Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach

Abstract

In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner’s optimal policy, we characterize it by necessary and sufficient stochastic Kuhn–Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect.

Appendix 1: Some Proofs and Technical Results

1.1 On the Proof of Proposition 3.3

In this section we prove Proposition 3.3. The proof is a generalization of Theorem 3.2 in [8] to the case of a multivariate optimal consumption problem with both monotone and classical absolutely continuous controls. Sufficiency easily follows from concavity of the utility functions $u^i$, $i=1,\ldots ,n$. On the other hand, the next Lemma accomplishes the proof of the necessity part. Necessity is proved by linearizing the original problem (3.2) around its optimal solution $(\underline{x}_{*}, \underline{C}_{*})$, by showing that $(\underline{x}_{*}, \underline{C}_{*})$ solves the linearized problem as well and that it satisfies some flat-off conditions as those of (3.8).

Recall the notation $x:=\sum _{i=1}^n x^i$ and $C:=\sum _{i=1}^n C^i$.

Lemma 6.1

Let Assumptions 1 and 2 hold and $(\underline{x}_{*}, \underline{C}_{*}) \in \mathcal {B}_w$ be optimal for problem (3.2) and set

$$\begin{aligned} \Psi _{*}(t):= E\biggl [\int _{t}^T e^{-\int _0^s r(u)\,du} \sum _{i=1}^n \gamma ^i\,u^{i}_c \big (x^i_{*}(s), C_{*}(s) \big )\,ds\Big |{\mathscr {F}}_{t}\biggr ]. \end{aligned}$$

(6.1)

Then $(\underline{x}_{*}, \underline{C}_{*})$

• i. solves the linear optimization problem

  $$\begin{aligned} \sup _{(\underline{x}, \underline{C}) \in \mathcal {B}_w} E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_x \big (x^i_{*}(t),C_{*}(t) \big ) x^i(t)dt + \int _0^T\Psi _{*}(t)dC(t)\biggr ];\nonumber \\ \end{aligned}$$

  (6.2)

• ii. satisfies

  $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \Big (e^{-\int _0^t r(s)ds} \gamma ^i u^i_x(x^i_{*}(t),C_{*}(t)) - M\psi _x(t)\Big )x^i_*(t) = 0 \quad \,\,P\otimes dt\text {-a.e.},\, i=1,\ldots ,n, \\ \displaystyle E\biggl [\int _0^T \Big (\Psi _{*}(t) - M \psi _c(t)\Big )dC_{*}(t)\biggr ] = 0, \end{array} \right. \end{aligned}$$

  (6.3)

  with

  $$\begin{aligned} M:={}&(P\otimes dt)\text {-}{{\mathrm{ess\,sup}}}\Biggl [\max \Biggl \{\frac{e^{-\int _0^t r(s)ds} \gamma ^i u^i_x(x^i_{*}(t),C_{*}(t))}{\psi _x(t)};\; i=1,\ldots ,n \Biggr \}\Biggr ] \nonumber \\ \vee \;&P\text {-}{{\mathrm{ess\,sup}}}\Biggl [\sup _{t\in [0,T]} \frac{\Psi _{*}(t)}{\psi _c(t)}\Biggr ]. \end{aligned}$$

  (6.4)

Proof

The proof splits into two steps.

• Step 1. Let $(\underline{x}_{*}, \underline{C}_{*}) \in \mathcal {B}_w$ be optimal for problem (3.2). For $(\underline{x}, \underline{C}) \in \mathcal {B}_w$ and $\epsilon \in [0,1]$, define the admissible strategy $(\underline{x}_{\epsilon }, \underline{C}_{\epsilon })$ with $\underline{x}_{\epsilon }(t):= \epsilon \underline{x}(t) + (1-\epsilon )\underline{x}_{*}(t)$ and such that $C_{\epsilon }(t)=\epsilon C(t) + (1-\epsilon )C_{*}(t)$. Notice that $\underline{x}_{\epsilon }(t)$ and $C_{\epsilon }(t)$ respectively converge to $\underline{x}_{*}(t)$ and $C_{*}(t)$ for all $t \in [0,T]$ a.s. when $\epsilon \downarrow 0$. Now, optimality of $(\underline{x}_{*}, \underline{C}_{*})$, Assumption 1.ii, concavity of $u^i$ and an application of Fubini’s Theorem allow us to write

