[Economics] Optimal Price and Quantity of Externality

Solution 1:

Your steps look correct. There is one small typo in the first $\frac{\partial \mathcal L}{\partial h}$: the term $-4h(h-10)$ should have been $-4(h-10)$.

The results also look reasonable: Regardless of who makes the offer, the Pareto optimal level of $h$ is produced. Given how the bargaining procedures are structured, whoever gets to make the offer gets to keep the surplus of $20$.

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P.S. There is an issue in part 2 of the question. It says: "If the firm rejects the offer it cannot produce so the firm's profit is 0." This statement is a contradiction. If the firm cannot produce ($h=0$), then profit would be $-80<0$; if its profit is $0$, then it must produce some positive amount of $h$. But this lack of rigor is on whoever wrote the question, not you.

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Solution 2:

Given that you already know the welfare maximizing level $h=5$ from your previous question, another approach would be to just consider that any optimal take-it-or-leave-it offer can be divided into two steps: First, maximize total surplus by setting $h=5$. Second, extract all surplus by maximizing the price subject to the other's participation constraint. The latter translates to setting utility or profit, respectively, to zero. The optimal price offer then follows immediately.