No lower bound, and the inventory holder's bound

(a) Reproduce the proof of Proposition 10.2 for an agent with no inventory, using the affine-payoff argument. (b) Now let the agent own $I > 0$ units in storage, so that $x \ge -I$. Show that an arbitrage exists if and only if $F_0 < S_0 e^{(r+u)T}$ and the agent places no value on holding the commodity between $0$ and $T$. (c) Explain why the bound in (b) is $F_0 \ge S_0 e^{(r+u-\ell)T}$ once the agent values the services of the marginal unit at yield $\ell$, and why different holders can have different $\ell$.