  $$\begin{aligned} 0\ge & {} \frac{1}{\epsilon }[U_{SP}(\underline{x}_{\epsilon }, \underline{C}_{\epsilon }) - U_{SP}(\underline{x}_{*}, \underline{C}_{*})] \nonumber \\\ge & {} E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x\big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )\big (x^i(t) - x^i_{*}(t)\big )\,dt\biggr ] \nonumber \\&+\, E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_c \big (x^i_{\epsilon }(t), C_{\epsilon }(t)\big ) \big (C(t) - C_{*}(t)\big )\,dt\biggr ]\nonumber \\= & {} E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big ) \big (x^i(t) - x^i_{*}(t) \big )\,dt\biggr ] \nonumber \\&+ E\biggl [\int _0^T \Phi _{\epsilon }(t) \big (dC(t) - dC_{*}(t) \big )\biggr ], \end{aligned}$$

  (6.5)

  where $\Phi _{\epsilon }(t):= \int _{t}^T e^{-\int _0^s r(u)\,du} \sum _{i=1}^n \gamma ^i\,u^{i}_c(x^i_{\epsilon }(s), C_{\epsilon }(s))\,ds$. One has

  $$\begin{aligned}&\liminf _{\epsilon \downarrow 0} E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )x^i(t) dt \biggr ] \\&\quad \ge E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{*}(t), C_{*}(t) \big )x^i(t) dt \biggr ] \end{aligned}$$

  and

  $$\begin{aligned} \liminf _{\epsilon \downarrow 0} E\biggl [\int _0^T \Phi _{\epsilon }(t) dC(t) \biggr ] \ge E\biggl [\int _0^T \Phi _{*}(t) dC(t) \biggr ], \end{aligned}$$

  with $\Phi _{*}:=\Phi _{0}$, by Fatou’s Lemma. We now claim (and we prove it later) that

  $$\begin{aligned}&\lim _{\epsilon \downarrow 0} E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )x^i_{*}(t) dt \biggr ] \nonumber \\&= E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{*}(t), C_{*}(t)\big )x^i_{*}(t) dt \biggr ] \end{aligned}$$

  (6.6)

  and

  $$\begin{aligned} \lim _{\epsilon \downarrow 0} E\biggl [\int _0^T \Phi _{\epsilon }(t) dC_{*}(t) \biggr ] = E\biggl [\int _0^T \Phi _{*}(t) dC_{*}(t) \biggr ]. \end{aligned}$$

  (6.7)

  Hence from (6.5)

  $$\begin{aligned}&E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{*}(t), C_{*}(t) \big )x^i(t) \,dt\biggr ] + E\biggl [\int _0^T \Phi _{*}(t)dC(t)\biggr ] \\&\quad \! \le E\biggl [\int _{0}^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{*}(t), C_{*}(t) \big )x^i_{*}(t) \,dt\biggr ] + E\biggl [\int _0^T \Phi _{*}(t)dC_{*}(t)\biggr ] \end{aligned}$$

  and by replacing $\Phi _{*}$ with its optional projection $\Psi _{*}$ as defined in (6.1) (cf. [22], Theorem 1.33) it follows that $(\underline{x}_{*}, \underline{C}_{*})$ is optimal for problem (6.2) as well. To conclude the proof we must prove (6.6) and (6.7). To prove (6.6) it suffices to show that the family $(\Gamma ^1_{\epsilon })_{\epsilon \in [0,\frac{1}{2}]}$ given by

  $$\begin{aligned} \Gamma ^1_{\epsilon }(t):= e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )x^i_{*}(t) \end{aligned}$$

  is $P\otimes dt$-uniformly integrable. Concavity of $u^i$ and the fact that $x^{i}_{\epsilon }(t) \ge \frac{1}{2}x^i_{*}(t)$ a.s. for $\epsilon \in [0,\frac{1}{2}]$ and every $t \in [0,T]$ lead to

  $$\begin{aligned}&\Gamma ^1_{\epsilon }(t) \le 2 e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_x( x^i_{\epsilon }(t),C_{\epsilon }(t))x^i_{\epsilon }(t) \nonumber \\&\quad \le 2 e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i \bigl [u^{i} \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )-u^{i}(0, C_{\epsilon }(t))\bigr ], \end{aligned}$$

  and the last term in the right-hand side above is $P\otimes dt$-uniformly integrable by Assumption 1.ii. Then (6.6) holds by Vitali’s Convergence Theorem. As for (6.7) note that by Fubini’s Theorem

  $$\begin{aligned} \int _0^T \Phi _{\epsilon }(t)dC_{*}(t) = \int _0^T e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_c \big (x^i_{\epsilon }(t), C_{\epsilon }(t) \big )C_{*}(t)dt. \end{aligned}$$

