DESCRIPCIÓN DE LOS DEPÓSITOS AÉREOS CON RECIPIENTE INTERNO

Proof of Lemma 1: Suppose there is contract which satisfies (5), and the agent defaults following outcome k. Then j ∈Ci receivesskj, instead of tkj, the transfer in the event that the agent does not default. Now the earlier contract can be replaced with one where ˜tkj = skj. This is feasible because P_j∈Ci

˜

tkj = P_j∈Ciskj = γ β_y

k ≤

γβyk+wi−d+σ(wi) the ex post wealth of the agent, as wi−d≥0, σ(wi)≥0. The agent will not default under this new contract as

X j∈Ci ˜ tkj = X j∈Ci skj =γβyk≤γβyk+p(wi−d+σ(wi)). (11) And each principalj ∈Ci earns the same payoff as before when default occurs, while the agent avoids the penalty associated with default. So the new contract ex post

Pareto-dominates the previous one, which is thus vulnerable to renegotiation. This establishes (i).

For (ii), note that ifd < wi, the downpaymentdcan be raised by some >0, with a corresponding reduction ofP

j∈Citkj and of P

j∈CiIj by. In the new contract the agent will also not want to default, as the mandated transfers fall by, while default penaltyp(wi−d+σ(wi)) falls by at most(because the slope ofp(.) is between 0 and 1). Then the agent’s payoff, as well as that of every j ∈ Ci, in state k is unaltered. This new contract is then payoff-equivalent to the previous one.

To show (iii), fixd=wi, then re-allocate transfers and contributions acrossj ∈Ci in such a way to leave their expected payoffs unchanged. The aggregate financial transfersvis-a-visthe agent remain unaltered, so do the agent’s incentives and payoffs. Specifically, let πj denote the expected operating profit of j, which must cover the overhead costfj (otherwise j would do better to leave the coalition):

πj ≡etsj+ (1−e)tf j−Ij ≥fj (12)

Let Π ≡ P

j∈Ciπj ≥ P

j∈Cifj denote aggregate operating profit. If Π > 0, define

δj ≡ πj Π and select ˜Ij = δj P l∈CiIl; ˜tkj = δj P

˜

πj = δjΠ = πj. If Π = 0, then we have Ij = etsj + (1−e)tf j for all j ∈ Ci: select arbitraryδj ≥0 with

P

j∈Ciδj = 1 and repeat the above construction. This completes the proof of Lemma 1.

Proof of Lemma 2:Sincei is viable, there exists a feasible contract (vs, vf, e, γ) for some coalitionCi including i. Let δ≡(wl+σ(wl))−(wi+σ(wi))>0. For agent

l we can select a coalition Cl consisting of l and the same principalsj that belong to

Ci. Then select the following contract: ˆvs = vs+δ,vˆf = vf +δ, combined with the samee and γ. By construction this contract satisfies (IC), (PCP) and (PCA). It also satisfies (LL): the increase in the left-hand-side of (LL) isδ, while the increase in the right-hand-side of (LL) is [σ(wl)−σ(wi)]−[p(σ(wl))−p(σ(wi))]≤σ(wl)−σ(wi)< δ.

Proof of Proposition 3: We proceed through a sequence of steps.

Step 1.1: A Walrasian allocation is stable. If not, there will exist an agent i that deviates with a coalition ˆCi and a contract (ˆvs,ˆvf,e,ˆ γˆ) which generates an expected utility larger than that in the A-optimal contract relative to π, and an expected operating profit for everyj ∈Cˆi that exceedsπ. This contradicts the definition of an A-optimal contract.

Step 1.2: In any stable allocation, all active principals attain the same (expected) operating profit. If this is false, there exist two active principals j, m with πj > πm,

πj ≥fj, πm ≥fm. Letji∈Ci. Then ican form a coalition with ˆCi ≡Ci\ {j} ∪ {m}. Letni denote the number of principals in Ci, and select any ∈(0,

πj−πm

ni+1 ). We can then select a contract for the deviating coalition which increasesvs, vf by , and also increases πh by for every h ∈ Cˆi. This contract is feasible and makes everyone in the deviating coalition better off.

Step 1.3: In any stable allocation, π≥fj for every active principal. If not, such a principal would be better off unmatched.

an unmatched principal j could make positive net profit by being matched with an active agent at the going profit rate ofπ. An argument analogous to Step 1.2 can now be used to show the allocation is not stable.

