Appendix

Proof of Lemma 1

The result follows directly from the inequality (7), and from the fact that the frequency

of bank trade isπmin{k_b,1}.

Proof of Lemma 2

A straightforward comparison of inequalities (7) and (17), when the constraints (14) and (15) are taken into account, reveals that the constraint set for bankers is strictly larger than that of nonbankers fork_n ≤ k_b < 1. In addition, the constraint for nonbankers impliespq < k_bfork_n< k_b.As a result,

(1−β)U_n=πpq[u(y_n)−y_n]< πk_b[u(y_b)−y_b] = (1−β)U_b

follows for the optimum choices ofy_n andy_b, givenk_n andk_b,which proves the

result.

Proof of Proposition 1

The nonbank constraint set is increasing inpand coincides with that of bankers if and only ifp= 1.It follows from the stationarity restrictions onpthat, ifπ <1,then p= 1only ifq= 0.Moreover, when participation constraints do not bind, although production levels are the same in both sectors, the frequency of trade is higher in the banking sector, since the stationary values ofp,such thatp=q,are less than1by a

difference that is decreasing inπ.

Proof of Lemma 3

Ifp₁denotes the current measure of producers without money and with deposits in the bank, thenp₁(1−πq)is the fraction of those making withdrawals in the next period, so thatq= 1−p+πpq+p₁(1−πq)in the steady state. Similarly, ifq₀is the measure of consumers with money andz= 0,thenp= 1−q+πpq+xq₀(1−πp), so thatp₁(1−πq) =xq₀(1−πp)must hold. Using the latter expression, together withm≡ ^p_p¹ = ^q_q¹ = 1−^q_q⁰, to solve form,yields equation (23). Equation (23) itself can be written in two different ways:

(x+ 1−x)q(1−πp) = 1−p+A_x(p, q)andp(1−πq) = 1−q+A_x(p, q), so that multiplying both sides of both equations by the denominator, call itD,of A_x(p, q),and rearranging terms, yields

[xq(1−πp)]²=D[1−p−q+πpq+xq(1−πp)]and[p(1−πq)]²=D(1−q). Since[DA_x(p, q)]²= [xq(1−πp)]²[p(1−πq)]², then

A_x(p, q) = (1−q)¹²[1−p−q+πpq+xq(1−πp)]¹²

indicates thatA_x(p, q) equals the composition of two strictly concave functions.

Therefore, (23) defines a concave function in the(p, q)plane.

Proof of Lemma 4

The first part of the proof shows that satisfying the deposit constraint impliesv_n0≥ βw_n1. Solving forv_n0andw_n1in (18–19), under the assumption thatw_n1−βv_n0 is a nonnegative constant, implies after some simple algebra, thatv_n0−βw_n1 is positive. Hence,w_n1−βv_n0≥0implies that the unique values solving (18–19) are all nonnegative, and thus the other participation constraints are satisfied.

To show that the deposit constraint is equivalent to (24), I proceed as follows. To save on notation below, I writeu = u(y_n), ρ = πp, ξ = πq, P = 1−δ+δρ, Q= 1−δ+δξandµ = det(M) = P Q−δρξ.Now (18–19) defines a system forv_n0andw_n1in two equations that can be written asdet(C)[v_n0w_n1]^T =CS, whereS=M[ρu ,−ξy_n]^T,

C=

µ+ (1−ξ)δP (1−ρ)βQ (1−ξ)βP µ+ (1−ρ)δQ

, anddet(C)>0. As a result,

det(C)(w_n1−βv_n0) =

(1−δ)β[−ξP −δξ(1−ρ)]µ S

=

−(1−δ)βξ µ Qρu−βρξy_n βρξu−P ξy_n

so thatw_n1≥βv_n0if and only ifβρuδρ(1−ξ)(1−δ)≥y_n[(1−δ)µ+δρµ−(1− δ)δρξ].Using nowµ=P Q−δρξ = (1−δ)[1−δ+δρ+δξ(1−ρ)]completes

the proof.

Proof of Lemma 5

Letp_bdenote the fraction of bank producers with capital, andq_bdenote the fraction of bank consumers without capital, both measured at the second round of meetings, when banks produce and consume. Then,k_b=p_b+1−q_b,and forq_b= 1−εequation (27) holds. Moreover, ifτis the probability that a bank producer with capital transfers capital to a nonbanker, thenp_b = (1−τ)˜k_b,so that (28) holds. Also, ifαis the probability that a bank consumer without capital receives capital from a nonbanker, then stationarity requiresq_b= (1−α)(1−p_b+p_bq_b)andp_b= (1−τ)(1−q_b+p_bq_b). These expressions can be rewritten as αq_b = (1−α)(1−p_b−q_b +p_bq_b) and τ p_b = (1−τ)(1−p_b−q_b+p_bq_b),so thatαq_b(1−τ) =τ p_b(1−α).This condition, together withq_b = (1−α)(1−p_b+p_bq_b),forq_b = 1−ε,implies (29). Finally, according to Lemma 3, the probability that a banker producer meets with a depositor is given by the third term on the right-hand side of (30).

Proof of Proposition 2

When participation constraints allowy_n = y^∗ and λ = 1, the welfare problem maximizespqsubject to (23) and (25), fork_n = 2.According to Lemma 2, askis

sufficiently reduced, maximizingmin{U_b, U_n}impliesk_b<1.Now, the level curves ofpq in the(p, q)plane are differentiable and strictly convex, while (23) defines a strictly concave constraint on the same plane. Since preferences are also continuous and differentiable, welfare varies continuously withkwhileλ= 1remains optimal.

Hence, there isk <¯ 3,such that fork= ¯k,there are two allocations, one withλ= 1, and another withλ= 0,that attain the same optimum welfare. Ifx >0in the latter allocation, there is nothing else to prove. If, on the contrary,x= 0in that allocation, which, by continuity, featuresk_n>1,then it also hasp=qbecause that maximizes pqand it is feasible withk_n >1.Thus, some nonbank capital remains idle in the hands of consumers with money and can be transferred to the bank sector. Since, again by continuity,k_b <1,this idle capital can be transferred to the bank sector, with a part used to support intermediation with a smallx >0,through a smallτand smallε, without reducing the measure of producers with capital, that is, such thatp_b remains the same. That reallocation of capital, by makingxpositive, increases both pandq.Asβhas been chosen sufficiently high so that participation constraints do not bind, welfare increases, contradicting the claim thatx= 0is optimal.

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Chapter 2. Financial fragility in small

open economies

international capital flows, and crises in small