  Hence, to have (6.7) it suffices to show that the family

  $$\begin{aligned} \Gamma ^2_{\epsilon }(t):= e^{-\int _0^t r(s)\,ds} \sum _{i=1}^n \gamma ^i u^{i}_c(x^i_{\epsilon }(t), C_{\epsilon }(t))C_{*}(t) \end{aligned}$$

  is $P\otimes dt$-uniformly integrable, but this follows by employing arguments similar to those used for $(\Gamma ^1_{\epsilon })_{\epsilon \in [0,\frac{1}{2}]}$.

• Step 2. We now show that the flat-off conditions (6.3) hold for any solution $(\hat{\underline{x}}, \hat{\underline{C}})$ of the linear problem (6.2). Then, by Step 1, they also hold for $(\underline{x}_{*},\underline{C}_{*})$. Notice that for every $(\underline{x}, \underline{C}) \in \mathcal {B}_w$ one has

  $$\begin{aligned}&E\biggl [\int _0^T \sum _{i=1}^n e^{-\int _0^t r(s)ds} \gamma ^i u^i_x(x^i_{*}(t),C_{*}(t)) x^i(t)dt + \int _0^T \Psi _{*}(t) dC(t)\biggr ] \nonumber \\&\quad \le M E\biggl [\int _0^T \sum _{i=1}^n \psi _x(t) x^i(t) dt + \int _0^T \psi _c(t)dC(t)\biggr ] = Mw \end{aligned}$$

  (6.8)

  by definition of M (cf. (6.4)). Obviously, if $(\underline{x}, \underline{C})$ satisfies (6.3) we then have equality in (6.8). On the other hand, if

  $$\begin{aligned}&\sup _{(\underline{x}, \underline{C}) \in \mathcal {B}_w} E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_x(x^i_{*}(t),C_{*}(t)) x^i(t)dt + \int _0^T \Psi _{*}(t) dC(t)\biggr ]\nonumber \\&\quad = Mw, \end{aligned}$$

  (6.9)

  then equality holds through (6.8) and we obtain (6.3). It therefore remains to prove (6.9). To this end take $K<M$ and define the investment strategies

  $$\begin{aligned} x^{i}_K(t):= {\left\{ \begin{array}{ll} \alpha &{} \text {if } e^{-\int _0^t r(s)ds} \gamma ^i u^i_x(x^i_{*}(t),C_{*}(t)) \ge K \psi _x(t), \\ 0 &{} \text {else} \end{array}\right. } \quad \text {and}\quad C_K(t):=\alpha \mathbbm {1}_{[\sigma _K,T]}(t), \end{aligned}$$

  with the stopping time

  $$\begin{aligned} \sigma _K:= \inf \{t \in [0,T): \Psi _{*}(t) \ge K \psi _c(t)\} \wedge T \end{aligned}$$

  and some $\alpha$ such that $E[\int _0^T \sum _{i=1}^n \psi _x(t)x^i_K(t) dt + \int _0^T \psi _c(t)dC_K(t)] =w$. Note that one can find such $\alpha$. Indeed, suppose to the contrary that for $\alpha =1$, $\max \{x^{i}_K(t);i=1,\ldots ,n\}=C_K(t)=0$ on $[0,T) P \otimes dt-$a.e. for some $K>0$. This would mean $M \le K$ by the definition of M (cf. (6.4)). We now have

