Question $2.(35$ points$)$ A perfectly competitive industry that produces soybeans consists of many farmers that can produce $45$ bushels per acre at a minimal average cost $\backslash (\backslash$$ $9\backslash )$ per bushel. The average size of a soybean field is $5,000$ acres. Each producer of soybeans must also pay shipping fees for its output, and the shipping fee $\backslash (\backslash$$ s $\backslash )$ per bushel. Assume that

$\backslash [$

s$=0\mathrm{.}8$ Q

$\backslash ]$

where $\backslash ($ Q $\backslash )$ is the total industry output in units of billions of bushels of soybeans. The demand for soybeans $($also measured in billions of bushels$)$ is given by

$\backslash [$

Q$=15\mathrm{.}75-$p

$\backslash ]$

where $\backslash ($ p $\backslash )$ is the price of one bushel of soybeans.

$($a$)$ Let the industry producing soybeans be in a long$-$run equilibrium. What is the equilibrium price of a bushel of soybeans? How many billions of bushels are produced? How many farmers are there in the industry? What is the shipping fee per bushel?

$($b$)$ Suppose that the demand for soybeans drops due to decreased import by China and becomes

$\backslash [$

Q$=15\mathrm{.}3-$p $.$

$\backslash ]$

In a new long run equilibrium, what is the equilibrium price of a bushel of soybeans? How many billions of bushels are produced? How many farmers are there in the industry? What is the shipping fee per bushel?

$($c$)$ Plot these two long$-$run equilibria in the market for soybeans. Calculate the change in the producers' surplus between the situations described in $($a$)$ and $($b$).$

$($d$)$ Show that the decrease in the producers' surplus equals to the decrease in the total shipping fees as the industry contracts incrementally from the equilibrium output in $($a$)$ to the equilibrium output in $($b$).$