if\ }p_{t}<\bar{P}{\rm\ and\ }i_{t}<\bar{I}\\\ r_{h}(p_{t},i_{t})&{\rm if\
}p_{t}=\bar{P}{\rm\ and\ }i_{t}<\bar{I}\\\ -r_{f}&{\rm if\
}i_{t}=\bar{I},\end{cases}$

where
$\mbox{E}\left[V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})\right]=\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]$.
For any $i_{t}^{+}\geq i_{t}$, we know that

$\displaystyle\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]=-r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}^{+}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}^{+}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle\
\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{+}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]$

where the first inequality holds because
$V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})\geq-r_{f}$ and
$r_{h}(\bar{P},i_{t}e^{\psi_{t}})\geq-r_{f}$ (this follows from the assumption
$r_{f}>c_{2}\bar{I}$), and the second inequality holds because
$r_{h}(\bar{P},i_{t}e^{\psi_{t}})\geq r_{h}(\bar{P},i_{t}^{+}e^{\psi_{t}})$
and the induction hypothesis together with $i_{t}^{+}e^{\psi_{t}}\geq
i_{t}e^{\psi_{t}}$ imply that
$\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]\geq\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{+}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]$
Consequently, $V_{t}(p_{t},i_{t},\mathcal{I}_{t})$ is also non-increasing in
$i_{t}$.

On the other hand, for any $p_{t}^{+}\geq p_{t}$, we know that

E
$\displaystyle[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]=-r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}-\ln_{p_{t}}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}-\ln_{p_{t}}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=-r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}-\ln_{p_{t}}^{+}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}-\ln_{p_{t}}^{+}}}^{\ln{\bar{P}-\ln_{p_{t}}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln_{p_{t}}^{+}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln_{p_{t}}^{+}}^{\ln{\bar{P}}-\ln_{p_{t}}}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\leq-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}-\ln_{p_{t}}^{+}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln_{p_{t}}^{+}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\leq-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}-\ln_{p_{t}}^{+}}}V_{t+1}(p_{t}^{+}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln_{p_{t}}^{+}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=$
$\displaystyle\mbox{E}[V_{t+1}(p_{t}^{+}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})]$
(34)

where the first inequality holds because $r_{h}(\bar{P},i_{t}e^{\psi_{t}})\geq
V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1})$ (this holds
due to the fact that the value function at a fixed impurity level can never
exceed the maximum harvesting reward achievable at that impurity level), and
the second inequality holds because of the induction hypothesis and
$p_{t}e^{\phi_{t}}\leq p_{t}^{+}e^{\phi_{t}}$. Consequently, we have
$V_{t}(p_{t},i_{t},\mathcal{I}_{t})$ is also non-decreasing in $p_{t}$.

Proof of Theorem 2

According to Theorem 2, if $\pi^{\star}(p_{t},i_{t}^{-},\mathcal{I}_{t})=H$,
then $\pi^{\star}(p_{t},i_{t}^{+},\mathcal{I}_{t})=H$ for any $i^{+}>i^{-}$.
Assume that $\pi^{\star}(p_{t},i_{t}^{-},\mathcal{I}_{t})=H$ but
$\pi^{\star}(p_{t},i_{t}^{+},\mathcal{I}_{t})=C$ as the contradiction
hypothesis. Then, it holds that

$\displaystyle r_{h}(p_{t},i_{t}^{-})$ $\displaystyle\geq-
c_{u}+\gamma\mbox{E}\left[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{-}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})\right]$
$\displaystyle r_{h}(p_{t},i_{t}^{+})$ $\displaystyle\leq-
c_{u}+\gamma\mbox{E}\left[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{+}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})\right].$

Thus, we have

$\displaystyle r_{h}(p_{t},i_{t}^{-})$ $\displaystyle-
r_{h}(p_{t},i_{t}^{+})\geq\gamma\mbox{E}\left[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{-}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})\right]-\gamma\mbox{E}\left[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}^{+}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})\right]$
$\displaystyle=$ $\displaystyle-\gamma
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{-}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}+\gamma
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}^{-}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}^{-}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}^{+}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}^{+}e^{\psi_{t}},\mathcal{I}_{t+1})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle\gamma
r_{f}\left[\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}-\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}\right]$
(35)
$\displaystyle+\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}\int_{-\infty}^{\infty}r_{h}(0,i_{t}^{-}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle-\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}\int_{-\infty}^{\infty}r_{h}(\bar{P},i_{t}^{-}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle\gamma
r_{f}\left[\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}-\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{+}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}\right]$
(36)
$\displaystyle+\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}r_{h}(0,i_{t}^{-}e^{\psi_{t}})f_{t}^{(i)}(\psi_{t})d\psi_{t}-\gamma\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}r_{h}(\bar{P},i_{t}^{-}e^{\psi_{t}})f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle=$ $\displaystyle\gamma
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}^{+}}}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}-\gamma
c_{1}\bar{P}\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}^{-}}}f_{t}^{(i)}(\psi_{t})d\psi_{t}$

which contradicts the condition, so that we have
$\pi^{\star}(p_{t},i_{t}^{+})=H$. Notice that, inequality (35) holds because:
(i) $V_{t+1}(0,i_{t}^{-}e^{\psi_{t}},\mathcal{I}_{t+1})\geq
r_{h}(0,i_{t}^{-}e^{\psi_{t}})$ by the definition of the value function (i.e.,
at any decision epoch, the value function at a particular state can never be
less than the harvesting reward at that state) and
$V_{t+1}(p_{t}e^{\phi_{t}},i_{t}^{-}e^{\psi_{t}},\mathcal{I}_{t+1})\geq
V_{t+1}(0,i_{t}^{-}e^{\psi_{t}},\mathcal{I}_{t+1})$ by the non-decreasing
behavior of the value function in $p_{t}$ for any protein growth-rate
realization $\phi_{t}$, and (ii)
$V_{t+1}(p_{t}e^{\phi_{t}},i_{t}^{+}e^{\psi_{t}},\mathcal{I}_{t+1})\leq
r_{h}(\bar{P},i_{t}^{+}e^{\psi_{t}})\leq r_{h}(\bar{P},i_{t}^{-}e^{\psi_{t}})$
as the reward function $r_{h}$ is non-increasing in $i_{t}$. Finally, the
inequality (36) holds because
$\ln{\bar{I}}-\ln{i_{t}^{+}}\leq\ln{\bar{I}}-\ln{i_{t}^{-}}$.