  $$\begin{aligned} Mw\ge & {} \sup _{(\underline{x}, \underline{C}) \in \mathcal {B}_w} E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_x \big (x^i_{*}(t),C_{*}(t) \big ) x^i(t)dt + \int _0^T \Psi _{*}(t) dC(t)\biggr ] \\\ge & {} E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_x \big (x^i_{*}(t),C_{*}(t) \big ) x^i_K(t)dt + \int _0^T \Psi _{*}(t) dC_K(t)\biggr ]\\\ge & {} K\,E\biggl [\int _0^T \sum _{i=1}^n \psi _x(t)x^i_K(t) dt + \alpha \psi _c(\sigma _K)\mathbbm {1}_{\{\sigma _K < T\}}\biggr ]\\\ge & {} K\,E\biggl [\int _0^T \sum _{i=1}^n \psi _x(t)x^i_K(t) dt + \int _0^T \psi _c(t)dC_K(t)\biggr ] = Kw, \end{aligned}$$

  which yields (6.9) by letting $K \uparrow M$.$\square$

We are now able to prove Proposition 3.3.

Proof of Proposition 3.3

Proof

Sufficiency follows from concavity of utility function $u^i$, $i=1,\ldots ,n$, (cf. Assumption 1). Indeed, for $(\underline{x}_{*}, \underline{C}_{*}) \in \mathcal {B}_w$ satisfying (3.8) and for $(\underline{x}, \underline{C})$ any other admissible policy we may write

$$\begin{aligned}&U_{SP}(\underline{x}_{*}, \underline{C}_{*}) - U_{SP}(\underline{x}, \underline{C}) \\&\quad \ge E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_{x} \big (x^i_{*}(t),C_{*}(t) \big )(x^i_{*}(t) - x^i(t))dt\biggr ] \\&\quad \quad + E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_{c} \big (x^i_{*}(t),C_{*}(t) \big )(C_{*}(t) - C(t)) dt\biggr ] \\&\quad = E\biggl [\int _0^T e^{-\int _0^t r(s)ds} \sum _{i=1}^n \gamma ^i u^i_{x} \big (x^i_{*}(t),C_{*}(t) \big )(x^i_{*}(t) - x^i(t))dt\biggr ] \\&\quad \quad + E\biggl [\int _0^T \biggl (\int _{t}^T e^{-\int _0^s r(u)\,du} \sum _{i=1}^n \gamma ^i u^{i}_c \big (x^i_{*}(s), C_{*}(s)\big )\,ds\biggr ) \big (dC_{*}(t) - dC(t)\big )\biggr ] \\&\quad \ge \lambda (w - w) = 0, \end{aligned}$$

where (3.8) and Fubini’s Theorem lead to the second inequality, whereas the last one is implied by the first and the fourth of (3.8) and by the budget constraint. Finally, Lemma 6.1 yields the proof of the necessity part. $\square$

1.2 A Useful Simple Lemma

Lemma 6.2

Let Assumption 1 hold and set $h^i(\psi ,c):=u_c^i(g(\psi ,c),c)$ for every $i=1,\ldots ,n$, $\psi ,c>0$, where $g^i(\cdot ,c)$ is the inverse of $u_x^i(\cdot ,c)$. Then $c \mapsto h^i(\psi ,c)$ is strictly decreasing for any $\psi >0$. Moreover, if also $u_{xc}\ge 0$, then $\psi \mapsto h(\psi ,c)$ is nonincreasing for any $c>0$.

Proof

$h^i$ is strictly decreasing in c by strict concavity of $u^i$. Indeed, since $u^i_x(g^i(\psi ,c),c)=\psi$ is constant in c for all $\psi >0$, by implicit differentiation $g^i_c(\psi ,c)=-u^i_{xc}(g^i(\psi ,c),c)/u^i_{xx}(g^i(\psi ,c),c)$. Then $h^i_c(\psi ,c)=u^i_{xc}(g^i(\psi ,c),c)g^i_c(\psi ,c)+u^i_{cc}(g^i(\psi ,c),c)=-\bigl [u^i_{xc}(g^i(\psi ,c),c)\bigr ]^2/u^i_{xx}(g^i(\psi ,c),c)+u^i_{cc}(g^i(\psi ,c),c)<0$, as the Hessian of $u^i$ is negative definite by Assumption 1.i.

