I took three days to work out the missing steps.
In this project, we argue that the flexible regime dominates the mandatory one. Intuitively the logic is simple. But the math is not straightforward. This is because two regimes differ substantially in objectives and hence policies, which defy the direct comparison.
We solve this problem by relating them to the second best. We do so in two steps. First, we define the common structure. The central notion here is the virtual cost. Second, we parameterize each regime by its deviation from the second best (in terms virtual cost).
This approach provides a unified framework to compare the flexible and mandatory regimes. It overcomes the structural difference by teasing out their the common core.
There are two caveats, though. First, in the flexible regime the agent decides production, whereas in the mandatory regime the principal decides. Second, although each the principal may not see all the types, we still define its payoff function for each type. This hypothetical treatment poses no harm.
Currently, PL is working on the case with additional budget constraint. That constraint is necessary for explaining the logic of exporting trade. It makes economic sense and numerically it works. Just wait to see the details.