Proof of Theorem 3

Note that $\mbox{E}[\widetilde{\Phi}_{t}]=\alpha_{t}^{(p)}$ and
$\mbox{E}[\widetilde{\Psi}_{t}]=\alpha_{t}^{(i)}$. We first consider the “non-
decreasing in $\alpha_{t}^{(p)}$” part, given the rest of the knowledge state
is fixed. By using the Bellman’s equation, we can do the proof through
backward induction. To start with, notice that

$\displaystyle
V_{\bar{T}}(p_{\bar{T}},i_{\bar{T}},\mathcal{I}_{\bar{T}})=\begin{cases}r_{h}(p_{\bar{T}},i_{\bar{T}}),~{}~{}&\mbox{~{}if~{}}i_{\bar{T}}<\bar{I}\\\
-r_{f},~{}~{}&\mbox{~{}if~{}}i_{\bar{T}}=\bar{I},\end{cases}$

is constant in $\alpha_{\bar{T}}^{(p)}$, and hence, non-decreasing in
$\alpha_{\bar{T}}^{(p)}$. As the induction hypothesis, assume that
$V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})$ is non-decreasing in
$\alpha_{t+1}^{(p)}$. At decision epoch $t<\bar{T}$, we have

$V_{t}(p_{t},i_{t},\mathcal{I}_{t})=\begin{cases}{\max}\left\\{r_{h}(p_{t},i_{t}),-c_{u}+\gamma\mbox{E}\left[V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})\right]\right\\}&{\rm
if\ }p_{t}<\bar{P}{\rm\ and\ }i_{t}<\bar{I}\\\ r_{h}(p_{t},i_{t})&{\rm if\
}p_{t}=\bar{P}{\rm\ and\ }i_{t}<\bar{I}\\\ -r_{f}&{\rm if\ }i_{t}=\bar{I}.\\\
\end{cases}$

Note that a change in the value of $\alpha_{t}^{(p)}$ affects the expectation
through the predictive distribution $f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})$;
and it also affects the knowledge state variables $\alpha_{t+1}^{(p)}$ and
$\beta_{t+1}^{(p)}$ through the knowledge states update, i.e., see (5). Let
$\mathcal{I}_{t+1}(\alpha_{t}^{(p)})$ denote the updated knowledge states.
Then, for any $\delta>0$, it holds that

$\displaystyle\mbox{E}\left[V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})\right]=\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))|\alpha_{t}^{(p)}]$
(37) $\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}-\delta}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}-\delta}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle\leq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
(38)
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}-\delta}V_{t+1}(p_{t}e^{\phi_{t}+\delta},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}-\delta}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$

where the inequality holds because $V_{t+1}(p,i,\mathcal{I})$ is non-
decreasing function of $p$, and
$V_{t+1}(p_{t}e^{\phi_{t}+\delta},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))\leq
r_{h}(\bar{P},i_{t}e^{\psi_{t}})$ for any $\phi_{t}$ and $\psi_{t}$.

Now consider a value $\alpha_{t}^{(p)+}$ such that
$\alpha_{t}^{(p)+}-\alpha_{t}^{(p)}=\delta\geq 0$, and let
$\phi_{t}^{\prime}=\phi_{t}+\delta$. It holds for the predictive density that

$\displaystyle f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})=$
$\displaystyle\pi^{-1/2}\dfrac{\Gamma\left(\lambda_{t}^{(p)}+\dfrac{1}{2}\right)}{\Gamma(\lambda_{t}^{(p)})}\left[\dfrac{\nu_{t}^{(p)}}{2\beta_{t}^{(p)}(1+\nu_{t}^{(p)})}\right]^{1/2}\left[1+\dfrac{\nu_{t}^{(p)}(\phi_{t}-\alpha^{(p)})^{2}}{2\beta_{t}^{(p)}(1+\nu_{t}^{(p)})}\right]^{-\lambda_{t}^{(p)}-1/2}$
$\displaystyle=$
$\displaystyle\pi^{-1/2}\dfrac{\Gamma\left(\lambda_{t}^{(p)}+\dfrac{1}{2}\right)}{\Gamma(\lambda_{t}^{(p)})}\left[\dfrac{\nu_{t}^{(p)}}{2\beta_{t}^{(p)}(1+\nu_{t}^{(p)})}\right]^{1/2}\left[1+\dfrac{\nu_{t}^{(p)}(\phi_{t}^{\prime}-\alpha_{t}^{(p)+})^{2}}{2\beta_{t}^{(p)}(1+\nu_{t}^{(p)})}\right]^{-\lambda_{t}^{(p)}-1/2}=f_{t}^{(p)}(\phi_{t}^{\prime}|\alpha_{t}^{(p)+}).$

Then, it holds for the RHS of (38) that

$\displaystyle{\rm RHS\ of\ }\eqref{ineq.theo9.1}\leq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}-\delta}V_{t+1}(p_{t}e^{\phi_{t}+\delta},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)+}))f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}-\delta}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t}|\alpha_{t}^{(p)})f_{t}^{(i)}(\psi_{t})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t})d\psi_{t}$
(39)
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}^{\prime}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)+}))f_{t}^{(p)}(\phi_{t}^{\prime}|\alpha_{t}^{(p)+})f_{t}^{(i)}(\psi_{t})d\phi_{t}^{\prime}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t}^{\prime}|\alpha_{t}^{(p)+})f_{t}^{(i)}(\psi_{t})d\phi_{t}^{\prime}d\psi_{t}$
$\displaystyle=$
$\displaystyle\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})|\alpha_{t}^{(p)+}].$

The inequality holds by induction assumption (since
$\alpha_{t+1}^{(p)+}\geq\alpha_{t+1}^{(p)}$ by the update rule (5) given that
$\alpha_{t}^{(p)+}\geq\alpha_{t}^{(p)}$), and (39) holds by performing
variable replacement $\phi_{t}^{\prime}=\phi_{t}+\delta$. Thus, we have
$V_{t}(p_{t},i_{t},\mathcal{I}_{t}(\alpha_{t}^{(p)}))\leq
V_{t}(p_{t},i_{t},\mathcal{I}_{t}(\alpha_{t}^{(p)+}))$.