On the other hand, $h^i(\psi ,c)=u^i_c(g^i(\psi ,c),c)$ is nonincreasing in $\psi$ if also $u^i_{xc}\ge 0$, since $g^i(\cdot ,c)$ is strictly decreasing like $u^i_x(\cdot ,c)$, which it is the inverse of. $\square$

1.3 Proof of Proposition 3.4

Proof

For any given $\lambda > 0$, set $X(t):=\lambda \psi _c(t)\mathbbm {1}_{\{t < T\}}$. Such a process vanishes at T, it is of class (D) and lower semicontinuous in expectation by Assumption 2. Moreover, we define the atomless, optional random Borel measure $\mu (\omega ,dt):= e^{-\int _0^t r(\omega ,s) ds}dt$ and the random field

$$\begin{aligned} f^{i}(\omega ,t,\ell ):= {\left\{ \begin{array}{ll} h^i(\frac{\lambda }{\gamma ^i}e^{\int _0^t r(\omega ,s) ds}\psi _x(\omega ,t), -\frac{1}{\ell }) &{} \text {if } \ell <0, \\ -\ell &{} \text {if } \ell \ge 0, \end{array}\right. } \end{aligned}$$

for some $\gamma ^i>0$. Notice that $(\omega ,t)\mapsto f^{i}(\omega ,t,\ell )$ is progressively measurable and $P\otimes \mu (dt)$ integrable for any given and fixed $\ell \in \mathbb {R}$. Moreover, since $c \mapsto h^i(\psi , c)$ is strictly decreasing (cf. Lemma 6.2) and, by assumption, satisfies the Inada conditions

$$\begin{aligned} \lim _{c \downarrow 0}h^i(\psi ,c)=+\infty , \qquad \lim _{c \uparrow \infty }h^i(\psi ,c)=0, \end{aligned}$$

(6.10)

then $f^{i}(\omega ,t,\cdot )$ is strictly decreasing from $+\infty$ to $-\infty$. All these properties are clearly inherited by the function $\sum _{i=1}^n \gamma ^i h^i(\psi , \cdot )$, being $\gamma ^i > 0$, $i=1,\ldots ,n$.

Following the arguments, e.g., in the proof of Proposition 3.4 of [21] (see also the proof of Theorem 2.4 in [9]), we can apply Theorem 3 of [5] to have existence of an optional signal process $l^*$ solving (3.11). Such a process is also upper right-continuous and therefore it is unique up to indistinguishability by [5], Theorem 1, and Meyer’s optional section theorem (see, e.g., [18], Theorem IV.86) (cf. again [21], proof of Proposition 3.4, or [7], proof of Theorem 1). $\square$

1.4 Proof of Proposition 5.4

Due to Theorem 4.4, to find the Nash equilibrium strategy of the public good contribution game (4.1) in our homogeneous and symmetric setting it suffices to solve backward equation (4.9).

Recall that $h^i(\psi ,c):=u^i_c(g^i(\psi ,c),c)$ with $g^i(\cdot , c)$ the inverse of $u^i_x(\cdot , c)$. For any $\lambda ^i > 0$, straightforward computations lead to $h^i(\lambda ^i e^{rt}\psi _x(t), C(t)) = \delta (\lambda ^i \mathcal {E}_x(t))^{\frac{\alpha }{\alpha -1}}C^{\frac{\alpha + \beta -1}{1 - \alpha }}(t)$, with $\delta :=\frac{\beta }{\alpha }\left( \frac{\alpha + \beta }{\alpha }\right) ^{\frac{1}{\alpha -1}}$. Set $\hat{C}^i(t)= \sup _{0\le s \le t}\hat{l}(s) \vee 0$ for some progressively measurable process $\hat{l}$ solving

$$\begin{aligned} E\biggl [\int _{\tau }^{\infty } \delta e^{-rs} (\lambda ^i \mathcal {E}_x(s))^{\frac{\alpha }{\alpha -1}} \Big ( n\sup _{\tau \le u \le s} \hat{l}(u)\Big )^{\frac{\alpha + \beta -1}{1 - \alpha }} ds \biggm |{\mathscr {F}}_{\tau }\biggr ] = \lambda ^i e^{-r\tau } \mathcal {E}_c(\tau ), \end{aligned}$$

i.e.,

$$\begin{aligned} E\biggl [\int _{0}^{\infty } \delta e^{-ru} (\lambda ^i)^{\frac{\alpha }{\alpha -1}} \frac{{\mathcal {E}_x}^{\frac{\alpha }{\alpha -1}}(u+\tau )}{\mathcal {E}_c(\tau )} \inf _{0 \le s \le u}\Big ( (n{\hat{l}})^{\frac{\alpha + \beta -1}{1 - \alpha }}(s+\tau )\Big ) du \biggm |{\mathscr {F}}_{\tau }\biggr ] = \lambda ^i.\nonumber \\ \end{aligned}$$

