Proposition (Investment behavior under weighted discounting.) With the notation above, the triple
solves the Bellman system with its boundary condition. Hence, $\hat u$ is an equilibrium stopping rule and $V^e$ is the corresponding value function.
Proof: From the boundary condition $w(0;r) = 0$, it is easy to see that $V (0) = 0$, which satisfies the boundary condition $V (0) = \max\lbrace 0,x − I\rbrace {|}_{x=0}$.
On the continuation region $(0,x_∗)$, since $V(x)=\int_0^\infty w(x;r)dF(r)$ and since $w(x;r)$ follows the ODE defined on continuation region
we can get the equation
which is equivalent to
This gives that $\mathcal{A}V(x)-\int_0^\infty rw(x;r) dF(r)=0$ in continuation region. But it still remains to check that $V(x)-G(x)>0$, i.e. $V(x)-(x-I)>0$ on continuation region and
on stopping region. Since $\mu(r)>1$ and $V(x)$ convex, the linearity of the payoff function, together with the value matching and smooth pasting conditions, yields inequality $V(x)-(x-I)>0$. As the second inequality
can be simplified as
Since $b<\inf\lbrace r\in [0,\infty)\mid F(r)>0\rbrace$, LHS is increasing in $x$, we only need to prove when $x=x_\ast$. If $b\leq0$, the fact that $\mu(r)>1$ and $x_\ast>I$ gives immediately the above inequality. If $b>0$, plug in the fact that when $x=x_\ast$, $x=A/(A-1)$ which is $\int_0^\infty\mu(r)dF(r)/(\int_0^\infty\mu(r)dF(r)-1)$. Let $\kappa=\int_0^\infty\mu(r)dF(r)$, we have
by taking derivatives, we can prove that
Hence, we obtain the desired inequality.
The triggering boundary $x_∗$ depends on the project return rate and volatility, the entry cost, and the discount function. In particular, the influence of these economic factors is independent from one another and determined by the function $\mu(r)$, the constant $I$, and the weighting distribution $F$, respectively. This observation allows the study of the impact of these factors on the investment decision separately. The following proposition lists several properties of the triggering boundary x∗with respect to the entry costs and payoff process.
The impact of the underlying process and entry cost on the triggering boundary
- $x_∗$ is greater than the entry cost $I$.
- $x_∗$ increases with the entry cost $I$.
- $x_∗$ increases with the return rate $b$ and volatility $\sigma$ of the project dynamics.
The impact of the discount factor on the triggering boundary
Proposition (Larger discount factors imply later investment) Consider weighted discount functions $h^F$ and $h^G$ with weighting distributions $F$ and $G$. Then,
where $x^G_∗$, $x^F_∗$ is the triggering boundary with weighting distribution $h^G$, $h^F$ respectively.
Note that the triggering boundary $x_∗$ is expressed in terms of the weighting distribution $F$ and not in terms of the corresponding weighted discount function $h^F$. Even though the relationship between $F$ and the group discount function $h^F$ is one-to-one, the actual correspondence for each and every case can be quite complicated. The proof’s main idea is to rewrite the intuitive expression for the triggering threshold, through a more complicated and less intuitive expression that, however, involves the original discount function.
Proposition (Decreasing impatience and later investment) Consider weighted discount function $h^F$ and $h^G$ with weighting distributions $F$ and $G$ and measures of decreasing impatience $P_F$ and $P_G$, respectively. Let the expectation of $F$ be no less than that of $G$ and $P_G(t) \geq P_F(t)$ $\forall t \geq 0$. Then $x^G_∗\geq x^F_∗$, where $x^G_∗$, $x^F_∗$ are the triggering boundary with discount function $h^G$, $h^F$.