On the other hand, to consider the “non-increasing in $\alpha_{t}^{(i)}$”
part, notice $V_{\bar{T}}(p_{\bar{T}},i_{\bar{T}},\mathcal{I}_{\bar{T}})$ is
non-increasing in $\alpha_{\bar{T}}^{(i)}$. As the induction hypothesis,
assume that $V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})$ is non-increasing in
$\alpha_{t+1}^{(i)}$. Similarly, change in the value of $\alpha_{t}^{(i)}$
affects the expectation through the predictive distribution
$f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})$; and it also affects the knowledge
state variables $\alpha_{t+1}^{(i)}$ and $\beta_{t+1}^{(i)}$ through the
knowledge states update. Let $\mathcal{I}_{t+1}(\alpha_{t}^{(i)})$ denote the
updated knowledge states. Then, for any $\delta>0$,

$\displaystyle\mbox{E}\left[V_{t+1}(p_{t+1},i_{t+1},\mathcal{I}_{t+1})\right]=\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))|\alpha_{t}^{(i)}]$
(40) $\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\infty}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}+r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\infty}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\infty}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
(41)
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}+\delta},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}+\delta})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$

where the second inequality holds because both $V_{t+1}(p,i,\mathcal{I})$ and
$r_{h}(p,i)$ is non-increasing function of $i$, and the first inequality holds
because

$\displaystyle
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle\geq$ $\displaystyle
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}-r_{f}f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}-r_{f}f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}-r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\ln{\bar{I}}-\ln{i_{t}}}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}=0$

since $-r_{f}\leq
V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}},\mathcal{I}_{t+1}(\alpha_{t}^{(p)}))\leq
r_{h}(\bar{P},i_{t}e^{\psi_{t}})$ for any $\phi_{t}$ and $\psi_{t}$.

Consider a value $\alpha_{t}^{(i)+}$ such that
$\alpha_{t}^{(i)+}-\alpha_{t}^{(i)}=\delta\geq 0$, and let
$\psi_{t}^{\prime}=\psi_{t}+\delta$. It holds for the predictive density that

$\displaystyle f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})=$
$\displaystyle\pi^{-1/2}\dfrac{\Gamma\left(\lambda_{t}^{(i)}+\dfrac{1}{2}\right)}{\Gamma(\lambda_{t}^{(i)})}\left[\dfrac{\nu_{t}^{(i)}}{2\beta_{t}^{(i)}(1+\nu_{t}^{(i)})}\right]^{1/2}\left[1+\dfrac{\nu_{t}^{(i)}(\psi_{t}-\alpha^{(i)})^{2}}{2\beta_{t}^{(i)}(1+\nu_{t}^{(i)})}\right]^{-\lambda_{t}^{(i)}-1/2}$
$\displaystyle=$
$\displaystyle\pi^{-1/2}\dfrac{\Gamma\left(\lambda_{t}^{(i)}+\dfrac{1}{2}\right)}{\Gamma(\lambda_{t}^{(i)})}\left[\dfrac{\nu_{t}^{(i)}}{2\beta_{t}^{(i)}(1+\nu_{t}^{(i)})}\right]^{1/2}\left[1+\dfrac{\nu_{t}^{(i)}(\psi_{t}^{\prime}-\alpha_{t}^{(i)+})^{2}}{2\beta_{t}^{(i)}(1+\nu_{t}^{(i)})}\right]^{-\lambda_{t}^{(i)}-1/2}=f_{t}^{(i)}(\psi_{t}^{\prime}|\alpha_{t}^{(i)+}).$

Then, it holds for the RHS of (41) that

$\displaystyle{\rm RHS\ of\ }\eqref{ineq.theo9.3}\geq$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}-\delta}^{\infty}f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}+\delta},\mathcal{I}_{t+1}(\alpha_{t}^{(i)+}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}-\delta}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}+\delta})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}|\alpha_{t}^{(i)})d\phi_{t}d\psi_{t}$
$\displaystyle=$ $\displaystyle-
r_{f}\int_{\ln{\bar{I}}-\ln{i_{t}}}^{\infty}f_{t}^{(i)}(\psi_{t}^{\prime}|\alpha_{t}^{(i)+})d\psi_{t}^{\prime}$
(42)
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{-\infty}^{\ln{\bar{P}}-\ln{p_{t}}}V_{t+1}(p_{t}e^{\phi_{t}},i_{t}e^{\psi_{t}^{\prime}},\mathcal{I}_{t+1}(\alpha_{t}^{(i)+}))f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}^{\prime}|\alpha_{t}^{(i)+})d\phi_{t}d\psi_{t}^{\prime}$
$\displaystyle+\int_{-\infty}^{\ln{\bar{I}}-\ln{i_{t}}}\int_{\ln{\bar{P}}-\ln{p_{t}}}^{\infty}r_{h}(\bar{P},i_{t}e^{\psi_{t}^{\prime}})f_{t}^{(p)}(\phi_{t})f_{t}^{(i)}(\psi_{t}^{\prime}|\alpha_{t}^{(i)+})d\phi_{t}d\psi_{t}^{\prime}$
$\displaystyle=$
$\displaystyle\mbox{E}[V_{t+1}(p_{t}e^{\widetilde{\Phi}_{t}},i_{t}e^{\widetilde{\Psi}_{t}},\mathcal{I}_{t+1})|\alpha_{t}^{(i)+}].$

The inequality holds by induction assumption (since
$\alpha_{t+1}^{(i)+}\geq\alpha_{t+1}^{(i)}$ by the update rule (5) given that
$\alpha_{t}^{(p)+}\geq\alpha_{t}^{(p)}$), and (42) holds by performing
variable replacement $\psi_{t}^{\prime}=\psi_{t}+\delta$. Thus, we have
$V_{t}(p_{t},i_{t},\mathcal{I}_{t}(\alpha_{t}^{(i)}))\geq
V_{t}(p_{t},i_{t},\mathcal{I}_{t}(\alpha_{t}^{(i)+}))$.

