A Supplemental material

A complete appendix with supplemental material is available online.

B Proofs

B.1 Lemma 1

Note that (2) and (5) implyλ^b=U_C^b

1−^β_β^b

. Using the latter, (6), (11), and the definition of βe_e, we get λ^e= ^U_R^CeL

1−^β_e^e

βe

.

If Supposeβ_b < β and β_e <βe_e. Then λ^b > 0 and λ^e >0, which implies that (4) and (9) are binding.

Section D.1provides a closed-form sequential solution for a unique steady state, where we set τ^D =τ^K = τ^L =τ^N = 0. As shown there, the binding collateral constraint is used to solve for L >0 conditional on K >0. The binding leverage constraint is then used to solve forD >0 conditional onL >0.

Only if Suppose there exists a unique steady state with D >0 andL >0. Sinceλ^b≥0 andλ^e≥0, we must haveβb ≤β andβe≤βee. If βb =β, then (5) is equivalent to λ^b = 0. Moreover, the complementary slackness conditions (7) are automatically satisfied. SinceL >0 by the premise, any D∈(0,(1−κ)L] can be part of an unstable steady state, which contradicts uniqueness. It follows that βb < β. An identical argument applied to (9) and (11) demonstrates that we must have βe<βee.

B.2 Lemma 2

Bankers Multiply both sides of (5) byD_t, multiply both sides of (6) byL_t, and subtract the former from the latter:

U_C,t^b (Lt−D_t) =β_bE_t[U_C,t+1^b (R^L_t+1L_t−R_tD_t)] +λ^b_t[(1−κ_t)Lt−D_t].

Using (3) and (7),

U_C,t^b (Lt−Dt) =βbE_t(U_C,t+1^b C_t+1^b ) +βbE_t[U_C,t+1^b (Lt+1−Dt+1)].

Iterating this equation forward, we obtain Lt−Dt= 1

U_C,t^b X∞ s=1

β_b^sE_t(U_C,t+s^b C_t+s^b ).

Entrepreneurs The argument is symmetric to the case of bankers. Multiply (11) byL_tand (12) byK_t, subtract the former from the latter and use (8), (10), and (13), noting thatF is Cobb—Douglas, to obtain

U_C,t^e (QtK_t−L_t) =β_eE_t(U_C,t+1^e C_t+1^e ) +β_eE_t[U_C,t+1^e (Qt+1Kt+1−Lt+1)].

Iterating forward, we get

Q_tK_t−L_t= 1 U_C,t^e

X∞ s=1

β_e^sE_t(U_C,t+s^e C_t+s^e ).

B.3 Lemma 3

The definition ofW_tⁱ implies β β−βi

(W_tⁱ−V_tⁱ) =E_t X∞ s=1

β^sV_t+sⁱ

!

=βE_t(V_t+1ⁱ ) +βE_t

"

E_t+1 X∞ s=1

β^sV_t+1+sⁱ

!#

=βE_t(V_t+1ⁱ ) +βE_t β

β−βi

(W_t+1ⁱ −V_t+1ⁱ )

. Hence,

W_tⁱ =V_tⁱ−βiE_t(V_t+1ⁱ ) +βE_t(W_t+1ⁱ )

=U_tⁱ+βE_t(W_t+1ⁱ )

=E_t X∞ s=0

β^sU_t+sⁱ

! .

B.4 Proposition 1

Defineλ^L_t ≡λ^L_1,t+λ^L_2,t[(1−κ_t)Lt−D_t]. The FOCs are C_t^b: 0 =ωbU_C,t^b −λ^Y_t −λ^C_t +λ^L_tU_CC,t^b Lt−1N(t)

β [λ^L_t−1βb(U_CC,t^b R^L_tLt−1+U_C,t^b ) +λ^e_t−1], C_t^e: 0 =ω_eU_C,t^e −λ^Y_t −λ^C_t,

C_t^w: 0 =ω_wU_C,t^w −λ^Y_t −λ^C_tW_C,tN_t−[λ^L_tβ_bE_t(U_C,t+1^b ) +βE_t(λ^C_t+1) +λ^e_t]R1,tU_CC,t^w D_t

−1^N(t)

