Delegation of information acquisition, information asymmetry, and outside option

Abstract

We study how a principal with an outside option optimally delegates information acquisition to an agent in a parsimonious environment in which the principal can observe neither the agent’s effort nor signal realizations. When the principal chooses an outside option, the true state is not revealed and thus not contractible. We precisely characterize an optimal contract for the principal, illustrating how to construct an optimal contract.

Appendices

A. Proof of Proposition 3.1

The proof is essentially identical to the proof of Proposition 4.1 in Han and Choi (2020), but we provide the proof for self-containedness. The following lemma states that there exist suitable thresholds and a solution of the differential equation as shown in Fig. 3.

Fig. 3

The blue solid line is ${\hat{v}}$ and the dashed line is $\psi$

Lemma A.1

There exist constants $\hat{q}_N \in (0,p)$ and $\hat{q}_I\in (p,1)$ and a twice differentiable function ${\hat{v}}:(0,1)\mapsto {\mathbb {R}}$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}{\hat{v}}=0,\\ {\hat{v}}(\hat{q}_N)=\psi (\hat{q}_N), \quad {\hat{v}}'(\hat{q}_N)=\psi '(\hat{q}_N), \\ {\hat{v}}(\hat{q}_I)=\psi (\hat{q}_I), \quad {\hat{v}}'(\hat{q}_I)=\psi '(\hat{q}_I), \end{array}\right. } \end{aligned}$$

(39)

where $\psi (q):=\max \{L(q),k \}$.

We omit the proof of Lemma A.1, since it is purely technical and identical to the proof of Lemma B.1 in Han and Choi (2020). Using the function ${\hat{v}}$ in Lemma A.1, we define $V:[0,1]\mapsto {\mathbb {R}}$ as

$$\begin{aligned} V(q)&:={\left\{ \begin{array}{ll} {\hat{v}}(q) &{}\text {if } q\in (\hat{q}_N,\hat{q}_I),\\ \psi (q) &{}\text {if } q\in [0,\hat{q}_N) \cup (\hat{q}_I,1]. \end{array}\right. } \end{aligned}$$

(40)

From construction, we observe that V has enough regularity to apply Ito formula and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}} V(q)=0 \text { and } V(q)>\psi (q) &{} \text {for }q\in (\hat{q}_N,\hat{q}_I),\\ {\mathcal {L}} V(q)<0 \text { and } V(q)=\psi (q) &{}\text {for } q\in [0,\hat{q}_N) \cup (\hat{q}_I,1]. \end{array}\right. } \end{aligned}$$

(41)

We apply Theorem 8.1 in Liptser and Shiryaev (2001) to (3) and obtain the SDE of $q_t$:

$$\begin{aligned} dq_t&= \frac{q_t(1-q_t)\mu }{\sigma ^2}\kappa _t(dY_t - \mu q_t dt). \end{aligned}$$

(42)

Note that we set $\kappa _t=1$ in the above SDE, since we are considering the first best case. For any stopping time $\tau$ and a constant $t\ge 0$, Ito formula and the above SDE produce

$$\begin{aligned}&e^{-r (t\wedge \tau )} V(q_{t\wedge \tau })- \int _0^{t\wedge \tau } e^{-rs } c\, ds\\&\quad = V(q_0)+ \int _0^{t\wedge \tau } e^{-rs} {\mathcal {L}} V(q_s) ds+ \int _0^{t\wedge \tau } e^{-rs} V'(q_s) dq_s\\&\quad \le V(q_0)+ \int _0^{t\wedge \tau } e^{-rs} V'(q_s) \tfrac{q_s(1-q_s)\mu ^2}{\sigma ^2}(1_{\{\theta =G\}} -q_s) ds + \int _0^{t\wedge \tau } e^{-rs} V'(q_s) \tfrac{q_s(1-q_s)\mu }{\sigma }dW_s, \end{aligned}$$

where the inequality is due to (41), and the inequality becomes equality in cases $\tau$ equals ${\hat{\tau }}$ in (14). We take expectation above, then the boundedness of $V'$ and the definition of $q_s$ in (3) imply

$$\begin{aligned} {{\mathbb {E}}}\left[ e^{-r (t\wedge \tau )} V(q_{t\wedge \tau })- \int _0^{t\wedge \tau } e^{-rs } c ds \right] \le V(q_0) = {{\mathbb {E}}}\left[ e^{-r (t\wedge {\hat{\tau }})} V(q_{t\wedge {\hat{\tau }}})- \int _0^{t\wedge {\hat{\tau }}} e^{-rs } c ds \right] . \end{aligned}$$

(43)

We let $t \rightarrow \infty$ above, then the boundedness of V, monotone convergence theorem, and (41) produce

$$\begin{aligned} {{\mathbb {E}}}\left[ e^{-r \tau }\psi (q_{\tau })- \int _0^{\tau } e^{-rs } c ds \right] \le V(q_0) = {{\mathbb {E}}}\left[ e^{-r {\hat{\tau }}}\psi (q_{{\hat{\tau }}})- \int _0^{ {\hat{\tau }}} e^{-rs } c ds \right] . \end{aligned}$$

(44)

Now we conclude that ${\hat{\tau }}$ in (14) is the optimal stopping strategy and V is the first best value, i.e., $V={\widehat{V}}$ in (15).

B. Proof of Proposition 4.1

($\Rightarrow$) Suppose that a contract $\gamma$ is ICIR. Define functions $\pi$ and $\pi '$ as

$$\begin{aligned} \pi (q)&:= {\mathcal {P}}(q,{\mathcal {D}}(q),G) q +{\mathcal {P}}(q,{\mathcal {D}}(q),B)(1-q), \end{aligned}$$

(45)

$$\begin{aligned} \pi '(q)&:= {\mathcal {P}}(q,{\mathcal {D}}(q),G) - {\mathcal {P}}(q,{\mathcal {D}}(q),B). \end{aligned}$$

(46)

It is straightforward to check (16). Substituting $\phi _\gamma (q,q)$ with $\pi (q)$ and $\phi _\gamma (q,{{\tilde{q}}})$ with $\pi ({{\tilde{q}}})+\pi '({{\tilde{q}}})(q-{{\tilde{q}}})$ in (7), we obtain the inequality $\pi (q)\ge \pi ({{\tilde{q}}})+\pi '({{\tilde{q}}})(q-{{\tilde{q}}})$, which implies that $\pi$ is convex and $\pi '$ is a subgradient of $\pi$.

