Proof

[Proposition 1] In a separating equilibrium, S would be able to infer B’s type by the signal (viz., the contract) she sends. Suppose a separating equilibrium exists such that UHNB≥UPB≥USNB‾. offers an UHNB=y(eNB)−pNB≥y(eNB)=USNB‾ contract and pNB\gt0 offers a S contract. The following condition should hold:

Transitivity implies that NB. By assumption, however, H, thus, B, which contradicts condition (1). Thus, S would profitably deviate from a NB contract by proposing an A contract. Now suppose that a separating equilibrium exists such that S offers a NB contract and S offers an B contract. NB knows that H never fulfills the promise, so he will reject any offer of an NB contract. Thus, λpNB would profitably offer a λ∈(0,1) contract. The same reasoning excludes any separating equilibrium for the two types of principal offering an A contract with different levels of price and/or effort.

Suppose that S proposes an NB contract by paying an installment H with S before that H supplies the required effort, in order to signal her type and discourage S to propose an NB contract. A will eventually pay the price promised, whereas y(eNB)−pNB\gty(eB)−(1+c)pB\gty(eNB)−λpNB. would lose the installment if she wants to signal to be an ∀-type. Therefore, the signal is credible if it is sufficiently high to discourage λ\lt1 from proposing an P contract in equilibrium and paying the installment. Assume that B will provide the effort requested after having received the installment, then the following condition must hold:

This condition never holds Bp≥12e2..

Proof

[Proposition 2] Consider an equilibrium where both types of P offer a P contract. UPB will accept a e contract if it satisfies his participation constraint:

pB=12β1+c22−β

has full bargaining power, so that substituting eq. (3) holding as an equality into UPB=2−β2β1+cβ2−βand UAB=0.’s utility function, NB, and maximizing with respect to A, we obtain S and P.

Thus, players’ utility will be:

This equilibrium exists because any deviation to an B contract is always rejected by A since he would believe that this deviation would come from type B. NB cannot also profitably deviate to any other S contract because she would get a lower payoff. Thus, the P equilibrium is always utility-maximizing. Hence, the following strategy combination and beliefs form a perfect Bayesian equilibrium in pure strategies:

1. B accepts any pB,eB contract satisfying condition (3), and rejects any P contract believing that comes from NB;

2. A offers the utility-maximizing NB contract (p≥12αe2.).

Consider now an equilibrium where both types of H offer a B contract. S will accept the H contract if and only if:

Since UHNB=eβ−12αe2\gt2−β2β1+cβ2−β=UPB., according to Proposition (1) if e,p(e) has no incentive to deviate to a NB contract, then it must also be true for NB. Therefore, substituting (4) as an equality into NB’s utility function, eNB,pNB, we can exclude that such a deviation is profitable if and only if:

Any couple (eNB=(αβ)12−β) satisfying condition (5) is an equilibrium because no deviation to another pNB=12αβ2−ββ22−β contract can occur. This class of S equilibria is non-empty if and only if condition (5) is satisfied at least for the UHNB=2−β2αββ2−β, USNB‾=αββ2−βand UANB=0. contract (UHNB) that maximizes NB. This utility-maximizing equilibrium exists with α\gt11+c=α_ and α\gtα_. Thus, since A will always renege on her promise, players’ utility will be:

Finally, substituting H from eq. (6) into condition (5), we find that the class of equilibria in α contracts is non-empty if and only if NB.

Hence, if S, the following strategy combination and beliefs form a perfect Bayesian equilibrium in pure strategies:

1. A accepts any B contract satisfying condition (4), believing that it comes from P with probability NB, and rejects any other H contract believing that it comes from B. pB,eB would also accept every FB contract satisfying condition (3).

2. A offers an P contract satisfying condition (4) and condition (5) that is fulfilled only by P, or the FB contract (A).

Proof

[Lemma 1] Suppose that a non-maximizing FB contract is proposed in equilibrium in a certain period α=1. UHNB is sure to be paid because any e would fulfill the contract. Nevertheless, S can profitably deviate to offering the utility-maximizing t contract, which t=T would be willing to accept. The utility-maximizing B equilibrium can be easily derived from Proposition 2 substituting t+1 into P and maximizing with respect to T.

Proof

[Lemma 2] The proofs of both parts (a) and (b) follow straightforward from the fact that α∈(0,1) has no interest to maintain reputation in period B if pB,eB, or if a NB contract is offered from period FB onwards.

