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

question 12

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

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Definitions:

Direct Labor Hours

The total number of working hours spent by employees directly involved in the production of goods or services.

Machine Hours

A measure of the amount of time a machine is operated, used in manufacturing to allocate costs to products based on machine usage.

Hooded Jacket

A type of apparel that includes a hood attached to it, often used for casual wear or protection against weather.

Departmental Overhead Rate

A calculation used to allocate overhead costs to specific departments, based on varying bases such as labor hours or machine hours.