(6.11)

Now take $\hat{l}(t):=\frac{\kappa }{n}{\mathcal {E}_{c}}^{\frac{1 -\alpha }{\alpha + \beta -1}}(t){\mathcal {E}_x}^{\frac{\alpha }{\alpha + \beta -1}}(t)$ for some constant $\kappa$ and use independence and stationarity of Lévy increments to rewrite (6.11) as

$$\begin{aligned} \kappa ^{\frac{\alpha + \beta - 1}{1 - \alpha }} E\biggl [\int _{0}^{\infty } \delta e^{-ru} \inf _{0 \le s \le u}\left( \mathcal {E}_c(s)\mathcal {E}_x^{\frac{\alpha }{\alpha -1}}(u-s)\right) du \biggr ] = {(\lambda ^i)}^{\frac{1}{1-\alpha }}. \end{aligned}$$

(6.12)

We claim (and we discuss later) that (cf. (5.17)) $A:= E[\int _{0}^{\infty }\delta e^{- r u} \inf _{0 \le s \le u} \left( \mathcal {E}_c(s) \mathcal {E}_x^{-\frac{\alpha }{1 - \alpha }}(u-s)\right) du]$ is finite. Then by solving (6.12) for $\lambda ^i$ one obtains

$$\begin{aligned} \lambda ^i= A^{1-\alpha } \kappa ^{\alpha + \beta - 1}=:\hat{\lambda }^i. \end{aligned}$$

But now $\hat{x}^i(t) = [\hat{\lambda }^i \left( \frac{\alpha + \beta }{\alpha }\right) \mathcal {E}_x(t) \hat{C}^{-\beta }(t)]^{\frac{1}{\alpha - 1}}$, and therefore

$$\begin{aligned} \hat{x}^i(t) = (\hat{\lambda }^i)^{-\frac{1}{1-\alpha }}\biggl [\left( \frac{\alpha + \beta }{\alpha }\right) \mathcal {E}_x(t) {\kappa }^{-\beta }\inf _{0 \le s \le t}\left( \mathcal {E}_c^{\frac{\beta (1-\alpha )}{1-\alpha - \beta }}(s)\mathcal {E}_x^{\frac{\alpha \beta }{1-\alpha - \beta }}(s)\right) \biggr ]^{-\frac{1}{1-\alpha }}; \end{aligned}$$

that is,

$$\begin{aligned} \hat{x}^i(t) = \kappa \gamma (t) \end{aligned}$$

(6.13)

with $\gamma (t)$ as in (5.14).

To determine $\kappa$ we use the budget constraint $\mathbb {E}[\int _{0}^{\infty }\psi _x(t) \hat{x}^i(t) dt + \int _{0}^{\infty }\psi _c(t) d\hat{C}^i(t)] = w$. Indeed, by (6.13) we have

$$\begin{aligned} \kappa E\biggl [\int _0^{\infty } \psi _x(t) \gamma (t) dt + \frac{1}{n}\int _{0}^{\infty }\psi _c(t)d\theta (t)\biggr ] = w, \end{aligned}$$

(6.14)

since $\hat{C}^i(t) = \sup _{0 \le s \le t}\hat{l}(s) = \frac{\kappa }{n} \sup _{0 \le s \le t}(\mathcal {E}_c^{-\frac{1 - \alpha }{1 - \alpha - \beta }}(s) \mathcal {E}_x^{-\frac{\alpha }{1 - \alpha - \beta }}(s)) = \frac{\kappa }{n} \theta (t)$ with $\theta (t)$ as in (5.15), and if $E[\int _0^{\infty } \psi _x(t) \gamma (t) dt + \frac{1}{n}\int _{0}^{\infty }\psi _c(t)d\theta (t)]<\infty$. Now the result follows by solving (6.14) for $\kappa$. Finally, arguments as those in the proof of Proposition 5.3 allow to show that under Assumption 3.4. all the quantities above are finite, thus completing the proof.