Proof of Theorem 4

We prove that the myopic policy is optimal by backward induction. To start
with, notice that, at time $\bar{T}-1$, the optimal action is to harvest if
and only if $(p_{\bar{T}-1},i_{\bar{T}-1})\in A$. Suppose the same applies at
any time $t+1$ less than $\bar{T}-1$, i.e., the optimal action is to harvest
if and only if $(p_{t+1},i_{t+1})\in A$. We want to show the optimal action at
time $t$ is to harvest if and only if $(p_{t},i_{t})\in A$.

For $(p_{t},i_{t})\in A$, it holds that

$\displaystyle V_{t}(p_{t},i_{t})$ $\displaystyle=$
$\displaystyle\max\\{r_{h}(p_{t},i_{t}),-c_{u}+\gamma
V_{t+1}(p_{t+1},i_{t+1})\\}$ (43) $\displaystyle=$
$\displaystyle\max\\{r_{h}(p_{t},i_{t}),-c_{u}+\gamma\mbox{E}[R(p^{\prime},i^{\prime};H)|p_{t},i_{t}]\\}$
$\displaystyle=$ $\displaystyle r_{h}(p_{t},i_{t}),$ (44)

where (43) follows from the fact that, first, under positive growth
assumption, we have $p_{t+1}\geq p_{t}$ and $i_{t+1}\geq i_{t}$, we have
$(p_{t+1},i_{t})\in A$ by applying condition (i), and we have
$(p_{t+1},i_{t+1})\in A$ by subsequently applying condition (ii), and then, by
the induction hypothesis that asserts the optimal action is to harvest when
$(p_{t+1},i_{t+1})\in A$. Furthermore, (44) holds because we know
$(p_{t},i_{t})\in A$.

On the other hand, for $(p_{t},i_{t})\notin A$, the policy that continues the
fermentation process one more time period and then takes the harvest decision
has an expected reward of

$-c_{u}+\gamma\,\mbox{E}[R(p^{\prime},i^{\prime};H)|p_{t},i_{t}],$

which is strictly greater than $r_{h}(p_{t},i_{t})$ (because
$(p_{t},i_{t})\notin A)$. Thus, we establish that
$V_{t}(p_{t},i_{t})=r_{h}(p_{t},i_{t})$ for $(p_{t},i_{t})\in A$, and
$V_{t}(p_{t},i_{t})>r_{h}(p_{t},i_{t})$ for $(p_{t},i_{t})\notin A$, and the
result follows.

Proof of Lemma 20

Suppose the condition (i) in Theorem 4 does not hold, i.e.,

$\displaystyle r_{h}(p,i)\geq-
c_{u}+\gamma\,\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i],$ $\displaystyle
r_{h}(p^{+},i)\leq-
c_{u}+\gamma\,\mbox{E}[R(p^{\prime},i^{\prime};H)|p^{+},i],$

implying that

$\dfrac{1}{\gamma}\left[r_{h}(p^{+},i)-r_{h}(p,i)\right]\leq\mbox{E}[R(p^{\prime},i^{\prime};H)|p^{+},i]-\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i].$
(45)

The left-hand side (LHS) of (45) is given by

${\rm LHS}=\dfrac{c_{1}}{\gamma}(p^{+}-p).$

On the other hand, it holds for the right-hand side (RHS) of (45) that

$\displaystyle{\rm
RHS}=c_{1}\mbox{Pr}(i^{\prime}<\bar{I}|i)\left(\mbox{E}[p^{\prime}|p^{+}]-\mbox{E}[p^{\prime}|p]\right)$
$\displaystyle=$ $\displaystyle
c_{1}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\Bigg{\\{}p^{+}e^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p^{+}}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right)+\bar{P}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p^{+}}-\mu_{c}^{(p)}}{\sigma_{c}^{(p)}}\right)$
$\displaystyle-
pe^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right)-\bar{P}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}}{\sigma_{c}^{(p)}}\right)\Bigg{\\}}$
$\displaystyle<$ $\displaystyle
c_{1}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i_{0}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\Bigg{\\{}p^{+}e^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right)+\bar{P}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}}{\sigma_{c}^{(p)}}\right)$
$\displaystyle-
pe^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right)-\bar{P}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}}{\sigma_{c}^{(p)}}\right)\Bigg{\\}}$
$\displaystyle=$ $\displaystyle
c_{1}(p^{+}-p)e^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i_{0}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right),$

where the inequality holds because $i_{0}\leq i$ and $p<p^{+}$. Therefore, we
can simplify the inequality (45) as

$\dfrac{1}{\gamma}<e^{\mu_{c}^{(p)}+\sigma_{c}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i_{0}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\mu_{c}^{(p)}-\sigma_{c}^{(p)2}}{\sigma_{c}^{(p)}}\right).$

However, this contradicts for $p\geq\underline{p}$ with $\underline{p}$ given
in Lemma 20. Thus, we know that Lemma 20 holds.

Proof of Lemma 2

Suppose the condition (ii) in Theorem 4 does not hold, i.e.,

$\displaystyle r_{h}(p,i)\geq-
c_{u}+\gamma\,\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i],$ $\displaystyle
r_{h}(p,i^{+})\leq-
c_{u}+\gamma\,\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i^{+}],$

implying that

$\dfrac{r_{h}(p,i)-r_{h}(p,i^{+})}{\gamma}\geq\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i]-\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i^{+}].$
(46)

The left-hand side of (46) is

${\rm LHS}=\dfrac{c_{2}}{\gamma}(i^{+}-i).$

On the other hand, it holds for the right-hand side of (46) that

$\displaystyle{\rm
RHS}=r_{f}[\mbox{Pr}(i^{\prime}\geq\bar{I}|i^{+})-\mbox{Pr}(i^{\prime}\geq\bar{I}|i)]+\left(c_{0}+c_{1}\mbox{E}[p^{\prime}|p]\right)[\mbox{Pr}(i^{\prime}<\bar{I}|i)-\mbox{Pr}(i^{\prime}<\bar{I}|i^{+})]$
$\displaystyle-
c_{2}\mbox{E}[i^{\prime}|i^{\prime}<\bar{I},i]\mbox{Pr}(i^{\prime}<\bar{I}|i)+c_{2}\mbox{E}[i^{\prime}|i^{\prime}<\bar{I},i^{+}]\mbox{Pr}(i^{\prime}<\bar{I}|i^{+})$
$\displaystyle>$ $\displaystyle
r_{f}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\right]$
$\displaystyle-
c_{2}ie^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)+c_{2}i^{+}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)$
$\displaystyle>$ $\displaystyle
r_{f}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\right]$
$\displaystyle-
c_{2}ie^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)\right]$
$\displaystyle\geq$ $\displaystyle
r_{f}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\right]$
$\displaystyle-
c_{2}\bar{I}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)\right],$

where the first inequality holds because
$\left(c_{0}+c_{1}\mbox{E}[p^{\prime}|p]\right)[\mbox{Pr}(i^{\prime}\leq\bar{I}|i)-\mbox{Pr}(i^{\prime}\leq\bar{I}|i^{+})]>0$,
the second inequality holds since $i<i^{+}$, and the last inequality holds
since $i\leq\bar{I}$. After plugging in the expressions of LHS and RHS, we
have