Since $\phi _\gamma (q,q)=\pi (q)$, according to Theorem 5.2.1 in Pham (2009), the unique bounded³ viscosity solution of (17) is $U_\gamma$ in (11). Due to the time homogeneity of the structure, for any $t\ge 0$,

$$\begin{aligned} U_\gamma (q_t)=\max _{\tau } {{\mathbb {E}}}\left[ e^{-r(\tau -t)} \phi _\gamma (q_\tau ,q_\tau ) - \int _t^\tau e^{-r(s-t)} c \, dt \, \Big | \, {\mathcal {F}}_t^Y \right] , \end{aligned}$$

(47)

where the maximum is taken over all stopping times $\tau \ge t$. Therefore, the condition (8) implies $U\ge 0$.

The restriction (4) and the expression of $\pi '$ in (46) imply that $\pi '(q)=0$ if ${\mathcal {D}}(q)=N$. Then, $\pi$ is minimized at $q\in {\mathcal {D}}^{-1}(N)$ by the the convexity of $\pi$.

($\Leftarrow$) Suppose such $\pi$ and $\pi '$ exist. Then, (16) produces $\phi _\gamma (q,q)=\pi (q)$ and $\phi _\gamma (q,{{\tilde{q}}})=\pi ({{\tilde{q}}})+\pi '({{\tilde{q}}})(q-{{\tilde{q}}})$. These equalities and the convexity of $\pi$ imply (7).

The unique bounded viscosity solution of (17) is $U_\gamma$ in (11) by Theorem 5.2.1 in Pham (2009). Therefore, the nonnegative of $U_\gamma$ and the characterization (47) imply (8).

Finally, if ${\mathcal {D}}(q)=N$, then (16) and $\pi '(q)=0$ produce (4). Therefore, we conclude that $\gamma$ is ICIR.

C. Some technical lemmas

For a given contract, the principal’s expected profit and the agent’s expected profit produce differential equations by the Feynman-Kac formula. In this section, we study some properties of the solution of the differential equations. In Appendix 1, these properties are used to prove Lemma 4.2 and Proposition 4.3.

The current appendix is rather technical and dense. To improve the readability, we first sketch why we care about the solutions of the differential equations and what properties we want to obtain. The Markovian structure of the optimal stopping problem enable us to focus on the hitting times of $(q_t)_{t\ge 0}$ only (see Lemma D.1 for details). Let us consider the case that the agent reports the belief $q_\tau$ when $q_t$ hits a or b, where $a<q_0<b$. In other words, let us consider $\tau = \inf \left\{ t\ge 0: \, q_t \notin (a,b) \right\}$. If the principal chooses N in case $q_\tau =a$ and I in case $q_\tau =b$ (i.e., ${\mathcal {D}}(a)=N$, ${\mathcal {D}}(b)=I$), then the sum of the principal’s and agent’s value functions, denoted by $v_{(a,b)}$, should satisfy the following differential equation with suitable boundary conditions:

$$\begin{aligned} {\mathcal {L}} v_{(a,b)}=0, \quad v_{(a,b)}(a)=k, \quad v_{(a,b)}(b)=L(b). \end{aligned}$$

(48)

We can find an explicit solution of (48), that is,

$$\begin{aligned} v_{(a,b)}(q)&:= -\tfrac{c}{r} + \tfrac{(c+r L(b))g(a)-(c+rk)g(b)}{r(f(b)g(a)-f(a)g(b))} \cdot f(q) + \tfrac{-(c+r L(b))f(a)+(c+rk) f(b)}{r(f(b)g(a)-f(a)g(b))} \cdot g(q), \end{aligned}$$

(49)

where the functions f, g and the constant $\beta$ are

$$\begin{aligned} f(q)&:= q \left( \tfrac{q}{1-q} \right) ^\beta , \quad g(q) := (1-q) \left( \tfrac{q}{1-q} \right) ^{-\beta }, \quad \beta : = \tfrac{1}{2}\left( \sqrt{1 + \tfrac{8 r \sigma ^2}{\mu ^2}} -1 \right) . \end{aligned}$$

In this case, the agent’s value function should satisfies the same differential equation with boundary condition that its derivative at $q=a$ equals zero. This boundary condition is due to Proposition 4.1 (3). Motivated by the agent’s value function, for $0<a<1$, $\rho , m\in {\mathbb {R}}$, we consider the same differential equation with boundary conditions at $q=a$ as below:

$$\begin{aligned}&{\mathcal {L}} u_{(a,\rho ,m)}=0, \quad u_{(a,\rho ,m)}(a)=\rho , \quad u'_{(a,\rho ,m)}(a)=m. \end{aligned}$$

(50)

The explicit solution of (50) is

$$\begin{aligned} u_{(a,\rho ,m)}(q)&:= -\tfrac{c}{r} + \tfrac{(\beta +a)(c+ r \rho )+a(1-a) r m}{r (1+2\beta )f(a)}\cdot f(q) + \tfrac{(\beta +1-a)(c+ r \rho )-a(1-a) r m}{r (1+2\beta )g(a)} \cdot g(q). \end{aligned}$$

(51)

Fig. 4

The left graph illustrates $v_{(a,b)}$ and the right graph illustrates $u_{(a,\rho ,m)}$

Lemma C.1 verifies that $v_{(a,b)}$ and $u_{(a,\rho ,m)}$ solve differential equations (48) and (50), respectively. Figure 4 illustrates $v_{(a,b)}$ and $u_{(a,\rho ,m)}$.

Intuitively, if we consider the smallest amount of the payment that the principal induces the agent to choose a as the left-hand boundary of the continuation interval for the stopping time, then in $u_{(a,\rho ,m)}$, we should set $\rho =0$ due to (8) and $m=0$ due to Proposition 4.1 (3). Therefore, in case the principal asks the agent to gather information, the principal’s maximization problem would be reduced to the following:

$$\begin{aligned} \max _{a,b\in {\mathbb {R}}\text { s.t. }a<q_0<b} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0). \end{aligned}$$

(52)

For fixed a, let h(a) be the point such that $v_{(a,h(a))}$ tangentially intersects with the graph of $\psi$. The existence and properties of the function h are studied in Lemma C.2. In Lemma C.3, it turns out that $v_{(a,h(a))}(q_0)\ge v_{a,b}(q_0)$ for all b, as Fig. 5 illustrates.

Fig. 5

The blue solid line is $v_{(a,h(a)}$ and the orange dashed line is $v_{(a,b)}$

Therefore, (52) becomes the following optimization problem with respect to one variable:

$$\begin{aligned} \max _{a<q_0} \quad v_{(a,h(a))}(q_0)-u_{(a,0,0)}(q_0). \end{aligned}$$

(53)

In Lemma C.4, we find a range of $q_0$ when the above maximum value is bigger than $\psi (q_0)$ (i.e., when it is better to ask the agent to gather information).

Now we move on to the detailed statements of the technical lemmas and their proofs.