β [λ^L_t−1β_bE_t−1(U_C,t^b ) +βE_t−1(λ^C_t ) +λ^e_t−1]R2,t−1U_CC,t^w D_t−1,

D_t: 0≥ −λ^b_t−λ^L_2,t[U_C,t^b −β_bE_t(U_C,t+1^b R^L_t+1)]Lt+λ^C_t −[λ^L_tβ_bE_t(U_C,t+1^b ) +βE_t(λ^C_t+1) +λ^e_t]Rt

+1N(t)

β (λ^L_t−1β_bU_C,t^b +λ^e_t−1), equality ifD_t>0,

Kt: 0 =−λ^C_t{Q2,t[Kt−(1−δ)ξtKt−1] +Qt}+λ^e_tmtE_t[(Q1,t+1Kt+Qt+1)ξt+1]−λ^Y_tI2,t

+βE_t[(λ^C_t+1+λ^Y_t+1)At+1FK,t+1ξt+1+λ^C_t+1{Qt+1(1−δ)ξt+1−Q1,t+1[Kt+1−(1−δ)ξt+1K_t]}

−λ^Y_t+1I1,t+1] + 1N(t)

β λ^e_t−1mt−1Q2,tξtKt−1,

L_t: 0 ={λ^b_t+λ^L_2,t[U_C,t^b −β_bE_t(U_C,t+1^b R^L_t+1)]Lt}(1−κ_t) +λ^L_tU_C,t^b −1N(t)

β (λ^L_t−1β_bU_C,t^b +λ^e_t−1), Nt: 0 =ωwU_N,t^w + (λ^C_t +λ^Y_t )AtFN,t−[λ^L_tβbE_t(U_C,t+1^b ) +βE_t(λ^C_t+1) +λ^e_t]R1,tU_CN,t^w Dt

−λ^C_t(WN,tNt+Wt)−1N(t)

β [λ^L_t−1βbE_t−1(U_C,t^b ) +βE_t−1(λ^C_t) +λ^e_t−1]R2,t−1U_CN,t^w Dt−1. The complementary slackness conditions are

0 =λ^b_t[(1−κt)Lt−Dt], λ^b_t ≥0,

0 =λ^L_1,t[U_C,t^b Lt−βbE_t(U_C,t+1^b Bt+1)], Dtλ^L_1,t≥0, 0 =λ^e_t[mtE_t(Qt+1ξt+1)Kt−E_t(Bt+1)], λ^e_t ≥0.

B.4.1 Constrained inefficiency

Follows from inspecting the planner’s analogs of (5) and (10)–(12). Consider them one-by-one.

Deposit supply The FOCs forC_t^b andD_timply U_C,t^b ≤β_bR_tE_t(U_C,t+1^b ) +λ^b_t

ω_b + Ψ^D_t , equality ifD_t>0, where

ω_bΨ^D_t ≡(β−β_b)RtE_t(ωbU_C,t+1^b ) +λ^Y_t −βR_tE_t(λ^Y_t+1) +λ^L_2,t[U_C,t^b −β_bE_t(U_C,t+1^b R^L_t+1)]Lt

−λ^L_t[U_CC,t^b +β_bR_tE_t(U_CC,t+1^b R^L_t+1)]Lt+βR_tE_t(λ^L_t+1U_CC,t+1^b Lt+1) +1N(t)

β λ^L_t−1β_bU_CC,t^b R^L_tLt−1. Loan demand IfD_t>0, the FOCs forC_t^e,D_t, andL_timply

U_C,t^e =βeE_t(U_C,t+1^e R^L_t+1) +λ^e_t

ω_eE_t(R^L_t+1) + Ψ^L_t, where

ω_eΨ^L_t = (β−β_e)E_t(ωeU_C,t+1^e R^L_t+1)−E_t[(βωeU_C,t+1^e +λ^e_t)(R^L_t+1−R_t)] +λ^Y_t −βR_tE_t(λ^Y_t+1)

−λ^L_t

"

U_C,t^b

1−κ_t−βbRtE_t(U_C,t+1^b )

#

+1N(t) β

κt

1−κ_t(λ^L_t−1βbU_C,t^b +λ^e_t−1).

If D_t = 0, we still have L_t > 0, so the FOC for L_t holds. To see this, note that the leverage constraint implies C_t+1^b +Lt+1−Dt+1 ≥ 0, and the inequality is strict ifC_t+1^b > 0. Provided that C_t+1^b > 0 with positive measure, which is guaranteed if bankers are risk averse and the Inada condition holds,D_t=L_t= 0 would contradict the constraint associated withλ^L_1,t. Note that the FOCs forC_t^bandC_t^eyield the following general relationship between the marginal utilities

ωbU_C,t^b =ωeU_C,t^e −λ^L_tU_CC,t^b Lt+1^N(t)

β [λ^L_t−1βb(U_CC,t^b R^L_tLt−1+U_C,t^b ) +λ^e_t−1].

