Assume an economy of two firms and two consumers. The two firms pollute. Firm one has a marginal savings function of \(M S_{1}(e)=5-e\) where \(e\) is the quantity of emissions from the firm. Firm two has a marginal savings function of \(M S_{2}(e)=8-2 e\). Each of the two consumers has marginal damage \(M D(e)=e\), where \(e\) is this case is the total amount of emissions the consumer is exposed to.

a. Graph the firm-level and aggregate marginal savings functions.

b. Graph the aggregate marginal damage function.

c. What is the optimal level of pollution, the appropriate Pigovian fee, and emissions from each firm?