Lemma C.1

The functions $v_{(a,b)}$ and $u_{(a,\rho ,m)}$ defined in (49) and (51) are the unique solutions of (48) and (50), respectively. Furthermore, for $0<a<q_0<b<1$, $v_{(a,b)}(q_0)$ has the following stochastic representation:

$$\begin{aligned} v_{(a,b)}(q_0)={\mathbb {E}}\left[ e^{-r\tau } \left( L(q_{\tau })\cdot 1_{\{q_{\tau }=b \}} +k \cdot 1_{\{q_{\tau }=a \}} \right) - \int _{0}^{\tau } e^{-r t} c \, dt \right] \quad \text {for} \quad \tau = \inf \left\{ t\ge 0 : \, q_t \notin (a,b) \right\} . \end{aligned}$$

(54)

Proof

Direct computations show that $v_{(a,b)}$ and $u_{(a,\rho ,m)}$ satisfy the differential equations with the boundary conditions. The solution is unique due to the uniform ellipticity of the operator ${\mathcal {L}}$ on every compact subset of (0, 1).

Since $v_{(a,b)}$ satisfies (48), we obtain the stochastic representation of $v_{(a,b)}$ by Proposition 7.2 and Lemma 7.4 in Karatzas and Shreve (1998). $\square$

In fact, the second-order differential equation ${\mathcal {L}}v=0$ has an explicit form of the general solution:

$$\begin{aligned} v(q)=-\tfrac{c}{r}+C_1 f(q)+ C_2g(q), \end{aligned}$$

(55)

where $C_1$ and $C_2$ are free constants. This implies that

$$\begin{aligned} \text {two different solutions of } {\mathcal {L}} v=0 \text { intersect at most once.} \end{aligned}$$

(56)

Lemma C.2

Assume that $0<a<p$. (1) There exists a constant ${\underline{m}}> - \tfrac{(\beta +a)(c+rk)}{a(1-a)r}$ such that

$$\begin{aligned} {\underline{m}} = \max \left\{ m\in {\mathbb {R}} \, : \, u_{(a,k,m)} \text { intersects with the graph of } q\mapsto L(q) \text { on } q\in (a,1) \right\} , \end{aligned}$$

(57)

and $u_{(a,k,{\underline{m}})}$ tangentially intersects with the graph of $q\mapsto L(q)$ at a single point, namely, at $q=h(a)\in (a,1)$. (2) h(a) above satisfies

$$\begin{aligned} u_{(a,k,{\underline{m}})}=v_{(a,h(a))}, \quad v_{(a,h(a))}(h(a))=L(h(a)), \quad v_{(a,h(a))}'(h(a))=L'(h(a)). \end{aligned}$$

(58)

(3) The function $h:[{\hat{q}}_N, p) \longrightarrow (p, {\hat{q}}_I]$ is continuous, bijective, and monotone decreasing. Therefore, its inverse function $h^{-1}: (p, {\hat{q}}_I]\longrightarrow [{\hat{q}}_N, p)$ is well-defined and continuous. Furthermore,

$$\begin{aligned} \begin{aligned}&v_{(a,h(a))}'(a)>0\quad \text {for}\quad a\in ({\hat{q}}_N, p), \quad v_{(a,h(a))}'(a)<0\quad \text {for}\quad a\in (0,{\hat{q}}_N). \end{aligned} \end{aligned}$$

(59)

Proof

Observe that $0<a<p$ implies that

$$\begin{aligned} u_{(a,k,m)}(a)=k>L(a). \end{aligned}$$

(60)

(1) Since ${\mathcal {L}} u_{(a,k,m)}=0$, the form of ${\mathcal {L}}$ in (18) implies that

$$\begin{aligned} u_{(a,k,m) \text { is strictly convex at } q} \quad \iff \quad \tfrac{c}{r}+u_{(a,k,m)}(q) > 0. \end{aligned}$$

(61)

By (51) and the fact that $\frac{f(q)}{g(q)}$ is an increasing function for $q\in (0,1)$, we observe that for $q\in (a,1)$ and $m> - \tfrac{(\beta +a)(c+rk)}{a(1-a)r}$, the inequality $\frac{c}{r}+u_{(a,k,m)}(q) >0$ holds. Therefore,

$$\begin{aligned} \text {if} \quad m> - \tfrac{(\beta +a)(c+rk)}{a(1-a)r}, \quad \text {then } u_{(a,k,m)}(q) \text { is strictly convex for } q\in (a,1). \end{aligned}$$

(62)

If $m> x-y$, then $u'_{(a,k,m)}(a)>x-y=L'(a)$. Together with (60) and (62), we obtain

$$\begin{aligned} \text {if} \quad m>x-y, \quad \text {then } u_{(a,k,m)}(q)>L(q) \text { for } q\in (a,1). \end{aligned}$$

(63)

On the other hand,

$$\begin{aligned} \lim _{m\downarrow - \frac{(\beta +a)(c+rk)}{a(1-a)r}} u_{(a,k,m)}(q) = -\tfrac{c}{r}+\tfrac{c+rk}{r g(a)} \cdot g(q) < L(q) \quad \text {for} \quad q \text { close enough to } 1. \end{aligned}$$

(64)

The observations (63) and (64) and the continuous dependence of $u_{(a,k,m)}$ on m imply the existence of ${\underline{m}}$ in (57). The strict convexity in (62) ensures that $u_{(a,k,{\underline{m}})}$ tangentially intersects with the graph of $q\mapsto L(q)$ at a single point, namely, at $q=h(a)\in (a,1)$.

(2) By construction, such h(a) satisfies

$$\begin{aligned} u_{(a,k,{\underline{m}})}(h(a))=L(h(a))\quad \text {and} \quad u_{(a,k,{\underline{m}})}'(h(a))=L'(h(a)). \end{aligned}$$

(65)

Comparing above condition with (48) and (50), we obtain (58).