WithD_t= 0, we haveU_C,t^b =β_bE_t(U_C,t+1^b R^L_t+1). The FOC forL_tthen impliesλ^L_tU_C,t^b = ^1N_β^(t)(λ^L_t−1β_bU_C,t^b + λ^e_t−1) att. Combining these results, ifD_t= 0, the wedge satisfies

ω_eΨ^L_t = (βb−β_e)E_t(ωeU_C,t+1^e R^L_t+1) +λ^L_t

U_CC,t^b L_t−U_C,t^b +β²_b

β E_t[(U_CC,t+1^b R^L_t+1L_t+U_C,t+1^b )R^L_t+1]

−βbE_t(λ^L_t+1U_CC,t+1^b Lt+1R^L_t+1)−1^N(t)

β λ^L_t−1βbU_CC,t^b R^L_tLt−1−β−β_b

β λ^e_tE_t(R^L_t+1).

Labor demand The FOCs forC_t^e,C_t^w, andN_tcombined with the definition ofW_timply W_t=A_tF_N,t+ Ψ^N_t ,

where

Ψ^N_t =(ωeU_C,t^e −ω_wU_C,t^w −λ^C_t)AtF_N,t−λ^C_tW_N,tN_t

ωwU_C,t^w +λ^C_t −U_CN,t^w U_CC,t^w

ω_wU_C,t^w −ω_eU_C,t^e +λ^C_t(1−W_C,tN_t) ωwU_C,t^w +λ^C_t .

Capital demand The FOCs forC_t^e andKt imply

That consumption insurance is generally imperfect follows immediately from inspecting the FOCs with respect to C_t^b, C_t^e, and C_t^w. The same applies to partial risk sharing between bankers and entrepreneurs.

Note that the FOCs forD_tandL_timply a steady-state relationshipλ^e=λ^L(β−β_b)U_C^b(C^b). Hence,λ^eand λ^L are either both zero or both positive.

Suppose workers have separable preferences U^w(C^w, N) = u(C^w)−v(N) and λ^e = λ^L = 0. In this

where the second equality is true if the steady-state profits of capital good producers are zero so that the worker’s budget constraint impliesC^w=W N+ (R−1)D. It follows thatωwu^′(C^w) =λ^Y +λ^C if and only if (−C^w)^u_u^′′′(C^(Cww)⁾ = 1 if and only ifu(·) = ln(·).

B.4.3 Indeterminacy and optimal steady state

SectionD.3 shows that the steady state construction reduces to considering two cases,λ^L= 0 andλ^L>0.

Ifλ^L= 0,D must satisfy the rearranged collateral constraint:

C^b+ (R−1)D+ max there is a generally infinite set of solutionsD∈[0,D] for some ¯¯ D >0. Since there is an uncountable infinity of steady states, each such steady state is unstable, and the FCEA is locally indeterminate. Numerical analysis under the baseline calibration demonstrates that each choice ofD yields either a unique solution to a nonlinear system or no solutions, and welfareW is strictly decreasing inD. The latter is related to the problem of finding an optimal steady state.

Consider the planner’s problem with no uncertainty, restricting attention to constant plans. An optimal plan of this sort will define the optimal steady state. In the steady state, R = _β¹, _K^I = Φ⁻¹[1−(1−δ)ξ],

conditional on (C^b, D). The optimal steady state is then a solution to

(C^b,C^e,Cmax^w,D,K,N)

X

i∈I

ωiUⁱ subject to

λ^C: 0 =AF(ξK, N)−Q[1−(1−δ)ξ]K−W(C^w, N)N−(R−1)D−C^b−C^e, λ^e: 0≤mQξK−[C^b+ (R−1)D+L],

λ^Y : 0 =AF(ξK, N)−X

i∈I

Cⁱ− I KK.

Conditional on C^b, L is a strictly increasing function of D, differentiable everywhere except at the kink.

