Hotelling with linear demand

Inverse demand is $p = a - q$ with choke price $a$, extraction is costless, the interest rate is $r$ and the reserve is $R_0$. (a) Show that the competitive price path is $p_t = a e^{r(t - T^*)}$ for $t \le T^*$ and derive the equation determining the exhaustion date $T^*$. (b) Solve numerically for $a = 100$, $r = 5\%$, $R_0 = 1000$. Report $T^*$, $p_0$ and $q_0$. (c) Show that $T^*$ increases with $R_0$ and that $p_0$ falls with $R_0$.