is not scarce, butπ <1, inside money, in connection with personalized credit, can insure nonbankers against the risk of unsuccessful trade attempts.8
in the first and fourth equations are written under the assumptions that consumers with good credit, who need money, agree to withdraw from the banker and have the record updated toz= 0; and that producers with bad credit, who were not able to spend money in the previous period, agree to deposit with the banker and have their record updated toz = 1.The participation constraints regarding these transactions are, respectively,
vn0≥βwn1 (20)
and
wn1≥βvn0. (21)
The first inequality assures that a nonbanker is willing to borrow when there is an opportunity, a constraint that is easily satisfied by a stationary allocation with discounting. I call the second inequality thedeposit constraint. It assures that a nonbanker is willing to deposit money with a banker, and to become a producer currently without money, just for the sake of improving his or her credit record.
It can be easily verified that this constraint is equivalent to the requirement that a producer with good credit is willing to produce in exchange for money, namely,
yn≤β(vn1−vn0). The participation constraint for producers with bad credit is
wn0≥0, (22)
which is the same as the inequalityβvn0 ≥ yn. Finally, there are participation constraints for consumers, requiring thatvn1andvn0be nonnegative, and which are again implied by the producer’s constraints.
Next, I discuss feasible measures of producers without money and consumers with money,pandq. It is intuitive that a credit allocation in this framework can increasebothmeasures. In order to keep the set of feasible measures tractable, I require that the distribution of deposits be constant and the same for both consumers and producers, so thatpp1 = qq1.With this additional requirement, I have the following lemma.
Lemma 3. The set of stationary measures(p, q, x), associated to the system (18-19), is fully described by the equality
q= 1−p+πpq+Ax(p, q), withp, q∈[0,1], (23) whereAx(p, q) =xq(1−πp)p(1−πq)/[xq(1−πp) +p(1−πq)]defines a concave function in the(p, q)plane.
The lemma shows that credit allows for an increase, when compared to equation (14), in the set of feasible distributions of potential producers and consumers. In the proof, I make use of the fact that a mass of producers, in proportion top(1−πq), fail to acquire money, but are able to make withdrawals at the next date because they
have good records. Likewise, a mass of consumers, in proportion toq(1−pπ), fail to spend their money holdings, and are able to make deposits at the next date in order to leave the bad-record state. Whenx= 1andp =q,half of the nonbank public holds a bad record, and the other half holds a good record.
The following lemma follows from the fact that the only potentially binding constraint for nonbanks is the deposit constraint (20).
Lemma 4. The nonbank productionyn satisfies the participation constraints (20–
22) and the system (18–19) if and only if
u(yn)≥yn β
(1−δ) 1
πp+δ+ det(M) (1−πq)δ(πp)2
. (24)
It is shown in the proof that the determinant of the matrixM,det(M),is positive and converges to zero asβapproaches one. Notice also thatxdoes not appear directly in the inequality (24). In fact, the deposit constraint is not affected byxbecausexshows up in equation (18) multiplyingwn1−βvn0,the term to be solved for when studying the deposit constraint. It follows that forβhigh enough, the deposit constraint does not bind foryn =y∗, that is, the value such thatu(y∗) = 1, the first-best level of production.
I now turn to the intermediation of capital. Again, to keep the analysis simple, I consider first allocations when there is no intermediation of capital by banks, either because capital is not scarce or because no deposit of money takes place. In these two cases there is no intermediation. I then discuss a small perturbation of an allocation with scarce capital and some deposits (smallx) that is achieved by letting banks transfer capital to nonbanks with a small probability. This approach avoids the need of additional notation as I have been able to do so far.
Inside money is destroyed in a credit allocation when a nonbanker producer with z = 0makes a deposit. The only reason this nonbanker has to actually make a deposit, instead of holding on to money and waiting to become a consumer in the next period, is the possibility of producing in the current period and acquiring more money (in the form ofz = 1). Without capital, the nonbanker will choose not to deposit. A necessary condition for a credit allocation to be implementable, therefore, is that depositors have access to capital.
I should now let pdenote the measure of producers with capital and without money, integrated over statesz.Regarding the set of consumers without money, it turns out that they will all be in statez= 0in the steady state, since the ones in state z = 1are able to withdraw from the bank and have money for the meetings with other nonbankers. As a result, the measure of consumers without money,1−q,is also understood to have capital. It is thus necessary to allocate capital at least to a measure ofpproducers and1−qconsumers. Without intermediation, the capital constraint is
p+ 1−q+λ(1−p+q)≤kn, (25) whereλis the measure of nonbankers with money and capital, andλ∈ {0,1}.
