with the property thatXs(θi mi σ)→0for some(θi σ)such thats(θi mi σ) >0.
LetU0:=U(αρ). This is the value afforded an agent who deposits all of his/her wealth
risk-free with the financial intermediary, b= −αρ. This value is finite,U0∈(−∞∞).
By continuity of the utility function, V(Cs) →V∗ ≥U
0. It follows that the product s(θi mi σ)U(Xs(θi mi σ))approaches a finite value. SinceU(Xs(θi mi σ))→ −∞,
the audit probabilitys(θi mi σ)approaches 0:
Let contractC∗ be defined as follows. All contract terms in C∗ are equal to those specified byCsaside from
X∗(θi mi σ)=(α+b)θi Z∗(mi σ)=0
By construction, this contract satisfies the ex post budget constraints (Constraint1). Consider the participation constraint. Under the new contract, this constraint is tightened by amount s(θi mi σ)Zs(mi σ). By (13), the audit probabilitys(θi mi σ)→0; by the ex post budget constraints (Constraint 1), the repayment allocation
Zs(m
i σ)is finite. It follows that the product s(θi mi σ)Zs(mi σ)→0and the par-
ticipation constraint remains satisfied after the perturbation.
Consider a binding incentive compatibility constraint for an agent earningθj > θi.
Under the new contract, this binding constraint is tightened by
s(θj mi σ)U
α+bs(θj−θi)+Xs(θi mi σ)
The utility allocation U((α+bs)(θj −θi)+Xs(θi mi σ)) →U((α+bs)(θj −θi)) ∈ (−∞∞). The probabilitys(θj mi σ)→0.26It follows that
s(θj mi σ)Uα+bs(θj−θi)+Xs(θi mi σ)→0 (14)
and, therefore, that the incentive compatibility constraint remains satisfied.
We have shown that the contractC∗is feasible and improves expected utility. But the new consumption bundle is now nonzero (X∗(θi mi σ)=(α+b)θi); therefore, the limit
of the sequenceCsdoes not specify a supremum.
(b) Upper limit of consumption.Consider a sequence of contracts with the property thatXs(θ
i mi σ)→ ∞for some(θi σ)such thats(θi mi σ) >0. It follows by the ex
post budget constraints (Constraint1) thatZs(m
i σ)→ −∞.
The financial intermediary’s participation constraint requires iσ s(θi σ)Zs(mi σ)≥bsρ+ i (θi)Qs(mi)κ
Borrowing bs is bounded below by −α. Expected audit costs
i(θi)Qs(mi)κ are
bounded by[0 κ]. By part (a) of this proof, combined with the ex post budget constraints (Constraint1), we also know that
Zsm j σ ≤α+bsθj ∀j σ Together, we have s(θi σ)Zs(mi σ)≥bsρ−α+bsE(θi) s(θi σ)≤ bsρ−α+bsE(θi) Zs(m i σ) 26Earlier we saw thats(θ
i mi σ)→0. This must mean either thatqsi→0ifσis an audit signal or that qsi→1ifσis the null signal. Either way, it follows thats(θj mi σ)→0∀j.
Consider the contractC∗defined with all terms equal to those specified byCsaside from
X∗(θi mi σ)=(α+b)θi Z∗(mi σ)=0
By construction, this contract satisfies the ex post budget constraints (Constraint1). The effect of this perturbation on the financial intermediary’s participation con- straint is positive, with expected repayments increasing bys(θi σ)(−Zs(mi σ)).
Expected utility decreases by
V∗−Vs=s(θ i σ) Uα+bsθi−Zs(mi σ) −Uα+bsθi ≤bsρ−α+bsE(θi) Uα+bsθi−Zs(mi σ)−Uα+bsθi Zs(m i σ)
Concavity of the utility function implies that asZs(m
i σ)→ −∞, the fraction Uα+bsθi−Zs(mi σ)−Uα+bsθi
Zs(m
i σ) →
0
Therefore, the perturbation deliversV∗= ˆV.