We can assume without loss of generality that the derivative at the kink is an average of the left and right derivatives. Suppose (C^b, C^e, C^w, D, K, N) is optimal, whereD >0. It must satisfy the FOC forD:

0 =−λ^C(R−1)−λ^e

R−1 + ∂L

∂D

.

Note thatR >1,λ^e≥0, and _∂D^∂L >0. If, moreover, λ^C>0, we have−λ^C(R−1)−λ^e R−1 + _∂D^∂L

<0, which is a contradiction. Therefore,D= 0 is optimal.

Intuitively,λ^C must be positive since it is the shadow value of wealth associated with the consolidated budget constraint of bankers and entrepreneurs. Assume separable preferences and combine the FOCs for C^wandN together with the definition ofW to obtain

λ^C= ωwu^′(C^w)(AFN −W) (WCN−1)AFN+W_NN+W.

By definition, WC > 0 and WN > 0. Hence, if N and C^w are less than in the first-best allocation, FN

must be greater andW less; therefore, AFN −W >0. A sufficient—but not necessary—condition for the denominator to be positive isWCN ≥1. If uhas constant relative risk aversion γw >0, as is the case in the quantitative analysis,W_C =−W^u_u^′′′(C^(Cww)⁾ = _C^Wwγ_w. Ifγ_w is large enough, we are done. Alternatively, if γw≈1 andD≈0, thenCw≈W N and WCN ≈1; therefore, (WCN−1)AFN ≈0. SinceWNN+W >0, we then haveλ^C>0.

B.5 Proposition 2

Bankers Note that the form ofT_t^b ensures that (3) is true in equilibrium. The Euler equation for deposits is now

U_C,t^b (1−τ_t^D)≤βbRtE_t(U_C,t+1^b ) +λ^b_t, equality ifDt>0.

Using (6)—which remains unchanged relative to the FCE—to solve forλ^b_t, the Euler equation for deposits can be rearranged as

U_C,t^b ≤βbRtE_t(U_C,t+1^b ) +U_C,t^b −βbE_t(U_C,t+1^b R^L_t+1) 1−κt

+τ_t^DU_C,t^b , equality ifDt>0.

As follows from SectionB.4.1, the right-hand side is equivalent to the one in the FCEA if and only if τ_t^D= 1

U_C,t^b

"

λ^b_t ωb

−U_C,t^b −β_bE_t(U_C,t+1^b R^L_t+1) 1−κt

+ Ψ^D_t

# .

Entrepreneurs The form ofT_t^eguarantees that (8) holds in equilibrium. Without loss of generality, let

λ^e_t

ωe denote the scaled Lagrange multiplier on the collateral constraint. The modified FOCs are (1 +τ_t^N)Wt=AtFN,t,

SectionB.4.1then immediately implies that we must set τ_t^N = −Ψ^N_t

W_t , τ_t^L= Ψ^L_t

U_C,t^e , τ_t^K= −Ψ^K_t U_C,t^e Q_t.

Ramsey equilibrium On the banker’s side, we can use the regulated deposit supply Euler equation to solve forτ_t^b in terms of allocations and prices. The remaining constraints are identical to those faced by the social planner in the definition of an FCEA. Similarly, on the entrepreneur’s side, we can use the regulated demand conditions for labor, loans, and capital to back out the corresponding tax rates τ_t^N, τ_t^L, and τ_t^K. Guessing that the private complementary slackness conditions associated with the collateral constraint are not binding, we are left with the entrepreneur’s budget constraint and the collateral constraint—the same set of constraints as in the FCEA definition. After solving for prices and the investment function as in the FCEA, the complete set of constraints faced by the Ramsey planner is identical to the one in the FCEA definition. Therefore, the FCEA is exactly the allocation that is part of the Ramsey equilibrium. Finally, we can verify that the individual entrepreneur’s complementary slackness conditions are indeed not binding because they are implied by the planner’s analogous complementary slackness conditions.

B.6 Lemma 4 4. An allocation-policy pair is part of a Ramsey equilibrium if—combined with the associated prices and Lagrange multipliers—it constitutes a regulated competitive equilibrium with the maximum level of welfare over all feasible allocation-policy pairs.