Step 1.5: In any stable allocation, every agent gets an A-optimal contract relative toπ, the common rate of operating profit earned by active principals.Otherwise, there exists a feasible contract for i which generates a higher expected payoff to i, while paying an operating profit rate of at leastπon each asset leased. We now argue there exists a contract in the neighborhood of this deviating contract which awards a profit rate greater than π for every asset leased, while still enabling the agent to attain a higher expected payoff compared with that in the stable allocation.

The original contract satisfied (PCA), so the deviating contract satisfied (PCA) with slack. We can thus ignore this aspect of feasibility in what follows.

If (LL) is slack in both states in the deviating contract, we can reduce vs, vf by some common but small , which preserves feasibility. In this case the argument is straightforward.

So consider the case where (LL) binds in some state. This must be statef, because

vs > vf is needed to induce e > 0, which in turn is necessary for feasibility ((PCP) requires the project to break-even in expectation, and this is not possible ife= 0 given thatyf < I). So we must havevf =σ(wi)−p(σ(wi)). Now holdingvf fixed at this level, consider varyingvsabovevf, witheadjusted according to the (IC): i.e., withe=e(vs) that solvesvs−{σ(wi)−p(σ(wi))}=D0(e). Let the corresponding aggregate profit for asset owners be denoted Π(vs)≡e(vs)[γβys−vs]+(1−e(vs))[γβyf−{σ(wi)−p(σ(wi))}], whereγ is the project scale in the deviating contract.

Define the P-optimal contract to be one where vs is selected to maximize Π(vs), subject tovs≥σ(wi)−p(σ(wi)). This problem can be restated as follows (replacing

eas the control variable): selecte ∈[0,1] to maximizee·[γβ(ys−yf)−D0(e)]. Noting thateD0(e) is strictly convex, the objective function is strictly concave, and thus has a unique solution. It is also evident thate >0 in the P-optimal contract.

We claim that in the original contract (in the stable allocation), the agent must have attained a utility at least as large as in the P-optimal contract. Otherwise, the agent obtained a smaller payoff, and the contract in the stable allocation was not P- optimal (as the P-optimal contract is unique, as shown above). So aggregate operating profit of the principals in Ci must be less than the profit in the P-optimal contract. Then the agent and principals in Ci could deviate to the P-optimal contract, which would make all of them strictly better off.

Since the agent received a higher payoff in the deviating contract compared with the contract in the stable allocation, it follows that the agent’s payoff in the deviat- ing contract is strictly higher than in the P-optimal contract. In both the deviating contract and in the P-optimal contract, we havevf =σ(wi)−p(σ(wi)), so they must differ in vs, with the deviating contract associated with a higher vs. It follows that as vs is lowered from that in the deviating contract to the level in the P-optimal contract, the agent’s payoff is (continuously) lowered while aggregate profit of the principals in Ci is (continuously) raised (owing to the strict concavity of aggregate profit with respect toe). Therefore we can find a contract with vs slightly below that in the deviating contract, which will allow a strictly higher aggregate profit, and a slightly lower payoff for the agent. This allows all members of Ci as well as i to be better off compared to the stable allocation, a contradiction. This completes the proof of Step 1.5.

The proof of Proposition 3 now follows from combining Steps 1.1–1.5 to infer that a stable allocation must be Walrasian.

Proof of Lemma 4:DefineS∗(π)≡γ∗βR(e∗)−e∗D0(e∗)−γ∗(I+π), where we drop the dependence of the first-best contract on π to avoid notational clutter. Then if

S∗(π) ≥ −p(σ(0)), the first-best satisfies (F) for all w ≥ 0, so we can set ¯w(π) = 0 in that case. Otherwise S∗(π) < −p(σ(0)) and there exists ¯w(π) > 0 such that

S∗(π) =−w−p(σ(w)), since −w−p(σ(w)) is decreasing inw and goes to −∞asw

only ifw≥w¯(π), which establishes (a).

(b) Suppose there existπ1 < π2, wealth wand choices of first-best contracts such

that γ2 =γ∗(π2, w)> γ1 =γ∗(π1, w). Then γ₁βR(e1)−D(e1)−γ1(I+π1)≥γ β 2R(e2)−D(e2)−γ2(I+π1) and γ₂βR(e2)−D(e2)−γ2(I+π2)≥γ1βR(e1)−D(e1)−γ1(I+π2)

whereem denotes e∗(πm, w), m = 1,2. Adding these two inequalities we obtain (γ2 −γ1)(I +π1)≥[γ2βR(e2)−D(e2)]−[γ1βR(e1)−D(e1)]≥(γ2−γ1)(I+π2)

which contradicts the hypothesis that γ2 > γ1, as π1 < π2. Hence, γ∗(π, w) is non-

increasing in π. This in turn implies e∗(π, w) is nonincreasing because it maximizes

γ∗βR(e)−D(e).