$\displaystyle\dfrac{c_{2}}{\gamma}(i^{+}-i)>$ $\displaystyle
r_{f}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}}\right)\right]$
$\displaystyle-
c_{2}\bar{I}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\left[\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}-\sigma_{c}^{(i)2}}{\sigma_{c}^{(i)}}\right)\right].$

However, this contradicts with the condition given in Lemma 2. Thus, we know
Lemma 2 holds.

Proof of Corollary 5

We let

$x=\dfrac{\ln{\bar{I}}-\ln{i}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}},~{}~{}x^{+}=\dfrac{\ln{\bar{I}}-\ln{i^{+}}-\mu_{c}^{(i)}}{\sigma_{c}^{(i)}},$

where $i^{+}>i$, $x>x^{+}$. Then we can rewrite the inequality in (21) as

$\dfrac{c_{2}\bar{I}}{\gamma}(e^{-\sigma_{c}^{(i)}x^{+}-\mu_{c}^{(i)}}-e^{-\sigma_{c}^{(i)}x-\mu_{c}^{(i)}})\leq
r_{f}[\Phi(x)-\Phi(x^{+})]-c_{2}\bar{I}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}[\Phi(x-\sigma_{c}^{(i)})-\Phi(x^{+}-\sigma_{c}^{(i)})].$
(47)

We apply the first-order Taylor series expansion on each exponential term in
the left-hand side and on the normal cumulative distribution functions in the
right-hand side of (47), and further approximate the expansion by only taking
the dominant elements, for example

$\displaystyle
e^{-\sigma_{c}^{(i)}x-\mu_{c}^{(i)}}=\sum_{n=0}^{\infty}\dfrac{(-\sigma_{c}^{(i)}x-\mu_{c}^{(i)})^{n}}{n!}\approx
1-\sigma_{c}^{(i)}x-\mu_{c}^{(i)}$

and

$\displaystyle\Phi(x)$
$\displaystyle=\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-z^{2}/2}dz=\dfrac{1}{2}+\dfrac{1}{2}\mbox{erf}(\dfrac{x}{\sqrt{2}})=\dfrac{1}{2}+\dfrac{1}{2}\cdot\dfrac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\dfrac{(-1)^{n}(x/\sqrt{2})^{2n+1})}{n!(2n+1)}\approx\dfrac{1}{2}+\dfrac{x}{\sqrt{2\pi}}$

where $\mbox{erf}(w)=\dfrac{2}{\sqrt{\pi}}\int_{0}^{w}e^{-z^{2}}dz$ is the
error function. Similar approximation can be applied to
$e^{-\sigma_{c}^{(i)}x^{+}-\mu_{c}^{(i)}}$, $\Phi(x^{+})$,
$\Phi(x-\sigma_{c}^{(i)})$ and $\Phi(x^{+}-\sigma_{c}^{(i)})$. Thus, we can
approximate (47) as

$\displaystyle\dfrac{c_{2}\bar{I}}{\gamma}\sigma_{c}^{(i)}(x-x^{+})\leq$
$\displaystyle\dfrac{r_{f}}{\sqrt{2\pi}}(x-x^{+})-\dfrac{c_{2}\bar{I}}{\sqrt{2\pi}}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}[(x-\sigma_{c}^{(i)})-(x^{+}-\sigma_{c}^{(i)})],$

which leads to

$\displaystyle\dfrac{c_{2}\bar{I}}{\gamma}\sigma_{c}^{(i)}(x-x^{+})\leq$
$\displaystyle\left[\dfrac{r_{f}}{\sqrt{2\pi}}-\dfrac{c_{2}\bar{I}}{\sqrt{2\pi}}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\right](x-x^{+}).$

Therefore, it follows that:

$\displaystyle\dfrac{r_{f}}{\sqrt{2\pi}}\geq$
$\displaystyle\dfrac{c_{2}\bar{I}}{\gamma}\sigma_{c}^{(i)}+\dfrac{c_{2}\bar{I}}{\sqrt{2\pi}}e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}$
$\displaystyle r_{f}\geq$ $\displaystyle
c_{2}\bar{I}\left[\dfrac{\sqrt{2\pi}\sigma_{c}^{(i)}}{\gamma}+e^{\mu_{c}^{(i)}+\sigma_{c}^{(i)2}/2}\right].$

Proof of Theorem 7

(i) We start with writing the function
$\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)$
explicitly:

$\displaystyle\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)=$
$\displaystyle c_{0}+c_{u}+c_{1}p-c_{2}i+\gamma
r_{f}\left[1-\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}}\right)\right]$
$\displaystyle-\gamma
c_{0}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}}\right)$
$\displaystyle-\gamma
c_{1}\bar{P}\left[1-\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\alpha^{(p)}}{\widetilde{\sigma}^{(p)}}\right)\right]\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}}\right)$
$\displaystyle-\gamma
c_{1}pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi\left(\dfrac{\ln{\bar{P}}-\ln{p}-\alpha^{(p)}-\widetilde{\sigma}^{(p)2}}{\widetilde{\sigma}^{(p)}}\right)\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}}\right)$
$\displaystyle+\gamma
c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi\left(\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}-\widetilde{\sigma}^{(i)2}}{\widetilde{\sigma}^{(i)}}\right)$
$\displaystyle=$ $\displaystyle c_{0}+c_{u}+c_{1}p-c_{2}i+\gamma
r_{f}[1-\Phi(x)]-\gamma c_{0}\Phi(x)-\gamma c_{1}\bar{P}[1-\Phi(y)]\Phi(x)$
$\displaystyle-\gamma
c_{1}pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})\Phi(x)+\gamma
c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)}),$

where the $x$ and $y$ are variable replacements defined as

$x=\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}},~{}~{}y=\dfrac{\ln{\bar{P}}-\ln{p}-\alpha^{(p)}}{\widetilde{\sigma}^{(p)}}.$