Consider a feasible policy {κt, m_t, τ_t^D, τ_t^N, τ_t^L, τ_t^K} ⊂[0,1]²×R⁴ and the corresponding regulated FCE allocation {C_t^b, C_t^e, C_t^w, Dt, Kt, Lt, Nt}. The policy is consistent with the construction in the lemma. If U_C,t^b > βbE_t(U_C,t+1^b R^L_t+1), then (6) implies thatλ^b_t >0, and thus the leverage constraint is binding, which impliesκt= 1−^D_L^t

t; otherwise,κt≥0 combined with the leverage constraint is equivalent toκt∈h

0,1−^D_L^t

t

i. The collateral constraint combined withm_t≤1 is equivalent tom_t∈h _E

t(R^L_t+1)L_t E_t(Q_t+1ξt+1)K_t,1i

. The tax rates are

consistent with the regulated analogs of (5) and (10)–(12). Moreover, as argued in Proposition2, the alloca-tion is feasible for the FCEA problem. SinceDt≤(1−κt)Lt≤LtandE_t(R^L_t+1)Lt≤mtE_t(Qt+1ξt+1)Kt≤ E_t(Qt+1ξt+1)Kt, the allocation is feasible for the relaxed problem.

Conversely, suppose an allocation {C_t^b, C_t^e, C_t^w, Dt, Kt, Lt, Nt} is feasible for the relaxed problem and construct the corresponding policy as described in the lemma. The construction ofκ_tensures that the FCE version of the bank leverage constraint and the private complementary slackness conditions are satisfied.

The construction of m_t guarantees that the FCE version of the collateral constraint is respected. The construction of the tax rates makes sure that the regulated analogs of (5) and (10)–(12) hold. The policy is feasible, that is,{κt, m_t, τ_t^D, τ_t^N, τ_t^L, τ_t^K} ⊂[0,1]²×R⁴. It follows that the allocation and the constructed policy—combined with the associated prices and Lagrange multipliers—constitute an FCE.

We have established that the two problems have identical feasible sets of allocation-policy pairs. Since the objective functions are equivalent, the two problems yield identical optimal allocation-policy pairs.

B.7 Lemma 5

The relaxed problem is

max

{C^b_t,C_t^e,C^w_t,Dt,Kt,Lt,Nt}

E₀ X∞ t=0

β^tX

i∈I

ωiU_tⁱ

!

subject to

λ^b_t: 0≤Lt−Dt,

λ^L_t : 0 =βbE_t[U_C^b(C_t+1^b )(C_t+1^b +Lt+1−Dt+1)]−U_C^b(C_t^b)(Lt−Dt), λ^D_t : 0≤U_C^b(C_t^b)−β_bR_tE_t(U_C^b(C_t+1^b )),

λ^C_t : 0 =A_tF(ξtKt−1, N_t)−Q(Kt−1, K_t, ξ_t)[Kt−(1−δ)ξtKt−1]−W(C_t^w, N_t)Nt+D_t

−Rt−1Dt−1−C_t^b−C_t^e,

λ^e_t: 0≤E_t(Q(Kt, Kt+1, ξt+1)ξt+1)Kt−E_t(C_t+1^b +Lt+1−Dt+1+RtDt),

λ^K_t : 0 =β_eE_t[U_C^e(C_t+1^e ){[At+1F_K(ξt+1K_t, N_t+1) +Q(Kt, K_t+1, ξ_t+1)(1−δ)]ξt+1K_t−C_t+1^b

−Lt+1+Dt+1−RtDt}]−U_C^e(C_t^e)(Q(Kt−1, Kt, ξt)Kt−Lt), λ^B_t : 0≤U_C^e(C_t^e)Lt−βeE_t[U_C^e(C_t+1^e )(C_t+1^b +Lt+1−Dt+1+RtDt)], λ^Y_t : 0 =AtF(ξtKt−1, Nt)−X

i∈I

C_tⁱ−I(Kt−1, Kt, ξt),

whereRt=R(U_C^w(C_t^w, Nt),E_t[U_C^w(C_t+1^w , Nt+1)]), the functionsW,R,Q, andI are as in Definition4.

In the absence of taxation on the banker’s side, (3)–(7) must be respected by the planner. As before, we can use (3) to solve for B_t ≡ R_t^LLt−1 = C_t^b+L_t−D_t+Rt−1Dt−1. Now we can use (5) to express λ^b_t =U_C,t^b −β_bR_tE_t(U_C,t+1^b ). Multiplying (5) byD_t and (6) byL_t, subtracting the former from the latter, and using the complementary slackness conditions (7), (6) can be expressed in terms of allocations only.