(3) The continuity of h is clear from the construction. Comparing (39) with (48) and (58), we obtain

$$\begin{aligned} h({\hat{q}}_N)={\hat{q}}_I \quad \text {and} \quad {\widehat{V}} (q)=v_{({\hat{q}}_N,{\hat{q}}_I)}(q) \quad \text {for} \quad q\in [{\hat{q}}_N,{\hat{q}}_I]. \end{aligned}$$

(66)

For $a\in ({\hat{q}}_N,p)$, we prove $h(a)< {\hat{q}}_I$ by contradiction. Suppose that $h(a)> {\hat{q}}_I$. Since $v_{(a,h(a))}$ and $v_{({\hat{q}}_N,{\hat{q}}_I)}$ are strictly convex for $q\in ({\hat{q}}_I,h(a))$ and touch the straight line L tangentially, we deduce that $v_{(a,h(a))}({\hat{q}}_I)>v_{({\hat{q}}_N,{\hat{q}}_I)}({\hat{q}}_I)$ and $v_{(a,h(a))}(h(a))<v_{({\hat{q}}_N,{\hat{q}}_I)}(h(a))$. These inequalities imply that $v_{(a,h(a))}$ and $v_{({\hat{q}}_N,{\hat{q}}_I)}$ intersect at least once for $q\in ({\hat{q}}_I,h(a))$. Using (56), we conclude that $v_{(a,h(a))}(q)>v_{({\hat{q}}_N,{\hat{q}}_I)}(q)$ for $q\in (0,{\hat{q}}_I)$. This contradicts to $v_{(a,h(a))}(a)=k<v_{({\hat{q}}_N,{\hat{q}}_I)}(a)$, where the inequality is due to $a>{\hat{q}}_N$ and (39), together with the strict convexity of $v_{({\hat{q}}_N,{\hat{q}}_I)}$. Therefore, we conclude that for $a\in ({\hat{q}}_N,p)$, $h(a)\le {\hat{q}}_I$. We can exclude the case $h(a)={\hat{q}}_I$, because the solution of ${\mathcal {L}}v=0$ with boundary conditions $v({\hat{q}}_I)=L({\hat{q}}_I)$ and $v'({\hat{q}}_I)=L'({\hat{q}}_I)$ is unique.

Let us check (59). For $a\in ({\hat{q}}_N,p)$, we have already concluded that $h(a)<{\hat{q}}_I$, so $v_{(a,h(a))}$ and $v_{({\hat{q}}_N,{\hat{q}}_I)}$ intersect for $q\in (h(a),{\hat{q}}_I)$. If $v_{(a,h(a))}'(a)\le 0$, then the convexity argument implies that $v_{(a,h(a))}$ and $v_{({\hat{q}}_N,{\hat{q}}_I)}$ also intersect for $q\in ({\hat{q}}_N,a)$, but this contradicts to (56). Therefore, we conclude that $v_{(a,h(a))}'(a)> 0$. The case of $a\in (0,{\hat{q}}_N)$ can be treated by the same way.

We repeat the same procedure as above using (56), and conclude that $p<h(a_2)<h(a_1)<{\hat{q}}_I$ for ${\hat{q}}_N< a_1<a_2<p$. Then, to complete the proof, it only remains to show that the function $h:[{\hat{q}}_N, p) \longrightarrow (p, {\hat{q}}_I]$ is surjective. Indeed, for any $b\in (p,{\hat{q}}_I)$, there exists a unique solution v of ${\mathcal {L}}v=0$ with boundary conditions $v(b)=L(b)$ and $v'(b)=L'(b)$. By the same procedure as above, there exists $a\in ({\hat{q}}_I,p)$ such that $v(a)=k$. Then, we conclude that $v=v_{(a,b)}$ and $h(a)=b$. $\square$

Figure 6 illustrates Lemma C.2.

Fig. 6

The left graph illustrates $v_{(a,h(a))}'(a)>0$ and $p<h(a)<{\hat{q}}_I$ for ${\hat{q}}_N<a<p$. The right graph illustrates $p<h(a_2)<h(a_1)<{\hat{q}}_I$ for ${\hat{q}}_N< a_1<a_2<p$

Lemma C.3

For $a\in (0,p)$, we define the function $w_a:(0,1)\longrightarrow {\mathbb {R}}$ as

$$\begin{aligned} w_a(q):=v_{(a,h(a))}(q)-u_{(a,0,0)}(q). \end{aligned}$$

(67)

(1) Suppose that $q_0\in ({\hat{q}}_N,p]$. Then, there exists $a^*\in ({\hat{q}}_N,q_0)$ such that

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le w_{a^*}(q_0) \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1). \end{aligned}$$

(2) Suppose that $q_0\in (p,{\hat{q}}_I)$. Then, one of the following holds:

(i) There exists $a^*\in ({\hat{q}}_N,h^{-1}(q_0))$ such that

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le w_{a^*}(q_0) \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1). \end{aligned}$$

(ii) Otherwise,

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le \psi (q_0) \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1). \end{aligned}$$

Proof

We start with several observations. The boundary condition in (50) and the observation (61) imply

$$\begin{aligned} u_{(a,0,0)}(q)>0 \quad \text {for}\quad 0<a<q<1. \end{aligned}$$

(68)

Since $q\mapsto \frac{f(q)}{g(q)}$ is a strictly increasing function, direct computations show that

$$\begin{aligned}&\tfrac{\partial }{\partial m} \Big ( u_{(a,\rho ,m)}(q) \Big ) = \tfrac{a(1-a)}{1+2\beta } \left( \tfrac{f(q)}{f(a)} -\tfrac{g(q)}{g(a)} \right) >0 \quad \text {for} \quad 0<a<q<1, \end{aligned}$$

(69)

$$\begin{aligned}&\tfrac{\partial }{\partial a} \Big ( u_{(a,0,0)}(q) \Big )= -\tfrac{c\beta (1+\beta )}{(1-a)a r(1+2\beta )} \left( \tfrac{f(q)}{f(a)} -\tfrac{g(q)}{g(a)} \right)<0 \quad \text {for} \quad 0<a<q<1. \end{aligned}$$

(70)

For $0<a<p$, the construction of ${\underline{m}}$ in Lemma C.2 implies that if $m>\underline{m}$, then $u_{(a,k,m)}$ does not intersects with the graph of $q\mapsto L(q)$ on $q\in (a,1)$. Therefore, by (48) and (50), there exists $m\le {\underline{m}}$ such that $v_{(a,b)}=u_{(a,k,m)}$. Together with (58) and (69), this implies

$$\begin{aligned} v_{(a,b)}(q)\le v_{(a,h(a))}(q) \quad \text {for}\quad 0<a<p, \,\, a<b<1,\,\, a<q<1. \end{aligned}$$

(71)

By the same argument as in the proof of Lemma C.2 (3) using (56), we obtain

$$\begin{aligned}&v_{(a,b)}(q_0)< \psi (q_0) \quad \text {for}\quad p<q_0<b,\,\, h^{-1}(q_0)<a<q_0. \end{aligned}$$

(72)

See Figure 7 for illustrations of (71) and (72).

