Suppose that demand for a product is Q $=1200-4$P and supply is Q $=-200+2$P$.$

Furthermore, suppose that the marginal external damage of this product is $$8$ per unit.

$($a$)$ How many more units of this product will the free market produce than is socially optimal?

$($b$)$ Calculate the deadweight loss associated with the externality.

$($c$)$ If the government uses a tax to address this externality, what is the optimal amount of tax to

be imposed?