Hence, the implementability conditions that go to the Ramsey problem from the banker’s side are 0≤(1−κ_t)Lt−D_t,

0 =βbE_t[U_C,t+1^b (C_t+1^b +Lt+1−Dt+1)]−U_C,t^b (Lt−Dt), 0≤U_C,t^b −βbRtE_t(U_C,t+1^b ),

0 = [U_C,t^b −β_bR_tE_t(U_C,t+1^b )][(1−κ_t)Lt−D_t].

Consider the entrepreneur’s problem. As before, we can use the regulated analog of (10) to solve for τ_t^N ≡ ^At_W^FN,t

t −1. Using (11), we can express λ^e_tE_t(Bt+1) = U_C,t^e Lt−βeE_t(U_C,t+1^e Bt+1). By multiplying (11) byLt and (12) byKt, subtracting the former from the latter, and using the complementary slackness conditions (13), (12) can be identically expressed in terms of allocations. Using the definition of Btbased

on (3), the implementability conditions from the entrepreneur’s side are

0 =AtF(ξtKt−1, Nt)−Qt[Kt−(1−δ)ξtKt−1]−WtNt+Dt−Rt−1Dt−1−C_t^b−C_t^e, 0≤m_tE_t(Qt+1ξt+1)Kt−E_t(Bt+1),

0 =β_eE_t[U_C,t+1^e {[At+1F_K(ξt+1K_t, N_t+1) +Q_t+1(1−δ)]ξt+1K_t−B_t+1}]−U_C,t^e (QtK_t−L_t), 0≤U_C,t^e Lt−βeE_t(U_C,t+1^e Bt+1),

0 = [U_C,t^e Lt−βeE_t(U_C,t+1^e Bt+1)][mtE_t(Qt+1ξt+1)Kt−E_t(Bt+1)].

The remaining implementability conditions are constituted in the functionsW,R,Q, andI, defined by (1), (2), (14), and (20), as well as the resource constraint obtained by combining (21) and (22).

The equivalence between the feasible sets of allocation-policy pairs that satisfy the implementability conditions above and the constraints of the relaxed problem follows from the arguments that are identical to the proof of Lemma 4. Now we have only one tax rate τ_t^N that can be constructed from the regulated version of (10), and both κt and mt are set such that the private complementary slackness conditions are satisfied.

B.8 Proposition 3

The FOCs for the problem of Lemma5 are

C_t^b : 0 =ωbU_C,t^b −λ^Y_t −λ^C_t −[λ^L_t(Lt−Dt)−λ^D_t]U_CC,t^b +1^N(t)

β {[λ^L_t−1(C_t^b+Lt−Dt)−λ^D_t−1Rt−1]βbU_CC,t^b +λ^L_t−1βbU_C,t^b −λ^e_t−1−(λ^K_t−1+λ^B_t−1)βeU_C,t^e }, C_t^e: 0 =ω_eU_C,t^e −λ^Y_t −λ^C_t −[λ^K_t (QtK_t−L_t)−λ^B_tL_t]U_CC,t^e

+1N(t)

β [λ^K_t−1R^K_t Qt−1Kt−1−(λ^K_t−1+λ^B_t−1)R^L_tLt−1]βeU_CC,t^e , C_t^w: 0 =ωwU_C,t^w −λ^Y_t −λ^C_tWC,tNt− {λ^D_tβbE_t(U_C,t+1^b ) + [βE_t(λ^C_t+1) +λ^e_t

+ (λ^K_t +λ^B_t)βeE_t(U_C,t+1^e )]Dt}R1,tU_CC,t^w −1^N(t)

β {λ^D_t−1βbE_t−1(U_C,t^b ) + [βE_t−1(λ^C_t) +λ^e_t−1 + (λ^K_t−1+λ^B_t−1)βeE_t−1(U_C,t^e )]Dt−1}R2,t−1U_CC,t^w ,

D_t: 0 =−λ^b_t+λ^L_tU_C,t^b +λ^C_t −[βE_t(λ^C_t+1) +λ^e_t+ (λ^K_t +λ^B_t)βeE_t(U_C,t+1^e )]Rt

+1^N(t)