Now we turn to incentive compatibility. First we consider the incentive of an agent receivingθitruthfully to report his/her income; then we turn to an agentθj=θi. Condi-
tional upon receiving shockθi, the expected utility of the agent decreases by
s(σ|θi) Uα+bsθi−Zs(mi σ) −Uα+bsθi =s(θi σ) (θi) Uα+bsθi−Zs(mi σ) −Uα+bsθi ≤ bsρ−α+bsE(θ i) (θi) ×U α+bsθi−Zs(mi σ) −Uα+bsθi Zs(m i σ)
Concavity of the utility function implies that asZs(m
i σ)→ −∞, the fraction Uα+bsθi−Zs(mi σ)−Uα+bsθi
Zs(m
i σ) →
0
Therefore, the expected utility attained by truth-telling for an agent receiving shockθiis
unchanged after the perturbation—incentive compatibility is retained.
For an agent receiving shockθj=θi, his/her expected repayment after reportingmi
is higher than before the perturbation,Z∗(mi σ) >Zˆ(mi σ). Therefore, without loss
of generality, the incentive compatibility constraint for agents receiving shockθj and
considering reportmiis relaxed after the perturbation.
(c) Upper limit of borrowing. First note that borrowingbis bounded below by−α, whereb= −αis feasible and impliesV=U(αρ). So we are concerned with establishing a closed upper bound on borrowingb.
By Proposition6, we know that nonnegative consumption implies the upper bound on leverage α+b α ≤ ρ ρ− θ1− i (θi)Q(θi)κ
Take a sequence of contracts with the property thatbs → ¯b, whereb¯ is the upper limit of borrowing. By Proposition 6, this implies that there exists a state (θi σ) where Xs(θ
i mi σ)→0and(θi σ) >0. It follows thatlimx→0U(x)= −∞. From here, we can
apply the same arguments from part (a) of this proof to show that the sequence does not converge to a supremum, and, therefore, that an interior optimal borrowing allocation exists.
Proof ofTheorem2. Audit technologies.Assumption1does not guarantee informa- tiveness of audits. Indeed, it is permissible that there may exist statesθi, θj with the
property that∀σ subject to(θi σ) >0∨(θj σ) >0,(σ|θi)=(σ|θj). If so, we say
that the audit technology isuninformativewith respect to revenue statesj,i.
Letldenote the lowest state for which all the audit technology is uninformative with respect to statel and all states greater thanl. If the audit signal were perfect, then l
would equaln, the number of states; if the audit signal were independent of the revenue state, thenlwould equal1.
We proceed to demonstrate that forκsufficiently low, standard debt contracts with
qi=1fori < landqi=0fori≥lare optimal. We start by demonstrating that there exists
an optimal contract that takes the form of standard debt; then we show that standard debt is exclusively optimal for small increases in audit costs.
Lemma2. Letκ=0. Then there exists an optimal contract that takes the form of a stan- dard debt contract with
q(θi)=
0 ∀i≥l
1 otherwise.
Proof ofLemma2. By Proposition5, an optimal contract exists. To see that optimal contracts exist withq(θi)=0fori≥l, refer to the first order condition for repayments
(Proposition1, (8)), which shows that if the audit signal σ is independent of the rev- enue state for revenue pairi,j, then repayments following reportsmi,mjare indepen-
dent of audit signals. It follows that repayments following reports for all statesi≥lare not dependent on audit signals, and these signals can be discarded (orq(θi)=0) for all i≥l.
For reportsi < l, note that audits do not impose a resource cost and, therefore, can be employed with probability1at no cost.
Lemma3. Following Lemma2, consider an optimal standard debt contract withq(θi)=0 ∀i≥land1otherwise. For eachi < l, there is no optimal contract withq(θi) <1.
Proof. By Proposition1, (8), we know that the optimal repayment allocations following reportiwill be contingent on the audit signalσif and only if there existsj > isuch that (i) the audit technology is informative with respect to revenue states i, j and (ii) the incentive compatibility constraint between the two revenue states is binding (μji>0).
(i) By construction ofl, the audit technology is informative with respect to revenue statesθl−1,θl. This means that the conditional distribution of audit signals differs
across these two revenue states. Letθibe a revenue state withi < l. The audit
technology must be informative with respect to either the pairθi,θl−1or the pair θi,θl, or both.
