Figure 1:

Action/Inaction boundary p_b along with the solution to the first-order optimality condition, i.e., $\overline{p}$, for (a) long-term focused planner (ρ₁ ≤ β − r) and (b) short-term focused planner (ρ₂ > β − r). Gray arrows indicate the vector field defined in the system of ODE’s (5)–(6) when W (t) = 1, t ≥ 0. In panel (a) where ρ₁ > β − r, observe that the infimum of the set of drug qualities for which it is economical to prescribe the drug for some p > 0 is zero (i.e., $\underset{\_}{q}=0$). In addition, since ${\overline{p}}^{\prime }(q)=-v_c^{\prime }(\overline{p}(q))/q{v}_c^{\prime \prime }(\overline{p}(q))$ converges to zero when $\overline{p}(q)$ is close to zero, $\overline{p}$ is tangent to the q axis. In contrast, in panel (b) where ρ₂ > β − r, it is $\underset{\_}{q}>0$. Similar to panel (a), since ρ₂ > 2β − 2r, the second derivative of v_c to diverges infinity when p approaches to zero. This causes ${\overline{p}}^{\prime }(\underset{\_}{q})=0$ leading to the tangency of $\overline{p}$ to the q-axis When ρ = ρ₃, however however, the second derivative of v_c is infinite at p = 0 and hence ${\overline{p}}^{\prime }(\underset{\_}{q})>0$. This case is shown in Fig. 9a in Appendix A.