Fig. 7

The left graph illustrates the inequality (71). The right graph illustrates the inequality (72)

By Lemma C.1, we observe that $w_a$ satisfies

$$\begin{aligned} - r w_a(q) +\tfrac{q^2(1-q)^2 \mu ^2}{2\sigma ^2} w_a''(q)=0, \quad w_a(a)=k. \end{aligned}$$

(73)

Furthermore, $u_{(a,0,0)}'(a)=0$ and (59) imply

$$\begin{aligned} w_a'(a)>0 \quad \text {for} \quad a\in ({\hat{q}}_N,p). \end{aligned}$$

(74)

(1) Suppose that $q_0\in ({\hat{q}}_N,p]$. Since $v_{(a,b)}$ and $u_{(a,0,0)}$ continuously depend on a and b, the map $a\mapsto w_a(q_0)$ is continuous due to the continuity of h in Lemma C.2. Since a continuous function has a maximum value on a compact interval, there exists $a^*\in [{\hat{q}}_N, q_0]$ such that

$$\begin{aligned} w_a(q_0)\le w_{a^*} (q_0) \quad \text {for}\quad a\in [{\hat{q}}_N, q_0] \end{aligned}$$

(75)

The stochastic representation (54) and the optimality of ${\hat{\tau }}$ in (14) imply that

$$\begin{aligned} \tfrac{\partial }{\partial a} \Big ( v_{(a,b)}(q_0) \Big ) \Big |_{(a,b)=({\hat{q}}_N,{\hat{q}}_I)}=0, \quad \tfrac{\partial }{\partial b} \Big ( v_{(a,b)}(q_0) \Big ) \Big |_{(a,b)=({\hat{q}}_N,{\hat{q}}_I)}=0. \end{aligned}$$

(76)

From the construction of h in Lemma C.2, we can see that

$$\begin{aligned} h(a)-h({\hat{q}}_N) = {\mathcal {O}}(a-{\hat{q}}_N), \end{aligned}$$

(77)

with standard big ${\mathcal {O}}$ notation. By (76) and (77), we obtain

$$\begin{aligned} v_{(a,h(a))}(q_0)-v_{({\hat{q}}_N,{\hat{q}}_I)}(q_0) = \zeta (a) \cdot (a-{\hat{q}}_N), \quad \text {for a function } \zeta \text { such that } \lim _{a\rightarrow {\hat{q}}_N} \zeta (a)=0. \end{aligned}$$

(78)

Combining (70) and (78), we conclude that $w_a(q_0)>w_{{\hat{q}}_N}(q_0)$ for a slightly bigger than ${\hat{q}}_N$, and this means that $a^* \ne {\hat{q}}_N$. Also, by (73) and (74), we observe that $w_a (q_0)>k=w_{q_0}(q_0)$ for $a\in ({\hat{q}}_N, q_0)$, and this implies that $a^* \ne q_0$. All in all, we improve (75) as

$$\begin{aligned} \text {there exists } a^*\in ({\hat{q}}_N, q_0) s.t. w_a(q_0)\le w_{a^*} (q_0) for a\in [{\hat{q}}_N, q_0], \quad \text {and } w_{a^*}(q_0)>k. \end{aligned}$$

(79)

Suppose that $a\in [{\hat{q}}_N, q_0)$ and $b\in (q_0,1)$. Then by (71) and (79), we obtain the desired inequality: $v_{(a,b)}(q_0)- u_{(a,0,0)}(q_0)\le w_a(q_0) \le w_{a^*}(q_0)$.

To complete the proof of part (1), we suppose that $a\in (0,{\hat{q}}_N)$ and $b\in (q_0,1)$. In case $v_{(a,b)}(q_0)\le k$, (68) and (79) imply $v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0)<k< w_{a^*}(q_0)$. In case $v_{(a,b)}(q_0)> k$, the argument we made for (71) implies that $v_{(a,b)}'(a)<0$ and $b>p\ge q_0$. By the intermediate value theorem, there exists ${\tilde{a}}\in ({\hat{q}}_N, q_0)$ such that $v_{(a,b)}({\tilde{a}})=k$. Since the solution of the differential equation (48) is unique, we conclude that $v_{(a,b)}=v_{({\tilde{a}}, b)}$. This observation, together with the inequalities (70) and (71), implies that

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0)<v_{({\tilde{a}},b)}(q_0) -u_{({\tilde{a}},0,0)}(q_0) \le w_{{\tilde{a}}}(q_0) \le w_{a^*}(q_0) \end{aligned}$$

where the last inequality is due to ${\tilde{a}}\in ({\hat{q}}_N, q_0)$ and (79).

(2) Suppose that $q_0\in (p,{\hat{q}}_I)$. By the same way as in the proof of part (1), we conclude that

$$\begin{aligned} \begin{aligned}&\text {there exists } a^*\in ({\hat{q}}_N, h^{-1}(q_0)] \text { such that }\\&v_{(a,b)}(q_0)- u_{(a,0,0)}(q_0) \le w_{a^*}(q_0) \text { for } a\in (0, h^{-1}(q_0)] \text { and } b\in (q_0,1). \end{aligned} \end{aligned}$$

(80)

Suppose that $a\in (h^{-1}(q_0),q_0)$ and $b\in (q_0,1)$. By (72), we obtain $v_{(a,b)}(q_0)<\psi (q_0)$. This inequality and (68) produce $v_{(a,b)}(q_0)- u_{(a,0,0)}(q_0)<\psi (q_0)$. Combining this observation and (80), we conclude that

$$\begin{aligned}&\text {there exists } a^*\in ({\hat{q}}_N, h^{-1}(q_0)] \text { such that }\\&v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le \max \left\{ w_{a^*}(q_0), \psi (q_0) \right\} \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1). \end{aligned}$$

To complete the proof, due to the above inequality, it is enough to observe that

$$\begin{aligned} w_{h^{-1}(q_0)}(q_0)=\psi (q_0)-u_{(h^{-1}(q_0),0,0)}(q_0)<\psi (q_0), \end{aligned}$$

where the inequality is due to (68). $\square$

Lemma C.4

There exists $q^\dagger \in (p,{\hat{q}}_I)$ such that the followings hold:

(1) If $q_0 \in ({\hat{q}}_N, q^\dagger )$, then there exist $q^*_N\in ({\hat{q}}_N, \min \{q_0,p\})$ and $q^*_I \in (\max \{q_0,p\},{\hat{q}}_I)$ such that

$$\begin{aligned}&v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le v_{(q^*_N,q^*_I)}(q_0)-u_{(q^*_N,0,0)}(q_0) \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1), \\&\psi (q_0)<v_{(q^*_N,q^*_I)}(q_0)-u_{(q^*_N,0,0)}(q_0), \quad 0<u_{(q^*_N,0,0)}(q_0), \quad \text {and} \quad v_{(q^*_N,q^*_I)}(q_0)<{\widehat{V}}(q_0). \end{aligned}$$

(2) If $q_0 \in [q^\dagger ,{\hat{q}}_I)$, then

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le \psi (q_0)\quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1). \end{aligned}$$