We know that:

$\displaystyle
ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\phi(x-\widetilde{\sigma}^{(i)})=ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\dfrac{e^{-(x-\widetilde{\sigma}^{(i)})^{2}/2}}{\sqrt{2\pi}}$
$\displaystyle=$ $\displaystyle
ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\dfrac{e^{-x^{2}/2+x\widetilde{\sigma}^{(i)}-\widetilde{\sigma}^{(i)2}/2}}{\sqrt{2\pi}}$
$\displaystyle=$ $\displaystyle
ie^{\alpha^{(i)}+x\widetilde{\sigma}^{(i)}}\dfrac{e^{-x^{2}/2}}{\sqrt{2\pi}}$
$\displaystyle=$ $\displaystyle ie^{\ln{\bar{I}}-\ln{i}}\phi(x)$
$\displaystyle=$ $\displaystyle\bar{I}\phi(x),$

where the equality (C) follows by plugging in
$x=\dfrac{\ln{\bar{I}}-\ln{i}-\alpha^{(i)}}{\widetilde{\sigma}^{(i)}}$.
Similarly, we have

$pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\phi(y-\widetilde{\sigma}^{(p)})=\bar{P}\phi(y).$

For function
$\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)$,
if we only consider the parts related to $\alpha^{(i)}$, we have

$\displaystyle\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)\propto$
$\displaystyle-r_{f}\Phi(x)-c_{0}\Phi(x)-c_{1}\bar{P}[1-\Phi(y)]\Phi(x)$
$\displaystyle-
c_{1}pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})\Phi(x)+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)}),$
(49)

Taking derivatives with respect to $\alpha^{(i)}$ leads to:

$\displaystyle\dfrac{\partial\widetilde{h}}{\partial\alpha^{(i)}}\propto~{}$
$\displaystyle-C\phi(x)\dfrac{\partial
x}{\partial\alpha^{(i)}}+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)})+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\phi(x-\widetilde{\sigma}^{(i)})\dfrac{\partial
x}{\partial\alpha^{(i)}}$ $\displaystyle\propto~{}$ $\displaystyle
C\phi(x)\dfrac{1}{\widetilde{\sigma}^{(i)}}+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)})-c_{2}\bar{I}\phi(x)\dfrac{1}{\widetilde{\sigma}^{(i)}}$
$\displaystyle\propto~{}$
$\displaystyle\left(C-c_{2}\bar{I}\right)\dfrac{\phi(x)}{\widetilde{\sigma}^{(i)}}+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)}),$

where

$C=r_{f}+c_{0}+c_{1}\bar{P}[1-\Phi(y)]+c_{1}pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)}).$

Since $r_{f}\geq c_{2}\bar{I}$, we have $C-c_{2}\bar{I}>0$, then
$\dfrac{\partial\widetilde{h}}{\partial\alpha^{(i)}}>0$.

(ii) On the other hand, if we only consider the parts related to
$\alpha^{(p)}$, we have

$\displaystyle\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)\propto$
$\displaystyle~{}c_{1}\bar{P}\Phi(y)\Phi(x)-c_{1}e^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})\Phi(x)$
$\displaystyle\propto$
$\displaystyle~{}\bar{P}\Phi(y)-pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)}).$
(50)

Taking derivatives with respect to $\alpha^{(p)}$ leads to:

$\displaystyle\dfrac{\partial\widetilde{h}}{\partial\alpha^{(i)}}\propto~{}$
$\displaystyle\bar{P}\phi(y)\dfrac{\partial
y}{\partial\alpha^{(p)}}-pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})-pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\phi(y-\widetilde{\sigma}^{(p)})\dfrac{\partial
y}{\partial\alpha^{(p)}}$ $\displaystyle\propto~{}$
$\displaystyle\bar{P}\phi(y)\dfrac{\partial
y}{\partial\alpha^{(p)}}-pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})-\bar{P}\phi(y)\dfrac{\partial
y}{\partial\alpha^{(p)}}$ $\displaystyle\propto~{}$ $\displaystyle-
pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})<0.$

(iii) If we only consider the parts related to $\widetilde{\sigma}^{(i)}$, we
have exactly the same formula of
$\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)$
as in (49). Taking derivatives with respect to $\widetilde{\sigma}^{(i)}$, we
have that

$\displaystyle\dfrac{\partial\widetilde{h}}{\partial\widetilde{\sigma}^{(i)}}\propto~{}$
$\displaystyle-C\phi(x)\dfrac{\partial
x}{\partial\widetilde{\sigma}^{(i)}}+c_{2}i\widetilde{\sigma}^{(i)}e^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)})+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\phi(x-\widetilde{\sigma}^{(i)})\left(\dfrac{\partial
x}{\partial\widetilde{\sigma}^{(i)}}-1\right)$ $\displaystyle\propto~{}$
$\displaystyle-C\phi(x)\dfrac{\partial
x}{\partial\widetilde{\sigma}^{(i)}}+c_{2}i\widetilde{\sigma}^{(i)}e^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\Phi(x-\widetilde{\sigma}^{(i)})+c_{2}\bar{I}\phi(x)\left(\dfrac{\partial
x}{\partial\widetilde{\sigma}^{(i)}}-1\right)$ $\displaystyle\propto~{}$
$\displaystyle-\left(C-c_{2}\bar{I}\right)\phi(x)\dfrac{\partial
x}{\partial\widetilde{\sigma}^{(i)}}+c_{2}ie^{\alpha^{(i)}+\widetilde{\sigma}^{(i)2}/2}\left[\widetilde{\sigma}^{(i)}\Phi(x-\widetilde{\sigma}^{(i)})-\phi(x-\widetilde{\sigma}^{(i)})\right].$

Similarly, we have $C-c_{2}\bar{I}>0$. Therefore, if the conditions in Theorem
7(iii) hold, then $\dfrac{\partial x}{\partial\widetilde{\sigma}^{(i)}}<0$,
and
$\widetilde{\sigma}^{(i)}\Phi(x-\widetilde{\sigma}^{(i)})-\phi(x-\widetilde{\sigma}^{(i)})>0$.
Thus, $\dfrac{\partial\widetilde{h}}{\partial\widetilde{\sigma}^{(i)}}>0$.

