$(6$ points$).$ This problem is based on the specific factors model. Consider the economy of the Shire that produces cereals and swords. Each industry employs Hobbits, but land is specific to the production of cereals, and metal is a specific factor in the sword industry. The amounts of land $($L$)$ and metal $($M$)$ are $L=1000$ and $M=40,$ and there are $100$ Hobbits in Shire. The production functions and the marginal products of Hobbits' labor are as follows:

$C=8(H_C{)}^{0\mathrm{.}5}(L{)}^{0\mathrm{.}5},MP{H}_C=4\frac{(L{)}^{0\mathrm{.}5}}{(H_C{)}^{0\mathrm{.}5}}$

$S=4(H_S{)}^{0\mathrm{.}5}(M{)}^{0\mathrm{.}5},MP{H}_S=2\frac{(M{)}^{0\mathrm{.}5}}{(H_S{)}^{0\mathrm{.}5}}$

where $C$ is the cereal output in pounds, $S$ is the number of swords produced, and $H_C$ and $H_S$ are the numbers of Hobbits employed in the cereal and sword sectors$,$ respectively.

$($a$)$ Assume that the price of $1$ pound of cereals is $$10$ and the price of a sword is $$200.$ Draw the equilibrium diagram $($the figure that is similar to Figure $3-4)$ for this economy.

$($b$)$ What is the wage rate paid to Hobbits in the Shire? How many Hobbits work in each sector?

$($c$)$ Because of the War of the Ring, the price of a sword went up to $$300.$ Calculate new wage and employment level in each sector.

$($d$)$ Do Hobbits gain or lose if the price of a sword goes up to $$300?($To answer the question, calculate their real wages before and after the price change$).$