Proof

For $a\in ({\hat{q}}_N,p)$, by (73) and (74), we observe that $w_a$ is strictly increasing and strictly convex for $q\in (a,1)$. The definition of $w_a$ in (67) and the positivity of $u_{(a,0,0)}$ in (68) imply that

$$\begin{aligned} w_a(h(a))<v_{(a,h(a))}(h(a))=\psi (h(a)). \end{aligned}$$

By the strict convexity and the intermediate value theorem, we conclude that there exists $q^\#$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} w_a(q)>\psi (q) &{} \text {for}\quad q\in (a,q^\#),\\ w_a(q)\le \psi (q) &{} \text {for}\quad q\in [q^\#,h(a)]. \end{array}\right. } \end{aligned}$$

Therefore, $w_a(q)<\psi (q)$ implies $w_a(q')<\psi (q')$ for $a<q'<q$ and $w_a(q+\epsilon )<\psi (q+\epsilon )$ for small enough $\epsilon >0$. This observation shows that there exists a constant $q^\dagger \in ({\hat{q}}_N, {\hat{q}}_I)$ such that

$$\begin{aligned} ({\hat{q}}_N, q^\dagger )=\Big \{ q\in ({\hat{q}}_N, {\hat{q}}_I): \,\, w_a(q)>\psi (q) \text { for some }a\in ({\hat{q}}_N,p) \text { such that } a<p<h(a) \, \Big \}, \end{aligned}$$

(81)

since the right-hand side is a connected open set.

By Lemma C.3 (1) and (79), if $q_0\in ({\hat{q}}_N,p]$, then there exists $a^*\in ({\hat{q}}_N,q_0)$ such that

$$\begin{aligned} v_{(a,b)}(q_0)-u_{(a,0,0)}(q_0) \le w_{a^*}(q_0) \quad \text {for} \quad a\in (0,q_0), \,\, b\in (q_0,1)\quad \text {and} \quad w_{a^*}(q_0)>\psi (q_0). \end{aligned}$$

This implies $q^\dagger >p$. We also observe that $q^\dagger <{\hat{q}}_I$, because

$$\begin{aligned} w_{{\hat{q}}_N}({\hat{q}}_N)=v_{({\hat{q}}_N,{\hat{q}}_I)}({\hat{q}}_I)-u_{({\hat{q}}_N,0,0)} ({\hat{q}}_I)<v_{({\hat{q}}_N,{\hat{q}}_I)}({\hat{q}}_I)=\psi ({\hat{q}}_I) \end{aligned}$$

and the continuous dependence of $w_a$ on a imply that $w_a(q)<\psi (q)$ for q and h(a) close enough to ${\hat{q}}_I$.

From the above construction of $q^\dagger \in (p,{\hat{q}}_I)$, we set $q^*_N:=a^*$ and $q^*_I:=h(a^*)$ in Lemma C.3 to obtain the desired result, except the inequality $v_{(q^*_N,q^*_I)}(q_0)<{\widehat{V}}(q_0)$. The inequality is derived from the proof of Lemma C.2 (3). $\square$

D. Proof of Lemma 4.2 and Proposition 4.3

For a given ICIR contract, we first characterize the optimal stopping time and the value of the agent’s maximization problem.

Lemma D.1

Let $\gamma$ be an ICIR contract and $\tau _{\gamma }$ be the corresponding optimal stopping time of the agent. Then, one of the following holds:

• (1) $\tau _{\gamma }= 0$ and $U_{\gamma }(q_0)=\phi _\gamma (q_0,q_0)$.

• (2) There exists two constants ${{\underline{q}}}$ and ${{\overline{q}}}$ such that $0<{{\underline{q}}}<q_0<{{\overline{q}}}<1$ and

  • (i) $\tau _{\gamma }=\inf \left\{ t\ge 0: \, q_t \notin ({{\underline{q}}}, {{\overline{q}}}) \right\}$.

  • (ii) $U_\gamma$ is a classical solution of ${\mathcal {L}}U=0$ on $({{\underline{q}}},{{\overline{q}}})$, and $U_\gamma (q) >\phi _\gamma (q,q)$ for $q\in ({{\underline{q}}},{{\overline{q}}})$.

  • (iii) $U_\gamma ({{\underline{q}}})\ge 0$, and $\lim _{q \downarrow {{\underline{q}}}} U_\gamma '(q)=0$ if ${{\underline{q}}}\in {\mathcal {D}}^{-1}(N)$.

Proof

By Proposition 4.1, there exists a convex function $\pi :[0,1]\longrightarrow {\mathbb {R}}$ such that the value function $U_\gamma$ is the unique viscosity solution of (11). It is well-known that (see, e.g., Chapter 5.2 in Pham (2009)) the optimal stopping time $\tau _\gamma$ is characterized as

$$\begin{aligned} \tau _{\gamma }=\inf \left\{ t\ge 0 : \, q_t \notin {\mathcal {C}} \right\} \quad \text {for}\quad {\mathcal {C}}:=\left\{ q\in (0,1) : U_\gamma (q)\ne \pi (q) \right\} \end{aligned}$$

(82)

and it is the unique optimal stopping time due to our tiebreak rule. One of the following holds:

(1) Suppose that $q_0\notin {\mathcal {C}}$. In this case, $\tau _\gamma =0$ and $U_\gamma (q_0)=\pi (q_0)=\phi _\gamma (q_0,q_0)$.

(2) Suppose that $q_0 \in {\mathcal {C}}$. In this case, since the set ${\mathcal {C}}$ is an open set due to the continuity of $U_\gamma$ and $\pi$, there exists ${{\underline{q}}}<q_0<{{\overline{q}}}$ such that ${{\underline{q}}}, {{\overline{q}}}\notin {\mathcal {C}}$ and $({{\underline{q}}},{{\overline{q}}}) \subset {\mathcal {C}}$. Then, $\tau _\gamma$ in (82) becomes $\tau _{\gamma }=\inf \left\{ t\ge 0: \, q_t \notin ({{\underline{q}}}, {{\overline{q}}}) \right\}$ and $U_\gamma (q) >\pi (q)=\phi _\gamma (q,q)$ for $q\in ({{\underline{q}}},{{\overline{q}}})$. By Lemma 5.2.2 in Pham (2009), $U_\gamma$ is a classical solution of ${\mathcal {L}}U=0$ on $({{\underline{q}}},{{\overline{q}}})$. Since the solution of ${\mathcal {L}}U=0$ is unbounded on (0, 1), we conclude that $0<{{\underline{q}}}<{{\overline{q}}}<1$.

Since $\gamma$ is ICIR, we have $U_\gamma ({{\underline{q}}})\ge 0$.