β [−λ^L_t−1β_bU_C,t^b +λ^e_t−1+ (λ^K_t−1+λ^B_t−1)βeU_C,t^e ],

Kt: 0 =−λ^C_t{Q2,t[Kt−(1−δ)ξtKt−1] +Qt}+λ^e_tE_t[(Q1,t+1Kt+Qt+1)ξt+1]−λ^Y_t I2,t

+λ^K_t {βeE_t[U_C,t+1^e {[At+1FKK,t+1ξt+1+Q1,t+1(1−δ)]ξt+1Kt+R^K_t+1Qt}]−U_C,t^e (Q2,tKt+Qt)}

+βE_t[(λ^C_t+1+λ^Y_t+1)At+1FK,t+1ξt+1+λ^C_t+1{Qt+1(1−δ)ξt+1−Q1,t+1[Kt+1−(1−δ)ξt+1Kt]}

−λ^K_t+1U_C,t+1^e Q1,t+1Kt+1−λ^Y_t+1I1,t+1] +1N(t)

β [λ^e_t−1+λ^K_t−1β_eU_C,t^e (1−δ)]Q2,tξ_tKt−1, Lt: 0 =λ^b_t−λ^L_tU_C,t^b + (λ^K_t +λ^B_t)U_C,t^e +1^N(t)

β [λ^L_t−1βbU_C,t^b −λ^e_t−1−(λ^K_t−1+λ^B_t−1)βeU_C,t^e ], N_t: 0 =ω_wU_N,t^w + (λ^C_t +λ^Y_t)AtF_N,t−λ^C_t(WN,tN_t+W_t)− {λ^D_t β_bE_t(U_C,t+1^b ) + [βE_t(λ^C_t+1) +λ^e_t

+ (λ^K_t +λ^B_t)βeE_t(U_C,t+1^e )]Dt}R1,tU_CN,t^w −1N(t)

β {λ^D_t−1β_bE_t−1(U_C,t^b ) + [βE_t−1(λ^C_t) +λ^e_t−1 + (λ^K_t−1+λ^B_t−1)βeE_t−1(U_C,t^e )]Dt−1}R2,t−1U_CN,t^w +1^N(t)

β λ^K_t−1βeU_C,t^e AtFKN,tξtKt−1.

The complementary slackness conditions are

0 =λ^b_t(Lt−Dt), λ^b_t≥0,

0 =λ^D_t [U_C,t^b −β_bR_tE_t(U_C,t+1^b )], λ^D_t ≥0, 0 =λ^e_t[E_t(Qt+1ξt+1)Kt−E_t(Bt+1)], λ^e_t ≥0, 0 =λ^B_t [U_C,t^e L_t−β_eE_t(U_C,t+1^e Bt+1)], λ^B_t ≥0.

Inspecting the FOCs forC_t^b,C_t^e, andC_t^w, we see that consumption insurance is generally imperfect.

Consider the steady state. The λ^L_t constraint impliesL−D = _1−β^βb_bC^b ≥0, which makes the relaxed bankers and entrepreneurs. Risk sharing is only approximate because generallyλ^L_t 6= 0 outside of the steady state. IfU^w(C^w, N) = ln(C^w)−v(N) and the steady-state profits of capital good producers are zero, the FOC forC_t^w impliesω_wU_C^w=λ^C+λ^Y, as in the proof of Proposition1.

SectionD.5constructs the steady state. The construction boils down to considering two cases: collateral constraint is slack or binding. Each case can be reduced to solving a system of three nonlinear equations.

Conditional on solving a nonlinear system, the sequential solution uniquely determines the steady state.

Since the problem reduces to a numerical one, we cannot claim that the steady state is unique. However, it is the case under the baseline calibration and other parameterizations considered in the analysis.

B.9 Proposition 4

C_t^w: 0 =ωwU_C,t^w −λ^Y_t −λ^C_tWC,tNt−[λ^L_tβbE_t(U_C,t+1^b ) +βE_t(λ^C_t+1) +λ^e_t]R1,tU_CC,t^w Dt−1^N(t)

The FOCs for C_t^e andKtimply

Risk sharing and steady state Risk sharing properties follow from inspecting the FOCs forC_t^b,C_t^e, andC_t^w. In particular, the latter now has the term that reflects the market power of retailers. Ifλ^e=λ^L= 0 and workers have separable preferences, the FOC forC^w in the steady state is

0 =ωwu^′(C^w)−λ^Y +λ^Cu^′′(C^w)

Sinceǫ <∞, even with logarithmic preferences, the worker’s steady-state Pareto-weighted marginal utility of consumption is less than that of bankers and entrepreneurs.