(iv) Further, if we only consider the parts related to
$\widetilde{\sigma}^{(p)}$, we will have the same formula as in (50). Taking
derivatives with respect to $\widetilde{\sigma}^{(p)}$, we have the following
holds,

$\displaystyle\dfrac{\partial\widetilde{h}}{\partial\widetilde{\sigma}^{(p)}}\propto~{}$
$\displaystyle\bar{P}\phi(y)\dfrac{\partial
y}{\partial\widetilde{\sigma}^{(p)}}-p\widetilde{\sigma}^{(p)}e^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})-pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\phi(y-\widetilde{\sigma}^{(p)})\left(\dfrac{\partial
y}{\partial\widetilde{\sigma}^{(p)}}-1\right)$ $\displaystyle\propto~{}$
$\displaystyle\bar{P}\phi(y)\dfrac{\partial
y}{\partial\widetilde{\sigma}^{(p)}}-p\widetilde{\sigma}^{(p)}e^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\Phi(y-\widetilde{\sigma}^{(p)})-\bar{P}\phi(y)\dfrac{\partial
y}{\partial\widetilde{\sigma}^{(p)}}+pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\phi(y-\widetilde{\sigma}^{(p)})$
$\displaystyle\propto~{}$ $\displaystyle
pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}\left[\phi(y-\widetilde{\sigma}^{(p)})-\widetilde{\sigma}^{(p)}\Phi(y-\widetilde{\sigma}^{(p)})\right].$

Since $pe^{\alpha^{(p)}+\widetilde{\sigma}^{(p)2}/2}>0$, we have
$\dfrac{\partial\widetilde{h}}{\partial\widetilde{\sigma}^{(p)}}<0$ if and
only if
$\widetilde{\sigma}^{(p)}\Phi(y-\widetilde{\sigma}^{(p)})-\phi(y-\widetilde{\sigma}^{(p)})>0$.

Proof of Theorem 8

Define the “harvest function" of myopic policy under perfect information as
following,

$\displaystyle
h(\mu^{(p)}_{c},\sigma^{(p)}_{c},\mu^{(i)}_{c},\sigma^{(i)}_{c};p,i)=$
$\displaystyle
r_{h}(p,i)+c_{u}-\gamma\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i]$
$\displaystyle
r_{h}(p,i)+c_{u}-\gamma\Big{\\{}-r_{f}\mbox{Pr}(i^{\prime}>\bar{I}|i,\mu^{(i)}_{c},\sigma^{(i)}_{c})$
$\displaystyle+\mbox{E}[r_{h}(p^{\prime},i^{\prime})|i^{\prime}\leq\bar{I},p,i,\mu^{(p)}_{c},\sigma^{(p)}_{c},\mu^{(i)}_{c},\sigma^{(i)}_{c}]\Big{\\}}$

then the harvest region in (19) can be written as
$A=\left\\{(p,i):h(p,i;\mu^{(p)}_{c},\sigma^{(p)2}_{c},\mu^{(i)}_{c},\sigma^{(i)2}_{c})\geq
0\right\\}$. Notice that under model risk,

$\displaystyle\widetilde{h}(\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)};p,i)=$
$\displaystyle
r_{h}(p,i)+c_{u}-\gamma\mbox{E}[R(p^{\prime},i^{\prime};H)|p,i,\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)}]$
$\displaystyle
r_{h}(p,i)+c_{u}-\gamma\Big{\\{}-r_{f}\mbox{Pr}(i^{\prime}>\bar{I}|i,\alpha^{(i)},\widetilde{\sigma}^{(i)})$
$\displaystyle+\mbox{E}[r_{h}(p^{\prime},i^{\prime})|i^{\prime}\leq\bar{I},p,i,\alpha^{(p)},\widetilde{\sigma}^{(p)},\alpha^{(i)},\widetilde{\sigma}^{(i)}]\Big{\\}},$

which can be obtained by replacing the unknown parameters $\mu^{(i)}_{c}$ and
$\sigma^{(i)2}_{c}$ with predictive mean $\alpha^{(i)}$ and predictive
standard deviation $\widetilde{\sigma}^{(i)}$ for the impurity growth rate;
and similarly, by replacing $\mu^{(p)}_{c}$ and $\sigma^{(p)2}_{c}$ with
$\alpha^{(p)}$ and $\widetilde{\sigma}^{(p)}$ for the protein growth rate. Our
objective is to show that the function
$\widetilde{h}(\alpha_{t}^{(p)},\widetilde{\sigma}_{t}^{(p)},\alpha_{t}^{(i)},\widetilde{\sigma}_{t}^{(i)};p,i)$
converges to its true counterpart
$h(\mu^{(p)}_{c},\sigma^{(p)}_{c},\mu^{(i)}_{c},\sigma^{(i)}_{c};p,i)$ as the
length of the historical data increases.

It is known that $\phi_{j}$, $j=1,2,\ldots,J_{t}$, are i.i.d. samples from the
underlying true protein growth rate distribution
$\mathcal{N}(\mu^{(p)}_{c},\sigma^{(p)2}_{c})$. According to the properties of
Gaussian sample mean and sample variance, we have
$\bar{\phi}\sim\mathcal{N}(\mu^{(p)}_{c},\dfrac{\sigma^{(p)2}_{c}}{J_{t}})$,
$\dfrac{\sum_{j=1}^{J_{t}}(\phi_{j}-\bar{\phi})^{2}}{\sigma^{(p)2}_{c}}\sim\chi^{2}(J_{t}-1)$,
and they are independent of each other, where
$\bar{\phi}=\frac{1}{J_{t}}\sum_{j=1}^{J_{t}}\phi_{j}$. According to the weak
law of large numbers (WLLN), we have sample mean converge to the true mean,
i.e., $\alpha_{t}^{(p)}\overset{p}{\to}\mu^{(p)}_{c}$ as $J_{t}\to\infty$. In
addition, it is known from Proposition 1(i) that

