$3.$ Consider an industry with demand $=1$ and supply $=.$ Suppose that each unit

of output implies one unit of CO$2$ added to the atmosphere and a marginal social cost

of $,$ where $$ is the total level of emissions.

a$.$ What is the level of CO$2$ emission at the market equilibrium?

b$.$ What is the socially optimal level of CO$2$ emissions?

c$.$ Determine the Pigou tax that achieves the social optimum.