As shown in Section D.6, the steady state construction parallels the FCEA, reducing to two cases—

whether λ^L = 0 or λ^L > 0. In both cases, the quantity of deposits is indeterminate, but conditional on choosing an admissible value ofD, there typically exists a unique steady state. The proof that the optimal steady state hasD= 0, provided thatλ^C>0, is identical to the FCEA in Proposition1.

Decentralization After replacingA_twithP_t^wA_tin the entrepreneur’s problem, the proof is identical to the proof of Proposition2.

λ^Ω_t : 0 = ǫ−1

The complementary slackness conditions are

0 =λ^b_t(Lt−Dt), λ^b_t≥0,

0 =λ^L_t[U_C,t^b L_t−β_bE_t(U_C,t+1^b B_t+1)], D_tλ^L_t ≥0, 0 =λ^e_t[E_t(Qt+1ξt+1)Kt−E_t(Bt+1)], λ^e_t≥0, 0 =λ^R_t[RtE_t(Πt+1)−R], λ^R_t ≥0.

The risk-sharing and steady-state properties follow from the proof of Proposition4after settingλ^L_2,t= 0, κt= 0, andmt= 1. The short-run inflation behavior is represented by the FOC for Πt. Section D.7shows that in the steady state, the FOC for Π is equivalent to

λ^R=Π−1

Therefore, sgn(λ^R) = sgn(Π−1). The complementary slackness conditions postulate that Π =βRifλ^R>0.

Hence, ifR≤_β¹, then Π = 1; ifR > _β¹, then Π =βR.

λ^R_t : 0≤RtE_t(Πt+1)−R,

Πt: 0 =λ^Ω_tPe^′(Πt)



ǫ−1 ǫ Y_t+

βθE_t

U_C,t+1^w Π^ǫ−1_t+1^Ω_e^1,t+1

Pt+1

U_C,t^w



+λ^∆_t ǫ

"

θΠ^ǫ−1_t ∆t−1−(1−θ)Pe^′(Πt) Pe_t^ǫ+1

#

−1^N(t)θΠ^ǫ−1_t Ω1,t

U_C,t^w U_C,t−1^w

"

λe^C_t−1∆t−1ǫ−λ^Ω_t−1Pet−1

Pet

ǫ−1 Πt

−Pe^′(Πt) Pet

!#

+1^N(t)

β λ^R_t−1Rt−1. The complementary slackness conditions are

0 =λ^b_t(Lt−D_t), λ^b_t≥0,

0 =λ^D_t [U_C,t^b −βbRtE_t(U_C,t+1^b )], λ^D_t ≥0, 0 =λ^e_t[E_t(Qt+1ξt+1)Kt−E_t(Bt+1)], λ^e_t ≥0, 0 =λ^B_t [U_C,t^e Lt−βeE_t(U_C,t+1^e Bt+1)], λ^B_t ≥0, 0 =λ^R_t[RtE_t(Πt+1)−R], λ^R_t ≥0.

The risk-sharing and steady-state properties follow from comparing the optimality conditions to those in the proof of Proposition3. The special case of approximate full insurance fails for the same reasons as in the proof of Proposition4. The short-run inflation behavior is represented by the FOC for Πt. SectionD.8 shows that in the steady state, the FOC for Π is equivalent to

λ^R=Π−1 Π

βθΠ^ǫ−1 1−βθΠ^ǫ

(ǫ−1)eλ^C+ǫλ^Y 1−θΠ^ǫ−1 βY.

Moreover,

(ǫ−1)eλ^C+ǫλ^Y =

ω_eU_C^e +λ^K

(R−1)(QK−L)U_CC^e +β_eRU_C^eα^ǫ−1_ǫ ǫ

v^′′(N)

u^′(C^w)N+W

+ω_wv^′(N)

v^′′(N)

u^′(C^w)N+W +¹_ǫ^A_∆FN

>0.

If the relaxed collateral constraint is slack so thatλ^K =λ^e= 0, the inequality follows immediately; otherwise, it can be verified numerically. Therefore, sgn(λ^R) = sgn(Π−1). The complementary slackness conditions postulate that Π =βRifλ^R>0. Hence, ifR≤_β¹, then Π = 1; ifR > _β¹, then Π =βR.

Im Dokument Financial constraints, risk sharing, and optimal monetary policy (Seite 44-58)