$\mbox{E}\left[\widetilde{\sigma}_{t}^{(p)2}\right]=\sigma^{(p)2}_{c}+\dfrac{(2J_{t}-1)\sigma^{(p)2}_{c}}{(J_{t}^{2}-2J_{t})},~{}~{}\mbox{Var}\left[\widetilde{\sigma}_{t}^{(p)2}\right]=\dfrac{2(J_{t}^{3}+J_{t}^{2}-J_{t}-1)\sigma^{(p)4}_{c}}{J_{t}^{4}-4J_{t}^{3}+4J_{t}^{2}}.$
(51)

As $J_{t}\to\infty$, we will have
$\mbox{E}\left[\widetilde{\sigma}_{t}^{(p)2}\right]\to\sigma^{(p)2}_{c}$ and
$\mbox{Var}\left[\widetilde{\sigma}_{t}^{(p)2}\right]\to 0$, so that
$\widetilde{\sigma}_{t}^{(p)2}\overset{p}{\to}\sigma^{(p)2}_{c}$. Similar
results also apply for impurity growth rate so that we also have
$\alpha_{t}^{(i)}\overset{p}{\to}\mu^{(i)}_{c}$ and
$\widetilde{\sigma}_{t}^{(i)2}\overset{p}{\to}\sigma^{(i)2}_{c}$. Thus,
through the continuous mapping theorem,
$\widetilde{\sigma}_{t}^{(p)}\overset{p}{\to}\sigma^{(p)}_{c}$ and
$\widetilde{\sigma}_{t}^{(i)}\overset{p}{\to}\sigma^{(i)}_{c}$, and we have
for any $(p_{t},i_{t})\in[p_{0},\bar{P}]\times[i_{0},\bar{I}]$ and
$m=1,2,\ldots,\bar{T}-t$,
$\widetilde{h}(\alpha_{t}^{(p)},\widetilde{\sigma}_{t}^{(p)},\alpha_{t}^{(i)},\widetilde{\sigma}_{t}^{(i)};p,i)\overset{p}{\to}{h}(\mu^{(p)}_{c},\sigma^{(p)}_{c},\mu^{(i)}_{c},\sigma^{(i)}_{c};p,i)$.
In other words, the decision boundary under model risk will converge to the
decision boundary under perfect information.

##  Appendix D Numerical Analysis on Costs and Rewards

We provide a sensitivity analysis with respect to (i) the unit reward obtained
per protein amount, i.e., $r_{h}(p,i)=5p-i$, $r_{h}(p,i)=10p-i$ and
$r_{h}(p,i)=15p-i$; and (ii) the failure penalty cost, i.e.,
$r_{f}=400,880,1000$. These different reward and cost configurations are
defined to capture a wide range of practically-relevant settings. Other
parameters remain the same as discussed in Section 6.1. The results are shown
in Table 2.

Table 2: The mean and standard deviation of the total reward achieved by
different strategies under various configurations on unit rewards and failure
costs.

|  | Fixed Threshold | RL ignoring MR | Myopic Policy | RL with MR | Perfect Info MDP
---|---|---|---|---|---|---
Protein Value | Failure Cost | $\widehat{\rho}^{c}(\pi)$ | $\widehat{\mbox{SD}}^{c}(\pi)$ | $\widehat{\rho}^{c}(\pi)$ | $\widehat{\mbox{SD}}^{c}(\pi)$ | $\widehat{\rho}^{c}(\pi)$ | $\widehat{\mbox{SD}}^{c}(\pi)$ | $\widehat{\rho}^{c}(\pi)$ | $\widehat{\mbox{SD}}^{c}(\pi)$ | $\widehat{\rho}^{c}(\pi)$ | $\widehat{\mbox{SD}}^{c}(\pi)$
5 | 400 | 29.68 | 142.86 | 66.22 | 58.89 | 67.84 | 59.08 | 71.28 | 33.90 | 72.20 | 34.01
10 | 400 | 140.60 | 184.12 | 155.00 | 135.88 | 175.80 | 122.88 | 181.34 | 110.70 | 185.71 | 87.15
15 | 400 | 251.51 | 226.89 | 298.34 | 140.28 | 285.80 | 173.85 | 299.47 | 123.00 | 314.25 | 105.63
5 | 880 | -13.52 | 279.42 | 61.42 | 102.12 | 60.88 | 102.09 | 71.28 | 33.90 | 71.80 | 33.87
10 | 880 | 97.40 | 318.19 | 143.47 | 196.39 | 168.21 | 166.31 | 171.63 | 126.68 | 177.23 | 125.86
15 | 880 | 208.31 | 358.48 | 268.61 | 227.41 | 273.80 | 228.98 | 281.01 | 194.74 | 295.97 | 152.93
5 | 1000 | -24.32 | 313.76 | 60.22 | 113.53 | 59.49 | 113.49 | 71.28 | 33.90 | 71.33 | 34.23
10 | 1000 | 86.60 | 352.19 | 139.87 | 215.71 | 165.18 | 181.73 | 170.43 | 137.03 | 176.03 | 136.27
15 | 1000 | 197.51 | 392.07 | 246.12 | 246.75 | 270.20 | 247.66 | 271.32 | 209.92 | 294.77 | 162.50

For each different revenue and cost configuration, Table 2 presents the mean
$\widehat{\rho}^{c}(\pi)$ and standard deviation
$\widehat{\mbox{SD}}^{c}(\pi)$ of total reward realized by $100$ batches. Each
column corresponds to a harvesting strategy. We considered the size of
historical data $J_{0}=10$ for Myopic, RL-ignoring-MR and RL-with-MR
strategies. For each strategy, we observe from Table 2 that the average and
standard deviation of the total reward increase as the unit reward of protein
increases; and the average reward decreases but the standard deviation
increases as the failure cost increases. This trend matches our intuition. For
the cases with lower protein rewards or/and lower failure costs, we observe
from Table 2 that the performance of the proposed RL-with-MR strategy (under
$J_{0}=10$) is closer to that of the perfect information setting PI-MDP. For
practitioners, this implies that the potential impact of model risk is higher
for high-revenue (and/or cost) drugs in comparison to their low-revenue
(and/or cost) counterparts.