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I am reading Demarzo, Fishman(RFS,2007). Any suggestion will be appreciated. My questions are the following:

(a) \begin{equation*} b_T^e(a) = \left\{\begin{array}{lll} -e^{(\gamma-r)(T^{+}-T)}a & for & a \ge 0\\ -\infty & for & a < 0 \end{array}\right. \end{equation*}

Paying the agent amount $a$ at time $T$ is the same as paying $e^{\gamma (T^{+}-T)}a$ at time $T^{+}$. We discount $e^{\gamma (T^{+}-T)}a$ back to time $T$ and get $e^{(\gamma-r)(T^{+}-T)}a$. Since $e^{(\gamma-r)(T^{+}-T)}a>a$, why don't we just pay the agent amount $a$ at time $T$?

(b) On page 2090, the paper says"the most efficient way ... is to pay the agent $e^{\gamma(T^{+}-T)}a$ in the following period $T^{+}$". In the same paragraph, it says"... while it is feasible to pay the agent after period T,..., such payments are (weakly) inefficient." Whether we should pay the agent before T, or at $T^{+}$?

(c) How does $b_t^{e}$ is plotted in figure 2? What is the intuition of $b_t^{e}$ when $a<a_t^L$ and $a>a_t^l$?

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