In order to establish (c), note the following.

Observation 3.1: In any A-optimal contract, γ(π, w) maximizesγβ_{R(e(π, w))−γ(I+}

π).Otherwise we can select a differentγ to raise the value ofγβR(e(π, w))−γ(I+π): this both raises the value of the objective function (AO), and helps relax the constraint (F).

Observation 3.2: In any first-best contract, γ∗(π) maximizes γβ_R(e∗_{(π))−γ(I+π).} Otherwise the value of the first-best objective function can be raised by switching to a differentγ, while leaving e∗(π) unchanged.

We now claim that ifw <w¯(π), then e(π, w)< e∗(π) for any selection of second- best and first-best contracts. If this is false, we can find contracts withe(π, w)≥e∗(π). Using Observations 3.1 and 3.2, it follows that corresponding second-best and first- best project scales satisfyγ(π, w)≥γ∗(π).

The hypothesis w < w¯(π) implies that the first-best cannot be achieved by w at profit rate π. This means that (γ∗(π), e∗(π)) violates (F), while by its very nature

(γ(π, w), e(π, w)) satisfies (F). This implies that

[γ(π, w)]βR(e(π, w))−e(π, w)D0(e(π, w))−γ(π, w)(I+π)

>[γ∗(π)]βR(e∗(π))−e∗(π)D0(e∗(π))−γ∗(π)(I+π) which in turn implies

{[γ(π, w)]βR(e(π, w))−γ(π, w)(I+π)} − {[γ∗(π)]β_R(e∗_(π))−γ∗_{(π)(I +π)}}

> e(π, w)D0(e(π, w))−e∗(π)D0(e∗(π))

> D(e(π, w))−D(e∗π))

the last inequality following from the fact that ∂eD_∂e0(e) = eD00(e) +D0(e) > D0(e). Then it must be the case that the second-best choices yield a higher utility than the first-best choice, which contradicts the definition of the first-best.

Therefore every second-best effort must always be less than any first-best effort. The rest of (c) now follows from Observations 3.1 and 3.2.

Proof of Lemma 5: Suppose there exist A-optimal efforts em ≡ e(π, wm), m= 1,2 such that e1 > e2. Then by Observation 3.1 corresponding A-optimal scales satisfy γ1 ≥γ2.

Since wm < w¯(π), the first-best is not achievable at (π, wm), m = 1,2 and con- straint (F) must be binding at (γm, em) for wm. Since w2 > w1, the contract (γ1, e1)

must satisfy (F) with slack at (π, w2). To see this, note that

[γ2]βR(e2) −e2D0(e2)−γ2(I+π)

=−w2−p(σ(w2)) <−w1−p(σ(w1))

This implies that

{[γ1]βR(e1)−γ1(I +π)} −{[γ2]βR(e2)−γ2(I+π)} ≥e1D0(e1)−e2D0(e2)

> D(e1)−D(e2)

the last inequality following from the hypothesis that e1 > e2. This implies that the

contract (γ1, e1) generates a higher payoff than (γ2, e2), contradicting the A-optimality

of the latter, since the former is feasible at the wealth w2. Hence e1 ≤ e2. The

remaining part of the result follows from Observation 3.1.

Proof of Lemma 6:The argument is virtually identical to that of Lemma 5 above, in the case that the first-best is not achievable in either situation. And if the first-best is achievable in either or both situations, we can use Lemma 4.

Proof of Lemma 7: Suppose the first-best payoff is achievable at (π, w) under rule

p1; then it is achievable under p2 and the A-optimal payoffs coincide. So suppose that

the first-best is not achievable at (π, w) under rule p1. If the first-best is achieved at

(π, w) under rule p2 then the result follows from Lemma 4. So we need to consider

the case where the first-best is not achieved under either rulep1 orp2.

Let el denote an A-optimal effort under rule pl. An argument analogous to that used in preceding Lemmas shows thate1 ≤e2, which in turn implies that γ1 ≤γ2.