Proof

[Proposition 3] This proof is provided by taking into account exclusively the α≤α_’s utility-maximizing contracts as defined in Corollary 1 and Lemma 1.

(a) Consider a backward induction procedure. Starting from period P, regardless of the value of NB, consider the α≤α_ equilibrium (UHNB≤UHB). As shown in Proposition 2 and Lemma 2, no deviation to the A or NB contracts occurs. The same reasoning applies to all periods S. We now prove that this equilibrium is unique if B. Consider then a putative equilibrium, where T proposes an T−1 contract. If S then A, thus NB would reject an FB contract because it would only come from α≤α_. Consequently, only the t contract applies in A. Consider now period B. Due to Lemma 2, pB,eB will always renege on her promise, therefore NB would refuse the FB or S contracts. A similar reasoning applies to all P. Thus, the equilibrium is unique.

Hence, if B the following strategy combination and beliefs form a perfect Bayesian equilibrium in pure strategies. In every period pB,eB:

1. FB accepts the t∗\ltT contract (NB) and rejects the NB or A contracts believing that come from B;

2. S offers the α>α_ contract (S).

(b) Consider the equilibrium where the FB contract is offered in each period until t+1 and the S contract is offered thereafter. Since breaking an FB contract would be punished by t by accepting only t∗UPFB+(T−t∗−1)UPNB+USNB‾\gt(t∗−1)UPFB+USFB‾+(T−t∗)UPB. contracts, then from Proposition 2α has no profitable deviation to breaking the contract in any period because β2+2−β2(T−t∗)11+cβ2−β2−β2(T−t∗−1)+12−ββ\ltα\ltβ2+2−β2(T−t∗−1)11+cβ2−β2−β2(T−t∗−2)+12−ββ\lt1..

Then, two conditions must hold. First, t∗ has a profitable deviation to breaking the T−2 contract in t∗=1. Second, α‾≡β2+2−β2(T−1)11+cβ2−β2−β2(T−2)+12−ββ. has no profitable deviation to breaking the t∗=T−1 contract in A. Thus, it must be

and

Conditions (7) and (8) hold contemporaneously if FB falls in the following interval:

The endpoints of the interval are increasing in β2+2−β211+cβ2−β2−ββ\ltα\lt1. and t∗=T−2 intervals exist with the lower endpoint for T−1 equal to:

Finally, if α, condition (7) does not apply because in no circumstance does t∗ accept the α‾\ltα\lt1 contract in the last period due to Lemma 2. Condition (8) applies, meaning that 1 should have no profitable deviation to breaking the FB contract in α\gtα‾. Therefore, condition (8) holds if:

As expected, the lower endpoint of this interval is equal to the upper endpoint of the interval in condition (9) when A. It follows that FB equilibria exist as pFB,eFB, with B monotone and increasing in pB,eB, and each equilibrium corresponds to different intervals of NB, which do not intersect with each other.^[27]

Hence, if FB the following strategy combination and beliefs form a perfect Bayesian equilibrium in pure strategies.

In every period pFB,eFB:

1. t\gtt∗ accepts the A contract (NB) or the pNB,eNB contract (B), and rejects the pB,eB contract believing that it comes from FB;

2. S offers and fulfils the P contract (NB).

In every period pNB,eNB:

1. S accepts the T contract (α_\ltα≤α‾) or the t∗ contract (FB), and rejects the NB contract believing that it comes from α\gtα_.

2. P offers the B contract (NB) and fulfils it except for type NB in period A.

(c) If B then no S exists satisfying condition (8); thus, the T contract is never offered in equilibrium, but the NB equilibrium is still feasible. From Proposition 2, the inequality α_\ltα≤α‾ implies that t has no profitable deviation to the A contract nor to another NB contract. Finally, since any breaking of the pNB,eNB contract would be punished by B by accepting only pB,eB contracts, it is easy to show that FB has no profitable deviation to breaking the contract in any period but S (see Lemma 2). Consequently, there exists an equilibrium in which the P contract applies in each period.

Hence, if NB the following strategy combination and beliefs form a perfect Bayesian equilibrium in pure strategies.

In every period pNB,eNB:

1. S accepts the T contract (pNB,eNB) or the B contract (pB,eB), and rejects the FB contract believing that it comes from S.

2. P offers the NB contract (pNB,eNB) and fulfils it except except for type S in period T.