Solve the Problem $70 \%$ Of Its Output To $\mathrm { M }$

Multiple Choice

Solve the problem.
-Suppose an economy consists of three sectors: Energy (E) , Manufacturing (M) , and Agriculture (A) Sector E sells $70 \%$ of its output to $\mathrm { M }$ and $30 \%$ to $\mathrm { A }$ .
Sector M sells $30 \%$ of its output to E, $50 \%$ to A, and retains the rest.
Sector A sells $15 \%$ of its output to E, $30 \%$ to $M$ , and retains the rest.
Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by $\mathrm { p } _ { \mathrm { e } } , \mathrm { p } _ { \mathrm { m } }$ , and $\mathrm { p } _ { \mathrm { a } }$ , respectively. If possible, find equilibrium prices that make each sector's income match its expenditures.
Find the general solution as a vector, with $\mathrm { p } _ { \mathrm { a } }$ free.

A)
$\left[ \begin{array} { l } p _ { e } \\ p _ { m } \\ p _ { a } \end{array} \right] = \left[ \begin{array} { c } 0.465 \mathrm { pa } ^ { 0.593 } \mathrm { pa } \\ p _ { a } \end{array} \right]$
B)
$$\left[ \begin{array} { l } \mathrm { p } _ { \mathrm { e } } \\\mathrm { p } _ { \mathrm { m } } \\\mathrm { p } _ { \mathrm { a } }\end{array} \right] = \left[ \begin{array} { c } 0.356 \mathrm { pa } \\0.686 \mathrm { pa } \\\mathrm { pa } _ { \mathrm { a } }\end{array} \right]$$

C)
$$\left[ \begin{array} { l } p _ { e } \\p _ { m } \\p _ { a }\end{array} \right] = \left[ \begin{array} { c } 0.607 \mathrm { pa } _ { a } \\0.481 \mathrm { pa } ^ { p _ { a } }\end{array} \right]$$
D)

$$\left[ \begin{array} { l } \mathrm { p } _ { \mathrm { e } } \\\mathrm { p } _ { \mathrm { m } } \\\mathrm { p } _ { \mathrm { a } }\end{array} \right] = \left[ \begin{array} { c } 0.308 \mathrm { p } _ { \mathrm { a } } \\0.716 \mathrm { pa } \\\mathrm { p } _ { \mathrm { a } }\end{array} \right]$$