Finally, assume that ${{\underline{q}}}\in {\mathcal {D}}^{-1}(N)$. The form of ${\mathcal {L}}U=0$ implies that $U_\gamma$ is convex for $q\in ({{\underline{q}}},{{\overline{q}}})$. Since $U_\gamma ({{\underline{q}}})=\pi ({{\underline{q}}})$ and $U_\gamma (q) >\pi (q)$ for $q\in ({{\underline{q}}},{{\overline{q}}})$, we conclude that

$$\begin{aligned} \alpha _1:=\lim _{q\downarrow {{\underline{q}}}} U_\gamma '(q) \ge \pi '({{\underline{q}}})=0, \end{aligned}$$

where the equality is due to the third statement in Proposition 4.1. Indeed, we can further check that $\alpha _1=0$. Suppose not, i.e., $\alpha _1 >0$. Let $\varphi (x):=U_\gamma ({{\underline{q}}})+\frac{\alpha _1}{2} (x-{{\underline{q}}})+\alpha _2 (x-{{\underline{q}}})^2$ for $\alpha _2\in {\mathbb {R}}$. Then $U_\gamma \ge \pi$ and $\pi '({{\underline{q}}})=0$ imply that $\varphi$ satisfies

$$\begin{aligned} \varphi ({{\underline{q}}})=U_\gamma ({{\underline{q}}}) \quad \text {and}\quad \varphi (x)\le U_\gamma (x)\text {for } x \text { near } {{\underline{q}}}. \end{aligned}$$

Since $\varphi \in C^2({\mathbb {R}})$, the above conditions imply that $\varphi$ can be used as a test function for the viscosity supersolution property of $U_\gamma$ (see, e.g., Chapter 5 in Pham (2009) for details):

$$\begin{aligned} {\mathcal {L}}\varphi ({{\underline{q}}})=-c - r U_\gamma ({{\underline{q}}}) +\frac{{{\underline{q}}}^2(1-{{\underline{q}}})^2 \mu ^2}{\sigma ^2} \cdot \alpha _2 \le 0. \end{aligned}$$

However, if we choose large enough $\alpha _2$ above, then the inequality is violated. Therefore, we reach a contradiction and conclude that $\alpha _1=0$. $\square$

According to Lemma D.1, we have

$$\begin{aligned} \Gamma = \Gamma _s \cup \Gamma _c,\quad \text {where} \quad {\left\{ \begin{array}{ll}\Gamma &{}:=\left\{ \text {ICIR contract} \right\} ,\\ \Gamma _s&{}:=\left\{ \gamma \in \Gamma \text { that satisfies the case (1) in Lemma D.1 } \right\} ,\\ \Gamma _c&{}:=\left\{ \gamma \in \Gamma \text { that satisfies the case (2) in Lemma D.1 } \right\} . \end{array}\right. } \end{aligned}$$

(83)

Considering the principal’s decision map ${\mathcal {D}}$, we can further decompose $\Gamma _c$ as

$$\begin{aligned} \Gamma _c=\bigcup _{d_1, d_2 \in \{N,I \}}\Gamma _c^{d_1 d_2}, \end{aligned}$$

(84)

where

$$\begin{aligned} \Gamma _c^{d_1 d_2}:=\left\{ \gamma \in \Gamma _c : \, \text { For } {{\underline{q}}}\text { and } {{\overline{q}}}\text { in the case (2) of Lemma D.1,} \, {\mathcal {D}}({{\underline{q}}})=d_1 \text { and } {\mathcal {D}}({{\overline{q}}})=d_2 \right\} \end{aligned}$$

(85)

Lemma D.2

For $d_1, d_2 \in \{ N,I \}$, let $\Gamma _c^{d_1 d_2}$ be as in (85). Then,

$$\begin{aligned} \max _{\gamma \in \Gamma \setminus \Gamma _c^{NI}} W_{\gamma }(q_0)\le \psi (q_0). \end{aligned}$$

(86)

Proof

Obviously, $W_{\gamma _0}(q_0)\le \psi (q_0)$ for $\gamma \in \Gamma _s$. To prove the lemma, by (83) and (84), it is enough to check $W_\gamma (q_0)\le \psi (q_0)$ for $\gamma \in \Gamma _c^{II}\cup \Gamma _c^{NN} \cup \Gamma _c^{IN}$. For $\gamma \in \Gamma _c^{II}\cup \Gamma _c^{NN} \cup \Gamma _c^{IN}$, there exist ${{\underline{q}}}$ and ${{\overline{q}}}$ satisfying the conditions in Lemma D.1 (2) case. Let l be the linear function whose graph is the line segment connecting two points $({{\underline{q}}},l({{\underline{q}}}))$ and $({{\overline{q}}},l({{\overline{q}}}))$, where

$$\begin{aligned} l({{\underline{q}}}):=L({{\underline{q}}})\cdot 1_{\{{\mathcal {D}}({{\underline{q}}})=I \}}+ k\cdot 1_{\{{\mathcal {D}}({{\underline{q}}})=N \}}, \quad l({{\overline{q}}}):=L({{\overline{q}}}) \cdot 1_{\{{\mathcal {D}}({{\overline{q}}})=I \}}+k\cdot 1_{\{{\mathcal {D}}({{\overline{q}}})=N \}}. \end{aligned}$$

(87)

The form of $\tau _\gamma$ described in Lemma D.1 (2) implies that

$$\begin{aligned} W_\gamma (q_0)&={\mathbb {E}} \left[ e^{-r \tau _{\gamma }} l( q_{\tau _{\gamma }}) \right] - U_\gamma (q_0) -{\mathbb {E}} \left[ \int _{0}^{\tau _{\gamma }} e^{-rt} c \, dt \right] \\&< {\mathbb {E}} \left[ l( q_{\tau _{\gamma }}) \right] = l(q_0), \end{aligned}$$

where the inequality is due to $r>0$, $c>0$ and $U_\gamma \ge 0$, and the second equality is due to the martingale property of $(q_t)_{t \ge 0}$ and the linearity of l. Finally, we complete the proof by observing that that $\gamma \in \Gamma _c^{II}\cup \Gamma _c^{NN} \cup \Gamma _c^{IN}$ implies $l(q) \le \psi (q)$ for $q\in [{{\underline{q}}},{{\overline{q}}}]$. $\square$

For constants a and b, we define $\Gamma _c^{NI}(a,b)$ as

$$\begin{aligned} \Gamma _c^{NI}(a,b):= \left\{ \gamma \in \Gamma _c : \, \text { In the case (2) of Lemma D.1,}\, \, a={{\underline{q}}}, \, b={{\overline{q}}}, \, {\mathcal {D}}(a)=N, {\mathcal {D}}(b)=I \right\} . \end{aligned}$$

(88)

By definition, we observe that

$$\begin{aligned} \Gamma _c^{NI}=\bigcup _{a\in (0,q_0), \, b\in (q_0,1)} \Gamma _c^{NI}(a,b). \end{aligned}$$

(89)

Lemma D.3

Assume that $0<a<q_0<b<1$. Then, $U_{\gamma }(q_0) \ge u_{(a,0,0)}(q_0)$ for $\gamma \in \Gamma _c^{NI}(a,b)$.

