.70) or equivalently : V_iB_,t=EtΛt,t+1 h (1−σB) +σB (1−κ)µL_i,t+1φ_iB_,t+1+κµs_i,t+1φB_i,t+1+νB_i,t+1iNB_i,t+1 Plug equation3.40: NB i,t+1 = (RLt+1−Rt+1)Li,t+ (Rst+1−Rt+1)QtSi,t+ Rt+1NBi,t into equationB.69above yields:

V_iB_,t=EtΛt,t+1Ωi,t+1 h RL_t+1−Rt+1 Li,t+ Rst+1−Rt+1 QtSi,t+Rt+1NBi,t i (B.71)

Finally, plugging Equation B.60; ₍VB

i,t=µLi,tLi,t+µsi,tQtSi,t+νiB,tNBi,t)

into equation (B.71) leads to three expressions forµL

i,t,µsi,tand νBi,t: µL_i,t= (νL_i,t−νB_i,t) =EtΛt,t+1Ωi,t+1 RL_t+1−Rt+1 (B.72) µs_i,t= (νs_i,t−νB_i,t) =EtΛt,t+1Ωi,t+1 Rst+1−Rt+1 (B.73) νB_i,t=EtΛt,t+1Ωi,t+1Rt+1 (B.74)

EquationsB.65and B.72, B.73, B.74 complete the banker’s solution and de-

termineφB_i,t and µL_i,t,µs_i,t, νB_i,tas a function of ΘB, RB_t+1−Rt+1, Rs_t+1−Rt+1 and Rt+1, which do not depend on bank specific factors. They are either exogenous

to banks or depend on economy wide variables. Therefore, they are common across all banks.

Since all banks chose the same leverage ratio and observe the same excess returns on their assets φB_t and µL_t, µs_t, νB_t, the bank balance sheet and leverage can be aggregated across banks.

NB_t +Bt =ABt =Lt+QtSt (B.75) φB_t = At NB_t = QtKt−NEt NB_t (B. 76)

where NE_t is firms’ net worth and QtKt total project cost. Aggregate bank net worth NB_t evolves as the sum of:

- net worth of old banks surviving from last period,NB_o,t+1 =σBNB_t+1,

- and the net worth of new onesNB_new,t+1=ξBABtRBt+1, since on aggregate new banks receive a transfer of ξB

1−σB from absconding ones

4

.

Aggregate net worth of banks is:

NB_t+1 =NB_new,t+1+NB_o,t+1

whereσBis the probability of the bank not absconding (surviving) next period. Evolution of net worth of banks not absconding at time ’t+1’ is obtained using equationB.88:

NB_o,t+1=σB RL_t+1Lt+Rs_t+1QtSt−Rt+1Bt

In addition a transfer ξb

1−σb of those 1−σb absconding transfer some net worth to new banks:

4

NB_new,t+1=ξBRBt+1ABt

Then aggregate net worth is:

NB_t+1 =σB

RL_t+1Lt+R_ts₊₁QtSt−Rt+1Bt

+ξBRBt+1ABt (B.77)

Intuitively, aggregate consumption of exiting banks is5

:

ΠB_t+1 = (1−σB)(RBt+1ABt −Rt+1Bt) −ξBRtB+1ABt (B.78)

For the sake of reference, the average return on securities Rs_t+1 is defined as: RS_t+1QtSt =EtRkt+1QtKt 1−Γ_ψ t+1 Xeq_t (B.79)

For the sake of reference I define the average return RB_t+1 on total bank assetsAB_t as: RB_t+1=RL_t+1Lt AB_t +R s t+1 QtSt AB_t (B. 80) 5 Since ₁₋ξ_σ

B portion of net worth is transferred by a fraction1−σB of exiting bankers to new

ones, then1−₁₋ξ_σ