$1.(32$ points$)$ Suppose the market supply and demand for guitars in Happy Valley are given by:

Supply: $\backslash ($ Q$=-60+4$ P $\backslash )$

Demand: $\backslash (\backslash$mathrm$\{$Q$\}=540-8\backslash$mathrm$\{$P$\}\backslash ).$

a$)(4$ points$)$ Calculate the equilibrium price and quantity of guitars.

$\backslash (\backslash$mathrm$\{$p$\}^\{*\}=\backslash )$

Q$*$

For the remaining questions, suppose a tax of $\backslash (\backslash$$ $\backslash$mathbf$\{24\}\backslash )$ per guitar is levied on the consumers.

b$)(6$ points$)$ How much will the consumers pay for guitars after the tax?

c$)(6$ points$)$ How much will producers receive for guitars after the tax? d$)(6$ points$)$ What proportion of the tax will be paid by the producers?

e$)(6$ points$)$ What quantity of the good will be bought$/$sold after the tax?

f$)(4$ points$)$ What is the deadweight loss created by the tax? $2.(6$ points$)$ Suppose there are two firms producing $6$ units of pollution each. The government decides that they want to reduce the total pollution to $6$ units, so they issue $3$ pollution permits to each firm. The cost of abating the pollution for each firm are given in the following table:

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|\}$

$\backslash$hline FIRM A & & FIRM B & $\backslash \backslash$

$\backslash$hline Unit of pollution & $\backslash$begin$\{$tabular$\}\{$l$\}$

Marginal Cost of $\backslash \backslash$

cleaning up this unit $\backslash \backslash$

of pollution

$\backslash$end$\{$tabular$\}$ & Unit of pollution & $\backslash$begin$\{$tabular$\}\{$l$\}$

Marginal Cost of $\backslash \backslash$

cleaning up this unit $\backslash \backslash$

of pollution

$\backslash$end$\{$tabular$\}\backslash \backslash$

$\backslash$hline $1$ & $\backslash (\backslash$$ $6\backslash )$ & $1$ & $\backslash (\backslash$$ $35\backslash )\backslash \backslash$

$\backslash$hline $2$ & $\backslash (\backslash$$ $14\backslash )$ & $2$ & $\backslash (\backslash$$ $70\backslash )\backslash \backslash$

$\backslash$hline $3$ & $\backslash (\backslash$$ $25\backslash )$ & $3$ & $\backslash (\backslash$$ $120\backslash )\backslash \backslash$

$\backslash$hline $4$ & $\backslash (\backslash$$ $40\backslash )$ & $4$ & $\backslash (\backslash$$ $180\backslash )\backslash \backslash$

$\backslash$hline $5$ & $\backslash (\backslash$$ $60\backslash )$ & $5$ & $\backslash (\backslash$$ $280\backslash )\backslash \backslash$

$\backslash$hline $6$ & $\backslash (\backslash$$ $90\backslash )$ & $6$ & $\backslash (\backslash$$ $390\backslash )\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

If the two firms work together to reduce the total cost of cleaning up the pollution, how many permits will be traded and which firm will sell the permits and which firm will buy the permits? $3.(12$ points$)$ Suppose a firm's short run production function is given in the following table:

Also, suppose the price of output, $\backslash (\backslash$mathrm$\{$P$\}\_\{\backslash$mathrm$\{$X$\}\}\backslash ),$ is $\backslash (\backslash$mathrm$\{$P$\}\_\{\backslash$mathrm$\{$X$\}\}=\backslash$$ $4\backslash ).$

Please complete the following table: