Question $3(12$ points$)$

Consider two firms which are polluting the air near some cily. The marginal damage costs of the pollution are $\backslash (\backslash$mathrm$\{$MDC$\}-2\backslash$mathrm$\{$c$\}\backslash )$ where c is the rate of emissions. The marginal costs of abatement, including some government information costs, are MAC $-120-$ c$.$

a$.$ Derermine the optinal level of emissions.

b$.$ Suppose that firzu $1$ is cmitting $60$ units as is firm $2.$ The marginal abatement costs for firm $1$ are $\backslash ($ M A C$\_\{$i$\}-1\mathrm{.}5\backslash$mathrm$\{$X$\}\_\{\backslash$text $\{$and $\}\}\backslash )$ and the marginal abatement costs for firm $2$ are given by $\backslash (\backslash$mathrm$\{$MAC$\}\_\{1\}-0\mathrm{.}5\backslash$mathrm$\{$X$\}\_\{2\}\backslash )$ where $\backslash (\backslash$mathrm$\{$X$\}\_\{\backslash$mathrm$\{$i$\}\}\backslash )$ is measured in units of emission reduction for firm $1$ and $\backslash (\backslash$mathrm$\{$X$\}\_\{3\}\backslash )$ is measured in units of emission reduction for tirm $2.$ Determine the cost eflective allocation of pollution between the two firms. Draw a diagram illustrating the allocation of pollution hetween the two firms. Suppose that the government decides that each firm should be allowed to emit only $\backslash (1/2\backslash )$ of the optimal level of emissions. Illustrate on your diagram, the costs associated with this policy.