(ii) Consider Proposition1, (8). Assumption3ensures that the ratio (θj|σ)
(θi|σ) is finite valued for allσ. It follows that if the incentive constraints ICCji were not bind-
ing for allj > i, then the repayment mappingZ(mi σ)would not be contingent
onσ. We also know that the signalmlis not audited by Lemma2. This generates a
contradiction: if repayments are not contingent on audit signals for either reports
mlormiandθl=θi, then the incentive constraint ICCliwould be binding for any
optimal repayment profile.
Combining (a) and (b) with Proposition1, we observe that the optimal repayment al- locations underq(θi)=1cannot be replicated with q(θi) <1for i < l. Therefore, no
optimal contract exists such thatq(θi) <1.
Proof of Theorem2(continued). LetCbe an optimal contract with some strictly positiveκ and at least oneifor which qi∈(01). Ifi≥l, then, similarly to the case
described in Lemma2, optimal repayments are not contingent on the signal revealed following reportmi, and the audit probabilityqican be set equal to 0 without having
any effect on expected utility or incentive compatibility, and relaxing the participation constraint. This generates a contradiction; the original contractCmust not have been optimal.
Fori < l, then we can decompose the welfare effects of an increase inqito1into two
components: a risk sharing benefit and a budgetary cost, denotedδ1andδ2in units of
expected welfare.
First, we perturb the contract to setqi=1for alli < l andqi∈(01). We then set
consumption allocations optimally conditional upon the new audit strategy, holding expected consumption constant. Denote this updated consumption schedule x. By Lemma3, the updated consumption schedule (i) leaves consumption allocations con- tingent on audit signals, (ii) for each i, leaves the incentive compatibility constraints ICCji binding for somej > i, and (iii) cannot be replicated with a lower audit proba-
bilityqi. It follows that the adjustment in repayment schedule must result in a strictly
positive welfare gain,δ1>0.
To obtain an upper bound on the welfare cost of the tightened participation con- straint, consider the allocationx:=minkσX(θk mk σ). It is clear from Assumption1
that theθkthat satisfies this minimand isθk=θ1. Consider a perturbation that funds
the increase in audit costs by reducing the consumption allocationx. This perturbation relaxes any initially binding incentive constraint ICCj1 for anyj, and leaves all other
binding incentive constraints unaffected. By Jensen’s inequality, the welfare cost of this perturbation is bounded above by
δ2< i|qi∈(01) (α+b)πi(1−qi)κU x− i|qi∈(01) πi(1−qi)κ
By Proposition5,xexists and is strictly positive; further, the utility allocationU(x)∈ (−∞∞). Therefore, whenκis small, δ2exists, and asκ→0, we haveδ2→0. It fol-
lows that for sufficiently lowκ,δ2< δ1, and there exists a standard debt contract that is superior to the initial contractC.
Proof ofProposition6. From the budget constraints (1), we have X(θj mi σ)= X(θi mi σ)+(α+b)(θj−θi). By assumption,(θi σ|qi) >0∀θi σ. If consumption
is nonnegative for all truth-telling states, X(θi mi σ)≥0∀θi σ, then it follows that
the expectation on the right hand side of the incentive compatibility constraint ICCji
is bounded below asσ(θj σ|qi)U(θj mi σ)≥U((α+b)(θj−θi)). Incentive compati-
bility requires σ (θj σ|qj)U(θj mj σ)≥ σ (θj σ|qi)U(θj mi σ)≥U (α+b)(θj−θi) which implies σ (θj σ|qj)X(θj mj σ)≥(α+b)(θj−θi)
by Jensen’s inequality. Using the budget constraint (1), we convert this inequality into an upper bound on repayments:
σ (θj σ|qj)(α+b)θj−Z(mj σ)≥(α+b)(θj−θi) σ (θj σ|qj)Z(mj σ)≤(α+b)θi
Now, substituting this into the participation constraint (6), iσ (θi σ|qi)Z(mi σ)≤(α+b)θ1 bρ+ n i=1 (θi)qi(α+b)κ≤(α+b)θ1
which can be rearranged to complete the proof.
F igure 1 . N u mer ical e xample .
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Co-editor Dilip Mookherjee handled this manuscript.
Manuscript received 28 June, 2016; final version accepted 8 January, 2019; available online 25 January, 2019.