Ifλ = 1,so that x = 1and the bank sector is not reallocating capital, then the capital constraint (25) requireskn = 2, that is, that all nonbankers hold capital.
As capital becomes scarce and falls below some critical point, the reduction in the amount of capital allocated to bankers that is required to keepλ= 1makes credit suboptimal. Hence, askis reduced continuously to the point at whichλ= 0becomes optimal, a point at whichkn>1andp=qis still feasible, then the extra capital that becomes available asλshifts from1to0can be allocated to the bank sector. Ifβis sufficiently high, so that the participation constraints do not bind withp=q, then bank intermediation with somex >0makes credit attain a higher welfare because of the increases inpandqallowed by havingAx(p, q)>0in equation (23).
I have now presented the main elements of the line of reasoning that shows that intermediation of capital can be desirable. Further characterization of the optimal xwould depend on how much intermediation imposes a cost on bankers since, as assumed in Section 2, bankers meeting with depositors are themselves producers.
Intermediation takes capital away from bank producers and tends to reduce bank welfare. That discussion would depend too much on details of the model and go beyond the scope of this paper. Also, the advantage of restricting attention toλ∈ {0,1}is that either all nonbankers with money hold capital, or none of them do. As a result, I do not need extra notation for distinguishing consumers with money and capital from those with money only. I present in a lemma below, for completeness, the full description of the allocation of capital in the bank sector when intermediation takes place.
I let the fraction of bank producers holding capital at the beginning of a period, before transfers to nonbankers take place, be denotedk˜b. If a request for capital from a depositor is agreed to with probabilityθ∈(0,1),then
x= ˜kbθ. (26)
The values ofk˜bandθconsistent with stationarity are as follows.
Lemma 5. Capital intermediation with probabilityθis feasible if, forpb, εandτin [0,1],
kb=pb+ε, (27)
˜kb= pb
1−τ, (28)
τ = ε(1−pb)
pb+ε(1−pb) (29)
and
τ=θ pq(1−πq)(1−πp)
p(1−πq) +xq(1−πp). (30) In the proof of the lemma, I make use of the fact that, with intermediation, bank capital needs to be split between a fraction of producers,pb, and a fraction of consumers, ε, because there is a constant flow of capital into the bank sector, which cannot
be transferred to producers in the same period. As a result, it is necessary to have p+ 1−q < kn in order to implement a small θ.9 Moreover, in order to keep these fractions stationary, the model requires a bank producer to transfer his capital with probabilityτ given by equation (29). Equation (30) is the requirement that τ coincides with the probability that depositors request capital,θ, multiplied the measure of depositors in a given period, which is given by the fraction in the right- hand side of (30).
Definition 3. A credit allocation(yb, yn, p, q)is implementable without intermedi- ation if (23–25) hold withk=kb+knandλ=x. A credit allocation(yb, yn, p, q) is implementable with intermediation if there existsθsufficiently small and (x,k˜b) such that (23–24) and (26–30) hold withp+ 1−q < knandk=kb+kn, when the capital allocated to bank producers is˜kb.
It is clear that the expressions forUbandUnremain unchanged. Hence, the optimum problem is stated as follows.
Welfare problem. Maximize min{Ub, Un} by choice of a credit allocation (yb, yn, p, q)with capitalk=kb+kn
Proposition 2. There exists an open intervalK⊂(0,3)of capital levels such that, if k∈Kandβis sufficiently high, then capital intermediation is essential, in the sense that bankers trade capital with nonbankers with positive probability in an optimum.
Although the proof of Proposition 2 is restricted to an improvement against the alterna- tive ofλ= 0, I believe that the argument holds more generally. BecauseAx(p, q)< q, then allocating a unit of capital to consumers with money creates less deposits than al- locating that unit to the bank sector. The difficulty with this more general discussion is that intermediation causes a cost to banks. That cost can be made arbitrarily small by choosingxclose to zero, so that the extra bank capital generates a welfare improve- ment when the alternative isλ= 0.If the alternative hasλ∈(0,1),further restrictions on the other parameters of the model may prove necessary.