Proof

Let $\gamma \in \Gamma _c^{NI}(a,b)$. According to Lemma D.1 and (50), there exist constants $\rho \ge 0$ and $m\ge 0$ such that $U_\gamma (q_0)=u_{(a,\rho ,m)}(q_0)$. Observe that

$$\begin{aligned}&\tfrac{\partial }{\partial \rho } \left( u_{(a,\rho ,m)}(q_0) \right) = \tfrac{\beta +a}{(1+2\beta ) f(a)}\cdot f(q_0) + \tfrac{\beta +1-a}{(1+2\beta )g(a)} \cdot g(q_0)>0. \end{aligned}$$

The above inequality and (69), together with the restrictions $\rho \ge 0$ and $m \ge 0$, produce the inequality $u_{(a,0,0)}(q_0)\le u_{(a,\rho ,m)}(q_0)= U_\gamma (q_0)$. $\square$

Now, we are ready to prove Lemma 4.2 and Proposition 4.3 at the same time. By the stochastic representation (54) and the definition of $U_\gamma$ and $W_\gamma$ in (11) and (12), we obtain

$$\begin{aligned} W_\gamma (q_0)=v_{(a,b)}(q_0)-U_\gamma (q_0)\quad \text {for} \quad \gamma \in \Gamma _c^{NI}(a,b), \,\, a\in (0,q_0),\,\, b\in (q_0,1). \end{aligned}$$

(90)

Let $q^\dagger \in (p,{\hat{q}}_I)$ be as in Lemma C.4. We consider two cases: $q_0 \in [q^\dagger ,{\hat{q}}_I)$ and $q_0 \in ({\hat{q}}_N, q^\dagger )$.

(1) Suppose that $q_0 \in [q^\dagger ,{\hat{q}}_I)$. By Lemma C.4, Lemma D.3 and Eq. (90), we obtain

$$\begin{aligned} W_\gamma (q_0)\le \psi (q_0)\quad \text {for} \quad \gamma \in \Gamma _c^{NI}(a,b), \,\, a\in (0,q_0),\,\, b\in (q_0,1). \end{aligned}$$

(91)

We combine the above inequality, Lemma D.2 and the decomposition (89) to conclude that $W_\gamma (q_0) \le \psi (q_0)$ for $\gamma \in \Gamma$. In this case, the principal does not propose a contract.

(2) Suppose that $q_0 \in ({\hat{q}}_N, q^\dagger )$. By Lemma C.4, Lemma D.3 and Eq. (90), there exist $q^*_N\in ({\hat{q}}_N, \min \{q_0,p\})$ and $q^*_I \in (\max \{q_0,p\},{\hat{q}}_I)$ such that

$$\begin{aligned}{} & {} W_\gamma (q_0) \le v_{(q^*_N,q^*_I)}(q_0)-u_{(q^*_N,0,0)}(q_0) \quad \text {for} \quad \gamma \in \Gamma _c^{NI}(a,b), \,\, a\in (0,q_0),\,\, b\in (q_0,1),\nonumber \\{} & {} \psi (q_0)<v_{(q^*_N,q^*_I)}(q_0)-u_{(q^*_N,0,0)}(q_0), \quad 0<u_{(q^*_N,0,0)}(q_0), \quad \text {and} \quad v_{(q^*_N,q^*_I)}(q_0)<{\widehat{V}}(q_0). \end{aligned}$$

(92)

Now we define $\gamma ^*=({\mathcal {D}}^*, {\mathcal {P}}^*)$ as

$$\begin{aligned} \begin{aligned} {\mathcal {D}}^*(q)&:={\left\{ \begin{array}{ll} I, &{}\text {if} \quad {\hat{\pi }}(q)>0 \\ N, &{}\text {if}\quad {\hat{\pi }}(q) = 0 \end{array}\right. },\\ {\mathcal {P}}^*(q, d, \theta )&:= \pi ^*(q) - (\pi ^*)'(q)q + (\pi ^*)'(q)\cdot 1_{\{\theta =G \}} \quad \text {for both } d=I \text { and } N,\\ \text {where} \quad \pi ^*(q)&:= \max \left\{ 0, u_{(q^*_N,0,0)} (q^*_I)+ u_{(q^*_N,0,0)}' (q^*_I) \big (q-q^*_I \big ) \right\} . \end{aligned} \end{aligned}$$

(93)

Figure 8 illustrates $u_{(q^*_N,0,0)}$ and $\pi ^*$.

Fig. 8

The functions $u_{( q^*_N,0,0)}$ and $\pi ^*$

The definition of $\pi ^*$ and the differential equation (50) imply that

$$\begin{aligned} \begin{aligned}&{\mathcal {L}}u_{(q^*_N,0,0)}(q)=0\quad \text {and} \quad u_{(q^*_N,0,0)}(q)> \pi ^* (q) \quad \text {for} \quad q\in (q^*_N, q^*_I),\\&u_{(q^*_N,0,0)}(q^*_N)=\pi ^*(q^*_N), \quad u_{(q^*_N,0,0)}(q^*_I)=\pi ^*(q^*_I),\quad u_{(q^*_N,0,0)}'(q^*_N)=(\pi ^*)'(q^*_N), \quad u_{(q^*_N,0,0)}'(q^*_I)=(\pi ^*)'(q^*_I). \end{aligned} \end{aligned}$$

(94)

Therefore, by Proposition 4.1, the corresponding value of the agent and the optimal stopping time are

$$\begin{aligned} U_{{\hat{\gamma }}}(q_0)=u_{(q^*_N,0,0)} (q_0), \quad \tau _{\hat{\gamma }}=\inf \left\{ t\ge 0 : \, q_t \notin (q^*_N, q^*_I) \right\} . \end{aligned}$$

By (54) and the above observation, we obtain

$$\begin{aligned} W_{{\hat{\gamma }}}(q_0)=v_{(q^*_N,q^*_I)}(q_0)-u_{(q^*_N,0,0)}(q_0). \end{aligned}$$

We combine the above equality, Lemma D.2, the decomposition (89) and the inequalities (92), and conclude that ${\hat{\gamma }}$ constructed in (93) is the optimal contract with the optimal stopping time (31) and the inequalities in (32)–(33) hold.

Obviously, (1) and (2) above imply Lemma 4.2.